The Ginzburg-Landau expansion and the slope of the upper critical field in disordered superconductors

It is shown that the slope of the upper critical field in superconductors with d pairing drops rapidly with increasing concentration of normal impurities, while in superconductors with anisotropic s pairing increases and reaches the well-known asymptotic level characteristic for the isotropic case. This difference makes it possible, in principle, to employ measurements of H _c 2 in disordered superconductors as an experimental method for determining the type of pairing in high-T _c superconductors and systems with heavy fermions. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 347–352 (10 March 1996)     

Magnetic field dependence ofT c and temperature dependence ofH c2 in layered superconductors with open normal state Fermi surface

A theoretical framework for treating the effects of magnetic fieldH on the pairing theory of superconductivity is considered, where the field is taken in an arbitrary direction with respect to crystal axes. This is applicable to closed, as well as open normal state Fermi surface (FS), including simple layered metals. The orbital effects of the magnetic field are treated semiclassically while retaining the full anisotropic paramagnetic contribution. Explicit calculations are presented in the limits |H| → |H _c2(T)|,T ∼ 0 andT →T _c(|H|), |H| ∼ 0. Effects of weak nonmagnetic impurity scattering, without vertex corrections, have also been taken into account in a phenomenological way. The final results for the case of open FS and layered materials are found to differ considerably from those of the closed FS. For example, an important parameter,h(T=0)=|H_c2(0)|/[-Tδ|H _c2 T|δT]_T{s0} for the case of a FS open ink _z-direction with thek _z-bandwidth, 4t ₃, very small compared to the Fermi energy,E _F, is close to 0.5906, compared to 0.7273 for the closed FS, in the clean limit. Analytical results are given for the magnetic field dependence ofT _c and the temperature dependence of H _c2 for a model of layered superconductors with widely open FS. For a set of band structure parameters for YBa₂Cu₃O₇ used elsewhere, we find reasonable values for the upper critical fieldH _c2(0), the slope (dH _c2/dT)_{T c0}, anisotropic coherence lengths ζ_i(T=0),i=x, y, z, and (dT _c/d|H|)_{|H| → 0}.     

Effect of the nonstoichiometric disorder on the temperature dependence of the upper critical field in Nd2? x Ce x CuO4+δ electron superconductors

The temperature dependences of the resistance of the Nd_2−x Ce _x CuO_4+δ single crystal with x = 0.15 and various disorder parameters δ have been investigated in various magnetic fields (B ‖ c, J ‖ ab) in the temperature range T = 1.8–40 K. It has been shown that the slope of the upper critical field, (dB _c2/dT)|T _c, in the electron superconductor decreases with an increase in the disorder parameter. This dependence makes it possible to experimentally distinguish between d-pairing superconductors and s-pairing superconductors. The dependence of the superconducting transition temperature on the disorder parameter in the system is investigated in the model of the superconductors with anisotropic impurity scattering. Original Russian Text ? T.B. Charikova, N.G. Shelushinina, G.I. Kharus, A.A. Ivanov, 2008, published in Pis’ma v Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2008, Vol. 88, No. 2, pp. 132–136.     

Pairing type change upon an increase in the cerium doping level in the Nd<Subscript>2 − <Emphasis Type="Italic">x</Emphasis></Subscript>Ce<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>CuO<Subscript>4 + δ</Subscript> electronic superconductor

We report on the results of analyzing the temperature dependences of the resistivity of the Nd_{2 − x} Ce _x CuO_{4 + δ} electronic superconductor with x = 0.14 (underdoped range), x = 0.15 (optimal doping), and x = 0.18 (overdoped range) and with various degrees of annealing in an oxygen-free atmosphere in magnetic fields up to H = 90 kOe (H ‖ c, J ‖ ab) in the temperature range T = (0.4–300) K. It is shown that the observed differences in the dependences of the slope of the upper critical field (dH_c2 /dT)|_Tc (dH_{c2} /dT)|_{T_c } on the degree of disorder in the Nd_{2 − x} Ce _x CuO_{4 + δ} system upon a change in the cerium doping level indicate a change in the symmetry of d-type pairing to anisotropic s pairing.     

The spin exchange interaction effect on Tc equation of anisotropic impure superconductors

We study the influence of spin exchange interaction of impurity scattering on critical temperature of anisotropic impure superconductors. The model of random non-magnetic and magnetic impurity are revised to cover the effect of spin exchange interaction. The sign of magnitude of the second-order Born scattering has been changed after consideration of the spin exchange interaction that also effects the form of T_c equation. We can get the general T_c equation that can be described well by anisotropic impure superconductors and covers all models done before.     

Effects of <Emphasis Type="Italic">d</Emphasis>-wave pairing in <Emphasis Type="Italic">n</Emphasis>-type high-temperature superconductors with anisotropic impurity scattering

The temperature dependences of the resistivity of single-crystal films of the Nd_{2 − x} Ce _x CuO_{4 + δ} n-type superconductors with x = 0.14 (underdoped region) and x = 0.15 (optimal doping region) and different degrees of disorder δ have been investigated in various magnetic fields (B ‖ c, J ‖ ab) in the temperature range 0.4–300 K. It has been demonstrated that there are differences in the behavior of the dependences of the slope of the upper critical field $ (dB_{c2} /dT)_{T \to T_c } $ (dB_{c2} /dT)_{T \to T_c } on the disorder parameter for the underdoped samples (x = 0.14) and the samples with the optimal doping (x = 0.15). The study of the dependence of the slope of the upper critical field on the degree of disorder has made it possible to discriminate experimentally between the superconductors with the d pairing and anisotropic s pairing. It has been revealed that the relative stability of the n-type superconductor with the optimal doping with respect to disordering is possibly due to the strong anisotropy of impurity scattering with symmetry of the d type.     

Electromagnetic response of unconventional superconductors

The magnetic field penetration depth, surface resistance and far infrared reflectivity are calculated for two anisotropic order parameters having either points or lines of nodes in the energy gap. Resonant impurity scattering is taken into account for a wide range of scattering rates. Comparison with experimental results on heavy Fermion superconductors shows that the order parameter cannot be deduced unambiguously from the temperature dependence of the penetration depth. Fits to surface resistance measurements on UBe₁₃ are best in the Born approximation, rather than the unitarity limit. Experiments on high-T _c materials are largely inconclusive, offering little support for the applicability of BCS theory and providing no evidence against novel pairing interactions leading to unusual order parameters.     

Spin-orbit effect on Tc of anisotropic superconductors

It is shown that spin-orbit impurity scattering brings about only a negligible change on the transition temperature T_c for anisotropic superconductors with an ellipsoidal Fermi surface.     

What is the highest Tc for phonon-mediated superconductivity?

We suggest that the high-temperature superconductivity can be attributed to the director-roles of the van Hove singularity between an electron-electron interaction and an electron-phonon interaction. The difference between the critical temperature and the pairing temperature is presented, and the Fermi arc, the d-wave symmetry and the poor conductivity, etc., are discussed. In particular, the non-s-wave symmetry is predicted to have the highest T _c for superconductors.      

Rare earth ion size dependence of the Tc of the “RE-123” HTSC within the (d + s)-wave description of their superconducting state

The dependence of the T_c's on the radius of the rare earth ions in the rare earth “123” high temperature superconductors (HTSC) is explained within the framework of a (d + s)-wave model of superconductivity. The appearance of a s-wave symmetry component in the order parameter of a spin fluctuation mediated d-wave superconductor is taken to be due to the orthorhombic distortion which occurs in these HTSC. It is assumed that the structural (orthorhombic) distortion induces a corresponding anisotropy in the pairing interaction, resulting in a nonseparable anisotropic interaction. It is shown that the T_c's are affected by the orthorhombic distortion which in turn is observed to depend on the size of the RE ions.     

Relation between the critical magnetic field and energy gap of a superconductor

A dynamical model is used to derive a relationship between the slope of the reduced critical magnetic field for T = T_c and the energy gap Δ_o of a superconductor. Good agreement with the relation |dh/dt|_t=1 = Δ_o/ k T_c proposed by Toxen is found up to Δ_o/k T_c = 4.     

Disordering effects in superconductors with anisotropic pairing: From Cooper pairs to compact bosons

In the weak-coupling BCS-theory approximation, normal impurities do not influence the superconducting transition temperature T _c in the case of isotropic s pairing. In the case of d pairing they result in a rapid destruction of the superconducting state. This is at variance with many experiments on the disordering of high-T _c superconductors, assuming that d pairing is realized in them. As the interelectronic attraction in a Cooper pair increases, the system transforms continuously from a BCS-type superconductor with “loose” pairs to a picture of superconductivity of “compact,” strongly coupled bosons. Near such a transition substantial deviations can be expected from the universal disorder dependence of T _c , as determined by the Abrikosov-Gor’kov equation, and T _c becomes more stable against disordering. Since high-T _c super-conducting systems fall into the transitional region from BCS-type pairs to compact bosons, these results can explain their relative stability against disordering. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 3, 258–262 (10 February 1997)     

Paraconductivity for a d-wave superconductor in short-wavelength fluctuation regime

A theoretical study of the fluctuation conductivity above T_c (paraconductivity) is reported for a d-wave superconductor with resonant scattering impurities. A d-wave system is modeled by tight-binding electrons in the two-dimensional squared lattice, and the impurity scattering is treated in the T-matrix approximation in a unitary limit. In calculating the Aslamazov–Larkin (AL) and the Maki–Thompson (MT) terms, we also consider effects of a short-wavelength cutoff in the fluctuation spectrum. The d-wave character in the AL and MT terms manifests itself to renormalization effects on the fluctuation amplitude and reduced temperature, whereas an anomalous-MT term is absent. The present calculations can describe fairly well experiments on the paraconductivity in zinc-doped cuprate superconductors provided that effects of a total-energy cutoff are taken into account.     

Theory of high-temperature superconductivity in cuprates

A microscopic theory of electronic spectrum and superconducting pairing in the high-temperature cuprate superconductors is presented. The theory is based on consideration of strong electron correlations within the Bogolyubov polar model. The Dyson equation is derived by using the equation of motion method for the thermodynamic Green functions in terms of the Hubbard operators. The self-energy is evaluated in the noncrossing approximation for electron scattering on spin and charge fluctuations induced by kinematic interaction. The theory demonstrates that a strong Coulomb repulsion results in the anomalous electronic spectrum and unconventional (d-wave) superconducting pairing with high T _c mediated by the antiferromagnetic exchange and spin fluctuations.     

Magnetic and superconducting properties of FeAs-based high-<Emphasis Type="Italic">T</Emphasis><Subscript>c</Subscript> superconductors with Gd

We report on successful synthesis under high pressure of a series of polycrystalline GdFeAsO_{1 − x} F _x high-T _c superconductors with different oxygen deficiency x = 0.12−0.16 and also with no fluorine. We have found that the high-pressure synthesis technique is crucial for obtaining the single-phase superconducting materials: by sythesizing the same compounds with no pressure in ampoules, we obtained non-superconducting materials with an admixture of incidental phases. Critical temperature for all the materials was in the range 40 to 53 K. The temperature derivative of the critical field dH _c2/dT is remarkably high, indicating potentially high value of the second critical field H _c2 ∼ 130 T.     

Superconducting properties of the atomically disordered MgB2 compound

The effect of disorder induced by neutron irradiation in a nuclear reactor (thermal neutron fluence 1×10¹⁹cm^?2) on the superconducting transition temperature T _c and the upper critical field H _c2 of polycrystalline MgB₂ samples was investigated. Despite the appreciable radiation-induced distortions (more than ten displacements per atom), the initial crystal structure (C32) was retained. The temperature T _c decreased from 38 to 5 K upon irradiation and was practically completely restored after the subsequent annealing at a temperature of 70°C. A weak change in the dH _c2/dT derivative upon irradiation is explained by the fact that the irradiated samples are described by the “pure” limit of the theory of disordered superconductors. The suppression of T _c upon disordering may be due to the isotropization of the originally anisotropic (or multicomponent) superconducting gap or to a decrease in the density of electronic states at the Fermi level.     

Pairing mechanisms and anisotropic superconductivity in layered crystals

A brief outline of the generalized BCS pairing theory is presented. The pairing of carriers can be due to the exchange of lattice-phonons or due to the exchange of electronic charge-density and spin-density excitations. It is argued that anisotropic physical properties in the normal as well as superconducting states in the new high-T_c materials are crucual in the development of any realistic theoretical approach, and in comparison of experimental results with correct BCS predictions involving superconductivity in layered crystals. The possibility of the break-down of the mean-field approximation is also discussed. As of now, the generalized BCS pairing approach is the only realistic microscopic theory available which may be applied to high-T_c superconductors.     

Some parameters of superconductors with an extreme singularity in the density of state: (s+d) model

We study some parameters of superconductors with δ-function type singularities in the electronic density of states (DOS), exhibiting (s+d)-wave symmetry of the order parameter. Starting with a pure s-wave pairing potential V_s, the critical temperature T_c at first slightly increases with increasing the d-wave interaction potential V_d, being determined by this interaction only for stronger V_d values. The ratio R=2|Δ(0)|/k_BT_c of the mean value of the zero temperature energy gap |Δ(0)| to T_c increases with increasing V_d, reaching a maximum which depends on the mixing interaction term. The maximum values of R are comparable with very high values obtained in some gap measurements. The jump in the specific heat at critical temperature is by a factor 2.4 higher for the extreme singularity of pure s-wave symmetry, as compared with the BCS theory with constant DOS. Such higher values of the jump are in agreement with the experimentally observed values, as well as with the calculations determined by extended saddle points in the electronic bands. By switching the d-wave channel, the value of the jump decreases. The results show the usefulness of calculations with δ-type singularities as a limiting case of very strong singularities in the DOS.     

Critical magnetic field H c2 and electron scattering in MgB2

The correlation between the upper critical field H _c2 and the residual resistivity ρ of the MgB₂ compound is studied in a wide range of ρ values. The slope ?dH _c2/dT of the temperature dependence of H _c2 near T _c is found to increase steadily with increasing residual resistivity to 100 μΩ cm. Over the range from 0 to 50 μΩ cm, the dependence of ?dH _c2/dT on ρ is fitted well by a linear function, which is typical of a single-band superconductor. The fit is performed using the electronic parameters of the two bands (π and σ) that form the Fermi surface. The field H _c2 is assumed to depend on the electronic parameters of the σ band only. Using this approximation, the following quantities are estimated: the relative contribution of σ electrons to the total conductivity along the boron planes (which turns out to be about 1/2), the ratio of the mean free paths of electrons in the σ and π bands l _σ/l _π ≈ 1.5, and the ratios between some other parameters describing electron scattering.     

Surface states and tunneling spectroscopy of high-Tcsuperconductors

Recent studies of high- T_csuperconductors have clarified new aspects of tunneling spectroscopy. The unconventional pairing states, i.e. d-wave symmetry in these materials have been established through various measurements. Differently from isotropic s-wave superconductors, d-wave pairing states have an internal phase of the pair potential. The internal phase modifies the surface states due to the interference effect of the quasiparticles. Along these lines, a novel formula of tunneling spectroscopy has been presented that fully takes into account of the anisotropy of the pair potential. The most essential difference of this formula from conventional ones is that it suggests the phase-sensitive capability of tunneling spectroscopy. The formula suggests that the symmetry of the pair potential is determined by the orientational dependence measurements of tunneling spectroscopy. Along these lines, several experiments have been performed on high-T_c superconductors. The observation of the zero-bias conductance peaks (ZBCP) on Y Ba₂Cu₃O_{7 − δ}strongly suggests the d_{x² − y²}-wave pairing states of hole-doped high-T_c superconductors. On the other hand, the absence of ZBCP on (electron-doped)Nd_1.85Ce_0.15CuO_{4 − δ}indicates that the pair potential of this material is a nodeless state. In this paper, recent developments of tunneling spectroscopy for anisotropic superconductors are reviewed both on theoretical and experimental aspects.