Investigation of Thermodynamic Properties of Zirconia Thin Films by Statistical Moment Method

The moment method in statistical (SMM) dynamics is used to study the thermodynamic quantities of ZrO2 thin films taking into account the anharmonicity effects of the lattice vibrations. The average lattice constant, thermal expansion coefficient and specific heats at the constant volume of ZrO2 thin films are calculated as a function of temperature, pressure and thickness of thin film. SMM calculations are performed using the Buckingham potential for the ZrO2 thin films. In the present study, the influence of temperature, pressure and the size on the thermodynamic quantities of ZrO2 thin film have been studied using three different interatomic potentials. We discuss temperature and thickness dependences of some thermodynamic quantities of ZrO2 thin films and we compare our calculated results with those of the experimental results.


Introduction
Oxide thin films are used for multi-layer optical coatings, and multi-layer optical thin film devices.Among oxide materials, pure and doped CeO 2 and ZrO 2 (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 [1] [2].These oxide thin films have been prepared by some con-V.Van  have been used produce CeO 2 thin films.For instance, Gerblinger et al. [3] have obtained CeO 2 thin films on Al 2 O 3 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 [4] [5] [6].Chemical vapor deposition (CVD) is a well-known technique for preparing thin films.For example, yttria doped CeO 2 (YDC) thin films have been grown on yttria doped zirconia (YSZ) single substrates by vacuum vapor deposition and slurry painting method [7].Different several chemical methods have also been applied to produce CeO 2 (or ZrO 2 ) thin films, such as sol-gel, aerosol-assisted MOCVD [8], mist microwave-plasma chemical vapour deposition (MPCVD) [9], atomic layer deposition (ALD) [10] and spray pyrolysis [11].Recently, the structural and optical properties of ZrO 2 thin films in relation to thermal annealing times, and properties of surface ceria-zirconia solid solution films were investigated [12] [13].ZrO 2 thin films were grown by thermal oxidation of metallic zirconium films deposited by sputtering of zirconium target by DC magnetron sputtering technique [14].
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 ZrO 2 , CeO 2 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) [15] [16] [17].The purpose of the present article is to investigate the temperature, pressure and thickness dependences of some thermodynamic properties of ZrO 2 thin films using the SMM.

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 Let us consider an oxide free standing thin film RO 2 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 RO 2 has the cubic fluorite structure, then here, r i is the equilibrium position of i-th atom, u i 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 u i of the atom i.
( ) 24 where i u α denotes α-Cartesian component of the atomic displacement of i-th atom, and the subscript eq means the quantities calculated at the equilibrium state. where 48 and Let us consider a quantum system given by the following Hamiltonian: where 0 Ĥ denotes the lattice Hamiltonian in the harmonic approximation, and the second term ∑ is added due to the anharmonicity of thermal lattice vibrations, i α denotes a parameter characterizing the anharmonicity of thermal lattice vibrations and ˆL V the related operator.The Helmholtz free energy of the system given by Hamiltonian ( 9) is formally written as [18] [19] where ˆi L V α expresses the expectation value at the thermal equilibrium with the (anharmonic) Hamiltonian Ĥ .Using Equations ( 3), ( 9) and ( 10) permits us to calculate the Helmholtz free energy of R atoms of the surface-layers as V. Van The Helmholtz free energy sur R Ψ for R atoms can be derived from the functional form of the potential energy of the above Equation (3) through the straightforward analytic integrations I 1 and I 2 , with respect to the two anharmonicity "variables" 1 γ and 2 γ .Firstly, for R atoms I 1 and I 2 are written in an integral form as Then the free energy of the sur R N atoms R of surface layers is given by ( ) where 0 R sur − Ψ denotes free energy in the harmonic approximation for the sur R N atoms R of surface layers which has the form as ( ) Using the expression of the second and fourth moments [17] [20], we calculate the anharmonicity contribution to the free energy, then the free energy 0 R sur − Ψ of the surface layer atoms R is given by where 2 2 , 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 RO 2 .Free energies of these layer-types respectively are where The number of atoms of internal-layers, and surface-layers of thin film RO 2 are respectively determined as , and . Free energy of thin film RO 2 and of one atom, respectively, are given by ( ) ( ) where c S is the configurational entropy of the system; , , Ψ are correspondingly the free energy of one R or O atom at surface-layers and internal-layers.
Using a as the average nearest-neighbor distance (NND) and b is the aver- age thickness of two-layers and h a 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 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, a sur and a int 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 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 K sur at temperature T 0 (for example T 0 = 0 K) are known.If the temperature T 0 is not far from T, then one can see that the vibration of a particle around a new equilibrium position (corresponding to T 0 ) is harmonic.Therefore, Equations ( 15) and ( 18) can be taken only to the second term, i.e.
( ) ( ) where ( ) Since pressure P is determined by , 3 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 RO 2 .The equation-of-states of these layer-types respectively are ( ) ( ) V. Van  are the vibration frequency of R (or O) atoms of surface-layers and internal-layers of thin film RO 2 at zero temperature (T = 0 K) and pressure P.
With the aid of the free energy formula E TS Ψ = − , we can find the ther- modynamic quantities of the system.Using Equation ( 29), it is easy to obtain the specific heats at constant volume C V of thin film RO 2 ; .

Results and Discussion
To calculate the thermodynamic quantities of ZrO 2 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 q i and q j are the charges of the i-th and the j-th ions, r is the distance between them and the parameters A ij , B ij and C ij are empirically determined by [21] [22] (listed in Table 1).
Firstly, we calculate the lattice parameters of ZrO 2 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 ZrO 2 thin films with a thickness d increases to about 100 Å (or number of layers n increases to 20), the lattice parameters of ZrO 2 thin films are similar to those of the bulk ZrO 2 .The full-potential linearized augmented-plane-wave (FLAPW) ab initio calculation of Jansen [23], based on the density functional theory in the local-density approximation (LDA), give a 0 = 5.03 Å, while Hartre-Fock calculations (the CRYSTAL code) give a 0 = 5.035 Å (both at zero K).The density functional theory (DFT) within the plane-wave pseudopotential (PWP) [24] and RIP give a 0 = 5.134 Å, and a 0 = 5.162 Å.These results and the CRYSTAL calculation [25] are larger than the experimental values.Our SMM calculations give a lattice parameter a 0 = 5.114 Å (using potential P1), a 0 = 5.081 Å (using P2), a 0 = 4.956 Å (using L-C potential) at zero temperature and are in best agreement with the experimental values a 0 = 5.086 Å [26] and FLAPW-DFT, LMTO and Hartree-Fock calculations.
The variation of the lattice parameter of ZrO 2 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 In Figure 8, we compare the SMM results of thermal expansion coefficient of ZrO 2 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 ZrO 2 ) [29].Temperature dependence of thermal expansion coefficient and specific heats at the constant volume of ZrO 2 thin films with different atomic layer number are similar for wider temperature range of 300 K -2900 K (as shown in Figure 8 and

Conclusion
In conclusion, it should be noted that the statistical moment method permits an investigation of the temperature, pressure and thickness dependences of ZrO 2 thin films.The SMM calculations are performed by using Buckingham potential for ZrO 2 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 ZrO 2 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 ZrO 2 (or CeO 2 ) 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 ZrO 2 thin films.
In general, we have found that the lattice constant, and thermal expansion coefficient of ZrO 2 thin films are decreasing functions of the pressure, and increasing functions of the temperature and thickness.We have calculated thermodynamic quantities for ZrO 2 thin films with different thickness using potentials 1 and 2 and Lewis-Catlow potential at various pressures, and these SMM calculated thermodynamic quantities of ZrO 2 thin films with enough large layer number (n increases to 20) are in good agreement with the experiments of bulk ZrO 2 .
of effective pair interaction energies for R and Oxygen atoms, respectively, ( )

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 ZrO 2 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 ZrO 2 thin film with three potentials presented in Figure6and Figure7.These Figures also shows the SMM calculations of lattice constants of ZrO 2 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 Figure7).Our SMM theory predicts the lattice parameters of the cubic fluorite thin film ZrO 2 to increase rapidly with temperature in agreement with those measured by experiments (for bulk ZrO 2 )[26] [27][28].

Figure 10 )
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 ZrO 2 thin films approach to the value of these thermodynamic quantities of the bulk material.

Figure 2 .
Figure 2. Thickness dependence of average lattice constant of thin film ZrO 2 with Lewis-Catlow, P1, and P2 potentials at zero temperature and pressure P = 0 GPa.

Figure 3 .
Figure 3. Atomic layers number dependence of average lattice constant of thin film ZrO 2 with Lewis-Catlow, and P1 potentials at zero pressure, and temperature T = 300 K, and 2200 K.

Figure 4 .
Figure 4. Thickness dependence of average lattice constant of thin film ZrO 2 with Lewis-Catlow, P1, and P2 potentials at room temperature and pressure P = 10 GPa.

Figure 5 .
Figure 5. Pressure dependence of average lattice constant of ZrO 2 thin film with atomic layer number n = 15 at room temperature using Lewis-Catlow, P1, an P2 potentials.

Figure 6 .
Figure 6.Temperature dependence of average lattice constant of ZrO 2 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 ZrO 2 [26] [27] [28].

Figure 9 .
Figure 9. Thickness dependence of specific heats at the constant volume of ZrO 2 thin film at room temperature and zero pressure using P1, P2 and Lewis-Catlow potentials.

Figure 10 .
Figure 10.Temperature dependence of specific heats at the constant volume of ZrO 2 thin films with different atomic layer numbers (n = 5, 10, 100, and 500) at pressure P = 10 GPa using potential P1.
Hung et al.

Table 1 .
The parameters of the Buckingham potential of ZrO 2 .