/10 Y. Mawatari, 150913 �������, ������� 1

� �� Ginzburg-Landau ���

The Time-Dependent Ginzburg-Landau Theory

�����������

����������#&���������� !$&�"%����, �������, 9/13/2015

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

/10 Y. Mawatari, 150913 �������, ������� 2

Time-Dependent Ginzburg-Landau Eqs.�

τ0∂∂t+ i 2π

φ0Φ

!

"##

$

%&&ψ = ξ0

2 ∇ − i 2πφ0A

!

"##

$

%&&

2

ψ + 1− ψ2!

"#

$%&ψ

!  τ0 : GL relaxation time !  ξ0 : coherence length !  λ0 : penetration depth !  Φ : scalar potential !  A : vector potential �!  φ0 : flux quantum

Nb� NbN�

Tc� 9K� 16K�

λ0� 85nm� 200nm�

ξ0� 40nm� 5nm�

�  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)

js =ψ

2

µ0λ02φ0

2π∇ϕ − A

"

#$

%

&',

jn =σ nE = −σ n∂A∂t+∇Φ

"

#$

%

&'

)

*

++

,

++

�  Current density, rot rot A /µ0 = js + jn

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

/10

�  Phenomenological derivation of the TDGL equation for slow relaxation of the order parameter ψ to the equilibrium state: !  Ginzburg-Landau free energy, FGL

Y. Mawatari, 150913 �������, ������� 3

Derivation and Limitation of the TDGL Equations�

γ∂ψ∂t

= −δFGLδψ*

γ∂∂t+ i 2π

φ0Φ

"

#$$

%

&''ψ = −

δFGLδψ*gauge 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).]

τ0 1+Γ2 |ψ |2( )

−1/2 ∂∂t+ i 2π

φ0Φ +

Γ 2

2∂ |ψ |2

∂t

#

$%%

&

'((ψ = ξ0

2 ∇ − i 2πφ0A

#

$%%

&

'((

2

ψ + 1− ψ2#

$%

&'(ψ

!  further extensions to: d-wave, muliband (mulitigap), Hall effect,...

ψ1 and ψ2 γ1 + i γ2

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

/10

�  Gauge invariance

Y. Mawatari, 150913 �������, ������� 4

Gauge for Vector A and Scalar Φ Potentials�

τ0∂∂t+ i 2π

φ0Φ

!

"##

$

%&&ψ = ξ0

2 ∇ − i 2πφ0A

!

"##

$

%&&

2

ψ + 1− ψ2!

"#

$%&ψ

�  Time-dependent Ginzburg-Landau (TDGL) equation:

ψ→ ψeiχ , A→ A+ (φ0 / 2π )∇χ , Φ→ Φ− (φ0 / 2π )∂χ ∂t

$

%&

'&

�  Gauge 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

/10

�  Electromagnetic energy, Fem

!  time derivative: Poynting’s theorem

Y. Mawatari, 150913 �������, ������� 5

Energy Balance of the Electromagnetic Energy�

Fem =12µ0

B2 +ε02E2

,

, 00

tt ∂∂

+=×∇∂∂

−=×∇

==

DjHBE

EDBH εµ

∂Fem∂t

= H ⋅∂B∂t

+ E ⋅ ∂D∂t

= −∇⋅ E ×H( )− E ⋅ j

Poynting’s vector�

/10

∂∂tFem + FGL( ) = −∇⋅ SE −W ,

W =σ nE2 + 2µ0Hc

2τ0∂∂t+ i 2π

φ0Φ

"

#$$

%

&''ψ

2

SE = E ×H + 2µ0Hc2ξ02 Re ∇ψ∗ + i 2π

φ0Aψ∗

$

%&&

'

())∂ψ∂t

+ i 2πφ0Φψ

$

%&&

'

())

+

,--

.

/00,

!  time derivative:

!  energy current

!  dissipation

dissipation due to the order parameter relaxation�Ohmic dissipation�

GL energy current�Poynting’s vector�

�  Free energy, Fem + FGL

Y. Mawatari, 150913 �������, ������� 6

Energy Balance of the EM and GL Free Energy�

!!"

#

$$%

&+−((

)

*++,

-−∇=

+=

422

0

20

20GL

202

0em

212

,22

1

ψψψϕπ

ξµ

εµ

A

EB

iHF

F

c Ginzburg-Landau free energy�

electromagnetic energy�

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

/10

�  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) expansion in powers of the mean flow speed v0

" equation of motion for vortices (derived from TDGL!): !  vortex-flow conductivity, σf = η/φ0B ( j = σf E )

vortex-flow dissipation, W = σf ‹E›2

Y. Mawatari, 150913 �������, ������� 7

2D Vortex Flow: Straight Vortices�

ηv0 = φ0 j × z

" vortex-flow Hall effect (γ → γ1 + i γ2): �ηv0 +αv0 × z = φ0 j × z [A. T. Dorsey, PRB 46, 8376 (1992).]�

σf = σn Bc2/B [J. Bardeen and M. J. Stephen, Phys. Rev. 140, A1197 (1965).]�

W =σ nE2 + 2µ0Hc

2τ0∂∂t+ i 2π

φ0Φ

"

#$$

%

&''ψ

2 substantial contribution from the order-parameter relaxation term [M. Tinkham, PRL 13, 804(1964).]�

∂ ∂t→ v0 ⋅∇

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

viscous drag force� Lorentz force�

/10

�  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!):

Y. Mawatari, 150913 �������, ������� 8

3D Vortex Flow: Bent Vortices�

η∂r0∂t

= ε1∂2r0∂z2

+φ0 j × z

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

viscous drag force�

vortex line tension�Lorentz force�

/10 Y. Mawatari, 150913 �������, ������� 9

HTS Modelling Workshop�

Lausanne (2010), Cambridge (2011), Barcelona (2012), and Bratislava (2014)�

A very complex behaviour is obtained when dealing with HTS materials due to high non-linearity and hysteresis, strong anisotropy, temperature dependence, high aspect ratio and complex 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.�

/10 Y. Mawatari, 150913 �������, ������� 10

Summary�

�Time-dependent Ginzburg-Landau (TDGL) theory: !  Although TDGL equations are strictly valid only in a state close to the

equilibrium, they give reasonable pictures for the wide variety of nonequilibrium 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 problem

of 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 a

magnetic 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).�