Flux vortices and transport currents in type II superconductors

This article is concerned with the mechanisms by which type II superconductors can carry currents. The equilibrium properties of the vortex lattice are described and the generalized driving force in gradients of temperature and field is derived using irreversible thermodynamics. This leads to expressions for thermal cross effects which can include pinning forces. The field distributions which occur in a range of situations are derived and a number of useful solutions of the critical state given. In particular, the distribution in a longitudinal field is obtained, and the conditions under which force-free configurations can break down by the cutting of vortices discussed. The effects of lattice rigidity on the summation of pinning forces is considered and it is shown that a summation based on statistical arguments uses the same approximations and leads to the same results as a dissipation argument. Theoretical expressions are derived for the vortex pinning interaction to a number of different metallurgical defects. The theoretical models are compared critically with experimental measurements of pinning forces and other related phenomena, such as flux creep, low amplitude vortex oscillations and vortex lattice defect effects. Finally, the implications for technological materials are assessed.     

Flux vortices and transport currents in type II superconductors

This article is concerned with the mechanisms by which type II super-conductors can carry currents. The equilibrium properties of the vortex lattice are described and the generalized driving force in gradients of temperature and field is derived using irreversible thermodynamics. This leads to expressions for thermal cross effects which can include pinning forces.

The field distributions which occur in a range of situations are derived and a number of useful solutions of the critical state given. In particular, the distribution in a longitudinal field is obtained, and the conditions under which force-free configurations can break down by the cutting of vortices discussed.

The effects of lattice rigidity on the summation of pinning forces is considered and it is shown that a summation based on statistical arguments uses the same approximations and leads to the same results as a dissipation argument. Theoretical expressions are derived for the vortex pinning interaction to a number of different metallurgical defects. The theoretical models are compared critically with experimental measurements of pinning forces and other related phenomena, such as flux creep, low amplitude vortex oscillations and vortex lattice defect effects. Finally, the implications for technological materials are assessed.     

Tunneling between thin film superconductors containing single flux vortices

Discrete conductance increments in the current-voltage characteristics of superconductor-insulator-superconductor tunnel junctions (of area $\text{?25 μ}\text{m}{}^2$) associated with single flux quanta (“vortices”) trapped in the thin film electrodes have been observed. These tunneling results were obtained for both Sn and Pb and provide, for the first time, a direct measure of the density of quasiparticle states corresponding to a single vortex. Simultaneous measurements of the Josephson critical current suppresion indicate that the vortex cores are on the average not aligned across the barrier on the scale of the coherence lenght. The measured vortex density of states is in disagreement with predictions of the normal cores model and previously published data.     

Interaction of vortices in type II superconductors and the behavior nearH c1 at arbitrary temperature

The interaction between well-separated Abrikosov vortices is calculated using Eilenberger's version of the Gorkov theory. Let λ and ξ be the decay lengths determining the asymptotic behavior of the magnetic field and the order parameter for an isolated vortex. Then the onset of attraction in the limit of large vortex separation is either given by λ=ξ or by λ becoming complex, whichever occurs first. It is shown that in general a first order transition atH _c1 takes place already in the region of asymptotic repulsion because of the appearance of a minimum in the interaction energy at finite vortex separation.     

On n-quantum vortices in superconductors

Taking into account terms of a higher order than those usually included in the expansion of free energy close to $\kappa = 1/\sqrt 2$ in the Ginzburg-Landau theory can lead either to the transition from the mixed to the intermediate state, as is commonly supposed, or to a chain of sequential transitions from one-to n-quantum vortices (n=2, 3, ...).     

Magnetic hysteresis in type II superconductors

The theories governing the magnetic behavior of Type II superconductors assume reversible magnetization curves. However, due to imperfections and inhomogeneities, many superconductors possess hysteretic curves. A theory is proposed which allows a ‘reversible’ curve to be found from the two irreversible curves. The ‘reversible’ curve is an approximation to the exact Abrikosov reversible curve for H_c1 ≤ H ≤ H_c2. The theory agrees closely with experimental results.     

Pinning of vortices by the domain structure in a two-layered type II superconductor-ferromagnet system

The effectiveness of magnetic pinning of vortices in a layered system formed by a uniaxial ferromagnet and type II superconductor is considered. It is shown that, irrespective of the saturation magnetization of the ferromagnet, the energy of pinning at the domain structure does not exceed, in order of magnitude, the energy of artificial pinning at a column-type defect. The limitation of pinning energy is caused by the interaction of external vortices with a spontaneous vortex lattice formed in the superconducting film when the magnetization of the ferromagnetic film exceeds the critical value.     

Muonic study of type II superconductors

A theory of μSR method is developed for uniaxial anisotropic high-T _c superconductors. In two extreme cases ofH _ext‖c andH _ext⊥c analytical formulas are obtained which makes it possible to determine from the position of van Hove singularities in the Fourier-spectrum of muonium polarization, the type and parameters of the vortex lattice, such as λ _ab (λ _c ). In caseH _ext≲H _c2 we obtained the shape of the Fourier-spectrum in analytical form and the simple method of determining the Ginsburg-Landau parameter κ. Convenient expressions for numerical calculations are obtained for the arbitrary orientation of the external field, and an algorithm is provided to compute the mean fieldB, the vortex lattice parameters and the bulk field distribution in anisotropic superconductor. The Fourier spectrum of polarization based on these calculations can be used to independently check the validity of the high-T _c parameters determination for “appropriate” orientations.     

Dynamics of quantum vortices in two-dimensional superconductors

The dynamics of rigid vortex translations in two-dimensional arrays of Josephson junctions is shown to be equivalent to the motion of a mass, proportional to the junction capacitance, in a periodic pinning potential. The quantum tunneling of the vortex through the potential barriers is predicted of importance in the existing Nb-arrays at very low temperatures. Above the vortex unbinding temperature there is a plasma resonance of the free vortices leading to an anomaly in the vortex frequency dependent dielectric constant, which could be observed via a radio frequency impedance measurement.     

Fluctuation-induced attraction of vortices in anisotropic superconductors

We study the fluctuation-induced attraction of vortices for continuous anisotropic superconductors by using quantum statistical physics. In the low temperature or quantum limit, only the low-lying modes are relevant, the induced short-range attractive potential is proportional to (k_BT)³K₁(ΓR/λ), while in high temperature or classical limit, the induced attractive potential reduced to long-range van der Waals type k_BT/R⁴ for two vortices separated by a distance R.     

Ac loss minimum in type II superconductors

Theoretical predictions with respect to a minimum in the ac losses in type II superconductors are compared with experimental results on Nb-50% Ti and Nb-10% Mo. Extensions of the theory are suggested to explain the deviations.     

Low-field peak effect in type II superconductors

The combined effect of a surface (edge) barrier and volume pinning on the dependence of critical current I _c on the magnetic field (I⊥H ₀) in bulk type II superconductors is investigated. In low magnetic fields, there is a portion of the curve I _c(H ₀) where I _c grows with H ₀, causing a nontrivial peak effect in this field range. Such behavior is explained by the combined effect of a surface (edge) barrier and volume pinning, the latter being rather sensitive to the transport current density distribution in a superconductor.     

Force free fields in type II superconductors

It seems possible that, under certain field and current conditions, the critical current of a type II superconductor is not determined simply by application of a Bean-London model, but by the joint use of the latter and a second equation.     

Amplification of longitudinal ultrasonic waves by a moving vortex structure in type II superconductors

A moving vortex structure can amplify (generate) longitudinal acoustic waves when the velocity of its motion exceeds a certain critical velocity. The critical velocity is determined by the logarithmic derivative of the viscosity coefficient of the vortex structure with respect to the magnetic field and may be much smaller than the speed of sound. In particular, this effect suggests an alternative explanation for the plateau observed in the current-voltage characteristic of superconducting bridges in a perpendicular magnetic field [S.G. Doettinger et al., Phys. Rev. Lett. 73, 1691 (1994)].