Linear and non-linear transport across a finite Kitaev chain: an exact analytical study

Abstract

We present exact analytical results for the differential conductance of a finite Kitaev chain in an N-S-N configuration, where the topological superconductor is contacted on both sides with normal leads. Our results are obtained with the Keldysh non-equilibrium Green's functions technique, using the full spectrum of the Kitaev chain without resorting to minimal models. A closed formula for the linear conductance is given, and the analytical procedure to obtain the differential conductance for the transport mediated by higher excitations is described. The linear conductance attains the maximum value of e 2 /h only for the exact zero energy states. Also the differential conductance exhibits a complex pattern created by numerous crossings and anticrossings in the excitation spectrum. We reveal the crossings to be protected by the inversion symmetry, while the anticrossings result from a pairing-induced hybridization of particle-like and hole-like solutions with the same inversion character. Our comprehensive treatment of the Kitaev chain allows us also to identify the contributions of both local and non-local transmission processes to transport at arbitrary bias voltage. Local Andreev reflection processes dominate the transport within the bulk gap and diminish for higher excited states, but reemerge when the bias voltage probes the avoided crossings. The non-local direct transmission is enhanced above the bulk gap, but contributes also to the transport mediated by the topological states.

Role of dephasing on the conductance signatures of Majorana zero modes

“…In the coherent regime, we point out contributions of the local Andreev reflection and the non-local transmissions toward the conductance signatures and that a half of the conductance quantum (e 2 /h) is indeed maintained. The half fraction, as pointed out in earlier works 42 is indeed due to a voltage divider formed at the two N-TS junctions, under a symmetric bias situation. The effect of contact broadening clearly shows the maintenance of the ZBCP at zero temperatures 9,40 .…”

“…Theoretical works have been based on minimal 1D models used to analyze the N-TS links [17][18][19][20][21][22][23][24][25] and have typically been based on a scattering formulation of quantum transport 35 in the coherent regime. In this work, we use the Keldysh non-equilibrium Green's function (NEGF) technique 27,[36][37][38][39][40][41][42] to analyze the transport across the N-TS-N junction, with the inclusion of channel interactions in the non-coherent regime [43][44][45][46][47] .…”

“…region flanked by two normal contacts and tunnel barriers. Such a geometry permits local and non-local transport measurements, with the local Andreev reflection, the crossed Andreev and direct transmissions respectively 16,41,42 . In the coherent regime, we point out contributions of the local Andreev reflection and the non-local transmissions toward the conductance signatures and that a half of the conductance quantum (e 2 /h) is indeed maintained.…”

“…Throughout this work, we work in the zero temperature limit with the thermal energy much less than the other energy scales involved in the system (k B T ∆). For reasons involving current conservation in systems featuring superconducting regions, well noted in previous works 39,42,52,53 , we resort to a symmetric voltage bias, i.e., µ L = qV /2 and µ R = −qV /2. However, the issue of current conservation can only be via a self consistent calculation of the order parameter as noted in 52 , or other number conserving methods beyond the meanfield description 54 .…”

“…10, 11 and 12. It is worth noting that, due to the symmetric biasing, f ee L = f hh R , and hence, the crossed Andreev transmission processes does not contribute to the current 42 , and the non-local current only consists of direct transmission contribution. Also, for the N-TS-N three terminal set up, the conductance is quantized at e 2 /h instead of 2e 2 /h due to the presence of two N-TS junctions in series and the resulting voltage divider 42,58 , as opposed to only one junction in a typical two terminal N-TS set up.…”