Topological superconductivity and the fractional Josephson effect in quasi-one dimensional wires on a plane

A time-reversal invariant topological superconductivity is suggested to be realized in a quasi-one-dimensional structure on a plane, which is fabricated by filling the superconducting materials into the periodic channel of dielectric matrices like zeolite and asbestos under high pressure. The topological superconducting phase sets up in the presence of large spin–orbit interactions when intra-wire s-wave and inter-wire d-wave pairings take place. Kramers pairs of Majorana bound states emerge at the edges of each wire. We analyze effects of the Zeeman magnetic field on Majorana zero-energy states. In-plane magnetic field was shown to make asymmetric the energy dispersion, nevertheless Majorana fermions survive due to protection of a particle–hole symmetry. Tunneling of Majorana quasiparticle from the end of one wire to the nearest-neighboring one yields edge fractional Josephson current with 4π-periodicity.     

Dominant Majorana bound energy and critical current enhancement in ferromagnetic-superconducting topological insulator

Among the potential applications of topological insulators, we theoretically study the coexistence of proximity-induced ferromagnetic and superconducting orders in the surface states of a 3-dimensional topological insulator. The superconducting electron-hole excitations can be significantly affected by the magnetic order induced by a ferromagnet. In one hand, the surface state of the topological insulator, protected by the time-reversal symmetry, creates a spin-triplet and, on the other hand, magnetic order causes to renormalize the effective superconducting gap. We find Majorana mode energy along the ferromagnet/superconductor interface to sensitively depend on the magnitude of magnetization m _zfs from superconductor region, and its slope around perpendicular incidence is steep with very low dependency on m _zfs . The superconducting effective gap is renormalized by a factor η(m _zfs ), and Andreev bound state in ferromagnet-superconductor/ferromagnet/ferromagnet-superconductor (FS/F/FS) Josephson junction is more sensitive to the magnitude of magnetizations of FS and F regions. In particular, we show that the presence of m _zfs has a noticeable impact on the gap opening in Andreev bound state, which occurs in finite angle of incidence. This directly results in zero-energy Andreev state being dominant. By introducing the proper form of corresponding Dirac spinors for FS electron-hole states, we find that via the inclusion of m _zfs , the Josephson supercurrent is enhanced and exhibits almost abrupt crossover curve, featuring the dominant zero-energy Majorana bound states.     

The invisible Majorana bound state at the helical edge

The presence of a Majorana bound state in condensed matter systems is often associated to a zero bias peak in conductance measurements. Here, we analyze a system were this paradigm is violated. A Majorana bound state is always present at the interface between a quantum spin Hall system that is magnetically gapped and a quantum spin Hall system gapped by proximity induced s-wave superconductivity. However, the linear conductance could be either zero or non-zero and quantized depending on the energy and length scales of the barriers. The transition between the two values is reminiscent of the topological phase transition in proximitized spin–orbit coupled quantum wires in the presence of an applied magnetic field. We interpret the behavior of the conductance in terms of scattering states at both zero and non-zero energy.     

Kondo-assistant Aharonov-Bohm transport in a quantum dot-Majorana wire system

We investigate the Kondo-assistant Aharonov-Bohm (AB) transport in a Quantum dot (QD) coupled with a topological Majorana wire. We noted that the conductance exhibits sensitive dependence on the phase factor of AB ring when the wire-QD coupling strength changes. The DOS resonance split when the coupling strength changes from small to large. The current is determined by the Kondo transport characteristics presented by the quantum dots (QDs). Also, the transport results show different p-dependence properties under parallel and anti-parallel leads alignment. We believe that these results can be helpful for understanding the Majorana-QD coupling properties as well as the detection of the Majorana bound states.     

Electronic transport through a Majorana bound state coupled to a T-shaped quantum-dot system with Coulomb interaction

Using the Green’s function technique, we respectively investigate the electron transport properties of two spin components through the system of a T-shaped double quantum dot structure coupled to a Majorana bound state, in which only one quantum dot is connected with two metallic leads. We explore the interplay between the Fano effect and the MBSs for different dot-MBS coupling strength λ, dot-dot coupling strength t, and MBS-MBS coupling strength ε_M in the noninteracting case. Then the Coulomb interaction and magnetic field effect on the conductance spectra are investigated. Our results indicate that G_↓(ω) is not affected by the Majorana bound states, but a “0.5” conductance signature occurs in the vicinities of Fermi level of G_↑(ω). This robust property persists for a wide range of dot-dot coupling strength and dot-MBS coupling strength, but it can be destroyed by Coulomb interaction in quantum dots. By adjusting the size and direction of magnetic field around the quantum dots, the “0.5” conductance signature damaged by U can be restored. At last, the spin magnetic moments of two dots by applying external magnetic field are also predicted.     

Quantized conductance at the Majorana phase transition in a disordered superconducting wire

Superconducting wires without time-reversal and spin-rotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zero-energy states is complicated by the proliferation of low-lying excitations in a disordered multimode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number.     

Thermoelectric effect in the Kondo dot side-coupled to a Majorana mode

We investigate the linear thermoelectric response of an interacting quantum dot side-coupled by one of two Majorana modes hosted by a topological superconducting wire. We employ the numerical renormalization group technique to obtain the thermoelectrical conductance L in the Kondo regime while the background temperature T, the Majorana-dot coupling Γ _m , and the overlap ε _m between the two Majorana modes are tuned. We distinguish two transport regimes in which L displays different features: the weak- (Γ _m <T _K ) and strong-coupling (Γ _m >T _K ) regimes, where T _K is the Kondo temperature. For an infinitely long nanowire where the Majorana modes do not overlap (ε _m = 0), the thermoelectrical conductance in the weak-coupling regime exhibits a peak at T ~ Γ _m <T _K . This peak is ascribed to the anti-Fano resonance between the asymmetric Kondo resonance and the zero-energy Majorana bound state. In the strong-coupling regime, on the other hand, the Kondo-induced peak in L is affected by the induced Zeeman splitting in the dot. For finite but small overlap (0 <ε _m <Γ _m ), the interference between the two Majorana modes restores the Kondo effect in a smaller energy scale Γ′ _m and gives rise to an additional peak in Γ ~ Γ′ _m, whose sign is opposite to that at T ~ Γ _m . In the strong-coupling regime this additional peak can cause a non-monotonic behavior of L with respect to the dot gate. Finally, in order to identify the fingerprint of Majorana physics, we compare the Majorana case with its counterpart in which the Majorana bound states are replaced by a (spin-polarized) ordinary bound state and find that the thermoelectric features for finite ε _m are the genuine effect of the Majorana physics.     

Majorana零模式的电导与低压振荡散粒噪声

Majorana零能量模式是自身的反粒子,在拓扑量子计算中有重要应用.本文研究量子点与拓扑超导纳米线混合结构,通过量子点的输运电荷检测Majorana零模式.利用量子主方程方法,发现有无Majorana零模式的电流与散粒噪声存在明显差别.零模式导致稳态电流差呈反对称,在零偏压处显示反常电导峰.电流差随零模式分裂能的增大而减小,随量子点与零模式耦合的增强而增大.另一方面,零模式导致低压散粒噪声相干振荡,零频噪声显著增强.分裂能导致相干振荡愈加明显且零频噪声减小,而量子点与零模式的耦合使零频噪声增强.当量子点与电极非对称耦合时,零模式使电子由反聚束到聚束输运,亚泊松噪声增强为超泊松噪声.稳态电流差结合低压振荡的散粒噪声能够揭示Majorana零模式是否存在.     

Magnetic flux control of chiral Majorana edge modes in topological superconductor

We study the transport of chiral Majorana edge modes (CMEMs) in a hybrid quantum anomalous Hall insulator-topological superconductor (QAHI-TSC) system in which the TSC region contains a Josephson junction and a cavity. The Josephson junction undergoes a topological transition when the magnetic flux through the cavity passes through half-integer multiples of magnetic flux quantum. For the trivial phase, the CMEMs transmit along the QAHI-TSC interface as without magnetic flux. However, for the nontrivial phase, a zero-energy Majorana state appears in the cavity, leading to that a CMEM can resonantly tunnel through the Majorana state to a different CMEM. These findings may provide a feasible scheme to control the transport of CMEMs by using the magnetic flux and the transport pattern can be customized by setting the size of the TSC.     

Aharonov–Bohm oscillations caused by non-topological surface states in Dirac nanowires

One intriguing fingerprint of surface states in topological insulators is the Aharonov–Bohm effect in magnetoconductivity of nanowires. We show that surface states in nanowires of Dirac materials (bismuth, bismuth antimony, and lead tin chalcogenides) being in non-topological phase, exhibit the same effect as amendment to magnetoconductivity of the bulk states. We consider a simple model of a cylindrical nanowire, which is described by the 3D Dirac equation with a general T-invariant boundary condition. The boundary condition is determined by a single phenomenological parameter whose sign defines topological-like and non-topological surface states. The non-topological surface states emerge outside the gap. In a longitudinal magnetic field B, they lead to Aharonov–Bohm amendment for the density of states and correspondingly for the conductivity of the nanowire. The phase of these magnetic oscillations increases with B from π to 2π.     

Observing Majorana bound states in p-wave superconductors using noise measurements in tunneling experiments

The zero-energy bound states at the edges or vortex cores of chiral p-wave superconductors should behave like Majorana fermions. We introduce a model Hamiltonian that describes the tunneling process when electrons are injected into such states. Using a nonequilibrium Green function formalism, we find exact analytic expressions for the tunneling current and noise and identify experimental signatures of the Majorana nature of the bound states to be found in the shot noise. We discuss the results in the context of different candidate materials that support triplet superconductivity. Experimental verification of the Majorana character of midgap states would have important implications for the prospects of topological quantum computation.     

Majorana–Kondo interplay in a Majorana wire-quantum dot system with ferromagnetic contacts*

We consider a single-level quantum dot(QD)and a topological superconducting wire hosting Majorana bound states at its ends.By the equation of motion method,we give the analytical Green’s function of the QD in the noninteracting and the infinite interacting case.We study the effects of QD energy level and the spin polarization on the density of states(DOS)and linear conductance of the system.In the noninteracting case,the DOS resonance shifts with the change of energy level and it shows bimodal structure at large spin polarization strength.In the infinite interacting case,the up-spin linear conductance first increases and then decreases with the increase of spin polarization strength,but the down-spin is stable.However,the DOS shows a splitting phenomenon in the large energy level with the increase of spin polarization strength.This provides an interesting way to explore the physical properties of such spin dependent effect in the hybrid Majorana QD systems.     

Effect of magnetic impurities in chiral <Emphasis Type="Italic">p</Emphasis>-wave superconducting nanoloops

The effect of many magnetic impurities in symmetric chiral p-wave superconducting nanoloops is investigated by numerically solving the BdG equations self-consistently. Two magnetic impurities can lead to the appearance of two impurity bound levels close to the Fermi level. The arising bound states can cross the Fermi level at the same impurity strength for the case of two independent midway impurities, while multiple zero-energy states can be obtained at two separated values of impurity strength when two independent edge impurities are present. Moreover, the multiple zero modes can only show up for appropriate relative positions between two edge impurities due to the quantum interference effect. Particularly, for some appropriate strength of two independent midway impurities, the impurity bound levels cross the Fermi level twice with increasing threaded flux, while the multiple zero modes can not emerge in the flux evolution.     

Majorana bound state of a Bogoliubov-de Gennes-Dirac Hamiltonian in arbitrary dimensions

We study a Majorana zero-energy state bound to a hedgehog-like point defect in a topological superconductor described by a Bogoliubov-de Gennes (BdG)-Dirac type effective Hamiltonian. We first give an explicit wave function of a Majorana state by solving the BdG equation directly, from which an analytical index can be obtained. Next, by calculating the corresponding topological index, we show a precise equivalence between both indices to confirm the index theorem. Finally, we apply this observation to reexamine the role of another topological invariant, i.e., the Chern number associated with the Berry curvature proposed in the study of protected zero modes along the lines of topological classification of insulators and superconductors. We show that the Chern number is equivalent to the topological index, implying that it indeed reflects the number of zero-energy states. Our theoretical model belongs to the BDI class from the viewpoint of symmetry, whereas the spatial dimension d of the system is left arbitrary throughout the paper.     

Finite-frequency noise in a topological superconducting wire

In this paper we study the finite-frequency current cross-correlations for a topological superconducting nanowire attached to two terminals at one of its ends. Using an analytic 1D model we show that the presence of a Majorana bound state yields vanishing cross-correlations for frequencies larger than twice the applied transport voltage, in contrast to what is found for a zero-energy ordinary Andreev bound state. Zero cross-correlations at high frequency have been confirmed using a more realistic tight-binding model for finite-width topological superconducting nanowires. Finite-temperature effects have also been investigated.     

Helical liquids and Majorana bound states in quantum wires

We show that the combination of spin-orbit coupling with a Zeeman field or strong interactions may lead to the formation of a helical electron liquid in single-channel quantum wires, with spin and velocity perfectly correlated. We argue that zero-energy Majorana bound states are formed in various situations when such wires are situated in proximity to a conventional s-wave superconductor. This occurs when the external magnetic field, the superconducting gap, or, most simply, the chemical potential vary along the wire. These Majorana states do not require the presence of a vortex in the system. Experimental consequences of the helical liquid and the Majorana states are also discussed.     

Aharonov-Bohm Interferometer in a T-Shaped Quantum Dot Embedded in Majorana Bound States *

We theoretically study the spin-dependent transport properties of anAharonov-Bohm (AB) interferometer composed by a T-shaped quantum dot (QD)embedded in Majorana bound states (MBS). We use the equation of motion method tocalculate the conductance across the interferometer. We note that the conductance exhibitssensitive dependence on the MBS-QD coupling strength as well as the polarization strengthof the leads when the phase factor of AB ring changes periodically. The conductance shows a transitionfrom resonance to anti-resonance when the MBS-QD coupling strength changes from small to large. Also, there is different p-dependence conductance when the leads alignment changesfrom parallel to anti-parallel. These findings suggest that such a model could be used for a sensitivedetection of MBS interactions, exploiting the high sensitivity of conductance to the AB phase in theinterferometer.     

Resonant Andreev reflection in a normal-metal/quantum-dot/superconductor system with coupled Majorana bound states

Andreev reflection(AR) in a normal-metal/quantum-dot/superconductor(N–QD–S) system with coupled Majorana bound states(MBSs) is investigated theoretically. We find that in the N–QD–S system, the AR can be enhanced when coupling to the MBSs is incorporated. Fano line-shapes can be observed in the AR conductance spectrum when there is an appropriate QD–MBS coupling or MBS–MBS coupling. The AR conductance is always e~2/2h at the zero Fermi energy point when only QD–MBSs coupling is considered. In addition, the resonant AR occurs when the MBS–MBS coupling roughly equals to the QD energy level. We also find that an AR antiresonance appears when the QD energy level approximately equals to the sum of the QD–MBS coupling and the MBS–MBS coupling. These features may serve as characteristic signatures for the probe of MBSs.     

Superconducting quantum interference devices made of Sb-doped Bi2Se3 topological insulator nanoribbons

We report the fabrication and characterization of superconducting quantum interference devices (SQUIDs) made of Sb-doped Bi₂Se₃ topological insulator (TI) nanoribbon (NR) contacted with PbIn superconducting electrodes. When an external magnetic field was applied along the NR axis, the TI NR exhibited periodic magneto-conductance oscillations, the so-called Aharonov-Bohm oscillations, owing to one-dimensional subbands. Below the superconducting transition temperature of PbIn electrodes, we observed supercurrent flow through TI NR-based SQUID. The critical current periodically modulates with a magnetic field perpendicular to the SQUID loop, revealing that the periodicity corresponds to the superconducting flux quantum. Our experimental observations can be useful to explore Majorana bound states (MBS) in TI NR, promising for developing topological quantum information devices.     

Effect of incommensurate potential on the resonant tunneling through Majorana bound states on the topological superconductor chains

We study the transport through the Kitaev chain with incommensurate potentials coupled to two normal leads by the numerical operator method. We find a quantized linear conductance of e ² / h, which is independent to the disorder strength and the gate voltage in a wide range, signaling the Majorana bound states. While the incommensurate potential suppresses the current at finite voltage bias, and then narrows the linear response regime of the I-V curve which exhibits two plateaus corresponding to the superconducting gap and the band edge, respectively. The linear conductance abruptly drops to zero as the disorder strength reaches the critical value 2g _s + 2Δ with Δ the p-wave pairing amplitude and g _s the hopping between neighbor sites, corresponding to the transition from the topological superconducting phase to the Anderson localized phase. Changing the gate voltage also causes an abrupt drop of the linear conductance by driving the chain into the topologically trivial superconducting phase, whose I-V curve exhibits an exponential shape.