We calculate the Pauli-limited upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) instability for {\it dirty} d-wave superconductors within the quasiclassical theory using the self-consistent $\hat{t}$-matrix approximation for impurities. We find that the phase diagram depends sensitively on the scattering rate and phase shift of nonmagnetic impurities. The transition into the superconducting state is always second order for weak (Born) scattering, while in the unitarity (strong) scattering limit a first-order transition into both uniform and spatially modulated superconducting states is stabilized. Contrary to general belief, we find that the FFLO phase is robust against disorder and survives impurity scattering equivalent to a $T_c$ suppression of roughly 40%. Our results bear on the search of FFLO states in heavy-fermion and layered organic superconductors.
Green function techniques for studying nonequilibrium processes in dirty two-band superconductors are discussed. Perturbation expansions and Green function equations are developed. A time dependent modification of the Usadel equation is given.
In this paper we study the effect of non-magnetic impurities on two-band $s_{\pm}$ superconductors by deriving the corresponding Ginzburg-Landau (GL) equation. Depending on the strength of (impurity-induced) inter-band scattering we find that there are two distinctive regions where the superconductors behave very differently. In the strong impurity induced inter-band scattering regime $T_c<<τ^{-1}_t$, where $τ_t\sim$ mean-life time an electron stays in one band the two-band superconductor behaves as an effective one-band dirty superconductor. In the other limit $T_c\geqτ^{-1}_t$, the dirty two…
Theories and experiments on dirty superconductors are sophisticated but important for both fundamentals and applications. It becomes more challenging when magnetic fields are present, because the field distribution, the electron density of states, and the superconducting pairing potentials are nonuniform. Here we present tunneling microspectroscopic experiments on NbC single crystals and show that NbC is a homogeneous dirty superconductor. When applying magnetic fields to the sample, we observe that the zero-energy local density of states and the pairing energy gap follow an explicit scale rel…
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.
Elastic forward scattering can lead to an anomaly in the ARPES spectrum of the cuprate superconductors. Here we discuss how this anomaly can be used to provide a measurement of the superconducting gap Delta_k for k values away from the Fermi surface.
Magnetic properties of dirty multiband superconductors near the upper critical field are studied. The parameter $κ_2$ characterizing magnetization slope is shown to have a significant temperature variation which is quite sensitive to the pairing interactions and relative strengths of intraband impurity scattering. In contrast to single-band superconductors the increase of $κ_2$ at low temperatures can be arbitrary large determined by the ratio of minimal and maximal diffusion coefficients in different bands. Temperature dependencies of $κ_2(T)$ in two-band MgB$_2$ and iron-based superconductor…
We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, Volovik has argued that the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution δC(H)\ltsim H^{1/2} . We show that the extended states continue to dominate the dirty limit, but lead to a H \log H behavior at the lowest fields, H_{c1}\ll H\ll H_{c2}. This crossover may explain recent discrepancies in specicific heat measurements at…
We discuss the symmetry property of a nodal superconductor that hosts robust flat-band zero-energy states at its surface under potential disorder. Such robust zero-energy states are known to induce the anomalous proximity effect in a dirty normal metal attached to a superconductor. A recent study has shown that a topological index ${\cal N}_\mathrm{ZES}$ describes the number of zero-energy states at the dirty surface of a $p$-wave superconductor. We generalize the theory to clarify the conditions required for a superconductor that enables ${\cal N}_\mathrm{ZES}\neq 0$. Our results show that ${…
We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution δC(H)\sim H^{1/2}. We show that the extended states continue to dominate the dirty limit, but lead to a H \log H behavior at the lowest fields, H_{c1}\ltsim H\ll H_{c2}. This crossover may explain recent discrepancies in specific heat measurements at low temperatures and fiel…
We study how quasiparticle interactions affect their localization properties in dirty superconductors with broken time reversal symmetry -- for example in a magnetic field. For SU(2) spin-rotation invariant (class C) systems, the only important coupling is the spin-spin triplet interaction, which we study within a renormalization group approach. Either an additional Zeeman coupling or a complete breaking of spin rotation symmetry renders all interactions irrelevant. These two situations realize, respectively, the non-interacting unitary Anderson and the ``thermal'' (class D) universality class…
The conductivity of vortex lattices in multiband superconductors with high concentration of impurities is calculated based on microscopic kinetic theory. Both the limits of high and low fields are considered, when the magnetic induction is close to or much smaller than the critical field strength $H_{c2}$, respectively. It is shown that in contrast to single-band superconductors the resistive properties are not universal but depend on the pairing constants and ratios of diffusivities in different bands. The low-field magneto-resistance can strongly exceed Bardeen-Stephen estimation in a quanti…
The proximity effect and the Josephson current in a superconductor-insulator-ferromagnet-superconductor (SIFS) junction are investigated within the framework of the quasiclassical Eilenberger equations. This investigation allows us to compare the dirty and the clean limits, to investigate an arbitrary impurity scattering, and to determine the applicability limits of the Usadel equations for such structures. The role of different types of the FS interface is analyzed. It is shown that the decay length and the spatial oscillation period of the Eilenberger function may exhibit a nonmonotonic depe…
An overview of the theory of the upper critical field in dirty two-gap superconductors, with a particular emphasis on MgB$_2$ is given. We focus here on the maximum $H_{c2}$ which may be achieved by increasing intraband scattering, and on the limitations imposed by weak interband scattering and paramagnetic effects. In particular, we discuss recent experiments which have recently demonstrated ten-fold increase of $H_{c2}$ in dirty carbon-doped films as compared to single crystals, so that the $H_{c2}(0)$ parallel to the ab planes may approach the BCS paramagnetic limit, $H_p[T] = 1.84T_c[K] \s…
The local differential tunneling conductance on a Zn impurity in a disordered d-wave superconductors is studied. Quantum interference between many impurities leads to definitive quasiparticle spectra. We suggest that an elaborate analysis on impurity-induced spectra with quantum interference effect included may be able to pin down the sign and strength of the scattering potential of a Zn impurity in low density limit. Numerical simulations calculated with appropriately determined impurity parameters are in satisfactory agreement with the observations from scanning tunneling microscopy (STM) ex…
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 b…
We consider the differential sum rule for the effective scattering rate $% 1/τ(ω)$ and optical conductivity $σ_{1}(ω) $ in a dirty BCS superconductor, for arbitrary ratio of the superconducting gap $% Δ$ and the normal state constant damping rate $1/τ$. We show that if $τ$ is independent of $T$, the area under $1/τ(ω)$ does not change between the normal and the superconducting states, i.e., there exists an exact differential sum rule for the scattering rate. For \textit{any} value of the dimensionless parameter $Δτ$, the sum rule is exhausted at frequencies controlled by $Δ$. %but the numerica…
We study the effects of the Lifshitz transition from closed to open Fermi surface in dirty topological insulators with the nematic superconductivity near the critical temperature. We solve linearized Gor'kov equations and find that the nematic superconductor with an open Fermi surface has a lower critical temperature and more susceptible to the disorder than the superconductor with the closed Fermi surface. We propose that correspondence between the critical temperature and stability against the disorder is the general feature of the superconductivity. We investigate the effects of the Lifshit…
Recent studies have emphasized the importance of impurity scattering for the optical Higgs response of superconductors. In the dirty limit, an additional paramagnetic coupling of light to the superconducting condensate arises which drastically enhances excitation. So far, most work concentrated on the periodic driving with light, where the third-harmonic generation response of the Higgs mode was shown to be enhanced. In this work, we additionally calculate the time-resolved optical conductivity of single- and two-band superconductors in a two-pulse quench-probe setup, where we find good agreem…
We study the structure of Abrikosov vortices in two-band superconductors for different external magnetic fields and different parameters of the bands. The vortex core size determined by the coherence lengths are found to have qualitatively different behaviour from that determined by the quasiparticle density of states spatial variation. These different vortex core length scales coincide near the upper critical field, while the discrepancy between them becomes quite significant at lower fields. Within the diffusive approximation we demonstrate several generic regimes in the field dependence of …
In this paper, a delocalization-localization transition within the superconducting state is explored. The symmetries of the Bogoliubov deGennes Hamiltonian endow the two associated superconducting phases - the thermal metal and the thermal insulator-, and the critical point between them with properties that are in stark contrast with analogous phases in normal systems. Here, for systems preserving spin rotational invariance and time reversal symmetry in three dimensions, the transition is shown to be of a different universality class from its normal partner by extracting and comparing the loca…
We developed microscopic theory of Josephson effect in point contacts between dirty two-band superconductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. This expression was used for calculation of current-phase relations and temperature dependences of critical current with application to MgB2 superconductor. Also we have considered influence on contact characteristics interband scattering effect appeared in case of dirty superconductors. It is shown that the correction to Josephson current due to the interband scattering depends on…
The Josephson effect in p-wave superconductor / diffusive normal metal / p-wave superconductor junctions is studied theoretically. Amplitudes of Josephson currents are several orders of magnitude larger than those in s-wave junctions. Current-phase (J-ϕ) relations in low temperatures are close to those in ballistic junctions such as J\propto\sin(ϕ/2) and J\proptoϕeven in the presence of random impurity potentials. A cooperative effect between the midgap Andreev resonant states and the proximity effect causes such anomalous properties and is a character of the spin-triplet superconductor juncti…
We discuss the Josephson current in superconductor / dirty normal conductor / superconductor junctions, where the superconductors have $d_{x^2-y^2}$ pairing symmetry. The low-temperature behavior of the Josephson current depends on the orientation angle between the crystalline axis and the normal of the junction interface. We show that the ensemble-averaged Josephson current vanishes when the orientation angle is $π/4$ and the normal conductor is in the diffusive transport regime. The $d_{x^2-y^2}$-wave pairing symmetry is responsible for this fact.
We investigate the phase diagram of dirty two-band superconductors. This paper primarily focuses on the properties and observability of the time-reversal symmetry-breaking $s+is$ superconducting states, which can be generated in two-band superconductors by interband impurity scattering. We show that such states can appear in two distinct ways. First, according to a previously discussed scenario, the $s+is$ state can form as an intermediate phase at the impurity-driven crossover between $s_{\pm}$ and $s_{++}$ states. We show that there is a second scenario where domains of the $s+is$ state exis…
A class of hydrogen-rich (H-rich) systems, consistent with Type II superconductivity, are known to have very high superconducting transition temperatures Tc and upper critical magnetic field Bc2 values (up to 288 K and 222 T respectively). By looking at all the experimental H-rich superconductors reported to date most are in the dirty limit, with only the highest Tc values being in the crossover to the clean regime. In this framework, there is a clear understanding of some previous unexplained behaviors: (1) A maximum in Bc2 as a function of Tc; (2) a clear change in slope in Tc as a function …
A scaling relation ρ_s \simeq 35σ_{dc}T_c has been observed in the copper-oxide superconductors, where ρ_s is the strength of the superconducting condensate, T_c is the critical temperature, and σ_{dc} is the normal-state dc conductivity close to T_c. This scaling relation is examined within the context of a clean and dirty-limit BCS superconductor. These limits are well established for an isotropic BCS gap 2Δand a normal-state scattering rate 1/τ; in the clean limit 1/τ\ll 2Δ, and in the dirty limit 1/τ> 2Δ. The dirty limit may also be defined operationally as the regime where ρ_s varies with…
We reconsider the force-balance relation on an isolated vortex in the flux flow state within the scheme of time-dependent-Ginzburg-Landau (TDGL) equation. We define force on the vortex by the total force on superconducting electrons in the region $S$ surrounding the vortex. We derive the local momentum balance relation of superconducting electrons and then find the force-balance relation on isolated vortex with taking account of the fact that the transport current in charged superconductors are inherently spatially varying with the scale of penetration depth $λ$. We also find that nature of th…
In multiband systems, such as iron-based superconductors, the superconducting states with locking and anti-locking of the interband phase differences, are usually considered as mutually exclusive. For example, a dirty two-band system with interband impurity scattering undergoes a sharp crossover between the $s_{\pm}$ state (which favors phase anti locking) and the $s_{++}$ state (which favors phase locking). We discuss here that the situation can be much more complex in the presence of an external field or superconducting currents. In an external applied magnetic field, dirty two-band supercon…
The contributions of superconducting fluctuations to the specific heat of dirty superconductors are calculated, including quantum and classical corrections to the `usual' leading Gaussian divergence. These additional terms modify the Ginzburg criterion, which is based on equating these fluctuation-generated contributions to the mean-field discontinuity in the specific heat, and set limits on its applicability for materials with a low transition temperature.
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.
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 $H_{c2}(θ)$ as a function of the angle $θ$ between ${\bf H}$ and the c-axis is obtained. It is shown that the shape of the $H_{c2}(T)$ curve essen…
We theoretically study the proximity effect in the thin-film layered ferromagnet (F) - superconductor (S) heterostructures in F$_1$F$_2$S design. We consider the boundary value problem for the Usadel-like equations in the case of so-called "dirty" limit. The "latent" superconducting pairing interaction in F layers taken into account. The focus is on the recipe of experimental preparation the state with so-called solitary superconductivity. We also propose and discuss the model of the superconducting spin valve based on F$_1$F$_2$S trilayers in solitary superconductivity regime.
The ultra-fast dynamics of superconducting vortices harbors rich physics generic to nonequilibrium collective systems. The phenomenon of flux-flow instability (FFI), however, prevents its exploration and sets practical limits for the use of vortices in various applications. To suppress the FFI, a superconductor should exhibit a rarely achieved combination of properties: weak volume pinning, close-to-depairing critical current, and fast heat removal from heated electrons. Here, we demonstrate experimentally ultra-fast vortex motion at velocities of 10-15 km/s in a directly written Nb-C supercon…
We investigate the vortex state in a two-band superconductor with strong intraband and weak interband electronic scattering rates. Coupled Usadel equations are solved numerically and the distributions of the pair potentials and local densities of states are calculated for two bands at different values of magnetic fields. The existence of two distinct length scales corresponding to different bands is demonstrated. The results provide qualitative interpretation of recent STM experiments on vortex structure imaging in MgB_2.
The s+- state in which order parameter has different signs in different bands is a leading candidate for the superconducting state in the iron based superconductors. We investigate a Josephson junction between s and s+- superconductors within microscopic theory. Frustration, caused by interaction of the s-wave gap parameter with the opposite-sign gaps of the s+- superconductor, leads to nontrivial phase diagram. When the partial Josephson coupling energy between the s-wave superconductor and one of the s+- bands dominates, s-wave gap parameter aligns with the order parameter in this band. In t…
The minimal magnetic field $H_{c2}$ destroying superconductivity in the {\it bulk} of a superconductor is smaller than the magnetic field $H_{c3}$ needed to destroy {\it surface} superconductivity if the surface of the superconductor coincides with one of the crystallographic planes and is parallel to the external magnetic field. While for a dirty single-band superconductor the ratio of $H_{c3}$ to $H_{c2}$ is a universal temperature-independent constant 1.6946, for dirty two-band superconductors this is not the case. I show that in the latter case the interaction of the two bands leads to a n…
We study the proximity-induced superconducting correlations as well as the local density of states of a ferromagnet, in a ferromagnet/s-wave superconductor heterostructure. We include the effects of spin-flip scattering, non-ideal interfaces, and the presence of impurities in the sample. We employ the quasiclassical theory of superconductivity, solving the Usadel equation with emphasis on obtaining transparent analytical results. As our main result, we report that in a certain parameter regime the spatial oscillations of the anomalous (superconducting) part of the Green's function induced in t…
We study theoretically the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in a two-band superconductor characterized by an equal-time s-wave interband pairing order parameter. The Fermi-Dirac statistics of electrons allows a spin-triplet s-wave pairing order as well as a spin-singlet s-wave order parameter due to the two-band degree of freedom. In a spin-singlet superconductor, $T_c$ is insensitive to the impurity concentration when we estimate the self-energy due to the random impurity potential within the Born approximation. On the other hand in …
Optical conductivity of a dirty current-carrying superconductor
L. D. Landau Institute for Theoretical Physics（列夫·朗道理论物理研究所）；Laboratory for Condensed Matter Physics, HSE University（高等经济学院凝聚态物理实验室）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Recent ARPES experiments show a narrow quasiparticle peak at the gap edge along the antinodal [1,0]-direction for the overdoped cuprate superconductors. We show that within weak coupling BCS theory for a d-wave superconductor the s-wave single-impurity scattering cross section vanishes for energies of the gap edge. This coherence effect occurs through multiple scattering off the impurity. For small impurity concentrations the spectral function has a pronounced increase of the (scattering) lifetime for antinodal quasiparticles but shows a very broad peak in the nodal direction, in qualitative a…
We discuss the stability of highly degenerate zero-energy states tha appear at the surface of a nodal superconductor preserving time-reversal symmetry. The existence of such surface states is a direct consequence of the nontrivial topological numbers defined in the restricted Brillouin zones in the clean limit. In experiments, however, potential disorder is inevitable near the surface of a real superconductor, which may lift the high degeneracy at zero energy. We show that an index defined in terms of the chiral eigenvalues of the zero-energy states can be used to measure the degree of degener…
I obtain the inverse of the correlation length exponent at the superfluid-Bose glass quantum critical point as a series in small parameter $\sqrt{d-1}$, with d being the dimensionality of the system, and compute the first two terms using a novel field-theoretic technique. For d=2 I find $ν_s = 0.81$ and $ν_c = 1.03$, for short-range and Coulomb interactions between bosons, respectively. When combined with the exact values of the dynamical critical exponents $z_s = d$ and $z_c = 1$, these results are in quantitative agreement with the experiments on onset of superfluidity in $^4 He$ in porous g…
We study the dissipative conductivity $σ_1$ of a dirty superconductor with a finite Dynes parameter $Γ$ under a dc-biased weak time-dependent field. The Usadel equation for the current-carrying state is solved to calculate the pair potential, penetration depth, supercurrent density, and quasiparticle spectrum. It is shown that, while the depairing current density $j_d$ for $Γ=0$ is coincident with the Kupriyanov-Lukichev theory, a finite $Γ$ decreases the superfluid density, resulting in a reduction of $j_d$. The broadening of the peaks of the quasiparticle density of states induced by a combi…
The formalism for analysing the magnetic field distribution in Pauli limited superconductors developed earlier is applied to the field dependence of the vortex lattice static linewidth measured in Muon Spin Rotation ($μ$SR) experiments. In addition to writing analytical formulae for the static linewidth for the vortex structure in the limit of independent vortices (i.e. moderate magnetic fields), we use Abrikosov's analysis to describe the field variations of the static linewidth at the approach of the superconductor to metal transition in the limit where the critical field is determined by Pa…
Disorder in two-band superconductors with repulsive interband interaction induces a frustrated competition between the phase-locking preferences of the various potential and kinetic terms. This frustrated interaction can result in the formation of an $s+is$ superconducting state, that breaks the time-reversal symmetry. In this paper we study the normal modes and their associated coherence lengths in such materials. We especially focus on the consequences of the soft modes stemming from the frustration and time-reversal-symmetry breakdown. We find that two-bands superconductors with such impuri…
The detailed optical properties of the multiband iron-chalcogenide superconductor FeTe$_{0.55}$Se$_{0.45}$ have been reexamined for a large number of temperatures above and below the critical temperature $T_c=14$ K for light polarized in the a-b planes. Instead of the simple Drude model that assumes a single band, above $T_c$ the normal-state optical properties are best described by the two-Drude model that considers two separate electronic subsystems; we observe a weak response ($ω_{p,D;1}\simeq 3000$ cm$^{-1}$) where the scattering rate has a strong temperature dependence ($1/τ_{D,1}\simeq 3…
This is a Comment on a recent Letter (Phys.Rev.Lett.93, 247002 (2004)) written by Dalidovich and Yang. It is pointed out that any Pauli-limited approach neglecting the vortices is inapplicable to describing the FFLO state in CeCoIn5.
I study the effects of disorder on the superconductivity of a three-band model with repulsive interband pairing. Such a model can support several possible superconducting order parameters, including a complex time-reversal symmetry breaking (TRSB) state. Impurity scattering suppresses the critical temperature of all these states, but the complex state survives, and remains a part of the phase diagram of the model even in the presence of moderate amount of disorder. This means that the TRSB states could be experimentally accessible in multiband materials like iron pnictides and chalcogenides.
We study the transport properties of ultrathin disordered nanowires in the neighborhood of the superconductor-metal quantum phase transition. To this end we combine numerical calculations with analytical strong-disorder renormalization group results. The quantum critical conductivity at zero temperature diverges logarithmically as a function of frequency. In the metallic phase, it obeys activated scaling associated with an infinite-randomness quantum critical point. We extend the scaling theory to higher dimensions and discuss implications for experiments.
We investigate the upper critical field in a dirty two-band superconductor within quasiclassical Usadel equations. The regime of very high anisotropy in the quasi-2D band, relevant for MgB$_{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 $T_{c}$, i…
We study the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in a two-band superconductor, where we assume the equal-time spin-singlet s-wave pair potential in each conduction band and the hybridization between the two bands as well as the band asymmetry. In the clean limit, the phase of hybridization determines the stability of two states: called $s_{++}$ and $s_{+-}$. The interband impurity scatterings decrease $T_c$ of the two states exactly in the same manner when the Hamiltonian preserves time-reversal symmetry. We find that a superconductor wi…
Diamagnetically coupled magnetic field can be used as a probe of nodal positions in unconventional superconductors. The heat capacity depends on the angle $ϕ_0$ between the magnetic field and the nodal directions. We show that the anisotropy $C(ϕ_0)$ persists even in systems with strong paramagnetic coupling to the electrons's spins.
We present a nearly perfect Pauli-limited critical field phase diagram for the anisotropic organic superconductor \$α$-(ET)$_2$NH$_4$(SCN)$_4$ when the applied magnetic field is oriented parallel to the conducting layers. The critical fields (${H_{c_2}$) were found by use of penetration depth measurements. Because ${H_{c_2}$ is Pauli-limited, the size of the superconducting energy gap can be calculated. The role of spin-orbit scattering and many-body effects play a role in explaining our measurements.
We investigate the multiquantum vortex states in type-II superconductor both in "clean" and "dirty" regimes defined by impurity scattering rate. Within quasiclassical approach we calculate self-consistently the order parameter distributions and electronic local density of states (LDOS) profiles. In the clean case we find the low temperature vortex core anomaly predicted analytically in G.E. Volovik, JETP Lett. {\bf 58}, 455 (1993) and obtain the interference patterns in LDOS distributions analogous to found previously in the framework of Bogolubov-de Gennes theory. In dirty regime the multiqua…
This work performs a comparative numerical study of the impurity average self-frequency $\tildeω$ in an unconventional superconducting alloy with non-magnetic impurities. Two methods are used: the Levenberg-Marquardt algorithm as a non-linear minimization problem, and a fixed-point iteration procedure. The unconventional superconducting renormalized by impurities \bw is a self-consistent complex non-linear equation with two varying parameters: the impurity concentration $Γ^+$ and the strength of the impurities c, for which its numerical solution is a computational challenge. Throughout this st…
