A Green’s function approach to topological insulator junctions with magnetic and superconducting regions

Abstract

This work presents a Green’s function approach, originally implemented in graphene with well-defined edges, to the surface of a strong 3D topological insulator with a sequence of proximitized superconducting (S) and ferromagnetic (F) surfaces. This consists of the derivation of the Green’s functions for each region by the asymptotic solutions method and their coupling by a tight-binding Hamiltonian with the Dyson equation to obtain the full Green’s functions of the system. These functions allow the direct calculation of the momentum-resolved spectral density of states, the identification of subgap interface states and the derivation of the differential conductance for a wide variety of configurations of the junctions. We illustrate the application of this method for some simple systems with two and three regions, finding the characteristic chiral state of the quantum anomalous Hall effect at the NF interfaces, and chiral Majorana modes at the NS interfaces. Finally, we discuss some geometrical effects present in three-region junctions such as weak Fabry–Pérot resonances and Andreev bound states.

Long-range Cooper pair splitting by chiral Majorana edge states

“…Green functions en TIs [80] For a planar junction with translational invariance along the y axis, the advanced (a) and retarded (r) Green functions are given by ĝr,a (E, x, x , y − y ) = dqe iq(y−y ) ĝr,a (E, x, x , q) /2π, (A1)…”

“…The transport properties of this type of nanostructure can be described by the Hamiltonian approximation, where each region of the system is coupled to adjacent regions through a tight-binding Hamiltonian [77,78] (see appendix D). In this formalism, the stationary transport properties can be expressed in terms of the equilibrium Green functions of the system, which are calculated by applying a Dyson equation to the Green functions of adjacent regions, and for planar junctions, these can be obtained analytically by the asymptotic solutions method [79,80] (see appendices A to C).…”

“…with i, j = (a, b) the contact indices, k, l = N, S the region indices ( dniσ = ĉiσ , dsiσ = biσ ), α, β = (+, −) the indices of the Keldysh temporal contour and T c is the Keldysh temporal ordering operator. Assuming normal electrodes on the surface of the TI, the hopping Hamiltonian D2 takes the form A3, and considering the two possible choices of the boundary conditions, the average current acquires a factor of 2 [80,83]. In a stationary situation, the average current in the electrode i can be expressed in the energy space as the average current can be written in terms of Green functions evaluated over a single type of region as )/π the density-of -states matrix of electrode i, being ĝr(a) i its equilibrium Green function.…”

“…In a stationary situation, the average current in the electrode i can be expressed in the energy space as the average current can be written in terms of Green functions evaluated over a single type of region as )/π the density-of -states matrix of electrode i, being ĝr(a) i its equilibrium Green function. Considering the following Dyson equation between the equilibrium and non-equilibrium Green functions involving the equilibrium local and non-local Green functions of the system (γ = +−, −+) [80,83] Ĝγ q,iS = Ĝr q,iS t † i ĝγ q,N ti Ĝa q,iS + Ĝr q,ijS ti ĝγ q,jS t † i Ĝa q,jiS , (D12) the total average current induced in the electrode i takes the form I i = I ii + I ij , where the local I ii and not local I ij contribution (i = j) are given by…”