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 melting temperatures of Si crystal with defects being in good agreement with
previous experiments.

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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.
Acknowledgment: This work is supported by the research project of NAFOSTED.
70
Pressure dependence of melting curve of silicon crystal with defects
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