Superconductivity

M.C. Chang

Dept of Phys

•Introduction

•Thermal properties

•Magnetic properties

•London theory of the Meissnereffect

•Microscopic (BCS) theory

•Flux quantization

•Quantum tunneling

A brief history of low temperature (Ref: 絕對零度的探索)

•1800 Charles and Gay-Lusac(from P-Trelationship) proposed

that the lowest temperature is -273 C (= 0 K)

•1877 Cailletetand PictetliquifiedOxygen (-183 C or 90 K)

•soon after, Nitrogen (77 K) is liquified

•1898 Dewar liquifiedHydrogen (20 K)

•1908 OnnesliquifiedHelium (4.2 K)

•1911 Onnesmeasured the resistance of metal at

such a low T. To remove residual resistance, he chose

mercury. Near 4 K, the resistance drops to 0.

ρ

T

Au

Hg

ρR

ρR

1913

G. Amontons

1700

1.14K

3.72K

7.19K

3.40K

1.09K

2.39K

4.15K

0.39K5.38K

0.55K

0.12K

9.50K

4.48K

0.92K

7.77K

0.01K1.4K

0.66K

0.14K

0.88K

0.56K0.51K

0.03K

1.37K1.4K

4.88K

0.20K0.60K

0.0003K

http://superconductors.org/Type1.htm

Tc'sgiven are for bulk, except for Palladium, which has been irradiated with

He+ ions, Chromium as a thin film, and Platinum as a compacted powder

Superconductivity in alloys and oxides

From Cywinski’slecture note

19101930195019701990

20

40

60

80

100

120

140

160

Superconducting transition temperature (K)

Hg

Pb

Nb

NbC

NbC

NbN

NbN

V3Si

V3Si

Nb3Sn

Nb3Sn

Nb3Ge

Nb3Ge

(LaBa)CuO

(LaBa)CuO

YBa2Cu3O7

YBa2Cu3O7

BiCaSrCuO

BiCaSrCuO

TlBaCaCuO

TlBaCaCuO

HgBa

2Ca2Cu3O9

HgBa2Ca2Cu3O9

HgBa

2Ca2Cu3O9

(under pressure)

HgBa2Ca2Cu3O9

(under pressure)

Liquid Nitrogen

temperature (77K)

Bednorz

Muller

1987

Applications of superconductor

•powerful magnet

•MRI, LHC...

•magnetic levitation

•SQUID (超導量子干涉儀)

•detect tiny magnetic field

•quantum bits

•lossless powerline

•…

•Introduction

•Thermal properties

•Magnetic properties

•London theory of the Meissnereffect

•Microscopic (BCS) theory

•Flux quantization

•Quantum tunneling

Thermal properties of SC: specific heat

For different superconductors,

at

CC

C

T

SN

N

C

−

~.143

The exponential dependence with Tis

called “activation”behavior and implies

the existence of an energy gapabove

Fermi surface.Δ

~ 0.1-1 meV(10-4~-5

EF

)

Tk

el

S

B

eC

)0(Δ−

∝

•Temperature dependence of Δ

(obtained from Tunneling)

Δ‘s scale with differentTc’s

2Δ(0) ~ 3.5 kBTc

Universal behavior of Δ(T)

•Connection between

energy gap and Tc

1/2

()

1.741 for

(0)

C

C

TT

TT

T

⎛⎞

Δ

=

−≈

⎜⎟

Δ

⎝⎠

•Entropy

H

S

CT

T

∂

⎛⎞

=

⎜⎟

∂

⎝⎠

2

2

H

H

H

F

S

T

CSF

TTT

∂

⎛⎞

=−

⎜⎟

∂

⎝⎠

⎛⎞

∂∂

⎛⎞

==−

⎜⎟

⎜⎟

∂∂

⎝⎠

⎝⎠

Less entropy in SC state:

more ordering

2nd order phase

transition

FN-FS

= Condensation energy

～10-8

eVper electron!

Al

Al

•free energy

More evidences of energy gap

•Electron tunneling

2Δsuggests excitations

created in “e-h”pairs

ν

==

2

Δ

h

480 GHz (microwave)

•EM wave absorption

Magnetic property of the superconductor

All curves can be collapsed onto

a similar curve after re-scaling.

()

{

}

2

()1

coc

HTHTT=−

()

{

}

2

()1

coc

HTHTT=−

sc

normal

•Superconductivity is destroyed by a strong magnetic field.

Hc

for metal is of the order of 0.1 Tesla or less.

•Temperature dependence of Hc(T)

Critical currents (no applied field)

Current

Radius, a

Magnetic field

Hi

The critical current density of a long

thin wire is therefore

jc~108A/cm2

for Hc=500 Oe, a=500 A

i

c

dH

π

4

=⋅

∫

so

cc

H

ca

i

2

=

a

cH

j

c

c

π

2

=

•Jc

has a similar temperature

dependence as Hc, and Tc

is similarly

lowered as Jincreases.

From Cywinski’slecture note

(thinner wire has larger Jc)

Cross-section through a

niobium–tin cable

Phys World, Apr 2011

Meissnereffect

(Meissnerand Ochsenfeld, 1933)

A SC is more than a perfect conductor

differentsame

not only dB/dt=0

but also B=0!

Perfect

diamagnetism

Lenz law

pure In

Superconductingalloy: type II SC

partial exclusion and remains superconducting at high B(1935)

(also called intermediate/mixed/vortex/Shubnikovstate)

•HC2

is of the order of 10~100 Tesla (called hard, or type II, superconductor)

STM image

NbSe2, 1T, 1.8K

Comparison between type I and type II superconductors

Hc2

B=H+4πM

Lead + (A) 0%, (B) 2.08%, (C) 8.23%, (D) 20.4%

Indium

Areas below the curves (=condensation energy)

remain the same!

Condensation

energy(for type I)

2

1

For a SC,

4

()(0)

8

S

SS

dFMd

dFHd

H

FH

H

H

F

π

π

=−⋅

=

→−=

2

()()

()(0) for nonmagnetic material

(0)

8

(0)

c

NcSc

NcN

NS

FHFH

FHF

FFF

H

π

=

=

∴

Δ=−=

(

)

is in Kittel

a

HB

(Magnetic energy density)

•Introduction

•Thermal properties

•Magnetic properties

•London theory of the Meissnereffect

•Microscopic (BCS) theory

•Flux quantization

•Quantum tunneling

()

2

2

2

s

s

s

s

s

S

ne

dB

J

dtmct

ne

J

ne

J

B

c

A

mc

m

φ

=

∂

∇×=−

∂

∇×=

−

−

+∇

It can be shown that

▽ψ=0 for simply

connected sample

(See Schrieffer)

1

Eq.(1)

B

E

ct

∂

+∇×=−

∂

2

2

22

4

s

L

ne

B

BB

mc

π

λ

∇=≡

London

proposed

()()

2

4

use and

s

BJ

c

vvv

π

∇×=

∇

×∇×=∇∇⋅−∇

London theory of the Meissnereffect

(Fritz London and Heinz London, 1934)

•Superfluiddensity ns

σ= ∞

•Normal fluid density nn

2

n

n

ne

m

τ

σ

=

nnn

sn

+

=

=

constant

Assume

2

(1)

(2)

ss

nn

dJneE

dtm

JE

σ

=

=

s

ss

nnn

J

env

J

env

=−

=−

where

like free

charges

nn

Tc

Carrier density

T

ns

Two-fluid model:

•Penetration length λL

Outside the SC, B=B(x) z

2

2

2

/

0

()

L

L

x

dB

B

dx

BxBe

λ

λ

−

=

→=

2

233

2

170 if =

4

10/cm

SL

S

m

ne

An

c

λ

π

=≈

/

0

4

44

L

s

x

sy

L

BJ

c

cB

cdB

Je

dx

λ

π

ππλ

−

∇×=

∴

=−=

also

decays

•Temperature dependence of λL

()

1/2

4

(0)

()

1/

C

T

TT

λ

λ

=

⎡

⎤

−

⎣

⎦

tin

•Higher T, smaller nS

Predicted λL(0)=340 A,

measured 510A

(expulsion of

magnetic field)

Coherence length ξ0

(Pippard, 1939)

•In fact, ns

cannot remain uniform near a surface.

The length it takes for ns

to drop from full value to

0is called

ξ

0

x

ns

surface

superconductor

ξ0

0

from BCS theory

FF

vv

p

ξ

π

≈=↔

ΔΔΔ

•The pair wave function (with range ξ0) is a

superposition of one-electron states with energies

within Δof EF

(A+M, p.742).

pp

m

Δ

≈

Δ

•Therefore, the spatial range of the variation of nS

ξ0

~ 1 μm>> λfor type I SC

•Microscopically it’s related to the range

of the Cooper pair.

Energy uncertainty

of a Cooper pair

Penetration depth, correlation length, and surface energy

•smaller ξ0, get more “negative”

condensation energy.

•smaller λ, cost more energy to

expel the magnetic field.

•

ξ0

> λ, surface energy is positive

From Cywinski’s lecture note

Type I superconductivity

•

ξ0

<

λ, surface energy isnegative

Type II superconductivity

•When ξ

0

>>λ(type I), there is a

net positive surface energy. Difficult

to create an interface.

•When ξ0

<<λ(type II), the surface

energy is negative. Interface may

spontaneously appear.

Vortex state of type II superconductor

(Abrikosov, 1957)

•the magnetic flux φin a vortex is

always quantized(discussed later).

•the vortices repel each other slightly.

•the vortices prefer to form a triangular

lattice (Abrikosovlattice).

Hc2

Hc1

H

0

-M

From Cywinski’slecture note

2003

•the vortices can move and dissipate energy

(unless pinned by impurity ←Flux pinning)

Normal

core

isc

Estimation of Hc1

and Hc2 (type II)

2

0

101

2

cc

HH

φ

πλφ

π

λ

≈→≈

•Near Hc1, there begins with a single

vortex with flux quantum φ0, therefore

•Near Hc2, vortex are as closely packed

as the coherence length allows, therefore

2

0

0202

2

0

cc

NHNH

φ

πξφ

π

ξ

≈→≈

2

2

10

Therefore,

c

c

H

H

λ

ξ

⎛⎞

≈

⎜⎟

⎝⎠

Typical values, for Nb3Sn,

ξ0

～34 A, λL

～1600 A

mercury

Origin of superconductivity?

•Metal X can (cannot) superconductbecause its atomscan

(cannot) superconduct?

Neither Au nor Bi is superconductor, but alloy Au2Bi is!

White tin can, grey tin cannot!(the only difference is lattice structure)

•goodnormal conductors (Cu, Ag, Au) are bad superconductor;

badnormal conductors are good superconductors, why?

•What leads to the superconducting gap?

•Failed attempts: polaron, CDW...

It is found that Tc

=const ×

M-α

α～1/2 for different materials

lattice vibration?

•Isotope effect(1950):

•Frohlich: electron-phonon interactionmaybe crucial.

•Reynolds et al, Maxwell: isotope effect

•Ginzburg-Landau theory: ρS

can be varied in space.

Suggested the connection

•1935 London: superconductivity is a quantum phenomenon

on a macroscopicscale. There is a “rigid”(due to the energy

gap) superconducting wave function Ψ.

•1950

2

()|()|

S

rr

ρψ

=

Brief history of the theories of superconductors

and wrote down the eq. for order parameter Ψ(r) (App. I)

2003

Ref: 1972 Nobel lectures by Bardeen, Cooper, and Schrieffer

•1956 Cooper pair: attractive interaction between electrons (with

the help of crystal vibrations) near the FS forms a bound state.

•1957 Bardeen, Cooper, Schrieffer: BCS theory

Microscopicwave function for the

condensation of Cooper pairs.

1972

Dynamic electron-lattice interaction →Cooper pair

Effective attractive interaction

between 2 electrons

(sometimes called overscreening)

～1 μm

+++

e

(range of a Cooper pair;

coherence length)

Cooper pair, and BCS prediction

•2 electrons with opposite momenta(p↑,-p↓)can form a bound

state with binding energy (the spin is opposite by Pauli principle)

Δ()

()

int

02

1

=

−

ω

D

DEV

e

F

, see App. H

•Fractionof electrons involved～kTc/EF

～10-4

•Average spacingbetween condensate electrons ～10 nm

•Therefore, within the volume occupied by the Cooper pair, there

are approximately (1μm/10 nm)3

～106

other pairs.

int

3

2

500,()1

5

/3

500

DF

c

KDEV

TKe

ω

−

≤≤

∴

≤=

(～upper limit of T

c)

2Δ(0) ~ 3.5 kBTc

kTe

BCD

DEV

F

=

−

113

1

.

()

int

ω

•These pairs (similar to bosons) are highly correlated

and form a macroscopic condensate state with (BCS result)

Energy gap and Density of states

~ O(1) meV

D(E)

•Electrons within kTC

of the FS have their energy lowered

by the order of kTC

during the condensation.

•On the average, energy difference (due to SC transition)

per electron is

8

4

1

0.110

10

C

BC

F

T

kTmeVeV

T

−

×

wiki

Families of superconductors

Conventional

BCS

Heavy fermion

Cuperate

(iron-based)

F. Steglich1978

T.C. Ozawa 2008

•Introduction

•Thermal properties

•Magnetic properties

•London theory of the Meissnereffect

•Microscopic (BCS) theory

•Flux quantization

•Quantum tunneling (Josephson effect, SQUID)

Flux quantizationin a superconducting ring

(F. London 1948 with a factor of 2 error, Byers and Yang, also Brenig, 1961)

**,

2

q

j

qe

mii

ψψψψ

⎛⎞

=

∇−∇=−

⎜⎟

⎝⎠

•Current density operator

•SC, in the presence of B

*

***

*

**

2

qqq

jAA

micic

ψψψψ

⎡

⎤

⎛⎞⎛⎞

=∇−+∇−

⎢

⎥

⎜⎟⎜⎟

⎢

⎥

⎝⎠⎝⎠

⎣

⎦

let =

||

and assume

||

var

y

slowl

y

with r

i

e

φ

ψψψ

2

2

2

then ||

ee

jA

mmc

φ

ψ

⎛⎞

=−∇+

⎜⎟

⎝⎠

London eq. with

•Inside a ring

0 jd

⋅

=

∫

22

cc

Add

ee

φ

φ

⇒⋅=−∇⋅=−Δ

∫∫

2

00

7

flux ||,

2

210 gauss-cm

2

hc

e

hc

nn

e

φφ

−

∴

Φ===×≡

•

φ

0

～the flux of the Earth's magnetic field

through a human red blood cell (~ 7 microns)

ns

=2

2

||

ψ

*

*

2

2

qe

mm

=

−

=

Single particletunneling

(Giaever, 1960)

•SIN

Ref: Giaever’s 1973

Nobel prize lecture

20-30 A thick

dI/dV

For T>0

(Tinkham, p.77)

•SIS

Josephson effect

(Cooper pairtunneling)

Josephson, 1962

1973

1) DC effect:

There is a DC current through SIS in the absence

of voltage.

ψψ

θ

11

1

=||ei

ψψ

θ

22

2

=||ei

()

()

12

1221

0

01

(/2)(/2)

()()

0

20

/2

2

sin

/,

iKxdiKxd

S

ii

Kd

S

Kd

S

nee

ien

jKeee

m

j

jenKem

θθ

θθθθ

ψ

δ

δθθ

−++−

−−

−

−

=+

=−+

≡

=

≡−

2/e

Δ

Giaever

tunneling

Josephson

tunneling

2) AC Josephson effect

Apply a DC voltage, then there is a rfcurrent oscillation.

1

()/

/

ˆ

1

()/(1,2)

NN

iEEt

it

iii

NNee

tti

μ

ψ

ψ

θμθ

−

−−

−

=−∝=

→=−+=

0

1

00

2

22

n

2

si

eVeV

tjjt

eV

δδ

μ

μ

δ

⎛⎞

∴

=+⇒

−=

=+

⎜⎟

⎝⎠

−

•An AC supercurrentof Cooper pairs with freq. ν=2eV/h, a

weak microwave is generated.

•

ν

can be measuredvery accurately, so tiny

ΔV as small as

10-15

Vcan be detected.

•Also, since V can be measured with accuracy about 1 part in

1010, so 2e/hcan be measured accurately.

•JJ-based voltage standard (1990):

1 V ≣

the voltage that produces ν=483,597.9 GHz (exact)

•advantage: independent of material, lab, time (similar to the

quantum Hallstandard).

(see Kittel, p.290 for an

alternative derivation)

VVt=

+

0

υ

ω

捯c

〰0

0

〰

0

2

獩ss楮

2

2

†††‽(1Φ獩n

†瑨=re猠=C×rre湴=

2

==

n

n

n

e

jjVtt

eV

e

jJt

V

e

n

n

t

υ

ωδ

ω

υ

ω

ω

δ

ω

⎡⎤

⎛⎞

=++

⎜⎟

⎢⎥

⎝⎠

⎣⎦

⎛⎞

⎛⎞

−−+

⎜⎟

⎜⎟

=

⎝⎠

⎝⎠

⇒

∑

Shapiro steps

(1963)

given I, measure V

3) DC+AC:

Apply a DC+ rfvoltage, then there is a DC current

•Another way of providing a voltage standard

SQUID

(Superconducting QUantumInterference Device)

00

0

sinsin

2cossin

22

ab

abab

jjj

j

δ

δ

δ

δδδ

=

+

−+

⎛⎞⎛⎞

=

⎜⎟⎜⎟

⎝⎠⎝⎠

1

2

11

22

Similar to

2

2e

We now have

c

2e

c

ba

C

ab

C

c

A

dd

e

Ad

Ad

θ

θ

θ

θ

θ

⋅

=−∇⋅

⋅=−

⋅=−

∫∫

∫

∫

0

max0

0

2

2

2

2cos

2

ab

C

e

Ad

c

jj

φ

δ

δπ

φ

πφ

φ

⇒−=⋅=

⎛⎞

∴

=

⎜⎟

⎝⎠

∫

The current of a SQUID

with area 1 cm

2

could

change from max to min

by a tiny ΔH=10-7

gauss!

For junction with finite thickness

Non-destructive testing

SuperConductingMagnet

MCG, magnetocardiography

MEG, magnetoencephlography

Super-sentitivephoton detector

科學人,2006年12月

semiconductor detector superconductor detector

Transition edge sensor

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