Relaxation of the Vortex Lattice in Type-II Superconductors

Abstract

When a type-II superconductor is put into a magnetic field H a between the lower (H c 1) and upper (H c 2) critical fields, magnetic flux can penetrate in form of flux lines or current vortices, each carrying a quantum of magnetic flux ø 0. These vortices arrange to a more or less perfect hexagonal flux-line lattice (FLL) with spacing a. Type-II superconductors are characterized by their Ginzburg-Landau parameter EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSMaey % ypa0Jaeq4UdWMaai4laiabe67a4jabgwMiZoaalyaabaGaaGymaaqa % amaakaaabaGaaGOmaaWcbeaaaaaaaa!4044! ]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\kappa = \lambda /\xi \geqslant {1 \mathord{\left/{\vphantom {1 {\sqrt 2 }}} \right.\kern-\nulldelimiterspace} {\sqrt 2 }}$$ where λ ≈.1...0.2μm is the magnetic penetration depth (the radius of the flux tubes) and ζ = λ/κ is the coherence length of the superconducting electron pairs (the radius of the vortex cores). Pure Nb has a small κ = 0.72, alloys (e.g. NbTi) exhibit κ ≈ 10...50, and the new high-temperature superconductors even κ ≈ 200; their (anisotropic) coherence length is, therefore very small, ζ ≈ ...30 Å. This means that the vortex cores do not overlap except at extremely high fields H a > 0.3H c 2 with μ 0 H c 2 = B c 2 ≈ 100 Tesla. The electrodynamics of the FLL in these new superconductors is, therefore, with good accuracy described by the London equation for the microscopic induction B(r, t): EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq % 4UdW2aaWbaaSqabeaacaaIYaaaaOGaey4bIe9aaWbaaSqabeaacaaI % YaaaaOGaamOqaiabgUcaRiaadkeacqGH9aqpcqaHvpGzdaWgaaWcba % GaaGimaaqabaGcdaaeqbqaamaapuaabaGaamizaiaadkhadaWgaaWc % baGaamODaaqabaaabeqab0GaeSyeUhTaey4kIipaaSqaaiaadAhaae % qaniabggHiLdGccqaH0oazdaWgaaWcbaGaaG4maaqabaGcdaqadaqa % aiaadkhacqGHsislcaWGYbWaaSbaaSqaaiaadAhaaeqaaaGccaGLOa % Gaayzkaaaaaa!54E0! ]></EquationSource><EquationSource Format="TEX"><![CDATA[$$ - {\lambda ^2}{\nabla ^2}B + B = {\phi _0}\sum\limits_v {\oint {d{r_v}} } {\delta _3}\left( {r - {r_v}} \right)$$ (1) where the sum is over all vortices, the integrals are along the vortex lines with (in general curved and time-dependent) core positions r v , and δ 3 is the three-dimensional delta function. The anisotropy of the material may be accounted for by replacing in (1) λ 2 by a tensor. In the following this anisotropy shall be disregarded for simplicity.KeywordsVortex CoreFlux TubeVortex LineVortex LatticeFlux LineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.