United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency

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1 Available at: ~ pub_off IC/2006/061 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS EQUATION OF STATES AND MELTING TEMPERATURE OF ZINC BLENDE SEMICONDUCTORS Vu Van Hung, Pham Thi Minh Hanh Hanoi National Pedagogic University, Km8 Hanoi- Sontay Highway, Hanoi, Vietnam and Nguyen Thanh Hai 1 Hanoi University of Technology, 01 Dai Co Viet Road, Hanoi, Vietnam and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract By the statistical moment method (SMM) established previously on the basis of the statistical mechanics, the equation of states and melting temperature of zinc-blende semiconductors are considered. In the application, the lattice constant, pressure-volume relation V/V o and melting temperature at various pressures are calculated numerically for the zinc-blende semiconductors as GaP, AlP, AlAs and compared with experimental and other theoretical results. MIRAMARE TRIESTE July Junior Associate of ICTP. hai@mail.hut.edu.vn

2 1. Introduction High pressure behaviour of semiconductors is a topic of some technological, as well as, scientific interest since it plays an important role in determining the residual stresses in solid-state devices. Over the last few years, the study of materials under high pressure has become an extremely important subject. This is primarily due to both theoretical and experimental developments which have at last facilitated such work. Experimentally, high-pressure studies are often confronted with practical difficulties which substantially complicate measurements [1-6]. In addition to these experimental advances, reliable computation methods for electronic structure and total energy calculation have made a substantial impact on highpressure physics [7-8]. These calculations provide important complementary data to the experimental work, determining which observed structures are thermodynamically stable indicating the assignment of electronic and vibrational excitation modes, and predicting stability of as yet undiscovered structures. Density-functional theory itself and its computational implementation have also been the subject of several detailed reviews [9-12]. Density functional theory (DFT) is a rigorous formulation of the many-body problem in which the fundamental variable is the (electronic) charge density. Hohenberg and Kohn (1964) proved that the total energy of the ground state is a unique function of the charge density and that this function is minimal for the correct density. One of the motivations for carrying out theoretical studies of materials under pressure is to investigate how well approximate functions, such as the local-density approximation (LDA) and generalized gradient approximation (GGA), perform in describing the wide range of chemical bonding that occurs in highpressure phases. First-principles (or ab initio) electronic structure computations have been performed on semiconductor compounds and the results compared with experiment. Methods for calculating the energies of solids within a quantum-mechanical framework have been under continuous development since Many important developments in methodology have been made [10] which, combined with the relentless increase in computing power, have led to the modern field of first-principles electronic structure computations in which the only experimental input is the atomic numbers of the atoms 2

3 involved. The possibility of using first-principles electronic structure computations to study high-pressure phases was first demonstrated by Yin and Cohen [7] who calculated the energies of several structural phases of Si, obtaining good agreement with the available experimental data. More recently a large number of high-quality calculations have been performed on group-iv, III-V, and II-VI materials. Modern calculations allow the accurate relaxation of structures to their minimum energy configurations and the incorporation of temperature effects. One can also study the melting of solids using firstprinciples molecular-dynamics methods [13], so that the entire P-T phase diagrams of materials could be investigated. Focher et al. [14] have used first-principles moleculardynamics methods to study phase transitions in Si. Such calculations are computationally expensive and, currently, simulations can only be run for periods of tens of picoseconds, which is not long enough for some of the processes of interest. However, first-principles molecular-dynamics is expected to play an important role in future theoretical studies of phase transitions. In order to determine theoretically the melting temperature of metals or semiconductors we must use the equilibrium condition of the liquid and solid phases. However, a definite expression of the melting temperature is not available in the literature. We note that the limiting temperature of the absolute stability of the crystalline state at finite pressure is not far from the melting one. Therefore, in order to determine the melting temperature the application of the condition of absolute stability for the crystalline phase was used in [15-18]. In the present paper, analogously as in [15-18], we use the limiting condition of absolute stability in order to find the corresponding temperature T S at various pressure (the limiting temperatures of absolute stability for the crystalline state) of zinc-blende semiconductors. Equation of states and expression for the limiting temperature T S are presented in Section 2. In Sec.3 we present the statistical moment method (SMM), calculated results and related discussions for GaP, AlAs, AlP semiconductors. 2. Equation of states and the limiting temperature of absolute stability To determine the Helmholtz free energy for zinc-blende semiconductors we used the statistical moment method. The theoretical scheme has been presented and explained in 3

4 our previous papers [19-22]. From the expression for the Helmholtz free energy of system the pressure P of the zinc-blende semiconductor can be written in the form a u 3γ θ 0 G P = +, (1) 3v a v γ G ω = = k m a 6k k x coth x a ; θ = k B T ; ; 0 hω u = E ; x = 0 i 2θ 2 E i k = ; 2 u ix where γ G is the Gruneisen constant, k B is the Boltzmann ' s constant, v is the atomic volume and E i the internal energy associated with atom i. In order to calculate the equation of states of zinc-blende semiconductor, we need to know the empirical interatomic potentials. The Stillinger-Weber [23] potentials are basically Lennard-Jones type potentials and are modified to include the short-range nature and the angle stiffness of the covalent bonding. The Stillinger-Weber potentials consist of two-body and three-body terms: 1 1 E = E = Φ (r, r ) + W (r, r,r ) 2. (3) i ij i j i, j,k i j k i i, j 3 i, j,k The two -body term is given by 1 d 4 ij Φ = ε ( ) {( ) } ij A Brij 1 exp rij b ; rij = (4) σ where d ij is the bond length between atom i and atom j, and ε - the cohesive energy per bond, b is a dimensionless parameter which represents the cutoff distance of the interaction. The three-body term corresponds to the angle distortion energy and is given by: 1 1 { η( r b) + η( r b) } 1 W = ελ exp cos ijk ij ik θ + (5) 3 where θ is the angle between bond ij and bond ik. The parameters in the above equations (4), (5) should be determined so that some basic properties of the material are reproduced from the potentials. Table 1 shows the obtained parameters together with the cohesive energy, the potential parameters A, B, σ and λ [24]. The cut off parameter b is taken to be1.8, and η to be 1.2, as originally proposed by Stillinger-Weber. eq ; 2 (2) 4

5 Using eq.(1), one can find the nearest neighbor distance a(p,t) at pressure P and temperature T. However, for numerical calculations, it is convenient to determine first the nearest neighbor distance a(p,0) at pressure P and at absolute zero temperature T=0K. For T= 0K temperature, eq. (1) is reduced to: 1 u 0 hω k Pv = a + (6) 3 a 4k a Using the Maple or Pascal programs, eq. (6) can be solved, we find the values of the nearest neighbor distance a (P,0). Then, we can find a(p,t) at pressure P and temperature T as: (P,T) = a(p,0) + y (P,T) (7) a 0 where y 0 (P,T) is the displacement of a particle from the equilibrium position at pressure P and temperature T. The thermally induced lattice expansion y 0 (P,T) is determined as in our previous papers [17,22,25] by the statistical moment method. P T V Using the thermodynamic relation = 1 T V V P P T condition of absolute stability for the crystalline state [15-18]: P P = 0, i.e. = 0 V T a T we find the corresponding expression of the limiting temperature: and the limiting a u o T T = P S +. (9) 9k γ a P B G a It is noted that the nearest neighbor distance a(p,t s ), the Gruneisen constant u 0 T γ G (P,T s ), and quantities of and in eq.(9) are calculated at pressure P and a P a temperature T S. With the aid of eqs. (1), (2), (3), (4), (5) and using the Maple or Pascal programs, eq.(9) can be solved, and we find the values of T S (P). (8) 5

6 3. Results and discussion It is noted that, at the same pressure value, the melting temperature T m and the limiting temperature of absolute stability T S are not so different [15-18]. Therefore, the melting temperature of crystal T m can be determined by an approximate expression T T m S. (10) Table 2 gives the SMM calculated melting temperature and experimental values at zero pressure P=0 for the GaP, AlP and AlAs. The deviation of our result from observed values is 9.7 % for GaP, 2.9 % for AlP and 3.3 % for AlAs. Figure 1 shows the melting temperature of the GaP, AlP and AlAs as a function of the pressure. The SMM calculated values of melting temperature at various pressures for GaP are very near to the experimental ones. The discrepancy of our results, in comparison with experiments, is some percent. The SMM calculated lattice constant a (P,300K), the pressure-volume relation V/V 0 0 at temperature T=300K and various pressures P for GaP, AlP and AlAs zinc-blende semiconductors are tabulated in Table 3 along with experimental and other theoretical results. Table 3 also shows that, the melting temperature of these semiconductors depends strongly on the pressure, and it is in good agreement with the corresponding experimental results. In conclusion, the statistical moment method allows us to investigate the equation of states and melting temperature of zinc-blende semiconductors under hydrostatic pressures. For GaP, we have obtained good agreement in the melting temperatures between the theoretical calculation and experimental results. Acknowledgments. This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 6

7 References 1. Nelmes R.J., Mc. Mahon M.I., Wright N.G., Allan D. R., Liu H. and Loveday J.S., Semicond. Semimet. 54 (1998) Nelmes R.J., Mc. Mahon M.I., Hatton P.D.,Crain J. and Piltz R.O., Phys. Rev. B 47 (1993) 35; Phys. Rev. B 48 (1993) Nelmes R. J. et all., Phys. Rev. B 48 (1993) Ohsumi et all., Phys. Rev. B 22 (1980) Prins A.D., Spain I.L. and Dunstan D.J., Semicond. Sci. Technol. 4 (1989) Jayaraman, Rev. Mod. Phys. 55 (1983) Yin M.T. and Cohen M.L., Phys. Rev. B 26 (1982) 3259; B 26 (1982) 5668; B 26 (1982) Car R. and Parrinello M., Phys. Rev. Lett., 55 (1985) Christensen N.E., Semicond. Semimet. 54 (1998) Payne M.C., Teter M. P., Allan D.C., Arias T.A. and Joannopoulos J. D., Rev. Mod. Phys. 64 (1992) Jones R.O. and Gunnarsson O., Rev.Mod. Phys. 61 (1989) Hemley R.J. and Ashcroft N.W., Phys. Today 51 (1998) Sugino O. and Car R., Phys. Rev. Lett. 74 (1995) Focher P., G.L. Chiarotti, M.Bernasconi, E. Tosatti, and M. Parrimello, Europhys. Lett. 26 (1994) Plakida N.M., Fiz. Tverd. Tela 11 (1969) Zubov V.I., Problemy statiticheskoi Kvantovoi Fiziki (UND, Moskva, 1978) (in Russian). 17. N.Tang and Vu Van Hung, Phys. Stat. Sol. (b) 162 (1990) 379; 162 (1990) Vu Van Hung and N.Thanh.Hai, J.of Phys. Soc. Jpn., 66 (11) (1997) N.Tang and Vu Van Hung, Phys. Stat.Sol. (b) 149 (1988) 511; 161 (1990) K.Masuda- Jindo, V.V.Hung and P.D.Tam, Phys.Rev. B 67 (2003) K.Masuda- Jindo, S.R. Nishitani and V.V.Hung, Phys. Rev. B 70 (2004) V.V.Hung, K.Masuda- Jindo, and P.T.M.Hanh, J. Phys.: Condens. Matter 18 (2006) Stillinger F.H. and Weber T.A., Phys. Rev. B 31 (1985) Ichimura M., Phys. Stat. Sol. (a) 153 (1996) V.V.Hung, K.Masuda- Jindo, J. Phys. Soc. Jpn. 69 (7) (2000) Dwight R.Lide, Handbook of Chemistry and Physics (1998)

8 28. P.K.Singh and Sadhna Singh, Phys. Rev. B 39 (1989) E.IU.Tonkov: Phase Trasformation Connection in High Pressure (Metalurghia, Moskva,1988 ) (in Russian). Table 1: The cohesive energy ε, and the obtained SW potential parameters A, B, σ and λ [24] Material ε(ev) A B σ(a 0 ) λ GaP GaAs AlP AlAs Table 2: SMM calculated melting temperature and experimental results at pressure P=0 for zincblende semiconductors Material GaP AlP AlAs T m (K) T m (K) Exp [ 26] T m (K) Exp [ 27]

9 Table 3: SMM calculated lattice constant, pressure - volume relation V/V o at temperature T=300 K and various pressures P and melting temperature T m along with observed data for GaP, AlP and AlAs Material P(GPa) a 0 (P,300K),A V/V 0 (SMM) GaP Other theory [28] T m (SMM) EXP [ 29 ] 2033± ± ± ± ± ± ± ± ± ±20 a 0 ( P,300K ), A AlP V/V T m a 0 ( P,300K ),A AlAs V/V T m

10 SMM 4000 SMM EXP [29] 3500 T ( K ) T ( K ) P ( GPa ) P ( GPa ) a) AlAs b) GaP 4000 SMM 3500 T ( K ) P ( GPa ) c)alp Fig. 1: Pressure dependence of melting temperature T m of zinc-blende semiconductors 10