In contrast to the normal pressure dependence of the noble metals in Fig.1, the calculated melting temperatures of Si, GaP, AlP and AlAs crystals show the negative pressure dependence in[r]
(1)Equation of states and melting temperatures of diamond cubic and zincblende semiconductors: pressure dependence
Vu Van Hung1, K Masuda-Jindo2, Pham Thi Minh Hanh1, and Nguyen Thanh
Hai3
1 Hanoi National Pedagogic University, Km8 Hanoi-Sontay Highway, Hanoi, Vietnam 2 Department of Material Science and Engineering, Tokyo Institute of Technology, Nagasuta, Midori-ku, Yokohama 226-8503, Japan
3 Hanoi University of Technology, 01 Dai Co Viet Road, Hanoi, Vietnam
Abstract: The pressure dependence of the melting temperatures of tetrahedrally coordinated semiconductors are studied using the equation of states derived from the statistical moment method, in comparison with those of the normal metals Using the general expressions of the limiting temperatures T , we calculate the “melting” temperatures of the semiconductorm crystals and normal metals as a function of the hydrostatic pressure The physical origins for the inverse pressure dependence of T observed for tetrahedrally coordinated semiconductorsm are also discussed
1 Introduction
Knowledge of the phase stabilities and phase diagrams under extreme (high pressure) conditions is of great importance for a better and sound understanding of a wide variety of physical phenomena [1-4] High pressure behaviour of materials is also of great interest from the viewpoints of geophysics [5] as well as of solid state sciences and technologies, e.g., in relation to the residual stresses in solid-state devices [6] It is the purpose of the present paper to study the pressure dependence of the melting temperatures of metals and semiconductors on the basis of the statistical moment method (SMM) [7-14]
In order to determine theoretically the melting temperatures of metals and semiconductors we will use the equilibrium condition of the solid phases In particular, we will use the limiting condition for the absolute crystalline stability 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 [7,8] Rigorously speaking, melting temperatures of the solid phases are defined as the temperature points at which the solid and the liquid states coexsist in the thermally equilibrium condition [4] Since the treatments of liquid phases are rather complicated [1-3], 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 [15,16], free energy of dislocation motions, or a simple order-disorder transition [4] The present SMM scheme has the virtue of providing the free energies of the system in the closed analytic forms on the basis of the exact moment formulas It is generally difficult to get a simple physical insight to the melting transitions of solids by using the highly sophisticated numerical simulation methods
(2)Equation of States and Melting Temperatures by SMM
To determine the Helmholtz free energy of tetrahedrally coordinated 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( , ),
tot vib vib
F V T =E V T +F V T +F V T (1)
where E is the internal energy, andtot and A vib vib
F F represent the harmonic and anharmonic vibrational contributions to the free energy, respectively Using the lattice dynamical model, the harmonic contribution is given by
( ) ( ) { ( ( ))}
0
,
, exp ,
vib j j
j
F V T =qåéëx + n - - x ùû
q
q l q (2)
where, xj( )q =hwj( )q q with q =k T wb j( )q is the atomic vibration frequencies, and it can be approximated in most cases (especially for obtaining the thermodynamic quantities at high temperatures near the melting temperature) to the Einstein frequency w given byE,
2 2 io E i ix k mw u j ổả = ỗ ữ ả ố ø å (3)
Here, m denotes the atomic mass, jio is the interatomic potential energy between the central 0th and ith sites, and u the atomic displacement of the ith atom in the x-direction.ix
The equation of states of the system at finite temperatures T is now obtained from Eqs.(1)-(3) and the pressure P of the system is given by the derivative of the free energy with respect to volume as
0
A
tot viv viv
T T T T
E F F
F P
V V V V
ỉ¶ ỉ¶
¶ ỉ
ổ
= -ỗ ữ = -ỗ ữ -ỗ ữ -ỗ ữ
ả ả ả ả
ố ø è ø è ø è ø (4)
Here, it is noted that the analytic expressions of the free energies [7-14 ] directly allow us to evaluate the hydrostatic pressure P From the limiting condition of the absolute stability for the crystalline phase,
0, i.e., 0,
T T P P V r ¶ ¶ ỉ = ổ = ỗả ữ ỗ ả ữ
è ø è ø (5)
one can find the limiting temperature T for the crystalline stability This limiting temperatures T cans be given in the harmonic approximation as
2
0
2
6
G
s B G
T
u a
T pv k a
r r
g g
é ù
ỉ ¶ ổ ổả ử
=ỗ + ả ữ ỗ ỗố ả ữứ - ữỳ
ố ứ è øû (6)
From the expression of the Helmholtz free energy, the pressure P of the diamond cubic and zincblende semiconductors can be written in the form
0 ,
3
G
u a P
v a v g q
¶
= - +
¶ (7)
where gG denotes the Grüneisen constant, kB the Boltzmann's constant, andn the atomic volume,
respectively
(3)0 9 s V B G u a T T P
k g a P
¶ ¶
ỉ ư ỉ ư
= ỗ ữ+ ỗ ữ
ả ốả ứ
è ø (8)
In case of P=0 it reduces to:
( )0 0
9 S S B G u a T T
k g a
ả
ổ ử
= ỗ ữ
¶
è ø (9)
After the nearest neighbour distance a is known [7-14] we can find both harmonic and anharmonic vibrational parameters K, , , ,K u0
a a
w ¶ ¶
¶ ¶ K Therefore, Eq.(8) permits us to determine the limiting
temperature of absolute stability TS( )0 at pressure P=0
We now introduce the following thermodynamic function f T P( , ) as
( , ) ,
3
G u
a f T P P
v a v
g q
¶
= +
-¶ (10)
from which we have
,
T P
v
T f f
P T P ¶ ¶ ¶ ổ ử ổ ử ổ ử = -ỗ ữ ỗ ữ ỗả ữ ốả ứ ốả ứ
ố ứ (11)
Here, it is straightforward to show that 1
T f P ả ổ ử = ỗả ữ
è ø and
2
0
3
3 27
2 9
8
G
G G B
P
u u
f a a
a k
T a a T a T a T T
g
g q q g
é ¶ ¶ ¶ ù
¶ ¶ ¶ ỉ
ỉ ư = - - - + +
ỗả ữ ả ả ả ả ả ỗ ả ữ ỳ
è ø ë è ø û (12)
Using Eqs (7), (10) and (11) one can derive the limiting temperature TS at pressure P as 0 27
3 9
S
B G G
G G B
u
a a P
T
k a u a u a
a k
a T a T a T T
g g
g q q g
¶ = + ¶ é ¶ ¶ - ỉ ¶ ư- ¶ + ỉ¶ ư+ ù ả ả ỗả ả ữ ả ỗ ả ữ ú è ø è ø ë û (13)
The above Eq.(13) can be used to find the limiting temperatureTS at pressure P in an iterative manner: Firstly, we calculate the limiting temperature at zero pressure and the second term of Eq.13, and evaluate the first approximation ( )1
s
T value at pressure P Then, ( )1
s
T value is used to obtain the improved limiting temperature ( )2
s
T using the right-hand side of Eq.(13)
3 Results and Discussions
(4)agreement with those of the empirical law by Kumari et al [17] For noble metals, we have also performed the similar calculations using the electronic many body potentials [11], and found that the calculated results are essentially the same as those by using the LJ potentials
Figure shows the melting temperatures of the diamond cubic Si and zincblende GaP, AlP and AlAs crystals, calculated by using the Stringer-Weber potentials [18], as a function of the pressure In contrast to the normal pressure dependence of the noble metals in Fig.1, the calculated melting temperatures of Si, GaP, AlP and AlAs crystals show the negative pressure dependence in agreement with experimental results (dashed lines in Fig.2) The negative or positive pressure dependence arises from the sign in the denomenater of the second term of Eq.13 The important factors are the thermal lattice expansion, Grüneisen constant gG and temperature dependence of gG Table gives the numerical values for the melting temperatures of zincblende semiconductors, as a function of the hydrostatic pressure P, in comparison with the available experimental results The calculated pressure dependence of lattice constants a0 (P, 300K), relative volumes V /V0 at temperature T = 300K for GaP AlP and AlAs zinc-blend semiconductors are also shown in Table together with available experimental results and other theoretical calculations [22]
Fig.1 Melting temperatures of noble metals (a) and rare gas solids Ar and Xe (b)
Fig.2 Melting temperatures of tetrahedral coordinated semiconductors; Si, GaP, AlAs and AlP crystals
(5)(K)=1685-3.8P (P in kbar) [20] The calculated melting temperatures of Si crystal deviate from the experimental results for higher hydrostatic pressure region The detailed discussions on this discrepancy will be given elsewhere
Table 1: Calculated pressure dependence of lattice constants a0(P,300K), relative volumes V V at0
temperatureT =300K and melting temperatureT for GaP, AlP and AlAs crystals.m
Material P(GPa) 10
a0(P, 300K) 5.4349 5.4166 5.3988 5.3815 5.3650 5.3489 5.3329 5.3176 5.3025 5.2878
V/V0 0.9897 0.9797 0.9701 0.9608 0.9520 0.9435 0.9350 0.9270 0.9191 0.9115
Ref.22 (cal) 0.9917 0.9833 0.9750 0.9667 0.9583 0.9542 0.9500 0.9417 0.9344 0.9292
0 m m
T T 0.9937 0.9843 0.9733 0.9613 0.9498 0.9372 0.9226 0.9079 0.8906 0.8697
-EXP [19-21] 2033±20 2009±20 1982±20 1992±20 1928±20 1901±20 1874±20 1847±20 1820±20 1793±20 GaP
0 m m
T T 0.9917 0.9800 0.9668 0.9717 0.9405 0.9273 0.9141 0.9010 0.8878 0.8746
a0(P, 300K) 5.4394 5.4207 5.4027 5.3850 5.3680 5.3515 5.3353 5.3194 5.3042 5.2894
V/V0 0.9893 0.9792 0.9694 0.9599 0.9509 0.9421 0.9336 0.9253 0.9174 0.9097
AlP
0 m m
T T 0.9926 0.9848 0.9764 0.9671 0.9578 0.9489 0.9381 0.9273 0.9165 0.9037
a0(P, 300K) 5.6439 5.6223 5.6015 5.5813 5.5617 5.5429 5.5246 5.5066 5.4894 5.4726
V/V0 0.9883 0.9770 0.9662 0.9558 0.9458 0.9362 0.9270 0.9160 0.9094 0.9011
AlAs
0 m m
T T 0.9902 0.9794 0.9676 0.9573 0.9440 0.9306 0.9162 0.9003 0.8828 0.8623
4 References
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