Dirty Superconductors Research Articles

Mixed state in dirty two-band superconductors: application to MgB 2

We investigate a two-band superconductor with strong intraband and weak interband electronic scattering rates in the framework of coupled Usadel equations. We calculate the upper critical field for different field orientation. Due to a very large difference between the c-axis coherence lengths in two-bands, the GL theory is applicable only in the extremely narrow temperature range. This leads to the strong temperature dependence of the H c2 anisotropy and large deviations of the angular dependence of H c2 from a simple effective-mass law. In the case of field along c-direction we solved coupled nonlinear Usadel equations numerically to find field evolution of the pair potentials and local densities of states for two-bands. The existence of two distinct length and field scales corresponding to different bands is demonstrated.

Magnetic and superconducting phase diagrams of single-crystalEr0.8R0.2Ni2B2C(R=Tb,Lu)andErNi1.9Co0.1B2C: Identification of pair-breaking mechanisms

We investigated the magnetism, superconductivity and their interplay in single crystals ${\mathrm{Er}}_{0.8}{R}_{0.2}{\mathrm{Ni}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ $(R=\mathrm{T}\mathrm{b},\mathrm{L}\mathrm{u})$ and ${\mathrm{ErNi}}_{1.9}{\mathrm{Co}}_{0.1}{\mathrm{B}}_{2}\mathrm{C}.$ In contrast to Co substitution, R substitutions induce considerable modifications in the magnetism of Er sublattice: e.g., Tb (Lu) substitution enhances (reduces) ${T}_{\mathrm{N}}$ and critical fields. Both R substitutions introduce size effects and pinning centers; the former modifies the magnon specific heat while the latter hinders the formation of a weak ferromagnetism. The superconductivity, on the other hand, is strongly (weakly) influenced by Tb and Co (Lu) substitution. Taking ${\mathrm{LuNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ as a nonmagnetic superconducting limit, we analyzed their superconductivities, as well as that of ${\mathrm{ErNi}}_{2}{\mathrm{B}}_{2}\mathrm{C},$ in terms of multiple pair breaking theory on dirty superconductors. Based on this analysis, many of their superconducting features can be explained: The breakdown of de Gennes scaling is due to the presence of multiple pair breakers, the anisotropy of ${H}_{\mathrm{c}2}(T)$ is related to the magnetic anisotropy, the absence of a structure in ${H}_{\mathrm{c}2}(T)$ at ${T}_{\mathrm{N}}$ of Lu substitution ${(T}_{\mathrm{N}}<{T}_{\mathrm{c}})$ is attributed to an alloying-induced destruction of phase space truncation, and the quasi parabolic temperature dependence of ${H}_{\mathrm{c}2}(T)$ of Tb and Co substitutions is in part due to a saturation of antiferromagnetic correlations. For Lu substitution, the strength of magnon mediated pair breaking process(es) is substantially reduced.

Andreev scattering and cotunneling between two superconductor-normal metal interfaces: the dirty limit

Crossed Andreev reflections and cotunneling occur between two neighbouring superconductor- normal metal or superconducting-ferromagnet interfaces. Previous works assumed a clean BCS superconductor. Here the calculation of the corresponding crossed conductance terms is generalized to a dirty superconductor. The range of the effect is shown to be the coherence length $\tilde{\xi} = \sqrt{\hbar D/\Delta}$, instead of the BCS coherence length $\xi_0$. Moreover, in three dimensions, the algebraic prefactor scales as $1/r$ instead of $1/r^2$. The calculation involves the virtual diffusion probability of quasiparticles below the superconducting gap, in the normal and the anomalous channel.

Upper critical field in dirty two-band superconductors: Breakdown of the anisotropic Ginzburg-Landau theory

We investigate the upper critical field in a dirty two-band superconductor within quasiclassical Usadel equations. The regime of very high anisotropy in the quasi-2D band, relevant for MgB$_{2}$, is considered. We show that strong disparities in pairing interactions and diffusion constant anisotropies for two bands influence the in-plane $H_{c2}$ in a different way at high and low temperatures. This causes temperature-dependent $H_{c2}$ anisotropy, in accordance with recent experimental data in MgB$_{2}$. The three-dimensional band most strongly influences the in-plane $H_{c2}$ near $T_{c}$, in the Ginzburg-Landau (GL) region. However, due to a very large difference between the c-axis coherence lengths in the two bands, the GL theory is applicable only in an extremely narrow temperature range near $T_c$. The angular dependence of $H_{c2}$ deviates from a simple effective-mass law even near $T_c$.

Differential sum rule for the relaxation rate in dirty superconductors

We consider the differential sum rule for the effective scattering rate $1/\ensuremath{\tau}(\ensuremath{\omega})$ and optical conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ in a dirty BCS superconductor, for arbitrary ratio of the superconducting gap $\ensuremath{\Delta}$ and the normal state constant damping rate $1/\ensuremath{\tau}.$ We show that if $\ensuremath{\tau}$ is independent of T, the area under $1/\ensuremath{\tau}(\ensuremath{\omega})$ does not change between the normal and the superconducting states, i.e., there exists an exact differential sum rule for the scattering rate. For any value of the dimensionless parameter $\ensuremath{\Delta}\ensuremath{\tau},$ the sum rule is exhausted at frequencies controlled by $\ensuremath{\Delta}.$ We show that in the dirty limit the convergence of the differential sum rule for the scattering rate is much faster then the convergence of the f-sum rule, but slower then the convergence of the differential sum rule for conductivity.

C-Axis Raman scattering spectra of MgB2: observation of a dirty-limit gap in the pi bands.

Raman scattering spectra from the ac face of thick MgB2 single crystals were measured in zz, xz, and xx polarizations. In zz and xz polarizations a threshold at around 29 cm(-1) forms in the below T(c) continuum but no pair-breaking peak is seen, in contrast to the sharp pair-breaking peak at around 100 cm(-1) in xx polarization. The zz and xz spectra are consistent with Raman scattering from a dirty superconductor while the sharp peak in the xx spectra argues for a clean system. Analysis of the spectra resolves this contradiction, placing the larger and smaller gap magnitudes in the sigma and pi bands and indicating that relatively strong impurity scattering is restricted to the pi bands.

Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors

Quasiclassic Uzadel equations for two-band superconductors in the dirty limit with the account of both intraband and interband scattering by nonmagnetic impurities are derived for any anisotropic Fermi surface. From these equations the Ginzburg-Landau equations, and the critical temperature $T_c$ are obtained. An equation for the upper critical field, which determines both the temperature dependence of $H_{c2}(T)$ and the orientational dependence of $H_{c2}(\theta)$ as a function of the angle $\theta$ between ${\bf H}$ and the c-axis is obtained. It is shown that the shape of the $H_{c2}(T)$ curve essentially depends on the ratio of the intraband electron diffusivities $D_1$ and $D_1$, and can be very different from the standard one-gap dirty limit theory. In particular, the value $H_{c2}(0)$ can considerably exceed $0.7T_cdH_{c2}/dT_c$, which can have important consequences for applications of $MgB_2$. A scaling relation is proposed which enables one to obtain the angular dependence of $H_{c2}(\theta)$ from the equation for $H_{c2}$ at ${\bf H}\| c$. It is shown that, depending on the relation between $D_1$ and $D_2$, the ratio of the upper critical field $H_{c2}^\|/H_{c2}^\perp$ for ${\bf H}\| ab$ and ${\bf H}\perp ab$ can both increase and decrease as the temperature decreases. Implications of the obtained results for $MgB_2$ are discussed.

Differential sum rule for the relaxation rate in the cuprates.

Motivated by recent experiments by Basov et al., we study the differential sum rule for the effective scattering rate 1/tau(omega). We show that, in a dirty BCS superconductor, the area under 1/tau(omega) does not change between the normal and the superconducting states. For magnetically mediated pairing, a similar result holds between T<T(c) and T>or=T(c), while, in the pseudogap phase, 1/tau(omega) is just suppressed compared to 1/tau(omega) in the normal state. We argue that this violation of the differential sum rule in the pseudogap phase is due to the absence of the feedback effects from the pairing.

Low temperature specific heat studies on the pairing states of high- Tc superconductors: a brief review

Low temperature specific heat (LTSH) data on a variety of high- T c superconductors, such as YBa 2Cu 3O 7, La 1− x Sr x CuO 4, Bi 2Sr 2CaCu 2O 8, and Ba 0.6K 0.4BiO 3 are reviewed. The agreement between the experimental data and theoretical predictions, such as T 2-dependence of specific heat at zero magnetic field and H 1/2-dependence of electronic specific heat is widely discussed within the scenario of d-wave superconductivity. Impurity scattering effects and scaling model on d-wave superconductivity are verified using Zn- and Ni-doped La 1− x Sr x CuO 4. The low energy quasiparticle density of states N( E)= N(0)+ E 1/2 are deduced from dirty d-wave superconductors. Absence of paramagnetic contribution to LTSH is found both in superconducting and non-superconducting underdoped samples suggesting that a mechanism beyond Kondo screening model maybe required to explain its magnetic property.

Gauge invariance and Hall terms in the quasiclassical equations of superconductivity

This paper presents a careful derivation of the quasiclassical equations of superconductivity so that a manifest gauge invariance is retained with respect to the space-time arguments of the quasiclassical Green's function $\hat{g}$. The terms responsible for the Hall effect naturally appear from the derivation. The equations are applicable to clean as well as dirty superconductors for an arbitrary external frequency much smaller than the Fermi energy. Thus, they will form a basis toward a complete microscopic understanding of the Hall effect in type-II superconductors.

Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One-dimensional examples

We introduce a strong-disorder renormalization group (RG) approach suitable for investigating the quasiparticle excitations of disordered superconductors in which the quasiparticle spin is not conserved. We analyze one-dimensional models with this RG and with elementary transfer matrix methods. We find that such models with broken spin rotation invariance {\it generically} lie in one of two topologically distinct localized phases. Close enough to the critical point separating the two phases, the system has a power-law divergent low-energy density of states (with a non-universal continuously varying power-law) in either phase, due to quantum Griffiths singularities. This critical point belongs to the same infinite-disorder universality class as the one dimensional particle-hole symmetric Anderson localization problem, while the Griffiths phases in the vicinity of the transition are controlled by lines of strong (but not infinite) disorder fixed points terminating in the critical point.

Density of states for dirtyd-wave superconductors: A unified and dual approach for different types of disorder

A two-parameter field theoretical representation is given of a 2-dimensional dirty d-wave superconductor that interpolates between the Gaussian limit of uncorrelated weak disorder and the unitary limit of a dilute concentration of resonant scatterers. It is argued that a duality holds between these two regimes from which follows that a linearly vanishing density of states in the Gaussian limit transforms into a diverging one in the unitary limit arbitrarily close to the Fermi energy.

Magnetic pair breaking in superconductingBa1−xKxBiO3investigated by magnetotunneling

The de Gennes and Maki theory of gapless superconductivity for dirty superconductors is used to interpret the tunneling measurements on the strongly type-II high-Tc oxide-superconductor Ba1-xKxBiO3 in high magnetic fields up to 30 Tesla. We show that this theory is applicable at all temperatures and in a wide range of magnetic fields starting from 50 percent of the upper critical field Bc2. In this magnetic field range the measured superconducting density of states (DOS) has the simple energy dependence as predicted by de Gennes from which the temperature dependence of the pair-breaking parameter alpha(T), or Bc2(T), has been obtained. The deduced temperature dependence of Bc2(T) follows the Werthamer-Helfand-Hohenberg prediction for classical type-II superconductors in agreement with our previous direct determination. The amplitudes of the deviations in the DOS depend on the magnetic field via the spatially averaged superconducting order parameter which has a square-root dependence on the magnetic field. Finally, the second Ginzburg-Landau parameter kappa2(T) has been determined from the experimental data.

Disordered 2d quasiparticles in class D: Dirac fermions with random mass, and dirty superconductors

Disordered noninteracting quasiparticles that are governed by a Majorana-type Hamiltonian -- prominent examples are dirty superconductors with broken time-reversal and spin-rotation symmetry, or the fermionic representation of the 2d Ising model with fluctuating bond strengths -- are called class D. In two dimensions, weakly disordered systems of this kind may possess a metallic phase beyond the insulating phases expected for strong disorder. We show that the 2d metal phase emanates from the free Majorana fermion point, in the direction of the RG trajectory of a perturbed WZW model. To establish this result, we develop a supersymmetric extension of the method of nonabelian bosonization. On the metallic side of the metal-insulator transition, the density of states becomes nonvanishing at zero energy, by a mechanism akin to dynamical mass generation. This feature is explored in a model of N species of disordered Dirac fermions, via the mapping on a nonlinear sigma model, which encapsulates a Z_2 spin degree of freedom. We compute the density of states in a finite system, and obtain agreement with the random-matrix prediction for class D, in the ergodic limit. Vortex disorder, which is a relevant perturbation at the free-fermion point, changes the density of states at low energy and suppresses the local Z_2 degree of freedom, thereby leading to a different symmetry class, BD.

Magnetotunneling and magnetic pair-breaking in superconducting Ba 1− xK xBiO 3

The gapless superconductivity theory for dirty superconductors has been used to interpret the tunneling measurements on the strongly type-II high- T c oxide-superconductor Ba 1− x K x BiO 3 in high magnetic fields up to 30 T . This theory describes the tunneling curves at all temperatures below T c not only near the upper critical field H c2 but in a wide range of magnetic fields H>0.5 H c2, yielding an estimation of the pair-breaking parameter α. This allows us to determine the temperature dependence of the upper critical field H c2( T) which is in agreement with the predictions of the classical Werthamer–Helfand–Hohenberg theory. Our results are comparable with H c2 data directly obtained from the tunneling experiment.

Keldysh action for disordered superconductors

Keldysh representation of the functional integral for the interacting electron system with disorder is used to derive microscopically an effective action for dirty superconductors. In the most general case this action is a functional of the 8 x 8 matrix Q(t,t') which depends on two time variables, and on the fluctuating order parameter field and electric potential. We show that this approach reproduces, without the use of the replica trick, the well-known result for the Coulomb-induced renormalization of the electron-electron coupling constant in the Cooper channel. Turning to the new results, we calculate the effects of the Coulomb interaction upon: i) the subgap Andreev conductance between superconductor and 2D dirty normal metal, and ii) the Josephson proximity coupling between superconductive islands via such a metal. These quantities are shown to be strongly suppressed by the Coulomb interaction at sufficiently low temperatures due to both zero-bias anomaly in the density of states and disorder-enhanced repulsion in the Cooper channel.

The second Josephson harmonic in the dirty limit

We calculate the Josephson current across a plane interface separating dirty superconductors up to second order in the transparency of the junction. The leading contribution to the second-order correction is caused by the suppression of the superconducting order parameter near the interface. The ratio of the correction to the Josephson current in the first order is about D(T)/l, where D is the transparency of the junction, (T) is the temperature-dependent coherence length, and l is the mean free path. This correction gives rise to the second harmonic in the current-phase relationship.

Charge carrier dynamics and the critical vortex velocity in high-Tc superconductors

A model is presented to explain the recently observed voltage jumps in Bi 2 Sr 2 CaCu 2 O 8+δ superconducting thin films. The present approach provides additional evidence of the validity of the flux-flow instability model. The field and temperature dependences of the critical vortex velocity were attributed to the physics of the vortex core in clean superconductors (high-T c superconductors) which is very different from the physics of the vortex core in dirty superconductors. The quasiparticle diffusion coefficient D qp was calculated based on the assumption that the quasiparticle motion is considered as a transport phenomenon on random walks. In addition, the effects of scattering of normal excitations by quasiparticles localized in the vortex core moving together with the vortex on the inelastic scattering time of quasiparticles τ in was also calculated. The predictions of the model and the experimental observations are in excellent agreement.

Quasiparticle density of states in dirty high-Tcsuperconductors

We study the density of quasiparticle states of dirty d-wave superconductors. We show the existence of singular corrections to the density of states due to quantum interference effects. We then argue that the density of states actually vanishes in the localized phase as $|E|$ or ${E}^{2}$ depending on whether time-reversal is a good symmetry or not. We verify this result for systems without time-reversal symmetry in one dimension using supersymmetry techniques. This simple, instructive calculation also provides the exact universal scaling function for the density of states for the crossover from ballistic to localized behavior in one dimension. Above two dimensions, we argue that in contrast to the conventional Anderson localization transition, the density of states has critical singularities which we calculate in a $2+\ensuremath{\epsilon}$ expansion. We discuss consequences of our results for various experiments on dirty high-${T}_{c}$ materials.

Quasiparticle Density of States of Clean and Dirtys-Wave Superconductors in the Vortex State

The quasiparticle density of states (DOS) in the vortex state has been probed by specific heat measurements under magnetic fields (H) for clean and dirty s-wave superconductors, Y(Ni1-xPtx)2B2C and Nb1-xTaxSe2. We find that the quasiparticle DOS per vortex is appreciably H-dependent in the clean-limit superconductors, while it is H-independent in the dirty superconductors as expected from a conventional rigid normal electron core picture. We discuss possible origins for our observations in terms of the shrinking of the vortex core radius with increasing H.