Lattice constant of ceria thin film: Temperature dependence

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.

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol 57, No 7, pp 79-87 This paper is available online at http://stdb.hnue.edu.vn LATTICE CONSTANT OF CERIA THIN FILM: TEMPERATURE DEPENDENCE Vu Van Hung, Nguyen Thi Hang1 and Le Thi Thanh Huong2 Faculty of Physics, Hanoi National University of Education Faculty 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 Introduction Cerium dioxide (or ceria) possesses a cubic fluorite structure with a lattice ˚ where in the unit cell the Ce4+ cations occupy the fcc lattice sites parameter of 5.411 A, 2− while the O 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] 2.1 Content 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 Ceria thin film with two free surface layers of cerium atoms The ceria free thin film consists of cerium free surface layers, 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 side Ψ = 2NCe Ψside Ce + 2NO ΨO + (n − 3)NO Ψinter + (n − 2)NCe Ψinter O Ce − T SC (2.1) where the numbers of cerium and oxygen ions of a layer are simply denoted by NCe inter side and NO = 2NCe , respectively, Ψside (or Ψinter ) denoting the free Ce (or ΨCe ) and ΨO O 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] Ψside Ce ≈ Ψside ≈3 O 80 side uCe + θ[xCe + ln(1 − e−2xCe )] (2.2) side uO + θ[xO + ln(1 − e−2xO )] (2.3) Lattice constant of ceria thin film: temperature dependence uside Ce = where i ϕCe−side (|ri |), anduside = O io i ϕO−side (|ri |) io (2.4) x = w/2θ with θ = kB T , w is the atomic vibration frequency, and it can be approximated in most cases to the Einstein frequency wE , given by k= i ∂ϕio ∂u2ix eq ≡ mwE2 , (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] Ψinter Ce ≈ Ψinter ≈3 O inter u + θ[xCe + ln(1 − e−2xCe )] Ce (2.6) inter u + θ[xO + ln(1 − e−2xO )] O (2.7) inter where uinter represent the sum of the effective pair interaction energies for Ce Ce and uO and O ions on the internal layers in ceria thin film uinter Ce = i ϕCe−inter (|ri |), anduinter = O io i O−inter ϕio (|ri |) (2.8) The average nearest-neighbor distance at T = K can be determined from experimentation or the minimum condition of the potential energy of the free surface side layer composed of NCe atoms Ce and NO atoms O, and that means ∂ ∂U r1 = 0, T,P,N leads to the following equation side ∂U side ∂UCe ∂UOsidei = + ∂r1 ∂r1 ∂r1 = NCe ∂ ∂r1 i ϕCe−side (|ri |) io + NO ∂ ∂r1 i ϕO−side (|ri |) io =0 or CCe ∂ ∂r1 i ϕCe−side (|ri |) io + CO ∂ ∂r1 i ϕO−side (|ri |) io = (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) = qi qj r Cij + Aij exp(− )− r Bij r (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 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 [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), ϕCe−side (|ri |) , and io i i = i i i ϕO−side (|ri |) have been summated by the Ewald method io ϕCe−side (|ri |) = io 2 qCe e erf c(αri )+ ri ϕO−side (|ri |) = io i i i ϕCe−side Ce−Ce (|ri |) + i ϕCe−side Ce−O (|ri |) ri qCe qO e2 erf c(αri ) + ACe−O exp − ri BCe−O ϕO−side O−Ce (|ri |) + i i i ri qO qO e2 erf c(αri ) + AO−O exp − ri BO−O + CCe−O ri6 (2.11) ϕO−side O−O (|ri |) qCe qO e2 ri erf c(αri ) + ACe−O exp − ri BCe−O = − − − CCe−O ri6 CO−O ri6 (2.12) From Eqs (2.11), (2.12) and (2.9) we obtain the following equation: CCe M + CO N = 82 (2.13) Lattice constant of ceria thin film: temperature dependence where 2 qCe e 8.2 √ erf c(αr1) + 5.erf c(αr2 ) a 2 4.4 4.12 q qO e √ erf c(αr1) + √ erf c(αr2 ) − Ce a 11 √ √ √ a 11 ACe−O √ a 3.exp − + 11.exp − − BCe−O 4BCe−O 4BCe−O     12  CCe−O  (2.14) + √ √ + 6 a7  11   M= − 4 √ qO2 e2 5.2erf c(αr ) + 2.erf c(αr2 ) a2 √ √ a AO−O a + 2.exp − exp − − BO−O 2BO−O 2BO−O       CO−O   CCe−O  + √ √ +  + 6 + a  a7   12   N= − − qCe qO e2 a2 − ACe−O BCe−O 10 √ 11 16 10.4 √ erf c(αr1 ) + √ erf c(αr2) 11 √ √ √ a a 11 10 √ + 3.exp − 11.exp − 4BCe−O 4BCe−O    6  (2.15) Minimizing the interaction potentials U inter of the internal layer with respect to ∂ U inter = 0, which leads to the the nearest-neighbor distance r1 , this means ∂ r1 T,P,N following equation CCe ∂ ∂r1 i ϕCe−inter (|ri |) io + CO ∂ ∂r1 i or CCe P + CO Q = ϕO−inter (|ri |) io =0 (2.16) 83 Vu Van Hung, Nguyen Thi Hang and Le Thi Thanh Huong 2 24 qCe e √ erf c(αr1 ) + 6.erf c(αr2 ) a2 qCe qO e2 32 4.24 √ erf c(αr1 ) + √ erf c(αr2) − a 11 √ √ √ a a 11 ACe−O √ + 11.exp − 3.exp − − BCe−O 4BCe−O 4BCe−O     24  CCe−O  (2.17) + √ √ + 6 a7  11   where P = − 4 qO2 e2 a2 24 12.erf c(αr1) + √ erf c(αr2 ) √ √ AO−O a a − 3.exp − + 2.exp − BO−O 2BO−O 2BO−O     CO−O  12  + √ + 6 a   21  Q= − 2 − qCe qO e a2 − ACe−O BCe−O + CCe−O a7 16 48 √ erf c(αr1) + √ erf c(αr2 ) 11 √ √ √ √ a a 11 3.exp − + 11.exp − 4BCe−O 4BCe−O      12  √ √ + 6  11   (2.18) Principle Eqs (2.13) and (2.16) permit us to find the nearest neighbor distance or r1inter (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 r1side (0) and r1inter (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 r1side (0) 84 side r1side (T ) = r1side (0) + CCe yCe (T ) + CO yOside (T ) (2.19) inter r1inter (T ) = r1inter (0) + CCe yCe (T ) + CO yOinter (T ) (2.20) Lattice constant of ceria thin film: temperature dependence side inter in which yCe (T ) (or yCe (T ) ) and yOside (T ) (or yOinter (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 ) = 2.2 d 2aside (T ) + (n − 3)ainter (T ) = n−1 n−1 (2.22) 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 shows good agreement between the SMM calculations of lattice constant at zero temperature T = 0K and the experimental results for CeO2 Table Potential parameters of CeO2 [8] Interaction potential O2− - O2− Ce4+ - O2− O2− - O2− Ce4+ - O2− O2− - O2− Ce4+ - O2− A(eV) ˚ B(A) ˚ 6) C(eV.A 9547.92 1809.68 9547.92 2531.5 22764.3 1986.83 0.2192 0.3547 0.2192 0.335 0.149 0.35107 32.00 20.40 32.00 20.40 43.83 20.40 Table Lattice parameter of bulk CeO2 ˚ Method ao (A) Potential Potential Simulation [8] 5.411 5.411 SMM 5.4107 5.4111 Expt [9] 5.411 5.353 Ab initio [10] Potential Potential Butler Butler 5.411 5.4099 In Figure 2, we present the thickness dependence of the lattice constant of ceria thin film using the potentials 1, and the Butler potential Figure 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 that the lattice constant increases with the thickness d, when the thickness d ≥ 500A0 (or the number n of layers of thin ˚ are film n ≥ 100) and the average lattice constant a(T ) of thin film (a(T ) ≈ 5.41 A) in agreement with the experimental results of bulk CeO2 One also sees in Figure the temperature dependence of the SMM lattice parameter of CeO2 thin films with different thickness using the potential Butler (a) potential (b) potential (c) the Butler potential Figure Thickness dependence of average lattice constant of ceria thin film using a, b, c 86 Lattice constant of ceria thin film: temperature dependence Figure Temperature dependence of average lattice constant a(T ) of ceria thin film using the Butler potential 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, 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 REFERENCES [1] L N Raffaella, G T Roberta, M Graziella and L F Ignazio, 2005 J Matter Chem., 15, pp 2328-2337 [2] A J Farah et al., 2009 J of Nuclear and Related Tech., Vol 6, No 1, p 183 [3] K N Rao, L Shivlingappa and S Mohan, 2003 Mater Scien and Engineering, B 98, pp 38-44 [4] V.V.Hung, B.D Tinh and Jaichan Lee, 2011 Modern Phys Letter B, Vol 25, No 12&13, pp 1001-1010 [5] V V Hung, L T M Thanh and K Masuda-Jindo, 2010 Comput Mat Science, 49, pp 355-358 [6] V V Hung and L T M Thanh, 2011 Physica B 406, pp 4014-4018 [7] V V Hung, J Lee and K Masuda-Jindo, 2006 J Phys Chem Sol 67, pp 682-689 [8] Shyam Vyas, Robin W Grimes, David H Gay and Andrew L Rohl, 1998 J Chem Soc., Faraday Trans., 94, pp 427-434 [9] L Gerwad and J Staun-Olsen, 1993 Powder diffr., 8, 127 [10] A Nakajima, A Yoshihara, M Ishigma, 1994 Phys Rev B 50, p 13297 87 ... Thickness dependence of average lattice constant of ceria thin film using a, b, c 86 Lattice constant of ceria thin film: temperature dependence Figure Temperature dependence of average lattice constant. .. present the thickness dependence of the lattice constant of ceria thin film using the potentials 1, and the Butler potential Figure shows the lattice constant of ceria thin film, calculated using... purpose of the present article is to investigate temperature and size dependences of the lattice constant of CeO2 thin film using SMM [7] 2.1 Content Theory Let us consider a ceria free thin film of