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The effect of weak anisotropic (momentum-dependent) impurity scattering in unconventional superconductors has been investigated. It is shown that the anisotropic scattering can lead either to a small reduction or a small enhancement of the isotropic pair-breaking effect. The influence of the anisotropy of the scattering potential becomes significant for the order parameters with large Fermi surface average values. In that case an unexpected enhancement (up to 10%) of the critical temperature over the critical temperature in the absence of impurities is predicted for a small impurity concentration., Comment: 12 pages, RevTeX, 3 PostScript figures

Transport properties of antiferromagnetic superconductors with T{sub N} < {Tc} were investigated. Detailed numerical, results are given for SmRh{sub 4}B{sub 4} by using the following model. The paramagnetic phase (T{sub N} < T < {Tc}) is described by using Abrikosov-Gorkov theory of magnetic ions in an ordinary superconductor. In the AF phase (T {le} T{sub N}), the effects of the molecular field H{sub Q}(T) and the elastic scattering of conduction electrons from spin fluctuations are included. Expressions for H{sub Q}(T) and the scattering rate from spin fluctuations were derived. The aim was to see if properties are enhanced or depressed by the AF ordering occurring below T{sub N}. It is found that whereas the electronic thermal conductivity K{sub s}, nuclear spin relaxation rate R{sub s}, and the longitudinal ultrasonic attenuation a{sub s}, are depressed by the AF ordering, the inverse of the magnetic penetration depth, ({lambda}(T)){sup {minus}1}, is enhanced below T{sub N}. The effect of all types of impurities is included in the study. Theoretical results for K{sub s} and ({lambda}(T)){sup {minus}1} agree with the experimental values for SmRh{sub 4}B{sub 4} (experimental data for other properties are not available in literature). In the above, T{sub N} is the Neel temperaturemore » and {Tc} is superconducting transition temperature.« less

The superconducting density of states ϱ S (ω) and the tunneling differential conductivity σ ns (V) for an antiferromagnetic superconductor (AFSC) are calculated both in the paramagnetic and in the antiferromagnetic phases. Our model for AFSC includes the effects of the molecular field, impurities and elastic scattering from spin-fluctuations. The experimental data on σ ns (V) for TbMO 6 S 8 is understood reasonably well.

Effects of all types of impurities (nonmagnetic, magnetic, and spin-orbit) on an antiferromagnetic superconductor (AFSC) have been investigated by studying the transition temperatureT c and the specific heat jump. We have assumed a one-dimensional electron band. The impurity scattering is treated within the self-consistent Born approximation. We find that: (a) the molecular fieldH Q and the magnetic impurities depress superconductivity of AFSC and their pair-breaking effect is additive; (b) the effect of spin-orbit impurities is the same as that of nonmagnetic impurities—these enhance superconductivity by screening the molecular field; and (c) in the extreme dirty limit, the AFSC is described in terms of an effective pair-breaking parameter given by 1/τeff=1/τ2+H Q 2 τ where 1/τ=1/τ1+2/3τso(1/τ1, 1/τ2, and 1/τso, respectively, are the scattering rates from nonmagnetic, magnetic and spin-orbit impurities).

We present a theory of combined nonmagnetic and magnetic impurity scattering in anisotropic superconductors accounting for the momentum-dependent impurity potential. Applying the model to the d-wave superconducting state, we obtain a quantitative agreement with the initial suppression of the critical temperature due to Zn and Ni substitutions as well as electron irradiation defects in the cuprates. We suggest, that the unequal pair-breaking effect of Zn and Ni may be related to a different nature of the magnetic moments induced by these impurities., 5 pages, 3 tables, RevTex, to be published in Phys. Rev. B

We discuss a case of a strong anisotropic impurity scattering within the model introduced in our previous paper [Phys. Rev. B54, 15463 (1996)] and clarify our former statement about a possible enhancement of the critical temperature in this scattering regime. We show that for the anisotropy of the impurity potential determined by the functions from the non-identity irreducible representations of the crystal point group, the critical temperature decreases with the impurity scattering rate which is consistent with a generally understood role of disorder in superconductors., Comment: 6 pages, RevTex

The specific heat jump at a normal-superconducting phase transition in an anisotropic superconductor with nonmagnetic impurities is calculated within a weak-coupling mean field approximation. It is shown that its dependence on the impurity concentration is remarkably different for d-wave and (d+s)-wave states. This effect may be used as a test for the presence of an s-wave admixture in the cuprates., 16 pages, RevTex, 8 PostScript figures, to be published in Phys. Rev. B (May 1, 1997)

A theory of nonmagnetic impurities in an anisotropic superconductor including the effect of anisotropic (momentum-dependent) impurity scattering is given. It is shown that for a strongly anisotropic scattering the reduction of the pair-breaking effect of the impurities is large. For a significant overlap between the anisotropy functions of the scattering potential and that of the pair potential and for a large amount of anisotropic scattering rate in impurity potential the superconductivity becomes robust vis a vis impurity concentration. The implications of our result for YBCO high-temperature superconductor are discussed., 14 pages, RevTeX, 5 PostScript figures, to be published in Phys. Rev. B (December 1, 1996)

The effect of a homogeneous magnetic field and nonmagnetic impurities on the critical temperature of an anisotropic superconductor has been investigated. The role of these pair-breakers in relation to the anisotropy of the order parameter is clarified., Comment: 7 pages, RevTeX, 1 PostScript figure

The doping and the temperature dependence of various normal-state properties of high-temperature superconductors are studied by applying the t-t'-J model. The slave-boson method is used to treat the strong correlations. The fermion quasiparticles carry both charge and spin and the volume of the Fermi surface satisfies the Luttinger theorem. The quasiparticle energy spectrum used in the calculation is obtained from a mean-field treatment of the t-t'-J model

The effect of a uniform magnetic field on a superconductor containing magnetic impurities has been investigated by treating the scattering of conduction electrons by magnetic impurities exactly (within the Shiba-Rusinov model). Our generalized equations defining the single-particle Green function reduce to earlier known results in appropriate limits. As an application of our equations we have calculated the transition temperature T c. For fixed values of the parameter ɛ0 (the normalized position of the bound state within the BCS gap), the parameter p (the normalized magnetic field), and the parameter α (=〈S z 2/S2), the detailed dependence of T c on the normalized impurity concentration ovc has been shown. The dependence of T c on p for fixed values of ɛ0, ovc, and α has also been shown. Our results are significantly different from the earlier known results.

Using the Anderson model in the nonmagnetic limit, we have calculated some properties of the superconducting virtual-bound-state alloys. The calculated properties are the transition temperature ${T}_{c}$, the jump in specific heat at ${T}_{c}$, $\ensuremath{\Delta}C$, the electronic density of states, and the tunneling conductance. Special attention is paid to the systematic variation of these properties with the resonance width $\ensuremath{\Gamma}$. It is found that the initial slope of the normalized $\ensuremath{\Delta}C$ versus the normalized ${T}_{c}$ curve has a maximum value of 3.638, which is the highest value reported in the literature for the impurity problem. Our tunneling study suggests that for an electron-tunneling search of bound states in alloys with relatively large $\ensuremath{\Gamma}$, an ultra-low-temperature experiment is desirable.

Generalized Eliashberg equations including the effects of Kondo impurities in strong coupling superconductors are given. The Kondo effect is treated using the Matsuura, Ichinose, Nagaoka (MIN) theory. The transition temperatureT c is calculated from the general equations by using the “square well” model for the electron-phonon interaction. Deviations of the present results from those of MIN are significant. In general, the depression ofT c by the impurities is softened.

Assuming a one-dimensional electron band and examining the free energy difference δF (=F S−F N) for an antiferromagnetic superconductor in the presence of a homogeneous magnetic field H0, it is shown that (i) for temperatureT=0, the spatially varying order parameter Δ Q (having the same period as the antiferromagnetic fieldH Q) and the BCS order parameter Δ coexist when |Δ−H Q≤H 0

The effect of paramagnetic impurities on superconductors with energy-gap anisotropy is studied using the Shiba-Rusinov theory and treating the ordinary and exchange interactions equally. The Abrikosov-Gor'kov-theory results are a special case of our calculations. The transition temperature ${T}_{c}$ and the jump in specific heat at ${T}_{c}$ are calculated. The emphasis is to clarify the roles of the ordinary and the exchange interactions in the electron-impurity scattering. The difference between the results of the Shiba-Rusinov and Abrikosov-Gor'kov theories is brought out.

A theory of the longitudinal electron spin susceptibility ${\ensuremath{\chi}}_{s}$ for an antiferromagnetic superconductor (AFS) has been given. For the AFS, we have assumed a homogeneous superconducting order parameter and a one-dimensional electron band that satisfies the nesting condition ${\ensuremath{\varepsilon}}_{\mathrm{k}}$=-${\ensuremath{\varepsilon}}_{\mathrm{k}+\mathrm{Q}}$, where Q is the wave vector characterizing the antiferromagnetic (AF) order. First we have studied the dependence of ${\ensuremath{\chi}}_{s}$ on the scattering rates for the scattering of conduction electrons from the nonmagnetic, spin-orbit, and magnetic impurities by regarding ${H}_{Q}$ as a parameter and neglecting the spin-fluctuation effects (${H}_{Q}$ is the AF field). The effect of impurities is found to be significant. Then we have investigated the temperature dependence of ${\ensuremath{\chi}}_{s}$ by taking ${T}_{N}$${T}_{c}$ and by including the spin-fluctuation effects and the temperature dependence of the AF field (${T}_{N}$ is the AF ordering temperature and ${T}_{c}$ is the superconducting transition temperature). The aim has been to see if ${\ensuremath{\chi}}_{s}$ is enhanced or depressed by the AF ordering occurring below ${T}_{N}$. We find that (1) ${\ensuremath{\chi}}_{s}$ increases with the increase in scattering rate from spin-orbit impurities both above and below ${T}_{N}$; (2) keeping other parameters fixed, the enhancement or depression of ${\ensuremath{\chi}}_{s}$ below ${T}_{N}$ depends on ${H}_{Q}$(0)---there is enhancement (depression) when ${H}_{Q}$(0) is larger (smaller) [${H}_{Q}$(0) is the zero-temperature value of ${H}_{Q}$]; (3) nonmagnetic impurities have a dramatic effect on ${\ensuremath{\chi}}_{s}$ in the AF phase. For cleaner (dirtier) superconductors, ${\ensuremath{\chi}}_{s}$ is enhanced (depressed) below ${T}_{N}$; (4) in ${\mathrm{SmRh}}_{4}$${\mathrm{B}}_{4}$, one expects a depression in ${\ensuremath{\chi}}_{s}$ by the AF ordering.

Applying the mean-field theory, we investigate the effect of an external homogeneous magnetic field \({\vec H_O}\), and the staggered magnetic field \({\vec H_Q}\), arising from antiferromagnetism on superconductors with one-dimensional incommensurate bands. The superconducting order parameter is spatially homogeneous (BCS state) for the weak Ho and oscillates in space (FF state) for the strong Ho. The superconducting domains on HQ−Ho plane as temperature decreases is shown by calculating the ground state and the second-order phase transition temperature. It is found that the interplay between \({\vec H_O}\) and \({\vec H_Q}\) leads to the crucial differences between the domain of \({\vec H_O}\)//\({\vec H_Q}\) and that of \({\vec H_O} \bot {\vec H_Q}\). Further, the effect of the one-dimensional fluctuation is examined by use of the bosonization method.

The effect of a uniform magnetic field on the specific heat jump\({\Delta C}\) of a superconductor containing paramagnetic impurities has been investigated. The impurities are described by the Shiba-Rusinov model. Taking the normalized position of the bound state within the BCS gap e0=0.8 and 0.0 and the parameter α(=〈S z 2 〉/S2)=1.0 and 1/3 as typical values, we find the detailed dependence of\({\Delta C}\) on the impurity concentration\(\bar c\) and the transition temperature Tc. In some cases\({\Delta C}\) diverges at certain values of\(\bar c\) or Tc, signaling a change of the normal-superconductor phase transition from second order to first order.

We have extended the McMillan tunneling model (MTM) of the superconducting proximity effect to the case of strongly coupled normal-superconducting composites. The generalization has been carried out by treating the tunneling Hamiltonian to all orders of self-consistent perturbation theory. The electronic density of states, the real and imaginary parts of the order parameter, the energy gap, and the transition temperatureT c calculated in the generalized model are significantly different from the MTM results. Generally, the peak inN N (E) is broader, shifted, and damped. The energy gap is lower and the depression inT c is softened.

The Shiba model for the paramagnetic impurities in a superconductor has been generalized to include the effect of gap anisotropy. The calculated properties are the transition temperatureTc and the specific heat jump atTc. The changes brought in these properties by anisotropy are significant.

By the treatment of the electron-impurity scattering exactly by means of at matrix, the effect of nonmagnetic impurities on antiferromagnetic superconductors has been investigated in detail. Our study is an extension of the recent work of Nass, Levin, and Grest to the strong-scattering case. The appearance of bound states due to impurities for H/sub Q/>..delta.. is discussed (here H/sub Q/ is the antiferromagnetic molecular field and ..delta.. is the superconducting order parameter). Investigating the density of states (N(..omega..)), we show the growth of the ''impurity band'' with the increase in impurity concentration. The ''dirty''-limit behavior of the system and the condition for the appearance of the gaplessness are discussed analytically.

The McMillan tunneling model of the superconducting proximity effect has been used to calculate the specific heat jumps on the S side and on the N side of a proximity composite. The case of the BCS interaction constant on the N side, λN being identically zero and the case λN ≠ 0 have both been studied. The detailed dependence of the normalized specific heat jumps on the McMillan parameters γS and γN and on the transition temperature of the composite has been computed. A reasonable agreement between our theoretical values and recent experimental data of Bevolo et al. is obtained.

The single-particle excitation spectrum and the density of states (DOS) for a superconducting bilayer composite are calculated by solving the Bogoliubov equations. The bilayer consists of two different superconducting films in contact. The superconducting order parameters of the films Δ1 and Δ2 are assumed constant. For various film thicknesses the dependence of the DOS on the ratio Δ1/Δ2 is shown. This dependence is most prominent for states with excitation energy ɛ less than Δ2.

The problem of nonmagnetic impurities in a two-band superconductor has been investigated for the case of strong scattering of a conduction electron from an impurity. We use the Green's function method and introduce a t-matrix to treat the scattering. We find that the transition temperature T/sub c/ calculated in the present theory differs significantly from its value obtained by using the lowest order Born approximation. In general, the present theory gives a softening of the depression in T/sub c/ by the impurities.

The reduced specific heat jumps for the superconducting (S) side and the normal (N) side of a strongly coupled N-S composite have been calculated by using the generalized McMillan tunneling model of the superconducting proximity effect. Both cases λ N = 0 and λ N ≠ 0 (where λ N is the BCS interaction constant) have been studied. The results obtained by using the generalized model are significantly different from the ones given by the McMillan tunneling model.

The ultrasonic longitudinal sound wave attenuation coefficient and the nuclear spin relaxation rate have been calculated for a superconducting alloy containing paramagnetic impurities with local states within the gap. The Green's function given by Shiba (1968) or Rusinov (1969), which is appropriate for the case of a strong exchange interaction between the conduction electron spin and the magnetic impurity spin is used. For temperatures near Tc and for a low impurity concentration the authors also include the Kondo effect in their calculations. They find significant deviations from calculations based on the Green's function given earlier by Abrikosov and Gor'kov (1961).

Extending some recent work by Shiba (see Prog. Theor. Phys., vol.50, p.50 (1973)), the authors have investigated the influence of transition metal impurities on the amplitude of the Josephson tunneling current, Js(c,T) (c is impurity concentration and T is temperature). A junction made up of two identical superconducting alloys is considered and T is assumed near the transition temperature Tc.

The critical factor curve of a transition of the second kind in an antiferromagnetic superconductor (AFS) with nonmagnetic impurities has been studied. The AFS is described by using the mean-field model given by Nass, Levin, and Grest and assuming a one-dimensional electron band. They find that the points on the critical curve satisfy the thermodynamic stability condition for 0 less than or equal to ..cap alpha../sub 1 tau//..delta../sub 0/ less than or equal to 5.04 and 0.49 less than or equal to H/sub Q tau//..delta../sub 0/ less than or equal to 1.64. Here ..cap alpha../sub 1/ is the inverse lifetime of a conduction electron for nonmagnetic impurity scattering, H/sub Q/ is the antiferromagnetic molecular field, ..delta../sub 0/ is the zero-temperature order parameter of a superconductor in the absence of H/sub Q/ and impurities. Further, ..cap alpha../sub 1 tau/ and H/sub Q tau/ denote the values of these quantities for points on the critical curve. For ..cap alpha../sub 1 tau//..delta../sub 0/ > 5.04 and H/sub Q tau//..delta../sub 0/ > 1.64, the phase transition from the superconducting to the normal state is always of the second kind. Some thermal properties of the system near the critical curve have also been investigated andmore » they find that these depends dramatically on the impurity concentration.« less

Role of charge density waves (CDW) on the superconducting (SC) transition temperature, T c , in a one-dimensional model is investigated by examining the relation between the insulating gap induced by the CDW and the band filling. In contrast to the fact that the gap destroys the SC state, it is shown that the CDW formation results in the T c enhancement due to the increase of the density of states when the gap edge exists near the Fermi point. Further the degree of the T c enhancement is examined in the presence of the CDW fluctuation.

The authors calculate the reduced electronic thermal conductivity of a superconducting alloy containing paramagnetic impurities with local states within the gap. The Green function given by Shiba or Rusinov (1968-69) (for s wave scattering) is used which is appropriate for the case when the exchange scattering of a conduction electron from a magnetic impurity is strong. The temperature dependence of (Ks/Kn) for various impurity concentrations is shown and for a typical value of epsilon 0 (=0.6; epsilon 0 is a parameter which determines the strength of the conduction electron-impurity interaction). Significant deviations are found, from the theoretical results based on Abrikosov-Gor'kov theory ( epsilon 0=1.0).

We show that the recent experimental data of Toplicar and Finnemore on the superconducting proximity effect in Pb--Cd thin films can be understood within the McMillan tunneling model by using our new transition temperature (T/sub c/) formula and determining the parameters self-consistently. Agreement between the theoretical and the experimental values of T/sub c/, the gap, and the general shape and the height of the electronic density of state N/sub S/(E) curves is very good. In agreement with experiment, we find no low-energy peak in the density of states.

We derive a formula for calculating the free-energy difference between the superconducting and the normal states of a strong-coupling superconductor with localized states within the gap (induced by magnetic impurities). The present formula is a generalization of the one given by Bardeen and Stephen (for the electron-phonon systems) to include the effect of Shiba-Rusinov impurities.

The superconducting order parameter Δ( T ) and the maximum Josephson tunneling current J s ( T ) for an antiferromagnetic superconductor (AFS) are calculated both in the paramagnetic and the antiferromagnetic phases. The effects of the molecular field, impurities and the elastic scattering from spin-fluctuations have been included. The enhancement in J s ( T ) and Δ( T ) by the antiferromagnetic (AF) ordering observed in SmRh 4 B 4 by Vaglio et al. [1] can be understood with a reasonable choice of parameters. We also predict depression in J s ( T ) and Δ( T ) by AF ordering in some materials.

The magnetic penetration depth in the local limit {lambda}({ital T}) for a high-{ital T}{sub {ital c}} superconductor has been calculated within the electron pairing induced by holes model proposed by us recently. The derivation of the formula for {lambda}({ital T}) is based on a new expression for the current-density operator which takes into account the nonparabolic band structure of the conduction electrons. Our numerical results for the temperature dependence of {lambda}({ital T}) when the hole concentration {delta}=0.263 (which corresponds to 90-K superconductors) are near to the BCS results for a conventional superconductor and also agree qualitatively with the experimental data on the single crystal of YBa{sub 2}Cu{sub 3}O{sub 7}. Our value for the zero-temperature penetration depth is also close to the experimental values for the single crystals of the above material.

We study the stability criterion for the critical point of a transition of the second kind in the T-H Q plane of an antiferromagnetic superconductor AFS (T is temperature and H Q is the staggered molecular field). The AFS is described by using the mean-field model given by Nass, Levin and Grest and we assume a one dimensional electron band. Some properties of the system near the critical point are also investigated.

The Shiba-Rusinov theory of magnetic impurities in a superconductor is investigated, with special attention paid to the role of the potential scattering term in the electron-impurity interaction. The meaning of Anderson's theorem in the Shiba-Rusinov theory is discussed.

We propose a new pairing mechanism for high- T c superconductivity in doped La 2 CuO 4 compounds. In this mechanism, the attractive interaction between the electrons in a Cooper pair is induced through holes which are spinless neutral bosons. The basic requirements of the model are the existence of CuO 2 planes containing a number of holes and a large intrasite Coulomb repulsion U . We find superconducting state with s -wave symmetry. For small values of the hole concentration δ, T c increases with an increase in δ and t (transfer integral). With δ = 0.15 and t = 1 eV, we obtain T c ∼ 75 K . We expect a maximum in the T c vs δ curve for some value of δ.

Using Anderson model in the non-magnetic limit, we have calculated the jump in specific heat at T c , (Δ C ), for superconducting virtual bound state alloys. It is found that the normalized (Δ C ) vs normalized T c curves deviate considerably from the BCS value. A more significant result of the present study is that the initial slope of such a curve has a maximum value of 3.638. It is the highest value reported in literature for the impurity problem and is much larger than obtained even in a large pair breaking situation like Kondo effect (2.481).

The problem of localized single-particle excitations for the geometry of a superconducting sphere embedded in another superconductor has been investigated. Assuming constant order parameters in both materials Bogoliubov equations are solved for the s-state. The dependence of the excitation energy on the particle size and the ratio of the order parameters is shown.

Shiba-Rusinov theory of magnetic impurities in isotropic superconductors beyond the s-wave scattering is generalized by using the Eliashberg formalism. The analytical expressions for the transition temperature T/sub c/ and the specific-heat jump ..delta..C are given using the square-well model for the electron-phonon interaction. Taken as a function of the impurity concentration, the quantities T/sub c//T/sub c0/ and ..delta..C/..delta..C/sub 0/ depend on the microscopic parameters lambda, ..mu..( and ..omega../sub D/ of the host material. However, this dependence is absent if one plots the above properties versus the normalized impurity concentration ..cap alpha../..cap alpha../sub cr/ or if ..delta..C/..delta..C/sub 0/ vs T/sub c//T/sub c0/ is studied. (T/sub c/0 and ..delta..C/sub 0/ are values of T/sub c/ and ..delta..C, respectively, in the absence of impurities; lambda is the electron-phonon interaction parameter, ..mu..( is the Coulomb pseudopotential, and ..omega../sub D/ is the Debye cutoff frequency; ..cap alpha.. is the spin-flip scattering rate; ..cap alpha../sub cr/ is the value of ..cap alpha.. for which T becomes zero.)

We have calculated some properties of the high-{ital T}{sub {ital c}} superconductors by using the electron pairing induced by holes model proposed by us recently. The predicted dependence of {ital T}{sub {ital c}} on the hole concentration {delta} agrees qualitatively with the experimental data of Shafer {ital et} {ital al}. (Phys. Rev. B 36, 4047 (1987)). The dependence of the normalized order parameter and the normalized gap energy on the reduced temperature are very close to the one given by the BCS theory of conventional superconductors. The ratio of the zero-temperature gap energy and {ital T}{sub {ital c}} is near 1.6. The dependence of the density of states for quasiparticles on energy is found to be asymmetric. The carrier effective mass for 90-K superconductors is {similar to}4{ital m}{sub {ital e}}.

The effect of an external homogeneous magnetic field, H 0 , and the staggered magnetic field, H Q (arising from antiferromagnetism) on a superconductor having one-dimensional like half-filled band has been investigated. We find that there are domains in the H Q -H o plane in which a variety of phase transitions are possible. At T = 0, there is a domain having gapless superconductivity with H o -dependent order parameter. At finite temperature, one domain has first order phase transition from metallic state to gapless superconducting state and another domain has second order phase transition from normal to superconducting state followed by a first order phase transition to the semiconducting state.