Ginzburg-Landau. Time-Dependent Ginzburg-Landau Eqs.

Transcription

1 ,, 9/13/15 Ginzburg-Landau The Time-Dependent Ginzburg-Landau Theory 1. Time-dependent Ginzburg-Landau (TDGL) Equations. Energy Balane 3. Vortex Flow 4. Marosopi Modelling Simulation Y. Mawatari, 15913, 1 /1 Time-Dependent Ginzburg-Landau Eqs. Time-dependent Ginzburg-Landau (TDGL) equation:! order parameter ψ = ψ exp(iϕ ); superfluid density ψ, phase ϕ! ψ normalized ( ψ = 1 for zero field)! t i π # Φ ψ = ξ! i π # A Current density, rot rot A /µ = j s j n ) *, j s = ψ µ λ # π ϕ A ', A j n = σ n E = σ n # t Φ ' [A. Shmid, Phys. Kondens. Materie 5, 3 (1966).] ψ! # 1 ψ ψ! : GL relaxation time! ξ : oherene length! λ : penetration depth! Φ : salar potential! A : vetor potential! φ : flux quantum Nb NbN T 9K 16K λ 85nm nm ξ 4nm 5nm Y. Mawatari, 15913, /1

2 Derivation and Limitation of the TDGL Equations [N. B. Kopnin, Theory of Nonequilibrium Superondutivity, Clarendon Press (Oxford 1).] Phenomenologial derivation of the TDGL equation for slow relaxation of the order parameter ψ to the equilibrium state:! Ginzburg-Landau free energy, F GL γ ψ t = δf GL γ δψ * t i π Φ ' ψ = δf GL δψ * gauge invariane Mirosopi onsideration leads to the limitation of the TDGL equations:! lose to T! deviations from the equilibrium are small! the quasipartile exitations are in equilibrium with the heat bath! gapless superondutivity generalized TDGL eqn. [Kramer and Watts-Tobin, PRL 4, 141 (1978).] ( ) 1/ 1 Γ ψ # t i π Φ Γ ψ φ t ( ψ = ξ # i π A ' φ ( '! further extensions to: d-wave, muliband (mulitigap), Hall effet,... ψ 1 and ψ γ 1 i γ ψ # 1 ψ (ψ ' Y. Mawatari, 15913, 3 /1 Gauge for Vetor A and Salar Φ Potentials Time-dependent Ginzburg-Landau (TDGL) equation:! t i π # Φ ψ = ξ! i π # A ψ! # 1 ψ ψ Gauge invariane ψ ψe iχ, A A (φ / π ) χ, ' Φ Φ (φ / π ) χ t Gauge hoies! Φ =! div A =! Φ = (1/µ σ n ) div A! Φ = (ω /µ σ n ) div A with ω! thin wire approximation, A = r B a / Y. Mawatari, 15913, 4 /1

3 Energy Balane of the Eletromagneti Energy Eletromagneti energy, F em F em = 1 µ B ε E! time derivative: Poynting s theorem F em t = H B t E D t = E H ( ) E j H = B µ, D = ε E B D E =, H = j t t Poynting s vetor Y. Mawatari, 15913, 5 /1 Energy Balane of the EM and GL Free Energy Free energy, F em F GL 1 ε Fem = B E, µ F GL = µ H ξ - π * i A( ψ, ϕ ) ψ! time derivative: ( F t em F GL ) = S E W,! energy urrent S E = E H µ H ξ Re ψ i π ' Aψ φ ) ψ ( t i π '. - Φψ φ ),- (/, Poynting s vetor! dissipation W = σ n E µ H t i π Φ ' ψ [A. Shmid, Phys. Kondens. Materie 5, 3 (1966).] eletromagneti energy 1 # 4 ψ!! Ginzburg-Landau free energy GL energy urrent Ohmi dissipation dissipation due to the order parameter relaxation Y. Mawatari, 15913, 6 /1

4 D Vortex Flow: Straight Vorties Two dimensional vortex flow without pinning for straight vorties! GL solution for steady state: ψ (r)! TDGL solution for uniform vortex flow: ψ (r, t) = ψ (r v t) expansion in powers of the mean flow speed v equation of motion for vorties (derived from TDGL!): ηv = φ j ẑ visous drag fore! vortex-flow ondutivity, σ f = η/φ B ( j = σ f E ) vortex-flow dissipation, W = σ f E [L. P. Gor kov and N. B. Kopnin, Sov. Phys. Usp. 18, 496 (1976).] Lorentz fore W = σ n E µ H t i π Φ ' ψ t v σ f = σ n B /B [J. Bardeen and M. J. Stephen, Phys. Rev. 14, A1197 (1965).] substantial ontribution from the order-parameter relaxation term [M. Tinkham, PRL 13, 84(1964).] vortex-flow Hall effet (γ γ 1 i γ ): ηv αv ẑ = φ j ẑ [A. T. Dorsey, PRB 46, 8376 (199).] Y. Mawatari, 15913, 7 /1 3D Vortex Flow: Bent Vorties Three dimensional vortex flow without pinning for bent vorties! GL solution for steady state: ψ (r)! TDGL solution for vortex flow: ψ (r, z, t) = ψ (r r (z,t)) expansion in powers of r / t and r / z equation of motion for vorties (derived from TDGL!): η r t = ε r 1 z φ j ẑ visous drag fore [L. P. Gor kov and N. B. Kopnin, Sov. Phys. Usp. 18, 496 (1976).] vortex line tension Lorentz fore Y. Mawatari, 15913, 8 /1

5 HTS Modelling Workshop A very omplex behaviour is obtained when dealing with HTS materials due to high nonlinearity and hysteresis, strong anisotropy, temperature dependene, high aspet ratio and omplex omposite struture of pratial wires and tapes. Suh a omplex behaviour raises new hallenges in the development of reliable modelling tools and requires a speialized researh effort to be effetively dealt with. Lausanne (1), Cambridge (11), Barelona (1), and Bratislava (14) Y. Mawatari, 15913, 9 /1 Summary Time-dependent Ginzburg-Landau (TDGL) theory:! Although TDGL equations are stritly valid only in a state lose to the equilibrium, they give reasonable pitures for the wide variety of nonequilibrium phenomena (e.g., vortex flow).! Poynting s theorem for the eletromagneti energy is naturally extended to the energy balane inluding the Ginzburg-Landau free energy (TDGL and Maxwell equations).! Dissipation due to the order parameter relaxation ontributes to the total dissipation, in addition to the Ohmi dissipation.! Equations of motion for vorties (without pinning enters) are naturally derived from the TDGL equations. Referenes:! N. B. Kopnin, Theory of Nonequilibrium Superondutivity, Clarendon Press (Oxford 1).! A. Shmid, A time dependent Ginzburg-Landau equations and its appliation to the problem of resistivity in the mixed state, Phys. Kond. Materie 5, 3 (1966).! L. P. Gor kov and N. B. Kopnin, Vortex motion and resistivity of type-ii superondutors in a magneti field, Sov. Phys. Usp. 18, 496 (1976).! A. T. Dorsey, Vortex motion and the Hall effet in type-ii superondutors: A time-dependent Ginzburg-Landau theory approah, PRB 46, 8376 (199). Y. Mawatari, 15913, 1 /1