Dressel.Gruner.Electrodynamics of Solids.2003

suppression of the absorption at low frequencies is due to the development of a pseudogap instead of a real gap. There are also arguments [Mot79] which suggest that the electronic states are localized only in the pseudogap region, but not at energies away from the pseudogap where the density of states is large. Measurements of the dc conductivity indeed show a well defined thermal gap; such a transition from localized to delocalized states as a function of frequency, however, is not obvious from the optical data. While there is a more or less well defined energy for the onset of appreciable absorption, called the absorption edge Ec, this edge may or may not correspond to the thermal gap. The subject is further complicated by the fact that a variety of different absorption features are found in different amorphous semiconductors and glasses. Often the absorption displays an exponential behavior above Ec, but often also a power law dependence

with n ranging from 1 to 3 is found. The reason for these different behaviors is not clear and is the subject of current research, summarized by Mott and Davis [Mot79].

References

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[Kit96] C. Kittel, Introduction to Solid State Physics, 7th edition (John Wiley & Sons, New York, 1996)

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Further reading

[Kli95] C.F. Klingshirn, Semiconductor Optics (Springer-Verlag, Berlin, 1995)

[Mit70] S.S. Mitra and S. Nudelman, eds, Far Infrared Properties of Solids (Plenum Press, New York, 1970)

[Mot87] N.F. Mott, Conduction in Non-Crystalline Materials (Oxford University Press, Oxford, 1987)

[Mot90] N.F. Mott, Metal–Insulator Transition, 2nd edition (Taylor & Francis, London, 1990)

[Nud69] S. Nudelman and S.S. Mitra, eds, Optical Properties of Solids (Plenum Press, New York, 1969)

[Par81] G.R. Parkins, W.E. Lawrence, and R.W. Christy, Phys. Rev. B 23, 6408 (1981) [Pei55] R.E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, 1955)

[Phi66] J.C Phillips in: Solid State Physics 18, edited by F. Seitz and D. Thurnbull (Academic Press, New York, 1966)

[Shk84] B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors, Springer Series in Solid-State Sciences 45 (Springer-Verlag, Berlin, 1984)

[Tau66] J. Tauc, ed., The Optical Properties of Solids, Proceedings of the International School of Physics ‘Enrico Fermi’ 34 (Academic Press, New York, 1966)

of the penetration depth, through the measurements of the radio frequency and microwave losses at frequencies below the superconductivity gap, to the evaluation of the single-particle absorption – and thus the gap – by optical studies. Experiments on the superconducting state of simple metals are a, more or less, closed chapter of this field, with the attention being focused on novel properties of the superconducting state found in a variety of new materials, which – in the absence of a better name – are called unconventional or non-BCS superconductors.

14.1.1 BCS superconductors

We first discuss the experimental results on superconductors for which the pairing is s-wave and the energy gap opens along the entire Fermi surface, with the gap anisotropy reflecting merely the subtleties of the band structure. Furthermore the weak coupling approximation applies, which leads to the BCS expressions for the various parameters, such as Eq. (7.1.15) for the ratio of the gap to the transition temperature. Most of the so-called conventional superconductors, i.e. simple metals with low transition temperatures, fall into this category.

We start with the penetration depth, one of the spectacular attributes of the superconducting state. Experiments are too numerous to survey here and are reviewed in several books [Pip62, Tin96, Wal64]. The London penetration depth derived in Section 7.2.1 can be written as

at finite temperatures, where Ns(T ) is the temperature dependent condensate density and mb is the bandmass of electrons; the expectation is that mb is the same as the bandmass determined via the plasma frequency in the normal state. The above expression holds in the limit where local electrodynamics applies and also in the clean limit where the mean free path is significantly larger than the penetration depth λL; in this limit λL = c/ωp at zero temperature. As discussed in Section 7.4, corrections to the above expressions are in order if the assumed conditions are not obeyed; we will return to this point later. One common method of measuring the penetration depth λ(T ) is to monitor the frequency of a resonant structure, part of which contains the material which becomes superconducting. In the majority of cases enclosed cavities are used which operate in the microwave spectral range (see Sections 9.3 and 11.3.3). The resonant frequency is (crudely speaking) proportional to the effective dimensions of the cavity, and this includes the surface layer over which the electromagnetic field penetrates into the material, i.e. the penetration depth. The canonical quantity which is evaluated is the surface reactance XS, related to the penetration depth by Eq. (7.4.24). The other quantity which is

measured is the surface resistance RS, i.e. the loss associated with the absorption of the electromagnetic field within the surface layer; this loss decreases the quality of the resonance. Both components of the surface impedance Zˆ S = RS + iXS can be evaluated, although often only one of the components is accessible in practice.

The temperature dependence of the real and imaginary parts of the surface impedance, measured in niobium at frequencies well below the gap frequency, is displayed in Fig. 14.1a. Both parameters approach their normal state values as the transition temperature Tc = 9 K is reached from below. At sufficiently low temperatures both parameters display an exponential temperature dependence, establishing the existence of a well defined superconducting gap. If the gap opens up along the entire Fermi surface, exp{− / kBT } is the leading term in the temperature dependence of RS(T ) and XS(T ); the correct expressions as derived from the BCS theory are given by Eqs (7.4.23),

and from the relative difference in the penetration depth (Eq. (7.4.25))

gap . A fit of the experimental data to the low temperature part of RS(T ) and XS(T ) gives 2 ≈ 3.7kBTc, suggesting that the material is close to the weak coupling limit, for which 2 / kBTc = 3.53. Often, only the ratio of the superconducting to the normal state impedance is evaluated, and this parameter is compared with the prediction of the BCS theory; the calculations by Mattis and Bardeen [Mat58] presented in Section 7.4.3 provide the theoretical basis. Note that the losses are approximately proportional to the square of the measuring frequency; this is in contrast to the normal state where, according to Eq. (5.1.18), in the Hagen–Rubens regime RS(ω) increases as the square root of the frequency. With simple (but somewhat misleading) arguments, we can understand the above expression for the surface resistance as follows: let us assume that the electrons which form the condensate and the electrons thermally excited across the single-particle gap form two quantum liquids which both respond to the applied electromagnetic field. At low temperatures the number of thermally excited electrons is small, and we also assume that for these electrons the Hagen–Rubens limit applies, i.e. σ1 σ2, the latter subsequently being neglected. For the condensate, on the other hand, we can safely disregard losses (σ1 = 0), and keep only the contribution to σ2, given by

Frequency f (GHz)

Fig. 14.1. (a) Temperature dependence of the surface reactance XS(T ) and surface resistance RS(T ) in superconducting niobium (Tc = 9 K) measured at 6.8 GHz. Rn and Xn refer to the normal state values. The change in surface reactance XS(T ) − XS(T = 0) (left axis) is proportional to the change in penetration depth λ(T ) − λ(T = 0) (right axis). The fit by the Mattis–Bardeen theory (full lines) leads to a superconducting gap of 2 = 3.7kBTc. (b) Frequency dependence of the surface resistance RS(ω) of niobium measured at T = 4.2 K. The dashed line indicates an ω2 frequency dependence, and the full line is calculated by taking a gap anisotropy into account (after [Tur91]).

where we have assumed that the concentration of superconducting carriers Ns is close to its T = 0 value and therefore the imaginary component is only weakly temperature dependent. Inserting this expression into Eq. (2.3.32a) for the surface resistance leads to RS ω3/2. Assuming further that the number of thermally excited electrons determines the conductivity (by assuming that the mobility is independent of temperature), we find that the temperature and frequency dependencies can be absorbed into an expression

The assumption of the two independent quantum fluids is not entirely correct, as we have to take into account mutual screening effects; this leads to a somewhat different frequency dependence, which is then derived on the basis of the correct Mattis–Bardeen expression [Hal71]. The temperature dependence of the surface resistance was first measured on aluminum by Biondi and Garfunkel [Bio59] at frequencies both below and above the gap frequency. A strong increase in RS with increasing frequency was found in the superconducting state. The temperature dependence as well as the frequency dependence observed can be accounted for by the Mattis–Bardeen theory; the experiments conducted at low frequencies and at low temperatures are in good semiquantitative agreement with Eq. (14.1.2a), and as such also provide clear evidence for a superconducting gap. More detailed data on niobium are displayed in Fig. 14.1b and show an RS ω1.8 behavior, close to that predicted by the previous argument. Note that the frequency dependence as given above follows from general arguments about the superconducting state, and is expected to be valid as long as the response of the condensate is inductive and the two-fluid description is approximately correct.

Other important consequences of the BCS theory are the so-called coherence factors introduced in Section 7.3.1. The case 2 coherence factor leads to the enhancement of certain parameters just below the superconducting transition; and the observation that the nuclear spin–lattice relaxation rate is enhanced [Heb57] provided early confirmation of the BCS theory. The real part of the conductivity σ1, which is proportional to the absorption, is also governed by the case 2 coherence factors; this then leads to the enhancement of σ1(T ) just below the superconducting transition [Kle94]. The temperature dependence of this absorption, calculated for frequencies significantly less than /h¯ , is displayed in Fig. 7.2. When evalu-

ated from the measured surface resistance and reactance1 such coherence factors become evident, as shown by the data for niobium, displayed in Fig. 14.2. In the figure the full lines are the expressions (7.4.22) based on the Mattis–Bardeen theory (see Section 7.4.3); the dashed line is calculated by assuming that niobium is a strong coupling superconductor – a notion which will be discussed later.

Another important observation by Biondi and Garfunkel was that for the highest frequencies measured the surface resistance does not approach zero as T → 0, but saturates at finite values, providing evidence that at these frequencies carrier excitations induced by the applied electromagnetic field across the gap are possible. The reflectivity R(ω) for a superconductor below the gap energy 2 approaches 100% as the temperature decreases towards T = 0, and this is shown in Fig. 14.3 for the example of niobium nitrate. Using a bolometric technique the absorptivity A(ω) was directly measured and clearly shows a drop below the gap frequency; this also becomes sharper as the temperature is lowered [Kor91]. The fringes in both data are due to multireflection within the silicon substrate. There is an excellent agreement with the prediction by the Mattis–Bardeen formalism (7.4.20), the consequences of which are shown in Fig. 7.5. The full frequency dependence of the electrodynamic response in the gap region has been mapped out in detail for various superconductors by Tinkham and coworkers [Gin60, Ric60], and in Fig. 14.4 we display the results for lead, conducted at temperatures well below Tc. The data, expressed in terms of the frequency dependent conductivity σ1(ω) and normalized to the (frequency independent) normal state value σn, have been obtained by measuring both the reflectivity from and transmission through thin films. There is a well defined threshold for the onset of absorption which defines the BCS gap; the conductivity smoothly increases with increasing frequencies for ω > 2 /h¯ , again giving evidence for the case 2 coherence factor as the comparison with Fig. 7.2 clearly demonstrates. The frequency for the onset of conductivity leads to a gap 2 /h¯ of approximately 22 cm−1 in broad agreement with weak coupling BCS theory, and the full line follows from the calculations of Mattis and Bardeen – as before, the agreement between theory and experiment is excellent. The data also provide evidence that the gap is well defined and has no significant anisotropy; if this were the case, the average overall orientations for the polycrystalline sample (such as the lead film which was investigated) would yield a gradual onset of absorption. With increasing temperature T < Tc the normal carriers excited across the gap become progressively important, causing an enhanced low frequency response. This is shown in Fig. 14.5, where σ1(ω) and σ2(ω), measured directly on a thin niobium film at various temperatures using a Mach–Zehnder interferometer, are displayed [Pro98]. Similar results have been

1Note that both parameters RS(T ) and XS(T ) have to be measured precisely in order to evaluate the conductivity σ1(T ) using Eqs (2.3.32).