Advances in Materials Science and Engineering

Volume 2017 (2017), Article ID 6413495, 9 pages

https://doi.org/10.1155/2017/6413495

## Surface Effects on the Properties of Screw Dislocation in Nanofilms

^{1}Department of Physics, Chongqing Three Gorges University, Chongqing 404100, China^{2}Institute for Structure and Function, Chongqing University, Chongqing 401331, China^{3}College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Xiaozhi Wu

Received 13 November 2016; Accepted 16 January 2017; Published 5 February 2017

Academic Editor: Santiago Garcia-Granda

Copyright © 2017 Lili Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The image dislocation method is used to construct the governing equation of dislocations in nanofilms. The classical Peierls-Nabarro equation can be recovered when the thickness of nanofilm is taken to be infinite. In order to determine the core width and Peierls stress of dislocations, the unstable stacking fault energies of Al and Cu nanofilms are calculated via the first-principle methods. It is found that surface effect can increase the Peierls stresses of screw dislocations in Al and Cu nanofilms.

#### 1. Introduction

Thin film materials are of great importance in modern technology. They have a wide range of applications in engineering systems, such as highly integrated electronic circuits and microelectromechanical devices. It is widely accepted that defects such as dislocations play a key role in determining the mechanical properties of materials [1]. The classical Peierls-Nabarro (P-N) dislocation model has been the most widely used model of dislocations in bulk materials [2, 3]. Although powerful in its simplicity, it cannot deal with the dislocations in nanofilm with free surfaces [4, 5]. Accounting for the free surface on the physical properties of dislocations in nanofilm is important, as the ratio of surface to volume is high. The effects of free surface on the dislocation properties cannot be neglected anymore; the dislocation behavior in thin films is different from that in bulk materials [6, 7]. Recently, Lee and Li constructed a half-space P-N model for dislocations near a free surface and they claimed that the mobility of the dislocation will be increased near the free surface [8, 9]. Cheng et al. investigated Peierls stress of a screw dislocation below a free surface via a self-consistent semidiscrete variational P-N model using image dislocation method [10]. It is found that the free surface increases the Peierls stress and in consistence with the results obtained by molecular dynamics simulation [11, 12]. In this letter, the image dislocation method is used to investigate the dislocation core structure and Peierls stress in nanofilm with double free surfaces. The image dislocations are introduced to satisfy the free surface boundary condition; namely, the stress field acting on the surface should vanish [13]. The governing equation for dislocations in nanofilm is constructed according to the procedure for P-N equation provided by Joós et al. [14, 15]. Furthermore, the unstable stacking fault (USF) energies of Al and Cu nanofilms are also calculated by first-principle methods to discuss the effect of surface [16].

#### 2. Dislocations Properties in Nanofilms

To facilitate the presentation we adopt the following conventions in all that follows (see Figure 1). In a Cartesian set of coordinates , the plane is the slip plane and -axis is normal to the slip plane. The -axis is the direction of the screw dislocation and the direction of the Burgers vector. In an isotropic crystal and elastic continuum theory, the screw dislocation in bulk material produces a stress field along its Burgers vector [1] is the shear modulus. In order to take into account the surface effect on the screw dislocation properties, a screw dislocation with thickness at a distance below the upper free surface is considered in a nanofilm (see Figure 1). In the typical way to count for the image effect, the two image screw dislocations with the opposite Burgers vector are distributed along a horizontal line at a distance above the upper free surface and below the lower free surface. These two image dislocations superpose new stresses which are equal and opposite to the already existing stresses on the upper and lower surfaces, respectively. However, the above (below) image screw dislocation leaves a residual stress acting on the lower (upper) free surface, so that an additional image is required, which in turn requires another image and so on. The result is an infinite set of image screw dislocations. The stress field of original screw dislocation in thin film combined with the surface effect can be equally produced by this infinite set of screw dislocations. Then, the stress field on the plane can be written asIt is helpful to rearrange the termsUsing identities,One obtainsIn the above equation is used. It is interesting to find that when . The first-term of (3) represents the shear stress on plane of array right-handed screw dislocations siting . The second-term of (3) represents the shear stress on plane of an array left-handed screw dislocations siting . It is also easy to verify that on and planes; that is to say the free surface boundary condition is satisfied.