Odd-frequency Spin-triplet Pairing Research Articles

Theory of Majorana Zero Modes in Unconventional Superconductors

Abstract Majorana fermions are spin-1/2 neutral particles that are their own antiparticles; they were initially predicted by Ettore Majorana in particle physics but their observation still remains elusive. The concept of Majorana fermions has been borrowed by condensed matter physics, where, unlike particle physics, Majorana fermions emerge as zero-energy quasiparticles that can be engineered by combining electrons and holes and have therefore been called Majorana zero modes. In this review, we provide a pedagogical explanation of the basic properties of Majorana zero modes in unconventional superconductors and their consequences in experimental observables, putting a special emphasis on the initial theoretical discoveries. In particular, we first show that Majorana zero modes are self-conjugated and emerge as a special type of zero-energy surface Andreev bound states at the boundary of unconventional superconductors. We then explore Majorana zero modes in 1D spin-polarized p-wave superconductors, where we address the formation of topological superconductivity and the physical realization in superconductor–semiconductor hybrids. In this part we highlight that Majorana quasiparticles appear as zero-energy edge states, exhibiting charge neutrality, spin-polarization, and spatial nonlocality as unique properties that can already be seen from their energies and wavefunctions. Next, we discuss the analytically obtained Green’s functions of p-wave superconductors and demonstrate that the emergence of Majorana zero modes is always accompanied by the formation of odd-frequency spin-triplet pairing as a unique result of the self-conjugate nature of Majorana zero modes. We finally address the signatures of Majorana zero modes in tunneling spectroscopy, including the anomalous proximity effect, and the phase-biased Josephson effect.

Fluctuation-induced odd-frequency spin-triplet pairing in a disordered electron liquid

We consider a two-dimensional disordered conductor in the regime when the superconducting phase is destroyed by the magnetic field. We analyze a combination of fluctuations of different origin which results in an effective interaction amplitude suitable for a spontaneous s-wave odd-frequency pairing instability. We observe that the end point of the superconductivity is a quantum critical point separating the conventional superconducting phase from a state with the odd-frequency spin-triplet pairing instability. We speculate that this could shed light on a rather mysterious insulating state observed in strongly disordered superconducting films in a broad region of the magnetic fields.

Unconventional Meissner screening induced by chiral molecules in a conventional superconductor

The coupling of a superconductor to a different material often results in a system with unconventional superconducting properties. A conventional superconductor is a perfect diamagnet expelling magnetic fields out of its volume, a phenomenon known as Meissner effect. Here, we show that the simple adsorption of a monolayer of chiral molecules, which are non-magnetic in solution, onto the surface of a conventional superconductor can markedly change its diamagnetic Meissner response. By measuring the internal magnetic field profile in superconducting Nb thin films under an applied transverse field by low-energy muon spin rotation spectroscopy, we demonstrate that the local field profile inside Nb is considerably modified upon molecular adsorption in a way that also depends on the applied field direction. The modification is not limited to the chiral molecules/Nb interface, but it is long ranged and occurs over a length scale comparable to the superconducting coherence length. Zero-field muon spin spectroscopy measurements in combination with our theoretical analysis show that odd-frequency spin-triplet states induced by the chiral molecules are responsible for the modification of the Meissner response observed inside Nb. These results indicate that a chiral molecules/superconductor system supports odd-frequency spin-triplet pairs due to the molecules acting as a spin active layer and therefore they imply that such system can be used as a simpler alternative to superconductor/ferromagnet or superconductor/topological insulator hybrids for the generation and manipulation of unconventional spin-triplet superconducting states.

Impact of impurity scattering on odd-frequency spin-triplet pairing near the edge of the Kitaev chain

We study a Kitaev chain model, which is the simplest model of topological superconductors hosting Majorana fermion, appearing as a zero-energy state at the edge. We analytically calculate the Green's function of the semi-infinite Kitaev chain with a delta-function-type impurity potential within the quasi-classical regime to obtain the spatial dependence of the induced odd-frequency pairing. It is found that if the position of the impurity is not far from the edge, the spatial profile of the local density of states (LDOS) and the odd-frequency spin-triplet $s$-wave pair amplitude is tunable as a function of the strength of the impurity potential. Moreover, the zero-energy LDOS and low-frequency odd-frequency pair amplitude are found to have the same spatial dependence. The spatial profile of the zero-energy LDOS is analyzed based on the wave function of Majorana fermions.

Large Josephson current in Weyl nodal loop semimetals due to odd-frequency superconductivity

Weyl nodal loop semimetals (WNLs) host a closed nodal line loop Fermi surface in the bulk, protected zero-energy flat band, or drumhead, surface states, and strong spin-polarization. The large density of states of the drumhead states makes WNL semimetals exceedingly prone to electronic ordering. At the same time, the spin-polarization naively prevents conventional superconductivity due to its spin-singlet nature. Here we show the complete opposite: WNLs are extremely promising materials for superconducting Josephson junctions, entirely due to odd-frequency superconductivity. By sandwiching a WNL between two conventional superconductors we theoretically demonstrate the presence of very large Josephson currents, even up to orders of magnitude larger than for normal metals. The large currents are generated both by an efficient transformation of spin-singlet pairs into odd-frequency spin-triplet pairing by the Weyl dispersion and the drumhead states ensuring exceptionally proximity effect. As a result, WNL Josephson junctions offer unique possibilities for detecting and exploring odd-frequency superconductivity.

Odd-frequency superconductivity near a magnetic impurity in a conventional superconductor

Superconductor-ferromagnetic heterostructures have been suggested as one of the most promising alternatives of realizing odd-frequency superconductivity. In this work we consider the limit of shrinking the ferromagnetic region to the limit of a single impurity embedded in a conventional superconductor, which gives raise to localized Yu-Shiba-Rusinov (YSR) bound states with energies inside the superconducting gap. We demonstrate that all the sufficient ingredients for generating odd-frequency pairing are present at the vicinity of these impurities. We investigate the appearance of all possible pair amplitudes in accordance with the Berezinskii $SP^{\ast}OT^{\ast} = -1$ rule, being the symmetry under the exchange of spin, spatial, orbital (in our case $O=+1$) and time index, respectively. We study the spatial and frequency dependence of of the possible pairing amplitudes, analyzing their evolution with impurity strength and identifying a reciprocity between different symmetries related through impurity scattering. We show that the odd-frequency spin-triplet pairing amplitude dominates at the critical impurity strength, where the YSR states merge at the middle of the gap, while the even components cancel out close to the impurity. We also show that the spin-polarized local density of states exhibits the same spatial and frequency behavior as the odd-$\omega$ spin-triplet component at the critical impurity strength.

Local density of states in two-dimensional topological superconductors under a magnetic field: Signature of an exterior Majorana bound state

We study quasiparticle states on a surface of a topological insulator (TI) with proximity-induced superconductivity under an external magnetic field. An applied magnetic field creates two Majorana bound states: a vortex Majorana state localized inside a vortex core and an exterior Majorana state localized along a circle centered at the vortex core. We calculate the spin-resolved local density of states (LDOS) and demonstrate that the shrinking of the radius of the exterior Majorana state, predicted in Ref. [R. S. Akzyanov et al., Phys. Rev. B 94, 125428 (2016)], under a strong magnetic field can be seen in LDOS without smeared out by non-zero-energy states. The spin-resolved LDOS further reveals that the spin of the exterior Majorana state is strongly polarized. Accordingly, the induced odd-frequency spin-triplet pairs are found to be spin-polarized as well. In order to detect the exterior Majorana states, however, the Fermi energy should be closed to the Dirac point to avoid contributions from continuum levels. We also study a different two-dimensional topological-superconducting system where a two-dimensional electron gas with the spin-orbit coupling is sandwiched between an s-wave superconductor and a ferromagnetic insulator. We show that the radius of an exterior Majorana state can be tuned by an applied magnetic field. However, on the contrary to the results at a TI surface, neither the exterior Majorana state nor the induced odd-frequency spin-triplet pairs are spin-polarized. We conclude that the spin-polarization of the Majorana state is attributed to the spin-polarized Landau level which is characteristic for systems with the Dirac-like dispersion.

Odd-frequency superconducting pairing and subgap density of states at the edge of a two-dimensional topological insulator without magnetism

We investigate the emergence and consequences of odd-frequency spin-triplet s-wave pairing in superconducting hybrid junctions at the edge of a two-dimensional topological insulator without any magnetism. More specifically, we consider several different normal-superconductor hybrid systems at the topological insulator edge, where spin-singlet s-wave superconducting pairing is proximity-induced from an external conventional superconductor. We perform fully analytical calculations and show that odd-frequency mixed spin-triplet s-wave pairing arises due to the unique spin-momentum locking in the topological insulator edge state and the naturally non-constant pairing potential profile in hybrid systems. Importantly, we establish a one-to-one correspondence between the local density of states (LDOS) at low energies and the odd-frequency spin-triplet pairing in NS, NSN and SNS junctions along the topological insulator edge; at interfaces the enhancement in the LDOS can directly be attributed to the contribution of odd-frequency pairing. Furthermore, in SNS junctions we show that the emergence of the zero-energy LDOS peak at the superconducting phase $\phi=\pi$ is associated purely with odd-frequency pairing in the middle of the junction.

Odd-frequency superconductivity induced in topological insulators with and without hexagonal warping

We study the effect of the Fermi surface anisotropy on the odd-frequency spin-triplet pairing component of the induced pair potential. We consider a superconductor/ ferromagnetic insulator (S/FI) hybrid structure formed on the 3D topological insulator (TI) surface. In this case three ingredients ensure the possibility of the odd-frequency pairing: (1) the topological surface states, (2) the induced pair potential, and (3) the magnetic moment of a nearby ferromagnetic insulator. We take into account the strong anisotropy of the Dirac point in topological insulators when the chemical potential lies well above the Dirac cone and its constant energy contour has a snowflake shape. Within this model, we propose that the S/FI boundary should be properly aligned with respect to the snowflake constant energy contour to have an odd-frequency symmetry of the corresponding pairing component and to insure the Majorana bound state at the S/FI boundary. For arbitrary orientation of the boundary, the Majorana bound state is absent. This provides a selection rule to the realization of Majorana modes in S/FI hybrid structures, formed on the topological insulator surface.

Odd-frequency spin-triplet pairing states in half-metal/d-wave superconductor junctions

Based on the extended Blonder–Tinkham–Klapwijk formalism, differential conductance spectrum of d-wave superconductor/half-metal junctions is investigated as a function of the interfacial spin–orbit coupling (SOC). It shows that the SOC modifies the interfacial spin-flip scattering, enabling spin-singlet and spin-triplet pairing conversion with the aid of equal-spin-band Andreev reflection. A zero-bias conductance peak has been found, which is closely related to the zero energy Andreev bound state of equal-spin pairing correlation induced by the broken electron-hole symmetry of model Hamiltonian and gives strong evidences for the odd-frequency spin-triplet pairing states.

Odd-frequency superconductivity on the Hubbard model using the FLEX approximation

It is an important issue to clarify whether the odd-frequency superconducting state can be derived from microscopic Hamiltonian or not, where gap function has an odd-parity in frequency. We study the instability of following four superconducting states: (1) even-frequency spin-singlet, (2) even-frequency spin-triplet, (3) odd-frequency spin-singlet and (4) odd-frequency spin-triplet. By using the fluctuation exchange (FLEX) approximation on a triangular and square lattice, we find that the odd-frequency spin-triplet pairing can become dominant at a certain region where the suppression of the antiferromagnetic fluctuation due to a geometric frustration becomes prominent.

Manifestation of the odd-frequency spin-triplet pairing state in diffusive ferromagnet/superconductor junctions

Using the quasiclassical Green's function formalism, we study the influence of the odd-frequency spin-triplet superconductivity on the local density of states (LDOS) in a diffusive ferromagnet (DF) attached to a superconductor. Various possible symmetry classes in a superconductor are considered which are consistent with the Pauli's principle: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity (OSO) state. For each of these states, the pairing state in DF is studied. Particular attention is paid to the study of spin-singlet s-wave and spin-triplet p-wave superconductors as the examples of ESE and ETO superconductors. For spin-singlet case the magnitude of the OTE component of the pair amplitude is enhanced with the increase of the exchange field in DF. When the OTE component is dominant at low energy, the resulting LDOS in DF has a zero energy peak (ZEP). On the other hand, in DF / spin-triplet p-wave superconductor junctions LDOS has a ZEP in the absence of the exchange field, where only the OTE paring state exists. With the increase of the exchange field, the ESE component of the pair amplitude induced in DF is enhanced. Then, the resulting LDOS has a ZEP splitting. We demonstrate that the appearance of the dominant OTE component of the pair amplitude is the physical reason of the emergence of the ZEP of LDOS.