Investigation Young’s modulus of metallic thin films by statistical moment method

111 JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0038 Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 111-121 This paper is available online at INVESTIGATION YOUNG’S MODULUS OF METALLIC THIN FILMS BY STATISTICAL MOMENT METHOD Duong Dai Phuong 1 and Nguyen Thi Hoa 2 1 Tank Armour Officers Training School, Tam Duong, Vinh Phuc 2Hanoi University of Transport and Communications Abstract. The Young’s modulus of metallic thin films with body-centered cubic (BCC) structure at zero pressure are investigated using the statistical moment method (SMM), including the anharmonicity effects of thermal lattice vibrations. The Helmholtz free energy, mean-square atomic displacements and Young’s modulus E are derived in closed analytic forms in terms of the power moments of the atomic displacements. Numerical calculations for Young’s modulus of W and Nb thin films are found to be in good and reasonable agreement with those of other theoretical results and experimental data. This research proves that Young’s modulus of thin films approach the values of bulk when the thickness of thin films is about 70 nm. Keywords: Metallic thin films, statistical moment method, thermodynamic properties, Young’s modulus. 1. Introduction Deformation is one of the most important considerations in structural applications of solid materials, especial in thin film. In recent years, this problem has interested many researchers [1, 2]. In the many approaches of thin films, investigating metallic thin film is one of the most important and it is enormous scientific interest. Metallic thin film is a common geometry for metallic applications, mainly due to their attractive novel properties for technological applications [3-5]. There are many ways to determine the behaviors deformed of thin film such as x-ray diffraction [6-9] and nanoindentation. However, very little is known about the thermomechanical properties of metallic free-standing thin films for both amorphous semiconductors and polycrystalline thin film. In addition, few theoretical or experimental systematic studies have been done to explain the thermomechanical properties of metallic free-standing thin films and there is a lack of experimental data. Most previous theoretical studies, however, are concerned with the materials properties of metallic thin film at low temperature, and temperature dependence has not been studied in detail. It is well known that the thermomechanical properties of thin films are strongly related to the film structure such as defects, network strain and dislocation but a complete study relating these properties has not yet been reported. The main purpose of this article is to provide an analysis of the Young’s modulus of metallic free-standing thin film using the analytic statistical moment method [10-12]. Received September 1, 2017. Accepted September 28, 2017. Contact Duong Dai Phuong, email: vanha318@yahoo.com Duong Dai Phuong and Nguyen Thi Hoa 112 The Young’s modulus is derived from the Helmholtz free energy, and the explicit expression of the Young’s modulus 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 Young’s modulus have also been studied. We compared the results of the present calculations with those of previous theoretical calculations as well as with available experimental results. 2. Content 2.1. Theory 2.1.1. The anharmonic oscillations of metallic thin films Let us consider a metal free standing thin film that has *n layers with the thickness d . We assume the thin film consists of two atomic surface layers, two next to surface atomic layers and ( *n 4 ) atomic internal layers. (see Fig. 1). Fig. 1. The free-standing thin film For internal layers atoms of thin films, we present the statistical moment method (SMM [12- 13]) formulation for the displacement of the internal layers atoms of the thin film try as a solution of the equation. 2 2 3 2 3 1 0tr trtr tr tr tr tr tr tr tr tr tr tr tr d y dy y y k y ( x coth x )y p , dp kdp              (1) where 2 2 , 02 1 ; ; , , 2 2 tr tr io tr i tr p tr B tr tr i i eq y u x k T k m u                  4 tr io 1tr 4 i i eq 1 , 48 u            4 tr io 2tr 2 2 i i i eq 6 , 48 u u                4 tr 4 tr io io tr 1tr 2tr4 2 2 i i i ieq eq 1 6 4 , 12 u u u                              Th ic kn es s (n *- 4 ) L ay er s d a a a ng ng 1 tr Investigation Young’s modulus of metallic thin films by statistical moment method 113 where Bk is the Boltzmann constant, T is the absolute temperature, 0m is the mass of atom, tr is the frequency of lattice vibration of internal layers atoms; trk , 1tr , 2tr , tr are the parameters of crystal depending on the structure of crystal lattice and the interaction potential between atoms; 0 tr i is the effective interatomic potential between 0 th and i th internal layers atoms; iu  , iu  , iu  are the displacements of i th atom from equilibrium position on direction ( , , ), ( , , ), ( , , )x y z x y z x y z        , respectively, and the subscript eq indicates evaluation at equilibrium. In the second approximation of the supplemental force, the solutions of the equation (1) can be expanded in the power series of the supplemental force p as [12, 13] 2 0 1 2 . tr tr tr try y A p A p   (2) Here, 0 try is the average atomic displacement in the limit of zero of supplemental force p . Substituting the above solution of Eq. (2) into the original differential Eq. (1), one can get the coupled equations on the coefficients 1 trA and 2 trA , from which the solution of 0 try is given as 2 tr tr 0 tr3 tr 2 y A , 3k    (3) where                2 2 3 3 4 4 5 5 6 6 tr tr tr tr tr trtr tr tr tr tr tr 1 2 3 4 5 64 6 8 10 12 tr tr tr tr tr A a a a + a a a , k k k k k with    tr a ( 1,2...,6) being the values of parameters of crystal depending on the structure of crystal lattice. Similar derivations can be also done for next surface layers atoms of thin film, and their displacement are solution of equations, respectively 2 1 12 3 1 1 1 1 1 1 1 1 1 1 12 1 3 1 0 ng ng ng ng ng ng ng ng ng ng ng ng ng ng d y dy y y k y ( x coth x )y p dp kdp              (4) 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 where ng ng iy u  is the solution of equation 2 2 1 0 i ng ing ng a ng i a ng a ng ng ng u k u u ( x coth x ) a , a m                      (5) where , , 2 ngng ng i a ng By u x k T           2 2 20 0 3 0 0 , 2 ng ng ng i ix i ng i k a m        Duong Dai Phuong and Nguyen Thi Hoa 114 3 3 3 2 , , , , , 1 . 4 ng ng io io ng ng i i ng i ng ieq eq u u u                                    In the second approximation of the supplemental force, the solutions of equation (5) can be expanded in the power series of the supplemental force p as 2 0 1 2 ng ngy y A a A a   , (6) here, 0 try is the average atomic displacement in the limit of zero of supplemental force p . The solution of 0 try is given as 0 2 ngng ng ng ng y x coth x . k     (7) Thus, the power moments of the atomic displacement can be got by using the statistical moment method. 2.1.2. Free energy of metallic thin films with body-centered cubic structure Usually, theoretical studies of the size effect have been carried out using the introducing the surface energy contribution in the continuum mechanics or by computational simulations reflecting the surface stress, or surface relaxation influence. In this paper, the influence of the size effect on thermodynamic properties of the 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, the free energy of these layers        2 2 2 1 0 22 3 2 2 2 1 1 24 3 2 3 1 1 3 2 6 4 1 2 2 1 1 3 2 2 trxtr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr N X U N x ln e X k N X X X X . k                                                (8)        1 2 2 1 1 1 11 2 1 0 1 1 2 1 12 1 3 1 1 12 2 2 1 1 1 1 1 1 2 1 14 1 3 2 3 1 1 3 2 6 4 1 2 2 1 1 3 2 2 ngx ng ng ngng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng ng N X U N x ln e X k N X X X X . k                                                     (9) In Eqs. (8), (9), using tr tr trX x cothx , 1 1 1ng ng ngX x cothx , and    11 10 0 , 0 0 , 1, , 2 2 ngtr tr ng ngtr i i tr i i ng NN U r U r    (10) where ri is the equilibrium position of i th atom, ui is its displacement of the i th atom from the equilibrium position; 0 tr i , 1 0 ng i , are the effective interatomic potential between the 0 th and i th internal layers atom, the 0 th and i th next surface layers atom, ; Ntr, Nng1 and are respectively the number of internal layers atoms, next surface layers atoms and of this thin film; 1 0 0, tr ngU U Investigation Young’s modulus of metallic thin films by statistical moment method 115 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]   20 3 1 ngxngng ng ngU N x ln e         (11) Let us assume that the system consists N atoms with *n layers. If the atom number on each layer is LN , then we have * * .L L N N n N n N    (12) The number of atoms of internal layers, next to surface layers and surface layer atoms are, respectively determined as  *tr L L L L N N n 4 N 4 N N 4N , N            * ng1 L LN 2N N ( n 2 )N    and * ng L LN 2N N ( n 2 )N .    The free energy of the system and of one atom, respectively, are given by  1 1 14 2 2tr tr ng ng ng ng c L tr L ng L ng cN N N TS N N N N TS ,               (13) 1 4 2 2 1 ctr ng ng* * * TS , N n n n N               (14) where cS is the entropy configuration of the system, ng , 1ng and tr are respectively the free energy of one atom at the surface layer, next to surface layer and internal layers. Using a as the average nearest-neighbor distance (NND) and b as the average thickness of two-layers, with body-centered cubic structure of the metal thin film, then we have 3 a b  . The thickness d of thin film can be given by      * * *ng ng1 tr a d 2b 2b n 5 b n 1 b n 1 . 3         (15) From equation (14), we derived * 31 1 . d d n b a     The average NND of thin film * 1 * 2 2 ( 5) . 1 ng ng tra a n a a n      (16) In above equation, nga , 1nga and tra are correspondingly the average NND between two intermediate atoms at the surface layer, next to surface layer and internal layers of thin film at a given temperature T. These quantities can be determined as Duong Dai Phuong and Nguyen Thi Hoa 116 1 0, 0 1 0, 1 0 0, 0, , , ng ng tr ng ng ng ng tr tra a y a a y a a y      (17) where 0,nga , 0, 1nga and 0,tra denote the values of nga , 1nga and tra at zero temperature which can be determined experimentaly or from the minimum condition of the potential energy. Substituting Eq. (15) into Eq. (14), we obtained the expression of the free energy per atom of the metal thin film with body-centered cubic structure as follows: Ψ .ctr ng ng1 TSd 3 3a 2a 2a N Nd 3 a d 3 a d 3 a            (18) 2.1.3. Young’s modulus of metallic thin films For surface layer atoms of metallic thin films, if 0,ng is the Helmholtz free energy for a volume element of the considered system in the case without external force p , ,p ng is the one in the case of external force p , ng is the elastic strain, ng is the stress, and the relation of these quantities can be written in approximate form [13], 2 0 0 2 2 ng ng ng ng p,ng ,ng ,ng E .          (19) In the above expression, using the relation of the stress ng and the strain ng for the Hookean deformation for a volume element of the surface layers atoms of thin metal films. ,ng ng ngE  (20) from Eq. (19), we have , , . p ng p ng ng ng ng ng ng ng E                (21) In the elastic deformation [13], from the definition of the elastic strain ng ng p,ng ng ng ng ng a a a , a a      (22) where, ,; ,ng P nga a are the nearest-neighbor of surface layer atoms of thin metal film at temperature T in the case without external force p and in the case of external force p can be determined as, respectively 0, 0 , 0 , . ng ng ng p ng p ng ng a a y a a y        (23) The relationship between the external force and the stress is given by 2 ,ng ng ng ngp S a    (24) because the difference between the nearest-neighbor of surface layer atoms of thin metal film at zero temperature in the case without external force 0,nga and in the case of external force 0 ,p nga is very small, so 0, 0 , 0, 0ng P ng nga a a    . Investigation Young’s modulus of metallic thin films by statistical moment method 117 where external force p small and unchanged, from Eqs. (22), (23), and (6), in approximate we can rewrite , 0 1 , ng ng p ng ng ng ng ng ng ng a a y y A p a a a       (25) in which, the 1 ngA has form as [11, 13]:   2 2 1 4 2 coth1 1 1 1 coth . 2 ng ng ngng ng ng ng ng x x A x x k k               (26) From Eqs. (23), (24), (25) and (20), we obtain the expression of Young ' s modulus for surface layers atoms of thin metal films  1 1 0, 0 1 1 1 . ng ng ng ng ng ng ng ng ng a E A p a A a y A        (27) A similar derivation can also be done for next to surface layer atoms and internal layer atoms of the Young ' s modulus, respectively   1 1 1 1 1 1 1 1 1 1 0, 1 0 1 1 1 , ng ng ng ng ng ng ng ng ng a E A p a A a y A        (28)  1 1 0, 0 1 1 1 ,tr trtr tr tr tr tr tr tr a E A p a A a y A        (29) in Eqs. (28) and (29), the values 1 1 ngA , 1 trA , have form   2 2 1 1 11 1 1 14 1 1 2 coth1 1 1 1 coth 2 ng ng ngng ng ng ng ng x x A x x k k               (30)   2 2 1 4 2 coth1 1 1 1 coth . 2 tr tr tr tr tr tr tr tr x x A x x k k              (31) The average Young ' s modulus of the metal thin film with body-centered cubic structure is determined as  *1 * 2 2 5 . 1 ng ng trE E n E E n      (32) 2.2 Numerical results and discussion In this section, the derived expressions in the previous section will be used to investigate the Young ' s modulus of metallic thin films with BCC structure for W and Nb at zero pressure. For the sake of simplicity, the interaction potential between two intermediate atoms of these thin films is assumed as the Mie-Lennard-Jones potential which has the form Duong Dai Phuong and Nguyen Thi Hoa 118   0 0( ) , n m r rD r m n n m r r                    (33) where D describes the dissociation energy, 0r is the equilibrium value of r, and the parameters n and m can be determined by fitting experimental data (e.g., cohesive energy and elastic modulus). The potential parameters , ,D m n and 0r of some metallic thin films are showed in Table 1 [15]. Table 1. Mie-Lennard-Jones potential parameters for W and Nb of metallic thin films [15] Metals n m 0 0 ,r A 0/ ,BD k K W 10.5 6.5 2.7365 11278.8 Nb 8.5 5.0 2.8648 8307.3 0 10 20 30 40 50 60 70 3.0850 3.0852 3.0854 3.0856 3.0858 3.0860 3.0862 3.0864 W thin film L a tt ic e c o n s ta n t ( 1 0 -1 0 m ) Thicknees (nm) Fig. 2: Dependence on thickness of the lattice constant at T=300K for W thin film 0 200 400 600 800 1000 3.075 3.080 3.085 3.090 3.095 3.100 3.105 L a tt ic e c o n s ta n t ( 1 0 -1 0 m ) T (K) 10 layers 30 layers 200 layers Fig.3. Temperature dependence of the lattice constant for W thin film Using the expression (16), the average NND of thin film can be determined as a function of thickness and temperature. In Fig. 2, the thickness dependence of the average lattice constant for W of thin film is presented using SMM. These results show that the average lattice constant for W thin film at room temperature increases with increasing thickness. As shown in Fig. 2, when the Investigation Young’s modulus of metallic thin films by statistical moment method 119 thickness is smaller range 70 nm the lattice constant increases with the thickness. On the other hand, when the thickness is larger than 70 nm, the lattice constant is almost independent of the thickness. In Fig. 3, the temperature dependence of the average lattice constant of W thin film is presented as a function of thickness and temperature. The theoretical calculations of the average lattice constant of W thin film are shown with various layer thicknesses. One can see that the value of the average lattice constant 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 strongly depends on the layer thickness, especially at high temperature. The obtained results of dependence on thickness show an agreement between our works and the results presented by X. Zhou, et al., in [14]. 0 500 1000 1500 2000 2500 12 14 16 18 20 22 24 26 28 30 32 34 36 Y o u n g 's m o d u lu s ( 1 0 1 0 G P a ) T (K) 10 layers 30 layers 150 layers Fig. 4. Temperature dependence of the Young ’ s modulus for W thin film The calculated dependence on temperature of Young’s modulus for W thin film at the various layer thicknesses by the SMM are presented in Fig. 4. As shown in Fig. 4, the Young’s modulus decrease with increasing absolute temperature T. These results are in good agreement with the laws of the bulk Young’s modulus of other authors and with the experimental results [14, 16]. Furthermore, at a constant temperature Young ’ s modulus of thin films is smaller than the bulk value of Young ’ s modulus. Our calculated results have been compared with the results presented in [13] for W thin film bulk and show in good agreement. 0 10 20 30 40 50 60 70 80 41.8 42.0 42.2 42.4 42.6 42.8 43.0 Y o u n g 's m o d u lu s (  1 0 1 0 P a ) Thicknees (nm) W - thin film Fig. 5. Dependence on thickness of Young’s modulus for W thin film at T=300K Duong Dai Phuong and Nguyen Thi Hoa 120 In Fig. 5, we present the thickness dependence of the Young’s modulus for W thin film at T=300K , As shown in Fig. 5, when the thickness is smaller than 30 nm the Young’s modulus increased with the thickness and depends strongly on th