Temperature Dependent Ultrasonic Study in Scandium Antimonide Semiconductor

In this paper analysis of wave propagation of elastic wave in scandium antimonide semiconductor was investigated. In scandium antimonide semiconductor, NaCl structure was found. Ultrasonic properties like ultrasonic attenuation, sound velocities, acoustic coupling constants, and thermal relaxation time have been investigated in cubic scandium antimonide semiconductor. Second and third order elastic constant have been computed for the evaluation of above said ultrasonic properties. Second and third elastic constant was studied at the various temperatures. Longitudinal and shear velocity was calculated by using the elastic constant. Longitudinal and shear velocity increase with increase the temperature. Ultrasonic attenuation either from longitudinal or shear wave propagation in cubic materials increase with increase the temperature.


Introduction
Ultrasonic's have become a powerful tool to characterize the material systems of different types like the interface between hard and soft solids and liquids [1][2] .Study of propagation behavior of ultrasonic is related to the microstructural phenomena in the materials and other thermo-physical properties of the materials [3][4] .Group III-V wide-gap semiconductor materials are very important because of their Opto-electronic technological applications as a commercial short wavelength light-emitting diode, laser diode, transparent conductors, solar cells, high-density optical memory and visual display [5].This importance arises from the role of the d-electrons in the valance band in hybridization, which tends to open a gap at the band crossing and make angular momentum labeling no longer suitable.
Within my best knowledge, less is known about ScSb and ScP compounds.Tabboune et al investigated the structural and electronic properties of ScSb and ScP III -V materials within a version of the first principles full potential linear muffin-tin orbital's method (FP-LMTO) 6 .They found that a transition from rock salt (RS) to cesium chloride (CsCl) structure is possible at high pressure (87.75 GPa for ScSb and 330GPa for ScP).
Scandium antimonide with NaCl-type structure has attracted much interest since the discovery of their anomalous physical properties 7 .The high pressure structural behavior of semiconductor scandium antimonide is very interested 8 .The temperature dependence behavior of this binary AB compound with a NaCl-type (B 1 ) structure has been popular topic on condensed matter research.This material is technologically important having many applications ranging from catalysis to microelectronics.Therefore we have chosen the materials for present investigation.
Aim of the study is to establish the theoretical approach for the determination of higher order elastic constants of the materials at high temperatures.Finally the theory will be established for the determination of ultrasonic attenuation and related parameters in these materials at high temperatures along different crystallographic directions.Also the characteristic behavior of ultrasonic propagation in the materials has been also discussed.

Experimental
Theory is categorized in two parts.

Second and Third order elastic constants (SOEC and TOEC):
The potential used for evaluation of SOEC and TOEC of the form: Where e is the electronic charge, r is the nearest neighbor distance and A is the strength parameter.Following Brugger's definition 9 of elastic constants at 0K and starting from nearest neighbor distance and hardness parameter of the substance and taking interaction effective up to second neighbor distance, according to lattice dynamics developed by Leibfried and Ludwig and Ghate 10,11 , lattice energy changes with temperature.Hence adding vibrational energy contribution to static elastic constants, one gets C IJ and C IJK at required temperature.
Where superscript 0 has been used to denote SOEC and TOEC at 0K (static elastic constants) and superscript vib.has been used to denote SOEC and TOEC at particular temperature.

Ultrasonic attenuation
In the second part of present investigation, Mason's theory [12][13] is still widely used successfully to study the ultrasonic attenuation at room temperature in solids.It is more reliable theory to study anhormonicity of the crystal as it involves elastic constants directly through non-linearity parameter (D) in the determination of ultrasonic attenuation (/f 2 ).According to the theory, ultrasonic attenuation due to phonon-phonon interactions at high temperatures can be given as: Where non-linearity parameter (acoustic coupling constant) is: Thermal relaxation time Debye average velocity of ultrasonic waves as: Thermo elastic loss is given by:

Evaluations of the results
The nearest neighbor distance (r 0 ) is taken 2.992Å and hardness parameter (b) is taken 0.310Å for ScSb 6,[14][15] .The present values of SOEC and TOEC are given in Table 1.The value of specific heat per unit volume (C V ) and thermal energy density (E 0 ) are determined by  tables in AIP Handbook 16 .The values of thermal relaxation time ( th ), thermal conductivity (K), specific heat per unit volume (C V ) and Debye average velocity ( V ) are presented in Table 2.The non-linearity parameter is a function of Grüneisen parameter   j i  , specific heat per unit volume (C V ), thermal energy density (E 0 ) and temperature (T).Grüneisen parameters are determined using elastic constants (SOEC and TOEC) [12][13] .The values of non-linearity parameters (acoustic coupling constants) are presented in Table 3. Finally ultrasonic attenuation coefficients .


due to phonon-phonon interaction and due to thermo-elastic are calculated using eqns.( 5) and ( 9) respectively, are shown in Figure 3.1-3.3.
In order to verify the validity of the present investigation the simple way to compare the calculated second and third order elastic constants, thermal relaxation time, acoustic coupling constants, ultrasonic attenuation coefficients due to phonon-phonon interaction and thermoelastic loss with the values for similar solids.

Thermal relaxation time
It can be seen from Tables 2 that order of thermal relaxation time ( th ) is 10 -14 sec.for ScSb semiconductor due to their different thermal conductivity values of the order of 10 2 erg/sec.cm.K [23][24][25] .
Table 2. Longitudinal and shear velocities (V l and V s ) in the unit of 10 5 cm/sec, Debye average velocity ( V ) in the unit of 10 5 cm/sec, thermal relaxation time ( th ) in the unit of 10 -14 sec, specific heat (C v ) in the unit of 10 6 erg/cm 3 K, internal energy (E 0 ) in the unit of 10 8 erg/cm 3 and thermal conductivity (K) in the unit of 10 2 erg/sec.cm.K of scandium antimonide in the temperature range 100-300K. Temp.

Acoustic coupling constants (D)
The variation of D with temperature is presented in Table 3.It can be seen that acoustic coupling constant decreases with the temperature and this behavior can be related with atomic number and atomic mass, as atomic number and atomic mass decreases with the value of acoustic coupling constant in these materials.The acoustic coupling constant ratio D l /D s along <100> lies between 14-15, D l /D s1 along <110> and shear wave polarized along <001> is about 7-10 and D s2 /D l along <110>, shear wave polarized along

attenuation due to phonon-phonon interaction and due to thermoelastic mechanism
In the present investigation it is pointed out that attenuation varies with temperature as α=α 0 T n where α 0 and n are constants.It is obvious from Figure 3.1-3.3that the ultrasonic attenuation due to p-p interaction (α/f 2 ) Akh.long.for longitudinal wave and (α/f 2 ) Akh.shear for shear wave increases with temperature in scandium antimonide.The values of (α/f 2 ) Akh.long.are greater than that of (α/f 2 ) Akh.shear along <100> direction and along <110> direction (shear wave polarized along <001> direction), while (α/f 2 ) Akh.long. is less than that of (α/f 2 ) Akh.shear values along <110> (shear wave polarized along   0 1 1 direction) and <111> direction (shear wave polarized along   10 1 direction), which has been observed experimentally in semiconductors [27][28][29] .It is clear from the Figures 3.1-3.3that the values of ultrasonic attenuation due to p-p interaction are found highest in ScSb due to their thermal conductivity values.Order of the attenuation in ScSb semiconductor is anomalous due to their thermal conductivity values.The ultrasonic attenuation of ScSb can be compared with the semiconductors [27][28][29] .There is good agreement between the attenuation of present material and other same type materials.Ultrasonic attenuation due to thermo elastic loss (α/f 2 ) th is negligible in comparison to attenuation due to p-p interaction.

Conclusion
On the basis of above analysis the following conclusions can be drawn.1.The method of calculation of SOEC/TOEC at different temperatures is justified and can be utilized for the determination of ultrasonic properties of these materials.2. Second and third order elastic constant changes with change the temperature.These elastic are also changed with according to the crystallographic direction of the materials.3. Theoretical approach utilized for the calculation of temperature dependency of ultrasonic attenuation in the materials is well established.4. Thermal conductivity of the material is the prominent contributor for the difference of ultrasonic attenuation Scandium antimonide semiconductor. 5.The results may be utilized for the materials processing in the industries.

Table 3 .
is comparable with that in experimental values of NaCl-type LiF26 and other like crystals27 .Acoustic coupling constants (D l for longitudinal wave, D S for shear wave along <100> orientation and ratio of D l & D S ), (D S1 * for shear wave polarized along <001> orientation, ratio of D l & D S1 , D S2 & D l ) along orientation, (D S $ for shear wave polarized along   10 1 orientation and ratio of D l & D S $ ) along <111> orientation of scandium antimonide in the temperature range 100-300K.
Table 1 that out of nine elastic constants, four i.e.C 12 , C 111 , C 123 , and C 166 are decreasing with temperature while others C 11 , C 44 , C 112 , and C 144 increase with temperature, while value of C 456 remains unchanged.This type of behavior has been found in other NaCl-type materials

Table 1 .
Second and third order elastic constants (SOEC and TOEC) in the unit of 10 11 dyne/cm 2 of scandium antimonide in the temperature range 100-300K.