Superconductors

• 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

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-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

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-0.07

-0.06

-0.05

-0.04

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-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

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-0.07

-0.06

-0.05

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-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

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-0.06

-0.05

-0.04

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-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

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-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

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-0.06

-0.05

-0.04

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-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

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-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

Magnetization Loop

• H = 0.00 Hc2

• A few fluxons remain as the inter-fluxon repulsion is lower than the surface 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