Dressel.Gruner.Electrodynamics of Solids.2003

transition rate apply. In the limit of long relaxation time τ , i.e. for h¯ /τ , the entire Drude spectral weight of the normal carriers collapses in the collective mode, giving

leading through Eq. (7.2.11b) to the London penetration depth λL. With increasing 1/τ , moving towards the dirty limit, the spectral weight is progressively reduced, resulting in an increase of the penetration depth. In the limit 1/τ 2 /h¯ , the missing spectral weight is approximately given by A ≈ σ1(2 /h¯ ) = 2N e2τ /hm¯ , which can be written as

Then the relationship between the area A and the penetration depth λ( ), which now is mean free path dependent as = vFτ , is given by λ2( ) = λ2Lh¯ π/(4τ ), which leads to the approximate expression

7.4.2 The electrodynamics for q = 0

We have discussed the electrodynamics and the transition processes in the q = 0 limit, the situation which corresponds to the local electrodynamics. However, this is not always appropriate, and under certain circumstances the non-local electrodynamics of the superconducting state becomes important. For the normal state, the relative magnitude of the mean free path and the skin depth δ0 determines the importance of non-local electrodynamics. For superconductors, the London penetration depth λL assumes the role of the skin depth, and the effectiveness concept introduced in Section 5.2.5 refers to the Cooper pairs; thus the relevant length scale is the correlation length ξ0, i.e. pairs with ξ0 much larger than λ0 cannot be fully influenced by the electrodynamic field (here we assume that is shorter than ξ0 and λL). The consequences of this argument have been explored by A. B. Pippard, and the relevant expression is similar to Chambers’ formula for normal metals (5.2.27).

The electrodynamics for finite wavevector q can, in principle, be discussed using the formalism developed in Chapter 4, with the electronic wavefunction

representing that of the superconducting case. This formalism will lead to the full q and ω dependence, including the response near 2kF.

Instead of this procedure we recall the simple phenomenological expression for the non-local response of metals, the Chambers formula (5.2.27), which reads

Note that, in view of σdc = N e2τ /m = N e2vl /m, the mean free path drops out from the factor in front of the integral. For superconductors, the vector potential A

where the function F(r) describes the spatial decay which has yet to be determined. For a vector potential of the form A(r) exp{iq · r}, the q dependent conductivity is written as

A more general definition of the kernel K is given in Appendix E.3.

Let us first assume that we deal with clean superconductors, and effects due to a

independent of the wavevector q. In general, K(q) is not constant, but decreases with increasing q. The argument, due to Pippard, that we have advanced before suggests that

We assume that at large distances F(r) exp{−r/R0} with R0 a (yet unspecified) characteristic distance. This then leads (for isotropic superconductors) to K (0) = λ−L 2 at q = 0 as seen before; for large q values

One can evaluate the integral in Eq. (7.4.9) based on this form of K(q), but we can also resort to the same argument which led to the anomalous skin effect for

normal metals. There we presented the ineffectiveness concept, which holds here for superconductors, with ξ0 replacing , and λ replacing δ0. The expression of the penetration depth – in analogy to the equation we have advanced for the anomalous skin effect regime – then becomes

thus, for large coherence lengths, the penetration depth increases, well exceeding the London penetration depth λL.

For impure superconductors the finite mean free path can be included by a further extension of the non-local relation between Js(r) and A(r). In analogy to the case for metals we write

J(r) dr r [r · A(r)] r−4 F(r) exp{−r/ } .

This includes the effect of the finite mean path, and additional contributions to the non-local relation, dependent on the coherence length, are contained in the factor F(r). If we take F(r) from Eq. (7.4.13), we find that

which reduces to Eq. (7.4.8) in the ξ0 > limit. This is only an approximation; the correct expression is [Tin96]

7.4.3 Optical properties of the superconducting state: the Mattis–Bardeen formalism

With non-local effects potentially important, calculation of the electromagnetic absorption becomes a complicated problem. An appropriate theory has to include both the coherence factors and the non-local relationship between the vector potential and induced currents. This has been done by Mattis and Bardeen [Mat58] and by Abrikosov [Abr59]. For finite frequencies and temperatures, the relationship between J and A takes, in the presence of an ac field, the following form:

J(0, t) exp{−iωt} dr r [r · A(r)] r−4 I (ω, r, T ) exp {−r/ } . (7.4.19)

Non-local effects are included in the kernel I (ω, r, T ) and in the exponential function exp {−r/ }. The relationship between J and A is relatively simple in two limits. For ξ0, the local dirty limit, the integral is confined to r ≤ , and one can assume that I (ω, r, T ) is constant in this range of r values. For λ ξ0, the Pippard or extreme anomalous limit, I (ω, r, T ) varies slowly in space with respect to other parts in the integral, and again it can be taken as constant. The spatial variation implied by the above expression is the same as that given by the Chambers formula, the expression which has to be used when the current is evaluated in the normal state. Consequently, when the complex conductivity σˆs(ω, T ) in the superconducting state is normalized to the conductivity in the normal state σˆn(ω, T ), numerical factors drop out of the relevant equations. The expressions derived by Mattis and Bardeen, and by Abrikosov, are then valid in both limits, and they read:

where for h¯ ω > 2 the lower limit of the integral in Eq. (7.4.20b) becomes − . At T = 0, σ1/σn describes the response of the normal carriers. The first term of Eq. (7.4.20a) represents the effects of thermally excited quasi-particles. The second term accounts for the contribution of photon excited quasi-particles; since it requires the breaking of a Cooper pair, it is zero for h¯ ω < 2 (T ). The expression for σ1(ω, T ) is the same as Eq. (7.3.9), which was derived on the basis of arguments relating to the electromagnetic absorption. This is not surprising, as

in the limits for which this equation applies. Figs 7.3 and 7.4 summarize the frequency and temperature dependence of σ1 and σ2 as derived from Eqs (7.4.20) assuming finite scattering effects /π ξ = 0.1 discussed in [Lep83]. It reproduces the behavior predicted for case 2 coherence factors (Fig. 7.2) inferred from symmetry arguments only. Although the density of states diverges at ± , the conductivity σ1(ω) does not show a divergency but a smooth increase which follows approximately the dependence σ1(ω)/σn 1 − (h¯ ω/ kBTc)−1.65. The temperature dependent conductivity σ1(T ) shows a peak just below Tc at low frequency. The height of the peak has the following frequency dependence: (σ1/σn)max log {2 (0)/h¯ ω}.

Above Tc the surface resistance shows the ω1/2 behavior of a Drude metal in the Hagen–Rubens regime (Eq. (5.1.18)). At low temperature, RS is vanishingly small and approaches an approximate ω2 dependence for ω → 0 (Fig. 7.6a). In contrast to metals, RS(ω) = −XS(ω) in the case of superconductors. By reducing the temperature to close to but below Tc, the surface reactance XS(T ) shows an enhancement before it drops. This is typically observed as a peak of XS(T )/Rn right below the transition temperature. The frequency dependence of XS can be evaluated from classical electrodynamics using Eq. (2.3.34b) and λ = c/ωk with

with a frequency dependence as shown in Fig. 7.6b. Equation (7.4.24) only holds for k n, which is equivalent to RS |XS|. It is immediately seen that the surface reactance is directly proportional to the penetration depth, which by itself is frequency independent as long as h¯ ω < . This is equivalent to Eq. (2.4.27) in the normal state, where λ is replaced by δ/2. The reason for the factor of 2 lies in the difference between the phase angle φ of the surface impedance, which is 90◦ for a superconductor (at T → 0) and 45◦ for a metal in the Hagen–Rubens regime. As λ is independent of the frequency for h¯ ω , XS is proportional to ω in this regime, as displayed in Fig. 7.6b. Even in the superconducting state the temperature dependence of the penetration depth can be calculated from the complex conductivity σˆ (ω) using Eq. (2.3.15b). In Fig. 7.1 (λ(T )/λ(0))−2 is plotted versus T / Tc for /π ξ0 = 0.01. In the London limit this expression is proportional to the density of superconducting carriers Ns as seen from Eq. (7.2.10). The energy gap 2 (T ) increases faster than the charge carriers condense. At low temperatures T < 0.5Tc the temperature dependence of the penetration depth λ(T ) − λ(0) can be well described by an exponential behavior:

This behavior is plotted in Fig. 7.1b.

7.5 The electrodynamics of density waves

Just as for superconductors, the full electrodynamics of the density wave states includes the response of the collective mode. The collective mode, i.e. the transla-

tional motion of the entire density wave, and absorption due to the quasi-particles, ideally occurs – in the absence of lattice imperfections – at zero frequency. At zero temperature, the onset for the quasi-particle absorption is set by the BCS gap 2 . The differences with respect to the superconducting case are due to the different coherence factors and the possibility of a large effective mass in the case of charge density wave condensates.

Additional complications may also arise: density wave condensates with a periodic modulation of the charge and/or spin density are incommensurate with the underlying lattice only in one direction; in other directions the lattice periodicity plays an important role in pinning the condensate to the underlying lattice. If this is the case, the collective mode contribution to the conductivity is absent due to the large restoring force exercised by the lattice.

7.5.1 The optical properties of charge density waves: the Lee–Rice–Anderson formalism

Both single-particle excitations across the density wave gap and the collective mode contribute to the frequency dependent response. For charge density waves this has been examined in detail by [Lee74]. When the effective mass is large, the collective mode contribution – centered, in the absence of impurities and lattice imperfections at zero frequency, see Eq. (7.2.15a) – is small, and consequently only minor modifications from those expected for a one-dimensional semiconductor are observed. For m = ∞, the collective mode spectral weight is zero, and the expression for the conductivity is identical to that given in Section 6.3.1 for a one-dimensional semiconductor.

This can be taken into account by utilizing the formalism outlined earlier, which leads to the BCS gap equation and to the quasi-particle excitations of the superconducting and density wave states. The formalism leads to the following expressions of the single-particle contribution to the conductivity for m = ∞:

where ζk = Ek − EF. Simply integrating the expression leads to Eq. (6.3.16).

In the presence of a finite mass associated with the collective mode, the spectral