1

Chapter Ten Superconductivity

An element, inter-metallic alloy or compound will conduct electricity without resistance below a certain temperature.

Temperature (K)

Res

ista

nce

(Ω)

Temperature (K)

Res

ista

nce

(mΩ

)

1987 U. of AlabamaOnnes H.K.,Commun. Phys. Lab. Univ. Leiden 124C (1911)

2

1988 Bi- 110K

1988 Tl- 125K

1993 Hg- 135K

The world record Tc

138K

Thallium-doped mercuric –cuprate

TlHgBaCaCuO

1994 NIST-Colorado

Under extreme pressure ~300,000 atmospheres,

Tc can be increased by 25 ~ 30 K.

3

References :

Kittel, Introduction to Solid State Physics, 8th Ed., Ch.10, 2005.

Aschroft and Mermin, Solid State Physics, Harcourt, Ch.34, 1976.

Duzer and Turner, Principles of Superconductive Devices and Circuits, Elservier, NY, 1981.

Tinkham, Introduction to Superconductivity, McGraw-Hill, 1975.

Orlando, Foundations of Superconductivity, Addison-Wesley, 1991.

4

The history of superconductors in brief :

1911 Superconductivity was first observed in Hg by Onnes of Leiden U.

1933 Meissner effect was discovered by Meissner and Ochsenfeld.

Superconducting materials will repel a magnetic field.

“Diamagnetism”

1935 Two fluid model was proposed to explain zero resistance behaviorand Meissner effect by Fritz and Heinz.

1950 “Phenomenological theories” (QM) were proposed to explain superconductivity by Ginzburg and Landau.

1957 The first widely-accepted theoretical understanding of superconductivity was advanced by Bardeen, Cooper, and Schrieffer. “BCS theory”

1913

1853~1926

19721908~1991 b: 1930 b: 1931

5

1957 Abrikosov predicted the existence of the other kind superconductor.(Type II superconductors)

• 1960 Tunneling experiment was performed by Giaever.Experimental measurement of the energy gap.

1962 Josephson effect was proposed by Josephson.A significant theoretical advancement. “SQUID”

1973

b: 1929

1973b: 1940

2003b: 1928 b: 1916Ginzburg

6

1980’s A decade of unrivaled discovery in the field of superconductivity.1964 Little suggested the possibility of organic superconductors.1980 Bechgard found the superconductor (TMTSF)2PF6. Tc=1.2K.1986 Müller and Bednorz created a brittle ceramic compound w/.

“insulator” Tc=30K.

1987

b: 1927 b: 1950

7

Phenomena Occurrence, properties

Theory Why superconductivity occur ?

Why different from ordinary metals ?

Applications Why scientifically interesting ?

Why technologically important ?

8

1.14K

3.72K

7.19K

3.40K

1.09K

2.39K4.15K

0.39K 5.38K

0.55K

0.12K

9.50K

4.48K

0.92K 7.77K

0.01K 1.4K 0.66K 0.14K

0.88K

0.56K0.51K

0.03K

1.37K 1.4K

New elements keep being added to the list.

In 2002, Li was shown to superconduct under pressures of 23 to 36GPa with critical temperature of 9 to 15K. V.V. Struzhkin et al., Science 298, 1213 (2002).

9

Superconducting alloys

PbSn 6.5K solder

NbTi 12K magnet wire

Nb3Al 18.9K

Nb3Sn 18.3K

Nb3Ge 23K Highest “low” Tc

A-15 compounds

LaSrCuO 30K 1st high Tc oxide

YBa2Cu3O7 92K 1st Tc>77K

HgBaCaCuO 134K Highest Tc

150K (under high pressure)

High Tc materials

10

Buckminsterfullerene(buckyball)

a closed sphere is formed by 60 carbon atoms and doped with Alkali metals

Organic superconductors

(TMTSF)2PF6 0.4~12K 1980

YPd2B2C 23 KLuNi2B2C 16.6 KYNi2B2C 15.5 KTmNi2B2C 11 K (resistance increases below Tc)ErNi2B2C 10.5 K (ferromagnetic)

(TMTSF)2PF6

UPt3

Ceramic superconductors

K3C60 18K 1991

Cs3C60 40K fulleride

Heavy FermionsHeavy fermions are compounds containing rare-earth elements such as Ce or Yb, or actinide elements like U.UPt3 0.48K CeRu2 6K UPd2Al3 2K

11

Basic properties :

A truly zero DC resistance

Onnes ρsc < 10-8 × ρ(300K) = 10-8 × 100µΩcm = 10-14 Ωm

Modern technique : decay of supercurrent in a loopρsc < 10-18 × ρ(300K) = 10-18 × 10µΩcm = 10-25 Ωm

(NMR or SQUID)

A phase transition at Tc

Transition is reversible and very narrow (<10-4 K)

broadened by impurities, fluctuations, and inhomogeneities

12

Magnetic properties more unusual than ρ=0

Superconductivity can be destroyed by magnetic field as temperature

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T/Tc

H/Hc

Superconductingstate

Normal state

−= 2

c

2

c TT1)0(HH(T)

100Gauss/K

constantT

)0(HT )0(H

c

c

cc

≈

=

∝Also,

13

Meissner effect : magnetic field is expelled. (1933)

T>TcNormal state

T<TcSuperconducting state

cooling down

In a constant H (< Hc)

When a specimen is placed in a magnetic field and is then cooled through the transition temperature for superconductivity,the magnetic flux originally present is ejected from the specimen.

A bulk superconductor in a weak magnetic field will act as a perfect diamagnet, with zero magnetic induction in the interior.

14

Diamagnetism

M 4HBrrr

π+=magnetization

total field

applied field

M is paralleled with H – paramagnetismM is anti-paralleled with H – diamagnetism

Applying magnetic field H to a magnetic sample, screening currents and alignment of magnetic moments (spins) alter the internal magnetic field.

EM theoryfor B=0

H

B

normal

superconductingHc

H

-4πM

normalsu

perconductin

g

Hc

superconductor

HM4

0Brr

r

=−

=

π

15

Diamagnetism can not be attributed to perfect conductivity (ρ=0).

Ohm’s law : Maxwell’s equations :

J Err

ρ=

tB E

∂∂

−=×∇r

r

0 tB hence, and 0E 0when =

∂∂

==r

rρ

constantB =r

Two different possible states for a perfect conductor.

Field kept out

Field trapped inside

Final state below Tc in the presence of H (<Hc) depends on history of the cooling process for a perfect conductor.

16

Superconductor is perfectly diamagnetic.Transition to superconducting state is perfect reversible.

Meissner effect

Field is expelled.

B=0

independent of history.

17

Type I and type II superconductors

In fact, perfect diamagnetism up to Hc is rare seen in less than 1% of known superconductors.

Type I Perfect diamagnets Most elements (Al, In, Sn, Pb,..)(Meissner 1933)

Type II Flux penetration Most alloys, films, dirty metals,…(Abrikosov 1957)

Meissner

Soft superconductor Hard superconductor

Type II superconductors stay superconducting at higher field than type I.

18

Hc1 : lower critical field < Hc

Hc2 : upper critical field >> Hc

H<Hc1, superconducting state (Meissner effect)

Hc1<H<Hc2, vortex state “fluxoids” in the superconductor

H>Hc2, normal state current vortices

Nb3Sn, NbTi Hc2 ~ 10T

Nb3Al.7Ge.3 41T

YBa2Cu3O7 100T

Pb PbIn alloys w/. weight % of In(B) 2.08%(C) 8.23%(D) 20.4%T=4.2K

19

Abrikosov vortexstate

Flux lattice in NbSe2 at 1000Oe at 0.2K

Magnetic flux enters sample and forms a triangular lattice of vortex lines.

H

Flux/Vortex

27

o

cm-Gauss1007.2 2

−×=

=Φeh

Flux Quantum

Density of vortex lines

o

BnΦ

= number/unit area

20

More reality

B=0 only in bulk of a type I superconductor

Screening currents flow within depth λ of surface and B decays in that. Penetration depth

J

superconductor

H

B, J

boundary

λ

x

21

Energy gap

Superconductors exhibit energy gap between the ground state and the lowest excited state.

filledEF

filled

∆

normal state superconducting state

~ 10-4EF

This gap is different from the energy gap of insulators(electron-lattice interaction)

The energy gap of superconductors is due to electron-electron interactionwhich order the electrons in k space with respect to the Fermi gas of electrons.

Electrons in the excited state above ∆ behave as normal electrons causing resistance. At DC they are shorted out by the superconducting electrons.

22

Heat capacity3T T C βγ +=Normal state :

electron Phonon : Debye T3 model

normal statejump(1) A heat capacity jump occurs at Tc.

A 2nd order phase transition

(2) Below Tc,

an exponential temperature dependence

Al

TkTaT

B

c

e e C∆

−−∝∝

N.E. Phillips, Phys. Rev. 114, 676 (1959).

Superconducting state : H=0

Normal state : H = 300 Gauss

43.1C

CC

n

ns =−

(3) At Tc,

can be explained by BCS theory.

23

Tc

Al

Tc

Al

Thermodynamic relations

Because of no structural change and the occurrence of Meissner effect, we can apply thermodynamics arguments to understand the superconducting transition.

2

2

T F

T S C

T F S

∂∂

=∂∂

=

∂∂

−=Entropy

Heat capacity

Fn(T) and Fs(T) merge at Tc.Fs(T) ≤ Fn(T) Ss(T) ≤ Sn(T)

Electrons are more ordered in the superconducting state than in the normal state.

24

TTcba

TClog

c

−=

γTkBe C

∆−

∝

must supply enough thermal energy kBT to excite carrier across gap.

# of carrier : normal ~ kBT/EF

superconducting ~ exp[-∆/kBT]

25

Electromagnetic absorption

hν ≥ ∆Photon energy is sufficient to create excitons

Ref

lect

ion

coef

ficie

nt

νh

2∆

normal

superconducting Threshold at 2∆ suggests excitations are created in pair.

Absorption is the same in normal and superconducting states for

ν >>∆/h.

( ) 480GHzHz108.4eVs106.1/1063.6

)eV10(22

5.3Tk

2

111934

3cB

=×=××

=∆

=

≈∆

−−

−

hυ

estimate 1 meV

26

Iosotope Effect

Tc varies with the isotope of the same element.

This effect is also found in some compounds.

27

Tc varies with isotope mass.

constantTM c =α

How does nuclear mass affect an electronic phenomena like SC ?Phonons (electron-phonon scattering)

Weak electron-phonon coupling

good normal conductor

bad superconductor

Strong electron-phonon coupling

good superconductor

0.08Nb3Sn0.50Hg

0.49Pb0.47Sn

0.33Mo0.32Cd

0.15Os0.45ZnExperimental values of α The original BCS model

gave 2/1Debyec MTT −∝∝

without the consideration of Coulomb interaction between electrons.

28

Theory of superconductivity

Phenomenological models :

Two-fluid model (normal + superfluid )

London equation (EM Meissner effect)

Ginzburg-Landau (Order parameter)

Microscopic theory : BCS theory

Key evidence :

(1) Energy gap Excitations

(2) Isotope effect Phonons

(3) Phase transition Thermodynamics

Phase transition occurs when free energy of SC state lower than that of normal state.

29

Thermodynamic relations

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T/Tc

H/Hc

dHdG Μ−=change in Gibbs free energy at constant T

costing free energy to expel magnetic field

∫

∫•=

•−=−

Hd4H

HdT)(0,GT)(H,G ss

rr

rr

π

Μ

for H≤Hc

8πHT)(0,GT)(H,G

2

ss =−

G

Gn=Gn(0)=constantnormal state

superconducting stateEc

Hc

H8πHT)(0,GT)(0,GE

2c

nsc =−=

Condensation energy

30

costing free energy to expel magnetic field (Meissener)

We can get Ec by measuring Hc.

o

2c

2c

c 2H

8πHE

µ==

(CGS) (SI)

Field energy density

Typical Type I superconductor

Hc ~ 100Gauss

Ec ~ 2.5×1014eV/cm3 carrier density n ~ 1022/cm3

Condensation energy per electron ~ 10-7eV/electron

Big or small ?

Fermi energy EF ~ 10eV

Superconducting energy gap ∆ ~ 10-3 eV

Condensation energy Ec ~ 10-7 eV

31

London equation and two fluid model

• Fritz and Heinz London found a “modified” set of Maxwells’ equations that have the Meissner effect as a solution.

• They started by assuming “two fluid model”.

Two fluid model

Superfluid density ns – supercurrent σ = ∞Normal state density nn – dissipation

mne2τσ = flow in parallel

Tc

T

ns

nn

Car

rier d

ensi

ty

constantnnn ns ==+

32

They postulated :

EJ (2)m

EendtJd )1(

nn

2ss

rr

rr

σ=

=

nnn

sss

venJ

venJrr

rr

−=

−=ns JJJrrr

+=where and

Eq.(1) suggests that the superfluid responds to electronic fluid in the same way that free charges do :

Eedtvdm

rr

−= ( ) m

Eendt

vend 2sss

rr

−=−

tB

c1E∂∂

−=×∇r

rrSubstituting Eq.(1) into

( ) tB

mcenJ

dtd 2

ss ∂

∂−=×∇

rrr

Bmc

enJ2

ss

rr−=×∇

Maxwell’s equation :

( ) Jc

4B s

×∇=×∇×∇

rrrrr π

London equation B

mcen 4B 2

2s2

rr π=∇Therefore,

33

This gives the “London Penetration Depth”

B1B 2L

2 rr

λ=∇

en 4mc

2s

2

L πλ =

and is also consistent with Meissner effect.An uniform field exists in a superconductor only when B=0.

where

superconductor

In the pure superconducting state the only field allowed is exponentially damped as we go in from an external surface.

For a semi-infinite superconductor in a vacuum under a homogeneous field B)0,0,(B =

r

0

B

−=

L

xB(0)expB(x)λ

Screening currents in the region of size ~ λLx

34

s2L

s2 J1J

rr

λ=∇ ( ) B

mcen B

mcenJ

2s

2s

s

rrrrrrr×∇−=

×∇−=×∇×∇

−=

Lss

x(0)expJ(x)Jλ

has the solution in the similar form

Sn 34nm

Pb 37nm

YBaCuO 150nm

λL(0)

Tc

λL Meissnereffect

Superconducting currents that screen the interior of superconductor against external fields also decay exponentially with distance into the superconductor.

Lower carrier density larger λL

en 4

mc2

s

2

L πλ =

Decay length for current and fields gives measure of superfluid fraction.

T

35

BCS theory

Electron pair via attractive electron-electron interaction mediated by phonon.

“Cooper Pair”

SC is a many body problem – cannot ignore interaction of electrons

Free electron model

Band theory electrons

Pairing lowers energy to give SC phase transition.

Pairing causes electrons (Fermions) to act like “Bosons”– allowing them to condensate into a “superfluid state”.

will not give SC

Three questions :

(1)What is attractive force – don’t electrons repel by Coulomb force ?

(2)How is energy lowered ?

(3)Nature of ground state and excitons ?

36

Attractive force

Microscopic theory in 1957 by Bardeen, Cooper, and Schrieffer

Ground state of e-s near EF in a superconductor occupy pair states-- equal and opposite momenta-- spin singlet

Motions of 2e- in “Cooper pair” are highly correlated.

Source of correlation is a phonon mediated attraction between e-snear EF within hωD of EF.

k ,k ~ pair ↓−↑Ψrr

+ + + + + + +

+ + + + + + +

+ + + + + + +

e-

e-

k ↑

-k ↓

step I

step II

step III

1st electron (k↑) passes ions

lattices are polarized

Before polarization of lattices disappears, the (-k ↓) e- get attracted.

37

Net attraction between e- mediated by lattice (phonon).

vF>>vphonon e- can respond to distortion before it propagates away

k 1 ↑r

k 2 ↓r

absorbs the phonon

emits the phononvirtual

k1

k2

k1’

k2’

q

Or in the opposite way

The lattice deformation reaches its maximum at a distance,

nm100m10s10cm/s10~2v 7138F ==× −−

Dωπ

Phonon vibration period

The two electrons correlated by the lattice deformation thus have an approximate separation about 100nm.

The “size” of a Cooper pair.

38

Condensation energy phase transition

Normal metal at T=0Kf(E)

Filled Fermi surface

1

T≠0K

EEF

Allow two electrons to be excited (costs Kinetic Energy)

And interact via attractive force (gains Potential Energy)

Cooper showed that this lowers system energy – normal ground state unstable to attractive force

Phase transition would occur.

Continue to allow more pairs to be excited and scatter until point of diminishing returns

KE (loss) > PE (gain)

39

Two electrons [k1, E(K1)] and [k2,E(K2)] are in states just above EF.

A weak attractive interaction between these two electrons is turned on in the form of phonon exchange.

All other electrons in the Fermi sea are assumed to be non-interacting, and on account of the Pauli exclusion principle.

constantkkkk '2

'121 =+=+

rrrrKr

Momentum conservation

kF

(outer)EF+hωD

EF-hωD(inner)

EF kx

ky

1kr

2kr

−kr

∆

The number of pairs k1, k2 is proportional to the cross volume in k-space and is maximum for K=0

40

Electron pairs with equal and opposite wave vectors

Two particle wave function 21 kkrr

−=)r,r( 21

rrΨ

( )( ) )r,r(2E

)r,r(E)r,r()r,r(V)r,r(m2

21F

2121212122

21

2

rr

rrrrrrrrh

Ψ+=

Ψ=Ψ+Ψ∇+∇−

ε

m2kE

2F

2

Fh

=

where ε is the energy of electron pair relative to the interaction-free state (V=0) in which two electrons at the Fermi sea would possess an energy

( ) ( ) ( )( )212211 rrkexp1rkexp1rkexp1 rrrrrrr−•

Ω=•

Ω•

ΩiiiNon-interacting

two electrons

( ) ( )( )∑ −•Ω

=−Ψk

2121 rrkexp)kg(1rrrrrrrr i

Two electrons with non-vanishing interaction V≠0

g(k) is the probability amplitudeprobability amplitude of finding the electron pair 2

)kg(r

probability

kkk 21

rrr=−=

( )k,krr

−

41

+><

=hh /)2m(Ek

kkfor 0g(k)

F

F

Dω

Substitute the wave function in Schrödinger equation

( ) )g(k2E)Vg(k'1g(k)mk

Fk'

'kk

22

ε+=∑Ω

+h

( )∫ •−−= r)'kki(drV(r)expV 'kkrrr

wheredescribes scattering of electron pair from (k,-k) to (k’,-k’)

+<<−=otherwise 0

Em2k,

m2kEfor VV F

2'222

Fok'k,

Dωhhh

In the simplest model,

∑Ω

−=

++−

k'

oF

22

)g(k'V

g(k)2Emk εh

ε++−

∑Ω−=

F

22k'

o

2Emk

)g(k'V

g(k)h

∑++−Ω

−=k

F

22o

2Emk

1V1

εh

42

∑Ω k

1Summing all states in k-space

( )∫−−

∫∇

Ω=

επ Fk

E3o 2EE2

dEE(k)

dS2

V1

( )dE

E(k)dS

2D(E)dE

constE(k) k

E3

∫

∇Ω

==π

( )F32

3

F E2

m2)D(E

hπ−=and

∫ −=

D

0Fo 'E2

dE')D(EV1ω

ε

h

( )

−−

=−=ε

εωε ω DFo0

Fo 22

)D(EV'E2

2)D(EV

1 D hll

h nn

where E’=E-EF energy respective to EF

−−=

)D(EV2exp2

FoDωε h

εωD

Fo

21)D(EV

2exp h−=

( ))D(EV/2exp12

Fo

D

−=

ωε h

For weak interaction VoN(EF)<<1

43

There thus exists a two-electron bound state, whose energy is lower than of the fully occupied Fermi sea by an amount ε = E-2EF <0.

Cooper pair ( )↓−↑ k ,krr

The two electron wavefunction is symmetric in spatial coordinate (r1,r2) under exchange of electrons 1 and 2, but the whole wavefunctionincluding spins must be antisymmetric .

consistent with the Pauli exclusion principle

−−=

)D(EV2exp2

FoDωε h

Phonon frequencye-ph coupling strength

density of state at EF

−−=+=

)D(EV2exp22E2EE

FoDFF ωε h

High phonon frequencyStrong e-ph couplingLarge density of states

Strong SC

44

The BCS ground state

dynamic states with electrons near Fermi surface scattering

k

-k

k’-k’

kx

ky

kF “annihilation” of a pair

“creation” of a pair

( )↓−↑ k ,krr

( )↓′−↑′ k ,krr

an energy reduction of Vkk’.the scattering of an electron pair with wavevectors to ( )k ,k

rr− ( k ,k −′

rr )′

okkkkiN ...),...kg(kii

φ∗↓−

∗↑

∗↓−

∗↑∑=Ψ

lll CCCC

N electrons with the Cooper pairing

vacuum stateN/2 pairs of creation operators

Grand canonical ensemble

45

( )∏=

∗↓−

∗↑+=Ψ

M1...kkkokkkkG vu φCC

BCS ground state

2kv the probability of the pair (k↑,-k↓) being occupied.

2k

2k v1u −= the probability of the pair (k↑,-k↓) being unoccupied.

∑∑ ↓−↑∗↓−

∗↑+=

llll

kkkk

kkk VnH CCCC

σσε

Pairing Hamiltonian

Determining the coefficients by Variational Method

0Vn Gk

kkkk

kkG =Ψ+Ψ ∑∑ ↓−↑∗↓−

∗↑

llll CCCC

σσεδ

-µN

l↑

-l↓k↑

-k↓

q

Set Ek’= Ek - EF single-particle energy relative to EF

∑∑ +=Ψ−Ψl

lllk

kkkk

2k

'kGG vuvuVvE2NH µ

46

the excitation energy of a quasi-particle of momentum hk( ) 2

k2

Fkk EE ∆+−=ε

∑−=∆l

lll vuVkk the minimum excitation energy, energy gap

−=∆

)D(EV2exp2

FoDωh

( )

2k

k

'k2

k

22Fk

Fk

k

'k2

k

v1E121u

EE

EE121E1

21v

−=

+=

∆+−

−−=

−=

ε

ε

47

Condensation energy

−=∆∆=−=

)D(EV2exp2 e wher)D(E

21EEE

FoD

2FSNC ωh

o

2

oFo

22

Fo

2

VPE

V)D(E211

V )D(E

21

VKE

∆−=∆

−

∆=∆−

∆=∆

1K)~ re temperatuTransition (T Tk ~ 100K)~ re temperatuDebye ( k

ccB

DDBD∆

ΘΘ=ωh EN

Es

KE PE

Ec

cost

gain

37.0~22)D(EV

DFo

∆

=ωh

ln

48

Electrons within ∆ of Fermi surface scatter # of electrons = D(EF)∆

Each pair contributes a decrease in energy by ∆

Condensation energy : 2F )D(E

21

∆

EF

∆

4

F

c

F

10~TT

E−≈

∆

Ek’=Ek-EF

Probability that e- state k is occupied.

Probability that pair state (k,-k) is occupied.

-hωD hωD

In reality, f(Tc)~vk2(0).

Electron distribution changes very little from Tc to 0K.

49

Excitation

(Ek-EF)/∆

εk/∆

1ElectronsHoles

Normal metalsEF

0

(Ek-EF)/∆

εk/∆

0

1

Superconductors

Electron-likeHole-likeEF

∆

( ) 2k

2Fkk EE ∆+−=ε

Quasi-particles

50

Ek’

0 ∆

D(Ek’)

1xDN Normal state

Density of quasi-particle state

( ) N22Fk

FkN

22'k

'k'

k DEE

EED

E

E)D(E

∆−−

−=

∆−=

States below the gap ∆ are pushed into divergence at the gap edge

Temperature dependent energy gap

T/Tc

∆(T

)/∆(0

) cBTk76.1)0( =∆

)0(TT1)T(

c

∆−=∆

51

Ginzburg-Landau Theory

phenomenological theory of phase transition

can be applied to the superconducting state

give equations for Ψ “the condensate wavefunction”

explain the difference between type I and type II superconductors

Concept of the order parameter Ψφiesn=Ψ

M=Ψ magnetizationsn=ΨΨ∗for superconductors,

for ferromagnets,

T/Tc

M(T

)/M(0

)

Ferromagnetic materials

Non-magnetic

Curie temperature

52

Expand free energy density near the transition in terms of Ψ

∫ •−Ψ

−∇−+Ψ+Ψ−=

H

0

242

ns 'dHMc

qA2m1

2(r)F(r)F hiβα

where α, β are phenomenological constants, α=αo(T-Tc), andβ >0 gives transition and stability

Fn(r) is free energy density of normal state, Ψ=0 as T>Tc

increases in energy due to spatial variation of Ψ-- like the kinetic energy density

2Ψ∇

q = -2e for an electron pair (Cooper pair), ns(r) : number density of Cooper pair

qA/c ensures that we have gauge invariance of the free energy

gives increase in F due to flux exclusion∫ •− 'dHM

53

( ) 0at 0FF ns =Ψ=Ψ∂−∂ ( )

ons at 0FF

Ψ=Ψ=Ψ∂−∂

Fs-Fn Fs-Fn

T>TcT<Tc

54

Minimize F wrt Ψ

.c.cc

qAc

qA2m1(r)F 2

s +Ψ

−∇−•Ψ

−∇−+ΨΨ+Ψ−= ∗δβαδ hh ii

Integrate by parts ( )( ) ( )∫∫ ∗∗ ΨΨ∇−=Ψ∇Ψ∇ δδ 2dVdV

If δΨ* vanishes on the boundaries, so that

.c.cc

qA2m1dV(r)dVF0

22*

s +

Ψ

−∇−+ΨΨ+Ψ−Ψ== ∫∫ hiβαδδ

= 0Ginzburg-Landau Equation

0c

qA2m1 2

2 =Ψ

−∇−+ΨΨ+Ψ− hiβα

Minimize F wrt A

( ) Amcq

2mq(r)j *

2*

s ΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗hi

BA and (r)jc

4H s

rrrr=×∇=×∇

π

55

This order parameter is closely related to the many body wavefunctionof superconducting state.

Apply these equations to (1) Hc2(T)

(2) coherence length ξ(3) penetration depth λ(4) flux quantization

(1) Set H=0 and let Ψ(r) be spatially constant

42ns 2(T)F(T)F Ψ+Ψ−=−

βαand from Ginzburg-Landau equation

so that

The earlier discussions gave

βα

=Ψ

βα2

FF2

ns −=−

π8HFF

2c

ns −=−

πβα

8H

2

2c

2

=

relates critical field to two of the phenomenological parameters

56

(2) Coherence length ξ

How does Ψ(r) vary?

02m

22

2 =Ψ∇−ΨΨ+Ψ−hβα

solution

=Ψξβ

α2

tanh)x( x

m2

2h

c

2

TT1~

)T(m2 −≡

αξ h

x

Ψ=0normal

Ψ(r)Spatiallyvarying

Ψ=(α/β)1/2

Super-conductor

x=0 x→∞

x/ξ

Ψ(x/ξ)

1 20

Ψ∞Normal

Superconducting

Ginzburg-Landau Equation :

B.C. Ψ=0 at x=0 in 1D

where the intrinsic coherence lengthαξ ≡

α(T) ~ T-Tc Hence, diverges at T=Tc

57

(3) Penetration depth λ

( ) Amcq

2mq(r)j *

2*

s

rhrΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗i

Assume that Ψ is real or φ=constant φiesn=ΨA

mcq

Amcq(r)j

22*

2

s

rrr Ψ−=ΨΨ−=

Amc

qB

4c(r)j

222

s

rr×∇

Ψ−=∇−=×∇

πB

mcq 4

B 2

222 Ψ

=∇π

απβλ 2

2

L q 4mc

= the London penetration depth

x/λ

B(x/λ)

1 20

HoSuperconducting

Normal

58

Dimensionless ratio κξλ

≡

πβ

α

απβ

ξλκ

2qmc

m2

q 4mc

2

2

2

hh==≡

21

=κ separates superconductors of type I and II.

κ>>1 Type II superconductorκ<<1 Type I superconductor

59

0.163.423Sn

1.023.93.8Nb

0.453.78.3Pb

0.011.6160Al

λo/ ξoλo (µm)ξo (µm)metal

(4) Flux quantization

( ) Amcq

2mq(r)j *

2*

s

rhrΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗i φiesn=Ψand

ns is constant in space but the phase of the order parameter is not, φ(r).

( )A

mcnq

mnq

Amc

nqnn2mq(r)j

s2

s

s2

sss

rh

rhr

−∇=

−∇+∇−= −−

φ

φφ φφφφ iiii eeieeii

vector potential

gradient of the phase of the order parameter

60

Ac

q)r(jnq

m ss

r

h

r

h+=∇φ

For Ψ to be single valued s 2 dc

πφ =∫ •∇ lr

If path is well inside the superconductor where js=0

( ) ∫ •=∫ •×∇∫ =•=∫ •∇ σσφ rr

h

rr

hlrr

hlr

dBc

qdAc

q dAc

q dcc

where dσ is the an element area on a surface bounded by c

where s is an integer

0dB =Φ=∫ • σrr

flux through the loop is zero

If path is well inside the superconductor where js=0

cqdB

cqs 2

h

rr

h

Φ=∫ •= σπ

so that the flux is quantized in units of 2πhc/q

2,...1,0,s re whes q

c2±±==Φ

hπ

On the basis of the BCS theory q=2e, 27-o cm-Gauss102.07

ec

×==Φhπ

61

Type I superconducting ringT<TcT>Tc

flux expelled from superconductor, get “trapped” flux in the center of ring, this trapped flux is quantized costing energy to remove flux

field penetrates metal uniform flux throughout

Type II superconductors H>Hc1

Hc1 Hc2

flux penetration

In flux penetration region,we must also quantize flux

62

c regions of flux penetration

∫ •

+==∫ •∇

cs

scdA

cqj

nqms 2 d l

rr

h

r

hlr

πφ

Minimum in energy when s=1 a “fluxoid” or “fluxon”

Field penetrates (Hc1<H<Hc2) by introducing “fluxoids” into the superconductor.

-- quantum of flux

63

Fluxoids

Field in the center of fluxoid is Hc1 when applied field is Hc1.

Field extends and decays away from the center over a distance ~λ.

“Core” or center of vortex is normal.

Radius of the core ~ ξ.

Hc2 is the field where cores overlap, i.e. all normal.

2c1o H πλ=Φ

2o

c1HπλΦ

=

2o

c2HπξΦ

= κ ~82Hc1

x

B

λ

Isolated Abrikosovvortex

64

Abrikosov vortexstate

Flux lattice in NbSe2 at 1000Oe at 0.2K

Magnetic flux enters sample and forms a triangular lattice of vortex lines.

The photo was taken using scanning tunneling microscope.

Hr

vortex core

Superconducting region

sensitive to density of states

65

Single-particle tunneling

Tunneling junction : metal -insulator- metal sandwich barrier : thin enough the e-s can tunnel through

but not leak throughMetal A Metal B Tunneling measurement in SC was first

performed by Giaever.

Metal A Metal B

A V

V

I

0

When one electrode is replaced by a superconductor, it becomes a SIN junction and the IV characteristic deviates from the straight line

due to the absence of states within the energy gap.

N22'

k

'k'

kS DE

E)(ED

∆−=

66

Cooper pair tunneling (Josephson superconductor tunneling)

Super-conductor

Ψ1

Super-conductor

Ψ2

For electron get from 1 to 2 (or 2 → 1)it must tunnel through the insulating barrier.

21s22s11 n and n θθ ii ee =Ψ=Ψ

Assume tunneling rate small,

aE t

aE t

1222

2111

Ψ+Ψ=∂Ψ∂

Ψ+Ψ=∂Ψ∂

∗h

h

i

iIn the absence of tunneling

222

111

Et

E t

Ψ=∂Ψ∂

Ψ=∂Ψ∂

h

h

i

i

a describes the small coupling between S1 and S2 due to tunneling.

a=ΨΨ=ΨΨ ∗1221 VV

2111s2s111s1

s1

s11 nanEnn2

n t

θθθθ θ iiii eeieeii +=

+=

∂Ψ∂ &&

hh

67

( )12

s1

s21

s1

s11 n

naE

2nn θθθ −+=+ iei &

h&h

( )21

s2

s12

s2

s22 n

naE

2nn θθθ −+=+ iei &

h&h

Real part( )

( )12s2s1s1

12s1

s211

sinnn2an

cosnnaE

θθ

θθθ

−=

−+=

h&

&h

Imaginary part

( )

( )12s2s1s2

12s2

s122

sinnn2an

cosnnaE

θθ

θθθ

−−=

−+=

h&

&h

tn

tn s2s1

∂∂

−=∂∂

∂∂

∂∂

tnor

tn s2s1Tunneling current I is determined by

12

s2s1o nn2I

θθδ −=

=h

aδsinI

tnI o

s1 =∂∂

≈ ?where

68

condition : small enough currents (ns1and ns2 are nearly constant.)

δsinII o= is the Josephson current relation

Supercurrent through junction depends on the phase difference across junction.

For identical superconductors s1 and s2 (ns1≅ ns2)

( ) 2121 EE −=−θθ &&h

( )hh

&& eV2EE 1212 −=

−=−= θθδ

dtd

teV2)0()t(h

−= δδ

where V is the voltage across junction

For a fixed V externally,

−= teV2)0(sinII o

hδ

the current oscillates with

frequency h

eV2=ω

AC Josephson effect

69

DC Josephson effect

As V=0, 0=dtdδ

and ( ))0(sinII o δ= constant

Supercurrent flow through the junction in the absence of bias voltage.

AC Josephson effect

−= teV2)0(sinII o

hδ

h

eV2−=

dtdδ

As V≠0, oscillatingand

Supercurrent oscillate through the junction with a frequency 2eV/h by applying a constant voltage across the junction.

Volt VMHz6.483eV2

µω ×==

h

70

Superconducting Quantum Interference

Jtot

insulator

1 2

Two Josephson junctions

No voltage is applied.

Two paths from point 1 to point 2 : a and ba

b

superconductor

In the presence of magnetic field and flux is ΦΦ+=

Φ−=

cec

e

ob

oa

h

h

δδ

δδba δδ =

∫ •+∆−=∫ •

+∇−=∫ •

2

1

2

s

2

1

2

s

2

1d

mc2e

mend

mc2e

mendJ l

rrhlrrh

lrr

AA xφφ

0, in the interior of SC∫ •−=∆2

1a d

c2e

lrr

hAφ ∫ •−=∆

2

1b d

c2e

lrr

hAφ

c2e

abh

Φ=∆−∆ φφ

In the absence of magnetic field

71

Therefore,

cecossin2J

cesin

cesinJJ

oo

oootot

h

hh

Φ=

Φ

−+

Φ

+=

δ

δδ

varies with magnetic flux Φ

has maxima when w/. integer s.πs=Φc

eh

T=39.5mG

T=16mG

72

Devices based on two junction loops are called SQUIDs.magnetometer

An applied field is measured by monitoring the current as the flux increases from zero and counting the maxima.

Flux can be measured with a precision much smaller than πhc/e. If the loop area is 1cm2, induction fields less than 10-11T can be measured.

SQUIDs are also used as sensitive ammeters and voltmeters.

Current noise Voltage noiseEnergy sensitivity 10-22 Joule

HzpV/HzpA/ 0.3

73

74

Magnetic Resonance Imaging (MRI) Biomagnetism

1905-1983

1952

1912-1997

By impinging a strong superconductor-derived magnetic field into the body, hydrogen atoms that exist in the body's water and fat molecules are forced to accept energy from the magnetic field. They then release this energy at a frequency that can be detected and displayed graphically by a computer.

NMR was actually discovered in 1946 by Felix Bloch and Edward Purcell. The first MRI exam on a human being was not performed until July 3, 1977.

-- took almost five hours to produce one image! Today's faster computers process the data in much less time.

75

Small- Scale Applications of superconductors

76

Applications of superconductors

Trains can be made to "float" on strong superconducting magnets.

MLX01

Bogie

Magnetic levitation

1996-2002 Yamanashi Maglev Test Line1999 April 14 MLX01 test line, reaching 581km/hr w/. seat capacity 68

ML-500

1977 Miyazaki Maglev Test Track

1979 December reaching 517km/hr

77

Ship Propulsion :5MW HTCS Motor March 2003

8 million projectONR

70 million projectONR

78

30m long, 30KV, 2,000A HTC power line in Copenhagen, Denmarksupply 50,000 households May 2001

500m HTCS power line in Yokosuka, Japan

79