Abstract. The coefficients of thermal expansion of thin metal film with
face-centered cubic structure at zero pressure are investigated using the statistical
moment method (SMM), including the anharmonic effects of thermal lattice
vibration. The Helmholtz free energy, mean-square atomic displacement and linear
thermal expansion coefficients are derived in closed analytic forms in terms of the
power moments of the atomic displacement. Numerical calculations for Al, Au
and Ag thin films are found to be in good and reasonable agreement with other
theoretical results and with the experimental data.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 109-116
This paper is available online at
COEFFICIENTS OF THERMAL EXPANSION OF THIN METAL FILMS
INVESTIGATED USING THE STATISTICAL MOMENT METHOD
Duong Dai Phuong1, Vu Van Hung2 and Nguyen Thi Hoa3
1Tank Armour Officers Training School, Tam Duong, Vinh Phuc
2Viet Nam Education Publishing House, Hanoi
3Hanoi University of Transport and Communications
Abstract. The coefficients of thermal expansion of thin metal film with
face-centered cubic structure at zero pressure are investigated using the statistical
moment method (SMM), including the anharmonic effects of thermal lattice
vibration. The Helmholtz free energy, mean-square atomic displacement and linear
thermal expansion coefficients are derived in closed analytic forms in terms of the
power moments of the atomic displacement. Numerical calculations for Al, Au
and Ag thin films are found to be in good and reasonable agreement with other
theoretical results and with the experimental data.
Keywords: Coefficients of thermal expansion, statistical moment method,
anharmonicity, thin film.
1. Introduction
Materials in the form of thin film have come under the spotlight in recent years.
They have been found to show physical, chemical and mechanical properties that differ
from corresponding bulk materials [3, 4]. Properties, limitations and advantages obtained
on thin film geometry have been widely studied and they were found to depend on
different factors: structure, size, film thickness and different substrates [3, 5]. Thin
film is used in a vast range of applications which include microelectromechanical and
nanoelectrome-chanical systems, sensors and electronic textiles [1].
Metallic thin film displays a common geometry and presents enormous scientific
interest, mainly due to their attractive and novel properties for technological applications
[1]. In the early twentieth century, the theory of size-dependent effects in thin metal layers
was developed by a number of scientists [5]. For ultrathin metal layers, both surface effects
and quantum size effects must be considered [8]. A large number of experimental and
Received September 20, 2013. Accepted October 30, 2013.
Contact Vu Van Hung, e-mail address: bangvu57@yahoo.com
109
Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa
theoretical studies have been carried out on the thermal behavior of metal and nonmetal
thin films.
Most of the previous theoretical studies, however, are concerned with the material
properties of thin metal film at low temperature, and the temperature dependence of
thermodynamic quantities has not been studied in detail. The purpose of the present
article is to investigate the temperature dependence and the thickness dependence of the
coefficients of thermal expansion of thin metal film with face-centered cubic structure
using the analytic statistical moment method (SMM) [6, 10].The coefficients for thermal
expansion are derived from the Helmholtz free energy, and the explicit expression of the
thermal expansion coefficient is presented taking into account the anharmonic effects of
the thermal lattice vibrations.
In the present study, the influence of surface and size effects on the coefficients
of thermal expansion have also been studied. We compared the results of the present
calculations with those of the previous theoretical calculations as well as with the available
experimental results.
2. Content
2.1. Theory
2.1.1. The anharmonic oscillations of thin metal film
Investigating a metallic free standing thin film includes n layers with film thickness
d. Suppose of the thin film consists of two surface layers of atoms, the next two deeper
layers of atoms and the (n-4) internal layers atoms (Fig. 1).
Figure 1. The thin metal film
110
Coefficients of thermal expansion of thin metal films investigated...
First of all, we present the statistical moment method (SMM [6, 10]) formulation
for the displacement of the surface layer atoms of the thin film as a solution of
the equation
γsurθ
2 d
2
dp2
+ 3γsurθ < u
sur
i >
d
dp
+ γsur < u
sur
i >
3 +ksur < u
sur
i >
+γsur
θ
ksur
(xsur coth xsur − 1) −p = 0, (2.1)
where
xsur =
~ωsur
2θ
; θ = kBT, (2.2)
ksur =
1
2
∑
i
(
∂2φinio
∂u2iα
)
eq
≡ m0ω2sur, (2.3)
γ1sur =
1
48
∑
i
(
∂4φsurio
∂u4iα
)
eq
, γ2sur =
6
48
∑
i
(
∂4φsurio
∂u2iβ∂u
2
iγ
)
eq
(2.4)
γsur =
1
12
∑
i
(∂4φsurio
∂u4iα
)
eq
+ 6
(
∂4φsurio
∂u2iβ∂u
2
iγ
)
eq
= 4 (γ1sur + γ2sur) ; α, β, γ = x, y, z.
(2.5)
Here, KB is the Boltzmann constant; T is the absolute temperature;m0 is the mass
of the atoms at the lattice node; ωsur is the frequency of lattice vibration of surface layer
atoms;
ksur, γ1sur, γ2sur and γsur are the parameters of crystals depending on the structure
of crystal lattice and the interaction potential between atoms at the node; is the effective
interatomic potential between 0th and ith surface layer atoms; φsuri0 , uiy and uiz are the
displacement of ith atom from the equilibrium position on direction x, y, z respectively;
and the subscript eq indicates the evaluation at equilibrium.
To determine the atomic displacement , the symmetry property
appropriate for cubic crystals is used
=< u
sur
iy >=< u
sur
iz >=< u
sur
i > . (2.6)
Then, the solutions of the nonlinear differential equation of Eq. (2.1) can be
expanded in the power series of the supplemental force p as
=< u
sur
i >0 +A1p+ A2p
2. (2.7)
111
Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa
Here, 0 is the average atomic displacement in the limit of zero of
supplemental force p. Substituting the above solution of Eq. (2.7) into the original
differential Eq. (2.1), one can get the coupled equations on the coefficients A1 and A2,
from which the solution of 0 is given as [6, 10]
0≈
√
2γsurθ2
3k3sur
Asur, (2.8)
where
Asur = a
sur
1 +
γ2surθ
2
k4sur
asur2 +
γ3surθ
3
k6sur
asur3 +
γ4surθ
4
k8sur
asur4 +
γ5surθ
5
k10sur
asur5 +
γ6surθ
6
k12sur
asur6 , (2.9)
with asurη (η = 1, 2, . . . , 6) being the values of parameters of crystals depending on the
structure of the crystal lattice [6, 10]. A similar derivation can be also done for atoms at
layers just below the surface and internal layers of atoms of thin film, their displacement
noted as 0 and < u
in
i >0, respectively.
2.1.2. Free energy of the thin metal film
In the present research, the influence of the size effect on the coefficients of thermal
expansion of the thin metal film is studied, introducing the surface energy contribution
in the free energy of the system atoms. Following the general formula in the SMM
formulation [6, 10], one can get the free energy of the surface layers atoms as
Ψsur ≈ {U sur0 + 3Nsurθ[xsur + ln(1− e−2xsur)]}+ 3Nsurθ
2
k2sur
{
γ2surX
2
sur − 2γ1sur3
(
1 + Xsur
2
)}
+6Nsurθ
3
k4sur
{
4
3
γ22sur(1 +
Xsur
2
)Xsur − 2 (γ21sur + 2γ1surγ2sur) (1 + Xsur2 )(1 +Xsur)
}
,
(2.10)
here, Xsur = xsurcthxsur.
The free energy of the layers of atoms just below the surface and internal layers of
atoms have fomulae similar to Eq. (2.10) and can be noted as Ψsur1,Ψin, respectively.
The free energy in the harmonic approximation for surface layers, layers of atoms
just below the surface and internal layers of atoms of thin film are given as
Ψsur ≈ 3Nsur
{
1
6
usur0 + θ[xsur + ln(1− e−2xsur)]
}
,
Ψsur1 ≈ 3Nsur1
{
1
6
usur10 + θ[xsur1 + ln(1− e−2xsur1)]
}
;
Ψin ≈ 3Nin
{
1
6
uin0 + θ[xin + ln(1− e−2xin)]
}
,
(2.11)
where
usur0 ≡
∑
i
φsuri0 , u
sur1
0 ≡
∑
i
φsur1i0 , u
in
0 ≡
∑
i
φini0 (2.12)
112
Coefficients of thermal expansion of thin metal films investigated...
Let us consider a system where N atoms consist of n layers, the number of atoms on
each layer are the same and and NL, ψsur denotes the free energy of the two surface layers
of atoms, ψsur1 denotes the free energy of the two layers of atoms just below the surface
and ψin denotes the free energy of the (n-4) internal layers atoms. The number of internal
layer atoms, atoms in layers just below the surface, and surface layer atoms, respectively
Nin = (n− 4)NL =
(
N
NL
− 4
)
NL = N − 4NL, (2.13)
Nsur1 = 2NL = N − (n− 2)NL and Nsur = 2NL = N − (n− 2)NL. (2.14)
Free energy of the system and of an atom, respectively, is given by
ψ = Ninψin+Nsur1ψsur1+Nsurψsur = (N − 4NL)ψin+2NLψsur1+2NLψsur. (2.15)
ψ
N
=
[
1− 4
n
]
ψin +
2
n
ψsur1 +
2
n
ψsur. (2.16)
Using a¯ as the average nearest-neighbor distance and b¯ as the average thickness of
two-layers and a¯c as the average lattice constant, respectively, we can easily calculate the
relation
b =
a¯√
2
and a¯c = 2b¯ =
√
2a¯. (2.17)
The thickness d of thin film can be given by
d = 2bsur+2bsur1+(n− 5) bin = (n− 1) b =
√
2asur+
√
2asur1+
n− 5√
2
ain = (n−1) a¯√
2
.
(2.18)
In Eq. (2.18), the average nearest-neighbor of surface layers atoms, atoms of layers
just below the surface and internal layer atoms in thin metal film at a given temperature T
can be determined as
asur = asur (0)+ < u
sur
i >0, asur1 = asur1 (0)+ < u
sur1
i >0, ain = ain (0)+ < u
in
i >0,
(2.19)
where asur (0), asur1 (0) and ain (0) denote the nearest-neighbor at zero temperature of
surface layers atoms, atoms of layers just below the surface and internal layer atoms,
respectively, which can be determined experimently or derived considering the minimum
condition of the potential energy of the system.
∂U sur0
∂asur
+
3~ωsur
4ksur
∂ksur
∂asur
= 0;
∂U sur10
∂asur1
+
3~ωsur1
4ksur1
∂ksur1
∂asur1
= 0;
∂U in0
∂ain
+
3~ωin
4kin
∂kin
∂ain
= 0,
(2.20)
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Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa
The average nearest-neighbor of thin film is determined as
a¯ =
2asur + 2asur1 + (n− 5)ain
n− 1 (2.21)
Finally, we obtain the expression of the free energy through the number of layers as
follows
ψ
N
=
d
√
2− 3a¯
d
√
2 + a¯
ψin +
2a¯
d
√
2 + a¯
ψsur +
2a¯
d
√
2 + a¯
ψsur1. (2.22)
2.1.3. The coefficients of thermal expansion
The average thermal expansion coefficient of metallic thin film can be calculated
as follows
α =
KB
a¯
da¯
dθ
=
dsurαsur + dsur1αsur1 + (d− dsur − dsur1)αin
d
, (2.23)
with
αin =
kB
ain
∂yin0 (T )
∂θ
;αsur =
kB
asur
∂ysur0 (T )
∂θ
;αsur1 =
kB
asur1
∂ysur10 (T )
∂θ
(2.24)
with dsur and dsur1 being the thickness of the surface layers and the layers just below the
surface, respectively.
2.2. Numerical results and discussion
In order to apply the theoretical calculation to the Au, Ag and Al of metallic
thin film, we use the pair interaction potential in the following form with the potential
parameters D,m, n and r0 in Table 1.
φ(r) =
D
(n−m)
[
m
(r0
r
)n
− n
(r0
r
)m]
. (2.25)
Using the expressions obtained in Section 2, we calculate the values of the average
lattice constant, a¯c, the thickness, d, and the linear thermal expansion coefficient, α for
Au, Ag and Al thin metal thin film at various temperatures. The calculated results are
presented in Figures 2-5.
Table 1. Lennars-Jones potential parameters for Au, Ag and Al of metallic thin film [7]
Metal n m r0, A0 D/kB, 0K
Au 10.5 5.5 2.8751 4683.0
Ag 11.5 5.5 2.8760 3325.6
Al 12.5 5.5 2.8541 2995.6
114
Coefficients of thermal expansion of thin metal films investigated...
In Figures 2 and 3, we present the thickness dependence of the average lattice
constant of Au thin metal film as a function of the thickness, and it has been compared
to gold splattered on glass presented in [3]. We obtained the results of an average lattice
constant for an Ag and Al decrease, also with increasing thickness.
In Figure 4, we present the temperature dependence of the thermal expansion
coefficients of thin metal film for Al as a function of the temperature. The thermal
expansion coefficients increase with absolute temperature T, and our calculated results
have been compared with the results presented in [2] for Al thin film, and Al bulk [9] are
shown to be in good agreement. In Figure 5, we also show the thickness dependence of
the thermal expansion coefficients of Au and Al thin film, and the results calculated show
that the thermal expansion coefficients for Au and Al decrease with increasing thickness.
When the thickness increases, the thermal expansion coefficients approach the bulk value
in agreement with the results presented in [9].
115
Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa
3. Conclusion
We have presented the SMM formalism using Lennard-Jones interaction potentials
and investigated the thermal expansion coefficients of Au, Ag and Al thin metal film. The
SMM is simple and physically transparent, and thermal expansion coefficients of thin
metal film with FCC structures can be expressed in closed forms within the fourth order
moment approximation of the atomic displacements. In general, we have obtained good
agreement in the thermal expansion coefficients between our theoretical calculations, and
other theoretical results and experimental values.
Acknowledgement: This research is funded by Vietnam National Foundation for Science
and Technology Development (NAFOSTED) under grant number 103.01-2011.16.
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