The topologically non-trivial band structure of A15 compounds has drawn attention owing to the possible realization of topological superconductivity. Here, we report a microscopic investigation of the superconducting ground state in A15 compound Ti$_{3}$X (X = Ir, Sb) by muon spectroscopy measurements. Zero field muon measurements have shown that time-reversal symmetry is preserved in these materials. Furthermore, specific heat and a transverse field muon spectroscopy measurement rule out any possibility to have a nodal or anisotropic superconducting gap, revealing a conventional s-wave nature in the superconducting ground state. This work classifies A15 compound Ti$_{3}$X (X = Ir, Sb) as a time-reversal preserved topological superconductor.
We study bulk electronic states of superconducting topological insulator, which is the promising candidate for topological superconductor. Recent experiments suggest that the three-dimensional Fermi surface evolves into two-dimensional one. We show that the superconducting energy gap structure on the Fermi surface systematically changes with this evolution. It is clarified that the bulk electronic properties such as spin-lattice relaxation rate and specific heat depend on the shape of the Fermi surface and the type of the energy gap function. These results serve as a guide to determine the pai…
In a superconducting topological insulator, superconducting vortex line may trap 1-dimensional topological band with Majorana zero modes (MZMs) localized at its ends. In this work, we study the effect of hexagonal warping to the vortex phase transition. We carry out both analytical calculations based on a semiclassical formula and numerical calculations based on full quantum mechanics of Bogliubov de Gennes equation. We find that the hexagonal warping term enlarges the topological region of the vortex line as the chemical potential changes, and leads to the MZMs even in the absence of topologi…
It was recently shown that a chiral topological phase emerges from the coupling of two twisted monolayers of superconducting Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ for certain twist angles. In this work, we reveal the behavior of such twisted superconducting bilayers with $d_{x^2-y^2}$ pairing symmetry in presence of applied magnetic field. Specifically, we show that the emergent vortex matter can serve as smoking gun for detection of topological superconductivity in such bilayers. Moreover, we report two distinct skyrmionic states that characterize the chiral topological phase, and provide full accoun…
Recently, the search for Majorana fermions (MFs) has become one of the most important and exciting issues in condensed matter physics since such an exotic quasiparticle is expected to potentially give rise to unprecedented quantum phenomena whose functional properties will be used to develop future quantum technology. Theoretically, the MFs may reside in various types of topological superconductor materials that is characterized by the topologically protected gapless surface state which are essentially an Andreev bound state. Superconducting doped topological insulators and topological crystal…
We consider a Rashba spin-orbit coupled nanowire with anisotropic spin-singlet superconducting pairing and time-reversal-invariant symmetry. We explore the evolution of the topological superconducting phases of this system due to geometric deformations for the representative case of a wire bent in a semielliptical shape. We find that when the system is in its topological superconducting phase, strong inhomogeneities in the profile curvature can produce a pair of localized eigenmodes, which can be attributed to a nonuniform topological phase. The curved geometric profile also allows to tune the…
We study electronic properties of a superconducting topological insulator whose parent material is a topological insulator. We calculate the temperature dependence of the specific heat and spin susceptibility for four promising superconducting pairings proposed by L. Fu and E. Berg (Phys. Rev. Lett. 105, 097001). Since the line shapes of temperature dependence of specific heat are almost identical among three of the four pairings, it is difficult to identify them simply from the specific heat. On the other hand, we obtain wide varieties of the temperature dependence of spin susceptibility for …
An external magnetic field is needed to drive a nanowire in proximity to an s-wave superconductor into a topological regime which supports Majorana end states. However, a magnetic field generally suppresses the proximity superconducting gap induced on the nanowire. In recent experiments using InSb nanowires coupled to Al, the induced proximity gap vanishes at magnetic fields B~1T. This results in a small superconducting gap on the wire and a narrow topological regime which is proportional to the strength of the magnetic field. In this work, we show that by placing nanowires in proximity to rec…
Although more than twenty years have passed since the discovery of high temperature cuprate superconductivity, the identification of the superconducting order parameter is still under debate. Here, we show that the nodal gap component is the best candidate for the superconducting order parameter. It scales with the critical temperature $T_c$ over a wide doping range and displays a significant temperature dependence below $T_c$ in both the underdoped and the overdoped regimes of the phase diagram. In contrast, the antinodal gap component does not scale with $T_c$ in the underdoped side and appe…
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 pea…
We present a unified description of a junction between $s$-wave (BCS) superconductors and a junction between $p$-wave superconductors in a topologically nontrivial phase, which relies on a scattering state expansion. We compute Josephson current and thermal noise in the two kinds of junction and exhibit some characteristic features for a junction of two topological superconductors hosting Majorana zero-energy modes.
We develop a theory of tunneling spectroscopy in superconducting topological insulator (STI), i.e. superconducting state of a carrier-doped topological insulator. Based on the quasi-classical approximation, we obtain an analytical expression of the energy dispersion of surface Andreev bound states (ABSs) and a compact formula for tunneling conductance of normal metal/STI junctions. The obtained compact formula of tunneling conductance makes the analysis of experimental data easy. As an application of our theory, we study tunneling conductance in Cu$_x$Bi$_2$Se$_3$. We reveal that our theory re…
We investigate characteristic features in the spin response of doped three-dimensional topological insulators with odd-parity unequal-spin superconducting pairing, which are predicted to have gapless Majorana surface modes. These Majorana modes contribute to the local spin susceptibility, giving rise to a characteristic temperature behavior of the Knight shift and the spin-lattice relaxation time in magnetic resonance experiments. Because of their different decay lengths, the Majorana modes can be observed and clearly distinguished from the Dirac modes of the topological insulator by local pro…
We study the surface state of a doped topological crystalline insulator in the superconducting state. Motivated by Sn$_{1-x}$In$_x$Te, we consider fully gapped pair potentials and calculate the surface spectral function. It is found that mirror-protected zero-energy Andreev bound states appear at the (001) surface and that these states can move along the mirror symmetric line on the surface Brillouin zone. We also show that the surface Andreev bound state changes systematically with doping due to the presence of the Dirac surface state in the normal state.
High-pressure synthesis techniques have allowed for the growth of Sn$_{1-x}$In$_x$Te samples beyond the ambient In-saturation limit of $x$ = 0.5 (T$_c \sim$ 4.5 K). In this study, we present measurements of the temperature dependence of the London penetration depth $Δλ(T)$ in this superconducting doped topological insulator for $x$ = 0.7, where T$_{c,onset}\approx 5$ K. The results indicate fully gapped BCS-like behavior, ruling out odd-parity $A_{2u}$ pairing; however, odd-parity $A_{1u}$ pairing is still possible. Critical field values measured below 1 K and other superconducting parameters …
Doping-controlled topological superconducting transition in misfit layer compounds
Institut des Nanosciences de Paris（巴黎纳米科学研究所）；Institut des Matériaux de Nantes Jean Rouxel（南特让·罗克塞尔材料研究所）；Universität Würzburg（维尔茨堡大学）；Pavol Jozef Šafárik University（帕沃尔·约瑟夫·沙法里克大学）
Using high-resolution angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy, the atomic and low energy electronic structure of the Sr-doped superconducting topological insulators (SrxBi2Se3) was studied. Scanning tunneling microscopy shows that most of the Sr atoms are not in the van der Waals gap. After Sr doping, the Fermi level was found to move further upwards when compared with the parent compound Bi2Se3, which is consistent with the low carrier density in this system. The topological surface state was clearly observed, and the position of the Dirac poin…
High-pressure synthesis techniques have allowed for the growth of samples on the indium-rich side of (Pb,In)Te, which have increased superconducting transition temperatures compared to lead-rich compounds. In this study we present measurements of the temperature dependence of the London penetration depth $Δλ(T)$ in the compound In$_{0.8}$Pb$_{0.2}$Te, which shows a bulk $T_{c,onset}$ of $\sim4.75$ K. The results indicate fully gapped BCS-like behavior, ruling out odd-parity, topologically nontrivial $A_{2u}$ pairing; however, odd-parity $A_{1u}$ pairing is still possible. Critical field values…
Recently it was demonstrated that the long-known transition between the gap and gapless superconducting states in the Abrikosov-Gor'kov theory of superconducting alloy with paramagnetic impurities is of the Lifshitz's type, i.e. at zero temperature this is the $2\frac12$ order phase transition. Since transitions of this kind in a normal metal are always associated to certain topological changes, then below we clarify the topological nature of the transition under consideration. Namely, we demonstrate that the topological invariant which in process of the transition undergoes the change is noth…
We investigate parity-mixing superconductivity in the two-dimensional Hubbard model with Rashba spin-orbit coupling, using Cellular Dynamical Mean-Field Theory (CDMFT). A superconducting state with mixed singlet $d$-wave and triplet $p$-wave character is found in a wide range of doping. The singlet component decreases with the amplitude of the Rashba spin-orbit coupling, whereas the triplet component increases, but both disappear at about 20\% doping. The effect of a Zeeman field is also investigated; it tends to suppress both types of superconductivity, but induces nontrivial topological prop…
Proximity-effect-induced superconductivity was studied in epitaxial topological insulator Bi2Se3 thin films grown on superconducting NbSe2 single crystals. A point contact spectroscopy (PCS) method was used at low temperatures down to 40 mK. An induced superconducting gap in Bi2Se3 was observed in the spectra, which decreased with increasing Bi2Se3 layer thickness, consistent with the proximity effect in the bulk states of Bi2Se3 induced by NbSe2. At very low temperatures, an extra point contact feature which may correspond to a second energy gap appeared in the spectrum. For a 16 quintuple la…
Spontaneously broken symmetry in the ground state of Bi-based topological materials can lead to promising Topological superconductors, such as Cu-Bi2Se3, Sr-Bi2Se3 and Nb-Bi2Se3. In recent studies, coexistence of multiple Fermi surface features and superconductivity was found in Nb-Bi2Se3. However, the resultant mutliple Fermi surface--charge density wave has not been experimentally reported yet. In this paper, we report possible evidence of co-occurence of a charge density wave (CDW) ground state and a superconducting (SC) ground state in Nb intercalated Bi2Se3. An intercalation induced Perio…
The superconductivity recently found in the doped topological insulator Cu_xBi_2Se_3 offers a great opportunity to search for a topological superconductor. We have successfully prepared a single-crystal sample with a large shielding fraction and measured the specific-heat anomaly associated with the superconductivity. The temperature dependence of the specific heat suggests a fully-gapped, strong-coupling superconducting state, but the BCS theory is not in full agreement with the data, which hints at a possible unconventional pairing in Cu_xBi_2Se_3. Also, the evaluated effective mass of 2.6m_…
We study topological vortex phases in iron-based superconductors. Besides the previously known vortex end Majorana zero modes (MZMs) phase stemming from the existence of a three dimensional (3D) strong topological insulator state, we show that there is another topologically nontrivial phase as iron-based superconductors can be doped superconducting 3D weak topological insulators (WTIs). The vortex bound states in a superconducting 3D WTI exhibit two different types of quantum states, a robust nodal superconducting phase with pairs of bulk MZMs and a full-gap topologically nontrivial supercondu…
Recently the doping of topological insulators has attracted significant interest as a potential route towards topological superconductivity. Because many experimental techniques lack sufficient surface sensitivity, however, a definite proof of the coexistence of topological surface states and surface superconductivity is still outstanding. Here we report on highly surface sensitive scanning tunneling microscopy (STM) and spectroscopy (STS) experiments performed on Tl-doped Bi$_2$Te$_3$, a three-dimensional topological insulator which becomes superconducting in the bulk at $T_{\rm C} = 2.3$\,K.…
We report an unexpected positive hydrostatic pressure derivative of the superconducting transition temperature in the doped topological insulator \NBS via $dc$ SQUID magnetometry in pressures up to 0.6 GPa. This result is contrary to reports on the homologues \CBS and \SBS where smooth suppression of $T_c$ is observed. Our results are consistent with recent Ginzburg-Landau theory predictions of a pressure-induced enhancement of $T_c$ in the nematic multicomponent $E_u$ state proposed to explain observations of rotational symmetry breaking in doped Bi$_2$Se$_3$ superconductors.
Using Andreev reflection (AR) as an experimental gauge of the superconducting proximity effect (PE), we assess the topological purity of the superconductivity that is induced by the c-axis PE between an s-wave superconductor and the topological insulators Bi$_{2}$X$_{3}$ (X=Se,Te). Point-contact AR spectroscopy is performed with Nb tips on Bi$_{2}$X$_{3}$ single crystals at 4.2 K. Scanning tunneling spectroscopy is also used, to locate the Fermi level $E_F$ relative to the Dirac point in the crystals. The AR data is analyzed with Blonder-Tinkham-Klapwijk theory, taking into account tip-induced…
We study the vortex bound states in three dimensional (3D) superconducting Dirac semimetals with time reversal symmetry. Assuming two Dirac points on the kz-axis and bulk s-wave superconductivity, with a quantum vortex line parallel to the z-direction, we find that the superconducting vortex line has a robust quasi-1D nodal phase. The nodal phase stems from the symmetry protected Dirac points in the normal state bands, and it can be characterized by a topological index (ν; n) at kz = 0 and kz = π, where νis the Z2 topological invariant for a 0D class-D system and n is the Z topological invaria…
We present a study of the superconducting properties of the candidate topological superconductor Ti_(3)Sb. Electrical transport measurements show zero resistance with a T_(c,onset) of ~ 5.9 K with a transition width ΔT_c~ 0.6 K. The superconducting phase boundaries as derived from magneto-transport and magnetic susceptibility measurements agree well. We estimate an upper critical field Bc2(0)~4.5 T. A Ginzburg-Landau (GL) analysis yields values of the coherence length and penetration depth of ζ = 6.2 nm and λ = 340 nm, respectively, and a GL parameter ~ 55, indicating extreme type-II behavior.…
We show that standard impurities, magnetic or nonmagnetic, weak or strong, can cause topological phase transitions inside the vortex cores of a conventional s-wave superconductor. Because of the nonzero angular momentum of Cooper pairs in the vortex cores, the vortex bound states in a two dimensional superconductor are sensitive to impurities in a way similar to the Yu-Shiba-Rusinov bound states induced by magnetic impurities. In three dimensional cases, the vortex bound states can be driven into topologically nontrivial phases by an impurity chain inside the vortex core. The system can host M…
Semi-Dirac fermions are known to exist at the critical points of topological phase transitions requiring fine-tuning of the parameters. We show that robust semi-Dirac points can appear in a heterostructure consisting of superconducting Sr2IrO4 and a t2g electron system (t2g-ES) without fine-tuning. They are topologically stable in the presence of the symmetries of the model, metallic t2g-ES and a single active band in Sr2IrO4. If the t2g metal is coupled to two different layers of Sr2IrO4 (effectively a multiband superconductor) in a three-layer-structure the semi-Dirac points can split into t…
We demonstrate that checkerboard magnet-superconductor hybrid systems possess a rich phase diagram exhibiting both strong topological superconducting (STSC) and topological nodal point superconducting (TNPSC) phases. We show that TNPSC phases exist both for ferromagnetic and antiferromagnetic systems, yielding a plethora of qualitatively different edge mode structures. Checkerboard MSH systems also facilitate the emergence of STSC phases which can be induced even in the limit of vanishing magnetization. Our results provide a new path for the quantum engineering of topological superconducting p…
The antiferromagnetic topological insulator (AFTI) is topologically protected by the combined time-reversal and translational symmetry $\mathcal{T}_c$. In this paper we investigate the effects of the $s$-wave superconducting pairings on the multilayers of AFTI, which breaks $\mathcal{T}_c$ symmetry and can realize quantum anomalous Hall insulator with unit Chern number. For the weakly coupled pairings, the system corresponds to the topological superconductor (TSC) with the Chern number $C=\pm 2$. We answer the following questions whether the local Chern numbers and chiral Majorana edge modes o…
It is a well known phenomenon in optics that spectroscopic resolution of a diffraction grating is much better compared to an interference device having just two slits, as in the Young's famous double-slit experiment. On the other hand, it is well known that a classical superconducting quantum interference device (SQUID) is analogous to the optical double-slit experiment. Here we report experiments and present a model describing a superconducting analogue to the diffraction grating, namely an array of superconducting islands positioned on a topological insulator (TI) film Bi0.8Sb1.2Te3. In the …
The Andreev reflection spectra were studied for the first time in the magnetic superconductor Dy0.6Y0.4Rh3.85Ru0.15B4. It is found that an external magnetic field (up to 0.7 Hc2) is a strong stimulator of superconductivity in contrast with traditional superconductors with a singlet pairing. The ratio 2Δ/kBTc \approx 4.0 obtained for some contacts is higher than the value 3.52 which is typical of conventional superconductors with a weak electron-phonon interaction. It has been proposed that in Dy0.6Y0.4Rh3.85Ru0.15B4 a triplet mechanism of superconducting pairing is realized.
We self-consistently study surface states of superconducting topological insulators. We clarify that, if a topologically trivial bulk s-wave pairing symmetry is realized, parity mixing of pair potential near the surface is anomalously enhanced by surface Dirac fermions, opening an additional surface gap larger than the bulk one. In contrast to classical s-wave superconductors, the resulting surface density of state hosts an extra coherent peak at the induced gap besides a conventional peak at the bulk gap. We also find that no such extra peak appears for odd-parity superconductors with a cylin…
We argue that the frustrated Josephson junction arrays may support a topologically ordered superconducting ground state, characterized by a non-trivial ground state degeneracy on the torus. This superconducting quantum fluid provides an explicit example of a system in which superconductivity arises from a topological mechanism rather than from the usual Landau-Ginzburg mechanism.
We theoretically study bilayer superconducting topological insulator film, in which superconductivity exists for both top and bottom surface states. We show that an in-plane magnetic field can drive the system into Larkin-Ovchinnikov (LO) phase, where electrons are paired with finite momenta. The LO phase is topologically non-trivial and characterized by a Z 2 topological invariant, leading to a Majorana zero mode chain along the edge perpendicular to in-plane magnetic fields.
The relation of topological insulators and superconductors and the field of nonlinear dynamics is widely unexplored. To address this subject, we adopt the linear coupling geometry of the Su-Schrieffer-Heeger model, a paradigmatic example for a topological insulator, and render it nonlinearly in the context of superconducting circuits. As a consequence, the system exhibits topologically-enforced bifurcations as a function of the topological control parameter, which finally gives rise to chaotic dynamics, separating phases which exhibit clear topological features.
We calculate tunneling conductance of normal metal and superconducting topological insulator CuxBi2Se3 junctions for the possible pairing symmetries. In the presence of gapless Andreev bound states (ABSs), the tunneling conductance shows a zero-bias peak even for the three-dimensional full-gap superconducting state. This zero-bias conductance peak stems from the enhancement of the surface density of states induced by the surface-state transition in momentum space.
The electronic bandstructure of Au$_2$Pb has a Dirac cone which gaps when undergoing a structural transition into a low temperature superconducting phase. This suggests that the superconducting phase ($T_c=1.1$ K) might hold topological properties at the surface. Here we make Scanning Tunneling Microscopy experiments on the surface of superconducting Au$_2$Pb. We measure the superconducting gap and find a sizeable superconducting density of states at the Fermi level. We discuss possible origins for this finding in terms of superconductivity induced into surface states.
Motivated by recent developments in the field of one-dimensional topological superconductors, we investigate the topological properties of s-matrix of generic superconducting junctions where dimension should not play any role. We argue that for a finite junction the s-matrix is always topologically trivial. We resolve an apparent contradiction with the previous results by taking into account the low-energy resonant poles of s-matrix. Thus no common topological transition occur in a finite junction. We reveal a transition of a different kind that concerns the configuration of the resonant poles…
How to distinguish the zero-bias peak (ZBP) caused by the Majorana fermions from that by the other effects remains a challenge in detecting the topological order of a quantum wire. In this paper we propose to distinguish the topological superconducting phase from the topologically trivial phase by making a Josephson junction of the quantum wire attached to a side lead and then measuring the tunneling conductance through it as the phase difference across the junction $ϕ$ varies. Even if the ZBPs exist in both phases, we can identify the topological superconducting phase by a conductance peak at…
Topological superconductors are an intriguing and elusive quantum phase, characterized by topologically protected gapless surface/edge states residing in a bulk superconducting gap, which hosts Majorana fermions. Unfortunately, all currently known topological superconductors have a very low transition temperature, limiting experimental measurements of Majorana fermions. Here we discover the existence of a topological Dirac-nodal-line state in a well-known conventional high-temperature superconductor, MgB2. First-principles calculations show that the Dirac-nodal-line structure exhibits a unique…
We derive an effective two-dimensional low-energy theory for thin superconducting films coupled to a three-dimensional fluctuating electromagnetic field. Using this theory we discuss plasma oscil- lations, interactions between charges and vortices and extract the energy of a vortex. Having found that the effective theory properly describes the long distance physics, we then use it to investigate to what extent the superconducting film is a topologically ordered phase of matter.
The monolayer FeSe with a thickness of one unit cell grown on a single-crystal SrTiO$_{3}$ substrate (FeSe/STO) exhibits striking high-temperature superconductivity with transition temperature $T_{c}$ over 65K reported by recent experimental measurements. In this work, through analyzing the distinctive electronic structure, and providing systematic classification of the pairing symmetry , we find that both $s$-and $p$-wave pairing with odd parity give rise to topological superconducting states in monolayer FeSe, and the exotic properties of $s$-wave topological superconducting states have clos…
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 m…
Angle resolved photoemission spectroscopy is used to observe changes in the electronic structure of bulk-doped topological insulator Cu$_x$Bi$_2$Se$_3$ as additional copper atoms are deposited onto the cleaved crystal surface. Carrier density and surface-normal electrical field strength near the crystal surface are estimated to consider the effect of chemical surface gating on atypical superconducting properties associated with topological insulator order, such as the dynamics of theoretically predicted Majorana Fermion vortices.
A composite system of Majorana-hosted semiconductor nanowire and superconducting flux qubit is inves- tigated. It is found that the coupling between these two subsystems can be controlled electrically, supplying a convenient method to implement π/8 phase gate of a Majorana-based topological qubit. We also present a scheme to transfer information from the flux qubit to the topological qubit using Landau-Zener transition. In addition, a structure named top-flux-flux is proposed to retrieve the information stored in the topological qubit. With the demonstration of the entanglement of two topologi…
The Cu-doped topological insulator Bi$_2$Se$_3$ has recently been found to undergo a superconducting transition upon cooling, raising the possibilities that it is the first known "topological superconductor" or realizes a novel non-Abelian superconducting state. Its true nature depends critically on the bulk and surface state band topology. We present the first photoemission spectroscopy results where by examining the band topology at many different copper doping values we discover that the topologically protected spin-helical surface states remain well protected and separate from bulk Dirac b…
It is argued that a superconducting instability appears in the electronic states on the surface of a topological insulator due purely to electromagnetic interactions. The discussion of this instability is based on the analysis of the effective Coulomb interaction and the assumption that the system behaves like a two dimensional Fermi liquid. It is shown that this superconducting instability appears in the p-wave channel (like in the pure QED$_{3}$ case) and an expression for the critical temperature of the transition is provided.
