Efficient Schrödinger – Poisson Solvers for Investigation of the Electrostatics of GeSn Heterostructures

The down-scaling of Complementary Metal Oxide Semiconductor (CMOS) devices has followed Moore’s law for decades, to obtain lower power consumption and higher computational speeds. With the limitations to scaling, there is a continued effort todevelop new technologies like non-planar devices and explore alternate material systems, like the Group-IV alloys. The properties of Si-Ge-Sn-Pb alloys allow electronic structure manipulation and bandgap engineering, thus making them suitable for applications in photonics, thermoelectrics, spintronics, quantum optics, etc. This opens up the possibility of seamless integration of CMOS with photonics. Hence, a robust model is needed to explore the electrical properties of devices made in the Group-IV alloy system.This work presents a robust tool capable of simulating the electrostatics of heterostructures with an arbitrary number of layers. It solves a coupled 1D Schr¨odinger – Poisson problem, which allows one to calculate the eigenvalues and the corresponding eigenvectors of a quasi-2D hole gas needed for future electrical transport studies.The theoretical model includes temperature-dependent material parameters, partial ionization of dopants, and Fermi-Dirac statistics to ensure accurate low temperature simulations. A theoretical framework for the solution of the 1D and 2D Schr¨odinger equation is developed and implemented to incorporate the effects of spatially varying and anisotropic effective masses. User-specified spatially varying doping profiles are used as inputs to the tool. The excellent agreement of the simulated temperature variation of the sheet hole density in Ge1-xSnx/Ge heterostructure with available experimental data for samples with different doping profiles validates the tool. This example also demonstrates the tool capability to simulate arbitrary heterostructures for a wide range of temperatures.