Effective theory approach to the SchrödingerPoisson problem in semiconductor Majorana devices
Abstract
We propose a method for solving the SchrödingerPoisson problem that can be efficiently implemented in realistic 3D tightbinding models of semiconductorbased Majorana devices. The method is based on two key ideas. (i) For a given geometry, the Poisson problem is only solved once (for each local orbital) and the results are stored as an interaction tensor; using this Green's function scheme, the Poisson component of the iteration procedure is reduced to a few simple summations. (ii) The 3D problem is mapped into an effective multiorbital 1D problem with molecular orbitals calculated selfconsistently as the transverse modes of an infinite wire with the same electrostatic potential as the local electrostatic potential of the finite 3D device. These two ideas considerably simplify the numerical complexity of the full 3D SchrödingerPoisson problem for the nanowire, enabling a tractable effective theory with predictive power. To demonstrate the capabilities of our approach, we calculate the response of the system to an external magnetic field, the dependence of the effective chemical potential on the work function difference, and the dependence of the effective semiconductorsuperconductor coupling on the applied gate potential. We find that, within a wide range of parameters, different lowenergy bands are characterized by similar effective couplings, which results in induced gap features characterized by a single energy scale. We also find that electrostatic effects are responsible for a partial suppression of the Majorana energy splitting oscillations. Finally, we show that a positiondependent work function difference can produce a nonhomogeneous effective potential that is not affected by the screening due to the superconductor and is only partially suppressed by the charge inside the wire. In turn, this potential can induce trivial lowenergy states that mimic the phenomenology of Majorana zero modes. Thus any positiondependent work function difference (even at the 1 % level) along the nanowire must be avoided through carefully engineered semiconductorsuperconductor interfaces.
 Publication:

Physical Review B
 Pub Date:
 July 2018
 DOI:
 10.1103/PhysRevB.98.035428
 arXiv:
 arXiv:1801.02630
 Bibcode:
 2018PhRvB..98c5428W
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics
 EPrint:
 28 pages, 25 figures