hydrogen in nb/ta superlattices

17
Z. Phys.B - CondensedMatter 74, 457-473(1989) Gondensed itse.r,. Matt for PhysikB Springer-Verlag 1989 Hydrogen in Nb/Ta superlattices A model system for a modulated lattice gas P.F. Miceli* and H. Zabel Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Received September 6, 1988 The structural and thermodynamic properties of hydrogen dissolved in Nb/Ta superlat- tices are studied by in-situ x-ray scattering techniques. From the x-ray satellite intensities, it is found that H induces a strain modulation exhibiting a Curie-Weiss temperature dependence. The intensities are analyzed in terms of a mean field model of the modulated lattice gas, yielding quantitative information on the hydrogen-metal and hydrogen-hydro- gen interaction energies. Critical behavior associated with a gas-liquid transition is also observed. Hydrogen density fluctuations with wavelengths shorter than a superlattice period are, however, suppressed by the superlattice, which represents a novel manifesta- tion of a coherent phase transition. These experiments provide new and fundamental insight into the role of spacially varying two body interactions in critical phenomena. I. Introduction Hydrogen-metal systems present a variety of interest- ing and important physical phenomena [1]. Atomic hydrogen resides in the interstitial lattice of the host metal and exhibits a high rate of diffusivity. The pres- ence of interstitial H causes an expansion of the host metal lattice, which provides for an elastically mediat- ed, long range, attractive H-H interaction. Thus, H- metal systems pose as ideal experimental realizations for the study of the physical and thermodynamic properties of a lattice gas with interactions. In this paper, we present a detailed x-ray scatter- ing study of the structural and thermodynamic prop- erties of hydrogen in periodic Nb/Ta superlattice sys- tems, which unfolds another dimension to the intrigu- ing properties of hydrogen in metals. Since both Nb and Ta dissolve H, the superlattice host metal pro- vides a modulation in the H-metal and H- H interac- tions. From measuring the superlattice satellite inten- sities we obtain information on the Fourier compo- nents of the modulated one-body and two-body inter- actions. In previous work we have demonstrated the existence of a temperature dependent strain modula- * New address: Bell Communications Research, Red Bank, NJ 07701,USA tion due to a hydrogen density modulation in the host metal superlattice [-2, 3]. Here we will expand on this work and also report on critical hydrogen fluctuations which occur for higher hydrogen concen- trations and for certain superlattices. Like any gas with attractive interactions, H in a metal can undergo a gas-liquid phase transition [4, 5], exhibiting regions of high (liquid) and low (gas) interstitial occupation density. We have found that the host metal superlattice has a profound impact on the critical fluctuation spectrum of the H density as a consequence of the modified spacial dependence of the two body (H-H) interaction [6]. While it is known that long range interactions suppress critical fluctuations, the role of spacially varying interactions in critical phenomena is not well understood. Since Ising ferromagnetism, real gases and the lattice gas belong to the same universality class [7], H-metal superlattices provide a unique opportunity to study, in general, the effects of spacially varying two body interactions on critical phenomena [-8, 9]. Short accounts of this work have been published previously in [2, 3, 6, 8]. The remainder of this paper is organized in the following way: In Sect. II experi- mental details on the superlattice growth and x-ray diffraction techniques are described, Sect. III contains a description of the hydrogen induced strain modula-

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Z. Phys. B - Condensed Matter 74, 457-473 (1989) Gondensed itse.r,. Matt for Physik B

�9 Springer-Verlag 1989

Hydrogen in Nb/Ta superlattices A m o d e l sy s t e m for a modula ted latt ice gas

P.F. Miceli* and H. Zabel Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Received September 6, 1988

The structural and thermodynamic properties of hydrogen dissolved in Nb/Ta superlat- tices are studied by in-situ x-ray scattering techniques. From the x-ray satellite intensities, it is found that H induces a strain modulation exhibiting a Curie-Weiss temperature dependence. The intensities are analyzed in terms of a mean field model of the modulated lattice gas, yielding quantitative information on the hydrogen-metal and hydrogen-hydro- gen interaction energies. Critical behavior associated with a gas-liquid transition is also observed. Hydrogen density fluctuations with wavelengths shorter than a superlattice period are, however, suppressed by the superlattice, which represents a novel manifesta- tion of a coherent phase transition. These experiments provide new and fundamental insight into the role of spacially varying two body interactions in critical phenomena.

I. Introduction

Hydrogen-metal systems present a variety of interest- ing and important physical phenomena [1]. Atomic hydrogen resides in the interstitial lattice of the host metal and exhibits a high rate of diffusivity. The pres- ence of interstitial H causes an expansion of the host metal lattice, which provides for an elastically mediat- ed, long range, attractive H - H interaction. Thus, H- metal systems pose as ideal experimental realizations for the study of the physical and thermodynamic properties of a lattice gas with interactions.

In this paper, we present a detailed x-ray scatter- ing study of the structural and thermodynamic prop- erties of hydrogen in periodic Nb/Ta superlattice sys- tems, which unfolds another dimension to the intrigu- ing properties of hydrogen in metals. Since both Nb and Ta dissolve H, the superlattice host metal pro- vides a modulation in the H-metal and H - H interac- tions. From measuring the superlattice satellite inten- sities we obtain information on the Fourier compo- nents of the modulated one-body and two-body inter- actions. In previous work we have demonstrated the existence of a temperature dependent strain modula-

* New address: Bell Communications Research, Red Bank, NJ 07701, USA

tion due to a hydrogen density modulation in the host metal superlattice [-2, 3]. Here we will expand on this work and also report on critical hydrogen fluctuations which occur for higher hydrogen concen- trations and for certain superlattices.

Like any gas with attractive interactions, H in a metal can undergo a gas-liquid phase transition [4, 5], exhibiting regions of high (liquid) and low (gas) interstitial occupation density. We have found that the host metal superlattice has a profound impact on the critical fluctuation spectrum of the H density as a consequence of the modified spacial dependence of the two body ( H - H ) interaction [6]. While it is known that long range interactions suppress critical fluctuations, the role of spacially varying interactions in critical phenomena is not well understood. Since Ising ferromagnetism, real gases and the lattice gas belong to the same universality class [7], H-metal superlattices provide a unique opportunity to study, in general, the effects of spacially varying two body interactions on critical phenomena [-8, 9].

Short accounts of this work have been published previously in [2, 3, 6, 8]. The remainder of this paper is organized in the following way: In Sect. II experi- mental details on the superlattice growth and x-ray diffraction techniques are described, Sect. III contains a description of the hydrogen induced strain modula-

458

tion, while Sect. IV deals with the temperature depen- dence of the strain modulation. This section also con- tains a theoretical outline for the description of a mo- dulated lattice gas. In Sect. V the critical behavior of the hydrogen density is described, including a dis- cussion of the importance of critical density fluctua- tions for the formation of domain boundaries. Finally, in Sect. VI we summarize our results.

II. Superlattice growth, hydrogen loading, and x-ray diffraction

The Nb/Ta Superlattices [10] were grown by molecu- lar beam epitaxy (MBE) along the [110] direction on [1120] sapphire substrates. Both Nb and Ta have body centered cubic lattices with nearly identical lat- tice parameters (3.302 ~ and 3.306 A, respectively), providing for a negligible intrinsic strain in the super- lattice. High quality crystalline films are obtained which exhibit x-ray mosaic widths of ~<0.1 degree, typical of metallic crystals, and structural coherence lengths equal to the film thickness perpendicular to the film, while longer coherence lengths are observed in the plane of the film.

The x-ray scattering results were obtained with a double-axis spectrometer using MoK~I characteris- tic radiation off a flat Si(111) single crystal mono- chromator for high resolution. The relatively short wavelength radiation minimized the effects of absorp- tion from these transition metal films (the absorption lengths are 6 gm and 65 gm for Ta and Nb, respec- tively, for MOK~A). High resolution is beneficial for superlattices with long modulation wavelengths where the satellite reflections are closely spaced. An- other advantage of high resolution is that the trans- verse width of the substrate reflection becomes very narrow as a results of the small mosaic (<0.01 ~ of sapphire convoluted with the narrow incident beam divergence. Thus, substrate reflections could easily be distinguished from those of the film, and a slight tilt of 0 relative to 2 0 dramatically reduced the measured substrate intensity in a longitudinal scan, while not affecting features due to the film.

Hydrogen loading of the superlattice was ac- complished in situ, using a high vacuum x-ray furnace which allowed experiments to be performed as a func- tion of temperature and H 2 pressure [9]. Typically, the superlattice was heated to 400 ~ C-500 ~ C in a vac- uum of better than 10 - 6 Tort before highly purified H2 from a Pd cell was introduced. When very low concentrations of hydrogen were desired it was possi- ble to absorb hydrogen from the residual H2 partial pressure of the vacuum system. This could be done because of the enormous solubility [2, 9, 11] of H in

thin films and the fact that in a metal vacuum system, the residual gas is mainly H2.

The lattice expansion due to H in bulk metals has cubic symmetry and is linear in the H concentra- tion [12]. For H in a metal superlattice or thin film, it is found [9, 13] that the expansion occurs only nor- mal to the film plane as a consequence of epitaxy to the substrate which hinders expansion in the plane of the film. For higher H concentrations, the one di- mensional lattice expansion becomes unstable, and phase separation occurs into ordered, hydride-like, phases. It is believed that this instability is associated with the critical strain for the formation of misfit dis- locations [13].

The volume change per metallic volume, A v/O, due to one H atom is 0.17 and 0.16, for Nb and Ta, respectively [12]. Here O is the atomic volume of the host lattice atom. For the superlattices, we can neglect the small difference between the volume changes and assume a single average volume expan- sion for both metals: (Av/t2)=0.165. In a thin film, the relative volume expansion, A V/V, for a H/metal ratio, c o, is given by A V/V=(A v/f•)Co, which is the same relationship as for bulk metals. In the film case we impose, however, the condition that A V/V= d d/d due to the one dimensional lattice expansion. Such a relationship is justified since it is generally observed that the volume expansion due to H is remarkably similar for a variety of metals and metal alloys, sug- gesting that the H defect has a well defined "size" [12]. As discussed elsewhere [11], the one dimension- al volume expansion is central to the enhanced solu- bility for H in these films.

lII. Hydrogen induced strain modulation

When hydrogen is dissolved into the superlattice two types of changes occur in the x-ray diffraction pattern. First, the average component of the H density causes a uniform lattice expansion, A d/d, according to:

Ad - cot0A0. (1)

d

Consequently, the entire superlattice diffraction pat- tern shifts to lower Bragg angles by the amount A 0. Secondly, a strain wave which is in-phase with the metallic superlattice periodicity causes an asymmetry in the intensities of the pairs of satellites on either side of the fundamental reflection. For small strain amplitudes, the asymmetry of the __ s(s = 1, 2 . . . . . ) sat- ellite intensities can be expressed as [9, 14] :

,(

459

to c) -- I00 X

r-

E

E 50 0 (..)

85~, .........

NblTa - H

Superlattice

- AT

-I

-4 -3 _...j

I

§

~ ,~_ +3 ~+4 015 16 T7 18 19

20 (deg) Fig. 1. X-ray scan of an 85 A superlattice (SL 18) with hydrogen at 500 ~ C. H introduces two types of strain in the superlattice. The average hydrogen concentration provides a uniform shift of the dif- fraction pattern to lower angles, while a hydrogen density modula- tion induces a strain modulat ion evidenced by the asymmetry of the satellite intensities belonging to the same harmonic. Since the + 1 satellite is larger than the - 1, it is concluded that the H favors the Nb layers

where e~ is the amplitude of the strain modulation, s refers to the harmonic index, A is the superlattice periodicity, and d is the atomic layer spacing. Fourier coefficients ~/s of the atomic form factor modulation are:

fTa - - fNb la~l, tl~- ao fT, + bo fNb

where fTa and fNb are the atomic form factors of tanta- lum and niobium, respectively, a o and b o are the frac- tions of the sublattices within a superlattice period (ao+bo= 1), and a~ are the Fourier coefficients for the composition modulation.

In an ideally unstrained superlattice, all satellite pairs (___ s) will exhibit equal intensities. However, if we introduce a strain wave of wavelength equal to that of the harmonic, the +__ s satellite intensities are no longer equal and (2) allows a quantitative assess- ment of both the compositional and strain modula- tion in the superlattice~ In this discussion we have neglected the contribution of H itself to the scattering, which for x-ray scattering is justified because of the negligible scattering amplitude.

Figure 1 shows an x-ray scan of a Nb/Ta superlat- tice with a periodicity of A = 85/~ and a Ta fraction of ao = 0.57 after hydrogen loading. This scan demon- strates that both types of strain are present in the superlattice. The shift of the diffraction pattern to lower angles indicates a homogeneous component of

Ad hydrogen induced strain, ~ - - e o = 0 . 0 1 . Using the

linear relation between volume expansion and hydro-

gen concentration, the hydrogen concentration is esti- mated to be Co ~0.06 [H/metal]. In addition to the angular shift of the diffraction pattern, there is a pro- nounced asymmetry between the (+_ s) satellite intensi- ties, indicating the presence of a strain modulation. From (2), the amplitude of the strain modulation for the first satellite is e1=0.0012, which is 12% of Co. Since there is more intensity in the s = + 1 satellite as compared to s = - 1, the strain wave has a maxi- mum in the Nb layers and a minimum in the Ta layers (the convention will be that strains are positive when the Nb has the larger lattice parameter). This result is consistent with known solubility data [15] for H in bulk Nb and Ta where the H-metal binding energy is found more attractive for Nb than for Ta by roughly 16 meV. Thus, there is a uniform distribu- tion of H, with an additional modulated component of the H density, induced by the superlattice periodici- ty. Converting the strain modulation to a hydrogen concentration amplitude, we obtain for the first Four- ier coefficient of this modulation: cl ~0.007.

A closer examination of the intensities before and after hydrogen loading reveals that the strain modula- tion is "in-phase" with the compositional modulation of the metal. According to (2), a strain modulation in-phase with the compositional modulation pro- duces relative changes of the satellite intensities I+, with a leading term linear in the strain. However, an out-of-phase strain modulation, corresponding to an accumulation of H at the interfaces, does not cause relative intensity changes of I+~ up to second order in the strain [9], while both satellite reflections would simultaneously change intensity relative to the funda- mental intensity. This intensity change would then appear as an effective change in the Fourier coeffi- cients of the compositional modulation, a~. The x-ray results of Fig. 1 yield exactly the same value for as before and after hydrogen loading of this superlattice. We are therefore led to the conclusion that the strain modulation is "in-phase" with the composition mod- ulation. In addition, we noticed from the satellite in- tensities that no interdiffusion between the Nb and Ta layers occurred during hydrogen loading at the temperature of 500 ~ C.

Before applying (2), the measured intensities were corrected for absorption and Lorentz polarization factors. Thermal Debye-Waller factors were taken to be those of bulk Nb and Ta, although this usually had a negligible effect on the results. The atomic form factors in q~ were evaluated at the appropriate value of sin(0)/2 according to [16].

Since the approximations leading to (2) require that both the strain and the scattering vector, Q, re- main small, all experiments have been performed on the satellite reflections centered about the (110) recip-

460

~ S = -I t'kl

~ 400

T=573 K

O V ~ , ~ " I T=2186K I ~ , ~ I , 14 16 18 20

20 Fig. 2. The relative intensities for the first harmonic of a 20 ~ super- lattice (SL 3) change with temperature. The asymmetry in these in- tensifies increases with decreasing temperature, indicating an in- crease in the strain modulation as the temperature is lowered

I 22

rocal lattice point. This has the additional advantage that the x-ray intensities are not drastically reduced by thermal and static Debye-Waller factors, atomic form factors, and the Lorentz polarization factor.

IV. Temperature dependence of the strain modulat ion

The s = + 1 satellite peaks of a 20 A superlattice con- taining hydrogen are shown in Fig. 2 for several tem- peratures. As the temperature decreases, the + 1 satel- lite intensity increases relative to the - 1 , indicating an increase in the strain modulation. The temperature dependence of the strain modulation is plotted in Fig. 3 a where the strain is calculated from the data according to (2). This result implies that more H re- sides in the Nb than in Ta layers as the temperature is lowered. The average hydrogen concentration, Co, remains constant for each of these temperatures, since the peak positions are found to be changing only with thermal expansion. Therefore, at high temperatures where the entropy dominates the free energy, we find a homogeneous distribution of the hydrogen density in both sublattices. However, as the temperature de- creases, the difference between the binding energies for H in Nb and Ta becomes noticeable, driving the hydrogen into the sublattice with the lower energy, which is the Nb lattice. The difference between these binding energies is equivalent to half of the room temperature thermal energy [15], and therefore it is reasonable to observe the hydrogen density modula- tion in a temperature range considered here.

In Fig. 3 b the results of Fig. 3 a are replotted in terms of a reciprocal strain versus the temperature. This graph reveals a linear relationship between e-1 and T, suggesting a Curie-Weiss type strain behavior

0.05

0.04

.E 0.03

o.oz

~ c 6 0 -

if) 40

20

I I [ I [

_~(a) 2o~ Nb/T.a Superlottice

I I I I I

(b)

0 [ [ I I I I00 200 300 400 500 600

T(K) Fig. 3. a The strain modulation is obtained from Fig. 2 and (2), showing that the strain modulation increases with decreasing tem- perature. In (b), the reciprocal strain is found to be linear with temperature and suggests a Curie-Weiss behavior similar to magnet- ic systems

analogous to paramagnetic susceptibilities. Therefore, we explore a lattice gas model in an external varying field which may describe the present system.

A. Curie- Weiss behavior: theory

Hydrogen in metals can be treated within the same theoretical framework as Ising ferromagnets since an interstitial site has two states, either occupied or unoc- cupied. In fact, the lattice gas problem can be directly mapped onto the Ising problem, and, instead of a spin variable having the values _ 2 x-, we use an occupa- tion variable having the values z=O, 1. The lattice gas Hamiltonian can be written as the sum of a one- body and a two-body interaction:

n = 2 Ua T~a"~ �89 Z Jab 75a Tb (3) a a, b

where the summations are over all sites. J~b is the H - H interaction energy between sites a and b when they are both occupied, and U, is the binding energy of the H to the interstitial site, a.

Since the elastically mediated H - H interaction in metals is of long-range, mean field theory is well suited for the present problem. Extremely short length scale fluctuations are difficult to treat and constitute only local changes. Therefore, we adopt a "coarse graining" procedure [7] which sacrifices spacial reso- lution of the observed occupancy in favor of a contin- uous description, replacing a small region of the sam- ple with the locally averaged parameters. Within this spirit, the Helmholtz free energy may be written in terms of an energy, E, and an entropy of mixing, S:

F/No = 1 ~d3r U(r) c(r)

1 1 ~ 3 1 - 3 , + ~ v j d r f f j d r c(r)J(r,r')c(r')

- T 1 ~ da r [-c (r) In (c (r)) + (1 - c (r)) In (1 - c (r)) ]

=E-- TS (3)

where c(r) is the H/metal ratio which can assume values continuously between 0 and 1. No is the number of available sites and V is the volume. It is the entropy term which gives us the best indication as to the validity limit of the course graining proce- dure, since, the concentration must be sufficiently slowly varing so that small regions can locally satisfy Stirling's approximation [17].

Since (3) is valid for any c(r), equilibrium is estab- lished when the free energy is minimized with respect to c(r), subject to the constraint of a constant number

1 OF of occupied sites. This requires - # , where # is the chemical potential: No 0c

(4)

In chemical equilibrium, # is a constant and is simply determined by the external hydrogen gas. Therefore, the detailed spacial variation of c(r) at high tempera- tures must be found from the potentials U and J.

By expanding the logarithmic term, (4) can be solved for small amplitude concentration waves about an average concentration, Co:

# = U o + J o c o - T l n ( ~ o ~ ) q=0, (5a)

= Cq q+O (5b) 0 gq-]- Jq+co(1Zc 0

461

Here we have introduced the Fourier transform of all spacially varying quantities:

U = l f d 3 r U(r)e -i"'~ q - v

jq = j q _ l fd3r J(r) e -iq'r

1 . 3 iq r Cq = ~ j d rc ( r )e - ". (6)

It is convenient to assume that J ( r , r ' ) = J ( r - r ' ) in the superlattice. Since Nb and Ta behave elastically similar, this assumption should not qualitatively affect the results. Equation (5 a) describes the average prop- erties of the H concentration, while (5 b) predicts a hydrogen density modulation,

Co (1 -- Co) Uq = Zq Uq, (7) cq= T-- Tq

where,

~ - -Co(i -Co) J. (8)

is a spinodal temperature. Therefore, the concentra- tion modulation is driven by the superlattice periodic- ity (Uq=0 for q + 0 in a homogeneous metal) where Uq acts as an "applied field" and Zq is the response function which is of the Curie-Weiss form. To make an analogy with magnetic systems, Uq can be thought of as an applied spacially modulated magnetic field with wavevector q, and cq as the sample magnetiza- tion which couples to the applied field through the "dressed" (corrected for the two-body interaction) susceptibility, )~q.

To describe the experimental result of Fig. 3 b, it is necessary to relate the concentration in (7) to the Fourier coefficient of the strain modulation. Since the lattice expansion due to H is very nearly the same in both Nb and Ta, it is reasonable (compared to the magnitude of the observed strain) to assume that the strain response is independent of the position c~(r) = e, and therefore:

e(r) = ~ c(r) (9a)

with the Fourier transform,

eq = o~ cq. (9 b)

The expansion coefficient c~ also enters the average Ad

lattice expansion as, ~ - - - e o = e Co. Therefore, com-

bining (7) and (9b) for small concentrations, Co ~ 1,

462

a result which is independent of both ~ and the aver- age concentration can be obtained:

% (lO) eq= -- T- - Tq "

This is just the Curie-Weiss form observed experimen- tally in Fig. 3 b, and we now further investigate the implications of (10).

B. Curie-Weiss behavior: exper iment

Ad Since the average lattice expansion, ~ - , can be ob-

tained from the shift of the Bragg peak upon H up- take, (10) provides a means to directly determine the fundamental interactions, Uq and Tq, from the temper- ature dependence of the strain modulation. For the

20 ~ sample in Fig. 3, using ~ = 0 . 0 1 9 and am =0.3,

we obtain Ul=170K and T ~ = - 2 4 0 K . The value of U1 seems to be in accordance with the difference in the binding energy between bulk Nb and Ta ob- tained from solubility data [15] while the negative T1 suggests that the H - H interaction is repulsive for the first harmonic. We have measured the temper- ature dependence of the strain modulation for several superlattices and the results are listed in Table 1. The physical significance of the determined potentials will be discussed further below. For now we will carefully examine the experimental validity of (10).

Figure 4a and b show the temperature depen- dence of the strain modulation for three satellites of a 85 ]~ period superlattice (SL 24) having a Ta frac- tion of ao = 0.37. The solid lines represent least square fits to the Curie-Weiss function in (10), demonstrating that the hydrogen density is well described by a mo- dulated lattice gas model.

In addition to the Curie-Weiss temperature de- pendence, (10) predicts that the amplitude of the strain modulation should be linear in the average lattice

Ad expansion, ~ - . To check this, a superlattice (SL 19)

was cut into two pieces: one for measuring the tem- perature dependence at constant concentration, and one for measuring the concentration dependence at constant temperature.

Figure 5 shows the concentration dependence of the strain modulation at the first superlattice satellite for a constant temperature of 450 ~ C. Indeed, the am- plitude of the strain modulation is linear in the aver-

Ad age lattice expansion, ~ - . Using a lattice expansion

Table 1. E x p e r i m e n t a l d a t a for the super la t t ices . A is the supe r l a t t i ce

per iodic i ty , A d / d refers to the a v e r a g e o n e - d i m e n s i o n a l la t t ice ex- p a n s i o n o f the super la t t i ces d u e to h y d r o g e n , s is the index o f the satell i te ref lect ions; as, U~, a n d T s refer to the F o u r i e r coefficients

of the c o m p o s i t i o n m o d u l a t i o n , the h y d r o g e n - m e t a l i n t e r a c t i o n a n d the h y d r o g e n - h y d r o g e n in t e r ac t ion , respect ively. The s ign of a , w a s

chosen b a s e d o n a s q u a r e wave c o m p o s i t i o n m o d u l a t i o n

S a m p l e A [ ~ ] A d [ a ~ - - s a~ [~ T~ [~ d

SL 3 20 0.019 0 0.6 . . . . . . 1 0.3 570 - 240

SL 7 20 0.020 0 0.6 . . . . . .

1 0.3 600 - 270

SL 18 85 0.010 0 0.57 . . . . . .

1 0.290 300 52 2 - 0.053 - 50 270

3 - 0 . 0 3 6 + -..

4 0.025 + ...

SL 19 86 0.008 0 0.17 . . . . . . 1 0.21 500 - 190

2 0.11 1100 - 4 0 0

3 0.04 + .--

SL 22 57 0.011 0 0.37 . . . . . . I 0 .310 320 - 2 8 9 2 0.087 840 - - 142

3 - 0 . 0 3 3 - 5 1 3 93 4 0.033 + ...

SL 23 86 0.008 0 0.37 . . . . . . 1 0.30 440 - 290

2 0.095 980 - 144 3 - -0 .034 - 2 6 5 260 4 - -0 .045 + ..-

5 - - 0.068 + ...

6 0.016 + ... 7 0.008 + .--

SL 24 113 0.009 0 0.34 . . . . . . 1 0.31 360 - 1 3 0

2 0.095 920 - - 120 3 - -0 .031 - 1000 - 135 4 - 0 . 0 4 4 360 216

Ad coefficient due to hydrogen of e=0.165, then ~ -

= 0.01 corresponds to a concentration of 0.06 H/met- al. This concentration satisfies the assumption we made earlier in the derivation of (10) that Co ~ 1.

An independent measurement of the temperature dependence of the strain modulation for the same su- perlattice is shown in Fig. 6 and gives a slope which is in excellent agreement with the slope of the e~/al

Ad versus ~ - plot in Fig. 5. Using T1 = - 190 K from

extrapolating the data in Fig. 6, we find from the slope of Fig. 5 for the Fourier component of the H-metal interaction U1 = 475 K, whereas from the slope of the temperature dependence in Fig. 6 we obtain U~ = 500 K an agreement to within 5%.

463

E

0.02

0.01

-0.01

- 0 . 0 2 0

400

(a)

I I 200 4 0 0 600 800 I000

T(K) I I I I

J 200

S = 2

Z o

- 2 0 0 -

(b) - 4 0 0 I I I I

0 200 400 600 800 I000 T(K)

Fig. 4. a Strain versus temperature for an 85/~ superlattice (SL 23). b demonstrates that the reciprocal strain follows a Curie-Weiss type temperature dependence for all of the satellites, each having a differ- ent slope and intercept

0.010 I I I /

0.008 - T = 450 ~ O / _

-'---- 0 . 0 0 6 -

UY 0.004

0.002 / / / ~

n - I I I - 0 O. 005 0.010 0.015 0.020

A__~d d

Fig. 5. The first Fourier component of the strain modulation (SL 19) plotted as a function of the average lattice expansion. The linear relationship is predicted by (10)

According to (10) all Fourier components of the strain modulation should be a linear function of the

Ad lattice expansion ~ - . We have tested this for the

first and second satellite reflection of sample SL 23, which is reproduced in Fig. 7a and b. Both slopes

300 I I I I j

I00

I I I I 0 200 400 600 800 I000

T(K) Fig. 6. The temperature dependence of the reciprocal strain modula- tion is shown for the same sample as in Fig. 5 (SL 19). The slope determined for the s = 1 satellite and the result of Fig. 5 are in excel- lent mutual agreement according to (10)

0.005 I I J [ |

(a) 0.004 - S = I

0.003 -

~- 0.0o2 -

0 . 0 0 1 -

0

(b) A ~ 0 .008 - S = 2

o .oo6

N

~o 0 .004

0.002

0 0 0.002 0 .004 0 .006 0 .008 0.010

ZXd d

Fig. 7a and b. Four ier components o f the strain modulat ion as a function of the average lattice expansion is plotted for the first (a) and second (b) harmonics (SL 23) and is in good agreement with the temperature dependent data in Fig. 4 and (10)

again agree very well with the slopes from tempera- ture dependent measurements of the strain modula- tion as already shown for the same sample in Fig. 4. These results are summarized in Table 2.

C. Determination of potentials

As discussed before, the Fourier components of H- metal binding energy, Uq, and H - H interaction, Tq,

464

a s

(I s I I I I Us[K] 0.4 - O0

0.2 50

0 . . . .

- 0 . 2 I I I I - 50 0 1.0 2.0 3,0 4.0 5.0

S

Fig. 8. Fourier coefficients of the compositional modulation, a,, (squares) and H-metal binding energy, U~, (circles) are plotted vs. harmonic index, s, for (SL 24). The binding energy follows the com- positional modulation qualitatively, however, quantitative differ- ences are apparent

Table 2. Comparison of experiments at constant temperature and constant hydrogen concentration

Sample Constant Constant temperature concentration

la• G T[K] s slope [K] T~ [K] ~ [K]

SL 19 720 1 0.52 480 -190 500 SL 23 800 1 0.40 440 --290 440

2 1.00 950 --140 980

can be obtained directly from the temperature depen- dence of the strain modulation. These were measured for several superlattices and are listed in Table 1. Due to the lower intensity of the higher order satellites, the number of Fourier coefficients for the modulation potential which can be obtained with reasonable ac- curacy is limited. The largest number has been mea- sured for sample SL 24, and the Fourier components of the metallic composition, as, as well as the Fourier components of the binding energy, Us, are plotted in Fig. 8. Since the x-ray satellite intensities provide only the magnitude of as, the sign of as has been chosen on the basis of a square-wave modulation, which is a good approximation for these small values of s. F rom Fig. 8 it becomes clear that the H-metal binding energy follows the metallic compositional modulation, since the qualitative dependence on the index s is very similar for both as and G.

Based on these results, it might be reasonable to assume that the H-metal binding energy is propor- tional to the local metallic composition:

U(r) = UT, nTa(r)d- UNb nNb(r) (11 a)

800 i i i

600

,,<, i.._J

o 400

2 0 0

~ I .... -2

0 I I I I 0 20 4 0 60 80 I00 o

tTo(A) Fi~~ 9. The first harmonic of the H-metal binding energy modulation depends linearly upon the absolute thickness of the Ta layers. The dashed lin e (180 meV) in dicat ~ s th e ~ expected ~ behavior i f the bind I

ing energy was simply proportional to the loeal composition

where the U's are the H-metal binding energies found in the bulk metals and n(r) is the composition profile of the metals. The corresponding Fourier coefficients of the H-metal binding energy in the superlattice are then given by:

U ~ = A u G , ( l l b )

where A u = U T a - - U N b ~ 1 6 m e V (or 180K) from known isotherms [15]. Thus, (11 b) would predict that the ratio of Us/a s should be a constant for all satellite orders and for all samples, however, looking back to Table 1, such a relationship is not observed.

Since G/as does not obey (11 b), several attempts were made to determine which variables U~ depend on. Plots of Us versus total film thickness, composition ao, superlattice period A, or compositional Fourier component a s showed no systematic behavior. Fig- ure 9 demonstrates that UjIasr appears to depend lin- early on the Ta layer thickness. It is rather surprising that the Fourier components depend strongly on the Ta thickness while no systematic dependence on the Nb layer thickness was observed. The results in Fig. 9 suggest that thinning the Ta layer thickness weakens the attraction for H in Ta, therefore, the difference between the Nb and Ta binding energies increases for decreasing Ta layer thickness. The dashed line in Fig. 9 indicates the prediction of (11 b).

The binding energy for H in the Ta layers should approach the bulk value as the Ta layers become very thick. An empirical description can be written as an exponential dependence of U on tr,:

G - [ UT, + U' e - tTa/L] __ UNb lasl

"~ A u + U'(1--tTa/L), (12)

465

which approaches the prediction of (11 b) in the limit of large Ta layer thickness. From Fig. 9, we obtain U '= 460 K (or 40 meV) and a characteristic length, L=62/~.

These results demonstrate that the Fourier com- ponents of the binding energy are quite sensitive to the surrounding environment. As shown previously in Fig. 8, U~ follows the composition profile, while from Fig. 9 it appears that the energy scale is set by the Ta layer thickness for thin layers.

The Fourier transform of the H - H interaction, Tq, is found to be generally repulsive, especially for T1, which is negative for all but one sample (SL 18). Since the H - H interaction is attractive in the bulk metals, the superlattice periodicity apparently pro- vides a repulsive contribution to Tq. Referring to Ta- ble 1, Tq is negative for low satellites and crosses over to positive values when one period of the harmonic, s, can fit inside the Ta layer; that is, when s ~ l / a o . Even sample SL 18, where Tt is positive but small in magnitude, shows an increase for T2. Based on the three samples for which data is available for s > 1/ao, Tq appears to approach a value on the order of 250 ~ at large s. This temperature is near the phase boundary of the bulk H-metal phase diagrams [18] for these concentrations, and, one might specu- late that the Tq's saturate to a constant value which is determined by the short range interactions respon- sible for the fl and ~ phases in Nb and the e phase in Ta [18]. In any case, the superlattice evidently has a dramatic effect on the long wavelength density waves, while the shorter wavelengths are probably affected only quantitatively.

D. Saturation of the temperature dependence

While some higher order Fourier components may display positive values of Tq, a positive intercept on a Curie-Weiss plot does not mean that the strain modulation will become infinite as the temperature approaches T~. We found that when the amplitude of the strain modulation becomes large, the Curie- Weiss temperature dependence saturates as is demon- strated in Fig. 10. Such saturation might be anticipat- ed since the Curie-Weiss behavior is a high tempera- ture approximation for small amplitudes, and, as the temperature is lowered, the concentration wave am- plitudes must not exceed the average concentration,

Ad Co. This amounts to requiring that ]es [<~- , which is well satisfied in Fig. 10.

At high temperatures, where the amplitude of the strain modulation is small, their Fourier components can be considered as being independent. At lower

0.04 I I I

o o o o

0 .02 o o S : 3 o o o

0.01

% to 0

-0.01 - ( a ) .

- 0 . 0 2 I I 0 2 0 0 4 0 0 6 0 0 8 0 0

T ( K )

5 0 0 I I I

S= I

�9 �9 �9 �9 �9 ~ ~ . t _ ~ , - H - - - ' ~ - - ' ~ o S : 3 o

0

" 0 0 0 i 0 200 400 600 Boo

T(K) Fig. 10a and b. Fourier coefficient (a) strain and Ca) reciprocal strain, demonstrating saturation of the Curie Weiss temperature depen- dence for SL 18 below 400 K

temperatures, however, the Fourier components of the strain modulation may couple in a nonlinear fash- ion. In any case, there must be a constraint on the amplitudes such that they are consistent with the av- erage concentration in the sample. A conservation "law" can be derived from the spacial average of the concentration waves by employing the inequality, c z (r) < c (r). This gives

Co (1 - Co) _-> <(c (r) - Co)2 L . > O, (12a)

where the Fourier components satisfy

Co(1-Co)~ ~ Ic~12~0. (12b) q:l:0

Equality on the left-hand side of (12) refers to a situa- tion of "complete modulation", where regions within the superlattice are either completely occupied or completely unoccupied by hydrogen atoms. Equality on the right-hand side corresponds to a uniform con- centration, %. Returning to Fig. 10, it is evident that the high spinodal temperature of the second satellite (T2 = 270 K) drives the saturation of all other Fourier components.

466

0.02

0.01

S--I

/ - - 0 . 0 1

~ I S=5

-0.02 I �9 0 200 400 600 800 I000 T(K)

Fig. 11. The second and fourth Fourier components (SL 24) of the strain modulation indicate a decrease of the strain amplitude with decreasing temperature below 500 ~ K. This decrease is due to the onset of critical fluctuations associated with a coherent phase transi- tion

Figure 11 shows the temperature dependence of the strain modulation in SL 24, where it can be seen that the amplitude actually decreases with decreasing temperature below 490 K for two out of the four satel- lites. As discussed in the next sections, this sample exhibits critical concentration fluctuations below 490 K, having wavelengths longer than the superlat- tice periodicity. The observed decrease in the strain modulation is a manifestation of the above conserva- tion law since the new concentration waves develop at the expense of the commensurate density modula- tion. It is not immediately obvious, however, why this decrease appears only for the even satellites, although, it may be related to the symmetry and magnitude of the strain modulation.

To summarize the results of this section, we have found excellent agreement, both qualitatively and quantitatively, of the Curie-Weiss type temperature dependence of the hydrogen density modulation in Nb/Ta superlattices and a mean-field lattice-gas mod- el in a modulated periodic potential provided by the host. Analysis of the superlattice satellite intensities yielded quantitative information on the H-metal and the H - H interaction energies. In agreement with H in bulk metals, we found that the H density modula-

tion shows preference for the Nb layers over the Ta layers. However, the H - H interaction is significantly modified, being repulsive for long wavelengths, at the lower order superlattice harmonics.

V. Critical behavior of the hydrogen density

A. Discussion of the role of fluctuations

In bulk metal-H systems, the long range, elastically mediated H - H interaction is responsible for a gas- liquid phase transition (lattice gas-liquid transition) where the H density separates into regions of high (liquid) and low (gas) density of interstitial occupa- tion. Real gases, the lattice gas, and Ising ferromag- netism belong to the same universality class of critical phenomena since these are comprised of an ensemble of interacting two state entities [7]. Within this class, atomic hydrogen dissolved on an interstitial host met- al lattice acts as a lattice gas and differs from real gases and magnetic systems in that the elastically me- diated two body (H--H) interaction is long range, allowing mean field theory to be applicable. In fact, the observed critical exponents for H in metal systems obey mean field theory better than those for real gases and magnetic systems [19, 20].

Critical fluctuations are essential to facilitate the transition from an initially homogeneous density to the final, two-phase state (gas-liquid). However, the formation of these fluctuations is strongly coupled to the details of the two body interaction which drives the phase transition. The two body interaction enters critical behavior through the spinodal temperature, Tq, which is proportional to the Fourier transform of the H - H interaction, and it determines the growth of mean square fluctuations according to [9, 21] :

((6Cq)2/x C0(1--C0) T No T-- ~ ' (13)

2re where q = 2 - is the wavevector. Therefore, the tem-

perature at which a fluctuation diverges depends on its wavelength, 2. It is important to emphasize that (13) refers to fluctuations driven by a thermodynamic instability and that it is of a very different physical orgin than the H density modulation given in (7), which results only from the H-metal interaction.

The wavelength dependence of critical fluctua- tions can be exemplified as follows. When q =0, To is equal to the critical temperature, T c. For short range interactions, Tq will fall off slowly with q. Thus, in the limit of infinitesimally short range interactions, T~ _~ T~ is a constant so that according to (13), all wave-

467

length fluctuations would develop when T - T~. Alter- natively, for long range interactions, Tq falls off quick- ly with q so that near T~, only long wavelength (small q) fluctuations develop while successively shorter wa- velength fluctuations develop as the temperature con- tinues to be lowered below To. Therefore, long range interactions suppress short wavelength fluctuations. In the case of modulated two body interactions, as in a superlattice, it is not immediately obvious what one should expect.

Because short wavelength fluctuations are needed in order to describe an interface between the gas and liquid phases, the suppression of these fluctuations by long range interactions will have important conse- quences on the phase separation so that the transition develops over a range of temperatures below T~. Such behavior is referred to as a "coherent" phase transi- tion [9, 22-25] as opposed to an "incoherent" transi- tion where the change from the homogeneous to the two-phase state occurs at T~. It has been shown that H-metal systems possess sufficiently long range inter- actions that the coherent phase transition displays macroscopic density modes which are sensitive to the boundary conditions on the sample [-22, 23, 26]. For H in a superlattice, we expect that a coherent phase transition can be observed on a much smaller, micro- scopic length scale, on the order of the superlattice periodicity. Such a system allows one to study the properties of a lattice gas in the presence of modulated interactions as well as provide insight on the role of the spacial dependence of the two-body interac- tions in critical phenomena.

Although Nb and Ta have many similar physical properties, an important difference is that H in Nb exhibits the lattice gas-liquid phase transition while no such behavior is observed in Ta [,-18]. Consequent- ly, a superlattice containing a small fraction of Nb may not exhibit the gas-liquid phase transition, while a superlattice having a large fraction of Nb could display critical behavior.

B. Observation of critical fluctuations

Figure 12 shows x-ray scans of the s= +__ 1 satellites about the (110) fundamental reflection for a 85/~ su- perlattice containing hydrogen and having a Ta frac- tion of 0.57. At the two extreme temperatures (773 K and 25 K) shown, there is no phase transition occur- ring over this temperature range, since, a broadening or splitting of these peaks is not observed. The only difference is the intensity of the satellite peaks due to the Curie-Weiss dependence of the strain modula- tion, discussed in the previous section.

200

to 150 _o • r

E I00

r

o 50

- - I I

�9 T= 7 7 3 K - o T = 2 5 K

16.5 17.0 17.5

20 (deg)

I

L 18.0

Fig. 12. X-ray scans of the • 1 satellites for an 85 A superlattice (SL 18; Ta fraction: a0=0.57) with hydrogen. No critical behavior is observed from 773 K to 25 K. The relative superlattice satellite intensities change with temperature due to changes in the strain modulat ion

Another 85 ~ superlattice with hydrogen, having a Ta fraction of 0.17, is shown in Fig. 13a and b where a dramatic change in the peak shape occurs upon lowering the temperature from 300 ~ C to 200 ~ C (572 K to 472 K). While the peak intensities have de- creased, the transverse peak width was found to in- crease so that the integrated intensities of the (110) fundamental peak are the same at both temperatures. The peak profile in Fig. 13b at 200~ does not change as the temperature is further lowered to room temperature. Therefore, these scans represent the state of the system before and after the phase transition. This critical behavior is reversible, since a subsequent rise of the temperature to 300 ~ C gave the same scat- tering profile as observed before the phase transition took place. Both samples in Figs. 12 and 13 contain the same H concentration of 0.06 H/metal, as deter-

mined from the lattice expansion of-~a~0.01.

The (110) reflection at 200 ~ C has been fit to three Gaussian curves (inset Fig. 13a) and it is evident that there are two distinct contributions to the line shape: a narrow peak (single Gaussian) which is only slightly broadened as compared to the (110) at 300~ and a much broader peak exhibiting a slight asymmetry towards lower angles (represented by two Gaussians). This asymmetry, along with the observation that the (110) peak shifted by much more than what thermal expansion would allow between 300 ~ C and 200 ~ C, suggests a connection to the lattice gas-liquid phase transition in bulk metals, where one would expect a splitting of the (110) reflection into two peaks in case of an incoherent transition [5]. Since the experi- mental resolution (<0.04 ~ in 20) is much narrower than the actual peak width and the peak profile does

468

3

__0 X "" 2-- E

"E

8~

o

I00

80- 0

x 6 0 - t-,- E

4 0 - t -

':3 0 (.3

2 0 -

0 16.5

I i I

- ~ T = 5 0 0 " C

�9 T = 2 0 0 " C

(o)

, , I 17.0 17.2

28

o T = 5 0 0 ~

�9 T = 2 0 0 ~

(b)

17.0 2O

I i i r i

T ~ 200~ .~

~:~ 6oo . .

'i 400 . ,

16.8 17.0 17,2 J7.4 176

2e

L I I , 17.4 17.6

k

17.5 18.0

Fig. 13a and b. Fundamental reflection (s=0) for an 85 ~ superlat- tice (SL 19; Ta fraction=0.17) containing H. A dramatic change in the peak width, indicative of critical behavior, is observed at 200 ~ C. There is no further change below this temperature. The small peak to the left is due to the substrate. The inset displays the narrow and broad contributions to the line shape at 200 ~ C. (b) is the same scan as (a) except the vertical scale is changed to show the _4-1 satellites. At and below 200 ~ C, there is a large diffuse scattering which exists only between the first satellites. This suggests that strain fluctuations have wavelengths longer than the superlat- tice periodicity

not change below 200 ~ C, we conclude that the phase transition is coherent without the formation of two incoherent phases.

Scattering experiments performed along the in- plane directions ((002) and (1i0)) at room temperature indicate pristine Nb(Ta) lattice parameters. Therefore, the one-dimensional lattice expansion is maintained inspite of the observed critical behavior. Furthermore, these in-plane peaks are symmetric and have longitu- dinal widths narrower than the out-of-plane reflec- tions. The transverse widths, however, broaden for all reflections. Thus, the critical fluctuations are plane waves normal to the film, in accordance with the one- dimensional lattice expansion.

An important feature of Fig. 13 b is that at 200 ~ C there is a large diffuse scattering between the funda- mental and the first order satellite reflections. The

I I --'- I / - - - - ~ 2 f . - q / ( . ~ )

( V Fig. 14. "Dispersion relation" for the spinodal temperature in the superlattice, schematically showing the variation of the spinodal temperature with wavevector in units of the superlattice periodicity. The solid circles represent the observed diffuse scattering between the fundamental and first satellites. The squares are obtained from the temperature dependence of the strain modulation where it is found that Tq is negative. The upper curve represents the unmodified spinodal temperature for a homogeneous metal. The main feature is that T~ drops rapidly as q approaches the first superlattice har- monic

diffuse scattering does not, however, extend beyond these satellites. This behavior suggests that hydrogen density fluctuations exist for wavelengths only longer than the superlattice periodicity, while fluctuations with wavelengths shorter than a superlattice period are suppressed. Therefore, the lattice gas-liquid phase transition can occur in the superlattice but with a novel modification: distinct gas and liquid phases do not form because the high frequency Fourier compo- nents necessary to construct a domain boundary are not present. The critical behavior in the superlattice consists of a transition to long wavelength density fluctuations only, having a cut-off wavelength on the order of the superlattice periodicity.

Independent experimental evidence supporting these conclusions comes from the Curie-Weiss depen- dence of the strain modulation, which revealed nega- tive (repulsive) spinodal temperatures, Tq, at least for the first few harmonics, implying that critical fluctua- tions at these wavelengths are not favorable at any temperature.

A schematic of how the superlattice modifies the Fourier transform of the H - H interaction is shown in Fig. 14, where the spinodal temperature is plotted versus Iq[- The observed diffuse scattering (Fig. 13b) is represented by the series of solid circles for wave- vectors between zero and the first superlattice har- monic. The squares represent the results of the strain versus temperature experiments (Fig. 6) performed at temperatures above the phase transition. Clearly, the curve must connect between the diffuse scattering and the first harmonic. This provides the essential feature

469

that, as compared to the monotonic behavior of a homogeneous metal, there is a sharp drop in Tq as the first satellite wavevector is approached so that the onset of critical fluctuations, described by (13), are suppressed for q larger than the first superlattice harmonic. The behavior of T o is just a manifestation of a coherent phase transition resulting from the peri- odic boundary conditions of the superlattiee and is analogous to the macroscopic density modes found in bulk metal-H systems [22, 23, 26]. However, these results are substantially different as compared to H in a homogeneous metal where the macroscopic den- sity modes would appear near q =0 and would be unresolved on the scale of Fig. 14. Furthermore, criti- cal fluctuations in the H-superlattice system suffer quite a different fate than in the homogeneous metal- H systems: the superlattice completely suppresses fluc- tuations having wavelengths shorter than the super- lattice period. Thus, phase separation does not occur and the critical behavior consists of a transition to a state of density fluctuations only.

The conclusion that the coherent nature of the phase transition dominates in a H-superlattice sys- tem, in contrast to homogeneous H-metal systems where the coherent part of the transition appears only near T~, can be illustrated both in real space and recip- rocal space. In real space, the superlattice exhibits many interfaces, or equivalently, the "surface to vol- ume ratio" is large (i.e. the boundary conditions are applied many times per unit volume). In general, the wavevectors describing critical fluctuations range

2re from 0 to - - , where a is on the order of interatomic

a

distances. For a homogeneous metal-H system, the 2re

macroscopic density modes appear for q ~ - where

L is on the order of the sample size. By comparison, the superlattice density modes extend to much higher

2re q values, on the order o f ~ - , thereby affecting a larger

portion of the available q space. In an effort to explore the interplay of the super-

lattice with the critical behavior of the hydrogen den- sity, several samples were studied with low hydrogen densities. The average lattice expansion due to hydro- gen was always below 0.01 so that there was no chance of encountering the structural instability which occurs at higher lattice expansions [13]. In to- tal, three samples exhibited the critical behavior, each with nearly the same total film thickness, ranging from 4500 ~ to 5200 ~.

A scan for an 86 ~ superlattice (SL 23; Ta frac- tion: ao=0.37) which displays critical behavior is shown in Fig. 15 for two temperatures, above and

1500

d

1000

g O o

5OO

- - 166oc

�9 2 1 1 o c

oM 1635 16.65

I I 17.15 17.65

28

18.15 18.65

Fig. 15. X-ray scan which includes the + 2 satellites of an 85 ~k superlattice (SL 23; a0=0.37) with hydrogen shows weak critical behavior where a small amoun t of diffuse scattering appears only between s = • 1 and the fundamental reflection (s = 0)

below the critical temperature. Diffuse intensity is found between the fundamental and first satellites ref- lections, however, the effect is much smaller than what was observed in Fig. 13. While the superlattice period- icity, A, is the same as for Fig. 13 (SL 19), the Ta fraction is larger, providing a thicker intervening Ta spacer as well as reducing the amount of Nb partici- pating in the phase transition. As demonstrated in Fig. 16a, the relative integrated intensity of the broad contribution to the total intensity is greatly reduced as compared to the SL 19 which is replotted in Fig. 16c.

The third sample (SL 24) which showed critical behavior had a periodicity of 113/~ and a Ta fraction of 0.37, giving a total Nb layer thickness equal to that of the first sample. A longitudinal scan through the (110) reflection, shown in Fig. 16b, revealed a sub- stantial peak broadening, together with a line shape asymmetry at low angles. The integrated intensity of the broad component relative to the total integrated intensity is the same as for SL 19. The peak width of the longer wavelength sample, however, is narrower

- a trend which is expected since fluctuations are sup- pressed at the first superlattice harmonic. This pro- vides direct evidence that the superlattice periodicity is responsible for modifying the critical behavior.

To determine the temperature dependence of the diffuse scattering, the minimum intensity between the s--0 and s= +1 reflections is plotted in Fig. 17 for sample SL 24. The phase transition occurs over a tem- perature range of 30 degrees with a well defined satu- ration below 200 ~ C. Below this temperature the peak broadening does not continue to develop into two peaks, indicating a coherent phase transition. The critical temperature for this sample is 220 ~ C, which is close to critical temperatures of the other two sam- ples, although an accurate determination of these

470

2500

2OOO K) O

x 1500

E

I000 == O c..)

5O0

(a)

16.85 17.65

I I I

- - i i

17.05 17.25 17.45 28

80C I I I I

(b)

m i

i,o 600 I .

O i i

X

"~ 400

c

o 200

0 16.8 17.0 17.2 17.4 7'7.6 17.8

20

8OO (c) i l i i

i 1

% 6 0 0 - - , �9 - -

X �9

E 4 0 0 - (.9

C

0 o 200-

0 J ' 16.8 17.0 17.2 17.4 17.6 17.8

20

Fig. 16a--c. The fundamental reflections are shown for three super- lattices which display critical behavior. The peaks are each fit to three Gaussian functions: one gives the narrow component to the line shape, while the other two combine to represent a broad, asym- metric component to the lineshape. Data points in the region of the substrate reflection have been omitted for clarity. In (a), the broad contribution is rather small, indicating rather weak critical behavior for this sample (SL 23) which has a thin Nb layer thickness. (b) shows the result for a 113 A. superlattiee (SL 24). (c) The inset of Fig. 13a is replotted (SL 19). This 85 A superlattice has the same Nb layer thickness as in (h), however, the width of the broad contri- bution to the lineshape is broader, in accordance with the depen- dence on superlattice periodicity as discussed in the text

12000

E 8000 E

c

O

o 4000

I i i l l l l l l l l l l l l l l l l l l I I i i t l l

�9 � 9

lll[IIlllIIIllIlllll lllIIII

0100 150 200 250 300 350 4 0 0

T(C) Fig. 17. Temperature dependence of the min imum intensity between the s = 0 and s = +1 reflections for SL 24. These results show that the critical temperature is 220~ and that the critical behavior does not continue below 190 ~ C

Table 3. Relationship between superlattice properties and critical behavior. B(20) is the full width at half max imum (degrees) of the broad contribution to the line shape

Sample A(•) ao t~b(A) IB,o,a B(20) B(20) x A /Total (1 --a0)

SL 19 86 0.17 71 0.64 0.18 19 SL 24 113 0.34 74 0.66 0.11 20 SL 23 86 0.37 54 0.23 0.10 14 SL 18 85 0.57 37 0.00 - - SL 22 57 0.37 36 0.00 -- -

temperatures was not attempted. The sample exhibit- ing the weakest critical behavior (SL 23) may have a slightly lower critical temperature, around 190_+20 ~ C. In any case, these critical temperatures are 10 to 40 degrees higher than those for bulk Nb. Furthermore, at these relatively low concentrations, the phase boundary in bulk Nb occurs at much lower temperatures of roughly 150 K than the presently ob- served critical temperatures. It is not clear whether this profound change in the phase diagram results from the superlattice periodicity or from the coupling of the film to the substrate. It is conceivable that through epitaxial constraints the critical concentra- tion in thin films might be shifted to values much lower than in the bulk.

The results for all samples are shown in Table 3. The "strength" of the critical behavior, as determined by the integrated intensity of the broad contribution relative to total integrated intensity of the fundamen- tal peak, depends strongly on the Nb layer thickness. The minimum layer thickness for which the phase

transition can still occur is somewhere between 37 and 54/~. It is certain that no phase transition occurs above 25 K for SL 18 with a Nb layer thickness of 37 ~, indicating that critical fluctuations diminish abruptly with decreasing Nb layer thickness. A possi- ble reason for the absence of critical behavior in thin Nb layers might be found in the interdiffusion of Nb with Ta at the interface, although the present x-ray results indicate that if interdiffusion exists it must be very small and on the order of 2 atomic layers. Non- etheless, should interdiffusion play a role, it is well known that the phase boundary becomes suppressed in alloy systems [27, 28]. It is also interesting to note that two samples having the same Nb layer thickness exhibit the same broad contribution to the Bragg in- tensities, even though both have different superlattice periodicities A.

C. The H - H interaction in the superlattices

Despite the absence of a lattice gas-liquid phase tran- sition for H in bulk Ta, the H - H interaction in Ta is attractive. The lack of this phase transition is due to a preceeding transition to an ordered phase and it is estimated that if the gas-liquid transition could be measured, the long range H - H interaction in Ta would yield a critical temperature roughly 200 K lower than in Nb [29]. Therefore, the Ta in the super- lattice can be thought of as being structurally similar to Nb while providing a modulation of the H - H interaction. As noted above, the absolute values of these interaction energies may be altered by epitaxial forces which are present in a thin film environment. Nevertheless, they may still serve as reference to ac- count for the difference between the H - H interaction energies in the Nb and Ta layers of the superlattice. Thus, the following discussion of the H - H interac- tion in the superlattices will be within this context.

The existence of critical behavior is determined by the average H - H interaction represented by its zero wavevector component, while the manner in which the phase transition proceeds (i.e. coherent ver- sus incoherent) is determined by the larger wavevector components of the H - H interaction. For a large Nb fraction, bo, the superlattice will average the Ta con- tribution for long wavelength fluctuations such that the H - H interaction will behave as in pure Nb. In the limit of an infinitesimally small Ta layer thickness (i.e. b0 ~ 1), we expect that hydrogen density fluctua- tions will encounter only a small perturbation, affect- ing Tq at wavevectors corresponding to the superlat- tice periodicity. This is schematically shown by the dashed curve in Fig. 18, Such behavior is analogous to the problem of an electron moving in a weak peri-

471

]-ql

[ I

I

l

1

Fig. 18. Limiting behavior of the spinodal temperature as the struc- ture approaches pure Nb (top solid curve). The lower curve is for the case of large Nb fraction (b0). As the Nb fraction increases, the supeflattice periodicity affects a smaller portion of q space. The dashed curve is for the limit of infinitesimally thin intervening Ta layers, where the H - H interaction is affected only at the superlat- tice harmonics

odic potential where a small gap appears at the Bril- louin zone boundary. Thus, another way of viewing the limit b o ~ 1 is to realize that the number of q modes which are significantly affected in reciprocal space will become smaller with decreasing Ta pertur- bation.

In this limit we propose a simple scaling procedure for the H - H interaction: any hydrogen density fluc- tuation of wavelength 2 in Nb, will, on the average, be increased by a factor of 1/b o due to the intervening Ta layers. To achieve the same amplitude of fluctua- tion, the wavelength must be rescaled: 2 ~ 2/bo. Refer- ring to Fig. 18, this rescaling provides a rough esti- mate of how the shape of Tq versus q changes as a function of bo at small q values, and predicts that the width of the diffuse scattering in reciprocal space should scale as bo/A. As shown in Table 3, the two samples which have the same Nb thickness are in excellent agreement with this scaling argument. The other sample (SL 23) shows a deviation which is most likely due to the fact that the critical fluctuations are rather weak. Here, a smaller than predicted width of the critical fluctuations occurs (Fig. 18) which is not quantitatively accounted for in this simple scaling ar- gument. However, the discrepancy is in the expected direction.

We now summarize the wavelength dependence of the H - - H interaction. At long wavelengths 2 >> A, Tq reflects the average interaction, while for wa- velengths 2 ~A, it is sensitive to the differences be- tween the two-body interaction of the Nb and Ta layers. At the higher superlattice harmonics, To should continue to depict the difference between the materi- als until the wavelength becomes shorter than the thinnest layer. These shorter wavelengths will then

472

probe the intra-layer interaction, therefore, reflecting the behavior of the bulk constituents. Thus, there are three regimes in which to consider the two-body inter- action for a fluctuation of wavelength, 2, in a superlat- tice with periodicity A:

{I} 2>> A. Long wavelengths represent a two-body interaction, Tq, mediated by the average host material.

{2} 2 ~< A. Intermediate wavelengths probe the dif- ferences in the H - H interaction between Nb and Ta.

{3} 2<~A, where ~ is the Nb fraction (b0) or the Ta fraction (ao), whichever is smallest. These short wavelengths reflect the interactions manifested in the bulk constituents since the Fourier transform probes within one layer.

These properties are consistent with the values of T~ measured in the high temperature experiments summarized in Table 1 and the parameters for the critical fluctuations are listed in Table 3. Most sam- ples have a negative T1 and many have negative high- er harmonics - all cross over to positive values on the order of 200 K when s> l /a o. The only sample which has a positive, but small value of T1 (SL 18) also has a majority Ta fraction, ao =0.57. In this case, the differential behavior probed by the first harmonic should, in fact, be positive.

VI. -Summary

Three thermodynamic states have been encountered for hydrogen in the superlattice. At high tempera- tures, the hydrogen density is in a para-elastic phase which is modulated by the "applied" superlattice field. Despite the modulation, this phase is akin to the homogeneous phase of hydrogen in bulk metals. At higher concentrations, the one-dimensional lattice expansion becomes unstable and the system phase separates into various orientationally distorted phases. The structural instability as well as the change in the phase diagram for the thin film, is a conse- quence of the film-substrate interaction and these ef- fects can be isolated from the influence of the superlat- tice periodicity on the lattice gas properties. Provided that the concentration remains low to avoid this in- stability, the hydrogen can exhibit a lattice gas-liquid phase transition which is coherent and is strongly affected by the superlattice properties, since, the hy- drogen density fluctuations interact with the artificial superlattice periodicity.

We have demonstrated that hydrogen in the su- perlattice behaves as a lattice gas subject to modulat- ed interactions and can be described by a modulated lattice gas model. X-ray scattering experiments in the para-elastic regime provide quantitative information

regarding the fundamental interactions in the system. Some superlattices containing hydrogen exhibit criti- cal behavior which originates from the long-range, elastically mediated, H - H interaction. However, this interaction is modified by the superlattice periodicity and suppresses critical fluctuations for wavelengths shorter than a superlattice period. Consequently, phase separation into distinct gas and liquid phases cannot occur. The spacial dependence of the H - H interaction obtained from the high temperature ex- periments is entirely consistent with the observed crit- ical behavior. These experiments clearly demonstrate the sensitive role which the spatial distribution of the two-body interaction plays in determining critical be- havior.

The critical behavior in the superlattice is also consistent with previous theoretical and experimental work on the coherent phase transition of H in bulk Nb. In the bulk system, a coherent gas-liquid transi- tion affects critical fluctuations having macroscopic wavelengths. For the superlattice, we find coherent critical behavior on the length scale of the superlattice period. An important difference is that the superlat- tice appears to completely suppress critical fluctua- tion having wavelengths shorter than the superlattice period, thus having the rather practical consequence of not allowing phase separation. Another way to look at the difference between the bulk and superlat- tice systems is to recognize that the wavcvector range

relevant to the problem is for 0<]q[__< z~, where a a

is the interatomic distance. For a bulk coherent tran- sition, all of the affected fluctuations appear relatively close to q--0 and comprise a small portion of q-space. In contrast, the superlattice bears its impact out to

much higher q, on the order of ~--~, therefore, more

profound consequences are expected.

We wish to thank J. Dura, J.E. Cunningham and S.M. Durbin for their skillful preparation of the MBE superlattices and C.P. Flynn for enlightening discussions. This work was supported by the U.S. Department of Energy, Division of Materials Sciences, under grant No. DE-AC02-76ER01198.

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H. Zabel Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1110 W. Green Street Urbana, IL 61801 USA

P.F. Miceli Bell Communications Research Red Bank, NJ 07701 USA