Dressel.Gruner.Electrodynamics of Solids.2003

both collective and single-particle excitations must be the same as the sum rule which is valid above the transition, in the metallic state. The so-called Tinkham– Ferrell sum rule for superconductors – discussed in Section 7.4.1 – has its origin in this condition, and analogous sum rule arguments can be developed for the density wave states (Section 7.5.3).

References

[Mer70] E. Merzbacher, Quantum Mechanics, 2nd edition (John Wiley & Sons, New York, 1970)

[Ste63] F. Stern, Elementary Theory of the Optical Properties of Solids, in: Solid State Physics 15, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1963), p. 299

Further reading

[Jon73] W. Jones and N.H. March, Theoretical Solid State Physics (John Wiley and Sons, New York, 1973)

[Mah90] G.D. Mahan, Many-Particle Physics, 2nd edition (Plenum Press, New York, 1990)

[Rid93] B.K. Ridley, Quantum Processes in Semiconductors, 3rd edition (Clarendon Press, Oxford, 1993)

[Smi85] D.Y. Smith, Dispersion Theory, Sum Rules, and Their Application to the Analysis of Optical Data, in: Handbook of Optical Constants of Solids, Vol. 1, edited by E.D. Palik (Academic Press, Orlando, FL, 1985), p. 35

[Woo72] F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972)

The discontinuity of its first derivative modifies the wave equation in the following way

Assuming that the scattering rate is independent of q, we can also utilize Boltzmann’s transport theory, and for the current density in the low temperature limit we get an expression similar to Eq. (5.2.13). In an even simpler approximation, we can apply the generalized Ohm’s law which connects the current and the electric field by the conductivity J(q, ω) = σ1(q, ω)E(q, ω). By neglecting the displacement term in Eq. (E.3) containing ω2/c2, we obtain for the electric field

As for the normal skin effect, also in the anomalous regime, the penetration of the electric field into the metal has approximately an exponential form, and we can define a characteristic length of how far the field penetrates the metal,

which has the same functional form as obtained from our phenomenological approach, Eq. (5.2.30), with γ ≈ 3π/2 ≈ 4.71. The electric field falls quickly with distance z so that the major part of the field is confined to a depth much less than. This rapid decrease is not maintained, however, and eventually, far from the surface, the field is approximately (z/ )−2 exp{−z/ }, so there is a long tail of small amplitude extending into the metal to a distance of the order of the mean free path .

From Eqs (2.4.23) and (2.4.25), the surface impedance of a material is given by

For an anomalous conductor the result can be obtained by inserting Eq. (E.7) into Eq. (E.9):

in the limit qvF |1 − iωτ |; the factor 8/9 drops for diffusive scattering at the surface [Reu48]. The most important feature of the anomalous conductors is that the surface resistance and surface reactance are not equal, but XS = −√3RS as displayed in Fig. E.1. Also the frequency dependence ω2/3 is different from that relating to the classical skin effect regime, where Zˆ S was found (Eq. (2.3.30)). A more rigorous derivation of the anomalous skin effect in conducting and superconducting metals given by Mattis and Bardeen [Mat58] leads to the same functional relations as obtained in the semiclassical approach.

Up to this point in the discussion of the non-local conductivity and anomalous skin effect, we have not taken into account effects of the band structure and nonspherical Fermi surface. In order to do so, we have to consider the range of integration in Chambers’ expression of J as introduced in Section 5.2.4 and in Eq. (E.4). Electrons passing through the volume element dV with wavevectors dk at the time t0 have followed some trajectory since their last collision. The distribution function f is obtained by adding the contributions from all electrons scattered into the trajectory at a time t prior to t0. The probability that scattering has not occurred

inthis period between t and t0 is given by the expression exp − !tt0 [τ (t )]−1dt ,

Fig. E.1. (a) Dependence of the surface resistance RS on the mean free path with R∞ = RS( → ∞). For large /δ (good conductor) the surface resistance goes to zero. (b) Ratio of imaginary and real parts of the surface impedance as a function of the inverse surface

resistance. For a perfect conductor ( = 2∞) the ratio approaches XS/RS = √3. The dashed lines correspond to the case of diffusive scattering and the solid line shows the specular scattering (after [Reu48]).

where the relaxation time in general depends on t through the dependence of the velocity v on the wavevector k. Thus, we obtain

It can be shown [Cha52, Kit63] that this expression is a solution of the Boltzmann equation (5.2.7) within the relaxation time approximation. Using Eq. (5.2.12), we can rewrite Eq. (E.4) as

For a detailed discussion see [Pip54b, Pip62] and other textbooks.

Anomalous skin effect Anomalous reflection

Fig. E.2. (a) Electron trajectories in the skin layer illustrating the nature of the interaction between photons and electrons in metals. The thick line represents the surface of the metal. The arrows indicate the incident light waves, the dashed line marks the skin layer, the zigzag line represents a path of the electron for successive scattering events, and the wavy line represents the oscillatory motion of the electron due to the alternating electric field. The optical characteristics in the four regions illustrated depend upon the relative value of the mean free path , the skin depth δ, and the mean distance traveled by an electron in a time corresponding to the inverse frequency of the light wave. (A) Classical skin effect, δ and vF/ω. (B) Relaxation regime, vF/ω δ. (D) Anomalous skin effect, δ and δ vF/ω. (E) Extreme anomalous skin effect, vF/ω δ .

(b) Logarithmic plot of the scattering time τ versus frequency ω showing the regions just described. Region C characterizes the transparent regime (after [Cas67]).

E.2 Extreme anomalous regime

The concept of the anomalous skin effect no longer holds if the distance the charge carriers travel per period of the alternating field E(ω) becomes comparable to the mean free path, i.e. the field direction alternates before the electrons have scattered. The absorption falls back to the classical value, and this observation is called the extreme anomalous skin effect. In order to understand this limit, we have to consider a third characteristic length scale of the problem, in addition to the mean free path and the skin depth δ: the distance vF/ω the electrons move during one period of the electromagnetic wave. The classical skin effect only holds if two conditions are met: the skin depth δ is larger than the mean free path , and the frequency is lower than the scattering rate. In this regime, the electrons suffer many collisions during the time they spend in the skin layer and during one period of the electromagnetic wave (Fig. E.2, region A). Thus the region is well described by a local, instantaneous relationship between the current and the total electric field. If the frequency is larger than the plasma frequency, the metal becomes transparent (region C), and in the intermediate spectral range between the scattering rate 1/τ and the plasma frequency ωp the absorption is frequency independent (Eq. (5.1.22)). In this so-called relaxation regime, many periods of the radiation fall between two scattering events; however, the electrons still experience a large number of collisions while they travel within the skin layer. The collisions become less important and the light basically experiences a layer of free electrons responding to the rapidly oscillating electric field (region B).

Similar considerations hold for the anomalous regime. The anomalous skin effect becomes important when the mean free path exceeds the skin depth δ (Fig. E.2, region D). The electrons leave the skin layer before they are scattered, and the collisions are of little importance. If in this case the frequency increases, we do not see any change if ωτ < 1, but a new regime is reached when vF/ω δ . During the time the electron spends in the skin layer, it experiences an increasing number of oscillations of the electric field (region E): δ/vF > 1/ω. The region is called the extreme anomalous skin effect or anomalous reflection. Since the electrons in the skin layer respond to the electric field as essentially free electrons, region E differs only slightly from region B. Most of the collisions happen at the surface. In the ωτ diagram of Fig. E.2b the five regimes are shown and the borders between them indicated [Cas67]. Fig. E.3 illustrates the relationship between the three different length scales.

E.3 The kernel

The non-local conductivity and the effects of a finite mean free path can be elegantly treated with the help of the non-local kernel Kµν . The kernel is a response

Fig. E.3. Schematic representation of the various skin depth regimes in the parameter space given by the three different length scales, the skin depth δ, the mean free path , and the distance the charge carriers travel during a period of the light wave vF/ω.

function which relates the current density J to the vector potential A; in general, it is a second order tensor. This approach is particularly useful for the discussion of impurity effects in superconductors.

In Eq. (4.1.16) we arrived at the Kubo formula for the conductivity as the most general description of the electrodynamic response. Here we want to reformulate this expression by defining a non-local kernel Kµν . We write the position of time dependent current as

=

Jip(r, t) indicates the ith component of the paramagnetic current density introduced in Eq. (4.1.6), ρ0 is the charge density, and refers to the step function

the details are discussed in [Sch83] and similar textbooks. The expectation value of the current density in an external field can be calculated from the fluctuationdissipation theorem by first order perturbation

Since σˆ (q, ω) = − 4iπc2ω K (q, ω), the kernel is directly related to the conductivity.

E.4 Surface impedance of superconductors

In the superconducting case, the kernel was studied intensively by [Abr63]. For the Pippard case λ ξ(0) we obtain for frequencies below the gap ω < 2 /h¯

and for h¯ ω > 2

Electromagnetic energy is absorbed only in the latter case. In order to relate the complex surface impedance Zˆ S to K , two cases have to be distinguished. For diffusive scattering we obtain

In general the ratio of the surface impedance Zˆ S(ω) to the real part of the normal state surface resistance Rn(ω) is considered. In the Pippard case we obtain

At T = 0 the integrals of the kernel can be evaluated by using the complete elliptic integrals E and K