Nvestigation of thermodynamic and mechanical properties of AlyGa1-YAs/GaAs systems by statistical moment method

Abstract. In this paper, the moment method in statistical dynamics (SMM) is used to study the thermodynamic quantities of AlyGa1-yAs/GaAs systems taking into account the anharmonicity effects of lattice vibrations. The nearest neighbor distance, lattice parameter, isothermal bulk modulus, specific heats at the constant volume and pressure of AlyGa1-yAs/GaAs systems are calculated as functions of the temperature and concentration of Al by using the many-body potential. The effects of temperature and concentration of Al on thermo-mechanical quantities of Al yGa1-yAs/GaAs systems will be discussed in detail.

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96 JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0036 Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 96-103 This paper is available online at INVESTIGATION OF THERMODYNAMIC AND MECHANICAL PROPERTIES OF AlyGa1-yAs/GaAs SYSTEMS BY STATISTICAL MOMENT METHOD Vu Thi Thanh Ha 1 , Vu Van Hung 2 and Vu Hong Nhat 2 1 University of Education Publishing House 2 University of Education, VNU Abstract. In this paper, the moment method in statistical dynamics (SMM) is used to study the thermodynamic quantities of AlyGa1-yAs/GaAs systems taking into account the anharmonicity effects of lattice vibrations. The nearest neighbor distance, lattice parameter, isothermal bulk modulus, specific heats at the constant volume and pressure of AlyGa1-yAs/GaAs systems are calculated as functions of the temperature and concentration of Al by using the many-body potential. The effects of temperature and concentration of Al on thermo-mechanical quantities of AlyGa1-yAs/GaAs systems will be discussed in detail. Keywords: Statistical moment method, anharmonicity, semiconductor superlattice. 1. Introduction Study of the thermodynamic and mechanical properties of semiconductor superlattice has become quite interesting in recent years since semiconductor superlattices play an important role in technological applications, especially in the electronic industry. The structural and thermo-mechanical properties of AlyGa1-yAs/GaAs systems were studied by using the lattice dynamics approach [1] and ab initio molecular dynamic simulations [2]. These physical properties of materials, in general, depend on temperature and their composition. The purpose of present article is to study the temperature effects on the thermodynamic and mechanical properties of the AlyGa1-yAs/GaAs semiconductor superlattice at zero pressure using the moment method in the quantum statistical mechanics, hereafter referred to as statistical moment method (SMM) [3-5]. The anharmonicity of thermal lattice vibrations of AlyGa1-yAs/GaAs systems has been considered by using the many-body interaction potential. Received August 7, 2017. Accepted August 30, 2017. Contact Vu Thi Thanh Ha, e-mail: thanhhadhsp@gmail.com. Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems 97 2. Content 2.1. Theory The AlyGa1-yAs/GaAs semiconductor superlattice consists of AlyGa1-yAs and GaAs layers, where AlyGa1-yAs and GaAs compounds are both in the zinc-blende form. Figure 1. Strucsture of AlyGa1-yAs/GaAs semiconductor superlattice We assume that the thickness of AlyGa1-yAs compound is d1 and this semiconductor layer consists of N1 atoms with NAl atoms of Al, NGa atoms of Ga and NAs atoms of As then 1 Al Ga AsN N N N   ,  1 1 1Al Ga As N y N N N ;N 1 y ;N 2 2 2     . (1) Similarly, GaAs compound layer is supposed to have the thickness d2 and consist of N2 atoms with *GaN atoms of Ga and * AsN atoms of As then * 2 GaN N N  * As , (2) So  1 2N N N n  . (3) where N is the number of atoms and n is the period of semiconductor superlattice. In order to investigate the thermodynamic and mechanical properties of the AlyGa1-yAs/GaAs semiconductor superlattice, we firstly consider the change of Helmholtz free energy  of AlyGa1-yAs when NGa atoms of Ga are replaced by Al atoms in GaAs crystal. The substitution of an atom Ga by an atom Al causes the change of the free Gibbs energy as f Ga Al 0 0g    (4) where Al0 is the free energy of an atom Al in the AlyGa1-yAs compounds and Ga 0 is the internal energy associated with atom Ga of the zinc-blende GaAs compound. In the following of paper, the interatomic potential is assumed to be the many-body potential [6] which consists of two-body and three-body terms as GaAs AlyGa1–yAs  d2  d1 Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat 98 i ij i j ijk i j k i,j i,j,k (r ,r ) W (r ,r ,r )     (5) with 12 6 0 0 ij i j ij i,j r r (r ,r ) 2 r r                    (6)       i j k i,j,k i j k 3 ij jk ki 1 3cos cos cos W r ,r ,r Z r r r      (7) where ij jk kir , r , r are respectively the bonds between the ith and jth atoms, jth and kth atoms, and kth and ith atoms; and i j k, ,   are the angles formed by three particles i, j and k. And ε, r0, and Z are the potential parameters which were fitted to the bond lengths of the dimer and trimer and the lattice parameter and cohesive energy of the structure [6]. It should be noted that the analytical expression of the free energy of an atom Ga or As in the zinc-blende GaAs compound in the harmonic approximation has the form as [4,7] * Ga As 2x 0 0 0 0 3 x n(1 e )            (8) where Ga As 0 0 0 i0 i ijk i i,j 1 1 (a ) W 2 3          , (9) and x 2    with Bk T  ,  is the atomic vibration frequency which can be approximated in most cases to the Einstein frequency as 2 2io 2 i ix eq 1 k m 2 u           , (10) with  Ga As 1 m m m 2   and io is the interatomic potential energy between the central 0th and ith sites. The free energy Al0 of an atom Al in AlyGa1-yAs system in the harmonic approximation has the form analogous to Eq. (8) as AlAl Al Al 2x 0 0 3 x n(1 e )          (11) where Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems 99 Al Al * k x 2m   , * Al As 1 m (m m ) 2   , Al 0 iAl i ijAl i i,j 1 1 (a ) W 2 3      , (12) and the pair interaction potential iAl between the Al atom and ith atom (i = As or Ga). Because the AlyGa1-yAs system is supposed to be built by substituting AlN atoms Al into the positions of Ga atoms of a zinc-blende GaAs system then the Gibbs free energy of system has an approximate form as   f 0 Al C * Ga Al 1 0 Al 0 0 C 1 G G N .g TS N N TS PV           (13) where P denotes the hydrostatic pressure, V1 is the volume of the AlyGa1-yAs system and SC is the configuration entropy. From Eqs. (3), (8) and (13), we derive the Gibbs free energy of AlyGa1-yAs/GaAs semiconductor superlattice as  * Ga AlSS 0 Al 0 0 CG N nN nTS PV       (14) where V is the volume of the AlyGa1-yAs/GaAs system. The Helmholtz free energy SS of the AlyGa1-yAs/GaAs system can be now derived  * Ga AlSS 0 Al 0 0 CN nN nTS       (15) and then the equation-of-states of the AlyGa1-yAs/GaAs systems at finite temperature T can be obtained from the Helmholtz free energy as SS SS T T a P V 3V a                 or  0 2 1 Al Al Al Al0 Al 2 2 1 1 y1 1 k Pv a 1 xcthx 3 d a 2k a 2 1 d y y 1 k x cthx d a d a2k 6 1 2 1 d d                                                          (16) where v is the atomic volume. By numerically solving the equation-of-state Eq. (16) one can determine the average nearest- neighbor distance (NND) between two intermediate atoms a(T) at temperature T. Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat 100 Let us now consider the isothermal compressibility of the AlyGa1-yAs/GaAs semiconductor superlattice. According to the definition of the isothermal compressibility T , it is given in terms of the volume V and pressure P as [3].   3 T 22 To SS 2 T a(P,T) 3 a(P,0)1 V V P a (P,T) 2P 3V(P,T) a                    (17) Where   2 2 2 Al SS 0 0 2 2 2 2 2T 1 1 22 2 2 2 2 Al Al 2 Al Al 2 2 1 y y1 1 N d da a a 2 1 2 1 d d xcthx k 1 x k 3 xcthx 2k aa 4k sh x y x cthx k 1 3 d 2k a 4 k2 1 d                                                                                   2Al 2Al Al Al 2 2 AlAl x k x cthx ash x                (18) Furthermore, the specific heat at constant volume CV, specific heat at constant pressure CP, and isothermal bulk modulus BT of AlyGa1-yAs/GaAs system are determined from the well-known thermodynamic relations as 2 V P V T T T E 9TV 1 C ; C C ; B T          , where E is the energy of a crystal AlyGa1-yAs/GaAs system which can be calculated by using the Gibbs-Helmholtz relation SSSSE .        2.2. Results and discussions In this section, the expressions derived in the previous section will be applied to numerically calculate self-consistently thermodynamic and mechanical properties of given AlyGa1-yAs and AlyGa1-yAs/GaAs systems including lattice parameter, bulk modulus and specific heats at constant volume and pressure. The potential parameters of binary GaAs and ternary AlGaAs systems are given in Table 1 [6]. The SMM calculations of the lattice parameter ah, isothermal bulk modulus BT and specific heat at constant pressure of the zinc-blende AlyGa1-yAs system at room temperature are shown in Table 2. As can be seen in this table, our results agree well with available experimental data [8]. The difference between theory and experiment is about 0.4% for lattice parameter and about 7-8% for bulk modulus and specific heat at constant pressure. From this table we can see that these thermodynamic properties of AlyGa1-yAs compound increase with the increase in composition of Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems 101 Al. The rapid increase bulk modulus BT indicates stronger rigidity properties of AlyGa1-yAs semiconductor when the Al composition increases. Table 1. Potential parameters of GaAs and AlGaAs systems [6] Quantity Ga-As Al-Ga Al-As Al-Ga-As AB (eV) 1.738 1.121 2.060 - 0ABr (Å) 2.448 2.490 2.430 - AABZ (eV.Å 9 ) 1900.0 2093.3 3000.0 - ABBZ (eV.Å 9 ) 4600.0 1955.3 5000.0 - ABCZ (eV.Å 9 ) - - - 2500.0 Table 2. Lattice parameter ah, isothermal bulk modulus TB and specific heat at constant pressure of the zinc-blende AlyGa1-yAs system at room temperature (T = 300 K) Composition ah (Å) SMM a (Å) Exp [8] BT (GPa) SMM BT (GPa) Exp [8] CP (cal/mol.K) SMM CP (cal/mol.K) Exp [8] 0 5.626 5.653 82.06 75.50 5.16 5.54 0.1 5.628 5.654 82.15 75.76 5.41 5.77 0.2 5.630 5.655 82.24 76.02 5.66 5.99 0.3 5.631 5.656 82.33 76.28 5.91 6.22 0.4 5.633 5.656 82.42 76.54 6.15 6.45 0.5 5.635 5.657 82.51 76.80 6.40 6.68 0.6 5.636 5.658 82.60 77.06 6.65 6.91 0.7 5.638 5.659 82.69 77.32 6.90 7.14 0.8 5.640 5.660 82.78 77.58 7.15 7.37 0.9 5.641 5.660 82.88 77.84 7.40 7.60 Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat 102 0.0 0.2 0.4 0.6 0.8 1.0 5.6262 5.6264 5.6266 5.6268 5.6270 5.6272 5.6274 5.6276 L a tt ic e c o n s ta n t (Å ) Composition (x) d2 = 10d1 d2 = 40d1 Fig 2. Composition dependence of the lattice parameter of the semiconductor superlattice AlyGa1-yAs/GaAs at room temperature 0.0 0.2 0.4 0.6 0.8 1.0 82.66 82.68 82.70 82.72 82.74 82.76 B u lk m o d u lu s ( G P a ) Composition (x) d2 = 10d1 d2 = 40d1 Fig 3. Composition dependence of the isothermal bulk modulus of the semiconductor superlattice AlyGa1-yAs/GaAs at room temperature In Figs 2 and 3, we present the lattice parameter and isothermal bulk modulus of the semiconductor superlattice AlyGa1-yAs/GaAs as functions of the composition Al at room temperature. As it can be seen from these two figures, the lattice parameter and bulk modulus of AlyGa1-yAs/GaAs systems are increasing functions of composition Al. Furthermore, when the thickness of GaAs rises, these thermo-mechanical properties increase rapidly. 200 300 400 500 600 700 800 4.6 4.8 5.0 5.2 5.4 5.6 C p ( c a l/ m o l. K ) T (K) x = 0.3; P = 0; d2 = 10d1 Fig 4. Temperature dependence of the specific heat at constant pressure CP of AlyGa1-yAs/GaAs systems 200 300 400 500 600 700 800 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 C v ( c a l/ m o l. K ) T (K) x = 0.3; P = 0; d2 = 10d1 Fig 5. Temperature dependence of specific heat at constant volume CV of AlyGa1-yAs/GaAs systems In Figs 4 and 5, we show the temperature dependences of the specific heats at constant pressure CP and constant volume CV of an AlyGa1-yAs/GaAs system with composition x = 0.3 of Al. The calculated specific heats CP and CV develop briskly when the temperature range is below 450 K. Beyond 450 K, when temperature increases the specific heat CP and CV of AlyGa1-yAs/GaAs system decreases gradually. Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems 103 3. Conclusion In conclusion, the SMM calculations have been performed to investigate the thermo- mechanical properties of AlyGa1-yAs/GaAs systems. Using the free energy formulas derived in the SMM, we have derived the analytical expressions of the lattice constant, bulk modulus, specific heats at the constant volume and constant pressure of the zinc-blende AlyGa1-yAs and AlyGa1-yAs/GaAs systems. Numerical calculations have been discussed and compared with those of experimental results. The moment method can be developed extensively for studying the atomistic structure and thermodynamic properties of other ternary and quaternary alloys as well. REFERENCES [1] P.S. Branicio, J.P. Rino, R.K. Kalia, A. Nakano and P. Vashishta, 2003. J. Appl. Phys. 94 3840. [2] S. Baron, S. de Giroconli, and P. Giannozzi, 1990. Phys. Rev. Lett. 65 84. [3] K. Masuda-Jindo, V.V. Hung, P.D. Tam, 2003. Phys. Rev. B67, 094301. [4] V.V. Hung, K. Masuda-Jindo, P.T.M. Hanh, 2006. J. Phys. Condens. Matter 18 (1), 283. [5] H.K. Hieu, V.V. Hung, Mod. 2011. Phys. Lett. B25, 1041. [6] S. Erkoc, Physics Reports, 1997. 278 (2), pp. 79-105. [7] V.T.T. 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