D-wave Pairing Research Articles

Enhanced Pair-Density-Wave Vertices in a Bilayer Hubbard Model at Half Filling.

Motivated by the pair-density-wave (PDW) state found in the one-dimensional Kondo-Heisenberg chain, we report on a determinant quantum MonteCarlo study of pair fields for a two-dimensional half-filled Hubbard layer coupled to an itinerant, noninteracting layer with one electron per site. In a specific range of interlayer hopping, the pairing vertex associated with PDW order becomes more attractive than that for uniform d-wave pairing, although both remain subdominant to the leading antiferromagnetic correlations at half filling. Our result sheds light on where one potentially may find a PDW state in such a model.

Evidences for d-wave symmetry of c-axis superconducting gap in atomically thin twisted flakes of bismuth-based HTS cuprates

c-axis Josephson tunnelling in atomically thin twisted flakes of quasi-two-dimensional cuprate superconductors is one of the experimental techniques which can be used to reveal the pairing symmetry in HTS cuprates. Until very recent experiments reported by Zhao et al. (Science382, 1422 (2023)), there was a widely accepted consensus that the c-axis Josephson currents, Ic(T), in twisted cuprate junctions manifest the s-wave gap symmetry for c-axis gap component. Here, I highlighted that this consensus was based on the analysis of the Ic(T) data by using the Ambegaokar-Baratoff equation, where the weak-coupling s-wave gap equation was utilized. In this study, I introduced a more accurate gap equation in the AB model and reanalyzed the Ic(T) data in atomically thin twisted Bi2Sr2-xLaxCuO6+y (Bi-2201) and Bi2Sr2CaCu2O8+x (Bi-2212) flakes. In the result, the analysis revealed evidences for d-wave pairing symmetry. Beside this, the analysis of the RnIc(T) data reported by Zhao et al. (Science382, 1422 (2023)) by a piecewise AB model, where a linear low-T dependence of the Ic(T) is used, and where the threshold temperature TM is a free-fitting parameter, confirmed the d-wave symmetry in these twisted Bi-2212 junctions.

Cyclic superconducting refrigerators using guided fluxon propagation

We propose cyclic refrigeration in solid state, employing a gas of magnetic field vortices in a type-II superconductor—also known as fluxons—as the cooling agent. Refrigeration cycles are realized by envisioning a racetrack geometry consisting of both adiabatic and isothermal arms, etched into a type-II superconductor. The guided propagation of fluxons in the racetrack is achieved by applying an external electrical current, in a Corbino geometry, through the sample. A gradient of magnetic field is set across the racetrack allowing one to adiabatically cool down and heat up the fluxons, which subsequently exchange heat with the cold and hot reservoirs, respectively. We characterize the steady state of refrigeration cycles thermodynamically for both s-wave and d-wave pairing symmetries, and present their figures of merit such as the cooling power delivered, and the coefficient of performance. Our cooling principle can offer significant cooling for on-chip microrefrigeration purposes, by locally cooling below the base temperatures achievable in a conventional dilution refrigerator. We estimate nW/mm2 of cooling power per unit area assuming a tunnel coupling with ∼MΩµm2 specific resistance. Integrating the fluxon fridge to quantum circuits can enhance their coherence time by locally suppressing thermal fluctuations, and improve the efficiency of single photon detectors and charge sensors. Published by the American Physical Society 2024

Manifestation of peak-dip-hump structure in tunneling spectra of cuprates: a study by extended d-wave pairing symmetry

Manifestation of peak-dip-hump structure in tunneling spectra of cuprates: a study by extended d-wave pairing symmetry

Solving local constraint condition problem in slave particle theory with the BRST quantization

With the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theory, we solve the long-standing difficult problem of the local constraint conditions, i.e. the single occupation of a slave particle per site, in the slave particle theory. This difficulty is actually caused by inconsistently dealing with the local Lagrange multiplier λ i which ensures the constraint: in the Hamiltonian formalism of the theory, λ i is time-independent and commutes with the Hamiltonian while in the Lagrangian formalism, λ i (t) becomes time-dependent and plays a role of gauge field. This implies that the redundant degrees of freedom of λ i (t) are introduced and must be removed by the additional constraint, the gauge fixing condition (GFC) ∂ t λ i (t) = 0. In literature, this GFC was missed. We add this GFC and use the BRST quantization of gauge theory for Dirac’s first-class constraints in the slave particle theory. This GFC endows λ i (t) with dynamics and leads to important physical results. As an example, we study the Hubbard model at half-filling and find that the spinon is gapped in the weak U and the system is indeed a conventional metal, which resolves the paradox that the weak coupling state is a superconductor in the previous slave boson mean field (MF) theory. For the t–J model, we find that the dynamic effect of λ i (t) substantially suppresses the d-wave pairing gap and then the superconducting critical temperature may be lowered at least a factor of one-fifth of the MF value which is of the order of 1000 K. The renormalized T c is then close to that in cuprates.

Theory of Josephson current on a lattice model of grain boundary in d-wave superconductors

Identifying the origins of suppression of the critical current at grain boundaries of high-critical-temperature superconductors, such as cuprates and iron-based superconductors, is a crucial issue to be solved for future applications with polycrystalline materials. Although the dominant factor of current suppression might arise during material fabrication and/or processing, investigating it due to an internal phase change of the pair potential is an important issue in understanding the threshold of the critical current. In this paper, we study the Josephson current on a symmetric [001]-tilt grain boundary (GB) of a d-wave superconductor on a lattice model. In addition to the suppression of the maximum Josephson current associated with the internal phase change of the d-wave pair potential which has been predicted in continuum models, we find a unique phase interference effect due to folding of the Fermi surface in the lattice model. In particular, the resultant maximum Josephson current at low-tilting-angle regions tends to be suppressed more than that in preexisting theories. Because similar suppressions of the critical current at GBs have been reported in several experimental works, the present model can serve as a guide to clarify the complicated transport mechanism in GBs.

Effect of Anisotropic Impurity Scattering in D-Wave Superconductors

In d x 2–y 2 -wave superconductors, the effect of s-wave point disorder has been extensively studied in literature. We study the anisotropic disorder in the form of with fk = cos(2θ) (θ the azimuthal angle of k), as proposed to be caused by apical oxygen vacancies in overdoped La-based cuprate films, under the Born approximation. The disorder self-energy and d-wave pairing affect each other and must be solved simultaneously self-consistently. We find that the self-energy is reduced at low frequencies and thus weakens the pair-breaking effect. This frequency dependence vanishes in the dirty limit for which the disorder is well described by a scattering rate . One consequence of the disorder effect is that the gap-to-T c ratio 2Δ(0)/T c is greatly enhanced by the d-wave disorder, much larger than the s-wave disorder and the clean BCS value 4.28. Lastly, we generalize the d-wave scattering rate to a general form around each nodal direction θ 0. We find the density of states ρ(ω)–ρ(0)∝|ω| (ω 2) for all α ≥ 1 (α < 1) in the limit of ω → 0. As a result, the superfluid density ρ s exhibits two and only two possible scaling behaviors: ρ s(0)–ρ s(T)∝ T (T 2) for α ≥ 1 (α < 1) in the low temperature limit.

Anomalous Josephson effect between d-wave superconductors through a two-dimensional electron gas with both Rashba spin–orbit coupling and Zeeman splitting

Based on the Bogoliubov–de Gennes equation and the extended McMillan’s Green’s function formalism, we study theoretically the Josephson effect between two d-wave superconductors bridged by a ballistic two-dimensional electron gas with both Rashba spin–orbit coupling and Zeeman splitting. We show that due to the interplay of Rashba spin–orbit coupling and Zeeman splitting and d-wave pairing, the current–phase relation in such a heterostructure may exhibit a series of novel features and can change significantly as some relevant parameters are tuned. In particular, anomalous Josephson current may occur at zero phase bias under various different situations if both time reversal symmetry and inversion symmetry of the system are simultaneously broken, which can be realized by tuning some relevant parameters of the system, including the relative orientations and the strengths of the Zeeman field and the spin–orbit field in the bridge region, the relative orientations of the a axes in two superconductor leads, or the relative orientations between the Zeeman field in the bridge region and the a axes in the superconductor leads. We show that both the magnitude and the direction of the anomalous Josephson current may depend sensitively on these relevant parameters.

Josephson effect in twisted cuprates

&lt;sec&gt;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 &lt;i&gt;d&lt;/i&gt;-wave superconductors twisted by 45°. This is in sharp contrast to persistent tunneling between two &lt;i&gt;s&lt;/i&gt;-wave superconductors. Experimentally, however, Josephson tunneling is observed in twisted bicrystalline cuprates at around 45°, against the expectation that cuprate superconductors possess &lt;i&gt;d&lt;/i&gt;-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 &lt;i&gt;s&lt;/i&gt;-, &lt;i&gt;d&lt;/i&gt;-, or &lt;i&gt;d&lt;/i&gt; + i&lt;i&gt;d&lt;/i&gt;-wave pairing. In the end, we propose possible directions for future exploration in this field.&lt;/sec&gt;&lt;sec&gt; 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 &lt;i&gt;d&lt;/i&gt;-wave or &lt;i&gt;s-&lt;/i&gt;wave superconductors, we summarize the predicted features from the emergent &lt;i&gt;d&lt;/i&gt;+i&lt;i&gt;d-&lt;/i&gt;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 &lt;i&gt;s&lt;/i&gt;-wave pairing and strong deviation from pure &lt;i&gt;d&lt;/i&gt;-wave pairing. This result is contrasted with another study on cryogenically stacked junctions. There, signatures of &lt;i&gt;d&lt;/i&gt;+i&lt;i&gt;d&lt;/i&gt;-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 &lt;i&gt;d &lt;/i&gt;+ i&lt;i&gt;d&lt;/i&gt;-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.&lt;/sec&gt;

Surface state of the interorbital pairing state in the Sr2RuO4 superconductor

We study the (001) surface state of a recently proposed $E_g$ symmetry inter-orbital-odd spin-triplet s-wave superconducting (SC) state in Sr$_2$RuO$_4$ (SRO). We confirm that this pair potential is transformed into a chiral $d$-wave pair potential and a pseudo-Zeeman field in the band basis for a low-energy range. Due to the chiral $d$-wave pair potential, the surface states appear near zero energy in the momentum range enclosed by the nodal lines of the chiral d-wave pair potential for each band at the (001) surface. Nevertheless, the pseudo-Zeeman field gives band splitting of the surface states, and its splitting energy is much smaller than the SC energy gap. The local density of states (LDOS) at the (001) surface of the SC state has a pronounced peak structure at zero energy due to the surface states near zero energy when the order of the resolution is lower than the splitting energy. This peak structure is robust under perturbations, such as an orbital Rashba coupling or an $E_u$ SC pair potential at the surface.

Local Kekulé distortion turns twisted bilayer graphene into topological Mott insulators and superconductors

Magic-angle twisted bilayer graphene displays at different fillings of the four flat bands lying around the charge neutrality point a wealth of notable phases that include magnetic Chern insulators, whose magnetization is mostly of an orbital nature and contiguous superconducting domes. Such a rich phase diagram is explained through the positive interplay of Coulomb repulsion and the electron coupling to a twofold optical mode that corresponds to Kekul\'e distortions localized into the small AA stacked regions of the moir\'e supercells. A static distortion stabilizes, at any integer filling of the flat bands, valence-bond insulators that carry finite Chern number away from charge neutrality. Similarly, a dynamic distortion that resonates between the two lattice vibrations leads to resonating-valence-bond topological insulators with built-in chiral d-wave pairs that have finite Chern number equal to the angular momentum, and thus are prone to turn superconducting upon doping away from integer filling.

Thermodynamics of the pseudogap in cuprates

The key thermodynamic characteristics of the pseudogap state in cuprate superconductors are reviewed. These include YBa2Cu3O7−δ, Y0.8Ca0.2Ba2Cu3O7−δ, YBa2Cu4O8, Bi2Sr2CaCu2O8+δ, La2−xSrxCuO4, and Tl2Ba2CuO4. The electronic specific heat was extracted using a differential technique, and the evolution of the specific-heat coefficient γ and electronic entropy S as a function of temperature, doping, and magnetic field reveals a canonical behavior summarized by the following. The normal-state gap which opens in the pseudogap domain apparently remains open to the highest temperatures investigated. The gap decreases in magnitude with increasing doping and closes abruptly at a critical doping of p ≈ 0.19 holes/Cu, independent of temperature. In this picture, the pseudogap is separated from the pseudogap-free region of the phase diagram by a vertical line similar to the vertical line separating the incoherent and coherent antinodal quasiparticle states found in ARPES. The important role of fluctuations is evident by a diverging enhancement of γ(T) on either side of Tc, and this enables extraction of the mean-field transition temperature Tcmf&gt;Tc, defining a crescent of parapairing above Tc(p) which extends across the entire superconducting phase diagram and which is quite distinct from pseudogap phenomenology. The data are consistent with d-wave pairing and the BCS ratios are extracted, revealing canonical near-weak-coupling behavior across the over-doped region with a sudden suppression occurring at p ≈ 0.19 when the pseudogap sets in.

Phase diagrams for two-component D-wave fermion superfluids with density imbalance

Finite-temperature phase structures for asymmetric two-component D-wave fermion superfluids in the weak coupling BCS regime are investigated. The Fulde–Ferrell–Larkin-Ovchinnikov (FFLO) state becomes non-degenerate with the orientation of Cooper pair momentum due to the anisotropic pairing gap for D-wave pairing. The new parameter θ 0, corresponding to the orientation of Cooper pair momentum, greatly enriches the phase structures. Especially, the phase separation (PS) region in the phase diagram is significantly reduced comparing to that for S-wave pairing. Theoretically, the stability condition of solutions is equivalent to the positive definite Hessian matrix of free energy in the hyperplane defined by the number density constraints in the parameter space spanned by the average chemical potential μ, relative chemical potential h, paring gap Δ, pair momentum q, and θ 0. In addition, the stability condition of the superfluid density is essentially included in the positive definite Hessian matrix. Moreover, the positiveness of imbalance number susceptibility is always equivalent to the positiveness of the smaller eigenvalue λ − of the number susceptibility matrix. While the criteria ∂2Ω/∂Δ2 and against PS in previous studies are equivalent to λ − only under the prerequisite that the solution is a local minimum in the corresponding hyperplane.

Two distinctive temperature dependences of the London penetration depth in high-Tc cuprate superconductors as support for the theory of Bose-liquid superconductivity

In the framework of the theory of Bose-liquid superconductivity, we show that in high-Tc cuprate superconductors the magnetic field penetration depth (known as the London penetration depth λL(T)) decreases first exponentially below the superconducting transition temperature Tc with decreasing the temperature T and then λL(T) decreases like a certain power of T at lower temperatures due to the vanishing of the energy gap Δg(T) in the excitation spectrum of a superfluid Bose-liquid at a characteristic temperature Tc⁎ lower than Tc. Below Tc, the London penetration depth λL(T) has an exponential temperature dependence down to Tc⁎, which is different from a BCS- like exponential temperature dependence, and then λL(T) will follow a power law temperature dependence down to T=0, which is also different from power-law temperature dependences predicted by some BCS-like (d-wave) pairing theories. In unconventional (bosonic) cuprate superconductors characterized by the strong interboson coupling constant (γB≳1), λL(T) exhibits the distinctly different temperature dependences in different temperature intervals 0≤T≤Tc⁎<<Tc and Tc⁎<T<Tc when the energy gap Δg(T) vanishes at a temperature Tc⁎ well below Tc. In the weak interboson coupling (γB<<1) the energy gap Δg(T) vanishes at a temperature Tc⁎ close to Tc and the reduced London penetration depth λL(T)/λL(0) has an unusual Tn power-law temperature dependence in a wide temperature range 0<T<Tc⁎ below Tc, i.e. the expression λL(T)/λL(0)≃[1−aL(T)(T/Tc)n]−1/2 (where n≳4 and aL(T)≲1) is valid only below Tc⁎ and is similar to the well-known empirical Gorter-Casimir law. Our theoretical predictions of the distinctive power-law and exponential temperature dependences of λL(T)/λL(0) are in good agreement with the experimental results reported for GdBa2Cu3O7−δ ceramics and YBa2Cu3O7−δ films. The peculiar temperature dependences of λL(T)/λL(0) observed in these high-Tc materials at low and high temperatures do not follow the predictions of the s- and d-wave BCS-like theories of Fermi-liquid superconductivity and give evidence for the predictions of the theory of Bose-liquid superconductivity.

Dynamical magnetic response in superconductors with finite-momentum pairs

We derive the dynamical magnetic response functions in the Fulde-Ferrell (FF) state of a superconductor with inversion symmetry. The pair momentum 2q is obtained by minimization of the condensation energy and the resulting quasiparticle states and spectral functions exhibit the segmentation into paired and unpaired regions due to the finite q. The dynamical magnetic susceptibility is then calculated in linear response formalism in the FF state with finite-q condensate resulting from s-wave or d-wave pairing. We show that quasiparticle excitations inside as well as between paired and unpaired segments contribute to the dynamical response. We discuss its dependence on frequency and momentum transfer which develops a characteristic symmetry-breaking parallel to q. Furthermore, we investigate the possible influence on Knight shift and in the case of d-wave pairing on the spin resonance formation in the FF state.

Effect of hopping anisotropy on the d-wave pairing in the Hubbard model: From two-leg ladder to square lattice

In the ladder-type cuprates, superconductivity is argued to be enhanced by increasing the electron hopping in the rung direction, while such an enhancement is not expected in the layered cuprates. Utilizing the quantum Monte Carlo calculations of the Hubbard model, we study the effect of the hopping anisotropy on the d-wave pairing, on the two-leg ladder, the square lattice and the coupled two-leg ladder lattice. On the square lattice, the hopping anisotropy monotonically suppresses the d-wave pairing; while on the two-leg ladder, an optimal hopping anisotropy for the d-wave pairing exists at tx/ty≈0.6. For the coupled two-leg ladder lattice, as the coupling strength ty′/ty decreases, the evolution of the d-wave pairing as a function of the hopping anisotropy behaves as a crossover from the square lattice to the two-leg ladder. Our results provide numerical proof for the possible enhancement of superconductivity in both the ladder-type and layered cuprates.

Enhanced d-wave pairing in the two-dimensional Hubbard model with periodically modulated hopping amplitudes

Utilizing determinant quantum Monte Carlo algorithm, the evolution of the d-wave pairing in the Hubbard model on the square lattice tuned by the periodically modulated hopping amplitudes is studied. The hopping amplitudes are homogeneous in the -direction, while in the -direction the hopping amplitudes are modulated with period P, where , , and the modulation period P equals 2, 3 and 4 lattice spacings. The latter two modulation periods are motivated by the observation of period-3 and period-4 stripe order in cuprate superconductors. For all the periods P, we find that the modulated hopping inhomogeneity enhances the d-wave pairing and an optimal inhomogeneity exists.

Interplay of magnetism and superconductivity in 2D extended Hubbard model

The interplay between magnetic and superconducting states on a square lattice is studied using the extended Hubbard model, which takes into account the attraction of electrons located at nearest neighbor sites. Ferro-, antiferro-, and spiral magnetic states with all possible sets of spiral wave vectors, as well as singlet superconducting states with s-wave and d-wave pairing order parameters, are considered. Formation of a state with the intermediate s+id-symmetry and phase separation between different phases are allowed. The results of the Hartree–Fock and slave boson approximations are compared in order to study the role of correlation effects. Both macroscopic phase separation and microscopic coexistence of the superconducting and magnetic phases are found under certain model parameters.

Diagnosis of pairing symmetry by vortex and edge spectra in kagome superconductors

Layered kagome metals AV3Sb5 (A=K, Rb, Cs) exhibit diverse correlated electron phenomena. It includes charge density wave formation and superconductivity the pairing symmetry of which, however, is controversial due to contradictory experimental evidence. Through calculations based on real-space lattice models at the mean-field level, we investigate the vortex and surface spectra of all competitive pairing propensities suggested for AV3Sb5 from a weak coupling analysis of unconventional superconductivity. Chiral p-wave pairing emerges as the only option to host Majorana bound states in the vortex core. We find chiral edge states for both p-wave and d-wave pairing, along with flat Andreev surface bound states for f -wave pairing. Our results expand the fingerprint of superconducting pairing, and thus will contribute to resolving the nature of superconductivity in AV3Sb5.

Nematic superconductivity in magic-angle twisted bilayer graphene from atomistic modeling

Twisted bilayer graphene (TBG) develops large moiré patterns at small twist angles with flat energy bands hosting domes of superconductivity. The large system size and intricate band structure have however hampered investigations into the superconducting state. Here, using full-scale atomistic modelling with local electronic interactions, we find at and above experimentally relevant temperatures a highly inhomogeneous superconducting state with nematic ordering on both atomic and moiré length scales. The nematic state has a locally anisotropic real-valued d-wave pairing, with a nematic vector winding throughout the moiré pattern, and is three-fold degenerate. Although d-wave symmetric, the superconducting state has a full energy gap, which we tie to a π-phase interlayer coupling. The superconducting nematicity is further directly detectable in the local density of states. Our results show that atomistic modeling is essential and also that very similar local interactions produce very different superconducting states in TBG and the high-temperature cuprate superconductors.