DSpace at VNU: MELTING CURVE OF SEMICONDUCTORS WITH DEFECTS: PRESSURE DEPENDENCE

March 13, 2012 12:10 WSPC/Guidelines-IJMPB S0217979212500506 International Journal of Modern Physics B Vol 26, No (2012) 1250050 (11 pages) c World Scientific Publishing Company DOI: 10.1142/S0217979212500506 Int J Mod Phys B 2012.26 Downloaded from www.worldscientific.com by GEORGE WASHINGTON UNIVERSITY on 01/25/15 For personal use only MELTING CURVE OF SEMICONDUCTORS WITH DEFECTS: PRESSURE DEPENDENCE VU VAN HUNG Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam bangvu57@yahoo.com LE DAI THANH Hanoi University of Science, Str Nguyen Trai, Hanoi, Vietnam Received 19 April 2011 Revised 12 June 2011 The high-pressure melting curve of semiconductors 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 We have derived a new equation for the melting curve of the defect semiconductors The melting was investigated at different high pressures, and the SMM calculated melting temperature of Si, AlP, AlAs and GaP crystals with defects being in good agreement with previous experiments Keywords: Melting temperature; defect semiconductors; statistical moment method Introduction Investigations on the nature of melting under extreme (high pressure) conditions are of great importance for a better and sound understanding of a wide variety of physical phenomena 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 Tm of crystals at high pressure was suggested by Kumari et al.2 On the theoretical side, in order to determine the melting temperature we must use the equilibrium condition of the liquid and solid phases (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 In order to determine theoretically the melting temperature of semiconductors 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 1250050-1 March 13, 2012 12:10 WSPC/Guidelines-IJMPB S0217979212500506 Int J Mod Phys B 2012.26 Downloaded from www.worldscientific.com by GEORGE WASHINGTON UNIVERSITY on 01/25/15 For personal use only V V Hung & L D Thanh find the melting temperatures Tm under the hydrostatic pressures We note that the limiting temperatures for the absolute crystalline stabilities of solid phases are very close to the melting temperatures.3 Since the treatments of liquid phases are rather complicated, the most of the previous studies have been performed on the basis of the properties of the solid phases, (starting with the Lindemann’s formula) theorized in terms of the lattice instablility,4 free energy of dislocation motions, or a simple order-disorder transition.5 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 experimental results are reviewed in Ref As is known, melting at high pressure can be measured mainly by means of in situ laser-heated diamond-anvil cells (DAC) and through shock-wave experiments Theoretical work based on density functional theory (DFT) supports the shock data.7–9 DFT calculated melting curve of molybdenum agrees well with experiment at ambient pressure and is consistent with shock data at high pressure, but does not agree with the high-pressure melting curve deduced from static compression experiments.10 In addition, many theoretical calculations and empirical laws have been developed to predict the melting curve of different materials under extreme compression However, still today, these different methods yield widely different results Recently numerical simulations have shown that correlated clusters of defects thermally excited play a central role in this process at the limit overheating.11 In addition, investigations revealed that various kinds of defects in solids, such as interfaces, grain boundaries, voids, impurities and other defects, also facilitate melting.12 The purpose of this paper is to discuss the effect of pressure and point defect on the melting temperature of semiconductors using statistical moment method.13–18 A P –V –T equation of states of Si, AlP, AlAS and GaP semiconductors is obtained, the pressure dependence of the melting temperature being estimated Theory 2.1 Equation of states and melting temperatures by SMM To determine the Helmholtz free energy of semiconductors, we will use the statistical moment method The Helmholtz free energy of the crystal at temperature T and of volume V is given in the sum of the three terms A F (V, T ) = Etot (V, T ) + Fvib (V, T ) + Fvib (V, T ) , (1) A where Etot is the internal energy, and Fvib and Fvib represent the harmonic and anharmonic vibrational contributions to the free energy, respectively Using the Stringer-Weber potentials which consist of two-body and three-body terms19 ϕi = φij (ri , rj ) + j Wijk (ri , rj , rk ) , j,k 1250050-2 (2) March 13, 2012 12:10 WSPC/Guidelines-IJMPB S0217979212500506 Melting Curve of Semiconductors with Defects: Pressure Dependence where and   rij    εA B σ φij (ri , rj ) =    0, rij −b σ − exp −1 , rij ρS , where ρ denotes the density and the subscripts l and s show liquid and crystal, respectively.29 For the negative-slope melting curve of bcc sodium, the first-principles molecular dynamics calculations showed that the liquid was more compressible than solid.30 The melting mechanism of pressure-induced drop in melting temperature is well explained in Refs 31 and 32 The Peierls mechanism has been reported to be the origin of the negative-slope melting curve in the fcc sodium.31 The fcc sodium forms a Peierls-like liquid by opening a pseudo-gap at the Fermi level after melting, thus the liquid gains a lower band energy than its solid, leading to the negative-slope of melting curve.31 It is known that the structure opens a pseudo-gap close to the Fermi level through some distortion and the distortion leads to the lowering of the band energy and increases the Coulomb repulsion between atoms This Peierls mechanism has been found to play an important role on the melting process in alkali metals,31 however, the physical orgin of the negative-slope melting curve of cI16 sodium is not related to the Peierls mechanism but the elastic constant softening.32 For the negative-slope melting curve of semiconductors the SMM calculations showed that the negative pressure dependence arises from the sign in the second Fig Pressure dependence of melting temperature of AlP and AlAs crystals 1250050-9 March 13, 2012 12:10 WSPC/Guidelines-IJMPB S0217979212500506 V V Hung & L D Thanh Int J Mod Phys B 2012.26 Downloaded from www.worldscientific.com by GEORGE WASHINGTON UNIVERSITY on 01/25/15 For personal use only term of Eq (13): dTm /dP < According to the Clausius–Clapeyron’s equation32 : dTm ∆V = Tm , dP ∆S where ∆V = Vl − Vs is the difference of molar volumes and ∆S = Sl − Ss is the difference of molar entropies, and assuming that the liquid entropy is bigger than the solid, if ∆V < this will leads to a denser liquid phase than solid phase, so we can come to the conclusion that the melting curve has a negative slope One can see in Fig that the calculated pressure dependence, decreasing rates, of the melting temperatures for zincblende AlP and AlAs crystals are very sensitive to the materials Tables and show that the SMM melting temperature values of Si, AlP, AlAs and GaP perfect crystals are considerably higher than the calculation results by the SMM of these crystals with defects The equilibrium vacancies concentration is very small at low temperature At high temperature being near the melting one the contribution of the vacancies on the melting temperature of semiconductor crystals is at some percent Conclusion The high-pressure melting curve of semiconductors has been studied using statistical moment method 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 the perfect semiconductor, Eqs (13) and (14), and semiconductor with defects, Eq (22) We have calculated melting curves for Si, AlP, AlAs and GaP semiconductors with defects and these calculated SMM melting curve are in good agreement with previous experiments The pressure dependence of the melting curve of Si, AlP, AlAs and GaP semiconductors with defects being estimated Acknowledgments This work is supported by the research project No 103.01.2011.16 of NAFOSTED References 10 11 F Simon et al., Z Phys Chem 6, 331 (1930) M 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J Iniguez, Phys Rev Lett 98, 055501 (2007) J.-Y Raty, E Schwegler and S A Bonev, Nature 449, 448 (2007) D Zhou et al., Phys Status Solidi B 248, 1143 (2011) 1250050-11 ... the melting temperature of semiconductors using statistical moment method.13–18 A P –V –T equation of states of Si, AlP, AlAS and GaP semiconductors is obtained, the pressure dependence of the melting. .. only Melting Curve of Semiconductors with Defects: Pressure Dependence Fig Fig Compressibility V /V0 of Si crystal at temperature T = 300K Compressibility V /V0 of GaP crystal at temperature... personal use only Melting Curve of Semiconductors with Defects: Pressure Dependence with the experimental results.25,26 The calculated melting temperature of Si crystal deviate from the experimental