Effective theory approach to the Schrödinger-Poisson problem in semiconductor Majorana devices

We propose a method for solving the Schr\"odinger-Poisson problem that can be efficiently implemented in realistic 3D tight-binding models of semiconductor-based 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 self-consistently 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 Schr\"odinger-Poisson 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 semiconductor-superconductor coupling on the applied gate potential. We find that, within a wide range of parameters, different low-energy 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 position-dependent 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 low-energy states that mimic the phenomenology of Majorana zero modes. Thus any position-dependent work function difference (even at the 1% level) along the nanowire must be avoided through carefully engineered semiconductor-superconductor interfaces.