Abstract. The moment method in statistical (SMM) dynamics is used to study the
lattice constant of CeO2 thin films taking into account the anharmonicity effects
of the lattice vibrations. The nearest neighbor distance and the lattice constant of
CeO2 thin films are calculated as a function of temperature. SMM calculations are
performed using the Buckingham potential for the CeO2 thin films. In the present
study, the influence of temperature and the size of the lattice constant of CeO2
thin film have been studied using three different interatomic potentials. We discuss
temperature and thickness dependence of the lattice constant of CeO2 thin films
and we compare our calculated results with those of the experimental results.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 79-87
This paper is available online at
LATTICE CONSTANT OF CERIA THIN FILM:
TEMPERATURE DEPENDENCE
Vu Van Hung, Nguyen Thi Hang1 and Le Thi Thanh Huong2
1Faculty of Physics, Hanoi National University of Education
2Faculty of Physics, Hai Phong University
Abstract. The moment method in statistical (SMM) dynamics is used to study the
lattice constant of CeO2 thin films taking into account the anharmonicity effects
of the lattice vibrations. The nearest neighbor distance and the lattice constant of
CeO2 thin films are calculated as a function of temperature. SMM calculations are
performed using the Buckingham potential for the CeO2 thin films. In the present
study, the influence of temperature and the size of the lattice constant of CeO2
thin film have been studied using three different interatomic potentials. We discuss
temperature and thickness dependence of the lattice constant of CeO2 thin films
and we compare our calculated results with those of the experimental results.
Keywords: Thin film, ceria, Lattice constant, statistical moment method.
1. Introduction
Cerium dioxide (or ceria) possesses a cubic fluorite structure with a lattice
parameter of 5.411 A˚, where in the unit cell the Ce4+ cations occupy the fcc lattice sites
while the O2− anions are located at the eight tetrahedral sites. Cerium dioxide (CeO2) is an
important oxide material used as high and low index films in multi-layer optical thin film
devices. CeO2 thin films have been deposited and characterized using different techniques
[1]. Among the oxide materials, CeO2 has attracted more and more attention because of its
desirable properties which includes high stability against mechanical abrasion, chemical
attack and high temperatures [2, 3].
Most previous theoretical studies were concerned with the material properties of
CeO2 bulk and thin film at absolute zero temperature while temperature dependence
of thermodynamic quantities and lattice constants have not been studied in detail.
Temperature and pressure dependences of the thermodynamic and elastic properties of
Received September 5, 2012. Accepted October 20, 2012.
Physics Subject Classification: 60 44 01.
Contact Vu Van Hung, e-mail address: bangvu57@yahoo.com
79
Vu Van Hung, Nguyen Thi Hang and Le Thi Thanh Huong
bulk cerium dioxide have been studied using the analytic statistical moment method
(SMM) [4, 5, 6]. The purpose of the present article is to investigate temperature and size
dependences of the lattice constant of CeO2 thin film using SMM [7].
2. Content
2.1. Theory
Let us consider a ceria free thin film of n cerium layers with film thickness d.
Suppose that two free surface area of ceria thin film are layers of cerium atoms.
Figure 1. Ceria thin film with two free surface layers of cerium atoms
The ceria free thin film consists of 2 cerium free surface layers, 2 oxygen next
free surface layers, and (n-3) oxygen internal layers and (n-2) cerium internal layers. The
general expression of the Helmholtz free energy Ψ of cerium dioxide thin film is given as
Ψ = 2NCeΨ
side
Ce + 2NOΨ
side
O
+ (n− 3)NOΨinterO + (n− 2)NCeΨinterCe − TSC (2.1)
where the numbers of cerium and oxygen ions of a layer are simply denoted by NCe
and NO = 2NCe , respectively, Ψ
side
Ce (or Ψ
inter
Ce ) and Ψ
side
O (or Ψ
inter
O ) denoting the free
energy of Ce and O ions on the free surface (or internal) layers, respectively, and SC - the
configurational entropies.
Here, it is noted that the analytic expression of the free energy of an atom of Ce and
O on the free surface layers in the harmonic approximation has the form [7].
ΨsideCe ≈ 3
{
1
6
usideCe + θ[xCe + ln(1− e−2xCe)]
}
(2.2)
ΨsideO ≈ 3
{
1
6
usideO + θ[xO + ln(1− e−2xO)]
}
(2.3)
80
Lattice constant of ceria thin film: temperature dependence
where usideCe =
∑
i
ϕCe−sideio (|ri|), andusideO =
∑
i
ϕO−sideio (|ri|). (2.4)
x = ~w/2θ with θ = kBT , w is the atomic vibration frequency, and it can be
approximated in most cases to the Einstein frequency wE , given by
k =
1
2
∑
i
(
∂ϕio
∂u2ix
)
eq
≡ mw2E , (2.5)
and ϕio is the interatomic potential energy between the central 0th and ith sites, and uix
is the atomic displacement of the ith atom in the x-direction.
The free energy of an atom of Ce or O on the internal layers in the harmonic
approximation has the form [7]
ΨinterCe ≈ 3
{
1
6
uinterCe + θ[xCe + ln(1− e−2xCe)]
}
(2.6)
ΨinterO ≈ 3
{
1
6
uinterO + θ[xO + ln(1− e−2xO)]
}
(2.7)
where uinterCe and u
inter
O represent the sum of the effective pair interaction energies for Ce
and O ions on the internal layers in ceria thin film.
uinterCe =
∑
i
ϕCe−interio (|ri|), anduinterO =
∑
i
ϕO−interio (|ri|). (2.8)
The average nearest-neighbor distance at T = 0 K can be determined from
experimentation or the minimum condition of the potential energy of the free surface
layer composed of NCe atoms Ce and NO atoms O, and that means
(
∂ Uside
∂ r1
)
T,P,N
= 0,
leads to the following equation
∂Uside
∂r1
=
∂UsideCe
∂r1
+
∂UsideiO
∂r1
=
NCe
2
∂
∂r1
(∑
i
ϕCe−sideio (|ri|)
)
+
NO
2
∂
∂r1
(∑
i
ϕO−sideio (|ri|)
)
= 0
or
CCe.
∂
∂r1
(∑
i
ϕCe−sideio (|ri|)
)
+ CO.
∂
∂r1
(∑
i
ϕO−sideio (|ri|)
)
= 0. (2.9)
81
Vu Van Hung, Nguyen Thi Hang and Le Thi Thanh Huong
where CCe = NCe/(NCe +NO) = 1/3, CO = NO/(NCe +NO) = 2/3.
Using Eq. (2.9), one can find the nearest neighbor distance at zero temperature
T = 0:K: r1(0). It’s known that the Buckingham potential has been very successfully used
to calculate the thermodynamic properties of CeO2. The atomic interactions are described
by a potential function which divides the forces into long-range interactions (described by
Coulomb’s Law and summated by the Ewald method) and short-range interactions treated
by a pairwise function of the Buckingham form
ϕij(r) =
qiqj
r
+ Aij exp(− r
Bij
)− Cij
r6
(2.10)
where qi and qj are the charges of ions i and j respectively, r is distance between
them and Aij , Bij and Cij are the parameters particular to each ion-ion interaction. In
Eq. (2.10), the exponential term corresponds to electron cloud overlap and the Cij/r
6
term to any attractive dispersion or Van der Waal’s force. Potential parameters Aij , Bij
and Cij have most commonly been derived by the procedure of ‘empirical fitting’, i.e.,
parameters are adjusted, usually by a least-squares fitting routine, so as to achieve the best
possible agreement between calculated and experimental crystal properties. The potential
parameters are listed in Table 1 [8].
Using the effective pair potentials of Eq. (2.10), and Eq. (2.4), it is straightforward
to get the interaction energy Uo in cerium dioxide. The terms of Eq. (2.9),(∑
i
ϕCe−sideio (|ri|)
)
, and
(∑
i
ϕO−sideio (|ri|)
)
have been summated by the Ewald method
∑
i
ϕCe−sideio (|ri|) =
∑
i
ϕCe−sideCe−Ce (|ri|) +
∑
i
ϕCe−sideCe−O (|ri|)
=
∑
i
q2Cee
2
ri
.erfc(αri)+
∑
i
[
qCeqOe
2
ri
.erfc(αri) + ACe−O.exp
(
− ri
BCe−O
)
− CCe−O
r6i
]
(2.11)
∑
i
ϕO−sideio (|ri|) =
∑
i
ϕO−sideO−Ce (|ri|) +
∑
i
ϕO−sideO−O (|ri|)
=
∑
i
[
qCeqOe
2
ri
.erfc(αri) + ACe−O.exp
(
− ri
BCe−O
)
− CCe−O
r6i
]
+
∑
i
[
qOqOe
2
ri
.erfc(αri) + AO−O.exp
(
− ri
BO−O
)
− CO−O
r6i
]
(2.12)
From Eqs. (2.11), (2.12) and (2.9) we obtain the following equation:
CCe.M + CO.N = 0 (2.13)
82
Lattice constant of ceria thin film: temperature dependence
where M = − q
2
Cee
2
a2
{
8.2√
2
.erfc(αr1) + 5.erfc(αr2)
}
− qCeqOe
2
a2
{
4.4√
3
.erfc(αr1) +
4.12√
11
.erfc(αr2)
}
− ACe−O
BCe−O
{√
3.exp
(
− a
√
3
4BCe−O
)
+ 3
√
11.exp
(
− a
√
11
4BCe−O
)}
+ 6.
CCe−O
a7
4(√3
4
)6 + 12(√
11
4
)6
(2.14)
N = − q
2
Oe
2
a2
{
5.2erfc(αr1) + 4.
√
2.erfc(αr2)
}
− AO−O
BO−O
{
5
2
.exp
(
− a
2BO−O
)
+ 4
√
2.exp
(
− a
√
2
2BO−O
)}
+ 6.
CO−O
a7
5(12)6 +
8(√
2
2
)6
+ 6.CCe−Oa7
4(√3
4
)6 + 10(√
11
4
)6
− qCeqOe
2
a2
{
16√
3
.erfc(αr1) +
10.4√
11
.erfc(αr2)
}
− ACe−O
BCe−O
{√
3.exp
(
− a
√
3
4BCe−O
)
+
10
4
√
11.exp
(
− a
√
11
4BCe−O
)}
(2.15)
Minimizing the interaction potentials U inter of the internal layer with respect to
the nearest-neighbor distance r1, this means
(
∂ U inter
∂ r1
)
T,P,N
= 0, which leads to the
following equation
CCe.
∂
∂r1
(∑
i
ϕCe−interio (|ri|)
)
+ CO.
∂
∂r1
(∑
i
ϕO−interio (|ri|)
)
= 0
or CCe.P + CO.Q = 0 (2.16)
83
Vu Van Hung, Nguyen Thi Hang and Le Thi Thanh Huong
where P = − q
2
Cee
2
a2
{
24√
2
.erfc(αr1) + 6.erfc(αr2)
}
− qCeqOe
2
a2
{
32√
3
.erfc(αr1) +
4.24√
11
.erfc(αr2)
}
− 2ACe−O
BCe−O
{√
3.exp
(
− a
√
3
4BCe−O
)
+ 3
√
11.exp
(
− a
√
11
4BCe−O
)}
+ 6.
CCe−O
a7
8(√3
4
)6 + 24(√
11
4
)6
(2.17)
Q = − q
2
Oe
2
a2
{
12.erfc(αr1) +
24√
2
.erfc(αr2)
}
− AO−O
BO−O
{
3.exp
(
− a
2BO−O
)
+ 6
√
2.exp
(
− a
√
2
2BO−O
)}
+ 6.
CO−O
a7
6(12)6 +
12(√
2
2
)6
− qCeqOe
2
a2
{
16√
3
.erfc(αr1) +
48√
11
.erfc(αr2)
}
− ACe−O
BCe−O
{√
3.exp
(
− a
√
3
4BCe−O
)
+ 3
√
11.exp
(
− a
√
11
4BCe−O
)}
+ 6.
CCe−O
a7
4(√3
4
)6 + 12(√
11
4
)6
(2.18)
Principle Eqs. (2.13) and (2.16) permit us to find the nearest neighbor distance
rside1 (0) or r
inter
1 (0) at zero temperature for the free surface layer (or internal layer).
Using the MAPLE program, Eqs. (2.13) and (2.16) can be solved and we find the values
of the nearest neighbor distances rside1 (0) and r
inter
1 (0). We assume that the average
nearest-neighbor distance of the free surface layers and internal layers for cerium dioxide
thin film at temperature T can be written as
rside1 (T ) = r
side
1 (0) + CCey
side
Ce (T ) + COy
side
O (T ) (2.19)
rinter1 (T ) = r
inter
1 (0) + CCey
inter
Ce (T ) + COy
inter
O (T ) (2.20)
84
Lattice constant of ceria thin film: temperature dependence
in which ysideCe (T ) (or y
inter
Ce (T ) ) and y
side
O (T ) (or y
inter
O (T )) are the atomic displacements
of Ce and O atoms from the equilibrium position in the free surface (or internal) layers.
In the above Eqs. (2.19) and (2.20), the atomic displacements of Ce and O atoms from the
equilibrium position are determined as [7].
The thickness d of thin film can be given by
d = 2aside(T ) + (n− 3)ainter(T ) (2.21)
where aside and ainter are the lattice constants of the free surface layer and internal layer,
respectively. Therefore, the average lattice constant a(T ) of thin film is determined as
a(T ) =
d
n− 1 =
2aside(T ) + (n− 3)ainter(T )
n− 1 (2.22)
2.2. Results and discussion
In this section we compare our lattice constant of internal layer for CeO2 thin film to
some experimental and other theoretical results. Table 2 shows good agreement between
the SMM calculations of lattice constant at zero temperature T = 0K and the experimental
results for CeO2
Table 1. Potential parameters of CeO2 [8]
Interaction
potential
A(eV) B(A˚) C(eV.A˚6)
O2− - O2− 9547.92 0.2192 32.00 Potential 1
Ce4+ - O2− 1809.68 0.3547 20.40
O2− - O2− 9547.92 0.2192 32.00 Potential 2
Ce4+- O2− 2531.5 0.335 20.40
O2− - O2− 22764.3 0.149 43.83 Butler
Ce4+ - O2− 1986.83 0.35107 20.40
Table 2. Lattice parameter of bulk CeO2
Method ao (A˚)
Potential 1 Potential 2 Butler
Simulation [8] 5.411 5.411 5.411
SMM 5.4107 5.4111 5.4099
Expt [9] 5.411
Ab initio [10] 5.353
In Figure 2, we present the thickness dependence of the lattice constant of ceria
thin film using the potentials 1, 2 and the Butler potential. Figure 2 shows the lattice
constant of ceria thin film, calculated using the Buckingham potentials, as a function of
85
Vu Van Hung, Nguyen Thi Hang and Le Thi Thanh Huong
the thickness d of thin film. One can see in Figure 2 that the lattice constant increases
with the thickness d, when the thickness d ≥ 500A0 (or the number n of layers of thin
film n ≥ 100) and the average lattice constant a(T ) of thin film (a(T ) ≈ 5.41 A˚) are
in agreement with the experimental results of bulk CeO2. One also sees in Figure 3 the
temperature dependence of the SMM lattice parameter of CeO2 thin films with different
thickness using the potential Butler.
(a) potential 1 (b) potential 2
(c) the Butler potential
Figure 2. Thickness dependence of average lattice constant
of ceria thin film using a, b, c
86
Lattice constant of ceria thin film: temperature dependence
Figure 3. Temperature dependence of average lattice constant a(T )
of ceria thin film using the Butler potential
3. Conclusion
In conclusion it should be noted that the statistical moment method really permits an
investigation into temperature and thickness dependences of CeO2 thin films. The results
obtained by this method are in good agreement with the experimental data. We have
calculated the lattice constant for CeO2 thin films of different thickness using potentials
1, 2 and the Butler potential and these calculated SMM lattice constants are in good
agreement with other calculations and experiments with bulk CeO2.
Acknowledgments. This research is funded by the Vietnam National Foundation for
Science and Technology Development (NAFOSTED), grant number 103.01-2011.16.
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