A Theoretical study of Debye Temperature Variations of Gallium Pnictides

A Theoretical study of Debye Temperature Variations of Gallium Pnictides

K.K.Mishra and K.S.Upadhyaya

Material Research Division, Dept of Physics, Nehru Gram Bharati University, Kotwa, Jamunipur, Allahabad 221505 (India)

Abstract: The Gallium Pnictides are one of the several groups III-V compounds with zinc-blende structure (ZBS) which have been widely studied because of their important semiconducting properties. A study of Debye temperature variations of Gallium Pnictides ( GaP, GaAs and GaSb) have been presented using an interionic potential, which consists of a long-range Coulomb and three-body interactions (TBI), short-range overlap repulsion and van der Waals (vdW) interactions. Our study shows a better agreement at lower temperature side. Slight disagreement at the higher temperature side may be ascribed to the non-inclusion of the anharmonic interactions in the present model. To conclude, we can say that our present model gives a better interpretation of the lattice dynamical studies of these crystals.

PACS No.: 63.20. e, 65.40.Ba, 78.30.-j

Index Terms Phonon; Debye Temperature, Combined densities of states, gallium antimonide, van der Waals interaction.

1. Introduction

   The compound semiconductors of zinc-blende structure (ZBS) crystals are the promising candidate materials for numerous experimental and theoretical investigations. These investigations are the

   (r) LR (r) SR (r) (1) where the first (r) consists of the long-range

   LR

   Coulomb and three-body interaction (TBI) energies given by

   consequence of efforts devoted to understand the

   interesting crystal property and interaction mechanisms exhibited by these compounds. Gallium

   LR (r)

   Zi Z j e 1

   2

   ij rij

   f (rik )

   k

   Pnictides are one of the several groups III-V

   i j k

   Z 2 e 2 4

   compounds which have been widely studied because of their important semiconducting properties. The lattice vibrations play an important role in

   M 1

   r

   f (r)

   Z

   (2)

   determining the dielectric and infrared optical properties of these crystals as well as their free carrier transport. Hence it is of considerable interest to study the phonon dispersion curves of Gallium Pnictides ( GaP, GaAs and and GaSb) whose experimental data [1 to 6 ] of different spectra are available. A good agreement of our results for phonon dispersion curves and Raman assignments for these crystals has been

   where is the Madelung constant (= 1.63805), the ionic charge of the i-th ion, the separation between ith and jth ions, and f(rik) the TBI parameter dependent on nearest neighbour distances (rik) and being a measure of ion size differences.

   The second term in equation (1) consist of the short-range energy contributions from the overlap repulsive and van der Waals interaction (VDWI) as

   shown in the previous paper published earlier [7and

   19]. Here a study of Debye temperatures variation of

   2

   SR (r) Nb

   r

   ij exp i

   rj rij

   GaP, GaAs and GaSb has been presented in terms of D and T(K). In addition, the results have been analysed with the aid of the Three Body Force Shell

   cij

   r

   6

   ij ij

   i, j 1

   ij

   dij

   r

   8

   ij

   (3)

   model (TSM) which have been found applicable to zincblende materials (R.K.Singh et al) [8 to 12]. For each material, a set of Debye temperatures variation has been obtained that are consistent with experimental data observed by various workers.

2. Theory

   1. Three Body Crystal Potential

      In order to describe the cohesion in ZBS semiconductors, we have employed a three- body potential to express the crystal energy for a particular lattice separation (r) as

      where the first term is the Hafemeister and Flygare (HF) potential [13] as used in Singh and coworkers[8 and 9] . The second and third order terms represent the energy due to van der Waals dipole- dipole (d-d) and dipole- quadrupole interaction, respectively.

      Using the crystal energy expression (1), the equations of motion of two cores and two shells can be written as:

      2M U = (R + Zm C' Zm) U + (T + Zm C' Ym) W (4)

      O = (TT + Ym C' Zm) U + (S + K + Ym C' Ym) W (5)

      Here U and W are vectors describing the ionic displacements and deformations, respectively. Zm and

      Hence, equation (9) can be written for zinc-blende structure type crystals, as

      Ym are diagonal matrices of modified ionic charges and shell charges, respectively; M is the mass of the core; T and R are repulsive Coulombian matrices,

      C 3NkB

      V 6000

      E(x)G( )d

      (12)

      respectively; C' and Ym are long-range interaction matrices, that include Coulombian and TBI respectively; S and K are core-shell and shell-shell repulsive interaction matrices, respectively and TT is the transpose of matrix T. The elements of matrix Zm consists of the parameter Zm giving the modified ionic charge.

      The contribution of each interval to the specific

      heat is obtained by multiplying an Einstein function corresponding to mid-point of each interval (say 0.1 THz) by its statistical weight. The statistical weight of the interval is obtained from the number of frequencies lying in that interval. The contributions of all such intervals when summed up

      Z m Z 1

      8 / Z

      f (r)

      (6)

      give

      E(x)G( )d

      . The specific heat CV is then

      The elimination of W from eqns. (4) and (5) leads to the secular determinant:

      calculated by expression (12).

      D q

      2 M I 0

      (7)

3. Result and Discussion:

   for the frequency determination. Here D (q) is the (6×6) dynamical matrix given by

   3.1 Gallium Phosphide (GaP):

   D q

   R Z m C Z m T Z m C Y m

   (8)

   Specific heat of GaP was measured from 3000K to

   S K Y m C Y m

   1. T T

      Y m C Z m

      100K by Tarassov and Demidenko [14]. Kushwaha

      The numbers of adjustable parameters have been largely reduced by considering all the short-range interactions to act only through the shells.

      2.2 Specific Heat and Debye Temperature: The specific heat at constant volume (CV), at temperature T is expressed as

      and Kushawaha [6] calculated CV values for this compound using bond-bending force model and obtained a good agrrement with experimental results at higher temperature. CV values calculated on the basis of present model have given a better agreement at low temperatures also upto 10K which shows predominant with the other zinc-blende semiconductor structure crystals. Calculated D Vs T curve for GaP as plotted in figure 1 shows a very

      m

      0

      CV 3NkB

      h

      k BT

   2. h

   e kBT

   m

   G( )d

   h 2

   e kBT 1

   (9)

   good agreement in the entire temperature range.

   G( )d

   0

   Where, is the maximum frequency, N is the Avogadros a number, h is the Plancks constant and kB is the Boltzmanns constant. The equation (9) can be written as a suitable form for a computational purpose as

   C 3Nk

   h

   kBT

   2 h

   e kBT

   G( )d

   (10)

   V B G( )d

   Where is the Einstein function, defined

   by E(x) x2

   exp( x)

   (11)

   With x

   Also,

   exp( x) 1 2

   h .

   kBT

   Fig. 1: Debye Temperature variation of GaP

   1. Gallium Arsenides (GaAs):

      Specific heat at constant pressure (CP) of this compound has been measured by Piesbergen [15] and

      G( )d

      =Total number of frquencies considered.

      = 6000 for zinc-blende structure.

      Lundin et al[16]. Specific Heat at constant volume (CV) has been calculated by Cetas et al [17] and Holste [18]. Specific Heat CV values calculated on

      the basis of present model have presented with better agreement at low temperatures also upto 10K which shows predominant with the other zinc-blende semiconductor structure crystals. Calculated D Vs T curve for GaP as plotted in figure 2 shows a very good agreement in the entire temperature range.

      Fig.2: Debye Tepmerature variation of GaAs

   2. Gallium Antimonide (GaSb):

   CV Values as obtained on the basis of present model have been shown in figure 3. The experimental data as obtained by Farr et al [4] has been included. The calculated D values differ slightly at high temperature from the experimental values, maximum discrepancy being 12% at 3000K.

   Fig.3: Debye Tepmerature variation of GaSb

4. Conclusion:

   Our study shows a better agreement at lower temperature side. Slight disagreement at the higher temperature side may be ascribed to the non- inclusion of the anharmonic interactions in the present model. To conclude, we can say that our

   present model gives a better interpretation of the lattice dynamical studies of these crystals.

5. Acknowledgement:

The authors are thankful to Prof. K.P.Mishra, V.C. and Dr. S.C. Tiwari ProV.C. , NGBU, Allahabad for providing the necessary facilties. One of us (KKM) is also thankful to Dr. G.K.Upadhyay, Director, Landmark Technical Campus, J.P.Nagar, India for granting leaves, constant encouragement and helpful attitude.

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