- Email: [email protected]
Elastic constants of CdSe at low temperature
J. Phys.
Chem.
Solids
Vol.
54, No.
2. pp.
209-212.
1993
Printedin Great Britain.
G
ELASTIC
CONSTANTS
0022-3697193 S6.00 + 0.00 I993 Pergamon PressLtd
OF CdSe AT LOW TEMPERATURE
B. BONELLOt and B. FERNANDEZ~ TDtpartement de Recherches Physiques (CNRS URA 71), Universitt Pierre et Marie Curie-Boite 136, 4, place Jussieu 75252 Paris cedex 05, France $Centro de Estudios de Semiconductores, Departamento de Fisica-Facultad de Ciencias, Universidad de Los Andes-Apartado de Correos 1 MCrida 5251, Venezuela (Received 3 September 1992; accepted 14 September 1992)
Abstract-By measuring the longitudinal and transverse ultrasonic sound velocities along different crystalline directions, the elastic constants C,, , Cj,, C, and C,, of CdSe have been determined in the temperature range O-300 K. From these measurements, the Debye temperature at 0 K is calculated to be 181.7 K. The possibility of using CdSe as an acousto-optic deflector is discussed. Keywords: CdSe, elastic constants, acousto-optic
interaction, Debye temperature.
1. INTRODUCTION This work was motivated suitable
by the search for materials
as acousto-optic
region. In fact, infrared
devices in the near infrared light modulators
tors are of great importance
and deflec-
in systems such as optical
communication devices or signal processors; correspondingly, knowledge of the elastic properties of the materials involved in these systems is essential to improve their efficiency. In the acousto-optic interaction, the main factor to consider is the factor of merit M. This parameter characterizes
the inherent
a light deflector
efficiency of the material
and is defined
M
_
as
by [l]:
n6p2 pv’ ’
where n, p, p and v are respectively, the refractive index, the elasto-optic coefficient, the density and the sound velocity in the solid. A high factor of merit will be obtained for materials with large refractive index and low sound velocity. Despite the fact that low sound velocity implies high sound attenuation, a compromise between high M and low acoustic loss can be reached [2]. Therefore, to discern the feasibility of using CdSe as an infrared acousto-optic device, a study of its acoustic properties must be performed. In the past few years, several papers have been devoted to the study of the electroacoustic properties [3], optical properties [4] or ultrasonic attenuation [5-71 in CdSe but, to the best of our knowledge, no attempt has been made to determine the full temperature dependence of the entire set of elastic constants of this compound. This situation could be due to the difficulty in preparing high quality CdSe large single
crystals. Nevertheless, partial data are available in the literature; for example Berlincourt et al. [8] and Cline et al. [9] independently determined the five components of the elastic tensor at room temperature whereas Lubashevskaya et al. [lo] examined the temperature dependence of C,, in the range IOO-300K. For this reason we have undertaken a complete acoustic characterization of CdSe in order to determine its application as an acousto-optic light deflector in the infrared. In the present work, the temperature dependence of the sound velocities for longitudinal and shear acoustic waves propagating along the sixfold and binary axes of CdSe single crystals are measured. From these results the temperature dependence of the elastic constants C,, , C,, , C,2 and C, are obtained. Using our data and a calculated value of the elastic constant C13, the Debye temperature BI, at OK of CdSe is determined. In a later paper the ultrasonic attenuation studies will be reported. As a by-product an analysis of the temperature dependence of 8n will be attempted.
2. EXPERIMENTAL
TECHNIQUE
The monocrystalline sample was grown by the Bridgman technique and shaped into a parallelepiped of 8.9 x 9.1 x 9 mm3. The main axes of the parallelepiped were parallel to a sixfold axis and to a twofold axis, respectively. To avoid edge effects in the ultrasonic measurements all faces were polished with parallelisms between opposite faces better than lo-‘rd. Thin plates of LiNbO, properly cut were used as transducers to generate both longitudinal and transverse acoustic waves in the sample at 65 and 209
210
B. BONELLO and B. FERNANDEZ Table 1. Relationship between sound velocities and elastic constants in crystals of hexagonal symmetry
Direction of propagation
Polarization
Poll
w11
PV2
P4 = Cf3+ e&l9
Normal to Wll
PO11
pv:
= c,
Normal to
Wll
Longitudinal
PGl = c,,
Normal to
Wll Normal to WI
PO11 [1011 [lo11
Wll
pv:
= C& + eslels/t,
Transverse Normal to
wo11 w101 Quasi longitudinal Quasi transverse
PO&=
1/2(C,,-Cd
pv; = l/4(& - c,, + 2C,) pv:,t=
WC,,+
pv;,t = l/4(&
c,,+ 2c.d+
m(c,,+
cd*+
1/4(C,, -
C,,J21”2
+ c,, + 2C,) - 1/2[(C,, + CM)*+ 1/4(C,, - C&“’
t Not corrected for piezoelectricity.
11 MHz, respectively. In the range 90-300 K, the transducers were bonded to the sample by means of Nonaq stop-cock grease. Below 90 K, a droplet of methane in liquid phase instilled between the transducer and the sample was used as a bonding agent; thus, as the temperature decreases, the methane solidifies ensuring bonding down to 4 K. The temperature was maintained within +0.2 K and the sound velocity measurements were made using either a pulse echoes overlap [l l] or an interferrometric method
WI. 3. RESULTS
CdSe is a semiconductor compound of the II-VI family which crystallizes in the hexagonal wurtzite structure (space group P6,mc). Hexagonal symmetry solids have five independent nonzero elastic constants: C,, , C,,, C,,, C,, and C,. These constants can be calculated from the density and the sound
velocities along the crystalline directions [loo], [OlO], [OOl] and (1011. In Table 1 is shown the relationship between C, and sound velocities for the propagation of an acoustic wave along these directions. The sound velocities for the longitudinal and shear waves as a function of temperature are plotted in Figs 1 and 2, respectively. As can be observed these values are rather low. The elastic constants C,, , &, C, and C, = $C,, - C,*) are calculated assuming a density of 5.684 gem-’ at room temperature. In order to obtain the elastic constants at temperatures different from room temperature, a correction to sample length and density due to thermal expansion has to be applied. This correction requires the knowledge of thermal expansion coefficient over the whole temperature range 4.2-300 K. Since such data for CdSe are not available, we have estimated this coefficient from room temperature values [13] LX,= 4.4 and clj = 2.45 (in 10m6K-l), tl, and clj being the expansion coefficients perpendicular and parallel to the C-axis,
1475
0
SO
100
160
Temperatur?lK) Fig. 1. Sound velocities for longitudinal waves along different crystalline directions as a function of temperature.
Ooo
soo
Fig. 2. Sound velocities for shear waves along different crystalline directions as a function of temperature.
211
Elastic constants of CdSe Table 2. Elastic constants and bulk modulus of CdSe at room temperature. Comparison of the present work with previous results. The bulk modulus of crystals of hexagonal symmetry is expressed by g = WC,, + Cu) - 2c:, C, I + Cl, + 2G, - 4CU
~rlincourt et al. [8] Cline ei ui. [9] Lubashevskaya et al. [IO] Present
respectively.
C,,
C,,
74.2 74.9
87.77 84.51 82.46 81.7
74.6
We assume a, and u3 to be constant
from
room temperature down to Debye temperature (0, = 181.7 K). Below &,, the thermal expansion
coefficient was assumed to vanish linearly with temperature. Since the sound velocities are weakly temperature dependent, the anharmonic effects in CdSe lattice seem to be small. This is corroborated by the low values of ct, and o/~and by the small variation observed in the sound velocities from room temperature down to 4 K. An additional correction due to the piezoelectric effect has to be applied to obtain C,, (see Table 1). In order to do this, the values at room temperature of the relative dielectric constant and of the piezoelectric constant: ej/c,, = 10.65 and e33= 0.347 Cms2, respectively, were used. These values were employed over the range 4.2-300 K to estimate the piezoelectric constant because no data are available at low temperatures. Nevertheless, negl~ting the tem~rature dependence of these parameters in the ratio &E, leads to an error in C,, of the order of 0.5 per cent. Considering all error sources, the absolute accuracy of the sound velocities reported in this paper is about 0.5 per cent and falls to about 2 per cent for the elastic constant. In Table 2 we summarize our results together with data published by other authors for the elastic constants and bulk modulus which for crystals of
CM C66 m units of GPa)
Cu
B
(’13.17 13.15
14.45 14.41
45.3 46.09
53.41 53.71
13.0
14.3
46.1
53.4
hexagonal symmetry is defined as: B
=
c33(Gl+ c,,
+
Cd Cl2
-
2G
2c,3 - 4c,, .
(2)
From this table it can be seen that the agreement with both Berlincourt et al. and Cline et al. is very good for the room temperature values of C,, , C, and C,, but a discrepancy for C,, is observed. This discrepancy might be ascribed to the neglect of the piezoelectric effect by these authors. The diagonal moduli C, i , C,, , C, and the cross coupling constant Cl2 are plotted in Figs 3 and 4, respectively. We note that the elastic moduli depend linearly on temperature for 300 K down to about 180 K. Moreover, the elastic moduli are weakly temperature dependent as confirmed by the low values of the temperature coefficients shown in Table 3. These two facts are characteristic of poor anha~onic effects in the CdSe lattice. Unlike Lubashevskaya’s work, no anomalous behaviour of the elastic moduli was observed in the interval 150-200 K neither a step in the sound velocity due to the acoustoelectric interaction [14] for the piezoactive modes (first and fourth modes in Table 1) was observed. The Debye temperature 0, at 0 K was calculated to be 181.7 K using the well-known expression: Q
= D
_ff,
9N
“3A
-l/3
k, ( 4nV )
A =r(-$+-$+$)
3. The diagonal elastic Constants as a function of temperature.
+
sin8d0,
(3b)
Fig. 4. Elastic constant C,, as a function of temperature.
B.
212
BONELLO
and B. FERNANDEZ
Table 3. T,.,,= (a log C,/aT) at room temperature in CdSe. The values of these parameters in CdS when reported in [S] are given for comparison Temperature coefficients at 300 K (in units of low6 I(-‘) ______~_ TCl, CdSe CdS
-221.2
T cu
T CM
T CM,
T Cl’
-210.5 -216
-111.9 -96
- 129.5
- 278.5
where V is the unit cell volume, N is the number of atoms in the unit ceil (N = 4 for CdSe). ci, cl, c3 are the velocities of the three acoustic modes. Since the evaluation of A requires the knowledge of C,, at 0 K, to estimate this value we have considered that the ratio Cn/Cn is not temperature dependent. As a matter of fact, by analogy with cadmium sulfide which has the same wurtzite symmetry and similar low temperature behaviour to that of CdSe [15], no anomaly of that ratio is expected in this compound. Moreover, CdSe is very close to be an isotropic material at room temperature and remains like that down to 4.2 K as asserted by the ratios C,/C, and C&/C,, . Therefore, the fact to consider C,2/C,, as a constant with temperature introduces only a negligible error. Good agreement is obtained between Bn from our elastic data and 0, from thermal measurements reported in the literature [16]. These values can only coincide near absolute zero temperature, at other temperatures the Debye theory yields only an effective value of 19~which departs from the elastic value reflecting the characteristics of the phonon spectra [17]. In the other hand, On elastic remains nearly constant in the whole temperature range following the same behaviour of the sound velocities up to room temperature. Similar results have been found in LiF [18] and ternary semiconductor compounds 1191. 4. CONCLUSION
In this work, we have shown that the sound velocity is rather low in any direction of propagation of pure waves in CdSe. This feature, combined with a high refractive index 1201,indicates that the factor of merit M defined by eqn (1) should be high in this compound. Moreover, the temperature coefficients
summarized in Table 3 show that the elastic constants are weakly temperature dependent. These properties are necessary for acousto-optic devices operating at low temperatures. To complete further the acoustic characterization of CdSe, ultrasonic attenuation measurements are underway. If this attenuation turns out to be low enough in the high frequency range, CdSe will be a promising candidate for acousto-optic applications in the infrared. Acknowledgments-The
authors wish to thank the CONICIT and the Consejo de Desarrolo Cientitico Humanistico y Tecnologico (CDCHT) of the Universidad de Los Andes in Venezuela and EDF for financial support to initiate the visits which successfully led to the collaboration program between our labs.
R EFERENC~S 1. Smith T. M. and Korpel A., IEEE .J, Quantum Electr. QE-1, 283 (1985). 2. Feigelson R., Jap. J. appl. Phys. 19 supp. 3, 371 (1980). 3. Westera W., Phys. Stat. Solidi a 75, 405 (1983). 4. Klassen N. V. and Ossipyan Y. A., J. de Phys. (Paris) 40, C6-91 (1979). 5. Kmita A. M., Morozov A. I. and Fedorets V. N., Fiz.
Tnerd. Tela 13, 1011 (1971) [English translation in Sov. Phys.-Solid State 13, 841 (1971)]. 6. Gitis M. B., Gulyaev Y. V. and Chaikovskii I. A., Zh. Eksp. Teo 28, 537 (1978) [English translation in Sov. Phys. JEPT Left. 28, 497 (197811. 7. Gitis M, 3. and Chaikovskii I. A., Zh. E&sp. & Theo. Fiz. 81, 263 (1981) [English translation in Sov. Phys. JEPT 54, 140 (19x1)]. 8. Berlincourt D., Jaffe H. and Shiozawa L. R., PIzys. Rev.
129, 1009 (1963). 9. Cline C. F., Dunegan H. L. and Henderson .I. appl. Phys. 38, 1944 (1967).
G. W.,
10. Lubashevskaya I. V., Komev S. A., Markov E. V., Zinov’ev V. E. and Biryukov V. M., Inorg. Mat. 24,773 09881.
11. Papadakis E. P., J. appl. Phys. 35, 1474 (1964). 12. Blume R. J.. Rev. Scient. Insr. 34. 1400 (1963). 13. Purlys R., Jakimavicius J. and Sirvaitisa A., Litov. Fiz. Sb. 19, 287 ( 1981) [English translation in Sov. Phys. Collect. 19, 86 (1981)]. 14. Hutson A. R. and White D. L., J. appl. Phys. 33, 40 (1962).
15. Gerlich D., J. Phys. Chetn. So&z%28, 2575 (1967). 16. Anderson 0. L., J. Phys. Gem. Solids 24, 909 (1963). 17. Rusakov A. P., Vekilov Y. K. and Kadishevich A. E., Sov. Phys.-Solid State 12, 2618 (1971). 18. Shark0 A. and Botaki A., Sov. Phys. J. 14, 1259 (1971). 19. Fernandez B. J.: Unpublished work. 20. Terekhova S. F., Onishchenko N. A., Talenskii 0. N. and Pendur S. A., Phys. Stat. Solidi b 131, 207 (1985).
Chem.
Solids
Vol.
54, No.
2. pp.
209-212.
1993
Printedin Great Britain.
G
ELASTIC
CONSTANTS
0022-3697193 S6.00 + 0.00 I993 Pergamon PressLtd
OF CdSe AT LOW TEMPERATURE
B. BONELLOt and B. FERNANDEZ~ TDtpartement de Recherches Physiques (CNRS URA 71), Universitt Pierre et Marie Curie-Boite 136, 4, place Jussieu 75252 Paris cedex 05, France $Centro de Estudios de Semiconductores, Departamento de Fisica-Facultad de Ciencias, Universidad de Los Andes-Apartado de Correos 1 MCrida 5251, Venezuela (Received 3 September 1992; accepted 14 September 1992)
Abstract-By measuring the longitudinal and transverse ultrasonic sound velocities along different crystalline directions, the elastic constants C,, , Cj,, C, and C,, of CdSe have been determined in the temperature range O-300 K. From these measurements, the Debye temperature at 0 K is calculated to be 181.7 K. The possibility of using CdSe as an acousto-optic deflector is discussed. Keywords: CdSe, elastic constants, acousto-optic
interaction, Debye temperature.
1. INTRODUCTION This work was motivated suitable
by the search for materials
as acousto-optic
region. In fact, infrared
devices in the near infrared light modulators
tors are of great importance
and deflec-
in systems such as optical
communication devices or signal processors; correspondingly, knowledge of the elastic properties of the materials involved in these systems is essential to improve their efficiency. In the acousto-optic interaction, the main factor to consider is the factor of merit M. This parameter characterizes
the inherent
a light deflector
efficiency of the material
and is defined
M
_
as
by [l]:
n6p2 pv’ ’
where n, p, p and v are respectively, the refractive index, the elasto-optic coefficient, the density and the sound velocity in the solid. A high factor of merit will be obtained for materials with large refractive index and low sound velocity. Despite the fact that low sound velocity implies high sound attenuation, a compromise between high M and low acoustic loss can be reached [2]. Therefore, to discern the feasibility of using CdSe as an infrared acousto-optic device, a study of its acoustic properties must be performed. In the past few years, several papers have been devoted to the study of the electroacoustic properties [3], optical properties [4] or ultrasonic attenuation [5-71 in CdSe but, to the best of our knowledge, no attempt has been made to determine the full temperature dependence of the entire set of elastic constants of this compound. This situation could be due to the difficulty in preparing high quality CdSe large single
crystals. Nevertheless, partial data are available in the literature; for example Berlincourt et al. [8] and Cline et al. [9] independently determined the five components of the elastic tensor at room temperature whereas Lubashevskaya et al. [lo] examined the temperature dependence of C,, in the range IOO-300K. For this reason we have undertaken a complete acoustic characterization of CdSe in order to determine its application as an acousto-optic light deflector in the infrared. In the present work, the temperature dependence of the sound velocities for longitudinal and shear acoustic waves propagating along the sixfold and binary axes of CdSe single crystals are measured. From these results the temperature dependence of the elastic constants C,, , C,, , C,2 and C, are obtained. Using our data and a calculated value of the elastic constant C13, the Debye temperature BI, at OK of CdSe is determined. In a later paper the ultrasonic attenuation studies will be reported. As a by-product an analysis of the temperature dependence of 8n will be attempted.
2. EXPERIMENTAL
TECHNIQUE
The monocrystalline sample was grown by the Bridgman technique and shaped into a parallelepiped of 8.9 x 9.1 x 9 mm3. The main axes of the parallelepiped were parallel to a sixfold axis and to a twofold axis, respectively. To avoid edge effects in the ultrasonic measurements all faces were polished with parallelisms between opposite faces better than lo-‘rd. Thin plates of LiNbO, properly cut were used as transducers to generate both longitudinal and transverse acoustic waves in the sample at 65 and 209
210
B. BONELLO and B. FERNANDEZ Table 1. Relationship between sound velocities and elastic constants in crystals of hexagonal symmetry
Direction of propagation
Polarization
Poll
w11
PV2
P4 = Cf3+ e&l9
Normal to Wll
PO11
pv:
= c,
Normal to
Wll
Longitudinal
PGl = c,,
Normal to
Wll Normal to WI
PO11 [1011 [lo11
Wll
pv:
= C& + eslels/t,
Transverse Normal to
wo11 w101 Quasi longitudinal Quasi transverse
PO&=
1/2(C,,-Cd
pv; = l/4(& - c,, + 2C,) pv:,t=
WC,,+
pv;,t = l/4(&
c,,+ 2c.d+
m(c,,+
cd*+
1/4(C,, -
C,,J21”2
+ c,, + 2C,) - 1/2[(C,, + CM)*+ 1/4(C,, - C&“’
t Not corrected for piezoelectricity.
11 MHz, respectively. In the range 90-300 K, the transducers were bonded to the sample by means of Nonaq stop-cock grease. Below 90 K, a droplet of methane in liquid phase instilled between the transducer and the sample was used as a bonding agent; thus, as the temperature decreases, the methane solidifies ensuring bonding down to 4 K. The temperature was maintained within +0.2 K and the sound velocity measurements were made using either a pulse echoes overlap [l l] or an interferrometric method
WI. 3. RESULTS
CdSe is a semiconductor compound of the II-VI family which crystallizes in the hexagonal wurtzite structure (space group P6,mc). Hexagonal symmetry solids have five independent nonzero elastic constants: C,, , C,,, C,,, C,, and C,. These constants can be calculated from the density and the sound
velocities along the crystalline directions [loo], [OlO], [OOl] and (1011. In Table 1 is shown the relationship between C, and sound velocities for the propagation of an acoustic wave along these directions. The sound velocities for the longitudinal and shear waves as a function of temperature are plotted in Figs 1 and 2, respectively. As can be observed these values are rather low. The elastic constants C,, , &, C, and C, = $C,, - C,*) are calculated assuming a density of 5.684 gem-’ at room temperature. In order to obtain the elastic constants at temperatures different from room temperature, a correction to sample length and density due to thermal expansion has to be applied. This correction requires the knowledge of thermal expansion coefficient over the whole temperature range 4.2-300 K. Since such data for CdSe are not available, we have estimated this coefficient from room temperature values [13] LX,= 4.4 and clj = 2.45 (in 10m6K-l), tl, and clj being the expansion coefficients perpendicular and parallel to the C-axis,
1475
0
SO
100
160
Temperatur?lK) Fig. 1. Sound velocities for longitudinal waves along different crystalline directions as a function of temperature.
Ooo
soo
Fig. 2. Sound velocities for shear waves along different crystalline directions as a function of temperature.
211
Elastic constants of CdSe Table 2. Elastic constants and bulk modulus of CdSe at room temperature. Comparison of the present work with previous results. The bulk modulus of crystals of hexagonal symmetry is expressed by g = WC,, + Cu) - 2c:, C, I + Cl, + 2G, - 4CU
~rlincourt et al. [8] Cline ei ui. [9] Lubashevskaya et al. [IO] Present
respectively.
C,,
C,,
74.2 74.9
87.77 84.51 82.46 81.7
74.6
We assume a, and u3 to be constant
from
room temperature down to Debye temperature (0, = 181.7 K). Below &,, the thermal expansion
coefficient was assumed to vanish linearly with temperature. Since the sound velocities are weakly temperature dependent, the anharmonic effects in CdSe lattice seem to be small. This is corroborated by the low values of ct, and o/~and by the small variation observed in the sound velocities from room temperature down to 4 K. An additional correction due to the piezoelectric effect has to be applied to obtain C,, (see Table 1). In order to do this, the values at room temperature of the relative dielectric constant and of the piezoelectric constant: ej/c,, = 10.65 and e33= 0.347 Cms2, respectively, were used. These values were employed over the range 4.2-300 K to estimate the piezoelectric constant because no data are available at low temperatures. Nevertheless, negl~ting the tem~rature dependence of these parameters in the ratio &E, leads to an error in C,, of the order of 0.5 per cent. Considering all error sources, the absolute accuracy of the sound velocities reported in this paper is about 0.5 per cent and falls to about 2 per cent for the elastic constant. In Table 2 we summarize our results together with data published by other authors for the elastic constants and bulk modulus which for crystals of
CM C66 m units of GPa)
Cu
B
(’13.17 13.15
14.45 14.41
45.3 46.09
53.41 53.71
13.0
14.3
46.1
53.4
hexagonal symmetry is defined as: B
=
c33(Gl+ c,,
+
Cd Cl2
-
2G
2c,3 - 4c,, .
(2)
From this table it can be seen that the agreement with both Berlincourt et al. and Cline et al. is very good for the room temperature values of C,, , C, and C,, but a discrepancy for C,, is observed. This discrepancy might be ascribed to the neglect of the piezoelectric effect by these authors. The diagonal moduli C, i , C,, , C, and the cross coupling constant Cl2 are plotted in Figs 3 and 4, respectively. We note that the elastic moduli depend linearly on temperature for 300 K down to about 180 K. Moreover, the elastic moduli are weakly temperature dependent as confirmed by the low values of the temperature coefficients shown in Table 3. These two facts are characteristic of poor anha~onic effects in the CdSe lattice. Unlike Lubashevskaya’s work, no anomalous behaviour of the elastic moduli was observed in the interval 150-200 K neither a step in the sound velocity due to the acoustoelectric interaction [14] for the piezoactive modes (first and fourth modes in Table 1) was observed. The Debye temperature 0, at 0 K was calculated to be 181.7 K using the well-known expression: Q
= D
_ff,
9N
“3A
-l/3
k, ( 4nV )
A =r(-$+-$+$)
3. The diagonal elastic Constants as a function of temperature.
+
sin8d0,
(3b)
Fig. 4. Elastic constant C,, as a function of temperature.
B.
212
BONELLO
and B. FERNANDEZ
Table 3. T,.,,= (a log C,/aT) at room temperature in CdSe. The values of these parameters in CdS when reported in [S] are given for comparison Temperature coefficients at 300 K (in units of low6 I(-‘) ______~_ TCl, CdSe CdS
-221.2
T cu
T CM
T CM,
T Cl’
-210.5 -216
-111.9 -96
- 129.5
- 278.5
where V is the unit cell volume, N is the number of atoms in the unit ceil (N = 4 for CdSe). ci, cl, c3 are the velocities of the three acoustic modes. Since the evaluation of A requires the knowledge of C,, at 0 K, to estimate this value we have considered that the ratio Cn/Cn is not temperature dependent. As a matter of fact, by analogy with cadmium sulfide which has the same wurtzite symmetry and similar low temperature behaviour to that of CdSe [15], no anomaly of that ratio is expected in this compound. Moreover, CdSe is very close to be an isotropic material at room temperature and remains like that down to 4.2 K as asserted by the ratios C,/C, and C&/C,, . Therefore, the fact to consider C,2/C,, as a constant with temperature introduces only a negligible error. Good agreement is obtained between Bn from our elastic data and 0, from thermal measurements reported in the literature [16]. These values can only coincide near absolute zero temperature, at other temperatures the Debye theory yields only an effective value of 19~which departs from the elastic value reflecting the characteristics of the phonon spectra [17]. In the other hand, On elastic remains nearly constant in the whole temperature range following the same behaviour of the sound velocities up to room temperature. Similar results have been found in LiF [18] and ternary semiconductor compounds 1191. 4. CONCLUSION
In this work, we have shown that the sound velocity is rather low in any direction of propagation of pure waves in CdSe. This feature, combined with a high refractive index 1201,indicates that the factor of merit M defined by eqn (1) should be high in this compound. Moreover, the temperature coefficients
summarized in Table 3 show that the elastic constants are weakly temperature dependent. These properties are necessary for acousto-optic devices operating at low temperatures. To complete further the acoustic characterization of CdSe, ultrasonic attenuation measurements are underway. If this attenuation turns out to be low enough in the high frequency range, CdSe will be a promising candidate for acousto-optic applications in the infrared. Acknowledgments-The
authors wish to thank the CONICIT and the Consejo de Desarrolo Cientitico Humanistico y Tecnologico (CDCHT) of the Universidad de Los Andes in Venezuela and EDF for financial support to initiate the visits which successfully led to the collaboration program between our labs.
R EFERENC~S 1. Smith T. M. and Korpel A., IEEE .J, Quantum Electr. QE-1, 283 (1985). 2. Feigelson R., Jap. J. appl. Phys. 19 supp. 3, 371 (1980). 3. Westera W., Phys. Stat. Solidi a 75, 405 (1983). 4. Klassen N. V. and Ossipyan Y. A., J. de Phys. (Paris) 40, C6-91 (1979). 5. Kmita A. M., Morozov A. I. and Fedorets V. N., Fiz.
Tnerd. Tela 13, 1011 (1971) [English translation in Sov. Phys.-Solid State 13, 841 (1971)]. 6. Gitis M. B., Gulyaev Y. V. and Chaikovskii I. A., Zh. Eksp. Teo 28, 537 (1978) [English translation in Sov. Phys. JEPT Left. 28, 497 (197811. 7. Gitis M, 3. and Chaikovskii I. A., Zh. E&sp. & Theo. Fiz. 81, 263 (1981) [English translation in Sov. Phys. JEPT 54, 140 (19x1)]. 8. Berlincourt D., Jaffe H. and Shiozawa L. R., PIzys. Rev.
129, 1009 (1963). 9. Cline C. F., Dunegan H. L. and Henderson .I. appl. Phys. 38, 1944 (1967).
G. W.,
10. Lubashevskaya I. V., Komev S. A., Markov E. V., Zinov’ev V. E. and Biryukov V. M., Inorg. Mat. 24,773 09881.
11. Papadakis E. P., J. appl. Phys. 35, 1474 (1964). 12. Blume R. J.. Rev. Scient. Insr. 34. 1400 (1963). 13. Purlys R., Jakimavicius J. and Sirvaitisa A., Litov. Fiz. Sb. 19, 287 ( 1981) [English translation in Sov. Phys. Collect. 19, 86 (1981)]. 14. Hutson A. R. and White D. L., J. appl. Phys. 33, 40 (1962).
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