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

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_{2}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 ZrO

_{2}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 ZrO

_{2}thin films. In the present study, the influence of temperature, pressure and the size on the thermodynamic quantities of ZrO

_{2}thin film have been studied using three different interatomic potentials. We discuss temperature and thickness dependences of some thermodynamic quantities of ZrO

_{2}thin films and we compare our calculated results with those of the experimental results.

KEYWORDS

1. 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 conventional methods. Many physical deposition techniques, such as pulsed laser deposition (PLD), magnetron sputtering, or molecular beam epitaxy (MBE), 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.

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 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.
${N}_{R}^{sur},{N}_{O}^{sur},{N}_{R}^{int},{N}_{O}^{int}$ 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
${N}_{R}^{sur}={N}_{R}^{int}={N}_{01}$ ,
${N}_{O}^{sur}={N}_{O}^{int}={N}_{02}$ , and
${N}_{01}/{N}_{02}=1/2$ . Then, the potential energy of the surface-layers with
${N}_{R}^{sur}$ atoms R, and
${N}_{O}^{sur}$ atoms O can be written as

${U}^{sur}={U}_{R}^{sur}+{U}_{O}^{sur}=\frac{{N}_{R}^{sur}}{2}{\displaystyle {\sum}_{i}{\phi}_{i0}^{R-sur}}\left(\left|{r}_{i}+{u}_{i}\right|\right)+\frac{{N}_{O}^{sur}}{2}{\displaystyle {\sum}_{i}{\phi}_{i0}^{O-sur}}\left(\left|{r}_{i}+{u}_{i}\right|\right),$ (1)

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;
${\phi}_{i0}^{R-sur},{\phi}_{i0}^{O-sur}$ 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.

$\begin{array}{l}{U}^{sur}={U}_{R}^{sur}+{U}_{O}^{sur}=\frac{{N}_{R}^{sur}}{2}{\displaystyle \underset{i}{\sum}\{{\phi}_{i0}^{R-sur}\left(\left|{r}_{i}\right|\right)+\frac{1}{2}{\displaystyle \underset{\alpha ,\beta}{\sum}{\left(\frac{{\partial}^{2}{\phi}_{i0}^{R-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}}\\ +\frac{1}{6}{\displaystyle \underset{\alpha ,\beta ,\gamma}{\sum}{\left(\frac{{\partial}^{3}{\phi}_{i0}^{R-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}\partial {u}_{i\gamma}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}{u}_{i\gamma}+\frac{1}{24}{\displaystyle \underset{\alpha ,\beta ,\gamma ,\eta}{\sum}{\left(\frac{{\partial}^{4}{\phi}_{i0}^{R-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}\partial {u}_{i\gamma}\partial {u}_{i\eta}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}{u}_{i\gamma}{u}_{i\eta}+\cdots \}\\ +\frac{{N}_{O}^{sur}}{2}{\displaystyle \underset{i}{\sum}\{{\phi}_{i0}^{O-sur}\left(\left|{r}_{i}\right|\right)+\frac{1}{2}{\displaystyle \underset{\alpha ,\beta}{\sum}{\left(\frac{{\partial}^{2}{\phi}_{i0}^{O-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}}+\frac{1}{6}{\displaystyle \underset{\alpha ,\beta ,\gamma}{\sum}{\left(\frac{{\partial}^{3}{\phi}_{i0}^{O-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}\partial {u}_{i\gamma}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}{u}_{i\gamma}\\ +\frac{1}{24}{\displaystyle \underset{\alpha ,\beta ,\gamma ,\eta}{\sum}{\left(\frac{{\partial}^{4}{\phi}_{i0}^{O-sur}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}\partial {u}_{i\gamma}\partial {u}_{i\eta}}\right)}_{eq}{u}_{i\alpha}{u}_{i\beta}}{u}_{i\gamma}{u}_{i\eta}+\cdots \},\end{array}$ (2)

where ${u}_{i\alpha}$ 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 RO_{2} 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 $\langle {u}^{m}\rangle $ of the atomic displacements and the harmonic vibrational parameter k, and three anharmonic parameters $\beta ,{\gamma}_{1}$ and ${\gamma}_{2}$ as

$\begin{array}{c}\langle {U}^{sur}\rangle ={U}_{0}^{R-sur}+{U}_{0}^{O-sur}+3{N}_{R}^{sur}[\frac{{k}_{R}^{sur}}{2}\langle {\left({u}_{R}^{sur}\right)}^{2}\rangle +{\gamma}_{1}^{R-sur}\langle {\left({u}_{R}^{sur}\right)}^{4}\rangle \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma}_{2}^{R-sur}{\langle {\left({u}_{R}^{sur}\right)}^{2}\rangle}^{2}]+3{N}_{O}^{sur}[\frac{{k}_{O}^{sur}}{2}\langle {\left({u}_{O}^{sur}\right)}^{2}\rangle +{\beta}_{O}^{sur}\langle {u}_{O}^{sur}\rangle \langle {\left({u}_{O}^{sur}\right)}^{2}\rangle \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma}_{1}^{O-sur}\langle {\left({u}_{O}^{sur}\right)}^{4}\rangle +{\gamma}_{2}^{O-sur}{\langle {\left({u}_{O}^{sur}\right)}^{2}\rangle}^{2}]+\cdots ,\end{array}$ (3)

where

${\gamma}_{1}^{R-sur}=\frac{1}{48}{{\displaystyle {\sum}_{i}\left(\frac{{\partial}^{4}{\phi}_{i0}^{R}}{\partial {u}_{i\alpha}^{4}}\right)}}_{eq},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\gamma}_{2}^{R-sur}=\frac{6}{48}{{\displaystyle {\sum}_{i}\left(\frac{{\partial}^{4}{\phi}_{i0}^{R}}{\partial {u}_{i\alpha}^{2}\partial {u}_{i\beta}^{2}}\right)}}_{eq},$ (4)

${\gamma}_{1}^{O-sur}=\frac{1}{48}{{\displaystyle {\sum}_{i}\left(\frac{{\partial}^{4}{\phi}_{i0}^{O}}{\partial {u}_{i\alpha}^{4}}\right)}}_{eq},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\gamma}_{2}^{O-sur}=\frac{6}{48}{{\displaystyle {\sum}_{i}\left(\frac{{\partial}^{4}{\phi}_{i0}^{O}}{\partial {u}_{i\alpha}^{2}\partial {u}_{i\beta}^{2}}\right)}}_{eq},$ (5)

${k}_{R}^{sur}=\frac{1}{2}{\displaystyle {\sum}_{i}{\left(\frac{{\partial}^{2}{\phi}_{i0}^{R}}{\partial {u}_{i\alpha}^{2}}\right)}_{eq},}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{O}^{sur}=\frac{1}{2}{\displaystyle {\sum}_{i}{\left(\frac{{\partial}^{2}{\phi}_{i0}^{O}}{\partial {u}_{i\alpha}^{2}}\right)}_{eq},}$ (6)

and ${\beta}_{O}^{sur}=\frac{1}{2}{\displaystyle {\sum}_{i}{\left(\frac{{\partial}^{3}{\phi}_{i0}^{O}}{\partial {u}_{i\alpha}\partial {u}_{i\beta}\partial {u}_{i\gamma}}\right)}_{eq},}$ (7)

with $\alpha \ne \beta \ne \gamma =x,y$ or z. ${U}_{0}^{R-sur}$ and ${U}_{0}^{O-sur}$ represent the sum of effective pair interaction energies for R and Oxygen atoms, respectively,

${U}_{0}^{R-sur}=\frac{{N}_{R}^{sur}}{2}{\displaystyle {\sum}_{i}{\phi}_{i0}^{R-sur}\left(\left|{r}_{i}\right|\right),}$

${U}_{0}^{O-sur}=\frac{{N}_{O}^{sur}}{2}{\displaystyle {\sum}_{i}{\phi}_{i0}^{O-sur}\left(\left|{r}_{i}\right|\right),}$ (8)

Let us consider a quantum system given by the following Hamiltonian:

$\stackrel{^}{H}={\stackrel{^}{H}}_{0}-{\displaystyle {\sum}_{i}{\alpha}_{i}{\stackrel{^}{V}}_{L}},$ (9)

where ${\stackrel{^}{H}}_{0}$ denotes the lattice Hamiltonian in the harmonic approximation, and the second term $\sum}_{i}{\alpha}_{i}{\stackrel{^}{V}}_{L$ is added due to the anharmonicity of thermal lattice vibrations, ${\alpha}_{i}$ denotes a parameter characterizing the anharmonicity of thermal lattice vibrations and ${\stackrel{^}{V}}_{L}$ the related operator. The Helmholtz free energy of the system given by Hamiltonian (9) is formally written as [18] [19]

$\Psi ={\Psi}_{0}-{\displaystyle {\sum}_{i}{\displaystyle \underset{0}{\overset{{\alpha}_{i}}{\int}}{\langle {\stackrel{^}{V}}_{L}\rangle}_{{\alpha}_{i}}\text{d}{\alpha}_{i},}}$ (10)

where ${\langle {\stackrel{^}{V}}_{L}\rangle}_{{\alpha}_{i}}$ expresses the expectation value at the thermal equilibrium with the (anharmonic) Hamiltonian $\stackrel{^}{H}$ .

Using Equations (3), (9) and (10) permits us to calculate the Helmholtz free energy of R atoms of the surface-layers as

${\Psi}_{R}^{sur}={U}_{0}^{R-sur}+{\Psi}_{0}^{R-sur}+{\displaystyle \underset{0}{\overset{{\alpha}_{i}}{\int}}{\langle {V}_{L}\rangle}_{{\alpha}_{i}}\text{d}{\alpha}_{i}.}$ (11)

The Helmholtz free energy
${\Psi}_{R}^{sur}$ 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”
${\gamma}_{1}$ and
${\gamma}_{2}$ . Firstly, for R atoms I_{1} and I_{2} are written in an integral form as

${I}_{1}={\displaystyle \underset{0}{\overset{{\gamma}_{1}}{\int}}\langle {u}^{4}\rangle \text{d}{\gamma}_{1},}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{2}={\displaystyle \underset{0}{\overset{{\gamma}_{2}}{\int}}{\langle {u}^{2}\rangle}_{{\gamma}_{1}=0}^{2}\text{d}{\gamma}_{2}.}$ (12)

Then the free energy of the ${N}_{R}^{sur}$ atoms R of surface layers is given by

$\begin{array}{c}{\Psi}_{R}^{sur}={U}_{0}^{R-sur}+{\Psi}_{0}^{R-sur}+3{N}_{R}^{sur}{\displaystyle \underset{0}{\overset{{\gamma}_{1}^{R-sur}}{\int}}\langle {\left({u}_{R}^{sur}\right)}^{4}\rangle \text{d}{\gamma}_{1}^{R-sur}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3{N}_{R}^{sur}{\displaystyle \underset{0}{\overset{{\gamma}_{2}^{R-sur}}{\int}}{\langle {\left({u}_{R}^{sur}\right)}^{2}\rangle}_{{\gamma}_{1}^{R-sur}=0}^{2}\text{d}{\gamma}_{2}^{R-sur}},\end{array}$ (13)

where ${\Psi}_{0}^{R-sur}$ denotes free energy in the harmonic approximation for the ${N}_{R}^{sur}$ atoms R of surface layers which has the form as

${\Psi}_{0}^{R-sur}=3{N}_{R}^{sur}\theta \left[{x}_{R}^{sur}+\mathrm{ln}\left(1-{\text{e}}^{-2{x}_{R}^{sur}}\right)\right].$ (14)

Using the expression of the second and fourth moments [17] [20] , we calculate the anharmonicity contribution to the free energy, then the free energy ${\Psi}_{0}^{R-sur}$ of the surface layer atoms R is given by

$\begin{array}{c}{\Psi}_{R}^{sur}\approx \left\{{U}_{0}^{R-sur}+{\Psi}_{0}^{R-sur}\right\}+\frac{3{N}_{R}^{sur}{\theta}^{2}}{{\left({k}_{R}^{sur}\right)}^{2}}\left\{{\gamma}_{2}^{R-sur}{\left({X}_{R}^{sur}\right)}^{2}-\frac{2{\gamma}_{1}^{R-sur}}{3}\left(1+\frac{{X}_{R}^{sur}}{2}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{6{N}_{R}^{sur}{\theta}^{3}}{{\left({k}_{R}^{sur}\right)}^{4}}\{\frac{4}{3}{\left({\gamma}_{2}^{R-sur}\right)}^{2}\left(1+\frac{{X}_{R}^{sur}}{2}\right){X}_{R}^{sur}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left({\left({\gamma}_{1}^{R-sur}\right)}^{2}+2{\gamma}_{1}^{R-sur}{\gamma}_{2}^{R-sur}\right)\left(1+\frac{{X}_{R}^{sur}}{2}\right)\left(1+{X}_{R}^{sur}\right)\},\end{array}$ (15)

where ${x}_{R}^{sur}=\frac{\hslash {\omega}_{R}^{sur}}{2\theta}=\hslash \frac{\sqrt{{k}_{R}^{sur}/m}}{2\theta}$ , ${X}_{R}^{sur}={x}_{R}^{sur}\mathrm{coth}{x}_{R}^{sur}$ , and m is the average atomic mass of the system, $m={C}_{R}{m}_{R}+{C}_{O}{m}_{O}$ .

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

$\begin{array}{c}{\Psi}_{O}^{sur}\approx \left\{{U}_{0}^{O-sur}+{\Psi}_{0}^{O-sur}\right\}+\frac{3{N}_{O}^{sur}{\theta}^{2}}{{\left({k}_{O}^{sur}\right)}^{2}}\left\{{\gamma}_{2}^{O-sur}{\left({X}_{O}^{sur}\right)}^{2}-\frac{2{\gamma}_{1}^{O-sur}}{3}\left(1+\frac{{X}_{O}^{sur}}{2}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{6{N}_{O}^{sur}{\theta}^{3}}{{\left({k}_{O}^{sur}\right)}^{4}}\{\frac{4}{3}{\left({\gamma}_{2}^{O-sur}\right)}^{2}\left(1+\frac{{X}_{O}^{sur}}{2}\right){X}_{O}^{sur}-2({\left({\gamma}_{1}^{O-sur}\right)}^{2}\end{array}$ (16)

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{\gamma}_{1}^{O-sur}\cdot {\gamma}_{2}^{O-sur})\left(1+\frac{{X}_{O}^{sur}}{2}\right)\left(1+{X}_{O}^{sur}\right)\}+3{N}_{O}^{sur}\theta [\frac{{\left({\beta}_{O}^{sur}\right)}^{2}}{6{K}_{sur}^{2}}\frac{{k}_{O}^{sur}}{{\gamma}_{0}^{O-sur}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\left({\beta}_{O}^{sur}\right)}^{2}}{6{K}_{sur}{\gamma}_{0}^{O-sur}}]+3{N}_{O}^{sur}{\theta}^{2}\left\{\frac{{\beta}_{O}^{sur}}{{K}_{sur}}\left[\frac{2{\gamma}_{0}^{O-sur}}{3{K}_{sur}^{3}}{\left(1+\frac{{X}_{O}^{sur}}{2}\right)}^{1/2}\right]\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3{N}_{O}^{sur}{\theta}^{3}\left\{-\frac{{\left({\beta}_{O}^{sur}\right)}^{2}}{9{K}_{sur}^{3}}\left(1+\frac{{X}_{O}^{sur}}{2}\right)+\frac{{\left({\beta}_{O}^{sur}\right)}^{2}}{9{K}_{sur}^{4}}\left(1+{X}_{O}^{sur}\right)+\frac{{\left({\beta}_{O}^{sur}\right)}^{2}}{6{K}_{sur}^{2}{k}_{O}^{sur}}\left({X}_{O}^{sur}-1\right)\right\},\end{array}$

$\begin{array}{c}{\Psi}_{R}^{int}\approx \left\{{U}_{0}^{R-int}+{\Psi}_{0}^{R-int}\right\}+\frac{3{N}_{R}^{int}{\theta}^{2}}{{\left({k}_{R}^{int}\right)}^{2}}\left\{{\gamma}_{2}^{R-int}{\left({X}_{R}^{int}\right)}^{2}-\frac{2{\gamma}_{1}^{R-int}}{3}\left(1+\frac{{X}_{R}^{int}}{2}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{6{N}_{R}^{int}{\theta}^{3}}{{\left({k}_{R}^{int}\right)}^{4}}\{\frac{4}{3}{\left({\gamma}_{2}^{R-int}\right)}^{2}\left(1+\frac{{X}_{R}^{int}}{2}\right){X}_{R}^{int}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left({\left({\gamma}_{1}^{R-int}\right)}^{2}+2{\gamma}_{1}^{R-int}\cdot {\gamma}_{2}^{R-int}\right)\left(1+\frac{{X}_{R}^{int}}{2}\right)\left(1+{X}_{R}^{int}\right)\},\end{array}$ (17)

$\begin{array}{c}{\Psi}_{O}^{int}\approx \left\{{U}_{0}^{O-int}+{\Psi}_{0}^{O-int}\right\}+\frac{3{N}_{O}^{int}{\theta}^{2}}{{\left({k}_{O}^{int}\right)}^{2}}\left\{{\gamma}_{2}^{O-int}{\left({X}_{O}^{int}\right)}^{2}-\frac{2{\gamma}_{1}^{O-int}}{3}\left(1+\frac{{X}_{O}^{int}}{2}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{6{N}_{O}^{int}{\theta}^{3}}{{\left({k}_{O}^{int}\right)}^{4}}\{\frac{4}{3}{\left({\gamma}_{2}^{O-int}\right)}^{2}\left(1+\frac{{X}_{O}^{int}}{2}\right){X}_{O}^{int}-2({\left({\gamma}_{1}^{O-int}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{\gamma}_{1}^{O-int}{\gamma}_{2}^{O-int})\left(1+\frac{{X}_{O}^{int}}{2}\right)\left(1+{X}_{O}^{int}\right)\}+3{N}_{O}^{int}\theta [\frac{{\left({\beta}_{O}^{int}\right)}^{2}}{6{K}_{int}^{2}}\frac{{k}_{O}^{int}}{{\gamma}_{0}^{O-int}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\left({\beta}_{O}^{int}\right)}^{2}}{6{K}_{int}{\gamma}_{0}^{O-int}}]+3{N}_{O}^{int}{\theta}^{2}\left\{\frac{{\beta}_{O}^{int}}{{K}_{int}}\left[\frac{2{\gamma}_{0}^{O-int}}{3{K}_{int}^{3}}{\left(1+\frac{{X}_{O}^{int}}{2}\right)}^{1/2}\right]\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3{N}_{O}^{int}{\theta}^{3}\left\{-\frac{{\left({\beta}_{O}^{int}\right)}^{2}}{9{K}_{int}^{3}}\left(1+\frac{{X}_{O}^{int}}{2}\right)+\frac{{\left({\beta}_{O}^{int}\right)}^{2}}{9{K}_{int}^{4}}\left(1+{X}_{O}^{int}\right)+\frac{{\left({\beta}_{O}^{int}\right)}^{2}}{6{K}_{int}^{2}{k}_{O}^{int}}\left({X}_{O}^{int}-1\right)\right\},\end{array}$ (18)

where

${x}_{O}^{sur}=\frac{\hslash {\omega}_{O}^{sur}}{2\theta}=\hslash \frac{\sqrt{{k}_{O}^{sur}/m}}{2\theta},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{X}_{O}^{sur}={x}_{O}^{sur}\mathrm{coth}{x}_{O}^{sur},$

${x}_{R}^{int}=\frac{\hslash {\omega}_{R}^{int}}{2\theta}=\hslash \frac{\sqrt{{k}_{R}^{int}/m}}{2\theta},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{X}_{R}^{int}={x}_{R}^{int}\mathrm{coth}{x}_{R}^{int},$

${x}_{O}^{int}=\frac{\hslash {\omega}_{O}^{int}}{2\theta}=\hslash \frac{\sqrt{{k}_{O}^{int}/m}}{2\theta},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{X}_{O}^{int}={x}_{O}^{int}\mathrm{coth}{x}_{O}^{int},$

The number of atoms of internal-layers, and surface-layers of thin film RO_{2} are respectively determined as
${N}_{R}^{sur}={N}_{R}^{int}={N}_{01}$ ,
${N}_{O}^{sur}={N}_{O}^{int}={N}_{02}$ , and
${N}_{01}/{N}_{02}=1/2$ . Free energy of thin film RO_{2} and of one atom, respectively, are given by

$\Psi =2{N}_{01}{\Psi}_{R}^{sur}+4{N}_{01}{\Psi}_{O}^{sur}+2\left(n-3\right){N}_{01}{\Psi}_{O}^{int}+\left(n-2\right){N}_{01}{\Psi}_{R}^{int}-T{S}_{c},$ (19)

$\frac{\Psi}{N}=\frac{2}{3n-2}{\Psi}_{R}^{sur}+\frac{4}{3n-2}{\Psi}_{O}^{sur}+\frac{2\left(n-3\right)}{3n-2}{\Psi}_{O}^{\mathrm{int}}+\frac{n-2}{3n-2}{\Psi}_{R}^{\mathrm{int}}-\frac{T{S}_{c}}{N},$ (20)

where ${S}_{c}$ is the configurational entropy of the system; ${\Psi}_{R}^{sur},{\Psi}_{O}^{sur},{\Psi}_{R}^{int}$ and ${\Psi}_{O}^{int}$ are correspondingly the free energy of one R or O atom at surface-layers and internal-layers.

Using $\stackrel{\xaf}{a}$ as the average nearest-neighbor distance (NND) and b is the average thickness of two-layers and ${\stackrel{\xaf}{a}}_{h}$ is the average lattice constant. Then we have

$\stackrel{\xaf}{b}=\frac{2}{\sqrt{3}}\stackrel{\xaf}{a};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{\xaf}{a}}_{h}=2\stackrel{\xaf}{b}=\frac{4}{\sqrt{3}}\stackrel{\xaf}{a}.$ (21)

The average nearest-neighbor distance of thin film at a given temperature T and pressure P can be determined as

$\stackrel{\xaf}{a}\left(P,T\right)=\frac{2{a}_{sur}\left(P,T\right)+\left(n-3\right){a}_{int}\left(P,T\right)}{n-1}.$ (22)

The thickness d of thin film can be given by

$d=\left(n-1\right)\stackrel{\xaf}{b}=\frac{2\left(n-1\right)}{\sqrt{3}}\stackrel{\xaf}{a},$ (23)

or $d=\frac{2}{\sqrt{3}}\left[2{a}_{sur}\left(P,T\right)+\left(n-3\right){a}_{int}\left(P,T\right)\right].$ (24)

From Equations (21) and (23), we derived

$n=1+\frac{\sqrt{3}}{2\stackrel{\xaf}{a}}d,$ (25)

and the average lattice constant of thin film RO_{2} is

${\stackrel{\xaf}{a}}_{h}\left(P,T\right)=\frac{4}{\sqrt{3}}.\frac{2{a}_{sur}\left(P,T\right)+\left(n-3\right){a}_{int}\left(P,T\right)}{n-1}.$ (26)

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

${a}_{sur}\left(P,T\right)={a}_{sur}\left(P,0\right)+{C}_{R}{y}_{0}^{R-sur}\left(P,T\right)+{C}_{O}{y}_{0}^{O-sur}\left(P,T\right),$ (27)

${a}_{int}\left(P,T\right)={a}_{int}\left(P,0\right)+{C}_{R}{y}_{0}^{R-int}\left(P,T\right)+{C}_{O}{y}_{0}^{O-int}\left(P,T\right),$ (28)

where
${y}_{0}^{R-sur}\left(P,T\right),{y}_{0}^{O-sur}\left(P,T\right),{y}_{0}^{R-int}\left(P,T\right),{y}_{0}^{O-int}\left(P,T\right)$ are the average displacements of R, and O atoms from the equilibrium position in the surface-layers or internal-layers of thin film RO_{2} at given temperature T and pressure P. The average displacements of R, and O atoms
${y}_{0}^{R-sur}\left(P,T\right),{y}_{0}^{O-sur}\left(P,T\right),{y}_{0}^{R-int}\left(P,T\right),{y}_{0}^{O-int}\left(P,T\right)$ have the analytic forms as [19] .

Substituting Equation (23) into Equation(20) we obtained the expression of the free energy per atom as follows

$\begin{array}{c}\frac{\Psi}{N}=\frac{2}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{\Psi}_{R}^{sur}+\frac{4}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{\Psi}_{O}^{sur}+\frac{\frac{\sqrt{3}}{2\stackrel{\xaf}{a}}d-1}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{\Psi}_{R}^{int}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2\left(\frac{\sqrt{3}}{2\stackrel{\xaf}{a}}d-2\right)}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{\Psi}_{O}^{int}-\frac{T{S}_{c}}{N}.\end{array}$ (29)

In the above Equations (15), (16), (17), and (18), the harmonic contributions to the Helmholtz free energies
${\Psi}_{0}^{R-sur},{\Psi}_{0}^{O-sur}$ are derived using the “Einstein” approximation. The results of Equations (15)-(18) permits us to find the free energies
${\Psi}_{R}^{sur},{\Psi}_{O}^{sur}$ at temperature T under the condition that the parameters
${k}_{R,O}^{sur},{\gamma}_{1}^{R,O-sur},{\gamma}_{2}^{R,O-sur},{\gamma}^{R,O-sur},{\beta}_{O}^{sur}$ 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.

${\Psi}_{R}^{sur}={U}_{0}^{R-sur}+{\Psi}_{0}^{R-sur}=3{N}_{R}^{sur}\left\{\frac{1}{6}{u}_{0}^{R-sur}+\theta \left[{x}_{R}^{sur}+\mathrm{ln}\left(1-{\text{e}}^{-2{x}_{R}^{sur}}\right)\right]\right\},$ (30)

${\Psi}_{O}^{sur}={U}_{0}^{O-sur}+{\Psi}_{0}^{O-sur}=3{N}_{O}^{sur}\left\{\frac{1}{6}{u}_{0}^{O-sur}+\theta \left[{x}_{O}^{sur}+\mathrm{ln}\left(1-{\text{e}}^{-2{x}_{O}^{sur}}\right)\right]\right\},$ (31)

where

${u}_{0}^{R-sur}={\displaystyle {\sum}_{i}{\phi}_{i0}^{R-sur}\left(\left|{r}_{i}\right|\right)}$

${u}_{0}^{O-sur}={\displaystyle {\sum}_{i}{\phi}_{i0}^{O-sur}\left(\left|{r}_{i}\right|\right).}$ (32)

Since pressure P is determined by

$P=-{\left(\frac{\partial \Psi}{\partial V}\right)}_{T}=-\frac{a}{3V}{\left(\frac{\partial \Psi}{\partial a}\right)}_{T},$ (33)

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

$Pv=-{a}_{sur}\left\{{C}_{R}\left[\frac{1}{6}\frac{\partial {u}_{0}^{R-sur}}{\partial {a}_{sur}}+\frac{{\omega}_{R}^{sur}\left(0\right)}{4{k}_{R}^{sur}}\frac{\partial {k}_{R}^{sur}}{\partial {a}_{sur}}\right]+{C}_{O}\left[\frac{1}{6}\frac{\partial {u}_{0}^{O-sur}}{\partial {a}_{sur}}+\frac{{\omega}_{O}^{sur}\left(0\right)}{4{k}_{O}^{sur}}\frac{\partial {k}_{O}^{sur}}{\partial {a}_{sur}}\right]\right\}.$ (34)

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

$Pv=-{a}_{int}\left\{{C}_{R}\left[\frac{1}{6}\frac{\partial {u}_{0}^{R-int}}{\partial {a}_{int}}+\frac{{\omega}_{R}^{int}\left(0\right)}{4{k}_{R}^{int}}\frac{\partial {k}_{R}^{int}}{\partial {a}_{int}}\right]+{C}_{O}\left[\frac{1}{6}\frac{\partial {u}_{0}^{O-int}}{\partial {a}_{int}}+\frac{{\omega}_{O}^{int}\left(0\right)}{4{k}_{O}^{int}}\frac{\partial {k}_{O}^{int}}{\partial {a}_{int}}\right]\right\}.$ (35)

In the above Equations (34) and (35),
${\omega}_{R}^{sur}\left(P,0\right),{\omega}_{O}^{sur}\left(P,0\right),{\omega}_{R}^{int}\left(P,0\right)$ and
${\omega}_{O}^{int}\left(P,0\right)$ 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
$\Psi =E-TS$ , we can find the thermodynamic 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}

$\begin{array}{c}{C}_{V}={\left[\frac{\partial E}{\partial T}\right]}_{V}=-T\frac{{\partial}^{2}\Psi}{\partial {T}^{2}}\\ =\frac{2}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{C}_{V}^{R-sur}+\frac{4}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{C}_{V}^{O-sur}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\frac{\sqrt{3}}{2\stackrel{\xaf}{a}}d-1}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{C}_{V}^{R-int}+\frac{2\left(\frac{\sqrt{3}}{2\stackrel{\xaf}{a}}d-2\right)}{1+\frac{3\sqrt{3}}{2\stackrel{\xaf}{a}}d}{C}_{V}^{O-int},\end{array}$ (36)

where

$\begin{array}{l}{C}_{V}^{R-sur}=-T\frac{{\partial}^{2}{\Psi}_{R}^{sur}}{\partial {T}^{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{V}^{O-sur}=-T\frac{{\partial}^{2}{\Psi}_{O}^{sur}}{\partial {T}^{2}};\\ {C}_{V}^{R-int}=-T\frac{{\partial}^{2}{\Psi}_{R}^{int}}{\partial {T}^{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{V}^{O-int}=-T\frac{{\partial}^{2}{\Psi}_{O}^{int}}{\partial {T}^{2}}.\end{array}$ (37)

3. 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

${\phi}_{ij}=\frac{{q}_{i}{q}_{j}}{r}+{A}_{ij}\mathrm{exp}\left(-\frac{r}{{B}_{ij}}\right)-\frac{{C}_{ij}}{{r}^{6}},$ (38)

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 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 Figure 6 and Figure 7. 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 Figure 7). 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] .

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 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.

Table 1. The parameters of the Buckingham potential of ZrO_{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. 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. 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. 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. 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 7. Temperature dependence of average lattice constant of ZrO_{2} 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 ZrO_{2} 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 ZrO_{2} [29] .

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. 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.

4. 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}.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Materials Sciences and Applications*,

**9**, 949-964. doi: 10.4236/msa.2018.912068.

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