Transport signatures of spin Berry phase and Majorana fermions in edge states of quantum spin Hall insulator

Adak, Vivekananda (2023) Transport signatures of spin Berry phase and Majorana fermions in edge states of quantum spin Hall insulator. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Vivekananda Adak (13RS031)) 13RS031.pdf - Submitted Version Restricted to Repository staff only Download (10MB)

Abstract

Topic of mesoscopic and nanoscopic physics has emerged as one of the most important fields of research in condensed matter physics in the last few decades due to tremendous advancement in device fabrication at meso- and nano-scales. A macroscopic system when scale down to the mesoscopic size starts revealing properties like quantization of physical observables, quantum confinement effects and quantum interference effect. In this regime, surface states of a material can make significant contributions to its physical properties with respect to the bulk. And one of the exotic properties that can make these contributions even more dramatic is the bulk-boundary correspondence which offers gapless surface states (edge states for 2-D materials) at the boundary of an insulating bulk. Specifically, these type of materials are known as the topological insulators or topological superconductors in literature which is a rapidly growing field of research. Most importantly, a large variety of these materials can be understood in terms of simple band Hamiltonian in presence of strong spin-orbit (SO) couplings. My thesis pertains to study of transport properties of edge states of 2-D topological insulators. Boundary states of topological materials is inspiring a new era for modern technologies owing to its direct connection to spintronics applications and its possible application in engineering building blocks for topological quantum computers. In particular the time reversal symmetry 2-D topological insulator, which hosts edge state at the boundary carrying a persistent spin current may find important application both in spintronic and topological quantum computers as it can host Majorana fermion (when proximitized to a S-wave superconductor). This edge state of time reversal symmetry 2-D topological insulator can be described by a spin-momentum locked massless Dirac like dispersion (helical edge state) which leads to a quantized spin Hall conductance in quantum transport measurements in low bias limit. In 2005, Kane and Mele first predicted that graphene in presence of spin-orbit (SO) interaction could lead to a quantized spin Hall (QSH) state. Unfortunately, at temperatures attainable with current technology, graphene is very unlikely to support a QSH state due to its extremely weak SO coupling. Then, in 2006, Bernevig, Hughes, and Zhang anticipated that quantum wells of mercury telluride sandwiched between cadmium telluride would give rise to a 2-D topological insulators, also known as the QSH insulators, with 1-D helical edge states at the boundary and these predictions were confirmed in 2007 by the group of Laurens W Molenkamp. In my thesis, I have explored the possible effect of various perturbations at the boundary of a QSH insulator through quantum transport calculations on a proposed device set-ups where these perturbations could lead to non-trivial effects like inducing a spin Berry (SB) phase or resulting in unique transport signatures of Majorana bound states (MBS). These effects have been first identified and estimated analytically using 1-D edge models and then follow up study is carried out using extensive numerical transport simulations of 2-D lattice model which confirms the feasibility of 1-D edge models. In this thesis we begin with a brief review of the quantum Hall effect. Then we describe the Bernevig-Hughes-Zhang (BHZ) model which has important implications in the simulation of 2-D lattice models of QSH insulator in the rest of the thesis. We then discuss numerical simulation to evaluate the conductance and also other physical quantities (like charge density and current density) for a given set-up using a lattice model. In this context, we introduce kwant package which is a numerical package in python commonly used to compute transport properties of the low-dimensional mesoscopic systems. Apart from these, we provide a brief introduction to the geometrical phase and majorana fermion which are going to be the main focus of our discussions in the following chapters. After this we consider study the effect of external electric field as a perturbation locally applied to the QSH edge, which induces Rashba type SO coupling at the edge. Spin quantization axis of a pristine helical edge is supposed to be uniform through out the edge as long as Sz symmetry is respected by the bulk Hamiltonian. But, in presence of external electric field acting locally on the edge, the induced Rashba type SO coupling can locally reorient the spin quantization axis and hence destroying the conservation of Sz symmetry. Interestingly, the Sz conservation of the system can be relaxed since the topological protection of edge state in a QSH system requires only the time-reversal symmetry. This provides a possibility for inducing SB phase that can arise from the evolution of spin on the edge in addition to the dynamical phase produced due to its propagation along the edge in a interferometric set-up. Our work provides a minimal framework to generate and detect these effects by employing spin-polarized leads. We show that spin-polarized leads could lead to resonances or anti-resonances in the two-terminal conductance of the interferometer. We further show that the positions of these anti-resonances get shifted as a function of energy of the incident electron which can be directly related to existence of SB phase in the system. At the end of this chapter, we present a realistic quantum transport simulations of a device setup in a lattice simulation of 2-D QSH insulator using kwant package where we obtain the conductance patterns which are almost identical to the previously obtained patterns from the low-energy effective edge model. Next, we carry out a model study of spin polarized voltage probe (SPVP) which could be used to read off spin resolved local voltages on the helical edge. Hence, at the very beginning of this chapter, we have proposed a possible configuration of SPVP specifically for the helical edge states where each SPVP is tunnel-coupled to the edge with multiple contacts (sub-probes) maintaining self-consistent relations among themselves. A comparative study with different number of sub-probes has been carried where we have estimated the magneto-resistance response even in presence of disorder in the local tunnelling strength and local spin polarisation. After that, we perform a numerical analysis to compute the Hall conductance (and also the longitudinal conductance) at the spin Hall edge considering a six-probe Hall bar setup (two current probes and four voltage probes) where four SPVP are implemented in a self-consistent fashion across a ferromagnetic barrier (local impurity) on the helical edge and we demonstrate that this model facilitates the evaluation of spin-resolved four-probe voltage drop across the ferromagnetic barrier. We also demonstrate the SPVP in a lattice simulation where we employ the edge state of a quantum anomalous Hall (QAH) system as SPVP which is tunnel-coupled over an optimized extended region of the helical edge. A 2-D lattice simulation for the quantum transport of a proposed device setup comprising a junction between QSH insulator and QAH insulator is considered and a feasibility study of using the edge of QAH insulator as an efficient spin-polarized voltage probe is carried out in presence of disorder. Finally, we discuss proximity effect of s-wave superconductivity on the QSH edge. The counter-propagating edge modes carry opposite spins and therefore, presence of a s-wave superconductor in close proximity of the QSH insulator can easily gapped out the edge via Cooper pairing. As a result, QSH system at its boundary acts like a spinless 1-D p-wave superconductor. One can expect the appearance of a pair of Majorana bound states (MBS) in a finite 1-D p-wave supercondctor at its two ends . Here we first explore the presence of MBS at the boundary of a proximity-induced QSH insulator within a 1-D edge model in context of quantum transport where we evaluate the Andreev reflection probability and obtain the corresponding electrical conductance. By using a ferromagnetic barrier at the end of a 1-D p-wave superconductor we can localized the Majorana mode on the edge. Existence of MBS ensures complete electron to hole conversion at zero energy across the ferromagnetic barrier for moderate barrier height and width. Hence, a resonance peak of 2e2/h at zero energy in the conductance vs incident energy of electron is expected to be observed and is a signature of MBS . Secondly, we carry out transport calculations in a 2-D QSH insulator and compute conductances numerically within a finite energy window using the BHZ model by employing the kwant package, which show a remarkable resemblance to the results obtained from the 1-D calculations of 1-D edge model. An unexplored region of the phase space involving the Zeeman-field induced boost of the helical edge state is proposed for the detection of MBS. Continuing the simulation of 2-D tight-binding system in presence of disorder confirms the robustness of our prediction. We also establish that chiral injection of electron (from a QAH insulator into the QSH system) is essential to detect the MBS in presence of disorder at a non-zero temperature (≈ 12 mK) in the previously mentioned transport calculations using kwant package.

Actions (login required)

View Item