small-polaron versus band conduction in some transition-metal oxides

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Small-polaron versus band conduction insome transition-metal oxidesA.J. Bosman a & H.J. van Daal aa Philips Research Laboratories, N.V. Philips' Gloeilampenfabrieken,Eindhoven, The Netherlands

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A D V A N C E S IN P H Y S I C S

VOLUME 19 1970 NUMBER 77

Smal l -polaron versus Band

Conduct ion in s o m e Transi t ion-metal Oxides

By A. J. BOSMAN and H. J. vAN DAAL

Philips I~esearch Laboratories, N.V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands

ABSTI~AOT

I n th i s p a p e r all a t t e m p t is m a d e to es tab l i sh t he n a t u r e of free charge carr iers a n d of charge carr iers b o u n d to cen t res in p - t y p e NiO, CoO a n d MnO a n d in n - t y p e M n O a n d a-Fe203.

~ o r free charge carriers , d.c. conduc t iv i ty , Seebeek coefficient a n d Hal l effect are considered. Ef fec ts ar is ing f rom i n h o m o g e n e o u s c o n d u c t i o n a n d i m p u r i t y c o n d u c t i o n are discussed. I m p u r i t y conduc t i on appea r s to h a v e a s t r ong inf luence on t r a n s p o r t p roper t i e s in the case of a-F%O3, less so in NiO, whereas no inf luence of th i s effect h a s been found in CoO a n d MnO. I t is s h o w n t h a t NiO a n d CoO do n o t exh ib i t t h e fea tu res charac te r i s t ic of smal l -po la ron c o n d u c t o r s b u t r a t h e r c a n be cons i s t en t ly conce ived of as i a rge-pola ron b a n d semiconduc to r s . I t is s u g g e s t e d t h a t m a g n e t i c res i s tance due to e x c h a n g e coupl ing b e t w e e n charge-car r ie r sp in a n d ca t ion spins p l ays a n i m p o r t a n t role. T h e a n o m a l o u s b e h a v i o u r of t h e Hal l effect in NiO a n d a-Fe203 is ex t ens ive ly discussed. I n con t rad i s t inc t ion to N iO , CoO a n d n - t y p e MnO, free charge carr iers in p - t y p e M n O seem to h a v e smal l -po la ron charac te r .

F o r charge carr iers b o u n d to eentres , dielectric loss, h i g h - f r e q u e n c y conduc- t i on a n d opt ical a b s o r p t i o n are considered. The dielectric loss d a t a re la te to Li or N a cen t res in NiO, CoO a n d MnO a n d to Ti, Zr, Sn, Ta , N b a n d p r e s u m a b l y o x y g e n v a c a n c y cen t res in a-F%O~. I t is conc luded f rom the dependence of dielectric loss on f r equency a n d t e m p e r a t u r e t h a t b o u n d charge carr iers are smal l po la rons . I t is s h o w n for t he eases of NiO a n d a-Fe2Oa t h a t a p a r t f r o m sma l l -po la ron effects, d isorder due to locally v a r y i n g electric fields d e t e r m i n e s t h e n a t u r e of dielectric loss. The smal l -po la ron cha rac t e r of b o u n d charge carr iers in NiO is cor robora ted b y t he b e h a v i o u r of h igh- f r e q ue n c y c o n d u c t i o n a n d opt leal abso rp t ion due to cen t res a n d also b y t h e m a g n i t u d e of i m p u r i t y conduc t ion .

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CONTENTS

PAGE

§ i. INTRODUCTION. 2

§ 2. SOME TIIEORETICAL ASPECTS. 6 2.1. D.C. Conductivity. 7

2.1.1. Large-polaron band conduction. 7 2.1.2. Small-polaron hopping conduction. 9 2.1.3. Small-polaron band conduction. 12

2.2. ]=[all Effect. 14 2.3. Seebeck Effect. 16 2.4. Dielectric Relaxation Loss. 18 2.5. Optical Absorpt ion. 21

§ 3. ]:~ESISTIVITY AND SEEBEC]~ COEFFICIENT ; DETERMINATION OF

DI%IFT IV[OB ILITY. 23

3.1. I11homogeneous Conduction. 24 3.2. Impurity Conduction. 27 3.3. Estimates of /~n from Resist ivity Data Solely. 31 3.4. Estimates of ~E from Resist ivity and Seebeck Coefficient. 40

§ 4. HALL COEFFICIENT AND I~ESISTIVITY ; INTEI%pI%ETATION OF

HALL AND DItIFT MOBILITY. 54 4.1. NiO. 54 4.2. CoO. 70 4.3. a-F%O 3. 75 4.4. MnO. 79

§ 5. DIELECTRIC RELAXATION LOSS AND OPTICAL ABSOI%I:'TION ;

MOBILITY ~ OF BOUND CtIA~GE CARRIERS. 82 5.1. Dielectric l~elaxation Processes ( 0 - 5 x l0 s Hz). 82

5.1.1. Ni0 . 83 5.1.2. CoO. 92 5.1.3. a-Fe~Oa. 95 5.1.4. MnO. 100

5.2. t i l th-frequency Conductivity in N i O ( ~ 2 × 10 l° Hz). 100 5.3. Optical Absorption in S i C and CoO (4 × 1011-6 × 10 ia Hz). 103

§ 6. DISCVSSION. 105 ACI~NOWLEDGMENTS. 112 I~ErF.RENCES. 112

§ 1. INTRODUCTION CONSDE~ABLE advances in physics have been made in recent years-- since about 1963--with respect to electronic phenomena in transition- metal oxides. Primarily, advances made in the experimental physics of this field will be discussed in this paper. On one hand progress was made by a sedulous study of the behaviour of electrical conductivity (~) and SeebecIc coej~cient (~) with varying temperature, impurity content and deviation from stoiehiometry. This led to conclusions quite different from those arrived at in an earlier period of invest igat ions from about 1950 unti l 1963. On the other hand, progress consisted in the successful measurement of effects which could not be determined before or were scarcely considered. The Hall coeJficient (RH), which was found before the year i963 in most eases to be immeasurably small and was even bel ieved by some authors to be essentially zero, has now been established to a considerable extent for these materials under quite varying eondi-

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Conduction in some Transition-metal Oxides

tions. Furthermore, data have been obtained from dielectric and mechanical loss measurements. A new field of research has been opened up by the experiments on optical absorption in the infra-red. Finally the first results from direct measurements of the charge-carrier drift mobility have become available. The main purpose of this paper is to summarize results of recent investigations--since about 1963--made at this laboratory on the electronic properties of some transition-metal oxides. We will restrict ourselves to the results obtained on the substances NiO, CoO, a-Fe203 and MnO. As far as these oxides are concerned, an a t tempt will be made to draw conclusions with regard to the nature of charge carriers, free or bound to centres, t~elevant data on these compounds obtained at other laboratories will be included in the discussion.

Reviews of modern developments in theoretical and/or experimental research on low-mobility materials had already been presented by Gerthsen, l~eik and Kauer (1966) and quite recently by Adler (1968), Appel (1968), Klinger (1968) and by Austin and Mott (1969). The present authors believe that there is still a need for a critical inspection of experimental data concerning a limited number of low-mobility materials with more or less analogous properties, and for a c()nfrontation of these data with the theory.

The arrangement in this paper of experimental facts and their interpretation are as follows : Firstly the determination of the drift-mobility behaviour of free carriers from d.c. electrical resistivity (~) and Seebeek coefficient is discussed (§ 3). Then the Hall coefficient data and an interpretation of I-Iall and drift-mobility behaviour are presented (§ 4). Section 5 considers the behaviour of charge carriers bound to eentres determined from a.c. resistivity, dielectric relaxation loss and optical absorption. Finally a discussion of the conclusions reached in §§ 3, 4 and 5 is given in § 6, with special attention to NiO. A recapitulation of the relevant theory can be found in § 2. Prior to this a short account will be given of the historical background.

Research on the mechanism of electrical conduction in NiO started about three decades ago with the work of Wagner (1933) and of de Boer and Verwey (1937). According to Verwey (1951), the resonance integral for electron transfer between neighbouring cations in NiO is significantly reduced, due to the small overlap of the 3d wave functions and by the presence of oxygen ions. An appreciable conductivity can only be achieved by the introduction of cations with different valency at erystallo- graphically equivalent lattice points. This situation can be brought about by introducing a deviation from stoiehiometry or by incorporating in the lattice appropriate foreign ions, with a fairly well fixed valency which differs from that of the substituents : this is the ' controlled valency ' method. A well-known example is that of Li in NiO. At low temperatures the holes, the Ni 3+ valency states, are bound by a Coulomb force to the nickel vacancy (VNi) or Li+ centre. At higher temperatures they are freed and then, according to Verwey, can wander through the lattice

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without the necessity of thermal activation. The same author also cautioned against the possible presence of counterdoping foreign elements in general and of high-ohmic grain-boundary layers in the case of ceramic material.

The first extensive s tudy of electrical transport properties of c~-Fe203 and NiO was made by Morin (1951, 1954). Conduction was achieved either by deviation from stoichiometry or by dope (~-F%Oa : 0"01 to 1 at. °/o Ti ; NiO : 0.01 to 1 at. ~o Li). From the experimental data for

and ~, the author arrived at the conclusion tha t the mechanism of conduction for the two substances is similar but differs fundamentally from that of broad-band semiconductors. Reasonable agreement between the charge-carrier concentration as derived from ~ and the impurity concentration known from chemical analysis could only be obtained if it was assumed that the heat of transfer (A kT) is zero and the density of states equals the concentration of cations (N o per cm3). This result suggested that conduction involves localized levels, or bands less than kT wide. Combination of the data for ~ and ~ led lV[orin to the conclusion that the drift mobility (/XD) has very low values, between about 10 -5 and 10 -~ cm2/v see at 500°K, and increases exponentially with temperature. Values of the activation energy (U) ranged from about 0.7 ev at low temperatures and low dopes to about 0.1 ev at high temperatures and high dopes. The thermally activated behaviour of ~ was found to be largely determined by /%, a minor part only being due to the concentration of electrons (holes). The large values of U were ascribed to an inhomogeneous distribution of impurities in the material, leading to local fluctuations of the charge-carrier potential : the ' wavy-band model '. The limiting value of 0-1 ev was felt to be fundamentally correlated with the conduction mechanism. Thus it appeared that, contrary to the expectations of Verwey, a charge carrier when freed from a centre still needs thermal activation for further motion.

~-F%0 a and NiO are antiferromagnetie with a Ndel temperature (TN) of 960 and 520°K, respectively. Between 250°K, for pure material, and 960°K, ~-Fe20 a is weakly ferromagnetic (~orin 1950). For the case of e-F%0 a up to 960°K, Morin discovered an anomalous behaviour of the Hall coefficient. According to Morin, no normal Hall effect could be detected either for e-F%O a or for NiO. This was concluded to be consistent with the low drift mobility.

Heikes and Johnston (1957) studied the behaviour of ~ for p-type ceramic samples, including NiO, CoO and MnO with Li contents up to the large value of about 20 at. %. These authors attached much importance to the fact tha t in all substances the activation energy (Q) for conduction at first rapidly decreases with increasing Li content up to 2 at. ~o and then , up to the highest dopes used, remains at an almost constant value, which varies for these materials between 0.1 and 0-6 ev. I t is argued that this residue of Q stems solely from the drift mobility, because at these high dopes there is no room for the holes to be freed

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from the Li centres. It is then postulated that for the lower dopes as well, Q is entirely a mobility activation energy, essentially all holes being free from the Li eentres in the whole temperature and concentration range considered. In the opinion of these authors tq) values are very low (10-9-10 -a em2/v see).

The origin of the activated character of/*n is sought in the polarization and the accompanying lattice distortion induced by the hole itself. A hole would be t rapped in its own polarization because it stays on a given lattice site for a much longer time than the vibrational lattice period. It follows that thermal activation is needed for the motion of the h01e. The model for conduction as described above has since become known as the ' thermally activated hopping mechanism '

From a revaluation of the relative energetic positions of the conduction levels as given earlier by Morin, van Houten (1960) concluded tha t p-type conduction in NiO is predominantly achieved by holes in the Ni 2+ levels. For the case of ceramic NiO doped with Li up to 20 at. %, it was inferred from data for ~ that the position of the Li-aceeptor level would only be about 0-03 ev above the Ni ~+ levels. This low value for the accepter ionization energy implied that exhaustion of the accepters would already be largely completed at room temperature. As a consequence values for t*D at room temperature derived from data for ~ were found to be very low : 2 × 10 -5 to 6 × 10 -4 em2/v sec. Furthermore, the relatively high values encountered for the activation energies for conduction ((2), between about 0-2 and 0-5 ev, were then at tr ibuted mainly to the thermally activated character of ~D"

A review of the situation before 1961 with respect to low-mobility transition-metal compounds has been presented by Jonker and van llouten (1961). From data for p obtained at room temperature as a function of dope, these authors arrived at the general conclusion that in quite a number of compounds (including NiO and ~-F%Os) the donors or accepters are exhausted. This led to the opinion that values for ~D at room temperature are very low : 10 -3 to I0 -s em2/v sec. The occurrence of relatively large values for Q was attributed to an exponential increase of/x D with T. From the behaviour of ~ as a function of dope, assuming exhaustion of the dope centres, it was concluded that the density of states in many compounds (including NiO and e-F%Oa) is already roughly equal to N o at room temperature. This was seen as a direct proof of the localized character of the conduction levels.

In the period briefly reviewed above many authors arrived at the conclusion, in contradistinction to the expectations of Verwey, tha t the drift mobility of charge carriers in compounds such as NiO and a-Fe20 a increases exponentially with temperature. Moreover it was concluded that g-n at room temperature has exceptionally low values. These features of the drift mobility a t t racted the attention of theoretieians, resulting in the development of the theory of ' small-polaron ' conductors (see § 2).

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However, results obtained since about 1963 have led to the conclusion that certainly as far as NiO and CoO are concerned, and probably ~-Fe20 a too, t~D at room temperature is not exceptionally small (of the order of 1 cm2/v sec) and does not exhibit a thermally activated behaviour (see § 3). Of the oxides considered the only exception probably is p-type MnO (see § 3).

The conclusions arrived at in the earlier period of research are partly invalid because insufficient account was taken of effects arising from grain boundaries, high dopes and of counter dope. Improper handling of ceramic material in many cases led to the formation of high-ohmic grain- boundary layers. The resulting high overall resistance led to wrong conclusions with respect to magnitude and temperature dependence of the drift mobility (see § 3.1). A too high dope level partly masked the properties of the pure materials, due to significant interference between the centres. The neglect of counter dope prevented many authors from considering impurity conduction as a possible alternative. This mechanism for conduction should certainly be taken into account for the cases of e-F%03 and NiO (see § 3.2).

The disappearance of the apparently clear small-polaron character of free charge carriers in NiO, CoO and ~-F%03 has not facilitated an understanding of the conduction mechanism in these materials. I t will become evident from what follows that it is very difficult in this respect to arrive at definite conclusions.

§ 2. SOME THEOI~ETICAL ASPECTS

In the ionic compounds considered in this review, charge carriers, free or bound to centres, should certainly be regarded as polarons. A polaron consists of the charge carrier and the distortion of the ionic lattice induced by the carrier itself. In this paper we will distinguish between large and small polarons. For the former the distortion of the lattice, induced around a charge carrier, extends over distances larger than, and for the latter over distances smaller than the lattice constant. Energetically, for the cases of large and small polarons, half the bandwidth is larger and smaller, respectively, than the maximum polaron binding energy. This latter quant i ty is the energy gained by an infinitely slow carrier (zero bandwidth) due to polarization and distortion induced in the lattice by the carrier itself. An extensive review of large and small-polaron theory has been presented by Appel (1968). A recapitulation of the theoretical outcome, as far as this seems to be relevant to the substances investigated, will successively be presented for the cases of d.c. conductivity, Seebeck and Hall coefficient, a.c. conductivity and dielectric relaxation loss, and finally for optical absorption.

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Conduction in some Transition-metal Oxides 7

2.1. D.C. Conductivity 2.1.1. Large-polaron band conduction

The description of large-polaron conduction is based essentially on a band model. For a one-dimensional lattice, with constant a, the energy of electrons as a function of the wave vector/c, when disregarding polaron effects, is given in a tight-binding approach by :

E(/C) = 2J(1 - cos/ca), (1)

where J is the resonance or electron-transfer integral. The band width A W = 4J. For the three-dimensional simple-cubic case A W = 12J. For small/c values the effective mass of the electrons is defined by :

E(/c) = h~/c2/2m * (2) with

m* = h2/2Ja 2 . . . . . . (3)

For a lattice constant a=3~ , a value for J of 0.3 ev corresponds to m ¢ _~ t o o .

I t is of importance in connection with the controversy ' small-polaron versus band conduction' to establish the relevant limits of the drift mobility. In a band model the lower limit of/~ depends on the width of the band compared with kT and on the criterion employed. In the case of broad energy bands, where the one-dimensional band width A W ( = 4 J ) is large compared with ½kT up to high temperatures, the criterion ~/r <~/CT (FrOhlieh and Sewell 1959, Herring 1960) leads with

i~ = er/m* (4) to

> 30 m~ 3OO T (cm~/v sec). (5)

The criterion l~/r <~/cT is equivalent to

1 > A, (6)

where 1 is the mean free path of the charge carriers and A is the de Broglie wavelength (2=h/mv). A less stringent criterion to the applicability of the band model is the condition

l>a, (7)

which in the same circumstances (4J> ½/cT) leads with/~ =er/m* to

~m°~l/2~300~l/2(cm2/v sec). (8)

< 1 In the case of narrow energy bands, where 4 J ~ ~/cT, the condition l> a is equivalent to h/r<~J. This equivalence follows from the relations 1 = (vr) and (v) =4Jaflrh, where (v) is the velocity of the carriers averaged

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over the states within the band. The mobility is described by the Einstein relation :

eD be =~-~, (9)

where D = (v2z) and ( v 2) = 2J2a2/h 2. The lower limit of the mobility is then given by :

~v ea 2 J be~ ~ U~' (lO)

where e a 2 / ~ _ l . 2 cm2/vsec in the case where a =3 4. A similar result has been derived by FrOhlieh and Sewell (1959).

In a broad-band semiconductor, e.g. with m* = m o (J=0-3 ev), at T -- 1200°K (]cT--- 0.1 ev) the criterion 1 ~> )~ leads to a lower limit (eqn. (5)) : be> 8 cme/v see. In the same circumstances the criterion l> a leads to (eqn. (8)) : be > 2.5 cm~/v see. I f the band width is an order of magnitude smaller (J=0.03 ev, m*=10mo) , at T=]200°K the criterion l > a in the broad-band approximation leads to be>0-8 em2/v see. In the same circumstances the criterion 1 > a leads in the narrow-band approximation to be > 0.5 cm2/v sec. Further reduction of the band width leads in the narrow-band approximation to a further decrease of the lower limit of be.

Large-polaron behaviour is determined by the energetic coupling between charge carriers and longitudinal optical modes. The relevant coupling constant is (Fr/~hlich 1954) :

where h~o o-- g denotes the characteristic temperature of the longitudinal optical phonons. Three regions of coupling strength can be defined, viz. weak (~<1), intermediate (1~<~<5) and strong (~>5). I f the large-polaron model applies to the substances considered in this paper, the intermediate-coupling region seems appropriate (see § 4). According to Lee et al. (1953, 1955), the polaron energy is a function of the wave vector k :

E (k ) = - ~hw o + h2k~/2mp *, (12)

where m,* is the polaron effective mass :

rap* •(1 -~ ~/6)m*. (13)

I f lattice scattering is dominant, the mobility of large polarons in the intermediate-coupling region is given by :

1 e ( m * ' f 0 be= 2~oJ o m* \ m y * / f(=) exp ~, (14)

where f(~) is about unity. The above formula is valid for T ~ 0.

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Conduct ion i n some Trans i t i on -me ta l Oxides 9

2.1.2. Smal l -po laron hopp ing conduct ion

Yamashita and Kurosawa (1958) were the first to derive a theoretical expression for smMl-polaron hopping conduction. In the case of semiconductors where the conduction band is sufficiently narrow, i.e. m* is large, and moreover, the electron-lattice interaction is strong, the small- polaron model applies. The electron is now so slow tha t it can be imagined to become self-trapped in its own polarization field generated in the ionic lattice, and thus to become localized at a certain site. A wave-like expansion of this localized state (' tunnelling ') should be possible throughout a perfectly periodic lattice within a certain time %(-~h/AW). Conduction would then be of the band type : ' s m a l l - polaron band ' conduction. However, phonon-induced transitions of the localized electron from ion to ion within a time ~ will dominate if r < % : ' smal l -polaron hopp ing ' conduction.

An upper l imi t for the mobility in the hopping region can be estimated. The mobility follows from the Einstein relation (eqn. (9)) with

and is given by : D = ½a'l -,

e a 2

1~--2T]c T . (15)

The condition for the occurrence of self-trapping :

v>> To, (16)

where Vo is the time period of a lattice vibration, leads to an upper limit :

ea ~ 1 ea 2 hco o

/x~2%kT 47r h kT" (17)

In the case where a ~ 3 X a n d k T ~ hw 0 it follows, with ea2//i ~ _ 1.2 cm2/v sec, tha t

/ ~ 0 - 1 cm2/v sec. (18)

The formula arrived at by Yamashita for small-polaron hopping mobility has been confirmed by later authors. This formula was derived on the assumption tha t the electron-transfer integral J , initiating motion of the localized small polaron, may be considered as a small perturbation. Furthermore, it is assumed that there is no dispersion present in the optical vibration spectrum. The electron-lattice interaction has been estimated with the aid of a continuum approximation for the energy of polarization induced by the electron in the lattice. The relation between the electron-latt ice coupl ing constant S defined by ¥amashi ta and the constant 7, as introduced later on by Holstein (1959), is :

S = 2 7 . . . . . . (19)

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In this paper the Holstein constant 7' will be used. Yamashita arrives at a coupling constant in the continuum approximation :

7'_~ - (20)

I n this approximat ion, 7' for the case of 1NiO (for values of the relevant parameters, see § 4) would have a value of about 2.5. Yamashita 's result for the mobility is :

-21r 1/2 ea 2 ( j u hOJo sinh

exp { - 27' tanh \7 ~-~]j. (21)

The above expression for the mobility could be obtained in explicit form with the aid of mathematical approximations valid only if > i T ~ ~0 and 27,> 1.

At high temperatures T > ½0, an approximate formula for the mobility reads :

= ~ ) Y \ ~ o / 7"~'~ \-~-T/ e~p (-½7"h~Oo/~T) (22)

o r

where

1 /~oc~-~ exp ( - U/kT), (23)

u = ½7'h~0. (24)

Holstein (1959), in his treatment of the small-polaron model, has specified the conditions for the validity of the formula for small-polaron hopping mobility, derived by Yamashita and Kurosawa. Firstly the condition for the existence of the small polaron should hold. In a one- dimensional model this condition reads :

2J < Eb, (25)

where 2J is half the rigid (one-dimensional) band width and E b is the maximum polaron binding energy (derived for J =0) :

E b = 7'hoJ o. (26)

Moreover, the employment of perturbation theory confines the validity of the formula for the mobility to J values with an upper limit of

j2 ~ ( UkT/Tr)l/2/&Oo/~r, (27)

or approximately of

J < hOJo. (28)

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This upper limit for J is at the same time the border line for non-adiabatic processes.

Holstein, however, stresses the extra limitations imposed on the validity of formula (21) for the mobility. The matrix elements for transition of a small polaron from one site to a neighbouring site can be classified according to whether they are diagonal or non-diagonal. In a diagonal transition no change is allowed in the phonon quantum numbers nq for all modes q, whereas in a non-diagonal transition some of the quantum numbers nq are changed: multi-phonon absorption and emission take place in this latter process. Diagonal transitions in a perfect lattice will lead at low temperatures to small-polaron band conduction, whereas at high temperatures the dominance of non-diagonal transitions leads to small-polaron hopping. Diagonal transitions cannot be handled with time-dependent perturbation techniques, because then they lead to terms divergent linearly with time. ¥amashi ta has by-passed this difficulty in the calculation of the transition probability by restricting the time of integration to the range of a lattice-vibration period. This procedure is correct as long as the diagonal compared to the non-diagonal transitions play a subordinate role, i.e. as long as the temperature is not too low and the coupling not too weak. Moreover, the time interval as chosen by Yamashita contains roughly all of the transition probability only in the case where sufficient dispersion is present in the optical vibration spectrum. When the phonon dispersion is expressed by :

(~0q 2 =0)02 +5012 COS (qa), (29)

where qa is the phonon vector times the lattice parameter, this condition reads :

2~(wl/OJo)¢ esch (½~c%/kT)>> 1. (30)

For the case where ~o1= 0.5w o, i.e. for a dispersion A~o/w 0 of 250 , and where the coupling is strong (~ ~ 10), this condition calls for temperatures even higher than the Debije temperature.

The mathematical approximations as used in the derivation of an explicit formula for the small-polaron hopping mobility make this formula relevant only at high temperatures for the strong-coupling case. Numerical calculations of the non-diagonal transition probability are indispensable for the case of relatively weak coupling and relatively small dispersion of the optical phonons. Numerical calculations have been performed by de Wit (1968). He arrives at the result tha t for a high coupling constant (y= 10), the introduction of small dispersion (AoJ/Wo~_lO°/o) does not change qualitatively the mobility behaviour obtained by Yamashita and by Holstein. Quantitatively the mobility increases by a factor of 2 at T = 0 and by a factor of 10 at T = ~-0. For a low coupling constant (~=3.5) the same small dispersion raises the mobility by a factor of 3.5 at T = 0 and a factor of 10 at T = ~0. For this

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case, in the hopping region the mobility becomes almost temperature- independent.

The adiabatic process of small-polaron hopping, for which

j 2 >> ( UlcT /Tr)l/2tiWo/~ ' (31)

or approximately

J > h~Oo, (32)

has been studied far less extensively than the non-adiabatic analogue. Emin and Holstein (1969) arrive at the result that

3 ea 2 ]ico o t ~ = ~ - ~ ~-~ exp {--(U-J)/I~T}, (33)

valid at high temperatures (T>½0) and relatively strong coupling. Emin stresses the point that one cannot arbitrarily let J have values approaching that of U, while preserving the small-polaron character of

< 1 the charge carrier. As a limit J ~ ~U is indicated. This limit is valid for a one-dimensional model. For the three-dimensional case (see § 5) the smal l -polaron condi t ion is :

J ~ U. (34)

2.1.3. Smal l -po laron band conduct ion

Yamashita and Kurosawa (1958) concluded that in the case of a small- polaron semiconductor at T = 0 , the bandwidth, which was already assumed to be small, is still further reduced due to the electron-lattice coupling. Moreover, with increasing temperature, the reduction was found to increase still further. These authors did not consider small- polaron band conduction, in the first instance, because the bandwidth in the presence of strong coupling is very narrow (see the end of this section).

Holstein, in contradistinction to Yamashita, emphasizes the importance of small-polaron band conduction at low temperatures (T<½0). The (one-dimensional) small-polaron bandwidth is :

where AW, = 4 J exp ( - - S T ) , (35)

~T/"

The corresponding effective small-polaron mass is :

my*= m* exp S T. (36)

I t is seen that the bandwidth is indeed very small at T = 0 (smaller than 0.1 mev if y = l0 and 4 J = 1 ev) and begins to decrease appreciably at temperatures T > ½0. In Holstein's model, the mobility of small polarons

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in this band is thought to be restricted, due to scattering by optical vibrations, i.e. due to non-diagonal transitions. For the non-adiabatic region

2J < Ebl/2( /iCOo) 1/2, (37)

the mobility behaviour is described by :

1 ea~ k<~o ( i~o~o~12 t~-7rll 2 h DT \~/ cseh 1_ -kT/ ( exp - 2 y csch ½- IcT]' (38)

if condition (30) is fulfilled. Polaron-band conduction dominates over hopping conduction as long as

]//~ < AWp. (39)

I t should be remarked that the relaxation time ~ for scattering of diagonal band states is in this model identical with the reciprocal transition probability for non-diagonM processes.

In Holstein's model the transition temperature T t marking the transition between polaron-band and hopping regime equals about 0-40 for the case of strong coupling (~ = 10). Such a value for T~ may be regarded as an upper limit. In reality a considerable reduction of this temperature may be expected. The numerical calculations of de Wit already point to an appreciable reduction of T~ in the presence of small phonon dispersion (Tt_0 .30 for ~ =10 and TriO.250 for ~ =3.5). Lang and Firsov (1962, 1963) conclude that, in the polaron-band regime, the mobility is significantly more hampered, due to scattering by optical phonons, than would follow from Holstein's estimate. The origin of the extra limitation imposed on the mobility lies in the occurrence of virtual transition processes in which the polaron jumps to a neighbouring site and back again, leading to a redistribution of some phonons over the vibration modes (Holstein, private communication, see also Friedman 1964). As a consequence of the extra scattering, the transition temperature for the case of strong coupling is reduced by a factor of about 2 (Tt~ 0/2 In ~). A further appreciable lowering of T t can be expected because of departures from linearity of the electron-phonon interaction (Holstein, private communication). According to ¥amashi ta and Kurosawa (1960, 1961) the sm~tll-polaron band model cannot be regarded as a suitable approximation when impurities are present to a sufficient degree. This is when fluctuations in potential energy between neighbouring sites caused by the randomly distributed impurities exceed the bandwidth. Because this width is extremely small, it is assumed that in practice the band is washed out entirely (see also § 6). In connection with the arguments given above, it seems to the present authors to be quite uncertain whether ever in practice a mobility behaviour will be found which is characteristic of small-polaron band conduction.

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2.2. Hall Effect

For the general case of an anisotropic substance, the non-diagonal components of the anti-symmetric part of the conductivity tensor, such a s

x~ ~ = ~ ( % ~ - %~), (40)

are an odd function of the magnetic induction B and thus are characteristic of the Hall effect. For the case of isotropy in a plane perpendicular to B (taken in the z direction), axy ~ reduces to ~xy( = - %x). The Hall mobility may then be expressed by :

~t~ = - e ~ y / B , ( 4 1 ) ~xx

where ffx~ and/~xx correspond to c%y/ne and ~xx/ne, respectively. For an n-type (p-type) broad-band semiconductor, f f ,y /B = + (- )r f fx~ 2

and ffI< = - ( + )rffx~. The Hall factor r in the presence of polar optical- mode scattering does not deviate much from unity. For the antiferromagnetic compounds considered in this paper, the Hall effect can be expected to have anomalous properties. This will be discussed in §4.1.

A theory of the Hall effect for the non-adiabatic hopping regime of a small-polaron semiconductor was first developed by Friedman and Holstein (1963). In their calculations these authors consider a two- dimensional lattice. For the case of an equilateral triangular lattice, where each atom belongs to a group of three mutually nearest neighbours, fin is found to be much larger than ffD (=fizz). In the classical limit (kT >> hw o and y >> 1), the transverse mobility ffxy turns out to be determined mainly by B exp ( - - } U / k T ) , while that of ffxx is given by exp ( - U/kT), leading to ff~o~exp ( - ½ U / k T ) . The expression derived for ffH is :

l V'2ea J ff~I~- if~} -~-&o-~o \-[-~] -~,-i~ exp ( - ½ U / k T ) , " (42)

valid under the same conditions imposed on the validity of formula (21) for fib" The ratio /~H//~D is found from eqns. (21) and (42) to be :

ff~iI~n ~- (1).~ ? exp (~UI~T). (43)

Thus for the non-adiabatic hopping regime in the presence of strong coupling, fin increases exponentially with temperature, albeit with an activation energy three times smaller than that valid for fib. At relatively low temperatures fftt is much larger than t~D, the ratio fig/fib decreasing with an increase of temperature. For a simple square or face-centred square lattice, which has only pairs of mutually nearest neighbours, fig has the same value as ffD"

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The Hall mobility for the adiabatic hopping regime at temperatures T>½0, has been evaluated by Emin and Holstein (1969). Particular features of the adiabatic Hall mobility are that it can be a decreasing function of temperature and that tz~i//ZD can have values smaller than unity. I t is concluded therefore that the absence of a thermally activated Hall mobility does not exclude the possibility of smMl-polaron conduction. The expression for the adiabatic Hall mobility in the case of a triangular lattice reads :

1 ea 2 Jim o /z~ =4rr // k-T F(T) exp { - (½U-J)/lcT}, (54)

where F(T)_~0.2, if h~%<J<2hoJ 0 and ½hwo<kT<hoJ o. According to Emin, the above result has physical significance only as long as J ~< ½U. This upper limit can be regarded as the border line for the small-polaron approach. For the situation J~_½U, tLg is a continuously decreasing function of temperature and becomes smaller than/z D above a temperature where J~_kT. It should, however, be remarked that the special features can be expected to become manifest only for the strong-coupling case. Combination of the condition for the adiabatic regime: j2>> (UlcT/~r)l/2 (]iwo/rr) and the three-dimensional small-polaron condition J ~ U leads to 7>10.

For the small-polaron band regime in the case of an equilateral triangular lattice, Friedman (1963) arrives at the result

where Jp

with

is the small-polaron transfer integral :

J , =J exp ( - S T )

~T/"

(46)

In the strong coupling case at not too low temperatures, Jp is very small compared to kT. Thus, in the small-polaron band regime too, t~H can be much larger than /x D. For a simple square or face-centred square lattice, /xg in the small-polaron band regime too is found to have the same value as/~n-

The Hall coefficient of a smMl-polaron semiconductor seems to have a negative sign for the case of a triangular lattice, irrespective of whether conduction is by electrons or holes, whereas in the case of a square lattice it would be negative or positive as in the normal case of an n-type or p-type conductor respectively.

Results for the hopping regime which differ--especiMly with regard to the temperature dependence of/zH from those given by Friedman and Holstein, were arrived at by Firsov (1963) and by Schnakenberg (1965).

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16 A . J . B o s m a n and H. J. van DaM on

Firsov concludes t ha t the t ransverse mobi l i ty t%y at high t empera tu res does not exhibi t any thermal ly ac t iva ted character and thus t h a t /x~ocexp ( + U/lcT). According to Holstein, the absence of a n y thermal ac t iva t ion in the t ransverse mobi l i ty is irreconcilable wi th the character of a small polaron in the hopping regime. Sehnakenberg arrives at the result t ha t tZxu varies wi th t empera tu re in the same manne r as /~xx, implying an almost t empera tu re - independen t behaviour of /z H. In a later paper (1968) Schnakenberg arrives a t a more general result for different values of the coupling constant , which includes his former result as well as t h a t ar r ived at by Firsov as special cases. According to Hols te in a t empera tu re - independen t behaviour of tL~i would erroneously follow from the calculations for the three-sites configurat ion if processes with non-coincident intermediate-s i te energy are allowed to cont r ibute to /xxy. Holstein and F r i edman (1968), f rom a re invest igat ion based on Kubo ' s formula for the t ransverse conduct ivi ty , i.e. using the same calculation technique as Firsov and Schnakenberg, essential ly confirmed their former results.

2.3. Seebeck Effect

The Seebeck coefficient of an n - type non-degenerate broad-band semiconductor , where a single conduct ion mechanism dominates , is given by the relat ion :

= - - + l n (47) e ~-T

The heat of transfer Q is connected wi th the kinetic electron energy :

Q = AkT , (48)

where A is a constant , depending on the dominant scat ter ing mechanism. I f the (broad) band model applies to the substances considered in this paper, scat ter ing of charge carriers is p robab ly de te rmined b y optical lat t ice vibrat ions and/or ion-spin disorder (see § 4). The corresponding value for A equals abou t 2.

The relat ion between the concentra t ion of free electrons or large polarons (n) and the effective densi ty of states of the conduct ion level (N c) is de te rmined by the absolute energy distance of the Fermi level (E~ > 0) below the b o t t o m of the conduct ion band :

n = N e exp ( - E,/lcT). (49)

The effective densi ty of states in the case of a large-polaron band semiconductor is :

No = 2( 27rmp* k T /h2) a/u, (50)

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where rap* is the polaron effective mass (see § 2.1.1). tween n and ~ can more convenient ly be described by :

The relat ion be-

2.a /e= - og (51)

I t follows from the above tha t E~ and ~ are interrelated as :

- eczT = E~ + A k T . (52)

For the case of a small-polaron semiconductor, eqn. (47) for the Seebeck coefficient is valid. In this case, however, the densi ty of states in general equals twice the concentrat ion of cations (2No) available as a site for the small polaron. The factor 2 arises, jus t as in the case of eqn. (50), because of spin degeneracy. We assume t h a t this factor of 2 applies ~lso to the t ransi t ion-metal oxides considered in this paper, a l though the cations iu these cases have a tota l spin. An indisputable expression for the hea t of t ranspor t in the case of a small-polaron conductor does not ye t seem to exist. I f i t is assumed t h a t Q is proport ional to the act ivat ion energy U :

Q = f l u . . . . . . . . (53)

expressions for the Seebeck coefficient in the hopping regime read :

e k-T + In 0 (54)

and

2N 0 exp (flUtisT) a ' = = - log (55)

2.3k/e n

- e e T = E~ + fiU . . . . . . (56)

A number of authors have contr ibuted to the discussion of the value for ft. Heikes ( 1962) arrives at fi ~_ 0-05, Chadda (1963) concludes to an anisotropy of /? , its value lying in the range of 0.1 to 0.4. Sewell (1963), Schotte (]966) and Klinger (1968) arrive a t f i= 0. A value for fi of zero implies, in the case of small-polaron hopping t ransport in the positive x direction, equal probabil i ty for the deformation-energy t ranspor t in the positive and negative x direction. For the small-polaron band regime, as long as the band width is small compared to kT, the heat of transfer m a y not be expected to contribute appreciably to the Seebeck effect.

An evaluat ion of the Seebeck coefficient as a function of tempera ture and dope can be made only if the relevant behaviour of charge-carrier concentrat ion is known. The equations for n, now presented, apply to large polarons as well as to small polarons. In the lat ter case Ar c has to

A.P. B

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be replaced by 2N 0. For a non-degenerate partially compensated n-type semiconductor (Blakemore 1962):

n(n+N•) _ N O e x p ( E D ) ND_N•_ n g ---k--~ , (57)

where N n is the donor concentration, N~ the acceptor concentration (N A < ND), E D the donor ionization energy (E D > 0) and g the degeneracy multiplicity of the donor centre. At low temperatures (n ~ (N D - NA) and n~Na), denoted by range I :

N n -- N A ~ E,= ED-IcT ln - ~ £ • J

At intermediate temperatures (N~ ~ n ~ ND), range II,

//N cND'~ 1/2 l n : ~ T j exp ( - ~ ) ,

H j (59) Ei = 1E D + +]cT In \ ND ] .

The situation I [ in most of the practical cases will not be encountered because it requires a very low compensation degree. At high temperatures, i.e. in the exhaustion range ( I I I ) :

I I I ( E f = / e T .

2.4. Dielectric Relaxation Loss The occurrence of dielectric loss has been established in a number of

transition-metal oxides. I t is at tr ibuted to the presence of charge carriers bound to centres. The centres correspond either to impurities or to native defects. Dielectric loss is observed at low temperatures, in some eases even at or below 4"2°K. In all cases considered it exhibits the features characteristic of Debije-type relaxation. In a quasi-classical description of the Debije relaxation loss, the equilibrium positions of the bound charge carrier are thought to be at the cation neighbours of the centre. These positions, all having the same energy in the absence of an external electric field (directional degeneracy), correspond to different dipole orientations. ' Partial alignment ' of these dipoles occurs when an electric field is applied. The relaxation time T for this process is determined by the probability of charge carrier transition between neighbouring

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cations. be described by a decay function (Fr~)hlieh 1958) :

~(t) = ~(0) exp ( - t/r).

The corresponding dielectric constant :

t(m) = q(m) + i%(m),

where

and

The response of the bound charge carrier to an electric field can

(61)

(62)

A t tl(m ) = coo + 1 + m2r - - - - - - ~ (63)

m r t2(m) = At 1 +m2r~" (64)

t~ denotes the dielectric constant at frequencies for which mr>> 1. Ae is the maximum contribution to the dielectric constant. The static value of the dielectric constant %(=¢oo+ At) is obtained at frequencies for which mr ~ 1. The imaginary par t %(m) of the dielectric constant exhibits a maximum at mr = 1. The corresponding real par t of the conductivity :

O) -l(m) = ~ t~(m) . . . . . . . (65)

at low frequencies (mT~ 1) is proportional to m 2 and at high frequencies (mr>> 1) is a constant : AE/47rr.

Fr0hlieh (1957)~ pointed out tha t the above classical interpretation of dielectric loss is not reconcilable in general with a quantum-mechanical description of the centre. The quantum states of a carrier bound to a centre are characterized by wave functions each of which is a linear combination of atomic orbitMs. These latter functions correspond to a charge carrier localized at one of the neighbouring cations of the centre. The ground state of the system is in general not degenerate, ruling out the possibility of Debije relaxation loss. If, however, the energies of all quantum states lie within an interval smM1 compared to kT, the t rapped charge carrier can be considered to be localized at one of the neighbouring cations of the centre and the above quasi-classical interpretation of Debije loss can again be employed. In general the energy spread for the quantum states is large compared to kT, even at high temperatures. However, in low-mobility semiconductors with a strong electron-phonon interaction the energy spread for the quantum states can be significantly reduced. I f the bound charge carrier is a small z)olaron, the energy spread can be reduced to such an extent tha t it is small e0mpared to / cT even at very low temperatures.

See also FrShlieh, Machlup and Mitra (1963).

B 2

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According to Sussmann (1962), Debije relaxation loss can occur in some eases without small-polaron formation. In these cases the specific s ymme t ry of the centre allows for electronic states wi th an electric dipole moment . These specific symmetries belong to the groups C(2~+1) v and Ta. Examples of practical importance are centres having three equi- dis tant or four te t rahedra l ly disposed equidistant nearest neighbours. Under certain conditions the corresponding 2-fold or 3-fold degenerate electronic states can be the ground state. In an electric field, t ransi t ions between dipole directions are caused by optical or acoustical phonon scattering processes. At temperatures T > 0, both scattering mechanisms lead to 1/Toe T 2. At temperatures T < 0, optieal-phonon scat ter ing leads to 1/z ocexp ( -/io)o/kT ) and acoustieal-phonon scattering to 1/T oc TL

Sewell (1963) considers a simple, idealized two-sites model for the small- polaron ease. In his model kT is assumed to be large compared to, among other things, the resonance-energy spread AW. The decay function found for the system is characterized by :

1 ~(t) o c ~ exp ( - t /w) cos (wrt), (66)

where the parameters T and m r have the properties :

(a) 1/T>oJ r a t T>Tc, ~ (67) (

(b) 1/T<c% at f < T o . j

The characteristic tempera ture T o is defined such t h a t motion of the small polaron is thermally activated at temperatures T > T o and proceeds via a tunnelling process at T < T o. The energy distance A W of the two electronic states for the system equals hoJr, while T determines the inertness of the system's response to an electric field. The decay function is formally identical wi th t h a t used in the description of dielectric resonance absorption (Frohlich 1958). In this case e2(~o ) is a ma x i mu m at

1 Wma x = - (1 +Wr2T2) 1/2. (68)

T

In general T m a y be expected to decrease with increasing temperature . The case of low temperatures where ~orT>> 1, i.e. COm~ x =(Z)r, corresponds to resonance absorption, whereas the case of high temperatures where c%T ~ 1, i.e. com~ X = l/w, corresponds to a Debije-type relaxatiou loss. In the lat ter ease ~2(w) exhibits a max imum at frequencies larger t han ~%. I t m a y then be concluded t h a t Sewell's model shows the features of Debije relaxation loss at high temperatures (T > To) and of resonance loss at low temperatures (T < Tc).

Klinger and Blakher (1969) have made an extensive s tudy of the bound small polaron. The centre consists of a central impur i ty ion or vacancy with z equivalent nearest neighbours. The energy levels of the bound

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small polaron have a spread AW which is small compared to/cT. Small- polaron behaviour is dominated by diagonal transitions at frequencies and temperatures below certain values. The frequency limit for diagonal transitions ultimately seems to be determined by h~o < AW, while for not too weak eleetron-phonon coupling the temperature limit is T<~O. Two eases for the ' band ' regime are considered :

A : All electronic states or some of them have a dipole moment. This case resembles closely the specific situation described by Sussmann. Dielectric loss is predominantly of the Debije-relaxation type. The characteristic time T represents the relaxation time for scattering between ' b a n d ' states by optical or acoustical phonons. The former process leads to 1/zocexp (-hWo/lCT) and the latter to 1/TooT 1, where 1 > 0.

B : None of the electronic states has a dipole moment. In this case, just as for the Sewell model, a dipole moment is induced by the electric field. Absorption is of the resonance type with a resonance frequency, for the two-sites centre, of wr=li/AW. I t seems that, just as for the Sewell model, Debije relaxation takes over from resonance absorption at high temperatures where non-diagonal processes dominate. In the ease A, due to the possibility of induced dipole moments, resonance absorption may occur too.

In connection with the above, two remarks can be made :

(1) In all cases considered, absorption either of the Debije or of the resonance type is found to be proportionM to 1/T. This is because in all eases kT is considered to be large compared to the resonance- energy splitting AW. If, however, kT becomes smaller than AW, resonance absorption should become independent of temperature, whereas Debije-relaxation loss with decreasing temperature should diminish and ultimately go to zero, unless the ground state is degenerate.

(2) In the presence of locally varying electric fields, introducing energy differences W D between neighbouring sites of the centre, ' b a n d ' formation will be prohibited if W D > A W. In tha t ease the small polaron at all temperatures remains localized at one of the sites of the centre, intra-centre motion being always thermally activated, leading to Debije-relaxation loss. Deviation from the 1/T dependence for the absorption can now be expected to occur at temperatures T < WD/lc.

2.5. Optical Absorption The absorption coefficient (K) due to charge carriers, free or bound, is

related to the real part of the corresponding conductivity :

I~e ~(~,T) K(~o,T)= el/~(co)%c.. (69)

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In this relation e(~o) is the corresponding dielectric constant, E 0 denotes the dielectric constant of vacuum and c is the velocity of light in vacuum.

For a large-polaron band semiconductor, with increasing frequency ~o, a continuous decrease of intraband free-carrier absorption can be expected. This dispersion takes place at frequencies for which ~oT = 1, where T is the relaxation time for scattering of band states. As long as h/w <kT and kT< hw0, 7 is longer than 70, where 70 is the lattice-vibration period. Therefore the dispersion occurs at frequencies ~o <~o 0 and certainly at frequencies co < AW/h, where AW is the band width. For the strong- coupling case (~> 5) of large polarons, at low temperatures (kT~hcOo) , a(co) exhibits resonances at frequencies w > w 0 (Feynman et al. 1962).

For a small-polaron semiconductor, in the hopping as well as in the band regime, at frequencies oJ >~o0, absorption due to free carriers increases with frequency, reaches a max imum at wma ~ -~ 2y~oo, i.e. hwm~ ~ _ 4 U, and then at higher frequencies decreases again (Klinger 1963, 1965, ]~eik 1963, 1967 and Reik and Heese 1967). Reik has pointed out that especially for the case of a small-polar0n semiconductor characterized by a moderate coupling strength, for which the behaviour of the d.c. mobility in the hopping regime does not have the thermally activated character, free-carrier absorption data might be decisive in order to establish the nature of the conduction mechanism.

In the calculatiol/ of a(w,T) the effect of diagonal transitions can be neglected at all temperatures. This is because the small-polaron band width corresponding to the diagonal transitions is small compared to &o o. An approximate expression for a(oJ,T), valid at high temperatures (T > ~-0) and strong coupling, reads (Rcik and Heese ]967) :

sinh \~ ~--~]

Re a(w,T)= a(0,T) ½/~w/kT -exp ( - w % q2) (70)

o r

where

and

Re (r@,T) sinh (2FoJ4)

a(0, T) 2F~o@ exp ( - w%~), (71)

sinh (½&oo/kT) v~2 =

4TCOO 2 (72)

F - 4~kT" (73)

Differentiating Re a(~o, T) with respect to w shows tha t it has a maximum at

~Omax'/J - F, (74) or at

Wm~x -~ 2yCOo, (75)

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so that l~e (~(Wmax, T) exp (P~)

( ~ ( O , T ) - 4F 2

At temperatures T>~ ½0 :

i.e. at T~_½0,

(76)

F ~ \~-~] , (77)

I t follows from the above that COm~ x is independent of temperature (see eqn. (75)). The same roughly applies to Re a(Wm~x,T ) (see eqn. (76)) if the concentration of free small polarons is independent of temperature, i.e. if the accepters (donors) are exhausted. In that case ~(0, T)= he, if(0, T)oct,(0, T). The temperature dependence of/z(0, T) can be seen from eqn. (23) to be roughly the inverse of that for the right-hand side of eqn. (76). An explicit expression for Re a(w, T) has also been given by l%eik for the low-temperature range (T-+0), valid at frequencies much higher than co0(w~OJm~). It turns out that at low temperatures too, COma x and ]~e (Y(COmax, T) are roughly temperature-independent.

Finally at very low temperatures, in the case of frequencies in the range w 0 to 3co0, fine structure in the absorption spectrum due to low-order phonon-sum transitions can be expected (i.e. in each process absorption of light is followed by emission of a few phonons). This fine structure will only be encountered under certain conditions concerning the wave- vector dependence of eleetron-phonon coupling and phonon dispersion.

Reik's theory, as briefly reviewed above, is concerned with absorption due to free small polarons. It may be expected (Klinger and Blakher 1969) that the typical behaviour of absorption, as a function of frequency and temperature, due to free small polarons will be encountered too for the case of small polarons bound to donor (accepter) eentres. However, in the ease of bound small polarons, s(0, T) in eqn. 70 has to be replaced by s(~or>~ l, T), where T is the reciprocal small-polaron transition probability. The conductivity s(c~T>~l, T) is that due to eentres at frequencies above the dipole-relaxation region (see § 2.4). It is conceivable that a bound charge carrier has a localized character, whereas the free charge carriers are deloealized. It is therefore of importance to establish by experiment whether the behaviour as found for optical absorption is due to free or to bound charge carriers.

§ 3. ~ESISTIVITY AND ~EEBECK COEFFICIENT ; DETERMINATION OF DRIFT MOBILITY

In this chapter a critical survey is presented of the data arrived at since about 1963 for the drift-mobility behaviour of free charge carriers in NiO, CoO, ~-F%O 3 and Mn0. The greater part of the data had

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already been published, but especially for the case of ~-F%Oa, as yet unpublished data obtained by Bosman and for the ease of MnO, not yet published work done by Creveeoeur and de Wit will be discussed. The results are deduced either from resistivity data solely (§ 3.3) or from combined resistivity and Seebeek-coefficient data (§ 3.4). An interpretation of the drift-mobility behaviour can be found in § 4, where Hall mobility as well as drift mobility of free charge carriers are considered. In the interpretation of resistivity and Seebeck coefficient, the possible influence on the experimental data of inhomogeneities in the sample and of conduction via impurities should be accounted for. Therefore, prior to the presentation of the experimental results, attention will be paid to the phenomena of inhomogeneous conduction (§ 3.1) and impurity con- duetion (§ 3.2).

3.1. Inhomogeneous Conduction

Inhomogeneous conduction is liable to be present in single crystalline as well as in polyerystalline (ceramic) material. An inhomogeneous distribution of dope throughout the material can lead to appreciably inhomogeneous conduction. This certainly applies to the substances considered here because the ionization energy of the dope centres is large and varies considerably with concentration. Conduction then becomes noticeably inhomogeneous at lower temperatures, i.e. below about room temperature. Similar effects arise due to the presence of other phases, with different electronic properties, in the material; for example CoaO 4 in CoO. Ceramic, compared to single crystalline material is easier to prepare and has the advantage that quite varying types and degrees of doping can be realized with a reasonably homogeneous distribution over the different grains. The difficulty with ceramic material, however, is to avoid, during the preparation of the samples, the formation of grain- boundary layers with a conductivity different from that of the grain bulk. In practice the presence of high-ohmic grain-boundary layers has been the origin of misinterpretation of experimental facts. Figure 1 is a demonstra- tion of the drastic influence of high-ohmic grain boundaries on the d.c. resistivity of samples of NiO (0.1 at. °/o Li), CoO (0.1 at. °/o Li) and ~-F%0 a (0.1 at. ~o Ti). The curves with the markings represent behaviour as a function of temperature for samples where grain-boundary effects have been avoided. The corresponding lines without markings have been taken from published investigations in which high-ohmic grain-boundary layers evidently play a decisive role. I t is seen that due to this effect d.c. resistivity at room temperature is enlarged several orders of magnitude, and moreover that the temperature dependence of is altered to a significant degree. I t will be obvious that neglect of this type of grain-boundary effect leads to too low estimates of the drift mobility.

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Fig. 1

T(~K) 4

10; 10' 10c" e~J')'cm) - 1000 S(~O

IOE /

10"

10 ~ -

10 ~

10 2 -

10

1 -

[ I I I I 1 O 0 1.0 2.0 3.0 4.0

_ _ . . . . ~(V, -~)

Influence of grain-boundary effects on the experimental value of the resistivity of ceramic material. All samples had been doped to the same amount of about 0"1 at. %. The curves with the markings (this work) represent the behaviour of the bulk, whereas the corresponding lines for NiO and ~-Fe203 (Morin 1954) and for CoO (I-Ieikes and Johnston 1957) probably resulted from the presence of high-ohmic grMn-boundary layers.

For the examples given in fig. 1, the high-ohmic grain-boundary layers were probably formed during the last stage of the preparation, viz. when the samples were cooled. A possible mechanism of the formation of high-ohmic grain boundaries will now be described. When heated in air above a certain temperature NiO and CoO oxidize, probably by formation of Ni and Co vacancies, whereas ~-F%Oa reduces, presumably by formation of oxygen vacancies. During the sintering at high temperature (_~ 1300°c) the substances reach equilibrium with the partial oxygen pressure in the gas atmosphere. Conduction in relatively low-doped samples is then mainly determined by the presence of cation or anion

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26 A. J. Bosman and H. J. van DaM on

vacancies. When the samples are cooled in the same atmosphere, the degree of oxidation (NiO, CoO) or reduction (~-F%O~) diminishes rapidly. In the first stage of cooling, where diffusion of oxygen is still sufficiently rapid, this process takes place throughout the whole sample. Conduction during this stage of cooling becomes dominated by the dope. Continued cooling can only introduce changes within grain-boundary layers which decrease in thickness as cooling proceeds. However, conduction remains determined throughout the whole sample by the dope. Further cooling below a critical temperature may at the grain boundaries even lead to the reverse process, i.e. reduction of NiO and CoO and oxidation of e-F%03, introducing extra partial compensation of the dope leading eventually to high-ohmic grain-boundary layers. I f the critical temperature is low enough, these grain-boundary layers have a negligible thickness, so that their effect on the overall resistivity is imperceptibly small. I t is of the utmost importance to keep this critical temperature as low as possible. Cooling samples of NiO and CoO in a lower partial oxygen pressure and samples of ~-F%O a in a higher one than tha t maintained during the sintering process, shifts this critical temperature to higher vMues. This eventually leads to the formation of relatively thick high- ohmic grain-boundary layers which dominate overail resistivity. In practice it has been found tha t if the samples are sintered in air, cooling of NiO and CoO in air or pure oxygen atmosphere and of ~-F%03 in nitrogen with a low partial oxygen pressure leads to a considerable suppression of grain-boundary effects.

The presence of high-ohmic grain-boundary layers in a ceramic sample can be demonstrated with the aid of measuring conductance and capacitance as a function of frequency (Koops 1951). Conductance as well as capacitance then prove to show dispersion with a relaxation time ~ = C1/Gb, where C 1 represents the capacitance of grain-boundary layer and G b the conductance of grain bulk. At frequencies co < ~-1, conductance and capacitance are more or less far below and above the grain-bulk value, respectively, whereas at frequencies co > r-1 both values approach to those representative of the bulk material. The temperature range in which the a.c. method can be employed as a check on the d.c. results is limited to both sides. With increasing temperature, bulk conductance increases and so too, therefore, does the frequency T -1 above which the measuring results are decisive for determining the real bulk properties. Therefore, at the high-temperature side the availability of high-frequency equipment is the limit to the a.c. method. At the low-temperature side the occurrence of dielectric loss can interfere with the a.c. check on the presence of high-ohmic layers.

As an example of an a.c. control of d.c. measurements, fig. 2 shows the results obtained on a sample of Ni0 ( _~ 0.1 at. ~o Li) which, after sintering in air, has been cooled in different atmospheres. Cooling in nitrogen, having a low partial oxygen pressure, results in d.c. resistivity values which are higher than those found when cooling the sample in air.

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Moreover, at any temperature in the relevant region a.c. values for both cases coincide with the d.c. curve obtained on the sample cooled in air. This demonstrates that grain-boundary effects are introduced when the sample is cooled in nitrogen atmosphere, and moreover that the d.c. data obtained after cooling in air are representative of the bulk properties.

2 ~

1

0

-1

-2

-3 o

Fig. 2

T (OK) ..,,~----- I]O

I I I I ,oi~ c~m) Ni,_xUxo /

/ / / /

_ / / / /

Sample d.c. ~c j 1 cocked in air []

2 ,, ,, nitrogen '~ ~r

1 2 3 Z~ 5 6 7 8 9 10 11 = -~-- (OK-l)

D.C. resistivity for a ceramic sample of NiO which, after sintering at 1200°o in air, has been cooled in nitrogen, introducing high-ohmic grain-boundary layers (sample 2) and d.c. resistivity for a similar sample when cooled in air (sample 1). A.C. resistivity was measured at frequencies such that its value was representative of the bulk. The result of these measurements demonstrates that for the samples cooled in air grain-boundary effects on d.c. resistivity are practically absent (Bosman and Crevecoeur 1966).

3.2. Impurity Conduction

The importance of the phenomenon of impurity conduction to an understanding of electrical transport properties of NiO was recognized for the first time by Ksendzov, Ansel'm, Vasileva and Latysheva (1963). Impuri ty conduction is brought about by the hopping of electrons from an occupied to a neighbouring unoccupied majority centre. The presence of minority centres, partially compensating the majority centres, is an essential prerequisite for the occurrence of this mechanism. Owing to locally

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varying electrical fields produced by neighbouring charged majori ty and minority centres, an energy difference exists between occupied and neighbouring unoccupied majority centres. Therefore the hopping process should be phonon-induced, implying a characteristic temperature dependence.

Miller and Abrahams (1960) have made calculations of this phonon- induced hopping process. These calculations are concerned with homo- polar broad-band semiconductors at low temperatures where acoustical one-phonon processes dominate. For low values of the degree of compensation K (K< 0-03 but not approaching zero) the essence of their outcome can be given as :

QimpOCexp (rmaj/ro)3/2 exp (E/]cT), (79) where

rmaj = (3/4wNm~ j)1/3 (80) and

E = (e2/ermaj)(1 - 1.35K1/a). (8~)

For higher values of K, the energy E has been tabulated in Miller's paper. In the limits of K approaching zero or unity, Qimp goes to infinity as may be expected because there are either no unoccupied or no occupied sites. Miller's result is valid as long as rm~j, half the average distance between neighbouring majority centres, is sufficiently large compared to r0, the latter quanti ty being a measure of the extension of the wave function of the electron bound to the majority centre. When rm~j/r o becomes too small (< 4, Mort and Twose 1961), impurity band conduction takes over from impurity conduction.

Schnakenberg (1968) has treated impurity conduction for the case of a small-polaron semiconductor. At low temperatures, the temperature dependence of aimp is found to be exactly the same as that calculated by Miller, but the magnitude of ain, p is a factor of exp (2y) smaller, y denoting the small-polaron coupling constant. In the low-temperature range, charge-carrier transport is an acoustical one-phonon assisted hopping process. At higher temperatures, optical two-phonon processes are found to dominate aimp" At temperatures T > 10, optical multi-phonon processes determine ainu,, which now becomes thermally activated : qimp oc T -3/2 exp ( - U/kT) with U = lyt~co0.

Some of the features of impurity conduction can be elucidated on the basis of fig. 3. This figure represents resistivity data obtained by Springthorpe et al. (1965) on single crystalline NiO samples with Li dopes varying between 0.2 and 3-2 at. ~o- A marked change of ~ behaviour for all samples occurs in the temperature range 50 to 100°K. Above this range, conduction is predominantly brought about by free charge carriers (holes), thermally excited from majority centres (Li acceptors). Below this range, transport of charge carriers is governed by impurity conduction. The high ionization energy of the Li acceptors (a few tenths of an

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ev), combined with the low drift mobility of free charge carriers, are the principal causes for the appearance of impurity conduction at relatively high temperatures. For the cases of Ge and Si, where the ionization energies of the dope centres are much lower, impurity conduction becomes noticeable at far more lower temperatures. In the high-temperature range @ increases in roughly inverse proportion to the dope, whereas in the low-temperature range (see fig. 3) @ increases by a factor of 106 when

Fig. 3

200

11

10

T(OK)~---- 5O

j ~

f l I ' ' " - ~ D

j f J E j f I ~

f 1 ~ T Nil_ x LixO

Ac'[ivatton Energies@V)

/

j/// / I l l

I '

20

200°K 25°K x A 0.28 0.006 ~0.002 B 0.28 0.006 ~0.003 C 0.16 0.008 ~ 0.016 D 0.12 0.007 ~ 0.026 E 0.10 0.007 ~ 0.032 I I l I I I

2/)0 300 400 = I~ (°K-l)

100 500

Impurity conduction in single-crystalline Li-doped NiO (0.2 to 3.2 at. %) at low temperatures according to Springthorpe, Austin and Smith (1965).

the dope is reduced by a factor of 16. In the paper by Austin and Mort (1969), almost the same data for Pimp appear to correspond to a variation of dope by a factor of only about 5. The dependence of Pimp on dope is in agreement with the exponential dependence of @imp on the average distance between the acceptors (see eqn. (79)). The variation of @imD with Li dope implies that r 0 is only about 3 £. Such a low value for r 0 is acceptable in view of the highly localized character expected for the charge carriers bound to ' deep ' acceptors and possibly also points to the small-polaron character of bound charge carriers. From the magnitude of sire,, it appears indeed that bound charge carriers are small polarons. For IqiO compared to p-type Ge (Fritsche and Cuevas 1960), inter-centre hopping mobility, extrapolated to values corresponding to the relevant

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30 A . J . Bosman and H. J. van DaM on

characteristic distances r 0 between majority centres, is roughly a factor of l0 s smaller. Such a large reduction of impurity-conduction mobility implies an extremely small value for the inter-centre electron-transfer integral and thus points to the small-polaron character of the charge carriers bound to the centres. The order of magnitude of this reduction : (JNio/JGe)2exp ( -2y ) , where J denotes ' r i g id ' resonance integrals, points to a value of the electron-lattice coupling constant for bound carriers of ~ ~ 6. The low value for r 0 implies tha t the onset of impurity band conduction can only be expected for large dopes, i.e. ~> 5 at. ~o. For the case, e.g., of Ge, impurity band conduction already occurs at much lower dopes because there r o is more than an order of magnitude larger. The activation energies for impurity conduction found by Springthorpe are quite low (_~ 0.007 ev) and fairly insensitive to dope. Miller's simpli- fied expression (eqn. (81)) leads to values which are an order of magnitude larger if one substitutes for the dielectric constant the static value of 12 for undoped NiO. However, according to Austin and Mort (1969) the contribution to the dielectric constant due to the presence of Li+-Ni s+ dipoles (Ae) should be taken into account. This contribution, which

Fig. l

12

10

8

6

0

-2 0

300 100 50

j)gtoia% :m) M%ILtx 0

-//, ~Fe1-x ~'x)2 3 / / / /

- ~ I I [ i I i I i i I P i i { 5 10 15 20

T COK) -,~---

J J J ~

l I 1 I

3O I

J

( I [ I I I I I 25 31) 35

This figure shows impurity conduction in NiO and ~-Fe20 s at low temperatures and the absence of this effect up to very high values of the resistivity for the cases of CoO and MnO. All samples had been heavily doped: ~-F%O a (2 at. ~o Ti), NiO (2"5 at. ~o Li), CoO (2.8 at. ~o Li) and MnO (5 at. % Li).

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has been found to be proportional to dope and reciprocal temperature in limited regions (see § 5.1), can indeed be appreciable. As an example, Ae_ 16 at 40°K for a Li dope of 0.i at. %. I t may be remarked tha t the formulas as given by Austin and Mort, e.g. their formula (50), seem to be incorrect, leading to a contribution to the dielectric constant arising from dipoles which is an order of magnitude too small. However, the experimental features of impurity conduction in NiO, viz. the relatively low values for the activation energy E and its decrease to lower temperatures, can indeed be reasonably explained in this way if the experimental values for Ae are used (see also § 5.1).

There is a clear distinction between MnO and CoO on the one hand and NiO and ~-F%0 a on the other, with respect to the occurrence of impurity conduction. This is demonstrated in fig. 4, where @ for fairly highly doped ceramic samples (2-5 at. °/o ) has been presented as a function of temperature. The sources of these data have been indicated in § 3.3. NiO and e-Fe20 a clearly exhibit the transition from normal to impurity conduction, whereas in CoO and MnO, down to the lowest temperatures considered, impurity conduction is absent. This different behaviour originates from the dissimilar properties of the relevant centres. For the eases of CoO and MnO; dielectric-loss measurements have revealed the existence of a high activation energy for movement of the bound hole around the Li ion (CoO : 0.2 ev, MnO : 0-3 ev), resulting at low temperatures in bound-carrier mobilities having values which are extremely low compared to those for the cases of NiO and e-F%0 a. I t may then indeed be expected that, compared to NiO and ~-F%Oa, impurity conduction in CoO and MnO is severely hampered.

Impuri ty conduction can have a marked influence on the temperature behaviour of Hall constant and Seebeek coefficient. A description for the latter case is given in § 3.4. I t will be shown there tha t the neglect of this effect has led in former times to wrong conclusions concerning the temperature behaviour of the free-carrier drift mobility. A careful study of impurity conduction as a phenomenon apart would still be desirable for the substances considered here.

3.3. Estimates of tLD from Resistivity Data Solely

Information with regard to the drift mobility of free charge carriers can be obtained from the observed (d.c.) resistivity behaviour as a function of temperature and dope. This information, although necessarily limited in its extent, contains some valuable facts which cannot easily be acquired in another way. Conductivity in the doped substances considered proves to be extrinsic in a large temperature range. In the low- temperature range conductivity increases exponentially with increasing temperature. The activation energy Q is in general quite high for the materials under consideration (0.1-0-6 ev). The old question can now be posed : to what extent is the temperature dependence of conductivity

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determined on the one hand by charge-carrier concentration and on the other by drift mobility ?

I t is by this time commonly accepted tha t the transition-metal oxides should be considered as partially compensated semiconductors, although the compensation degree can in some eases be very low. Therefore, below a certain temperature the variation of the charge-carrier concentration n is prescribed by the statistics of a partially compensated semiconductor (eqn. (58)). I f the compensation degree is low an intermediate temperature range exists where the variation of n is governed by the statistics of a non-compensated semiconductor (eqn. (59)). How- ever, for the major part of the samples considered here, compensation is not low enough to make this type of temperature dependence distinguish- able. At higher temperatures exhaustion of the majority eentres in principle leads to a temperature-independent behaviour of n. I t is in this region that valuable information concerning drift mobility can be obtained from resistivity data solely if tzD may be assumed to be independent of dope. Additional information on the degree of compensation should, however, be available. Moreover difficulties can arise due to the extra contribution to n from anion or cation vacancy centres or from intrinsic effects. In the exhaustion range a is proportional to the dope in the ease of negligibly small compensation. The proportionality constant then yields the value of the drift mobility,

Formerly the resistivity behaviour viewed in two aspects led some authors to the conclusion tha t conduction is achieved by hopping of small polarons (see § 1) :

(1) I t was concluded in many eases tha t at room temperature a is proportional to the dope, implying tha t exhaustion of the dope is already completed. The temperature dependence of ~ at and above room temperature would then be determined exclusively by the drift mobility. In this way it was inferred from the behaviour of ~ tha t / z D at room temperature would have very low values and would increase exponentially with temperature.

I t will be shown, however, that exhaustion of dope occurs only at temperatures far above room temperature.

(2) At high dopes the activation energy Q has relatively high values. I t was argued tha t in the case of broad-band semiconductors doped to the same degree this energy is zero. This different behaviour was seen as evidence tha t the remaining activation energy stems from the exponential drift-mobility behaviour.

I t will be shown, however, that the above line of reasoning has no firm basis.

In figs. 5-8, resistivity as a function of temperature and dope is presented for the substances NiO, CoO, e-Fe20a and MnO. The data were obtained by Bosman and Crevecoeur (NiO, 1966 ; CoO, 1969), ]3osman (~-Fe2Oa, unpublished) and Creveeoeur and de Wit (MnO, to be published).

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10 3

10

Fig. 5

T (OK) - 1000 500 /4-00 300

- ~l_xLixO ~ /

lO 2 x /' ~ - ~ 0.o05 , _.o--~-~ , ~ ~../ /

0.07 1 ~.~-,~-~:~-==-"-- L..o--'-'~ / o.o9/ ~ ~ ~.~

/ 2.5 ~ 7 " ~ ' ~

/ l '

J

= I E

I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ~.3 (OK-l)

The behaviour of resistivity as a function of reciprocal temperature and Li dope in the ease of NiO (Bosman and Creveeoeur 1966). The broken line represents @ behaviour of non-deliberately doped single crystalline NiO in equilibrium with 1 al)m of O~ (Mitoff 1961).

The measurements were performed primarily on ceramic samples. The absence or near absence of grain-boundary effects was checked with the aid of a.c. techniques (see § 3.1). In the case of ~-F%O a at lower temperatures, where an influence of these effects on d.c. values cannot easily be avoided, only a.c. results representative of the bulk properties could be obtained. All samples of NiO, CoO and MnO were doped with Li and are p-type conducting. The ~-F%O 3 samples were doped with (tetra- valent) Zr or Ti and (pentavalent) Nb, introducing n-type conduction.

Conduction for all samples in the low-temperature region (T~< 300°K) can be characterized by an activation energy which depends on dope. The characterization with a single activation energy goes only as far as impurity conduction can be neglected (see § 3.2). In fig. 9 the activation energy of conduction (Q) has been plotted as a function of Nm~j 1/3, where Nm~ j denotes the dope concentration. At a certain dope concentration,

A.P. C

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Fig. 6

T(oK) ~. 1ooo 300 200 125

-I I I ~ !

e,

;

°2! 2"761: ~ 9.0- I

o 1 2 3 # 5 6 3 7 8

Resistivity of Li-doped CoO as a function of reciprocal temperature. Open markings represent d.c. values and black markings a.c. values (Bosman and Creveeoeur 1969).

for the sequence MnO, CoO, NiO, a-F%O a, the value of Q decreases. I t may be noted tha t Nb compared to Zr (or Ti) as an impurity in ~-F%0 a introduces a higher activation energy. Furthermore, at very low Li dopes in NiO (< 0.01 at~ %), Q shows higher values than expected from extrapolation from higher dopes. Probably at these low dopes ionization of a deeper lying centre dominates conduction (see also § 5). I t is seen in fig. 9 that Q decreases linearly with Nm~j 1/3 up to concentrations between 0.5 and 1 at. %, in roughly the same manner for all substances. Above this range Q values tend to become constant.

A reliable interpretation of the activation energy cannot be given on the basis of resistivity data only. The ~ behaviour at higher temperatures, however, suggests that at least a considerable part of Q should be attributed to the ionization energy of the dope eentres. Further considerations, including Seebeek-eoeffieient behaviour (see § 3.4), show tha t for NiO and

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Fig. 7

T (OK).,

35

0 2 /~ 6 8 10 12 10 3 OK-1 )

Resistivity as a function of reciprocal temperature for Zr or Nb-doped ~-F%0a. Open markings represent d.c. values and black markings a.c. values. The different 0 behaviour of the Nb-doped samples I and I' is due to different counterdope, which is the smallest in sample I'. In sample I', the 0 behaviour as shown is determined by partially compensated V 0 donors and in sample I by partially compensated Nb donors.

CoO and probably also for ~-F%03, Q is almost entirely determined at all dopes considered by the ionization energy of the acceptor and donor centres, respectively. Mn0 seems to be the only exception where the ionization energy of the Li acceptor as welt as the activation energy of the drift mobility of free carriers contribute to Q to about the same amount (see § 3.4).

I t follows from the above that the dope independence of Q at concentrations above 1 at. % cannot in general be attributed to a thermally activated character of the drift mobility. The decrease of Q, i.e. of the ionization energy (E) of the majority centres, with increasing dope is most probably due to an increase of charged impurities (Debije and

C 2

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36 A . J . B o s m a n and I-I. J . v a n Daal on

12

11

Fig. 8

T (OK) ~ - - • 00 300 200

l~ .e( .n .~m ) I

- - t Hnl_xkix (

/

¥

/

/

1 2 3 4 3 5 6 - - ~ ~ : (OK-l)

Resistivity as a function of reciprocal temperature of Li-doped ceramic and single-crystalline MnO : • : single crystal, I) : 0.03 at. % Li, © : 0.1 at. % Li, ~ : 5 at. % Li (Crevecoeur and de Wit, to be published).

Conwell 1954). I n a par t i a l ly compensa t ed semiconductor a t low tempera tu res , where Q values are de termined, the concent ra t ion of charged impuri t ies is p ropor t iona l to the minor i ty centre concent ra t ion (N~in). I n this model E varies as :

or wi th Nmi n = K N m a j :

0 e 2

E = E 0 - - Nmin 1/a . . . . . . (82) E

(3,~,=2 1 3 E = E o - - - K / N m a y a (83)

E

where E 0 is the ionizat ion energy for the case of ve ry low dopes, C a constant , e the s ta t ic dielectric constant , K the compensa t ion degree and

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0.01 0.',

-q (#"](eV) -I O.E

O. ~.

0.4

0.3

05

0.1

0.1 I

Fig. 9 ~ dope conce~ratlon(at %)

0.5 1 2 5 I I I I

- , I I i I i i , = I i E I J i i , t

5

Mn1_ x LixO O ~

".-Col_~LixO

Ni l_xLix 0

a'~_~Mx)2 03 (M = Zr,TI,Nb)

I I I I [ I I I I

Activation energy of the conductivity Q(~), determined at temperatures below 300°K, as a function of the reciprocal average distance between majori ty centres in p- type NiO, CoO and MnO and in n-type ~-F%0 a.

Nma j the majority centre concentration. Assuming that e is a constant, the behaviour of E versus Nm~j I/3 with K--0"10 up to l at. % of dope is quite similar to that found in broad-band semiconductors. A decrease of K with increasing dope above 1 at. % would then account for the behaviour of E in this concentration region. However, due to the dipole contribution (Ae) to the dielectric constant arising from neutral majority centres (see §§ 3.2 and 5), E within the relevant temperature region is not independent of dope concentration. Austin and Mott (1969), assuming E0= e2/Ed, where d is the distance between nearest-neighbour cations, ascribe the decrease of E with dope to the increase of e due to the dipole contribution. Their final formula (28), however, does not contain terms deriving from this dipole contribution, but closely resembles eqn. (83), describing the influence on E solely from the presence of charged centres. According to the present authors, the influence of the dipole contribution as proposed by Austin would imply that

E

E = ~ + Ac E°" (8~:)

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38 A. J. Bosman and H. J. van DaM on

Substitution in eqn. (84) of the Ae values as determined from dielectric- loss measurements (see § 5.1.1) leads to a relation between E and Nm~j 1/3 which is different from tha t presented in fig. 9.

I t seems to the present authors that eqn. (83) can be appropriate for describing the behaviour of E versus dope. The experimental behaviour of E versus Nm~j 1/3 suggests that the increase of • due to the dipole contribution mainly affects the term (ce2/e)K1/aNm~j 1/3. I t may be remarked that in this ease the dipole contribution itself does not lead to a decrease of E but on the contrary counteracts the effect of charged impurities. I t then follows immediately that with increasing dope, above a concentration where the dipole contribution becomes appreciable, the dependence of E on Nm~jl/3 weakens, as found experimentally.

On the other hand, in the situation where charge carriers when bound to eentres are small polarons and when free are large polarons, the activation energy for intra-eeutre mobility can be expected to contribute to a residual value of Q. Such a situation seems realized in the cases of NiO and a-F•203 (see §§ 5.1.1 and 5.1.3). For the ease of CoO, such a contribution to Q can be expected to be extra large, due to the low-spin state of Co a+ when neighbofiring a Li+ ion (see § 5.1.2).

Above 300°K with increasing temperature, the temperature dependence of ~ for almost all samples decreases. This effect is quite marked for NiO and a-F%03. In these cases ~ eventually becomes nearly temperature-independent. In this region (T_~ 1000°I~) exhaustion of the dope centres is practically complete. I f one supposes that NiO and a-F%0 a are non-adiabatic small-polaron conductors, upper limits at T _ ~ 1000°K for the activation energy U (eqn. (23)) of 0.i5 and 0.10 ev, respectively, should apply. Such a simple estimate of an upper limit for U is not possible for CoO and MnO because the exhaustion region there is not clearly defined. Especially for the low-doped samples of MnO and CoO in the vicinity of 1000°K, ionization of deeper acceptor levels, corresponding to cation vacancies, leads to a marked increase of the temperature dependence of ~. A similar effect can be seen above 1000°2 for the case of low-doped Ni0. The change of Q behaviour in a -F%Q above ]000°K, however, is due to the onset of intrinsic conduction (see § 3.4).

An estimate of/~D can be made from the dependence of ~ in the exhaustion region on dope. The assumption underlying this estimate is that p (or n) equals the dope concentration. This assumption is reasonable because eounterdope for all samples proves to be low. Furthermore, the value of p (or n) at a certain temperature in the exhaustion region is somewhat lower than the uncompensated dope, because exhaustion may not be expected to be totally completed. Therefore the value derived for t~D will be an under limit. The dependence of ~ on dope is shown in fig. 10 for all substances at a temperature of 1200°I{. I t is seen that, as expected, ~ depends linearly on dope except at low dopes where cation vacancies (NiO, CoO, MnO) or intrinsie effects (~-F%Os) contribute to conduction. The values of /x D at 1200°~, derived from fig. 10, are

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0.02 cm~/v sec for MnO and 0.1, 0.25 and 0.4 cm~/v see for ~-1~%08, CoO and NiO, respectively. The values derived for the latter three substances are not so far apart, whereas/z D for MnO is an order of magnitude lower. Out of these four substances, p-type MnO seems therefore most appropriate to show the characteristic features of a small-polaron conductor. Actually, /z D at 1200°K in the oxides Ni0, CoO and ~-F%0 a has values which are not small compared with the upper limit of 0.1 cm2/v see for smMl-polaron hopping mobility (see § 2.1.2).

~pC~,cm) -I

~0

Fig. 10

T= t200 °g

[ I I I I I T I I [ I I I I 1 ~ I r i I r , , l J . . . . .

10-3 10-2 ~0 -~ t t(3 - - -w dope concentration ~t. %)

l~esistivity in the exhaustion region at 1200°K as a function of Li dope for Ni0, CoO and lVInO and as a function of Zr dope (open squares) and twice the Nb dope (black squares) for ~-F%0a. Deviations from the linear behaviour at low dopes are due either to the presence of cation vacancies (Ni0, CoO and 3/[nO) or to intrinsic effects (ct-F%0a).

Incorporation of Nb compared to Zr (Ti) as an impuri ty in ~-F%03 not only leads to higher donor-ionization energies but also to an increase of conduction at high temperatures by a factor of 2 (see fig. 7). This latter fact demonstrates tha t te travalent Zr (or Ti) corresponds to a donor holding one electron and pentavalent Nb to a donor holding two electrons.

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Dielectric-loss measurements (see § 5.1.3) have shown that the donor centre corresponding presumably to a neutral oxygen vacancy (Vo) has an ionization energy which has a value in between that corresponding to Zr and neutral Nb donors. I t therefore depends on the amount of counterdope whether conduction in Nb-doped samples at low temperatures is determined either by V 0 donors or by Nb donors. At high temperatures Nb donors always dominate conduction if [Nb] > [ V0].

The resistivity data for NiO, CoO and ~-F%03 extend over a temperature region including the N6el temperatures of these substances : 520, 290 and 960°K, respectively. The N6el temperature of MnO (120°K) lies below the temperature range considered. The resistivity behaviour for the three substances mentioned is not influenced noticeably by the magnetic transition, except only for the higher doped CoO samples (see fig. 6). In this latter case the effective density of states of the valence band is probably influenced by the magnetic transition. Changes in the temperature dependence of ~ have been observed for the case of single crystalline NiO at 520°K by Austin, Springthorpe, Smith and Turner (1967) and at 400°~ and 500°K by Vernon and Lovell (1967). A discussion of the influence of magnetic effects on conduction will be postponed until §4.

3.4. Estimates of/ .9 from Resistivity and Seebeclc CoeJficient

Further information with regard to drift mobility of free charge carriers can be obtained from Seebeck-coefficient data. The Seebeck coefficient of an extrinsic semiconductor is a measure of the free charge carrier concentration (see eqns. (51) and (55)). Therefore, combination of ~ and

data in principle allows the determination of/*n and its temperature dependence. When taking as a basis the magnitude of /*D, calculated from ~ data in the high-temperature exhaustion range (see preceding section), an estimate of/*D can be given in the whole temperature range considered. Complications in the estimate of/*D behaviour arise due to the other parameters that determine ~, viz. the density of states of the conduction level and the heat of transfer (see eqns. (51) and (55)). More- over, influences of impurity conduction (see § 3.2) and inhomogeneities (see § 3.1) in the sample may make a solution in terms o f / .9 behaviour impossible. A critical review of the present state of affairs in this respect, will be presented in this section.

Figure 11 presents in one graph the behaviour of resistivity and Seebeek coefficient as a function of temperature for NiO, CoO, ~-F%0 a and MnO. For the sake of convenience, the reduced Seebeck coefficient is shown (e/(2.3k~e)), which will be denoted by ~ ' ; 2.3k/e is equivalent to 198.4/,V/°K. All samples have been doped roughly to the same amount of 0.1 at. °/o. I t may be remarked that the choice of dope level is, within the framework of the present technical abilities, the best possible for the analysis in terms of drift mobility. Higher dopes in general may be

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expected to introduce effects due to interference of dope centres, and in the cases of Ni0 and e-F%O 3, to a further increase of impurity-conduction level. Lower dopes introduce extra difficulties with the control of

homogeneity.

Fig. 11

T (oK) q

1000 300 200 150 100 00 [ i I I [

i ) (,l'~¢m) MnO ~ / //o / , / / o-re 03

/ / / / / /

11

/ f-..~io / / \

/

\

0 1 2 3 ~ 5 6 7 8 1

9 10 11 12 13 m. ~ - 3 (o K-l)

A comparison between the behaviour of log 9 and the absolute value of the reduced Seebeck coefficient as a function of reciprocal temperature for Li-doped (_0.1 at. %) p-type NiO, CoO and MnO and Nb-doped ( ___0.1 at. %) n-type ~-F%Oa.

The important feature of ~ and @ data presented in fig. 11 is on the one hand the close similarity in temperature dependence of e and log 9 for the cases of CoO and Ni0, in the lat ter case only at temperatures above 160°K, and on the other hand the dissimilarity of e and log 9 behaviour for ~-~e203 and Mn0.

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where

In the case of NiO, below 160°~, and of e-F%Oa, in almost the whole temperature range, the diverging behaviour of e and log ~ is thought to be due to the occurrence of impur i ty conduction. No indications have been found in NiO and ~-F%Oa, of an influence on e behaviour from second- phase inclusions. For the case of CoO, however, the presence of CoaO 4 as a second phase has been found to have a marked influence on ~ behaviour (see page 49 of this paper). In the simultaneous presence of (partially compensated) n-type conduction (o-n) and impurity conduction (aimp), the Seebeek coefficient is determined by the position of the Fermi level with respect to the two conduction levels and by the relative contributions of charge transport :

O'tot,

o-tot = an + o-ling" (86)

In the ease of impurity conduction the heat of transfer has been assumed to be negligibly small. In the temperature region, where conduction is dominated by ~n, a is determined by the first term on the right-hand side of eqn. (85). At temperatures below the exhaustion region (see § 2.3), - a will rise with decreasing temperature. At low temperatures, where o-im, dominates conduction, ~ is determined by the second term on the right-hand side of eqn. (85). In this region ~ may be expected to have a low, temperature-independent value which can have a positive or a negative sign, depending on compensation degree and multiplicity factor of the aceeptor level (see eqn. (58)). I t follows from the above tha t in the intermediate temperature range, - e passes through a maximum. This situation is found for n-type ~-F%O a at about 125°K and for p-type NiO at 140°K (see fig. 11). In a-F%O 3 compared to NiO, the larger contribution to charge transport from impurity conduction (see § 3.2) is already apparent at temperatures around about 300°K in a divergent behaviour of ~ and log ~. In the ease of NiO this divergence becomes noticeable only below 160°K. A further analysis of ~ and log Q behaviour for the case of a-F%O 3 does not make much sense because of the strong influence on ~ from impurity conduction. An unambiguous separation of the two contributions to conduction seems as yet impossible because the temperature dependence of impurity conduction, and possibly of normal conduction too, is influenced by the temperature-dependent dipole contribution to the dielectric constant (see 993.2 and 3.3). Furthermore, the small-polaron character of bound charge earners (see § 5.1.3) will give rise to an appreciably activated character of o-~mp at temperatures approaching ½0 (see § 3.2). I t would be highly desirable to have n-type ~-Fe20 a samples where the donors are very weakly compensated so that the influence of impurity conduction has been sufficiently suppressed. The absence of a noticeable influence of impurity conduction

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in Ni0 above 160°K and in CoO and MnO at all temperatures considered (see § 3.2) allows a further analysis of ~ and log ~ for these cases.

The activation energies determining ~ and ~ behaviour between 200°x and 500°K (see fig. 11) are 0-30 and 0.31 ev, respectively, for the ease of Ni0 (0.088 at. °/o Li) and 0.40 and 0.385ev, respectively, for CoO (0-08 at. °/o Li). The small, non-systematic difference between these quantities implies that tZD between 200 and 500°K is approximately independent of temperature. Anticipating the extensive discussion on the ' smMl-polaron versus band conduction ' alternative presented in § 6, the possibility of small-polaron conduction in Ni0 and CoO will be briefly discussed here. On the assumption of a constant density of states, the and ~ data presented above for NiO and CoO lead to a tzD behaviour as shown in fig. 14. I t is seen that below 500°K, I~D is indeed eonstant or even decreases with increasing temperature. Above 500°K, tzD shows a more or less thermally activated behaviour. I t will be shown (see § § 4.1 and 4.2) that it does not seem possible to account for this latter behaviour on the basis of the existing theory for adiabatic smatl-polaron hopping, because the value of ~D is too large and, certainly in the ease of Ni0, the coupling constant is too small. In the same context it will be shown tha t the assumption of non-adiabatic smMl-polaron hopping seems "inap- propriate for explaining the thermally activated behaviour of tLD above 500°K solely. The assumption tha t /z D below 500°K is determined by small-polaron band conduction does not seem acceptable because the transition temperature between hopping and band regime would be expected at appreciably lower temperatures, i.e. at T<~O~_200°K (see § 2.1.3). In the above interpretation of e and ~ it is assumed, in conformity with the theoretical predictions, that the heat of transfer (flU) due to drifting small polarons is negligibly small (fl=0, see § 2.3). A heat-of-transfer f lu with fi~ 1 seems indeed improbable because one would then expect in the exhaustion range (T--1000°K) a continuous decrease of e with temperature, characterized by an activation energy U. No such behaviour of ~ has been found. In the approach to the exhaustion region ~ behaves in accordance with log ~ (see fig. 11).

In contradistinction to the similar behaviour of ~ and log ~ versus l IT found for Ni0 and CoO in a considerable temperature region, a strikingly different behaviour of these quantities has been found for the ease of MnO (see fig. 11). For MnO (0.1 at. % Li), Q(~) = 0.2 ev (200-1000°K), whereas Q(~) = 0.5 ev (200-400°K). From this different behaviour of ~ and log e, Crevecoeur and de Wit have concluded tha t the drift mobility of holes in MnO, which already proved to be exceptionally small at high temperatures (/x D _~ 0.02 cm~/v sec at 1200°K, see § 3.3), decreases roughly exponentially when the temperature is lowered from 1000°K to 200°K. Thus p-type conduction in MnO seems to be effected by thermally activated hopping of small polarons. Crevecoeur and de Wit ascribe the small-polaron character of holes in MnO to an extra interaction between hole and lattice vibrations brought about by the Jahn-Teller deformation of the Mn a+ ion

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44 A . J . Bosman a n d H. J. van DaM o n

and the surrounding octahedron of neighbouring O 2- ions. I f this con- elusion is correct, the case of p-type MnO demonstrates the above- mentioned proposition, viz. small-polaron hopping remains dominant over small-polaron band conduction down to temperatures far below T = ½0_~ 400°K.

A further analysis of the Seebeck-coefficient data will be made from a plot of le~zT{ versus T for the four substances considered, all samples being doped to about 0.1 at. °/o (see fig. 12). In the extrinsic range in principle three regions can be distinguished (see § 2.3), viz. the ' partially

0.7 - efodT (eV)

t 0.6

0 . 5 - -

0.4

0.3 - t ~

0.2 ; /

0.1

00 I

Fig. 12

N i 6 (~xx×~-~ ~x~ x× y

- Fe 203

i I

MnO

" ' ">oo' . . . . . . . . . . . . i l o o - T ( O K )

The variation of e l~ 1T with temperature for the samples corresponding to fig. 11.

compensated region' (I), the 'uncompensated region' (II) and the ' exhaustion region' (III), each region being characterized by a different slope of e ~ T versus T. In NiO, the regions I and I I [ are clearly visible. At the low-temperature side (below 170°K) impurity conduction lowers ec~T below the normal extrinsic value. In CoO regions I and I I I are visible too, although the latter is not clearly pronounced because of the influence on ~ at the high-temperature side of deeper aeeeptor levels (Co-vacancies). In the ease of CoO an influence of impurity conduction at the low-temperature side has not been found. MnO shows only one region, which can be classified either as I or II. Crevecoeur's analysis, given below, decides in favour of I I ; no influence from impurity conduction is detectable. In the ease of ~-F%O 3 region I I I is only partially recognizable. At the high-temperature side a deeper donor level (Nb +)

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comes into play. No certainty about the presence of other regions can be obtained because of the strong influence on ~ from impuri ty conduction.

A quantitative analysis of the ~ behaviour as a function of temperature allows in principle for the determination only of the ionization energy of the majori ty centre (E A for aceeptors, E n for donors) and of the majori ty and minority-centre concentration. The (effective) density of states N v (or Ne), the heat-of-transfer constant A and the degeneracy-multiplicity of the majori ty centre g cannot be solved separately. These latter constants always present themselves in combinations of the type N v exp A or g exp A. Supplementary data on one of these latter constants would only enable a complete solution for all constants. In region I I I of a p-type conductor the relation

eaT =]cT In {NveA/(N ~ - ND) } (87)

holds if exhaustion of one type of acceptor predominantly determines conduction (see § 2.3). In what follows arguments will be adduced in support of a low compensation degree for all samples considered, so that in eqn. (87), N n may be neglected. Estimates of Nv cA, obtained, from the slope e~T versus T and the dope concentration are : 0.35N 0 (Ni0), 0-5N 0 (COO), 0.8N 0 (e-F%03) and 3.5N o (MnO), where N o denotes the concentration of cations.

A more accurate estimate of N v eA in the exhaustion region can be obtained from a plot of ~', at a certain temperature in this region, versus dope. The relation between a' and dope for a p-type conductor is given by :

' NveA log N~'eA (88) c~ = log N~ - N D -~ N A

I t is seen in fig. 13 tha t for each substance at 1000°K, c~' is indeed proportional to log (Nm~j) -1, Nma j denoting the different dope concentrations. For the ease of Nb-doped ~-F%03 samples the dope plotted represents twice the Nb concentration because in the exhaustion region the Nb donors are twofold ionized. Extrapolat ion of the linear relation between e' and log (dope) to ~ '= 0 gives values for Nve A. These values can be somewhat too high if, at 1000°~, exhaustion of the majori ty eentres is not totally completed and counterdope is not completely negligible. The value of 1000°K has been selected because at tha t temperature in all samples considered, exhaustion is next to completion and an influence on from ionization of cation vacancies or from intrinsic conduction is barely noticeable. The results for N.~e ~ are : 0-3N o (NiO and COO), 0.9N 0 (~-F%0~) and 2N o (MnO). I t turns out again that there is a distinct difference between, on the one hand, NiO, CoO and to a less extent also ~-Fe20 a and on the other, MnO. For the case of MnO, Nve ~ roughly equals the max imum value for the density of states : 2N0, whereas its value is definitely lower for the other substances. There are thus three

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46 A . J . Bosman and It. J. van DaM on

facts in Crevecoeur's data on ~ and 9 for Mn0 which point to its small- polaron character : (1) t~D at high temperatures (~ 1200°K) is relatively small (___0.02 em2/

V see). (2) /~n with decreasing temperature down to 200°K, starting at 1200%(,

diminishes roughly exponentially. (3) N,~eA~_2No, which, within current small-polaron theory (A=0),

means tha t indeed AT v ___ 2N 0. On the contrary, in NiO and CoO, and to a less degree also in a-F%O~ : (1) /~D at high temperatures (~ 1200°K) is relatively large (0.4, 0-25 and

0.1 em2/v see). (2) ~D with decreasing temperature down to 200°K does not diminish

exponentially. (3) Nve A at 1000°x equals 0.3N 0 (NiO, CoO) and 0.9N o (e-FegO3), which

is already significantly below the maximum value of 2N o. The last-mentioned fact suggests that NiO, CoO and a-F%O s can be considered to be band semiconductors, where N v is an effective density of states which is smaller than the maximum value 2N o and where the heat of transfer A/cT has a value different from zero. In a broad-band semiconductor AT, increases with temperature in proportion to T ~/2. An important indication for an increase of N , has been found for the cases of NiO and CoO in the temperature range 1000°K to 1300°K, where increases with temperature (see Bosman and Crevecoeur 1966, fig. 4 and Bosman and Crevecoeur 1969, fig. 2). In a band semiconductor, for most types of charge-carrier scattering, values of A lie between 2 and 4. I t will be shown in § 4 that, within the assumption of a large-polaron band model, the drift mobility of holes in NiO and CoO is determined by polar optical-mode scattering (intermediate coupling strength) and/or spin- disorder scattering. A value for A close to 2 is compatible with this situation. We then arrive at values for N~ at 1000°K of 0.04N 0 (NiO and CoO) and presumably of -~0.1N o (e-F%Oa). A value for N~, such as found for NiO and CoO, corresponds to a density-of-states effective polaron mass rap*___6 m o (see cqn. (50)), i.e. to a resonance integral Jp ~_ 0.05 ev and a large-polaron band width AWp _~ 0-6 ev.

Further information concerning the parameters determining transport properties can be obtained from the behaviour of ec~T versus T outside the exhaustion region. The ' partially compensated region ' encountered in the cases of NiO and CoO (see fig. 12) is described by the relation (see § 2.3) :

eaT= E a - l e T In \ ND geA l. (89)

Extrapolation versus T = 0 of the linear part of the curves leads to values for E a of 0.31 ev (Ni0) and 0-385 ev (COO). Extra information concerning the compensation degree K ( = N D / N a ) is wanted to permit a further interpretation of the a behaviour. Chemical determination of the ratio

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[Nia+]/[Li +] leads to the es t imate K = 4% for the NiO sample of fig. 12. F r o m an a l ternat ive method, in which the e da ta were analysed in the t rans i t ion range between regions I and I I I , i t was es t imated tha t K_~ 10%. F o r CoO, informat ion with regard to the a mo u n t of counterdope could be ob ta ined from dielectric-loss measurements on very l ightly doped samples. I f the same counterdope is present in the CoO sample of fig. 12, it can be

Fig. 13

lal

T= 1000 OK

Mn1_xLix 0

%-xLixO ~ "l:lx xX " x

x 4 ~ , , %.~, "%'%

0 ~ _ ' ' ' I F I I ' , , , , , , 1 , , , , ~ , , , I ` ` ' , , , , i ~ ' , ,

lo-~- 10 -1 1 10 102 dope concentration (at %)

The absolute value of the reduced Seebeck coefficient in the exhaustion region at 1000°K as a function of dope for the cases of p-type NiO, CoO and MnO and n-type a-F%Oa. For the ease of Nb-doped ~-Fe20 a the data have been plotted versus twice the dope concentration.

character ized by K = 5 ° . Wi th K = 5% and A = 2 the slopes of e~T/kT for N i 0 and CoO in region I lead to g_~ 1. The inaccuracy in the determina t ion of the compensat ion degree allows for an appreciable uncer ta in ty in this value of g.

A fur ther analysis for the ease of ~-Fe20 a outside the exhaus t ion region seems bare ly significant because of the marked influence of impur i ty conduct ion on e behaviour .

The ~ behaviour found for M n 0 (see fig. 12) can be character ized ei ther as ' par t ia l ly compensa ted ' or as ' non-compensa ted '. In the former

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case this calls for a high and in the latter for a low counterdope. In the ' non-compensated region ' holds :

eaT = ½E~ + ½/cT In {ge~Nv~ (90) \ N A ] "

I f the MnO sample, down to the lowest temperature (_~ 200°K), may be considered to be ' non-compensated ', extrapolation of eaT versus T = 0 leads to a value of 0.2 ev for ½E A and thus to an ionization energy of 0.4 ev for the Li aeceptor. With A = 0 and N v = 2N0, the slope of ea t versus T then leads to g - 40, a value which is rather high compared with that estimated for the Li acceptor in NiO and CoO t. Because there are no indications at the low-temperature side of eaT versus T of an onset to the 'par t ia l ly compensated region ', it should be concluded that, within this interpretation, the compensation degree K is certainly smaller than 0.01%. I f on the other hand the partially compensated ease applies (see eqn. (89)) it follows that the ionization energy of the Li acceptor is only 0.2 ev. Furthermore, quite high degrees of compensation, i.e. of about 90% or more, would then be involved. Crevecoeur and de Wit decided in favour of the ' non-compensated ' situation because results from chemical analysis and dielectric loss measurements are not compatible with a large compensation. Furthermore, according to these authors, exhaustion of the accepters would be expected at relatively low temperatures, which is not observed.

In connection with the rejection of the highly compensated alternative, a discussion of the interpretation of Mn0 data seems appropriate at this point. I t should be recalled that two of the three facts mentioned above in favour of the small-polaron character of free holes in Mn0 are weakened or even become invalid if allowance is made for a high compensation degree (_~ 90%). In tha t case, p in the exhaustion region equals N A - N D instead of NA, so tha t /d. D at 1200°K is an order of magnitude larger (_~ 0.2 cm2/v see) and Nve A an order of magnitude smaller (_~0.2N0). The latter consequence especially would not be consistent with the small- polaron character of the charge carriers. An appreciably lower compensation degree, for instance of about 10%, would not introduce the above-mentioned inconsistency. However, such a compensation degree requires an impossibly high value of about 2500 for the multiplicity factor. Therefore, the available experimental facts can only be consistently interpreted if it is assumed that effects due to counterdope are absent. In the ' non-compensa ted ' case, for the sample doped with 0.1 at. °/o of Li, the compensation degree was found to be smaller than 0.01% and thus the counterdope smaller than 5x 1015 cm -a. In the opinion of the present authors, confirmation of the low value arrived at for the eounterdope is desirable.

t A discussion of the multiplicity factor g, in connection with the small- po]aron character of bound charge carriers in NiO, will be found in § 6.

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The possibility of the presence of effects having an influence especially on the behaviour of a should be carefully considered. I t seems to be justified to assume that in MnO impurity conduction has a negligible influence on ~ (see § 3.2). Another effect might correspond to the presence of second-phase inclusions due to oxydation of the material (MnaOa). For the case of CoO a spurious effect has been discovered which leads indeed to a deviation from the normal ~ behaviour similar to that brought about by impurity conduction, without it having any perceptible influence on the ~ behaviour (Bosman and Crevecoeur 1969, fig. 3). This effect is brought about by oxidation of CoO, resulting in the introduction of small amounts of CoaO 4 into the sample. The effect can be suppressed again by reduction of the sample. The origin of this effect is not yet understood. In Li-doped single crystals of MnO, All, Fridman, Denayer and Nagels (1968) measured a ~ behaviour quite analogous to that found by Crevecoeur, whereas a was observed to be almost temperature-independent. Furthermore, partial oxidation of the material could even result in a decrease of a and an increase of ~. Crevecoeur and de Wit observed that the introduction of Mn304 regions into an MnO sample, brought about by oxidation of the material, did'not introduce any change in the values of a and ~ at room temperature. However, the above-mentioned discrepancy in the results obtained for by different authors remains unsatisfactory.

The conclusions arrived at in this and the former section concerning parameters determining transport properties allow for an estimate of/~n behaviour as a function of temperature. A study of resistivity as a function of temperature, dope and counterdope led to reliable values of/z D in the exhaustion region (~ 1200°K). These values are the basis for the further valuation of ~D at lower temperatures. Data for ~ led to the inference that the density of states is a maximum in the case of MnO but is two orders of magnitude below the maximum for the cases of NiO and CoO and one order for that of ~-Fe20 a. Evidence was found in favour of an increase of the density of states for NiO and CoO at T > 1000°K. It was concluded that MnO can be characterized as a small-po]aron semiconductor while NiO, CoO and probably also ~-F%03 can be conceived of as band semiconductors. It seems then most appropriate in the interpretation of a at temperatures below 1000°K to let N v he independent of temperature for the case of MnO and to allow for a further decrease of N v for NiO and CoO. For reasons mentioned above, a-F%0 a will be left out of further valuations. It seems justified to assume IVy ocTal2 for NiO and CoO, because the relatively low value for the density of states at 1000°K implies a band width at that temperature which is an order of magnitude larger than ]cT. Evaluation of ~D as a function of T can now be made from the experimental data for ~ and employing the relations ~oclog (Nv/p) and ~oc(1/p/~n). The results can be seen in fig. 14. The resulting /~D behaviour has also been indicated for l~iO and CoO on the assumption that N v is a constant. It will be

A.P. D

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50 A . J . B o s m a n and H. J . v a n Daa l on

obvious f rom the above tha t , in the opinion of the presen t authors , there is no clear evidence in f avour of the l a t t e r behav iou r as far as the presen t da t a for a and ~ are concerned~. I t is seen t h a t wi th decreasing t e m p e r a - ture down to room t e m p e r a t u r e / x D for NiO and CoO rises cont inuous ly

lOOO ~! l u i ( cm ' / ' '~c)

Fig. 14

r ("K) 300

' I . . . . . . . . ---NiO(NvaT ~)

" . _ _ _ c o c , ( , ¢ ~ , T * ' ~

..,..,.

10- ~ i ~ I I i I r

\ \

,,,~no (t,4,'= c)

\ \

I f I l [ I I I I f ' ~ 4-

q I k I I r t I I

5 6 . ..~-3 (o K -'1)

The behaviour of the drift mobi l i ty /% as a function of reciprocal temperature, deduced from a and @ data on the assumption either of a constant density of states (Nv=C) : p-type NiO, CoO and 1VfnO, or of a density of states NvocTam: p- type NiO and CoO. In all cases Li-doped samples ( _ 0" 1 at. °/o ) were used.

to values of a b o u t 5 and 0-7, respect ively, whereas for MnO it diminishes to a va lue of 3 x 10 -5 cm2/v sec. I n the l a t t e r case FD is t h e r m a l l y ac t iva t ed wi th an ac t iva t ion energy U = 0.3 ev : /% oc T -a/~ exp ( - U/kT).

The d a t a of ~ and @ considered in the foregoing, re la ted to the extr insic region only. I t should, however , be ascer ta ined t h a t conduct ion a t high t empe ra tu r e s is indeed pure ly extr insic and t h a t intrinsic effects have no

The alternative interpretation of c~ and 9 data in terms of small-polaron conduction will be considered in the discussion (see § 6) and in § 4.

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influence. Otherwise, the presence of free electrons as well as holes will severely hamper the interpretation of experimental facts in terms of mobility. While conduction in the lighter doped samples of all materials at high temperatures is determined by deviations from stoichiometry, in some of them (~-Fe203 and MnO) it can be influenced at the same time by intrinsic effects. One can discriminate between these two effects by means o f ' Jonker's Pear ' construction (Jonker 1969). This construction consists of a plot of ~ versus log ~, where all data have been obtained at one and the same temperature. The quantities are varied by altering either the dope or the deviation from stoichiometry. I t has been shown by Jonker that in the presence of intrinsic effects a pear-shaped curve results. Under certain conditions the shape of the pear is determined by the parameter (Eg/kT + A++ A- ) solely, i.e. by the energy gap (Eg) and the heats of transfer for holes (A +) and electrons (A-). These conditions are tha t for electrons as well as for holes, density of states,

Fig. 15

/.- MnO(1000°K) i

_ NiO(1000°K) ,-,/ ~ NiO~00OK)

400 "~...,,~NiO (I&50°K _ C°O (I000 °K)/~/o//~

200 ." / / ~---Mn00370~K)

] //// /

O- / - ... CC Fe203(1400OK)

. %

-re O (1 OOOK)

~ - Fe2%0000 OK)

r I I ~H I ( r f i r t H , I / i [ ( ( I ,

10 -3 10 -2 10 -1 1 10 10 2 10 3 , , p (,.n.orn)

The relation between the Seebeck ooeifioient ~ and the resistivity Q at different temperatures for ~TiO, CoO, MnO and ~-F%O 3. The data for MnO were obtained by Creveooeur and de Wit (1968 a and to be published). Data marked with ~TiO 1) are from Bransky and Tallan (1969).

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52 A. J. Bosman and H. J. van Daal on

heat of transfer and mobility are independent of dope. An example of such a pear is presented in fig. 15 for the ease of MnO at 1370°K (Creveeoeur and de Wit 1968). The two straight lines forming the low-resistivity part of the pear intersect at a point slightly below the ~ = 0 line, from which it can be concluded tha t ~-XV-e A- is only slightly larger than F+N+e ~+. Of these two straight lines, the upper one represents the purely extrinsic p-type region, intersecting the ~ -- 0 line (or log p = 0 line) at - (+ ) log (N+eA+eF+). The point on the upper line where a deviation from the linear behaviour begins marks the onset of intrinsic effects. I t can thus easily be checked, for a set of (~, ~) points at a certain temperature, where the influence of intrinsic effects begins to be noticeable. From the shape of the pear for MnO at 1370°K, Crevecoeur arrives at an upper limit for Eg of 1.8 ev at this temperature. In fig. 15 examples of partial pears for the case of ~-F%Os at different temperatures are presented. This fig. 15 also shows from the ~ versus log p behaviour for NiO and CoO at 1000°K tha t conduction is extrinsic down to the lowest dopes applied.

Finally we would like to consider the very high temperature region of l~iO, CoO and MnO. I t has been claimed by a number of authors that evidence exists for the small-polaron character of free holes in the above- mentioned materials at 1000°c ~ T ~< 1500°c. I f the mobility is described by ~ocT -3/~ exp ( - U/kT), values arrived at for U are 0-43 ev for NiO (Bransky and Tallan 1968), 0.36 ev for CoO (Fisher and Tannhauser 1966) and 0.57 ev for MnO (Hed and Tannhauser 1967). These conclusions were based either on a and 9 behaviour (1NliO) or on p behaviour and gravimetrie data (CoO and MnO). The above inferences with regard to mobility behaviour are in disagreement with those drawn from a and Q behaviour in the relatively low-temperature region (T~< 1000°c) as described in the major part of this chapter. A fundamental difference between the FD behaviour above and below 1000°c seems scarcely conceivable. In our opinion the conclusions arrived at by the authors mentioned above with regard to ~D behaviour in the very-high-temperature range are disputable either because of inconsistency of the experimental data or because their interpretation of these data is questionable. I t should be realized tha t in the very-high-temperature region, where the average value of let is large (~_0.15 ev), while the value of 1/T only slightly changes, small deviations from the true a and p behaviour or a slight mistake in the interpretation of the data easily lead to an appreciable activation energy in the drift mobility.

Bransky's results for NiO are not supported by the experimental data presented in this paper. Extrapolation of his data for p and a, taken at 1 at. of 0~, to 1000°K (his figs. 3 and 6, respectively) leads to an (a, p) point which, for the purely extrinsic case, strongly deviates from our data (see fig. 15). Furthermore, according to Bransky, a at 1176°c shows a non-systematic, quite small variation with oxygen pressure (at the most 50 Fv/°c, see his table II), whereas in the same circumstances 9 changes

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by an order of magnitude (see his fig. 4). The corresponding (~, ~) points have been plotted in our fig. 15. Such a behaviour of ~ and Q is not consistent with extrinsic conduction. The interpretation as given by Bransky of ~ and ~ data, however, was based on the assumption tha t conduction behaves purely extrinsically. I t might be suggested tha t a spurious effect has influenced Bransky's data for ~. An indication of a possible error in his evaluation of ~ is the fact that the thermal e.m.f. does not go to zero in the absence of a thermal gradient (see his fig. 5).

Fisher's conclusion as to the activated character of the drift mobility of holes in CoO between 1000°c and 1350°c was based on observations of conductance and deviation from stoichiometry (Co vacancies) with varying oxygen pressure. I t is claimed tha t within a narrow range, of about 0.25 to 0.60 mol °/o excess oxygen, the Co vacancies are predominantly singly ionized, whereas above and below this range neutral and doubly- ionized vacancies become important, respectively. I t is observed that with increasing temperature at a certain amount of excess oxygen, within the narrow range mentioned, conductance increases by about 35~o (see her fig. 4). I t is concluded tha t this increase is due solely to the drift mobility. I t seems to the present authors, however, tha t at" least part of the relatively small increase of the conductivity might still be due to an increase of the ionization degree of the Co vacancies. This is because the existence region of singly-ionized vacancies is small. Addi- tional evidence for the activated character of the mobility therefore seems to be desirable. Data for the Seebeck coefficient, presented in Fisher's paper, however, do not corroborate her interpretation of gravi- metric and electric measurements. Combination of Fisher's data for and ~ at constant 08 pressure leads to the conclusion tha t Nve~tLD decreases with increasing temperature. With constant ~V v and A this implies a decrease of tLD with rising temperature. A marked temperature dependence of A could introduce a rise of tLD with temperature. How- ever, such an assumption, as made by Fisher, seems to have no firm basis.

Hed arrived at a large value for the activation energy of the drift mobility of holes in MnO between 1100 and 1550°c from an interpretation of the measured relation between conductance and deviation from stoichiometry. In his paper the parameter q denotes the ratio between the partial CO 2 and CO pressures around the sample. I t could be concluded that the contribution to conduction from holes (pet%) in a fairly large range of q values, up to q_~ 50, is determined by doubly-ionized Mn vacancies. A comparison at constant g value of the excess oxygen concentration (x) and conductance (G) as a function of temperature led to the inference that/~p behaves as exp ( - U/kT) with U = (0.37 _+ 0-07) ev. This method of deriving t% behaviour is allowed as long as ~t%/nt~n at constant g is independent of temperature. This condition seems to be reasonably fulfilled. In that case a comparison between the temperature behaviour of x and G at different q values is allowed, From such a

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comparison for x at q = 20 and G at q = 1 (Hed's fig. 7), taking into account the inaccuracy of the data, we arrive at a t% behaviour characterized roughly by exp ( - U/leT) with U___ 0.15 ev, which value for the activation energy is significantly lower than that arrived at by Hed.

§ 4. HALL COEFFICIENT AND RESISTIVITY; INTERPRETATION OF HALL AND DRIFT MOBILITY

In this chapter data for the Hall coefficient obtained on NiO (§ 4.1), CoO (§ 4.2), ~-F%O 8 (§ 4.3) and MnO (§ 4.4) are presented. Particular attention is paid to the occurrence of an anomaly shown by the Hall coefficient of NiO and ~-F%03. The behaviour of the Hall mobility as a function of temperature and dope is discussed. An interpretation is presented of Hall and drift mobility, the latter having been discussed in the preceding chapter.

I t will become clear from what follows that reliable results for/z H are obtained on single crystalline as well as on polycrystalline (ceramic) material. In the case of isotropic material the need to have Hall-effect data on single crystals is sometimes exaggerated in the literature. I t might be emphasized that both types of material are liable to suffer from inhomogeneities. In the presence of inhomogeneous conduction in general the Hall mobility is found to have too low values. This has been demonstrated, e.g. for single crystals of SiC (van Daal 1965). In the case of ceramic material an improper preparation mostly leads to relatively thin high-ohmic grain-boundary layers (see § 3). I t has been shown by Volger (1950) that in this case the measured value of R H is very near to the bulk value, whereas the measured value of Q is much higher. The combination of these results leads to a much too low value for IzH. By proper preparation, grain-boundary effects of this type can be suppressed to a very large extent. Due to the low value in these materials of the free-path length for charge carriers compared to the size of the grain, the correct value of/z~ is then obtained.

4.1. NiO

Data concerning Hall mobility of holes in NiO presented in literature, are summarized in table l, together with the main conclusions drawn by the various authors from these data.

I t is seen from table 1 that in the historical development the value of /ztt at 300°K consolidates around a value of about + 0.3 cm2/v sec for single crystals as well as for polycrystalline material.

The origin of the amazingly high values for t~H following from the data given by Wright and Andrews (1949) is mysterious. The dominance of spurious thermal effects in their measurements can be excluded because, according to Wright, the Hall voltage measured was equivalent to a

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change in temperature difference of 100°c. The presence of free Ni in the samples, according to Wright, is improbable and moreover would lead to a negative (anomalous) contribution to R m

With regard to the results of Fujime, Murakami and Hirahara (1961), the recorded extremely low value for tLE is probably not due to a too low value for R E but to a too high value for @. No information is given by Fujime about the nature of his NiO sample.

The results obtained by Nachman, Popescu and Rut te r (1965) are quite probably erroneous. In their determination of t ~ the possible influence of current-contact resistance has not been eliminated. Furthermore spurious thermal effects probably dominate their results. In the Hall- effect measurements these effects can arise because of a change in temperature difference across the voltage probes induced by reversal or switching on of the magnetic field. In d.c. experiments the corresponding Seebeck voltage adds algebraically to the Hall voltage. The alteration of temperature difference may be due either to movement of the heating system or of the sample. Obviously this effect can be suppressed by rigorously positioning heating system and sample. Residual thermal effects due to movement of the heating system can be eliminated by averaging data obtained by reversing the direction of electrical current and of magnetic field. This averaging procedure Seems to have been neglected by Nachman. Thermal effects due to movement of the sample cannot be eliminated by the above-mentioned averaging of data. However, a check can be made to see whether the positioning is rigorous enough. For that purpose the current electrodes are short-circuited directly at the sample. No voltage effect should then be detectable when performing a similar type of measurement as in the Hall-effect experiment. The spurious thermal effects may be accompanied by a noticeable inertness in the response of the voltage output upon reversal of the magnetic field. The corresponding relaxation time depends on the heat capacity and resistance of the system. The ' magnetic viscosity ' as observed b y Naehman below 500°K and the ' t ime dependence ' as observed by Austin (1966) above 500°K can be explained in this way (see also Austin 1967). The occurrence in both cases of a limiting temperature for the thermal effects near to the NSel point (520°K) can be considered to be accidental. The ideal solution for the elimination of thermal effects is the application of a method of the double a.e. type (current and magnetic field) and was realized for the case of NiO by Tallan and Tannhauser (1968).

An interesting, not yet explained, phenomenon is the sign reversal of /~tt at about 600°K. Zhuze and ShelyCk (1963) already reported a rapid decrease with increasing T of t ~ above 400°K. They state that this effect is probably related to the onset of mixed conduction. All samples, except one, were measured only up to at most 600°K. The exception was an undoped polyerystalline sample for which R E data are given between 670 and 830°K. I t is strange that the sign of R E in this sample was not reported by Zhuze to deviate from that found in the other samples.

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Tab

le 1

. H

all

Mob

ilit

y N

iO

Date

1949

1961

1963

1963

1964

Au

tho

rs

Wri

gh

t an

d

An

dre

ws

Fu

jim

e et

al.

Kse

nd

zov

et

al.

Zh

uze

an

d

Seh

elyc

k

Roi

los

and

N

agel

s

Nac

hm

an e

t al

.

Ex

per

imen

tal

s. :

sa

mp

le,

c. :

co

nta

cts,

m

. :

met

ho

d

Tem

per

a-

ture

s.

no

t in

ten

tio

nal

ly d

op

ed

poly

crys

tall

ine,

ox

yd

ized

m

etal

str

ip

c.

Ag

m

. d.

c. ;

B

=ll

.5-k

a

s.

pre

sum

ably

po

lycr

yst

alli

ne

m.

a.c.

cu

rren

t (1

025

HZ

), d.

e.

fiel

d (5

kG

?)

s.

Li

do

ped

(1-

20 a

t. %

) po

lycr

ysta

llin

e, d

ensi

ty

86

-93

%

c.

curr

ent

Ag,

vo

ltag

e P

t m

. d.

c. ;

B

<2

0

kG

s.

no

n-d

elib

erat

ely

do

ped

si

ngle

cry

stal

s an

d

po

lycr

yst

alli

ne

mat

eria

l ;

Li-

do

ped

(3-

5%)

poly

- cr

ysta

llin

e, d

ensi

ty 8

0-8

5%

c.

A

g

m.

d.c.

, B

= 3

0 kG

s.

Li-

do

ped

sin

gle

cry

stal

s (0

.2%

) c.

ce

rro

low

+ 1

0~o

T1

m.

d.c.

, B

= 8

kG

s.

non-

deli

bera

l~el

y d

op

ed

poly

crys

tall

ine,

den

sity

87

%

c.

Ag

m

. d.

c.,

B=

20

kG

ran

ge

(°K

)

400-

700

290

150-

500

250-

500

170-

300

520-

660

Beh

avio

ur

of ~

H (

cme/

v se

e)

+ 5

00

(40

0°K

),

+ 1

00 (

700°

K)

+3

"7 x

10

-4

Incr

ease

wit

h d

ope

: 29

3°K

:

+0

.00

5

(1%

L

i),

+0

.02

(1

0%

Li)

m

axim

um

at

--

275

°K

T >

275

°K :

/X

H o

cexp

(0"

06 e

v/k

T)

Sin

gle

cry

stal

+

0.2

(300

°x)

po

lycr

yst

alli

ne

+0

.07

(3

00°~

) de

crea

se w

ith

tem

per

atu

re,

incr

ease

of

tem

per

atu

re

dep

end

ence

ab

ov

e 40

0°K

+ 0-

25 (

300°

K)

slig

ht d

ecre

ase

wit

h d

ecre

asin

g t

emp

erat

ure

+0

'06

(5

00°K

), a

bo

ve

500°

K

ccex

p (-

-0"6

ev/

kT).

Rm

be

low

50

0°K

, d

epen

ds

mar

ked

ly o

n m

agn

etic

fie

ld.

Th

ere

also

' m

agn

etic

vis

cosi

ty '

ob

serv

ed

Con

clus

ions

fro

m H

all-

effe

ct d

ata

Ex

trin

sic

con

du

ctio

n,

du

e to

im

pu

riti

es a

t lo

w T

an

d N

i v

acan

cies

at

hig

h T

, b

y h

oles

in

a re

lati

vel

y b

road

val

ence

ban

d

Ver

y s

mal

l v

alu

e o

f/~

m

ay b

e co

mp

atib

le w

ith

co

nce

pt

of h

op

pin

g

mo

del

Var

iati

on

of

a ab

ov

e 20

0°K

mai

nly

d

eter

min

ed b

y c

har

ge-

carr

ier

con

cen

trat

ion

. D

ecre

ase

of/~

H

abo

ve

200°

K d

ue

to l

atti

ce s

catt

erin

g

of h

oles

in

a n

arro

w b

and

Var

iati

on

of

a at

all

tem

per

atu

res

mai

nly

du

e to

ch

arg

e-ca

rrie

r co

nce

ntr

atio

n.

No

ho

pp

ing

ch

arac

ter

of m

ob

ilit

y.

Pro

bab

ly

con

du

ctio

n i

n n

arro

w b

and

by

ch

arg

e ca

rrie

rs w

ith

a h

eav

y

effe

ctiv

e m

ass

Car

rier

den

sity

in

crea

ses

wit

h

tem

per

atu

re

Ab

ov

e 50

0°K

:

ho

pp

ing

-ho

le

mec

han

ism

of

con

du

ctio

n d

om

inan

t.

Bel

ow 5

00°K

:

pre

sen

ce o

f an

om

alo

us

anti

ferr

om

agn

etie

Hal

l ef

fect

¢-~

©

O2

q~

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1965

1966

1967

1967

1967

1968

Bo

sman

et

al.

Au

stin

et

al.

Au

stin

et

al.

Kse

nd

zov

et

al.

van

Daa

l an

d

Bo

sman

Tal

lan

an

d

Tan

nh

ause

r

s.

poly

erys

tall

ine,

Li

dope

(0

.00

5-0

.1%

) c.

P

t-A

u

pas

te

m.

d.c.

, B

= 3

0 k

o

s.

Li-

do

ped

(0

.1-2

%)

sing

le

cry

stal

s m

. d.

c.

s.

Un

do

ped

an

d L

i-d

op

ed

(0-0

3-0

.5%

) si

ngle

cry

stal

s c.

P

t m

. d.

c.,

B=

18

kG

8,

Un

do

ped

an

d L

i-d

op

ed

(0.0

2-1

% )

sin

gle

-cry

stal

fi

lms

s.

poly

erys

tall

ine,

0

.00

5-0

.1%

L

i, d

ensi

ty

93

%,

sing

le c

ryst

al

(<0

.00

1%

L

i)

c.

Pt-

Au

p

aste

m

. d.

c.,

B=

30

kG

s.

Un

do

ped

sin

gle

cry

stal

c.

P

t-A

u

pas

te

m.

a.e.

cu

rren

t (5

10 t

tz),

a.e

. fi

eld

(1.5

-2.5

Hz)

of

7 kG

p

eak

to

pea

k

300-

1100

200-

750

200-

500

170-

600

200-

1500

820-

1280

+ 0.

3 (3

0O°K

), d

ecre

ase

wit

h

incr

easi

ng

T,

sign

rev

ersa

l at

_

600°

K

--0"

004

(T>

60

0°x

) at

all

te

mp

erat

ure

s in

dep

end

ent

of d

ope

+0

.1,

+0

.15

(3

00°K

), s

ign

reve

rsal

at

500°

K

--0

-05

(T

> 5

00°~

). R

m

abo

ve

500°

K,

sho

ws

' tim

e d

epen

den

ce '

30

0°K

: +

0"1

(0

"5%

Li)

, +

0"4

(0

"2%

Li)

max

imu

m

at

~2

00

°K

(0"2

an

d 0

"5%

L

i) a

bo

ve

200°

K

ocex

p (0

.09

ev/I

cT)

(0.0

3%

Li)

300°

K :

+

0"1

5 (

1% L

i),

+0

"25

(0

'05

%

Li)

A

bo

ve

200°

K (

0'0

5%

L

i)

ocex

p (0

.073

/kT

) M

axim

um

at

300°

K (

1%

Li)

In

dep

end

ence

of

Li

dope

l~es

ults

sim

ilar

to

tho

se p

rese

nte

d

by

Bo

sman

et

al.

(196

5)

At

1280

°K:

--2

×1

0 -

2 (1

0 -s

at.

O~)

<

--2"

10 -4

(1

at.

Oe)

Sig

n re

ver

sal

of ~

H a

t --

600°

K

con

nec

ted

wit

h m

agn

etic

str

uct

ure

Sm

all-

po

laro

n c

on

du

ctio

n i

n a

nar

row

ban

d

No

evi

denc

e fo

r th

erm

ally

act

ivat

ed

ho

pp

ing

co

nd

uct

ion

. S

mal

l-p

ola

ron

co

nd

uct

ion

in

nar

row

ban

d.

An

om

alo

us/

~H

beh

avio

ur

may

be

a p

rop

erty

of

nar

row

ban

ds

(an

thra

cen

e)

Co

nd

uct

ion

in

a re

lati

vel

y b

road

ban

d

(m*

-- l

m0)

. S

catt

erin

g

pre

do

min

antl

y b

y o

ptic

al p

ho

no

ns.

:N

o in

flue

nce

of c

har

ged

im

pu

rity

sc

atte

rin

g.

An

om

alo

us

Hal

l ef

fect

:

R 0

(1-

4taX

), w

ith

th

e co

nst

ant

a su

ch t

hat

4~

aX

= 1

at

TN

An

om

alo

us

beh

avio

ur

of t

~H q

ual

ita-

ti

vel

y i

n ag

reem

ent

wit

h

Mar

anza

na'

s m

odel

. R

esu

lts

belo

w

400°

~ m

ilit

ate

agai

nst

val

idit

y o

f h

op

pin

g m

odel

. E

ven

qu

esti

on

able

w

het

her

sm

all-

po

laro

n m

od

el a

ppli

es

~¢[i

xed

con

du

ctio

n w

ith

~H

= 0

for

ho

les

in a

gre

emen

t ei

ther

wit

h

Mar

anza

na'

s m

od

el o

r w

ith

th

e sm

all-

po

laro

n c

har

acte

r of

hol

es

C~

C~ g~

¢¢°

¢Jt

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The sign reversal of tLH at 600°K, see fig. 16, as reported by Bosman, van Daal and Knuvers (1965), was not reflected in the behaviour of resistivity and Seebeek coefficient up to the highest temperature considered (1300°K). For temperatures above as well as below 600°K, /~tt was found to be independent of magnetic and electrical field strength and of dope. An explanation in terms of normal mixed conduction was rejected because of the large value for the energy gap of NiO ( _~ 3.5 ev). The anomalous behaviour of t ~ was suggested to be connected with the magnetic structure.

Various explanations for this sign reversal have been proposed by different authors. Fisher and Wagner (1966) suggested that in (p-type) NiO, owing to the presence of minority centres (donors), the ratio of hole to electron concentration may decrease with increasing temperature such that, if the electron mobility is much higher than the hole mobility, the Hall effect becomes dominated by electrons. Van Daal and Bosman (1966), however, have emphasized that the normal possibility at 600°K that mixed conduction will have a noticeable effect on the Hall coefficient is excluded by the large energy gap of ~iO, while, moreover, the ratio between electron and hole mobility cannot be very large owing to spin- disorder scattering of charge carriers. The presence of minority centres can be considered in this respect to be of minor importance.

The qualitative similarity between the behaviour of t ~ for holes in NiO and Cu20 in the temperature region of 300-1000°K--~ for Cu20 is a factor of 100 larger than for NiO--has tempted McKinzie and O'Keeffe (1967) to suggest tha t the reversal of sign is not to be associated with the antiferromagnetic-paramagnetie transition in NiO but has to be ascribed to the onset of intrinsic conduction. I t is obvious, however, tha t this similarity can only be accidental, because at 1000°c the energy gap for Cu20 is only 1.4 ev compared to about 3.5 ev for NiO.

Austin et al. (1967) suggest that the sign reversal of ~H as found in the vicinity of the N~el point in the eases of p-type MnTe (Wasscher, Seuter and Haas 1964), •iO and ~-F%O 3 (van Daal and Bosman 1967) is a property of narrow bands. Indeed, Friedman (1964 a) has shown that in very narrow bands, much less than ]cT wide, the sign of the Hall effect may be opposite to that normally expected. An example of this anomaly is p-type anthracene (Delacote and Sehott 1966). However, p-type MnTe is a broad-band semiconductor. Moreover, the coincidence of sign reversal and magnetic transition should be considered either to be fortuitous or be incorporated in Friedman's model. Finally, Friedman's anomaly brings with it a large ratio for It~H/t~D], being essentially determined by lcT/AW, where AW is the bandwidth . In the case of anthracene, experiments have shown that at room temperature tLH/t~D_~- 25. However, for ~-F%O 3 and NiO above TN, t~i~/tL D is small ( -0 .25) and very small ( - 0.01), respectively.

The present authors (van Daal and Bosman 1967) have mentioned with respect to the Hall-effect anomaly the possible influence of n-type surface

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Conduct ion in some Transi t ion-metal Oxides 59

layers or inclusions in the case of p-type bulk. I t is known, bu t not yet understood in all cases, that in this situation ~ can have the anomalous (negative) sign whereas no anomaly is found in the behaviour of a or/and @. Examples are UOe (Nagels, private communication), Si (Putley and Mitchell 1958) and InAs (Ruppreeht 1958, Dixon 1959). The anomaly or the onset to the anomaly in NiO, however, is present quite reproducibly in ceramic as well as in single crystals made by entirely different preparation techniques. Where both groups of material are enormously different as to effective surface, the role of inversion (grain-boundary) surface layers seems not to be of importance. Moreover the anomaly remains present at high temperatures, where the material reaches equilibrium with the oxygen pressure of the gas atmosphere within a short time. Although it does not seem likely that n-type inclusions are at the root of the anomalous /~ behaviour, a closer s tudy of the role of inversion layers and n-type inclusions would, however, be desirable (see also at the end of this section).

Ksendzov, Avdeenko and Makarov (1967) present a phenomenological description of an anomalous Hall effect in anti ferromagnets on the analogy of the already known ferromagnetic model. Ksendzov states tha t he has not observed a sign reversal--probably because data were not taken at the relevant temperature region--and actually his quantitative elaboration of the model leads only to a zero point of the Hall coefficient at the Ndel temperature. The Hall resistivity @H is defined as :

@~I=E ±/JII . . . . . . (91)

where E ± is the transverse electric field induced in a direction perpendicular to that of the current density J ~L and the applied magnetic induction B, the latter being applied perpendicular to J ll" Because of the low value for the magnetic susceptibility )/ of an antiferromagnet, B = H and the magnetization M = xH, where H is the external applied magnetic field. Then

@H = (Rnorm + RAF4~X) H, (92)

where R.orm is the normal Hall coefficient, r/(pe) with the Hall factor r of the order of unity, and RAF the anomalous antiferromagnetie Hall coefficient. On the analogy of ferromagnets, with the spontaneous magnetization zero, it is then postulated that

RAF = - - aRnorm , (93)

Thus the measured Hall where a is a constant with a positive value. coefficient (R E = @H/H) :

R K ---- (1 + f~)Rnorm, (94)

where fi = - 4~rax . . . . . . (95)

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If now, as done by Ksendzov, the value of a is chosen such tha t 4rraX = 1 at the N6el point, where X is a maximum, R• is zero at this point, but has a positive finite value at all other temperatures. In an alternative choice of the value of a, viz. such that 47tax = l at a temperature T < TN, R H indeed has the reversed sign above that temperature and remains so until again, at higher temperatures, the same condition is fulfilled. An implication of Ksendzov's model is that RH/Rnorm ~_ 0.4 at T < T N. The same reduction holds for the ratio ~tH/~D , if the Hall factor r = 1.

Maranzana (1967) has presented a theory for the anomalous Hall effect in ferromagnetic and antiferromagnetic metals and semiconductors. In his model electrical conduction is achieved by quasi-free s electrons, characterized by plane-wave eigenfunctions. The magnetic properties of his model substance are determined by d electrons localized at the cations. The s-d exchange interaction between free-electron spin and cation spins (which contributes also to an indirect coupling between the cation spins) is responsible, in the presence of cation-spin disorder, for the magnetic contribution to resistivity. I t is assumed tha t s-d exchange interaction dominates resistivity. The Hamiltonian for s-d interaction between a conduction electron with spin $ at r and a particular cation with spin $ at R is :

~ss : - - F ~ ( r - R ) s . S, (96)

where F is the exchange integral, expressed in energy times volume, and the 8-function localizes the interaction at the position of the cation. Another interaction mechanism is now introduced which, in cooperation with s-d exchange coupling, leads to an anomalous Hall effect. This mechanism is the energetic coupling between the orbital angular momentum 1 of the s electron around a particular cation and the magnetic moment of that cation. The orbital angular momentum 1 = ( r - R) ×/ik, where k is the wave vector of the electron. The Hamiltonian for this orbit (1)-spin ($) interaction is :

~f ls = e g~B S . 1 m* It- a[3' (97)

where e and m* are the charge (with a negative value) and effective mass of the electron and g is the Land6 factor, ~B the Bohr magneton (with a negative value). The t tamiltonian ~flS is antisymmetric with respect to mirroring along a plane containing S and k. Therefore the corresponding matrix element for scattering of the electron by the cation spin is antisymmetric with respect to interchange of k and k', the initial and final s-electron-wave vector. This property of ~ ] s introduces the possibility of skew scattering, i.e. of the occurrence of a transverse Hall current. The complete Hamiltonians ~ s s and 5~f~s are obtained by summation over the N cation spins per unit volume. The total Hamiltonian ( ~ s s + 5~f~s) introduces transitions between states characterized by s-electron-wave vector, s-electron spin and the cation-spin lattice,

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Conduction in some Transi t ion-metal Oxides 61

For the calculation of transition probabilities between these states, perturbation theory is adopted on the assumption tha t the spin-Spin t and the Spin-orbit coupling are weak compared with the kinetic energy of the charge carriers. Two types of transition probability are considered namely one containing twice the spin-Spin coupling constant F and one containing I" twice and the orbit-Spin coupling constant once.

The first transition probability will in general contain interference terms in which two cation spins (1 and 2)--not necessarily nearest neighbours-- are involved. In order to estimate this ' two-sites ' transition probability W(2), it may be remarked that in view of the antisymmetric character of ~ s , the two mixed terms in ~fss and ~f ls cancel, while the term J/f is z can be neglected with respect to the term J/Css 2. I t is then plausible tha t W(2)acF2J~(2), i.e. proportional to the square of the exchange integral and to the average of the product of the deviations for the cation sites 1 and 2 of their magnetic moment from the average value : J£/(2) = <(m I - <m>)(m 2- <m>)>. With the aid of the expression for W(2) a formula for spin-disorder resistivity can be derived.

The second transition probability contains interference terms in which three cation spins 1, 2 and 3 are involved. For the calculation of" this ' three-s i tes ' transition probability W(3), the terms ~ s s a, 5/fls a and ~ s s ~ l s 2 are neglected, while one term containing ~¢t°~s2J£Zls is retained. This term describes a process where two of the three sites contribute to scattering via exchange coupling and only one of the three via orbit-Spin coupling. The overall effect is a skew scattering probability W(3) proportional to F 2 and to J£/(3), the average of the product of the deviations from the average moment of the three sites : J//(3) = ((m 1 - <m>)(m2- (m>)(m a - (m>)>. Actually, Maranzana calcu- lates W(2) and W(3) as resulting from scattering due to one and the same cation spin, thus d//(2) and d//(3) are expressed in terms of ( ( m - (m>)2> and ( ( m - (m>)a>, respectively.

I t follows from the above tha t the combined presence of spin-Spin and orbit-Spin coupling is a necessary prerequisite for the existence of an anomalous Hall effect. For, if e.g. spin-Spin coupling is replaced by electron-phonon coupling, W(3) is proportional to Jg/(1) = ((m - (m})> = 0 and thus W(3) = 0.

Transport calculations lead to an expression for the anomalous Hall resistivity, which in the case of an n-type semiconductor with n electrons per unit volume is approximately :

1 ~ ~- - - Nm*ah-nr2gl~BkTJ/[(3 ). (98)

%e

I t is seen that QH is proportional to the normal Hall coefficient 1~(he), which has a negative value for the case of electrons. For the case of an

The term ' Spin ' with a capital S denotes the cation spin.

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62 A . J . Bosman a n d H. J . van Daal o n

a n t i f e r r o m a g n e t , with H parallel to the axis of easy magnet iza t ion Maranzana has eva lua ted J~(3) in the molecular-field approx imat ion and arrives a t the resul t t ha t J / / (3 )= 0 at low tempera tures , with increasing t empera tu re becomes posit ive in the vic ini ty of the Ndel point and then above T N remains roughly constant . At T N :

d / (3 ) = - 2 S a g I ~ B H / ( / c T N ) . (99)

I t should be no ted t ha t ~B has a negat ive value. The case of H perpendicular to the axis of easy magnet iza t ion was not t rea ted . I t is probable, however, t h a t similar results will be obta ined in t h a t case for d4(3) a t T > T N because the susceptibi l i ty there does not behave differently f r o m the former case. At t empera tu res T > TN, the anomalous Hal l res is t iv i ty has the o p p o s i t e s i g n f rom tha t of the normal Hall resist ivity, i r respect ive of whether conduct ion is b y electrons or holes. Thus the anomalous Hal l effect can in principle compensate or even overcompensa te the normal effect.

The expression for the Hal l coefficient a t the Ndel point , due to the combined normal and anomalous Hal l effect, reads :

R H = (1 + fi)Rnorm , where

fi ~- - 2-Nm*ah-6F2g21~B~S4 (at T = TN). (100)

A discussion of this resul t in connect ion with NiO will now be presented. Le t us assume tha t the Maranzana model can be employed in the case of p- type NiO, which should then be regarded as a band semiconductor where the holes are character ized b y plane-wave eigenfunctions and where dr i f t mobi l i ty is de termined by Spin-disorder scat ter ing only. Wh e n disregarding the na tu re of the eigenfunctions of holes, the above condit ions are not unreasonable : p - type NiO can be conceived of as a large-polaron band semiconductor with a densi ty-of-states effective mass of abou t 6 m o and a band width of about 0.6 ev (see § 3.4), while drif t mobi l i ty can be de termined by Spin-disorder scattering. In t h a t case the exchange coupling constant F equals abou t 18 ev A a (see the end of this section). A rough est imate of fi a t T N can now be made for NiO. Subs t i tu t ion of N = 5 x 1083 cm -a, g = 2, S = 1, as applies to Ni 2+ cations, leads to :

fi "~ - l O - 7 ( m ~ * / m o ) a F 2. (101)

Wi th m p * = 6 m 0 and F = l S e v A a this leads to f i_0 .007 at T•. A sign reversal of R E at T u due to the ant i fer romagnet ic cont r ibut ion can only be expected if fi ~ - 1. I t follows then tha t , wi thin the f ramework of the present model, the anomalous cont r ibut ion to R~ is two orders of magnitude too weak.

The too low resul t for the calculated value of fi might be due to the description of charge carriers in terms of plane-wave eigenfunctions. In

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a more realistic model, where the charge carriers are described by a tight- binding combination of proper atomic orbitals, Spin-orbit coupling will be significantly stronger, leading to appreciably larger values for ]Pl (see eqn. (97)). According to Maranzana, a refinement of his model in the indicated direction seems promising. I t should, furthermore, be remarked that, in the above estimate, F has a value of about 0.6 ev per atomic volume, which value is comparable to the band width. Therefore spin-Spin interaction cannot be regarded as a small perturbation. More- over, in the present model the total Hall coefficient is taken to be the sum of the normal and the anomalous contribution. This implies that fl should haye a value close to - 1 at temperatures T > T• if R~I, at these temperatures changes sign and remains small, as has been found for the cases of 5~i0, a-FelOn (see § 4.3) and MnTe (Wasscher et al. 1964). How- ever, in the Maranzana model a value for fl of - 1 should be seen as accidental This seems to be an unsatisfactory feature of this model.

In conclusion, the outcome of Maranzana's theory contains in principle the features necessary for a description of the anomalous behaviour of/x~. Quantitatively, however, his model seems to be inadequate. I t has already been suggested by Maranzana that in a more adequate model, conduction perhaps has to be achieved by the magnetic analogue of polarons, the so-called ' magnarons ' : charge carriers accompanied by magnons (spin waves). A direct calculation of the total Hall effect within this model, orbit-Spin coupling included, seems, however, to be a long way off from being realized.

Adler (1968) rejects Maranzana's theory apparently mainly on the ground of the experimental situation for paramagnetic semiconductors. For such materials, according to Adler, an anomaly of the type where Hall and Seebeck coefficients have opposite sign at all temperatures has thus far never been observed. This argumentation can be criticized on fundamental grounds : a pure paramagnet cannot exist if s-S coupling and thus indirect S-S coupling is present (Maranzana, private communication). However, the free-electron concentration in these semiconductors is in general too low to induce an appreciable coupling between the cation spins. In practice, therefore, Maranzana's anomalous Hall effect could be noticeable in the case of a paramagnet. In the limit of negligible S-S coupling, the outcome for the ferromagnet as well as for the anti-ferromagnet is the same, viz. eqn. (100), and applies to the case of the paramagnet. I t should be emphasized, however, that an anomaly need not be so drastic that R~ has the reversed sign. Appreciable effects can only be expected if f i g - 0.5. Finally, it should be remarked that, as far as the present authors know, the relevant data concerning paramagnets are too sparse to allow any pronouncement upon the occurrence of anomalies.

An important contribution to the experimental specification of the anomalous ~ behaviour in NiO is due to Tallan and Tannhauser (1968). Their result is of two kinds :

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64 A . J . Bosman and H. J . van Daal on

(1) At t empera tu res above 820°K, /~tt in samples measured in a gas a tmosphere of 1 at. O 8 behaves essentially in the same way as found earlier by the present authors (see fig. 16), wi th the impor t an t discrepancy t h a t a t 1280°K, the highest t empera tu re considered, ~H < -- 2 × 10 -a cmS/v sec. This result suggests t h a t t~H (again) goes to zero at about 1280°K. The exper imenta l point ob ta ined at 1400°K b y the present authors (see fig. 16) might be erroneous due to residual the rmal effects. These effects were r igorously excluded b y the double a.c. me thod employed b y Tallan.

(2) At 1280°K, wi th decreasing oxygen par t ia l pressure, the absolute value of tLH increases in propor t ion to the square of the resist ivi ty.

The second resul t is explained by Tallan in terms of mixed conduct ion on the assumption that/~i~ for holes (t%H) is essentially zero. According to Tal lan a justif ication for this assumption might be found ei ther in the occurrence of a Maranzana anomaly with fi exac t ly equal to - 1 for T > T N or in the small-polaron character of the holes. The la t ter just if ication is wi thout theoret ical foundat ion. I t will be shown t h a t the assumpt ion tzpH = 0 is consistent with the second bu t inconsistent wi th the first resul t obta ined by Tallan.

The Hal l mobi l i ty for mixed conduction is given b y :

~H = ~ttpH -- e2Ki~2~p~Cn 8, (102

where t%H denotes the Hal l mobi l i ty of holes and t% and t~n are the dr i f t mobilities of holes and electrons. The equat ion for t ~ has been der ived assuming t h a t conduct ion is still extrinsic, i.e. pt%>~nt~n, and t h a t ~nH----~n" The former assumption has been verified to hold t rue for the case even of l ightly doped NiO up to ve ry high tempera tures . K i ( = p n ) is the intrinsic equil ibrium constant , de termined b y the effective densities of states No for the conduct ion band and N v for the valence band and b y the energy gap E~ according to : K i = N c N ~ exp ( - Eg/kT). A m a x i m u m value is obta ined for K i by taking for No and N v the ex t reme values of 108a cm -3. The value of E~ can be es t imated f rom the value of abou t 3.7 ev for Eg at room t empera tu re (Newman and Chrenko 1959, Ksendzov and Drabkin 1965) and assuming t ha t dE~/dT has the commonly found value of abou t - 4 × 10 -4 ev/degree. We arr ive then at m a x i m u m values for K i of _~ 1028 cm -G ( E ¢ - ~ 3.50 ev) and ___ 10 a8 cm -~ ( E g ~ 3.30 ev) at 800°K and 1300°K, respectively. I f /%H = 0 and if t% and t~n are independen t of dope it follows indeed t h a t a t a certain t empera tu re ~H = - C~ 8, where C is a constant equal to e~'Kit%t~n 8. An est imate m a y then be made of the value for t~H at 1280°K in the case of 1 at. O 8. Fo r this case :

= 3 ~ cm, K i _ 103~ and ~p ___ 0.5 cm2/v sec. Wi th a reasonable es t imate for tZn (1 cmS/v sec) i t is found t h a t a t 1280°K in 1 at. of 08, ~H----- --10-5 cm2/V sec. This value for t~H is in accordance wi th the ex t rapo la ted value obta ined b y Tallan. Thus it seems probable t h a t the assumption t%it = 0 holds t rue a t 1280°K. However , the exper imenta l behaviour of tZH wi th

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Fig. 16 T( ° K ) ~ - - 1500 1000 600 400 300 ;tOO

. . . . , I I 7 " I I I[ /

Ntl-x Lix 0 // ~ / 6 lOOx rnclrkin~ / f

Z ceramic O. 005 • 0

5 " " o.o, ~ - + / ~ I I I ,, o. o9 o . - * /

xv .,.o,. o.~.~ <ooo, t'. " / /

moo p (D.,) I / / 5 ," / '

- I / / ' ' / ° ` 2 ( ,0. lR.;(.,fic ) I J/

( -o , , ) ¢ , - 3

~ " ,o~ #.(c,.?/V.oo)

1 i

0

-I

-2

-:3

&

A . P .

. . . . . . . . . . . . . , , . . . . . , , . ,

1 2 3 4 5

Resistivity @, Hall coefficient Rl t , and Hall mobility FH for p-type Li-doped ceramic and non-deliberately-doped single-crystalline NiO as a function of reciprocal temperature. A sign reversal of R H and FH takes place at about 600%:. Values of e for ceramic samples have been indicated by solid lines solely. With regard to the resistivity of single crystals at high temperatures, the black circles and squares represent different values for a thin disk (0.4 mm thickness) measured when cooling the sample slowly and rapidly, respectively; the broken line (IV) results from a rapid run of measurements performed on a thick bar (10 x 4 x 4 mma). The Ndel temperature T N has been marked (van Daal and Bosman 1967).

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decreasing temperature at 1 at. 02 as found by the present authors and by Tallan too, is in fiat contradiction with the model of Taltan. When going from 1300°K down to 800°K, K i decreases by a factor of ~_ 101°, whereas ~2 increases by a factor of ~ 104 (Bransky and Tallan 1968), /% and tLn being quite probably nearly constant. Then, according to Tallan's model, It~HI would decrease by a factor of _ l06, whereas experiment shows that in reality It~nl increases by a factor of _ l0 s, thus leaving an ultimate discrepancy of a factor of - l 0 s. Therefore, although the assumption t%H=0 may hold true at a temperature of about 1300°K--and also at 600°K where the first zero point of R H has been found--t% H definitely cannot be zero below 1300°K--and above 600°K--but should have there an anomalous negative value.

In a speculative explanation, anticipating the results of a type of Maranzana model adapted to the situation for NiO, it might be concluded that f l = - 1.000 at 600°K and 1300°K and that for temperatures in between it has some value like - 1.01.

I t might be asked whether n-type inclusions, if present in the material, can determine R~ behaviour as found. Suppose that the sign reversal of/~tt at 600°K is indeed due to the presence of small n-type inclusions, which dominate R~ because of a negligibly small value of t%H. Then, when increasing the temperature of the sample in 1 at. 03, some temperature will be reached--say 1300°K--where these inhomogeneities, due to a sufficiently large diffusion coefficient of oxygen through the sample, disappear, so that the Hall mobility goes to zero. However, a drastic decrease of the oxygen pressure at 1300°K, as realized by Tallan, might then lead again to an increasing influence of n-type inclusions. The behaviour of/~n in that case will be describable in terms of mixed conduction. I t seems to the present authors that the experimental situation for NiO is not yet mature enough to permit a definite exclusion of this spurious effect.

We shall now discuss what information can be drawn from the Hall- effect data concerning the nature of the conduction mechanism. Almost all authors agree that in NiO conduction does not seem to be brought about by non-adiabatic hopping of small polarons (see table 1). This conclusion is based on the decrease of t~H with temperature (t~nOcexp ( + 0.08 ev//~T)) in the range 200 to 400°K, and on the magnitude of t~H//~D which is estimated to be smaller than or equal to unity, see fig. 17. The assumption that the temperature dependence of t~E in this temperature region is not influenced appreciably by the anomaly which emerges at higher temperatures, seems reasonable. In Maranzana's model, for

~T N. The example tLH behaves almost normally at temperatures < 1 phenomenological description of the anomaly as given by Ksendzov leaves the temperature dependence of/~K for T < ½T N almost unchanged but leads to a constant reduction of about 2. I t may be remarked that a decrease of t~H with increasing temperature and a value for ~H/ittD smaller than unity could be compatible with small-polaron hopping if the

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adiabatic case were adequate (see § 2.2). I f i t is supposed t h a t NiO is an adiabat ic small-polaron conductor , the ac t iva t ion energy ( U - J ) at high t empera tu res would be at the most 0 . 1 0 e v (see § 3.3). The small- polaron condit ion J ~ U then implies a coupling constant 7 with a value

Fig. 17

T (OK) ~,--- 1000 500 300 200

! ~ (cm2/Qsec) I

t l l l 0 I

I I I I q I I I I I I I J I I I I I I r 2 3 4. 5 6

~-~ (OK-') The drift mobility/xD, derived from a and ~ data, and Hall mobility/x H for

Li-doped (_~0-1 at. %) p-type NiO. The Hall mobility changes of sign at 600°K. The curve marked K represents /x H behaviour as obtained by Ksendzov et al. (1967) on Li-doped single crystalline NiO. Curve (1) for/x D corresponds to N v = constant and curve (2) to N v oc T a/2. Curves A and B represent t~D behaviour in the presence of different scattering mechanisms if the large-polaron band model applies. Curve (A) represents/x D behaviour in the ease of polar optical-mode scattering and curve (B) shows/x D behaviour in the case of spin-disorder scattering if the magnetic exchange coupling constant equals 0.6 ev.

of a t the most 3. I t m a y then be concluded t h a t there is no room for the adiabat ic small-polaron case (see § 2.2). The same ma x i mu m value of abou t 4 for the coupling constant ~ would follow from ~ behaviour a t high temperatures , if the non-adiabatic ease of small-polaron hopping were to apply (U <~ 0.15 ev). The drif t mobil i ty, der ived on the assump- t ion t h a t N v is constant , shows at 500°K a change of t empera tu re dependence (see fig. 17). The supposit ion t h a t at 500°g small-polaron band

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68 A . J . Bosman and I t . J . van Daal on

conduction takes over from hopping does not seem relevant. In a small- polaron conductor one would expect the hopping mechanism to be dominant down to at least ~0_ 200°K (§ 2.2). Furthermore, the presence of locally varying electric fields due to charged eentres would not seem favourable for the formation of the essentially narrow small-polaron band (see § 6). The temperature dependence of the Hall mobility in the region 200 to 400°K seems also to be irreconcilable with the non-adiabatic small- polaron model.

A definite conclusion concerning the ratio t~ii//XD still seems to be impossible. Makarov, Ksendzov and Kruglov (1967), from direct measurements of/x n at 300°K arrive at a value of 0.3 em2/v see for holes and of 0.14 cm2/v see for electrons. I t is felt by the present authors, however, that these values can only be considered to be a lower limit. In Makarov's experiments the measured transit time for the cross-over through the sample of charge carriers generated by a light pulse at one end of the sample is assumed to be inversely proportional to the product of driving electric field and drift mobility. The driving electric field has been taken to be determined by the d.c. voltage applied across the sample. I f the contact resistances are not negligibly small compared to the real material resistance, however, the driving electric field has been over-estimated by Makarov and consequently the drift mobility underestimated. I t would be desirable to have data as a function of the thickness of the sample in order to eliminate possible influences of contact resistance. Furthermore, trapping of the charge carriers cannot be ruled out as an influence on the measured transit time and, if effective, will lead to a reduction of this time and thus also to a too low estimate for the intrinsic value of ~ . Moreover, trapping can have a quite different effect for electrons and holes so that the ratio of the measured transit times need not be representative of the intrinsic ratio/~i,//z~. The important feature of Makarov's outcome seems therefore to be tha t at 300%: the drift mobility of holes and electrons cannot have a lower absolute value than 0.3 cm2/v sec and 0-14 cm2/v see, respectively. In so far Makarov's result is consistent with the estimate of /~n arrived at by Bosman and Creveeoeur (1966) from resistivity and Seebeck-coeffieient data for holes at 300°K: 0.5 ~</x D ~< 5 cm2/v sec (see also § 3). For the Hall mobility of holes, the value of 0.3 cm2/v sec arrived at by various authors (see table 1) can quite probably be considered to be a lower limit for the real intrinsic value too. For the samples considered the upper limit does not seem to be so far above this value as in the case of the drift mobility. Therefore it can be said that/zn//~ D ~< 1 at 300°K. This conclusion too leaves little room for the applicability of the non-adiabatic small-polaron model for which ~ / ~ D ~> 1.

Among the various authors no agreement prevails with regard to positive findings concerning the nature of the conduction mechanism. The conclusions vary from the one extreme, viz. small-polaron conduction in a narrow band (Austin), to the other, viz. normal conduction in a

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relatively broad band (Ksendzov). In our opinion the first mentioned extreme can be excluded because smMl-polaron band conduction can only be expected to be dominant in the very low temperature region (T < 200°K, see § 2), where almost no experimental data for R~, p and exist. The other extreme has been arrived at by Ksendzov et al. (1967) from an analysis of R~, a and p behaviour for one sample of Li-doped single crystalline material, leading to a density-of-states effective mass of the order of unity. However, disregarding the confusion in Ksendzov's paper about the value of/z D at high temperatures (-~ 1200°K), which is 0.4 cm2/v see and not 0.03 cm2/v see as stated there, the uncertainty in the outcome for m*/m o admits values appreciably larger than unity.

Arguments have been presented in § 3.4 of this paper in favour of a band model for p-type conduction in NiO. As a preliminary estimate for the density-of-states effective mass mp*/m o we arrived at a value of about 6, which in the simple tight-binding approximation corresponds to a band width of about 0.6 ev. On the basis of such a band model the behaviour of the drift mobility was derived from ~ and ~ data, assuming that the density of states varies as T 3/2 (see fig. 17). Drift mobility was found to increase continuously from 0.4 em2/v sec at 1200°K to about 4 cm2/v sec at 300°K. An interpretation of this behaviour of/i'D is presented below.

Two scattering mechanisms seem most appropriate for determining mobility behaviour, namely polar optical-mode scattering and spin- disorder scattering. The coupling constant ~ for the former scattering mechanism, is given by relation (11). For NiO, according to Plendl, Mansur, Mitra and Chang (]969), %pt=5"7, %t~t=ll '75 and 0=840%: (_~ 0.07 ev). Substitution of these values leads to :

~ 1.2(m*/mo)l/2. (103)

Values for the parameters e and m* can be deduced from relations (103) and (13) if one takes for rap* the value of 6m 0, derived for the effective density of states of the large-polaron band. On the basis of such a value for m,*, we arrive at m*=4"3m 0 and ~__2-5. The value for ~ implies that the energetic coupling between holes and longitudinal optical phonons is fairly strong, but still of intermediate strength. The formula for large-polaron mobility, at temperatures T ~ 0, derived from the intermediate coupling theory is presented in § 2.1.1 (eqn. (14)). The theoretical temperature dependence of tz(oc exp ( + 0.07ev/kT)) is in reasonable agreement with that found experimentally for the drift mobility and moreover with the measured behaviour of t~t~ between 200 and 400°K (see fig. 17, see also Ksendzov 1967). The theoretieM magnitude of/z at 3000K, for a ' b a r e ' mass value m*/m o of 4-3 and a corresponding polaron-mass value mp/m o of about 6, is found from eqn. (14) to be about 5 em2/v see. This value agrees quite reasonably with the experimentM value for /~D at 300°K of about 4 cm2/v see. I t seems therefore that, if the use of the band model is admitted, drift-mobility behaviour can be explained in terms of optical-mode lattice scattering. I t remains, howevert to be

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explained within the above model why /z~ is an order of magnitude smaller than/z D. Such a difference between/z H and/z D might be due to an anomalous antiferromagnetic contribution to R E and to a specific (valence) band structure leading to a Hall factor smaller than unity.

The second possible scattering mechanism, spin-disorder scattering, leads according to Haas (1968) for the case of an antiferromagnetic broad- band semiconductor to a mobility :

I~ = (2~r)~/~(NglxB)21el/i4m*-5/2(P')-2{2(T)}-~(kT)-8/2, (104)

where F' is the exchange-coupling constant expressed in ev and 2(T) is the average susceptibility expressed in e.m.u./g. Data for 2(T) of NiO have been presented by Johnston, Heikes and Sestrich (1958). The theoretical temperature dependence of ~ in this case is determined by T-a/2{2(T)}-1, which in the whole temperature region considered (the N~el temperature included) does not deviate much from the experimental behaviour of ~D (see fig. 17). I t may be remarked tha t the dependence of/~ on 2(T), according to Haas' formula, is only slightly reflected in its behaviour around the N6el temperature. This is due to the smoothing effect of the T a/2 factor. The results obtained in the case of NiO for t~D behaviour with constant density-of-states (see curve 1 in fig. 17), which in fact represents t~D behaviour for the band case multiplied by T a/2, suggest a correlation between mobility and susceptibility behaviour. I f it is assumed tha t spin-disorder scattering solely determines mobility, F' should be taken to have a value of about 0-6 ev in order to obtain with mp*/m o = 6 agreement between theoretical and experimental value. Such a value for F' does not seem to be unreasonable. I t should be noted that the exchange-coupling constant F' would be about equal to the band width. I t is questionable, therefore, whether Haas' theory may be employed for such a relatively strong coupling.

The results obtained above suggest that p-type NiO can be conceived of as a ' broad ' band semiconductor, where the drift mobility of holes is determined by optical-mode lattice scattering and/or spin-disorder scattering.

4.2. CoO

Data concerning the Hall mobility of holes in CoO are summarized in table 2. The conclusions drawn by the various authors either from the t ~ behaviour solely or from the consideration o f ~ n and t~D behaviour together are also given in table 2. See also fig. 18.

The main facts with regard to/~H behaviour are (see fig. 19) :

(1) tLH has the positive sign in the antiferromagnetic region (T< TN~290°x) as well as in the paramagnetic region. For the case of low dopes, no marked effects have been detected in the vicinity of the Ndel point. In this respect CoO behaves quite

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8

7

6

5

4

3

0.4 0.2 0.1

2

1

0

-1

-2

Fig. 18

300[" r(°K/~ 1500 1000 500 J ' " ' I i , , *

J - / C01-x Lix 0

100x markings / I 0.02 +. /¢t /,~ II o.1 i_~ ~ o /J/ ~°/J~ 6

T / / j ,_T , / ~/:" / ,o~. % cc.Tc) ,og ~o ;ncm) j / ( " / ~

/ I~ 2

,, J / 0 . <o

If -I

.~:'~ 12 ) -2

-3 y* C()=)

T HH :¢,,YV,,,o.)

200

. 0 1 , , , , , , , , , , , , , . . , . , , , , i , . , , . . . . I 2 3 4 ~ ~-'(°fC) 5

Resistivity g, Hall coefficient R•, and Hall mobility / ~ as a function of reciprocal temperature for p- type Li-doped ceramic CoO. Measure- ments were made in technical Zq 2 (10 -4 arm of 02). Some data obtained at high teraperatures in 1 arm of 02 are also presented. The Ndel temperature T~r has been indicated (van Daal and Bosman 1967).

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72 A . J . Bosman a n d H. J. van Daal o n

differently f rom NiO. Changes in the t empera tu re dependences of ~, and RH, at TN, as found especially a t higher dopes, are p robab ly related to an ab rup t change of the densi ty of states a t T N and no t to a change of the mobi l i ty (Bosman and Crevecoeur 1969).

1

i ~ ~

I d 2 T. T t T C

Fig. 19 r(~)*--

500 300 200 l

CoO i ~ f

i ~ i T ~ T ~ r ~ I ~ ~ T r T 1 v f i ! *~ 3 5

~ (°K -~)

Drift mobility, derived from a and 9 data, and Hall mobility (/~u) for Li- doped ( _~0.1 at. %) p-type CoO. Curve (1) for /x D was obtained on the assumption : N v = constant, curve (2) is valid for N v oc T 3/2. Curves A and B represent riD behaviour determined by different scattering mechanisms if the large-polaron band model can be employed. Curve (A) represents the behaviour of ~D expected in the case of polar optical- mode scattering, curve (B) in the case of spin-disorder scattering if the magnetic coupling constant is 0"6 ev.

(2) ~H is r emarkab ly insensit ive to var ia t ion of t empera tu re in a wide range. Even the slight decrease with decreasing t empera tu re of ~H below T N is most p robab ly only an apparen t decrease. Determina- t ion of d.c. and a.c. values for ~, on samples similar to those considered in fig. 18 has shown th a t g ra in-boundary effects have a slight influence on the d.c. behaviour of e below 300°K (see also fig. 6). A correct ion for these effects leads to a constant value of - 0.l cm2/v sec down to the lowest t empera tu re considered. Above

500°K, /~H diminishes slightly with tempera ture . For NiO, in the same region, the sign reversal of ~H had been found (see § 4.1). In § 4.1, the presence of n - type inclusions has been ment ioned as a possible source of an effective lowering or even sign reversal o f /x H. In this case too, the presence of such inclusions, a l though it cannot be definitely excluded, seems improbable. This is because in the ve ry high t empera tu re region (1000-]500°K) and at different oxygen pressures, the slight reduct ion of the value of /x~ compared to t h a t at lower t empera tures (T < 500°K) remains present.

(3) In the low- tempera ture range, where a comparison can be made for ~ I in the cases of CoO and NiO, ~H has a lower value for CoO. Roughly the same applies, a t all t empera tures considered, to the corresponding values for the drif t mobi l i ty (see figs. 17 and 19).

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Tab

le 2

. H

all

Mob

ilit

y C

oO

Date

1966

1966

1967

1967

Au

tho

rs

Zh

uze

an

d

Sh

ely

kh

Au

stin

et

al.

Au

stin

et

al.

van

Daa

l an

d

Bo

sman

Ex

per

imen

tall

y

8.

: S

an]p

le,

(3.

: co

nta

cts,

m.

: m

eth

od

s.

un

do

ped

sin

gle

cry

stal

s m

. d.

c.,

B=

23

kG

s.

Li-

do

ped

sin

gle

cry

stal

(1%

) m

. d.

c.

s.

Li-

do

ped

sin

gle

cry

stal

s (0

.04

-0.5

%)

c.

Ag

pas

te

m.

d.c.

, B

=1

8k

G

s.

Li-

do

ped

cer

amic

mat

eria

l (0

.02

-0.1

%)

den

sity

93%

c.

P

t-A

u p

aste

m

. d.

c.,

B=

30

kG

Tem

per

a-

ture

ra

ng

e (°

K)

300-

900

250-

400

270-

500

210-

1500

Beh

avio

ur

of

~H

(em

2/v

see)

+0

.04

to

+

0.0

7 (

300°

K),

ris

es

wea

kly

wh

en t

emp

erat

ure

is

rais

ed,

pra

ctic

ally

in

dep

end

ent

of

tem

per

atu

re a

t h

igh

te

mp

erat

ure

s, ~

I~

~D (

at h

igh

te

mp

erat

ure

s)

+ 0

'02

(300

°K)

incr

ease

wit

h

tem

per

atu

re

+ 0

"06

(300

°K)

pas

ses

thro

ug

h

shal

low

max

imu

m

+ 0

-1 (

300°

K)

alm

ost

co

nst

ant

in

the

who

le t

emp

erat

ure

ran

ge

and

p

ract

ieM

ly i

nd

epen

den

t o

f d

op

e at

h

igh

tem

per

atu

res

~H//~

D--

~ 0'2

5

Con

clus

ions

fro

m H

all-

effe

ct d

ata

Sm

all-

po

laro

n c

on

du

ctio

n i

n th

e in

term

edia

te r

egio

n b

etw

een

ho

p-

pin

g a

nd

ban

d m

od

el.

Ho

pp

ing

no

t v

ery

pro

no

un

ced

bec

ause

of

eith

er

insu

ffic

ient

ly s

mal

l el

ectr

on

ic b

and

w

idth

or

insu

ffic

ient

ly s

tro

ng

el

ectr

on

-ph

on

on

co

up

lin

g

Incr

ease

of/

xH

wit

h t

emp

erat

ure

m

ay i

nd

icat

e a

ho

pp

ing

mec

han

ism

No

ev

iden

ce o

f th

erm

ally

act

ivat

ed

ho

pp

ing

co

nd

uct

ion

. In

terp

reta

tio

n

in t

erm

s o

f sm

all-

po

laro

n b

and

co

nd

uct

ion

su

gg

este

d

Dat

a do

no

t su

pp

ort

val

idit

y o

f h

op

pin

g m

od

el,

even

see

m t

o

mil

itat

e ag

ain

st s

up

po

siti

on

th

at

CoO

is

a sm

all-

po

laro

n s

emic

on

du

cto

r

(%

(%

(%

C~

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(4) In the high-temperature range (_~ 1000°K), where /£D values can be unambiguously determined (see § 3.3),/x~i has a value which is a factor of 3 to 4 smaller than that of/£D (see fig. 19). A similar conclusion is arrived at by Shelykh, Artemov and Shvaiko-Shvaikovskii (1966).

A discussion of/*H and/*D behaviour will now be presented (see fig. 19). Arguments had already been adduced in disfavour of the /z D behaviour represented by curve 1 in fig. 19 (see § 3.4). Such a behaviour of/x D results from ~ and ~ data if the density of states is taken to be temperature independent and would correspond to a type of small-polaron model. Austin and Mort (1969) find this behaviour of/z D (curve 1) and tha t of /ztt consistent with adiabatic small-polaron theory. This is because this theory, which applies to the hopping regime, admits of an exponential increase of /£D and a simultaneous decrease of /~t with temperature, while /£H//ID c a n be smaller than unity (see § 2.2). Such a situation is indeed found at temperatures above 500°K for the experimental behaviour of/z D (curve 1) and/*H" However, it seems to the present authors tha t adiabatic small-polaron transport cannot determine /z behaviour. The thermally activated behaviour of/z D (curve 1) above 500°K would correspond at the most to an energy (U-J)___0.2 ev, implying with J ~ U a coupling constant ~ ~ 6. For such a coupling constant and at the temperatures considered there is no room for the adiabatic ease (see § 2.2). Furthermore, the experimental value of/~n at 1200°K is a factor of 10 larger than would follow from the adiabatic small-polaron theory (eqn. (33)). An explanation of the divergent behaviour of/x D (curve 1) and /xi~ above 500°K cannot be given either on the basis of non-adiabatic small-polaron theory. I t would, furthermore, be expected that, within the small-polaron model, conduction remains dominated by the hopping mechanism at temperatures appreciably lower than 500°K.

Arguments have been given in § 3.4 in favour of a /d, D behaviour as represented by curve 2 in fig. 19. This curve has been deduced from and ~ data on the assumption tha t N v varies as T a/~. A behaviour of /£D as represented by curve 2 may be expected if p-type conduction is achieved on the basis of a band model. From a preliminary estimate, this band, just as for the case of NiO, can be characterized by a density- of-states effective mass mp*/m o___ 6 and a corresponding width of about 0.6 ev. A treatment of the possible scattering mechanisms determining drift-mobility behaviour, namely polar optical-mode and spin-disorder scattering, proceeds analogously to that presented in the preceding section.

The parameters determining optical-mode lattice scattering are (Plendl et al. 1969) : %pt = 5"3, ~st~t= 13 and 0----785°K. The coupling constant (see eqn. (11)) is :

o~ ~_ 1.5(m* /mo) 1/2. (105)

A value of 6 for the polaron effective mass then leads to e = 3 and m* _~ 4m 0. Intermediate coupling theory therefore seems appropriate. The theo- retically expected variation of/£D with temperature differs, but not too

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much, from that of curve 2 in fig. 19. The theoretical magnitude of it at 300°K, for a ' bare ' mass m*/m o-~ 4 and a polaron mass mp*/m o-~ 6 is somewhat larger (_~ 3 cm2/v sec) than the corresponding value of P'D in curve 2 (___0.7 cm2/v see). I t may be concluded tha t this scattering mechanism can determine feD behaviour.

For the evaluation of spin~lisorder scattering, use was made of the data for X of undoped polycrystalline CoO as presented by Johnston (1958). This scattering mechanism (see eqn. (104)) leads to a temperature behaviour of/~ which roughly resembles tha t of curve 2 in fig. 19. With a value of 6 for the polaron effective mass, the theoretical magnitude of t* equMs the experimental value if a not unreasonable value of 0.6 ev is taken for F'. I t seems therefore tha t spin-disorder scattering too can determine drift-mobility behaviour.

I t might be concluded tha t p-type conduction in CoO can be consistently described on the basis of a simple band model where drift mobility is determined by optical-mode lattice scattering and/or spin-disorder scattering. The two scattering mechanisms considered are characterized by a weak dependence of the relaxation time for scattering on electron energy. This implies tha t the heat-of-transfer constant A ~ 2. This result had been used in the estimate of N v at 1000°K (see § 3.4).

An explanation for the behaviour of tZH within the band model seems difficult to give at the moment. Maranzana's model (see § 4.1) does not seem to be applicable because no relevant effect in tzi~ is found in the vicinity of the N6el temperature. I t has been remarked (van Daal and Bosman 1967) that CoO, in comparison to the other compounds considered, is an exceptional case because Co s+ when it is in the low-spin state has no spin at all. I f the charge carriers in CoO corresponded to this ion state, the Maranzana effect would be absent. However, the concept of the ion-spin state applies to localized charge carriers and seems in- compatible with the band model, where carriers should be expected to have the normal free electron (hole) spin with s = ½. The small ratio of tzH//ZD is possibly due to a specific band structure.

4.3. ~-F%0 a

Hall-effect data for the case of ~-F%03 are sparse. In pure material, in the region between the Ndel temperature (960°K) and the Morin temperature (250°K), the Hall coefficient shows the features of anomalous ferromagnetic behaviour (Morin 1950). In that temperature region, pure material is weakly ferromagnetic (N@el 1949). The magnetic structure is predominantly antiferromagnetic, the spins being aligned in planes perpendicular to the hexagonal c axis (corundum structure), subsequent planes coupled antiferromagnetically. However, the spins are canted slightly out of these planes. Below 250°K, c~-~%0 a is an antiferromagnet, the spins having directions parallel to the c axis. The Morin transition between the weakly ferromagnetic and the antiferromagnetic state is

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suppressed to very low temperatures in the presence of sufficient dope. Acket and Volger (1966) observed an anisotropic R E behaviour in the weakly ferromagnetic region of Sn-doped ( _~ 0.6 at. °/o ) single crystals. I t seems to be very hard to obtain information on electrical transport properties from Hall-effect data in the weakly ferromagnetic region.

Hall-effect data in the paramagnetic region up to 1500°K were obtained by van DaM and Bosman (1967) for the case of Zr (0.01 to 1 at. ~o) and Nb (0.12 to 0.25 at. °/o ) doped ceramic material (see fig. 20). For these n-type samples the Hall coefficient proved to have the positive sign. A sign reversal of R H was found at or just below the N6el temperature. The situation is analogous to that for the case of NiO, where RI~ has the anomalous sign above 600°K in the paramagnetie region (see § 4.1). In the ease of ~-F%O a compared to that of NiO, a sign reversal of R H due to intrinsic effects seems to be more probable. In fig. 15 the data for e as presented in fig. 20 have been plotted versus those obtained for log ~ at different temperatures. The linear behaviour at 1000°K of ~ versus log ~ and its negative value imply that conduction at this temperature for all dopes considered is purely n-type extrinsic. At higher temperatures the influence of mixed conduction becomes noticeable for the lighter doped samples. Values for the energy gap Eg can be estimated from the (~, ~) graphs (Jonker 1969) if they are assumed to be symmetric with regard to the line ~=0. One arrives at 1.3 ev<~Eg~< 1-7 ev at 1200°K and 1.1 eg ~<Eg~< 1.6 ev at ]400°K. The upper limit corresponds to a value for the heat-of-transfer constant A =0, the lower to A = 2. I t follows then that the energy gap for e-F%08 has an appreciably lower value than tha t for NiO (_~ 3.5 ev). I t will be shown, however, tha t in the case of ~-F%0 a too, the sign reversal of R n cannot be due to the intrinsic effect.

In the extrinsic region where nt~n >>p/xp, the Hall mobility is given by :

~c]~ I = - - / X n i t + e 2 K i Q 2 / X n / X p / X p n • (lO6)

Equation (106) is the analogue of eqn. (102) given for the case of NiO. I t is valid at 1000°K for all dopes considered. I t should be noticed that the second term on the right-hand side of eqn. (106), i.e. C~ 2 with C = e2Kitznlxpt%~, varies by a factor of 1.6 x 10 a in the ~ range considered at 1000°K. The experimental Hall mobility at 1000°K, however, proves to have a constant positive value, independent of ~. One may then conclude :

(1) At 1000°K intrinsic effects have no influence on tzE either, the term C~ ~ remaining small at all dopes.

(2) The Hall mobility for electrons has the anomalous positive sign and is independent of dope.

An upper limit for the product t%tZpH at 1000°K can be derived. An extrapolation of the values for E~ obtained at 1400°K and 1200°K leads

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10

1

Fig. 20

q T(°K ) 1600 1400 1200 1000

, , , -800 ~_Fe203 t - - 1

t C~ op)

- - - 200

1 i

- = ]

"~T ~ " ~ S "~"'~-! . . . . .

6 7 8 9 10 11

~10 -2

+10 -3

Seebeek coefficient a, resistivity ~, Hall coefficient R H, and Hall mobility/x n for ceramic n-type a-F%0 3 samples as a funetion of reciprocal temperature. A sign reversal of R n and /~H occurs just below the N6el temperature. The samples numbered 1 to 5 have been doped with 0.01 at. % Zr (I), 0"1 a t . 0~0 Z r (2), 0.12 at. % Nb (3), 0.25 at. % Nb (4), and 1 at. % Zr (5). The black markings represent values obtained in 1 arm of O 2, the open markings indicate values resulting from measurements in 0.01 arm partial pressure of O 2 (van DaM and Bosman 1967).

to 1.35 < Eg < 1"75 at 1000°t{. Wi th these values for Eg and the experi- menta l value N o e ~ _ N o ~ - 4 x l O 2 ~ c m -a (see §3.4) one arrives a t K i ~ 2 . 5 x 1 0 3 6 c m -6 at 1000°K. I t follows then from the condit ion IC~I <l/xnHI at the highest value for e, wi th /~n~0"l cm2/vsee, t h a t I/%/%~[ ~ 5 x 10 -2 (cm2/v sec) 2 a t 1000°K. I t can fur thermore be concluded from the t empera tu re - independen t behaviour of ~ and/~H in the extr insic region, i.e. for the higher dopes, t h a t a t all t empera tures considered /% and /xnn are roughly constant , having values of about - 0 . 1 and + 0.025 cm2/v sec, respectively.

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In format ion with regard to/xp and/xpL ~ can be ob ta ined f rom ~, ~ and /x~ in the region of mixed conduction. For the l ightest doped sample, measured at 1 at. of 02, n - p _ 1019 em -3 and np=Kl~_ 1039 em -6 at 1400°K. The effective dope, consisting of Zr and presumably Vo's, has been es t imated with the aid of the value of ~ in the exhaus t ion region. The value of K i corresponds to the est imates arr ived at for E~. One finds then t h a t a t 1400°K, n _ 3 . 7 x 1019 em -3 and p___2.7 x 10 TM em -3. Wi th these values for n and p and those measured for ~ and ~ one arr ives a t / % _~ 0.2 em2/v see, on the reasonable assumption t h a t N - e x - ~_ N+e A+. This result implies t ha t pl%/nl% ~ _ 1.5 at 1400°K. I t follows then t h a t a t 1400°K the exper imenta l ly established reduct ion of/x~i b y a fac tor of 2.5 with respect to/~=I~ is de termined by the relat ion :

n/~n /x K _ /Xng , (107)

p/zp -]- n/~ n

the t e rm ]Pt%t%gl being small compared to n/Xn/Xng. The upper l imit for ]t%/~pgl found above, f rom da ta a t 1000°K, is t he reby fu r the r reduced to ]/xp/Xpn I ~ 3 x 10 -3. I t can then be concluded, wi th /x p ___ 0.2 em2/v see, t ha t I/xpri]~0.01 cm2/vsec. One thus arrives a t the inference t h a t I/%~/t%1 ~ 1. I t would be desirable to establish the sign of/xpi ~ f rom Rtt da ta for p- type mater ia l a t these tempera tures because, just as in the ease of electrons, the Hal l mobi l i ty of holes might have the anomalous sign.

The da ta obta ined in the h igh- tempera ture region f o r / % will be discussed on the basis of a band model, assuming t h a t the drif t mobi l i ty is de termined by spin-disorder scattering. The magnet ic susceptibi l i ty of ~-F%03 between 300 and 950°K has an almost constant value of abou t 2 x 10 a e.m.u./g (Ndel 1950). We assume tha t this value for X also applies at higher tempera tures . Subst i tu t ion of the value of 0.1 em2/v see for/~n at 1200°K in the mobi l i ty formula (104) for the ease of spin-disorder scattering leads to the es t imate (ms*/mo)a/2(F')2~ 70 (ev) 2. Given a value of 0.5 ev for the exchange coupling constant F', i.e. about the same value as proposed for NiO and CoO, mp*/m o ~- 9.5. A value such as mp*/m o = 9.5 for the densi ty-of-states effective polaron mass corresponds to a value for N v at 1000°K of 4"4x1021em -3. Wi th a value for A of abou t 2, as expected in the presence of spin-disorder scattering, it follows then t h a t Nve A at 1000%: has a value of about 0.8N 0. This value is quite near to t ha t es t imated from the behaviour of ~ as a funct ion of dope (0.9N0, see § 3.4). A value for mp* of about 10m 0 corresponds to a resonance integral Jp ~ 0.03 ev and a large-polaron band width A W p --- 0.4 ev. F o r e-F%03 compared to NiO and CoO, within the assumption of large- polaron band conduction, the smaller value for t~D at high t empera tu res results f rom a smaller band width and a greater influence f rom spin- disorder scattering.

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The sign reversal of/~i~ in the vicinity of the N6el temperature might be thought to be connected with the presence of small p-type inclusions embedded in n-type material or of p-type grain-boundary layers (see also § 4.1). At temperatures T < T N the relatively large 'weakly ferromagnet ic ' Hall effect with a negative sign would dominate, its disappearance at T = T N revealing the presence of a small spurious p-type effect. However, the presence of p-type grain-boundary layers seems improbable at very high temperatures (_~ 1500°K), where equilibrium is reached with the surrounding atmosphere, and at low oxygen partial pressures. Nevertheless in such a situation R H remains to have the positive sign. The anomaly in /~H may therefore be considered to be an intrinsi c property of antiferromagnetic a-F%0 a. The anomaly will be discussed on the basis of Maranzana's model (see § 4.1). For a-F%O 3 compared to NiO the parameter fi, determining the anomalous antiferromagnetic contribution to R~ has an appreciably larger value at the N@I temperature. This is because the total spin S has a value of 5/2 for the Fe 3+ ion compared to a value of 1 for the Ni 2+ ion, while fi is proportional to S ~. I t follows from eqn. (100) that

f i~ --3 × lO-6(mp*/mo)aI ~2 (T= TN) (lOS)

for the case of a-F%08. The values mj,*/mo ~_ 9.5 and F'_~0.5 ev, i.e. F_~lSev$, 3, mentioned above lead to f i ~ - 0 . 6 . This result for fl, being near to the critical value of - 1, might be considered as an indication tha t Maranzana's model applies to a-Fe203. However, in the light of the relevant discussion presented in § 4.1, a definite conclusion does not yet seem justified.

4.4. MnO

The Hall mobility of holes in Li-doped ( ~ 0.1 at. %) single crystalline MnO was firstly established by Nagels and Denayer (1967) to have a low, positive value of about 7 x 10 -3 cm2/v see at 500°K < T < 825°K, diminish- ing to about 5.5× 10 -8 at T_~425°K. This result was concluded to be not in favour of small-polaron hopping conduction.

A transition to a Hall coefficient with a negative sign was found by de Wit and Crevecoeur (1967) to occur in Li-doped (0.05 and 1 at. %) ceramic samples at high temperatures. In the region where R H has the negative sign, ~H proved to reach fairly high values of about 10 cm2/v sec. I t was shown tha t the sign reversal of R~ is due to intrinsic effects. At constant temperature, the relation between tLH and ~, determined for undoped and doped samples at varying oxygen partial pressures, is con- formal to that expected in the presence of mixed conduction. Values for t~n~t of 11"3 and 9"3 cm~/v sec were established at temperatures of 1273 and 1473°K, respectively. I t was later on established by Crevecoeur and de Wit (1968), from a s tudy of ~ and ~, that the energy gap E~ ~< 1.8 ev

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80 A . J . B o s m a n and H. J. v a n Daa l on

a t 1373°K (see § 3.4). Such a value of the energy gap combined wi th the large ra t io Fn~//%I~ (-~ 1000) (see fig. 21) is compat ib le wi th the influence of intrinsic effects on R n. Fo r the case of T i -doped n - t ype samples /xn} I was found to increase f rom a value of abou t l0 cme/v sec a t 1400°K to abou t 20 to 40 cm2/v sec a t room t empera tu re .

Fig. 21

10

1~ 2

10 -3 _

1000 500

- I g ~

_3(9. ~_../ /

~ . / /

/ I®', I Jg®

v ',I

T (OK) 4 - - 300 200

I

)

I@

f

~-NiO 1,1' I ~-CoO 2,2' Tr

n - c{- Fe203 3 11T p-MnO ~ n-MnO ¥

l l l l l l I l ITII o I 2 4- 5 3

~ (°K-~)

\ \

I I I ~ I I ~ I 3

Survey of the data obtained for Hall mobility (/x~) and drift mobility (FD) in the cases of NiO, CoO, a-F%03 and MnO. Data relating to MnO were obtained by Crevecoeur and de Wit (to be published). The sign of /z D aS well as of t ~ has been indicated. Curves 1 and 2 were obtained from e and @ data on the assumption NvocTs/~, curves 1', 2' and 4 on the assumption N v = constant.

F r o m Hall-effect m e a s u r e m e n t s a t 1216°K on a single crys ta l of MnO, Gvishi, Tal lan and T a n n h a u s e r (1968), while confirming the va lue of abou t 10 cm2/v sec for /Xnm ar r ived a t the different conclusion t h a t

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t%n~_0.2 cm2/v sec. Exper imen ta l values for tZH, in the region where R n has the negat ive sign, were ob ta ined as a funct ion of the rat io q =p(CO2)/p(CO)) , determining the oxygen part ial pressure of the atmosphere around the sample. The value for/%i~ was der ived from an analysis of tLtt as a funct ion of q, assuming t h a t p~p /n l z n =q2/3. This assumption is val id only if a t the re levant t empera tu re all manganese vacancies are twofold ionized.

Nagels, Denayer , de Wit and Crevecoeur (1968) poin ted out t h a t the exper imenta l value of 7 x 10 .3 cm~/v sec for /%tt above 500°K has been well established fo r the ease of single crystalline as well as ceramic material . Fur the rmore , in de Wit ' s and in Gvishi's exper imenta l results, the behaviour of tLt~ in the region of negat ive Hall coefficient as a funct ion of q proves to be the same. These au thors therefore suggest t h a t the assumption p = 2[V~a"] under lying Gvishi 's analysis of t~n versus q is incorrect, leading to an erroneous es t imate o f / % m

The Hal l -mobi l i ty behaviour for holes, according to Crevecoeur and de Wi t (to be published), see fig. 21, does not contradict the postula te t h a t p - type MnO is a smMl-polaron semiconductor . At room t empera tu re /xpt ~ is appreciably larger ( _~ 1.5 x 10 -a em2/v see) t h a n /x D (_~ 3 x 10 -5 em2/v see, see § 4.4), while wi th increasing t empera tu re bo th quant i t ies increase and reach a value of about 0.01 cme/v see at high t empera tu res (_~ 1000°K). Such a behaviour might indeed be expec ted for a small- polaron semiconductor in the hopping regime, where the Hall coefficient is de termined by processes corresponding to the three-sites configuration. I t m a y be no ted t ha t in an ant i fer romagnet ie p - type small-polaron conductor , such as MnO seems to be, exper iment shows t h a t the Hal l coefficient has the posit ive sign (see § 2.2).

An in te rpre ta t ion of the Hal l mobi l i ty of electrons in MnO can be given on the basis of a band model where scat ter ing of charge carriers is determined by spin disorder (see eqn. (104)). MnO is ant i fer romagnet ic wi th a N6el t empera tu re of 120°K. The susceptibi l i ty equals about 7 x 10 -a e.m.u./g a t room tempera ture . The value for/Zni t of 20 to 40 cm2/v see at room tempera tu re then leads to an effective mass of about 1.5m 0, if the exchange-coupling constant F' is assumed to have the same value of 0.5 ev as proposed for the other oxides. I t would then follow t h a t n - type MnO is a b road-band semiconductor (AW e -- 2.5 ev).

Anomalies in the Hal l effect such as encountered in NiO and a-Fe20 ~ above the N6el t empera tu re seem to be absent in the case of MnO. Wi th in Maranzana 's model (see § 4.1), the pa ramete r fi, de termining the anomalous ant i fer romagnet ie cont r ibut ion to R~I at t empera tu res T > T N, is given by eqn. (108), val id also for e-F%03. The Maranzana model cer ta in ly does not apply to a small-polaron semiconductor such as p - type MnO seems to be, For the case of n - type MnO, the relat ively low value for the effective mass implies a negligibly small value for ft. This might be considered as an indicat ion t ha t anomalies cannot have an appreciable effect on R~I where n - type mater ia l is concerned.

A.P. F

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82 A . J . Bosman and It . J . van Daal on

§ 5. DIELECTI~IC I~ELAXATION Loss AND OPTICAL /~BSOI~PTION ; ' ~¢~OBILITY ' OF BOUND CHARGE CARRIERS

The behaviour of charge carriers bound to centres can be studied by measuring %o as a function of frequency and temperature. The objectives of this research may be :

(1) to determine the nature and concentration of eentres capable of binding charge carriers. Information of this type can be of great help in explaining d.c. conduction properties. This will be demonstrated in the case of ~-Fe20~,

(2) to ascertain the mechanism for transport of bound charge carriers within the centre,

(3) to establish the relation, if any, between the behaviour of bound and free charge carriers.

The frequency range considered in most of the investigations is confined to one of the following regions (Kz) : (a) 0 - S x l 0 s , (b) 5 x 10~-3 x 10 l°, (c) 3x 10n-Sx 101~, the latter mentioned being the optical region (' far infra-red ' to ' visible '). Some of the investigations extend over regions (a) and (b).

In § 5.1 of this chapter new data, obtained in frequency region (a), will be presented for NiO and a-Fe20 a. Furthermore, results acquired in the same frequency region for CoO and M_nO will be discussed. In § 5.2 data presented in literature for NiO in the frequency range (b) and in § 5.3 literature data for NiO and CoO in the frequency range (c) will be briefly outlined. An a t tempt will be made to discern the connection between these results and those obtained in region (a).

5.1. Dielectric Relaxation Processes (0-5 x l0 s Hz)

The occurrence of dielectric relaxation can be at tr ibuted in general to the presence either of macroscopic or of microscopic inhomogeneities in the sample. Macroscopic inhomogeneities, having dimensions that are large compared to a lattice constant, can be located either at the surface of the sample, e.g. high-ohmic layers directly underneath the measuring electrodes, or in the bulk, e.g. well-conducting regions embedded in an insulating matrix or high-ohmic grain-boundary layers. Relaxation effects due to this type of inhomogeneities, see Koops (1951) and Volger (1959), are not of interest for the present purpose. One should, however, be able to recognize these effects. Surface effects manifest themselves in an apparent dielectric constant which is a function of the thickness of the sample, whereas in the case of bulk effects the dielectric constant is independent of this parameter. Bulk effects due to macroscopic inhomogeneities in general have the features that the frequency Vmax, at which dielectric loss is a maximum, is a function of dope concentration,

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while the magnitude of dielectric loss is independent of temperature. This is in contrast with the behaviour found in general for the case of bulk effects due to microscopic inhomogeneities. Our attention will be focussed on these phenomena. Microscopic inhomogeneities consist either of single eentres or of small groups of centres. These centres are introduced into the material by incorporation of impurities or correspond to native defects. The relaxation process for a single-centre system occurs because of the ' mobility ' of the bound charge carrier within each centre. I t is characterized in sufficiently dilute systems at a certain temperature by a single ~relaxation time which in general depends on the type of dope but is independent of the dope concentration. The magnitude of the relaxation process is expected to increase linearly with dope concentration and to vary as I /T, except at very low temperatures. In non-dilute systems, interference between centres leads in general to broadening of the relaxation-time spectrum. Moreover, in partially compensated non-dilute systems, movement of a ' bound ' charge carrier from an occupied to a neighbouring unoccupied centre leads to a relaxation process characterized by a broad spectrum of relaxation times, corresponding to the distribution of distances between such pairs of centres. Movement of ' bound ' charge carriers within a pair of centres is identical with impurity conduction, albeit on a limited scale.

5.1.1. NiO

Prior to investigations on dielectric loss, the existence of mechanical relaxation processes correlated to Li centres ( _ 1 0 a t . %) in NiO was established (van I~outen 1962). Later on, measurements of this type were extended to smaller dopes (~ 0.1 at. °/o ), and simultaneously dielectric-loss data were obtained for Li dopes between 10 -3 and 1 at. % (van t touten and Bosman 1964, Bosman and van I-Iouten 1965): The occurrence of mechanical loss in Li-doped samples can be considered as a strong indication for the small-polaron character of the charge carrier bound to the Li centre. I t will become clear from what follows, however, tha t quantitative conclusions on the basis of available data do not seem justified because these data were taken at fixed frequency and varying temperature, i.e. with a simultaneous change of relaxation time(s) and magnitude of the relaxation effect. Two different mechanisms for mechanical relaxation were found to be correlated with Li centres. One of these was identical with the dielectric relaxation process of single Li centres, the other having no dielectric analogue. This latter mechanism was assumed to correspond to pairs of Li centres with an antiparallel position of the electric dipoles.

New data will now be presented for the complex dielectric constant (Bosman, unpublished) at frequencies between 102 and 5× 106 Hz of samples doped with Li or Na. In the frequency range considered, dielectric loss could only be observed at low temperatures. Dielectric

F 2

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loss proved to be of the Debije relaxation type . Therefore a cont r ibut ion to the real pa r t of the dielectric constant due to resonance processes is concluded to be absent.

In fig. 22, the dielectric constant %- -de t e rmined at frequencies sufficiently low so t h a t ~ o T ~ l - - h a s been p lo t ted as a funct ion of 1/T for samples doped wi th 0.1 at. % Li or Na. The t empera tu re range ex tends

Fig. 22

T(OK~ - - 200 100 50 30

c, | I I ' _ $ NiO doped with Li or Na

/ / /

-~0.1at % Na / ~ / 2E \ "/

2C ~ ' ~ " ~ ~0"1at °/° Li

.+o ,~ ~ O ' 0 9 a t % Lt

1 ~,,j4~,~. -

I I I I I I I I ) I I i I I i I I I I I I I i E I I r [ I I I {

0 10 20 30 40 3

" ~ PK -1)

Static dielectric constant e s of Li or Na-doped NiO as a function of reciprocal temperature.

only down to 30°K. Ex t r apo la t i on of E s to ]/T->O gives the value of 12, which is equal to t h a t for undoped material . The increase of e s due to dope (Ae) can also be described as the difference be tween e+ and %, where Eo~ is the dielectric constant a t frequencies such t h a t oJ~>)1. I t follows f rom the above t h a t the magni tude of the re laxat ion process is proportional to I /T. I f the in ternal electric field equals the applied field,

then

4~/Vp 2 AE= bk----T--' (109)

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where N is the concentration of dipoles, p the dipole moment and b a constant having values between 1 and 3 for the cases where the dipole can have only two or a large number of different orientations, respectively. The dipole moment p equals ed, where e is the absolute value of the electron charge and d is assumed to be the distance between nearest-neighbour Ni ions (2.9 4). Substitution in eqn. (109) of the values for N (0.1 at. %) and p leads to the experimental relation between AE and l IT if b has a value of about 1.5. The supposition tha t dielectric relaxation results from the presence of dipoles formed by a LP + ion and a hole (Ni a+ ion) localized at one of the nearcst-neighbour cation sites thus seems to be reasonable.

I t could not be ascertained whether eqn. (109) still applies at temperatures below 30°K because, in the frequency range employed, dielectric relaxation was still found to be present at the lowest frequencies (see also figs. 25 and 26). Correspondingly, the dielectric constant was found to increase down to this range. Experiments at frequencies lower than

Fig. 23

25

20

15

-t I

T=77.# °K Nil_ Lix0

y

0 0.1 0.2 O.3 0.~

Effl(a- %)

The contribution Ae to the static dielectric constant of NiO at 77"4°K due to neutral Li centres.

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86 A . J . Bosman and I-I. J. van Daal on

100 Hz are needed to obtain information about the behaviour of AE below 30°K.

The linear dependence of AE on Li-dope concentration is demonstrated in fig. 23 for different samples at 77.4°K. This behavlour of Ae as a function of dope and tha t described above as a function of temperature seem to substantiate the close relation between dielectric relaxation and Li (Na) dope.

Partial compensation o£ the Li acceptors by donors leads to partial suppression of the relaxation process. This is shown in fig. 24 for the case of Li-doped samples (0.1 and 0.3 at. %) with different amounts of Ga as a counterdope. I t is seen that, with increasing Ga concentration, Ae at 77"4:°K, first decreases rapidly--about proportional to three times the Ga concentration--and at higher counterdopes much more slowly. This behaviour suggests tha t in the beginning Ga is incorporated as an interstitial Ga 3+ ion, each Ga ion compensating three Li ions. At a later stage incorporation proceeds probably substitutionally.

2' : i 8

2£

] ~0.1 at %Li

0 T',-'~-~.

Fig. 2¢

T: 77./~- ~< ] N'[O do }ed with Li and Ga

~ 3 at % Li

0 0.1 0.2 0.3 - - - [G~](~ O~o)

The effect o£ counterdoping with Ga on Ae of Li-doped NiO at 77"4:°K.

In contrast to the relatively simple behaviour of the magnitude of the relaxation process (Ae) as a function of dope and temperature, the relaxation itself has a quite complicated character. This will be demonstrated for the case of a Na-doped sample (0.1 at. %). Figures 25 and 26 show the behaviour for this sample of the real part of the dielectric constant ~1 as a function of temperature at different frequencies and of the imaginary part of the dielectric constant E 2 as a function of frequency at various temperatures, respectively. I t may be concluded from these figures that the relaxation process cannot be described by a single relaxation time. For comparison, in fig. 26 an ideal Debije curve has been inserted. l~elaxation is evidently characterized by a large spread of relaxation times which even increases towards lower temperatures.

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Fig. 25

-t \ \

I Nil-xNax0

frequency (sec-1)_ o t0 z

A 10 ~

m 10 h

• 105

V 106

• 3=10 6

87

I I I I I i 20 40 60 80 100 120

T(oK)

The real part el of the dielectric constant of Na-doped NiO ( _~0.1 at. %) as a function of temperature at different frequencies. The broken line represents the behaviour of the static dielectric constant if Ae is proportional to 1/T.

Fig. 26

!

/

Nil_xNaxC

Y / ,-/- / # / ./

I I r l l l l ~ q 1 1 1 1 1 1 [ I J I I I I I

10 s 10':' frequency (sec -~)

The imaginary par t e2 ( = el tan 8) of the dielectric constant of Na-doped NiO ( ~ 0.1 at. %) as a function of frequency at different temperatures. The broken line represents a Debije curve with its maximum at 107 Hz.

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88 A . J . Bosman a n d H. J. van Daal on

Some details of the broad re laxat ion- t ime spect rum become visible at ve ry low tempera tures for differently doped samples. This is i l lustrated in fig. 27 for a t empera tu re of 4.2°K and Li dopes between 0.01 and 0.3 at. %. At dopes lower than 0.I at. % a broad peak is s i tua ted ei ther a round about 2 x 105 t{z (> 0.04 at. %) or at abou t 1.5 x 106 tzz ( < 0.04 at. °/o ). The re laxat ion process corresponding to the la t te r peak is fu r thermore character ized at 77.4°~ by a value of Ae per uni t of dope which is about two t imes smaller t han t h a t for the former peaks. This dissimilar behaviour suggests a difference of re levant eentres. I t has a l ready been concluded (see § 4.3) t ha t in samples with a Li dope below about 0.01 at. % the electrically active centre is not t h a t corresponding to Li bu t p robab ly to a Ni vacancy, which has a dis t inct ly higher ioniza- t ion energy. I t m a y then be presumed tha t the peaks at 2 x 105 t{z and

Fig. 27

,c - ~ Nil.xLix0 T= 4~2 °K

T

- i .-~ t ~ I ~ - . ~,. ~- z- - - - t . ~ ^ ' , ~ , ~ I "*-

10 2 t0 3 10 z~ 10 s 10 6 10 7 fre,tuency (sec 4)

The imaginary part e2 ( = q tan 5) of the dielectric constant of Li-doped NiO at 4"2°K as a function of frequency. The Li contents (at. %) are about : 0"01 (1), 0.04 (2), 0.07 (3), 0.1 (4), 0.15 (5), 0.2 (6) and 0"3 (7). The samples numbered 8 to 11 contain 0.1 at. % of Li and have a Ga counterdope (at. %) of 0-01 (8), 0.02 (9), 0"05 (10) and 0"09 (11). The peak at about 2 × 105 ~z (curves 2 and 3) probably corresponds to isolated Li eentres.

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Conduction in some Transit ion-metal Oxides 89

1.5 x l0 G ~z are connected wi th Li and Ni -vaeancy centres, respec t ive ly (see § 5.2). At Li dopes > 0.07 at. % a n u m b e r of peaks in e 2 versus v become visible, each p e a k being re la t ive ly b road and a t higher dopes becoming less resolved. The peak a t a b o u t 2 x 103 Rz p r o b a b l y corresponds to a deeper centre. I t can be seen t h a t this peak and t h a t a t 2 x 10 a ~Iz (' L i -peak ') r ema in present , a l though of significantly reduced magni tude , in a v e r y heav i ly compensa t ed sample (see fig. 27). The enormous broadness of the r e l axa t ion - t ime spec t rum a t higher dopes

300 100 50

i NiO

Fig. 28

T P K)'~,--" 30 20 15 12.5

. / ° ~ ' ~

~ o j

10-9

16'

s

I I I I q I i I i I i I I I I I

10 20 30 40

! I t

I I I I I [ I I I I I I i ~ L I

50 60 70 80 103 o -1 T ( K )

The behaviour of relaxation times z for dipole relaxation in NiO as a function of reciprocal temperature. Data below 50°K relate to different peaks in the dielectric loss. Curves 1 and 2 are thought to be related to Na and Li-dipole centres, respectively. The open square at 40°K represents an upper limit. Further information on Na and Li-dipole relaxation can be found in figs. 26 and 27, respectively. Curves 3 and 4 were obtained on Li-doped samples with Ga as a counterdope. Curve 3 corresponds to the peak at 4"2°K in E2 at a frequency of about 103 t{z (see fig. 27, curve 11). The symbols S and K denote values obtained by Snowden and Saltsburg (1965) on undoped single crystals and values obtained by Kabashima and Kawakubo (1968) on Li-doped single erystMs. The curve marked A was calculated from data for CrFi (Austin, Clay and Turner 1968) and AE (this work) in the ease of Li- doped NiO.

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90 A . J . Bosman and It. J. van Daal on

(>0.07 at. %) suggests interference between the eentres and probably also the presence of relaxation processes corresponding to pairs of centres. An estimate of the integrated loss at 4.2~K leads to a value of AE which is an order of magnitude smaller than that which one would expect if the 1/T dependence, found above 30°K, were to hold down to this temperature. Furthermore, a linear relation between Li dope and Ae does not seem to be present at 4"2°K. Although relaxation times can be barely defined, due to the broadness of the peaks, an impression of their temperature dependence can be obtained from the shift of E 2 at the peaks versus T. The result is shown in fig. 28 for some of these peaks. In the temperature region considered, the various relaxation times can be seen to be weakly dependent on temperature. I t may be noted tha t if relaxation times behave as shown in fig. 28, measurements of dielectric loss made at constant frequency v and varying temperature can lead to an erroneous interpretation of the data. This is because in such experiments, relaxation processes for which r - l < v do not contribute to the dielectric loss.

A full interpretation of the above data seems difficult to give at the moment. Some facts, however, are of importance. The magnitude of Ae and its linear dependence on dope and reciprocal temperature for T > 30°K can be satisfactorily described on the basis of eqn. 109. This implies that corrections due to internal-field effects can be neglected. The linear dependence of Ae on 1/T down to low temperatures should then be explained by assuming tha t the spread of the energy levels representing the dipole centre is at most equal to k T with T_~ 30°x. The Li centre, consisting of the Li + ion and its nearest cation neighbours, has a centre of symmetry in the purely cubic (T> TN) as well as in the slightly rhombohedrally distorted phase (T< T~). In the former case the Li+ ion has 12 and in the latter 6 equivalent nearest neighbours. The disappearance of the 1/T dependence of Ae below 30°K at a critical temperature T c implies an energy spread of the order of 1 mev. Such a small value for this energy spread points to the smaU-polaron character of the bound charge carrier (see § 2.4). I f the energy spread kT e is interpreted as the small-polaron ' band width ' for the centre (AW), one can arrive at an estimate for the intra-centre electron-transfer integral J . Assuming A W = 4 J e x p (--ST)~_4Jex p (--7) (see § 2.1) and 7_~6, then for the centre J_~ 0.1 ev. The value for 7 was derived from optical absorption data in the near infra-red for Li centres (see § 5.3).

The above interpretation of AE behaviour will be shown to be inconsistent with the nature of the relaxation process observed at temperatures lower than T e. In general, i.e. in the case of a non-degenerate ground state, one would expect at temperatures T < A W / k resonance absorption to take over from Debije relaxation loss, the latter occurring at temperatures T > AW/k. This would imply tha t at low temperatures (T < AW/k) the contribution to AE due to centres could totally disappear only at the resonance frequency h/AW~ 10 n ~z. However, it is observed

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at temperatures T < To that this contribution to AE disappears at relatively low frequencies ( g l0 s Hz) via a relaxation process of the Debije type. Therefore, it should be concluded that the energy spread for the centre is not determined by the ' band width ' AW but is probably an energy spread due to disorder (WI)), originating from internal locally varying electrical fields, abolishing the equivalence of the sites at the centre. The formation of a type of small-polaron ' b a n d ' for the centre and thus the occurrence of resonance absorption seems to be prohibited by disorder. I f WD~_kTo>AW, the bound charge carrier in a semi- classical way can be thought to be localized at one of the cations neighbouring the Li+ ion. This situation gives occasion predominantly to Debije relaxation, resonance absorption at the frequency WD/h being of minor importance if (AW/WD)2~ 1. The estimates given above for AW and also for J should then be regarded as upper limits.

Additional information concerning the small-polaron character of the bound charge carriers can be obtained from the relaxation-time behaviour as a function of temperature. As a consequence of the arguments presented above, small-polaron ' b a n d conduction ', i.e. diagonal transitions, cannot be expected to have an influence on r behaviour down to the lowest temperatures. I t remains determined by phonon-assisted transitions, i.e. by non-diagonal processe s . The mechanism for transport of bound charge carriers within the centre in the presence of locally varying electric fields may be expected to resemble that of small-polaron impurity conduction (Schnakenberg 1969, see § 3.2). There is indeed a remarkable resemblance between the temperature dependence of impurity conduction, i.e. of inter-centre mobility, and the temperature dependence of l /r , i.e. of intra-centre mobility. In addition, intra-centre mobility (ed2/~-lcT) proves to be of the same order of magnitude as inter-centre mobility if the latter is extrapolated to the situation where jump distances of about one lattice constant are involved. According to Schnakenberg, small-polaron impurity conduction in the low-temperature region leads to an inter-centre mobility which is a factor of exp (2y) smaller than the corresponding case where small-polaron effects are absent. At very high temperatures, however, impurity conductivity seems in both cases to approach to about the same value. The same conclusions may be assumed to apply to the mechanism of intra-centre charge-carrier transport in the presence of disorder. Therefore the total variation of with temperature may be expected to be of the order of exp (2~). The experimental behaviour of T is in satisfactory agreement with the above small-polaron model. The low-temperature data (4"2-50°K), derived from dielectric loss measurements, correspond to the region where acoustical one-phonon assisted transitions dominate. The spread in the relaxation times for different centres can probably be at tr ibuted to a difference of parameters determining the small-polaron character of bound carriers. The high-temperature data (200-500°K), derived from optical absorption due to centres in the far infra-red, correspond to the

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regions where optical two-phonon (T < ½0) or optical multi-phonon transitions ( T > 10) dominate. The latter region is the thermally activated hopping regime. I t can be seen in fig. 28 that the total variation of with temperature for the Li centre is a factor of about 105, which corresponds to a coupling constant y_~ 6. I t may then be concluded that , disregarding interference effects between centres, dielectric loss in NiO may consistently be at tr ibuted to bound charge carriers being small polarons.

5.1.2. CoO

Dielectric relaxation as observed in Li(Na)-doped CoO (Bosman and Creveeoeur 1968) differs appreciably from that encountered in the investigations on NiO. The data obtained on CoO can be at t r ibuted consistently to a relaxation process involving a single type of dipole only (Li+-Co 3+ or Na+-CoS+). The different dopes considered were in the range 0.01 to 0.5 at. ~o, the frequency range between 60 and 2 x 106 nz and the main part of the temperature region between 100 and 200°K. Crevecoeur and de Wit (to be published) have extended the experimental range down to temperatures of 60°K and frequencies of 0.01 ~z.

The behaviour of the complex dielectric constant of CoO can be con- trasted with that of NiO :

(1) The magnitude Ae of the relaxation process for Li(Na) dopes around about 0.1 at. °/o is two times smaller than that of NiO and can be described with eqn. 109 (b_~ 3).

(2) Li(Na)-dipole relaxation results in a single well-defined peak of E 2 versus v, its width being much smaller than that of the NiO peaks (see fig. 29). Actually the Li-dipole peaks for CoO are only slightly broader than the width expected for an ideal Debije relaxation process. Thus a relaxation time can be very well determined.

(3) At any temperature the relaxation time z proves to be much longer than that characteristic of the comparable situation in NiO, while its value decreases strongly with temperature (see fig. 30) :

~ = % e x p (Q/]~T), (110)

where Q = 0.20 ev (0.25 ev) for Li(Na) dipoles and z0~ 5 × 10 -13 see. The former fact implies tha t dielectric loss occurs at higher temperatures in CoO than in NiO, the latter fact tha t in CoO the temperature dependence of Ae is negligible compared with that of z. Thus reasonably accurate values can be obtained for z from a plot of E 2 versus temperature at one and the same frequency. I t may be emphasized that the activation energy of the relaxation time is quite constant over the whole temperature range, i.e, from 200°K down to 60°K,

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Fig. 29

100

8~

60

40

20

0 10

10 3 fan 6

t

/

/ /

J / ' /

/ /

, , - - 4 ~ i I L I I I I

102

\

I I I I I I I I [ I t t l l [ I

10 3 10 4.

CoO

\ \ \ \ k

i ( I 1 1 1 ( i i i l l J

10 5 10 6 .~ frequency (sec -I)

Dipole-relaxation loss of Li-doped CoO as a function of frequency at 121°K, Li concentration: 0"08 at. %. The dashed line represents a Debije curve (Bosman and Creveeoeur 1968).

Fig. 30

T(OK)~-- 200 150 100 O0

~g F(se:) , o ~/_ - ~ L / o /

Mnl-xLix O, 'o ff J /i

-2 / / , / / ' . l_xkzx 0

-4 f /

-6 f Y

-8 ' I I I I 4- 6 8 10 12 1/.1.

,,. ~ (OK-)

The relaxation time T obtained from dielectric-loss measurements on Li-doped Mn0 (Crevecoeur and de Wit 1968 b) and Li-doped CoO (Bosman and Crevecoeur 1968, Creveeoeur and de Wit) as a function of reciprocal temperature.

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The mobility within the ]~i(Na) centre of bound holes can be evaluated by substitution of the values of r in eqn. (15). A comparison between free-carrier and bound-carrier mobility can now be made valid at a temperature of, e.g. 200°K, where relevant experimental data are available (see § 3.4). I t seems justified to conclude tha t bound holes behave quite differently from free holes :

(1) At T___ 200°K the mobility of holes bound to a Li(Na) centre has the very low value of 10 -6 (10 -7) cm2/v sec compared with that of about 1 em2/v sec for free holes.

(2) In the temperature region around 200°K, the mobility of bound holes increases markedly with temperature in contrast to the almost temperature-independent behaviour of free-hole mobility.

I t may be concluded that, just as in the ease of NiO, the existence and behaviour of dielectric loss down to temperatures of at least 60°K points to a small energy spread of the levels corresponding to the centre. In CoO, however, T has a thermally activated character with a constant activation energy down to 60°K. This behaviour is in contrast with tha t encountered in NiO. This different r behaviour suggests that in CoO, effects other than those due to small-polaron formation at the centre dominate the relaxation process. Such an effect will be present if the Co 8+ ion neighbouring a Li 1+ or Na 1+ ion is in the low-spin state. I t might be suggested tha t the small-polaron effect in this case initiates the necessary localization of the bound charge carrier. As a consequence of the low-spin state of the Co 3+ ion, charge transfer between this ion and one of the neighbouring Co 2+ ions, being in the high-spin state, will be severely hampered. This is because it is a multi-electron process, which is possible only via an excitation of the Co 3+ ion into the high-spin state. The activation energy for transfer of a bound hole (Co~+-ion state) between neighbouring Co ions may be expected to be identical with the necessary excitation energy. Energy differences between a low-spin and a high- spin Co 3÷ ion are known to have values between 0.01 and 0.8 ev. The values found for Li-doped (0.2 ev) and Na-doped CoO (0.25 ev) thus seem to be reasonable. The low-spin state of a Co ~+ ion near to a Li+ ion is favoured because crystal-field splitting is enlarged, due to the extra polarization of the oxygen ions by the Li+ ion. Because Li+ ions are somewhat smaller and Na + ions larger than Co 2+ ions, an extra enlargement of crystal-field splitting may be expected in the case of a Na + ion. In NiO, the spin state of the Ni 8+ ion (3d v) in the Li centre is of no importance, charge transfer between either high-spin or low-spin NP + on the one hand and high-spin Ni 2+ on the other being a simple one-electron process without the necessity of an excited state. An important consequence of the large activation energy needed for intra-centre movement of a bound hole is tha t inter-eentra motion will also be hampered severely. Thus d.c. impurity conductivity can be expected to have very low values and to be markedly temperature dependent. I t has already been

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remarked (see § 3.2) that in concurrence with the above, down to the lowest temperatures considered for CoO, not a trace of d.c. impurity conduction could be found.

5.1.3. ~-Fe203 The occurrence of dielectric relaxation due to the presence of dipoles

was established for the first time in reduced ~-Fe20 a (Volger 1957), at temperatures below about 30°K. The relaxation time proved to vary only weakly with temperature. Further data for reduced ~-F%Oa (Bosman and van Houten 1965), while affirming the above findings below 30°t<, pointed to a larger temperature dependence of r above this temperature. New data concerning dipole-relaxation processes will be presented for ~-Fe2Oa doped with V, Ti, Zr, Sn, Ta or Nb (Bosman, unpublished). Incorporation of these elements into ~-Fe203 was already known to give rise to donor centres, with the exception of V (gonker 1962).

Fig. 31

6{ i ~103tan 6 a-~203

T=4.,2OK

Nb

/ / /

m,, frequency (s~¢ -1)

Dielectric relaxation loss at 4"2°K as a function of frequency in undoped and doped ( ~0.1 at. %) ~-F%03.

Figure 31 represents for a number of samples the observed behaviour of tan 3(= E2/E1) at 4.2°X versus frequency. The starting material con- tained Ni as the main impurity (___0.05 at. %). One should be aware tha t incorporation of Ni into ~-F%O3 leads to acceptors and thus to (partial) compensation of donors. The samples were prepared at 1200°c in 1 at. of 02 and thereafter rapidly quenched to room temperature. Within the frequency range considered, no peak was found for non- deliberately doped and V-doped samples. Furthermore, peaks were

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found at a f requency of abou t 5 x 103 ~z for Ti, Zr and Sn-doped samples, at 2 × 105 ~z for Ta and at 5 × 105 ~z for Nb. I t will first of all be shown tha t the peak at 5 × 103 nz does not correspond to dipole re laxat ion of a Ti, Zr or Sn centre bu t is due to the presence of a native-defect centre, presumably an oxygen vacancy (Vo). Undoped samples, p repared from much purer s tar t ing material , did show dielectric re laxat ion at 4-2°K with a peak in t an 3 versus v at the v e ry same position (see fig. 32). The magni tude of this peak could be var ied in a reproducible way by altering the deviat ion from stoichiometry, i.e. b y changing the oxygen- vacancy concentrat ion. An increase of the prepara t ion tempera ture , at constant oxygen pressure, as well as a decrease of oxygen pressure, at constant prepara t ion tempera ture , led to an enlargement of the peak. Doping of these samples, originating from purer s tar t ing material , with Ti, Zr or Sn did no t in t roduce any ext ra re laxat ion effect. On the other hand, doping with Ta or Nb resul ted in bo th cases in two peaks, one

Fig. 32

-Io 3 ~n 6' ia_Fe=%

-- ! T=4. .oK TC c)I oo 1 oo - I /~ 'l 02 a ir

_ Do .

3O

"-..2 I I I I I I I I I I I ! I I I I

I( 10 5 frequency (sec -1)

Dielectric relaxation loss ~t 4"2°K ~S a function of frequency in reduced ~-F%0 ~. The samples, either pure or impure ( _~ 0.05 ~t. % Ni), were he~t-treated at different temperatures and in different atmospheres .

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Conduction in some Transit ion-metal Oxides 97

being ckaracteristic of the V o centre, the other of the Ta or Nb centre. For material containing 0.05 at. °/o 1Ni as a counterdope, the V 0 peak could be evoked also for undoped samples and samples doped with Ta or I~b, by preparing them in highly reducing circumstances.

I t m a y b e concluded from the above tha t out of the centres investigated, only those corresponding to V 0, Ta and Nb lead to dipole-relaxation effects within the frequency range considered. For the samples containing 0.05 at. % Ni as an impurity, the data presented in fig. 31 can be explained if the V 0 concentration is smaller than the 1Ni concentration and if ionization energies of the donors increase in the sequence Ti (Zr or Sn), V 0, Nb (Ta). These conditions are fulfilled. The V o concentration in these samples is about 0.02 at. %. This estimate stems from a comparison of dielectric-loss peaks due to V 0 centres and to a known concentration of ~{b centres. A similar estimate has been obtained from the behaviour of d.c. resistivity at 1000°K as a function of dope (see § 3.3). Estimates of the ionization energies, derived from d.c. resistivity behaviour, are 0.18 ev (Ti, Zr, Sn), 0.20 ev (Vo) and 0.24 ev (Ta, Nb), valid at donor concentrations of about 0.1 at. ~/o (see § 3.3).

I t follows then, for the samples with 0.05 at. % Ni acceptors, tha t in the absence of any other effective dope, the V 0 donors are fully compensated by the Ni acceptors and cannot give rise to dipole relaxation. Introduction of donors, such as corresponding to Ti (Zr or Sn), with a shallower level than tha t of the V0 donor, removes the compensation of the V o donors, enabling the occurrence of V0-centre dipole loss. The situation in this case is analogous to tha t encountered for low Li dopes in NiO (see § 5.1.1). On the other hand, introduction of donors, such as corresponding to Ta or Nb, with a deeper energy level than tha t of the V0 donor, leaves this latter donor completely compensated, a Ta or Nb dielectric-loss peak only being present.

The absence of dielectric loss correlated with Ti, Zr or Sn-donor eentres is a remarkable phenomenon. First of all, one has to make sure tha t at 4.2°K the relevant dielectric loss does not occur at frequencies beyond the range considered. I f this loss were present at 4"2°K below 102 ~z, increase of temperature would eventually bring the loss within the frequency range considered. This has not been observed to be the case. Alternatively, if this loss were to occur at 4"2°K above 5 × 106 ~z, it would be accompanied by a contribution to e 1 at frequencies below 5 x 106 ~z. No clear indications for such an effect could be found. Heavy doping only with Zr or Ti led to a relatively small value of AE at low frequencies. I t might therefore be concluded tha t these centres at not too high concentrations do not give rise at all to dipole-relaxation effects.

The absence in a-F%0s at low temperatures of dipole relaxation for donors capable of holding only one electron, as is the case for the Ti, Zr or Sn donor, quite probably relates to the specific (hexagonal) crystal structure of this material. In a-1%20 a each cation has only one nearest- neighbour cation. This implies tha t if incorporation, e.g. of Ti into

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a-F%Oa results in a donor centre with the Ti 4+ ion on a cation site and a Fe 2+ ion on the nearest-neighbour cation site, the corresponding dipole (Ti~+-Fe 2+) has only one configuration of stable equilibrium. In that case it will be clear tha t dipoles of this type cannot give rise to dipole- relaxation loss. On the other hand, it should be concluded tha t Ta or Nb-donor centres do give rise to relaxation loss because these donors are capable of holding two electrons. The behaviour of d.c. conduction for Ta (Nb) compared to Ti (Zr, Sn)-doped samples corroborates this conclusion (see § 3.3). Incorporation, e.g. of Ta results in the formation of a centre TaS+-2Fe ~+, of which one of the Fe 2+ ions has the single nearest- neighbour position and the other either one of three second-neighbour or one of the six third-neighbour positions. The position of this latter Fe 2+ ion gives rise to a dipole having more than one configuration of stable equilibrium. An interesting consequence of this model is that the resulting dipoles should give rise to non-isotropic dielectric loss. I t would be desirable to have dielectric-loss measurements made on Ta or Nb-doped single crystals of a-F%0 a with the electric field either parallel or perpendicular to the hexagonal c axis. One would expect dielectric-relaxation loss to be zero in the former and maximum in the latter case.

Dielectric loss corresponding to Ta, Nb and V o centres has the following features :

(1) The magnitude of dielectric loss has been checked for the cases of Ta and Nb dope to be proportional to concentration up to 0.5 at. %.

(2) The shape of the peaks representative of Ta, Nb and V 0 centres only slightly deviates from the ideal case of Debije relaxation loss. In this respect dielectric relaxation in a-Fe20 a is similar to tha t in CoO but at variance with tha t occurring in NiO.

(3) The relaxation time for reorientation of the dipoles increases in the sequence Nb, Ta, V 0 and is weakly temperature-dependent up to

20°K (see fig. 33). Above this temperature, the relaxation time characteristic of the V 0 centre shows an increasing temperature dependence. At the highest temperatures considered (77°K), the increase of 1/T with temperature can be described with an activation energy of about 0.04 ev. The ~ behaviour shows a similarity to tha t encountered in the case of NiO.

(4) The magnitude of dielectric loss in the greater part of the temperature range considered (4"2--77°K) depends only weakly on temperature. At the high-temperature side the onset to an 1/T dependence is seen. This is illustrated for the Nb and V 0 centre in fig. 34.

(5) With increasing frequency the contribution from centres to the real part of the dielectric constant disappears totally by a Debije relaxation process, pointing to the absence of resonance processes.

I t may be remarked that de Vos and Volger (1967 a, b) found dielectric loss in smoky quartz with similar features, especially with regard to points 3 ° and 4 ° mentioned above, as in a-F%0 a.

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The above-described dielectric-relaxation behaviour can be considered, just as in the case of NiO, to point to the small-polaron character of bound charge carriers. The absence of resonance processes points to the pre- dominan t influence of disorder on the spread of the energy levels corresponding to the relevant centres. The relaxat ion-t ime behaviour as a funct ion of tempera ture suggests a coupling constant 7 of about the same

Fig. 33

T(°K) 10_ 4 100 20 10 5 h

10-s

10-s

10-7

:z" (sec) I

f / f

OL" F 'e203

Ta

" 'Nb

0 - 8 I I f I I T l l l r f I f 1 I I l f l I

0 50 100 150 200 2E 0 103/T (°K-l)

The relaxation time z as a function of reciprocal temperature for different centres in ¢-F%0~. For the V o centre, black markings resulted from measurements at constant temperature whereas open markings were obtained from measurements at constant frequency. All other data were obtained at constant temperature.

100 100 ~ i ~ j

j~3 n a'~

80

6O

2O I / /

0 I I I 0 2 0 4 0

Fig. 34

20 T(OK)~ I, ~, 1o

a-Fe203 I

[] I

I

1 1 i 1 i

r l ! z 6 0 8 0 100

. -~ (oK -~)

The maximum value of tan 8, obtained from measurements of tan ~ as a function of frequency, in a-F%0 a corresponding to the V 0 centre and the contribution Ae to the static dielectric constant of ~-F%Oa due to the Nb centre as a function of reciprocal temperature.

G 2

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magnitude as proposed for NiO. The behaviour of Ae as a function of temperature is qualitatively similar to that found for NiO. However, deviation from an 1/T dependence starts at considerably higher temperatures. This suggests tha t in a centre the spread of energy levels due to disorder is larger than tha t in NiO. In a quantitative analysis of Ae behaviour one should take account of the complicated character of the centres, which contain more than one electron and can have anisotropic properties.

5.1.4..MnO

Dielectric relaxation has been observed in Li-doped MnO (0.02 to 0.5 at. °/o ) at frequencies in the range 0.01 to 105 ~z and temperatures between 100 and 200°K (Creveeoeur and de Wit 1968 and to be published). I t has properties similar to those of Li-doped CoO :

(1) The contribution of Li+-Mn a+ dipoles to Ae is described by eqn. (109) with b _~ 3.

(2) The peaks in tan 8 versus v conform reasonably well to the peak corresponding to Debije relaxation loss.

(3) The relaxation time for reorientation of the dipoles varies as described by eqn. (110) with r0~_5xl0-1~sec and Q=o .29ev (see fig. 3o).

The mobility of bound carriers is thereby established to be thermally activated. The activation energy has been attributed to removal of the Jahn-Teller deformation of the oxygen octahedron surrounding the Mn 8+ ion. The mobility of the bound hole at 200°K, calculated by substituting the measured value of r in equation (15), amounts to 7 x 10 -1° cm~/v sec, which is almost three orders of magnitude lower than the corresponding value derived for the free hole. Thus there is a significan~ difference between free and bound-carrier mobility although the relevant activation energies have about the same values.

5.2. High-frequency Conductivity in NiO (v ~<2 x 10 l° tIZ) Snowden and Saltsburg (1965) measured the conductivity of non-

deliberately doped single crystals of NiO at room temperature in the frequency range 10 a to 10 l° Hz. In the range 106 to 5 × 10 s ~z, conduc- t ivity increased in proportion to v 2, becoming independent of frequency above 5 × 10 s ~z. Such a behaviour can be expected if conductivity is determined by reorientation of dipoles, the process being characterized mainly b y a single relaxation time. The observed variation of conduc- t ivity with frequency is the equivalent of the occurrence of ideal Debije relaxation peaks in the imaginary part ~ of the dielectric constant versus frequency (see § 2.4). For two crystals of different origin, similar frequency behaviour of a was found, corresponding to a relaxation time of

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3 x 10 -10 see (see fig. 28). This led Snowden to the assumption that the dipoles are probably correlated with Ni vacancies. The concentrations of these vacancies (N) were derived from the observed values of a(v) on the assumption that the dipoles have a moment corresponding to the nearest-neighbour cation distance d. At frequencies such that o~>> 1 :

Need 2 . . . . . . (III)

3~kT"

Substitution of the known values for a(wv>~ 1) and ~ leads to concentrations of l0 TM and 10 ~° cm -a. I t is not readily possible to check these relatively large values from the values given for d.c. conduct ivi ty: about 10 -9 and 10 -l° (~ cm) -1, respectively. The situation is confused because of the opposite signs of the Seebeck coefficient of the two samples. Snowden considers the observed dispersive conduction behaviour at one temperature as direct evidence of the hopping of bound carriers around lattice imperfections and even as an indirect support for the assumption of hopping motion of free carriers.

The temperature behaviour of high-frequency conduction in NiO was studied by Kabashima and Kawakubo (1968). These authors determined a(v) at frequencies between 3-3 x 107 and 2.4 x 101° tIz and at temperatures in the range from 110 to 450°K on Li-doped single crystals (1-3 x 10 -a, 8.5x 10 -a and 1.4x 10-2at. % ). Below 200°K for the two lightest doped samples, and only below 125°K for the highest doped, a(v) proved to be independent of temperature at all frequencies considered. I t was concluded that the frequency behaviour of a, within the temperature ranges indicated above, was similar to tha t determined by dipole relaxation with a single temperature-independent relaxation time of 2.2 x 10 -l° sec. On the basis of this result the inference was drawn that charge carriers bound to Li centres do not have a small-polaron character.

Some of the conclusions arrived at by Kabashima are disputable. First of all, d.c. conductivity in the two lightest doped samples is most likely not determined by Li centres, because its activation energy is much too high (0.50 ev). The relevant centres in these eases are probably niclcel vacancies. The Li acceptors as introduced in the samples only reduce the compensation degree of the deeper V~i acceptors, thus leading to an increase of conductivity which remains, however~ determined by the nickel vacancies (see § § 5.1.1 and 4.3). The conclusions drawn concerning a.e. conduction therefore probably apply to this centre. An accurate estimate of ~ from the data given for a(v) b y Kabashima is in our opinion hindered by the spread of a(v) values at the highest frequencies. In any case the temperature-independent value as derived by Kabashima for between I00°K and 200°K is much the same as that estimated by Snowden for the case of nickel vacancies at 300°K. The VNi concentration estimated with the aid of eqn. (111) is also about the same (_~ 10 i9 em-a). I t would appear from the above that the dipole-relaxation process correlated with nickel vacancies between 100°~: and 300°K is characterized by

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a constant relaxation time of about 3 × 10 -1° sec. As a consequence of a constant relaxation time, the temperature-independent a(V ) behaviour then suggests tha t the magnitude of the relaxation process is likewise insensitive to variation of temperature. I t follows then that the estimate of the concentration of VN~ centres with the aid of equ. (111) can be incorrect. The fact that at relatively high temperatures (_~ 300°K) the relaxation mechanism corresponding to nickel-vacancy centres is totally independent of temperature is at variance with the behaviour of Li- centre dipole relaxation. The almost temperature-independent behaviour of Ae as well as of ~ up to relatively high temperatures suggests tha t for the neutral VNi centre disorder leads to an energy spread which is much larger than that found for the Li centre. The marked influence of disorder is probably inherent in the presence of two equivalent bound charge carriers in one centre. I t will furthermore be seen in § 5.3 that at temperatures above 100°K the V~¢ i relaxation process is slow compared to tha t corresponding to a Li centre.

D.C. conduction in the highest Li-doped sample considered by Kabashima is quite probably determined by Li centres because its activation energy has a value near to that expected (0.36 ev, see § 3.3). Evi- dently in this case the Li content has surpassed the counterdope, which is then about 0.01 at. %. I t is to be expected, however, tha t a.c. conduction will be determined by (uncompensated) nickel-vacancy centres as well as by (partially compensated) Li centres. I t can be seen in Kabashima's results that the behaviour of a.c. conduction for this sample above 125°K deviates from that of the other two. I t might even be suggested that this different behaviour is due to the presence of two relaxation processes with different relaxation times and dissimilar contributions to the relaxation effect. These different processes would then be correlated with the two types of centres present.

Aiken and Jordan ( 1968) investigated the b ehaviour of a.c. conductivity up to frequencies of l0 s HZ in non-deliberately doped single crystals of NiO prepared either by halide decomposition or by a flame-fusion method. A.C. conductivity of the former type of crystals varied in proportion to v 2 and between 20°e and 500°e had a temperature-independent value which was about the same as tha t found by Kabashima at the corresponding frequencies for his lightest-doped sample. In this case conductivity is attributed to the presence of isolated nickel-vacancy centres. These results suggest tha t the VN~-dipole relaxation mechanism remains temperature-independent up to very high temperatures. A.C. conduc- t ivity of crystals prepared by flame fusion was investigated up to a frequency of l0 s HZ and at temperatures between 150 and 300°K. In these crystals a(v) proved to vary only weakly with frequency, pointing to the occurrence of relaxation processes with a large spread of relaxation times. I t is shown by Aiken tha t these processes quite probably correspond to hopping o f ' bound ' charge carriers predominantly between pairs of nickel vacancies. The varying distance between the centres of these

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pairs brings about the spread of relaxation times. In the theoretical model as given by Aiken the presence of partially compensating donors is essential. I t may be noted that in this model a(v) increases linearly with counterdope. The inverse relation has been observed for the case of Li-dipole relaxation (see § 5.1.1).

5.3. Optical Absorption in NiO and CoO (4 × 1011-6 × I014 HZ)

An important contribution to knowledge of the very-high-frequency behaviour of non-deliberately doped and Li-doped NiO and CoO crystals has been made by Austin and co-workers. In their first paper (Austin, Clay, Turner and Springthorpe 1968) they present optical-absorption data, obtained from transmission measurements in the wavelength region 0.5 to 700 ~m, for an undoped NiO sample and a Li-doped CoO sample at 290°K and for Li-doped NiO (0.2 at. °/o ) at temperatures between 100 and 475°K. In comparison with undoped material, an extra absorption is found in Li-doped samples, which rises with frequency up to a maximum at 1.2/~m (_ 1 ev) followed by a further increase at smaller wavelengths. The behaviour of Li-induced absorption, up to and including the maximum, is concluded to be in agreement with that expected for small polarons with a binding energy (Eb) of 0.5 eV and a coupling constant y of 6 (see § 2.5). I t is then stated that charge carriers in NiO and CoO are small polarons, some of the carriers being bound to impurity centres.

In a subsequent paper (1968) Austin, Clay and Turner present absorption data in the same wavelength region for undoped and Li-doped NiO (0.03 to 0.54at. %) and CoO (0-13 and 0.39 at. %) at temperatures between 100 and 570°K. In the interpretation of absorption-coefficient (K) data, Austin distinguishes between two regions : (1) The far-infra-red region (KFI ; 100-700tzm), at frequencies suffi-

ciently far below the optical lattice-vibration region (_ 10 TM ~z). (2) The near-infra-red region (KNI ; 0.5-10/~m) at frequencies above

that of the optical lattice vibrations. The relevant absorption coefficients KF~ and KNI can be converted into conductivities asi and asi with the aid of relation (69) (see § 2.5). I t is concluded that the Li-induced part of aFI for NiO as well as CoO is nearly frequency-independent and increases with temperature and Li concentration. The frequency-independent behaviour of aFi is explained as being due to dielectric relaxation of Li-centre dipoles with a single relaxation time ~ at frequencies such ~hat w~>> 1. The increase of aFi upon rise of temperature should then [~e due to the diminution of • (see eqn. (111)) pointing to a thermally activated character of Li-centre dipole reorientation. Ionization of the Li centres can only be expected to become noticeable at the highest temperatures considered. The behaviour of dipole relaxation strictly determines aFi-aac. A plot of this quantity versus 1/T, with aFI taken at a wavelength of 700 ~m, leads to activation energies 0.1-0.16 ev and 0-2-0.3 ev for NiO between 200 and

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300°K and CoO between 300 and 500°K, respectively. The higher limit as indicated for the activation energies corresponds to lower Li concentrations. The decrease of activation energy upon increase of Li dope is suggested to be due to interaction between the centres. The thermally activated charact~er of Li-centre dipole reorientation is considered as evidence of the small-polaron character of charge carriers bound to Li

> 1 centres in NiO and CoO. The high-temperature ( T ~ 0 ) activation energies U are estimated to have values some 30-50% higher than those mentioned above (see § 2.1).

The behaviour of near-infra-red conductivity aNi is explained by Austin as being roughly consistent with that of far-infra-red conductivity ari. I t is argued that if the Li-induced part of GEl is dominated by hopping of small polarons around Li centres, Li-induced (rji tOO should be determined by this conduction mechanism. Within the validity of the model as given by l~eik (see § 2.5) one should then expect to find that aNi shows a max imum at a frequency such that }l¢0ma x ---- 4 U , while aNI ' max varies directly as the Li concentration and is fairly insensitive to a change of temperature. A maximum in Li-induced aNi for Ni0 is indeed found at ]/OJmax=4U=0"8-1"0 ev (1.5-1.2/zm), while the further behaviour of aNi is roughly consistent with the model as given by Reik. A confirmation of the validity of the model employed is derived from the observation that in NiO at temperatures above 350°K, aNi ' max/Tadc is thermally activated with an energy of 0.2 ev, i.e. equal to the hopping energy U. The general conclusion is then reached that optical absorption in the whole frequency region considered is primarily due to small polarons hopping around Li centres with high-temperature activation energies of 0.20-0.25 ev and 0.30-0.40 ev for NiO and CoO, respectively.

In the opinion of the present authors, the optical absorption data for the case of NiO provide convincing evidence in favour of the small- polaron character of charge carriers bound to Li centres. In some respects, however, the interpretation of optical absorption data, as presented by Austin, may be subjected to criticism.

I t follows directly from the known electrical transport properties of Li-doped NiO that, up to the highest temperatures considered in the optical absorption measurements, the concentration of free carriers remains small compared to that of bound carriers. Therefore the above- mentioned evidence has reference only to bound charge carriers. Employ- ment of Reik's formula, giving a relation between a j i ' m a x and ado, iS allowed only if the two quantities refer to the very same charge carriers (see § 5.2). Significant information from the experimental relation between aNi and a d o for the case of Li-doped NiO where aNi is determined by bound and aac by free carriers, cannot therefore be obtained.

Li-induced optical absorpt ion in NiO reaches a (weak) maximum at 1.2 tLm, but then at shorter wavelengths sharply increases again. Such a behaviour is not in conformity with that expected for small-polaron conduction. In that case absorption beyond the maximum should

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decrease continuously. I t is felt therefore that alternative explanations of the frequency behaviour of absorption should be considered carefully. Ksendzov et al. (1967) have shown that ' Li-induced absorption ' in the near-infra-red and visible region depends markedly on the circumstances under which the crystals are prepared. Therefore, the value for the small-polaron hopping energy of bound carriers, derived from the position of a Li-induced absorption peak in the near infra-red, may be subject to appreciable error.

Using Austin's absorption data for NiO with a Li dope of 0.2 at. %, we conclude to a value for the activation energy determining (aFi-aae) behaviour which is about a factor of 2 lower than that arrived at by Austin valid for temperatures down to 200°K. Analysis of (aF~--%~) data below 200°K is severely hampered by contributions to conduction originating from centres other than the Li centre. These centres probably correspond to nickel vacancies, their contribution to aFI being not very much different from those reported by Snowden and by Kabashima (see § 5.2). A combination of data obtained from far-infra-red-optical absorption and dielectric-loss measurements can yield an estimate of the relaxation-time behaviour for the Li-dipole centre. I t seems reasonable to assume that at the relatively high temperatures considered ( > 200°K), Li-dipole relaxation is determined by a single relaxation time r. The estimate of ~ behaviour may therefore be based on the equation :

Ae . . . . . . (112)

~Tr (YFI"

The result of this estimate is shown in fig. 28. I t has already been remarked (see § 5.1.1) that for the Li centre, T behaviour determined from optical absorption and dielectric-loss data in the whole temperature region (4"2-300°K), is consistent with small-polaron hopping of bound charge carriers. I t can furthermore be seen in fig. 28 that, where there are equal concentrations of Li and VNi centres, aFi below 100°~ will be dominated by nickel vacancies and above 1005~ by Li centrcs.

The r values derived above allow an estimate of the mobility of charge carriers bound to the Li-centre (eqn. (15)). We arrive at a value of about 10 -2 cm~/v see at 300°K, which is two orders of magnitude below the corresponding value for free-carrier mobility in Ni0 (see § 3.4).

§ 6. ~)ISOUSSION

In the following discussion we will deal mainly with the alternative : ' small-polaron versus large-polaron conduction ', paying special attention to RiO. This is because a greater variety of data is available on Ni0 than on the other oxides. In this paper we have arrived at the conclusion that transport properties of free charge carriers in Ni0 can be described on the basis of a lar~le-polaron band model~ whereas charge carriers bound

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to centres have to be considered as small polarons. In the following, a critical discussion of this conclusion is presented.

The assumption tha t conduction in p-typ~ NiO can be described on the basis of a semiconductor model, where the width of the valence band is sufficiently large compared to k T in the range considered, does not in advance seem unreasonable. Data concerning optical absorption (Newman and Chrenko 1959), photoconductivity and intrinsic conduction (Ksendzov and Drabkin 1965) lead to the inferences tha t

(1) NiO behaves as a semiconductor with an energy gap of about 3.7 ev at 300°K ;

(2) the total width of the bands involved in intrinsic conduction has a value of some electron-volts.

Electroreflectance studies (McNatt 1969) suggest tha t the valence band, i.e. the band involved in the intrinsic absorption process starting at an energy of 3.7 ev, has some critical points in the density of states and extends over more than 1 ev. Photoemission measurements (Chcrkashin, Vilosov, Keier and Bulgakov 1969) show tha t the edge of the valence band, interpreted as a 3d nickel band, lies 4.9 ev below the vacuum level, while its width is some ev. The band edge found at 9 ev below the vacuum level is at tr ibuted to a 2p oxygen band.

The fact tha t NiO behaves as a semiconductor and not as a metal can be explained in terms of electron-correlation effects, which lead to a splitting of the partially filled total nickel 3d band. The theoretical aspects of band structure have been reviewed by Adler (1968).

The features of transport properties of free carriers tha t led us to the assumption of the large-polaron band model will now be critically surveyed.

(1) At high temperatures (_~1200°K), the drift mobility /~D could be firmly established to have a value of about 0.4 cm~/v sec. In a ' broad-band ' model, such a value for ~D although marginally small, seems still admissible (see § 2.1.1).

(2) The density of states of the valence band N v at high temperatures ( -~ 1000°K), determined from the Seebeck coefficient a, is significantly smaller than the maximum value of 2N 0. Moreover, the temperature dependence of a at temperatures above 1000°K points to an increase of N v with temperature.

(3) On the basis of a simple isotropie band model the assumptions tha t IVv oc T 3/~ and tha t the heat-of-transfer constant A ~ 2 lead to the estimates for the large-polaron effective mass mp*_ 6m o and for the large-polaron band width Wp_0.6 ev. The corresponding width of the rigid band AW~0.8 ev. Larger values for the band width would result if the effective mass were anisotropic. A value of about 1 ev for the width AW of the valence band is not at variance with the optical data mentioned above. An estimate of the electron-transfer integral J from antiferromagnetic coupling (Appel 1966) leads to 0.1 ev~<J<0.4 ev and to a rigid band width 1.2 ev~<AW~<4.8 or.

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(4) As a consequence of the above assumption, N v oc T 3/2, the value of ~ obtained from a and ~ data, increases continuously from 0.4 cm~/v sec at 1200°K to 4 cm2/v sec at 300°K. With a value of 6 for mp*, the criterion 1 > a in the broad-band approximation leads to a lower limit for ~D of about 2 cm2/v sec at room temperature (see eqn. (8)). This illustrates the applicability of the broad-band model.

(5) The magnitude as well as the temperature dependence of/z D can be satisfactorily described in terms of polar optical-mode scattering and/ or spin-disorder scattering if it is assumed that the kinetic effective mass equals the density-of-states effective polaron mass rap*___ 6m 0. A value for the heat-of-transfer constant A _ 2 is indeed consistent with these types of scattering. For the former mechanism electron- lattice coupling is of intermediate strength (a___ 2.5), for the latter the exchange coupling constant F' should have a value of about 0-6 ev. Such a value for F' seems not unreasonable in view of the value of about 0.4 ev derived for the exchange integral for antiferromagnetic coupling (Appel 1966).

Within the band approximation, magnetic exchange interaction between charge-carrier spin and cation spins is relatively strong and even augments in the cases of CoO and a-F%0 a. I t seems therefore not improbable that a final explanation of ~t D behaviour in these oxides will be based on the magnaron rather than on the small or nearly-small polaron concept. I t is then also conceivable tha t the relatively low value found for/~H compared to that of tZD, for NiO below 400°K and for CoO at all temperatures considered, is inherent in the situation of large effective masses and strong exchange interaction between charge-carrier spin and cation spins.

In the above quantitative interpretation of ~ and ~ in terms of a ' broad-band ' model, a decisive role is played by the proposed behaviour for the density of states : N v oc T a/2, as well as by the assumption for the heat-of-transfer constant : A _ 2. First of all, it should be unambiguously ascertained from direct experimental methods whether /~D has indeed a value of about 4 cm~/v sec at room temperature, which value is a direct consequence of the assumption : N v oc T a/~. Furthermore, it should be remarked tha t the assumption A _~ 2 can be disputed on the basis of data obtained in the partially compensated region. I t was concluded from ~ behaviour in this temperature region that, for compensation degrees K of 4 and 10%, the assumption A _~ 2 leads to a multiplicity factor for the Li acceptor of g_~ 1. The former estimate for K stems from chemical analysis while the latter value has been derived from an analysis of ~ data in the high-temperature region. A value of about ~unity for the g factor seems to be at variance with the conclusion drawn from dielectric- loss measurements, viz. tha t the spread of the energy levels corresponding to the Li centre is of the order of 1 mev. Such a small spread of the energy levels, pointing to the small-polaron character of the bound holes, implies tha t at the temperatures considered in the ~ measurements, the Li centre

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should be considered to be degenerate. The degeneracy is determined by the number of equivalent cations neighbouring the Li ion and by spin multiplicity. One may thus expect g ~o have a value of the order of magnitude 10. For such a value of g and with a compensation degree of 5 to 10~o it would then follow that A is practically zero. A solution A _~ 2 would only be consistent with a compensation degree K of 0.5 to l°/o . For our samples there are no indications that K would have such low values. Chemical analysis cannot easily lead to accurate values for K, while the analysis of a data at high temperatures can lead to erroneous results due to neglect of the effect of excited states for the Li centre. There is, however, another indication for a value of about 10~o for K. I t was concluded from the behaviour of the activation energy for conduction Q as a function of dope tha t for ILl] ~0.01 at. ~o, the Li aceeptor is totally compensated. This implies that the donor concentration is about 0.01 at. %. I f the same donor concentration is present in samples with higher Li dopes, then for the sample considered here, which had been doped with 0.1 at. o/o Li, one arrives at a value K_~10%. An unambiguous determination of the compensation degree would still be desirable. In this respect it may be remarked tha t photoemission measurements (Cherkashin 1969) can lead to an estimate of K. Using this method on his Li-doped samples, Cherkashin arrived at values for K smaller than 1~o. In another method, K may be estimated from counterdoping experiments. A careful study of ~ behaviour at low temperatures in samples whose dope and counterdope have been well established, should lead eventually to a reliable value for the combined quanti ty g exp A.

The above considerations may certainly be regarded as a criticism of the employment of the band model. A value A = 0 would mean that transport of kinetic energy is negligible, implying in the relevant temperature range (T > 200°Z) tha t the band width is small compared with kT. This would also dispose of the assumption NvocT a/u. I t is, however, remarkable that in the range 200°K < T < 330°K the activation energy for the Hall coefficient E(RH) is significantly larger (~< 0.05 ev) than tha t for the Seebeck coefficient E(a). This difference in activation energies can be accounted for if N v oc T a/~. In that case, in the partially compensated region, where RH-locpocNv and ~ocln (Nv/p) , E(RH) should be apparently enlarged by about 0-03 ev. Alternative explanations of this different behaviour of R H and ~ cannot, however, be excluded. The anomalous Hall coefficient, dominating at temperatures above 500°K, might contribute even below 330°K to an enhancement of E(RH). Furthermore, it remains possible that impurity conduction induces a decrease of E(a) in the range of 170°K to 330°K.

With regard to charge carriers bound to centres, it has been concluded tha t these carriers should be considered as small polarons. This conclu- ion was based on a number of facts ;

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(1) The existence of dielectric loss of the Debije relaxation type due to the presence of neutral centres and the behaviour of this loss as a function of temperature.

(2) The occurrence, due to neutral centres, of optical absorption in the near infra-red characteristic of small polarons.

(3) The relatively low value for impurity conduction.

I t was deduced from dielectric-relaxation data tha t the energy spread of the levels corresponding to the Li centre is of the order of 1 mev. The absence down to the lowest temperature of a contribution to dielectric loss due to ~'esonance absorption led to the inference tha t this energy spread arises from locally varying electric fields. The spread of the levels due to resonance should then be smaller than 1 mev. This led to the estimate tha t the intra-centre electron-transfer integral J is less than 0.1 ev. Furthermore the three facts mentioned above led to about the same value for the intra-centre electron-lattice coupling constant, viz. ~ _ 6 .

I t may be wondered whether, on the one hand, bound charge carriers can behave as small polarons whereas, on the other, free charge carriers can have large-polaron character. Such a proposition does not seem unfeasible having regard to the different situations for the charge carriers :

(1) At the centre the wave function of the charge carrier becomes markedly polarized by the presence of the effective negatively charged Li + ion. This can be imagined to introduce an appreciable reduction of the intra-centre electron-transfer integral J .

(2) The energetic condition for the formation of a small polaron at the centre seems to be less stringent than that for the free charge carrier. For the free charge carrier this condition in a one-dimensional model is < 1 J N ~ U . For the three-dimensional case, where the number of nearest neighbours is a factor of 3 larger, this condition becomes more stringent : J ~ U. On the other hand, for the charge carrier bound to the Li centre, where the number of nearest neighbours is a factor of 3 less than in the free situation, the small-polaron condition becomes correspondingly less stringent.

In the case of free charge carriers, the assumption of large-polaron band conduction led to a rigid electron-transfer integral J > 0 . 0 7 e v . An electron-lattice coupling constant 7_~ 6, i.e. the same value as found for bound charge carriers, would correspond to U_~ 0.2 ev. I t follows then tha t J should be smaller than or equal to 0.02 ev if the small-polaron condition J ~ U would hold for free carriers too. The large-polaron description of free charge carriers seems therefore consistent with the small-polaron character of bound carriers, because in the former case the small-polaron condition is not fulfilled.

The exclusively small-polaron character of bound charge carriers would imply that photo-ionization of the Li centres takes place at a higher energy than for the corresponding thermal process. For Li-doped NiO

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an absorption at 0.43 ev has been interpreted by Ksendzov (1968) as being due to photo-ionization of the Li centres. This process would then call for an energy about 0.1 ev larger than for thermal ionization. I t would be recommendable, on the one hand, to establish whether the absorption observed does in fact lead to photo-conduction, and, on the other, to have an adequate model predicting the magnitude of this energy difference in terms of small-polaron binding energy.

We shall now consider how far the features of free-carrier transport properties are compatible with a small-polaron model. We should then distinguish between adiabatic and non-adiabatic small-polaron transport. The condition for adiabatic processes: J2>~(UkT/~r)l/zhOJo/~r combined with the small-polaron condition: J ~ U requires an electron-lattice coupling constant y which is definitely larger than 10, i.e. appreciably larger than the value of 6 found for charge carriers bound to the Li centre. There are no indications of such a strong coupling. I t seems therefore that the non-adiabatic small-polaron model only is adequate for a confrontation with the features of transport properties : (1) A value for ~D of 0"4 cm2/V see at 1200°K seems to be too high to be

compatible with non-adiabatic small-polaron transport. High- temperature data for ~ and ~ (T>500°K) allow for an activation energy U ~< 0.15 ev. This means with hw 0_~ 0.07 ev tha t 7 ~ 4. The small-polaron condition J ~ U implies tha t J~<0.02 ev. I t then follows from numerical calculations (see § 2.1.2) that t~D at 1200°K can have a value which is at the most an order of magnitude lower than the experimental value of 0.4 cm2/v sec. The same result follows if one applies the upper limit for the hopping model at this temperature (see eqn. (17)) : ~ D ~ 0 - 1 em2/v see.

(2) Within the small-polaron model (A=0), the experimental result Nve ~ ~ 0-3iV o at 1000°K means that N v _ 0.3N o. I t might be thought that the decrease of iV v below the maximum value of 2iV 0 is due to energy differences between the sites for the small polaron. The maximum spread of the energy levels should then be at least 0.1 ev. One would then also expect an increase of/V v with temperature, in agreement with experimental indications. The energy spread can be thought to be connected with locally varying electric fields induced by randomly distributed charged Li centres, exhausted at high temperatures, and by the charged counterdope centres. In a more or less analogous model, excited states of the Li centre would lead to an effective reduction of N v. Some facts, however, seem to militate against the validity of such a model. Firstly, MnO, which seems to be a small-polaron semiconductor, does not show an appreciable reduction of N v below the maximum value. Secondly, it might be expected within such a model that 1V v would have the maximum value (2/Y0) at very low dope and then decrease with increasing dope. However, in the case of NiO and also in CoO, the

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(3)

value for N v proves to be independent of dope within a fairly wide range : 0.005 at 5 at. %. I t seems therefore that the reduction of N v below the maximum value is not consistent with the small- polaron concept. I t might be thought that the temperature dependence of /~D, as derived from a and Q with constant Nv, can be described in terms of small-polaron hopping above 500°K and of small-polaron band conduction below this temperature (see fig. 17). A similar description would then hold for CoO (see fig. 19). In such a description one should then: neglect the observed increase of 1V v with temperature and one should assume that the value of J exceeds the limit of abou~ 0.02 ev prescribed by the small-polaron condition (see point (1)), i.e. J_~ hWo _~ 0.07 ev. The high-temperature hopping regime would then imply that y _ 4 (see point (1)). A transition to a small- polaron band regime below 500°K seems impossible for two reasons :

(a) I f small-polaron conduction were possible, the transition temperature between hopping and band regime cannot be expected to lie at T > ½0 ~_ 400°K but rather at T < ~0_~ 200°K (see § 2.1.3).

(b) Even at the relatively low value of the coupling constant, small- polaron band conduction seems impossible due to locally varying electric fields between neighbouring sites. The average spread for the energy levels of neighbouring sites can be estimated from data on impurity conduction. In the temperature range considered (T>200°K), the average energy W D between majori ty centres with a concentration of 0.1 at. % varies between about 30 and 60 mev for compensation degrees between 1 and 10% (see § 3.2). This implies tha t the average energy W n between neighbouring sites varies between 1 and 3 mev. One should compare these values for WD with the small-polaron transfer integral J9 between neighbouring sites: J p = J exp (--ST) with S T = y eoth (½~O~o/kT). One arrives at values for Jp of 0.2 and 0.7 mev at temperatures of 500 and 300%~, respectively. I t appears that even for the exaggerated values of J , the ratio Jp/WD is significantly smaller than unity.

I t may be remarked that the value of 1 to 3 mev arrived at for W D between neighbouring lattice sites, corroborates the conclusion from dielectric-loss measurements that a W D term of such a magnitude should be realized at the centres. No appreciable decrease of W D due to an enlargement of the dielectric constant by the dipole contribution arising from neutral centres need be expected because, in the relevant temperature and dope range, this contribution is still small (see § 5.1.1). At higher temperatures (_~ 500°K) the effect on W D of an increase of charged majori ty eentres is probably cancelled by screening effects due to the enhanced charge-carrier concentration.

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I t may finally be concluded that NiO and CoO too do not exhibit the features characteristic of a small-polaron semiconductor. As regards MnO, on the contrary, available experimental facts are compatible with the small-polaron model. On the other hand, NiO and CoO can be conceived of, without serious difficulties, as large-polaron band semiconductors. However, definite evidence for this proposition does not yet exist. The final solution should probably be a compromise between the large and the small polaron, viz. the nearly-small polaron. According to Eagles (1966), in his t reatment of the adiabatic nearly-small polaron, there exists a critical value for the electron-transfer integral J such that for J > J ~ the weak-coupling large-polaron and for J < J o the nearly- small-polaron situation will be realized. The energetic condition for the nearly-small polaron : J ~ U is less stringent than that for the small polaron: J ~ U. An important distinction between the nearly-small and the large-polaron case is the fact that the polaron band width in the former case is narrowed to a considerable degree. I t is therefore of paramount importance to establish the value for the heat-of-transfer constant A, which one would expect to be zero in the nearly-small polaron case. I t is furthermore highly desirable to have an unambiguous direct experimental determination of the drift mobility at room temperature. In either the large or nearly-small polaron solution, the effect of a strong magnetic interaction between charge carriers and cations should be taken into account.

ACKNOWLEDGMENTS

The writing of this paper was activated by Professor Sir Nevill Mort, F.R.S. who asked the present authors to summarize all that had recently been done at this laboratory in the field of semiconducting transition-metal oxides. This paper, however, deals only with a part of this research. I t is hoped that the summary as presented here will be of general value.

Instructive discussions with Professor D. Polder and Dr. F. E. Maranzana are gratefully acknowledged. Thanks are due to Dr. S. van Houten and to Dr. C. Crevecoeur for their support with regard to the preparation of the samples.

The authors wish to express their gratitude to Dr. C. Crevecocur and Dr. H. J. de Wit for making available the results of their measurements on NnO before publication.

~EFE~ENCES

ACKET, G. A., and VOLOER, J., 1966, Physica, 32, 1543, Electric transport in n-type F%0 8.

ADLER, D., 1968, Solid State Physics, 21, 1, Insulating and metallic states in transition metal oxides, edited by F. Seitz, D. Turnbull and H. Ehrenreich (New York : Academic Press).

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

21:

43 2

2 M

ay 2

012


AIKEN, J. G., and JORDAN, A. G., 1968, J. Phys. Chem. Solids, 29, 2153, Electrical transport properties of single crystal nickel oxide.

APPEL, J., 1966, Phys. Rev., 141, 506, Remark on the hopping transition probability of polaron holes in Ni0 ; 1968, Solid State Physics, 21, 193, Polarons, edited by F. Seitz, D. Turnbull and H. Ehrenreich (New York : Academic Press).

ALI, M., ~FRIDMAN, M., DENAYER, M., and I~AGELS, P., 1968, Phys. Stat. Sol., 28, 193, Electrical conduction in Li-doped MnO single crystals.

AUSTL'¢, I. G., CLxY, B. D., and TURNER, C. E., 1968, J. Phys. C (Proe. phys. Sou.), 1, 1418, Optical absorption of small polarons in semiconducting Ni0 and CoO in the near and far infra-red.

AUSTIN, I. G., CLAP, B. D., TURNER, C. E., and SPRI~GT~ORP~, A. J., 1968, Solid St. Commun., 6, 53, Near and far infra-red absorption by small polarons in semiconducting i~i0 and CoO.

AUSTIN, I. G., and MOTT, N. F., 1969, Adv. Phys., 18, 41, Polarons in crystalline and non-crystalline materials.

AUSTIN, I. G., Sr~INGT~OR~E, A. J., and SMIT~, B. A., 1966, Physics Lett., 21, 20, Hopping and narrow-band polaron conduction in Ni0 and CoO.

AUSTIN, I. G., SPI~INGTtIORI~E, A. J., SMITH, B. A., and TURNER, C. E., 1967, Proe. phys. Sou., 90, 157, Electronic transport phenomena in single- crystal iNiO and CoO.

BOER, J. H. DE, and VERWE¥, E. J. W., 1937, Proc. phys. Soc. Lond., 49, 59, Semi-conductors with partially and with completely filled 3d-l~ttice bands.

BLAKEMOaE, J. S., 1962, Semiconductor Statistics (New York : Pergamon Press, Inc.).

BOSMAN, A. J., and CREVnCOEUR, C., 1966, Phys. Rev., 144, 763, Mechanism of the electrical conduction in Li-doped NiO ; 1968, J. Phys. Chem. Solids, 29, 109, Dipole relaxation losses in CoO doped with Li or Na ; 1969, Ibid., 30, 1151, Electrical conduction in Li-doped CoO.

BOSMAN, A. J., DAAL, H. J. VAN, and KNUVERS, G. F., 1965, Physics Lett., 19, 372, Hall effect between 300°K and 1100%: in NiO.

BOSMAN, A. J., and HOUTEN, S. VAN, 1965, Physics of Semiconductors, Proc. 7th Int. Conf., Paris 1964, p. 1203, Mechanical and dielectric rela:~ation in transition metal oxides.

BRANSK¥, I., and TALLAN, N. M., 1968, J. chem. Phys., 49, 1243, High-temperature defect structure and electrical properties of iNiO.

CHADDA, M. M., and SIN~A, A. P., 1963, Indian J. pure appl. Phys., 1, 161, Theory of thermoelectric power in low-mobility semiconductors.

CHERKASHIN, A. E., VILESOV, F. I., KEIER, N. P., and BULGAKOV, N. l~., 1969, Soviet Phys. solid St., 11, 506, Investigations of the spectrum of energy states of nickel oxide and LizNil_xO solutions by the photo- electric emission method.

CREV~COEUR, C., and WIT, H. J. DE, 1968 a, Solid St. Commun., 6, 295, Thermo- electric force versus electrical resistivity of MnO at ll00°C. An example of ' Jonker's pear ' ; 1968 b, Ibid., 6, 843, Dipole relaxation in MnO, doped with Li.

DA~m, H. J. vA~, 1965, Philips t~es. Pep. Suppl. No. 3, Mobility of charge carriers in silicon carbide.

DAAL, H. J. VAN, and BOSMAN, A. J., 1966, Physics Lett, 23, 525, Influence of native defects on transport properties of Li-doped l~iO, 1967, Phys. Rev., 158, 736, Hall effect in CoO, NiO and a-F%0 3.

DELACOT~, G., and SCHOTT, M., 1966, Solid St. Commun., 4, 177, Hall mobility of holes and valence band width in anthracene.

A.P. H

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

21:

43 2

2 M

ay 2

012

114 A . J . Bosman and H. J. van Dual on

DIxoN, J. R., 1959, J. appl. Phys., 30, 1412, Anomalous electrical properties of p-type indium arsenide.

EMIN, D., and HO]~STEIN, T., 1969, Ann. Phys., 53, 439, Studies of small polaron motion. IV : Adiabatic theory of the Hall effect.

EAGLES, D. M., 1966, Phys. Per., 145, 645, Adiabatic theory of nearly small polarons.

FEYNMAN, ~. P., HELLWARTH, I~. W., IDDINGS, C. K., and PLATZ~AN, P. M., 1962, Phys. t~ev., 127, 1004, Mobility of slow electrons in a polar crystal.

Fmsov, Yu. A., 1964, Soviet Phys. solid St., 5, 1566, Theory of the Hall effect in low-mobility semiconductors.

FISH~R, B., and W~G~ER, Jr, J. B., 1966, Physics Lett., 21, 606, On the problem of doping semiconducting oxides.

FISHER, B., and TAN~AUSER, D. S., 1966, J. chem. Phys., 44, 1663, Electrical properties of cobalt monoxide.

F~IEDMA~, L., 1963, Phys. Rev., 131, 2445, Hall effect in the polaron-band regime ; 1964 a, Ibid., A, 133, 1668, Transport properties of organic semiconductors ; 1964 b, Ibid., A, 135, 233, Density matrix formulation of small-polaron motion.

FRIEDMAN, L., and HOLSTEIN, T., 1963, Ann. Phys., 21, 494, Studies of polaron motion. Part I I I : The Hall mobility of the small polaron.

FRITSC~n, H., and CuEvAs, M., 1960, Phys. Rev., 119, 1238, Impurity conduction in transmutation-doped p-type germanium.

FRSHLICE, H., 1954, Adv. Phys., 3, 325, Electrons in lattice fields; 1957, Archives des Sciences, 10, 5, Pertes Debije duns les solides ioniques ; 1958, Theory of Dielectrics (Oxford University Press).

F~SHLICH, H., M~cm~uP, S., and M~T~A, T. K., 1963, Phys. Kondens Mater., l , 359, Low mobility materials and Debije loss due to electrons.

FR51tLICH, H., and SEWELL, G. L., 1959, Proc. phys. Soc. Zond., 74, 643, Electric conduction in semiconductors.

F~xIME, S., MURAKAMI, M., and Hn~nA~A, E., 1961, J. phys. Soc. Japan, 16, 183, Hall effect on some of the magnetic compounds.

GERTHSE~, P., KAUER, E., and REIK, H. G., 1966, FestlcSrperprobleme, Band V, 1 (Braunschweig : Friedr. Vieweg & Sohn GmbH), Halbleitung einiger ~bergangsmetalloxide im Polaronenbild.

GVISHI, ~., T~LA~, ~. M., and'TAN~HAUSER, D. S.~ 1968, Solid St. Commun., 6, 135, The Mall mobility of electrons and holes in MnO at high temperatures.

HAAS, C., 1968, Phys. l~ev., 168, 531, Spin-disorder scattering and magneto- resistance of magnetic semiconductor.

HED, A. Z., and TANNHAUSER, D. S., 1967, J. chem. Phys., 47, 2090, High- temperature electrical properties of manganese monoxide.

HEIKES, R. R., 1961, Thermoelectricity, edited by R. R. Heikes and W. U. Ure (New York: Interscienee), Chap. IV, ~arrow band semiconductors, ionic crystals and liquids.

HWIKES, R. R., and JOHNStOn, W. D., 1957, J. chem. Phys., 26, 582, Mechanism of conduction in Li-substituted transition-metal oxides.

HERRING, C., 1960, Proceedings of the International Conference on Semiconductor Physics, Prague, 1960, p. 60, The current state of transport theory.

HOLSTERS, T., 1959 a, Ann. Phys., 8, 325, Studies of polaron motion. Part I : The molecular-crystal model ; 1959 b, Ibid., 8, 343, Studies of polaron motion. Part II : The ' small ' polaron.

HOLSTEIN, T., and FRIED~A~V, L., 1968, Phys. Rev., 165, 1019, Hall mobility of the small polaron. II.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

21:

43 2

2 M

ay 2

012


HOUTEN, S~ VAN, 1960, J. Phys. Chem. Solids, 17, 7, Semiconduction in LixNi(l_x)O ; 1962, Ibid., 23, 1045, Mechanical losses in Li-doped NiO semiconductors.

HOUTEN, S. v~-~, and BOSMAN, A. J., 1964, Informal Proceedings of the 1st Buhl International Conference on Transition Metal Compounds, Pittsburgh, 1963, p. 123, Mechanical and dielectric relaxation in transition metal oxides.

JOHNSTON, W. D., ttEIKES, R. R., and SESTRIC~, D., 1958, J. Phys. Chem. Solids, 7, 1, The preparation, crystallography, and magnetic properties of the LixCol_x0 system.

JONKE~, G. H., 1962, Physics of Semiconductors, Proceedings of the 6th Inter- national Conference, Prague, 1962, p. 864 ; Energy levels of impurities in transition metal oxides; 1968, Philips Res. Re19., 23, 131; The application of combined conductivity and Seebeck-effcct plots for the analysis of semiconductor properties.

JONKER, G. H., and I-IouTEN, S. van, 1961, Halbleiterlgrobleme V1 (Braunschweig : Verlag Friedr. Vieweg & Sohn), p. 118, Semiconducting properties of transition metal oxides.

KAB.~SP, I ~ , S., and K~wAxv~O, T., 1968, J. l~hys. Soc. Ja2an, 24, 493, High frequency conductivity of Ni0.

K~.INGER, ~. 1., 1963, Physics Lett., 7, 102, Quantum theory of non-steady-state conductivity in low-mobility solids; 1965, Phys. Stat. Sol., 11, 499, On the theory of transport phenomena in low-mobility crystals (I) ; 1968, Rep. Prog. Phys., 30, 225, Electrons in low-mobility and disordered semiconductors ; theory of transport and related phenomena.

KLIN(~R, M. I., and BLAX_~rR, E., 1969, Phys. Stat. Sol., 31, 515, On specific mechanism of absorption of electromagnetic waves by impurity polarons.

Koors, C. G., 1951, Phys. Rev., 83, 121, On the dispersion of resistivity and dielectric constant of some semiconductors at audiofrequencies.

KSENDZOV, YA. M., ANSEL'M, N. L., Vxsrsv.vA, L. L., and LAT:ZS~EVA, V. M., 1963, Soviet Phys. solid St., 5, 1116, Investigation of the carrier mobility in Ni0 with Li impurity.

KS~NDZOV, YA. ~., AW)~.ENXO, B. K., and M_~a~ov, V. V., 1967~ Soviet Phys. solid St., 9, 828, Semiconductor properties of single crystals of nickel oxide.

KS~NDZOV, Y~. M., and DRABKI~, I. A., 1965, Soviet Phys. solid St., 7, 1519, Forbidden-band width of nickel monoxide.

LA~G, I. G., and Fnzsov, ¥u. A., 1963, Soviet Phys. solid St., 16, 1301, Kinetic theory of semiconductors with low mobility ; 1964, Ibid., 5, 2049, Mobi- lity of small polarons at low temperatures.

LEE, T. D., Low, F. E., and PINES, D., 1953, Phys..Rev., 90, 297, The motion of slow electrons in a polar crystal.

Low, F. E., and PriEs, D., 1955, Phys. Bey., 98, 414, Mobility of slow electrons in polar crystals.

McK~zI~, H. L., and O'KEEFFE, M., 1967, Physics Lett., A, 24, 137, High temperature Hall effect in cuprous oxide.

MCN~TT, J. L., 1969, Phys. Rev. Lett., 28, 915, Electroreflectance study of NiO. MAKe_ROY, V. V. KSENDZOV, YA. M., and KRVGLOV, V. I , 1967, Soviet Phys.

solid St., 9, 512, Drift mobility of carriers in Ni0. MARANZXNA, F. E., 1967, Phys. Rev., 16O, 421, Contributions to the theory of

the anomalous Hall effect in ferro- and anti-ferromagnetic materials. MILLV.R, A., and ABRXgAMS, E., 1960, Phys. Rev., i20, 745, Impurity conduc-

tion at low concentrations. MZTOFF, S. P., 1961, J. chem. Phys., 35, 882, Electrical conductivity and thermo-

dynamic equilibrium in Ni0. H 2

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

21:

43 2

2 M

ay 2

012

116 A . J . :Bosman and H. J. van Daal on

Mo~IN, F. J., 1950, Phys. Rev., 78, 819, Magnetic susceptibility of ~-F%08 and ~-F%0 3 with added titanium; 1951, Ibid., 83, 1005, Electrical properties of ~-F%0 8 and ~-F%O 3 containing titanium ; 1954 a, Ibid., 93, 1195, Electrical properties of ~-F%Oa; 1954 b, Ibid., 93, 1199, Electrical properties of NiO.

MOTT, N. F., and Twos•, W. D., 1961, Adv. Phys., 10, 107, The theory of impurity conduction.

I~,~OHMAN, M., P0~]~scu, F. G., and RVTT~.R, J., 1965, Phys. Stat. Sol., 10, 519, Hall effect in NiO.

:NAGELS, P., and DENAY~R, M., 1967, Solid St. Commun., 5, 193, Hall-effect in Li-doped MnO.

NAGELS, P , DENAY~R , M., DE WIT, H. J., and CREV~CO~UR, C., 1968, Solid St: Commun., 6, 695, Comments on ' The Hall m0bility of electrons and holes in MnO at high temperatures ' by Gvishi et al.

N]~EL, L., 1949, Ann. Phys. (Paris), 4, 249, Essai d'interprdtation des propridtds magn6tiques du sesquioxyde de fer rhomboddrique ; 1952, C. r. hebd. Sganc. Acad. Sci., Paris, 234, 2172, Etude thermomagn~tique d'un monocristal de F%Oa~.

:N]~wMA~, R., and CHRENKO, R. M., 1959, Phys. l~ev. , 114, 1507, Optical properties of nickel oxide.

PL~NDn, J. N., MANSU~, L. C., MITRe, S. S., and C~ANG, I. F., 1969, Solid St. Commun., 7, 109, Reststrahlen-Spectrum of MnO.

PUTLEr, E. H., and I~TCH]~LL, W. H., 1958, Proc. phys. Soc. Lond., 72, 193, The electrical conductivity and Hall effect of silicon.

REIK, H. G., 1963, Solid St. Commun., 1, 67, Optical properties of small polarons in the infra-red; 1967, Z. Phys., 203, 346, Optical effects of small polarons at high frequencies with an application to reduced strontium- titanate.

R~IK, H. G., and HEESE, D., 1967, J. Phys. Chem. Solids, 28, 581, :Frequency dependence of the electrical conductivity of small polarons for high and low temperatures.

ROLLOS, M., and NAGELS, P., 1964, Solid St. Commun., 2, 285, Electrical measurements on CoO and NiO single crystals.

RUePRV.CHT, H., 1958, Z. Naturf. A, 13, 1094, Zum anomalen Temperaturverlauf des Hall-Koeffizienten yon schwach p-dotierten InAs.

SC~NXKV,~B]~RG, J., 1965, Z. Phys., 185, 123, The Hall coefficient of the small polaron ; 1968 a, Ibid., 208, 165, Temperature dependence of the small polaron Hall mobility; 1968 b, Phys. Stat. Sol., 28, 623, Polaron impurity hopping conductivity.

SCHOTT~, K. D., 1966, Z. Phys., 196, 393, The thermoelectric properties of the small polaron.

S]~W]~LL, G. L., 1963, Phys. Rev., 129, 597, Model of thermally activated hopping motion in solids.

S~]~LYY~, A. I., ARTE~OV, K. S., and S~V~IKo-SHw~KOVSK~I, V. E., 1966, Soviet Phys. solid St., 8, 706, Electrical properties of cobaltous oxide single crystals at high temperatures and their dependence on the partial pressure at oxygen.

S~OWD]~N, D. P., and S~SBUR~, H., 1965, Phys. Rev. Lett., 14, 497, Hopping conduction in NiO.

SPRINGTI~O~]~, A. J., AUSTIN, I. G., and S~ITH, B. A., 1965, Solid St. Commun., 3, 143, Hopping conduction in LixNii_xO crystals at low temperatures.

SUSSM~N, J. A., 1962, Proc. phys. Sot., 79, 758, Electric dipoles due to trapped electrons.

TXLLX~, ~. M., and T~HXVS~R, D. S., 1968, Physics Lett. A, 26, 131, The Hall mobility of NiO at elevated temperatures.

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nloa

ded

by [

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ford

Uni

vers

ity L

ibra

ries

] at

21:

43 2

2 M

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012


VERNON, M. W., and LOVELL, M. C., 1966, J. Phys. Chem. Solids, 27, 1125, Anomalies in the electrical conductivity of nickel oxide above room temperature.

VEaWEY, E. J. W., 1951, Semiconducting Materials (London: Butterworths Scientific Publications, Ltd,), p. 151, Oxidic semiconductors.

VOLGE•, J., 1950, Phys. Rev., 79, 1023, :Note on the Hall potential across an inhomogeneous conductor; 1957, Discuss. Earaday Soc., 23, 63, Dielectric loss in insulating solids caused by impurities and colour eentres.

Vos, W. J. DE, and VoLo]~, J., 1967 a, Physiea, 34, 272, A new investigation of dielectric relaxation processes in smoky quartz crystals; 1967 b, Physics Lett. A, 24, 539, Dielectric relaxation processes in smoky quartz crystals at very low temperatures.

WAG~ER, C., 1933, Z. Phys. chem. B, 22, 181, Theorie der geordneten Misch- phasen III. Fehlordnungserscheinungen in polaren Verbindungen Ms Grundlage ffir Ionen- und Elcktronenleitung.

WASSCHER, J. D., SEUT]~R, A. M. J. H., and H)~AS, C., 1964, Proceedings of the International Conference on Semiconductor Physics, Paris (Paris : Dunod Cie.), p. 1269, Spin-disorder scattering, anomalous behaviour of the Hall coefficient and magnon drag in anti-ferromagnetic MnTe.

WIT, H. J., DE, 1968, Philips Res. Rep., 23, 449, Some numerical calculations on Holstein's small polaron theory.

WIT, H. J. DE, and CaEV~CO~UR, C., 1967, Physics Left. A, 25, 393, n-type Mall effect in MnO.

WaIGHT, 1~. W., and A~D~EWS, J. P., 1949, Proe. phys. Soc. Lond. A, 62, 446, Temperature variation of the electrical properties of nickel oxide.

YAMASHI~A, J., and KVROSAWA, T., 1958, J. Phys. Chem. Solids, 5, 34, On electronic current in :NiO ; 1960, J. phys. Soc. Japan, 15, 802, Heitler- London approach to electrical conductivity and application to d-electron conduction; 1961, J. appl. Phys., Suppl., 32, 2215, Heitler-London approach to electrical conductivity.

ZHvzE, V. P., and S~EzY~I~, A. I., 1963, Soviet Phys. solid St., 5, 1278, Mall effect in nickel oxide ; 1966, Ibid., 8, 509, Hall effect in cobaltous oxide single crystals.

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small-polaron versus band conduction in some transition-metal oxides

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