Lattice vibrational, thermal and mechanical properties of CuInSe2

Solar Cells, 16 (1986) 399 - 418

399

LATTICE VIBRATIONAL, T H E R M A L AND MECHANICAL PROPERTIES OF CuInSe 2

H. NEUMANN

Sektion Physik, Karl-Marx-Universitat, DDR- 7010 Leipzig (G.D.R.) (Received June 5, 1985)

Summary Experimental data for the lattice vibrational properties, thermal properties such as thermal expansion, specific heat capacity and thermal conductivity and mechanical properties of CuInSe 2 are critically reviewed and partly completed by recent measurements. In those cases where experimental measurements are missing theoretical estimates of corresponding material parameters are discussed with regard to their reliability.

1. Introduction The lattice vibrational properties, thermal properties such as thermal expansion, specific heat capacity and thermal conductivity and mechanical properties of a given c o m p o u n d are closely interrelated because all of these material parameters are essentially determined by the nature of the interatomic forces [1, 2]. For this reason, the present paper pursues two objectives. The first is to give a critical review of existing experimental data regarding these material properties for CuInSe2. In some cases these results are completed by our own recent measurements. The second objective is to consider to what extent it is possible to derive interatomic force models which, on the one hand, explain the experimental results and, on the other hand, enable us to estimate material parameters hitherto unknown. It is shown that the thermal properties of CuInSe2 reported by various authors agree with each other and can be interpreted in terms of the same lattice anharmonicity model for the interatomic forces. With regard to the vibrational properties of CuInSe2 the situation is less satisfactory. The main aim of the present paper is to discuss in detail the discrepancies in the results published in the literature and to determine reliable values for the frequencies of the high-energy zone-centre optical phonons. It is shown t h a t theoretical calculations based on a simplified version of the valence-force field model of Keating may be helpful in assigning observed structures in the phonon spectra. Finally, some remarks are made concerning the measured or calculated mechanical properties of CuInSe: reported so far. 0379-6787/86/$3.50

© Elsevier Sequoia/Printed in The Netherlands

400 2. Lattice vibrations 2. 1. F u n d a m e n t a l lattice m o d e s The primitive unit cell of the tetragonal chalcopyrite lattice (space group D2a 12) of CuInS% contains two formula units, meaning that there are three acoustic and twenty-one optical phonon branches. A complete group theoretical analysis of the lattice vibrations of ternary chalcopyrite compounds, including all important symmetry points and lines of the Brillouin zone, has been given in ref. 3. Since all the experimental data reported for CuInS% relate to the long-wavelength vibrational modes at the centre of the Brillouin zone we restrict our considerations to this case. Then the corresponding irreducible representation of the phonon normal modes is given by

Pac = 1B 2 + 1E for the acoustic modes and Fo~t = 1A1 + 2A2 + 3B1 + 3B2 + 6E for the optical modes. Of the optical modes the A 1, B1, B2 and E modes are Raman active, the B 2 and E modes are infrared active for incident radiation polarized parallel and perpendicular to the tetragonal c axis of the crystal respectively and the A2 modes are inactive. So far, most of the experimental information regarding the optical vibrational modes of CulnSe2 comes from infrared optical measurements and is therefore related to the B2 and E modes. Before discussing existing experimental results, the experimental conditions required to obtain reliable vibrational mode data will be considered. In numerous experiments with CulnS 2 [4] and CuInSe2 we have found that mode leakage effects due to imperfect polarization conditions in the optical measurements can result (i) in incorrect mode assignments, and (ii) in mode frequency shifts in those cases where modes with nearly the same oscillator strength and frequency are present in both polarization directions. Reproducible results for the anisotropy of vibrational mode behaviour can only be obtained when single crystals with the tetragonal e axis oriented parallel to the reflecting surface plane are used and when the measurements are performed with radiation polarized perpendicularly to the plane of incidence. If these conditions are fulfilled the reflectivity spectra of CuInSe z shown in Fig. I are obtained. Obviously, these spectra enable us to determine the parameters for all three B 2 modes, but only for three of the six E modes expected from group theory. If measurements are made with single crystals having that (112) face as the reflecting surface plane, the rather strong low-energy B 2 mode appears also in the E mode spectra [ 5]. Furthermore, in measurements with unpolarized radiation or with polycrystalline samples it becomes difficult to resolve the two highenergy E modes. To determine the transverse and longitudinal optical mode frequencies ~TO and ~LO, the oscillator strengths sj and the damping constants 7j of the modes and the high-frequency dielectric constant e=, the reflectivity spectra

401 i

i

I

I

I

[

E 200

I

06 f 0.4

9.2,[O 96 94 02 0

r

100

(cm-1)

300

Fig. 1. Typical polarized infrared reflectivity spectra of CuInSe 2 at 300 K.

TABLE 1 Parameters of the B 2 and E modes in CuInSe 2 derived from the spectra of Fig. 1. Mode

/~TO ( cm-1 )

/~LO(cm-1)

sj

7i

B21 B22 B23 e~ll = E1

214 181 64

232 193 65

0.89 1.69 0.40

0.025 0.098 0.019

213 207 179

229 212 183

0.38 1.02 0.66

0.024 0.032 0.102

7.80

E2 E3 e ~ l = 7.59

a r e u s u a l l y a n a l y s e d in t e r m s o f a m u l t i h a r m o n i c using a complex dielectric function of the form e(~) = e~ + j=l L

sj~f ~j2__ ~2__

damped

oscillator model

(1)

w h e r e n is t h e n u m b e r o f l a t t i c e o s c i l l a t o r s . I f t h e s p e c t r a o f F i g . 1 a r e a n a l y s e d in t h i s m a n n e r w e o b t a i n t h e m o d e p a r a m e t e r s g i v e n in T a b l e 1. M e a s u r e m e n t s o n five d i f f e r e n t c a r e f u l l y p r e p a r e d c r y s t a l s s h o w e d t h a t t h e m o d e p a r a m e t e r s g i v e n in T a b l e 1 a r e r e p r o d u c i b l e w i t h i n -+ 1 c m -1 f o r t h e f r e q u e n -

402 cies I~TO,LO, within -+10% for sj, within -+ 20% for 7i and within -+ 4(/~ for e~. T h e r e f o r e , we c o n c l u d e t h a t the m o d e frequencies o f Table I are the m o s t reliable values available at present. Table 2 gives a c o m p i l a t i o n o f the frequencies o f the B 2 m o d e s and the t h r e e high-energy E m o d e s of CuInSe 2 published to date. T o facilitate the s u b s e q u e n t discussion, d i f f e r e n t m o d e s o f the same s y m m e t r y are m a r k e d with an u p p e r i n d e x which increases with decreasing f r e q u e n c y . Considering the frequencies o f the B~ 1, B23, E 1 and E 2 m o d e s we see t h a t e x c e p t for the data r e p o r t e d b y Gan e t aI. [7] the results o f d i f f e r e n t a u t h o r s agree within the possible a c c u r a c y in the d e t e r m i n a t i o n o f m o d e frequencies f r o m the optical spectra. In the case o f the B22 and E 3 m o d e s the results are always in t h e same f r e q u e n c y range b u t the scatter in the data is larger than for the o t h e r modes. T o clarify the origin o f this e f f e c t we carried o u t special test e x p e r i m e n t s with single crystals o f d i f f e r e n t o r i e n t a t i o n and r e c o r d e d the spectra with polarized and u n p o l a r i z e d radiation. We f o u n d t h a t the structures in the reflectivity spectra caused b y these m o d e s are particularly sensitive to deviations f r o m ideal polarization c o n d i t i o n s and t h a t f r e q u e n c y shifts o f + 6 cm -1 f r o m the values given in Table 1 can be easily o b t a i n e d by changing the crystal o r i e n t a t i o n , whilst the changes in the frequencies o f the o t h e r m o d e s are o n l y in the range o f +- 2 cm -1. T h e frequencies for the remaining t h r e e low-energy E m o d e s and for the A 1 and B 1 m o d e s have been d e t e r m i n e d b y Gan e t al. [7] f r o m polarized R a m a n scattering spectra. However, f r o m an e x a m i n a t i o n o f the data compiled in Tables 1 and 2 it is evident t h a t only the B22 and E 3 m o d e frequencies o f ref. 7 are in the e x p e c t e d range whilst the frequencies o f the B21, E 1 and B23 m o d e s are well above those f o u n d to be characteristic o f CuInSe 2. F u r t h e r m o r e , the frequencies of the B21 and E 1 m o d e s r e p o r t e d in ref. 7 (see Table 2) are very close to those f o u n d in CuGaSe2 (~TO,Lo(B21) = 2 5 4 / 2 7 8 cm -1, ~WO,Lo(E l) = 2 5 0 / 2 7 6 cm -1 [11] ) and AgGasSe 2 (~ro,Lo(Bz 1) = 2 4 8 / 2 6 9 cm -1, ~TO,Lo(E 1) = 2 5 0 / 2 7 4 cm -1 [12] ). To a c c o u n t for this fact and for the general trends in the lattice vibrational p r o p e r t i e s o f the I III VI 2 comp o u n d s [13] it must be c o n c l u d e d t h a t Gan e t al. [7] have m a d e their meas u r e m e n t s n o t with CuInSe 2 b u t with CuGaSe 2 single crystals. F u r t h e r evid e n c e for this supposition c o m e s f r o m Table 3. We see t h a t every B 2 and E m o d e f r e q u e n c y f o u n d by Gan e t al. [7] and ascribed to CuInSe 2 has its c o u n t e r p a r t in the vibrational s p e c t r u m o f CuGaSe;. However, the experim e n t a l d a t a r e p o r t e d b y Gan e t al. [7] for the (CuInSe2)l_x(2ZnSe)~ m i x e d crystal system can be successfully e m p l o y e d to gain additional i n f o r m a t i o n regarding the vibrational p r o p e r t i e s o f CuInSe 2. No essential changes in the optical m o d e frequencies in this system are e x p e c t e d for x = 0 to x = 0.08 for CuInSe> Indeed, to a c c o u n t for the e x p e r i m e n t a l observation t h a t the E 1 and E 2 m o d e s are usually n o t resolved in u n p o l a r i z e d reflectivity spectra, the infrared d a t a for (CuInSe2)0.92(2ZnSe)0.0 s given in ref. 7 (see the last row in Table 2) are in g o o d a g r e e m e n t with the m e a s u r e m e n t s for CuInSe> This result e n c o u r a g e d us to analyse also the polarized R a m a n scattering spectra for this m i x e d crystal c o m p o s i t i o n (see Fig. 3 o f ref. 7). T h e frequencies esti-

403

t'--

¢,D

r~

r-~ - ~ ~0 r--4 ¢X~ t'-t~ r'~

C'q

¢q Cq

¢.D O C-I

O CO Cq

t--- C'q C',l

~L ¢ q cO

r~

u~ ¢D

¢q

?

O t'--

~L

~9

¢q

r,D tt~

r'~

t'--

r'~

-.... O0 O0

r'~ O0 r-~

¢,D cO ~'q cO

O

=:o d~

0

d~ ¢D ¢J

C'q

<

¢j

?

404 m a t e d f r o m the s t r u c t u r e s o b s e r v e d in these s p e c t r a are c o m p i l e d in T a b l e 4. When t h e d a t a o f T a b l e 4 are c o m p a r e d with t h o s e o f Tables 1 and 2 we see t h a t the R a m a n s c a t t e r i n g values o f t h e t h r e e B 2 m o d e f r e q u e n c i e s and the t w o h i g h - e n e r g y E m o d e f r e q u e n c i e s are in g o o d a g r e e m e n t with t h e infrared data. T h e t w o l o w - e n e r g y s t r u c t u r e s at 51 and 73 c m -1 a p p e a r i n g o n l y in t h e x ( y z ) 2 s p e c t r u m should be d u e to the E 6 and E s m o d e s respectively, since t h e E 4 m o d e is e x p e c t e d to have a f r e q u e n c y n e a r 135 c m -1 [ 1 3 ] . T h e s t r o n g p e a k at 178 c m -1 o b s e r v e d in x(zz)2 and z(yy)~ is e v i d e n t l y d u e to the A 1 m o d e , whilst t h e w e a k s t r u c t u r e at 165 c m -1 f o u n d o n l y in t h e z(yy)~ spect r u m is c e r t a i n l y d u e to a B 1 m o d e . A c o m p a r i s o n w i t h t h e o r e t i c a l l y calculated f r e q u e n c i e s o f these m o d e s (Table 5) c o n f i r m s these assignments. Finally, the a p p e a r a n c e o f s t r u c t u r e s at 63 and 236 c m -1 in x(zz)2 can o n l y be e x p l a i n e d b y d e p o l a r i z a t i o n effects, and p r o b a b l y the s a m e a r g u m e n t applies to t h e s t r u c t u r e s at 63 and 241 c m -1 o b s e r v e d in z(yy)~. H o w e v e r , it c a n n o t be ruled o u t t h a t the s t r u c t u r e at 241 c m -1 is in f a c t d u e to a B~ m o d e as e x p e c t e d f r o m the scattering g e o m e t r y b e c a u s e the t h e o r e t i c a l freq u e n c y o f t h e B11 m o d e (see T a b l e 5) is very close to this value. Thus, we can state t h a t reliable m o d e p a r a m e t e r s are n o w available f r o m infrared r e f l e c t i v i t y m e a s u r e m e n t s for all the t h r e e B 2 m o d e s and the t h r e e high-energy E m o d e s o f CuInSe2. With regard t o t h e A~ m o d e and the t w o highe n e r g y B1 m o d e s it can be s h o w n t h a t their calculated f r e q u e n c i e s are v e r y close to t h o s e derived f r o m R a m a n s c a t t e r i n g s p e c t r a o f (CuInSe2)0.92(2ZnSe)0.0s. T h e r e f o r e , it s e e m s t h a t t h e f r e q u e n c i e s o f these m o d e s are c e r t a i n l y c o r r e c t

TABLE 3 Comparison of the results of ref. 7 ascribed to CuInSe2 with experimental data reported for CuGaSe 2 [11, 14]

Mode

AI B21 B22 B2a E1 E3 E4 E5 E6 aR, Raman scattering. bIR, Infrared reflectivity.

Mode frequencies (cm -1) Reference 7

References 11 and 14

186 (R) a 248/274 (IR) u 252]275 (R) 188/196 (IR) --/194 (R) 96 (R) 248/274 (IR) 252/275 (R) 188/191 (IR) --/191 (R) 153 (R) 78 (R) 61 (R)

187 (R) 254/278 (IR) 178/196 (IR) 96 (a) 250/276 (IR) 249/274 (R) 170/190 (IR) 154 (R) 86 (R) 60 (R)

405 TABLE 4 Mode frequencies estimated from the Raman scattering spectra of (CuInSe2)0.92(2ZnSe)0.0s reported in ref. 7

Scattering geometry

Expected mode symmetry

Mode frequencies (cm-1)

x(yz)~ z(xy)~ x(zz)~ z(yy)~

E B2 A1 A1, B1

51, 63, 63, 63,

73, 182, 236 185, 236 178, 236 165, 178, 241

TABLE 5 Theoretical estimates for some mode frequencies in CuInSe2 according to ref. 13 Mode P(cm-1)

A1 174

A21 196

A22 149

Bll 239

B12 165

E4 135

within + 5 cm -1. Further Raman scattering, and perhaps also neutron scattering measurements are necessary to determine the frequencies of the three low-energy E modes and the low-energy B~ mode. Finally, we note that the high-frequency dielectric constants determined from the infrared reflectivity spectra (Table 1) are in good accord with the value of e . = 7.78 obtained from optical transmission spectra of thin polycrystalline films of CuInSe 2 [15].

2.2. T w o - p h o n o n absorption spectra Recent studies of the two-phonon spectra of CuInS2 [4] and of some II-IV-V 2 chalcopyrite compounds [16] have shown that they can give additional information concerning the frequencies of fundamental vibrational modes, especially the silent A2 modes and the Raman active B1 modes which usually have low scattering efficiencies. In our previous investigations no polarization dependence of the two-phonon absorption spectra of CuInSe2 could be found and only five structures could be clearly identified in the spectra [17]. When measurements were repeated with carefully oriented single crystals having the c axis within + 2 ° parallel to the surface plane of the samples, we were able to reveal the polarization dependence of the twophonon absorption s p e c t r a . T h e resulting room temperature spectra (Fig. 2) exhibit nine structures for EII c" and seven structures for E J_~. At 80 K we found (i) an overall decrease of the absorption coefficient compared to that at 300 K and (ii) a shift of all structures to higher frequencies by about 5 cm -1. This result indicates that all the structures observed can be ascribed to phonon modes.

406 'zgr I

q i

:

!

~ i !!

'; I \1

5°i i

i

,~.j

1 i

o, , /-~ \i

i

/

i

i ,a

I\,

]

y

,

P,

'1

/ ,

7 [ i

?O:

350

_ _ &,30

A__ &50

9 c~Ell

Fig. 2. Polarized two-phonon absorption spectra of CuInSe2 at 300 K. For clarity the curve for E II~is shifted to higher &values by A& = 30 cm-I. To assign the structures in the absorption spectra we have to account for the selection rules for infrared active two-phonon combination modes. Since our knowledge is restricted to zone-centre phonons, only combination modes of these phonons will be considered. Then the selection rules are A1 × B2, A2 × B1 and E X E for/~[[ 5, and A1, 2 X E and B1, 2 X E f o r / ~ ± c ' . Using the fundamental mode frequencies given in Tables 1 and 5 we were able to explain all the structures observed in the absorption spectra f o r / ~ II ~" (Table 6) and / ~ ± ~ (Table 7). It can be seen from Table 6 that the two-phonon spectra for E II ~" give no new information about the A1, A2 and B1 modes because all the structures can be explained by E × E combination modes. The most interesting result is that the structures 2 and 3 can only be interpreted by E × E combinations which involve the E 4 mode. In view of the good agreement between the calculated and experimental frequencies of these structures (see Table 6) it can be concluded that the calculated frequency of the E 4 mode ~Table 5) is certainly correct within + 2 cm-< In the ease of the spectra for E ± c-*, the structures 1, 2 and 7 can only be explained if A2 X E and B1 × E combination modes are included. For the structures 1 and 7 the estimated frequencies of both the Bx2 and B11 modes (Table 5) have the right magnitude, whilst it follows from structure 2 that the frequency of either A21 or A2 2 mode has been evaluated correctly. There is no additional information regarding the frequency of the A 1 mode proposed in the previous section. Although A 1 X E combination modes may contribute to the structures 4 and 5 in the spectra f o r / ~ ± ~ (see Table 7) these structures can be also explained by B~ X E combination modes. Summarizing the results, we see that the analysis of the two-phonon absorption spectra confirms the assumptions made in the previous section with regard to the frequencies of the E 4, B11, B12 and A21 or A22 modes. The rapid

407 TABLE 6 F r e q u e n c i e s and p r o p o s e d assignments of the structures in the t w o - p h o n o n absorption spectra of CuInSe2 for ETII c+

Structure

Pexp(crn -1)

Assignment

Pcalc(Cm-1)

1

316

2 3

340 349

4

395

5 6

411 427

7

438

8

443

9

460

A22 + B12 E 4 -}- ETO 3 E4 + ELO3 E4 + ETO2 E 4 + ELO2 E 4 + ETO 1 ELO3 + ELO2 ELO3 + ETO 1 ELO3 + ELO 1 ELO2 + ETO 1 2ETo 1 A21 + BtI ETO2 + ELOI ELO2 + ELO 1 ETO 1 + ELO 1 2ELo 1

314 314 318 342 347 348 395 396 412 425 426 435 436 441 442 458

TABLE 7 Frequencies and proposed assignments of the structures in the t w o - p h o n o n absorption spectra of CuInSe2 for 1~±~

Structure

Pexp(Cm-1 )

Assignment

Pcalc(cm-1)

1 2

300 332

3

347

4

385

5

405

6

428

7

465

B2 + E4 A21 + E 4 A22 + ELO 3 Bt 2 + ELO3 B2TO1 + E 4 A! + ELO2 A 1 + ETO 1 B2TO2 + ETO2 A1 + ELO 1 B2LO2 + ELO2 B2LO2 + ETO 1 A21 + ELO 1 B2TO1 + ELO2 B2TO1 + ETO 1 BI l + ELO 1

297 331 332 348 349 386 387 388 403 405 406 425 426 427 468

increase o f the absorption coefficient due to the fundamental B21 and E 1 modes prevents the det ect i on o f any structures in the spectra below about 280 cm -1. Therefore, we are unable to extract any i n f o r m a t i o n regarding the low-energy B13, E 5 and E 6 modes from t he t w o - p h o n o n absorption-spectra

4o8 because the frequencies of these modes are expected to have values below about 80 cm -1.

3. Thermal properties

3.1. Thermal expansion The thermal expansion behaviour of CuInSe2 has been studied by various authors and thermal expansion coefficients are now available for the temperature range 30 - 1200 K. The experimental methods used are dilatomerry on polycrystalline samples [18 - 23], measurements of the change in density with temperature [24], the X-ray powder technique [25- 28] and the X-ray Bond technique [29]. Before discussing the experimental data we should mention that the chalcopyrite lattice has axial symmetry, which means that the lattice parameters a and c vary independently with temperature [30]. Thus, a complete characterization of the thermal expansion behaviour of CuInSe: requires the determination of the two principal thermal expansion coefficients (~a and ac of the lattice parameters a and c respectively. Most of the reported experimental results apply to the temperature range above 300 K. Measurements over sufficiently large temperature ranges have clearly shown that the thermal expansion coefficients remain temperature dependent up to the highest temperatures investigated [20 - 23, 27]. According to the fairly accurate X-ray diffraction studies of Kistaiah et al. [27], performed in the temperature range 301 - 839 K, only ~a varies with temperature according to the relation C~a(K-1) = 8.3182 X 10 -6 + 3.3891 X 10-9(T - To) + 7.7734 X 10-12(T-- T0) 2

(2) with T in K and To = 273.15 K; ac remains constant and equal to c~¢ = 7.89 X 10 -6 K -1 over the temperature range considered. The temperature variation of c~a results in a corresponding temperature dependence of the average linear thermal expansion coefficient ~L = (2~a + C~¢)/3 which has been confirmed in dilatometric measurements on polycrystalline samples [ 2 0 - 2 3 ] . In many other investigations over less extended temperature ranges it has usually been assumed that the thermal expansion coefficients are independent of temperature at T >~ 300 K [18, 19, 25, 26, 28]. To facilitate comparison of the results of various authors a compilation of all existing experimental data for the temperature range above 300 K is given in Table 8. In those cases where a variation of the thermal expansion coefficients with temperature has been proved, the corresponding oq intervals are given. In ref. 25 the temperature range covered in the measurements has not been specified, and therefore could not be included in Table 8. A comparison of the results of the X-ray measurements, taking into account the usual error of about -+ 10% in thermal expansion coefficient determinations,

409 TABLE 8 Thermal expansion coefficients (all values in 10-6 K) of CuInSe2 at T/> 300 K Method

Temperature range

aa

ac

aL

Reference

7.0 - 7.3 6.6 5.8 - 7.9 9.1-10.7 8.6

[18] [ 19 ] [20] [21-23] [24]

10.5 10.1 8.2-11.0 9.8

[25] [26] [27] [28]

(K) Dilatometry on polycrystalline samples

Density change with temperature X-ray powder technique

293 - 673 293 - 343 303 - 1073 293-1000 300 - 1200 >~ 300 300 - 573 301-839 293 - 713

11.4 11.0 8.4-12.7 10.8

8.6 8.4 7.89 7.7

s h o w s t h e a g r e e m e n t o f t h e d a t a is surprisingly g o o d . Also, t h e aLValues f r o m refs. 2 1 - 2 3 o b t a i n e d f r o m d i l a t o m e t r i c m e a s u r e m e n t s o n large-grain p o l y crystals agree with t h e X-ray d a t a . Slightly l o w e r aL values have b e e n f o u n d in refs. 18 - 20. T h e s e d e v i a t i o n s are p r o b a b l y caused b y t h e f a c t t h a t p o l y c r y s t a l line s a m p l e s m a y c o n t a i n w e a k l y b o n d e d particles, since small particle size a n d i s o t r o p y can p r o d u c e smaller aL values t h a n in t h e case o f single crystals [ 3 1 ] . With regard to t h e a n i s o t r o p y o f t h e t h e r m a l e x p a n s i o n o f CuInSe2 all X-ray m e a s u r e m e n t s s h o w t h a t aa ~ c~¢ o v e r t h e w h o l e t e m p e r a t u r e range a b o v e 3 0 0 K (see T a b l e 1). A similar result has b e e n f o u n d f o r all t h e o t h e r C u - I I I - V I 2 c o m p o u n d s (see ref. 32 and r e f e r e n c e s cited therein). T h e p h a s e d i a g r a m studies p u b l i s h e d b y various a u t h o r s h a v e s h o w n that CuInSe 2 undergoes a polymorphous phase transition from the chalcopyrite s t r u c t u r e t o t h e s p h a l e r i t e s t r u c t u r e at 1083 K and m e l t s at 1262 K [33 38]. In r e c e n t d i l a t o m e t r i c m e a s u r e m e n t s [22, 23] it has b e e n established t h a t at t h e p o l y m o r p h o u s p h a s e t r a n s i t i o n t e m p e r a t u r e a s h a r p p e a k a p p e a r s in t h e curve f o r t h e t e m p e r a t u r e d e p e n d e n c e o f t h e average linear t h e r m a l e x p a n s i o n coefficient. T h e p e a k m a x i m u m is at ~L ~ 8 × 10 -s K -1. A b o v e this t e m p e r a t u r e t h e t h e r m a l e x p a n s i o n c o e f f i c i e n t r e m a i n s c o n s t a n t u p to 1 2 0 0 K and is a p p r o x i m a t e l y equal to 2 × 10 -5 K -1 [22, 23]. Finally, a volu m e e x p a n s i o n c o e f f i c i e n t o f a b o u t a v = 7 X 10 -5 K -1 f o r t h e liquid p h a s e o f C u I n S e 2 can be e s t i m a t e d f r o m t h e d a t a r e p o r t e d in ref. 24. Thermal expansion coefficient determinations for temperatures below 300 K have b e e n r e p o r t e d in refs. 21 - 23, 26, 29. T h e m o s t reliable d a t a are t h o s e o f D e u s e t al. [29] based o n precision lattice p a r a m e t e r d e t e r m i n a t i o n s d o w n t o 30 K using t h e X - r a y B o n d m e t h o d . T h e aa versus T and ac versus T curves o b t a i n e d in this investigation are r e p r e s e n t e d in Fig. 3. It can be seen t h a t b o t h oLa and a c d e c r e a s e b e l o w 3 0 0 K and b e c o m e negative at 60 and 80 K respectively. This result is c o m p l e t e l y a n a l o g o u s t o t h a t f o r C u I n T e 2 w h e r e negative t h e r m a l e x p a n s i o n c o e f f i c i e n t s have also b e e n f o u n d at t e m p e r a -

410

tures below about 50 K [39, 40]. On the basis of the low-temperature behaviour of the thermal expansion coefficients in CuInSe2 some conclusions may be drawn regarding the lattice mode Gri~neisen parameters. In the case of the uniaxial chalcopyrite lattice there are two independent principal Grfineisen parameters 7a and ~'c which are related to the principal thermal expansion coefficients aa and a0 via Vm

a/a = ~

up

(3a)

[ ( C l l s ~- Cl2S)oh nk cI3SOLc]

and Ym

s

~'c = ~ [ 2 c 1 3 oh + ca2oh]

(3b)

where Vm is the molar volume, Cp the molar specific heat capacity at constant pressure and the c u s are the adiabatic elastic stiffnesses [30]. No experimental measurements or theoretical estimates of c u have yet been reported for CuInSe 2 which prevents a quantitative analysis of the temperature dependence of ~/a and 7¢- However, assuming that the inequality Cll > C33 > C12 ~ Ci3 > 0

(4)

is valid for chalcopyrite compounds [40] it follows from eqns. (3a) and (3b), Fig. 3, and the cq and a c values of Table 8 that, firstly, 7a > 7c over the whole temperature range for which anisotropic thermal expansion coefficients are available. Because the same inequality 7a > 7¢ has been found to be valid also in CuInTe: [40], AgGaS: and CdGeAs 2 [41], it may be suggested that all chalcopyrite compounds behave similarly with regard to their

1}r

-

~c

i

T

110 ""T

T-

T-

•

"

/

•

% ,~I

Y

!

"f

•

'I l

,,

I i

I i .

,c~

•

•

.r c~ 100

/

I l

L

10',3

2LC T[K]

L

7)0

3~

I

•

400

I

•

!,:,

~ T:LK;

Fig. 3. T e m p e r a t u r e d e p e n d e n c e o f t h e principal t h e r m a l e x p a n s i o n c o e f f i c i e n t s o f C u l n Se 2 b e l o w 3 0 0 K. Fig. 4. T e m p e r a t u r e d e p e n d e n c e of t h e m o l a r h e a t c a p a c i t y at c o n s t a n t pressure of CuInSe 2 a b o v e 300 K.

411 principal Gri~neisen parameters. Secondly, we see from eqns. (3a), (3b) and Fig. 3 that both 7a and 7c are positive above 80 K and negative below 60 K. Since 7a and % are weighted averages of the respective principal mode Grfineisen parameters ~/ai and %i, in which the contribution of the ~/ji of each normal mode to 7j is weighted by its (always positive) contribution to the total specific heat [30], the occurrence of negative values of ~a and 7c at low temperatures implies the existence of low-energy lattice vibrational modes with negative mode Gri]neisen parameters in CuInSe 2. This result is not unexpected for two reasons. Firstly, it is well known that CuInSe2 undergoes a pressure-induced phase transition to the denser rock salt structure in the pressure range 5 - 6 GPa [42, 43]. It can therefore be deduced that lowenergy soft modes must be present in the crystal [44]. Secondly, in studying the pressure dependence of the lattice vibrational properties of CuGaS2 [44, 45] and AgGaS2 [44] it has been established that besides the acoustic modes, the E mode with lowest energy also has a negative mode Grflneisen parameter. Unfortunately, since our knowledge regarding the low-energy branches of the phonon spectrum of CuInSe2 is as yet rather incomplete (see Section 2), no more detailed discussion of this point can be given at present.

3.2. Specific heat capacity To the best of my knowledge, the only study of the specific heat capacity of CuInSe2 reported in the literature to date is that of Bachmann et al. [46]. In that paper the heat capacity at constant pressure of CuInSe2 has been given for the temperature range 1 - 40 K. The data have been used to determine the Debye temperature, which has been found to be 0 D = 221.9 K, and to estimate the standard energies and entropies of the compound. From a consideration of the thermal expansion data of refs. 20 - 23, 27 (see Table 8 and eqn. (2)), it is evident that higher-order anharmonic contributions to the interaction potential of the lattice atoms play an important role in establishing the thermal properties of CuInSe2. Indeed, if the potential energy is, as is usually done, approximated by a power series in the displacements of the atoms from their equilibrium positions, it follows that terms up to the eighth order in the displacements must be included to obtain a quadratic dependence of the thermal expansion coefficient on temperature [1, 2, 47]. If the same model is used to derive the specific heat capacity at constant pressure, the contribution to the heat capacity due to anharmonic forces is, in general, given by a polynomial which contains terms up to the third order in the temperature [47]. Therefore, it should be expected that the specific heat capacity at constant pressure of CuInSe 2 follows a relation of the type Cp(T) = 12R[F(OD/T) + clT + c 2 T 2 + c3 T3]

(5)

and remains temperature dependent up to high temperatures in accordance with the temperature variation of the thermal expansion coefficients given by eqn. (2). In eqn. (5), R is the molar gas constant, F(XD) with XD = OD/T is the Debye function describing the temperature dependence of the specific

412 heat in t h e h a r m o n i c a p p r o x i m a t i o n [ 1] XD

F(XD) =

3 f -xD 3

x4eXdx (e x-

1) 2

(6)

and the coefficients c i are rather complicated expressions of the force constants introduced to describe the interatomic potential [47]. To verify the validity of eqn. (5) we have measured the specific heat capacity at constant pressure of CuInSe 2 in the temperature range 300-500 K (experimental details are published in ref. 48 and will not be repeated here); the resulting Cp versus T curve is shown in Fig. 4. The broken line represents the harmonic c o n t r i b u t i o n Cph = 12R F(OD/T) t o the t o t a l m o l a r h e a t c a p a c i t y Cp(T) calculated with 0D = 2 2 1 . 9 K. T h e n , analysing t h e a n h a r m o n i c part ACp = C p Cph o f the heat c a p a c i t y we f o u n d t h a t t h e t e m p e r a t u r e d e p e n d e n c e o f AC, can i n d e e d o n l y be described b y a p o l y n o m i a l c o n t a i n i n g t e r m s u p to at least t h e third o r d e r in the t e m p e r a t u r e . T h e values o f the coefficients c~ resulting f r o m a least-squares fit o f eqn. (5) t o the e x p e r i m e n t a l Cp versus T data are cl = - - 7 . 6 7 2 × 10 -4 K -1, c2 = 4 . 0 5 8 × 10 -6 K -2 and c3 = - - 4 . 2 9 3 × 10 .9 K -3. Thus, we see t h a t the t e m p e r a t u r e d e p e n d e n c e s o f b o t h the t h e r m a l e x p a n s i o n coefficients and the heat c a p a c i t y o f CuInSe~ are c o n s i s t e n t with each o t h e r regarding the i n f l u e n c e o f a n h a r m o n i c c o n t r i b u t i o n s to the intera t o m i c potentials. F r o m a c o n s i d e r a t i o n o f this result, it c a n n o t be ruled o u t t h a t in m o r e detailed calculations o f the vibrational p r o p e r t i e s o f CuInSe2 o n e m u s t a c c o u n t f o r this r a t h e r strongly p r o n o u n c e d lattice a n h a r m o n i c i t y . Finally, it is interesting to n o t e t h a t t h e c o e f f i c i e n t cl o f eqn. (5) is negative. This result is in a g r e e m e n t with p r e d i c t i o n s m a d e o n the basis of quite general considerations c o n c e r n i n g the influence o f a n h a r m o n i c forces on the thermal p r o p e r t i e s o f solids [49].

3. 3. Thermal conductivity T h e t e m p e r a t u r e d e p e n d e n c e o f the t h e r m a l c o n d u c t i v i t y X has been m e a s u r e d in n - t y p e CuInSe2 in t h e t e m p e r a t u r e range T = 4 - 300 K [50], in p - t y p e CuInSe 2 in the range T = 80 - 300 K [51, 52] and in CuInSe 2 polycrystals o f unspecified c o n d u c t i v i t y t y p e in the range T = 300 - 550 K [21, 53]. T h e results o f these a u t h o r s are p r e s e n t e d in Fig. 5. It can be seen t h a t b o t h the low- and the h i g h - t e m p e r a t u r e m e a s u r e m e n t s m a d e in d i f f e r e n t research groups agree very well. In all cases a r o o m t e m p e r a t u r e value o f X = 0 . 0 8 6 W cm -I K -1 has been obtained. F u r t h e r m o r e , it has b e e n s h o w n t h a t t h e variation o f X with t e m p e r a t u r e in the range T ~ 300 K can be e x p l a i n e d in t e r m s o f a t h e o r e t i c a l m o d e l including lattice a n h a r m o n i c i t y effects and p h o n o n scattering b y p o i n t defects [50 - 52]. T h e r e f o r e , we c o n c l u d e t h a t t h e results s h o w n in Fig. 5 are r e p r e s e n t a t i v e o f the lattice t h e r m a l conductivity o f CuInSe 2. In t w o earlier investigations c o n s i d e r a b l y l o w e r values o f X = 0 . 0 2 9 W cm -1 K -1 [19] and X = 0.037 W cm -1 K -1 [54] were r e p o r t e d for t h e t h e r m a l c o n d u c t i v i t y o f CuInSe 2 at r o o m t e m p e r a t u r e . Since these m e a s u r e m e n t s

413 101 ,

"

oO o Oo

2? Oo

o

o

2

o% °o o

10 o

E

°o

q

i

! I

2.51!

°o

s

i @e

10-4 °D

5

2.1

%

2 1.9

10-2

i

5

10~

i

i

2

5 T(KI

I

10 z

2

300

400

500 T(K)

Fig. 5. Temperature dependence of the thermal conductivity of CuInSe2. The experimental data points are: ©, from ref. 50; o, from refs. 51 and 52; ~, from refs. 21 and 53. Fig. 6. Temperature dependence of the Griineisen parameter: o, calculated from the data of refs. 21 and 53 using eqn. (7); , calculated using eqn. (8).

were probably made on small-grain polycrystalline samples, the reduction in the thermal conductivity compared to the value derived from Fig. 5 may be due to additional scattering of phonons at grain boundaries. If the contribution of lattice anharmonicity to the thermal conductivity is described in terms of an average Gri~neisen parameter 7 and if the correct Debye temperature 0D = 221:9 K (see Section 3.2) is inserted in the corresponding relations, a value of 7 = 1.85 is obtained from the data for p-type CuInSe2 [51, 52] and one of ~ = 2.32 from the data for n-type CuInSe2 [50]. If we make the simplifying assumption that the thermal conductivity is determined by lattice anharmonicity alone, we have [55] 12 ( 4 V)I/31~/I(hOD) 3 )~ = 5

h3T(7 + 1/2) 2

(7)

where V is the average volume occupied by one atom of the c r y s t a l , / ~ the average atomic mass, h the Boltzmann constant and h the Planck constant. The thermal conductivity is obtained in W cm -1 K -1 if, in eqn. (7), ~" is inserted in cm 3, 3~ in g and T and 0 D in K. Using eqn. (7) and the above mentioned room temperature value for ~, we obtain 7 = 1.91 at 300 K, which is close to the value calculated from the more detailed theoretical analysis of the thermal conductivity in p-type CuInSe~. Using eqn. (7) to describe the experimental data of refs. 21, 53 (data points in Fig. 5) we found that agreement between (7) and these data can only be achieved if a temperature dependent Griineisen parameter is introduced (Fig. 6) which follows the relation 7 = 1.746 + 6.5455 × 1 0 - 3 ( T - To) -- 1.1559 X 1 0 - 5 ( T - T0) 2

(8)

414 with T in K and T O= 273.15 K. This result is not unexpected in view of the strongly p r o n o u n c e d lattice anharmonicity derived from the temperature dependence o f the thermal expansion coefficients and the heat capacity at high temperatures (see eqns. (2) and (5)). The average Gri]neisen parameter 7, the volume thermal expansion coefficient av = 2aa + ~c, the molar specific heat at constant pressure Cp and the isothermal compressibility XT are related to each o th er via the Grfineisen equation 7 = O~v V m / C p × T

(9)

where Vm is the molar volume. Then, from a consideration of the temperature dependences of a v and Cp found experimentally, and to account for the fact that the lattice anharmonicity model explaining this variation o f a v and Cp with t e m p e r a t u r e also predicts a quadratic t e m p e r a t u r e dependence for XT [56] it must be expected t hat ~/ t o o will exhibit at least a quadratic temperature dependence. With regard to the absolute magnitude of the ~ values shown in Fig. 6 it should be not ed t hat t hey represent upper bounds of the Griineisen parameters because p h o n o n scattering at any t ype of defect has been neglected in their calculation.

4. Mechanical properties The experimental data on the mechanical properties of CuInSe2 are rather scarce. Only for the microhardness have data been report ed in the literature but the results differ widely from aut hor to author. For polycrystalline material, values of 2.55 × 109 N m -2 [18], 2.46 × 109 N m -2 [19] and 1.81 × 109 N m -2 [ 57] have been measured, but these values are likely to be t oo small for the same reasons as discussed in connect i on with the thermal expansion coefficient measurements on polycrystalline samples. A considerably higher value of 3.7 × 109 N m -2 for the microhardness has been published for single crystalline samples [58]. Our own measurements on the (112) face of a CuInSe2 single crystal yielded 3.2 × 109 N m -2 in acceptable agreement with the latter value. Theoretical estimates based on semiempirical models gave values of 1.57 × 109 N m -2 [59] and 1.38 X 109 N m -2 [60] for the microhardness of CuInSe 2 which are below even the lowest experimental data for polycrystalline material. There are as ye t no direct experimental determinations of the elastic constants and the compressibility of CuInSe 2. In ref. 46 a compressibility of × = 2.4 × 10 -11 m 2 N -1 has been estimated from the Debye temperature. However, if we compare the × values for some ot her I-III-VI 2 c o m p o u n d s given in ref. 46 with those measured directly in high-pressure experiments or evaluated from elastic constants (Table 9) it seems t hat the compressibilities estimated in ref. 46 are generally t oo large. A n o t h e r way to estimate the compressibility is to use the Grfineisen relation (9). Inserting the r o o m temperature values of ~v, Cp and ~/ given in Section 3 we find × ~ 8 × 10 -12 m 2 N -1. This value must be considered as a lower bound for the compressibility

415 TABLE 9 C o m p a r i s o n o f e x p e r i m e n t a l a n d e s t i m a t e d c o m p r e s s i b i l i t i e s o f s o m e I - I I I - V I 2 compounds

Compound

Experimental X 10 -H

Reference

Estimated X 10-H ( m 2 N -1)

( m 2 N -1)

CuA1S 2 CuGaS 2 CuGaSe 2 CuGaTe 2 AgGaS 2 CuInSe2

1.06 1.06 1.04 1.40 1.80 1.67 1.50

45 45 61 62 62 61 63

Reference 46

Reference 64

1.5

1.03 1.04

2.2 4.6 2.1

1.31 1.81 1.29

2.4

1.46

of CuInSe 2 because the Grfineisen parameters given in Section 3.3 are certainly overestimated. Recently, a model essentially based on surface energy calculations has been applied to evaluate the compressibility of all I-III-VI: compounds [64]. It can be seen from Table 9 that the × values estimated in this manner are in good agreement with the experimental ones. Therefore, it seems that the compressibility of CuInSe2 obtained in this way (see Table 9) is the most reliable one at present. This point of view is supported if the empirical relation between the microhardness and the compressibility of chalcogenide compounds, discussed in detail in refs. 65, 66, is used to calculate the compressibility of CuInSe: from the experimentally measured microhardness. Even if the large scatter in the microhardness data is taken into account we find that the compressibility should be in the range × = (1.4 + 0 . 1 ) × 10 -11 m 2 N -1 which is in good agreement with the value estimated in ref. 64 (see Table 9). The only other mechanical parameter reported in the literature is the longitudinal sound velocity. An experimental value of 3.43 X 105 cm s -1 measured on polycrystaUine material has been given in ref. 19, while a theoretical estimate based on the relation between the sound velocity and the Debye temperature yields a value of 2.18 × 105 cm s-1 [50].

5. Conclusions Our knowledge regarding the lattice vibrational properties of CuInSe2 is restricted to the long-wavelength optical phonons at the centre of the Brillouin zone, and even here the experimental data are still incomplete. Considering the difficulties arising in the unambiguous identification of the lowenergy zone-centre optical modes of E and B 1 symmetry even in extensively investigated I-III-VI2 compounds such as CuGaS2 and AgGaS2 [67], it must be concluded that further progress in completing the picture of the vibra-

416 tional properties o f C u I n S e 2 can p r o b a b l y be achieved o n l y by n e u t r o n scattering e x p e r i m e n t s . It c a n n o t be e x p e c t e d that t h e o r e t i c a l calculations based, for instance, on the valence-force field m o d e l of Keating will be helpful in solving this problem. As with the b i n a r y I I I - V and I I - V I c o m p o u n d s , a k n o w ledge o f all z o n e - c e n t r e optical p h o n o n s as well as the elastic c o n s t a n t s is the necessary p r e c o n d i t i o n for d e t e r m i n i n g a sufficiently a c c u r a t e set o f the Keating m o d e l p a r a m e t e r s [ 13, 68 ]. A n e x a m i n a t i o n o f the e x p e r i m e n t a l data p r e s e n t e d for the t h e r m a l properties o f C u I n S e 2 shows that, within the a c c u r a c y attainable in such m e a s u r e m e n t s , t h e results r e p o r t e d b y various research g r o u p s agree with each o t h e r and, t h e r e f o r e , can be c o n s i d e r e d as sufficiently reliable. Furthermore, if higher o r d e r a n h a r m o n i c c o n t r i b u t i o n s to t h e i n t e r a t o m i c potentials are t a k e n into a c c o u n t , a c o n s i s t e n t t h e o r e t i c a l i n t e r p r e t a t i o n o f the experim e n t a l d a t a and t r e n d s can be given. A t present t h e r e is no reliable e x p e r i m e n t a l i n f o r m a t i o n c o n c e r n i n g the m e c h a n i c a l p r o p e r t i e s o f C u I n S e 2. Here, we can o n l y h o p e t h a t these properties will be elucidated in the near future. The present state o f t h e art in the t e c h n o l o g y o f growing C u I n S e 2 crystals s h o u l d m a k e it possible to prepare sufficiently large and h o m o g e n e o u s single crystal specimens for t h e appropriate m e a s u r e m e n t s to be made.

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