Breznay Superconductivity

Beyond Zero Resistance – Phenomenology of Superconductivity

Nicholas P. Breznay

SASS Seminar – Happy 50th!

SLAC

April 29, 2009

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• Motivation / Paradigm Shift

• Normal State behavior

• Hallmarks of Superconductivity– Zero resistance

– Perfect diamagnetism

– Magnetic flux quantization

• Phenomenology of SC– London Theory, Ginzburg-Landau Theory

– Length scales: and – Type I and II SC’s

Physics of Metals - Introduction

• Atoms form a periodic lattice

• Know (!) electronic states key for the behavior we are interested in

• Solve the Schro …

… in a periodic potential

EH

)(2

)( 22

rVm

rH

)()( KrVrV

K is a Bravais lattice vector

K

Wikipedia

Physics of Metals – Bloch’s Theorem

• Bloch’s theorem tells us that eigenstates have the form …

… where u(r) is a function with the periodicity of the lattice …

ErVm

)(2

22

Em

H 22

2

)()( ruer rki

)()( Kruru

rkiAer )(

Free particle Schro

Wikipedia

Physics of Metals – Drude Model

• Model for electrons in a metal– Noninteracting, inertial gas

– Scattering time

• Apply Fermi-Dirac statistics

)(

)(tp

Eqtpdt

d damping

term

H

E

k

E

k

EfEf

m

kE

2

22

http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg

Physics of Metals – Magnetic Response

• Magnetism in media

• Larmor/Landau diamagnetism– Weak anti-// response

• Pauli paramagnetism– Moderate // response

• Typical values –– Cu~ -1 x 10-5

– Al~ +2 x 10-5

minimal response to B fields– r ~ 1 B = 0H

)(0 MHB in SI

linear response

familiarly

H

E

k

E

k

EfEf

HM

H

H

H

r

0

0 )1(B

Physics of Metals – Drude Model Comments

• Wrong!– Lattice, e-e, e-p, defects,

– ~ 10-14 seconds MFP ~ 1 nm

• Useful!– DC, AC electrical conductivity

– Thermal transport• Lorenz number T

– Heat capacity of solids

Wikipedia

Em

neJ

2

m

nep

p

0

22

2

2

,1)(

)(

)(tp

Eqtpdt

d

3ATTCv

Electronic contribution

Lattice

1~'sfe

meas

28

2

22

1044.23 K

W

e

k

TL B

8106.21.2 measL

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Hallmark 1 – Zero Resistance

• Metallic R vs T– e-p scattering (lattice interactions) at high temperature

– Impurities at low temperatures

R

Temperature

ResidualResistance

(impurities)

Impure metal

Electrical resistance

R0

Lattice (phonon)interactions

Pure metal

TD/3


• Superconducting R vs T

R

Temperature

R0

Superconductor

Tc“Transition temperature”


• Hard to measure “zero” directly

• Can try to look at an effect of the zero resistance

• Current flowing in a SC ring– Not thought experiment –

standard configuration for high-field laboratory magnets (10-20T)

• Nonzero resistance changing current changing magnetic field

• One such measurement

SuperconductorCirculating

supercurrent

Magnetic (dipole) field

From Ustinov “Superconductivity” Lectures (WS 2008-2009)

I

1810Cu

SC

Hallmark 1 – Zero Resistance Notes

• R = 0 only for DC

• AC response arises from kinetic inductance of superconducting electrons– Changing current electric field

• Model: perfect resistor (normal electrons), inductor (SC electrons) in parallel

• Magnitude of “kinetic inductance”:

At 1 kHz, NormalRL 1210~

Vac

L

R

http://www.apph.tohoku.ac.jp/low-temp-lab/photo/FUJYO1.png

Hallmark 2 – Conductors in a Magnetic Field

Normal metal

Field off

Applyfield

t

EJB

t

BE

B

E

0

0

jE

)1(~)( /0

teBtB

RL /

Hallmark 2 – Conductors in a Magnetic Field

Applyfield

Perfect (metallic) conductor SuperconductorNormal metal

Cool Cool

Field off

Applyfield

Applyfield

Hallmark 2 – Meissner-Oschenfeld Effect

Superconductor

CoolApplyfield

• B = 0 perfect diamagnetism: M = -1

• Field expulsion unexpected; not discovered for 20 years.

HHM

MHB

0)(0

B/0

H

-M

HHc Hc

Hallmark 3 – Flux Quantization

27150 102~

2102~

2cmG

e

hcsV

e

h

Earth’s magnetic field ~ 500 mG, so in 1 cm2 of BEarth there are ~ 2 million 0’s.

first appearance of h in our description; quantum phenomenon

0nAdB

Total flux (field*area) is integer multiple of

Hallmark 3 – Flux Quantization

Apply uniform field

Measure flux

Aside – Cooper Pairing

• In the presence of a weak attractive interaction, the filled Fermi sphere is unstable to the formation of bound pairs electrons

• Can excite two electrons above Ef, obtain bound-state energy < 2Ef due to attraction

• New minimum-energy state allows attractive interaction (e-p scattering) by smearing the FS

The physics of superconductors Shmidt, Müller, Ustinov

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SC Parameter Review

g(H)

HHc

gnormal state

gsc state

2

2

0cHg

• Magnetic field energy density

• Extract free energy difference between normal and SC states with Hc

• Know magnetic response important; use R = 0 + Maxwell’s equations … ?

London Theory – 1

• Newton’s law (inertial response) for applied electric field

SJdt

dE 2en

m

s

en

J

dt

dmeE

s

S svdt

dmF

sss evnJ

dt

dJ

m

Een Ss 2

dt

Jd

m

Een Ss

2

dt

Jd

dt

Bd

m

en Ss

2

02

B

m

enJ

dt

d sS

Supercurrent density is

Bm

enJ sS

2

We know B = 0 inside superconductors

Faraday’s law

Fritz & Heinz London, (1935)

London Theory – 2

SJdt

dE 2en

m

s

Bm

enJ sS

2

London Equations

t

EJB

000

JB

0

Bm

enBB s

2

02

Bm

enB s

2

02

Ampere’s law

=0; Gauss’s law for

electrostatics

Magnetic Penetration Depth -

B(z)

z

20

2

en

m

s BB

2

2 1

• Screening not immediate;

characteristic decay length

• Typical ~ 50 nm

• m,e fixed – uniquely specifies the superconducting electron density ns

Sometimes called the “superfluid

density”

/0)( zeBzB

B0

SC

Ginzburg-Landau Theory - 1

42

2 ns ff

• First consider zero magnetic field

• Order parameter

• Associate with cooper pair density:

• Expand f in powers of ||2

To make sense, > 0, (T)

Free energy ofsuperconducting state

Free energy ofnormal state

2sn

Need > -Infinity; B > 0

Free energy of SC state

~ # of cooper pairs


42

2 ns ff

42

2 ns ff

02

2 ns ffd

d

2

• For < 0, solve for minimum in fs-fn …

http://commons.wikimedia.org/wiki/File:Pseudofunci%C3%B3n_de_onda_(teor%C3%ADa_Ginzburg-Landau).png

• Know that fn-fs is the condensation energy:


242

2 ns ff

2

2

ns ff

2

02

1csn Bff

sn ff

2

02

1cB

2

0cB


qAip

• Momentum term in H:

• Now – include magnetic field

• Classically, know that to include magnetic fields …

0

2242

22

2

1

2

BeAi

mff ns

ipVm

pH ,

2

2

0

2

2B

fmagnetic

42

2 ns ff


• Free Energy Density 0

2242

22

2

1

2

BeAi

mff ns

02

22

1

2 0

2242

dVB

eAim

0F

022

1 22 eAim

eAim

eJ 2Re

2 *


022

1 22 eAim

Take real,normalize

2 0

22

23

m

0

2)(32

2

mT

Define

mTT

2)()(

2

0)(

22

2 T

Linearize in

Superconducting coherence length -

x

(x)Vacuum SC

Superconductor

022

1 22 eAim

0)(

22

2 T

• Characteristic length scale for SC wavefunction variation

• London Theory magnetic penetration depth

• Ginzburg-Landau Theory coherence length

two kinds of superconductors!

Pause

Surface Energy and “Type II”

H(x)

x

(x)H(x)

x

(x)

Surface Energy: H(x)

(x)

gmagnetic(x)

2

2

0c

cond

Hg

2

2

0c

cond

Hg

energy penalty for excluding B

energy gain for being in SC state

gsc(x)

SC


(x)

gmagnetic(x)

2

2

0c

cond

Hg

2

2

0c

cond

Hg



net energy penalty at a surface / interface

gnet(x)

gsc(x)

SC


(x)

gmagnetic(x)

2

2

0c

cond

Hg

2

2

0c

cond

Hg



net energy gain at a surface / interfacegnet(x)

gsc(x)

SC

Type I Type II

H(x)

(x)

gmagnetic(x)

gnet(x)

gsc(x)

H(x)

(x)

gmagnetic(x)

gnet(x)

gsc(x)

• predicted in 1950s by Abrikosov• elemental superconductors

2

1

2

1

nm (nm) Tc (K) Hc2 (T)

Al 1600 50 1.2 .01

Pb 83 39 7.2 .08

Sn 230 51 3.7 .03

nm (nm) Tc (K) Hc2 (T)

Nb3Sn 11 200 18 25

YBCO 1.5 200 92 150

MgB2 5 185 37 14

Type II Superconductors

H

Normal state cores

Superconducting region

http://www.nd.edu/~vortex/research.html

• London Theory magnetic penetration depth

• Ginzburg-Landau Theory coherence length

two kinds of superconductors

The End

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Breznay Superconductivity