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

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

c World Scientific Publishing Company

DOI: 10.1142/S0217979212500506

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 calcu-lations 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.

1 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 crit-ical 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 ob-tained in this way In order to determine theoretically the melting temperature of semiconductors we will use the equilibrium condition of the solid phases In par-ticular, we will use the limiting condition for the absolute crystalline in order to

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 cal-culations are applied to the investigations of melting curve, and these theoretical and experimental results are reviewed in Ref.6 As is known, melting at high pres-sure can be meapres-sured 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 9DFT 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.10In 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

2 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

F (V, T ) = Etot(V, T ) + Fvib0 (V, T ) + FvibA (V, T ) , (1) where Etot is the internal energy, and F0

vib and FA

vib 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=X

j

φij(ri, rj) +X

j,k

Wijk(ri, rj, rk) , (2)

φij(ri, rj) =











εA

"

B rij σ

−4

− 1

# exp

"

 rij

σ − b

−1#

, rij

σ < b

σ ≥ b

(3)

and

Wijk(ri, rj, rk) = ελ exp

"

γ rij

σ − b

−1

+ γ rik

σ − b

−1 cos θijk+1

3

2# , (4)

where θijk is the angle between bond ij and bond ik The effective interatomic potentials (internal energy) of the systems are given by

Etot=X

i

ϕi =1 2 X

i,j

φij(ri, rj) +1

3 X

i,j,k

Wijk(ri, rj, rk) , (5)

where ϕi is the internal energy associated with atom i

Using the lattice dynamical model the harmonic contribution is given by

F0 vib(V, T ) = θX

q,j

[xj(q) + ln{1 − exp(−2xj(q))}] , (6)

where xj(q) = ~wj(q)/2θ with θ = kBT , wj(q) is the atomic vibration frequencies, and it can be approximated in most cases (especially for obtaining the thermo-dynamic quantities at high temperatures near the melting temperature) to the Einstein frequency, given by

k = 1 2 X

i

 ∂φi0

∂u2 ix



eq

≡ mw2

Here, m denotes the atomic mass, φi0 is the interatomic potential energy between the central 0th and ith sites, and uixis the atomic displacement of the ith atom in the x-direction

The equation of states of the system at finite temperature T is now obtained from Eqs (1), (5), and (6) and the pressure P of the system is given by the derivative

of the free energy with respect to volume as

P = − ∂F

∂V



T

= − ∂Etot

∂V



T

− ∂F

0 vib

∂V



T

− ∂F

A vib

∂V



T

Here, it is noted that the analytic expressions of the free energy directly allow us

to evaluate the hydrostatic pressure P From the expression of the Helmholtz free energy in the harmonic approximation, the pressure P of the diamond cubic and zinc-blende semiconductors can be written in the form

P = − r 3v

∂ϕ0

∂r +

3γGθ

where γG is the Gr¨uneisen constant, v is the atomic volume

From Eq (9) one can find the average nearest-neighbor distance (NND) of atoms

in crystal r(P, T ) at pressure P and temperature T However, for numerical calcu-lations, it is convenient to determine first the NND of crystals r(P, 0) at pressure

P and at absolute zero temperature T = 0 K For T = 0 K temperature, Eq (9) is reduced to:

P v = −a 1

3

∂ϕ0

∂r +

~ω 4k

∂k

∂r



Equation (10) can be solved using a computational program to find out the values of the NND r(P, 0) of the semiconductors 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 ) , (11) where y0(P, T ) is the displacement of an atom from the equilibrium position at pressure P and temperature T 13,14,17

Using the above formula of NND r(P, T ), we can find the change of the crystal volume under pressure P at a given temperature T as

V

V0

= r

3(P, T )

From the limiting condition of the absolute stability for the crystalline phase, (∂P /∂V )T = 0, i.e., (∂P /∂r)T = 0 and Eq (9), we can find the expression of the limiting temperature as

TS = r 9kBγG

 ∂ϕ0

∂r

 + ∂T

∂P



V

In the case of P = 0 it reduces to

TS(0) = r

9kBγG

 ∂ϕ0

∂r



Equation (14) permits us to determine the limiting temperature of absolute sta-bility TS(0) at pressure P = 0 because the melting temperature is less different from the limiting temperature of absolute stability at same pressure value There-fore, the melting temperature of crystals Tmcan be determined by an approximate expression:

2.2 Effect of point defect on the melting temperature

The melting temperature of the crystal with defects, TV

m(P, nV) is the function of the equilibrium vacancy concentration nV and pressure P In first-order approxi-mation the melting temperature TV

m of the defect crystal at the pressure P can be expanded in term of as

TmV(P, nV) = Tm(0) + ∂T

∂P



n V ,V

· P + ∂T

∂nV



V,P

· nV + · · · , (16)

TV

m(P, nV) ≈ Tm(P ) + ∂T

∂nV



V,P

where the melting temperature Tm(P ) at the pressure P of the perfect crystal is calculated by Eq (13)

From the minimization condition of the Gibbs free energy of the crystal with point defects, we obtain the equilibrium concentration of the vacancies as20,21

nV = exp



−g

f(P, T ) θ



where gf is the change in the Gibbs free energy due to the formation of a vacancy and can be given by

It should be noted that pressure affects the diffusivity through both the free ener-gies, F∗, 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

In the above Eq (19), ϕ0 = (1/2)P

iφi0(|r|) + (1/3)P

i,jWij0(ri, rj, r0) rep-resent the internal energy associated with the 0th atom, and φi0 the effective in-teraction energies between the 0th and ith atoms, ∆F∗ denotes the change in the 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

where F∗ 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,21 we take the average value for C as

C ≈ 1 + ϕ0

Using the derivative of the equilibrium vacancy concentration nV of Eq (18) with respect to temperature T and Eq (17), we obtain

TmV(P, nV) = Tm(P ) + T

2

m(P )

Tm(P )∂g

f V

∂θ −

gfV

kB

3 Results and Discussion

In this section we compare our melting curve, Eqs (13) and (22), for Si, AlP, AlAs and GaP semiconductors to some experimental melting curves Tables 1 and 2 show the good agreement between the SMM calculations of melting temperature Tm(P )

at various pressures and experimental results for Si and GaP semiconductors Our

Table 1 SMM calculated pressure dependence of melting temperature for GaP, AlP and AlAs perfect crystals, T m (P ); and GaP, AlP and AlAs crystals with defects,

T V

m (K).

T m (P )/T m (0) 1 0.990 0.980 0.957 0.933 0.904 0.866

T V

m (K) 1741 1725 1710 1674 1636 1590 1529

T V m(P )/T V m(0) 1 0.991 0.982 0.962 0.940 0.913 0.878 Exp (±20) 25

2050 2033 2009 1955 1901 1847 1793

1 0.992 0.980 0.954 0.927 0.901 0.875

T m (P )/T m (0) 1 0.992 0.984 0.966 0.948 0.926 0.903

T V

m (K) 1870 1856 1842 1812 1781 1744 1703

T V m(P )/T V m(0) 1 0.993 0.985 0.969 0.952 0.933 0.911

T m (P )/T m (0) 1 0.990 0.979 0.957 0.931 0.900 0.862

T V

m (K) 1774 1758 1740 1705 1661 1612 1549

T V m(P )/T V m(0) 1 0.991 0.981 0.961 0.936 0.909 0.873

Table 2 SMM calculated pressure dependence of melting tem-perature for Si perfect crystal, T m (P ), and Si crystal with defect,

T V

m (K).

T m (P )/T m (0) 1 0.985 0.974 0.966 0.960 0.955

T V

m (K) 1640 1602 1585 1572 1563 1553

T V m(P )/T V m(0) 1 0.977 0.966 0.959 0.953 0.947 Exp 24

1687 1647 1609 1571 1533 1495

1 0.977 0.955 0.932 0.910 0.887

calculated zero-pressure melting temperature for Si crystal with defect (1640 K) is

in good agreement with the experimental value of 1687 K.22 We note that DFT calculations of zero-pressure melting point for Si using the local density approxima-tion (LDA) predict values in the range 1300–1350 K,23,24and 1492 ± 50 K when a generalized-gradient approximation (GGA) is used instead.24

In Figs 1 and 2, we present the pressure dependence of the change of the crystal volume (compressibility V /V0) of Si and GaP semiconductors, by dashed lines, in comparison with the corresponding experimental results27and other calculation,28

by solid lines Figures 3 and 4 show the SMM melting temperatures (dashed lines) of the diamond cubic Si and zincblende GaP crystals, calculated by using the Stringer– Weber potentials,19 as a function of the pressure One can see in Figs 3 and 4 that the melting temperatures decrease with the hydrostatic pressure, in agreement

Fig 1 Compressibility V /V0 of Si crystal at temperature T = 300K.

Fig 2 Compressibility V /V 0 of GaP crystal at temperature T = 300 K.

Fig 3 Pressure dependence of melting temperature of Si crystal.

Fig 4 Pressure dependence of melting temperature of GaP crystal.

with the experimental results.25,26The calculated melting temperature of Si crystal deviate from the experimental results for higher hydrostatic pressure region The detailed discussions on this discrepancy will be given elsewhere

The melting temperatures Tm of a bulk stable Si and GaP decrease as the pressure P increases, or dTm/dP < 0 since ρl> ρS, where ρ denotes the density and the subscripts l and s show liquid and crystal, respectively.29For the negative-slope melting curve of bcc sodium, the first-principles molecular dynamics calculations showed that the liquid was more compressible than solid.30The 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 5 Pressure dependence of melting temperature of AlP and AlAs crystals.

term of Eq (13): dTm/dP < 0 According to the Clausius–Clapeyron’s equation32:

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, and assuming that the liquid entropy is bigger than the solid, if ∆V < 0 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 5 that the calculated pressure dependence, decreasing rates,

of the melting temperatures for zincblende AlP and AlAs crystals are very sensitive

to the materials Tables 1 and 2 show that the SMM melting temperature values

of Si, AlP, AlAs and GaP perfect crystals are considerably higher than the calcula-tion 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

4 Conclusion

The high-pressure melting curve of semiconductors has been studied using statisti-cal moment method In agreement with experiments and with DFT statisti-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

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