Chapter 10. Superconductivity Two basic properties of superconductivity 1. Zero resistivity Below a certain temperature, the critical temperature Tc (property of the superconductor), resistivity of a superconductor will become exactly zero. The first superconductor was mercury, discovered by Onnes in 1911. 2. Perfect diagmagnetism A superconductor expels magnetic field completely when it is in superconducting phase (T<Tc). This phenomenon was discovered by Meissner (and Ochsenfeld) in 1933, so it is called the Meisner effect.

T

ρ

Tc

T<Tc T>Tc

3. Missing any one of these two properties will make the superconducting phase thermodynamically unstable. Hence it needs both properties to prove a material is a superconductor. 4. In measuring resistance of a superconductor, if contact resistance >> normal resistance of the superconductor, strict four points measurement is needed: For I to be constant, R>>Sample resistance + contact resistance. Two fluid model 1. Assume there are two kind of carriers – normal and superconducting. In here, let the normal carriers form component 1, and the superconducting carriers form component 2. 2. Superconducting carriers are in a condensed state, they are at the lowest energy state and they carry no entropy. 3. As a result, there is no scattering for the superconducting carriers and there is no resistance for them – they cause the phenomenon of superconductivity. 4. When T>Tc, all carriers are normal. When T=0, all carriers will be superconducting. When 0<T< Tc, ω = ns/N will be superconducting and (1-ω) will be normal. ω can be considered as an order parameter. We want now to determine the value of ω for the equilibrium between the two components.

V

Constant I

R

5.

To fit the experimental data, Gorter and Casimir replaced (1-ω) in above

expression by (1-ω)1/2:

πω−⎟

⎠⎞

⎜⎝⎛ γ−ω=

ω+⎟⎠⎞

⎜⎝⎛ γ−ω=

ω+ω=∴

π−=

π−===

γ=

γ−=γ−=−=

==

π=⇒

π=

∫∫

8)0(H

T21 )-(1

)0,0(fT21 )-(1

)0,T(f)0,T(f )-(1 )0,T(f:is phase ctingsupercondu theofenergy Free

8)0(H

8

)0(H)0,0(f )0,0(f constant )0,T(f

:effectMeissner of because and entropy, no hascomponent ctingsupercondu TheT eunit volumper heat specific where

T21TdTdTC)0,0(f)0,T(f

0)0,0(f assume and 0H For:component neutral theofenergy Free

BdH41-SdT-du df BH

41-TS-ufenergy free Define

2c2

22

21s

2c

2c

122

2en11

1

( )4

c

2c

2c

c

2

2c

2

2c

2

2c

2

2c2s

2c2

s

TT1 )0(HT2

0 ,TTAt )0(H

T21

)0(HT2

4)0(H

T21

-1

08

)0(HT

21

-121 0

fm,equilibriu At

8)0(H

T21 )-(1 )0,T(f

⎟⎟⎠

⎞⎜⎜⎝

⎛−=ω⇒=πγ∴

=ω=

⎟⎟⎠

⎞⎜⎜⎝

⎛ πγ−=ω⇒

πγ=

π

⎟⎠⎞

⎜⎝⎛ γ

=ω⇒

=π

−⎟⎠⎞

⎜⎝⎛ γ

ω⇒=

ω∂∂

πω−⎟

⎠⎞

⎜⎝⎛ γ−ω=

Thermodynamics of superconductor 1. Superconducting state is an ordered state, so its free energy and entropy are lower than the free energy and entropy of the normal state. 2. Applied magnetic field can destroy conductivity. A superconducting state will become normal when H > Hc(T). Define free energy f=U-TS-HB/4π 3. Along the blue line in the above figure,

At the phase boundary,

)0,T(f)H,T(f effect Meissner

8H)0,T(f)H,T(f

ss

2

nn

=⇒

π−=

π−===∴

=

8H

)0,T(f)H,T(f )H,T(f)0,T(f

)H,T(f)H,T(f 2

cncncss

cscn

⎟⎟⎠

⎞⎜⎜⎝

⎛+

γ−=

γ+

γ−

γ−=

ππγ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ γ−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

π⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ γ−⎟⎟

⎠

⎞⎜⎜⎝

⎛=∴

⎟⎟⎠

⎞⎜⎜⎝

⎛=ω

πω−⎟

⎠⎞

⎜⎝⎛ γ−ω=

2c2

c

4

42

c

2c4

2c

2c

4

c

22

c

2c

4

c

22

cs

4

c

2c2

s

TTT

4

TT4

14T

TT2

1

8T2

TT-1 - T

21

TT

8)0(H

TT-1 - T

21

TT)0,T(f

TT-1 with

8)0(H

T21 -1)0,T(f

Tc

Hc(0)

Superconducting state

Normal state 2

c

2

c

c

TT1

)0(H)T(H

−=

3. Along the red line (T-axis) in above figure:

The two curves have the same slope and join together at T=Tc, hence the transition is second order. 4. From two fluid model:

T/Tc

π−γ−=

π−γ−=

π−=⇒

π=−

γ=

=γ−=γ−=γ−=−= ∫∫

8)T(H

T21

8)T(H

T21)0,0(f

8)T(H

)0,T(f)0,T(f8

)T(H)0,T(f)0,T(f

T eunit volumper heat specific where

)0)0,0(f( T21T

21)0,0(fTdT)0 ,0(fdTC)0 ,0(f)0 ,T(f

2c2

2c2

n

2c

ns

2c

sn

n22

nnennn

H f

fn

fs

Hc H

π8H 2

c

Electrodynamics of superconductors – London equation 1. Semiclassical equation of motion: 2. Faraday’s Law:

3. Introduce penetration depth λL:

2

2c

2

c

c

2c

2

c

c

2c

2

2c

2c

2c

2

4c

4

2c

2c

22c2

c

42

c

2c22

c2c

4

2c2

c

4

s

t-1h have we

TT tand

)0(H)T(H

hquantity normalized write weI

TT1

)0(H)T(H

TT21

)0(H)T(H

TT21

TT

T2)T(H

T4TTT2)T(H

8)T(H

T21T

TT

4

TTT

4 )0,T(f

=

==

−=⇒

−≈⇒

−⎟⎟⎠

⎞⎜⎜⎝

⎛+=

πγ⇒

πγ−⎟⎟⎠

⎞⎜⎜⎝

⎛+πγ=⇒

π−γ−=⎟

⎟⎠

⎞⎜⎜⎝

⎛+

γ−∴

⎟⎟⎠

⎞⎜⎜⎝

⎛+

γ−=

2s

s

s2s

ss

enm e wher )J(

t E

J t

en

m E

) ven(ten

m Ee - vmt

F

=ΛΛ∂∂

=⇒

∂∂

=⇒

∂∂

=⇒∂∂

=

vv

vv

vvvv

Jc- B

0 cB J

t

0 tB

c1 J

t 0

tB

c1 E

vv

vv

vv

vv

Λ×∇=⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛+Λ×∇

∂∂

⇒

=∂∂

+⎥⎦⎤

⎢⎣⎡ Λ∂∂

×∇⇒=∂∂

+×∇

2s

2

L2

2L

2s en4

mc c

4en

mπ

=λ⇒πλ

==Λ

4. Combine London’s second equation with Maxwell equation:

5. London Gauge Canonical momentum:

For a ground state system, 6. Original London’s equation can also be derived from this result:

{( ) ( )

( )

B1 B

c4

c

Bc

4B-

)HB(with cB

c4B

J c

4H c

J4tDH

2L

2

2

2L

2

0

vv

vv

vvv

v

vvvv

v

λ=∇⇒

πλπ

−=∇⇒

=Λ

π−=×∇×∇⇒

×∇π

=×∇×∇⇒π

=∂∂

−×∇

=

Ace vm pvvv +=

GaugeLondon -- 0A

0J Since

A c

1Amc

en venJ

Amce v 0p

s

2s

ss

s

=⋅∇∴

=⋅∇Λ

−=−>=<>=<∴

−><⇒>=<

v

v

vvvv

vvv

equation) eLondon' (second Bc

1J

)AB( A c

1J A c

1J

equation) eLondon'(first E 1Jt

)EAt

c1- E B

t

c1 that (note A

t

c1J

t A

c1J

s

ss

s

ss

vv

vvvvvv

vv

vvvvvvvv

Λ−=×∇⇒

×∇=×∇Λ

−=×∇⇒Λ

−=

Λ−=

∂∂

⇒

=∂∂

⇒×−∇=∂∂

∂∂

Λ−=

∂∂

⇒Λ

−=

7. Example:

Ginzburg-Landau (GL) equation (1950) 1. Ginzburg-Landau equation is a general phenomenological theory for phase transition by introducing an order parameter Ψ to describe the more ordered state. In the case of superconductor, the superconducting carrier density we used in the two fluid model can be used as the order parameter: ns= |Ψ|2 2. The GL equation is mostly valid when T~Tc. Although it is valid only in a short temperature range, but it is extremely powerful when the superconductor becomes inhomogeneous when Ψ(x) is a function of position. Examples when superconductor becomes inhomogeneous: (i) at surface of type I superconductor in an external field, when the field penetrate the superconductor according to London’s equations. (ii) when a type II superconductor is in its mixed or vortex state. 3. GL (time independent) equation looks like Schroedinger equation: In general, α<0 and β>0. Ψ(x) now plays the role of a wave function and it can be complex.

L

L

x/-

x/-

2L

2

2

2L

2

B(0)e B B(0)B 0,At x

Ke B

B1 Bx

B1 B

λ

λ

λλ

=∴

===⇒

=∂∂

⇒=∇vv

Vacuum Superconductor

B(0)

z

x

λL

(T) - || Ac*e - i

*m21

mlinear ter-Non

22

Ψ=ΨΨ+Ψ⎟⎠⎞

⎜⎝⎛ ∇− αβ 43421

vh

The supercurrent is given by Ψ(x): 4. Experimentally it was determined that e*=2e and m*=2m. 5. Example 1 (proximity effect) With no magnetic field (A=0), GL equation becomes: Now introduce coherence length ξL:

( ) A c*m

*e- * - **2mh*ie- J 2

22

s

vΨΨ∇ΨΨ∇Ψ=

Superconductor Normal metal

Ψ

x

0 ff fdxd

a*m2

0 faaa fdxda

*m2

f aLet

0 ||a(T) - dxd

*m2

0a and -aLet

0 ||(T) *m2

32

22

3

3

2

22

22

22

222

=+−−⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛+Ψ−−∴

=Ψ

=ΨΨ+ΨΨ−

>=

=ΨΨ+Ψ+Ψ∇−

h

h

h

h

ββ

ββ

β

β

α

βα

2x tanh a

2 tanh f :Solution

0 ||(T) dxd

x Let

a*m2

a*m2

22

2L

L

22

L

ξβ

η

βα

ξη

ξξ

=Ψ

=

=ΨΨ−Ψ+Ψ∴

=

=⇒=hh

6. GL equation introduces a new length scale: coherence length ξL. It is a measure on how fast Ψ drops to 0 at the boundary between normal metal and superconductor. ξL depends on temperature, because a depends on temperature: 7. α(T) varies linearly with T: α(T) ~ T - Tc Note that ξL diverges (to infinity) as T →Tc . 8. Now we have two length scales: penetration depth λ and coherence length ξL. The relative magnitude of these two length scales will affect how the normal region is formed as the magnetic field penetrates the superconductor. For example, if λ >> ξL, one can expect the field can penetrate deep inside the superconductor with the formation of small normal region (~ξL).

x ξ 2ξ 3ξ

βa

Ψ

21

(T)*m2

a(T)*m2)T(L

αξ hh

==

TT1~)T(c

L−

∴ξ

Superconductor Normal metal B(x) ξL

9. Define GL parameter κ: According to Abrikosov theory (1957), κ define two types of superconductors: κ determines how the magnetic energy is distributed between surface and volume. For type I superconductor, most energy is stored in the surface. If λ is large, it is more efficient to store energy in “tubes” of ξL in diameter. These tubes are called vortices, occur only in type II superconductors. 10. Type-I superconductor London’s equations are followed when it is in the superconducting state (H< Hc). Most elemental superconductors are type-I superconductors. Type-II superconductors.

H

-4πM

Hc

L

ξλκ =

ctorsupercondu II Type- 2

1

ctor supercondu I Type- 2

1

>

<

κ

H

-4πM

Hc1 Hc2

Meissner state

Mixed state

Normal state Hc (thermodynamic

critical field)

12. Vortex formation in type II superconductor: Top view: ` 13. If we write the order parameter as

Flux quanta Φs

Superconductor

Vortex

B

Vortex – normal region with bundle of magnetic field (1 flux quanta Φs) penetrating the sample.

Superconducting area Vortices forming hexagonal structure

)r(ie )r( vv θφ=Ψ

φ∇φ+θ∇φ−=Ψ∇Ψ

φ∇+θ∇φ−=Ψ∇

φ∇φ+θ∇φ=Ψ∇Ψ

φ∇+θ∇φ=Ψ∇∴

θ

θ

2

i-

2

i

i * e ) i( *

i * e ) i(

Φs is the flux quanta of a vortex. Note that Φs = Φ0 /2. BCS Theory (1957) 1. Sincee*=2e and m*=2m. ∴electrons form pairs. Pairing allows fermion to form boson (after pairing) and Bose-Eisntein condensation to occur. 2. BCS theory is the only successful microscopic theory so far that explains how the electron pair is formed. Yet there are many exotic superconductors with strange behavior that cannot be fully explained yet. 3. Isotopic effect: the Tc of a superconductor depends on the nucleus mass (of the same element). Frolich theory ⇒ M1/2Tc = constant This is not necessary precisely correct. In general, we have MαTc = constant with α to be determined experimentally. The importance of isotopic effect is that the phenomenon of superconductivity is related to the atoms in the superconductors, not just the free electrons alone. 4. The exchange of virtual phonons produces a small attraction between the electrons. The free electron Fermi sphere is unstable against this small attraction, and a new ground state is defined.

( )

s

s

2s

2

2

2s

2

2s

22

2s

22

s

n *ehn A

0J

A - *ehn

J

*ec*m

Ac*m

*e - *m*e2

J

2 i.e.

2 of multiple a bemust path closed a around phase of

Ac*m

*e - *m

*e J

A c*m

*e- *m

*e J

A c*m

*e- * - **2m

*ie- J

s

Φ=⎟⎠⎞

⎜⎝⎛=⋅=⋅

=

⋅⎟⎠⎞

⎜⎝⎛=⋅⇒

⋅=⋅∴

=⋅∇

⋅⋅∇=⋅⇒

∇=⇒

ΨΨ∇ΨΨ∇Ψ=

∫∫

∫∫

∫∫

∫

∫∫∫

Φ

cddB

If

dcd

dnd

nd

nIntegratio

ddd

lvvvv

v

lvv

321

lv

v

lvvh

lv

vlv

lvv

lvh

lv

v

vhv

vhv

σ

φ

πφ

πθ

πθ

θφ

φθφ

Dωh

5. . From this, BCS postulated that the electron pairs are mediated by the vibration of atoms (i.e. phonons) – electrons from pairs by exchange of virtual phonons. Virtual phonons are phonons with wavelength λ >> distance between the electrons forming the pair – this simply means distortion of lattice. The idea is that as an electron moves around in the lattice, its interaction with the ions will cause a small distortion in the lattice and this distortion will attract another electron into the region. Only electrons within a shell thickness can form pairs. ωD is the Debye temperature. 5. BCS approximations (i) For the lowest energy state, electrons form pair so that their total momentum and total spin are zero, i.e. Only consider pairing between electrons with (k, ↑) and (-k,↓). (ii) Assume isotropic potential, because of phonon mediated interaction.

This isotropic potential gives rise to “s-wave” superconductors. Meaning of “s-wave” superconductors: Pairing of two spins.

Distortion of lattice (formed by another electron)

e-

Dωh

⎩⎨⎧ ω+≤≤ω

= otherwise 0

E E -E if V- V DFkDF

'k,k

hh vvv

s=0 ⇒ symmetric wavefunction. Angular momentum =s, d, g, i, ……. s=1 ⇒ antisymmetric wavefunction. Angular momentum =p, f, h, ……. Most simple superconductors, as required by BCS theory, are s-wave superconductor. The spatial wave function is isotropic. Pairing of superfluid He3 is p-wave (note that He-4 does not form pair because He-4 is a boson by itself). 7. By forming pairx, the total energy (2EF) is lowered is lowered by an amount of 2∆. 2∆ is known as the energy gap of the superconductor. ∆(T) can also be considered as the order parameter, related to Ψ. 8. ∆(0) depends on ωD and V:

N(E=0) is the density of states at the Fermi energy. 9. ∆(0) determines Tc:

Some superconductors may have a significant larger value. For example, for Ph,

These are known as strong coupling superconductors. This does not mean that these superconductors are not BCS like or phonon mediated. 10. Pippard coherence length ξ0. In superconductor, there is another coherence length, called the Pippard coherence length ξ0. Only electron within 2∆ on the Fermi surface can contribute to superconductivity, even at T=0K.

( )

( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⇒=

⎪⎪

⎭

⎪⎪

⎬

⎫

↓↑>+↑↓>

↓>↓↑↑>

⇒=↓↑>−↑↓>

=↓>+↑>symmetric.ion wavefunctofpart Spin stateTriplet 1.s

||2

1 | |

ric.antisymmetion wavefunctofpart Spin stateSinglet 0.s ||2

1

| |

V)0(N1

De2 ~ )0(−

ω∆ h

3.5 ~ Tk

)0(2 ctor,supercondu couplingFor weak B

∆

4.3 ~ Tk

)0(2

B

Pb∆

This corresponds to an uncertainty in time of

In this time, the electron will travel a distance of

ξ0 can be more precisely derived in BCS theory as:

ξ0 is a constant independent of T, and ξ0 ~ ξ(T) (GL coherence length) when T<<Tc. 10. ∆(T) depends on temperature. Assuming an isotropic ∆, it is given by the equation:

)Tk

1( d tanh

V)0(N

1

B22

2221

0

D =βζ∆+ζ

∆+ζβ= ∫

ωh

2∆

4

t 2

~ 2t∆

=δ⇒∆⋅δhh

4v

t v FF0 ∆

=δ=ξh

v

F0 ∆π=ξh

As T → Tc, we know that But

Therefore we expect More accurately, according to BCS: 11. Thermal excitations. At T>0, thermal excitations occur. Electrons in a superconductor can be considered as a Fermi liquid. For this reason, thermal excitation can be described as vreation of quasiparticle. A quasiparticle at state k ≡ no pair between (k, ↑) and (-k,↓). Quasiparticles have lifetime. They behave pretty much like an electron at state k (Landau Fermi liquid theory). They follow Fermi-Dirac distribution:

12. BCS theory gives the density of states of the quasiparticle as:

TT1~)T(c

L−

ξ

v

F0 ∆π=ξh

cc T~Tat TT~)T( −∆

cc

T~Tat TT1~

)0()T(

−∆∆

1e1 )E(f Tk/Ek Bk +

= vv

⎪⎩

⎪⎨⎧

∆<

∆>∆−=

)(E 0

)(E E

E

)0(N)E(N

22S

Energy spectrum of quasiparticle:

Hole like quaiparticle: Electron like quasiparticle:

Josephson Tunneling 1. Josephson tunneling is often called a weak link between two superconductors. Types of weaklinks:

Josephson junction

Bridge

Point contact

Soldering junction

2. dc Josephson effect: Weak links can conduct supercurrent at zero voltage. The critical current of the weak link is much smaller than the bulk critical current. Time dependent Schroedinger equation:

With coupling by the weak link K:

If there is a potential difference V across the junction, when a pair “tunnel“ through the junction, there is an energy change of 2eV. We can phenomenologically write H1 as eV and H2 as –eV at the junction:

Since |Ψ|2 =n, we can write

x

Ψ Ψ1

` S1

Ψ2

Weak link

H t

i and H t

i 222

111 Ψ=

∂Ψ∂

Ψ=∂Ψ∂

hh

K H t

i

and KH t

i

1222

2111

Ψ+Ψ=∂Ψ∂

Ψ+Ψ=∂Ψ∂

h

h

K eV- t

i

KeV t

i

122

211

Ψ+Ψ=∂Ψ∂

Ψ+Ψ=∂Ψ∂

∴

h

h

en and en 21 i22

i11

θθ =Ψ=Ψ

)(i211111

i2

i11

i11

i

121

1

12

2111

ennKneV t

innt2

1i

enKeneV t

enint

en2

1i KeV t

i

θ−θ

θθθθ

+=⎥⎦⎤

⎢⎣⎡ θ

∂∂

+∂∂

⇒

+=⎥⎥⎦

⎤

⎢⎢⎣

⎡θ

∂∂

+∂∂

⇒Ψ+Ψ=∂Ψ∂

h

hh

Equating real and imaginary parts: In other words, It is more common to use Josephson current than current density. I=JA, and we can write similarly, I = Ic sin ∆θ

)(i212122

i1

i22

i22

i

212

2

21

1222

ennKneV t

innt2

1i

enKeneV- t

enint

en2

1i K eV- t

i

θ−θ

θθθθ

+−=⎥⎦⎤

⎢⎣⎡ θ

∂∂

+∂∂

⇒

+=⎥⎥⎦

⎤

⎢⎢⎣

⎡θ

∂∂

+∂∂

⇒Ψ+Ψ=∂Ψ∂

h

hh

h

hhhhhh

h

h

h

h

h

h

2eVt

n ~n If

cosnnKeV cos

nnKeV

t

n(4)

n(2)

sinnnKnt2

(3)

sinnnKnt2

(1)

If

(4)- )cos(nnKneVt

n

(3)- )sin(nnKnt2

(2)- )cos(nnKneVt

n

(1)- )sin(nnKnt2

21

2

1

1

2

21

212

211

12

2121222

21212

1221111

12211

=θ∆∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡θ∆+−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡θ∆+=θ∆

∂∂

⇒−

θ∆−=∂∂

⇒

θ∆=∂∂

⇒

θ−θ=θ∆

θ−θ+−=θ∂∂

−

θ−θ=∂∂

θ−θ+=θ∂∂

−

θ−θ=∂∂

nn4KJ sinJ J

sinnn4K

) sinnnK( - sinnnK2 )nn(dtde J

:(3) and (1) From

21cc

21212121

h

hh

=θ∆=∴

θ∆=θ∆−θ∆=−=

Josephson super current ⇔ phase difference across the junction dc-Josephson super current ⇔ constant phase difference across the junction

3. dc-Josephson effect: When a non-zero voltage is applied across the Josephson junction, the phase difference across the two sides will oscillate and this gives rise to an oscillating supercurrent.

I

V

Ic

Weak link is superconducting and there is a constant phase difference across the two sides (dc-Josephson effect).

I = Ic sin ∆θ

ac-Josephson effect occurs when V≠0. <I>=0 for ac-Josephson effect, and this is not observable in this dc-characteristic.

ac-Josephson super current ⇔ oscillating phase difference across the junction

GHz 3.037 10037.3 05510.1

10602.110122eV V,1 V If

2eV frequency gOscillatin

2eVtsin I sin I I

V)(constant 2eVt 2eV t

934

196

cc

=×=××××

==ωµ=

=ω=

=θ∆=∴

=θ∆⇒=θ∆∂∂

−

−−

h

h

h

hh

e2/ωh

ac-Josephson is observed as microwave radiation from the junction (caused by the oscillating supercurrent) as a constant voltage is applied across the junction. reverse, if microwave of frequency ω is shined on a Josephson junction, voltate steps will occur and the step width equals to . 4. Small junction and large junction When a strong enough magnetic field is applied, it will penetrate the superconductor at the junction area first: The junction is small if the cross sectional dimensions Lx, Lx < λJ. The junction is large if the cross sectional dimensions Lx, Lx >> λJ. Below argument applied only to small junctions. 5. Quantum interference Current density (and phase θ) across the junction will not be uniform if a magnetic field is applied parallel to the junction:

y

x

z Bz(y)

Ax(y) Ax(y)

Js

d

a

SC1 SC2

SC1 SC2

λJ λ1 λ2

d

)(J8hc )T(

21c

2

J de ++=

λλπλ

yz

For SC1, along the path ABCD,

areajunction in the field uniform theis B

d)21 |y(| iyB- A A B

k (y)B B

0

0

z

≤=⇒×∇=

=vvv

v

y

Ay

SC1 SC2

y

Bz

SC1 SC2

SC1

Js

y

x

SC2

A B

C

A’ B’

C’ D’ θ10 θ20

θ1(x)

D

z

θ2(x)

A1∞ A2∞

Similarly, for SC2, along the path A’B’C’D’, Subtracting these two equations, A1∞+ A2∞ can be calculated from the combined loop:

1001s

1

01101s

-x)(

AD

0

DC

)x-x(A

CB

0

BA

s

s

2

)x-x(A2 x)(

)x-x(A ]-x)([2

0 0 A

0 A 0 A 0B ctor,supercondu aFor

2

2

*ec A A

c*m*e

*m*e

10101

θπθ

θθπ

θθθθπ

θπ

θθ

θθ

+Φ

=⇒

=Φ

⇒==⋅

=⋅⇒=×∇⇒=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅∇+⋅∇+⋅∇+⋅∇Φ

=

⋅∇Φ

=

⋅∇=⋅⇒⋅=⋅∇

∞

∞

=

→

=

→

−=

→

=

→

∫

∫

∫∫∫∫

∫

∫∫∫∫

∞

lvv

lvvvv

43421

lv

43421

lv

43421

lv

43421

lv

lv

lvh

lvv

lvv

lvh

d

d

dddd

d

dddd

2002s

2 )x-x(A2 x)( θπθ +Φ

−= ∞

( )xAA2

)x-x(A2 )x-x(A2x)(-x)( )x(

21s

0

2002s

1001s

21

∞∞

∞∞

+Φ

+=

⎥⎦

⎤⎢⎣

⎡+

Φ−−⎥

⎦

⎤⎢⎣

⎡+

Φ==∆

πφ

θπθπθθθ

( ) ( ) 021s

20100 xAA2- ∞∞ +Φ

−=πθθφ

s

sc

cc

c

s0

s0

2121

sin)0(I

dzdx )x( sinJ )(I

)x(sin JJ Now

xa

2 x a

2 (x)

a)A(A a)A(A dA

ΦΦ

⎟⎟⎠

⎞⎜⎜⎝

⎛ΦΦ

=

∆=Φ∴

∆=

⎟⎟⎠

⎞⎜⎜⎝

⎛ΦΦ

+=⎟⎠⎞

⎜⎝⎛ Φ

Φ+=∆∴

Φ=+⇒+=⋅=Φ

∫

∫ ∞∞∞∞

π

π

θ

θ

πφπφθ

lvv

This gives rise to the Fraunhofer pattern for Josephson junction: 6. SQUID From above we know that measurement of Ic allows accurate determination of Φ (like using interference to determine small distance). The device using this method to measure Φ is known as Superconducting QUantum Interference Device (SQUID). There are two types of SQUID: ac-SQUID (or rf-SQUID) and dc-SQUID.

Φ/Φ0

I

1 2 3 4 0