Cooper Pair Wave Function Research Articles

Evidence for vertical line nodes in Sr2RuO4 from nonlocal electrodynamics

By determining the superconducting lower and upper critical fields Hc1(T) and Hc2(T), respectively, in a high-purity spherical Sr2RuO4 sample via ac-susceptibility measurements, we obtain the temperature dependence of the coherence length ξ and the penetration depth λ down to 0.04Tc. Given the high sample quality, the observed T2 dependence of λ at low temperatures cannot be explained in terms of impurity effects. Instead, we argue that the weak type-II superconductor Sr2RuO4 has to be treated in the nonlocal limit. By comparing our data with existing theory in that limit, the penetration depth in Sr2RuO4 agrees with a gap structure having vertical line nodes, while horizontal line nodes cannot account for the observation. The work highlights the potential benefits of purifying other unconventional superconductors in order to access the fascinating nonlocal regime in more materials and to determine their Cooper pair wave functions. Published by the American Physical Society 2024

Engineering low-temperature proximity effect in clean metals by spectral singularities

The present study investigates the behavior of the Cooper pair wave function in a normal metal (NM) near superconductor-NM-junctions, specifically focusing on the ballistic regime at zero temperature. It is widely assumed that the wave function follows a power-law decay, with the decay exponents dependent on the system’s dimensionality. Our work reveals that the multiband nature of a compound significantly influences the damping degree of pair amplitudes in an NM, rendering it sensitive to the position of the Fermi level. To explore this phenomenon, we employ the numerical method of self-consistent Bogoliubov–de Gennes equations, utilizing a nanowire as a model for an electronic multiband system. By analyzing the obtained pair amplitudes, we extract relevant lengths and exponents that characterize the leakage of superconducting correlations. We further examine this phenomenon by varying the sample’s cross-sectional size and the superconducting coupling constant. Consequently, our findings demonstrate that the properties of a superconducting/NM junction’s proximity effect can be manipulated not only through temperature, total impurity and defect density, but also by controlling the position of the Fermi level. This tunability enables the transition from a long-range regime to a short-range one, providing valuable insights for designing and understanding such junctions in practical applications.

Boundary superconductivity in the BCS Model

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg–Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

Multiband Superconductors: Two Characteristic Lengths for Each Contributing Condensate.

The interference of multiple condensates coexisting in one system may lead to unconventional coherent behavior. This is expected when the spatial lengths of the condensates are essentially different. Traditionally, the characteristic spatial length of a superconducting condensate is associated with the gap function. However, the broader readership is more familiar with the concept of the Cooper-pair wave function. For conventional single-band superconductors, the gap function coincides with the center-of-mass Cooper-pair wave function up to the coupling constant, and the corresponding gap and wave function characteristic lengths are the same. Surprisingly, we find that in two-band superconductors, these lengths are the same only near the critical temperature. At lower temperatures, they can significantly deviate from each other, and the fundamental question of which of these lengths should be preferred when specifying the spatial scale of a band-dependent condensate in multiband superconducting materials arises.

Time reversal symmetry and the structure of Cooper pair wavefunction in topological superconductor UTe2

Recently discovered topological superconductor UTe2 reveals unusual symmetry features, namely, chiral nonunitary triplet ESP (equal spin pairing) SOP (superconducting order parameter) with point nodes. However, the description of these symmetry features in a framework of crystal point group D2h is not sufficient. In the present work the space-group approach to the wavefunction of a Cooper pair, based on Mackey-Bradley theorem, is extended, making use of Anderson ansatz, to the pair functions in Shubnikov group symmetry and applied to UTe2. A nonunitary triplet SOP was obtained for Shubnikov group m′m′m on the basis of one-electron functions. Irreducible corepresentations of group m′m′m for triplet ESP Cooper pairs, whose nodal structure corresponds to all experimentally established symmetry features, namely point nodes, nonunitarity and chirality, of SOP of UTe2 are identified.

Electronic structure and superconductivity of the non-centrosymmetric Sn4As3

In a superconductor that lacks inversion symmetry, the spatial part of the Cooper pair wave function has a reduced symmetry, allowing for the mixing of spin-singlet and spin-triplet Cooper pairing channels and thus providing a pathway to a non-trivial superconducting state. Materials with a non-centrosymmetric crystal structure and with strong spin–orbit coupling are a platform to realize these possibilities. Here, we report the synthesis and characterisation of high quality crystals of Sn4As3, with non-centrosymmetric unit cell (R3m). We have characterised the normal and superconducting states using a range of methods. Angle-resolved photoemission spectroscopy shows a multiband Fermi surface and the presence of two surface states, confirmed by density-functional theory calculations. Specific heat measurements reveal a superconducting critical temperature of Tc ∼ 1.14 K and an upper critical magnetic field of μ0Hc ≳ 7 mT, which are both confirmed by ultra-low temperature scanning tunneling microscopy and spectroscopy. Scanning tunneling spectroscopy shows a fully formed superconducting gap, consistent with conventional s-wave superconductivity.

Proximity-Induced Superconductivity in Monolayer MoS2.

Proximity effects in superconducting normal (SN) material heterostructures with metals and semiconductors have long been observed and theoretically described in terms of Cooper pair wave functions and Andreev reflections. Whereas the semiconducting N-layer materials in the proximity experiments to date have been doped and tens of nanometers thick, we present here a proximity tunneling study involving a pristine single-layer transition-metal dichalcogenide film of MoS2 placed on top of a Pb thin film. Scanning tunneling microscopy and spectroscopy experiments together with parallel theoretical analysis based on electronic structure calculations and Green's function modeling allow us to unveil a two-step process in which MoS2 first becomes metallic and then is induced into becoming a conventional s-wave Bardeen-Cooper-Schrieffer-type superconductor. The lattice mismatch between the MoS2 overlayer and the Pb substrate is found to give rise to a topographic moiré pattern. Even though the induced gap appears uniform in location, the coherence peak height of the tunneling spectra is modulated spatially into a moiré pattern that is similar to but shifted with respect to the moiré pattern observed in topography. The aforementioned modulation is shown to originate from the atomic-scale structure of the SN interface and the nature of local atomic orbitals that are involved in generating the local pairing potential. Our study indicates that the local modulation of induced superconductivity in MoS2 could be controlled via geometrical tuning, and it thus shows promise toward the integration of monolayer superconductors into next-generation functional electronic devices by exploiting proximity-effect control of quantum phases.

Induced Representation Method in the Theory of Electron Structure and Superconductivity

It is shown that the application of theorems of induced representations method, namely, Frobenius reciprocity theorem, transitivity of induction theorem, and Mackey theorem on symmetrized squares, makes simplifying standard techniques in the theory of electron structure and constructing Cooper pair wavefunctions on the basis of one-electron solid-state wavefunctions possible. It is proved that the nodal structure of topological superconductors in the case of multidimensional irreducible representations is defined by additional quantum numbers. The technique is extended on projective representations in the case of nonsymmorphic space groups and examples of applications for topological superconductors UPt3 and Sr2RuO4 are considered.

Group Theoretical Lines of Nodes in Triplet Chiral Superconductor Sr2RuO4

Cooper pair wavefunctions for chiral triplet (p-wave) superconductor Sr2RuO4 were considered making use of the spin-space group approach, and the states similar to A and B phases of 3He belonging to IRs (irreducible representations) of D4h group are constructed making use of projection operator technique. Group theoretical lines of nodes were obtained for space-group analog of B phase. In this case nodal and nodeless one-dimensional states belong to the same IR, and differ by spin projection on vertical plane. Four chiral states, similar to A phase and belonging to complex IR Eu, are classified by the label of one-dimensional even IR. These states are non-unitary and differ by nodal structures and momentum projections on vertical axis. It was obtained that for each vertical plane of symmetry nodal and nodeless chiral states of Eu symmetry are possible. This means that there are no so called symmetry protected lines of nodes on vertical planes for chiral Eu states, but experimental vertical lines of nodes in Sr2RuO4 do not contradict with general group theoretical consideration. The results are in agreement with experimental nodal structure and may be used as alternative initial basis in microscopic and phenomenological theories of superconductivity in Sr2RuO4.

Localization of pairing correlations in nuclei within relativistic mean field models

We analyze the localization properties of two-body correlations induced by pairing in the framework of relativistic mean field (RMF) models. The spatial properties of two-body correlations are studied for the pairing tensor in coordinate space and for the Cooper pair wave function. The calculations are performed both with Relativistic-Hatree-Bogoliubov (RHB) and RMF+Projected-BCS (PBCS) models and taking as examples the nuclei $^{66}$Ni, $^{124}$Sn and $^{200}$Pb. It is shown that the coherence length have the same pattern as in previous non-relativistic HFB calculations, i.e., it is maximum in the interior of the nucleus and drops to a minimum in the surface region. In the framework of RMF+PBCS we have also analysed, for the particular case of $^{120}$Sn, the dependence of the coherence length on the intensity of the pairing force. This analysis indicates that pairing is reducing the coherence length by about 25-30 $\%$ compared to the RMF limit.

Vestigial nematic order and superconductivity in the doped topological insulator CuxBi2Se3

If the topological insulator Bi2Se3 is doped with electrons, superconductivity with Tc = 3–4 K emerges for a low density of carriers (n = 1020 cm−3) and with a small ratio of the superconducting coherence length and Fermi wave length: ξ/λF = 2…4. These values make fluctuations of the superconducting order parameter increasingly important, to the extend that the Tc-value is surprisingly large. Strong spin–orbit interaction led to the proposal of an odd-parity pairing state. This begs the question of the nature of the transition in an unconventional superconductor with strong pairing fluctuations. We show that for a multi-component order parameter, these fluctuations give rise to a nematic phase at Tnem > Tc. Below Tc several experiments demonstrated a rotational symmetry breaking where the Cooper pair wave function is locked to the lattice. Our theory shows that this rotational symmetry breaking, as vestige of the superconducting state, already occurs above Tc. The nematic phase is characterized by vanishing off-diagonal long range order, yet with anisotropic superconducting fluctuations. It can be identified through direction-dependent para-conductivity, lattice softening, and an enhanced Raman response in the Eg symmetry channel. In addition, nematic order partially avoids the usual fluctuation suppression of Tc.

Quasiclassical theory for the superconducting proximity effect in Dirac materials

We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations to first order in quantities small compared to the Fermi energy, valid for Dirac edge and surface electrons with spin-momentum locking, as relevant for topological insulators. We discuss in detail several of the key technical points and assumptions of the derivation, and provide a Riccati-parametrization of the equations. Solving first the equilibrium equations for S/N and S/F bilayers and Josephson junctions, we study the superconducting proximity effect in Dirac materials. Similarly to related works, we find that the effect of an exchange field depends strongly on the direction of the field. Only components normal to the transport direction lead to attenuation of the Cooper pair wavefunction inside the F. Fields parallel to the transport direction lead to phase-shifts in the dependence on the superconducting phase difference for both the charge current and density of states in an S/F/S-junction. Moreover, we compute the differential conductance in S/N and S/F bilayers with an applied voltage bias, and determine the dependence on the length of the N and F regions and the exchange field.

Spin-Triplet Pairing Induced by Spin-Singlet Interactions in Noncentrosymmetric Superconductors

In noncentrosymmetric superconductors, we examine the effect of the difference between the intraband and interband interactions, which becomes more important when the band splitting increases. We define the difference ΔVμ between their coupling constants, i.e., that between the intraband and interband hopping energies of intraband Cooper pairs. Here, the subscript μ of ΔVμ indicates that the interactions scatter the spin-singlet and spin-triplet pairs when μ = 0 and μ = 1,2,3, respectively. It is shown that the strong antisymmetric spin–orbit interaction reverses the target spin parity of the interaction: it converts the spin-singlet and spin-triplet interactions represented by ΔV0 and ΔVμ>0 into effective spin-triplet and spin-singlet pairing interactions, respectively. Hence, for example, triplet pairing can be induced solely by the singlet interaction ΔV0. We name the pairing symmetry of the system after that of the intraband Cooper pair wave function, but with an odd-parity phase factor excluded. The ...

Giant Mesoscopic Fluctuations and Long-Range Superconducting Correlations in Superconductor-Ferromagnet Structures.

The fluctuating superconducting correlations emerging in dirty hybrid structures under the conditions of the strong proximity effect are demonstrated to affect the validity range of the widely used formalism of Usadel equations at mesoscopic scales. In superconductor-ferromagnet structures these giant mesoscopic fluctuations originating from the interference effects for the Cooper pair wave function in the presence of the exchange field can be responsible for an anomalously slow decay of superconducting correlations in a ferromagnet even when the noncollinear and spin-orbit effects are negligible. The resulting sample-to-sample fluctuations of the Josephson current in superconductor-ferromagnetic-superconductor junctions and the local density of states in superconductor-ferromagnetic hybrid structures can provide an explanation of the long-range proximity phenomena observed in mesoscopic samples with collinear magnetization.

Spin-polarized neutron matter: Critical unpairing and BCS-BEC precursor

We obtain the critical magnetic field required for complete destruction of $S$-wave pairing in neutron matter, thereby setting limits on the pairing and superfluidity of neutrons in the crust and outer core of magnetars. We find that for fields $B \ge 10^{17}$ G the neutron fluid is non-superfluid -- if weaker spin-1 superfluidity does not intervene -- a result with profound consequences for the thermal, rotational, and oscillatory behavior of magnetars. Because the dineutron is not bound in vacuum, cold dilute neutron matter cannot exhibit a proper BCS-BEC crossover. Nevertheless, owing to the strongly resonant behavior of the $nn$ interaction at low densities, neutron matter shows a precursor of the BEC state, as manifested in Cooper-pair correlation lengths {being} comparable to the interparticle distance. We make a systematic quantitative study of this type of BCS-BEC crossover in the presence of neutron fluid spin-polarization induced by an ultra-strong magnetic field. We evaluate the Cooper pair wave-function, quasiparticle occupation numbers, and quasiparticle spectra for densities and temperatures spanning the BCS-BEC crossover region. The phase diagram of spin-polarized neutron matter is constructed and explored at different polarizations.

Nuclear pairing from two-body microscopic forces: analysis of the Cooper pair wavefunctions

In a recent paper we studied the behavior of the pairing gaps $\Delta_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations employed realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. In this contribution, after a detailed description of the numerical method we employed in the solution of the BCS equations, we will show a preliminary analysis of the Cooper pair wavefunctions.

Nuclear pairing from two-body microscopic forces: analysis of the Cooper pair wavefunctions

In a recent paper we studied the behavior of the pairing gaps $\Delta_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations employed realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. In this contribution, after a detailed description of the numerical method we employed in the solution of the BCS equations, we will show a preliminary analysis of the Cooper pair wavefunctions.

BCS-BEC crossovers and unconventional phases in dilute nuclear matter

The authors examine the phase diagram of neutron-rich nuclear matter, allowing for four competing phases. The resulting phase diagram features two tricritical points as well as crossovers from the BCS phase to a Bose-Einstein deuteron condensate plus a neutron gas. The physics of the individual phases and the transition from weak to strong coupling is traced by examining the Cooper-pair wave function and related quantities.

Asymptotic form of neutron Cooper pairs in weakly bound nuclei

Asymptotic form of neutron Cooper pair penetrating to the exterior of nuclear surface is investigated with the Bogoliubov theory for the superfluid Fermions. Based on a two-particle Schr\"{o}dinger equation governing the Cooper pair wave function and systematic studies for both weakly bound and stable nuclei, the Cooper pair is shown to be spatially correlated even in the asymptotic large distance limit, and the penetration length of the pair condensate is revealed to be universally governed by the two-neutron separation energy $S_{2n}$ and the di-neutron mass $2m$.

Zero-bias conductance peak in detached flakes of superconducting 2H-TaS2probed by scanning tunneling spectroscopy

We report an anomalous tunneling conductance with a zero bias peak in flakes of superconducting 2H-TaS$_2$ detached through mechanical exfoliation. To explain the observed phenomenon, we construct a minimal model for a single unit cell layer of superconducting 2H-TaS$_2$ with a simplified 2D Fermi surface and sign-changing Cooper pair wavefunction induced by Coulomb repulsion. Superconductivity is induced in the central $\Gamma$ pocket, where it becomes nodal. We show that weak scattering at the nodal Fermi surface, produced by non-perturbative coupling between tip and sample, gives Andreev states that lead to a zero bias peak in the tunneling conductance. We suggest that reducing dimensionality down to a few atom thick crystals could drive a crossover from conventional to sign changing pairing in the superconductor 2H-TaS$_2$.