 #&   !$&"%,   , 9/13/2015  Ginzburg-Landau  
The Time-Dependent Ginzburg-Landau Theory

1. Time-dependent Ginzburg-Landau (TDGL) Equations
2. Energy Balance
3. Vortex Flow
4. Macroscopic Modelling Simulation

Y. Mawatari, 150913

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Time-Dependent Ginzburg-Landau Eqs.

[A. Schmid, Phys. Kondens. Materie 5, 302 (1966).]
Time-dependent Ginzburg-Landau (TDGL) equation:! order parameter ψ = |ψ | exp(iϕ ); superfluid density |ψ |2, phase ϕ ! ψ normalized (|ψ | = 1 for zero field)
2
!∂
!
2$!
2π $
2π $
2
τ 0 ## + i Φ &&ψ = ξ 0 ##∇ − iA&& ψ + #1− ψ &ψ
"%
φ0 %
φ0 %
" ∂t"



Current density, rot rot A /µ0 = js + jn2)
ψ " φ0%+ js =∇ϕ − A',$2+&µ 0 λ0 # 2 π*" ∂A%++, j n = σ n E = −σ n $# ∂t +∇Φ '&Y. Mawatari, 150913

Nb

NbN

Tc

9K

16K

λ0

85nm

200nm

ξ040nm

5nm

τ0 : GL relaxation timeξ0 : coherence length! λ0 : penetration depth! Φ : scalar potential! A : vector potential ! φ0 : flux quantum!
!

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Derivation and Limitation of the TDGL Equations

[N. B. Kopnin, “Theory of Nonequilibrium Superconductivity,” Clarendon Press (Oxford 2001).]

Phenomenological derivation of the TDGL equation
for slow relaxation of the order parameter ψ to the equilibrium state:! Ginzburg-Landau free energy, FGL"∂δFδF∂ψ
2π %γ= − GLγ+iΦ ''ψ = − GL$$*∂tφ0 &δψδψ *# ∂tgauge invarianceMicroscopic consideration leads to the limitation of the TDGL equations:! close to Tc! deviations from the equilibrium are small! the quasiparticle excitations are in equilibrium with the heat bath! gapless superconductivity" generalized TDGL eqn. [Kramer and Watts-Tobin, PRL 40, 1041 (1978).]2#−1/2 # ∂2&#2π
Γ 2 ∂ | ψ |2 &2π &222%% + i Φ +((ψ = ξ 0 %%∇ − i
τ 0 1+ Γ | ψ |A(( ψ + %1− ψ (ψ$'φ02 ∂t 'φ0 '$$ ∂t(!

)

further extensions to:d-wave, muliband (mulitigap), Hall effect,...
ψ1 and ψ2
γ1 + i γ2Y. Mawatari, 150913

Gauge for Vector A and Scalar Φ Potentials


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Time-dependent Ginzburg-Landau (TDGL) equation:
2!∂!
2$!
2π $
2π $2
τ 0 ## + i Φ &&ψ = ξ 0 ##∇ − iA&& ψ + #1− ψ &ψ
"%
φ0 %
φ0 %
" ∂t"



Gauge invariance$ ψ → ψ ei χ ,&% A → A + (φ0 / 2π )∇χ ,&' Φ → Φ − (φ0 / 2π )∂χ ∂tGauge choices
Φ=0! div A = 0! Φ = – (1/µ0σn) div A! Φ = – (ω /µ0σn) div A with ω ≥ 0! thin wire approximation, A = r × Ba / 2!

Y. Mawatari, 150913

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Energy Balance of the Electromagnetic EnergyElectromagnetic energy, Fem
Fem =!

1 2 ε0 2
B + E
2µ 02time derivative: Poynting’s theorem∂Fem∂B∂D=H⋅+E⋅∂t∂t∂t= −∇ ⋅ E × H − E ⋅ j(H = B µ0 , D = ε 0 E∇× E = −∂B∂D, ∇× H = j +∂t∂t)

Poynting’s vector

Y. Mawatari, 150913

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Energy Balance of the EM and GL Free EnergyFree energy, Fem + FGL
1 2 ε0 2Fem =B + E ,2 µ02&

[A. Schmid, Phys. Kondens. Materie 5, 302 (1966).]

electromagnetic energy2π *FGL = µ0 H c2 $ξ02 ++ ∇ − iA (ψϕ 0 ()$ ,%21 4#2−ψ + ψ !
2!"

Ginzburg-Landau free energy

time derivative:∂F + F = −∇ ⋅ S E −W ,∂t em GL! energy current+$'$ ∂ψ 2π'.
2π
S E = E × H + 2µ0 H c2ξ 02 Re -&& ∇ψ ∗ + iAψ ∗ ))&&+ i Φψ ))0,
φ0
φ0-,%(% ∂t(0/!

(!

)

Poynting’s vector
2dissipation"%∂
2π
W = σ n E 2 + 2µ0 H c2τ 0 $$ + i Φ ''ψ
φ0 &# ∂t
Ohmic dissipation

Y. Mawatari, 150913

GL energy currentdissipation due to the order parameter relaxation ,  6 /10

2D Vortex Flow: Straight Vortices

[L. P. Gor’kov and N. B. Kopnin, Sov. Phys. Usp. 18, 496 (1976).]


Two dimensional vortex flow without pinning for straight vortices
GL solution for steady state: ψ0 (r)! TDGL solution for uniform vortex flow: ψ (r, t) = ψ0 (r – v0t)
∂ ∂t → v0 ⋅ ∇expansion in powers of the mean flow speed v0!

" equation of motion for vortices (derived from TDGL!):ηv0 = φ0 j × ẑ
viscous drag force!

σf = σn Bc2/B

Lorentz force

vortex-flow conductivity, σf = η/φ0B ( j = σf E )vortex-flow dissipation, W = σf ‹E›2
W = σnE2"%∂
2π+ 2µ0 H c2τ 0 $$ + i Φ ''ψ
φ0 &# ∂t2

[J. Bardeen and M. J. Stephen,Phys. Rev. 140, A1197 (1965).]

substantial contribution from theorder-parameter relaxation term
[M. Tinkham, PRL 13, 804(1964).]

" vortex-flow Hall effect (γ → γ1 + i γ2): [A. T. Dorsey, PRB 46, 8376 (1992).]
ηv0 + α v0 × ẑ = φ0 j × ẑY. Mawatari, 150913

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 ,  3D Vortex Flow: Bent Vortices[L. P. Gor’kov and N. B. Kopnin, Sov. Phys. Usp. 18, 496 (1976).]


Three dimensional vortex flow without pinning for bent vortices! GL solution for steady state: ψ0(r)! TDGL solution for vortex flow: ψ (r, z, t) = ψ0(r – r0(z,t))
expansion in powers of ∂r0/∂t and ∂r0/∂z" equation of motion for vortices (derived from TDGL!):

η∂r0
∂2 r
= ε1 20 + φ0 j × ẑ∂t∂z

viscous drag force

Lorentz force

vortex line tension

Y. Mawatari, 150913

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HTS Modelling Workshop

A very complex behaviour is obtained when dealing with HTS materials due to high nonlinearity and hysteresis, strong anisotropy, temperature dependence, high aspect ratio andcomplex composite structure of practical wires and tapes. Such a complex behaviour
raises new challenges in the development of reliable modelling tools and requires a
specialized research effort to be effectively dealt with.

Lausanne (2010), Cambridge (2011), Barcelona (2012), and Bratislava (2014)
Y. Mawatari, 150913

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SummaryTime-dependent Ginzburg-Landau (TDGL) theory:Although TDGL equations are strictly valid only in a state close to theequilibrium, they give reasonable pictures for the wide variety ofnonequilibrium phenomena (e.g., vortex flow).
! Poynting’s theorem for the electromagnetic energy is naturally extended
to the energy balance including the Ginzburg-Landau free energy(TDGL and Maxwell equations).
! Dissipation due to the order parameter relaxation contributes to the total
dissipation, in addition to the Ohmic dissipation.
! Equations of motion for vortices (without pinning centers) are naturally
derived from the TDGL equations.
!

References:! N. B. Kopnin, “Theory of Nonequilibrium Superconductivity,” Clarendon Press (Oxford 2001).
! A. Schmid, “A time dependent Ginzburg-Landau equations and its application to the problemof resistivity in the mixed state,” Phys. Kond. Materie 5, 302 (1966).
! L. P. Gor’kov and N. B. Kopnin, “Vortex motion and resistivity of type-II superconductors in amagnetic field,” Sov. Phys. Usp. 18, 496 (1976).
! A. T. Dorsey, “Vortex motion and the Hall effect in type-II superconductors: A time-dependent
Ginzburg-Landau theory approach,” PRB 46, 8376 (1992).
Y. Mawatari, 150913

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