Lattice constant of ceria thin film: Temperature dependence

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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