Institute of Solid State PhysicsTechnische Universität Graz

18. Superconductivity

Dec. 3, 2018

YBa2Cu3Ox

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(Sn5In)Ba4Ca2Cu10Oy Tc = 212 K

SuperconductivityCritical temperature Tc

Critical current density Jc

Critical field Hc

2 2 210 2 22B c c c

mn nk T H nmv Jne

Superconductivity

Perfect diamagnetism

Jump in the specific heat like a 2nd order phase transition, not a structural transition

Superconductors are good electrical conductors but poor thermal conductors, electrons no longer conduct heat

There is a dramatic decrease of acoustic attenuation at the phase transition, no electron-phonon scattering

Dissipationless currents - quantum effect

Electrons condense into a single quantum state - low entropy.

Electron decrease their energy by but loose their entropy.

Probability current

21 ( )2

i i qA Vt m

Schrödinger equation for a charged particle in an electric and magnetic field is

write out the term 2( )i qA

ie i ie i e

22 2 22i i i ie i e i e e

2 2 2 21 -2

i i qA ihq A q A Vt m

write the wave function in polar form

Probability current

22 2 2

2 2

1 - 22

i i it t m

i qA i ihq A q A V

Schrödinger equation becomes:

Real part:

222-

2q A V

t m

Imaginary part:

22 21 - 2 22

i qA q At m

Probability current

Imaginary part:

22 21 - 2 22

i qA q At m

Multiply by || and rearrange

2 2 0q At m

This is a continuity equation for probability

0P St

2 qS Am

The probability current:

Probability current / supercurrent

2 qS Am

The probability current:

All superconducting electrons are in the same state so

2cp

e

e n ej Am

2 cpj en S

In superconductivity the particles are Cooper pairs q = -2e, m = 2me, ||2 = ncp.

London gauge 0 2 22 cp s

e e

n e n ej A Am m

ns = 2ncp

This result holds for all charged particles in a magnetic field.

1st London equation

First London equation:

2s

e

n ej Am

2 2s s

e e

n e n edj dA Edt m dt m

dA Edt

2s

e

n edj Edt m

s

dv m djeE mdt n e dt

Classical derivation:

2s

e

n edj Edt m

Heinz & Fritz

2nd London equation

Second London equation:

2s

e

n ej Am

2s

e

n ej Am

2s

e

n ej Bm

Meissner effect

London penetration depth:

20s

e

n eB Bm

Combine second London equation with Ampere's law

0B j

20s

e

n em

2s

e

n ej Bm

2B B B

2 2 B B

Helmholtz equation:

Meissner effect2

0s

e

n em 2 2 B B

superconductor

0 ˆexp xB B z

Al = 50 nmIn = 65 nmSn = 50 nmPb = 40 nmNb = 85 nm

0

0

ˆexpB xj y

0B j

solution to Helmholtz equation:

B0

z

x

Flux quantization

2cp

e

e n ej Am

For a ring much thicker than the penetration depth, j = 0 along the dotted path.

20 e A

Integrate once along the dotted path.

2 2 2 2

S S

e e e edl A dl A ds B ds

magnetic fluxStokes' theorem

Flux quantization

0

22 en

n = ... -2, -1, 0, 1, 2 ...

0n

Superconducting flux quantum

15 20 2.0679 10 [W = T m ]

2he

22 edl n

Flux quantization:

Flux quantization

0 2nhn

e

Flux is quantized through a superconducting ring.

STS image of the vortex lattice in NbSe2.(630 nm x 500 nm, B = .4 Tesla, T = 4 K)

http://www.insp.upmc.fr/axe1/Dispositifs%20quantiques/AxeI2_more/VORTICES/vortexHD.htm

Vortices in Superconductors

Type I and Type II

Superconductors are perfect diamagnets at low fields. B=0 inside a bulk superconductor.

0B H M

-M-M

Ha Ha

Type I and Type II

Vortices in Superconductors

F q E v B

1F j Bn

j nqv

dVdt

Lorentz force

Faraday's law

j

Defects are used to pin the vortices

Superconducting Magnets

Whole body MRI

Magnets and cables

Maglev trains

ITER

Magnet wire

Nb3Sn Magnet

Superconducting magnets

Largest superconducting magnet, CERN21000 Amps

ac - Josephson effect

10 V standard

Brian Josephson

http://www.nist.gov/pml/history-volt/superconductivity_2000s.cfm

DOI: 10.1140/epjst/e2009-01050-6

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0 2d nhfV n fdt e