Topological Superconducting State Research Articles

Moiré-enabled topological superconductivity in twisted bilayer graphene

Twisted van der Waals materials have risen as highly tunable platforms for realizing unconventional superconductivity. Here we demonstrate how a topological superconducting state can be driven in a twisted graphene multilayer at a twist angle of approximately 1.6 degrees proximitized to other 2D materials. We show that an encapsulated twisted bilayer subject to induced Rashba spin–orbit coupling, s-wave superconductivity, and exchange field generates a topological superconducting state enabled by the moiré pattern. We demonstrate the emergence of a variety of topological states with different Chern numbers, that are highly tunable through doping, strain, and bias voltage. Our proposal does not depend on fine-tuning the twist angle, but solely on the emergence of moiré minibands and is applicable for twist angles between 1.3 and 3 degrees. Our results establish the potential of twisted graphene bilayers to create topological superconductivity without requiring ultraflat dispersions.

Majorana fermions in a two-band two-dimensional Fermi system

Majorana fermions represent potential candidates for realizing fault-tolerant topological quantum computation. We obtain the exact mapping between a two-band hybridized model and a model with a topological superconducting state supporting Majorana fermions. The antisymmetric hybridization employed resembles that of spin–orbit coupling. We consider the situation where the model Hamiltonian comprises only inter-band s-wave pairing, i.e. the one in which the superconducting pairing is formed by one fermion from one of the bands and another fermion of opposite spin and momentum from the other band. From this we obtain an effective Hamiltonian of the px+ipy superconductor and calculate the topological invariant that distinguishes the weak and strong pairing phases. We also obtain the gap parameter and chemical potential of the Fermi system under spin–orbit coupling beyond the mean-field approximation. We show that the spin–orbit coupling substantially enhances the density of states at the Fermi surface.

Nodal line topological superconducting state in quasi-one-dimensional A2Cr3As3 (A=K,Rb,Cs) superconductors

Nodal line topological superconducting state in quasi-one-dimensional A2Cr3As3 (A=K,Rb,Cs) superconductors

Spin-dependent resonant Andreev tunneling in hybrid ferromagnetic metal–magnetic quantum dot–superconductor nanostructures

There exists a variety of theoretical proposals to transform states induced by magnetic nanoparticles inside a superconducting gap into Majorana fermion states. The main challenge in this route is a conclusive proof and undoubted distinguishing between topologically trivial subgap Andreev bound states and topologically nontrivial magnetically polarized Majorana bound states. This motivated us to investigate a nonequilibrium electrons tunneling through a ferromagnetic normal metal–magnetic quantum dot–s-wave superconductor (F-mQD-SC) nanostructure, where the mQD’s discrete levels are spin splitted. By using the Keldysh Green’s function method, the expressions for a tunnel current and probability of the Andreev reflection (AR) versus energy are derived and studied. We find that the system’s resonant ARs conductance exhibits different kinds of peaks depending on a spin splitting of the mQD levels, the spin polarization magnitude of the F-lead current, the gate voltage, and an external magnetic field magnitude. The nanostructure’s conductance versus a bias voltage exhibits extra peaks which at some combination of its parameters can mimic ones expected for Majorana modes in a topological superconducting state. The distinguishing transport characteristics of a F-mQD-SC nanoscale structure being in non-topological state are discussed. We suggest that the results obtained can provide helpful clarification for understanding recent experiments in superconductor–ferromagnet hybrid nanostructures with topologically protected excitations.

Magnetic‐Field Regulation of Nodal Topological Superconducting States in Monolayer Ising Superconductor NbSe2

The nodal topological superconducting (NTSC) state with Majorana flat bands (MFBs) is an exotic matter of state hosting Majorana fermions that obey non‐Abelian statistics. Recently, monolayer Ising superconductor NbSe2 is shown to be an ideal platform for realizing an NTSC state. Through tight‐binding calculations based on the Bogoliubov–de–Gennes Hamiltonian, it is demonstrated that the in‐plane magnetic field B combined with superconducting pairing Δs, Rashba spin–orbit coupling (SOC) Vr, and chemical potential μ can regulate the topological properties of the NTSC state. First, B larger than Δs effectively induces an NTSC state with nodal points (NPs) located on the high‐symmetry line of the Brillouin zone. The length of the resulting MFBs in the ribbon increases with B. Second, Rashba SOC greatly affects the number and locations of NPs by tilting bands near Fermi level (EF), leading to unidirectional Majorana edge states. Finally, when B > Δs, μ affects the bulk band structure at EF, resulting in an NTSC state with 6, 12, 18, 24 NPs and 0, 2, 3, 4 MFBs. These results are important to clarifying the behavior of the NTSC state under a B and designing a quantum computing platform based on exotic Majorana fermions.

Realizing a Superconducting Square-Lattice Bismuth Monolayer

Interplay of crystal symmetry, strong spin-orbit coupling (SOC), and many-body interactions in low-dimensional materials provides a fertile ground for the discovery of unconventional electronic and magnetic properties and versatile functionalities. Two-dimensional (2D) allotropes of group 15 elements are appealing due to their structures and controllability over symmetries and topology under strong SOC. Here, we report the heteroepitaxial growth of a proximity-induced superconducting 2D square-lattice bismuth monolayer on superconducting Pb films. The square lattice of monolayer bismuth films in a C4 symmetry together with a stripey moiré structure is clearly resolved by our scanning tunneling microscopy, and its atomic structure is revealed by density functional theory (DFT) calculations. A Rashba-type spin-split Dirac band is predicted by DFT calculations to exist at the Fermi level and becomes superconducting through the proximity effect from the Pb substrate. We suggest the possibility of a topological superconducting state in this system with magnetic dopants/field. This work introduces an intriguing material platform with 2D Dirac bands, strong SOC, topological superconductivity, and the moiré superstructure.

Topological superconductivity driven by correlations and linear defects in multiband superconductors

There have been several proposals for platforms sustaining topological superconductivity in high temperature superconductors, in order to make use of the larger superconducting gap and the expected robustness of Majorana zero modes towards perturbations. In particular, the iron-based materials offer relatively large $T_c$ and nodeless energy gaps. In addition, atomically flat surfaces enable the engineering of defect structures and the subsequent measurement of spectroscopic properties to reveal topological aspects. From a theory perspective, a materials-specific description is challenging due to the correlated nature of the materials and complications arising from the multiband nature of the electronic structure. Here we include both aspects in realistic interacting models, and find that the correlations themselves can lead to local magnetic order close to linear potential scattering defects at the surface of the superconductor. Using a self-consistent Bogoliubov-de Gennes framework in a real-space setup using a prototype electronic structure, we allow for arbitrary magnetic orders and show how a topological superconducting state emerges. The calculation of the topological invariant and the topological gap allows us to map out the phase diagram for the case of a linear chain of potential scatterers. While intrinsic spin-orbit coupling is not needed to enter the topological state in presence of spin-spiral states, it enlarges the topological phase. We discuss the interplay of a triplet component of the superconducting order parameter and the spin spiral leading effectively to extended spin orbit coupling terms, and connect our results to experimental efforts on the Fe(Se,Te) system.

Quantum Phase Diagram and Spontaneously Emergent Topological Chiral Superconductivity in Doped Triangular-Lattice Mott Insulators.

The topological superconducting state is a highly sought-after quantum state hosting topological order and Majorana excitations. In this Letter, we explore the mechanism to realize the topological superconductivity (TSC) in the doped Mott insulators with time-reversal symmetry (TRS). Through large-scale density matrix renormalization group study of an extended triangular-lattice t-J model on the six- and eight-leg cylinders, we identify a d+id-wave chiral TSC with spontaneous TRS breaking, which is characterized by a Chern number C=2 and quasi-long-range superconducting order. We map out the quantum phase diagram with by tuning the next-nearest-neighbor (NNN) electron hopping and spin interaction. In the weaker NNN-coupling regime, we identify a pseudogaplike phase with a charge stripe order coexisting with fluctuating superconductivity, which can be tuned into d-wave superconductivity by increasing the doping level and system width. The TSC emerges in the intermediate-coupling regime, which has a transition to a d-wave superconducting phase with larger NNN couplings. The emergence of the TSC is driven by geometrical frustrations and hole dynamics which suppress spin correlation and charge order, leading to a topological quantum phase transition.

Antiferromagnetism-driven two-dimensional topological nodal-point superconductivity

Magnet/superconductor hybrids (MSHs) hold the promise to host emergent topological superconducting phases. Both one-dimensional (1D) and two-dimensional (2D) magnetic systems in proximity to s-wave superconductors have shown evidence of gapped topological superconductivity with zero-energy end states and chiral edge modes. Recently, it was proposed that the bulk transition-metal dichalcogenide 4Hb-TaS2 is a gapless topological nodal-point superconductor (TNPSC). However, there has been no experimental realization of a TNPSC in a MSH system yet. Here we present the discovery of TNPSC in antiferromagnetic (AFM) monolayers on top of an s-wave superconductor. Our calculations show that the topological phase is driven by the AFM order, resulting in the emergence of a gapless time-reversal invariant topological superconducting state. Using low-temperature scanning tunneling microscopy we observe a low-energy edge mode, which separates the topological phase from the trivial one, at the boundaries of antiferromagnetic islands. As predicted by the calculations, we find that the relative spectral weight of the edge mode depends on the edge’s atomic configuration. Our results establish the combination of antiferromagnetism and superconductivity as a novel route to design 2D topological quantum phases.

Ferromagnetic impurity induced Majorana zero mode in iron-based superconductors

Recent experiments reported the puzzling zero energy modes associated with ferromagnetic impurities in some iron-based superconductors with topological band structures. Here, we show that the sufficiently strong exchange coupling between a ferromagnetic impurity and substrate can trigger a quantum phase transition, beyond which, the phase of the topological surface superconducting order parameter around the impurity acquires a sign-change. In such a case, we prove that a Kramers degenerate pair of Majorana modes can be induced at the boundary separating the two sign-change regimes and trapped around the impurity in the topological surface superconducting state. Furthermore, we show that our theory can explain the controversial observations and confusing features of the zero energy modes from recent experiments in some iron-based superconductors.

Disorder suppression in topological semiconductor-superconductor junctions

Disorder in a proximitizing bulk superconductor can scatter quasiparticles in a putative topological superconductor and eventually destroy the topological superconducting state. We use a scattering approach and a random-matrix calculation to estimate the disorder scattering time in a topological Josephson junction. We find that the disorder scattering rate from the bulk of the superconductor, even in the strong coupling limit, is suppressed in the ratio of Fermi momenta between the semiconductor and superconductor. This suppression of disorder scattering is accompanied by near perfect Andreev reflection at such semiconductor/superconductor interfaces, which can be used as a signature of such clean proximity effect. We also find that these results can be understood by a semiclassical estimate of scattering. We discuss limits in other systems such as the semiconductor nanowire where disorder scattering is suppressed according to similar classical estimates.

Contact of the intrinsic magnetic topological insulator Mn(Bi,Sb)2Te4 with a superconducting Pb film

Intrinsic magnetic topological insulators (MTIs), such as an ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ family, have proved to be an essential platform for the study of various quantum effects and can be used for a range of applications from information storage and dissipationless spin and charge transfer to quantum computers. The latter requires the creation of Majorana fermions, which are expected to emerge when a superconductor is contacted with an MTI surface. Therefore the study of the features arising at the interface between the MTI and the superconductor is of great interest. For this purpose in this paper, the gradual growth of the elemental superconductor (Pb) film on the surface of $\mathrm{Mn}{({\mathrm{Bi}}_{0.7}{\mathrm{Sb}}_{0.3})}_{2}{\mathrm{Te}}_{4}$, which has a Dirac point near the Fermi level, was experimentally investigated by ARPES and XPS methods. We observed that the shape of the Dirac cone state remains almost unchanged up to coverage of $\ensuremath{\sim}1.5$ Pb monolayers, thus remaining topological properties of the material. It was shown that Pb atoms deposition results in surface modification. They detached upper Te atoms and form TePb alloy on the surface. Moreover, Pb deposition leads to disorder of Te, Bi(Sb) layers closest to the surface. As a result, the localization of the Dirac cone state shifts towards the bulk. All these features should be taken into account in the implementation of a topological superconducting state based on the Pb/Mn(Bi,${\mathrm{Sb})}_{2}{\mathrm{Te}}_{4}$ systems.

Impurity-induced excitations in a topological two-dimensional ferromagnet/superconductor van der Waals moiré heterostructure

The emergence of a topological superconducting state in van der Waals heterostructures provides a new platform for exploring novel strategies to control topological superconductors. In particular, impurities in van der Waals heterostructures, generically featuring a moir\'e pattern, can potentially lead to the unique interplay between atomic and moir\'e length scales, a feature absent in generic topological superconductors. Here we address the impact of nonmagnetic impurities on a topological moir\'e superconductor, both in the weak and strong regime, considering both periodic arrays and single impurities in otherwise pristine infinite moir\'e systems. We demonstrate a fine interplay between impurity-induced modes and the moir\'e length, leading to radically different spectral and topological properties depending on the relative impurity location and moir\'e lengths. Our results highlight the key role of impurities in van der Waals heterostructures featuring moir\'e patterns, revealing the key interplay between length and energy scales in artificial moir\'e systems.

Topological superconductivity in a two-dimensional Weyl SSH model

We study the emergence of topological superconductivity in a two-dimensional Weyl system, composed of stacked Su-Schrieffer-Heeger (SSH) chains. A previous analysis of the model showed that the addition of an attractive Hubbard interaction between spinful electrons leads to a superconducting state that has an intricate pairing structure, but is topologically trivial. Here we consider a pairing interaction that couples spinless fermions on opposite sublattices within the same unit cell. We observe that this physically motivated, momentum-independent pairing interaction induces a topological superconducting state, characterized by a gap function with a nontrivial phase, as well as Majorana and Fermi arc edge states along the edge perpendicular to the direction of the SSH dimerization. In addition, we observe a transition as a function of pairing interaction strength and chemical potential, indicated by a change in the sign of the topological charge carried by each of the four Bogoliubov-Weyl nodes.

Anisotropic topological superconductivity in Josephson junctions

We investigate the effects of magnetic and crystalline anisotropies on the topological superconducting state of planar Josephson junctions (JJs). In junctions where only Rashba spin-orbit coupling (SOC) is present, the topological phase diagram is insensitive to the supercurrent direction, but exhibits a strong dependence on the magnetic field orientation. However, when both Rashba and Dresselhaus SOCs coexist, the topological phase diagram strongly depends on both the magnetic field and junction crystallographic orientations. We examine the impact of the magnetic and crystalline anisotropy on the current-phase relation (CPR), energy spectrum, and topological gap of phase-biased JJs, where the junction is connected in a loop and the superconducting phase difference is fixed by a loop-threading magnetic flux. The anisotropic CPR can be used to extract the ground-sate phase (i.e., the superconducting phase difference that minimizes the system free energy) behavior in phase-unbiased JJs with no magnetic flux. Under appropriate conditions, phase-unbiased JJs can self-tune into or out of the topological superconducting state by rotating the in-plane magnetic field. The magnetic field orientations at which topological transitions occur strongly depend on both the junction crystallographic orientation and the relative strength between Rashba and Dresselhaus SOCs. We find that for an optimal practical application, in which the junction exhibits topological superconductivity with a sizable topological gap, a careful balancing of the magnetic field direction, the junction crystallographic orientation, and the relative strengths of the Rashba and Dresselhaus SOCs is required. We discuss the considerations that must be undertaken to achieve this balancing for various junction types and parameters.

Majorana zero modes induced by superconducting phase bias

Majorana zero modes in condensed matter systems have been the subject of much interest in recent years. Their non-Abelian exchange statistics, making them a unique state of matter, and their potential applications in topological quantum computation, earned them attention from both theorists and experimentalists. It is generally understood that in order to form Majorana zero modes in quasi-one-dimensional topological insulators, time-reversal symmetry must be broken. The straightforward mechanisms for doing so—applying magnetic fields or coupling to ferromagnets—turned out to have many unwanted side effects, such as degradation of superconductivity and the formation of sub-gap states, which is part of the reason Majorana zero modes have been eluding direct experimental detection for a long time. Here we review several proposal that rely on controlling the phase of the superconducting order parameter, either as the sole mechanism for time-reversal-symmetry breaking, or as an additional handy knob used to reduce the applied magnetic field. These proposals hold practical promise to improve Majorana formation, and they shed light on the physics underlying the formation of the topological superconducting state.

Phase-Manipulation-Induced Majorana Mode and Braiding Realization in Iron-Based Superconductor Fe(Te,Se).

A recent experiment reported the evidence of dispersing one-dimensional Majorana mode trapped by the crystalline domain walls in FeSe_{0.45}Te_{0.55}. Here, we perform the first-principles calculations to show that iron atoms in the domain wall spontaneously form the ferromagnetic order in line with orientation of the wall. The ferromagnetism can impose a π phase difference between the domain-wall-separated surface superconducting regimes under the appropriate width and magnetization of the wall. Accordingly, the topological surface superconducting state of FeSe_{0.45}Te_{0.55} can give rise to one-dimensional Majorana modes trapped by the wall. More interestingly, we further propose a surface junction in the form of FeSe_{0.45}Te_{0.55}-ferromagnet-FeSe_{0.45}Te_{0.55}, which can be adopted to create and fuse the Majorana zero modes through controlling the width or magnetization of the interior ferromagnetic barrier. The braiding and readout of Majorana zero modes can be realized by the designed device. Such surface junction has the potential application in the superconducting topological quantum computation.

Fulde-Ferrell-Larkin-Ovchinnikov pairing induced by a Weyl nodal line in an Ising superconductor with a high critical field

Superconducting and topological states are two quantum phenomena attracting much interest. Their coexistence may lead to topological superconductivity sought-after for Majorana-based quantum computing. However, there is no causal relationship between the two, since superconductivity is a many-body effect due to electron-electron interaction while topology is a single-particle manifestation of electron band structure. Here, we demonstrate a novel form of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing, induced by topological Weyl nodal lines in Ising Bardeen-Cooper-Schrieffer (IBCS) superconductors. Based on first-principles calculations and analyses, we predict that the nonmagnetic metals of MA$_2$Z$_4$ family, including ${\alpha}_1$-TaSi$_2$P$_4$, ${\alpha}_1$-TaSi$_2$N$_4$, ${\alpha}_1$-NbSi$_2$P$_4$, ${\alpha}_2$-TaGe$_2$P$_4$, and ${\alpha}_2$-NbGe$_2$P$_4$ monolayers, are all superconductors. While the intrinsic IBCS paring arises in these non-centrosymmetric systems, the extrinsic FFLO pairing is revealed to be evoked by the Weyl nodal lines under magnetic field, facilitating the formation of Cooper pairs with nonzero momentum in their vicinity. Moreover, we show that the IBCS pairing alone will enhance the in-plane critical field $B_c$ to ~10-50 times of Pauli paramagnetic limit $B_p$, and additional FFLO pairing can further triple the $B_c/B_p$ ratio. It therefore affords an effective approach to enhance the robustness of superconductivity. Also, the topology induced superconductivity renders naturally the possible existence of topological superconducting state.

Moiré-Enabled Topological Superconductivity.

The search for artificial topological superconductivity has been limited by the stringent conditions required for its emergence. As exemplified by the recent discoveries of various correlated electronic states in twisted van der Waals materials, moiré patterns can act as a powerful knob to create artificial electronic structures. Here, we demonstrate that a moiré pattern between a van der Waals superconductor and a monolayer ferromagnet creates a periodic potential modulation that enables the realization of a topological superconducting state that would not be accessible in the absence of the moiré. The magnetic moiré pattern gives rise to Yu–Shiba–Rusinov minibands and periodic modulation of the Majorana edge modes that we detect using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). Moiré patterns and, more broadly, periodic potential modulations are powerful tools to overcome the conventional constraints for realizing and controlling topological superconductivity.

Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems

We discuss a pairing mechanism in interacting two-dimensional multipartite lattices that intrinsically leads to a second order topological superconducting state with a spatially modulated gap. When the chemical potential is close to Dirac points, oppositely moving electrons on the Fermi surface undergo an interference phenomenon in which the Berry phase converts a repulsive electron–electron interaction into an effective attraction. The topology of the superconducting phase manifests as gapped edge modes in the quasiparticle spectrum and Majorana Kramers pairs at the corners. We present symmetry arguments which constrain the possible form of the electron–electron interactions in these systems and classify the possible superconducting phases which result. Exact diagonalization of the Bogoliubov-de Gennes Hamiltonian confirms the existence of gapped edge states and Majorana corner states, which strongly depend on the spatial structure of the gap. Possible applications to vanadium-based superconducting kagome metals AV3Sb5 (A = K, Rb, Cs) are discussed.