D-wave Superconductors Research Articles

D-Wave Superconducting Gap Symmetry as a Model for Nb1−xMoxB2 (x = 0.25; 1.0) and WB2 Diborides

Recently, Pei et al. (National Science Review2023, nwad034, 10.1093/nsr/nwad034) reported that ambient pressure β-MoB2 (space group: R3¯m) exhibits a phase transition to α-MoB2 (space group: P6/mmm) at pressure P~70 GPa, which is a high-temperature superconductor exhibiting Tc=32 K at P~110 GPa. Although α-MoB2 has the same crystalline structure as ambient-pressure MgB2 and the superconducting critical temperatures of α-MoB2 and MgB2 are very close, the first-principles calculations show that in α-MoB2, the states near the Fermi level, εF, are dominated by the d-electrons of Mo atoms, while in MgB2, the p-orbitals of boron atomic sheets dominantly contribute to the states near the εF. Recently, Hire et al. (Phys. Rev. B2022, 106, 174515) reported that the P6/mmm-phase can be stabilized at ambient pressure in Nb1−xMoxB2 solid solutions, and that these ternary alloys exhibit Tc~8 K. Additionally, Pei et al. (Sci. China-Phys. Mech. Astron. 2022, 65, 287412) showed that compressed WB2 exhibited Tc~15 K at P~121 GPa. Here, we aimed to reveal primary differences/similarities in superconducting state in MgB2 and in its recently discovered diboride counterparts, Nb1−xMoxB2 and highly-compressed WB2. By analyzing experimental data reported for P6/mmm-phases of Nb1−xMoxB2 (x = 0.25; 1.0) and highly compressed WB2, we showed that these three phases exhibit d-wave superconductivity. We deduced 2Δm(0)kBTc=4.1±0.2 for α-MoB2, 2Δm(0)kBTc=5.3±0.1 for Nb0.75Mo0.25B2, and 2Δm(0)kBTc=4.9±0.2 for WB2. We also found that Nb0.75Mo0.25B2 exhibited high strength of nonadiabaticity, which was quantified by the ratio of TθTF=3.5, whereas MgB2, α-MoB2, and WB2 exhibited TθTF~0.3, which is similar to the TθTF in pnictides, A15 alloys, Heusler alloys, Laves phase compounds, cuprates, and highly compressed hydrides.

Monolayer Kagome metals AV3Sb5

Recently, layered kagome metals AV3Sb5 (A = K, Rb, and Cs) have emerged as a fertile platform for exploring frustrated geometry, correlations, and topology. Here, using first-principles and mean-field calculations, we demonstrate that AV3Sb5 can crystallize in a mono-layered form, revealing a range of properties that render the system unique. Most importantly, the two-dimensional monolayer preserves intrinsically different symmetries from the three-dimensional layered bulk, enforced by stoichiometry. Consequently, the van Hove singularities, logarithmic divergences of the electronic density of states, are enriched, leading to a variety of competing instabilities such as doublets of charge density waves and s- and d-wave superconductivity. We show that the competition between orders can be fine-tuned in the monolayer via electron-filling of the van Hove singularities. Thus, our results suggest the monolayer kagome metal AV3Sb5 as a promising platform for designer quantum phases.

Evidence for chiral superconductivity on a silicon surface

Tin adatoms on a Si(111) substrate with a one-third monolayer coverage form a two-dimensional triangular lattice with one unpaired electron per site. These electrons order into an antiferromagnetic Mott-insulating state, but doping the Sn layer with holes creates a two-dimensional conductor that becomes superconducting at low temperatures. Although the pairing symmetry of the superconducting state is currently unknown, the combination of repulsive interactions and frustration inherent in the triangular adatom lattice opens up the possibility of a chiral order parameter. Here we study the superconducting state of Sn/Si(111) using scanning tunnelling microscopy, scanning tunnelling spectroscopy and quasiparticle interference imaging. We find evidence for a doping-dependent superconducting critical temperature with a fully gapped order parameter, the presence of time-reversal symmetry breaking and a strong enhancement in zero-bias conductance near the edges of the superconducting domains. Although each individual piece of evidence could have a more mundane interpretation, our combined results suggest the possibility that Sn/Si(111) is an unconventional chiral d-wave superconductor.

The magnetic field driven superconductor–metal transition in disordered hole-overdoped cuprates

By solving the Bogoliubov–de Gennes equations for a d-wave superconductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor–metal transition of the hole-overdoped cuprates. For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero. We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli–de Gennes–Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.

Josephson effect in twisted cuprates

<sec>To tunnel, or not to tunnel, that is the question for a Josephson junction constructed by superconductors with unidentified pairing symmetry. Theoretically, Josephson tunneling is forbidden between two <i>d</i>-wave superconductors twisted by 45°. This is in sharp contrast to persistent tunneling between two <i>s</i>-wave superconductors. Experimentally, however, Josephson tunneling is observed in twisted bicrystalline cuprates at around 45°, against the expectation that cuprate superconductors possess <i>d</i>-wave pairing. Due to technical uncertainties, the early studies on twisted bulk cuprates were not widely recognized. The recent advent of van der Waals stacking has allowed a fresh look at this problem. Indeed, twisted thin flakes of cuprates have been realized and the corresponding pairing symmetry has been revisited both experimentally and theoretically. In this work, we overview the recent development on twisted cuprates. After summarizing the theoretical treatment and recent proposals, we introduce the technical progress of making the twisted cuprate junctions in van der Waal stacking, and discuss the recent experimental results of <i>s</i>-, <i>d</i>-, or <i>d</i> + i<i>d</i>-wave pairing. In the end, we propose possible directions for future exploration in this field.</sec><sec> This paper has three major sections: theories on twisted cuprates in Section 1, techniques of realizing twisted cuprates in Section 2, and experimental results on twisted cuprates in Section 3. Specifically, in Section 1, both the early theory and the latest theoretical proposals are introduced. After discussing the calculated angular dependence of Josephson tunneling between two <i>d</i>-wave or <i>s-</i>wave superconductors, we summarize the predicted features from the emergent <i>d</i>+i<i>d-</i>wave pairing. They include unconventional temperature dependence of the critical Josephson current, doubling in frequency of the Fraunhofer pattern or Shapiro steps, and spontaneous Kerr rotation or emergence of Josephson diode effect. In Section 2, the technological progress of van der Waals stacking of cuprate superconductors is presented. Ultrathin twisted Josephson junctions of cuprates can be realized by either dry stacking together with oxygen post-annealing or cryogenic stacking at tens of degrees below 0 °C. In Section 3, the recent experimental results on van der Waals stacked twisted cuprates are reviewed. Tunneling in twisted underdoped cuprates realized by post-annealing indicates the existence of <i>s</i>-wave pairing and strong deviation from pure <i>d</i>-wave pairing. This result is contrasted with another study on cryogenically stacked junctions. There, signatures of <i>d</i>+i<i>d</i>-wave pairing, such as fractional Shapiro steps, are reported. Still, our recent experiments on 45°-twisted junctions with ultraclean interfaces, which are also realized by cryogenic stacking, show standard Fraunhofer patterns and AC Josephson effect with only integer steps, indicating the absence of <i>d </i>+ i<i>d</i>-wave pairing. These results have far-reaching influence on understanding the pairing symmetry of twisted cuprates. Future efforts to study the twisted cuprates may include: extending to a wider pool of materials, pushing the thickness to the atomic limit, and adopting other characterization tools. The twisted cuprates may also find applications in high temperature superconducting quantum bit as well as Josephson diodes.</sec>

Functional renormalization group study of the two-dimensional Su-Schrieffer-Heeger-Hubbard model

We study the Hubbard model on the square lattice coupled in addition to the optical Su-Schrieffer-Heeger (SSH) phonons, using the singular-mode functional renormalization group method. At half-filling and in the absence of the Hubbard interaction $U$, we find the degenerate spin-density-wave (SDW)/charge-density-wave (CDW)/s-wave superconductivity (sSC) state at smaller electron-phonon coupling strength $\lambda$ and higher phonon frequency $\omega$, and the valence bond solid (VBS) state at larger $\lambda$ and lower $\omega$. After switching on a positive $U$, the VBS state is suppressed, while the SDW state is enhanced. At finite doping, the SSH phonon is found to favor sSC at $U=0$. With increasing positive $U$, we find d-wave superconductivity (dSC) and incommensurate SDW states. In a narrow window of moderate $U$ and $\lambda$, we also find the incommensurate VBS state. The sSC and dSC here can be naturally related to the CDW and SDW fluctuations, both of which can be triggered by the SSH phonons. In contrast, the repulsive interaction $U$ enhances SDW but suppresses CDW fluctuations. Our results at half filling are consistent with quantum Monte Carlo (QMC), and provide insights at finite doping where QMC may suffer from the minus sign problem.

Critical Current Density in d-Wave Hubbard Superconductors.

In this work, the Generalized Hubbard Model on a square lattice is applied to evaluate the electrical current density of high critical temperature d-wave superconductors with a set of Hamiltonian parameters allowing them to reach critical temperatures close to 100 K. The appropriate set of Hamiltonian parameters permits us to apply our model to real materials, finding a good quantitative fit with important macroscopic superconducting properties such as the critical superconducting temperature (Tc) and the critical current density (Jc). We propose that much as in a dispersive medium, in which the velocity of electrons can be estimated by the gradient of the dispersion relation ∇ε(k), the electron velocity is proportional to ∇E(k) in the superconducting state (where E(k)=(ε(k)-μ)2+Δ2(k) is the dispersion relation of the quasiparticles, and k is the electron wave vector). This considers the change of ε(k) with respect to the chemical potential (μ) and the formation of pairs that gives rise to an excitation energy gap Δ(k) in the electron density of states across the Fermi level. When ε(k)=μ at the Fermi surface (FS), only the term for the energy gap remains, whose magnitude reflects the strength of the pairing interaction. Under these conditions, we have found that the d-wave symmetry of the pairing interaction leads to a maximum critical current density in the vicinity of the antinodal k-space direction (π,0) of approximately 1.407236×108 A/cm2, with a much greater current density along the nodal direction (π2,π2) of 2.214702×109 A/cm2. These results allow for the establishment of a maximum limit for the critical current density that could be attained by a d-wave superconductor.

Characterization of YBa2Cu3O7−δ coplanar resonator for microwave kinetic inductance detectors

We demonstrate an improved YBa2Cu3O7−δ -based microwave kinetic inductance detector with a quality factor and noise equivalent power, at 10 K. Zero field cooled (ZFC) and field cooled (FC) measurements of the magnetic field dependence of the resonance characteristics, show substantially different behavior, indicating that both the screening currents and vortices play a role. The ZFC measurements exhibit a sharp decrease of the resonance frequency, , and at low fields, up to the full penetration field, revealing the dominant role of the screening currents. In contrast, the FC measurements exhibit a moderate decrease of and with field, revealing the role of vortices and reflecting the field dependence of the penetration depth in a d-wave superconductor.

Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires.

In most naturally occurring superconductors, electrons with opposite spins form Cooper pairs. This includes both conventional s-wave superconductors such as aluminium, as well as high-transition-temperature, d-wave superconductors. Materials with intrinsic p-wave superconductivity, hosting Cooper pairs made of equal-spin electrons, have not been conclusively identified, nor synthesized, despite promising progress1-3. Instead, engineered platforms where s-wave superconductors are brought into contact with magnetic materials have shown convincing signatures of equal-spin pairing4-6. Here we directly measure equal-spin pairing between spin-polarized quantum dots. This pairing isproximity-induced from an s-wave superconductor into a semiconducting nanowire with strong spin-orbit interaction. We demonstrate such pairing by showing that breaking a Cooper pair can result in two electrons with equal spin polarization. Our results demonstrate controllable detection of singlet and triplet pairing between the quantum dots. Achieving such triplet pairing in a sequence of quantum dots will be required for realizing an artificial Kitaev chain7-9.

Determination of spin polarization of metals and gap of superconductors by spin-polarized Andreev reflection spectroscopy

Andreev reflection spectroscopy (ARS) is a unique technique that can determine the spin polarization of a metal as well as the gap of a superconductor but d-wave superconductors are seldomly used in ARS due to the lack of quantitative model to extract parameters from experiments. Here we calculate spin-polarized ARS based on the modified Blonder-Tinkham-Klapwijk model for both s- and d-wave superconductors of a three-dimensional (3D) interface with a point angle. We show that the 3D interface and the point angle can drastically affect the spectra for both s- and d-wave ARS. For s-wave, the 3D effect is similar to that of an interfacial barrier but it can generate a variety of rich features in d-wave ARS and induce Andreev bound states (ABS). The spin polarization can linearly suppress both s- and d-wave spectra including ABS. We show that the ABS states can be controlled by the spin polarization, the point angle and the interfacial barrier but very high polarization is required to completely suppress the ABS. Furthermore, the Fermi wavevector mismatch as well as the carrier effective mass mismatch can cause effects very similar to that of an interfacial barrier for both s- and d-wave ARS.

Self-consistent theory of current injection into d and d + is superconductors

We present results for the steady-state nonlinear response of a superconducting film connected to normal-metal reservoirs under voltage bias, allowing for a subdominant s-wave component appearing near the interfaces. Our investigation is based on a current-conserving theory that self-consistently includes the non-equilibrium distribution functions, charge imbalance, and the voltage-dependencies of order parameters and scalar impurity self-energies. For a pure d-wave superconductor with [110] orientation of the interfaces to the contacts, the conductance contains a zero-bias peak reflecting the large density of zero-energy interface Andreev bound states. Including a subdominant s-wave pairing channel, it is in equilibrium energetically favorable for an s-wave order parameter component to appear near the interfaces in the time-reversal symmetry breaking combination d + is. The Andreev states then shift to finite energies in the density of states. Under voltage bias, we find that the non-equilibrium distribution in the contact area causes a rapid suppression of the s-wave component to zero as the voltage . The resulting spectral rearrangements and voltage-dependent scattering amplitudes lead to a pronounced non-thermally broadened split of the zero-bias conductance peak that is not seen in a non-selfconsistent Landauer–Büttiker scattering approach.

Identification of a nematic pair density wave state in Bi2Sr2CaCu2O8+x

Electron-pair density wave (PDW) states are now an intense focus of research in the field of cuprate correlated superconductivity. PDWs exhibit periodically modulating superconductive electron pairing that can be visualized directly using scanned Josephson tunneling microscopy (SJTM). Although from theory, intertwining the d-wave superconducting (DSC) and PDW order parameters allows a plethora of global electron-pair orders to appear, which one actually occurs in the various cuprates is unknown. Here, we use SJTM to visualize the interplay of PDW and DSC states in Bi2Sr2CaCu2O8+x at a carrier density where the charge density wave modulations are virtually nonexistent. Simultaneous visualization of their amplitudes reveals that the intertwined PDW and DSC are mutually attractive states. Then, by separately imaging the electron-pair density modulations of the two orthogonal PDWs, we discover a robust nematic PDW state. Its spatial arrangement entails Ising domains of opposite nematicity, each consisting primarily of unidirectional and lattice commensurate electron-pair density modulations. Further, we demonstrate by direct imaging that the scattering resonances identifying Zn impurity atom sites occur predominantly within boundaries between these domains. This implies that the nematic PDW state is pinned by Zn atoms, as was recently proposed [Lozano etal., Phys. Rev. B 103, L020502 (2021)]. Taken in combination, these data indicate that the PDW in Bi2Sr2CaCu2O8+x is a vestigial nematic pair density wave state [Agterberg etal. Phys. Rev. B 91, 054502 (2015); Wardh and Granath arXiv:2203.08250].

Floquet topological superconductivity induced by chiral many-body interaction

Non-equilibrium engineering is becoming a seminal way for realising novel quantum phases that are unimaginable in equilibrium. In particular, Floquet theory applied to quantum mechanics revealed that we can even control the band topology in semimetallic/insulating systems, while the straightforward application to topological superconductivity fails for typical superconductors because the supercondicting gap function does not couple to the electromagnetic field in a direct manner. Here we show that we can overcome this difficulty by taking account of correlation effects. Namely, we study how a d-wave superconductivity is changed when illuminated by circularly-polarised light (CPL) in the repulsive Hubbard model in the strong-coupling regime. We adopt the Floquet formalism for the Gutzwiller-projected effective Hamiltonian with the time-periodic Schrieffer-Wolff transformation. We find that CPL induces a topological superconductivity with a d + id pairing, which arises from the chiral spin coupling and the three-site term generated by the CPL. The latter term remains significant even for low frequencies and low intensities of the CPL. This is clearly reflected in the obtained phase diagram against the laser intensity and temperature for various frequencies red-detuned from the Hubbard U, with the transient dynamics also examined. The phenomenon revealed here can open a novel, dynamical way to induce a topological superconductivity.

Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities

The spatial modulation of the Fermi velocity for gapless Dirac electrons in quantum materials is mathematically equivalent to the problem of massless fermions on a certain class of curved spacetime manifolds. We study null geodesic lensing through these manifolds, which are dominated by curvature singularities, such as nematic singularity walls (where the Dirac cone flattens along one direction). Null geodesics lens across these walls, but do so by perfectly collimating to a local transit angle. Nevertheless, nematic walls can trap null geodesics into stable or metastable orbits characterized by repeated transits. We speculate about the role of induced one-dimensionality for such bound orbits in 2D dirty d-wave superconductivity.

Generalising Holographic Superconductors

In this paper we propose a generalised holographic framework to describe superconductors. We first unify the description of s-, p-, and d-wave superconductors in a way that can be easily promoted to higher spin. Using a semianalytical procedure to compute the superconductor properties, we are able to further generalise the geometric description of the hologram beyond the AdS-Schwarzschild Black Hole paradigm and propose a set of higher-dimensional metrics which exhibit the same universal behaviour. We then apply this generalised description to study the properties of the condensate and the scaling of the critical temperature with the parameters of the higher-dimensional theory, which allows us to reproduce existing results in the literature and extend them to include a possible description of the newly observed f-wave superconducting systems.

Degenerate plaquette physics as key ingredient of high-temperature superconductivity in cuprates

We study the physics of high-temperature cuprate superconductors starting from the highly degenerate four-site plaquette of the t-t^{prime} -U Hubbard model as a reference system. The degeneracy causes strong fluctuations when a lattice of plaquettes is constructed. We show that there is a large binding energy between holes when a set of four plaquettes is considered. The next-nearest-neighbour hopping t^{prime} plays a crucial role in the formation of these strongly bound electronic bipolarons whose coherence at lower temperature could be the explanation for superconductivity. A complementary approach is cluster dual fermion starting from a single degenerate plaquette, which contains the relevant short-ranged fluctuations from the beginning. It gives d-wave superconductivity as the leading instability under a reasonably broad range of parameters. The origin of the pseudogap is also discussed in terms of the coupling of degenerate plaquettes. Thus, some of the essential elements of cuprate superconductivity appear from the local plaquette physics.

Superconductivity from repulsion in the doped 2D electronic Hubbard model: an entanglement perspective

We employ the momentum space entanglement renormalization group (MERG) scheme developed in references (Mukherjee et al 2021 J. High Energy Phys. JHEP04(2021)148; Patra and Lal 2021 Phys. Rev. B 104 144514) for the study of various insulating, superconducting and normal phases of the doped and the undoped 2D Hubbard model on a square lattice found recently by us (Mukherjee and Lal 2020 New J. Phys. 22 063007; Mukherjee and Lal 2020 New J. Phys. 22 063008). At each MERG step, disentanglement of particular degrees of freedom, transforms the tensor network representation of the many-particle states. The MERG reveals distinct holographic entanglement features for the normal metallic, topologically ordered insulating quantum liquid and Neél antiferromagnetic symmetry-broken ground states of the 2D Hubbard model at half-filling, clarifying the essence of the entanglement phase transitions that separates the three phases. An MERG analysis of the quantum critical point of the hole-doped 2D Hubbard model reveals the evolution of the many-particle entanglement of the quantum liquid ground state with hole-doping, as well as how the collapse of Mottness is concomitant with the emergence of d-wave superconductivity.

Standard measurement method for normal state resistance and critical current of resistively shunted Josephson junctions

An important parameter of Josephson junctions (JJs) is the product of normal state resistance (R n) and critical current (I c) for designing superconductor analogue devices or digital circuits. Determination of R n and I c from voltage–current (U–I) characteristic curves often faces difficulties; in particular I c is considerably reduced by intrinsic thermal or extrinsic electrical noises. Here, we propose a standard measurement method of R n and intrinsic critical current (I ci) for high-T c superconductor (HTS) grain boundary JJs operated in liquid nitrogen and low-T c superconductor (LTS) multilayer superconductor/normal-conductor/superconductor (SNS) JJs in liquid helium. The applicable condition of this method is that both HTS and LTS JJs have U–I curves compatible with resistively-shunted junction (RSJ) model. Both R n and I ci values are extracted by combining a geometric mean criterion to select a data set and a least-squares fitting method with the RSJ model, eliminating two distortion effects on U–I curves: noise-rounding and self-heating. The combined method ensures relative standard uncertainty values of 1.9% for R n and 8% for I ci or better, when the users follow the standard protocol. It is demonstrated that the combined method is valid for d-wave HTS JJs near 77 K, regardless of peculiarities such as 0–π junction transition in d-wave superconductors at lower temperatures, and s-wave LTS SNS JJs with a wide range of junction parameters. This work is the first step to facilitate standardization for superconductor electronics with JJs.

Unconventional spectral signature of Tc in a pure d-wave superconductor.

In conventional superconductors, the phase transition into a zero-resistance and perfectly diamagnetic state is accompanied by a jump in the specific heat and the opening of a spectral gap1. In the high-transition-temperature (high-Tc) cuprates, although the transport, magnetic and thermodynamic signatures of Tc have been known since the 1980s2, the spectroscopic singularity associated with the transition remains unknown. Here we resolve this long-standing puzzle with a high-precision angle-resolved photoemission spectroscopy (ARPES) study on overdoped (Bi,Pb)2Sr2CaCu2O8+δ (Bi2212). We first probe the momentum-resolved electronic specific heat via spectroscopy and reproduce the specific heat peak at Tc, completing the missing link for a holistic description of superconductivity. Then, by studying the full momentum, energy and temperature evolution of the spectra, we reveal that this thermodynamic anomaly arises from the singular growth of in-gap spectral intensity across Tc. Furthermore, we observe that the temperature evolution of in-gap intensity is highly anisotropic in the momentum space, and the gap itself obeys both the d-wave functional form and particle-hole symmetry. These findings support the scenario that the superconducting transition is driven by phase fluctuations. They also serve as an anchor point for understanding the Fermi arc and pseudogap phenomena in underdoped cuprates.

Quantum criticality of fermion velocities and critical temperature nearby a putative quantum phase transition in the d-wave superconductors

Quantum critical behaviors induced by a putative quantum phase transition are vigilantly investigated, which separates a d-wave superconducting state and d-wave superconducting+X state below the superconducting dome of the d−wave superconductors with tuning the non-thermal doping variable. Within the framework of renormalization group approach, we start with a phenomenological effective theory originated from the Landau-Ginzburg-Wilson theory and practice one-loop calculations to construct a set of coupled flows of all interaction parameters. After extracting related physical information from these coupled evolutions, we address that both fermion velocities and critical temperatures exhibit critical behaviors, which are robust enough against the initial conditions due to strong quantum fluctuations. At first, the evolution of Yukawa coupling between X-state order parameter and nodal fermions in tandem with quantum fluctuations heavily renormalize fermion velocities and generally drive them into certain finite anisotropic fixed point at the lowest-energy limit, whose concrete value relies upon the very quantum phase transition. In addition, these unique properties of fermion velocities largely reshape the fate of superfluid density, giving rise to either an enhancement or a dip of critical temperature. Moreover, we find that fermion-fermion interactions bring non-ignorable quantitative corrections to quantum critical behaviors despite they are subordinate to quantum fluctuations of order parameters.