IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of ``Brownian motors.'' In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications.

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Artificial Brownian motors: Controlling transport on the nanoscale

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...

Flux vortices and transport currents in type II superconductors

Large and randomly arranged pinning centers cause a strong deformation of a flux line lattice, so that each pinning center acts on the lattice with a maximum force. The average force for such single-particle pinning can be inferred from a simple summing procedure and has a domelike dependence on magnetic field. Pinning centers of average force, such as clusters of dislocations, strongly deform the flux line lattice only in weak fields and in fields close to the critical field, where there is a peak in the dependence of the critical current on magnetic field. In the range of intermediate fields there is a weak collective pinning. A large concentration of weak centers leads to collective pinning in all fields. In this case, near the critical field a critical current peak should be observed. To explain this peak and to define the boundaries between the regions of collective and single-particle pinning the possible break-off of the flux line lattice from the lines of magnetic force should be taken into consideration, which leads to extra softening of the lattice.

Pinning in type II superconductors

Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux phi 0=h/2e. These tiny vortices of supercurrent tend to arrange themselves in a triangular flux-line lattice (FLL), which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-Tc superconductors (HTSCs) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSCs the FLL is very soft mainly because of the large magnetic penetration depth lambda . The shear modulus of the FLL is c66~1/ lambda 2, and the tilt modulus c44(k)~(1+k2 lambda 2)-1 is dispersive and becomes very small for short distortion wavelengths 2 pi /k<< lambda . This softness is enhanced further by the pronounced anisotropy and layered structure of HTSCs, which strongly increases the penetration depth for currents along the c axis of these (nearly uniaxial) crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may `melt` the FLL and cause thermally activated depinning of the flux lines or ofthe two-dimensional `pancake vortices` in the layers. Various phase transitions are predicted for the FLL in layered HTSCs. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSCs as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

https://iopscience.iop.org/article/10.1088/0034-4885/58/11/003/pdf

The flux-line lattice in superconductors

Current density, magnetic field, penetrated magnetic flux, and magnetic moment are calculated analytically for a thin strip of a type-II superconductor carrying a transport current [ital I] in a perpendicular magnetic field [ital H][sub [ital a]] Constant critical current density [ital j][sub [ital c]] is assumed The exact solutions reveal interesting features of this often realized [ital perpendicular] geometry that qualitatively differs from the widely used Bean critical state model: At the penetrating flux front the field and current profiles have vertical slopes; the initial penetration depth and penetrated flux are [ital quadratic] in [ital H][sub [ital a]] and [ital I]; the initial deviation from a linear magnetic moment is [ital cubic] in [ital H][sub [ital a]]; the hysteresis losses are proportional to the [ital fourth] power of a small ac amplitude; the current density [ital j] is [ital finite] over the entire width of the strip even when flux has only partly penetrated; in thin films, as soon as the direction of the temporal change of [ital H][sub [ital a]] or [ital I] is reversed, [ital j] falls below [ital j][sub [ital c]] [ital everywhere], thus stopping flux creep effectively; the Lorentz force can drive the vortices uphill'' againstmore » the flux-density gradient These analytical results are at variance with the critical-state model for longitudinal geometry and explain numerous experiments in a natural way without the assumption of a surface barrier« less

Type-II-superconductor strip with current in a perpendicular magnetic field.

The magnetic moment, flux and current penetration, and creep in type-II superconductors of nonzero thickness in a perpendicular applied magnetic field are calculated. The presented method extends previous one-dimensional theories of thin strips and disks to the more realistic case of arbitrary thickness, including as limits the perpendicular geometry (thin long strips and circular disks in a perpendicular field) and the parallel geometry (long slabs and cylinders in a parallel field). The method applies to arbitrary cross section and arbitrary current-voltage characteristics E(J) of conductors and superconductors, but a linear equilibrium magnetization curve B=${\mathrm{\ensuremath{\mu}}}_{0}$H and isotropy are assumed. Detailed results are given for rectangular cross sections 2a\ifmmode\times\else\texttimes\fi{}2b and power-law electric field E(J)=${\mathit{E}}_{\mathit{c}}$(J/${\mathit{J}}_{\mathit{c}}$${)}^{\mathit{n}}$ versus current density J, which includes the Ohmic (n=1) and Bean (n\ensuremath{\rightarrow}\ensuremath{\infty}) limits. In the Bean limit above some applied field value the lens-shaped flux- and current-free core disconnects from the surface, in contrast to previous estimates based on the thin strip solution. The ideal diamagnetic moment, the saturation moment, the field of full penetration, and the complete magnetization curves are given for all side ratios 0b/a\ensuremath{\infty}. \textcopyright{} 1996 The American Physical Society.

Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs.

The determination of the magnetic penetration depth \ensuremath{\lambda} in type-II superconductors from the field distribution n(B) measured by muon-spin rotation (\ensuremath{\mu}SR) is discussed. Particular stress is put on high-${T}_{c}$ oxide superconductors (coherence length \ensuremath{\xi}\ensuremath{\ll}\ensuremath{\lambda}; upper critical field much greater than the applied field; anisotropy and pinning). Fitting of the highly asymmetric n(B) by a Gaussian or Lorentzian, as done in existing \ensuremath{\mu}SR experiments, yields for a perfect vortex lattice \ensuremath{\lambda} values which are too large, and for a strongly distorted (due to pinning) lattice \ensuremath{\lambda} values which are too small. An improved evaluation of \ensuremath{\mu}SR data is suggested.

Flux distribution and penetration depth measured by muon spin rotation in high- T c superconductors

The current density in type-II superconductor circular disks of arbitrary thickness, or cylinders of finite length, in an axial magnetic field is calculated from first principles by treating the superconductor as a conductor with nonlinear resistivity or with linear complex resistivity, both caused by thermally activated depinning of Abrikosov vortices. From these currents follows the magnetic field inside and outside the specimen and the magnetic moment, which in its turn determines the nonlinear and linear ac susceptibilities. The magnetization loops and nonlinear ac susceptibilities are obtained directly by time integration of an integral equation for the current density, which does not require any cutoff or approximation of the magnetic field outside the cylinder. With increasing thickness the results go over from the recently obtained solutions for thin disks in a perpendicular field to the classical behavior of long cylinders in a parallel field. Here this direct method is applied to homogeneous disks with constant thickness, but it applies to any axially symmetric superconductor with arbitrary cross section and inhomogeneity.

Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves

The thermal fluctuation of the vortex positions in type-II superconductors with large Ginzburg-Landau parameter \ensuremath{\kappa} and weak pinning is drastically increased when calculated from the correct nonlocal elasticity of the vortex lattice. As compared to the usual (local) elastic result, the mean-square shear strain and vortex displacements increase by a factor of \ensuremath{\approxeq}(B\ifmmode\bar\else\textasciimacron\fi{}/${\mathit{B}}_{\mathit{c}1}$${)}^{1/2}$/(1-B\ifmmode\bar\else\textasciimacron\fi{}/${\mathit{B}}_{\mathit{c}2}$${)}^{1/2}$ (B\ifmmode\bar\else\textasciimacron\fi{} is the flux density, ${B}_{c1}$ and ${B}_{c2}$ are the lower and upper critical fields). The melting temperature of the vortex lattice is reduced by the same factor.

Ernst Helmut Brandt

Papers

Type-II-superconductor strip with current in a perpendicular magnetic field.

Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs.

Flux distribution and penetration depth measured by muon spin rotation in high- T c superconductors

Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves

Thermal fluctuation and melting of the vortex lattice in oxide superconductors.