Abstract. The moment method in statistical dynamics (SMM) is used to study of thermal
expansion coefficients of metallic thin films with body-centered cubic (BCC) structure
taking into account the anharmonicity effects of the lattice vibrations and hydrostatic
pressures. The explicit expressions coefficients of thermal expansion of metallic thin films
are derived in closed analytic forms in terms of the power moments of the atomic
displacements. Numerical calculations of the coefficients of thermal expansion have been
performed for Fe and W thin films are found to be in good and reasonable agreement with the
laws of other authors and approach the experimental values of bulk. The effective pair
potentials work well for the calculations of BCC metallic thin films.
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106
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0035
Natural Sciences 2018, Volume 63, Issue 6, pp. 106-118
This paper is available online at
TEMPERATURE AND PRESSURE-DEPENDENT OF THERMAL EXPANSION
COEFFICIENTS OF METALLIC THIN FILMS
WITH BODY-CENTERED CUBIC STRUCTURE
Duong Dai Phuong
Military College of Tank Armour Officer
Abstract. The moment method in statistical dynamics (SMM) is used to study of thermal
expansion coefficients of metallic thin films with body-centered cubic (BCC) structure
taking into account the anharmonicity effects of the lattice vibrations and hydrostatic
pressures. The explicit expressions coefficients of thermal expansion of metallic thin films
are derived in closed analytic forms in terms of the power moments of the atomic
displacements. Numerical calculations of the coefficients of thermal expansion have been
performed for Fe and W thin films are found to be in good and reasonable agreement with the
laws of other authors and approach the experimental values of bulk. The effective pair
potentials work well for the calculations of BCC metallic thin films.
Keywords: Moment method, thin films, hydrostatic pressures, equation of state.
1. Introduction
Film materials have become interesting for recent years. The knowledge about the
thermodynamic properties of metallic thin film, such as heat capacity, coefficient of thermal
expansion is of great important to determine the parameters for the stability and reliability of the
manufactured devices [1, 2].
There are many ways to determine the behaviors deformed of thin film such as x-ray
diffraction [3-5]. However, rarely research has been known about the thermodynamic properties
of metallic free-standing thin films. Most of the previous theoretical studies, however, are
concerned with the materials properties of metallic thin film at low temperature, temperature and
pressure dependence of the thermodynamic quantities has not been studied in detail. In general,
the investigating dependence on pressure of thin film almost at low-pressure. There are many
studies about the dependence on pressure of the thin film deposited on a substrate [6, 7]. Most of
the researches of thin film are used experiment methods [4-7] but very few studies of them are
used theoretical method. The influence of oxygen pressure on the growth of (Ba0.02Sr0.98)TiO3 thin
film on MgO substrate by pulsed laser deposition techniques have been investigated in the oxygen
pressure range from 40 to 10
-3
Pa [8]. At lower oxygen pressure, more high energy particles will
arrive to the substrate. Most previous studies of thin films have been done in non-metal thin films
Received August 10, 2018. Revised August 22, 2018. Accepted August 29, 2018.
Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com
Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films
107
and not free-standing metallic thin films. In order to understand the pressure dependence of the
of thermal expansion coefficient of metallic thin films, it is highly desirable to establish an
analytical method which enables us to evaluate the free energy of the system taking into account
both the anharmonicity and quantum mechanical effect of the lattice vibration.
In this article, we present an analysis which can be used to extract metallic thin films, as a
function of the pressure. The explicit expressions of the thermal expansion coefficient are
derived in terms of the hydrostatic pressure P and the temperature T. The numerical calculations
are performed for Fe and W metallic thin films and compared with the laws of other authors
and the bulk values.
2. Content
2.1. Theory
2.1.1. Pressure versus volume relation
Let us consider a metal free standing thin film has
*n layers with the film thickness d.
Suppose of the thin film has been consisting two atomic surface layers, two next surface atomic
layers and (
*n 4 ) atomic internal layers, with Nng surface layer atoms, Nng1 next surface layer
atoms and Ntr internal layers atoms (see Figure 1).
Figure 1. The metallic free standing thin film
For the surface layer atoms of thin film, we will derive the pressure versus volume relation of
crystals limiting only quadratic terms in the atomic displacements. The expression for the free
energy in the harmonic approximation is determined as [9, 10].
ng2xng
ng ng 0 ng
ng ng
0 i0 i ,ng
i
1
3N u x ln(1 e ) ,
6
u a .
(1)
Th
ic
kn
es
s
(n
*-
4
) L
ay
er
s
d
a
a
a
ng
ng
1
tr
Duong Dai Phuong
108
The hydrostatic pressure P is determined by [9, 10]
3
ng ng ng
ng ng ngT T
a
P .
V V a
(2)
From Eqs. (1) and (2), we obtain equation of state denotes the pressure versus volume
relation of the lattice as
ng
ng0
ng ng ng ng
ng ng ng
ku1 1
PV a x coth x ,
6 a 2k a
(3)
where,
2
ng ng ng
ng ng ng
x x k
a k a
,
ng
ngx
2
, Bk T and ng is the atomic volume
ng
ng
ng
V
N
of
the crystal, for the BCC lattice of the metal thin films
34
3 3
ng
ng
a
. Using eq. (3), one can find the
nearest neighbour distance nga at pressure P and temperature T. However, for numerical
calculations, it is convenient to determine firstly the nearest neighbour distance nga ( P,0 ) at
pressure P and at absolute zero temperature T = 0. For T = 0 temperature, eq. (3) is reduced to
ng
ng ng0
ng ng
ng ng ng
ku1
PV a .
6 a 4k a
(4)
For simplicity, we take the effective pair interaction energy in metal systems as the power
law, similar to the Lennard-Jones potential
n m
o or rDr m n ,
( n m ) r r
(5)
where D, m, n, r0 are determined by fitting the experimental data (e.g., cohesive energy and elastic
modulus). For BCC of the metallic thin films we take into account the first nearest and second
nearest neighbour interactions.
Using the effective pair potentials of Eq. (5), it is straitforward to get the interaction energy
ng
0u and the parameter ngk in the crystal as [9, 10]
n m
ng o o
0 ng ,n ng ,m
ng ng
r rD
u mA nA
( n m ) a a
(6)
2
ng
ng 2
i i eq
1
k
2 u
2 2
ix ix
n m
a a 2o o
ng ,n 4 ng ,n 2 ng ,m 4 ng ,m 2 0 ng2
ng ng ng
r rDnm
( n 2 )A A ( m 2 )A A m ,
2a ( n m ) a a
(7)
Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films
109
where
2 2 4 4 2 2 2 2
ng ,ix ng ,ix ng ,ix ng ,ix ng ,ix ng ,iy ng ,ix ng ,iya a a a a a a a
ng ,n ng ,m ng ,n ng ,m ng ,n ng ,m ng ,n ng ,mA ,A ,A ,A ,A ,A ,A ,A , ... are the structural sums of
surface layers atoms for the given crystal and given by [9, 10]
ng ng
i i
ng ,n ng ,mng ,n ng ,m
i ii i
Z Z
A ; A ;
2 2
ng ,ix ng ,ix
ng 2 ng 2
a ai ,x ng ,ix i ,x ng ,ix
ng ,n ng ,m2 ng ,n 2 ng ,m
i ing i ng i
Z a Z a1 1
A ; A ;
a a
(8)
4 4
ng ,ix ng ,ix
ng 4 ng 4
a ai ,x ng ,ix i ,x ng ,ix
ng ,n ng ,m4 ng ,n 4 ng ,m
i ing i ng i
Z a Z a1 1
A ; A ;
a a
2 2 2 2
ng ,ix ng ,iy ng ,ix ng ,iy
ng 2 2 ng 2 2
a a a ai ,xy ng ,ix ng ,iy i ,xy ng ,ix ng ,iy
ng ,n ng ,m4 ng ,n 4 ng ,m
i ing i ng i
Z a a Z a a1 1
A ; A ;
a a
here
ng
iZ is the coordination number of i-th nearest neighbour atoms of surface layers with radius
ng
ir (for BCC lattice of thin films)
ng ng
k k ngr a .
Similar for next surface layers atoms and internal layers atoms of the system, we determine
the interaction energy
1
0 0,
ng tru u , the parameter 1,ng trk k and structural sums
2 2 4 4 2 2 2 2
ng 1,ix ng 1,ix ng 1,ix ng 1,ix ng 1,ix ng 1,iy ng 1,ix ng 1,iy
22 2 4 4
tr ,ix tr ,tr ,ix tr ,ix tr ,ix tr ,ix
a a a a a a a a
ng1,n ng1,m ng1,n ng1,m ng1,n ng1,m ng1,n ng1,m
a aa a a a
tr ,n tr ,m tr ,n tr ,m tr ,n tr ,m tr ,n
A ,A ,A ,A ,A ,A ,A ,A ,
A ,A ,A ,A ,A ,A ,A
2 2 2
iy tr ,ix tr ,iya a
tr ,m,A .
(9)
For surface layers and next surface layers, we have considered to the surface effect - the
defect on surface layers atoms and next surface layers atoms of the thin films. Using for two first
coordination sphere, we obtain the structural sums as follows:
, 2 , 22 22 2
, ,
5 5
8 , 8 ,
4 4
3 3
ng ng
k k
ng n ng mn mn m
k kng k ng k
Z Z
A A
2 2
, ,2 2
, 4 , , 4 ,4 42 4 2 4
, ,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k ng ng k
Z Z
A a A a
a a
2 2
, ,2 2
, 6 , , 6 ,6 62 6 2 6
, ,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k tr ng k
Z Z
A a A a
a a
, 4 , 44 44 4
, ,
5 5
8 , 8 ,
4 4
3 3
ng ng
k k
ng n ng mn mn m
k kng k ng k
Z Z
A A
Duong Dai Phuong
110
4 4
, ,4 4
, 8 , , 8 ,8 84 8 4 8
, ,
1 8 32 1 8 32
, ,
9 94 4
9 9
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k ng ng k
Z Z
A a A a
a a
2 2 2 2
, , , ,
2 2 2 2
, , , , , ,
, 8 , 84 8 4 8
, ,
1 8 1 8
, .
9 9
ng kx ng ky ng kx ng ky
ng ng
a a a ak xy ng kx ng ky k xy ng kx ng ky
ng n ng mn m
k kng ng k ng ng k
Z a a Z a a
A A
a a
1 1
1, 2 1, 22 22 2
1, 1,
5 5
8 , 8 ,
4 4
3 3
ng ng
k k
ng n ng mn mn m
k kng k ng k
Z Z
A A
2 2
1 , 1 ,
1 1
2 2
1, 4 1, 1, 4 1,4 42 4 2 4
1 1, 1 1,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k ng ng k
Z Z
A a A a
a a
2 2
1 , 1 ,
1 1
2 2
1, 6 1, 1, 6 1,6 62 6 2 6
1 1, 1,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k tr ng k
Z Z
A a A a
a a
1 1
1, 4 1, 44 44 4
1, 1,
5 5
8 , 8 ,
4 4
3 3
ng ng
k k
ng n ng mn mn m
k kng k ng k
Z Z
A A
4 4
1 , 1 ,
1 1
4 4
1, 8 1, 1, 8 1,8 84 8 4 8
1 1, 1 1,
1 8 32 1 8 32
, ,
9 94 4
9 9
3 3
ng kx ng kx
ng ng
a akx kx
ng n ng kx ng m ng kxn mn m
k kng ng k ng ng k
Z Z
A a A a
a a
2 2 2 2
1, 1, 1, 1,
1 2 2 1 2 2
, 1, 1, , 1, 1,
1, 8 1, 84 8 4 8
1 1, 1 1,
1 8 1 8
, .
9 9
ng kx ng ky ng kx ng ky
ng ng
a a a ak xy ng kx ng ky k xy ng kx ng ky
ng n ng mn m
k kng ng k ng ng k
Z a a Z a a
A A
a a
, 2 , 22 22 2
, ,
6 6
8 , 8 ,
4 4
3 3
tr tr
k k
tr n tr mn mn m
k ktr k tr k
Z Z
A A
2 2
, ,2 2
, 6 , , 6 ,6 62 6 2 6
, ,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
tr kx tr kx
tr tr
a akx kx
tr n tr kx tr m tr kxn mn m
k ktr tr k tr tr k
Z Z
A a A a
a a
Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films
111
2 2
, ,2 2
, 4 , , 4 ,4 42 4 2 4
, ,
1 8 8 1 8 8
, ,
3 34 4
3 3
3 3
tr kx tr kx
tr tr
a akx kx
tr n tr kx tr m tr kxn mn m
k ktr tr k tr tr k
Z Z
A a A a
a a
, 4 , 44 44 4
, ,
6 6
8 , 8 ,
4 4
3 3
tr tr
k k
tr n tr mn mn m
k ktr k tr k
Z Z
A A
4 4
, ,4 4
, 8 , , 8 ,8 84 8 4 8
, ,
1 8 32 1 8 32
, ,
9 94 4
9 9
3 3
tr kx tr kx
tr tr
a akx kx
tr n tr kx tr m tr kxn mn m
k ktr tr k tr tr k
Z Z
A a A a
a a
2 2 2 2
, , , ,
2 2 2 2
, , , , , ,
, 8 , 84 8 4 8
, ,
1 8 1 8
, .
9 9
tr kx tr ky tr kx tr ky
tr tr
a a a ak xy tr kx tr ky k xy tr kx tr ky
tr n tr mn m
k ktr tr k tr tr k
Z a a Z a a
A A
a a
From eqs. (4), (6), and (8) we obtain equation of state for surface layers of the metallic thin
films with body-centered cubic structure at zero temperature.
n m
0 0
ng ng ,n ng ,m
ng ng ng 0
r rDnm 1 Dnm
PV A A .
6( n m ) a a a 2( n m )4 m
2 2
, ,
2 2
, ,
0 0
, 4 , 2 , 4 , 2
0 0
, 4 , 2 , 4 , 2
( 2) ( 2) ( 2) ( 2)
.
( 2) ( 2)
ng ix ng ix
ng ix ng ix
n m
a a
ng n ng n ng m ng m
ng ng
n
a a
ng n ng n ng m ng m
ng ng
r r
n n A A m m A A
a a
r r
n A A m A A
a a
.
m
(10)
Eq. (10) can be transformed to the form
n 4 m 4
3ng ng 4ng ng3 n 3 m 3
0 1ng ng 2ng ng
n m
5ng ng 6 ng ng
c y c y4
P r c y c y .
3 3 c y c y
(11)
where
0
ng
ng
r
y ,
a P,0
1ng ng ,n
Dnm
c A ,
6( n m )
2ng ng ,m
Dnm
c A ,
6( n m )
2
ng ,ixa
3ng ng ,n 4 ng ,n 2
00
Dnm 1
c ( n 2 ) ( n 2 )A A ,
2( n m ) r4 m
2
ng ,,ixa
4ng ng ,m 4 ng ,m 2
00
Dnm 1
c ( m 2 ) ( m 2 )A A ,
2( n m ) r4 m
Duong Dai Phuong
112
2
ng ,ixa
5ng ng ,n 4 ng ,n 2c n 2 A A ,
2
ng ,ixa
6ng ng ,m 4 ng ,m 2c m 2 A A .
Similar derivation can be also done for next surface layers atoms and internal layers atoms of
the equation of state as
n 4 m 4
3ng1 ng1 4ng1 ng13 n 3 m 3
0 1ng1 ng1 2ng1 ng1
n m
5ng1 ng1 6 ng1 ng1
c y c y4
P r c y c y ,
3 3 c y c y
(12)
n 4 m 4
3 n 3 m 3 3tr tr 4tr tr
0 1tr tr 2tr tr
n m
5tr tr 6tr tr
c y c y4
P r c y c y .
3 3 c y c y
(13)
In principle Eqs. (11), (12) and (13) permit to find the nearest neighbour distance nga ( P,0 ) ,
ng1a ( P,0 ) and tra ( P,0 ) for surface layers, next surface layers and internal layers atoms at zero
temperature and pressure P .
2.1.2. Thermal expansion coefficient of metallic thin films
For surface layers of metallic thin films, the calculation of the lattice spacing of metallic thin
films at finite temperature and pressure P , the fourth order vibrational constants ng and ngk at
pressure P and T = 0K are defined by [9, 11]
4 ng 4 ng
io io
ng 1ng 2ng4 2 2
i i i ieq eq
1
6 4 ,
12 u u u
(14)
4 ng
io
1ng 4
i i eq
1
,
48 u
(15)
4 ng
io
2ng 2 2
i i i eq
6
'
48 u u
(16)
2 ng
2io
ng 0 ng2
i i eq
1
k m .
2 u
(17)
Using the effective pair potentials of eq. (5), the parameter ng 1ng 2ng ng, , ,k of the BCC
metallic thin films have the form
4 2 2
, , ,
2
,
4 2 2
, , ,
, 8 , 8
0
4
, 6 , 4
, 8 , 8
4
,
2 4 6 61
( )
12
18( 2)( 4) 9( 2)
2 4 6 61
12
18( 2)( 4)
ng ix ng ix ng iy
ng ix
ng ix ng ix ng iy
a a a
ng n ng n
n
ng
ang ng
ng n ng n
a a a
ng m ng m
ng
ng m
n n n A A rDmn
n m a a
n n A n A
m m m A ADmn
n m a
m m A
2
,
0
6 , 4
( ) ,
9( 2)ng ix
m
a ng
ng m
r
a
m A
(18)
Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films
113
2 2
, ,0 0
, 4 , 2 , 4 , 22
2 2 .
2
ng ix ng ix
n m
a a
ng ng n ng n ng m ng m
ng ng ng
r rDmn
k n A A m A A
n m a a a
(19)
Similar derivation can be also done for next surface layers atoms and internal layers atoms,
we obtain the values of
ng1 ng1,k , tr tr,k .
Using the obtained results of nearest neighbour distance for surface layers atoms nga ( P,0 )
and Eqs. (18) and (19), we find the values of parameters ngk ( P,0 ) , and ng( P,0 ) at pressure P
and T = 0 K.
Similar derivation can be also done for next surface layers atoms and internal layers atoms,
we find the values of parameters ng1 trk ( P,0 ),k ( P,0 ), and ng1 tr( P,0 ), ( P,0 ), at pressure P
and T = 0K.
For surface layers of metallic thin films, the thermally induced lattice expansion
ng
0y ( P,T )
at pressure P and temperature T is given in a closed formula using the force balance criterion of
the fourth order moment approximation as [10, 11]
2ngng
0 ng3
ng
2 P,0
y P,T A ( P,T ),
3k ( P,0 )
(20)
where
2 2 3 3
ng ngng ng ng
ng 1 2 34 6
ng ng
4 4 5 5 6 6
ng ng ngng ng ng
54 68 10 12
ng ng ng
P,0 P,0
A P,T a a a
k P,0 k P,0
P,0 P,0 P,0
+ a a a .
k P,0 k P,0 k P,0
(21)
In Eq. (21), using ng ng ngX x cothx , one can find the values of parameters as [9]
ngng ng 2 3
ng ng ng1 2
X 13 47 23 1
a 1 ; a X X X ,
2 3 6 6 2
ng 2 3 4
ng ng ng ng3
25 121 50 16 1
a X X X X ,
3 6 3 3 2
ng 2 3 4 5ng ng ng ng ng4
43 93 169 83 22 1
a X X X X X ,
3 2 3 3 3 2
ng 2 3 4 5 6
ng ng ng ng ng ng5
103 749 363 391 148 53 1
a X X X X X X ,
3 6 2 3 3 6 2
ng 2 3 4 5 6 7ng ng ng ng ng ng ng6
561 1489 927 733 145 31 1
a 65 X X X X X X X ,
2 3 2 3 2 3 2
ng
ng
( P,0 )
x
2
,
ng
ng
0
k ( P,0 )
( P,0 )
m
. (22)
Similar derivation can be also done for next surface layers atoms and internal layers atoms of
the average atomic displacement.
Duong Dai Phuong
114
2ng1ng1
0 ng13
ng1
2 P,0
y P,T A ( P,T ),
3k ( P,0 )
(23)
2trtr
0 tr3
tr
2 P,0
y P,T A ( P,T ).
3k ( P,0 )
(24)
So, for surface layers atoms we can find the nearest neighbour distance
nga ( P,T ) at
pressure P and temperature T as
ng
ng ng 0a ( P,T ) a ( P,0 ) y ( P,T ) . (25)
Thus, for next surface layers atoms and internal layers atoms the nearest neighbour distance
are determined as
ng1
ng1 ng1 0a ( P,T ) a ( P,0 ) y ( P,T ) (26)
tr
tr tr 0a ( P,T ) a ( P,0 ) y ( P,T ) . (27)
The average nearest neighbor distance of thin film at pressure P , and temperature T and
zezo temperature are determined as [10, 12]
*1
*
2 , 2 , ( 5) ,
( , ) .
1
ng ng tra P T a P T n a P T
a P T
n
(28)
*1
*
2 ,0 2 ,0 ( 5) ,0
( ,0) .
1
ng ng tra P a P n a P
a P
n
(29)
The average thermal expansion coefficient of metallic thin films can be calculated as [10, 11]
1 1 1,
,
,0
ng ng ng ng ng ng trB
d d d d da P Tk
a P d d
(30)
where
10 0 0
1
1
, , ,
; ;
,0 ,0 ,0
tr ng ng
B B B
tr ng ng
tr ng ng
y P T y P T y P Tk k k
a P a P a P
(31)
here
ngd and 1ngd are the surface layers and next surface layers thickness.
One can now apply the above formular to study thermal expansion coefficient of BCC
metallic thin films under hydrostatic pressures.
2.2. Numerical results and discussion
In order to check the validity of the present moment method for study of thermal expa