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PHILOSOPHICAL MAGAZINE A 2001 VOL 81 NO 1 109plusmn 124
The eŒect of stress on cubic-to-tetragonal phasetransitions in Mg2TiO4 and Mg2GeO4 spinel reg lms
ST SENZW BLUMy and D HESSEzMax-Planck-Institut fuEgrave r Mikrostrukturphysik Weinberg 2 D-06120 Halle
Germany
[Received 14 May 1999 accepted 29 February 2000 ]
ABSTRACTEpitaxial thin reg lms of the spinels Mg2TiO4 and Mg2GeO4 are grown by a
solid-state reaction between MgO(001) single-crystal substrates and vapours orthin reg lms of the oxides TiO2 and GeO2 respectively Cubic-to-tetragonal phasetransitions in the Mg2TiO4 reg lms are studied by transmission electron microscopyincluding in-situ observations at high temperatures The transition kinetics areshown to be inmacr uenced by the presence of stress Compared with bulk transitionsin thin reg lms the kinetics are slowed Moreover the kinetics become sensitive tothe crystallographic orientation of the tetragonal Mg2TiO4 grains with respect tothe reg lm plane Stress develops owing to the mismatch of the thermal expansionbetween reg lm and substrate during the cooling of the reg lms from the growthtemperature of about 1270K as well as during the phase transition owing tolattice parameter mismatches The latter depend on the crystallographicorientation of the tetragonal grains which results in certain orientations thatare more stable than others and also in a characteristic superstructureoccurring in the electron diŒraction patterns of Mg2TiO4 reg lms Surprisinglyvery similar superstructures occur in the electron diŒraction patterns ofMg2GeO4 reg lms although no tetragonal phase is known of this normal spinelThis unexpected phenomenon is discussed in terms of a cation site ordering inthe Mg2GeO4 reg lms enabled by a non-zero inversion degree
1 INTRODUCTIONPhase transitions in spinels are most interesting to investigate In particular
cubic tetragonal transitions in the inverse spinel Mg2TiO4 (Wechsler andNavrotsky 1984 Wechsler and Von Dreele 1989) and spinel-to-olivine transitions
in Mg2GeO4 (Lauterjung and Will 1986 Ross and Navrotsky 1987 Rubie and
Champness 1987) have attracted the attention of various groups Phase transitions
in Mg2GeO4 are signireg cant since this spinel has been introduced as a model systemfor the investigation of deep earthquakes The origin of the latter is attributed to
olivine-to-spinel transitions of the mineral (Mg Fe)2SiO4 prevailing in the deep
Earth mantle (Burnley and Green 1989 Green and Burnley 1989) While the transi-
tion of (Mg Fe)2SiO4 occurs at temperatures between 1500 and 1900 K and at high
pressures of 8plusmn 18 GPa the olivine-to-spinel transition in Mg2GeO4 is easier to
achieve in the laboratory at a transition temperature of 1100K and at normal
Philosophical Magazine A ISSN 0141plusmn 8610 printISSN 1460-6992 online 2001 Taylor amp Francis Ltd
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Present address AMD Saxony Manufacturing GmbH Dresden GermanyzEmail hessempi-hallede
pressure (valid for bulk samples) In spite of many investigations performed there
has not been any agreement as to the details of the mechanism of the olivine-to-
spinel transition in Mg2GeO4 under high pressures (Lauterjung and Will 1986 Rubieand Champness 1987) More investigations on the inmacr uence of pressure (or stress) on
phase transitions in spinels are thus highly desirable
A rather unconventional way of introducing high pressures into solid materials
opens up if thin reg lms are studied Stresses in thin reg lms may reach several gigapascals
(HoŒmann 1981 Fabis et al 1990) and are easy to attain under laboratory condi-tions Epitaxial reg lms are favourable if the crystallography of the transition is to be
considered We prepared epitaxial Mg2TiO4 and Mg2GeO4 thin reg lms by solid-state
reactions a route previously described in some detail (Sieber et al 1997 Blum and
Hesse 1997) While Mg2TiO4 reg lms form by a topotaxial reaction between a TiO2
vapour and the (001) surface of a MgO single crystal Mg2GeO4 reg lms were formedfrom a thin solid GeO2 reg lm deposited on to the MgO single-crystal substrate The
heating of this MgOplusmn GeO2 couple is accompanied by a sequence of chemical reac-
tions occurring with the temperature increasing forming thin reg lms of the product
phases MgGeO3 Mg2GeO4 and reg nally Mg28Ge10O48 This sequence corresponds
to the increasing Mg content owing to the excess MgO originating from the bulk
substrate crystal (Blum and Hesse 1997) If this process is stopped in due time anepitaxial reg lm of Mg2GeO4 will have formed Previous investigations of reg lms of this
kind showed some inmacr uence of stress on the reg lm structure Thus the stability of the
Mg2GeO4 spinel phase with respect to the transition to olivine is extended by more
than 170 K up to temperatures above 1270 K owing to the thermal stress inherent in
the reg lms (Blum and Hesse 1997) Moreover electron diŒraction and high-resolutiontransmission electron microscopy (HRTEM) revealed a superstructure of the
Mg2GeO4 spinel lattice in these reg lms the nature of which indicated some similarity
to the well known tetragonal phase of Mg2TiO4 which formed because of cation
ordering (Blum 1997 Blum et al 1998) This result is most surprising in view of the
fact that no tetragonal phase of Mg2GeO4 is known so far The present study on theinmacr uence of stress on cubic-to-tetragonal phase transitions in Mg2TiO4 and
Mg2GeO4 thin reg lms has been encouraged by these observations
2 CATION ORDERING AND INVERSION DEGREESuperstructures forming during phase transitions because of some ordering of
cations or cation vacancies have been observed in several spinels The type of these
transitions was predicted on the basis of symmetry considerations (Haas 1965)
Examples are the formation of tetragonal structures by the ordering of cations in
the inverse spinel Mg2TiO4 (Wechsler and Von Dreele 1989) and by the ordering of
iron vacancies in the spinel g-Fe2O3 (Greaves 1983) The cubic spinel Mg2TiO4
(lattice parameter aTic ˆ 84376hellip5dagger pm) (Wechsler and Von Dreele 1989) is a nearly
ideal inverse spinel Half the Mg and all the Ti cations are randomly distributed over
the octahedral sites Below the transition temperature the ordering of these octahe-
dral Mg and Ti cations occurs with up to two thirds of all cations involved As aresult of this ordering a tetragonal low-temperature phase of Mg2TiO4 forms A
structure rereg nement of this phase performed by neutron powder diŒraction yielded
the space group P4122 (No 91) and the lattice parameters aTit ˆ 59748hellip5dagger pm and
cTit ˆ 8414hellip7dagger pm (Wechsler and Von Dreele 1989) which obey the conditions
cTit lt aTic lt 21=2aTit hellip1dagger
110 St Senz et al
It should be pointed out that the ordering process and the concomitant formation of
the tetragonal phase are only possible as Mg2TiO4 is an inverse spinel The non-zero
inversion degree is the prerequisite to this type of ordering which renders a con-sideration of the inversion degree of Mg2GeO4 particularly important if the
observed superstructure of this nominally normal spinel is to be investigated and
discussed
The inversion degree of Mg2GeO4 (aGec ˆ 82496hellip6dagger pm) was reported to be zero
for olivine samples grown at a Mg-to-Ge ratio of 20 and converted into a spinel byannealing at 05 GPa and 1120 K (Von Dreele et al 1977) However the minimum of
the structure rereg nement was not exactly at zero Rather a tetrahedral site occupancy
of 099 Ge and 001 Mg resulted in the lowest value of the variance In another
experiment (Ross and Navrotsky 1987) the Mg2GeO4 spinel was synthesized at
02 GPa and at temperatures between 973 and 1048 K Raman spectra and X-raydiŒractograms again proved the spinel close to normal These workers compared
experimental and theoretical data of the X-ray diŒraction (XRD) intensity ratio
Ihellip422dagger=Ihellip400dagger revealing a Mg fraction of about 005 on the tetrahedral sites A
similar value of 007 can be deduced from the work of Guyot et al (1986) if the
data given by Ross and Navrotsky (1987) are taken into account Following all the
above results stoichiometric Mg2GeO4 seems to be entirely or almost a normalspinel However up to now no data have been published about a possible solid
solution of either GeO2 or MgO in the Mg2GeO4 spinel An oV-stoichiometric spinel
could at least be partially inverse Moreover unusual lines in the infrared spectrum
of Mg2GeO4 formed under pressure at a high temperature of 1273 K were inter-
preted in terms of a non-zero inversion degree (Jeanloz 1980)
3 EXPERIMENTAL DETAILSThe spinel reg lms have formed by either a gasplusmn solid (Mg2TiO4) or a solidplusmn solid
(Mg2GeO4) topotaxial chemical reaction In a high-vacuum chamber a polished
MgO(001) crystal of 10 mm pound 10 mm pound 1 mm size was heated to temperatures
between 800 and 1300 K A target of the respective oxide that is TiO2 or GeO2
was evaporated using an electron-beam evaporator O2 was introduced into thevacuum system through a needle valve to maintain an O2 background pressure up
to 1 pound 10iexcl2 Pa which for Mg2TiO4 was necessary to avoid the formation of the
spinel MgTi2O4 in which Ti has a valency of 3 Owing to the diŒerent sticking
coe cients of the two oxides at 1300K the Mg2TiO4 spinel has been formed by a
chemical reaction of the MgO substrate with the Tiplusmn O vapour at 1300 K whereas
under the same conditions no Mg2GeO4 reg lm had formed from the Geplusmn O vapourTherefore Mg2GeO4 reg lms were grown by depositing an amorphous GeO2 reg lm on to
the MgO substrate at a low temperature (773 K) and subsequently reacting the reg lm
in air at 1300 K
For the X-ray investigations a Philips XrsquoPert MRD X-ray diŒractometer was
used which was equipped with a Eulerian cradle and Cu Ka radiation The stresswas measured using a thin reg lm attachment (parallel plate collimator and graphite
secondary monochromator 2sup3 resolution 0358) High resolution was achieved
using a Ge(220) four-remacr ection channel-cut monochromator and a similar analyser
Specimens for transmission electron microscopy (TEM) were prepared by standard
techniques namely grinding dimpling and ion milling A Philips CM20 twin elec-tron microscope was used with a double-tilt heating stage for in-situ observations up
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 111
to 1300 K High-resolution images were obtained with a JEOL 4000 EX high-resolu-
tion transmission electron microscope
4 R ESULTS
41 Overview
Both Mg2TiO4 and Mg2GeO4 spinel reg lms were grown epitaxially as shown byXRD electron diŒraction and HRTEM all revealing the orientation relationship
hellip001daggerspinelkhellip001daggerMgO permil100Šspinelkpermil100ŠMgO hellip2dagger
This orientation relationship results in low lattice misreg t along the spinelplusmn MgO inter-
face namely only 02 for Mg2TiO4 plusmn MgO and 2 for Mg2GeO4 plusmn MgO
Corresponding to these misreg t values the perfection of the reg lm orientation is diŒer-ent for the two spinels While the Mg2TiO4 rocking curve peaks display a very small
full width at half-maximum (FWHM) of the order of 0058 which is almost identical
with the FWHM of the substrate peaks the tilt FWHM of the Mg2GeO4(004)
rocking curve is about 158 (A typical MgO substrate consists of two to reg ve
small-angle grains with tilt FWHMs between 00078 and 00168 of the individual
grains The grains are tilted against each other and the envelope of the tilt distribu-tion has a FWHM between 0028 and 0078 The Mg2TiO4 reg lm reproduces this tilt
distribution thus the upper limit of its tilt FWHM is in fact only about 0018)Typical defects of the spinel reg lms are cation antiphase boundaries in Mg2TiO4
(Hesse and Bethge 1981 Hesse 1987 Sieber et al 1997) and small-angle grain
boundaries in Mg2GeO4 Figure 1 (a) shows a diŒraction pattern and reg gure 1 (b) aplan-view dark-reg eld image of a Mg2TiO4 reg lm reg gures 1 (c) and (d) are TEM cross-
section images of a Mg2TiO4 reg lm and reg gure 1 (e) a TEM cross-section image of a
Mg2GeO4 reg lm Electron diŒraction patterns (reg gure 2 (a)) display weak superstruc-
ture remacr ections of the Mg2GeO4 reg lm HRTEM lattice plane images Fourier-reg ltered
for the noise reduction show corresponding weak contrast features (reg gure 2 (b)) Theinset in reg gure 2 (b) presents the computer-generated Fourier transform of the high-
resolution image which also shows the weak superstructure remacr ections
42 Stress-induced distortions of the cubic phase
At room temperature the spinel reg lms are tetragonally distorted because of the
diŒerence between the thermal expansion coe cients of reg lm and substrate as wasrevealed by precise XRD measurements determining and comparing the lattice
plane spacings of those remacr ections which should show equal spacings in a cubic
phase DiŒerent 931 remacr ections and diŒerent 862 and 10 2 0 remacr ections respec-
tively were compared revealing a tetragonal distortion of the reg lm lattice The lattice
parameters ak and a were measured (reg gure 3) that is the lattice parameters inthe directions parallel and perpendicular respectively to the substrate surface The
pseudocubic parameter apc was calculated from ak and a (Blum 1997) assuming a
reasonable value for Poissonrsquo s ratio of cedil ˆ 035 (as given by Hellwege (1969) for the
MgAl2O4 spinel) The values thus obtained are shown in table l together with the
distortions calculated according to
ˆak iexcl apc
apc
hellip3dagger
112 St Senz et al
The negative sign of in table 1 shows that the distortion is a biaxial compression
in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2
(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)
and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical
compression after cooling the reg lms from the growth temperature of 1300 K to
room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than
the theoretical values part of the strain is obviously relaxed
Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio
cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from
the strains (Brantley 1973) which results in the experimentally determined stresses
frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp
Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in
the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and
frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will
be subject to the inmacr uence of a rather high stress
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 113
Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
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973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
pressure (valid for bulk samples) In spite of many investigations performed there
has not been any agreement as to the details of the mechanism of the olivine-to-
spinel transition in Mg2GeO4 under high pressures (Lauterjung and Will 1986 Rubieand Champness 1987) More investigations on the inmacr uence of pressure (or stress) on
phase transitions in spinels are thus highly desirable
A rather unconventional way of introducing high pressures into solid materials
opens up if thin reg lms are studied Stresses in thin reg lms may reach several gigapascals
(HoŒmann 1981 Fabis et al 1990) and are easy to attain under laboratory condi-tions Epitaxial reg lms are favourable if the crystallography of the transition is to be
considered We prepared epitaxial Mg2TiO4 and Mg2GeO4 thin reg lms by solid-state
reactions a route previously described in some detail (Sieber et al 1997 Blum and
Hesse 1997) While Mg2TiO4 reg lms form by a topotaxial reaction between a TiO2
vapour and the (001) surface of a MgO single crystal Mg2GeO4 reg lms were formedfrom a thin solid GeO2 reg lm deposited on to the MgO single-crystal substrate The
heating of this MgOplusmn GeO2 couple is accompanied by a sequence of chemical reac-
tions occurring with the temperature increasing forming thin reg lms of the product
phases MgGeO3 Mg2GeO4 and reg nally Mg28Ge10O48 This sequence corresponds
to the increasing Mg content owing to the excess MgO originating from the bulk
substrate crystal (Blum and Hesse 1997) If this process is stopped in due time anepitaxial reg lm of Mg2GeO4 will have formed Previous investigations of reg lms of this
kind showed some inmacr uence of stress on the reg lm structure Thus the stability of the
Mg2GeO4 spinel phase with respect to the transition to olivine is extended by more
than 170 K up to temperatures above 1270 K owing to the thermal stress inherent in
the reg lms (Blum and Hesse 1997) Moreover electron diŒraction and high-resolutiontransmission electron microscopy (HRTEM) revealed a superstructure of the
Mg2GeO4 spinel lattice in these reg lms the nature of which indicated some similarity
to the well known tetragonal phase of Mg2TiO4 which formed because of cation
ordering (Blum 1997 Blum et al 1998) This result is most surprising in view of the
fact that no tetragonal phase of Mg2GeO4 is known so far The present study on theinmacr uence of stress on cubic-to-tetragonal phase transitions in Mg2TiO4 and
Mg2GeO4 thin reg lms has been encouraged by these observations
2 CATION ORDERING AND INVERSION DEGREESuperstructures forming during phase transitions because of some ordering of
cations or cation vacancies have been observed in several spinels The type of these
transitions was predicted on the basis of symmetry considerations (Haas 1965)
Examples are the formation of tetragonal structures by the ordering of cations in
the inverse spinel Mg2TiO4 (Wechsler and Von Dreele 1989) and by the ordering of
iron vacancies in the spinel g-Fe2O3 (Greaves 1983) The cubic spinel Mg2TiO4
(lattice parameter aTic ˆ 84376hellip5dagger pm) (Wechsler and Von Dreele 1989) is a nearly
ideal inverse spinel Half the Mg and all the Ti cations are randomly distributed over
the octahedral sites Below the transition temperature the ordering of these octahe-
dral Mg and Ti cations occurs with up to two thirds of all cations involved As aresult of this ordering a tetragonal low-temperature phase of Mg2TiO4 forms A
structure rereg nement of this phase performed by neutron powder diŒraction yielded
the space group P4122 (No 91) and the lattice parameters aTit ˆ 59748hellip5dagger pm and
cTit ˆ 8414hellip7dagger pm (Wechsler and Von Dreele 1989) which obey the conditions
cTit lt aTic lt 21=2aTit hellip1dagger
110 St Senz et al
It should be pointed out that the ordering process and the concomitant formation of
the tetragonal phase are only possible as Mg2TiO4 is an inverse spinel The non-zero
inversion degree is the prerequisite to this type of ordering which renders a con-sideration of the inversion degree of Mg2GeO4 particularly important if the
observed superstructure of this nominally normal spinel is to be investigated and
discussed
The inversion degree of Mg2GeO4 (aGec ˆ 82496hellip6dagger pm) was reported to be zero
for olivine samples grown at a Mg-to-Ge ratio of 20 and converted into a spinel byannealing at 05 GPa and 1120 K (Von Dreele et al 1977) However the minimum of
the structure rereg nement was not exactly at zero Rather a tetrahedral site occupancy
of 099 Ge and 001 Mg resulted in the lowest value of the variance In another
experiment (Ross and Navrotsky 1987) the Mg2GeO4 spinel was synthesized at
02 GPa and at temperatures between 973 and 1048 K Raman spectra and X-raydiŒractograms again proved the spinel close to normal These workers compared
experimental and theoretical data of the X-ray diŒraction (XRD) intensity ratio
Ihellip422dagger=Ihellip400dagger revealing a Mg fraction of about 005 on the tetrahedral sites A
similar value of 007 can be deduced from the work of Guyot et al (1986) if the
data given by Ross and Navrotsky (1987) are taken into account Following all the
above results stoichiometric Mg2GeO4 seems to be entirely or almost a normalspinel However up to now no data have been published about a possible solid
solution of either GeO2 or MgO in the Mg2GeO4 spinel An oV-stoichiometric spinel
could at least be partially inverse Moreover unusual lines in the infrared spectrum
of Mg2GeO4 formed under pressure at a high temperature of 1273 K were inter-
preted in terms of a non-zero inversion degree (Jeanloz 1980)
3 EXPERIMENTAL DETAILSThe spinel reg lms have formed by either a gasplusmn solid (Mg2TiO4) or a solidplusmn solid
(Mg2GeO4) topotaxial chemical reaction In a high-vacuum chamber a polished
MgO(001) crystal of 10 mm pound 10 mm pound 1 mm size was heated to temperatures
between 800 and 1300 K A target of the respective oxide that is TiO2 or GeO2
was evaporated using an electron-beam evaporator O2 was introduced into thevacuum system through a needle valve to maintain an O2 background pressure up
to 1 pound 10iexcl2 Pa which for Mg2TiO4 was necessary to avoid the formation of the
spinel MgTi2O4 in which Ti has a valency of 3 Owing to the diŒerent sticking
coe cients of the two oxides at 1300K the Mg2TiO4 spinel has been formed by a
chemical reaction of the MgO substrate with the Tiplusmn O vapour at 1300 K whereas
under the same conditions no Mg2GeO4 reg lm had formed from the Geplusmn O vapourTherefore Mg2GeO4 reg lms were grown by depositing an amorphous GeO2 reg lm on to
the MgO substrate at a low temperature (773 K) and subsequently reacting the reg lm
in air at 1300 K
For the X-ray investigations a Philips XrsquoPert MRD X-ray diŒractometer was
used which was equipped with a Eulerian cradle and Cu Ka radiation The stresswas measured using a thin reg lm attachment (parallel plate collimator and graphite
secondary monochromator 2sup3 resolution 0358) High resolution was achieved
using a Ge(220) four-remacr ection channel-cut monochromator and a similar analyser
Specimens for transmission electron microscopy (TEM) were prepared by standard
techniques namely grinding dimpling and ion milling A Philips CM20 twin elec-tron microscope was used with a double-tilt heating stage for in-situ observations up
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 111
to 1300 K High-resolution images were obtained with a JEOL 4000 EX high-resolu-
tion transmission electron microscope
4 R ESULTS
41 Overview
Both Mg2TiO4 and Mg2GeO4 spinel reg lms were grown epitaxially as shown byXRD electron diŒraction and HRTEM all revealing the orientation relationship
hellip001daggerspinelkhellip001daggerMgO permil100Šspinelkpermil100ŠMgO hellip2dagger
This orientation relationship results in low lattice misreg t along the spinelplusmn MgO inter-
face namely only 02 for Mg2TiO4 plusmn MgO and 2 for Mg2GeO4 plusmn MgO
Corresponding to these misreg t values the perfection of the reg lm orientation is diŒer-ent for the two spinels While the Mg2TiO4 rocking curve peaks display a very small
full width at half-maximum (FWHM) of the order of 0058 which is almost identical
with the FWHM of the substrate peaks the tilt FWHM of the Mg2GeO4(004)
rocking curve is about 158 (A typical MgO substrate consists of two to reg ve
small-angle grains with tilt FWHMs between 00078 and 00168 of the individual
grains The grains are tilted against each other and the envelope of the tilt distribu-tion has a FWHM between 0028 and 0078 The Mg2TiO4 reg lm reproduces this tilt
distribution thus the upper limit of its tilt FWHM is in fact only about 0018)Typical defects of the spinel reg lms are cation antiphase boundaries in Mg2TiO4
(Hesse and Bethge 1981 Hesse 1987 Sieber et al 1997) and small-angle grain
boundaries in Mg2GeO4 Figure 1 (a) shows a diŒraction pattern and reg gure 1 (b) aplan-view dark-reg eld image of a Mg2TiO4 reg lm reg gures 1 (c) and (d) are TEM cross-
section images of a Mg2TiO4 reg lm and reg gure 1 (e) a TEM cross-section image of a
Mg2GeO4 reg lm Electron diŒraction patterns (reg gure 2 (a)) display weak superstruc-
ture remacr ections of the Mg2GeO4 reg lm HRTEM lattice plane images Fourier-reg ltered
for the noise reduction show corresponding weak contrast features (reg gure 2 (b)) Theinset in reg gure 2 (b) presents the computer-generated Fourier transform of the high-
resolution image which also shows the weak superstructure remacr ections
42 Stress-induced distortions of the cubic phase
At room temperature the spinel reg lms are tetragonally distorted because of the
diŒerence between the thermal expansion coe cients of reg lm and substrate as wasrevealed by precise XRD measurements determining and comparing the lattice
plane spacings of those remacr ections which should show equal spacings in a cubic
phase DiŒerent 931 remacr ections and diŒerent 862 and 10 2 0 remacr ections respec-
tively were compared revealing a tetragonal distortion of the reg lm lattice The lattice
parameters ak and a were measured (reg gure 3) that is the lattice parameters inthe directions parallel and perpendicular respectively to the substrate surface The
pseudocubic parameter apc was calculated from ak and a (Blum 1997) assuming a
reasonable value for Poissonrsquo s ratio of cedil ˆ 035 (as given by Hellwege (1969) for the
MgAl2O4 spinel) The values thus obtained are shown in table l together with the
distortions calculated according to
ˆak iexcl apc
apc
hellip3dagger
112 St Senz et al
The negative sign of in table 1 shows that the distortion is a biaxial compression
in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2
(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)
and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical
compression after cooling the reg lms from the growth temperature of 1300 K to
room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than
the theoretical values part of the strain is obviously relaxed
Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio
cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from
the strains (Brantley 1973) which results in the experimentally determined stresses
frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp
Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in
the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and
frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will
be subject to the inmacr uence of a rather high stress
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 113
Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
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elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
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ctio
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ken
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Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
It should be pointed out that the ordering process and the concomitant formation of
the tetragonal phase are only possible as Mg2TiO4 is an inverse spinel The non-zero
inversion degree is the prerequisite to this type of ordering which renders a con-sideration of the inversion degree of Mg2GeO4 particularly important if the
observed superstructure of this nominally normal spinel is to be investigated and
discussed
The inversion degree of Mg2GeO4 (aGec ˆ 82496hellip6dagger pm) was reported to be zero
for olivine samples grown at a Mg-to-Ge ratio of 20 and converted into a spinel byannealing at 05 GPa and 1120 K (Von Dreele et al 1977) However the minimum of
the structure rereg nement was not exactly at zero Rather a tetrahedral site occupancy
of 099 Ge and 001 Mg resulted in the lowest value of the variance In another
experiment (Ross and Navrotsky 1987) the Mg2GeO4 spinel was synthesized at
02 GPa and at temperatures between 973 and 1048 K Raman spectra and X-raydiŒractograms again proved the spinel close to normal These workers compared
experimental and theoretical data of the X-ray diŒraction (XRD) intensity ratio
Ihellip422dagger=Ihellip400dagger revealing a Mg fraction of about 005 on the tetrahedral sites A
similar value of 007 can be deduced from the work of Guyot et al (1986) if the
data given by Ross and Navrotsky (1987) are taken into account Following all the
above results stoichiometric Mg2GeO4 seems to be entirely or almost a normalspinel However up to now no data have been published about a possible solid
solution of either GeO2 or MgO in the Mg2GeO4 spinel An oV-stoichiometric spinel
could at least be partially inverse Moreover unusual lines in the infrared spectrum
of Mg2GeO4 formed under pressure at a high temperature of 1273 K were inter-
preted in terms of a non-zero inversion degree (Jeanloz 1980)
3 EXPERIMENTAL DETAILSThe spinel reg lms have formed by either a gasplusmn solid (Mg2TiO4) or a solidplusmn solid
(Mg2GeO4) topotaxial chemical reaction In a high-vacuum chamber a polished
MgO(001) crystal of 10 mm pound 10 mm pound 1 mm size was heated to temperatures
between 800 and 1300 K A target of the respective oxide that is TiO2 or GeO2
was evaporated using an electron-beam evaporator O2 was introduced into thevacuum system through a needle valve to maintain an O2 background pressure up
to 1 pound 10iexcl2 Pa which for Mg2TiO4 was necessary to avoid the formation of the
spinel MgTi2O4 in which Ti has a valency of 3 Owing to the diŒerent sticking
coe cients of the two oxides at 1300K the Mg2TiO4 spinel has been formed by a
chemical reaction of the MgO substrate with the Tiplusmn O vapour at 1300 K whereas
under the same conditions no Mg2GeO4 reg lm had formed from the Geplusmn O vapourTherefore Mg2GeO4 reg lms were grown by depositing an amorphous GeO2 reg lm on to
the MgO substrate at a low temperature (773 K) and subsequently reacting the reg lm
in air at 1300 K
For the X-ray investigations a Philips XrsquoPert MRD X-ray diŒractometer was
used which was equipped with a Eulerian cradle and Cu Ka radiation The stresswas measured using a thin reg lm attachment (parallel plate collimator and graphite
secondary monochromator 2sup3 resolution 0358) High resolution was achieved
using a Ge(220) four-remacr ection channel-cut monochromator and a similar analyser
Specimens for transmission electron microscopy (TEM) were prepared by standard
techniques namely grinding dimpling and ion milling A Philips CM20 twin elec-tron microscope was used with a double-tilt heating stage for in-situ observations up
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 111
to 1300 K High-resolution images were obtained with a JEOL 4000 EX high-resolu-
tion transmission electron microscope
4 R ESULTS
41 Overview
Both Mg2TiO4 and Mg2GeO4 spinel reg lms were grown epitaxially as shown byXRD electron diŒraction and HRTEM all revealing the orientation relationship
hellip001daggerspinelkhellip001daggerMgO permil100Šspinelkpermil100ŠMgO hellip2dagger
This orientation relationship results in low lattice misreg t along the spinelplusmn MgO inter-
face namely only 02 for Mg2TiO4 plusmn MgO and 2 for Mg2GeO4 plusmn MgO
Corresponding to these misreg t values the perfection of the reg lm orientation is diŒer-ent for the two spinels While the Mg2TiO4 rocking curve peaks display a very small
full width at half-maximum (FWHM) of the order of 0058 which is almost identical
with the FWHM of the substrate peaks the tilt FWHM of the Mg2GeO4(004)
rocking curve is about 158 (A typical MgO substrate consists of two to reg ve
small-angle grains with tilt FWHMs between 00078 and 00168 of the individual
grains The grains are tilted against each other and the envelope of the tilt distribu-tion has a FWHM between 0028 and 0078 The Mg2TiO4 reg lm reproduces this tilt
distribution thus the upper limit of its tilt FWHM is in fact only about 0018)Typical defects of the spinel reg lms are cation antiphase boundaries in Mg2TiO4
(Hesse and Bethge 1981 Hesse 1987 Sieber et al 1997) and small-angle grain
boundaries in Mg2GeO4 Figure 1 (a) shows a diŒraction pattern and reg gure 1 (b) aplan-view dark-reg eld image of a Mg2TiO4 reg lm reg gures 1 (c) and (d) are TEM cross-
section images of a Mg2TiO4 reg lm and reg gure 1 (e) a TEM cross-section image of a
Mg2GeO4 reg lm Electron diŒraction patterns (reg gure 2 (a)) display weak superstruc-
ture remacr ections of the Mg2GeO4 reg lm HRTEM lattice plane images Fourier-reg ltered
for the noise reduction show corresponding weak contrast features (reg gure 2 (b)) Theinset in reg gure 2 (b) presents the computer-generated Fourier transform of the high-
resolution image which also shows the weak superstructure remacr ections
42 Stress-induced distortions of the cubic phase
At room temperature the spinel reg lms are tetragonally distorted because of the
diŒerence between the thermal expansion coe cients of reg lm and substrate as wasrevealed by precise XRD measurements determining and comparing the lattice
plane spacings of those remacr ections which should show equal spacings in a cubic
phase DiŒerent 931 remacr ections and diŒerent 862 and 10 2 0 remacr ections respec-
tively were compared revealing a tetragonal distortion of the reg lm lattice The lattice
parameters ak and a were measured (reg gure 3) that is the lattice parameters inthe directions parallel and perpendicular respectively to the substrate surface The
pseudocubic parameter apc was calculated from ak and a (Blum 1997) assuming a
reasonable value for Poissonrsquo s ratio of cedil ˆ 035 (as given by Hellwege (1969) for the
MgAl2O4 spinel) The values thus obtained are shown in table l together with the
distortions calculated according to
ˆak iexcl apc
apc
hellip3dagger
112 St Senz et al
The negative sign of in table 1 shows that the distortion is a biaxial compression
in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2
(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)
and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical
compression after cooling the reg lms from the growth temperature of 1300 K to
room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than
the theoretical values part of the strain is obviously relaxed
Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio
cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from
the strains (Brantley 1973) which results in the experimentally determined stresses
frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp
Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in
the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and
frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will
be subject to the inmacr uence of a rather high stress
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 113
Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
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BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
to 1300 K High-resolution images were obtained with a JEOL 4000 EX high-resolu-
tion transmission electron microscope
4 R ESULTS
41 Overview
Both Mg2TiO4 and Mg2GeO4 spinel reg lms were grown epitaxially as shown byXRD electron diŒraction and HRTEM all revealing the orientation relationship
hellip001daggerspinelkhellip001daggerMgO permil100Šspinelkpermil100ŠMgO hellip2dagger
This orientation relationship results in low lattice misreg t along the spinelplusmn MgO inter-
face namely only 02 for Mg2TiO4 plusmn MgO and 2 for Mg2GeO4 plusmn MgO
Corresponding to these misreg t values the perfection of the reg lm orientation is diŒer-ent for the two spinels While the Mg2TiO4 rocking curve peaks display a very small
full width at half-maximum (FWHM) of the order of 0058 which is almost identical
with the FWHM of the substrate peaks the tilt FWHM of the Mg2GeO4(004)
rocking curve is about 158 (A typical MgO substrate consists of two to reg ve
small-angle grains with tilt FWHMs between 00078 and 00168 of the individual
grains The grains are tilted against each other and the envelope of the tilt distribu-tion has a FWHM between 0028 and 0078 The Mg2TiO4 reg lm reproduces this tilt
distribution thus the upper limit of its tilt FWHM is in fact only about 0018)Typical defects of the spinel reg lms are cation antiphase boundaries in Mg2TiO4
(Hesse and Bethge 1981 Hesse 1987 Sieber et al 1997) and small-angle grain
boundaries in Mg2GeO4 Figure 1 (a) shows a diŒraction pattern and reg gure 1 (b) aplan-view dark-reg eld image of a Mg2TiO4 reg lm reg gures 1 (c) and (d) are TEM cross-
section images of a Mg2TiO4 reg lm and reg gure 1 (e) a TEM cross-section image of a
Mg2GeO4 reg lm Electron diŒraction patterns (reg gure 2 (a)) display weak superstruc-
ture remacr ections of the Mg2GeO4 reg lm HRTEM lattice plane images Fourier-reg ltered
for the noise reduction show corresponding weak contrast features (reg gure 2 (b)) Theinset in reg gure 2 (b) presents the computer-generated Fourier transform of the high-
resolution image which also shows the weak superstructure remacr ections
42 Stress-induced distortions of the cubic phase
At room temperature the spinel reg lms are tetragonally distorted because of the
diŒerence between the thermal expansion coe cients of reg lm and substrate as wasrevealed by precise XRD measurements determining and comparing the lattice
plane spacings of those remacr ections which should show equal spacings in a cubic
phase DiŒerent 931 remacr ections and diŒerent 862 and 10 2 0 remacr ections respec-
tively were compared revealing a tetragonal distortion of the reg lm lattice The lattice
parameters ak and a were measured (reg gure 3) that is the lattice parameters inthe directions parallel and perpendicular respectively to the substrate surface The
pseudocubic parameter apc was calculated from ak and a (Blum 1997) assuming a
reasonable value for Poissonrsquo s ratio of cedil ˆ 035 (as given by Hellwege (1969) for the
MgAl2O4 spinel) The values thus obtained are shown in table l together with the
distortions calculated according to
ˆak iexcl apc
apc
hellip3dagger
112 St Senz et al
The negative sign of in table 1 shows that the distortion is a biaxial compression
in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2
(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)
and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical
compression after cooling the reg lms from the growth temperature of 1300 K to
room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than
the theoretical values part of the strain is obviously relaxed
Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio
cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from
the strains (Brantley 1973) which results in the experimentally determined stresses
frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp
Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in
the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and
frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will
be subject to the inmacr uence of a rather high stress
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 113
Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
The negative sign of in table 1 shows that the distortion is a biaxial compression
in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2
(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)
and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical
compression after cooling the reg lms from the growth temperature of 1300 K to
room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than
the theoretical values part of the strain is obviously relaxed
Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio
cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from
the strains (Brantley 1973) which results in the experimentally determined stresses
frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp
Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in
the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and
frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will
be subject to the inmacr uence of a rather high stress
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 113
Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
114 St Senz et al
Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image
Figure 3 Schematic diagram of the measured lattice parameters
Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th
for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms
ac ak a apc exp th
(pm) (pm) (pm) (pm) ( ) ( )
Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic
distortions of the tetragonal grains
Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the
cubic-to-tetragonal phase transition mentioned in 2 which can be induced by
annealing at a temperature of 873 K In principle these grains may have three
diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix
because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a
and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third
orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns
are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of
a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-
tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase
High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is
distorted but also that the grains of the tetragonal spinel phase are distorted by the
thermally induced stresses occurring after the samples had been cooled to room
temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the
MgO(002) peak ( is the angle between the incident beam and the sample surface
and 2sup3 is the angle between the incident beam and the detector Here the measure-
ment was performed using a Ge(220) channel-cut monochromator in the primary
beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the
measurement was performed with a coupled scan The measured XRD intensity of
the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6
shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around
ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 115
Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
116 St Senz et al
Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4
Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
geneous in spite of the strain as indicated by the interference fringes between the
reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the
reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness
most probably owing to the formation of the tetragonal grains of diŒerent orienta-
tions (In another experiment we could restore the thickness interference fringes by
heating the sample to above 1200 K ie a temperature su cient for a fast reverse
transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the
average lattice parameter a due to the presence of both the cubic and the tetragonal
spinel in the reg lm and in particular their distortion at room temperature
The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel
were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the
tetragonal spinel with their c axes parallel and perpendicular to the substrate surface
can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for
grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains
of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c
grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7
the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-
plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak
intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a
linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to
three which is typical of linear growth in three dimensions with no further nuclea-
tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 117
Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo
nuclei within the cubic phase secondly the maximum tetragonal volume portion
during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the
reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)
Further XRD investigations conreg rmed that at room temperature also the tetra-
gonal grains are distorted owing to the thermal expansion mismatch According
to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic
lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the
thermally induced tetragonal distortion of the cubic matrix of our reg lms after
cooling them from the deposition temperature to room temperature namely
Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now
dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-
tion of the in-plane grains has been conreg rmed by the fact that remacr ections which
coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for
the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm
but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger
(311)t remacr ections because of intensity problems due to the small fraction of c grains
in the sample Thus it has been proven that the distortions of the matrix and the
tetragonal grains have the same sign The lattice parameters of the in-plane grains
parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of
coherent tetragonal grains As is shown in appendix A the coherent c grains are
under a higher elastic strain than the coherent in-plane grains are which is a result
that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition
44 Tetragonal-to-cubic phase transition
The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the
tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view
TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-
to-tetragonal phase transition and prepared to enable the in-situ study of the reverse
transition For room temperature the TEM dark- and bright-reg eld images (reg gures
8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal
grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with
several grains three patterns are superimposed one for each orientation (reg gures 5 (d)
and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-
nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do
the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K
shows superlattice remacr ections of the remaining relatively stable in-plane grains but no
remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-
peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))
118 St Senz et al
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 119
Fig
ure
8
(a)
TE
Md
ark
-regel
dim
age
usi
ng
are
macrec
tion
from
the
tetr
agon
alsp
inel
(b
)a
bri
gh
t-reg
eld
image
of
tetr
ago
nalsp
inel
gra
ins
ina
Mg
2T
iO4
reglm
at
roo
mte
mper
atu
re(d
)E
lect
ron
diŒ
ract
ion
at
roo
mte
mper
atu
re
show
ing
the
remacr
ecti
ons
of
tetr
agon
algra
ins
of
thre
eo
rien
tati
on
sth
ecu
bic
spin
elm
atr
ix
an
dth
eM
gO
sub
stra
te
(e)
Ele
ctro
nd
iŒra
ctio
nta
ken
at
973
Ksh
ow
ing
that
only
the
tetr
ago
nalgra
ins
of
the
in-p
lan
eori
enta
tio
nare
left
(c
)B
right-
regel
dim
age
an
d(f
)el
ectr
on
diŒ
ract
ion
taken
at
1173
K
wit
hth
etr
an
siti
on
toth
ecu
bic
spin
elreg
nis
hed
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
5 DISCUSSION
51 Mg2TiO4 reg lms
The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even
slower than observed before in bulk samples Wechsler and Von Dreele (1989) have
shown that for powder samples annealed at 773 K 1 month was necessary to satu-
rate the superstructure remacr ection intensity whereas at 873 K it took only a few days
The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained
material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal
phase at 873 K took several weeks This transition rate is considerably lower than
that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data
is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition
arises as the latter is a pure cation ordering process with no need for any extensive
rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via
purely three-dimensional growth with no further nucleation occurring Stress arises
as early as during the nucleation of the tetragonal grains resulting in distortion and
a corresponding elastic enthalpy term He With the growth of tetragonal grains
the elastic enthalpy increases implying a further decrease in the overall driving force
of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and
He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of
tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-
tive and hence the transition stops In the reg ne-grained powder samples used by
Wechsler and Von Dreele most probably the stress is relaxed by shape distortions
of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way
Unlike the work of Wechsler and Von Dreele in our experiment there is a
tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive
stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent
orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to
the substrate surface so that the formation of the tetragonal in-plane grains is
favoured The distortion of the latter (see table A 1 in appendix A) as well as their
elastic enthalpy is lower than that of the c grains As a consequence during the
reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the
c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and
the tetragonal-to-cubic transitions are aŒected by the stress present causing strong
deviations from the bulk transition kinetics
52 Mg2GeO4 reg lms
The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains
may help to explain the unexpected occurrence of superlattice remacr ections in the
Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4
spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane
120 St Senz et al
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an
ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the
superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4
The low non-zero degree of inversion assumed to enable an ordering process to
occur in Mg2GeO4 might be the result of the growth conditions the large MgO
excess certainly promotes the formation of a Mg-rich spinel In such a spinel some
of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature
(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that
our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice
parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our
reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg
ions should occupy tetrahedral sites
In principle cation ordering on these tetrahedral sites is also possible but the
crystallography of such an ordering process would considerably diŒer from that of
the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering
on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the
octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal
(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron
diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-
tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider
an octahedral site ordering most probable in our Mg2GeO4 thin reg lms
With respect to the signireg cance of a partial inversion as a prerequisite to the
assumed octahedral site ordering the following should be pointed out In the litera-
ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of
Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-
tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2
reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the
spinel formed directly with no olivine phase forming by the reaction of MgGeO3
with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm
samples a spinel forms that is not entirely normal A diŒerent formation route may
indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra
of the Mg2GeO4 spinel showed additional lines if the spinel was produced by
annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we
should like to say that whatever the reg nal reason may be there seems to be a partial
inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering
reg nally resulting in the tetragonal superstructure observed
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 121
There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
53 Consequences for both spinel reg lms
Mechanical stress strongly aŒects the transition kinetics between the cubic and
tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics
are inmacr uenced by the crystallographic orientation of the tetragonal grains
There are two sources of stress in the spinel reg lms One type of stress develops
owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid
MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase
within the single-crystal matrix of the other phase owing to the diŒerent lattice
parameters Both these types of stress superpose and interact Unlike bulk powder
samples with only the second type of stress occurring (which can relax by changes in
the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way
The amount of stress depends on the orientation of the tetragonal grains with
respect to the reg lm plane which reg nally results in the above-mentioned orientation
sensitivity of the transition kinetics As a consequence of these factors tetragonal
Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others
Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction
patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm
plane
Surprisingly a superstructure very similar to the latter is observed in the electron
diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the
above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the
observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to
their diŒraction superstructures allows the conclusion to be drawn that a cation
ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the
precondition for this octahedral site ordering As discussed above this condition
is most probably fulreg lled in our thin reg lms
6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from
cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared
with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As
a consequence grains of a certain crystallographic orientation of the tetragonal
phase are more stable than others The existence of only the more stable grains in
the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns
Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising
because this spinel is considered to have no tetragonal phase at all The detailed
discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-
sion caused by the special conditions under which the thin reg lms were grown leads to
the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may
122 St Senz et al
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
cause conditions that enable a tetragonal phase to be formed in a spinel which under
normal bulk conditions has no tetragonal phase
ACKNOWLEDGEMENTS
The authors are grateful to Dr P Werner for the HRTEM images and to
Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of
this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany
A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of
1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently
annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the
MgO substrate and to be stressed by the diŒerence between the thermal expansion
coe cients of spinel and MgO The same linear thermal expansion coe cient was
assumed to be valid for both the cubic and the tetragonal spinel Three values of this
coe cient were used in succession for the calculations reg rstly the literature value of
10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-
lated stress of the cubic matrix at room temperature thirdly the value of
118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature
The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c
grains are under higher elastic strain than the in-plane grains irrespective of which of
the thermal expansion coe cients is valid
REFERENCESAVRAMI M 1939 Chem Phys 7 1103
BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg
BLUM W and HESSE D 1997 Solid St Ionics 95 41
Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 123
Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not
In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO
hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO
ahellip10iexcl6
Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t
10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114
BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-
als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik
Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542
124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4
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