Patterson, Bailey - Solid State Physics Introduction to theory

498 8 Superconductivity

Fig. 8.20. Gap in single-particle excitations near the Fermi energy

If we further suppose that N(E) constant N(0) ≡ the density of states at the Fermi level, then (8.211) becomes

This equation can be written as

500 8 Superconductivity

perturbation theory because with ordinary perturbation theory, we cannot reproduce the effect of an essential singularity in the perturbation. This is similar to what happened when we discussed a single Cooper pair.

Our discussion has only been valid for weakly coupled superconductors.

Roughly speaking, these have (T /θ )2 > (500)−2. Pb, Hg, and Nb are strongly cou-

c D ~

pled, and for them (Tc /θD)2 ≥ (300)−2. Alternatively, the electron–phonon coupling parameter is about three times larger than is a typical weak coupling superconductor. A result for the strong coupling approximation is given below.

8.5.4Remarks on the Nambu Formalism and Strong Coupling Superconductivity (A)

The Nambu approach to superconductivity is presented by matrices and diagrams. The Nambu formalism includes the possibility of Cooper pairs in the calculation from the beginning via two component field operators. This approach allows for the treatment of retardation effects that need to be included for the strong (electron lattice) coupling regime. An essential step in the development was taken by Eliashberg and this leads to his equations. The Eliashberg strong coupling calculation of the superconducting transition temperature gives with a computer fit (via McMillan):

θD is the Debye temperature, and for definitions of λ (the coupling constant) and μ* (the Coulomb pseudopotential term) see Jones and March [8.17]. They also

give a nice summary of the calculation. Briefly λ = N(0)Vphonon, μ = N(0)Vcoulomb where V in (8.218) is Vphonon − Vcoulomb, and

Usually λ empirically turns out to be not much larger than 5/4 (or smaller).

q

k – q

k

Fig. 8.21. Lowest-order correction to self-energy Feynman diagram (for electrons due to phonons)

502 8 Superconductivity

8.8 High-Temperature Superconductors (EE, MS, MET)

It has been said that Brazil is the country of the future and always has been as well as always will be. A similar comment has been made about superconductors. The problem is that superconductivity applications have been limited by the fact that liquid helium temperatures (of order 4 K) have been necessary to retain superconductivity. Liquid nitrogen (which boils at 77 K) is much cheaper and materials that superconduct at or above the boiling temperature of liquid nitrogen would open a large range of practical applications. Particularly important would be the transport of electrical power.

Just finding a high superconducting transition temperature Tc, however, does not solve all problems. The critical current can be an important limiting factor. Thermally activated creep of fluxoids (due to J × B) can lower Jc (the critical current) as the current interacts with the fluxoid and causes energy loss when the fluxoid becomes unpinned and thus creeps (can move). This is important in the high-Tc superconductors considered in this section.

Until 1986, the highest transition temperature for a superconductor was Tc = 23.2 K for Nb3Ge. Then Bednortz and Müller found a ceramic oxide (product of clay) of lanthanum, barium, and copper became superconducting at about 35 K. For this work they won the Nobel prize for Physics in 1987. Since Bednortz’s pioneering work several other high-Tc superconductors have been found.

The “1-2-3” compound YBa2Cu3O7, has a Tc of 92 K. The “2-1-4” compound (e.g. BaxLa2–xCuO4–y) are another class of high-Tc superconductors.

Tl2Ba2Ca2Cu3O10 has a remarkably high Tc of 125 K.

The high-Tc materials are type II and typically have a penetration depth to coherence length ratio K ≈ 100 and typically have a very large upper critical field. As we have mentioned, thermally activated creep of fluxoids due to the J × B force may cause energy dissipation and limit useable current values. For real materials, the critical current (Jc), critical temperature (Tc), and critical magnetic field (Bc) vary, but can be conveniently represented as shown in Fig. 8.22. As mentioned, the high-temperature superconductors (HTSs) are typically type II and also their Jc parallel to the copper oxide sheets (mentioned below) ≈ 107 A/cm2, while perpendicular to the sheets Jc can be about 107 A/cm2. A schematic of J, Bc, and Tc is shown in Fig. 8.22 for type I materials. For HTS, the representation of Fig. 8.22 is not complex enough. In Table 8.1 we list selected superconductor elements and compounds along with their transition temperature.

For HTS, we are faced with a puzzle as to what causes some ceramic copper oxide materials to be superconductors at temperatures well above 100 K. In conventional superconductors, we talk about electrons paired into spherically symmetric wave functions (s-waves) due to exchange of virtual phonons. Apparently, lattice vibrations cannot produce a strong enough coupling to produce such high critical temperatures. It appears parallel Cu-O planes in these materials play some very significant but not yet fully understood role. Hole conduction in these planes is important. As mentioned, there is also a strong anisotropy in electrical conduction. Although there seems to be increasing evidence for d-wave pairing, the exchange

504 8 Superconductivity

J

Bc

B

Tc

T

Fig. 8.22. J, B, T surface separating superconducting and normal regions

8.9 Summary Comments on Superconductivity (B)

1.In the superconducting state E = 0 (superconductivity implies the resistivity ρ vanishes, ρ → 0).

2.The superconducting state is more than vanishing resistivity since this would imply B was constant, whereas B = 0 in the superconducting state (flux is excluded as we drop below the transition temperature).

3.For “normal” BCS theory:

a)An attractive interaction between electrons can lead to a ground state separated from the excited states by an energy gap. Most of the important properties of superconductors follow from this energy gap.

b)The electron–lattice interaction, which can lead to an effective attractive interaction, causes the energy gap.

c)The ideas of the penetration depth (and, hence, the Meissner effect—flux exclusion) and the coherence length follow from the theory of superconductivity.

4.Type II superconductors have upper and lower critical fields and are technically important because of their high upper critical fields. Magnetic flux can penetrate between the upper and lower critical fields, and the penetration is quantized in units of hc/|2e|, just as is the magnetic flux through a superconducting ring. Using a unit of charge of 2e is consistent with Cooper pairs.

506 8 Superconductivity

In Table 8.2 we give a subjective “Top Ten” of superconductivity research.

Table 8.2 Top 10 of superconductivity (subjective)