Helical Majorana Fermions Research Articles

Thermal transport in multiple Majorana edge states of hybrid topological superconductor junctions

The hybrid topological superconductor (TSC) with multiple Majorana edge states can be formed in a system where a magnetic topological insulator (MTI) thin film is coupled with two s-wave superconductors in proximity. The band structure and thermal transport properties of the hybrid TSC hosting chiral and helical Majorana edge states are investigated. By using the nonequilibrium Green’s function method combined with the Landauer–Büttiker formula, the scattering coefficients (i.e., the normal tunneling T, the local Andreev reflection TLAR, the crossed Andreev reflection TCAR) and two-terminal electric thermal conductance κe are calculated. With the change of the exchange field Mz, the system goes through a series of topological phase transitions. The chiral Majorana edge states, the helical Majorana edge states and the multiple Majorana edge states possessing both chiral and helical edge modes are induced at the boundary of TSC. At the boundary of the central TSC, a single Majorana fermion edge state is equivalent to half an ordinary fermion to carry heat, leading to the generation of a half-integer quantized thermal conductance plateau. Thus, half-integer (i.e., 1/2, 3/2) and integer (i.e., 1, 2) quantized thermal conductance plateaus appear in different topological phases. These quantized plateaus that survive well over a certain range of temperatures can be used experimentally to detect chiral and helical Majorana fermions.

Thermal transport of helical Majorana edge modes in a superconducting antiferromagnetic quantum spin Hall insulator

In the system composed of a two-dimensional antiferromagnetic quantum spin Hall insulator (AFQSHI) nanoribbon covered by s -wave superconductors, possessing helical Majorana edge mode (HMEM), and preserving effective time-reversal symmetry. Based on such the AFQSHI-SC hybrid system, we investigate the band structure and thermal transport properties. The two-terminal thermal conductance κ e l and electric conductance G are calculated by using the Landauer-Büttiker formula and the nonequilibrium Green’s function method. When a pair of intersecting edge states exists in the bulk band gap, a helical topological superconductor with Z 2 invariant ν = ± 1 is formed, which has a pair of Majorana edge states with opposite chirality, that is Majorana Kramer pairs, leading to a integer quantized thermal conductance plateau. We find that the quantized thermal conductance plateau is well preserved in a certain range of temperature. The presence and number of Majorana Kramer pairs in helical topological superconductor can be confirmed with the help of the quantized thermal conduction plateau, which is promising for the experimental detection of helical Majorana fermions. • In this manuscript, based on the Landauer-Buttiker formula and the nonequilibrium Green’s function method, we investigate the properties of helical Majorana fermions edge states thermal transport in a hybrid AFQSHI-SC structure. • When a pair of Majorana edge states with opposite chirality residing at the edge of the central topological superconductor region, a integer quantized thermal conductance plateau is discovered. • This quantized plateau can be used as strong evidence for the experimental detection of helical topological superconductors.

Intrinsic Time-Reversal-Invariant Topological Superconductivity in Thin Films of Iron-Based Superconductors.

We establish quasi-two-dimensional thin films of iron-based superconductors (FeSCs) as a new high-temperature platform for hosting intrinsic time-reversal-invariant helical topological superconductivity (TSC). Based on the combination of Dirac surface state and bulk extended s-wave pairing, our theory should be directly applicable to a large class of experimentally established FeSCs, opening a new TSC paradigm. In particular, an applied electric field serves as a "topological switch" for helical Majorana edge modes in FeSC thin films, allowing for an experimentally feasible design of gate-controlled helical Majorana circuits. Applying an in-plane magnetic field drives the helical TSC phase into a higher-order TSC carrying corner-localized Majorana zero modes. Our proposal should enable the experimental realization of helical Majorana fermions.

Chiral and counter-propagating Majorana fermions in a p-wave superconductor

Chiral and helical Majorana fermions are two archetypal edge excitations in two-dimensional topological superconductors. They emerge from systems of different Altland–Zirnbauer symmetries and characterized by and topological invariants respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate in a single system without symmetry breaking. Here, we explore the peculiar behaviors of Majorana edge modes in topological superconductors with an additional ‘mirror’ symmetry which changes the bulk topological invariant to type. A theoretical toy model describing the proximity structure of a Chern insulator and a px-wave superconductor is proposed and solved analytically to illustrate a direct transition between two topologically nontrivial phases. The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by mirror-graded Chern number and hosts a pair of counter-propagating Majorana fermions protected by the mirror symmetry. The edge theory is worked out in detail, and implications to braiding of Majorana fermions are discussed.

Helical Majorana fermions and flat edge states in the heterostructures of iridates and high- TC cuprates

Majorana bound states exist inside a core of half-quantum vortex in spin-triplet superconductors. Despite intense efforts, they have not been discovered in any spin-triplet superconductor candidate materials. After the success of topological insulator, another route to achieve Majorana fermions has been suggested: heterostructures of s-wave superconductors and topological insulator or semimetal with strong spin-orbit coupling provide an effective spinless $p+ip$ pairing which supports Majorana bound states in a single vortex. This theoretical observation has stimulated both experimental and theoretical communities to search for Majorana fermions, and recently a localized Majorana state at the end of one-dimensional wire has been reported. Here we study the two-dimensional interface of Weyl semimetal and d-wave superconductor which promotes a pair of helical Majorana fermions propagating along the edge of the interface. We suggest that Iridium oxide layer IrO$_2$ classified as two-dimensional Weyl semimetal in close proximity to high temperature Cuprates would be a best example to explore these helical Majorana modes.

Andreev reflection in two-dimensional relativistic materials with realistic tunneling transparency in normal-metal/superconductor junctions

The Andreev conductance across 2d normal metal (N)/superconductor (SC) junctions with relativistic Dirac spectrum is investigated theoretically in the Blonder-Tinkham-Klapwijk formalism. It is shown that for relativistic materials, due to the Klein tunneling instead of impurity potentials, the local strain in the junction is the key factor that determines the transparency of the junction. The local strain is shown to generate an effective Dirac $\delta$-gauge field. A remarkable suppression of the conductance are observed as the strength of the gauge field increases. The behaviors of the conductance are in well agreement with the results obtained in the case of 1d N/SC junction. We also study the Andreev reflection in a topological material near the chiral-to-helical phase transition in the presence of a local strain. The N side of the N/SC junction is modeled by the doped Kane-Mele (KM) model. The SC region is a doped correlated KM t-J (KMtJ) model, which has been shown to feature d+id'-wave spin-singlet pairing. With increasing intrinsic spin-orbit (SO) coupling, the doped KMtJ system undergoes a topological phase transition from the chiral d-wave superconductivity to the spin-Chern superconducting phase with helical Majorana fermions at edges. We explore the Andreev conductance at the two inequivalent Dirac points, respectively and predict the distinctive behaviors for the Andreev conductance across the topological phase transition. Relevance of our results for the adatom-doped graphene is discussed.

Helical Majorana fermions in d(x2-y2) + id(xy)-wave topological superconductivity of doped correlated quantum spin Hall insulators.

There has been growing interest in searching for exotic self-conjugate, charge-neutral low-energy fermionic quasi-particles, known as Majorana fermions (MFs) in solid state systems. Their signatures have been proposed and potentially observed at edges of topological superconcuctors with non-trivial topological invariant in the bulk electronic band structure. Much effort have been focused on realizing MFs in odd-parity superconductors made of strong spin-orbit coupled materials in proximity to conventional superconductors. In this paper, we propose a novel mechanism for realizing MFs in 2D spin-singlet topological superconducting state induced by doping a correlated quantum spin Hall (Kane-Mele) insulator. Via a renormalized mean-field approach, the system is found to exhibits time-reversal symmetry (TRS) breaking -wave (chiral d–wave) superconductivity near half-filling in the limit of large on-site repulsion. Surprisingly, however, at large spin-orbit coupling, the system undergoes a topological phase transition and enter into a new topological phase protected by a pseudo-spin Chern number, which can be viewed as a persistent extension of the quantum spin Hall phase upon doping. From bulk-edge correspondence, this phase is featured by the presence of two pairs of counter-propagating helical Majorana modes per edge, instead of two chiral propagating edge modes in the d + id′ superconductors.

Two-impurity helical Majorana problem

We predict experimentally accessible signatures for helical Majorana fermions in a topological superconductor by coupling to two quantum dots in the local moment regime (corresponding to spin-$1/2$ impurities). Taking into account RKKY interactions mediated by bulk and edge modes, where the latter cause a long-range antiferromagnetic Ising coupling, we formulate and solve the low-energy theory for this two-impurity helical Majorana problem. In particular, we show that the long-time spin dynamics after a magnetic field quench displays weakly damped oscillations with universal quality factor.

Experimental studies of the topological superconductor CuxBi2Se3

CuxBi2Se3 was recently elucidated to be the first example of a time-reversal-invariant topological superconductor characterized by the gapless surface quasiparticle states consisting of helical Majorana fermions. The evidence for topological superconductivity in CuxBi2Se3 was obtained by the point-contact spectroscopy that indirectly detected the itinerant massless Majorana state on the surface. In this paper, we show new point-contact spectroscopy data which supplement the previous conclusion. We also present the anisotropic normal-state resistivity data for superconducting samples of CuxBi2Se3 which strongly suggest that not only the effective mass but also the scattering rate is anisotropic in this material by a factor of ~3.

Anomalous Josephson current in superconducting topological insulator

We investigate the effect of helical Majorana fermions at the surface of superconducting topological insulators (STI) on the Josephson current by referring to possible pairing states of Cu-doped Bi2Se3. The surface state in the present STI has a spin helicity because the directions of spin and momentum are locked to each other. The Josephson current-phase relation in an STI/s-wave superconductor junction shows robust sin(2{\phi}) owing to mirror symmetry, where \phi denotes the macroscopic phase difference between the two superconductors. In contrast, the maximum Josephson current in an STI/STI junction exhibits a nonmonotonic temperature dependence depending on the relative spin helicity of the two surface states. Detecting these features qualifies as distinct experimental evidence for the identification of the helical Majorana fermion in STIs.

Anomalous suppression of the superfluid density in the CuxBi2Se3superconductor upon progressive Cu intercalation

CuxBi2Se3 was recently found to be likely the first example of a time-reversal-invariant topological superconductor accompanied by helical Majorana fermions on the surface. Here we present that progressive Cu intercalation into this system introduces significant disorder and leads to an anomalous suppression of the superfluid density which was obtained from the measurements of the lower critical field. At the same time, the transition temperature T_c is only moderately suppressed, which agrees with a recent prediction for the impurity effect in this class of topological superconductors bearing strong spin-orbit coupling. Those unusual disorder effects give support to the possible odd-parity pairing state in CuxBi2Se3.

Theory of tunneling conductance and surface-state transition in superconducting topological insulators

We develop a theory of the tunneling spectroscopy for superconducting topological insulators (STIs), where the surface Andreev bound states (SABSs) appear as helical Majorana fermions. Based on the symmetry and topological nature of parent topological insulators, we find that the SABSs in the STIs have a profound structural transition in the energy dispersions. The transition results in a variety of Majorana fermions, by tuning the chemical potential and the effective mass of the energy band. We clarify that Majorana fermions in the vicinity of the transitions give rise to robust zero bias peaks in the tunneling conductance between normal metal/STI junctions.

Topological defects and gapless modes in insulators and superconductors

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians $\mathcal{H}(\mathbf{k},\mathbf{r})$ that vary slowly with adiabatic parameters $\mathbf{r}$ surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.