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

Abstract

We propose a method for solving the Schrödinger-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. These two ideas considerably simplify the numerical complexity of the full 3D Schrödinger-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 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.

Superconducting size effect in thin films under electric field: Mean-field self-consistent model

“…The modulation of screening by superconductivity will also appear in proximity systems, e.g. in semiconductor/superconductor hybrids 27,52 recently considered as Majorana fermion platforms.…”

Spectroscopic Visualization of a Robust Electronic Response of Semiconducting Nanowires to Deposition of Superconducting Islands

“…This value corresponds to a surface electron density of about 10 12 cm −2 , which is just about the value at which Fermi-level pinning is known to set in [1,2]. At such densities the surface states become so dense that The presence of surface states with high density, which is ubiquitous to III-V semiconductors but was overlooked in recent modellings of nanowire-metal interfaces [8][9][10], relieves the susceptibility of the semiconducting nanowire to various surface perturbations (including invasiveness of the STM tip) and specifically to the introduction of metallic islands [12].…”

The self-consistent quantum-electrostatic problem in strongly non-linear regime

“…In Sec. 4 we have argued that the Poisson approximation is generally accurate. There is, nonetheless, a caveat to this argument.…”

“…This leads to the Thomas-Fermi approximation. One could also use the adiabatic approximation such as in [4] where the 3D LDOS is replaced by the solution of 2D problems that depend on the third dimension. Iterative methods such as the Kernel Polynomial Method (KPM) are also natural approaches for obtaining the ILDOS [33].…”