Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films with body-centered cubic structure

106 HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0035 Natural Sciences 2018, Volume 63, Issue 6, pp. 106-118 This paper is available online at TEMPERATURE AND PRESSURE-DEPENDENT OF THERMAL EXPANSION COEFFICIENTS OF METALLIC THIN FILMS WITH BODY-CENTERED CUBIC STRUCTURE Duong Dai Phuong Military College of Tank Armour Officer Abstract. The moment method in statistical dynamics (SMM) is used to study of thermal expansion coefficients of metallic thin films with body-centered cubic (BCC) structure taking into account the anharmonicity effects of the lattice vibrations and hydrostatic pressures. The explicit expressions coefficients of thermal expansion of metallic thin films are derived in closed analytic forms in terms of the power moments of the atomic displacements. Numerical calculations of the coefficients of thermal expansion have been performed for Fe and W thin films are found to be in good and reasonable agreement with the laws of other authors and approach the experimental values of bulk. The effective pair potentials work well for the calculations of BCC metallic thin films. Keywords: Moment method, thin films, hydrostatic pressures, equation of state. 1. Introduction Film materials have become interesting for recent years. The knowledge about the thermodynamic properties of metallic thin film, such as heat capacity, coefficient of thermal expansion is of great important to determine the parameters for the stability and reliability of the manufactured devices [1, 2]. There are many ways to determine the behaviors deformed of thin film such as x-ray diffraction [3-5]. However, rarely research has been known about the thermodynamic properties of metallic free-standing thin films. Most of the previous theoretical studies, however, are concerned with the materials properties of metallic thin film at low temperature, temperature and pressure dependence of the thermodynamic quantities has not been studied in detail. In general, the investigating dependence on pressure of thin film almost at low-pressure. There are many studies about the dependence on pressure of the thin film deposited on a substrate [6, 7]. Most of the researches of thin film are used experiment methods [4-7] but very few studies of them are used theoretical method. The influence of oxygen pressure on the growth of (Ba0.02Sr0.98)TiO3 thin film on MgO substrate by pulsed laser deposition techniques have been investigated in the oxygen pressure range from 40 to 10 -3 Pa [8]. At lower oxygen pressure, more high energy particles will arrive to the substrate. Most previous studies of thin films have been done in non-metal thin films Received August 10, 2018. Revised August 22, 2018. Accepted August 29, 2018. Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films 107 and not free-standing metallic thin films. In order to understand the pressure dependence of the of thermal expansion coefficient of metallic thin films, it is highly desirable to establish an analytical method which enables us to evaluate the free energy of the system taking into account both the anharmonicity and quantum mechanical effect of the lattice vibration. In this article, we present an analysis which can be used to extract metallic thin films, as a function of the pressure. The explicit expressions of the thermal expansion coefficient are derived in terms of the hydrostatic pressure P and the temperature T. The numerical calculations are performed for Fe and W metallic thin films and compared with the laws of other authors and the bulk values. 2. Content 2.1. Theory 2.1.1. Pressure versus volume relation Let us consider a metal free standing thin film has *n layers with the film thickness d. Suppose of the thin film has been consisting two atomic surface layers, two next surface atomic layers and ( *n 4 ) atomic internal layers, with Nng surface layer atoms, Nng1 next surface layer atoms and Ntr internal layers atoms (see Figure 1). Figure 1. The metallic free standing thin film For the surface layer atoms of thin film, we will derive the pressure versus volume relation of crystals limiting only quadratic terms in the atomic displacements. The expression for the free energy in the harmonic approximation is determined as [9, 10].   ng2xng ng ng 0 ng ng ng 0 i0 i ,ng i 1 3N u x ln(1 e ) , 6 u a .              (1) Th ic kn es s (n *- 4 ) L ay er s d a a a ng ng 1 tr Duong Dai Phuong 108 The hydrostatic pressure P is determined by [9, 10] 3 ng ng ng ng ng ngT T a P . V V a                     (2) From Eqs. (1) and (2), we obtain equation of state denotes the pressure versus volume relation of the lattice as ng ng0 ng ng ng ng ng ng ng ku1 1 PV a x coth x , 6 a 2k a            (3) where, 2 ng ng ng ng ng ng x x k a k a      , ng ngx 2    , Bk T  and ng is the atomic volume ng ng ng V N   of the crystal, for the BCC lattice of the metal thin films 34 3 3 ng ng a   . Using eq. (3), one can find the nearest neighbour distance nga at pressure P and temperature T. However, for numerical calculations, it is convenient to determine firstly the nearest neighbour distance nga ( P,0 ) at pressure P and at absolute zero temperature T = 0. For T = 0 temperature, eq. (3) is reduced to ng ng ng0 ng ng ng ng ng ku1 PV a . 6 a 4k a           (4) For simplicity, we take the effective pair interaction energy in metal systems as the power law, similar to the Lennard-Jones potential   n m o or rDr m n , ( n m ) r r                    (5) where D, m, n, r0 are determined by fitting the experimental data (e.g., cohesive energy and elastic modulus). For BCC of the metallic thin films we take into account the first nearest and second nearest neighbour interactions. Using the effective pair potentials of Eq. (5), it is straitforward to get the interaction energy ng 0u and the parameter ngk in the crystal as [9, 10] n m ng o o 0 ng ,n ng ,m ng ng r rD u mA nA ( n m ) a a                     (6) 2 ng ng 2 i i eq 1 k 2 u          2 2 ix ix n m a a 2o o ng ,n 4 ng ,n 2 ng ,m 4 ng ,m 2 0 ng2 ng ng ng r rDnm ( n 2 )A A ( m 2 )A A m , 2a ( n m ) a a                                    (7) Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films 109 where 2 2 4 4 2 2 2 2 ng ,ix ng ,ix ng ,ix ng ,ix ng ,ix ng ,iy ng ,ix ng ,iya a a a a a a a ng ,n ng ,m ng ,n ng ,m ng ,n ng ,m ng ,n ng ,mA ,A ,A ,A ,A ,A ,A ,A , ... are the structural sums of surface layers atoms for the given crystal and given by [9, 10] ng ng i i ng ,n ng ,mng ,n ng ,m i ii i Z Z A ; A ;      2 2 ng ,ix ng ,ix ng 2 ng 2 a ai ,x ng ,ix i ,x ng ,ix ng ,n ng ,m2 ng ,n 2 ng ,m i ing i ng i Z a Z a1 1 A ; A ; a a     (8) 4 4 ng ,ix ng ,ix ng 4 ng 4 a ai ,x ng ,ix i ,x ng ,ix ng ,n ng ,m4 ng ,n 4 ng ,m i ing i ng i Z a Z a1 1 A ; A ; a a     2 2 2 2 ng ,ix ng ,iy ng ,ix ng ,iy ng 2 2 ng 2 2 a a a ai ,xy ng ,ix ng ,iy i ,xy ng ,ix ng ,iy ng ,n ng ,m4 ng ,n 4 ng ,m i ing i ng i Z a a Z a a1 1 A ; A ; a a     here ng iZ is the coordination number of i-th nearest neighbour atoms of surface layers with radius ng ir (for BCC lattice of thin films) ng ng k k ngr a . Similar for next surface layers atoms and internal layers atoms of the system, we determine the interaction energy 1 0 0, ng tru u , the parameter 1,ng trk k and structural sums 2 2 4 4 2 2 2 2 ng 1,ix ng 1,ix ng 1,ix ng 1,ix ng 1,ix ng 1,iy ng 1,ix ng 1,iy 22 2 4 4 tr ,ix tr ,tr ,ix tr ,ix tr ,ix tr ,ix a a a a a a a a ng1,n ng1,m ng1,n ng1,m ng1,n ng1,m ng1,n ng1,m a aa a a a tr ,n tr ,m tr ,n tr ,m tr ,n tr ,m tr ,n A ,A ,A ,A ,A ,A ,A ,A , A ,A ,A ,A ,A ,A ,A 2 2 2 iy tr ,ix tr ,iya a tr ,m,A . (9) For surface layers and next surface layers, we have considered to the surface effect - the defect on surface layers atoms and next surface layers atoms of the thin films. Using for two first coordination sphere, we obtain the structural sums as follows: , 2 , 22 22 2 , , 5 5 8 , 8 , 4 4 3 3 ng ng k k ng n ng mn mn m k kng k ng k Z Z A A                           2 2 , ,2 2 , 4 , , 4 ,4 42 4 2 4 , , 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k ng ng k Z Z A a A a a a                          2 2 , ,2 2 , 6 , , 6 ,6 62 6 2 6 , , 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k tr ng k Z Z A a A a a a                          , 4 , 44 44 4 , , 5 5 8 , 8 , 4 4 3 3 ng ng k k ng n ng mn mn m k kng k ng k Z Z A A                           Duong Dai Phuong 110 4 4 , ,4 4 , 8 , , 8 ,8 84 8 4 8 , , 1 8 32 1 8 32 , , 9 94 4 9 9 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k ng ng k Z Z A a A a a a                          2 2 2 2 , , , , 2 2 2 2 , , , , , , , 8 , 84 8 4 8 , , 1 8 1 8 , . 9 9 ng kx ng ky ng kx ng ky ng ng a a a ak xy ng kx ng ky k xy ng kx ng ky ng n ng mn m k kng ng k ng ng k Z a a Z a a A A a a          1 1 1, 2 1, 22 22 2 1, 1, 5 5 8 , 8 , 4 4 3 3 ng ng k k ng n ng mn mn m k kng k ng k Z Z A A                           2 2 1 , 1 , 1 1 2 2 1, 4 1, 1, 4 1,4 42 4 2 4 1 1, 1 1, 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k ng ng k Z Z A a A a a a                          2 2 1 , 1 , 1 1 2 2 1, 6 1, 1, 6 1,6 62 6 2 6 1 1, 1, 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k tr ng k Z Z A a A a a a                          1 1 1, 4 1, 44 44 4 1, 1, 5 5 8 , 8 , 4 4 3 3 ng ng k k ng n ng mn mn m k kng k ng k Z Z A A                           4 4 1 , 1 , 1 1 4 4 1, 8 1, 1, 8 1,8 84 8 4 8 1 1, 1 1, 1 8 32 1 8 32 , , 9 94 4 9 9 3 3 ng kx ng kx ng ng a akx kx ng n ng kx ng m ng kxn mn m k kng ng k ng ng k Z Z A a A a a a                          2 2 2 2 1, 1, 1, 1, 1 2 2 1 2 2 , 1, 1, , 1, 1, 1, 8 1, 84 8 4 8 1 1, 1 1, 1 8 1 8 , . 9 9 ng kx ng ky ng kx ng ky ng ng a a a ak xy ng kx ng ky k xy ng kx ng ky ng n ng mn m k kng ng k ng ng k Z a a Z a a A A a a          , 2 , 22 22 2 , , 6 6 8 , 8 , 4 4 3 3 tr tr k k tr n tr mn mn m k ktr k tr k Z Z A A                           2 2 , ,2 2 , 6 , , 6 ,6 62 6 2 6 , , 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 tr kx tr kx tr tr a akx kx tr n tr kx tr m tr kxn mn m k ktr tr k tr tr k Z Z A a A a a a                          Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films 111 2 2 , ,2 2 , 4 , , 4 ,4 42 4 2 4 , , 1 8 8 1 8 8 , , 3 34 4 3 3 3 3 tr kx tr kx tr tr a akx kx tr n tr kx tr m tr kxn mn m k ktr tr k tr tr k Z Z A a A a a a                          , 4 , 44 44 4 , , 6 6 8 , 8 , 4 4 3 3 tr tr k k tr n tr mn mn m k ktr k tr k Z Z A A                           4 4 , ,4 4 , 8 , , 8 ,8 84 8 4 8 , , 1 8 32 1 8 32 , , 9 94 4 9 9 3 3 tr kx tr kx tr tr a akx kx tr n tr kx tr m tr kxn mn m k ktr tr k tr tr k Z Z A a A a a a                          2 2 2 2 , , , , 2 2 2 2 , , , , , , , 8 , 84 8 4 8 , , 1 8 1 8 , . 9 9 tr kx tr ky tr kx tr ky tr tr a a a ak xy tr kx tr ky k xy tr kx tr ky tr n tr mn m k ktr tr k tr tr k Z a a Z a a A A a a          From eqs. (4), (6), and (8) we obtain equation of state for surface layers of the metallic thin films with body-centered cubic structure at zero temperature. n m 0 0 ng ng ,n ng ,m ng ng ng 0 r rDnm 1 Dnm PV A A . 6( n m ) a a a 2( n m )4 m                      2 2 , , 2 2 , , 0 0 , 4 , 2 , 4 , 2 0 0 , 4 , 2 , 4 , 2 ( 2) ( 2) ( 2) ( 2) . ( 2) ( 2) ng ix ng ix ng ix ng ix n m a a ng n ng n ng m ng m ng ng n a a ng n ng n ng m ng m ng ng r r n n A A m m A A a a r r n A A m A A a a                                                               . m       (10) Eq. (10) can be transformed to the form n 4 m 4 3ng ng 4ng ng3 n 3 m 3 0 1ng ng 2ng ng n m 5ng ng 6 ng ng c y c y4 P r c y c y . 3 3 c y c y          (11) where   0 ng ng r y , a P,0  1ng ng ,n Dnm c A , 6( n m )   2ng ng ,m Dnm c A , 6( n m )   2 ng ,ixa 3ng ng ,n 4 ng ,n 2 00 Dnm 1 c ( n 2 ) ( n 2 )A A , 2( n m ) r4 m          2 ng ,,ixa 4ng ng ,m 4 ng ,m 2 00 Dnm 1 c ( m 2 ) ( m 2 )A A , 2( n m ) r4 m          Duong Dai Phuong 112   2 ng ,ixa 5ng ng ,n 4 ng ,n 2c n 2 A A ,      2 ng ,ixa 6ng ng ,m 4 ng ,m 2c m 2 A A .    Similar derivation can be also done for next surface layers atoms and internal layers atoms of the equation of state as n 4 m 4 3ng1 ng1 4ng1 ng13 n 3 m 3 0 1ng1 ng1 2ng1 ng1 n m 5ng1 ng1 6 ng1 ng1 c y c y4 P r c y c y , 3 3 c y c y          (12) n 4 m 4 3 n 3 m 3 3tr tr 4tr tr 0 1tr tr 2tr tr n m 5tr tr 6tr tr c y c y4 P r c y c y . 3 3 c y c y         (13) In principle Eqs. (11), (12) and (13) permit to find the nearest neighbour distance nga ( P,0 ) , ng1a ( P,0 ) and tra ( P,0 ) for surface layers, next surface layers and internal layers atoms at zero temperature and pressure P . 2.1.2. Thermal expansion coefficient of metallic thin films For surface layers of metallic thin films, the calculation of the lattice spacing of metallic thin films at finite temperature and pressure P , the fourth order vibrational constants ng and ngk at pressure P and T = 0K are defined by [9, 11]   4 ng 4 ng io io ng 1ng 2ng4 2 2 i i i ieq eq 1 6 4 , 12 u u u                              (14) 4 ng io 1ng 4 i i eq 1 , 48 u            (15) 4 ng io 2ng 2 2 i i i eq 6 ' 48 u u             (16) 2 ng 2io ng 0 ng2 i i eq 1 k m . 2 u            (17) Using the effective pair potentials of eq. (5), the parameter ng 1ng 2ng ng, , ,k   of the BCC metallic thin films have the form               4 2 2 , , , 2 , 4 2 2 , , , , 8 , 8 0 4 , 6 , 4 , 8 , 8 4 , 2 4 6 61 ( ) 12 18( 2)( 4) 9( 2) 2 4 6 61 12 18( 2)( 4) ng ix ng ix ng iy ng ix ng ix ng ix ng iy a a a ng n ng n n ng ang ng ng n ng n a a a ng m ng m ng ng m n n n A A rDmn n m a a n n A n A m m m A ADmn n m a m m A                                 2 , 0 6 , 4 ( ) , 9( 2)ng ix m a ng ng m r a m A            (18) Temperature and pressure-dependent of thermal expansion coefficients of metallic thin films 113       2 2 , ,0 0 , 4 , 2 , 4 , 22 2 2 . 2 ng ix ng ix n m a a ng ng n ng n ng m ng m ng ng ng r rDmn k n A A m A A n m a a a                                   (19) Similar derivation can be also done for next surface layers atoms and internal layers atoms, we obtain the values of ng1 ng1,k , tr tr,k . Using the obtained results of nearest neighbour distance for surface layers atoms nga ( P,0 ) and Eqs. (18) and (19), we find the values of parameters ngk ( P,0 ) , and ng( P,0 ) at pressure P and T = 0 K. Similar derivation can be also done for next surface layers atoms and internal layers atoms, we find the values of parameters ng1 trk ( P,0 ),k ( P,0 ), and ng1 tr( P,0 ), ( P,0 ),  at pressure P and T = 0K. For surface layers of metallic thin films, the thermally induced lattice expansion ng 0y ( P,T ) at pressure P and temperature T is given in a closed formula using the force balance criterion of the fourth order moment approximation as [10, 11]     2ngng 0 ng3 ng 2 P,0 y P,T A ( P,T ), 3k ( P,0 )    (20) where                                      2 2 3 3 ng ngng ng ng ng 1 2 34 6 ng ng 4 4 5 5 6 6 ng ng ngng ng ng 54 68 10 12 ng ng ng P,0 P,0 A P,T a a a k P,0 k P,0 P,0 P,0 P,0 + a a a . k P,0 k P,0 k P,0 (21) In Eq. (21), using ng ng ngX x cothx , one can find the values of parameters as [9]       ngng ng 2 3 ng ng ng1 2 X 13 47 23 1 a 1 ; a X X X , 2 3 6 6 2            ng 2 3 4 ng ng ng ng3 25 121 50 16 1 a X X X X , 3 6 3 3 2      ng 2 3 4 5ng ng ng ng ng4 43 93 169 83 22 1 a X X X X X , 3 2 3 3 3 2              ng 2 3 4 5 6 ng ng ng ng ng ng5 103 749 363 391 148 53 1 a X X X X X X , 3 6 2 3 3 6 2        ng 2 3 4 5 6 7ng ng ng ng ng ng ng6 561 1489 927 733 145 31 1 a 65 X X X X X X X , 2 3 2 3 2 3 2 ng ng ( P,0 ) x 2    , ng ng 0 k ( P,0 ) ( P,0 ) m   . (22) Similar derivation can be also done for next surface layers atoms and internal layers atoms of the average atomic displacement. Duong Dai Phuong 114     2ng1ng1 0 ng13 ng1 2 P,0 y P,T A ( P,T ), 3k ( P,0 )    (23)     2trtr 0 tr3 tr 2 P,0 y P,T A ( P,T ). 3k ( P,0 )    (24) So, for surface layers atoms we can find the nearest neighbour distance nga ( P,T ) at pressure P and temperature T as ng ng ng 0a ( P,T ) a ( P,0 ) y ( P,T )  . (25) Thus, for next surface layers atoms and internal layers atoms the nearest neighbour distance are determined as ng1 ng1 ng1 0a ( P,T ) a ( P,0 ) y ( P,T )  (26) tr tr tr 0a ( P,T ) a ( P,0 ) y ( P,T )  . (27) The average nearest neighbor distance of thin film at pressure P , and temperature T and zezo temperature are determined as [10, 12]      *1 * 2 , 2 , ( 5) , ( , ) . 1 ng ng tra P T a P T n a P T a P T n      (28)      *1 * 2 ,0 2 ,0 ( 5) ,0 ( ,0) . 1 ng ng tra P a P n a P a P n      (29) The average thermal expansion coefficient of metallic thin films can be calculated as [10, 11]      1 1 1, , ,0 ng ng ng ng ng ng trB d d d d da P Tk a P d d            (30) where            10 0 0 1 1 , , , ; ; ,0 ,0 ,0 tr ng ng B B B tr ng ng tr ng ng y P T y P T y P Tk k k a P a P a P                (31) here ngd and 1ngd are the surface layers and next surface layers thickness. One can now apply the above formular to study thermal expansion coefficient of BCC metallic thin films under hydrostatic pressures. 2.2. Numerical results and discussion In order to check the validity of the present moment method for study of thermal expa