A BCS (Bardeen-Cooper-Schrieffer) superconductor, which is placed out of equilibrium, can develop quantum instabilities, which manifest themselves in oscillations of the superconductor's order parameter (pairing amplitude $Δ$). These instabilities are a manifestations of the Cooper instability. Inelastic collisions are essential in resolving those instabilities. Incorporating the quantum instabilities and collisions in a unified approach based on Richardson's exact solution of the pairing Hamiltonian, we find that a BCS superconductor may end up in a state in which the spectrum has more than one gap.
We develop a squeezed-field path-integral representation for BCS superconductors utilizing a generalized completeness relation of squeezed-fermionic coherent states. We derive a Grassmann path integral of fermionic quasiparticles that explicitly includes the collective degrees of freedom of the order-parameter dynamics governed by the classical Anderson pseudospin model. Based on this method, we analyze the spectral function of the single-particle excitations, and show that the squeezed-field path integral for the BCS Hamiltonian describes a bosonic sideband branch that corresponds to the Higg…
We derive effective dynamical theories for metals and BCS superconductors, based on the effective action formalism. Both the metallic regime $T\gtrsim T_{C}$ and the superconducting regime $T\ll T_{C}$ are studied in the clean and dirty limit. Furthermore, we consider the effect of particle-hole asymmetry in the band structure. Using gauge invariance, the electrodynamics of the problem is formulated in a transparent way. The effective actions are useful starting points for treating dynamical problems involving BCS superconductors.
For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only h…
We investigate the physics of an adiabatic transition from a BCS superconductor to a Fermi liquid for an exponentially slow decreasing pairing interaction. In particular, we show that the metal keeps memory of the parent BCS state so it is possible to reverse the dynamics and go back to the original state similarly to a spin/photon echo experiment. Moreover, we study the evolution of the order parameter phase phi in transforming the BCS superconductor to a conventional metal. Since the global phase is the conjugate variable of the density we explicitly show how to use the dynamics of phi toget…
We discuss the evolution from BCS to Bose superconductivity versus carrier density in gated two-dimensional s-wave superconductors. We show that the transition from the normal to the superconducting state is controlled by the Berezinskii-Kosterlitz-Thouless vortex-antivortex pairing mechanism, and that the evolution from high carrier density (BCS pairing) to low carrier density (Bose pairing) is just a crossover in s-wave systems. We compare our results to recent experiments on the superconductor LixZrNCl, a lithium intercalated layered nitride, and obtain very good quantitative agreement at l…
Ultrafast responses of BCS superconductor Nb1-xTixN films in a nonadiabatic excitation regime were investigated by using terahertz (THz) pump-THz probe spectroscopy. After an instantaneous excitation with the monocycle THz pump pulse, a transient oscillation emerges in the electromagnetic response in the BCS gap energy region. The oscillation frequency coincides with the asymptotic value of the BCS gap energy, indicating the appearance of the theoretically-anticipated collective amplitude mode of the order parameter, namely the Higgs amplitude mode. Our result opens a new pathway to the ultraf…
Based on a simple model of a layered superconductor with strong attractive interaction, we find that the separation of the pair-condensation temperature from the pair-formation temperature becomes more remarkable as the interlayer hopping gets smaller. We propose from this result the BCS-BEC crossover induced by the change in dimensionality, for instance, due to insertion of additional insulating layers or application of uniaxial pressure. The emergence of a pseudogap in the electronic density of states, which supports the idea of the dimensionality-induced BCS-BEC crossover, is also verified.
Within the weak-coupling BCS scheme we derive a general form of the coefficients in the Ginzburg-Landau expansion of the free energy of a superconductor for the case of a Fermi level close to a van Hove singularity (VHS). A simple expression for the influence of the VHS on the specific heat jump is then obtained for the case where gaps for different bands are distinct but nearly constant at the corresponding sheets of the Fermi surface.
We consider the interface between a Bardeen-Cooper-Schrieffer superconductor and non-superconducting band insulator. We show that under certain conditions, such interfaces can have an elevated superconducting critical temperature, without increasing the strength of the pairing interaction at the interface. We identify the regimes where the interface critical temperature exceeds the critical temperature associated with a superconductor-vacuum interface.
We develop a description of the mixed state of type II superconductivity valid within a wide range of temperatures and external magnetic fields. It is based on the quasiclassical version of microscopic BCS theory and employs an expansion in the "distance" from the Hc2(T) line. We prove that in a clean metal with spherical Fermi surface and isotropic pairing interaction the superconducting condensation always produces a hexagonal vortex lattice.
The Bardeen-Cooper-Schrieffer (BCS) condensation and the Bose-Einstein condensation (BEC) are the two extreme limits of the ground state of the paired fermion systems. We report crossover behavior from the BCS condensation to the BEC realized in the two-dimensional (2D) superconductor, electron doped layered material ZrNCl. The phase diagram, established by simultaneous experiments of resistivity and tunneling spectra under the ionic gating, demonstrates the pseudogap phase at the low doping regime. In the low carrier density limit, $T_{\rm BKT}$ (Berezinskii-Kosterlitz-Thouless transition tem…
A universal relation connecting Fermi surface (FS) to the symmetry of the gap function in BCS-like superconductors is derived. It is found that the shape of the FS can be deduced directly from the symmetry of the superconducting gap function, and is also influenced by the next nearest-neighbor overlapping. The application of this relation to cuprate superconductors is discussed. There is observed an interesting property that Luttinger's theorem perfectly holds for the tight-binding band while it is violated by inclusion of the next nearest-neighbor overlapping integral.
In three-band BCS superconductors with repulsive interband interactions, frustration between the bands can lead to an inherently complex gap function, arising out of a phase difference between the bands in the range 0 and π. Since the complex conjugate of this state is also a solution, the ground state is degenerate, and there appears a time-reversal-symmetry-broken state. In this paper we investigate the existence of this state as a function of interband coupling strength and show how a new phase transition appears between the TRSB and conventional BCS states.
In a superconductor electrons form pairs for which the end states of normal and umklapp scattering may overlap. This cuts electron pairing off at a phonon frequency, $ω_c$, low compared with the Debye frequency, $ω_D$. Using this insight, correct values of $2Δ/k_BT_c$ (average error 8.9%) for 12 superconductive metals, including Hg and Pb, are achieved from simple BCS formalism with an average $ω_c/ω_D$ of 0.148: Landau's idea of a Fermi liquid may cover strong-coupling superconductors. The cancellation between normal and umklapp scattering may be more important than a strong electron-phonon i…
The dissipation process in two dimensional BCS superconductor due to the quasiparticles in the core of moving vortex is studied from a quantum mechanical point of view, especially paying attention to the spectral flow of the bound state energies in the vortex core. In order to clarify the nature of the spectral flow, we performed a numerical study of a finite system and, by extending the analysis to include the effect of impurities, we discuss the quantum mechanical origin of the dissipation in the vortex core.
It is analyzed what fundamental new information for the properties of the superconductors can be obtained by systematic investigation of the Bernoulli effect. It is shown that it is a tool to determine the effective mass of Cooper pairs, the volume density of charge carriers, the temperature dependence of the penetration depth and condensation energy. The theoretical results for disordered and anisotropic gap superconductors are systematized for this aim. For clean-anisotropic-gap superconductors is presented a simple derivation for the temperature dependence of the penetration depth
We study order parameter fluctuation effects in the superconducting state as a possible precursor to the pseudogap phenomena. Using a low-energy effective theory in the d-wave BCS model, we self-consistently calculate the single-particle properties. We find that the fluctuations reduce the spectral gap, and the shape of the gap is deformed, showing reduced slope of the gap at the nodes. We also show the angle-dependence of the quasiparticle lifetime due to the fluctuations.
The transport of heat and charge in the overdoped cuprate superconductor Tl_2Ba_2CuO_(6+delta) was measured down to low temperature. In the normal state, obtained by applying a magnetic field greater than the upper critical field, the Wiedemann-Franz law is verified to hold perfectly. In the superconducting state, a large residual linear term is observed in the thermal conductivity, in quantitative agreement with BCS theory for a d-wave superconductor. This is compelling evidence that the electrons in overdoped cuprates form a Fermi liquid, with no indication of spin-charge separation.
Motivated by the recent experiment of the non-BCS scaling relation of the condensation energy $ΔCE$ vs. $T_c$ ($ΔCE \sim T_c ^β, β\approx 3.5$) [PRB 89 140503 (2014)] for the Fe-based superconductors, we studied the CE and $T_c$ of the multiband BCS superconductors. We showed that the experimentally observed anomalous scaling relation $ΔCE \sim T_c ^{3.5}$ is well reproduced by the two-band BCS superconductor paired by a dominant interband interaction ($V_{inter} > V_{intra}$). Our result implies that this seemingly non-BCS-like scaling behavior, on the contrary to the common expectation, is i…
According to the BCS model the probability of filling of pair electron states vk2 as a function of kinetic energy is smeared i.e. is not a step function. This means that part of energy of connection through virtual phonons is consumed to increase the kinetic energy in an electron system at the transition to the superconducting state. In this work a modified mechanism of vk2 formation is proposed which takes into account the dynamics of fluctuations near the phase transition. The estimations of superconducting gap and isotopic effect for a matrial with parameters of high-temperature superconduc…
In this paper we study a model of s-wave marginal Fermi liquid superconductor at two dimensions. Besides the usual Bardeen-Cooper-Schrieffer(BCS) point attractive interaction, the fermions in our system also interact {\em via} a long range repulsive interaction $v(q)=g_{D}/q^2$. We find that although superconductivity is not destroyed by the long-ranged repulsion interaction (demonstrated by the existence of Meissner effect), nevertheless the BCS pairing function $Δ_{BCS}$ is strongly suppressed resulting in power-law correlation $<Δ_{BCS}(\vec{x})Δ_{BCS}(0)>\sim|\vec{x}|^{-2γ}$. This observat…
We review the magnetic form factor deduced by Delrieu from the Gorkov's equation for a Bardeen-Cooper-Schrieffer (BCS) type-II superconductor near its Bc2 phase boundary, i.e. when its magnetization is small. A numerical study of the form factor, field map, and field distribution follows. The characteristics of the transition from the low-temperature BCS to the high-temperature Ginzburg-Landau vortex lattices is studied. The exotic shape of the component field distribution and the form factor at low temperature and as a function of the external field intensity are discussed. Our numerical work…
The results of the muon-spin rotation experiments on BCS superconductors NbB_2+x (x = 0.2, 0.34) are reported. Both samples, studied in the present work, exhibit rather broad transitions to the superconducting state, suggesting a distribution of the volume fractions with different transition temperatures (T_c)'s. By taking these distributions into account, the dependence of the inverse squared zero-temperature magnetic penetration depth (λ_0^{-2}) on T_c was reconstructed for temperatures in the range 1.5K<T_c<8.0K. λ_0^{-2} was found to obey the power law dependence λ_0^{-2}\propto T_c^{3.1(1…
We reveal that due to the presence of disorder oscillations of the order parameter amplitude called the Higgs mode can be effectively excited by the external electromagnetic radiation in usual BCS superconductors. This mechanism works for superconductors with both isotropic s-wave and anisotropic, such as d-wave, pairings. The non-linear response in the presence of impurities is captured by the quasiclassical formalism. We demonstrate that analytical solutions of the Eilenberger equation with impurity collision integral and external field drive coincide with the exact summation of ladder impur…
We performed high-resolution angle-resolved photoemission spectroscopy on triple-layered high-Tc cuprate Bi2Sr2Ca2Cu3O10+delta. We have observed the full energy dispersion (electron and hole branches) of Bogoliubov quasiparticles and determined the coherence factors above and below EF as a function of momentum from the spectral intensity as well as from the energy dispersion based on BCS theory. The good quantitative agreement between the experiment and the theoretical prediction suggests the basic validity of BCS formalism in describing the superconducting state of cuprates.
According to the BCS theory the superconducting condensate develops in a single quantum mode and no Cooper pairs out of the condensate are assumed. Here we discuss a mechanism by which the successful mode inhibits condensation in neighboring modes and suppresses a creation of noncondensed Cooper pairs. It is shown that condensed and noncondensed Cooper pairs are separated by an energy gap which is smaller than the superconducting gap but large enough to prevent nucleation in all other modes and to eliminate effects of noncondensed Cooper pairs on properties of superconductors. Our result thus …
We consider a superconductor hosting mobile impurities, which locally change the superconducting transition temperature. The BCS interaction at the impurity is different both in magnitude and in sign from the BCS interaction in the bulk. It is shown that due to the attraction between impurities, they tend to form more condensed state. We also consider the distribution of mobile impurities with local BCS attraction or repulsion at the vicinity of the superconductor-normal metal interface.
Although cuprate superconductors have been intensively studied for the past decades, there is no consensus regarding the microscopic origin of their superconductivity. In this work, we measure the low-energy electrodynamic response of slightly underdoped and overdoped La$_{2-x}$Ce$_x$CuO$_4$ thin films using time-domain terahertz (THz) spectroscopy to determine the temperature and field dependence of the superfluid spectral weight. We show that the temperature dependence obeys the relation \textit{n$_s$} $\propto$ $1-(T/T_c)^2$, typical for dirty limit BCS-like $d$-wave superconductors. Furthe…
In this review article we consider theoretically and give experimental support to the models of the Fermi-Bose mixtures and the BCS-BEC crossover compared with the strong-coupling approach, which can serve as the cornerstones on the way from high-temperature to room-temperature superconductivity in pressurized metallic hydrides. We discuss some key theoretical ideas and mechanisms proposed for unconventional superconductors (cuprates, pnictides, chalcogenides, bismuthates, diborides, heavy-fermions, organics, bilayer graphene, twisted graphene, oxide hetero-structures), superfluids and balance…
We present a unified description of a junction between $s$-wave (BCS) superconductors and a junction between $p$-wave superconductors in a topologically nontrivial phase, which relies on a scattering state expansion. We compute Josephson current and thermal noise in the two kinds of junction and exhibit some characteristic features for a junction of two topological superconductors hosting Majorana zero-energy modes.
Superconductivity in iron-based, magnesium diborides, and other novel superconducting materials has a strong multi-band and multi-gap character. Recent experiments support the possibillity for a BCS-BEC crossover induced by strong-coupling and proximity of the chemical potential to the band edge of one of the bands. Here we study the simplest theoretical model which accounts for the BCS-BEC crossover in a two-band superconductor, considering tunable interactions and tunable energy separations between the bands. Mean-field results for condensate fraction, correlation length, and superconducting…
It was recently shown that the BCS formalism leads to several solutions for the energy gap and the equilibrium quasiparticle distribution, with a phase transition temperature which depends on the position of the chemical potential within the attraction band (the attraction band AB is defined as the single-particle energy interval in which the pairing interaction is manifested). Moreover, in some cases, the phase transition may be of the first, not of the second order. Here I will find two sets of solutions for any temperature below the phase transition temperature. I will also show that, when …
In the context of the Anderson theory of high T_c cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular T_c predicted by the BCS equation for Luttinger liquids is quite higher than the usual T_c for Fermi liquids.
Ultranarrow superconductors are in the strong quantum confinement regime with formation of multiple coherent condensates associated with the many subbands of the electronic structure. Here we analyze the multiband BCS-BEC crossover induced by the chemical potential tuned close to a subband bottom, in correspondence of a superconducting shape resonance. The evolution of the condensate fraction and of the pair correlation length in the ground state as functions of the chemical potential demonstrates the tunability of the BCS-BEC crossover for the condensate component of the selected subband. The…
We develop a multi-step workflow for the discovery of conventional superconductors, starting with a Bardeen Cooper Schrieffer inspired pre-screening of 1736 materials with high Debye temperature and electronic density of states. Next, we perform electron-phonon coupling calculations for 1058 of them to establish a large and systematic database of BCS superconducting properties. Using the McMillan-Allen-Dynes formula, we identify 105 dynamically stable materials with transition temperatures, Tc>5 K. Additionally, we analyze trends in our dataset and individual materials including MoN, VC, VTe, …
We propose a model of quasiparticle tunneling across the high-$T_c$ superconductor-insulator-normal metal junction based on the different mechanisms for tunneling of electrons at positive bias and dissociating polaronic Cooper pairs and large polarons at negative bias, and the gap inhomogeneity (i.e., multi-gap) picture. We show that the main features of the tunneling spectra such as low-bias U- and V-shaped features, asymmetry and high-bias dip-hump features, their temperature and doping dependences, and shoulders inside the conductance peaks observed in high-$T_c$ cuprates arise naturally fr…
We investigate the ground state properties of a disordered superfluid Fermi gas across the BCS-BEC (Bose Einstein condensate) crossover. We show that, for weak disorder, both the depletion of the condensate fraction of pairs and the normal fluid density exhibit a nonmonotonic behavior as a function of the interaction parameter $1/k_Fa$, reaching their minimum value near unitarity. We find that, moving away from the weak coupling BCS regime, Anderson's theorem ceases to apply and the superfluid order parameter is more and more affected by the random potential.
We systematically explore the exquisiteness of Bardeen-Cooper-Schriefer(BCS) theory where the BCS-type electron-phonon interaction is unambiguously reinforced as the only viable superglue in cuprate superconductors because phonon-induced scattering is effectively nil for Cooper pairs (in its original form), and also phonons are never required to Bose-condense. Here, we prove that (i) the Cooper-pair binding energy can be strengthened to obtain high superconductor transition temperature (Tsc) and (ii) the existence of a generalized electron-phonon potential operator that can induce the finite-t…
Since the discovery of superconductivity in the cuprates two decades ago, it has been firmly established that the CuO_2 plane is consequential for high T_C superconductivity and a host of other very unusual properties. A new family of superconductors with the general composition of LaFeAsO_(1-x)F_x has recently been discovered but with the conspicuous lacking of the CuO_2 planes, thus raising the tantalizing questions of the different pairing mechanisms in these oxypnictide superconductors. Intimately related to pairing in a superconductor are the superconducting gap, its value, structure, and…
The free energy, non-gradient terms of the Ginzburg-Landau expansion, and the jump of the specific heat of a multiband anisotropic-gap clean BCS superconductor are derived in the framework of a separable-kernel approximation. Results for a two-band superconductor, d-wave superconductor, and some recent models for MgB_2 are derived as special cases.
This course has a dual purpose. First we review the successes of the weak-coupling BCS theory in describing new classes of superconductors discovered since 1979. They include the heavy-fermion superconductors, high-Tc cuprate superconductors, organic superconductors, Sr2RuO4, etc. Second, we present the quasiclassical approximation introduced by Volovik, which we extend to describe the thermodynamics and the thermal conductivity of the vortex state in nodal superconductors. This approach provides the most powerful tool to identify the symmetry of the energy gap function Delta(k) in these new s…
In this communication we report the interplay of the normal electron-phonon (EP) interaction, dynamic Jahn-Teller (DJT) distortion and superconductivity in high temperature superconductors in presence of a static lattice strain. This model consists of a degenerate two orbital band separated by Jahn-Teller (JT) energy modified by the DJT interaction in the conduction band. The superconductivity is assumed to be s-wave type present in the same band. The interaction Hamiltonian is solved by Green's function method and a modified BCS gap equation is obtained with a modified conduction band energy …
The confinement of a superconductor in a thin film changes its Fermi-level density of states and is expected to change its critical temperature $T_c$. Previous calculations have reported large discontinuities of $T_c$ when the chemical potential coincides with a subband edge. By solving the BCS gap equation exactly, we show that such discontinuities are artifacts and that $T_c$ is a continuous function of the film thickness. We also find that $T_c$ is reduced in thin films compared with the bulk if the confinement potential is lower than a critical value, while for stronger confinement $T_c$ i…
We present a mean field approach based on pairs of fermionic atoms to describe condensates in the BEC-BCS crossover regime. By introducing an effective potential, the mean field equation allows us to calculate the chemical potential, the equation of states and the atomic correlation function. The results agree surprisingly well with recent quantum Monte Carlo calculations. We show that the smooth crossover from the bosonic mean field repulsion between molecules to the Fermi pressure among atoms is associated with the evolution of the atomic correlation function.
Through the Higgs mechanism, the long-range Coulomb interaction eliminates the low-energy Goldstone phase mode in superconductors and transfers spectral weight all the way up to the plasma frequency. Here we show that the Higgs mechanism breaks down for length scales shorter than the superconducting coherence length while it stays intact, even at high energies, in the long-wavelength limit. This effect is a consequence of the composite nature of the Higgs field of superconductivity and the broken Lorentz invariance in a solid. Most importantly, the breakdown of the Higgs mechanism inside the s…
A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at T=0. Taking into account the density fluctuations at the outset we get the effective action where the density $ρ$ is the conjugated momentum to the phase $θ$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time dependent non-linear Schrödinger equation (TDNLS) for the field $Ψ(x)=\sqrt{ρ/2} e^{iθ}$. The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (…
Superconductivity mostly appears in high carrier density systems and sometimes exhibits common phase diagrams in which the critical temperature Tc continuously develops with carrier density. Superconductivity enhanced in lightly doped regime has seldom been reported, although it is an ideal direction towards the crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) limits, where the behavior of Cooper pairs changes dramatically. Here we report transport properties and superconducting gaps in single-crystalline lithium-intercalated layered nitrides (LixHfNCl and L…
Superconducting condensation energy $U_0^{int}$ has been determined by integrating the electronic entropy in various iron pnictide/chalcogenide superconducting systems. It is found that $U_0^{int}\propto T_c^n$ with $n$ = 3 to 4, which is in sharp contrast to the simple BCS prediction $U_0^{BCS}=1/2N_FΔ_s^2$ with $N_F$ the quasiparticle density of states at the Fermi energy, $Δ_s$ the superconducting gap. A similar correlation holds if we compute the condensation energy through $U_0^{cal}=3γ_n^{eff}Δ_s^2/4π^2k_B^2$ with $γ_n^{eff}$ the effective normal state electronic specific heat coefficien…
The BCS model of an isolated superconductor initially prepared in a nonequilibrium state, predicts the existence of interesting dynamical phenomena in the time-dependent order parameter as decaying oscillations, persistent oscillations and overdamped dynamics. To make contact with real systems remains an open challenge as one needs to introduce dissipation due to the environment in a self-consistent computation. Here, we reach this goal with the use of the Keldysh formalism to treat the effect of a thermal bath. We show that, contrary to the dissipationless case, all dynamical phases reach the…
We report on the structure and physical properties of bulk Palladium Tellurium superconductor, which is synthesized via quartz vacuum encapsulation technique at 750 C. The as synthesized compound is crystallized in hexagonal crystal structure. Magnetization and Magneto-transport measurements provided the values of lower and upper critical field to be 250 and 1200 Gauss respectively at 2 Kelvin. The Coherence length and GL parameter are estimated from the experimentally determined upper and lower critical fields, which are 45 nm and 1.48 respectively. The jump in Cp(T) at Tc is found to be 1.33…
There is a need to better understand the intrinsic limit of radiofrequency (RF) surface impedance that determines the performance of superconducting RF cavities in particle accelerators. Here we present a field-dependent derivation of Mattis-Bardeen (M-B) theory of the RF surface impedance of BCS superconductors based on the shifted Density of States (DoS) resulting from coherently moving Cooper pairs [1].The surprising reduction in resistance with increasing field is explained to be an intrinsic effect. Using this analysis coded in MathematicaTM, survey calculations have been completed which …
We analyze the qualitative features in the transition spectra of a model superconductor with multiple energy gaps, using a simple extension of the Mattis-Bardeen expression for probes with case I and case II coherence factors. At temperature T = 0, the far infrared absorption edge is, as expected, determined by the smallest gap. However, the large thermal background may mask this edge at finite temperatures and instead the secondary absorption edges found at Delta_i+Delta_j may become most prominent. At finite T, if certain interband matrix elements are large, there may also be absorption peak…
Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition as a single critical point where the gap function or, equivalently, the order parameter vanishes uniformly in the entire system. We demonstrate that in superconductors described by standard BCS models, the superconducting gap survives near the sample boundaries at higher temperatures than superconductivity in the bulk. Therefore, conventional superconductors have multiple critical points associated with separate phase transitions at the boundary and in the bulk. We show this by revising the Caroli-De Gennes-Matricon …
We performed resonant microwave measurements on superconducting titanium (Ti) down to temperatures of 40~mK, well below its critical temperature $T_\mathrm{c} \approx 0.5$~K. Our wide frequency range 3.3-40~GHz contains the zero-temperature energy gap $2Δ_0$ and allows us to probe the full electrodynamics of the superconducting state, including excitations across the gap and the low-frequency responses of superfluid condensate and thermal quasiparticles. The observed behavior follows the predictions of the BCS-based Mattis-Bardeen formalism, which implies that superconducting Ti is in the dirt…
