Coefficients of thermal expansion of metallic thin films with body-centered cubic structure

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0039 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 112-121 This paper is available online at COEFFICIENTS OF THERMAL EXPANSION OF METALLIC THIN FILMS WITH BODY-CENTERED CUBIC STRUCTURE Duong Dai Phuong1, Nguyen Xuan Viet1, Nguyen Thi Hoa2 and Doan Thi Van3 1Tank Armour Officers Training School, Tam Duong, Vinh Phuc 2Hanoi University of Transport and Communications 3Faculty of Nursing, Hanoi Medical Colleges Abstract. The coefficients of thermal expansion of metallic thin films with body-centered cubic (BCC) structure at ambient conditions were investigated using the statistical moment method (SMM), including the anharmonicity effects of thermal lattice vibrations. The analytical expressions of Helmholtz free energy, lattice constant, and linear thermal expansion coefficients were derived in terms of the power moments of the atomic displacements. Numerical calculations of the quantities were performed for Fe and W thin films and found to be in good and reasonable agreement with other theoretical results and experimental data. This research proves that thermal expansion coefficients of thin films approach the values of bulk when thin film is about 70 nm thick. Keywords: Metallic thin film, statistical moment method, thermodynamic properties, anharmonicity. 1. Introduction Knowledge of thermodynamic properties of metallic thin film, such as heat capacity and coeffcient of thermal expansion, are of great importance when determining parameters for the stability and reliability of manufactured devices. In many cases, the thermodynamic properties of metallic thin film are not well known or differ from the values for corresponding bulk materials. A large number of experimental and theoretical studies have been carried out on the thermodynamic properties of metal and nonmetal thin film [1-4]. Most of them describe the method used for measuring the thermodynamic properties of crystalline thin films on the substrates [5-9]. There are many ways to determine the behaviors deformed of thin film such as x-ray diffraction [4, 6-8] and nanoindentation [9]. However, rarely has research been done on the thermodynamic properties of metallic free-standing thin film. Most of the previous theoretical studies were concerned with the material properties of metallic thin film at low temperature while temperature dependence of the 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 metallic thin films with a Received September 28, 2016. Accepted October 22, 2016. Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com 112 Coefficients of thermal expansion of metallic thin films with body-centered cubic structure body-centered cubic structure using the analytic statistical moment method (SMM) [10-13]. The coefficients of thermal expansion are derived from the Helmholtz free energy, and the explicit expression of the thermal expansion coefficient is presented taking into account the anharmonicity 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 metallic thin films Let us consider a metalic free standing thin film that has n∗ layers with the thickness d. We assume that the thin film consists of two atomic surface layer, two next surface atomic layers and (n∗ − 4) atomic internal layers. (see Figure 1). Figure 1. The free-standing thin film For internal layer atoms of thin films, we present the statistical moment method (SMM [12-13]) formulation for the displacement of the internal layer atoms of the thin film ytr is solution of equation. γtrθ 2d 2ytr dp2 + 3γtrθytr dytr dp + γtry 3 tr + ktrytr + γtr θ ktr (xtr coth xtr − 1)ytr − p = 0 , (2.1) where ytr ≡p;xtr = ~ωtr 2θ ; θ = kBT, (2.2) ktr = 1 2 ∑ i ( ∂2ϕtrio ∂u2iα ) eq ≡ m0ω2tr, (2.3) 113 Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van γ1tr = 1 48 ∑ i ( ∂4ϕtrio ∂u4iα ) eq , (2.4) γ2tr = 6 48 ∑ i ( ∂4ϕtrio ∂u2iβ∂u 2 iγ ) eq , (2.5) γtr = 1 12 ∑ i  (∂4ϕtrio ∂u4iα ) eq + 6 ( ∂4ϕtrio ∂u2iβ∂u 2 iγ ) eq   = 4 (γ1tr + γ2tr) , (2.6) where kB is the Boltzmann constant, T is the absolute temperature,m0 is the mass of the atom, ωtr is the frequency of lattice vibration of internal layer atoms; ktr , γ1tr , γ2tr , γtr are the parameters of crystal depending on the structure of crystal lattice and the interaction potential between atoms; ϕtri0 is the effective interatomic potential between 0 th and ith internal layers atoms; uiα , uiβ , uiγ are the displacements of ith atom from equilibrium position in the direction α(α = x, y, z), β(β = x, y, z), γ(γ = x, y, z), respectively, and the subscript eq indicates evaluation at equilibrium. In the second approximation of the supplemental force, the solutions of the nonlinear differential equation of Eq. (2.1) can be expanded in the power series of the supplemental force p as [12, 13]. ytr = y tr 0 +A tr 1 p+A tr 2 p 2. (2.7) Here, ytr0 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 for the coefficients Atr1 and A tr 2 , from which the solution of y tr 0 is given as [13] ytr0 ≈ √ 2γtrθ2 3k3tr Atr, (2.8) where Atr = a tr 1 + γ2trθ 2 k4tr atr2 + γ3trθ 3 k6tr atr3 + γ4trθ 4 k8tr atr4 + γ5trθ 5 k10tr atr5 + γ6trθ 6 k12tr atr6 . (2.9) with atrη (η = 1, 2..., 6) being the values of parameters of crystal depending on the structure of crystal lattice. Similar derivation can be also done for next surface layer atoms of thin film, their displacement are the solution of equations, respectively γng1θ 2d 2yng1 dp2 +3γng1θyng1 dyng1 dp +γng1y 3 ng1+kng1yng1+γng1 θ kng1 (xng1 coth xng1−1)yng1−p = 0 (2.10) For surface layers atoms of thin films, we present the statistical moment method (SMM) formulation for the displacement of the surface layers atoms of the thin film yng =< u ng i > is solution of equation kng < u ng i >a + γng [ < ung i >2a +θ ∂〈ungi 〉a ∂a + θ mω2ng (xng coth xng − 1) ] − a = 0 (2.11) 114 Coefficients of thermal expansion of metallic thin films with body-centered cubic structure where yng =< u ng i >a, xng = ~ωng 2θ , θ = kBT (2.12) kng = 3 2 ∑ i [( 02ϕngi0 ) a2ix + (0ϕ ng i0 ) ] = mω2ng, (2.13) γng = 1 4 ∑ i,α,β,γ α6=β  ( ∂3ϕngio ∂u3iα,ng ) eq + ( ∂3ϕngio ∂u2iα,ng∂u ng iγ ) eq  . (2.14) In the second approximation of the supplemental force, the solutions of equation (2.11) can be expanded in the power series of the supplemental force p as yng = y ng 0 +A1a+A2a 2. (2.15) Here, ytr0 is the average atomic displacement in the limit of zero of supplemental force p. The solution of ytr0 is given as yng0 = − γngθ k2ng xng cothxng. (2.16) Using the statistical moment method, we can get power moments of the atomic displacement. 2.1.2. Free energy of metallic thin films Usually, theoretical studies on size effect are done by considering the surface energy contribution in continuum mechanics or by using computational simulations reflecting surface stress or a surface relaxation influence. In this paper, the influence of size effect on thermodynamic properties of metal thin film is studied by looking at the surface energy contribution in the free energy of the system atoms. For the internal layers and next surface layers. In articles [11, 12], free energy of these layers Ψtr = { U tr0 + 3Ntrθ [ xtr + ln ( 1− e−2xtr)]}+ 3Ntrθ2 k2tr { γ2trX 2 tr − 2γ1tr 3 ( 1 + Xtr 2 )} 6Ntrθ 3 k4tr { 4 3 γ22tr ( 1 + Xtr 2 ) Xtr − 2 ( γ21tr + 2γ1trγ2tr )( 1 + Xtr 2 ) (1 +Xtr) } . (2.17) Ψng1 = { Ung10 + 3Nng1θ [ xng1 + ln ( 1− e−2xng1)]} + 3Nng1θ 2 k2ng1 { γ2ng1X 2 ng1 − 2γ1ng1 3 ( 1 + Xng1 2 )} + 6Nng1θ 3 k4ng1 { 4 3 γ22ng1 ( 1 + Xng1 2 ) Xng1−2 ( γ21ng1 + 2γ1ng1γ2ng1 )( 1 + Xng1 2 ) (1 +Xng1) } ; (2.18) In Eqs. (2.17), (2.18), using Xtr = xtrcothxtr , Xng1 = xng1cothxng1; and U tr0 = Ntr 2 ∑ ϕtri0 (ri,tr);U ng1 0 = Nng1 2 ∑ ϕng1i0 (ri,ng1); (2.19) 115 Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van where ri is the equilibrium position of the ith atom, ui is its displacement of the ith atom from the equilibrium position; ϕtri0, ϕ ng1 i0 , are the effective interatomic potential between the 0 th and ith internal layer atoms and the 0th and ith and next to surface layer atoms, Ntr, Nng1 and are respectively the number of internal layers atoms, next surface layer atoms and of this thin film; U tr0 , U ng1 0 represent the sum of effective pair interaction energies for internal layer atoms and next to surface layer atoms, respectively. For the surface layers, the Helmholtz free energy of the system in the harmonic approximation is given by [12] Ψng = { Ung0 + 3Nngθ [ xng + ln ( 1− e−2xng)]} (2.20) Let us assume that the system consists of N atoms with n∗ layers, the atom number on each layer is NL, we then have N = n∗NL ⇒ n∗ = N NL . (2.21) The number of atoms in the internal layers, next to surface layers and surface layers are, respectively determined as Ntr = (n ∗ − 4)NL = ( N NL − 4 ) NL = N − 4NL, Nng1 = 2NL = N − (n∗ − 2)NLandNng = 2NL = N − (n∗ − 2)NL. (2.22) The free energy of the system and of one atom, respectively, are given by Ψ = Ntrψtr +Nng1ψng1 +Nngψng − TSc = (N − 4NL)ψtr + 2NLψng1 + 2NLψng − TSc, (2.23) Ψ N = [ 1− 4 n∗ ] ψtr + 2 n∗ ψng1 + 2 n∗ ψng − TSc N , (2.24) where Sc is the entropy configuration of the system and ψng, ψng1 and ψtrare respectively the free energy of one atom at surface layers, next surface layers and internal layers. Using a¯ as the average nearest-neighbor distance (NND) and b¯ as the average thickness two-layers, we have b = a¯√ 3 . (2.25) The thickness d of thin film can be given by d = 2bng + 2bng1 + (n ∗ − 5) btr = (n∗ − 1) b = (n∗ − 1) a√ 3 . (2.26) From equation (25), we derived n∗ = 1 + d b = 1 + d √ 3 a . (2.27) The average NND of thin film is a¯ = 2ang + 2ang1 + (n ∗ − 5)atr n∗ − 1 . (2.28) 116 Coefficients of thermal expansion of metallic thin films with body-centered cubic structure In above equation, ang, ang1 and atrare correspondingly the average NND between two intermediate atoms at surface layers, next surface layers and internal layers of thin film at a given temperature T. These quantities can be determined as ang = a0,ng + y ng 0 , ang1 = a0,ng1 + y ng1 0 , atr = a0,tr + y tr 0 , (2.29) where a0,ng, a0,ng1 and a0,tr denote the values of ang, ang1 and atr at zero temperature which can be determined from experiment or from the minimum condition of the potential energy of the system. Substituting Eq. (2.27) into Eq. (2.24), we obtain the expression of the free energy per atom as follows Ψ N = d √ 3− 3a¯ d √ 3 + a¯ Ψtr + 2a¯ d √ 3 + a¯ Ψng + 2a¯ d √ 3 + a¯ Ψng1 − TSc N (2.30) 2.1.3. The coefficients of thermal expansion The average thermal expansion coefficient of thin metal films can be calculated as α = kB a¯0 da¯ dθ = dngαng + dng1αng1 + (d− dng − dng1)αtr d , (2.31) where dng and dng1 are the thickness of surface layers and next to surface layers, and αtr = kB a0,tr ∂ytr0 (T ) ∂θ ;αng = kB a0,ng ∂yng0 (T ) ∂θ ;αng1 = kB a0,ng1 ∂yng10 (T ) ∂θ . (2.32) 2.2. Numerical results and discussion In this section, the derived expressions in previous section will be used to investigate the thermodynamic as well as mechanical properties of metallic thin films with BCC structure for Fe and W at zero pressure. For the sake of simplicity, the interaction potential between two intermediate atoms of these thin films is assumed to have a Mie-Lennard-Jones potential which has the form ϕ(r) = D (n−m) [ m (r0 r )n − n (r0 r )m] (2.33) where D describes the dissociation energy; r0 is the equilibrium value of r; and the parameters n and m can be determined by fitting in the experimental data (e.g., cohesive energy and elastic modulus). The potential parameters D,m,n and r0 of some metallic thin films are shown in Table 1 [14]. Table 1. Mie-Lennard-Jones potential parameters for Fe and W of metallic thin film [14] Metal n m r0, (A 0) D/kB , (K) Fe 8.26 3.58 2.4775 12576.70 W 8.58 4.06 2.7365 25608.93 117 Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van Figure 2. Temperature dependence of the average NND for Fe thin film Figure 3. Thickness dependence of the average NND for Fe thin film 118 Coefficients of thermal expansion of metallic thin films with body-centered cubic structure Using the expression (2.28), we can determine the average NND of thin film as a function of thickness and temperature. In Figure 2, we present the temperature dependence of the average NND for Fe of thin film using SMM. One can see that the value of the average NND increases with the increasing of absolute temperature T. These results show that the average NND for Fe increases with increase in thickness. In Figure 3, we present the thickness dependence of the average NND for Fe thin film at room temperature. The average NND increases when the thickness increases. We realized that for Fe thin film when the thickness value is larger than 70 nm, the average NND approaches the bulk value. Figure 4. Temperature dependence of the thermal expansion coefficients for Fe thin film In Figure 4, we present the temperature dependence of the thermal expansion coefficients of Fe thin film as a function of thickness and temperature. We showed the theoretical calculations of thermal expansion coefficients of Fe thin film with various layer thickneses. The experimental thermal expansion coefficients [15] of bulk material have also been reported for comparison. One can see that the value of the thermal expansion coefficient increases with the increase of absolute temperature T. It can also be noted that, at a given temperature, the lattice parameter of thin film is not a constant but rather, it strongly depends on the layer thickness, especially at high temperature. In Figure 5, we present the thickness dependence of the thermal expansion coefficients for Fe thin film at room temperature, the thermal expansion coefficients decreasing with the increasing thickness, approaching the bulk value. The obtained results of dependence on thickness show agreement between our works and the results presented in [10]. By using above scheme of SMM theory, the thermodynamic properties of other metallic thin films can be determined analogously. In Table 2, we reported the values of some thermodynamic quantities as functions of temperature at ambient pressure of W thin film. 119 Duong Dai Phuong, Nguyen Xuan Viet, Nguyen Thi Hoa and Doan Thi Van Table 2. The thermodynamic quantities for W thin film Thermodynamic T(K) quantities Layers 200 500 800 1500 2000 2500 10 2.6712 2.6743 2.6774 2.6848 2.6902 2.6957 a, ( A0 ) SMM 30 2.6787 2.6815 2.6844 2.6913 2.6963 2.7014 70 2.6838 2.6863 2.6890 2.6955 2.7003 2.7050 200 2.6848 2.6875 2.6902 2.6966 2.7013 2.7060 10 0.5055 0.6469 0.6697 0.6843 0.6902 0.6958 30 0.4116 0.4823 0.4942 0.5037 0.5089 0.5145 α× 10−5, (K−1) SMM 70 0.3506 0.3754 0.3802 0.3864 0.3912 0.3967 200 0.3355 0.3489 0.3519 0.3573 0.3620 0.3674 [15] Bulk 0.41 0.46 0.48 0.56 0.64 — Figure 5. Thickness dependence of the thermal expansion coefficients for Fe thin film 3. Conclusion The SMM calculations are performed using the effective pair potential for Fe and W thin metal films. We used simple potentials because the purpose of the present study was to gain a general understanding of the effects of the anharmonicity of the lattice vibration and temperature on thermodynamic properties for BCC thin metal films. 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