Strain in semiconductor core-shell nanowires

Abstract

We compute strain distributions in core-shell nanowires of zinc blende structure. We use both continuum elasticity theory and an atomistic model, and consider both finite and infinite wires. The atomistic valence force-field (VFF) model has only few assumptions. But it is less computationally efficient than the finite-element (FEM) continuum elasticity model. The generic properties of the strain distributions in core-shell nanowires obtained based on the two models agree well. This agreement indicates that although the calculations based on the VFF model are computationally feasible in many cases, the continuum elasticity theory suffices to describe the strain distributions in large core-shell nanowire structures. We find that the obtained strain distributions for infinite wires are excellent approximations to the strain distributions in finite wires, except in the regions close to the ends. Thus, our most computationally efficient model, the finite-element continuum elasticity model developed for infinite wires, is sufficient, unless edge effects are important. We give a comprehensive discussion of strain profiles. We find that the hydrostatic strain in the core is dominated by the axial strain-component, εZZ . We also find that although the individual strain components have a complex structure, the hydrostatic strain shows a much simpler structure. All inplane strain components are of similar magnitude. The non-planar off-diagonal strain-components (εXZ and εY Z ) are small but nonvanishing. Thus the material is not only stretched and compressed but also warped. The models used can be extended for study of wurtzite nanowire structures, as well as nanowires with multiple shells.

Strain in crystalline core-shell nanowires

“…The present calculation does not reproduce the inhomogeneity of the in-plane strain which is calculated for a hexagonal NW, but it was already noted [37] that the central values of strain are quite similar in hexagonal and circular cores. This was confirmed in the very detailed study of reference [6], where NWs with hexagonal and circular cross sections are compared. Indeed the results of the present model compare fairly well to the results of numerical calculations made for hexagonal NWs.…”

“…It can be done on the c ijkl tensor, or directly in the Voigt notation using the rotation rules described in reference [13]. We take the basis defined by the three vectors x = [110], y = [112], z = [1 1 1], identical to that in reference [6] but different from reference [26]. Then [8] …”

“…NWs with either a circular or a hexagonal crosssections have been modeled by Grönqvist et al [6] using both the valence force field model and a finite element treatment of the continuum elasticity theory. Other core-shell configurations with hexagonal cross-sections are described in reference [31].…”

“…The six components are not independent since they can be expressed using the three coefficients c 11 , c 12 and c 44 relevant for the cubic symmetry [6]: …”

“…However, the crystal structure results in anisotropic elastic properties, the core and shell materials have different values of the stiffness constants, and the NW shape can deviate from a e-mail: joel.cibert@neel.cnrs.fr the ideal cylinder with a circular base and for instance feature facetting. As a result, calculating the strain configuration in a real semiconductor core-shell NW is not an easy task: quantitative descriptions usually imply to compute numerically the local strain, either using a microscopic model such as the valence force field model, or performing a finite element treatment of the continuum elasticity theory [6]. Nevertheless, an analytical description, such as what has been developed and reviewed in reference [7] for the case of quantum dots and NWs embedded in an infinite or semi-infinite material, remains the best starting point for an implementation of the strainrelated mechanisms governing the electronic properties, through deformation potentials and piezoelectric fields.…”