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.     

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.     

Topological superfluid in one-dimensional ultracold atomic system with spin-orbit coupling

We propose a one-dimensional Hamiltonian H _1D which supports Majorana fermions when d _{x² ? y²}-wave superfluid appears in the ultracold atomic system and obtain the phase diagrams both for the time-reversal-invariant (TRI) case and time-reversal-symmetry-breaking (TRSB) case. From the phase diagrams, we find that the Majorana doublets and the single Majorana fermions exist in the topological superfluid (TSF) regions for the TRI case and the TRSB case, respectively, and we can reach these regions by tuning the chemical potential μ and spin-orbit coupling α _R . Importantly, the spin-orbit coupling has been realized in ultracold atoms by the recent experimental achievement of synthetic gauge field, therefore, our one-dimensional ultra-cold atomic system described by H _1D is a promising platform to find the mysterious Majorana fermions.     

Transport through a quantum dot coupled to two Majorana bound states

We investigate electron transport inside a ring system composed of a quantum dot (QD) coupled to two Majorana bound states confined at the ends of a one-dimensional topological superconductor nanowire. By tuning the magnetic flux threading through the ring, the model system we consider can be switched into states with or without zero-energy modes when the nanowire is in its topological phase. We find that the Fano profile in the conductance spectrum due to the interference between bound and continuum states exhibits markedly different features for these two different situations, which consequently can be used to detect the Majorana zero-energy mode. Most interestingly, as a periodic function of magnetic flux, the conductance shows 2π periodicity when the two Majorana bound states are nonoverlapping (as in an infinitely long nanowire) but displays 4π periodicity when the overlapping becomes nonzero (as in a finite length nanowire). We map the model system into a QD–Kitaev ring in the Majorana fermion representation and affirm these different characteristics by checking the energy spectrum.     

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.     

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.     

Realization of a Double-Slit SQUID Geometry by Fermi Arc Surface States in a WTe<Subscript>2</Subscript> Weyl Semimetal

We experimentally study electron transport between two superconducting indium leads, coupled to the WTe₂ crystal surface. WTe₂ is characterized by presence of Fermi arc surface states, as a predicted type-II Weyl semimetal candidate. We demonstrate Josephson current in unprecedentedly long 5 µm In–WTe₂–In junctions, which is confirmed by I–V curves evolution with temperature and magnetic field. The Josephson current is mostly carried by the topological surface states, which we demonstrate in a double-slit superconducting quantum interference device geometry, realized by coupling the opposite WTe₂ crystal surfaces.     

Coulomb repul sion of hole s and competition between $${d_{{x^2} - {y^2}}}$$ -wave and s-wave paring s in cuprate superconductor s

The effect of the Coulomb repulsion of holes on the Cooper instability in an ensemble of spin–polaron quasiparticles has been analyzed, taking into account the peculiarities of the crystallographic structure of the CuO₂ plane, which are associated with the presence of two oxygen ions and one copper ion in the unit cell, as well as the strong spin–fermion coupling. The investigation of the possibility of implementation of superconducting phases with d-wave and s-wave of the order parameter symmetry has shown that in the entire doping region only the d-wave pairing satisfies the self-consistency equations, while there is no solution for the s-wave pairing. This result completely corresponds to the experimental data on cuprate HTSC. It has been demonstrated analytically that the intersite Coulomb interaction does not affect the superconducting d-wave pairing, because its Fourier transform V _q does not appear in the kernel of the corresponding integral equation.     

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.     

The competition and coexistence of antiferromagnetism and <Emphasis Type="Italic">d</Emphasis>-wave superconductivity: probing coupled thermal fluctuations in a two dimensional minimal model

We consider the coexistence of antiferromagnetism and d-wave superconductivity, motivated by what one observes in the quasi-two dimensional organic salts. We study an electronic model that approximates some features of the Hubbard model, e.g., a repulsion that promotes local moments and Neel order, and an attractive intersite density–density coupling that promotes d-wave superconductivity. Staying at half-filling and a fixed attractive interaction we probe the effect of varying repulsion, using mean field theory for the ground state but retaining the full O(3) × U(1) spectrum of classical fluctuations at finite temperature. The ground state is superconducting at weak repulsion, a Neel ordered insulator at large repulsion, and a coexistence of the two orders in the intermediate regime. We observe four distinct kinds of thermal behaviour depending on the strength of repulsion. Starting with weak repulsion these are, first, a d-wave superconductor renormalised by magnetic fluctuations, second, a d-wave state transiting to an antiferromagnetic insulator and then to the normal state, third, a coexistent state transiting to the antiferromagnetic insulator and then the normal state, and, fourth, a Neel ordered insulator with weak pairing fluctuations. The low temperature state is either “nodal” or gapped, due to long range order, and the low energy spectral weight generally increases monotonically with temperature. At intermediate repulsion, however, the transition from the d-wave state to Neel antiferromagnet causes a loss of low energy weight which is gradually regained only at high temperature.     

Screening properties of multiply connected ferromagnet–superconductor hybrid structures

Superconducting phase transition temperature T _c of a ferromagnet/superconductor (SF) hybrid structure consisting of a hollow superconducting (S) cylinder (shell) with the central part (core) filled with a ferromagnetic (F) metal has been analyzed on the basis of linearized Usadel equations. It has been shown that the proximity effect between the S and F metals, as well as the exchange interaction, may induce an inhomogeneous superconducting state with Δ ~ exp(iLθ + ipz), which is characterized by nonzero circulation of phase L and wavenumber p describing the Larkin–Ovchinnikov–Fulde–Ferrell (LOFF) instability along the cylinder axis. The transitions between the states with different values of L and p, which are accompanied by a nonmonotonic dependence of superconducting transition temperature T _c and effective magnetic field penetration depth Λ into the SF structure on the characteristic size of the ferromagnetic region, have been investigated.     

Nonlinear generation of vorticity in thin smectic films

The energy levels of the fermions bound to the vortex are considered for vortices in the superfluid/superconducting systems that contain the symmetry protected plane of zeroes in the gap function in bulk. The Caroli–de Gennes–Matricon branches with different approach zero energy level at p_z → 0. The density of states of the bound fermions diverges at zero energy giving rise to the \(\sqrt \Omega \) dependence of the density of states in the polar phase of superfluid ³He rotating with the angular velocity Ω and to the \(\sqrt B \) dependence of the density of states for superconductors in the (d_xz + id_yz)-wave pairing state.     

Cascade of Quantum Transitions and Magnetocaloric Anomalies in an Open Nanowire

A sequence of magnetocaloric anomalies occurring with the change in a magnetic field H is predicted for an open nanowire with the Rashba spin–orbit coupling and the induced superconducting pairing potential. The nature of such anomalies is due to the cascade of quantum transitions related to the successive changes in the fermion parity of the nanowire ground state with the growth of the magnetic field. It is shown that the critical H_c values fall within the parameter range corresponding to the nontrivial values of the Z₂ topological invariant of the corresponding 1D band Hamiltonian characteristic of the D symmetry class. It is demonstrated that such features in the behavior of the open nanowire are retained even in the presence of Coulomb interactions.     

Tunneling conductance study of a metal-superconductor junction in the presence of Rashba spin orbit coupling

The tunneling conductance for a junction device consisting of a normal metal and a singlet superconductor is studied with Rashba spin orbit coupling (RSOC) being present in the metallic lead and the interface separating the two regions via an extended Blonder-Tinkham-Klapwijk (BTK) formalism. Interesting interplay between the RSOC and a number of parameters that have experimental significance, and characterize either the junction or the superconducting leads, such as the barrier transparency, quasiparticle lifetime, Fermi wavevector mismatch, an in-plane magnetic field and their effects on the tunneling conductance are investigated in details for both a s-wave and a d-wave superconductor. In an opaque barrier, in presence of a quasiparticle lifetime, a Fermi wavevector mismatch or an external in-plane magnetic field, RSOC enhances the conductance corresponding to low biasing energies, that is, at energies lesser than the superconducting gap, while the reverse is noted for energies exceeding the magnitude of the gap. Further, there are exciting anomalies noted in the conductance spectrum for the d-wave gap which can be understood by incorporating the interplay between the superconducting gap and the angle of incident of the charge carriers.     

Extended s-wave pairing symmetry on the triangular lattice heavy fermion system

We investigate the pairing symmetry of the Kondo-Heisenberg model on triangular lattice, which is believed to capture the core competition of Kondo screening and local magnetic exchange interaction in heavy electron compounds. On the dominant background of the heavy fermion state, the introduction of the Heisenberg antiferromagnetic interaction (J _H ) leads to superconducting pairing instability. Depending on the strength of the interactions, it is found that the pairing symmetry favours an extended s-wave for small J _H and high conduction electron density but a chiral \(d_{x^2 - y^2 } + id_{xy}\)-wave for large J _H and low conduction electron density, which provides a phase diagram of pairing symmetry from the calculations of the ground-state energy. The transition between these two pairing symmetries is found to be first-order. Furthermore, we also analyze the phase diagram from the pairing strengths and find that the phase diagram obtained is qualitatively consistent with that based on the ground-state energy. In addition, we propose an effective single-band BCS Hamiltonian, which is able to describe the low-energy thermodynamic behaviors of the heavy fermion superconducting states. These results further deepen the understanding of the antiferromagnetic interaction which results in a geometric frustration for the model studied. Our work may provide a possible scenario to understand the pairing symmetry of the heavy fermion superconductivity, which is one of active issues in very recent years.     

Magnetic susceptibility and superconductivity of (Pb<Subscript>0.2</Subscript>Sn<Subscript>0.8</Subscript>)<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>In<Subscript>x</Subscript>Te alloys as a function of in content

Experimental data on the superconductivity of In-doped Pb_zSn_1?zTe alloys (z=0.2) are discussed. The superconducting transition was detected from simultaneous measurements of the resistivity and magnetic susceptibility of a series of samples with different indium contents (2–12 mol % InTe). The superconducting transition detected by the magnetic susceptibility was observed at a temperature which was, on the average, 0.1 K below that determined from the resistivity. The increase in the superconducting transition temperature T _c with increasing indium content is of a threshold character, with T _c being proportional to the inverse electronic density of states at the Fermi level. The observed features in the experimental data are accounted for in terms of indium impurity resonance states in the material.     

Anisotropy of the heat capacity of a mixed-state superconducting Nd<Subscript>1.85</Subscript>Ce<Subscript>0.15</Subscript>CuO<Subscript>4</Subscript> single crystal for various magnetic field orientations with respect to the crystallographic axes

The anisotropy in the superconducting properties of single-crystal Nd_1.85Ce_0.15CuO₄ was studied from measurements of the heat capacity within the temperature interval 2–40 K in zero magnetic field and in a magnetic field of 8 T. We report on the first observation of heat capacity jumps occurring at the superconducting transition for various magnetic field orientations with respect to the crystallographic axes and on a strong anisotropy of the magnetic contribution to heat capacity in magnetic fields oriented in the a-b plane and perpendicular to it. These measurements yielded the anisotropy in the electronic heat capacity coefficient γ_n(H) and in the superconducting transition temperature T_c(H). The angular dependence of the Sommerfeld coefficient γ_n in the a-b plane observed in a magnetic field of 8 T exhibits four-lobe symmetry and zero gap direction of the order parameter. A comparison of the results obtained on the Nd_1.85Ce_0.15CuO₄ single crystal with the data available for La_1.85Sr_0.15CuO₄ permits one to conclude that the mechanisms of superconductivity in the electron-and hole-doped superconductors are similar.     

Quantum interference effects induced by multiple magnetic impurities on a triangular lattice f-wave superconductor

The effects of multi-impurity quantum interference on triangular lattice f-wave superconductors are studied by self-consistently solving Bogoliubov-de Gennes equations within the t?t′?J?V model. An overall phase diagram is presented, which shows that f-wave superconductivity dominates near 0.3 doping. Rich phenomena are induced by quantum interference effects, such as periodic modulations in charge orders, pyramid frustum structures, and a magnetic moment reverse transition, which are qualitatively different from the single-impurity case. We also examine the local density of states to show how localized quasiparticle states are created at or near the impurity sites, which can be directly measured by scanning tunneling microscopy experiments.     

Analytical assessment of some characteristic ratios for <Emphasis Type="Italic">s</Emphasis>-wave superconductors

We evaluate some thermodynamic quantities and characteristic ratios that describe low- and high-temperature s-wave superconducting systems. Based on a set of fundamental equations derived within the conformal transformation method, a simple model is proposed and studied analytically. After including a one-parameter class of fluctuations in the density of states, the mathematical structure of the s-wave superconducting gap, the free energy difference, and the specific heat difference is found and discussed in an analytic manner. Both the zero-temperature limit T = 0 and the subcritical temperature range T ? T_c are discussed using the method of successive approximations. The equation for the ratio R₁, relating the zero-temperature energy gap and the critical temperature, is formulated and solved numerically for various values of the model parameter. Other thermodynamic quantities are analyzed, including a characteristic ratio R₂, quantifying the dynamics of the specific heat jump at the critical temperature. It is shown that the obtained model results coincide with experimental data for low-T_c superconductors. The prospect of application of the presented model in studies of high-T_c superconductors and other superconducting systems of the new generation is also discussed.