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A Method for Calculating Surface Stress and Elastic Constants by Molecular Dynamics

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A Method for Calculating Surface Stress and Elastic Constants by Molecular Dynamics A METHOD FOR CALCULATING SURFACE STRESS AND ELASTIC CONSTANTS BY MOLECULAR DYNAMICS Satoshi Izumi Dept. of Mechanical Engineering, Univ. of Tokyo., JAPAN Shotaro Hara Dept. of Mechanical Engineering, Univ. of Tokyo., JAPAN Tomohisa Kumagai Dept. of Mechanical Engineering, Univ. of Tokyo., JAPAN Shinsuke Sakai Dept. of Mechanical Engineering, Univ. of Tokyo., JAPAN Abstract 1 Introduction For nano-scale thin film, effects of The films used in the manufacture surface and interface, which can be ig- of semiconductors are less than 10 nm nored on a macro scale, become impor- thick. At this thickness, the effects of tant. For example, surface energy and surface and interface, which can be ig- stress are key parameters for predicting nored on a macro scale, become impor- the intrinsic stress of thin films. Sev- tant. For example, surface (interface) eral researchers have reported that the energy and stress are key parameters elastic constants of thin films are dif- for predicting the intrinsic stress of thin ferent from those of bulks. We have films, though the stress depends also on recently proposed new definitions and the microstructures and growth mode at calculation methods regarding surface atomic level[1][2]. However, it is difficult stress and elastic constants for thin films to obtain values of surface energy and by extending Martin's method, which surface stress experimentally. Therefore, is useful for obtaining the internal dis- numerical evaluation by molecular sim- placement and elastic constants within ulation has been attempted. In addi- the framework of the molecular dynam- tion, several researchers have reported ics method. We applied our method that the elastic properties (e.g. Young to nano-scale thin films of crystal and module) of thin films are different from amorphous silicon. The effects of sur- those of bulks[3][4]. In particular, the face reconstructions and the width of the elastic properties of films with thickness surface are also investigated. in the range of several nm may exceed the range of continuum approximation. Keywords: Molecular Dynamics, Therefore, it has become very important Elastic constants, Surface Energy, to investigate the limit of continuum ap- Surface Stress proximation and to predict unique phe- nomena in that thickness range. In cal- Free culating the elastic properties of inho- surface mogeneous surface structures, the effect of internal displacement, which is non- Periodic boundary linear atomic displacement in response to deformation, must be taken into ac- Periodic boundary count. We have recently proposed new z y definitions and calculation methods re- x garding surface stress and surface elas- Surface model tic constants for thin films with free sur- faces through an extension of Martin's Figure:1 Surface model method[5][6]. This method is useful for obtaining the internal displacement and −1 ^ ^ ^ elastic constants within the framework F = hh0 , where the subscript 0 means of the molecular dynamics. We applied the state before deformation. In-plane our method to nano-scale thin films of Green-Lagrange strain η^ can be written crystal and amorphous silicon. In order in Eq. (2). to investigate the depth of the surface h h effect, the local atomic elastic constants ^ = 11 12 (1) h12 h22 are newly defined. Those effects on the elastic properties of whole thin films are ¤ ¢ − ¢ t t 1 ¡ 1 1 £ also discussed. ^ = ( ^ ^ − ^£ ) = (^ ^ ^ − ^); 2 2 0 0 t 2 The definitions and calculation (G^ = h^ h^) (2) methods of surface energy, sur- face stress and surface elastic The definitions are represented for- constants mally by three-dimensional notations, i.e. Eq. (3). Definition of surface In order to define the surface of a F^11 F^12 0 η^11 η^12 0 ¢ molecular dynamics system, we pre- ¡ ^ = F^21 F^22 0 ; ^ = η^21 η^22 0 pared a thin film model with a free 0 0 0 ! 0 0 0 ! boundary condition for the z-direction (3) and two periodic boundary conditions Definition of internal displace- for the x- and y-directions as shown in ment vector Fig. 1. Therefore, the evaluation area Although in the elastic theory the dis- is made up of two surfaces that face placements of material points become each other. The film must be enough linear to a deformation, this is not the thick to prevent interference of these two case with atomic displacements. In surfaces. It should be noted that the an atomic system, each atom moves to surface reconstructions greatly influence its most stable point in response to a the surface properties. deformation. This difference, between Definition of strain the displacement of continuum body ap- Since the z-direction is not subject proximation and that of the atoms, is to a periodic boundary condition, the called internal displacement. shape matrix of a MD cell must be writ- Distance vector rαβ between atoms α ten in Eq. (1). Therefore in-plain de- and β after a deformation can be writ- formation gradient tensor is defined by ten by the sum of the homogeneous de- formation term (the first term on the Definition of surface stress right-hand side of Eq. (4)) and the Surface stress is the variation in total relative displacement term (the second surface energy (Eγ) per unit area with term), where the subscript 0 symbolizes respect to the strain η^ij . This variation the state before deformation. corresponds to the stress generated by creation of the surface. ¢ αβ ^ αβ β α ¡ = 0 + ¡ − : (4) Now internal displacement vector is 1 d(A(η^)γ(η^)) α t f (η^) = ; ^ ^ α ij 0 ξ F u ¡ 0 newly defined by = as a rota- A0 dη^ij ¢ ^ ^ =0; =0 tional invariant variable. By using this (i; j = 1; 2) (7) definition, rotational invariant distance αβ αβ t αβ 0 s = (r ) r can be represented as η^ means that all components except follows: for the differentiating component are set to zero. Unlike thin-film stress, sur- β α face stress does not depend on thickness, ¡ αβ αβ t αβ αβ ¢ ¢ ^ £ ^ − ^ s = ( 0 ) (2 + ) 0 + 2( ) 0 since the surface effect is not divided by β α − β α + (ξ^ − ξ^ )(2η^ + I) 1(ξ^ − ξ^ ): volume but by surface area as in Eq. (7). Therefore, it can be said that sur- (5) face stress is an intrinsic property of the From Eq. (5), the variation in the in- surface. teratomic distance in response to the de- If γ(") is not dependent on the strain, formation F^ (averaged strain η^) can be which is the case with liquid, derived. ¡ ¡ It should be noted that displacements ¡ 1 fij (^) = · A0δij γ(^) = γ(^)δij : (8) in the z-direction are included in the in- A0 ^α ternal displacement vector ξ . If γ(") is dependent on the strain, Definition of surface energy which is the case with solid, Surface energy γ is the energy varia- ¡ tion per unit area due to the surface cre- ¡ ¡ surf ¡ A(^) dγ(^) ation. It is defined by Eq. (6). E fij (^) = γ(^)δij + ¡ : (9) A d^ and Ebulk are the energy of the system 0 ij with and without the surface, respec- The second term is the characteristic tively. The latter value must be esti- term for a solid only. mated separately by a bulk model. A is When we apply this definition to the surface area of two free surfaces. molecular dynamics, Eq. (10) is ob- The strain is defined so that the cell tained by using Eγ. size in the zero-strain state is equal to that of the zero-stress state of a bulk; γ γ ¡ 1 dE 1 @E fij (^) = that is, the strain corresponding to the = 0 0 ¡ ¡ 0 0 ¢ A0 dη^ij ¢ ^ A0 @η^ij ^ lattice parameter at a bulk stress of zero ^ ; =0 ^ ; =0 is defined as zero. surf ¡ 1 @E (^) = 0 ¡ 0 A0 @η^ij ¢ ^ ^ ; =0 ¡ ¡ ¡ 1 surf bulk − bulk ¡ γ(^) = ¡ E (^) E (^) @E (^) A(^) − 0 ¡ ¡ 0 γ @η^ij ¢ ^ E (^) ^ ; =0! = ¡ (6) ¡ ¡ A(^) = f surf (^) − f bulk(^) (i; j = 1; 2): (10) ij ij Since the stress does not depend on d2Ebulk(η^) − 0 the internal displacement[5], total differ- dη^ dη^ ^ 0 ^ ij kl η ;ξ =0! entiations can be replaced by partial dif- surf η − bulk η ferentiations. = dijkl (^) dijk l (^) (14) bulk fij (η^) must be estimated separately by the bulk model. In general, the stress ∗ tensor of the bulk model is defined by dsurf = (d0 )surf + (d )surf using the shape matrix shown below. ijkl ijkl ijkl (15) h11 h12 h13 ∗ 1 surf α αβ β = h h h (11) (dijkl) = − DijmgmnDkln (16) 12 22 23 A0 h13 h23 h33 ! 2 surf 0 surf 1 @ E A different strain tensor η and a differ- (dijkl) = 0 (17) α A0 @η^ @η^ ^ 0 ^ ent internal displacement vector ξ are ij kl η ;ξ =0 defined by using h. However, since the 2 surf α @ E stress is not dependent on the internal Dijk = (18) ^α 0 @η^ij @ξ ^ 0 ^ displacement, the definition by using η k η ;ξ =0 and that by using η^ become equivalent 2 surf αβ @ E to each other in the case of i; j = 1; 2. Eij = ^α ^β 0 @ξ @ξ ^ 0 ^ i j η ;ξ =0 αβ β γ αγ bulk ; gij Ej k = δ δik (19) ¡ 1 @E f bulk(^) ; (i; j = 1; 2) ij = 0 ¡ 0 A0 @η^ij ¢ ^ ^ ; =0 0 surf ∗ surf (dijkl) and (dijkl) are referred 1 @Ebulk to as local and relaxation term, respec- = (12) αβ ¡ 0 α A0 @ηij 0 ¢ ; =0 tively. Dijk and Eij correspond to the piezoelastic constants and the force Therefore, the stress tensor can be de- constants between atoms, respectively. fined by Eq. (13), where we assume that Since the elastic constants depend on the volume is V0 = A0Lz=2 and m indi- the internal displacement, and since dif- cates bulk or surf.
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