Temperature and thickness-dependent thermodynamic properties of metal thin films

TẠP CHÍ KHOA HỌC − SỐ 14/2017 29 
TEMPERATURE AND THICKNESS-DEPENDENT 
THERMODYNAMIC PROPERTIES OF METAL THIN FILMS 
Nguyen Thi Hoa1(1), Duong Dai Phuong2 
1Fundamental Science Faculty, University of Transport and Communications 
2Fundamental Science Faculty, Tank Armour Officers Training School, Vinh Phuc 
Abstract: The thermodynamic properties of metal thin films with body-centered cubic 
(BCC) structure at ambient conditions were investigated using the statistical moment 
method (SMM), including the anharmonic effects of thermal lattice vibrations. 
The analytical expressions of Helmholtz free energy, lattice constant, linear 
thermal expansion coefficients, specific heats at the constant volume and those at the 
constant pressure, VC and PC were derived in terms of the power moments of the atomic 
displacements. Numerical calculations of thermodynamic quantities have been perform 
for W and Nb thin films are found to be in good and reasonable agreement with those of 
the other theoretical results and experimental data. This research proves that 
thermodynamic quantities of thin films approach the values of bulk when the thickness of 
thin films is about 150 nm. 
Keywords: thin films, thermodynamic 
1. INTRODUCTION 
The knowledge about the thermodynamic properties of metal thin film, such as heat 
capacity, coefficient of thermal expansion, are of great important to determine the 
parameters for the stability and reliability of the manufactured devices. 
In many cases of the thermodynamic properties of metal thin film are not well 
known or may differ from the values for the corresponding bulk materials. A large 
number of experimental and theoretical studies have been carried out on the 
thermodynamic properties of metal and nonmetal thin films [1-5]. Most of them 
describe the method for measuring the thermodynamic properties of crystalline thin 
films on the substrates [6-9]. 
(1) Nhận bài ngày 20.02.2017; chỉnh sửa, gửi phản biện và duyệt đăng ngày 20.3.2017 
 Liên hệ tác giả: Nguyễn Thị Hòa; Email: hoanguyen1974@gmail.com 
30 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
The main purpose of this article is to provide an analysis of the thermodynamic 
properties of metal free-standing thin film with body-centered cubic structure using 
the analytic statistical moment method (SMM) [10-12]. The major advantage of our 
approach is that the thermodynamic quantities are derived from the Helmholtz free 
energy, and the explicit expressions of the thermal lattice expansion coefficient, 
specific heats at constant volume and those at the constant pressure CV, CP and the 
coefficient of thermal expansion α are 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 thermodynamic properties have also been studied. 
2. THEORY 
2.1. The anharmonic oscillations of thin metal films 
Let us consider a metal free standing thin film has *n layers with the thickness d . We 
assume the thin film consists of two atomic surface layers, two next surface atomic layers 
and ( *n 4− ) atomic internal layers. (see Fig. 1). 
Fig. 1. The free-standing thin film 
For internal layers atoms of thin films, we present the statistical moment method 
formulation for the displacement of the internal layers atoms of the thin film try is 
solution of equation [11] 
2
2 3
2
3 1 0tr trtr tr tr tr tr tr tr tr tr tr tr
tr
d y dy
y y k y ( x coth x )y p ,
dp kdp
θ
γ θ γ θ γ γ+ + + + − − = (1) 
T
hi
ck
ne
ss
(n
*-
 4
) 
La
ye
rs
d
a
a
a
ng
ng
1
tr
TẠP CHÍ KHOA HỌC − SỐ 14/2017 31 
where 
 ,
; ; ,
2
tr
tr i tr p tr By u x k T
ω
θ
θ
≡ = =


2 tr
2io
tr 0 tr2
i i eq
1
k m ,
2 u α
ϕ
ω
 ∂
= ≡ 
∂ 
∑ (2) 
4 tr
io
1tr 4
i i eq
1
,
48 u α
ϕ
γ
 ∂
=  
∂ 
∑
4 tr
io
2tr 2 2
i i i eq
6
,
48 u uβ γ
ϕ
γ
 ∂
=   ∂ ∂ 
∑
4 tr 4 tr
io io
tr 4 2 2
i i i ieq eq
1
6
12 u u uα β γ
ϕ ϕ
γ
   ∂ ∂ = +     ∂ ∂ ∂     
∑
(3)
where Bk is the Boltzmann constant, T is the absolute temperature, 0m is the mass of 
atom, trω is the frequency of lattice vibration of internal layers atoms; trk , 1trγ , 2trγ , trγ 
are the parameters of crystal depending on the structure of crystal lattice and the interaction 
potential between atoms; 0
tr
iϕ is the effective interatomic potential between 0
th and ith 
internal layers atoms; iu α , iu β , iu γ are the displacements of i
th
 atom from equilibrium 
position on direction , , ( , , , , )x y zα β γ α β γ = , respectively, and the subscript eq indicates 
evaluation at equilibrium. 
The solutions of the nonlinear differential equation of Eq. (1) can be expanded in the 
power series of the supplemental force p as [11] 
2
0 1 2 .
tr tr tr
try y A p A p= + + 
(4) 
Here, 0
try is the average atomic displacement in the limit of zero of supplemental force 
p . Substituting the above solution of Eq. (4) into the original differential Eq. (1), one can 
get the coupled equations on the coefficients 1
trA and 2
trA , from which the solution of 0
try is 
given as [10] 
2
tr tr
0 tr3
tr
2
y A ,
3k
γ θ
≈
(5) 
where 
γ θ γ θ γ θ γ θ γ θ
= + + + +
2 2 3 3 4 4 5 5 6 6
tr tr tr tr tr trtr tr tr tr tr
tr 1 2 3 4 5 64 6 8 10 12
tr tr tr tr tr
A a a a + a a a .
k k k k k
 (6)
32 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
with η η =
tra ( 1,2...,6) are the values of parameters of crystal depending on the structure of 
crystal lattice [10]. 
Similar derivation can be also done for next surface layers atoms of thin film, their 
displacement are solution of equations, respectively 
2
1 12 3
1 1 1 1 1 1 1 1 1 1 12
1
3 1 0ng ngng ng ng ng ng ng ng ng ng ng ng
ng
d y dy
y y k y ( x cothx )y p
dp kdp
θ
γ θ γ θ γ γ+ + + + − − =
(7) 
For surface layers atoms of thin films, the displacement of the surface layers atoms of 
the thin f ... energy contribution in the continuum mechanics or by the computational 
simulations reflecting the surface stress, or surface relaxation influence. In this paper, the 
TẠP CHÍ KHOA HỌC − SỐ 14/2017 33 
influence of the size effect on thermodynamic properties of the metal thin film is studied 
by introducing the surface energy contribution in the free energy of the system atoms. 
For the internal layers and next surface layers. Free energy of these layer are 
( ){ }
( ) ( )
2
2 2 1
0 22
3
2 2
2 1 1 24
3 2
3 1 1
3 2
6 4
1 2 2 1 1
3 2 2
trxtr tr tr tr
tr tr tr tr tr
tr
tr tr tr
tr tr tr tr tr tr
tr
N X
U N x ln e X
k
N X X
X X . 
k
θ γ
θ γ
θ
γ γ γ γ
−    Ψ = + + − + − + +      
    
+ − + + +    
    
(12) 
( ){ }
( ) ( )
1
2
2 1 1 1 11 2
1 0 1 1 2 1 12
1
3
1 1 12 2
2 1 1 1 1 1 1 2 1 14
1
3 2
3 1 1
3 2
6 4
1 2 2 1 1
3 2 2
ngx ng ng ngng
ng ng ng ng ng
ng
ng ng ng
ng ng ng ng ng ng
ng
N X
U N x ln e X
k
N X X
X X ; 
k
θ γ
θ γ
θ
γ γ γ γ
−     Ψ = + + − + − + +       
     
+ + − + + +    
      
(13) 
In Eqs. (12), (13), using tr tr trX x cothx= , 1 1 1ng ng ngX x cothx= ; and 
 ( ) ( )11 10 0 , 0 0 , 1; ;2 2
ngtr tr ng ngtr
i i tr i i ng
NN
U r U rϕ ϕ= =∑ ∑ (14) 
where ri is the equilibrium position of i
th atom, ui is its displacement of the i
th atom from 
the equilibrium position; 0
tr
iϕ , 
1
0
ng
iϕ , are the effective interatomic potential between the 0
th 
and ith internal layers atom, the 0th and ith next surface layers atom, ; Ntr, Nng1 and are 
respectively the number of internal layers atoms, next surface layers atoms and of this thin 
film; 10 0,
tr ngU U represent the sum of effective pair interaction energies for internal layers 
atom, next surface layers atom, respectively. 
For the surface layers, the Helmholtz free energy of the system in the harmonic 
approximation given by [11] 
( ){ }20 3 1 ngxngng ng ngU N x ln e θ − Ψ = + + −  (15) 
Let us assume that the system consists N atoms with *n layers, the atom number on 
each layer is LN , then we have 
* * .L
L
N
N n N n
N
= ⇒ =
The number of atoms of internal layers, next surface layers and surface layers atoms 
are , respectively determined as 
34 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
( )*tr L L L
L
N
N n 4 N 4 N N 4N ,
N
 
= − = − = − 
 
*
ng1 L LN 2N N ( n 2 )N= = − − and 
*
ng L LN 2N N ( n 2 )N .= = − − (17) 
Free energy of the system and of one atom, respectively, are given by 
( )tr tr ng1 ng1 ng ng c L tr L ng1 L ng cN N N TS N 4N 2N 2N TS ,= + + − = − + + −Ψ ψ ψ ψ ψ ψ ψ (18) 
c
tr ng1 ng* * *
TS4 2 2
1 ,
N Nn n n
 = − + + −  
Ψ
ψ ψ ψ
 (19) 
where cS is the entropy configuration of the system; ngψ , 1ngψ and trψ are respectively the 
free energy of one atom at surface layers, next surface layers and internal layers. 
Using a as the average nearest-neighbor distance and b is the average thickness two-
layers and ca is the average lattice constant. Then we have 
3
a
b = and 
2
3
ca a= . (20) 
The thickness d of thin film can be given by 
 ( ) ( ) ( )
* * *
ng ng1 tr
a
d 2b 2b n 5 b n 1 b n 1
3
= + + − = − = − (21) 
From equation (21), we derived 
 *
3
1 1 .
d d
n
b a
= + = + 
The average nearest-neighbor distance of thin film 
*
1
*
2 2 ( 5)
.
1
ng ng tra a n aa
n
+ + −
=
−
 (23) 
In above equation, nga , 1nga and tra are correspondingly the average 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 
1
0, 0 1 0, 1 0 0, 0, , ,
ng ng tr
ng ng ng ng tr tra a y a a y a a y= + = + = + 
(24) 
where 0,nga , 0, 1nga and 0,tra denotes the values of nga , 1nga and tra at zero temperature 
which can be determined from experiment or from the minimum condition of the potential 
energy of the system. 
TẠP CHÍ KHOA HỌC − SỐ 14/2017 35 
Substituting Eq. (22) into Eq. (19), we obtained the expression of the free energy per 
atom as follows 
.ctr ng ng1
TSd 3 3a 2a 2a
N Nd 3 a d 3 a d 3 a
Ψ
Ψ Ψ Ψ
−
= + + −
+ + + 
(25) 
2.3. Thermodynamic quantities of the thin metal films 
The average thermal expansion coefficient of thin metal films can be calculated as 
( )1 1 1
0
,
ng ng ng ng ng ng trB
d d d d dk da
a d d
α α α
α
θ
+ + − −
= =
 (26) 
where ngd and 1ngd are the thickness of surface layers and next surface layers, and 
( ) ( ) ( )10 0 0
1
0 , 0 , 0 , 1
; ;
tr ng ng
B B B
tr ng ng
tr ng ng
y T y T y Tk k k
a a a
α α α
θ θ θ
∂ ∂ ∂
= = =
∂ ∂ ∂
. (27) 
The specific heats VC at constant volume temperature T is derived from the free 
energy of the system and has the form 
2
1
2
3 3 2 2
,
3 3 3
tr ng ng
V V V V
V
W d a a a
C T C C C
T T d a d a d a
Ψ∂ ∂ − = = − = + + ∂ ∂ + + + 
 (28) 
where 
2 3 4 4 2
1 1
2 22 2 2 4 2
22
3 2 2
3 3
tr tr tr tr tr tr tr tr tr
V B tr tr
tr tr tr tr tr
x x cothx x x coth x
C k N .
sinh x k sinh x sinh x sinh x
γ γθ
γ γ
    = + + + − +    
      
(29) 
The isothermal compressibility Tλ is given by 
3
0
2 222
0 1
2 2 2
1
3
1
3 3 2 2
2
3 3 3 3
T
T ng ngtr
tr ng ng T
a
aV
V P a d a a a
P
V a a ad a d a d a
 
 
∂   λ = − = ∂  ∂ Ψ ∂ Ψ  ∂ Ψ−
+ + +  ∂ ∂ ∂+ + +  
(30) 
Furthermore, the specific heats at constant pressure PC is determined from the 
thermodynamic relations 
2
29 .P V V T
P T
V P
C C T C TV B
T V
α
∂ ∂   = − = +   ∂ ∂   
 (31) 
36 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
3. 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 Nb and W at zero pressure. For the sake of simplicity, the interaction potential 
between two intermediate atoms of these thin films is assumed as the Mie-Lennard-Jones 
potential which has the form as 
( )
0 0( )
n m
r rD
r m n
n m r r
ϕ
    
= −    −      
 (32) 
where D describing dissociation energy; 0r is the equilibrium value of r; and the 
parameters n and m can be determined by fitting experimental data (e.g., cohesive energy 
and elastic modulus). The potential parameters , ,D m n and 0r of some metallic thin films 
are showed in Table 1. 
Table 1. Mie-Lennard-Jones potential parameters for Nb of thin metal films [12] 
Metal n m 00 , ( )r A / , ( )BD k K 
Nb 7.5 1.72 2.8648 21706.44 
W 8.58 4.06 2.7365 25608.93 
0 200 400 600 800 1000 1200 1400 1600 1800
2.755
2.760
2.765
2.770
2.775
2.780
2.785
2.790
2.795
2.800
2.805
a
 (
A
°)
T (K)
 10 layers
 20 layers
 70 layers
 200 layers
Fig. 2. Dependence on thickness of the nearest-neighbor distance for Nb thin film 
TẠP CHÍ KHOA HỌC − SỐ 14/2017 37 
Using the expression (23), we can determine the average nearest-neighbor distance of 
thin film as functions of thickness and temperature. In Fig. 2, we present the temperatures 
dependence of the average nearest-neighbor distance of thin film for Nb using SMM. One 
can see that the value of the average nearest-neighbor distance increases with the 
increasing of absolute temperature T. These results showed the average nearest-neighbor 
distance for Nb increases with increasing thickness. We realized that for Nb thin film when 
the thickness larger value 150 nm then the average nearest-neighbor distance approach the 
bulk value. The obtained results of dependence on thickness are in agreement between our 
works with the results presented in [14]. 
0 200 400 600 800 1000 1200 1400 1600 1800
0.6
0.7
0.8
0.9
1.0
1.1
α
 (
1
0-
5
K
-1
)
T (K)
 10 layers
 20 layers
 70 layers
 200 layers
 [17] bulk
Fig. 3. Temperature dependence of the thermal expansion coefficients for Nb thin film 
In Fig. 3, we present the temperature dependence of the thermal expansion coefficients 
of Nb thin fillm as functions of thickness and temperature. We showed the theoretical 
calculations of thermal expansion coefficients of Nb 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 thermal expansion 
coefficient increases with the increasing of absolute temperature T. It also be noted that, at 
a given temperature, the lattice parameter of thin film is not a constant but strongly 
depends on the layer thickness, especially at high temperature. Interestingly, the thermal 
expansion coefficients decreases with increasing thickness and approach the bulk value. 
[15] bulk 
38 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
0 200 400 600 800 1000 1200 1400 1600 1800
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
6.2
6.4
λ T
 (
1
0
-1
2
 P
a
)
T (K)
 10 layers
 20 layers
 70 layers
 200 layers
 [17] bulk
Fig. 4. Temperature dependence of the isothermal compressibility for Nb thin film 
In Fig. 4, we present the temperature dependence of the isothermal compressibility of 
the Ag films as a function of the temperature in various thickneses and the bulk Nb [15] by 
the SMM. We realized that also, it increases with absolute temperature T. When the 
thickness increases, the average of the isothermal compressibility approach the bulk 
values. These results are in agreement with the laws of the bulk isothermal compressibility 
depends on the temperature of us [10]. 
The specific heat at constant pressure PC is one of important thermodynamic 
quantities of solid. Its dependence on thickness and temperature was showed in Fig. 5 for 
Nb thin film. Experimental data of PC of Nb bulk crystal were also displayed for 
comparison [15]. It is clearly seen that at temperature range below 700 K, the specific heat 
PC of thin film follows very well the value of bulk material. When temperatures and the 
thickness of thin film increase, the specific heat at constant pressure increase with the 
absolute temperature, therefore the specific heat PC depends strongly on the temperature. 
In Fig. 6, we presented SMM results of the specific heats at constant volume of Nb 
thin film with various thickness as functions of temperature. It is clearly seen that at 
temperature in range T<300K, the specific heat at constant volume VC depends strongly on 
the temperature. It increases robustly with the increasing of absolute temperature. In 
[ 5] lk 
TẠP CHÍ KHOA HỌC − SỐ 14/2017 39 
temperature range T >300 K, the specific heat VC reduces and depends weakly on the 
temperature. The thicker thin film is the less dependent on temperature specific heat VC 
becomes. In our SMM calculations, when the thicknesses of Nb and W thin films are larger 
than 150 nm, the specific heats VC are almost independent on the layer thickness and reach 
the values of bulk materials. 
0 200 400 600 800 1000 1200
3.5
4.0
4.5
5.0
5.5
6.0
6.5
C
p
 (
C
a
l/m
o
l.K
)
T (K)
 10 layers
 20 layers
 70 layers
 200 layers
 [17] bulk
Fig. 5. Temperature dependence of the specific heats at constant pressure for Nb thin film 
0 200 400 600 800 1000 1200 1400 1600 1800
3.5
4.0
4.5
5.0
5.5
6.0
C
v 
(C
a
l/m
ol
.K
)
T (K)
 10 layers
 20 layers
 70 layers
 200 layers
 [17] bulk
Fig. 6. Temperature dependence of the specific heats at constant volume for Nb thin film 
5 
[15] bulk 
40 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
4. CONCLUSIONS 
The SMM calculations are performed by using the effective pair potential for the W 
and Nb thin metal films. The use of the simple potentials is due to the fact that the purpose 
of the present study is to gain a general understanding of the effects of the anharmonic of 
the lattice vibration and temperature on the thermodynamic properties for the BCC thin 
metal films. 
In general, we have obtained good agreement in the thermodynamic quantities 
between our theoretical calculations and other theoretical results, and experimental values. 
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CÁC TÍNH CHẤT NHIỆT ĐỘNG HỌC PHỤ THUỘC ĐỘ DÀY 
VÀ NHIỆT ĐỘ CỦA MÀNG MỎNG KIM LOẠI 
Tóm tắt: Ứng dụng phương pháp thống kê mô men vào nghiên cứu tính chất nhiệt động 
của màng mỏng kim loại với cấu trúc lập phương tâm khối. Quá trình nghiên cứu có kể 
đến đóng góp của hiệu ứng phi điều hòa trong dao động mạng tinh thể. Đã thu được các 
biểu thức giải tích cho phép tính năng lượng tự do Helmholtz của hệ, các hàng số mạng, 
hệ số dãn nở nhiệt của màng mỏng, Các kết quả nghiên cứu lý thuyết được áp dụng 
tính số với màng mỏng kim loại Nb và so sánh với số liệu thực nghiệm và các kết quả tính 
bằng phương pháp khác cho thấy có sự phù hợp tốt. 
Từ khóa: màng mỏng, nhiệt động lực học