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A finite-size scaling of the nanoscale magnetization m on size averaging R of a single vortex in d -wave bulk superconductor is developed using quasiclassical Eilenberger equations. Nanoscaling is anchoring around the linear London approximation for bulk superconductors. Comparing the results with those obtained in local nonlinear approach demonstrated the importance of the nonlocal contribution. Temperature dependences of two-point correlation function χ ( T , R 1 , R 2 ) = m ( T , R 2 ) / m ( 0 , R 2 ) − m ( T , R 1 ) / m ( 0 , R 1 ) with R 2 > R 1 and one-point function χ ( T, R 1 → ∞, R 2) are calculated. It is found that χ ( T, R 1, R 2), R 2 > R 1, is a nonmonotonous function of temperature and changes sign at high temperatures. This nonmonotonous temperature dependence can be understood as a result of competition between various effects i) Volovik effect and nonlocal corrections to superconducting electron density dominating in low temperature range, and ii) current-induced suppression of the order parameter dominating at high temperatures. The introduced nonmagnetic disorder greatly suppresses the low temperature nonlocal and nonlinear effects, leaving the order parameter effects to prevail in the whole temperature range. Nonlocal pairing and tunneling effects are investigated at the superconductor - normal metal border by considering a d -wave superconducting dot ( d -dot) inside a normal diffusive metal. These effects result in a suppression of the supercurrent in the vortex core and are essential in nanodots with relatively small sizes. At sizes larger than a temperature dependent characteristic length the nanoscale physics transforms into bulk solution.

The generalized London equation in the mixed state of high- κ s -wave pairing superconductors with impurities is considered as a projection of the quasiclassical nonlocal nonlinear Eilenberger theory. Only one fitting parameter – the cutoff parameter ξ h – is used in the theory. The distribution of the magnetic field is calculated self-consistently. Both nonlocal effects originated from extended states between the vortices and bound Andreev states in the vortex are taken into account. Comparison with different analytical nonlocal linear approaches (the Kogan–Gurevich, Amin–Franz–Affleck, Kogan–Zhelezina models) including only extended states is done. The importance of the Kramer–Pesch nonlinear effect and the field dependence of the cutoff parameter is emphasized and their strong influence on the variance of the magnetic field is found. The influence of the impurities on the ratio of the cutoff parameter ξ h and the Ginzburg–Landau coherence length ξ c 2 is considered. Quasiparticle scattering by impurities and lowering of the temperature reduces the value of ξ h to the values much less than ξ c 2 . This is different from the prediction of the local Ginzburg–Landau theory where ξ h is scaled by ξ c 2 . It is found that impurities influence by different way on the cutoff parameter ξ h and the order parameter coherence length ξ 1 . The ξ h decreases monotonously with the impurity scattering time in contrast to the nonmonotonous behavior of ξ 1 . The results can be used for analysis of the μ SR experimental data.

Hall effect and flux pinning in YBa 2 Cu 3 O 6 + x (YBCO) thin films doped with BaZrO 3 (BZO) nanoparticles is investigated. The results show that sign reversal of the Hall coefficient from positive hole-like to negative electron-like occurs in vortex-liquid regime of undoped and BZO-doped YBCO films. With increasing BZO concentration the amplitude of the negative Hall effect is suppressed while the temperature position of the anomalous Hall effect does not depend significantly on doping level. In addition, it is shown that Hall conductivity increases non-monotonically with increasing BZO doping. These results support a model where BZO at low doping concentration induces point pinning centres turning to strong columnar pinning defects in films doped with 4% BZO.

Angular dependence of the resistivity, the irreversibility field and the critical current of undoped and BaZrO 3 -(BZO)-doped YBa 2 Cu 3 O 6+ x (YBCO) thin films are investigated using pulsed magnetic fields up to 30 T. Doping with BZO forms a dense network of anisotropic c -axis oriented columnar defects, which lead, in comparison with undoped material, to clearly improved values of the pinning potential, the critical current density and the irreversibility field in the whole angular range. The results are explained by point defect scaling theory using a renormalized anisotropy parameter in connection with linear defects. In oblique magnetic field a porous vortex matter is formed by means of kinks on the columnar defects. Our results demonstrate the efficiency of the columnar defects in thin YBCO films in magnetic fields much above the matching field, opening up a possibility to use this material in various high field applications.

The Hall effect in the II–V group semiconductor n-CdSb doped with In is investigated in pulsed magnetic fields up to B = 25 T and temperatures between T = 2 and 77 K for samples oriented along the [0 1 0] and [0 0 1] crystallographic axes. The Hall coefficient, R (B, T), exhibits a sequence of an almost flat region followed by a descending interval and an upturn when B is increased. The decrease of R (B, T) is interpreted by the presence of two groups of electrons with concentrations n2(T) > n1(T) and mobilities μ2(T) < μ1(T). Analysis of n1(T) and n2(T) demonstrates that the high-mobility carriers n1 are the conduction band (CB) electrons, whereas the low-mobility carriers n2 are itinerant electrons of a lower resonant impurity band (IB), having at low T energies of Ei ~ 3–4 meV above the CB edge. In addition, near the CB edge lies a higher IB containing only localized electron states. The IBs are split by spin to states differing by an energy ΔEi ≈ 0.9 meV. The upturn of R (B, T) in the high-field region is explained by the redistribution of the electrons between the IBs due to the decrease of ΔEi when B is increased. The mobility of the CB electrons is determined presumably by strong anisotropic scattering on neutral impurity centres, accompanied at high T by isotropic scattering on acoustic phonons, whereas scattering from ionized impurities is small.

Field dependence of the Labusch parameter α L v ( B ) is calculated for vortex arrays assuming planar pinning defects parallel to the vortices, taking into account the discreteness of the vortex lattice, i.e. beyond the framework of the elasticity theory. Considering both compressional and shear vortex oscillations the distribution of the vortex displacements are obtained. For the compressional case a region of instability is found near the vortex pinning layers. This region is important for determination of α L v ( B ) . In contrast, the distribution of the vortex displacements for the shear oscillations is smooth and independent of the field. This results in saturation of α L v ( B ) at high fields. Considering also nonlinear effects the dependence of α L v on the vortex displacement is obtained up to displacements where depinning of the vortices occurs.

Magnetophonon resonance in quantum wells in a tilted magnetic field B is investigated. Measurements of the Hall coefficient and correspondingly of the carrier concentration as a functions of magnetic field and temperature are simultaneously performed. It is shown that the experimental data can be interpreted in terms of a great sensitivity to the effects of varying the two dimensional carrier concentration ns in a certain concentration interval. In other words, the observed angular dependence of the MPR amplitudes is a manifestation of dependence of ns on the magnitude of the magnetic field B. We believe that such a dependence can be relevant in general for the interpretation of magnetotransport in nondegenerate 2D electron gas.

The Hall effect in the anisotropic II–V group semiconductor p-CdSb is investigated at temperatures between T = 3.6 and 200 K and pulsed magnetic fields up to B = 25 T in unintentionally doped samples oriented along the crystallographic axes [100] and [010]. The Hall coefficient, R(B, T), with B ∥ [001] exhibits in low fields a flat region followed by a descending interval when B is increased. This behaviour is attributed to the presence of two groups of holes with concentrations p2(T) > p1(T) and mobilities μ2(T) Tcr p1 and p2 are related to the holes activated to the light- and heavy-hole bands, respectively. The analysis of μ1(T) and μ2(T) confirms the existence of the heavy-hole band or a non-equivalent maximum and two equivalent maxima of the light-hole valence band.

The resistivity of the anisotropic semiconductor p-CdSb is investigated in a wide temperature range of T = 1.9–300 K in pulsed magnetic fields up to B = 25 T. Two not intentionally doped single-crystalline samples oriented along the crystallographic axes [100] (no. 1) and [010] (no. 2) are used for measurements of the resistivity, ρ, in transversal magnetic field configuration. Below the resistivity follows the laws for B Bc, where the coefficients C and S do not depend on T, and . These are characteristics of nearest-neighbour hopping conductivity. The coefficients C and S depend on the direction of B, and their ratios agree completely with the values calculated with the components of the hole effective mass. The acceptor concentrations, and for no. 1 and no. 2, respectively, are relatively close to the critical value of the metal–insulator transition (MIT), . This leads to enhancement of the mean localization radii, in no. 1 and in no. 2, with respect to the value of far from the MIT determined by an asymptote of the wavefunction at a large distance from the impurity centre.

The ac response of a slab of a clean type-II superconductor in the mixed state is theoretically investigated taking into account the bulk and surface pinning. In extremely clean superconductors sharp resonances of the ac response are predicted. Pinning-induced resonance is found only for an asymmetric ac magnetic field h(ω) pattern with the direction of h(ω) being opposite on opposite sides of the superconducting slab. The contribution of the surface pinning to the resonance frequency depends on the tilt modulus of the flux line lattice. A resonance at the cyclotronic frequency is found only in samples thinner than the London penetration depth. In the universal flux-flow regime, predicted for d-wave superconductors, wide maxima are found in the frequency dependence of the ac absorption.

A narrow dip is observed around zero magnetic field in magnetization curves M(B) of superconducting YBCO films with about 10 nm thickness. This anomaly occurs in the same field range with an anomaly of ac surface impedance Z(B) found recently in thin YBCO films. Because the thickness of our films is considerably less than the London penetration depth, two-dimensional limit of the critical state model is applied. In the framework of this model the magnetic field dependence of the critical current density jc(B) is found. The obtained jc(B) function agrees well with that found in the ac surface impedance investigation.

The effects of the pairing symmetries (${s}_{\ifmmode\pm\else\textpm\fi{}}$ and ${s}_{++}$) on the cutoff parameter of field distribution, ${\ensuremath{\xi}}_{h}$, in stoichiometric (like LiFeAs) and nonstoichiometric (like doped ${\mathrm{BaFe}}_{2}{\mathrm{As}}_{2}$) iron pnictides have been investigated using Eilenberger quasiclassical equations. Magnetic field, temperature, and impurity scattering dependencies of ${\ensuremath{\xi}}_{h}$ have been calculated. Two opposite behaviors have been discovered. The ${\ensuremath{\xi}}_{h}/{\ensuremath{\xi}}_{c2}$ ratio is less in ${s}_{\ifmmode\pm\else\textpm\fi{}}$ symmetry when the intraband impurity scattering (${\ensuremath{\Gamma}}_{0}$) is much larger than 1 and is much larger than the interband impurity scattering (${\ensuremath{\Gamma}}_{\ensuremath{\pi}}$), i.e., in nonstoichiometric iron pnictides. In contrast, the value ${\ensuremath{\xi}}_{h}/{\ensuremath{\xi}}_{c2}$ is higher in the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ case and the field-dependent curve is shifted upward from the ``clean'' case (${\ensuremath{\Gamma}}_{0}={\ensuremath{\Gamma}}_{\ensuremath{\pi}}=0$) for stoichiometric iron pnictides (${\ensuremath{\Gamma}}_{0}={\ensuremath{\Gamma}}_{\ensuremath{\pi}}\ensuremath{\ll}1$). Results can be tested in muon spin rotation measurements.

The problem of transformation of the quasiclassical Eilenberger theory to the one-parameter London model is considered for the mixed state of clean type-II superconductors. Numerical computations demonstrate that introduction of a field dependent coherence length ${\ensuremath{\xi}}_{h}(B)$ is enough for description of the magnetic field distribution in the vortex core of a superconductor. It is found that the shape of ${\ensuremath{\xi}}_{h}(B)$ has strong temperature dependence. The comparison between $s$- and $d$-wave pairing symmetry is done. Different behavior of ${\ensuremath{\xi}}_{h}(B)$ in triangular and square vortex lattices is found.

We extend the Ginsburg-Landau solution for cutoff function in London equation to low temperatures by solving numerically the quasiclassical Eilenberger equations in mixed state of $s$-wave superconductors. As a result the nonlocal generalized London equation (NGLE) is obtained. The magnetic field and temperature dependence of the cutoff function parameter ${k}_{1}(B,T)$ are calculated. Due to Kramer-Pesch effect ${k}_{1}$ decreases strongly at low temperatures. It is also found that ${k}_{1}$ has a minimum at a value of magnetic field depending on temperature. We reduce the NGLE model to an effective local model and calculate the value of an effective penetration depth ${\ensuremath{\lambda}}_{\mathrm{eff}}(B,T)$. The sublinear field dependence of ${\ensuremath{\lambda}}_{\mathrm{eff}}$ is predicted that agrees with experimental $\ensuremath{\mu}$SR results for the penetration depth of magnetic field in the $s$-wave superconductor ${\mathrm{V}}_{3}\mathrm{Si}$ and ${\mathrm{NbSe}}_{2}$.

Dependence of the critical current on applied magnetic field ${j}_{c}({B}_{a})$ is investigated in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ (YBCO) films prepared by pulsed laser deposition from pure YBCO target and from those doped with 2.9 or $9.0\phantom{\rule{0.3em}{0ex}}\mathrm{wt}.\phantom{\rule{0.2em}{0ex}}%$ ${\mathrm{BaZrO}}_{3}$. All targets were sintered from nanophase powders made by a citrate-gel process. Transmission electron microscopy images show distributions of ${\mathrm{BaZrO}}_{3}$ particles with typical diameters of $5--10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ in the doped films. In comparison with the pure YBCO film the value of ${j}_{c}(0)$ is slightly smaller in the film deposited from the target doped with $2.9\phantom{\rule{0.3em}{0ex}}\mathrm{wt}.\phantom{\rule{0.2em}{0ex}}%$ ${\mathrm{BaZrO}}_{3}$ and only half in case of $9.0\phantom{\rule{0.3em}{0ex}}\mathrm{wt}.\phantom{\rule{0.2em}{0ex}}%$ doping. Two interesting phenomena in the behavior of ${j}_{c}({B}_{a})$ are observed for the doped films. First, the low-field plateau of ${j}_{c}$ is significantly lengthened and second, in fields above the accommodation field ${B}^{*}$ marking the end of this plateau the rate of decrease of ${j}_{c}$ is strongly reduced. These results are well fitted with a critical state model of superconductors taking into account the self field effects at low applied magnetic fields.

Quasiclassical Eilenberger equations are solved numerically for a case of an isolated two-dimensional vortex in a $d$-wave superconductor taking the screening into account self-consistently by introducing the vector potential in the equations. The field distribution is found in the whole coordinate range and compared with that obtained from the nonlocal (NGLE) and local generalized London equations with different cutoff functions. A good agreement can be obtained if the arguments of the cutoff functions are taken to be temperature dependent. It is also shown that the anisotropy of the field distribution can be qualitatively described by NGLE.

Quasiclassical Eilenberger equations are solved numerically for the case of an isolated two-dimensional vortex in a d-wave superconductor. The asymptotical behavior of the amplitude and the phase of the pairing potential,superconducting current, and superconducting electron density at long distances from the vortex core are obtained. The local Doppler-shift (DS) method is found to work reasonably well for description of the superconducting electron density at distances longer than the coherence length at low temperatures. Nonlocal effects are important inside the vortex core and for a description of the effects of the fourfold vortex symmetry outside the core. It is also shown that at higher temperatures the DS method should be modified by including a pairing potential calculated self-consistently.

Temperature dependence of single vortex magnetic moment in nanosize superconducting particles is investigated in the framework of quasiclassical Eilenberger approach. Such nanoparticles can be used for preparation of high-quality superconducting thin films with high critical current density. In contrast to bulk materials where the vortex magnetic moment is totally determined by flux quantum, in nano-sized specimens (with characteristic size, D, much less than effective penetration depth, λeff) the quantization rule is violated and magnetic moment is proportional to D2/λ2eff(T). Due to strong repulsion between vortices in nanoparticles only a single vortex can be trapped in them. Because of small size of particles the screening current of the vortex is located near the vortex core where the current is quite high and comparable to depairing currents. Therefore, the superconducting electron density, ns, depends on the current value and the distance from the vortex core. This effect is especially important for superconductors having gap nodes, such as YBCO. The current dependence of ns in nanoparticles is analogous to the Volovik effect in flux-line lattice in bulk samples. The magnitude of the effect can be obtained by comparing the temperature dependence of magnetic moment in the vortex and in the Meissner states. In the last case the value of screening current is small and superconducting response to the external field is determined by London penetration depth. Because of importance of nonlinear and nonlocal effects, the quantum mechanical Eilenberger approach is applied for description of the vortex in nanoparticles. The flattening of 1/λ2eff(T) dependence has been found. A comparison of the theoretical results with experimental magnetization data in Meissner and mixed states of YBCO nanopowders has been done. The presence of nonlinear and nonlocal effects in vortex current distribution is clearly visible. The obtained results are important for the description of pining in nanostructured high-Tc thin films.

Influence of the order parameter symmetries on the cutoff parameter ξh and vortex core size ξ2 (the distance from the vortex center at which the current density reaches the maximum value) in the mixed state are investigated in the framework of Eilenberger theory for unconventional superconductors. The cutoff parameter determines the field distribution in the generalized London equation obtained as a projection of the quasiclassical theory. It is used for the fitting of the μSR and small-angle neutron scattering experimental data. Anisotropic dx2−y2 and isotropic s±-wave superconducting pairings are studied. These pairing symmetries can be realized in iron pnictide superconductors. Conventional s++ pairing mediated electron-phonon interaction is also considered. Temperature, field, and impurity scattering dependences of ξh/ξc2 are obtained. It is found that normalized ξ2/ξc2(B/Bc2) dependence is increasing with pair breaking impurity scattering (interband scattering for s±-wave and intraband impurity scatt...

Quasiclassical Eilenberger equations are solved for s±-wave superconductors in the mixed state. This symmetry has been proposed for multiband superconductors as pnictides. This mechanism can be realized because of Umklapp scattering between the electron and the hole Fermi surface pockets resulting in opposite sign of pairing gap in these pockets. The applicability of the phenomenological Hao-Clem theory is investigated. Magnetic, temperature and impurity scattering rate dependencies of vortex core size are calculated. It is found that the accuracy of the effective London model gets better with the presence of the impurity scattering and even near the second critical field it is below 6%. The model with the parameters of intraband and interband impurity scattering, describing well superfluid density in BaFe2As2, is also considered.

Influence of impurities on coherence length Ξh in the mixed state of s-wave superconductors is investigated in framework of quasiclassical Eilenberger theory. The increasing of impurity scattering rate results in decreasing of Ξh. The obtained field dependence of Ξh for clean superconductors has a minimum and it is similar to that in Hao-Clem and Miranovic-Ichioka-Machida theories for order parameter of coherence length. It is found that growing behavior of Ξh with magnetic field in dirty superconductors is different from order parameter coherence length determining by pairing potential near with vortex core. The magnetic field dependence of coherence length in normalized units, Ξh/Ξc2(B/Bc2), is nonuniversal and depends on impurity scattering potential.

The resistivity of YBa2Cu3O7−δ (YBCO) films deposited by a pulsed laser from nanograined targets doped with various amounts of BaZrO3 (BZO) is investigated in the thermally activated flux-flow (TAFF) regime, where the activation energy, U, has a linear temperature dependence. It is found that the magnetic field dependence of the irreversibility temperature, Tirr(B), has a maximum at some optimal BZO concentration, which depends on the applied magnetic field. Our results show that the superconducting materials used in different magnetic fields need optimization of the dopant concentration to achieve the best properties for various applications.