Centrosymmetric Superconductors Research Articles

Lifting of Gap Nodes by Disorder in Tetragonal FeSe_{1-x}S_{x} Superconductors.

The observation of time-reversal symmetry breaking and large residual density of states in tetragonal FeSe_{1-x}S_{x} suggests a novel type of ultranodal superconducting state with Bogoliubov Fermi surfaces (BFSs). Although such BFSs in centrosymmetric superconductors are expected to be topologically protected, the impurity effect of this exotic superconducting state remains elusive experimentally. Here, we investigate the impact of controlled defects introduced by electron irradiation on the superconducting state of tetragonal FeSe_{1-x}S_{x} (0.18≤x≤0.25). The temperature dependence of magnetic penetration depth is initially consistent with a model with BFSs in the pristine sample. After irradiation, we observe a nonmonotonic evolution of low-energy excitations with impurity concentrations. This nonmonotonic change indicates a transition from nodal to nodeless, culminating in gapless with Andreev bound states, reminiscent of the nodal s_{±} case. This points to the accidental nature of the possible BFSs in tetragonal FeSe_{1-x}S_{x}, which are susceptible to disruption by the disorder.

Unequivocal Identification of Spin-Triplet and Spin-Singlet Superconductors with Upper Critical Field and Flux Quantization.

Spin-triplet superconductors play central roles in Majorana physics and quantum computing but are difficult to identify. We show the methods of kink-point upper critical field and flux quantization in superconducting rings can unequivocally identify spin-singlet, spin-triplet in centrosymmetric superconductors, and singlet-triplet admixture in noncentrosymmetric superconductors, as realized in γ-BiPd, β-Bi_{2}Pd, and α-BiPd, respectively. Our findings are essential for identifying triplet superconductors and exploring their quantum properties.

Topological skyrmion phases of matter

We introduce topological phases of matter defined by skyrmions in the ground state spin—or pseudospin—expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr2RuO4 with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when atomic spin–orbit coupling is non-negligible and there is a suitable additional degree of freedom. This contradicts the ‘flat band’ limit assumption important in use of entanglement spectrum and Wilson loops, and in construction of the ten-fold way classification scheme of topological phases of matter. We furthermore predict two kinds of bulk-boundary correspondence signatures—one for measurements which execute a partial trace over degrees of freedom other than spin, which yields quantized transport signatures—and a second resulting from skyrmions trapping defects with their own non-trivial topology that is discussed in a second work, which yields generalizations of unpaired Majorana zero-modes.

Evidences of Topological Surface States in the Nodal-Line Semimetal SnTaS2 Nanoflakes.

Exploring the topological surface state of a topological semimetal by the transport technique has always been a big challenge because of the overwhelming contribution of the bulk state. In this work, we perform systematic angular-dependent magnetotransport measurements and electronic band calculations on SnTaS2 crystals, a layered topological nodal-line semimetal. Distinct Shubnikov-de Haas quantum oscillations were observed only in SnTaS2 nanoflakes when the thickness was below about 110 nm, and the oscillation amplitudes increased significantly with decreasing thickness. By analysis of the oscillation spectra, together with the theoretical calculation, a two-dimensional and topological nontrivial nature of the surface band is unambiguously identified, providing direct transport evidence of drumhead surface state for SnTaS2. Our comprehensive understanding of the Fermi surface topology of the centrosymmetric superconductor SnTaS2 is crucial for further research on the interplay of superconductivity and nontrivial topology.

Enhanced critical field and anomalous metallic state in two-dimensional centrosymmetric 1T′−WS2

Layered transition-metal chalcogenide material (TMD) offers a platform to investigate two-dimensional (2D) superconductivity. Here, we report an electrical transport study of 2D centrosymmetric superconductor $1{T}^{\ensuremath{'}}\text{\ensuremath{-}}\mathrm{W}{\mathrm{S}}_{2}$. For a typical five-layer sample, the 2D superconductivity is revealed in Berezinskii-Kosterlitz-Thouless (BKT) transition with the characteristic temperature ${T}_{BKT}\ensuremath{\sim}5\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The superconductivity is observed with a strong resist against the in-plane magnetic field. We find that the critical in-plane magnetic field is \ensuremath{\sim}3 times the Pauli paramagnetic limit ${B}_{p}$, with effective Zeeman-type spin-orbital coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{{\ensuremath{\beta}}_{\mathrm{SO}}}\ensuremath{\sim}2.8\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$. In addition, we find the magnetoresistance trend to saturate at low temperature, suggesting anomalous quantum metallic states. These findings might stimulate further investigations on the correlation between superconductivity and symmetry of 2D superconductors, as well as the quantum metallic states at low temperature with finite magnetic field.

Symmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal pairing

Multiband effects can lead to fundamentally different electronic behavior of solids, as exemplified by the possible emergence of Fermi surfaces of Bogoliubov quasiparticles in centrosymmetric superconductors which break time-reversal symmetry. We extend the analysis of possible pairing symmetries, the corresponding nodal structure, and the Bogoliubov Fermi surfaces in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than the effective angular momentum of length $j=3/2$ examined so far. Since our main focus is on the Bogoliubov Fermi surfaces we concentrate on even-parity pairing. The required symmetry analysis is illustrated for several examples, as a guide for the reader. We find that the inclusion of nonlocal pairing leads to a much larger range of possible pairing symmetries. For infinitesimal pairing strength, we find a simple yet powerful criterion for nodes in terms of a scalar product of form factors.

Linking structures of doubly charged nodal surfaces in centrosymmetric superconductors

In topological semimetals and nodal superconductors, band crossings between occupied and unoccupied bands form stable nodal points/lines/surfaces carrying quantized topological charges. In particular, in centrosymmetric systems, some nodal structures at the Fermi energy $E_F$ carry two distinct topological charges, and thus they are called doubly charged nodes. Here we show that doubly charged nodal surfaces of centrosymmetric superconductors in three-dimensions always develop peculiar linking structures with nodal points or lines formed between occupied bands below $E_F$. Such linking structures can naturally explain the inherent relationship between the charge of the node below $E_F$ and the two charges of the nodal surfaces at $E_F$. Based on the Altland-Zirnbauer (AZ)-type ten-fold classification of nodes with additional inversion $\mathcal{I}$ symmetry, which is called the AZ$+\mathcal{I}$ classification, we provide the complete list of linking structures of doubly charged nodes in centrosymmetric systems. The linking structures of doubly charged nodes clearly demonstrate that not only the local band structure around the node but also the global band structure play a critical role in characterizing gapless topological phases.

Distortional weak-coupling instability of Bogoliubov Fermi surfaces

Centrosymmetric multiband superconductors which break time-reversal symmetry generically have two-dimensional nodes, i.e., Fermi surfaces of Bogoliubov quasiparticles. We show that the coupling of the electrons to the lattice always leads to a weak-coupling instability of such a state towards spontaneous breaking of inversion symmetry at low temperatures. This instability is driven by a Cooper logarithm in the internal energy but the order parameter is not superconducting but distortional. We present a comprehensive symmetry analysis and introduce a measure that allows to compare the strengths of competing distortional instabilities. Moreover, we discuss the instability using an effective single-band model. This framework reveals a duality mapping of the effective model which maps the distortional order parameter onto a superconducting one, providing a natural explanation for the Cooper logarithm and the weak-coupling nature of the instability. Finally, we consider the possibility of a pair-density wave state when inversion symmetry is broken. We find that it can indeed exist but does not affect the instability itself.

Evidence of topological nodal lines and surface states in the centrosymmetric superconductor SnTaS2

The discovery of signatures of topological superconductivity in superconducting bulk materials with topological surface states has attracted intensive research interest recently. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, here, we demonstrate the existence of topological nodal-line states and drumheadlike surface states in centrosymmetric superconductor $\mathrm{Sn}\mathrm{Ta}{\mathrm{S}}_{2}$, which is a type-II superconductor with a critical transition temperature of about 3 K. The valence bands from Ta $5d$ orbitals and the conduction bands from Sn $5p$ orbitals cross each other, forming two nodal lines in the vicinity of the Fermi energy without the inclusion of spin-orbit coupling (SOC), protected by the spatial-inversion symmetry and time-reversal symmetry. The nodal lines are gapped out by SOC. The drumheadlike surface states, the typical characteristics in nodal-line semimetals, are quite visible near the Fermi level. Our findings indicate that $\mathrm{Sn}\mathrm{Ta}{\mathrm{S}}_{2}$ offers a promising platform for exploring the exotic properties of the topological nodal-line fermions and helps in the study of topological superconductivity.

Two-gap time reversal symmetry breaking superconductivity in noncentrosymmetric LaNiC2

We report a $\mu$SR investigation of a non-centrosymmetric superconductor (LaNiC$_2$) in single crystal form. Compared to previous $\mu$SR studies of non-centrosymmetric superconducting polycrystalline and powder samples, the unambiguous orientation of single crystals enables a simultaneous determination of the absolute value of the magnetic penetration depth and the vortex core size from measurements that probe the magnetic field distribution in the vortex state. The magnetic field dependence of these quantities unambiguously demonstrates the presence of two nodeless superconducting energy gaps. In addition, we detect weak internal magnetic fields in the superconducting phase, confirming earlier $\mu$SR evidence for a time-reversal symmetry breaking superconducting state. Our results suggest that Cooper pairing in LaNiC$_2$ is characterized by the same interorbital equal-spin pairing model introduced to describe the pairing state in the centrosymmetric superconductor LaNiGa$_2$.

Superconductivity with broken time-reversal symmetry inside a superconducting s-wave state

In general, magnetism and superconductivity are antagonistic to each other. However, there are several families of superconductors, in which superconductivity may coexist with magnetism, and only a few examples are known, when superconductivity itself induces spontaneous magnetism. The most known compounds are Sr$_2$RuO$_4$ and some noncentrosymmetric superconductors. Here, we report the finding of a narrow dome of a novel $s+is'$ superconducting (SC) phase with broken time-reversal symmetry (BTRS) inside the broad $s$-wave SC region of the centrosymmetric multiband superconductor Ba$_{\rm 1-x}$K$_{\rm x}$Fe$_2$As$_2$ ($0.7 \lesssim x \lesssim 0.85$). We observe spontaneous magnetic fields inside this dome using the muon spin relaxation ($\mu$SR) technique. Furthermore, our detailed specific heat study reveals that the BTRS dome appears very close to a change in the topology of the Fermi surface (Lifshitz transition). With this, we experimentally demonstrate the emergence of a novel quantum state due to topological changes of the electronic system.

Higher-order topological superconductivity of spin-polarized fermions

We study the superconductivity of spin-polarized electrons in centrosymmetric ferromagnetic metals. Due to the spin-polarization and the Fermi statistics of electrons, the superconducting pairing function naturally has odd parity. According to the parity formula proposed by Fu, Berg, and Sato, odd-parity pairing leads to conventional first-order topological superconductivity when a normal metal has an odd number of Fermi surfaces. Here, we derive generalized parity formulae for the topological invariants characterizing higher-order topology of centrosymmetric superconductors. Based on the formulae, we systematically classify all possible band structures of ferromagnetic metals that can induce inversion-protected higher-order topological superconductivity. Among them, doped ferromagnetic nodal semimetals are identified as the most promising normal state platform for higher-order topological superconductivity. In two dimensions, we show that odd-parity pairing of doped Dirac semimetals induces a second-order topological superconductor. In three dimensions, odd-parity pairing of doped nodal line semimetals generates a nodal line topological superconductor with monopole charges. On the other hand, odd-parity pairing of doped monopole nodal line semimetals induces a three-dimensional third-order topological superconductor. Our theory shows that the combination of superconductivity and ferromagnetic nodal semimetals opens up a new avenue for future topological quantum computations using Majorana zero modes.

Bogoliubov Fermi surfaces stabilized by spin-orbit coupling

It was recently understood that centrosymmetric multiband superconductors that break time-reversal symmetry generically show Fermi surfaces of Bogoliubov quasiparticles. We investigate the thermodynamic stability of these Bogoliubov Fermi surfaces in a paradigmatic model. To that end, we construct the mean-field phase diagram as a function of spin-orbit coupling and temperature. It confirms the prediction that a pairing state with Bogoliubov Fermi surfaces can be stabilized at moderate spin-orbit coupling strengths. The multiband nature of the model also gives rise to a first-order phase transition, which can be explained by the competition of intra- and interband pairing and is strongly affected by cubic anisotropy. For the state with Bogoliubov Fermi surfaces, we also discuss experimental signatures in terms of the residual density of states and the induced magnetic order. Our results show that Bogoliubov Fermi surfaces of experimentally relevant size can be thermodynamically stable.

Nonhelical spin texture in the normal states of the centrosymmetric superconductor β−PdBi2

A nonhelical spin texture was observed in the centrosymmetric superconductor beta-PdBi2 using an imaging-type spin- and angle-resolved spectroscopy. The observed surface and bulk states near the Fermi energy are found to be spin polarized with nonhelical textures. First-principles calculations and effective models show that the inter-and intralayer hoppings account for the complex spin texture and thus play a significant role in the spatial distribution of the layer-locked spin states. Our work not only provides insights into the local spin states in inversion-symmetric systems but also paves the way to the identification of the nature of superconducting pairing in the presence of a complex spin texture in beta-PdBi2 and the related centrosymmetric superconductors.

First-principles investigation of the mechanical and thermodynamic properties of the antiperovskite centrosymmetric platinum-based superconductors APt3P (A=Ca, Sr, La and Ce)

The elastic, anisotropy, Vicker's hardness, Debye temperature θD and the minimum thermal conductivity kmin of APt3P (A = Ca, Sr, La and Ce) have been reported by using the first-principles calculations. The calculated ground state lattice parameters of APt3P agree will with the reported experimental results. The elastic constants are predicted and compared with published theoretical results. The calculation results show that the APt3P compounds are soft materials, and they are mechanically stable and behave in a ductile manner. In addition, the elastic anisotropic have been analyzed by plotting the three-dimensional (3D) surface of the modulus and the universal anisotropy index Au. Finally, the θD and kmin were predicted. The results demonstrate that they have the same variation tendency and both of them increases along the CePt3P→SrPt3P→LaPt3P→CaPt3P sequence.

Are AuPdTM (T = Sc, Y and M = Al, Ga, In), Heusler Compounds Superconductors without Inversion Symmetry?

Heusler compounds with 2:1:1 stoichiometry either have a centrosymmetric CuMnAl structure or an LiAgSb structure without a centre of inversion. The centrosymmetry is always lost in quaternary Heusler compounds with 1:1:1:1 stoichiometry and LiMgPdSn structure. This presents the possibility of realizing non-centrosymmetric superconductors in the family of Heusler compounds. The objective of this study is to search for and investigate such quaternary derivatives of Heusler compounds, particularly with respect to superconductivity. Several compounds were identified by carrying out calculations from first principles and superconductivity was observed in experiments conducted on AuPdScAl and AuPtScIn at the critical temperatures of 3.0 and 0.96 K, respectively. All investigated compounds had a valence electron count of 27, which is also the case in centrosymmetric Heusler superconductors.

Classification of topological crystalline superconducting nodes on high-symmetry lines: Point nodes, line nodes, and Bogoliubov Fermi surfaces

Recent development in exact classification of a superconducting gap has elucidated various unconventional gap structures, which have not been predicted by the classification of order parameter based on the point group. One of the important previous results is that all symmetry-protected line nodes are characterized by nontrivial topological numbers. Another intriguing discovery is the gap structures depending on the angular momentum $j_z$ of normal Bloch states on threefold and sixfold rotational-symmetric lines in the Brillouin zone. Stimulated by these findings, we classify irreducible representations of the Bogoliubov-de Gennes Hamiltonian at each $\boldsymbol{k}$ point on a high-symmetry $n$-fold ($n = 2$, $3$, $4$, and $6$) axis for centrosymmetric and paramagnetic superconductors, by using the combination of group theory and $K$ theory. This leads to the classification of all crystal symmetry-protected nodes (including $j_z$-dependent nodes) on the axis that crosses a normal-state Fermi surface. As a result, it is shown that the classification by group theory completely corresponds with the topological classification. Based on the obtained results, we discuss superconducting gap structures in SrPtAs, CeCoIn$_5$, UPt$_3$, and UCoGe.

Full-gap superconductivity in spin-polarised surface states of topological semimetal \u03b2-PdBi2

A bulk superconductor possessing a topological surface state at the Fermi level is a promising system to realise long-sought topological superconductivity. Although several candidate materials have been proposed, experimental demonstrations concurrently exploring spin textures and superconductivity at the surface have remained elusive. Here we perform spectroscopic-imaging scanning tunnelling microscopy on the centrosymmetric superconductor β-PdBi2 that hosts a topological surface state. By combining first-principles electronic-structure calculations and quasiparticle interference experiments, we determine the spin textures at the surface, and show not only the topological surface state but also all other surface bands exhibit spin polarisations parallel to the surface. We find that the superconducting gap fully opens in all the spin-polarised surface states. This behaviour is consistent with a possible spin-triplet order parameter expected for such in-plane spin textures, but the observed superconducting gap amplitude is comparable to that of the bulk, suggesting that the spin-singlet component is predominant in β-PdBi2.

Superconductivity protected by spin–valley locking in ion-gated MoS2

Symmetry-breaking has been known to play a key role in noncentrosymmetric superconductors with strong spin-orbit-interaction (SOI). The studies, however, have been so far mainly focused on a particular type of SOI, known as Rashba SOI, whereby the electron spin is locked to its momentum at a right-angle, thereby leading to an in-planar helical spin texture. Here we discuss electric-field-induced superconductivity in molybdenum disulphide (MoS2), which exhibits a fundamentally different type of intrinsic SOI manifested by an out-of-plane Zeeman-type spin polarization of energy valleys. We find an upper critical field of approximately 52 T at 1.5 K, which indicates an enhancement of the Pauli limit by a factor of four as compared to that in centrosymmetric conventional superconductors. Using realistic tight-binding calculations, we reveal that this unusual behaviour is due to an inter-valley pairing that is symmetrically protected by Zeeman-type spin-valley locking against external magnetic fields. Our study sheds a new light on the interplay of inversion asymmetry with SOI in confined geometries, and its unprecedented role in superconductivity.

Topologically protected surface states in a centrosymmetric superconductor β-PdBi2.

The topological aspects of electrons in solids can emerge in real materials, as represented by topological insulators. In theory, they show a variety of new magneto-electric phenomena, and especially the ones hosting superconductivity are strongly desired as candidates for topological superconductors. While efforts have been made to develop possible topological superconductors by introducing carriers into topological insulators, those exhibiting indisputable superconductivity free from inhomogeneity are very few. Here we report on the observation of topologically protected surface states in a centrosymmetric layered superconductor, β-PdBi2, by utilizing spin- and angle-resolved photoemission spectroscopy. Besides the bulk bands, several surface bands are clearly observed with symmetrically allowed in-plane spin polarizations, some of which crossing the Fermi level. These surface states are precisely evaluated to be topological, based on the Z2 invariant analysis in analogy to three-dimensional strong topological insulators. β-PdBi2 may offer a solid stage to investigate the topological aspect in the superconducting condensate.