Parametric numerical study of dislocation density distribution in Czochralski-grown germanium crystals

Numerical and experimental investigation of the effect of the solid–liquid interface shape on grown-in defects in a silicon single crystal

“…∆z 973 represents the distance from the position of 973 K. This results in 12 levels of temperature distribution with three levels of interface shape and four levels of G . set In the CTP, the heat flux in the crystal is considered; 30) therefore, it is close to the actual temperature distribution of the crystal.…”

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).…”

“…where ρ is the density, c p -specific heat capacity, and λ-thermal conductivity. Heat transfer by convection is not included in the temperature equation, but is considered by the temperature update in each time step according to the shifts in the mesh nodes [25]. We chose to solve the local temperature Equation (1) instead of simply interpolating the whole field from the global results, in order to obtain a more accurate T distribution on a finer local mesh and take into account the transient effects (at least partially, since temperature at the crystal surface is still taken from the steady-state global results).…”

“…For simplicity, the stretching mesh approach is used in the local model, i.e., without changing the number of points and without mesh regeneration. Additional aspects related to the mesh movement, such as the update of the temperature field to model the heat transfer by convection, are described in [25].…”

“…The coupling is considered in one direction, i.e., the temperature at the surface of the crystal is transferred from the output of the global simulation to the boundary conditions in the local simulation. For the global simulation, we use the open-source software Elmer, and for the local one the open-source software MACPLAS [24], which has been already applied to a parametric study of dislocations in Ge crystals [25]. The coupled approach presented here is general and can be used for any Czochralski process.…”