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• Gor’kov and Eliashberg found the time-dependent G-L equations using microscopic theory:
• These equations are solved numerically
0212
11
22
20
e
itD
e
iA
*2
0 0 0
1 1Re 2
2e
e i t
AA A
•Model superconductors
Fluxons moving into a Superconductor Coated With a Normal Metal, = 5 as the applied
magnetic field increases
• H = 0.05 Hc2
• The material is in the Meissner state
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
• H = 0.10 Hc2
• The material is in the Meissner state
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
Magnetization Loop
• H = 0.15 Hc2
• The material is in the mixed state
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.20 Hc2
• The material is in the mixed state
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.25 Hc2
• The material is in the mixed state
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.30 Hc2
• Note the nucleation of fluxons at the superconductor-normal boundary
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.35 Hc2
• Note the nucleation of fluxons at the superconductor-normal boundary
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.40 Hc2
• Note the nucleation of fluxons at the superconductor-normal boundary
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.45 Hc2
• Note the nucleation of fluxons at the superconductor-normal boundary
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.50 Hc2
• Note the nucleation of fluxons at the superconductor-normal boundary
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.55 Hc2
• In the reversible region, one can determine
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.60 Hc2
• In the reversible region, one can determine
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.65 Hc2
• In the reversible region, one can determine
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.70 Hc2
• In the reversible region, one can determine
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.75 Hc2
• In the reversible region, one can determine
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.80 Hc2
• The core of the fluxons overlap and the average value of the order parameter drops
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.85 Hc2
• The core of the fluxons overlap and the average value of the order parameter drops
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.90 Hc2
• The core of the fluxons overlap and the average value of the order parameter drops
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.95 Hc2
• The core of the fluxons overlap and the average value of the order parameter drops
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 1.00 Hc2
• Eventually the superconductivity is destroyed
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.95 Hc2
• Superconductivity nucleates
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.90 Hc2
• Superconductivity nucleates
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.85 Hc2
• Superconductivity nucleates
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.80 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
netiz
aion
M (Hc
2)
Magnetization Characteristic100 80, = 5
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.75 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
netiz
aion
M (Hc
2)
Magnetization Characteristic100 80, = 5
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.70 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
netiz
aion
M (Hc
2)
Magnetization Characteristic100 80, = 5
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.65 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
netiz
aion
M (Hc
2)
Magnetization Characteristic100 80, = 5
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.60 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.55 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.50 Hc2
• Note the Abrikosov flux-line-lattice with hexagonal symmetry
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.45 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.40 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.35 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.30 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.25 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.20 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.15 Hc2
• The flux-line-lattice is defective
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.10 Hc2
• The magnetisation is positive because of flux pinning
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5
Magnetization Loop
• H = 0.05 Hc2
• The magnetisation is positive because of flux pinning
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
Applied field H (H c 2)
Mag
neti
zaio
n M
(Hc
2)
Magnetization Characteristic100 80, = 5