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 surfaces through an extension of Martin’s Figure:1 Surface model method[5][6]. This method is useful for obtaining the internal displacement and −1 ˆ ˆ ˆ 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 ηˆ 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 ˆ = 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. ˆ = ( ˆ ˆ − ˆ£ ) = (ˆ ˆ ˆ − ˆ), 2 2 0 0 t 2 The definitions and calculation (Gˆ = hˆ hˆ) (2) methods of surface energy, surface 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

Fˆ11 Fˆ12 0 ηˆ11 ηˆ12 0 ¢ molecular dynamics system, we pre- ¡ ˆ = Fˆ21 Fˆ22 0 , ˆ = ηˆ21 ηˆ22 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 deformation 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 ηîj . This variation the state before deformation. corresponds to the stress generated by

creation of the surface.

αβ ˆ αβ β α ¡ = 0 + ¡ − . (4) Now internal displacement vector is 1 d(A(ηˆ)γ(ηˆ)) α t f (ηˆ) = , ˆ ˆ α ij 0 ξ F u ¡ 0 newly defined by = as a rota- A0 dηîj ¢ ˆ ˆ =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 ηˆ 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

¢ ¢ ˆ £ ˆ − ˆ s = ( 0 ) (2 + ) 0 + 2( ) 0 since the surface eﬀect is not divided by β α − β α + (ξˆ − ξˆ )(2ηˆ + I) 1(ξˆ − ξˆ ). 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(ε) is not dependent on the strain, formation Fˆ (averaged strain ηˆ) can be which is the case with liquid,

derived.

¡ ¡ It should be noted that displacements ¡ 1 fij (ˆ) = · A0δij γ(ˆ) = γ(ˆ)δij . (8) in the z-direction are included in the in- A0 ˆα ternal displacement vector ξ . If γ(ε) 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(ˆ) dγ(ˆ)

ation. It is defined by Eq. (6). E fij (ˆ) = γ(ˆ)δij + ¡ . (9) A dˆ 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 (ˆ) =

that is, the strain corresponding to the = 0 0 ¡

¡ 0 0 ¢ A0 dηîj ¢ ˆ A0 ∂ηîj ˆ lattice parameter at a bulk stress of zero ˆ , =0 ˆ , =0 is defined as zero. surf ¡ 1 ∂E (ˆ) = 0

¡ 0

A0 ∂ηˆij ¢ ˆ ˆ , =0

¡ ¡ ¡ 1 surf bulk − bulk ¡ γ(ˆ) = ¡ E (ˆ) E (ˆ) ∂E (ˆ) A(ˆ) −

0 ¡

¡ 0

γ ∂ηˆij ¢ ˆ E (ˆ) ˆ , =0!

= ¡ (6) ¡

¡ A(ˆ) = f surf (ˆ) − f bulk(ˆ) (i, j = 1, 2). (10) ij ij Since the stress does not depend on d2Ebulk(ηˆ) − 0 the internal displacement[5], total differ- dηˆ dηˆ ˆ 0 ˆ ij kl η ,ξ =0! entiations can be replaced by partial dif- surf η − bulk η ferentiations. = dijkl (ˆ) dijk l (ˆ) (14) bulk fij (ηˆ) 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 ∂ηˆ ∂ηˆ ˆ 0 ˆ 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) ˆα 0 ∂ηîj ∂ξ ˆ 0 ˆ displacement, the definition by using η k η ,ξ =0 and that by using ηˆ become equivalent 2 surf αβ ∂ E to each other in the case of i, j = 1, 2. Eij = ˆα ˆβ 0 ∂ξ ∂ξ ˆ 0 ˆ i j η ,ξ =0 αβ β γ αγ bulk ; gij Ej k = δ δik (19)

¡ 1 ∂E f bulk(ˆ) , (i, j = 1, 2) ij = 0

¡ 0

A0 ∂ηˆij ¢ ˆ ˆ , =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. ferent deﬁnitions of strain and internal displacement are used between the m bulk

bulk and surface models, d /Lz is not ¡ m ¡ 1 dE m ijkl σ (ˆ) = = 2f (ˆ)/Lz ij 0 ij

¡ 0 equal to the usual elastic constants, un-

V0 dηˆij ¢ ˆ ˆ , =0 like the case with stress. (13) Deﬁnition of surface elastic constants d2Ebulk d2Ebulk =6 (20)

Surface elastic constants are deﬁned 0 ¡

0 ¡ 0 ¢ dηˆij dηˆkl ¢ ˆ dηij dηkl ˆ , =0 0, =0 as second-order derivatives of surface energy with respect to the strain ηˆij , which represents the variation in elastic prop- Lz dbulk =6 C , erties due to surface creation. ijkl 2 ijkl 1 d2Ebulk

Cijkl = (21)

¡ ¡

2 ¡ 0

V0 dηij dηkl 0 ¢

¡ 1 d (A(ˆ)γ(ˆ)) , =0! dijkl(ˆ) = 0

¡ 0 A0 dηˆij dηˆkl ˆ ¢ ˆ , =0 bulk dijkl corresponds to the elastic con- 2 surf 1 d E (ηˆ) stants under the two-dimensional plane- = 0 A0 dηˆijdηˆkl ˆ 0 ˆ stress condition (σ = γ = γ ≡ 0). η ,ξ =0 z xz yz

bulk For example, dijkl can be written by Eq. method (relaxed surface) or by thermal (22)-(25) in the orthotropic elastic body, annealing for 4ns at 1200K and 2ns where the Voigt notation is used. at 800K (well-annealed surface). This recipe is determined so that the state of 2 the bulk does not change by annealing. bulk C13 d11 = Lz C11 − (22) Three samples are produced. C33 Surface energy, surface stress 2 bulk C23 and surface elastic constants d22 = Lz C22 − (23) C33 The surface energy γ, surface stress f (in-plane f11 and f22 ) and surface elas- bulk C12C13 d12 = Lz C12 − (24) tic constants dIJ are shown in Tables 1 C33 and 2. bulk d44 = LzC44 (25) Table:1 Surface energy γ (J/m2) and From the definition of Eq. (14), it is surface stress f (N/m) of the surface of shown that the surface elastic constants crystal silicon and amorphous silicon, are coefficients between surface stress whose evaluation area is 14.6 nm2. The fij and strain ηˆkl as shown in Eq.(26). x-direction of the (2×1) surface model fij(0) is the surface stress at the zero corresponds to the dimer-bonding strain state, which originates in the fact direction. γ f , f that the strain is defined on the basis of 11 22 (100) 1×1 2.27 -0.88 the state of the bulk. (100) 2×1 1.48 0.40, -1.34 a-Si(relaxed) 1.620.06 -0.500.19 a-Si(well-ann.) 1.070.06 1.380.19 dijkl(ηˆ)ηˆkl = fij(ηˆ) − fij(0) (26)

3 Results The surface energy of crystal decreases from 2.27 J/m2 to 1.48 due to the (2×1) Analysis condition reconstruction. The compressive in- We applied our method to nano-scale plane surface stress also decreases, from thin ﬁlms of crystal and amorphous sili- -0.88 to -0.50 N/m. The well-annealed con. Two kinds of crystal surfaces, i.e., amorphous surface has the lowest sur- (100)1×1 and (100)2×1, and two kinds face energy, 1.07 J/m2, and a large ten- of amorphous surfaces, i.e., relaxed and sile surface stress, 1.38 N/m. These well-annealed, are prepared. The Tersoﬀ variations are caused by a large recon- potential is used to model the silicon[7]. struction of the amorphous surface. The For the amorphous model, we coordination number and ring statistics prepared a bulk model including are shown in Tables. 3 and 4, respec- 1000 atoms and with overall size of tively. From the comparison of the re- 2.7×2.7×2.7 nm. First the structures laxed amorphous surface with the well- of the bulk amorphous silicon were pro- annealed one, we can see that annealing duced by a melt-quench method. Then resulted in the disappearance of the 2- the structures are annealed at 1600K coordination number and the increase of for 10 ns. The surfaces are formed the 4-coordination number. Increases in by removing the periodic boundary the number of six- and seven-membered condition of the z-direction. After the rings are also observed. It is thought surfaces are created, the systems are that the strong structures of the bond relaxed by the conjugated gradient network are constructed by the surface Table:2 Surface elastic constants dIJ (N/m) of the surface of crystal silicon and amorphous silicon d11, d22 d12 d44 (100) 1×1 -8.1 0.28 -3.6 (100) 2×1 -17.8, -0.7 -0.55 -0.40 a-Si(relaxed) -11.2 2.3 -5.7 1.8 -2.4 0.5 a-Si(well-annealed) -7.1 4.2 -4.9 0.4 -1.2 2.7

α reconstruction, and that those effects in- N and A0 are the number of atoms fluence the surface properties greatly. and the surface area per atom, respec- While the coordination number of a tively. 0α 0α well-annealed amorphous surface is al- The distributions of d11 and d44 along most the same as that of the crystal sil- the direction of thickness are shown in icon surface (100)(2×1), the ring statis- Fig. 2 and 3 for (100) 2×1 and relaxed tics show a different tendency, i.e. the a-Si surfaces. Both ends of the x-axis amorphous surface has a larger num- correspond to the surface area. ber of five- and seven-membered rings. In contrast to the continuum approx- This reflects the difference in the bond imation, the elastic constants of an network between amorphous and crystal atomic system show not a homogeneous surfaces. distribution but an inhomogeneous one. The negative surface elastic constants The scattering region of atomic elastic also decreased due to the surface recon- constants reaches about four atomic lay- struction. This indicates that the elastic ers. Therefore, negative (soft) surface properties of thin films approach those of elastic constants are generated mainly bulks. within three or four atomic layers from the surface, which corresponds to a 4 Discussion width of 0.2-0.5 [nm]. Distribution of atomic elastic constants 40 In order to investigate the depth of 20 the surface effect, the local atomic elas- 0 0α surf tic constants, (dijkl) , are newly de- -20

fined by Eq. (27), which expresses qual- -40 itatively the contribution of the local ef- -60 fect to the whole system. Therefore, its -80 d11 definition is intrinsically different from d44 0 surf -100 (dijkl) . atomic local surface elastic constants [N/m] 0 0.5 1 1.5 2 z-coordination [nm] Figure:2 Distribution of atomic local elastic constants along the z-direction. d0α = (d0α )surf − (d0α )bulk ijkl ijkl ijkl (001)2×1 surface model surf αβ A0 ∂fij ∂r 0α bulk = α αβ − (dijkl) , A0 ∂r ∂ηij Dependence of surface elastic β X constants on the thickness 1 d0α = d0 (27) The dependence of the surface stress N ijkl ijkl α ! and that of the surface elastic constants X Table:3 Ratio of the coordination number within 5.5nm from the surface. The standard deviation of amorphous results is in the range of 0 to 2 % N2 N3 N4 N5 Nave (100) 1×1 22.2 0 77.8 0 3.56 (100) 2×1 0 22.2 77.8 0 3.78 Crystal(Bulk) 0 0 100 0 4 a-Si(relaxed) 8.8 20.7 65.9 3.8 3.63 a-Si(well-annealed) 0.0 22.6 74.8 2.5 3.79 a-Si(Bulk) 0 0.1 96.2 3.7 4.03

Table:4 Ring statistics within 5.5nm from surface R3 R4 R5 R6 R7 (100) 1×1 0 0 0 1.33 0 (100) 2×1 0 0 0.22 1.44 0.22 Crystal(Bulk) 0 0 0 2 0 a-Si(relaxed) 0.00 0.03 0.32 0.69 0.66 a-Si(annealed) 0.00 0.03 0.40 0.81 0.80 a-Si(Bulk) 0.00 0.03 0.40 1.03 1.03

80 -6 60 -8 40 -10 20 0 -12 d11(1*1) d11(2*1) -20 -14 -40 -16 -60 d11 d44 -18

-80 surface elastic constants d [N/m] -100 -20 atomic local surface elastic constants [N/m] 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 z-coordination [nm] thickness Lz [nm] Figure:3 Distribution of atomic local Figure:4 Surface elastic constants d11 elastic constants along the z-direction. as a function of ﬁlm thickness Lz for Relaxed amorphous surface model c-Si surface.

Deviation from the continuum approximation Since the surface elastic constants in- clude only the surface effect and do not on the thickness have been investigated. depend on the film thickness, the elastic The results of (100)1×1 and (100)2×1 constants of the whole thin film grad- surfaces are shown in Fig. 4 for d11. ually approach those of bulks as the The values of elastic constants d11 are thickness increases. Therefore, the in- not dependent on the film thickness. Be- plane strain of a thin film caused by low 1nm thickness, a small variation of the same surface stress decreases as the d11 appears. These are caused by the in- thickness decreases, due to the softening terference of surfaces, since the surface effect of the surface elastic constants. In effect reaches about 0.5nm as shown in the isotropic case, in-plane strain ηˆfilm Fig. 2. caused by surface stress at the zero- strain state is written by Eq. (28) from 50 f surf = 0. 45 ij 40 35 30 film f11 25 ηˆ = surf surf (28) d11 + d12 20 Deviation [%] 15 On the other hand, if the film is ap- 10 proximated by a continuum body, in- 5 bulk 0 plane strain ηˆ can be written by re- 0 2 4 6 8 10 12 surf bulk placing d into d in Eq. (28). Film thickness [nm] Therefore, Eq. (28) can be written in Figure:5 Deviation of in-plane strain as a surf bulk simple form by using dij = dij + dij function of film thickness (well-annealed and Eq. (22)-(25). surface of amorphous silicon)

f the framework of the molecular dynam- ηˆfilm = (C11+2C12)(C11−C12) ics method. We applied our method d11 + d12 + Lz C11 (29) to nano-scale thin films of crystal and The deviation of ηˆfilm from ηˆbulk can amorphous silicon. The effects of surface reconstructions were also investi- be written by film thickness Lz and softening parameter C∗ which we have gated. The well-annealed surface has re- newly defined. markably different properties compared with the relaxed surface. The width of the surface is also discussed in terms of film bulk ηˆ − ηˆ 1 newly defined atomic-level elastic prop- dev = bulk = − ∗ , ηˆ 1 + LzC erties. It is found that the width of the ∗ (C11 + 2C12)(C11 − C12) surface is approximately 0.5nm. C = . (30) C11(d11 + d12) References As the thickness increases, the devi- 6 ation approaches zero. The curve of a [1] R. Koch, J. Phys. Condens. Mater., well-annealed surface of amorphous sil- (1994), 9519-9550. icon is shown in Fig. 5 as a function [2] S. Hara, S. Izumi, S. Sakai, Proceed- ings of the 2002 International Confer- of thickness L . The deviation from the z ence on Computational Engineering & continuum approximation becomes less Sciences, 131 (2002) than 5% when the thickness exceeds ap- [3] K. E. Petersen, C. R. Guarnieri, J. proximately 5nm. These results lead us Appl. Phys. 50 (1979), 6761 to conclude that continuum approxima- [4] S. Matsuoka, K. Miyahara, N. Na- tion can be established in such a thin gashima, K. Tanaka, Trans. Jpn. Soc. film. Mech. Eng., (in Japanese), 62 (1996), 134 5 Conclusions [5] J. W. Martin, J. Phys. C, 8 (1975), 2858 We have proposed new definitions and [6] S. Izumi, S. Sakai, JSME International calculation methods regarding surface Ser. A, submitted stress and elastic constants for thin films [7] J. Tersoff, Phys. Rev. B, 38 (1988), by extending Martin’s method, which 9902 is useful for obtaining the internal displacement and elastic constants within