We study the effects of anisotropic order parameters on the temperature dependence of London penetration depth anisotropy $gamma_lambda(T)$. After MgB$_2$, this dependence is commonly attributed to distinct gaps on multi-band Fermi surfaces in superc
onductors. We have found, however, that the anisotropy parameter may depend on temperature also in one-band materials with anisotropic order parameters $Delta(T,k_F)$, a few such examples are given. We have found also that for different order parameters, the temperature dependence of $Delta(T)/Delta(0)$ can be represented with good accuracy by the interpolation suggested by D. Einzel, J. Low Temp. Phys, {bf 131}, 1 (2003), which simplifies considerably the evaluation of $gamma_lambda(T)$. Of particular interest is mixed order parameters of two symmetries for which $gamma_lambda(T)$ may go through a maximum for a certain relative weight of two phases. Also, for this case, we find that the ratio $Delta_{max}(0)/T_c$ may exceed substantially the weak coupling limit of 1.76. It, however, does not imply a strong coupling, rather it is due to significantly anisotropic angular variation of $Delta$.
The anisotropic London equations taking into account the normal currents are derived and applied to the problem of the surface impedance in the Meisner state of anisotropic materials. It is shown that the complex susceptibility of anisotropic slab de
pends on the orientation of the applied microwave field relative to the crystal axes. In particular, the anisotropic sample in the microwave field is subject to a torque, unless the field is directed along with one of the crystal principle axes.
We study vortex current distributions in narrow thin-film superconducting strips. If one defines the vortex core ``boundary as a curve where the current reaches the depairing value, intriguing features emerge. Our conclusions based on the London appr
oach have only qualitative relevance since the approach breaks down near the core. Still, the main observation which might be useful is that the core size near the strip edges is smaller than in the rest of the strip. If so, the Bardeen-Stephen flux-flow resistivity should be reduced near the edges. Moreover, at elevated temperatures, when the depairing current is small, the vortex core may extend to the whole strip width, thus turning into an edge-to-edge phase-slip line.
We show on a few examples of one-band materials with spheroidal Fermi surfaces and anisotropic order parameters that anisotropies $gamma_H$ of the upper critical field and $gamma_lambda$ of the London penetration depth depend on temperature, the feat
ure commonly attributed to multi-band superconductors. The parameters $gamma_H$ and $gamma_lambda$ may have opposite temperature dependencies or may change in the same direction depending on Fermi surface shape and on character of the gap nodes. For two-band systems, the behavior of anisotropies is affected by the ratios of bands densities of states, Fermi velocities, anisotropies, and order parameters. We investigate in detail the conditions determining the directions of temperature dependences of the two anisotropy factors.
The dissipative currents due to normal excitations are included in the London description. The resulting time dependent London equations are solved for a moving vortex and a moving vortex lattice. It is shown that the field distribution of a moving v
ortex looses it cylindrical symmetry, it experiences contraction which is stronger in the direction of the motion, than in the direction normal to the velocity $bm v$. The London contribution of normal currents to dissipation is small relative to the Bardeen-Stephen core dissipation at small velocities, but approaches the latter at high velocities, where this contribution is no longer proportional to $v^2$. To minimize the London contribution to dissipation, the vortex lattice orients as to have one of the unit cell vectors along the velocity, the effect seen in experiments and predicted within the time-dependent Ginzburg-Landau theory.
Effective demagnetizing factors that connect the sample magnetic moment with the applied magnetic field are calculated numerically for perfectly diamagnetic samples of various non-ellipsoidal shapes. The procedure is based on calculating total magnet
ic moment by integrating the magnetic induction obtained from a full three dimensional solution of the Maxwell equations using adaptive mesh. The results are relevant for superconductors (and conductors in AC fields) when the London penetration depth (or the skin depth) is much smaller than the sample size. Simple but reasonably accurate approximate formulas are given for practical shapes including rectangular cuboids, finite cylinders in axial and transverse field as well as infinite rectangular and elliptical cross-section strips.
A model of a clean two-band s-wave superconductor with cylindrical Fermi surfaces, different Fermi velocities v_{1,2}, and a general 2x2 coupling matrix V_{alpha beta} is used to study the order parameter distribution in vortex lattices. The Eilenber
ger weak coupling formalism is used to calculate numerically the spatial distributions of the pairing amplitudes Delta_1_ and Delta_2_ of the two bands for vortices parallel to the Fermi cylinders. For generic values of the interband coupling V_{12}, it is shown that, independently of the couplings V_{alpha beta}, of the ratio v_1 /v_2, of the temperature, and the applied field, the length scales of spatial variation of Delta_1 and of Delta_2 are the same within the accuracy of our calculations. The only exception from this single length-scale behavior is found for V_{12} --> 0, i.e., for nearly decoupled bands.
A two-band model with repulsive interband coupling and interband {it transport} (potential) scattering is considered to elucidate their effects on material properties. In agreement with previous work, we find that the bands order parameters $Delta_{1
,2}$ differ and the large is at the band with a smaller normal density of states (DOS), $N_{n2}<N_{n1}$. However, the bands energy gaps, as determined by the energy dependence of the DOS, are equal due to scattering. For each temperature, the gaps turn zero at a certain critical interband scattering rate, i.e. for strong enough scattering the model material becomes gappless. In the gapless state, the DOS at the band 2 is close to the normal state value, whereas at the band 1 it has a V-shape with non-zero minimum. When the normal bands DOS are mismatched, $N_{n1} e N_{n2}$, the critical temperature $T_c$ is suppressed even in the absence of interband scattering, $T_c(N_{n1})$ has a dome-like shape. With increasing interband scattering, the London penetration depth at low temperatures evolves from being exponentially flat to the power-law and even to near linear behavior in the gapless state, the latter being easily misinterpreted as caused by order parameter nodes.
It is shown that the order parameter $Delta$ induced in the normal part of superconductor-normal-superconductor proximity system is modulated in the magnetic field differently from vortices in bulk superconductors. Whereas $Delta$ turns zero at vorte
x centers, the magnetic structure of these vortices differs from that of Abrikosovs.
By evaluating the upper and thermodynamic critical fields, $H_{c2}$ and $H_c$, and their ratio $H_{c2}/H_c $ at arbitrary temperatures, we argue that situations are possible when a type-II material is transformed to type-I by adding magnetic impurities.
