Coefficients of thermal expansion of thin metal films investigated using the statistical moment method

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 109-116 This paper is available online at COEFFICIENTS OF THERMAL EXPANSION OF THIN METAL FILMS INVESTIGATED USING THE STATISTICAL MOMENT METHOD Duong Dai Phuong1, Vu Van Hung2 and Nguyen Thi Hoa3 1Tank Armour Officers Training School, Tam Duong, Vinh Phuc 2Viet Nam Education Publishing House, Hanoi 3Hanoi University of Transport and Communications Abstract. The coefficients of thermal expansion of thin metal film with face-centered cubic structure at zero pressure are investigated using the statistical moment method (SMM), including the anharmonic effects of thermal lattice vibration. The Helmholtz free energy, mean-square atomic displacement and linear thermal expansion coefficients are derived in closed analytic forms in terms of the power moments of the atomic displacement. Numerical calculations for Al, Au and Ag thin films are found to be in good and reasonable agreement with other theoretical results and with the experimental data. Keywords: Coefficients of thermal expansion, statistical moment method, anharmonicity, thin film. 1. Introduction Materials in the form of thin film have come under the spotlight in recent years. They have been found to show physical, chemical and mechanical properties that differ from corresponding bulk materials [3, 4]. Properties, limitations and advantages obtained on thin film geometry have been widely studied and they were found to depend on different factors: structure, size, film thickness and different substrates [3, 5]. Thin film is used in a vast range of applications which include microelectromechanical and nanoelectrome-chanical systems, sensors and electronic textiles [1]. Metallic thin film displays a common geometry and presents enormous scientific interest, mainly due to their attractive and novel properties for technological applications [1]. In the early twentieth century, the theory of size-dependent effects in thin metal layers was developed by a number of scientists [5]. For ultrathin metal layers, both surface effects and quantum size effects must be considered [8]. A large number of experimental and Received September 20, 2013. Accepted October 30, 2013. Contact Vu Van Hung, e-mail address: bangvu57@yahoo.com 109 Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa theoretical studies have been carried out on the thermal behavior of metal and nonmetal thin films. Most of the previous theoretical studies, however, are concerned with the material properties of thin metal film at low temperature, and the temperature dependence of thermodynamic quantities has not been studied in detail. The purpose of the present article is to investigate the temperature dependence and the thickness dependence of the coefficients of thermal expansion of thin metal film with face-centered cubic structure using the analytic statistical moment method (SMM) [6, 10].The coefficients for thermal expansion are derived from the Helmholtz free energy, and the explicit expression of the thermal expansion coefficient is presented taking into account the anharmonic effects of the thermal lattice vibrations. In the present study, the influence of surface and size effects on the coefficients of thermal expansion have also been studied. We compared the results of the present calculations with those of the previous theoretical calculations as well as with the available experimental results. 2. Content 2.1. Theory 2.1.1. The anharmonic oscillations of thin metal film Investigating a metallic free standing thin film includes n layers with film thickness d. Suppose of the thin film consists of two surface layers of atoms, the next two deeper layers of atoms and the (n-4) internal layers atoms (Fig. 1). Figure 1. The thin metal film 110 Coefficients of thermal expansion of thin metal films investigated... First of all, we present the statistical moment method (SMM [6, 10]) formulation for the displacement of the surface layer atoms of the thin film as a solution of the equation γsurθ 2 d 2 dp2 + 3γsurθ < u sur i > d dp + γsur < u sur i > 3 +ksur < u sur i > +γsur θ ksur (xsur coth xsur − 1) −p = 0, (2.1) where xsur = ~ωsur 2θ ; θ = kBT, (2.2) ksur = 1 2 ∑ i ( ∂2φinio ∂u2iα ) eq ≡ m0ω2sur, (2.3) γ1sur = 1 48 ∑ i ( ∂4φsurio ∂u4iα ) eq , γ2sur = 6 48 ∑ i ( ∂4φsurio ∂u2iβ∂u 2 iγ ) eq (2.4) γsur = 1 12 ∑ i (∂4φsurio ∂u4iα ) eq + 6 ( ∂4φsurio ∂u2iβ∂u 2 iγ ) eq  = 4 (γ1sur + γ2sur) ; α, β, γ = x, y, z. (2.5) Here, KB is the Boltzmann constant; T is the absolute temperature;m0 is the mass of the atoms at the lattice node; ωsur is the frequency of lattice vibration of surface layer atoms; ksur, γ1sur, γ2sur and γsur are the parameters of crystals depending on the structure of crystal lattice and the interaction potential between atoms at the node; is the effective interatomic potential between 0th and ith surface layer atoms; φsuri0 , uiy and uiz are the displacement of ith atom from the equilibrium position on direction x, y, z respectively; and the subscript eq indicates the evaluation at equilibrium. To determine the atomic displacement , the symmetry property appropriate for cubic crystals is used =< u sur iy >=< u sur iz >=< u sur i > . (2.6) Then, the solutions of the nonlinear differential equation of Eq. (2.1) can be expanded in the power series of the supplemental force p as =< u sur i >0 +A1p+ A2p 2. (2.7) 111 Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa Here, 0 is the average atomic displacement in the limit of zero of supplemental force p. Substituting the above solution of Eq. (2.7) into the original differential Eq. (2.1), one can get the coupled equations on the coefficients A1 and A2, from which the solution of 0 is given as [6, 10] 0≈ √ 2γsurθ2 3k3sur Asur, (2.8) where Asur = a sur 1 + γ2surθ 2 k4sur asur2 + γ3surθ 3 k6sur asur3 + γ4surθ 4 k8sur asur4 + γ5surθ 5 k10sur asur5 + γ6surθ 6 k12sur asur6 , (2.9) with asurη (η = 1, 2, . . . , 6) being the values of parameters of crystals depending on the structure of the crystal lattice [6, 10]. A similar derivation can be also done for atoms at layers just below the surface and internal layers of atoms of thin film, their displacement noted as 0 and < u in i >0, respectively. 2.1.2. Free energy of the thin metal film In the present research, the influence of the size effect on the coefficients of thermal expansion of the thin metal film is studied, introducing the surface energy contribution in the free energy of the system atoms. Following the general formula in the SMM formulation [6, 10], one can get the free energy of the surface layers atoms as Ψsur ≈ {U sur0 + 3Nsurθ[xsur + ln(1− e−2xsur)]}+ 3Nsurθ 2 k2sur { γ2surX 2 sur − 2γ1sur3 ( 1 + Xsur 2 )} +6Nsurθ 3 k4sur { 4 3 γ22sur(1 + Xsur 2 )Xsur − 2 (γ21sur + 2γ1surγ2sur) (1 + Xsur2 )(1 +Xsur) } , (2.10) here, Xsur = xsurcthxsur. The free energy of the layers of atoms just below the surface and internal layers of atoms have fomulae similar to Eq. (2.10) and can be noted as Ψsur1,Ψin, respectively. The free energy in the harmonic approximation for surface layers, layers of atoms just below the surface and internal layers of atoms of thin film are given as Ψsur ≈ 3Nsur { 1 6 usur0 + θ[xsur + ln(1− e−2xsur)] } , Ψsur1 ≈ 3Nsur1 { 1 6 usur10 + θ[xsur1 + ln(1− e−2xsur1)] } ; Ψin ≈ 3Nin { 1 6 uin0 + θ[xin + ln(1− e−2xin)] } , (2.11) where usur0 ≡ ∑ i φsuri0 , u sur1 0 ≡ ∑ i φsur1i0 , u in 0 ≡ ∑ i φini0 (2.12) 112 Coefficients of thermal expansion of thin metal films investigated... Let us consider a system where N atoms consist of n layers, the number of atoms on each layer are the same and and NL, ψsur denotes the free energy of the two surface layers of atoms, ψsur1 denotes the free energy of the two layers of atoms just below the surface and ψin denotes the free energy of the (n-4) internal layers atoms. The number of internal layer atoms, atoms in layers just below the surface, and surface layer atoms, respectively Nin = (n− 4)NL = ( N NL − 4 ) NL = N − 4NL, (2.13) Nsur1 = 2NL = N − (n− 2)NL and Nsur = 2NL = N − (n− 2)NL. (2.14) Free energy of the system and of an atom, respectively, is given by ψ = Ninψin+Nsur1ψsur1+Nsurψsur = (N − 4NL)ψin+2NLψsur1+2NLψsur. (2.15) ψ N = [ 1− 4 n ] ψin + 2 n ψsur1 + 2 n ψsur. (2.16) Using a¯ as the average nearest-neighbor distance and b¯ as the average thickness of two-layers and a¯c as the average lattice constant, respectively, we can easily calculate the relation b = a¯√ 2 and a¯c = 2b¯ = √ 2a¯. (2.17) The thickness d of thin film can be given by d = 2bsur+2bsur1+(n− 5) bin = (n− 1) b = √ 2asur+ √ 2asur1+ n− 5√ 2 ain = (n−1) a¯√ 2 . (2.18) In Eq. (2.18), the average nearest-neighbor of surface layers atoms, atoms of layers just below the surface and internal layer atoms in thin metal film at a given temperature T can be determined as asur = asur (0)+ < u sur i >0, asur1 = asur1 (0)+ < u sur1 i >0, ain = ain (0)+ < u in i >0, (2.19) where asur (0), asur1 (0) and ain (0) denote the nearest-neighbor at zero temperature of surface layers atoms, atoms of layers just below the surface and internal layer atoms, respectively, which can be determined experimently or derived considering the minimum condition of the potential energy of the system. ∂U sur0 ∂asur + 3~ωsur 4ksur ∂ksur ∂asur = 0; ∂U sur10 ∂asur1 + 3~ωsur1 4ksur1 ∂ksur1 ∂asur1 = 0; ∂U in0 ∂ain + 3~ωin 4kin ∂kin ∂ain = 0, (2.20) 113 Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa The average nearest-neighbor of thin film is determined as a¯ = 2asur + 2asur1 + (n− 5)ain n− 1 (2.21) Finally, we obtain the expression of the free energy through the number of layers as follows ψ N = d √ 2− 3a¯ d √ 2 + a¯ ψin + 2a¯ d √ 2 + a¯ ψsur + 2a¯ d √ 2 + a¯ ψsur1. (2.22) 2.1.3. The coefficients of thermal expansion The average thermal expansion coefficient of metallic thin film can be calculated as follows α = KB a¯ da¯ dθ = dsurαsur + dsur1αsur1 + (d− dsur − dsur1)αin d , (2.23) with αin = kB ain ∂yin0 (T ) ∂θ ;αsur = kB asur ∂ysur0 (T ) ∂θ ;αsur1 = kB asur1 ∂ysur10 (T ) ∂θ (2.24) with dsur and dsur1 being the thickness of the surface layers and the layers just below the surface, respectively. 2.2. Numerical results and discussion In order to apply the theoretical calculation to the Au, Ag and Al of metallic thin film, we use the pair interaction potential in the following form with the potential parameters D,m, n and r0 in Table 1. φ(r) = D (n−m) [ m (r0 r )n − n (r0 r )m] . (2.25) Using the expressions obtained in Section 2, we calculate the values of the average lattice constant, a¯c, the thickness, d, and the linear thermal expansion coefficient, α for Au, Ag and Al thin metal thin film at various temperatures. The calculated results are presented in Figures 2-5. Table 1. Lennars-Jones potential parameters for Au, Ag and Al of metallic thin film [7] Metal n m r0, A0 D/kB, 0K Au 10.5 5.5 2.8751 4683.0 Ag 11.5 5.5 2.8760 3325.6 Al 12.5 5.5 2.8541 2995.6 114 Coefficients of thermal expansion of thin metal films investigated... In Figures 2 and 3, we present the thickness dependence of the average lattice constant of Au thin metal film as a function of the thickness, and it has been compared to gold splattered on glass presented in [3]. We obtained the results of an average lattice constant for an Ag and Al decrease, also with increasing thickness. In Figure 4, we present the temperature dependence of the thermal expansion coefficients of thin metal film for Al as a function of the temperature. The thermal expansion coefficients increase with absolute temperature T, and our calculated results have been compared with the results presented in [2] for Al thin film, and Al bulk [9] are shown to be in good agreement. In Figure 5, we also show the thickness dependence of the thermal expansion coefficients of Au and Al thin film, and the results calculated show that the thermal expansion coefficients for Au and Al decrease with increasing thickness. When the thickness increases, the thermal expansion coefficients approach the bulk value in agreement with the results presented in [9]. 115 Duong Dai phuong, Vu Van Hung and Nguyen Thi Hoa 3. Conclusion We have presented the SMM formalism using Lennard-Jones interaction potentials and investigated the thermal expansion coefficients of Au, Ag and Al thin metal film. 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