Schrödinger - Poisson Calculations for Majorana Devices

This thesis is concerned with modelling the electrostatic environment in devices used for realizing Majorana zero modes (Majorana devices). A simulation model based on the SchrödingerPoisson method is developed and applied to two geometries. The first geometry is a planar semiconductor-metal heterostructure contacted to a gate electrode and with translational invariance in two directions. We show how to set up the Schrödinger-Poisson method and include hybridization between the metal and semiconductor in the Schrödinger equation. Simulation results based on an InAs-Al junction are then presented, and it is found that the electrostatic interaction creates wells at the various heterointerfaces. These results are then compared to an approximation, where the hybridization is neglected, and it is found that the two approaches produce comparable results with some deviation at the semiconductor-metal interface. Subsequently, the effects of this deviation on the wave functions of the hybrid system is investigated and found to be negligible with our parameters. The second geometry is a cross section of a hexagonal semiconductor nanowire with translational invariance in the longitudinal direction. The nanowire is covered by a metallic superconductor on two facets and placed on a dielectric layer in contact to a gate electrode. We show how set up the Schrödinger-Poisson method and argue that including hybridization between the semiconductor and superconductor is too computationally demanding. Consequently a simpler approach is used where the Schrödinger equation is solved only in the semiconductor region. Simulation results based on an InAs nanowire with Al as the superconductor are presented and good agreement with the results from the planar geometry is found. Acknowledgements First and foremost I would like to thank Karsten Flensberg for his supervision during this project. Though technically not my main supervisor, Karsten showed interest in my project from early on and has helped me throughout the whole process. Whether the problems I faced were related to mathematical derivations, numerical details or physical understanding, Karsten was always willing to discuss them and could usually explain things in a language that was easy to understand. Secondly, I would like to thank my main supervisor Peter Krogstrup for getting me interested in this project and giving me many opportunities for developing intuition of the experimental side of things. I would also like to Panagiotis Kotetes for helping me during the times, where Karsten was abroad, as well as in the final month of the thesis writing. Panagiotis always had a nice way of telling me when what I did was complete bullshit. A special thanks goes to my beloved girlfriend Clara, who has been invaluable to me during the final months of this project. Without her encouragement and support this process would have been a lot tougher. Lastly a big thanks goes to all the people in QDev, who have been a part of my life for the past two years. You made my time here a lot more enjoyable.