Dressel.Gruner.Electrodynamics of Solids.2003

Fig. 7.1. (a) Temperature dependence of the superconducting penetration depth λ(T ) in comparison with the order parameter (T ). According to Eq. (7.2.10), (λ(T )/λ(0))−2 is proportional to the density of superconducting charge carriers Ns. (b) Temperature dependence of the penetration depth λ(T ) calculated by Eq. (7.4.25).

(retarded) electron–phonon interaction and the cutoff is associated with the energy of the relevant phonons h¯ ωP. In case of density wave states such retardation does not play a role, and the cutoff energy is the bandwidth of the metallic state. As this is usually significantly larger than the phonon energies, the transition temperatures for density waves are, in general, also larger than the superconducting transition temperatures. Another difference lies in the character of the state: if we calculate the electronic density for superconductors, we find that it is constant and independent of position, this is due to the observation that the total momentum of the Cooper pairs is zero – the superconducting gap opens at zero wavevector. In

These differences between the superconducting and density wave ground state lead to responses to the electromagnetic fields which are different for the various ground states.

7.2.1 London equations

We write for the acceleration of the superconducting current density Js in the presence of an external field E

where Ns is the superfluid particle density, which can be taken as equal to the particle density in the normal state N . In addition, using Maxwell’s equation (2.2.7a), this relation can be written as

Flux expulsion, the so-called Meissner effect, is accounted for by assuming that not only the time derivative in the previous equation, but the function in the bracket itself, is zero, × (mc/Nse2)Js + B = 0. With Eq. (2.1.2) the expression reduces to

Thus, the superconducting current density is proportional to the vector potential A instead of being proportional to E as in the case of normal metals. We can also utilize Maxwell’s equation × B = 4cπ Js (neglecting the displacement term and the normal current) to obtain for the magnetic induction B

with the so-called London penetration depth

where ωp is the well known plasma frequency. As this equation describes the spatial variation of B, λL characterizes the exponential decay of the electromagnetic field. As such, λL is the equivalent to the skin depth δ0 we encountered in metals (Eq. (2.3.16)). There are, however, important differences. δ0 is inversely proportional to ω−1/2, and thus it diverges at zero frequency. In contrast, the penetration depth is frequency independent; there is a decay for B, even for dc applied fields. The temperature dependence of the penetration depth follows the

temperature dependence of the condensate density. Ns(T ) can be also calculated using the BCS formalism, and it is displayed in Fig. 7.1. For time-varying fields, Eq. (7.2.7) also gives the 1/ω dependence of the imaginary part of the conductivity as

Through the Kramers–Kronig relation (3.2.11a) the real part of the conductivity is given by

7.2.2 Equation of motion for incommensurate density waves

The energy related to the spatial and temporal fluctuations of the phase of the density waves is described by the Lagrangian density, which also includes the potential energy due to the applied electric field. The equation of motion of the phase condensate is

where m is the mass ascribed to the dynamic response of the density wave condensates. If the interaction between electrons is solely responsible for the formation of density wave states, m is the free-electron mass – or the bandmass in the metallic state, out of which these density wave states develop. This is simply because the kinetic energy of the moving condensate is 12 N mv2, with no additional terms included. This is the case for spin density waves. For charge density waves, however, the translational motion of the condensate leads also to oscillations of the underlying lattice; this happens because of electron–phonon coupling. The motion of the underlying ions also contributes to the overall kinetic energy, and this can be expressed in terms of the increased effective mass m . The appropriate expression is [Gru94]

where λP is the dimensionless electron–phonon coupling constant, and ωP is the relevant phonon frequency, i.e. the frequency of the phonon mode which couples the electrons together with an electron–hole condensate. As the single-particle gap 2 ωP as a rule, the effective mass is significantly larger than the bandmass mb.

The frequency and wavevector dependent conductivity of the collective mode σˆ coll(q, ω) is obtained by a straightforward calculation, and one finds that

In general, non-local effects are not important and the q dependence can be neglected. The real part of the optical conductivity is

and the imaginary part is evaluated from Eq. (7.2.14) as

These are the same expressions as those we derived above for the superconducting case; this becomes evident by inserting the expression for the penetration depth λL into Eqs (7.2.11). Note, however, that because of the (potentially) large effective mass, the total spectral weight associated with the density wave condensate can be small.

7.3Coherence factors and transition probabilities

7.3.1Coherence factors

One of the important early results which followed from the BCS theory was the unusual features of the transition probabilities which result as the consequence of an external perturbation. Let us assume that the external probe which induces transitions between the various quasi-particle states has the form

,k ,σ,σ

In a metal or semiconductor, we will find that the various transitions between the states k and k proceed independently; as a consequence in order to obtain the total transition probability we have to add the squared matrix elements. In superconductors – and also for the other broken symmetry ground states – the situation is different. Using the transformation (7.1.10) we have utilized before, one can show that two transitions, (ak+,σ ak,σ ) and (a−+k,−σ a−k ,−σ ), connect the same quasi-particle states. In the case where σ = σ , for example,

The last two terms in both equations describe the creation of (or destruction of) pairs of quasi-particles, and the process requires an energy which exceeds the gap; here the combinations vkuk − ηukvk appear in the transition probabilities. The first two terms describe the scattering of the quasi-particles, processes which are important for small energies h¯ ω < . For these processes the coefficients which determine the transition probabilities are uk uk + ηvk vk . These coefficients are the so-called coherence factors, with η = +1 constructive and with η = −1 destructive interference between the two processes. Whether η is positive or negative depends upon whether the interaction changes sign by going from k to k .

and the interaction depends on the direction of the momentum. In this case, the transition probabilities add, and η = +1. The coherence factors can be expressed in terms of quasi-particle energies and gap value: after some algebra for scattering processes one obtains the form [Tin96]

When summing over the k values, the second term on the right hand side becomes zero, and the coherence factor can be simply defined as

In the case of density waves a transformation similar to that performed for the superconducting case defines the quasi-particles of the system separated into two categories: the rightand left-going carriers. After some algebra, along the lines performed for the superconducting case, one finds that the case 1 and case 2 coherence factors for density waves are the opposite to those which apply for the superconducting ground state.

7.4 The electrodynamics of the superconducting state

The electrodynamics of superconductors involves different issues and also varies from material to material. The reason for this is that different length scales play important roles, and their relative magnitude determines the nature of the superconducting state, and also the response to electromagnetic fields. The first length scale is the London penetration depth

determined through the plasma frequency ωp by the parameters of the metallic state and derived in Section 7.2.1. The second length scale is the correlation length

describing, crudely speaking, the spatial extension of the Cooper pairs. This length scale can be estimated if we consider the pair wavefunctions to be the superpositions of one-electron states within the energy region around the Fermi level; then the corresponding spread of momenta δp is approximately | | = δ( p2/2m) ≈ vFδp/2, where vF is the Fermi velocity. This corresponds to a spatial range of ξ0 ≈ h¯ /δp. The correct expression, the BCS coherence length, is given by Eq. (7.4.2) at zero temperature. This is the length scale over which the wavefunction can be regarded as rigid and unable to respond to a spatially varying electromagnetic field. This parameter is determined through the gap , by the strength of the coupling parameter. The third length scale is the mean free path of the uncondensed electrons

set by the impurities and lattice imperfections at low temperatures; vF is the Fermi velocity, and τ is the time between two scattering events. For typical metals, the three length scales are of the same order of magnitude, and, depending on the relative magnitude of their ratios, various types of superconductors are observed; these are referred to as the various limits. The local limit is where ξ, λ. More commonly it is referred to as the dirty limit defined by /ξ → 0. In the opposite, so-called clean, limit /ξ → ∞, it is necessary to distinguish the following two cases: the Pippard or anomalous limit, defined by the inequality λ ξ, (type I superconductor); and the London limit for which ξ λ, (type II superconductor).2

We first address the issues related to these length scales and describe the modifications brought about by the short mean free path; these are discussed using

2 See Fig. E.6 and Appendix E.4 for further details.

spectral weight arguments. The relationship between λL and ξ can be cast into the form which is similar to the relationship between the skin depth δ0 and the mean free path in metals; here also we encounter non-local electrodynamics in certain limits. Finally we recall certain expressions for the conductivity and surface impedance as calculated by Mattis and Bardeen [Mat58]; the formalism is valid in various limits, as will be pointed out.

7.4.1 Clean and dirty limit superconductors, and the spectral weight

The relative magnitude of the mean free path with respect to the coherence length has important consequences for the penetration depth, and this can be described using spectral weight arguments. The mean free path is given by ≈ vFτ , whereas the coherence length ξ ≈ vF/ , and consequently in the clean limit 1/τ < and 1/τ > is the so-called dirty limit. In the former case the width of the Drude response in the metallic state, just above Tc, is smaller than the frequency corresponding to the gap 2 /h¯ , and the opposite is true in the dirty limit.

The spectral weight associated with the excitations is conserved by going from the normal to the broken symmetry states. While we have to integrate the real part of the normal state conductivity spectrum σ1n(ω), in the superconducting phase there are two contributions: one from the collective mode of the Cooper pairs σ1coll(ω), and one from the single-particle excitations σ1sp(ω); thus

assuming that all the normal carriers condense. Because of the difference in the coherence factors for the superconducting and density wave ground states, the conservation of the spectral weight has different consequences. The arguments for superconductors were advanced by Tinkham, Glover, and Ferrell [Glo56, Fer58], who noted that the area A which has been removed from the integral upon going through the superconducting transition,

∞

0+

is redistributed to give the spectral weight of the collective mode with σ1(ω = 0) = Aδ{ω}. A comparison between this expression and Eq. (7.2.11b) leads to

connecting the penetration depth λ to the missing spectral weight A. This relationship is expected to hold also at finite temperatures and for various values of the mean free path and coherence length – as long as the arguments leading to the