Continuum modelling of semiconductor heteroepitaxy: an applied perspective

Abstract

Semiconductor heteroepitaxy involves a wealth of qualitatively different, competing phenomena. Examples include threedimensional island formation, injection of dislocations, mixing between film and substrate atoms. Their relative importance depends on the specific growth conditions, giving rise to a very complex scenario. The need for an optimal control over heteroepitaxial films and/or nanostructures is widespread: semiconductor epitaxy by molecular beam epitaxy or chemical vapour deposition is nowadays exploited also in industrial environments. Simulation models can be precious in limiting the parameter space to be sampled while aiming at films/nanostructures with the desired properties. In order to be appealing (and useful) to an applied audience, such models must yield predictions directly comparable with experimental data. This implies matching typical time scales and sizes, while offering a satisfactory description of the main physical driving forces. It is the aim of the present review to show that continuum models of semiconductor heteroepitaxy evolved significantly, providing a promising tool (even a working tool, in some cases) to comply with the above requirements. Several examples, spanning from the nanometre to the micron scale, are illustrated. Current limitations and future research directions are also discussed.

Dislocation-Free SiGe/Si Heterostructures

“…The evolution displayed in Figure 6 was subsequently simulated [33] by also taking into account surface-energy anisotropy [23], yielding results even closer to the experimental ones. It has been further demonstrated that merging can also be obtained directly during growth (by raising the growth temperature) [21,31]. The suspended film is found [18] to still profit of the ability of the underlying pillars to release thermal strain by tilting.…”

“…The suspended film displayed in Figure 6 was obtained [30] by prolongated high-temperature annealing of vertical Ge crystals on Si pillars. Temporal snapshots of the evolution are also displayed, together with a corresponding continuum simulation [31] (panel b). The latter was performed exploiting a diffusion-equation approach, implemented in a phase-field framework [22], where the only contribution to the chemical potential µ (determining material flow) was the Mullins [32] term µ = −kγ, where k is the local curvature and γ the surface-energy density.…”

Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model

“…A semi-implicit time discretization scheme is used in order to solve the set of equations defined in (4) and (5), and it is reported in detail in Appendix B 1. It consists of solving four second-order partial differential equations (PDEs) for each amplitude function.…”

“…In the past few decades, phase-field (PF) models have been used extensively for modeling the ordering of nanoand micro-structures. Such models provide a suitable framework for the investigation of a wide range of phenomena such as solidification processes, grain growth, surface diffusion, heteroepitaxy, and even dislocation dynamics [1][2][3][4][5]. Despite their versatility, strong limitations arise for PF models when looking at material properties closely related to atomic arrangement and periodicity.…”

Thin Film Evolution Equation for a Strained Anisotropic Solid Film on a Deformable Isotropic Substrate

“…The central issue is that the dynamics of the surface morphology is coupled to the elastic strain in the system, so dynamic models require solving the elasticity problems throughout the film and substrate at each time step and are limited by storage limitations for 3-dimensional problem. Simulations involving the full elasticity problem are limited to one or few islands [23] [24]. Only one recent work [25] explored a large number of islands using a large-scale 3-dimensional calculation.…”