  We propose a method for solving the Schrodinger-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 multi-orbital 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. 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 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 non-homogeneous 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.
