Results from BCS Theory (Bardeen, Cooper, Schrieffer)
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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 electron- phonon interaction energy V: 1
½ D(E ) V F
1 Eg 3.5 kBTc kBTc kBD exp( )
The maximum achievable critical temperature Tc is given by the prefactor kBD , 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 =
½ mass, see Lect. 11). The M 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 DD (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.)
2 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/) n(z) [1 exp(z/)]
z z normal superconducting normal superconducting
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 super- conductivity compete with each other (compare Lect. 21, p. 4).
London equations: (1) describes vanishing resistance and ballistic motion of pairs (Newton’s F= mv/t) (2) describes decay of the magnetic field the Meissner effect (Lect 29, Slides 2,3), and
2 ½ (1) E = + ∙ μ0 ∙ ∂J/∂t = penetration depth n
2 (2) B = ∙ 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!)
2 2 2 2 (iħ + 2eA) /4m ∙ Ψ + β∙|Ψ| ∙Ψ = ħ /4m ∙ ξ ∙ Ψ ξ = coherence length
2 2 Generalized current density: J = 2e/2m ∙ |Ψ| ∙(ħφ + 2eA) Pair density: n = |Ψ|
The pair wave function Ψ = eiφ ∙|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).
3 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: Pure materials, such as Al, Pb.
Type II: 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. D) Type I: Josephson junctions, SQUIDs, magnetic shielding.
Type II: Superconducting magnets.
The coherence length x 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.
4 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.
5 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: 1 f /VDC = 2e/h = 0 483.6… 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).
6 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 1015 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 6 sensitivity of 10 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.
(per SQUID area) 0
7 SQUID Magnetometer
1 mm
8 Magnetometer Gradiometer
SQUID SQUID
A transformer concentrates the The background field is eliminated magnetic flux from time-dependent by two pickup coils, wound in signals, such as brain activity. opposite directions
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