Results from BCS theory (Bardeen, Cooper, Schrieffer)
The BCS theory gives a microscopic explanation of the many-electron interaction that
binds the electrons into pairs. It makes predictions of the key parameters, such as the
critical temperature Tc and the value of the gap Eg = 2. They are determined by the
Debye temperature D , the density of states at the Fermi level D(EF), and the electronphonon interaction energy V:
Eg  3.5  kBTc
kBTc  kBD  exp(
1
)
½ D(EF) V
The maximum achievable critical temperature Tc is given by the prefactor kBD , which
corresponds to a typical phonon energy (D =Debye temperature, Lect. 13, p. 2). And the
phonon energy is given by ħ = ħ( f /M ) ½ ( f = force constant, M = mass, see Lect. 11).
The M ½ dependence produces an isotope effect on Tc which proves that phonons are
involved in superconductivity. TC is maximized by a high phonon frequency, together
with a large product D(EF)V, similar to the Stoner criterion for ferromagnetism
D̃ (EF)I > 1 . In high temperature superconductors the phonons are replaced by bosons
with higher frequencies which allow higher transition temperatures.
Superconducting transition temperature Tc versus
time. It took nearly 50 years from the discovery to
its explanation by the BCS theory. The second
rise in the 80s comes from high temperature
superconductors, but that has reached a plateau as
well. There is no accepted theory of high
temperature superconductors yet, but theorists did
not have 50 years to think about it either.
(Notice the stretched scales in the figure below.)
1
Length scales
1. The penetration depth  is the decay constant of the magnetic field B. It appears in
the London equations which augment the Maxwell equations in superconductors.
2. The coherence length  is the decay constant of the pair density n. It is described by
the Ginzburg-Landau equation, which is similar to the Schrödinger equation, but
for Cooper pairs instead of single electrons.
B
n
B(z)  exp(z/)
normal
superconducting
n(z)  [1  exp(z/)]
z
normal
superconducting
z
When going inside a superconductor the magnetic field B decreases while the density of
superconducting pairs n increases. This shows once more that magnetism and superconductivity compete with each other (compare Lect. 21, p. 4).
London equations:
(1) describes vanishing resistance and ballistic motion of pairs (Newton’s F= mv/t)
(2) describes decay of the magnetic field the Meissner effect (Lect 29, Slides 2,3), and
(1) E = +2 · μ0 · ∂J/∂t
 = penetration depth  n ½
(2) B = 2 · μ0 · ×J
(similar to a screening length)
These equations describe the time- and space-dependence of the superconducting current
density J and the pair density n (in the presence of electric and magnetic fields E and B) .
Ginzburg-Landau equation:
Like the Schrödinger equation, but for pairs.
(A = vector potential, 2m,2e  pairs!)
(iħ + 2eA)2 / 4m · Ψ + β·|Ψ|2 ·Ψ = ħ2/4m · ξ2 · Ψ
Generalized current density: J = 2e/2m · |Ψ|2 ·(ħφ + 2eA)
ξ = coherence length
Pair density: n = |Ψ|2
The pair wave function Ψ = e i φ ·|Ψ| is called the order parameter of a superconductor.
Its phase φ determines the DC current density J across a Josephson junction (see p. 5).
2
Type I versus Type II superconductors
A) Type I:
>> 
Type II:
<< 
B) Type I:
Single critical field Hc (rather low).
Type II: Two critical fields Hc1 ,Hc2 . Between Hc1 and Hc2 the magnetic field
penetrates part of the superconductor forming flux tubes (= vortices).
C) Type I:
Type II:
D) Type I:
Type II:
Pure materials, such as Al, Pb.
Alloys containing pinning centers for the magnetic flux, for example
NbTi , Nb3 Sn, and high temperature superconductors. The latter are
extreme Type II, where ξ shrinks down to nearly the size of a unit cell.
Josephson junctions, SQUIDs, magnetic shielding.
Superconducting magnets.
The coherence length  is so short in Type II superconductors that the pair density
recovers between the flux tubes. This leads to the “Swiss cheese” structure of a Type II
superconductor, where holes created by the flux tubes are completely surrounded by the
superconductor, thereby keeping the superconducting regions connected. The diameter of
the holes is comparable to . In Type I superconductors  is so large that the holes
overlap. The whole solid loses superconductivity at the same (rather low) magnetic field.
3
Magnetic Flux Quantization
The magnetic flux is quantized in units of h/2e in a superconductor. Compared to the
flux quantum seen in the quantum Hall effect, one has 2e instead of e. This shows that
electron pairs with charge 2e are involved.
Regular array of flux quanta (white dots) crossing a thin film of a Type II
superconductor. This is a STM image taken with a very small applied voltage (mV). The
superconductor surrounding the flux quanta appears dark, because electrons at the Fermi
level cannot tunnel into the superconducting gap. The magnetic field of the vortices
eliminates the gap and produces a normal metal, which allows tunneling at any voltage.
4
Superconducting Devices
Josephson Junction
Superconductor-Insulator-Superconductor junction, where electrons tunnel as pairs
across the insulator. (Distinguish that from single electron tunneling from metal to
superconductor, which is used to determine the gap Eg from dI/dV.) The pair wave
functions are all coherent and exhibit quantum interference phenomena that become
amplified to macroscopic dimensions due to the coherence of the superconducting pairs.
1. DC Josephson effect: IDC without V
Permanent current caused by a phase difference  across the junction. The phase 
is that of the pair wave function  in the Ginzburg-Landau equation (p. 2).
2. AC Josephson effect: IAC from VDC
The oscillation frequency f of the current is determined by setting the electrostatic
energy 2e VDC equal to hf (Planck’s energy quantum).
The AC effect makes a precise voltage-to-frequency converter:
f /VDC = 2e/h = 01  483.6… MHz /V
The frequency can be measured very accurately with an atomic clock. The result is a
voltage standard. The image shows a chip containing many Josephson junctions in
series to obtain a sizeable voltage. (Fabricated at the National Bureau of Standards
NBS, now National Institute for Standards and Technology NIST).
5
SQUID
(Superconducting Quantum Interference Device)
Two Josephson junctions in parallel. The current through this device oscillates as a
function of the magnetic flux through the loop, analogous to Young’s double-slit
interference. Each oscillation corresponds to a single magnetic flux quantum 0 = h/2e
= 2  1015 Tm2 , which is extremely small. In addition, one can determine the position of
the sharp minima very accurately by modulation techniques, where only the noise at a
specific frequency enters the measurement. Noise from any other frequency is filtered
out. The combination makes the most sensitive magnetometer (Lect. 30, last two slides).
Schematic of a DC SQUID (left)
and a typical I(B) curve (below).
A sensitivity of 106 0 /Hz can
be achieved by modulating B
around one of the sharp minima
with a well-defined frequency and
detecting the current response in a
narrow frequency band with a
lock-in amplifier.
0 (per SQUID area)
6
SQUID Magnetometer
1 mm
7
Magnetometer
Gradiometer
SQUID
A transformer concentrates the
magnetic flux from time-dependent
signals, such as brain activity.
SQUID
The background field is eliminated
by two pickup coils, wound in
opposite directions
8