We compute the spectral function $A({\bf k}, ω)$ of a model two-dimensional high-temperature superconductor, at both zero and finite temperatures $T$. We assume that an areal fraction $c_β$ of the superconductor has a large gap $Δ$ ($β$ regions), while the rest has a smaller $Δ$ ($α$ regions), both of which are randomly distributed in space. We find that $A({\bf k}, ω)$ is most strongly affected by inhomogeneity near the point $\mathbf k = (π, 0)$ (and the symmetry-related points). For $c_β\simeq 0.5$, $A({\bf k}, ω)$ exhibits two double peaks (at positive and negative energy) near this k-point if the difference between $Δ_α$ and $Δ_β$ is sufficiently large in comparison to the hopping integral. The strength of the inhomogeneity required to produce a split spectral function peak suggests that inhomogeneity is unlikely to be the cause of a second branch in the dispersion relation. Thermal fluctuations also affect $A({\bf k}, ω)$ most strongly near $\mathbf k = (π,0)$. Typically, peaks that are sharp at $T = 0$ become reduced in height, broadened, and shifted toward lower energies with increasing $T$; the spectral weight near $\mathbf k = (π, 0)$ becomes substantial at zero energy for $T$ greater than the phase-ordering temperature.
Scanning tunneling spectroscopy at very low temperature on homogeneously disordered superconducting Titanium Nitride thin films reveals strong spatial inhomogeneities of the superconducting gap $Δ$ in the density of states. Upon increasing disorder, we observe suppression of the superconducting critical temperature $T_c$ towards zero, enhancement of spatial fluctuations in $Δ$, and growth of the $Δ/T_c$ ratio. These findings suggest that local superconductivity survives across the disorder-driven superconductor-insulator transition.
The inhomogeneities inherent to the random distribution of Sr dopants in La_{2-x}Sr_xCuO_4 superconductors are probed by measuring the x-ray diffraction linewidths and the Meissner transition widths, and then consistently explained on the grounds of a simple model in which the local Sr content is calculated by averaging over distances close to the in-plane electronic mean free path. By taking into account these intrinsic bulk inhomogeneities with long characteristic lengths (much larger than the superconducting coherence length amplitudes), the precursor diamagnetism measured above Tc, a finge…
We present a novel technique for the investigation of local variations of the critical current density in large bulk superconductors. In contrast to the usual Hall probe scanning technique, the sample is not magnetized as a whole before the scan, but locally by a small permanent magnet, which is fixed near the Hall probe, during the scanning process. The resulting signal can be interpreted as a qualitative measure of the local shielding currents flowing at the surface.
The two-component model of high temperature superconductors in its real space version has been solved using Bogoliubov-de Gennes equations. The disorder in the electron and boson subsystem has been taken into account. It strongly modifies the superconducting properties and leads to local variations of the gap parameter and density of states. The assumption that the impurities mainly modify boson energies offers natural explanation of the puzzling positive correlation between the positions of impurities and the values of the order parameter found in the scanning tunnelling microscopy experiment…
The measurements of temperature dependence of AC magnetic susceptibility were used to study the influence of inhomogeneities on intragranular superconducting properties of YBa2Cu3-xMxOy (M=Fe,x=0.005; Al,x=0.01; Cr,x=0.01), Bi1.7Pb0.2Sb0.1Sr2Ca2Cu3Oy and Bi1.5Pb0.3Sb0.2Sr2Ca2Cu3Oy ceramics. It is found that step-like regions are revealed on AC magnetic field dependence curves Tmg(ho) of hysteresis losses peak temperature for superconducting granuls. The steps at temperature scale depend on sample's composition as well as on the type of atoms sabstituted for the copper. They range from 1 to 2 K…
We perform calculations to obtain the $H_{c2}$ curve of high temperature superconductors (HTSC). We consider explicitly the fact that the HTSC possess intrinsic inhomogeneities by taking into account a non uniform charge density $ρ(r)$. The transition to a coherent superconducting phase at a critical temperature $T_c$ corresponds to a percolation threshold among different superconducting regions, each one characterized by a given $T_c(ρ(r))$. Within this model we calculate the upper critical field $H_{c2}$ by means of an average linearized Ginzburg-Landau (GL) equation to take into account the…
We show that inhomogeneities in the spatial distribution of Cooper pairs and in the phase of the local superconducting order parameter in the vicinity of a superconductor-normal state transition (SNT) in two dimensions can be highly sensitive to a perpendicular magnetic field. We focus on the role of orbital effects in the field-dependence of local superfluid stiffness and superconducting phase disorder in homogeneously-disordered two-dimensional superconductor thin films. The relative importance of these orbital effects is analyzed in different physical regimes dominated by Coulomb blockade, …
The magnetic field dependence of the magnetization was measured above the superconducting transition in a high-Tc underdoped cuprate La1.9Sr0.1CuO4 and in a low-Tc alloy (Pb55In45). Near the superconducting transition [typically for (T-Tc)/Tc<0.05] and under low applied magnetic field amplitudes [typically for H/Hc2(0)<0.01, where Hc2(0) is the corresponding upper critical field extrapolated to T=0 K] the magnetization of both samples presents a diamagnetic contribution much larger than the one predicted by the Gaussian Ginzburg-Landau (GGL) approach for superconducting fluctuations. These ano…
We study a short coherence length d-wave superconductor with finite density of unitary scatterers using the Bogoliubov-deGennes technique. We find that the low-energy density of states is reduced, the superfluid stiffness is significantly larger and off-diagonal long range order is more robust than the self-consistent T-matrix prediction. These results are a consequence of the inhomogeneous pairing amplitude in the ground state and of the low-lying excitations formed by hybridized impurity resonances. These features, with their nontrivial spatial structure, cannot be adequately described withi…
Type II superconductors, consisting of superconducting domains embedded in a normal or insulating matrix, undergo a rounded phase transition. Indeed, the correlation length cannot grow beyond the spatial extent of the domains. Accordingly, the thermodynamic properties will exhibit a finite size effect. It is shown that the specific heat and penetration depth data of a variety of type II superconductors, including cuprates, exhibit the characteristic properties of a finite size effect, arising from domains with nanoscale extent. The finite size scaling analysis reveals essential features of the…
We discuss the Meissner response to a known field source of superconductors having inhomogeneities in their penetration depth. We simplify the general problem by assuming that the perturbations of the fields by the penetration depth inhomogeneities are small. We present expressions for inhomogeneities in several geometries, but concentrate for comparison with experiment on planar defects, perpendicular to the sample surfaces, with superfluid densities different from the rest of the samples. These calculations are relevant for magnetic microscopies, such as Scanning Superconducting Quantum Inte…
We analyze numerically how the voltage-current (V-I) characteristics near the so-called Berezinskii-Kosterlitz-Thouless (BKT) transition of 2D superconductors are affected by a random spatial Gaussian distribution of critical temperature inhomogeneities with long characteristic lengths (much larger than the in-plane superconducting coherence length amplitude). Our simulations allow to quantify the broadening around the average BKT transition temperature of both the exponent alpha in V I^alpha and of the resistance V/I. These calculations reveal that strong spatial redistributions of the local …
We review some aspects of the interplay between inhomogeneities at long length scales and the intrinsic fluctuations of Cooper pairs above Tc in HTSC. These inhomogeneities do not directly affect the thermal fluctuations, but they may affect the measured behaviour of any observable around Tc. The emphasis is centered on the role played by the presence of Tc-inhomogeneities on the in-plane transport properties. For completeness, we will also summarize some results on this interplay when various types of inhomogeneities (i.e., structural and stoichiometric, uniformly and non-uniformly distribute…
Nanoscale electronic inhomogeneity in optimally doped single-layer Bi_2Sr_{1.6}L_{0.4}CuO_{6+delta}, (L-Bi2201, L=La and Gd) as well as bilayer Bi_2Sr_2CaCu_2O_{8+delta} (Bi2212), superconductors has been investigated by the low-temperature scanning-tunneling-microscopy and spectroscopy. The inhomogeneous gap structures are observed above and below Tc on Gd-Bi2201. Increasing the out-of-plane disorder which is controlled by mismatch of the ionic radius between Sr and Ln results in the enhancement of pseudogap region with suppressed coherence peaks in the conductance spectrum, making the gap av…
Holographic duality is a powerful tool to investigate the far-from equilibrium dynamics of superfluids and other phases of quantum matter. For technical reasons it is usually assumed that, after a quench, the far-from equilibrium fields are still spatially uniform. Here we relax this assumption and study the time evolution of a holographic superconductor after a temperature quench but allowing spatial variations of the order parameter. Even though the initial state and the quench are spatially uniform we show the order parameter develops spatial oscillations with an amplitude that increases wi…
A superconductor is influenced by an applied magnetic field. Close to the transition temperature $T_{c}$ fluctuations dominate and the correlation length $ξ$ increases strongly when $T_{c}$ is approached. However, for nonzero magnetic field $H$ there is another length scale $L_{H}=\sqrt{Φ_{0}/aH}$ where $a$ is a universal amplitude. It is comparable to the average distance between vortex lines. We show that the correlation length is bounded by this length scale, so that $ξ$ cannot grow beyond $L_{H}$. This implies that type II superconductors in a magnetic field do not undergo a phase transiti…
We analyze the complex interplay of the strong correlations and impurities in a high temperature superconductor and show that both the nature and degree of the inhomogeneities at zero temperature in the local order parameters change drastically from what are obtained in a simple Hartree-Fock-Bogoliubov theory. While both the strong electronic repulsions and disorder contribute to the nanoscale inhomogeneity in the population of charge-carriers, we find them to compete with each other leading to a relatively smooth variation of the local density. Our self-consistent calculations modify the spat…
We review briefly several approaches used to investigate the stability of stripe phases in high temperature superconductors, where charge inhomogeneities arise from competing kinetic and magnetic energies. The mechanism of stripe formation, their consequences for the normal state and enhancement of pairing interaction triggered by charge inhomogeneities are briefly summarized. Finally, we demonstrate that orbital degeneracy ($i$) leads to a more subtle mechanism of stripe formation, and ($ii$) plays an important role and determines the symmetry of the superconducting state in pnictides.
The microscopic nature of an insulating state in the vicinity of a superconducting state, in the presence of disorder, is a hotly debated question. While the simplest scenario proposes that Coulomb interactions destroy the Cooper pairs at the transition, leading to localization of single electrons, an alternate possibility supported by experimental observations suggests that Cooper pairs instead directly localize. The question of the homogeneity, granularity, or possibly glassiness of the material on the verge of this transition is intimately related to this fundamental issue. Here, by combini…
The number of bound states due to inhomogeneities in a BCS superconductor is usually established either by variational means or via exact solutions of particularly simple, symmetric perturbations. Here we propose estimating the number of sub-gap states using the Birman-Schwinger principle. We show how to obtain upper bounds on the number of sub-gap states for small normal regions and derive a suitable Cwikel-Lieb-Rozenblum inequality. We also estimate the number of such states for large normal regions using high dimensional generalizations of the Szego theorem. The method works equally well fo…
Proper characterisation of investigated samples is vital when studying superconductivity as impurities and doping inhomogeneities can affect the physical properties of the measured system. We present a method where a polarised neutron imaging setup utilises the precession of spin-polarised neutrons in the presence of a trapped field in the superconducting sample to spatially map out the critical temperature for the phase transition between superconducting and non-superconducting states. We demonstrate this method on a superconducting crystal of the prototypical high-temperature superconductor …
Scanning tunneling spectroscopy applied to the high-$T_{c}$ cuprates has revealed significant spatial inhomogeneity on the nanoscale. Regions on the order of a coherence length in size show variations of the magnitude of the superconducting gap of order $\pm20%$ or more. An important unresolved question is whether or not these variations are also present in the bulk, and how they influence superconducting properties. As many theories and data analyses for high-$T_{c}$ superconductivity assume spatial homogeneity of the gap magnitude, this is a pressing question. We consider the far-infrared op…
Thermodynamic and transport properties of normal disordered conductors are strongly influenced by the proximity of a superconductor. A cooperation between mesoscopic coherence and Andreev scattering of particles from the superconductor generates new types of interference phenomena. We introduce a field theoretic approach capable of exploring both averaged properties and mesoscopic fluctuations of superconductor/normal-metal systems. As an example the method is applied to the study of the level statistics of a SNS-junction.
We show that quantum spin fluctuations in inhomogeneous ferromagnets drastically affect the Andreev reflection of electrons and holes at a ferromagnet-superconductor interface. As a result a strong long-range proximity effect appears, associated with electron-hole spin triplet correlations and persisting on a lenght scale typical for non-magnetic materials, but anomalously large for ferromagnets.
We consider the Nernst and Hall effects in fluctuation regime of chiral superconductors above transition temperatures, that are raised not by conventional Lorentz force, but by asymmetric scattering due to fluctuations of the Berry phase of the Bogoliubov-de Gennes Hamiltonian. It is found that these effects can be more significant than conventional ones for cleaner samples, exhibiting qualitatively distinct behaviors. The results provide systematic and comprehensive understanding for recent experimental observations of the Nernst effect in a clean URu$_2$Si$_2$ sample, which is suggested to b…
We study the influence of thermal fluctuations on the magnetic behavior of square mesoscopic superconductors. The strength of thermal fluctuations are parameterized using the Ginzburg number, which is small ($G_i \approx 10^{-10}$) in low-$T_c$ superconductors and large in high-$T_c$ superconductors ($G_i \approx 10^{-4}$). For low-$T_c$ mesoscopic superconductors we found that the meta-stable states due to the surface barrier have a large half-life time, which leads to the hysteresis in the magnetization curves as observed experimentally. A very different behavior appears for high-$T_c$ mesos…
The effect of smooth inhomogeneities near a superconductor boundary on the magnetic penetration depth $λ$ is studied with emphasis on the proximity-induced spatial dependence of the Cooper pair amplitude. The influence of surface pair breaking or pair formation on $λ$ is described within the Ginzburg-Landau theory, with no model assumptions, for both strongly type-II and strongly type-I homogeneous superconductors. Generic values of $λ$, which can differ greatly from the London penetration depth, are identified and demonstrated to be induced by large-scale inhomogeneities, when superconductivi…
In this study we perform numerical calculations of the spatial and time variation of the magnetic induction, current density and temperature in an infinite superconducting slab with macroscopic inhomogeneities. We also modeled an homogeneuos superconductor as a referential material, considering characteristic magnetothermal properties of a high-T$_c$ single crystal. The theoretical formalism is enclosed in the dynamical regime and a continuum electrodynamic approach. In particular, we studied the correlation between the magnetic jumps and temperature rise occurrence with the presence of macros…
The magnetization around the superconducting transition was measured in a Tl$_{0.5}$Pb$_{0.5}$Sr$_2$CaCu$_2$O$_7$ crystal affected by a considerable reduction ($\sim$55%) of its effective superconducting volume fraction but still with a relatively sharp low-field Meissner transition, a behaviour that may be attributed to the presence of structural inhomogeneities. By taking into account these inhomogeneities just through the Meissner fraction, the observed diamagnetism may still be explained, consistently above and below the superconducting transition, in terms of the conventional Ginzburg-Lan…
We calculate transport properties of disordered 2D d-wave superconductors from solutions of the Bogoliubov-de Gennes equations, and show that weak localization effects give rise to a finite frequency peak in the optical conductivity similar to that observed in experiments on disordered cuprates. At low energies, order parameter inhomogeneities induce linear and quadratic temperature dependencies in microwave and thermal conductivities respectively, and appear to drive the system towards a quasiparticle insulating phase.
We studied the influence of the amplitude fluctuations of a non-Fermi superconductor on the energy spectrum of the 2D Anderson non-Fermi system. The classical fluctuations give a temperature dependence in the pseudogap induced in the fermionic excitations.
It is shown that the specific heat jump at the superconducting transition temperature in pnictide and cuprate superconductors is produced by ferroelastic fluctuations just below the transition temperature. The amplitude of the corresponding lattice distortion is estimated. It is shown that a contribution from ferroelastic fluctuations to the specific heat jump is also present in low-temperature superconductors.
We investigate the effects of mesoscopic inhomogeneities on the metal-superconductor transition occurring in several two-dimensional electron systems. Specifically, as a model of systems with mesoscopic inhomogeneities, we consider a random-resistor network, which we solve both with an exact numerical approach and by the effective medium theory. We find that the width of the transition in these two-dimensional superconductors is mainly ruled by disorder rather than by fluctuations. We also find that "tail" features in resistivity curves of interfaces between LaAlO3 or LaTiO3 and SrTiO3 can ari…
When a finite superconductor is in contact with a 1D normal conductor, superconducting phase fluctuations lead to power-law response of the normal subsystem. As a result, the charge fluctuations on the superconductor at zero temperature have logarithmic correlator and a 1/ωpower spectrum (1/f-noise). At higher temperatures 1/ωis pushed to the high frequency region, and 1/ω^2 behavior prevails.
A theory of fluctuations in two-band superconductor MgB$_{2}$ is developed. Since the standard Ginzburg-Landau (GL) approach fails in description of its properties, we generalize it basing on the microscopic theory of a two-band superconductor. Calculating the microscopic fluctuation propagator, we build up the nonlocal two-band GL functional and the corresponding time-dependent GL equations. This allows us to calculate the main fluctuation observables such as fluctuation specific heat and conductivity.
Based on recent experimental results for electron-doped cuprate oxides and ferromagnetic superconductors, it is shown that antiferromagnetic fluctuations always develop in the superconducting phase of both low- and high-temperature superconductors. The relation between the magnitude of the antiferromagnetic pseudogap and the characteristic temperature of the antiferromagnetic pseudogap opening is obtained. The characteristic temperature of the antiferromagnetic pseudogap opening for metal superconductors is estimated.
Iron-based superconductors can exhibit different magnetic ground states and are in a critical magnetic region where frustrated magnetic interactions strongly compete with each other. Here we investigate the longitudinal modes of spin fluctuations in an unified effective magnetic model for iron-based superconductors. We focus on the collinear antiferromagnetic phase (CAF) and calculate the behavior of the longitudinal modes when different phase boundaries are approached. The results can help to determine the nature of the magnetic fluctuations in iron-based superconductors.
Magnetic field fluctuations are inevitable in practical applications of superconductors and it is often necessary to estimate the AC losses these fluctuations induce. If the fluctuation wavelength is greater than the size of a superconductor, known estimates for an alternating uniform external magnetic field can be employed. Here we consider the opposite case and analyze, using a model critical-state problem, penetration of spatially nonuniform fluctuations into type-II superconductors. Numerical simulation is based on a variational formulation of the Bean model. The analytical solutions, foun…
We study the effects of low-energy nodal quasiparticles on the classical phase fluctuations in a two-dimensional d-wave superconductor. The singularities of the phase-only action at T\to 0 are removed in the presence of disorder, which justifies using an extended classical XY-model to describe phase fluctuations at low temperatures.
We study the effects of quantum vortex fluctuations in two-dimensional superconductors using a dual theory of vortices, and investigate the relevance to underdoped cuprates where the superconductor-insulator transition (SIT) is possibly driven by quantum vortex proliferation. We find that a broad enough phase fluctuation regime may exist for experimental observation of the quantum vortex fluctuations near SIT in underdoped cuprates. We propose that this scenario can be tested via pair-tunneling experiments which measure the characteristic resonances in the zero-temperature pair-field susceptib…
Spin fluctuations are widely believed to play an important role in the superconducting mechanisms of unconventional high-temperature superconductors. Spin fluctuations have been observed in iron-based superconductors as well. However, in some iron-based superconductors such as LaFePO$_{0.9}$, they have not been observed by inelastic neutron scattering (INS). LaFePO$_{0.9}$ is an iron-based superconductor with a low superconducting transition temperature ($T_{\rm c}$= 5 K), where line nodes are observed in the superconducting gap function. The line-node symmetry typically originates from sign r…
Thermal fluctuation conductivity for a layered superconductor in perpendicular magnetic field is treated in the frame of the self-consistent Hartree approximation for an arbitrarily strong in-plane electric field. The simultaneous application of the two fields results in a slightly stronger suppression of the superconducting fluctuations, compared to the case when the fields are applied individually.
The review is devoted to the detailed description of the fluctuation phenomena in superconductors. Developing phenomenological methods through the first five sections we deal with the direct fluctuation pair contributions only. The indirect fluctuation effects are discussed in the last two sections, devoted to the microscopic justification of the time dependent Ginzburg-Landau equation, the description of the microscopic theory of fluctuations, and discussion of their manifestations in various physical properties of superconductors. The first six sections are written in detail, so they can ser…
The effect of fluctuation induced weakly first order phase transition known for three dimensional (3D) type I superconductors appears in a modified and strongly enhanced variant in thin (quasi-2D) superconducting films. The unusual thermodynamic properties of this new type of first order phase transitions and the possibility for an experimental verification of the effect are established and discussed.
We study the superconducting order parameter fluctuations near the phase transition into the Larkin-Ovchinnikov-Fulde-Ferrell state in the clean limit at zero temperature. In contrast to the usual normal metal-to-uniform superconductor phase transition, the fluctuation corrections are dominated by the modes with the wave vectors away from the origin. We find that the superconducting fluctuations lead to a divergent spin susceptibility and a breakdown of the Fermi-liquid behavior at the quantum critical point.
It is shown that a ferroelastic lattice distortion is associated with the superconducting transition both in low- and high-temperature superconductors. In layered tetragonal superconductors with a sufficiently high transition temperature, a ferroelastic distortion is improper and is caused by an out-of-plane charge ordering. The relation between the out-of-plane charge gap and the maximum superconducting transition temperature for these superconductors is obtained. It is shown also that the specific heat jump at the superconducting transition temperature in pnictide and cuprate superconductors…
Spin subsystem of the copper oxides within the polar Jahn-Teller centers model corresponds to a singlet-triplet magnet where the local boson movement accompanied by the induced longitudinal spin fluctuations. These fluctuations determine the features of the magnetic inelastic scattering in the high-Tc cuprates.
We study the order parameter phase fluctuation effects in cuprate superconductors near T=0, using a quasi-two-dimensional d-wave BCS model. An effective phason theory is obtained which is used to estimate the strength of the fluctuations, the fluctuation correction to the in-plane penetration depth, and the pair-field susceptibility. We find that while the phase fluctuation effects are difficult to observe in the renormalization of the superfluid phase stiffness, they may be observed in a pair tunneling experiment which measures the pair-field susceptibility.
Within the phase fluctuation picture for the pseudogap state of a high-$T_{c}$ superconductor, we incorporate the phase fluctuations generated by the classical XY model with the Bogoliubov-de Gennes formalism utilizing a field-theoretical method. This picture delineates the inhomogeneous characteristics of local order parameters observed in high-$T_{c}$ superconductors above $T_{c}$. We also compute the local density of states near a non-magnetic impurity with a strong scattering potential. The resonance peak smoothly evolves as temperature increases through $T_{c}$ without showing any sudden …
Recent measurements of fluctuation diamagnetism in high temperature superconductors show distinct features above and below Tc, which can not be explained by simple gaussian fluctuation theory. Self consistent calculation of magnetization in layered high temperature superconductors, based on the Ginzburg-Landau-Lawrence-Doniach model and including all Landau levels is presented. The results agree well with the experimental data in wide region around Tc, including both the vortex liquid below Tc and the normal state above Tc. The gaussian fluctuation theory significantly over estimates the diama…
The observation of a strong nonlinear diamagnetic signal well above T_c in several cuprate superconductors constitutes clear evidence of the persistence of local superconducting correlations.
If phase coherence determines the superconducting transition temperature Tc in the cuprate oxides, it is of great interest to understand the role that dynamics of the phase fluctuation plays in bringing about depletion of the superconducting condensate. We will show that a phase correlation function can be calculated that allows us to describe depletion of superconducting condensate as a result of the quantum fluctuation. In two dimension, dynamic phase fluctuation or pair fluctuation gives rise to a condensate depletion linear with temperature as T$\to O$ in superconductors with nodes at the …
We report neutron inelastic scattering measurements on polycrystalline LaFePO and Sr2ScO3FeP, two members of the iron phosphide families of superconductors. No evidence is found for any magnetic fluctuations in the spectrum of either material in the energy and wavevector ranges probed. Special attention is paid to the wavevector at which spin-density-wave-like fluctuations are seen in other iron-based superconductors. We estimate that the magnetic signal, if present, is at least a factor of four (Sr2ScO3FeP) or seven (LaFePO) smaller than in the related iron arsenide and chalcogenide supercond…
We investigate the microwave absorption arising from inhomogeneity in the superfluid density of a model high-T_c superconductor. Such inhomogeneities may arise from a wide variety of sources, including quenched random disorder and static charge density waves such as stripes. We show that both mechanisms will inevitably produce additional absorption at finite frequencies. We present simple model calculations for this extra absorption, and discuss applications to other transport properties in high-T_c materials. Finally, we discuss the connection of these predictions to recent measurements by Co…
