Application of the Alexander–Haasen Model for Thermally Stimulated Dislocation Generation in FZ Silicon Crystals

Abstract

Numerical simulations of the transient temperature field and dislocation density distribution for a recently published silicon crystal heating experiment were carried out. Low- and high-frequency modelling approaches for heat induction were introduced and shown to yield similar results. The calculated temperature field was in very good agreement with the experiment. To better explain the experimentally observed dislocation distribution, the Alexander–Haasen model was extended with a critical stress threshold below which no dislocation multiplication occurs. The results are compared with the experiment, and some remaining shortcomings in the model are discussed.

A Coupled Approach to Compute the Dislocation Density Development during Czochralski Growth and Its Application to the Growth of High-Purity Germanium (HPGe)

“…The local model uses the previously developed open-source solver package MACPLAS [24] which is based on the open-source finite element library deal.II [33]. MACPLAS has recently been applied to analyse thermally induced dislocation generation in floating-zone Si crystals [34] and dislocation dynamics in a parametric model for Czochralski Ge growth [25]. For the present study, the MACPLAS crystal growth solver has been extended with an option to import external temperature boundary conditions T(r, z, t).…”

“…Non-zero homogeneous dislocation density distribution and zero plastic strain are set at t = 0. For additional details regarding the AH model, we refer the reader to our prior works [25,34].…”

“…, where T ref is the reference temperature and ᾱ is the averaged thermal expansion coefficient (calculated from the differential expansion coefficient α L ; see [34]). Currently, only the isotropic model with H = 0 (i.e., the influence of the growth direction is not considered) is implemented in MACPLAS:…”