Elsevier

Thin Solid Films

Volume 467, Issues 1–2, 22 November 2004, Pages 253-260
Thin Solid Films

A method for calculating surface stress and surface elastic constants by molecular dynamics: application to the surface of crystal and amorphous silicon

https://doi.org/10.1016/j.tsf.2004.03.034Get rights and content

Abstract

For nano-scale thin film, effects of surface and interface, which can be ignored on a macroscale, become important. For example, surface energy and surface stress are key parameters for predicting the intrinsic stress of thin films. Several researchers have reported that the elastic constants of thin films are different from those of bulks. We have recently proposed new definitions and calculation methods regarding surface stress and elastic constants for thin films by extending Martin's method, which is useful for obtaining internal displacement and elastic constants within the framework of the molecular dynamics (MD) method. We applied our method to nano-scale thin films of crystal and amorphous silicon. It is found that elastic constants of thin film depend on both the surface elastic constants and on film thickness. Those values are significantly affected by the surface reconstructions. The width of the surface and the limit of continuum homogeneous approximation are also investigated.

Introduction

The films used in the manufacture of semiconductors are less than 10 nm thick. At this thickness, the effects of surface and interface, which can be ignored on a macroscale, become important. For example, surface (interface) energy and surface stress are key parameters for predicting the intrinsic stress of thin films, since the stress depends also on the microstructures and growth mode at atomic level [1], [2]. However, it is difficult to obtain values of surface energy and surface stress experimentally. Therefore, numerical evaluation by molecular simulation has been attempted. In addition, several researchers have reported that the elastic properties (e.g. Young module) of thin films are different from those of bulks [3], [4], [5]. In particular, the elastic properties of films with thickness in the range of several nanometers may exceed the range of continuum approximation. Therefore, it has become very important to investigate the limit of continuum homogeneous approximation and to predict unique phenomena in that thickness range. In calculating the elastic constants of inhomogeneous surface structures, the effect of internal displacement, which is nonlinear atomic displacement in response to deformation, must be taken into account. We have recently proposed new definitions and calculation methods regarding surface stress and surface elastic constants for thin films with free surfaces through an extension of Martin's method[6], [7]. This method is useful for obtaining the internal displacement and elastic constants within the framework of the molecular dynamics (MD). We applied our method to nano-scale thin films of crystal and amorphous silicon. It should be noted that the surface reconstructions greatly in influence the surface properties. Therefore, special care has been taken to clarify those effects. In order to investigate the depth of the surface effect, local atomic elastic constants are newly defined. That effect on the elastic constants of whole thin films is also discussed.

In Section 2, we define and describe the calculation of surface energy, surface stress and surface elastic constants. Section 3 presents the calculation results for crystalline and amorphous surfaces of silicon. Section 4 discusses the effect of surface reconstruction and the limit of continuum homogeneous approximation.

Section snippets

Definition of surface

In order to define the surface of a molecular dynamics system, we prepared a thin film model with a free boundary condition for the z-direction and two periodic boundary conditions for the x- and y-directions as shown in Fig. 1. Therefore, the evaluation area is made up of two surfaces that face each other. The film must be enough thick to prevent interference of these two surfaces.

Since no force can act on the surface, stress components related to the z-direction must be zero. That is,

Analysis condition

We applied our method to nano-scale thin films of crystal and amorphous silicon. Two kinds of crystal surfaces, i.e. (100)1×1 and (100)2×1, and two kinds of amorphous surfaces, i.e. unrelaxed and well-relaxed, are prepared. The Tersoff potential [8] (T3) is used for interatomic potential, which has been used in a broad range of studies of crystal and amorphous silicon and is known for its strong ability to express the physical properties of the bulk crystalline phase [9], the bulk amorphous

Distribution of atomic elastic constants

In order to investigate the depth of the surface effect, the local atomic elastic constants, (dijkl0α)surf, are newly defined by Eq. (28), which expresses qualitatively the contribution of the local effect to the whole system. Therefore, its definition is intrinsically different from (dijkl0)surf.dijkl=(dijkl)surf−(dijkl)bulk=A0A0αβfijsurfrαβrαβηij−(dijkl)bulk,1Nαdijkl=dijkl0N and A0α are the number of atoms and the surface area per atom, respectively.

The distributions of d110α

Conclusions

We have proposed new definitions and calculation methods regarding surface stress and elastic constants for thin films by extending Martin's method, which is useful for obtaining the internal displacement and elastic constants within the framework of the molecular dynamics method. We applied our method to nano-scale thin films of crystal and amorphous silicon. It is found that elastic constants of thin film depend both on the surface elastic constants and on film thickness. The effects of

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