Abstract. In this paper, the moment method in statistical dynamics (SMM) is used to study the
thermodynamic quantities of AlyGa1-yAs/GaAs systems taking into account the anharmonicity
effects of lattice vibrations. The nearest neighbor distance, lattice parameter, isothermal bulk
modulus, specific heats at the constant volume and pressure of AlyGa1-yAs/GaAs systems are
calculated as functions of the temperature and concentration of Al by using the many-body
potential. The effects of temperature and concentration of Al on thermo-mechanical quantities
of Al
yGa1-yAs/GaAs systems will be discussed in detail.

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96
JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0036
Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 96-103
This paper is available online at
INVESTIGATION OF THERMODYNAMIC AND MECHANICAL PROPERTIES
OF AlyGa1-yAs/GaAs SYSTEMS BY STATISTICAL MOMENT METHOD
Vu Thi Thanh Ha
1
, Vu Van Hung
2
and Vu Hong Nhat
2
1
University of Education Publishing House
2
University of Education, VNU
Abstract. In this paper, the moment method in statistical dynamics (SMM) is used to study the
thermodynamic quantities of AlyGa1-yAs/GaAs systems taking into account the anharmonicity
effects of lattice vibrations. The nearest neighbor distance, lattice parameter, isothermal bulk
modulus, specific heats at the constant volume and pressure of AlyGa1-yAs/GaAs systems are
calculated as functions of the temperature and concentration of Al by using the many-body
potential. The effects of temperature and concentration of Al on thermo-mechanical quantities
of AlyGa1-yAs/GaAs systems will be discussed in detail.
Keywords: Statistical moment method, anharmonicity, semiconductor superlattice.
1. Introduction
Study of the thermodynamic and mechanical properties of semiconductor superlattice has
become quite interesting in recent years since semiconductor superlattices play an important role
in technological applications, especially in the electronic industry.
The structural and thermo-mechanical properties of AlyGa1-yAs/GaAs systems were studied
by using the lattice dynamics approach [1] and ab initio molecular dynamic simulations [2]. These
physical properties of materials, in general, depend on temperature and their composition.
The purpose of present article is to study the temperature effects on the thermodynamic and
mechanical properties of the AlyGa1-yAs/GaAs semiconductor superlattice at zero pressure using
the moment method in the quantum statistical mechanics, hereafter referred to as statistical
moment method (SMM) [3-5]. The anharmonicity of thermal lattice vibrations of AlyGa1-yAs/GaAs
systems has been considered by using the many-body interaction potential.
Received August 7, 2017. Accepted August 30, 2017.
Contact Vu Thi Thanh Ha, e-mail: thanhhadhsp@gmail.com.
Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems
97
2. Content
2.1. Theory
The AlyGa1-yAs/GaAs semiconductor superlattice consists of AlyGa1-yAs and GaAs layers,
where AlyGa1-yAs and GaAs compounds are both in the zinc-blende form.
Figure 1. Strucsture of AlyGa1-yAs/GaAs semiconductor superlattice
We assume that the thickness of AlyGa1-yAs compound is d1 and this semiconductor layer
consists of N1 atoms with NAl atoms of Al, NGa atoms of Ga and NAs atoms of As then
1 Al Ga AsN N N N ,
1 1 1Al Ga As
N y N N
N ;N 1 y ;N
2 2 2
.
(1)
Similarly, GaAs compound layer is supposed to have the thickness d2 and consist of N2 atoms
with *GaN atoms of Ga and
*
AsN atoms of As then
*
2 GaN N N
*
As , (2)
So
1 2N N N n . (3)
where N is the number of atoms and n is the period of semiconductor superlattice.
In order to investigate the thermodynamic and mechanical properties of the
AlyGa1-yAs/GaAs semiconductor superlattice, we firstly consider the change of Helmholtz free
energy of AlyGa1-yAs when NGa atoms of Ga are replaced by Al atoms in GaAs crystal. The
substitution of an atom Ga by an atom Al causes the change of the free Gibbs energy as
f Ga Al
0 0g (4)
where Al0 is the free energy of an atom Al in the AlyGa1-yAs compounds and
Ga
0 is the internal
energy associated with atom Ga of the zinc-blende GaAs compound.
In the following of paper, the interatomic potential is assumed to be the many-body potential
[6] which consists of two-body and three-body terms as
GaAs AlyGa1–yAs
d2
d1
Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat
98
i ij i j ijk i j k
i,j i,j,k
(r ,r ) W (r ,r ,r ) (5)
with
12 6
0 0
ij i j
ij i,j
r r
(r ,r ) 2
r r
(6)
i j k
i,j,k i j k 3
ij jk ki
1 3cos cos cos
W r ,r ,r Z
r r r
(7)
where ij jk kir , r , r are respectively the bonds between the ith and jth atoms, jth and kth atoms, and
kth and ith atoms; and i j k, , are the angles formed by three particles i, j and k. And ε, r0, and
Z are the potential parameters which were fitted to the bond lengths of the dimer and trimer and
the lattice parameter and cohesive energy of the structure [6].
It should be noted that the analytical expression of the free energy of an atom Ga or As in the
zinc-blende GaAs compound in the harmonic approximation has the form as [4,7]
* Ga As 2x
0 0 0 0 3 x n(1 e )
(8)
where
Ga As
0 0 0 i0 i ijk
i i,j
1 1
(a ) W
2 3
,
(9)
and x
2
with Bk T , is the atomic vibration frequency which can be approximated in
most cases to the Einstein frequency as
2
2io
2
i ix eq
1
k m
2 u
,
(10)
with Ga As
1
m m m
2
and io is the interatomic potential energy between the central 0th and
ith sites.
The free energy Al0 of an atom Al in AlyGa1-yAs system in the harmonic approximation has
the form analogous to Eq. (8) as
AlAl Al Al 2x
0 0 3 x n(1 e )
(11)
where
Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems
99
Al
Al
*
k
x
2m
, * Al As
1
m (m m )
2
,
Al
0 iAl i ijAl
i i,j
1 1
(a ) W
2 3
,
(12)
and the pair interaction potential iAl between the Al atom and ith atom (i = As or Ga).
Because the AlyGa1-yAs system is supposed to be built by substituting AlN atoms Al into the
positions of Ga atoms of a zinc-blende GaAs system then the Gibbs free energy of system has an
approximate form as
f
0 Al C
* Ga Al
1 0 Al 0 0 C 1
G G N .g TS
N N TS PV
(13)
where P denotes the hydrostatic pressure, V1 is the volume of the AlyGa1-yAs system and SC is the
configuration entropy.
From Eqs. (3), (8) and (13), we derive the Gibbs free energy of AlyGa1-yAs/GaAs
semiconductor superlattice as
* Ga AlSS 0 Al 0 0 CG N nN nTS PV (14)
where V is the volume of the AlyGa1-yAs/GaAs system.
The Helmholtz free energy SS of the AlyGa1-yAs/GaAs system can be now derived
* Ga AlSS 0 Al 0 0 CN nN nTS (15)
and then the equation-of-states of the AlyGa1-yAs/GaAs systems at finite temperature T can be
obtained from the Helmholtz free energy as
SS SS
T T
a
P
V 3V a
or
0
2
1
Al Al
Al Al0
Al
2 2
1 1
y1 1 k
Pv a 1 xcthx
3 d a 2k a
2 1
d
y y 1 k
x cthx
d a d a2k
6 1 2 1
d d
(16)
where v is the atomic volume.
By numerically solving the equation-of-state Eq. (16) one can determine the average nearest-
neighbor distance (NND) between two intermediate atoms a(T) at temperature T.
Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat
100
Let us now consider the isothermal compressibility of the AlyGa1-yAs/GaAs semiconductor
superlattice. According to the definition of the isothermal compressibility T , it is given in terms
of the volume V and pressure P as [3].
3
T 22
To SS
2
T
a(P,T)
3
a(P,0)1 V
V P a (P,T)
2P
3V(P,T) a
(17)
Where
2 2 2 Al
SS 0 0
2 2 2
2 2T
1 1
22 2
2 2 2
Al Al 2 Al
Al 2
2
1
y y1
1
N d da a a
2 1 2 1
d d
xcthx k 1 x k
3 xcthx
2k aa 4k sh x
y x cthx k 1
3
d 2k a 4 k2 1
d
2Al 2Al
Al Al
2 2 AlAl
x k
x cthx
ash x
(18)
Furthermore, the specific heat at constant volume CV, specific heat at constant pressure CP, and
isothermal bulk modulus BT of AlyGa1-yAs/GaAs system are determined from the well-known
thermodynamic relations as
2
V P V T
T T
E 9TV 1
C ; C C ; B
T
,
where E is the energy of a crystal AlyGa1-yAs/GaAs system which can be calculated by using the
Gibbs-Helmholtz relation SSSSE .
2.2. Results and discussions
In this section, the expressions derived in the previous section will be applied to numerically
calculate self-consistently thermodynamic and mechanical properties of given AlyGa1-yAs and
AlyGa1-yAs/GaAs systems including lattice parameter, bulk modulus and specific heats at constant
volume and pressure. The potential parameters of binary GaAs and ternary AlGaAs systems are
given in Table 1 [6].
The SMM calculations of the lattice parameter ah, isothermal bulk modulus BT and specific
heat at constant pressure of the zinc-blende AlyGa1-yAs system at room temperature are shown in
Table 2. As can be seen in this table, our results agree well with available experimental data [8].
The difference between theory and experiment is about 0.4% for lattice parameter and about 7-8%
for bulk modulus and specific heat at constant pressure. From this table we can see that these
thermodynamic properties of AlyGa1-yAs compound increase with the increase in composition of
Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems
101
Al. The rapid increase bulk modulus BT indicates stronger rigidity properties of AlyGa1-yAs
semiconductor when the Al composition increases.
Table 1. Potential parameters of GaAs and AlGaAs systems [6]
Quantity Ga-As Al-Ga Al-As Al-Ga-As
AB (eV) 1.738 1.121 2.060 -
0ABr (Å) 2.448 2.490 2.430 -
AABZ (eV.Å
9
) 1900.0 2093.3 3000.0 -
ABBZ (eV.Å
9
) 4600.0 1955.3 5000.0 -
ABCZ (eV.Å
9
) - - - 2500.0
Table 2. Lattice parameter ah, isothermal bulk modulus TB and specific heat at constant
pressure of the zinc-blende AlyGa1-yAs system at room temperature (T = 300 K)
Composition
ah (Å)
SMM
a (Å)
Exp [8]
BT (GPa)
SMM
BT (GPa)
Exp
[8]
CP
(cal/mol.K)
SMM
CP
(cal/mol.K)
Exp [8]
0 5.626 5.653 82.06 75.50 5.16 5.54
0.1 5.628 5.654 82.15 75.76 5.41 5.77
0.2 5.630 5.655 82.24 76.02 5.66 5.99
0.3 5.631 5.656 82.33 76.28 5.91 6.22
0.4 5.633 5.656 82.42 76.54 6.15 6.45
0.5 5.635 5.657 82.51 76.80 6.40 6.68
0.6 5.636 5.658 82.60 77.06 6.65 6.91
0.7 5.638 5.659 82.69 77.32 6.90 7.14
0.8 5.640 5.660 82.78 77.58 7.15 7.37
0.9 5.641 5.660 82.88 77.84 7.40 7.60
Vu Thi Thanh Ha, Vu Van Hung and Vu Hong Nhat
102
0.0 0.2 0.4 0.6 0.8 1.0
5.6262
5.6264
5.6266
5.6268
5.6270
5.6272
5.6274
5.6276
L
a
tt
ic
e
c
o
n
s
ta
n
t
(Å
)
Composition (x)
d2 = 10d1
d2 = 40d1
Fig 2. Composition dependence of the lattice
parameter of the semiconductor superlattice
AlyGa1-yAs/GaAs at room temperature
0.0 0.2 0.4 0.6 0.8 1.0
82.66
82.68
82.70
82.72
82.74
82.76
B
u
lk
m
o
d
u
lu
s
(
G
P
a
)
Composition (x)
d2 = 10d1
d2 = 40d1
Fig 3. Composition dependence of the isothermal
bulk modulus of the semiconductor superlattice
AlyGa1-yAs/GaAs at room temperature
In Figs 2 and 3, we present the lattice parameter and isothermal bulk modulus of the
semiconductor superlattice AlyGa1-yAs/GaAs as functions of the composition Al at room
temperature. As it can be seen from these two figures, the lattice parameter and bulk modulus of
AlyGa1-yAs/GaAs systems are increasing functions of composition Al. Furthermore, when the
thickness of GaAs rises, these thermo-mechanical properties increase rapidly.
200 300 400 500 600 700 800
4.6
4.8
5.0
5.2
5.4
5.6
C
p
(
c
a
l/
m
o
l.
K
)
T (K)
x = 0.3; P = 0; d2 = 10d1
Fig 4. Temperature dependence of
the specific heat at constant pressure CP
of AlyGa1-yAs/GaAs systems
200 300 400 500 600 700 800
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
C
v
(
c
a
l/
m
o
l.
K
)
T (K)
x = 0.3; P = 0; d2 = 10d1
Fig 5. Temperature dependence of
specific heat at constant volume CV
of AlyGa1-yAs/GaAs systems
In Figs 4 and 5, we show the temperature dependences of the specific heats at constant
pressure CP and constant volume CV of an AlyGa1-yAs/GaAs system with composition
x = 0.3 of Al. The calculated specific heats CP and CV develop briskly when the temperature range
is below 450 K. Beyond 450 K, when temperature increases the specific heat CP and CV of
AlyGa1-yAs/GaAs system decreases gradually.
Investigation of thermodynamic and mechanical properties of AlyGa1-yAs/GaAs Systems
103
3. Conclusion
In conclusion, the SMM calculations have been performed to investigate the thermo-
mechanical properties of AlyGa1-yAs/GaAs systems. Using the free energy formulas derived in the
SMM, we have derived the analytical expressions of the lattice constant, bulk modulus, specific
heats at the constant volume and constant pressure of the zinc-blende AlyGa1-yAs and AlyGa1-yAs/GaAs
systems. Numerical calculations have been discussed and compared with those of experimental
results. The moment method can be developed extensively for studying the atomistic structure and
thermodynamic properties of other ternary and quaternary alloys as well.
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