Theoretical Study of Mechanical, Elastic and Phonon Frequency Spectrum Properties for GaAs at High Pressure Текст научной статьи по специальности «Физика»

УДК 517.9

Mechanical, Elastic and Phonon Properties for GaAs at High Pressure

Raed H. AL-Saqa*

Tatwaan Secondary School Directorate General of Education Province of Nineveh Iraq

Siham J. AL-Taief

Alnoor University college Iraq

Received 25.12.2018, received in revised form 22.02.2019, accepted 26.03.2019

The calculations for variations of bulk modulus (B), compression volume (V^, lattice constant and phonon frequency spectrum (pfs) under high pressure up to 17 GPa at room temperature has been carried out. Three equations of state (EOSs) including (Birch- Murnaghan, Vinet and modified Lennard-Jouns) are used in the calculations. The variations of these properties under such high pressure for GaAs are obtained and we got suitable EOS for it.

Keywords: equation of state, Bulk modulus, compression, lattice parameter, phonon frequency spectrum. DOI: 10.17516/1997-1397-2019-12-3-371-378.

Theoretical Study of Frequency Spectrum

Introduction

The evaluation of pressure-volume (P-V) phonon frequency spectrum and lattice constant are very importance in basic and applied science (Recio et al. [12]), EOSs are fundamentally important in study the properties of solid state at high pressure, where the study of (P-V) EOSs of relevant material is one of the most basics are needed for pressure calibration. The Vinet, Birch-Murnaghan (BM) and modified Lennard-Jouns (mL-J) EOSs are intended to accounting volumetric properties for solid which has structural configurations that vary with pressure and temperature (Anderson [1]), BM EOS was derived from the assumption that the strain energy of a solid undergoing compression can be expressed as a Taylor series in the finite strain. (Birch [3]). The (mL-J) EOS was derived from interatomic potential (Jiuxun [6]) but Vinet EOS from relationship between variables (P, V) (Tripathi et. al. [15]). Phonon frequency spectrum curves for most solids shifts up slightly to the higher frequency values (Cwler & Cwler [8]).

In present work, the effect of pressure on bulk modulus (B), compression volume (jp

V Vo

lattice constant and phonon frequency spectrum (PFS) for GaAs has been achieve using different EOSs.

* [email protected] 1 [email protected] © Siberian Federal University. All rights reserved

1. Theoretical Details 1.1. Equations of State (EOSs)

Birch—Murnaghan EOS. Suggested EOS for solids (Birch [2]) is given as:

Pb-m (V) = 2Bo

GT -GT] {' - (4)(* - "o){( %

tO -1

(1)

where

Bo — thermal bulk modulus at atmosphere pressure, B'o — pressure bulk modulus derivative, Vo — volume at atmospheric pressure, Vp — volume at pressure P.

Vinet EOS. This EOS describe the relationship between the variables (P, V) and it was derived from a general inter-atomic potential (Griineisen [5]). (Vinet el al. [16]) give suitable EOS as:

PVinet

3Bo

(tf"{i - '■}] -[(> - 4 - (£>'"

(2)

Modified Lennard-Jones EOS. (mL-J EOS) (Jiuxun [6])

PmL-J = ( TT""

1

dB

where n = — Bo; Bo = ——.

3

dp

(vp) i\.Vp^)

^ -1

(3)

1.2. Bulk modulus

In general, bulk modulus given by

= -v(dP)

dV)r

B=V

(4)

from derivation of Eqs. (1), (2) and (3) with respect to volume and substiting in Eq. (4) we get the bulk modulus equation as function of pressure which are given as:

3Bo

3-M -

(7 )n-7/3 - 3- 4 (b'o - 4)n-3 +7 (B'O - 4)n-7/3 +4(B'- 4) n

-5/3

(5)

2 / /3

n

n

2

B

Vinet

2Bo (n-2/°3 - n-1/3) + Bon-1/3 + 2 Bo {b'o - l) (n-1/3 - l)

xe

[{I {b'0-i) }(iV

/3

(6)

B

mL-J = Bo

*( K V

2 ^ -1

(7)

X

n

1.3. Lattice constant

Radi et al. [11] determined the variation of lattice constant with the pressure according to Murnaghan EOS (Murnaghan [10]).

_ i

aP = a^ 1 + B'0B-^j 3B , (8)

where a0 is Lattice parameter at atmosphere pressure, ap is Lattice parameter at pressure P. From Murnaghan EOS

Vo T ' o B, Then Eq. (8) can be written as:

i

Ï-0 + BO^ (9)

aP = ao( ^ 3- (10)

1.4. Phonon frequency spectrum (PFS)

The pressure applied on the solid causes many changes in the internal atomic forces. Therefore, the properties of the solid under high pressure can vary (Dlouha [4]). In this study, the effect of pressure on the lattice constant and the density of state for GaAs calculated by using Grbneisen approximation to calculate the variation of the frequency where variation of mode density due to the specific volumes change of the crystal using the above equations of state (1), (2) and (3).

2. Calculations and Results

2.1. Calculations of compression volume ( f°r GaAs under high pressure.

(t)

On substituting the values BO, BO tabulated in Tab. 1 in to Eqs. (1), (2) and (3). Table 1. Values of bulk modulus (BO), its derivative (BO) and lattice constant aO for GaAs

Bo 74.7GPa ( Hanfland, et al. [9])

BO 4.67 ( Hanfland, et al. [9])

ao 5.650 A (Levenshtein and Rumyantsev [7])

Obtained results for ^ variations under high pressures are shown in Fig. 1, in comparison with experimental data published in the scientific literatures.

2.2. Calculations of variations bulk modulus Bo with high pressure for GaAs using B.M, Vinet, and mL.J EOSs

On substituting the values of B0,B'0 tabulated in Table (I), into Eqs. (5), (6) and (7). Obtained results for variations of bulk modules, for GaAs, are showing Fig. 2 with high-pressure

give us the values of bulk modulus according to the values of ^ V^ at different values of pressure

by using different EOS (Fig. 2).

Fig. 1. Variation of (V^ for GaAs at different values of pressure, using BM, Vinet, mL.J EOSs comparison with other experiment data

Fig. 2. Variation of bulk modulus Bo for GaAs at different values of pressure, using BM, Vinet & mL-J EOSs

2.3. Calculations of lattice constant variations, for GaAs, under high pressure

On using Eq.(10) with (ao = 5.650A, for GaAs) (Levenshtein and Rumyantsev [7]), and

' \

p I data from Fig.1. Present result for variations of lattice constant, for GaAs, with pressure

o

are shown in Fig. 3. on comparison with experimenter data of (Ruoff et al. [14]).

Fig. 3. Variation of lattice parameter for GaAs with high pressure using different EOSs

2.4. Evaluations of (PFS), for GaAs, under high pressure using (BM, Vinet, mL.J) EOSs

Fig. 4 shows phonon frequency spectrum for GaAs at atmospheric pressure and room temperature

Fig. 4. Variation of frequencies of GaAs at atmosphere pressure (Fu & Maguchi [17])

Since any change in produce, a change in the equilibrium position of lattice point and

it will produce a change in the frequency spectrum, which is given as (Dlouha [4]):

■Jp — vo

(11)

7: Grneisen parameter at atmosphere pressure

g (u,vp)= T7 g Uo,Vo]

(12)

uo — frequency at atmospheric pressure, vp — frequency at pressure P.

From Eqs. (11) and (12), and putting 7 = 1.73, for GaAs, (Resul and Irving [13]), and using shown in Fig. 1. Obtained results for variation of GaAs phonon frequency spectrum under

Vo

high pressure by using different equation of state are shown in Figs. 5-7.

Frequency'

Fig. 5. Variation of frequencies with pressure for GaAs using Birch-Murnaghan EOS

Frequency THz

Fig. 6. Variation of frequencies with pressure for GaAs using mL.J EOS

14

p^i-Gpa p=8Gpa p-12Gpa p=16Gpa

m o a

0

0 2 4 G 8 10 12 14 1G 18

Frequency THz

Fig. 7. Variation of frequencies with pressure for GaAs using Vinet EOS

3. Discussion and Conclusion

Variations under high pressure of I —p ), Bulk modulus, lattice constant for GaAs have been

evaluated by using three different EOSs given in Eqs. (1), (2) and (3). All the three different equations give an excellent agreement between their results in comparison with experimenter data. On evaluating variations of (pfs) for GaAs under high pressure B.M EOS (Eq. (1)) and mL-J EOS (Eq. (3)) give a reasonable shift to higher frequencies in agreement with theoretical interpretation for the effect of high pressure on pfs for solids. While Vinet EOS (Eq. (2)) cannot give a coincidence results with B.M and mL-J EOSs results.

References

[1] O.L.Anderson, Equation of state for Geophysical and Ceramic Science, Oxford University. Press, New York, 1995.

[2] F.Birch, Finite elastic strain of cubic crystal, Phys. Rev., 71(1947), 809-824.

[3] F.Birch, Equation of State and Thermodynamic Parameters of NaCl to 300 kbar in the High -Temperature Domain, J. Geophy. Res., 91(1968), no. B5, 4949-4954.

[4] J.Dlouha, The influence of pressure on the Messbauer effect, Czech. J. Phys., B 14(1964), 571-579.

[5] E.Griineisen, (1912), Cited in I.O Radi, M.A.Abdulsattar, A.M.Abdul-Lettif, Phys. Status Solidi b, 244(2007), no. 4, 1304-1317.

[6] S.Jiuxun, A modified Lennard-Jones type equation of state for solids strictly satisfying the spinodal condition, J. Phys.: Condens. Matter, 17( 2005), L103-L111.

[7] M.E.Levinshtein, S.L.Rumyantsev, Series on Semiconductor parameters, World Scientific, London, Handbook, Vol. 1, 2010, 77-103.

[8] M.Culer, E.Ciiler, Theoretical Analysis of Elastic, Mechanical and Phonon Properties of Wurtzite Zinc Sulfide under Pressure, Crystals, 7(2017), no. 6, 161.

[9] M.Hanfland, K.Syassen, N.Christensen, Volume dependence of optical transitions in GaAs: Photomodulated reflectivity, Journal de Physique Colloques, 45(1984), no. C8, 57-60.

10] F.D.Murnaghan, Finite deformations of an elastic solid, Am. J. Math. 49(1937), 235-260.

11] I.O.Radi, M.A.Abdulsatter, M.Abdul-Lettif, Semiemperical LUC- INDO calculations on the effect of pressure on the electronic structure of diamond. Phys. Stat. Sol. b, 244(2007), no.4, 1304-1317.

12] J.M.Recio, A.M.Pendas, E.Francisco, M.Florez, V.Luana, Low- and high- pressure ab initio equations of state for the alkali chlorides.Phys. Rev. B, 48(1993), no. 9, 5891-5901.

13] E.Resul, P.Irving, Lattice properties of strained GaAs, Si, and Ge using amodified bondcharge model. Phys. Rev. (B), 53(1996), no. 12, 7775-7785.

14] A.L.Ruoff, M.A.Baublitz, Volume dependence of optical transitions in GaAs: Photomodulated reflectivity, J. Appl. Phys., 53(1982), 6179.

15] P.Tripathi, G.Misra, S.Goyal, Equation of state for group IV-IV semiconductors; Solid state communication, 139(2006), 132-137.

16] P.Vinet, J.Ferrante, J.Rose, J.Smith, Compressibility of solids, J. Geophys. Res., 92(1987a), 9319-9325.

17] Z.Fu, M.Yamaguchi, Coherent Excitation of Optical Phonons in GaAs by Broadband Terahertz Pulses, Scientific Reports, 6(2016), no. 38264. DOI: 10.1038/srep38264.

Теоретическое исследование механических, упругих и фононных частотных свойств GaAs при высоком давлении

Раед Х. Аль-Сака

Главное управление образования средней школы Татваан в провинции Ниневия

Ирак

Сихам Дж. Аль-Тайе

Альнуур университетский колледж

Ирак

Представлены расчеты для изменения объемного модуля (B), объем сжатия yV^J, постоянная решетки и спектр фононных частот при высоком давлении до 17 ГПа при комнатной температуре. В 'расчетах используются три уравнения состояния EOS, в том числе Берч-Мурнаган, Винет и модифицированный Леннард-Джунс. Получены вариации этих свойств при высоком давлении для GaAs и подходящие EOS для них.

Ключевые слова: уравнение состояния, объемный модуль, сжатие, параметр решетки, спектр фононных частот.