Materials Sciences and Applicatio ns, 2011, 2, 1225-1232
doi:10.4236/msa.2011.29166 Published Online September 2011 (http://www.SciRP.org/journal/msa)
1225
Thermodynamic Properties of Semiconductors
with Defects
Vu Van Hung, Le Dai Thanh
Department of Physics, Hanoi National Univ ersity of Education, Hanoi, Vietnam.
Received April 13th, 2011; revised May 22nd, 2011; accepted June 3rd, 2011.
ABSTRACT
Thermodynamic properties of diamond cubic and zinc-blende semiconductors with point defects are considered by the
statistical moment method (SMM). The thermal expan sion coefficient, the specific h eats at constan t volume and those at
constant pressure, CV and CP, and the isothermal compressibility are derived analytically for semiconductors with de-
fects. The SMM calculated thermodynamic quantities of the Si, and GaAs semiconductors with defects are in good
agreement with the exp erimental results.
Keywords: Anharmonic Defective Semiconductor, Statistical Moment Method
1. Introduction
Recently, there has been a great interested in the study of
bulk semiconductor, semiconductor heterostructures and
nanodevices [1-4] since they provide us a wide variety of
academic problems as well as the technological applica-
tions. The physical characteristics of semiconductors are
determined both by the properties of the host crystal and
by the presence impurities and crystalline defects. Crys-
tal lattice defects or other impurities also modify the
properties of the semiconductor and thus may make a
semiconductor unsuitable for its intended applications.
The point defects in semiconductors including the va-
cancies play an important role in many properties of
material. Understanding these defects will lead to im-
proved semiconductor devices for the technological ap-
plications.
First principles (or ab initio) electronic structure com-
putations have been performed on semiconductor com-
pounds and the results compared with experiment [5-7].
More recently a large number of high-quanlity calcula-
tions have been performed on group-IV, III-V, and II-VI
materials. Modern calculations allow the accurate re-
laxation of structures to their minimum energy configu-
rations and the incorporation of temperature effects. One
can also study the melting of solids and phase transitions
in Si using first-principles molecular-dynamics method
[8,9]. Such calculations are computationally expensive
and, currently, simulations can only be run for periods of
tens of picoseconds, which is not long enough for some
of the processes of interest.
In our previous papers [10,11] the statistical moment
method was used to investigate the thermodynamic
quantities of the elemental perfect semiconductors, tak-
ing into account the anharmonicity effects of thermal
lattice vibrations. The thermal expansion coefficients,
elastic moduli, specific heats at constant volume and
those at constant pressure, CV and CP, are derived ana-
lytically for diamond cubic semiconductors.
The purpo se of the present article is to investigate the
temperature dependence of the thermodynamic proper-
ties of the semiconductors with defects using the analytic
statistical moment method (SMM) [12-16]. The ther-
modynamic quantities are derived from the Helmholtz
fnee energy of semiconductors with defects.
2. Theory
2.1. Atomic Displacements of Semiconductor
To derive the temperature dependence of the thermody-
namic properties of semiconductors, we use the statisti-
cal moment method. This method allows us to take into
account the anharmonicity effects of thermal lattice vi-
brations on the thermodynamic quantities in the analytic
formulations.
The essence of the SMM scheme can be summarized
as follows: for simplicity, we derive the thermodynamic
quantities of crystalline materials with cubic symmetry,
taking into account the higher (fourth) order anharmonic
contributions in the thermal lattice vibrations going be-

Thermodynamic Properties of Semiconductors with Defects
1226
yond the quasi-Ha monic (QH) approximation. The exten-
tions for the SMM formalism to non-cubic systems is
straightforward. The basic equations for obtaining ther-
modynamic quantities of the given crystals are derived in
a following manner: the equilibrium thermal lattice ex-
pansions are calculated by the force balance criterion and
then the thermodynamic quantities are determinded for
the equilibrium lattice spacings. The anharmonic contri-
butions of the thermodynamic quantities are given ex-
plicitly in terms of the power moments of the thermal
atomic displacements.
Let us first define the lattice displacements We demote
il the vector defining the displacement of the ith atom
in the lth unit cell, from its equilibrium po sition. The po-
tential energy of the whole crystal is expressed
in terms of the positions of all the atoms from the sites of
the equilibrium lattice. We use the theory of small atomic
vibrations, and expand the potential energy Uas a
power series in the cartesian components,
u

il
Uu
i

u of the
displacement vector around this point.
il
For the evaluation of the anharmonic contributions to
the free energy
u
,

we consider a quantum system,
which is influenced by supplemental forces i

in the
space of the generalized coordinates .
i For simplicity,
we only discuss monatomic systems and hereafter omit
the indices on the sublattices Then, the Hamiltonian
of the crystalline system is given by
q
l


0ˆ
ii
i
H
Hq


 (1)
where

0
H
denote the crystalline Hamiltonian without
the supplementary forces i

and upper huts

repre-
sent operrators. The supplementary forces i

are acted
in the direction of the generalized coordinates The
thermodynamic quantities of the anharmonic crystal
(harmonic Hamiltonian) will be treated in the Einstein
approximation.
.
i
q
After the action of the supplementary forces i the
system passes into a new equilibrium state. If the 0th
atom in the lattice is affected by a supplementary force
p
p

, then the total force acting on it must be zero, and one
gets the force balance relation as
23
00
,,,
40
,,,
11
24
10
12
ii
ii
ii
ii iii
eq eq
iiii
iiiii
uu
uu uuu
uuu p
uuuu
i
u



 
 
 


 

 
 
 
 
 









(2)
The thermal averages of the atomic displacements
ii
uu


 and iii uuu


 (called as second and third-
order moments) at given site i can be expressed in
terms of the first moment i
R
u

with the aid of the re-
curence formula [12-14]. Then Equation (2) is trans-
formed into the new differential equation:


2
2
d y
32
22
2
31
10
dy
y
yXy
dp
dy kyp
dp m

 

 

dp
yX

m (3)

 
where j
P
yu, coth
X
xx

, and ;
B
kT

2
x


.
In the above Equation (3), ,k

and

are defined
by
22
0
6
i
eq
uu

0
2
44
0
2 2
30
1
2
1
12
i
iieq
i
iiii
eq
i
ijx jy
u
u
uu










4
eq
jz
km
u











 




 

 





 











 



(4)
In deriving Equation (3) we have imposed the symme-
try criterion for the thermal averages in the diamond cu-
bic lattices as
jj j
pppp
uu uy
 j
u


Let us introduce the n ew variable y in the above Equa-
tion (3)
3
yy


 (5)
Then we have the new differential equation instead of
Equation (3)

2
32
2
dd
31
dd
0
yy
yy X
k
pp
Ky p

 


 y





(6)
where

2
2
2
;;
3
212 1
273 3
Kkp pK
k
KX
kk


 



 







(7)
For higher temperatures, the relation xcothx1 holds
and Equation (6) is reduced to

2
23
2
dd
30
dd
yy
yyKyp
pp
 





 (8)

Thermodynamic Properties of Semiconductors with Defects 1227
The nonlinear differential equ ation of Equation (8) can
be solved in the following manner: We expand the solu-
tion in terms of the “force” up to the second
order as
y* p
*
01 2
*2
y
yApAp

  (9)
where 1
A
and 2
A
are the constants [12]. The above
Equation (8) is solved as
2
03
2
3
y
K

A (10)
with
2233 44
123
468
55 66
56
10 12
4
A
aaa
KKK
aa
KK
 
 


a
(11)
and 12 34 32; 332;51;4583;aa aa 
56
1589 3;1633.aa
Here, 0 represents the atomic displacement for the
case when the force is zero. The general solution of
Equation (3) is solved as
y*
p
*
00
22 2
04
3
16 12
1
332
ppK
yy y
k
yK
K7k



 





 


 
 

(12)
Once the thermal expansion 0 of the lattice is found,
one can get the Helmholtz free energy of the system in
the following form
y
00
U1

  (13)
where 0

denotes the free energy in the harmonic ap-
proximation and 1

the anharmonic contribution to the
free energy. The Helmholtz free energy of our system can
be derived from the Hamiltonian H of the following
form:


0
H
HV



where

0
H
denote the Hamiltonian of the harmonic ap-
proximation,

the parameter and V

the anharmonic
vibrational contributions. Following exactly the general
formular in the SMM formulation [12], one can get the
free energy

of the system as

00
0
dUV



 
 (14)
where

V

represents the Hamiltonian corresponding t
o the anharmonicity contribution. Then the free energy
of the system is given by



23
21
00 2
24
22
2112
3
3
3
2
2
2
2
2 32
13
32
412211
32 2
33
27
311
1(1)3
27
36
NX
UX
kk
XX
N
X
X
k
NM kKN M
kk
X
KKK k
kK
NM

K


 











 






 



 
 



 




 

 


 

 
  
 

  



2
2
332
23
24
2
21
3
2
22
111
3(
33 6
2
31
27 9
231
3(1)
18
31
6
k
NX
KKk K
kk k
NM M
KK
k
NaX
Kk K
k
NM K

















 



 
 







 







 


 

 







1)
(15)
where
1
2
1
3
2
3
a
MK
 


, and the second term of above
Equation (15) denotes the harmonic contribution to the
free energy

2
03ln1e
x
Nx 






 (16)
with the aid of the “real space” free energy formula
,ETS


 one can find the thermodynamic quantities
of given systems. The thermodynamic quantities such as
specific heats and elastic modul at temperature T are
directly derived from the free energy

of the system.
2.2. Thermodynamic Properties of
Semiconductors with Defects
The Gibbs free energy of crystals consisting of N atoms
and vacancies has the form:
nN




0
,,
f
V
GTPG TPngTS
C
(17)
where


0,GTP is the Gibbs free energy of the perfect
crystals consisting N atoms,

,
f
V

g
TP is the Gibbs en-
ergy change on forming a single vacancy, -the en-
tropy of mixing: C
S

!
ln !!
CB
Nn
Sk Nn

 (18)

Thermodynamic Properties of Semiconductors with Defects
1228
From the minimization condition of t
rgy of the crystal with point defectshe Gibbs free ene , we obtain the eq
uilibrium concentration of the vacancies as [15,16]

,
exp f
g
PT
n









(19)
where
f
g

is the change in the Gibbs fre
the foon of a vacancy and can be given by
e energy due to
mati

*
00
,
f
V
g
TP uPV    (20)
It should be noted that pressure affects the diffusivity
through both the free energies, *
0

, and
change, resulting from the formatof the point defect,
In
the volume
ion
V. This change is due to the PV work done by the
pressure medium against the volume change associated
with defect formation and migrati o n.
the above Equation (20),


00
11
() ,,
iiijoij o
uWrrr  

r rep
,
23
ii j
resent the inter-
nal energy associated with atom 0 and io

the effective
interaction energies between the oth and ith atoms, *
o


denotes the change in the Helmholtzenergy free of the
central atom which creates the vacancy, by moving itself
to a certain sink site in the crystal, and is given by

**
1
oo
C

  (21)
where 0

 denotes the free energy of the central atom
after mo a certain sink sites i
simply regarded as a numerical factor. In the previous
7]
oving tn the crystal, C is
paper [1, we t a ke the ave rag e value for C as
*
12o
o
u
C

 (22)
From Equations (20), (21), and (22
pression of the Gibbs energy change
vacancy:
) we obtain the ex-
on forming a single

0
,
f
V
u
2
g
TP PV (23)
Form Equations (17) and (23) it is easy to ob
expression for the Helmholtz free energ y o
defects:
tain the
f crystals with
0
def C
u
nN TS    (24)
2
Applying the Gibbs-Helmholtz relation and using
Equation (24) we find the expression for t
crystal with defects and so the specific heat at constant
vo
he energy of a
lume CV has the form
00
200
2
2
022
2
def V
VV
Nn
CC

11
ff f
fVVV
V
ff
VV
ff
V
fV
V
B
gu gg
gTu
TT
uugg
TT
T
gg
g
uk







 
 

 


   
 


  






 











 







(25)
In the case of zero pressure,

0
,0 2
f
V
u
gT 
def
V
Cof crystal with
, the
specific heat at constant volume
fe
de-
cts has the simple form:
00 00
00
2
20000
2
2
00 00
22
2
def VB
VV
Nn k
CC
1(1
)
22 2
2
1
22
uu uu
uu
uuuu
uu uu

  




 
  
 



 
 




 





 

 





(26)
where CV is the specific heat at constant volume of per-
fect crystal [10].
The equation of states of the system with defects at fi-
nite temperature T is now obtained from Equation (24)
and the pressure P of the system is given by the deriva-
tive of the free energy with respect to volume as
3
def def
TT
a
PVVa
 
 
 
 

 
or
.3
def
T
a
Pv Na
 

 


(27)
where v is the atomic volume.
From the Equations (24) and (27)
equation of states of the crystal with defects at zero pres-
ion
one can find the
sure in the harmonic approximat

Thermodynamic Properties of Semiconductors with Defects 1229
00
1
22
V
TT
Nn uu
aa


  


  

  
(28)
From the Equation (28) one can find the average near-
est-neighbor distance (NND), of ato
tal at zero pressure and temperuation (28) can
be
(0, )aT
ature T. Eq
ls with
ms in crys-
solved using a computational program to find out the
values of the NND of the crystadefects, (0, )aT .
Let us now consider the compressibility of the solid
phase (diamond and zinc-blende structures). The iso-
thermal compressibility can be given as
3
2
(,)
3(,0)
1
T
aPT
aP
V



  
 
 2
2
(,)
23( ,)
oTdef
T
VP aP
T
PVPT a

  



(29)
where
22
200
22 2
2
00
1
22
1
1
24
def V
TT
T
V
T
Nn uu
aa a
Nn uu
a





 

   
 


 

 


 

 

 
(30)
In the case of zero pressure, the expression of t
thermal compressibility for crystals with defects is given
as
he iso-
3
def
T

0
22
2
2
200
00
2
2
2
3
3
1
111
24
2
T
V
T
T
T
a
a
a
Va
Nn uuuu
a
a
a

















  


 


 


 






(31)
or

2
20000
2
2
2
1
11
2
2
def
T
T
V
T
T
T
Nnuu uu
a
a
a


1
4












  












(32)
with is the isothermal compressibility of perfective
crystals at zero pressure
T

3
0
22
2
3
3
T
T
a
a
a
Va









(33
The specific heat at constant pressure,
)
P
C of cr
ef ystal with dects is determined from the well known thermo-
dynamic relations

2
9
def defdef
PV def
T
TV
CC


 (34)
where the thermal expansion coefficient def

of defec-
tive crystal is given as
22
0
33
def
def BT a
ka
aV a





  
 (35)
3. Results and D
def
iscussion
To calculate the thermodynamic quanti
GaAs crystals with defects, we will use the many-body
bo
tions
ties of Si and
potential [18], which include both the two-dy and the
three-body atomic interac


,,,
12 6
0
,
,
ij ijk
ij ijk
ij
ijjk ki
W
r

0
2
ij ij
rr
r

3
13cos coscos.
ij k
ijk
WZ rr r





 

 













(36)
The parameters were fitted to the bond lengths of the
dimer and trimer and the lattice parameters and cohesive
energy of the diamond structure. Parameters of the many-
body potential for monoatomic (A), binary (A-B
are given in Tables 1 and 2, respectively.
In Table 3, we compare the calculation results of the
specific heats at constant pressure, CP of Si crystal with
defects obtained by using the SMM analytic formula with
the experimental results of Ref. [19]. Here, it should be
no
0
) systems
ted that the equilibrium concentration of the vacancies
is very small at low temperature. At high temperature
being near the melting one the contribution of the vacan-
Table 1. Paramete rs of many body potential r, Z, and  for
Si [18].
Quantity Si

AA (eV) 2.81
r0AA (A0) 2.295
ZAAA(eVA09) 3484.0

Thermodynamic Properties of Semiconductors with Defects
1230
Table 2. Paramete rs of many body potential r0, Z, and
 for
GaAs [18].
Quantity GaAs

AA (eV) 1.738
r0AA (A
0) 2.448
Z) 0.0
Z)9) .0
Z0)9)
AAB (eVA09
(eV(A0190
460
ABB
AA (eV(A 0
1826.
A4
P.
In Table 3, we also present the SMM calculations of the
spe at constant volume, CV and
V
for Si crystal with defects
present the SMM results quilib-
riumof the vacancies, the speats at
con CV and Vmpare
thcalculation results of th at constant
al. The Figure
1 shown that thonable
va
pendence of the
[19,20], by solid lines. Figures 2
an
cies on the specific heats at constant volume, C
V and
those at constant pressure C of Si crystal is about 05%.
cific heats
def

VV
a
CC C 
In e . Tble 4, wof the e
concentration ecific h
stant volume,def
VV
CC C ,
e specific heats
and co
e
pressure, CP for GaAs defective crystal with the experi-
mental results [20].
The linear thermal expansion coefficient of Si crystal
is calculated using the many-body potenti
e many-body potential gives reas
Figure 1. Temperature dependence of the linear thermal
expansion coefficient of Si crystal with defects.
the many-body potentials, as a function of the tempera-
ture. One can see in Figures 2 and 3 that the specific
heats at constant pressure, CP increase with the tempera-
ture, in agreement with th e experimental results [19,20].
lues of thermal expansion coefficient compared with
the experimental results [19]. In Figures 2 and 3, we
present the temperature despecific heat at
constant pressure CP of Si and GaAs crystals with defects,
by dashed lines, in comparison with the corresponding
experimental results
4. Conclusions
The thermodynamic properties of semiconductors with
defects have been studied using statistical moment
method. We have presented the SMM formulation for the
thermodynamic quantities of diamond cubic and zinc-
blende semiconductors with defects taking into account
d 3 show the SMM specific heats at constant pressure,
CP (dashed lines) of the diamond cubic Si and zinc-
blende GaAs crystals with defects, calculated by using
Table 3. SMM calculated temperature dependence of
T(K) 300 400 500 600 700 800
thermodynamic quantities for Si crystal with de fec t s.
900 1000 1100 1200 1300 1400 1500
6.3 6.8 4.5 7.3 1.5 7.7 1.7 2.0 1.5 8.1 3.4 1.2 3.3
nv 1038 1025 1020 1017 1014 1013
7.1 3.2 1.1 1.0 1.3 4.5
CV 1026 1018 1013 1010 108 107
def
V
C 2.369 3.577 4.318 4.793 5.1075.326
f
1011 1010 109 109 108 107 107
6.9 5.9 3.3 1.4 4.5 1.2 2.8
106 105 104 103 103 102 102
5.4845.603 5.696 5.768 5.826 5.868 5.895
2.39 3.63 4.41 4.92 5.28 5.53 5.73 5.88 6.01 6.12 6.21 6.30 6.36
de
P
C
P
Cexp.[ 6.70 16] - 5.33 5.63 5.83 5.98 6.10 - 6.30 - 6.47 - -
Table 4M calculateperature ddenf thnc quantities foAs cal wifect
T(K) 300 500 600 700 800 900 1000 1200
. SMd temepence oermodyamir Garystth des.
400
1.0 .7 2.84 2.0 5 7.6 2.3.9 1 2.5 1.
nv 10 10 10 107 10
6.7 712.
CV10 1085 102 10
5. 93 6.24 6.36.55 6.76
23 1018 14 1012 109 108 106
0
1.3 1.1.9 5.1 5. 3.6 1.5 1
17 1011 106 104 103 102
de
P
Cf 22 5.675.6.11 6 6.48
P
Cexp.[16] 5.65 5.79 5.93 6.07 6.21 6.35 6.49 6.63 6.91

Thermodynamic Properties of Semiconductors with Defects 1231
Figure 2. Temperature dependence of the specific heat at
constant pressure CP of Si crystal with defects.
Figure 3. Temperature dependence of the specific heat at
constant pressure CP of GaAs crystal with defects.
the higher order (fourth order) anharmonic vibrat
terms in the Helholtz free energy and derived the various
thermodynamic quantities in closed analytic forms. Th
lattice constants, linear thermal expansion coefficients,
specific heats at constant volume and those at co
pressure, C
V and CP, have been calculated successfully
for the Si and GaAs crystals with defects. We have cal
M
Twork is supporty therch p No.
1.2609AFOS .
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