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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