Abstract. The coefficients of thermal expansion of metallic thin films with body-centered
cubic (BCC) structure at ambient conditions were investigated using the statistical moment
method (SMM), including the anharmonicity effects of thermal lattice vibrations. The
analytical expressions of Helmholtz free energy, lattice constant, and linear thermal
expansion coefficients were derived in terms of the power moments of the atomic
displacements. Numerical calculations of the quantities were performed for Fe and W thin
films and found to be in good and reasonable agreement with other theoretical results and
experimental data. This research proves that thermal expansion coefficients of thin films
approach the values of bulk when thin film is about 70 nm thick.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0039
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 112-121
This paper is available online at
COEFFICIENTS OF THERMAL EXPANSION OF METALLIC THIN FILMS
WITH BODY-CENTERED CUBIC STRUCTURE
Duong Dai Phuong1, Nguyen Xuan Viet1, Nguyen Thi Hoa2 and Doan Thi Van3
1Tank Armour Officers Training School, Tam Duong, Vinh Phuc
2Hanoi University of Transport and Communications
3Faculty of Nursing, Hanoi Medical Colleges
Abstract. The coefficients of thermal expansion of metallic thin films with body-centered
cubic (BCC) structure at ambient conditions were investigated using the statistical moment
method (SMM), including the anharmonicity effects of thermal lattice vibrations. The
analytical expressions of Helmholtz free energy, lattice constant, and linear thermal
expansion coefficients were derived in terms of the power moments of the atomic
displacements. Numerical calculations of the quantities were performed for Fe and W thin
films and found to be in good and reasonable agreement with other theoretical results and
experimental data. This research proves that thermal expansion coefficients of thin films
approach the values of bulk when thin film is about 70 nm thick.
Keywords: Metallic thin film, statistical moment method, thermodynamic properties,
anharmonicity.
1. Introduction
Knowledge of thermodynamic properties of metallic thin film, such as heat capacity and
coeffcient of thermal expansion, are of great importance when determining parameters for the
stability and reliability of manufactured devices.
In many cases, the thermodynamic properties of metallic thin film are not well known
or differ from the values for corresponding bulk materials. A large number of experimental and
theoretical studies have been carried out on the thermodynamic properties of metal and nonmetal
thin film [1-4]. Most of them describe the method used for measuring the thermodynamic
properties of crystalline thin films on the substrates [5-9].
There are many ways to determine the behaviors deformed of thin film such as x-ray
diffraction [4, 6-8] and nanoindentation [9]. However, rarely has research been done on the
thermodynamic properties of metallic free-standing thin film. Most of the previous theoretical
studies were concerned with the material properties of metallic thin film at low temperature while
temperature dependence of the 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 metallic thin films with a
Received September 28, 2016. Accepted October 22, 2016.
Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com
112
Coefficients of thermal expansion of metallic thin films with body-centered cubic structure
body-centered cubic structure using the analytic statistical moment method (SMM) [10-13]. The
coefficients of thermal expansion are derived from the Helmholtz free energy, and the explicit
expression of the thermal expansion coefficient is presented taking into account the anharmonicity
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 metallic thin films
Let us consider a metalic free standing thin film that has n∗ layers with the thickness d. We
assume that the thin film consists of two atomic surface layer, two next surface atomic layers and
(n∗ − 4) atomic internal layers. (see Figure 1).
Figure 1. The free-standing thin film
For internal layer atoms of thin films, we present the statistical moment method (SMM
[12-13]) formulation for the displacement of the internal layer atoms of the thin film ytr is solution
of equation.
γtrθ
2d
2ytr
dp2
+ 3γtrθytr
dytr
dp
+ γtry
3
tr + ktrytr + γtr
θ
ktr
(xtr coth xtr − 1)ytr − p = 0 , (2.1)
where
ytr ≡p;xtr = ~ωtr
2θ
; θ = kBT, (2.2)
ktr =
1
2
∑
i
(
∂2ϕtrio
∂u2iα
)
eq
≡ m0ω2tr, (2.3)
113
Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van
γ1tr =
1
48
∑
i
(
∂4ϕtrio
∂u4iα
)
eq
, (2.4)
γ2tr =
6
48
∑
i
(
∂4ϕtrio
∂u2iβ∂u
2
iγ
)
eq
, (2.5)
γtr =
1
12
∑
i
(∂4ϕtrio
∂u4iα
)
eq
+ 6
(
∂4ϕtrio
∂u2iβ∂u
2
iγ
)
eq
= 4 (γ1tr + γ2tr) , (2.6)
where kB is the Boltzmann constant, T is the absolute temperature,m0 is the mass of the atom, ωtr
is the frequency of lattice vibration of internal layer atoms; ktr , γ1tr , γ2tr , γtr are the parameters
of crystal depending on the structure of crystal lattice and the interaction potential between atoms;
ϕtri0 is the effective interatomic potential between 0
th and ith internal layers atoms; uiα , uiβ , uiγ
are the displacements of ith atom from equilibrium position in the direction α(α = x, y, z), β(β =
x, y, z), γ(γ = x, y, z), respectively, and the subscript eq indicates evaluation at equilibrium.
In the second approximation of the supplemental force, the solutions of the nonlinear
differential equation of Eq. (2.1) can be expanded in the power series of the supplemental force p
as [12, 13].
ytr = y
tr
0 +A
tr
1 p+A
tr
2 p
2. (2.7)
Here, ytr0 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 for the coefficients Atr1 and A
tr
2 , from which the solution of y
tr
0 is given
as [13]
ytr0 ≈
√
2γtrθ2
3k3tr
Atr, (2.8)
where
Atr = a
tr
1 +
γ2trθ
2
k4tr
atr2 +
γ3trθ
3
k6tr
atr3 +
γ4trθ
4
k8tr
atr4 +
γ5trθ
5
k10tr
atr5 +
γ6trθ
6
k12tr
atr6 . (2.9)
with atrη (η = 1, 2..., 6) being the values of parameters of crystal depending on the structure of
crystal lattice.
Similar derivation can be also done for next surface layer atoms of thin film, their
displacement are the solution of equations, respectively
γng1θ
2d
2yng1
dp2
+3γng1θyng1
dyng1
dp
+γng1y
3
ng1+kng1yng1+γng1
θ
kng1
(xng1 coth xng1−1)yng1−p = 0
(2.10)
For surface layers atoms of thin films, we present the statistical moment method (SMM)
formulation for the displacement of the surface layers atoms of the thin film yng =< u
ng
i > is
solution of equation
kng < u
ng
i >a + γng
[
< ung
i
>2a +θ
∂〈ungi 〉a
∂a
+
θ
mω2ng
(xng coth xng − 1)
]
− a = 0 (2.11)
114
Coefficients of thermal expansion of metallic thin films with body-centered cubic structure
where
yng =< u
ng
i >a, xng =
~ωng
2θ
, θ = kBT (2.12)
kng =
3
2
∑
i
[(
02ϕngi0
)
a2ix + (0ϕ
ng
i0 )
]
= mω2ng, (2.13)
γng =
1
4
∑
i,α,β,γ
α6=β
( ∂3ϕngio
∂u3iα,ng
)
eq
+
(
∂3ϕngio
∂u2iα,ng∂u
ng
iγ
)
eq
. (2.14)
In the second approximation of the supplemental force, the solutions of equation (2.11) can
be expanded in the power series of the supplemental force p as
yng = y
ng
0 +A1a+A2a
2. (2.15)
Here, ytr0 is the average atomic displacement in the limit of zero of supplemental force p. The
solution of ytr0 is given as
yng0 = −
γngθ
k2ng
xng cothxng. (2.16)
Using the statistical moment method, we can get power moments of the atomic displacement.
2.1.2. Free energy of metallic thin films
Usually, theoretical studies on size effect are done by considering the surface energy
contribution in continuum mechanics or by using computational simulations reflecting surface
stress or a surface relaxation influence. In this paper, the influence of size effect on thermodynamic
properties of metal thin film is studied by looking at the surface energy contribution in the free
energy of the system atoms.
For the internal layers and next surface layers. In articles [11, 12], free energy of these layers
Ψtr =
{
U tr0 + 3Ntrθ
[
xtr + ln
(
1− e−2xtr)]}+ 3Ntrθ2
k2tr
{
γ2trX
2
tr −
2γ1tr
3
(
1 +
Xtr
2
)}
6Ntrθ
3
k4tr
{
4
3
γ22tr
(
1 +
Xtr
2
)
Xtr − 2
(
γ21tr + 2γ1trγ2tr
)(
1 +
Xtr
2
)
(1 +Xtr)
}
. (2.17)
Ψng1 =
{
Ung10 + 3Nng1θ
[
xng1 + ln
(
1− e−2xng1)]}
+
3Nng1θ
2
k2ng1
{
γ2ng1X
2
ng1 −
2γ1ng1
3
(
1 +
Xng1
2
)}
+
6Nng1θ
3
k4ng1
{
4
3
γ22ng1
(
1 +
Xng1
2
)
Xng1−2
(
γ21ng1 + 2γ1ng1γ2ng1
)(
1 +
Xng1
2
)
(1 +Xng1)
}
;
(2.18)
In Eqs. (2.17), (2.18), using Xtr = xtrcothxtr , Xng1 = xng1cothxng1; and
U tr0 =
Ntr
2
∑
ϕtri0 (ri,tr);U
ng1
0 =
Nng1
2
∑
ϕng1i0 (ri,ng1); (2.19)
115
Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van
where ri is the equilibrium position of the ith atom, ui is its displacement of the ith atom from
the equilibrium position; ϕtri0, ϕ
ng1
i0 , are the effective interatomic potential between the 0
th and
ith internal layer atoms and the 0th and ith and next to surface layer atoms, Ntr, Nng1 and are
respectively the number of internal layers atoms, next surface layer atoms and of this thin film;
U tr0 , U
ng1
0 represent the sum of effective pair interaction energies for internal layer atoms and next
to surface layer atoms, respectively.
For the surface layers, the Helmholtz free energy of the system in the harmonic
approximation is given by [12]
Ψng =
{
Ung0 + 3Nngθ
[
xng + ln
(
1− e−2xng)]} (2.20)
Let us assume that the system consists of N atoms with n∗ layers, the atom number on each layer
is NL, we then have
N = n∗NL ⇒ n∗ = N
NL
. (2.21)
The number of atoms in the internal layers, next to surface layers and surface layers are,
respectively determined as
Ntr = (n
∗ − 4)NL =
(
N
NL
− 4
)
NL = N − 4NL,
Nng1 = 2NL = N − (n∗ − 2)NLandNng = 2NL = N − (n∗ − 2)NL. (2.22)
The free energy of the system and of one atom, respectively, are given by
Ψ = Ntrψtr +Nng1ψng1 +Nngψng − TSc = (N − 4NL)ψtr + 2NLψng1 + 2NLψng − TSc,
(2.23)
Ψ
N
=
[
1− 4
n∗
]
ψtr +
2
n∗
ψng1 +
2
n∗
ψng − TSc
N
, (2.24)
where Sc is the entropy configuration of the system and ψng, ψng1 and ψtrare respectively the free
energy of one atom at surface layers, next surface layers and internal layers.
Using a¯ as the average nearest-neighbor distance (NND) and b¯ as the average thickness
two-layers, we have
b =
a¯√
3
. (2.25)
The thickness d of thin film can be given by
d = 2bng + 2bng1 + (n
∗ − 5) btr = (n∗ − 1) b = (n∗ − 1) a√
3
. (2.26)
From equation (25), we derived
n∗ = 1 +
d
b
= 1 +
d
√
3
a
. (2.27)
The average NND of thin film is
a¯ =
2ang + 2ang1 + (n
∗ − 5)atr
n∗ − 1 . (2.28)
116
Coefficients of thermal expansion of metallic thin films with body-centered cubic structure
In above equation, ang, ang1 and atrare correspondingly the average NND between two
intermediate atoms at surface layers, next surface layers and internal layers of thin film at a given
temperature T. These quantities can be determined as
ang = a0,ng + y
ng
0 , ang1 = a0,ng1 + y
ng1
0 , atr = a0,tr + y
tr
0 , (2.29)
where a0,ng, a0,ng1 and a0,tr denote the values of ang, ang1 and atr at zero temperature which
can be determined from experiment or from the minimum condition of the potential energy of the
system.
Substituting Eq. (2.27) into Eq. (2.24), we obtain the expression of the free energy per atom
as follows
Ψ
N
=
d
√
3− 3a¯
d
√
3 + a¯
Ψtr +
2a¯
d
√
3 + a¯
Ψng +
2a¯
d
√
3 + a¯
Ψng1 − TSc
N
(2.30)
2.1.3. The coefficients of thermal expansion
The average thermal expansion coefficient of thin metal films can be calculated as
α =
kB
a¯0
da¯
dθ
=
dngαng + dng1αng1 + (d− dng − dng1)αtr
d
, (2.31)
where dng and dng1 are the thickness of surface layers and next to surface layers, and
αtr =
kB
a0,tr
∂ytr0 (T )
∂θ
;αng =
kB
a0,ng
∂yng0 (T )
∂θ
;αng1 =
kB
a0,ng1
∂yng10 (T )
∂θ
. (2.32)
2.2. Numerical results and discussion
In this section, the derived expressions in previous section will be used to investigate the
thermodynamic as well as mechanical properties of metallic thin films with BCC structure for
Fe and W at zero pressure. For the sake of simplicity, the interaction potential between two
intermediate atoms of these thin films is assumed to have a Mie-Lennard-Jones potential which
has the form
ϕ(r) =
D
(n−m)
[
m
(r0
r
)n
− n
(r0
r
)m]
(2.33)
where D describes the dissociation energy; r0 is the equilibrium value of r; and the parameters
n and m can be determined by fitting in the experimental data (e.g., cohesive energy and elastic
modulus). The potential parameters D,m,n and r0 of some metallic thin films are shown in Table
1 [14].
Table 1. Mie-Lennard-Jones potential parameters for Fe and W of metallic thin film [14]
Metal n m r0, (A
0) D/kB , (K)
Fe 8.26 3.58 2.4775 12576.70
W 8.58 4.06 2.7365 25608.93
117
Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van
Figure 2. Temperature dependence of the average NND for Fe thin film
Figure 3. Thickness dependence of the average NND for Fe thin film
118
Coefficients of thermal expansion of metallic thin films with body-centered cubic structure
Using the expression (2.28), we can determine the average NND of thin film as a function
of thickness and temperature. In Figure 2, we present the temperature dependence of the average
NND for Fe of thin film using SMM. One can see that the value of the average NND increases with
the increasing of absolute temperature T. These results show that the average NND for Fe increases
with increase in thickness. In Figure 3, we present the thickness dependence of the average NND
for Fe thin film at room temperature. The average NND increases when the thickness increases.
We realized that for Fe thin film when the thickness value is larger than 70 nm, the average NND
approaches the bulk value.
Figure 4. Temperature dependence of the thermal expansion coefficients for Fe thin film
In Figure 4, we present the temperature dependence of the thermal expansion coefficients
of Fe thin film as a function of thickness and temperature. We showed the theoretical calculations
of thermal expansion coefficients of Fe thin film with various layer thickneses. The experimental
thermal expansion coefficients [15] of bulk material have also been reported for comparison. One
can see that the value of the thermal expansion coefficient increases with the increase of absolute
temperature T. It can also be noted that, at a given temperature, the lattice parameter of thin film is
not a constant but rather, it strongly depends on the layer thickness, especially at high temperature.
In Figure 5, we present the thickness dependence of the thermal expansion coefficients for Fe
thin film at room temperature, the thermal expansion coefficients decreasing with the increasing
thickness, approaching the bulk value. The obtained results of dependence on thickness show
agreement between our works and the results presented in [10].
By using above scheme of SMM theory, the thermodynamic properties of other metallic
thin films can be determined analogously. In Table 2, we reported the values of some
thermodynamic quantities as functions of temperature at ambient pressure of W thin film.
119
Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van
Table 2. The thermodynamic quantities for W thin film
Thermodynamic T(K)
quantities Layers 200 500 800 1500 2000 2500
10 2.6712 2.6743 2.6774 2.6848 2.6902 2.6957
a,
(
A0
)
SMM 30 2.6787 2.6815 2.6844 2.6913 2.6963 2.7014
70 2.6838 2.6863 2.6890 2.6955 2.7003 2.7050
200 2.6848 2.6875 2.6902 2.6966 2.7013 2.7060
10 0.5055 0.6469 0.6697 0.6843 0.6902 0.6958
30 0.4116 0.4823 0.4942 0.5037 0.5089 0.5145
α× 10−5, (K−1) SMM 70 0.3506 0.3754 0.3802 0.3864 0.3912 0.3967
200 0.3355 0.3489 0.3519 0.3573 0.3620 0.3674
[15] Bulk 0.41 0.46 0.48 0.56 0.64 —
Figure 5. Thickness dependence of the thermal expansion coefficients for Fe thin film
3. Conclusion
The SMM calculations are performed using the effective pair potential for Fe and W thin
metal films. We used simple potentials because the purpose of the present study was to gain a
general understanding of the effects of the anharmonicity of the lattice vibration and temperature
on thermodynamic properties for BCC thin metal films.
In the present study, we used effective pair potentials for metallic thin film atoms to
demonstrate the utility of the present theoretical scheme based on the moment method in statistical
dynamics. The method is simple and physically transparent, and thermodynamic quantities of
metallic thin films with BCC 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 thermodynamic quantities between our theoretical calculations, and other theoretical results and
experimental values.
120
Coefficients of thermal expansion of metallic thin films with body-centered cubic structure
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