Pressure dependence of melting curve of silicon crystal with defects

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 65-71 PRESSURE DEPENDENCE OF MELTING CURVE OF SILICON CRYSTAL WITH DEFECTS Vu Van Hung(∗) Hanoi National University of Education Le Dai Thanh Hanoi University of Science (∗)E-mail: bangvu57@yahoo.com Abstract. The high-pressure melting curve of silicon crystal with defects has been studied using statistical moment method (SMM). In agreement with experiments and with DFT calculations we obtain a negative slope for the high-pressure melting curve of silicon crystal. SMM calculated melt- ing temperatures of Si crystal with defects being in good agreement with previous experiments. Keywords: Melting curve, silicon crystal with defects, pressure, statistical moment method. 1. Introduction In 1930, the melting curve of the crystals were described by the empirical Simon equation, but this simple law breaks at high pressure [1]. A new empirical law for the melting temperatures of crystals at high pressure was suggested by M. Kumari et al. in 1987 [2]. Melting of a solid is known as a first-order discontinuous phase transformation occurring at a critical temperature at which Gibbs free energies of the solid and the liquid states are equal [3]. However, a clear expression of the melting temperature is not yet obtained in this way. The Lindemann and dislocation-mediated melting models, molecular dynamics (MD) and ab initio quantum mechanical calculations are applied to the investigations of melting curve, and these theoretical and experi- mental results are reviewed in ref. [4]. Recently numerical simulations have shown that correlated clusters of defects thermally excited play a central role in this process at the limit overheating [5]. In addition, investigations revealed that various kinds of defects in solids, such as interfaces, grain boundaries, voids, impurities and other defects, also facilitate melting [6]. 65 Vu Van Hung and Le Dai Thanh The purpose of this present paper is to discuss the effect of pressure and point defect on the melting temperature of semiconductors using statistical moment method (SMM) [7-9]. Using many-body potentials, a PV T equation of states of Si semi-conductors is obtained, the pressure dependence of the melting temperature being estimated. 2. Content 2.1. Theory 2.1.1. Effect of vacancies on the melting temperature The melting temperature of the crystal with point defect, T Vm (P, nV ) is the function of the equilibrium vacancy concentration nv and pressure P . In first-order approximation the melting temperature T Vm of the defect crystal at the pressure P can be expanded in term of the equilibrium vacancy concentration nv as T Vm (P, nV ) = Tm(0) + ( ∂T ∂P ) nv,V .P + ( ∂T ∂nV ) V,P .nV + , (2.1) or T Vm (P, nV ) ≈ Tm(P ) + ( ∂T ∂nV ) V,P .nV (2.2) where Tm(P ) is the melting temperature at the pressure P of the perfect crystal Tm(P ) = Tm(0) + ( ∂T ∂P ) nV ,V .P From the minimization condition of the Gibbs free energy of the crystal with the point defect, we obtain the equilibrium concentration of the vacancies as [10,11]: nv = exp { −g f v (P, T ) θ } , (2.3) where gfυ is the change in the Gibbs free energy due to the formation of a vacancy and can be given by gf v = −ϕ0 +∆F ∗0 + P∆V (2.4) It should be noted that pressure affects the diffusivity through both the free energies, F ∗0 and the volume change, resulting from the formation of the point defect, ∆V. This change is due to the P∆V work done by the pressure medium against the volume change associated with defect formation and migration. 66 Pressure dependence of melting curve of silicon crystal with defects In eq. (2.4), ϕ0 = 1 2 ∑ i φi0(|~r|) + 1 3 ∑ i,j Wijo(ri, rj , ro) represent the internal energy associated with 0th atom and φio effective interaction energies between 0 th and ith atoms, ∆Fo∗ denotes the change in Helmholtz free energy of the central atom which creates the vacancy, by moving itself to a certain sink site in the crystal, and is given by ∆Fo∗ = (C − 1)F ∗o , (2.5) where F ∗0 denotes the free energy of the central atom after moving to a certain sink sites in the crystal, C is simply regarded as a numerical factor. In the previous paper [11], we take the average value for C as C ≈ 1 + ϕ0 2F ∗0 (2.6) Using the derivative of the equilibrium vacancy concentration nV of eq. (2.3) with respect to temperature T and eqs. (2.2) and (2.4), we obtain T Vm (P, nV ) ≈ Tm(P ) + T 2m(P ) Tm(P ) ∂gf V ∂θ − g f V kB (2.7) In order to determine theoretically the melting temperature of perfect semi- conductors, Tm(P ), we will use the equilibrium condition of the solid phases. Since the treatments of liquid phases are rather complicated, most of the previous studies have been performed on the basis of the properties of the solid phases, (starting with the Lindemanns formula) theorized in terms of the lattice instability [12], free energy of dislocation motions, or a simple order-disorder transition [13]. In the following sub-section, we calculate the melting temperature Tm(P ) of perfect semiconductor using many-body potentials. 2.1.2. Equation of states and melting temperature of perfect semicondu- -ctors by SMM From the expression of the Helmholtz free energy in the harmonic approxima- tion, the pressure P of the diamond cubic and zinc-blende semi-conductors can be written in the form [14] P = − r 3v ∂ϕ0 ∂r + 3γGθ v (2.8) where γG is the Gru¨neisen constant, v is the atomic volume. From eq. (2.8) one can find the average nearest-neighbour distance (NND) of atoms in crystal r (P, T ) at pressure P and temperature T. However, for numerical calculations, it is convenient to determine firstly the NND of crystals r (P, 0) at 67 Vu Van Hung and Le Dai Thanh pressure P and at absolute zero temperature, T = 0K. For T = 0K temperature, eq. (2.8) is reduced to: Pv = −r [ 1 3 ∂ϕ0 ∂r + ~ω 4k ∂k ∂r ] (2.9) Eq. (2.9) can be solved using a computer programme to find out the values of the NND r (P, 0) of the perfect semi-conductors. From the obtained results of NND r (P, 0) we can find r (P, T ) at pressure P and temperature T as: r (P, T ) = r (P, 0) + y0 (P, T ) , (2.10) where y0 (P, T ) is the displacement of an atom from the equilibrium position at pressure P and temperature T [7, 15]. Using the many-body potentials which consist of two-body and three-body terms [16] ϕi = ∑ j φij(ri, rj) + ∑ j,k Wijk(ri, rj, rk) (2.11) where φij = ε [( r0 rij )12 − 2 ( r0 rij )6] (2.12) Wijk = Z (1 + 3 cos θi cos θj cos θk) (rijrikrkj)3 (2.13) with rij is the distance between the i-th atom and j-th atom in crystal; θi, θj , θk are the inside angles of a triangle to create from three atoms i, j and k; and the potential parameters ε, r0, Z are taken from ref. [17]. These parameters are determined so as to fit the experimental lattice constants and cohesive properties of Si crystal. Using the many-body potentials of eqs. (2.12), and (2.13), we obtained the expression of the parameter k and internal energy ϕ0 and then the eq. (2.9) can be solved to find out the values of the NND r (P, 0) of the perfect silicon. In order to determine theoretically the melting temperature of semi-conductors we will use the equilibrium condition of the solid phases. In particular, we will use the limiting condition for the absolute crystalline in order to find the melting tem- peratures under the hydrostatic pressures. We note that the limiting temperatures for the absolute crystalline stabilities of solid phases Ts are very close to the melting temperatures Tm [3]. From the limiting condition of the absolute stability for the crystalline phase,( ∂P ∂V ) T = 0, i.e. ( ∂P ∂r ) T = 0 and eq. (2.8), we find the expression of the limiting 68 Pressure dependence of melting curve of silicon crystal with defects temperature as TS(P ) = r 9kBγG ( ∂ϕ0 ∂r ) + ( ∂T ∂P ) V P (2.14) In the case of P = 0, it reduces to TS(0) = r 9kBγG ( ∂ϕ0 ∂r ) (2.15) Eq. (2.15) permits us to determine the limiting temperature of absolute sta- bility TS(0) at pressure P = 0. Because of the melting temperature is less different from the limiting temperature of absolute stability at same pressure value. There- fore, the melting temperature of crystals Tm can be determined by an approximate expression: Tm(P ) ≈ TS(P ) (2.16) 2.2. Results and discussion In this section we compare our melting curve, eq. (2.7), for Si crystal to experimental melting curves. Table 1 shows the good agreement between the SMM calculations of melting temperature Tm(P ) at various pressures and experimental results for Si semiconductor. Our calculated zero-pressure melting temperature for Si crystal with defect (1640 K) is in good agreement with the experimental value of 1687 K [18]. We note that density functional theory (DFT) calculations of zero- pressure melting point for Si using the local density approximation (LDA) predict values in the range 1300 - 1350 K [19, 20] and 1492 ± 50 K when a generalized- gradient approximation (GGA) is used instead [20]. Table 1. SMM calculated pressure dependence of melting temperature, T Vm (K) and contribution of the vacancies on the melting temperature, ∆T (K) for Si diamond cubic crystal P (GPa) 0 1 2 3 4 5 ∆T (K) - 81 - 93 - 92 - 91 - 90 - 90 T Vm (K) 1640 1602 1585 1572 1563 1553 Exp.[21] 1685 1647 1609 1571 1533 1495 Table 1 shows that the contribution of the vacancies on the melting temper- ature of silicon crystal with defects, ∆T = T Vm − Tm, is about 5% ÷ 6%, and the SMM melting temperature values of Si perfect crystal are considerably higher than the calculation results by the SMM of this crystal with defects. The equilibrium va- cancies concentration is very small at low temperature. At high temperature being 69 Vu Van Hung and Le Dai Thanh near the melting one the contribution of the vacancies on the melting temperature of semi-conductor crystals is some percent. In this paper we use the statistical moment method (SMM) to study the pressure dependence of melting temperature of crystalline semi-conductors. We use the many-body potentials which consist of two-body and three-body terms for calculations of melting temperature and the potential parameters used in the present study are taken from ref. [17]. We notice that the melting temperature of Si crystal in Table 1 shows a small deviation from the SMM results to the experimental ones with increasing pressure: Our calculated zero-pressure melting temperature for Si crystal with defect (1640 K) is in good agreement with the experimental value of 1687 K (with 2.7% deviation), and at pressure P = 5 Gpa melting temperature for Si crystal with defect (1553 K) is also in good agreement with the experimental value of 1495 K (with 3.8% deviation). To achieve more accurate results for Si under high pressure we can choose the better parameters of potential or use the different potential parameters for various pressure, and the detailed discussions on this discrepancy will be given elsewhere. For the negative-slope melting curve of semi-conductors the SMM calculations showed that the negative pressure dependence arises from the sign in the second term of eq. (2.16): dTm/dP < 0. Since the melting slope dTm/dP is equal by the Clausius-Clapeyron in relation to dTm dP = Tm ∆V ∆S , where ∆V = Vl − Vs is the difference of molar volumes and ∆S = Sl − Ss is the difference of molar entropies, respectively, and assuming that the liquid entropy is bigger than the solid, if the melting curve has a negative dTm/dP < 0, this will lead to the conclusion that a denser liquid phase than solid phase: ∆V < 0. 3. Conclusion The high-pressure melting curve of silicon crystal has been studied using sta- tistical moment. In agreement with experiments and with DFT calculations we obtain a negative slope for the high-pressure melting curve. We have derived a new equation for the melting curve of semiconductor with defects, eq.(2.7). We have cal- culated melting curves for Si crystal with defects and these calculated SMM melting curve are in good agreement with previous experiments. 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