Oxide thin films are used for multi-layer optical coatings, and multi-layer optical thin film devices. Among oxide materials, pure and doped CeO2 and ZrO2 (bulk and thin films) have attracted more attention because of its desirable properties, such as high stability against mechanical abrasion, chemical attack and high temperatures   . These oxide thin films have been prepared by some conventional methods. Many physical deposition techniques, such as pulsed laser deposition (PLD), magnetron sputtering, or molecular beam epitaxy (MBE), have been used produce CeO2 thin films. For instance, Gerblinger et al.  have obtained CeO2 thin films on Al2O3 substrate by sputter process, or the deposition of cerium dioxide thin films on Ni textured substrates by metallorganic chemical vapor deposition (MOCVD) has been reported    . Chemical vapor deposition (CVD) is a well-known technique for preparing thin films. For example, yttria doped CeO2 (YDC) thin films have been grown on yttria doped zirconia (YSZ) single substrates by vacuum vapor deposition and slurry painting method  . Different several chemical methods have also been applied to produce CeO2 (or ZrO2) thin films, such as sol-gel, aerosol-assisted MOCVD  , mist microwave-plasma chemical vapour deposition (MPCVD)  , atomic layer deposition (ALD)  and spray pyrolysis  . Recently, the structural and optical properties of ZrO2 thin films in relation to thermal annealing times, and properties of surface ceria-zirconia solid solution films were investigated   . ZrO2 thin films were grown by thermal oxidation of metallic zirconium films deposited by sputtering of zirconium target by DC magnetron sputtering technique  .
Recently, extensive studies of elastic and thermodynamic properties of oxide materials appear because of their important applications in high-frequency resonators. These materials are systematically fabricated by film deposition techniques in devices and their elastic constants are definitely required. Knowledge of mechanical and thermodynamic properties of these oxide thin films is essential to design Micro-Electro-Mechanical Systems (MEMS) devices. It is known that the size effect of phonon frequency is attributed to phonon confinement, surface pressure, or interfacial vibration effects. Therefore, understanding the size effect of elasticity and thermodynamic properties and their theoretical mechanism is important.
Most previous theoretical studies were concerned with the material properties of ZrO2, CeO2 bulk and thin films at absolute zero temperature or low temperature while temperature and pressure dependences of thermodynamic quantities have not been studied in detail. Recently, temperature and pressure dependences of thermodynamic and elastic properties of bulk cerium dioxide have been studied using the analytic statistical moment method (SMM)    . The purpose of the present article is to investigate the temperature, pressure and thickness dependences of some thermodynamic properties of ZrO2 thin films using the SMM.
2. Theoretical Approach
Theoretical explanations for the size effect are made by introducing the surface energy contribution in the continuum mechanics or the computational simulations reflecting the surface stress, or the surface relaxation influence. In this present research, the influence of the size effect on thermodynamic properties of zirconia thin film is studied by introducing the surface energy contribution in the free energy of zirconium and oxygen atoms of surface layers.
Let us consider an oxide free standing thin film RO2 has n atomic layers of R atom (R = Zr) and (n − 1) atomic layer O with the thickness d as shown in Figure 1. We assume that the thin film consists of two zirconium surface-layers, two oxygen next surface-layers, (n − 3) oxygen internal-layers and (n − 2) zirconium internal-layers. are the number of R or O atoms on surface-layers, next-surface-layers and internal-layers of this thin film, respectively. The thin film RO2 has the cubic fluorite structure, then , , and . Then, the potential energy of the surface-layers with atoms R, and atoms O can be written as
here, ri is the equilibrium position of i-th atom, ui is its displacement of the i-th atom from the equilibrium position; are the effective interatomic potential between 0-th R atom and i-th atom, the 0-th O atom and i-th atom, respectively.
First, we expand the potential energy of the system in terms of the atomic (ionic) displacements ui of the atom i.
where denotes α-Cartesian component of the atomic displacement of i-th atom, and the subscript eq means the quantities calculated at the equilibrium state.
Figure 1. Free standing thin film RO2 with two R surface-layers.
Using Equation (2), the thermal average of the crystalline potential energy of the system is given in terms of the power moments of the atomic displacements and the harmonic vibrational parameter k, and three anharmonic parameters and as
with or z. and represent the sum of effective pair interaction energies for R and Oxygen atoms, respectively,
Let us consider a quantum system given by the following Hamiltonian:
where denotes the lattice Hamiltonian in the harmonic approximation, and the second term is added due to the anharmonicity of thermal lattice vibrations, denotes a parameter characterizing the anharmonicity of thermal lattice vibrations and the related operator. The Helmholtz free energy of the system given by Hamiltonian (9) is formally written as  
where expresses the expectation value at the thermal equilibrium with the (anharmonic) Hamiltonian .
Using Equations (3), (9) and (10) permits us to calculate the Helmholtz free energy of R atoms of the surface-layers as
The Helmholtz free energy for R atoms can be derived from the functional form of the potential energy of the above Equation (3) through the straightforward analytic integrations I1 and I2, with respect to the two anharmonicity “variables” and . Firstly, for R atoms I1 and I2 are written in an integral form as
Then the free energy of the atoms R of surface layers is given by
where denotes free energy in the harmonic approximation for the atoms R of surface layers which has the form as
Using the expression of the second and fourth moments   , we calculate the anharmonicity contribution to the free energy, then the free energy of the surface layer atoms R is given by
where , , and m is the average atomic mass of the system, .
Similar derivation can be also done for the free energy of R and O atoms of the surface-layers and internal-layers of thin film RO2. Free energies of these layer-types respectively are
The number of atoms of internal-layers, and surface-layers of thin film RO2 are respectively determined as , , and . Free energy of thin film RO2 and of one atom, respectively, are given by
where is the configurational entropy of the system; and are correspondingly the free energy of one R or O atom at surface-layers and internal-layers.
Using as the average nearest-neighbor distance (NND) and b is the average thickness of two-layers and is the average lattice constant. Then we have
The average nearest-neighbor distance of thin film at a given temperature T and pressure P can be determined as
The thickness d of thin film can be given by
From Equations (21) and (23), we derived
and the average lattice constant of thin film RO2 is
From Equation (26), one can find that the average lattice constant of the thin film depends on the thickness d of the thin film. When the number of crystalline layers n is large enough, the thickness of the thin film reaches a certain limit, the average lattice constant approaches the value of lattice constant of the bulk material.
In above equation, asur and aint are correspondingly the average NND between two intermediate atoms on the surface-layers, and internal-layers of thin film at a given temperature T and pressure P. These quantities can be determined as
where are the average displacements of R, and O atoms from the equilibrium position in the surface-layers or internal-layers of thin film RO2 at given temperature T and pressure P. The average displacements of R, and O atoms have the analytic forms as  .
Substituting Equation (23) into Equation(20) we obtained the expression of the free energy per atom as follows
In the above Equations (15), (16), (17), and (18), the harmonic contributions to the Helmholtz free energies are derived using the “Einstein” approximation. The results of Equations (15)-(18) permits us to find the free energies at temperature T under the condition that the parameters and Ksur at temperature T0 (for example T0 = 0 K) are known. If the temperature T0 is not far from T, then one can see that the vibration of a particle around a new equilibrium position (corresponding to T0) is harmonic. Therefore, Equations (15) and (18) can be taken only to the second term, i.e.
Since pressure P is determined by
from Equations (30), and (31), it is easy to take out an equation-of-states of a thin film system consists of one R surface-layer, and one oxygen next surface-layer at zero temperature T = 0 K and pressure P
Similar derivation can be also done for the equation-of-states of internal-layers of thin film RO2. The equation-of-states of these layer-types respectively are
In the above Equations (34) and (35), and are the vibration frequency of R (or O) atoms of surface-layers and internal-layers of thin film RO2 at zero temperature (T = 0 K) and pressure P.
With the aid of the free energy formula , we can find the thermodynamic quantities of the system. Using Equation (29), it is easy to obtain the specific heats at constant volume CV of thin film RO2
3. Results and Discussion
To calculate the thermodynamic quantities of ZrO2 thin film, we will use three different potentials, which include the electrostatic Coulomb interactions and two body terms to describe the short-range interactions. The two body terms arise from the electronic repulsion and attractive van der Waals forces, and they are described by a Buckingham potential form
where qi and qj are the charges of the i-th and the j-th ions, r is the distance between them and the parameters Aij, Bij and Cij are empirically determined by   (listed in Table 1).
Firstly, we calculate the lattice parameters of ZrO2 thin films at zero temperature and 293 K (room temperature) using the three different potentials (P1, P2 and L-C potentials). One can see in Figures 2-4 that the calculated lattice parameter increases with the increasing thickness. In the case of ZrO2 thin films with a thickness d increases to about 100 Å (or number of layers n increases to 20), the lattice parameters of ZrO2 thin films are similar to those of the bulk ZrO2. The full-potential linearized augmented-plane-wave (FLAPW) ab initio calculation of Jansen  , based on the density functional theory in the local-density approximation (LDA), give a0 = 5.03 Å, while Hartre-Fock calculations (the CRYSTAL code) give a0 = 5.035 Å (both at zero K). The density functional theory (DFT) within the plane-wave pseudopotential (PWP)  and RIP give a0 = 5.134 Å, and a0 = 5.162 Å. These results and the CRYSTAL calculation  are larger than the experimental values. Our SMM calculations give a lattice parameter a0 = 5.114 Å (using potential P1), a0 = 5.081 Å (using P2), a0 = 4.956 Å (using L-C potential) at zero temperature and are in best agreement with the experimental values a0 = 5.086 Å  and FLAPW-DFT, LMTO and Hartree-Fock calculations.
The variation of the lattice parameter of ZrO2 thin films as a function of pressure in the present work by SMM calculations using potentials 1, 2 and L-C potential (as presented in Figure 5). One can see in Figures 2-6 that the lattice parameters calculated by using potentials 1 and 2 are very similar. The small difference between the two calculations simply comes from the difference in zirconium-oxygen interaction potentials, since the ionic Coulomb contribution and the oxygen-oxygen potential are the same for potentials 1 and 2. The calculated lattice parameters of ZrO2 thin film by potentials 1 and 2 are almost identical, while the L-C potential gives somewhat smaller values, shifted upwards about 2% at wider temperature range of 300 K - 2900 K and under pressure range of 0 GPa - 50 GPa.
Temperature dependence of the lattice parameters of ZrO2 thin film with three potentials presented in Figure 6 and Figure 7. These Figures also shows the SMM calculations of lattice constants of ZrO2 thin film as an increasing function of temperature. Temperature dependence of the lattice parameters of ZrO2 thin film with different atomic layer numbers is similar for wider temperature range (see Figure 7). Our SMM theory predicts the lattice parameters of the cubic fluorite thin film ZrO2 to increase rapidly with temperature in agreement with those measured by experiments (for bulk ZrO2)    .
In Figure 8, we compare the SMM results of thermal expansion coefficient of ZrO2 thin films with different atomic layer number (n = 5, 10, 15, 20 and 100) using potential P1 at zero pressures with the experimental results (for bulk ZrO2)  . Temperature dependence of thermal expansion coefficient and specific heats at the constant volume of ZrO2 thin films with different atomic layer number are similar for wider temperature range of 300 K - 2900 K (as shown in Figure 8 and Figure 10). One can see in Figures 8-10 that when the thickness d of thin film increases to about 100 Å (or number of layers n increases to 20), thermal expansion coefficient and specific heats at the constant volume of ZrO2 thin films approach to the value of these thermodynamic quantities of the bulk material.
Table 1. The parameters of the Buckingham potential of ZrO2.
Figure 2. Thickness dependence of average lattice constant of thin film ZrO2 with Lewis-Catlow, P1, and P2 potentials at zero temperature and pressure P = 0 GPa.
Figure 3. Atomic layers number dependence of average lattice constant of thin film ZrO2 with Lewis-Catlow, and P1 potentials at zero pressure, and temperature T = 300 K, and 2200 K.
Figure 4. Thickness dependence of average lattice constant of thin film ZrO2 with Lewis-Catlow, P1, and P2 potentials at room temperature and pressure P = 10 GPa.
Figure 5. Pressure dependence of average lattice constant of ZrO2 thin film with atomic layer number n = 15 at room temperature using Lewis-Catlow, P1, an P2 potentials.
Figure 6. Temperature dependence of average lattice constant of ZrO2 thin film with atomic layer number n = 20 at zero pressure (P = 0 GPa) using P1, P2 and Lewis-Catlow potentials. The red empty squares, black squares, and blue dimonds are the results from the SMM calculations, the royal and magenta circles and olive stars are the experimental results of bulk ZrO2    .
Figure 7. Temperature dependence of average lattice constant of ZrO2 thin film with different atomic layer numbers (n = 5, 10, 20, and 100) at zero pressure (P = 0 GPa) using P1 potential.
Figure 8. Temperature dependence of thermal expansion coefficient of ZrO2 thin films with different atomic layer numbers (n = 5, 10, 15, 20 and 100) at zero pressure using potential P1. The navy stars are the experimental results of bulk ZrO2  .
Figure 9. Thickness dependence of specific heats at the constant volume of ZrO2 thin film at room temperature and zero pressure using P1, P2 and Lewis-Catlow potentials.
Figure 10. Temperature dependence of specific heats at the constant volume of ZrO2 thin films with different atomic layer numbers (n = 5, 10, 100, and 500) at pressure P = 10 GPa using potential P1.
In conclusion, it should be noted that the statistical moment method permits an investigation of the temperature, pressure and thickness dependences of ZrO2 thin films. The SMM calculations are performed by using Buckingham potential for ZrO2 thin films. Our development is establishing and solving equation of state to get the pressure dependence of the lattice constant, and then is the derivation of the analytical expressions of pressure dependence for some thermodynamic quantities of ZrO2 thin films as specific heats at the constant volume, as well as for the thermal expansion coefficient. The present formalism takes into account the higher-order anharmonic vibrational terms in the Helmholtz free energy of ZrO2 (or CeO2) thin films and it enables us to derive the various thermodynamic quantities in closed analytic forms. Using the free energy formulas derived in the statistical moment method scheme, we have studied the temperature and thickness dependences of thermodynamic quantities of ZrO2 thin films. In general, we have found that the lattice constant, and thermal expansion coefficient of ZrO2 thin films are decreasing functions of the pressure, and increasing functions of the temperature and thickness. We have calculated thermodynamic quantities for ZrO2 thin films with different thickness using potentials 1 and 2 and Lewis-Catlow potential at various pressures, and these SMM calculated thermodynamic quantities of ZrO2 thin films with enough large layer number (n increases to 20) are in good agreement with the experiments of bulk ZrO2.