The moment method in statistical (SMM) dynamics is used to study the thermodynamic quantities of ZrO 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.
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 [
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) [
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
U s u r = U R s u r + U O s u r = N R s u r 2 ∑ i φ i 0 R − s u r ( | r i + u i | ) + N O s u r 2 ∑ i φ i 0 O − s u r ( | r i + u i | ) , (1)
here, ri is the equilibrium position of i-th atom, ui is its displacement of the i-th atom from the equilibrium position; φ i 0 R − s u r , φ i 0 O − s u r 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.
U s u r = U R s u r + U O s u r = N R s u r 2 ∑ i { φ i 0 R − s u r ( | r i | ) + 1 2 ∑ α , β ( ∂ 2 φ i 0 R − s u r ∂ u i α ∂ u i β ) e q u i α u i β + 1 6 ∑ α , β , γ ( ∂ 3 φ i 0 R − s u r ∂ u i α ∂ u i β ∂ u i γ ) e q u i α u i β u i γ + 1 24 ∑ α , β , γ , η ( ∂ 4 φ i 0 R − s u r ∂ u i α ∂ u i β ∂ u i γ ∂ u i η ) e q u i α u i β u i γ u i η + ⋯ } + N O s u r 2 ∑ i { φ i 0 O − s u r ( | r i | ) + 1 2 ∑ α , β ( ∂ 2 φ i 0 O − s u r ∂ u i α ∂ u i β ) e q u i α u i β + 1 6 ∑ α , β , γ ( ∂ 3 φ i 0 O − s u r ∂ u i α ∂ u i β ∂ u i γ ) e q u i α u i β u i γ + 1 24 ∑ α , β , γ , η ( ∂ 4 φ i 0 O − s u r ∂ u i α ∂ u i β ∂ u i γ ∂ u i η ) e q u i α u i β u i γ u i η + ⋯ } , (2)
where u i 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.
Using Equation (2), the thermal average of the crystalline potential energy of the system is given in terms of the power moments 〈 u m 〉 of the atomic displacements and the harmonic vibrational parameter k, and three anharmonic parameters β , γ 1 and γ 2 as
〈 U s u r 〉 = U 0 R − s u r + U 0 O − s u r + 3 N R s u r [ k R s u r 2 〈 ( u R s u r ) 2 〉 + γ 1 R − s u r 〈 ( u R s u r ) 4 〉 + γ 2 R − s u r 〈 ( u R s u r ) 2 〉 2 ] + 3 N O s u r [ k O s u r 2 〈 ( u O s u r ) 2 〉 + β O s u r 〈 u O s u r 〉 〈 ( u O s u r ) 2 〉 + γ 1 O − s u r 〈 ( u O s u r ) 4 〉 + γ 2 O − s u r 〈 ( u O s u r ) 2 〉 2 ] + ⋯ , (3)
where
γ 1 R − s u r = 1 48 ∑ i ( ∂ 4 φ i 0 R ∂ u i α 4 ) e q , γ 2 R − s u r = 6 48 ∑ i ( ∂ 4 φ i 0 R ∂ u i α 2 ∂ u i β 2 ) e q , (4)
γ 1 O − s u r = 1 48 ∑ i ( ∂ 4 φ i 0 O ∂ u i α 4 ) e q , γ 2 O − s u r = 6 48 ∑ i ( ∂ 4 φ i 0 O ∂ u i α 2 ∂ u i β 2 ) e q , (5)
k R s u r = 1 2 ∑ i ( ∂ 2 φ i 0 R ∂ u i α 2 ) e q , k O s u r = 1 2 ∑ i ( ∂ 2 φ i 0 O ∂ u i α 2 ) e q , (6)
and β O s u r = 1 2 ∑ i ( ∂ 3 φ i 0 O ∂ u i α ∂ u i β ∂ u i γ ) e q , (7)
with α ≠ β ≠ γ = x , y or z. U 0 R − s u r and U 0 O − s u r represent the sum of effective pair interaction energies for R and Oxygen atoms, respectively,
U 0 R − s u r = N R s u r 2 ∑ i φ i 0 R − s u r ( | r i | ) ,
U 0 O − s u r = N O s u r 2 ∑ i φ i 0 O − s u r ( | r i | ) , (8)
Let us consider a quantum system given by the following Hamiltonian:
H ^ = H ^ 0 − ∑ i α i V ^ L , (9)
where H ^ 0 denotes the lattice Hamiltonian in the harmonic approximation, and the second term ∑ i α i V ^ L is added due to the anharmonicity of thermal lattice vibrations, α i denotes a parameter characterizing the anharmonicity of thermal lattice vibrations and V ^ L the related operator. The Helmholtz free energy of the system given by Hamiltonian (9) is formally written as [
Ψ = Ψ 0 − ∑ i ∫ 0 α i 〈 V ^ L 〉 α i d α i , (10)
where 〈 V ^ L 〉 α i expresses the expectation value at the thermal equilibrium with the (anharmonic) Hamiltonian H ^ .
Using Equations (3), (9) and (10) permits us to calculate the Helmholtz free energy of R atoms of the surface-layers as
Ψ R s u r = U 0 R − s u r + Ψ 0 R − s u r + ∫ 0 α i 〈 V L 〉 α i d α i . (11)
The Helmholtz free energy Ψ R s u r 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” γ 1 and γ 2 . Firstly, for R atoms I1 and I2 are written in an integral form as
I 1 = ∫ 0 γ 1 〈 u 4 〉 d γ 1 , I 2 = ∫ 0 γ 2 〈 u 2 〉 γ 1 = 0 2 d γ 2 . (12)
Then the free energy of the N R s u r atoms R of surface layers is given by
Ψ R s u r = U 0 R − s u r + Ψ 0 R − s u r + 3 N R s u r ∫ 0 γ 1 R − s u r 〈 ( u R s u r ) 4 〉 d γ 1 R − s u r + 3 N R s u r ∫ 0 γ 2 R − s u r 〈 ( u R s u r ) 2 〉 γ 1 R − s u r = 0 2 d γ 2 R − s u r , (13)
where Ψ 0 R − s u r denotes free energy in the harmonic approximation for the N R s u r atoms R of surface layers which has the form as
Ψ 0 R − s u r = 3 N R s u r θ [ x R s u r + ln ( 1 − e − 2 x R s u r ) ] . (14)
Using the expression of the second and fourth moments [
Ψ R s u r ≈ { U 0 R − s u r + Ψ 0 R − s u r } + 3 N R s u r θ 2 ( k R s u r ) 2 { γ 2 R − s u r ( X R s u r ) 2 − 2 γ 1 R − s u r 3 ( 1 + X R s u r 2 ) } + 6 N R s u r θ 3 ( k R s u r ) 4 { 4 3 ( γ 2 R − s u r ) 2 ( 1 + X R s u r 2 ) X R s u r − 2 ( ( γ 1 R − s u r ) 2 + 2 γ 1 R − s u r γ 2 R − s u r ) ( 1 + X R s u r 2 ) ( 1 + X R s u r ) } , (15)
where x R s u r = ℏ ω R s u r 2 θ = ℏ k R s u r / m 2 θ , X R s u r = x R s u r coth x R s u r , 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 RO2. Free energies of these layer-types respectively are
Ψ O s u r ≈ { U 0 O − s u r + Ψ 0 O − s u r } + 3 N O s u r θ 2 ( k O s u r ) 2 { γ 2 O − s u r ( X O s u r ) 2 − 2 γ 1 O − s u r 3 ( 1 + X O s u r 2 ) } + 6 N O s u r θ 3 ( k O s u r ) 4 { 4 3 ( γ 2 O − s u r ) 2 ( 1 + X O s u r 2 ) X O s u r − 2 ( ( γ 1 O − s u r ) 2 (16)
+ 2 γ 1 O − s u r ⋅ γ 2 O − s u r ) ( 1 + X O s u r 2 ) ( 1 + X O s u r ) } + 3 N O s u r θ [ ( β O s u r ) 2 6 K s u r 2 k O s u r γ 0 O − s u r − ( β O s u r ) 2 6 K s u r γ 0 O − s u r ] + 3 N O s u r θ 2 { β O s u r K s u r [ 2 γ 0 O − s u r 3 K s u r 3 ( 1 + X O s u r 2 ) 1 / 2 ] } + 3 N O s u r θ 3 { − ( β O s u r ) 2 9 K s u r 3 ( 1 + X O s u r 2 ) + ( β O s u r ) 2 9 K s u r 4 ( 1 + X O s u r ) + ( β O s u r ) 2 6 K s u r 2 k O s u r ( X O s u r − 1 ) } ,
Ψ R i n t ≈ { U 0 R − i n t + Ψ 0 R − i n t } + 3 N R i n t θ 2 ( k R i n t ) 2 { γ 2 R − i n t ( X R i n t ) 2 − 2 γ 1 R − i n t 3 ( 1 + X R i n t 2 ) } + 6 N R i n t θ 3 ( k R i n t ) 4 { 4 3 ( γ 2 R − i n t ) 2 ( 1 + X R i n t 2 ) X R i n t − 2 ( ( γ 1 R − i n t ) 2 + 2 γ 1 R − i n t ⋅ γ 2 R − i n t ) ( 1 + X R i n t 2 ) ( 1 + X R i n t ) } , (17)
Ψ O i n t ≈ { U 0 O − i n t + Ψ 0 O − i n t } + 3 N O i n t θ 2 ( k O i n t ) 2 { γ 2 O − i n t ( X O i n t ) 2 − 2 γ 1 O − i n t 3 ( 1 + X O i n t 2 ) } + 6 N O i n t θ 3 ( k O i n t ) 4 { 4 3 ( γ 2 O − i n t ) 2 ( 1 + X O i n t 2 ) X O i n t − 2 ( ( γ 1 O − i n t ) 2 + 2 γ 1 O − i n t γ 2 O − i n t ) ( 1 + X O i n t 2 ) ( 1 + X O i n t ) } + 3 N O i n t θ [ ( β O i n t ) 2 6 K i n t 2 k O i n t γ 0 O − i n t − ( β O i n t ) 2 6 K i n t γ 0 O − i n t ] + 3 N O i n t θ 2 { β O i n t K i n t [ 2 γ 0 O − i n t 3 K i n t 3 ( 1 + X O i n t 2 ) 1 / 2 ] } + 3 N O i n t θ 3 { − ( β O i n t ) 2 9 K i n t 3 ( 1 + X O i n t 2 ) + ( β O i n t ) 2 9 K i n t 4 ( 1 + X O i n t ) + ( β O i n t ) 2 6 K i n t 2 k O i n t ( X O i n t − 1 ) } , (18)
where
x O s u r = ℏ ω O s u r 2 θ = ℏ k O s u r / m 2 θ , X O s u r = x O s u r coth x O s u r ,
x R i n t = ℏ ω R i n t 2 θ = ℏ k R i n t / m 2 θ , X R i n t = x R i n t coth x R i n t ,
x O i n t = ℏ ω O i n t 2 θ = ℏ k O i n t / m 2 θ , X O i n t = x O i n t coth x O i n t ,
The number of atoms of internal-layers, and surface-layers of thin film RO2 are respectively determined as N R s u r = N R i n t = N 01 , N O s u r = N O i n t = N 02 , and N 01 / N 02 = 1 / 2 . Free energy of thin film RO2 and of one atom, respectively, are given by
Ψ = 2 N 01 Ψ R s u r + 4 N 01 Ψ O s u r + 2 ( n − 3 ) N 01 Ψ O i n t + ( n − 2 ) N 01 Ψ R i n t − T S c , (19)
Ψ N = 2 3 n − 2 Ψ R s u r + 4 3 n − 2 Ψ O s u r + 2 ( n − 3 ) 3 n − 2 Ψ O int + n − 2 3 n − 2 Ψ R int − T S c N , (20)
where S c is the configurational entropy of the system; Ψ R s u r , Ψ O s u r , Ψ R i n t and Ψ O i n t 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 average thickness of two-layers and a ¯ h is the average lattice constant. Then we have
b ¯ = 2 3 a ¯ ; a ¯ h = 2 b ¯ = 4 3 a ¯ . (21)
The average nearest-neighbor distance of thin film at a given temperature T and pressure P can be determined as
a ¯ ( P , T ) = 2 a s u r ( P , T ) + ( n − 3 ) a i n t ( P , T ) n − 1 . (22)
The thickness d of thin film can be given by
d = ( n − 1 ) b ¯ = 2 ( n − 1 ) 3 a ¯ , (23)
or d = 2 3 [ 2 a s u r ( P , T ) + ( n − 3 ) a i n t ( P , T ) ] . (24)
From Equations (21) and (23), we derived
n = 1 + 3 2 a ¯ d , (25)
and the average lattice constant of thin film RO2 is
a ¯ h ( P , T ) = 4 3 . 2 a s u r ( P , T ) + ( n − 3 ) a i n t ( P , T ) 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, 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
a s u r ( P , T ) = a s u r ( P , 0 ) + C R y 0 R − s u r ( P , T ) + C O y 0 O − s u r ( P , T ) , (27)
a i n t ( P , T ) = a i n t ( P , 0 ) + C R y 0 R − i n t ( P , T ) + C O y 0 O − i n t ( P , T ) , (28)
where y 0 R − s u r ( P , T ) , y 0 O − s u r ( P , T ) , y 0 R − i n t ( P , T ) , y 0 O − i n t ( P , T ) 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 y 0 R − s u r ( P , T ) , y 0 O − s u r ( P , T ) , y 0 R − i n t ( P , T ) , y 0 O − i n t ( P , T ) have the analytic forms as [
Substituting Equation (23) into Equation(20) we obtained the expression of the free energy per atom as follows
Ψ N = 2 1 + 3 3 2 a ¯ d Ψ R s u r + 4 1 + 3 3 2 a ¯ d Ψ O s u r + 3 2 a ¯ d − 1 1 + 3 3 2 a ¯ d Ψ R i n t + 2 ( 3 2 a ¯ d − 2 ) 1 + 3 3 2 a ¯ d Ψ O i n t − T S c N . (29)
In the above Equations (15), (16), (17), and (18), the harmonic contributions to the Helmholtz free energies Ψ 0 R − s u r , Ψ 0 O − s u r are derived using the “Einstein” approximation. The results of Equations (15)-(18) permits us to find the free energies Ψ R s u r , Ψ O s u r at temperature T under the condition that the parameters k R , O s u r , γ 1 R , O − s u r , γ 2 R , O − s u r , γ R , O − s u r , β O s u r 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.
Ψ R s u r = U 0 R − s u r + Ψ 0 R − s u r = 3 N R s u r { 1 6 u 0 R − s u r + θ [ x R s u r + ln ( 1 − e − 2 x R s u r ) ] } , (30)
Ψ O s u r = U 0 O − s u r + Ψ 0 O − s u r = 3 N O s u r { 1 6 u 0 O − s u r + θ [ x O s u r + ln ( 1 − e − 2 x O s u r ) ] } , (31)
where
u 0 R − s u r = ∑ i φ i 0 R − s u r ( | r i | )
u 0 O − s u r = ∑ i φ i 0 O − s u r ( | r i | ) . (32)
Since pressure P is determined by
P = − ( ∂ Ψ ∂ V ) T = − a 3 V ( ∂ Ψ ∂ a ) 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
P v = − a s u r { C R [ 1 6 ∂ u 0 R − s u r ∂ a s u r + ω R s u r ( 0 ) 4 k R s u r ∂ k R s u r ∂ a s u r ] + C O [ 1 6 ∂ u 0 O − s u r ∂ a s u r + ω O s u r ( 0 ) 4 k O s u r ∂ k O s u r ∂ a s u r ] } . (34)
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
P v = − a i n t { C R [ 1 6 ∂ u 0 R − i n t ∂ a i n t + ω R i n t ( 0 ) 4 k R i n t ∂ k R i n t ∂ a i n t ] + C O [ 1 6 ∂ u 0 O − i n t ∂ a i n t + ω O i n t ( 0 ) 4 k O i n t ∂ k O i n t ∂ a i n t ] } . (35)
In the above Equations (34) and (35), ω R s u r ( P , 0 ) , ω O s u r ( P , 0 ) , ω R i n t ( P , 0 ) and ω O i n t ( P , 0 ) 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 Ψ = E − T S , 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
C V = [ ∂ E ∂ T ] V = − T ∂ 2 Ψ ∂ T 2 = 2 1 + 3 3 2 a ¯ d C V R − s u r + 4 1 + 3 3 2 a ¯ d C V O − s u r + 3 2 a ¯ d − 1 1 + 3 3 2 a ¯ d C V R − i n t + 2 ( 3 2 a ¯ d − 2 ) 1 + 3 3 2 a ¯ d C V O − i n t , (36)
where
C V R − s u r = − T ∂ 2 Ψ R s u r ∂ T 2 ; C V O − s u r = − T ∂ 2 Ψ O s u r ∂ T 2 ; C V R − i n t = − T ∂ 2 Ψ R i n t ∂ T 2 ; C V O − i n t = − T ∂ 2 Ψ O i n t ∂ T 2 . (37)
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
φ i j = q i q j r + A i j exp ( − r B i j ) − C i j r 6 , (38)
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 [
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 [
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
Temperature dependence of the lattice parameters of ZrO2 thin film with three potentials presented in
In
Interactions | A (eV) | B (Å) | C (eV. Å6) | |
---|---|---|---|---|
O2−-O2− Zr4+-O2− | 9547.96 1502.11 | 0.2192 0.3477 | 32.0 5.1 | Potential 1 |
O2−-O2− Zr4+-O2− | 9547.96 1502.11 | 0.224 0.345 | 32.0 5.1 | Potential 2 |
O2−-O2− Zr4+-O2− | 22.764 1453.8 | 0.149 0.35 | 112.2 0.0 | L-C Potential |
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.
The authors declare no conflicts of interest regarding the publication of this paper.
Van Hung, V., Huong, L.T.T. and Hai, D.T. (2018) Investigation of Thermodynamic Properties of Zirconia Thin Films by Statistical Moment Method. Materials Sciences and Applications, 9, 949-964. https://doi.org/10.4236/msa.2018.912068