Scattering in quantum wires and junctions of quantum wires with edge states of quantum spin Hall insulators

An integral part of scattering theory calculations in quantum systems involves identifying appropriate boundary conditions in addition to writing down the correct Hamiltonian. Even in the simplest problem of scattering in one dimension, scattering due to an onsite potential and scattering due to an unequal bond give different results for scattering amplitudes. In continuum model, we introduce a new parameter $c$ in boundary condition. Using the mapping between continuum and lattice models as a tool, we understand the local boundary conditions and their implications on scattering in (i)~normal metal quantum wires, and (ii)~junction between a normal metal quantum wire and a one dimensional edge of quantum spin Hall insulator (QSHI). In the case of junction between normal metal quantum wire and edge of QSHI, we identify the boundary condition that permits maximum transmission. The problem of transport between four channels of spinful normal metal quantum wire and two channels of QSHI edge is not well defined. We rectify this situation by formulating the scattering problem in terms of a junction of a semi-infinite normal metal quantum wire with infinite edge of QSHI, gapping out one semi-infinite section of the QSHI edge by a Zeeman field and applying the appropriate boundary condition at the junction. We calculate scattering amplitudes for the electrons incident on the junction from quantum wire.Comment: 5 pages, 2 figure

Nonadiabatic charge pumping by oscillating potentials in one dimension: results for infinite system and finite ring

We study charge pumping when a combination of static potentials and potentials oscillating with a time period T is applied in a one-dimensional system of non-interacting electrons. We consider both an infinite system using the Dirac equation in the continuum approximation, and a periodic ring with a finite number of sites using the tight-binding model. The infinite system is taken to be coupled to reservoirs on the two sides which are at the same chemical potential and temperature. We consider a model in which oscillating potentials help the electrons to access a transmission resonance produced by the static potentials, and show that non-adiabatic pumping violates the simple \sin \phi rule which is obeyed by adiabatic two-site pumping. For the ring, we do not introduce any reservoirs, and we present a method for calculating the current averaged over an infinite time using the time evolution operator U(T) assuming a purely Hamiltonian evolution. We analytically show that the averaged current is zero if the Hamiltonian is real and time reversal invariant. Numerical studies indicate another interesting result, namely, that the integrated current is zero for any time-dependence of the potential if it is applied to only one site. Finally we study the effects of pumping at two sites on a ring at resonant and non-resonant frequencies, and show that the pumped current has different dependences on the pumping amplitude in the two cases.Comment: 14 pages including 9 figures; significantly expanded -- this is the published versio

Nonadiabatic charge pumping across two superconductors connected through a normal metal region by periodically driven potentials

Periodically driven systems exhibit resonance when the difference between an excited state energy and the ground state energy is an integer multiple of $\hbar$ times the driving frequency. On the other hand, when a superconducting phase difference is maintained between two superconductors, subgap states appear which carry a Josephson current. A driven Josephson junction therefore opens up an interesting avenue where the excitations due to applied driving affect the current flowing from one superconductor to the other. Motivated by this, we study charge transport in a superconductor-normal metal-superconductor (SNS) junction where oscillating potentials are applied to the normal metal region. We find that for small amplitudes of the oscillating potential, driving at one site reverses the direction of current at the superconducting phase differences when difference between the subgap eigenenergies of the undriven Hamiltonian is integer multiple of $\hbar$ times the driving frequency. For larger amplitudes of oscillating potential, driving at one site exhibits richer features. We show that even when the two superconductors are maintained at same superconducting phase, a current can be driven by applying oscillating potentials to two sites in the normal metal differing by a phase. We find that when there is a nonzero Josephson current in the undriven system, the local peaks and valleys in current of the system driven with an amplitude of oscillating potential smaller than the superconducting gap indicates sharp excitations in the system. In the adiabatic limit, we find that charge transferred in one time period diverges as a powerlaw with pumping frequency when a Josephson current flows in the undriven system. Our calculations are exact and can be applied to finite systems. We discuss possible experimental setups where our predictions can be tested.Comment: 9 pages, 9 figures. Published versio

Nonequilibrium Josephson diode effect in periodically driven SNS junctions

In typical Josephson junctions, the Josephson current is an odd function of the superconducting phase difference. Recently, diode effect in Josephson junctions is observed in experiments wherein the maximum and the minimum values of the Josephson current in the current-phase relation do not have the same magnitude. We propose a superconductor-normal metal-superconductor (SNS) junction where Josephson diode effect manifests when the normal metal region is driven. Time reversal symmetry and inversion symmetry need to be broken in the SNS junction for the diode effect to show up. We calculate long time averaged current and show that the system exhibits diode effect for two configurations of the driven SNS junction - one in which inversion symmetry is broken in the undriven part of the Hamiltonian and the other wherein both the symmetries are broken by the driving potential. In the latter configuration, a nonzero current known as anomalous current appears at the junction in absence of phase bias. In the proposed setup, the diode effect vanishes in the adiabatic limit.Comment: 8 pages, 7 captioned figure

Anomalous Josephson effect and rectification in junctions between Floquet topological superconductors

Periodically driven Kitaev chains are known to exhibit novel Floquet Majorana fermions and anomalous Floquet end modes. The fact that `quasienergy in Floquet systems is periodic' poses a difficulty in defining the ground state in periodically driven systems. To overcome this problem, we start with the ground state of the undriven Kitaev chain and gradually switch on the periodic driving in chemical potential over a timescale $\tau$. The extent to which the single particle eigenstates of the undriven system get distributed among the Floquet states upon driving can be characterized by inverse participation ratio. In a Josephson junction between two periodically driven Kitaev chains not differing in phase of the pair potentials but differing in phases of the driving potentials, a net average current flows from one superconductor to the other. We term such a current anomalous current. Further, we study current phase relation in junctions between two periodically driven superconductors and find that the system exhibits nonequilibrium Josephson diode effect. The Floquet Majorana end modes and anomalous Floquet end modes whenever present contribute significantly to the anomalous current and the diode effect. Further, the anomalous current and the nonequilibrium Josephson diode effect survive when the periodic driving is adiabatically switched on.Comment: 9 pages, 10 captioned figures. Published versio

Magnetic field induced Fabry-P\'erot resonances in helical edge states

We study electronic transport across a helical edge state exposed to a uniform magnetic ({$\vec B$}) field over a finite length. We show that this system exhibits Fabry-P\'erot type resonances in electronic transport. The intrinsic spin anisotropy of the helical edge states allows us to tune these resonances by changing the direction of the {$\vec B$} field while keeping its magnitude constant. This is in sharp contrast to the case of non-helical one dimensional electron gases with a parabolic dispersion, where similar resonances do appear in individual spin channels ($\uparrow$ and $\downarrow$) separately which, however, cannot be tuned by merely changing the direction of the {$\vec B$} field. These resonances provide a unique way to probe the helical nature of the theory.Comment: v1: 5 pages, 5 figure

Tunnel Magnetoresistance scan of a pristine three-dimensional topological insulator

Though the Fermi surface of surface states of a 3D topological insulator (TI) has zero magnetization, an arbitrary segment of the full Fermi surface has a unique magnetic moment consistent with the type of spin-momentum locking in hand. We propose a three-terminal set up, which directly couples to the magnetization of a chosen segment of a Fermi surface hence leading to a finite tunnel magnetoresistance (TMR) response of the nonmagnetic TI surface states, when coupled to spin polarized STM probe. This multiterminal TMR not only provides a unique signature of spin-momentum locking for a pristine TI but also provides a direct measure of momentum resolved out of plane polarization of hexagonally warped Fermi surfaces relevant for $Bi_2Te_3$, which could be as comprehensive as spin-resolved ARPES. Implication of this unconventional TMR is also discussed in the broader context of 2D spin-orbit (SO) materials.Comment: Version accepted in Phys. Rev. B (Rapid Communications