We recall expressions (26.19) for the expectation value of the reduced Hamiltonian

Let us calculate the energy required to add an electron to the system in a state |k↓〉 assuming its companion state | –k↓〉 is empty. In order to do this we must: (i) account for the energy increase due to removing the amplitude of the pair associated with this wave vector k and (ii) add the energy of the lone electron introduced into the state | k〉. When we remove the bound pair from the ground state, according to (27.1) the energy of the system changes by an amount

(the factor 2 in the second term arises because the chosen pair state (k, – k) occurs twice since we have a double sum). Using Eq. (26.23) we may write the form (27.2) as

Adding to (27.3) the energy ξk of one (unbound) electron then yields the quasiparticle excitation energy,

where we used Eqs. (26.24) and (26.27). Thus the energy needed to add an electron in state k↓ is σk. If we calculate the energy required to remove an electron in a state – k↓ we also obtain σk. Note the minimum excitation energy is Ak; i.e., the excitation spectrum has an energy gap.

A system which is disturbed from its equilibrium state will relax toward equilibrium when the associated perturbation is removed. This relaxation process has two characteristic forms in general: (i) damped oscillations, or (ii) monotonic decay. In this section we will be concerned primarily with the latter type of decay (oscillatory responses of bulk superconductors will be discussed in Sec. 54 and are termed collective modes).

Thermodynamic systems have characteristic microscopic (rapid) relaxation times and the relaxation of the vast majority of their internal degrees of freedom proceeds on these time scales. There are two important exceptions: (i) modes involving degrees of freedom for which conservation laws exist; and (ii) additional modes involving a broken symmetry of the system. The conservation laws are those involving mass (or charge), energy (or entropy) and momentum (which, being a vector quantity, involves an equation for each component). For a liquid this leads to the existence of five low frequency (also called hydrodynamic) modes involving: a heat conduction mode (1), transverse viscous shear waves (2), and longitudinal sound (2). The number in parentheses denotes the number of such modes, there being five in all, corresponding to the five conservation laws. For a derivation of this mode structure see Landau and Lifshitz (1987) or Miyano and Ketterson (1979). Additional equations of motion exist when the system spontaneously breaks some symmetry, and we now list some examples.

Although our discussion in this section will be largely phenomenological we must anticipate the major result of the microscopic theory which will be developed in Parts II and III; in particular we must make use of the fact that superfluidity in Fermi systems arises from the formation of a special kind of bound state in which electrons (or He quasiparticles) on (or near) the Fermi surface having oppositely directed momenta, + p and – p, form (collectively) bound pairs.

For the vast majority of superfluid electron systems (metals), the pairs form in a state with no net orbital angular momentum, an l = 0 or s-wave state which we will refer to as conventional fermion superfluidity; by the Pauli principle the electrons must then be in a singlet spin state (s = 0).

In the decade or so following the development of the BCS theory, and especially after the experimental discovery of the superfluid phases of liquid He, theorists explored various types of pairing in systems with a more complicated orbital and spin structure in which the gap can vary, in both magnitude and phase, with position on the Fermi surface. In the case of odd l pairing, the pairs form (again due to the Pauli principle) in a triplet s = 1 rather than a singlet spin state. The energy gap associated with such states may have zeros (i.e., vanish) at points or lines on the Fermi surface, and as a result the number of excitations at low temperatures varies as a power of the temperature, rather than exponentially.

Dichalcogenides of transition metals such as NbSe2, superconductor/insulator superlattices, and high Tc oxide superconductors all have a layered structure. Most of these systems are Type II superconductors with relatively large K values. One of the fascinating characteristics of the layered superconductors is their strongly anisotropic magnetic properties. Usually, the coherence length perpendicular to the layer plane (ξ⊥ is much smaller than that parallel to the layer (ξ∥). The anisotropy can be characterized by an anisotropy ratio, γ, defined as γ = ξ∥ξ⊥. Other length scales of interest are the penetration length, λ, and the scale of intrinsic inhomogeneities.

Depending on the relative size of ξ⊥ and the layer spacing, s, we may identify three different regimes for the vortex structure in the layered, high-K superconductors: (1) If ξ⊥(T) > > s, then the layered structure is largely irrelevant and the superconductor may be regarded as threedimensional: anisotropic, but uniform. Since the coherence length diverges as Tc is approached, this regime will always occur sufficiently close to Tc. The vortex structure can be described using the London or G–L theories by introducing an anisotropic mass tensor as discussed in Sec. 11. (2) With decreasing temperature, we may have a regime where ξ⊥(T) < < s (especially in high Tc oxides). If both regimes can be entered by sweeping the temperature a 3D–2D crossover will occur at some temperature.