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Electrodynamics of metallic superconductors.

1. Introduction

1.1. Historical Developments. The history of superconductivity in the 1950s is one of the best examples of how the interplay of experiment and theory advances science towards a profound understanding. After having moved to Urbana in 1951 and even more intense after having received the Nobel Prize for the development of the transistor in 1956, John Bardeen was pressing to solve the longstanding problem of superconductivity. During these years he was very well aware of the experimental facts and recent developments but also triggered experimental investigations and pushed certain measurements.

The presence of an energy gap in the density of states of a superconductor (and thus in the excitation spectrum) can be inferred from the temperature dependence of the specific heat that had been measured already in Heike Kamerlingh Onnes' laboratory in Leiden [1]; however, it was Bardeen who seriously discussed this idea in more detail in 1955 [2,3] based on considerations proposed by Welker [4] years earlier. He assumed "that in the superconducting state a finite energy gap [DELTA] [approximately equal to] [k.sub.B] [T.sub.c] is required to excite electrons from the surface of the Fermi sea." Cooper developed this concept further when he introduced the electron pairs in the following year [5].

In the seminal BCS paper of 1957 [6], Bardeen, Cooper and Schrieffer finally derived the fundamental ratio 2[DELTA]/[k.sub.B][T.sub.c] = 3.5 and the temperature dependence of the energy gap. In their calculation of the matrix elements, they explicitly refer to the ongoing NMR relaxation-rate experiments by Hebel and Slichter [7, 8], ultrasound-attenuation measurements by Morse and Bohm [9], and microwave-absorption experiments by the Tinkham group [10-12].

Concerning the high-frequency absorption, in an article submitted 1955 [13] Buckingham suggested that (for consistency reasons the notation is transformed):

"a gap of width 2[DELTA](0) in the electron energy spectrum in the ground state, its width, 2[DELTA], at finite temperatures will decrease, eventually vanishing at the transition temperature, Tc. The normal state is regarded as one in which the energy spectrum is without a gap and is independent of temperature, the system representing the superconducting state becoming identical with the normal one for T > [T.sub.c]. The electromagnetic properties of such a system will depend strongly on the size of the gap, but at high frequencies and near the transition temperature, when [??][omega])/2[DELTA] is large, will approach those of the normal state (without gap)."

Looking at the experimental results of Blevins et al. who measured the surface resistance of tin in the millimeter-wave range [14], he suggested some scaling of frequency and transition temperature close to [T.sub.c]. This was further developed by Tinkham [11], based on his far-infrared transmission measurements on thin lead films [10] where an anomalously high transmission was observed (cf. Figure 1(b)). He suggested two contributions (superconducting and normal electrons) becoming relevant in the limiting cases of low and high frequencies:

"We consider a model of a superconductor at absolute zero in which a gap of width [??][[omega].sub.g] [approximately equal to] 3[k.sub.B][T.sub.c] appears in the spectrum of one-electron energy states. [...] The lossy conductivity [[sigma].sub.i] will be zero for photon energies hw < [??][[omega].sub.g]. For w > [[omega].sub.g], [[sigma].sub.1] will rise gradually with [omega] as an increasing fraction of the states originally within [??][omega] of the occupied states below the Fermi level are still available for excitation. If the gap merely excluded the states within [??][[omega].sub.g] leaving all else unchanged, this simple picture would suggest a rise as [[sigma].sub.1] = [[sigma].sub.n] (1-[[omega].sub.g]/[omega]). The experimental rise seems to be faster, cutting off at least as fast as (1 - [[omega]sup.2.sub.g]/[[omega].sup.2]).This might correspond to the states displaced from the gap just "piling up" on either side, so that the change averages to zero more rapidly for [omega][much greater than] [[omega].sub.g]."

These ideas are further developed and extended in later papers [12, 15], where the size of the superconducting energy gap and the shape [[sigma].sub.1]([omega]) at [[omega].sub.g] were analyzed using data of different materials [16].

The situation was more or less settled in 1958, when the Westinghouse group of Biondi et al. summarized the situation on the superconducting energy gap in a review [17]. Previously they had published several thorough microwave and millimeter-wave results on superconducting aluminum and tin [18-20], and several nice papers followed [21-23]. At the same time Tinkham gave an overview on the work of his group [24]. While Biondi and Garfunkel always measured the temperature dependence for certain frequencies, Tinkham and his collaborators focussed at the spectral properties at fixed temperatures. This allowed him to make the sum-rule argument introduced in the previous quote [11], which was finalized by Ferrell et al. in 1959 [25, 26]. In Figure 1 the frequency dependence of the electrodynamical properties of a superconductor at T = 0 is sketched. Upon decreasing the frequency the real part of the conductivity [[sigma].sub.1]([omega])/[[sigma].sub.n] gradually vanishes at the gap frequency [[omega].sub.g] = 2[DELTA]/[??], while the imaginary part [[sigma].sub.2]([omega])[[omega].sub.n] diverges as [omega] [right arrow] 0. The maximum in the relative transmission [Tr.sub.s]/[Tr.sub.n] is observed right at the gap (with values well above the unity) before it goes to zero for [omega] = 0. There is also a maximum of the absorption above the superconducting gap, but no absorption takes place below the gap 2[DELTA]. It poses a real challenge to measure the reflectivity of a metallic superconductor since it approaches unity at [[omega].sub.g], but in general it is well above 99% already in the normal state (Figure 1(c)).

The problem remaining was the determination of the superconducting conductivity, that is, the actual frequency and temperature dependence of [[sigma].sub.1]([omega]) and [[sigma].sub.2] ([omega]) individually without the assumption of any model or Kramers-Kronig relation. Figure 2 displays temperature and frequency dependent conductivity ollan as it is calculated from Mattis Bardeen theory. ollan features a minimum at [??][omega]/2[DELTA](T) = 1 which shifts to lower frequencies and is smeared out as T approaches [T.sub.c] from below. Also seen is the enormous enhancement of the conductivity [[sigma].sub.1]([omega])/[[sigma].sub.n] for low frequencies observed right below the superconducting gap: it became known as the coherence peak Despite continuous efforts over the years [30-32], only in 1968 Palmer and Tinkham reached this goal of measuring the real and imaginary parts of the electrodynamic response independently [33]; simultaneous measurements of the far-infrared transmission through and reflection off thin films of lead made it possible to calculate the conductivity unambiguously. The status is summarized in several reviews [34, 35]. The precise measurement of the temperature-dependent conductivity right below the superconducting transition, however, was achieved only a quarter of a century later in the group of Graner [36, 37]. Previous efforts by the Cambridge group of Pippard and others were less convincing [38-40]. The so-called coherence peak known from the NMR experiments of Hebel and Slichter [7, 8] is also seen in the microwave conductivity of lead, niobium, and eventually even in alumimum [29] where for the first time the complete frequency evolution of the coherence peak was measured; exactly fifty years after it was calculated. Still ultrasound absorption remains a desideratum in the exploration of the coherence effects: a peak is predicted above 2[DELTA], but no frequency-dependent ultrasound absorption measurements on superconductors have been carried out in the vicinity of the energy gap.

Even today, Tinkham's book [41-43] gives the best introduction to the electrodymical properties of conventional superconductors. More detailed descriptions can be found by Rickayzen [44] and Parks [45] or more recent textbooks [46].

1.2. Recent Developments. Having understood the bulk properties of metallic superconductors fairly well, there are still many open questions when going to the vortex lattice, very thin films, nanostructures, inhomogeneous materials, granular superconductors, highly disordered films and so forth, which are subject of today's research efforts. Thin films of metallic superconductors are of both technological and academic interests.

As we will see, the properties of interacting electron systems are governed by the electronic density of states [47], which changes upon reducing spatial dimensions. Thus, dimensional reduction from three dimensions to two dimensions (3D [right arrow] 2D) may affect optical, electronic, and thermodynamic properties of the system. In superconductivity, studying quasi-two-dimensional thin-film systems turns out to be a particularly fruitful field and leads to remarkable findings such as insulators featuring a superconducting gap [48], pseudogap in conventional s-wave superconductors [49], or extremely strong-coupling superconductivity in Kondo lattices [50]. NbN has recently gained attention as a model system for the superconductor-insulator transition; a topic intensively investigated in thin films of Bi, InO, or TiN: tunneling studies [51] reveal a gapped density of states slightly above [T.sub.c] in highly disordered films as it has been found for TiN films [49]. This peculiar state resembles the well established pseudogap in high-[T.sub.c] cuprate superconductors [52, 53]. In case of NbN, however, [T.sub.c] is about one order of magnitude higher compared to TiN. Thus, measurements on NbN in the vicinity of the superconductor-insulator transition (SIT) are much less demanding [54, 55]. Consequently, results obtained for NbN might serve as a key to understand the still puzzling pseudogap state in high-[T.sub.c] cuprates.

Besides pure academic interest, thin-film superconductors play a key role in many state-of-the-art applications such as superconducting quantum interference devices (SQUID), thermal and electrical switches using Josephson junctions, or microwave resonators [56]. Contrary to nonmetallic systems such as Mg[B.sub.2] or cuprates, thin and ultrathin films of NbN and TaN can be produced of very high quality, for example, by reactive dc-magneton sputtering [57, 58]. They are of particular interest due to their high critical temperature that makes them suited for novel approaches to build single-photon detectors, for example [57, 59-63].

2. Theory

2.1. Mattis-Bardeen Theory. Basedonthe BCStheory[6], the description of the electrodynamic properties of a superconductor was given by Mattis and Bardeen in their seminal paper [27]. Miller applied these considerations on the surface impedance [64]. In parallel Abrikosov and coworkers [65, 66] also described the electromagnetic absorption of a superconductor, including the coherence factors and the nonlocal relationship between vector potential and induced currents. The celebrated Mattis-Bardeen equations read



where for [??][omega] > 2[DELTA] the lower limit of the integral in (2) becomes -[DELTA].Here f(E,T) = [(exp{E/[K.sub.B]T} -l).sup.-1] is the usual Fermi-Dirac distribution function. The expression of [[sigma].sub.1]/[[sigma].sub.n] describes the response of the normal carriers, that is, the dissipative part. The first term of (1) represents the effects of thermally excited quasiparticles, which vanishes for T = 0. The second term mainly accounts for the contribution of photon-excited quasiparticles; since it requires the breaking of a Cooper pair, it is basically zero for [??][omega] < 2[DELTA](T).However, there is a small contribution possible of quasiparticles thermally excited across the gap for T [??] [T.sub.c].

The sketch in Figure 3 illustrates the electronic density of states for a normal metal (panel a), a superconductor at finite temperatures (panel b) and the superconductor at T = 0 (panel c). Single-particle excitations only take place from occupied states (dark colors) to empty states (light colors). In the case of a metal (T > [T.sub.c]) this is possible for any finite photon energy few leading to a Drude response limited only by the scattering rate 1/[tau]. When the superconducting gap opens in the density of states for T < [T.sub.c], the states from the gap region pile up above and below [+ or -] [DELTA](T) making a large number of excitations possible. We have to distinguish excitations across the gap (solid arrows) for [??][omega] > 2[DELTA](T) from those above (and correspondingly below) the gap between occupied and unoccupied states due to the finite-temperature Fermi distribution f(E,T).The latter ones have no lower-energy limit and dominate the low frequency response since the joint density of states is largest for [??][omega] [right arrow] 0. These thermally activated contributions to the optical conductivity [first term in (1)] are shown by red dashed lines in Figure 4. They are actually observed as the coherence peak in the temperature dependent conductivity right below [T.sub.c] (see Section 3.1 below). These quasiparticle excitations are limited by the occupation given by the Fermi distribution. The across-gap excitations, on the other hand, always have a clear-cut lower-frequency limit [[omega].sub.g] determined by the gap 2[DELTA]/[??]; this is the energy required to break a Cooper pair. In Figure 4 these excitations (second term in (1)) are plotted as blue dashed lines. Due to the case-2 coherence factor describing the quasiparticle excitations in a superconductor, the conductivity only smoothly increases for [omega] > 2[DELTA]/[??], despite the divergency in the density of states right above and below the gap [46]. For T = 0 these are the only quasiparticle contributions to [[sigma].sub.1]([omega]).The superconducting condensate is observed as a [delta]-peak at [omega] = 0 and dominates the imaginary part [[sigma].sub.2]([omega]).A generalized Mattis-Bardeen theory which accounts for quasiparticle scattering is derived by Zimmermann et al. [28]. The inset of Figure 4 displays how a finite scattering rate modifies [[sigma].sub.1]([omega]) and [[sigma].sub.2]([omega]).

An interesting aspect was elucidated by Chang and Scalapino [67-69], who found an enhancement of the superconducting energy gap and a change in the quasiparticle density of states upon microwave irradiation, in addition to a redistribution of quasiparticles by the microwaves. Eventually it extends to the field of nonequilibrium superconductivity that has been a widely studied field for half a century [70,71].

Already from the London equations, a relation between the imaginary part of the conductivity and the London penetration depth is obtained [46]:

[[sigma].sub.2]([omega]) = N[e.sup.2].m[omega] = [c.sup.2]/4[pi][[lambda].sup.2.sub.L][omega], (3)

where the London penetration [[lambda].sub.L] = [([mc.sup.2]/4[pi][N.sub.s][e.sup.2]).sup.1/2] is determined by the superconducting charge carrier mass and density, m and [N.sub.S]. Through the Kramers-Kronig relation the real part of the conductivity is then given by

[[sigma].sub.1]([omega]) = [pi]/2] [N.sub.s][e.sub.2]/m [delta] {[omega] = 0} = [c.sup.2]/ 8[[lambda].sup.2.sub.L] [delta] {[omega] = 0}, (4)

where [delta][omega] = 0 denote a delta peak centered at [omega] = 0.

Figures 5 and 6 summarize the frequency and temperature dependence of [[sigma].sub.1] and [[sigma].sub.2] as derived from (1) and (2) assuming the finite scattering effects [pi][xi](0)/l = 10 discussed in [28, 73-77]. Although the density of states diverges at [+ or -] [DELTA], the conductivity [[sigma].sub.1]([omega]) does not show a divergency but a smooth increase which follows approximately the dependence [[sigma].sub.1]([omega])/[[sigma].sub.n] [varies] 1 -[([??][omega]/[K.sub.B][T.sub.c]).sup.-1.65]. The temperature dependent conductivity [[sigma].sub.1](T) shows a peak just below [T.sub.c] at a low frequency; the maximum occurs around 0.8[T.sub.c] and slightly shifts up with frequency. It is called the coherence peak because it does not just reflect the enhanced density of states but is also an expression of the quantum mechanical factor relevant for these excitations. The so-called coherence factor F([E.sub.k], [E.sub.k']) describes the scattering of a quasiparticle from a state k with energy [E.sub.k] to a state k' = k + q with energy [E.sub.k'] = [E.sub.k] + [??][omega] upon absorption of a photon with energy [??][omega] and momentum q. If summed over all k values, it reads [41-43, 46]

F([DELTA],E, E') = 1/2 (1 + [[DELTA].sup.2]/EE'). (5)

Only for energies below the gap 2[DELTA],this factor is appreciable: F [approximately equal to] 1 for [??][omega] [much less than] 2[DELTA].For [??][omega] [greater than or equal to] 2[DELTA], the coherence factors are reversed, and F vanishes in the present case. For large energies, the coherence effects become negligible since E, E' [much greater than] 2[DELTA] and F [approximately equal to] 1/2. Hence the coherence peak is seen as a maximum in [[sigma].sub.1](T) in the low-frequency limit; it becomes smaller with increasing frequency and shifts to higher temperatures. The height of the peak has the following frequency dependence:

[([[sigma].sub.1]/[[sigma].sub.n]).sub.max] ~ log {2[DELTA](0)/[??][omega]}. (6)

The peak disappears completely for [??][omega] [greater than or equal to] [DELTA]/2 (well below 2[DELTA]). At T = 0 and [omega] < 2[DELTA]/[??] the complex part of the conductivity [[sigma].sub.2]/[[sigma].sub.n] describes the response of the Cooper pairs and is related to the gap parameter through the expression


2.2. Scattering Effects, Length Scales, and Sum Rule. The Equations (1) and (2) were derived by Mattis and Bardeen [27] assuming the infinite scattering time [tau] [right arrow] [infinity].In 1967 Nam showed [76, 77], however, that these equations are more general and also valid for [tau] [right arrow] O. He also extended the application towards the strong coupling regime: 2[DELTA]/[K.sub.B][T.sub.c] > 3.53. Nevertheless, in order to describe the electrodynamics of superconductors, different length scales have to be considered as their relative magnitude determines the nature of the super conducting state and also the response to electromagnetic fields. Te first length scale is the London penetration depth

[lambda] = [(m[c.sup.2]/4[pi]N[e.sup.2]).sup.1/2] = c/[[omega].sub.p] (8)

determined through the plasma frequency [[omega].sub.p] = [(4[pi]N[e.sup.2]/m).sup.1/2] by the parameters of the metallic state, that is, the density of charge carriers, the carrier mass, and the electronic charge -e. The second length scale is the correlation length

[[xi].sub.0] = [??][v.sub.F]/[pi][DELTA] (9)

describing, crudely speaking, the spatial extension of the Cooper pairs. This is the length scale over which the wave function can be regarded as rigid and cannot respond to a spatially varying electromagnetic field. This parameter is determined through the energy gap [DELTA], which is linked to critical temperature by the coupling parameter [alpha] = 2[DELTA]/[k.sub.B][T.sub.c]. The third length scale is the mean free path of the uncondensed electrons

l = [v.sub.F][tau] (10)

set by the impurities and lattice imperfections at low temperatures; [v.sub.F] is the Fermi velocity; [tau] is the time between two scattering events. For typical metals in the superconducting state, the three length scales are of the same order of magnitude, and, depending on the relative magnitude various types of superconductors are observed, these are referred to as the various limits. The local limit is where l [much less than] [xi], [lambda]; more commonly it is referred to as the dirty limit defined by l/[xi] [right arrow] 0. In the opposite, so-called clean limit l/[xi] [right arrow] [infinity], it is necessary to distinguish the following two cases: the Pippard or anomalous limit, defined by the inequality A < I (type I superconductor), and the London limit for which [xi] [much less than] [lambda], l (type II superconductor). Figure 7 summarizes the relation between these different length scales and the respective limiting cases.

The relative magnitude of the mean free path with respect to the coherence length has important consequences on the penetration depth, and this can be described using spectral weight arguments. The mean free path is given by 1 [approximately equal to] [v.sub.F][tau], while for the coherence length we get [xi] [approximately equal to] [v.sub.F]/[DELTA] and consequently in the clean limit 1/[tau] < [DELTA]/[??], while 1/[tau] > [DELTA]/[??] is the so-called dirty limit. In the former case the width of the Drude response in the metallic state, just above [T.sub.c], is smaller than the frequency corresponding to the gap 2[DELTA]/[??], 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 [[sigma].sup.n.sub.1]([omega]),in the superconducting phase there are two contributions: one from the condensate of the Cooper pairs [sigma].sup.coll.sub.1]([omega]) and one from the single-particle excitations [[sigma].sup.sp.sub.1([omega]); thus


assuming that all the normal carriers condense. The arguments for superconductors were advanced by Ferrell et al. [25, 26] who noted that the area which has been removed from the integral upon going through the superconducting transition

[[integral].sup.[infinity].sub.0+] [[[sigma].sup.n.sub.1] - [[sigma].sup.s.sub.1]]d[omega] = A (12)

is redistributed to give the spectral weight of the collective mode with [[sigma].sub.1]([omega] = 0) = A[delta]{[omega]}.By comparison of the following expression with the Cooper pair response derived in (4):

[[[sigma].sub.1]([omega]), = [c.sup.2]/8[[lambda].sup.2.sub.L] [delta] {[omega] = 0}, (13)

we see that the missing spectral weight is connected to the penetration depth [lambda] by

[lambda] = c/[square root of (8A)] or A = [c.sup.2]/8[lambda]. (14)

This relation is expected to hold also at finite temperatures and also for various values of the mean free path and coherence length. In the limit of long relaxation time, that is, for [??]/[tau] [much less than] [DELTA], all the Drude spectral weight of the normal carriers collapses in the collective mode, giving

[[sigma].sub.1] ([omega] = 0) = [pi][N.sub.s][e.sup.2]/2m] [delta] {[omega] = 0} (15)

and through (4) leading to the London penetration depth [lambda]L. With increasing 1/[tau], moving towards the dirty limit, the spectral weight is progressively reduced, resulting in an increase of the penetration depth. In the limit 1/[tau] [much greater than] 2[DELTA]/[??], the missing spectral weight is approximately given by A [approximately equal to] [[sigma].sub.1](2[DELTA]/[??]) = 2N[e.sup.2][tau][DELTA]/[??]m which can be written as

A = [c.sup.2]/2[pi]] [tau][DELTA]/[??]. (16)

Then the relation between the area A and the penetration depth [lambda](l)--which now is mean free path dependent as 1 = [v.sub.F][tau]--is given by [[lambda].sup.2](l) = [[lambda].sup.2.sub.L][??][pi]/(4[tau][DELTA]), which leads to the approximate expression

[lambda](l) [approximately equal to] [[lambda].sub.L][square root of ([[xi].sub.0]/l])]. (17)

3. Experimental Results

3.1. Coherence Peak. In order to determine the temperature dependence of the complex conductivity at frequencies well below the superconducting gap, Klein et al. [37] have performed precise measurements of the complex surface impedance of Nb using cavity perturbation methods [78-80]. A bulk Nb sample of 99.9% purity was used to replace the endplate of a 60 GHz cylindrical copper cavity. An abrupt drop of the bandwidth was observed upon cooling through the 9.3 K transition temperature. In Figure 8(a) the temperature dependence of the surface resistance [R.sub.s] and surface reactance [X.sub.s] normalized to the normal state value (defined as the value when the temperature of the sample is slightly above [T.sub.c], i.e., ~10 K): [R.sub.n] = (0.0538[OMEGA])/[Z.sub.0](where [Z.sub.0] is the free space impedance),corresponding to [[sigma].sub.n](T = 10 K) = 0.85 x [10.sup.6][([OMEGA]cm).sup.-1]. Below [T.sub.c], the surface resistance [R.sub.s](T) drops rapidly with decreasing temperature, as shown in the logarithmic plot of Figure 8(a). The length scale for the surface reactance is determined by the skin depth in the normal state [delta] = 2270 [Angstrom]. [X.sub.s](T) saturates at low temperatures, yielding a penetration depth [lambda](0) = 440 [Angstrom]. Using the measured values of [R.sub.s] and [X.sub.s], the temperature dependence of the complex conductivity was evaluated and displayed in Figure 8(b). The coherence peak can be seen as a wide peak below [T.sub.c]. The solid curves represent the Mattis-Bardeen results [(1) and (2)], while the dashed line corresponds to the Eliashberg theory for strong coupling [37].

Only recently, broadband microwave measurements on Al films succeeded to probe the coherence peak of a superconductor (as shown in Figure 2) both as a function of temperature and frequency. Steinberg et al. [29] employed a broadband microwave Corbino spectrometer [81] where basically a coaxial microwave line is terminated by the sample, in this case, a 550 [Angstrom] aluminum film on a sapphire substrate. Using a vector network analyzer, the complex reflection coefficient is measured in the spectral range from 45 MHz to 60 GHz, allowing the calculation of the real and imaginary parts of the conductivity after also the calibration measurements have been performed with high precision. The method is most sensitive when the film impedance is close to the wave guide impedance of 50[OMEGA] .Forthickerorthinner films different geometries are possible [82]. The specimen is cooled down to 1.1 K and even lower temperatures are available [83]. Figure 9(a) shows data (taken at different frequencies on a film with thickness 500 [Angstrom] and aroom temperature resistivity of 8.7 x [10.sup.-5] [OMEGA]cm) of the temperature dependence of the real part of the conductivity. [[sigma].sub.1] is governed by thermally excited charge carriers above and below the gap; their density of states diverges at the gap edge as depicted in Figure 3. Right below [T.sub.c], [DELTA](T) is small: both the thermal energy and the photon energy h" are sufficient to break up Cooper pairs. The imaginary part [[sigma].sub.2]([omega])/[[sigma].sub.n] displayed in the lower frame basically describes the temperature dependence of energy gap, in full agreement with (7).

In accord with theory, the coherence peak is seen as a maximum in [[sigma].sub.1](T) at approximately 0.8[T.sub.C] in the low frequency limit; it becomes smaller with increasing frequency and shifts to higher temperatures in according to (6); this is clearly observed in Figure 9(a). Above 28 GHz [corresponding to 0.2 x [DELTA](0)] the peak is completely suppressed, and for higher frequencies [[sigma].sub.1](T) monotonously drops below [T.sub.c].

3.2. Energy Gap. In order to explore the energy gap in a superconductor the most straight-forward experiment is a transmission measurement through metallic films in the relevant frequency range around [[omega].sub.g], and to vary the temperature from above to below [T.sub.c], as first has been done by Glover III and Tinkham on lead films in 1956 [10]. In Figure 10(a) the transmission through a NbN film at different temperatures as indicated normalized to the transmission in the metallic state just above [T.sub.c] = 13.5 K[84]. With decreasing temperature a maximum in T[r.sub.s]/T[r.sub.n] is observed that becomes more pronounced and shifts to higher frequencies. As sketched in Figure 1(b) it marks the superconducting energy gap 2[DELTA]; in this case 2[DELTA]/hc = 35 [cm.sup.-1].

The effect of the scattering rate on the transmission through superconducting NbN films has been explored experimentally [86] and analyzed on the basis of Leplae's theory [73] that describes strong-coupling materials. By reducing the film thickness from 100 to 20 nm, the critical temperature [T.sub.c] is lowered from 17 to 14 K, but, more importantly, the energy gap 2[DELTA](0)/hc is reduced from 52[cm.sup.-1] to 39[cm.sup.-1]. Also the transmission maximum becomes significantly lower. Recently, it was shown [87, 88] that when the superconducting state is suppressed by a magnetic field, this transmission maximum broadens and shifts to lower temperatures. This is in agreement with the fact that the maximum in transmission corresponds to the energy gap. Using the transmission of 13.5[cm.sup.-1] gas laser the transmission through a 80 nm NbN film on a Si substrate was measured as a function of temperature for different external fields (Figure 11). With higher frequencies, the transmission peak first becomes more pronounced and shifts to lower temperatures; however, it eventually vanishes at [[omega].sub.g] [approximately equal to] 2[DELTA]/[??]. When a magnetic field is applied parallel to the propagation direction (Faraday geometry), the electric field vector is perpendicular to the external magnetic field. For the Voigt geometry, the external field is chosen parallel to the electric field vector. The different behavior reflects the vortex dynamics in this constraint geometry determined by the thin superconducting film. The high-frequency response of type-II superconductors in the mixed state was extensively studied by Coffey and Clem [89, 90].

The absorption of electromagnetic radiation by a superconductor was directly investigated by measuring the temperature increase due to the deposited energy (bolometric absorption). A sputtered thin film of NbN on a silicon substrate (0.39 mm) was mounted at the bottom of a highly reflecting copper light cone, supported by four strands of 25 [micro]m diameter gold wire of about 1.5 mm long. This gives the bolometer a thermal response time constant of about 30 ms at 10.8 K. The temperature is measured by a small doped silicon thermometer glued to the back of the silicon substrate. The resistance of this thermometer is exponentially decreasing with increasing temperature [85]. In Figure 10(b) the relative power absorption spectra are plotted for a 250 nm NbN superconducting film ([T.sub.c] = 13.3 K) on a 0.39 mm silicon substrate. The curves are taken at 5.3 K and 10.8 K and each normalized to a normal state spectrum taken at T = 14 K. With decreasing frequency, the absorption increases and exhibits a maximum around the superconducting energy gap, before it rapidly drops to zero; the behavior is more pronounced for lower temperatures. On the same thin film, reflection measurements have been performed using Fourier transform spectrometer with a resolution of 0.5 [cm.sup.-1]. Figure 10(c) shows the reflectivity of the NbN film on Si substrate above and below [T.sub.c]. The reflectivity in the normal state is relatively low (due to the particular film thickness), making the superconductivity behavior very evident. While in the normal state at 15.5 K the reflectivity levels off around R [approximately equal to] 0.93, a pronounced dip is observed in the superconducting state before the reflectivity rapidly rises to 1.0 [+ or -] 0.01 for frequencies below 40 [cm.sup.-1], clearly indicating the superconducting gap frequency. In any case the high-resolution reflection spectra exhibit pronounced interference fringes due to multi reflection within the Si substrate. In particular, one can see in the inset that the phase of the fringes at 5.3 K changes with respect to those at 15.5 K near 46 [cm.sup.-1] .Above 46 [cm.sup.-1] the fringes are in phase, while below they are 180[degrees] out of phase. From the lower panel (Figure 10(d)) we see that this is the frequency where the real and imaginary parts of the conductivities are equal at 5.3 K. The metallic reflection turns into a dielectric reflection, governed by the out-of-phase response of the Cooper pairs, leading to a sign change in the reflection coefficient. Using Fresnel's equations in order to describe the complete system of film and substrate, the Fabry-Perot pattern allows the calculation of the real and imaginary parts of the conductivity as displayed in Figure 10(c).No Kramers-Kronig transformation is used here, and thus no extrapolation or other model is required. The results agree very well with the theoretical predictions by Mattis and Bardeen [27], except slightly above [[omega].sub.g]. The superconducting gap is larger than expected from weak-coupling mean-field theory: 2[DELTA](0)/[K.sub.B][T.sub.c] = 3.9; this is in agreement with the strong coupling nature of the Nb compounds.

While the previous experiments have been performed utilizing Fourier-transform spectrometers, Gorshunov et al. employed a THz spectrometer on the basis of tunable coherent monochromatic backward wave oscillators [93-95] to measure 115 [Angstrom] thin NbN films on sapphire substrate [91, 92]. They demonstrated that for frequencies below 2[DELTA] the error bars are too large to draw conclusions about the superconducting state and in-gap states, for instance. However, if the substrate is covered with a superconducting film on both sides (double-sided film), the Fabry-Perot resonances get much sharper (increase in quality factor) and the conductivity can be evaluated with a much larger precision, allowing even to observe the coherence peak right below the transition temperature. The raw data if the THz transmission are plotted in Figures 12(a) and 12(b) for single and double sided films; the corresponding conductivity spectra are plotted in panel (c) for two different frequencies close to and well below the superconducting gap 2[DELTA].

3.3. Complex Conductivity. Very similar results have been achieved by Nuss et al. [96] using THz time-domain spectroscopy. The amplitude transmitted through a thin Nb film ([T.sub.c] = 6.25 K) is plotted in Figure 13 as a function of time. Both the amplitude and phase of the transients changeas the film becomes superconducting. In accord with the observations of Glover III and Tinkham [10,12] and others, the transmission in the superconducting state is higher than in the normal state. Since time-domain spectroscopy actually probes the electric field intensity and the shift in time, the real and imaginary parts of the conductivity can be calculated directly as plotted in Figure 13(b). The agreement with the BCS theory [27] shown by the dashed line is very good.

Alternatively a Mach-Zehnder interferometer allows one to measure both the change in transmission and phase shifts when the electromagnetic radiation passes through a metallic and superconducting film. This method was intensively explored by Gorshunov and collaborators [93-95]. Figure 14 displays the results of a transmission experiment [97] performed on a thin niobium film (thickness 150 [Angstrom]) on a 0.45 mm thick sapphire substrate using a Mach-Zehnder interferometer. The substrate acts as a Fabry-Perot resonator due to multi reflection leading to pronounced interference fringes in the spectra; this effect enhances the interaction of the radiation with the film considerably and thus improves the sensitivity and accuracy. In Figures 14(a) and 14(b) the transmission T[r.sub.F] and relative phase shifts [phi]v, respectively, through this composite sample are shown as a function of frequency. As the temperature decreases below the superconducting transition [T.sub.c] = 8.3 K, the transmitted power and phase shifts are modified significantly. Since the properties of the dielectric substrate do not vary in this range of frequency and temperature, the changes observed are due to the electrodynamic properties of the superconductor [97]. When cooling from T = 9 to 6 K, the overall transmission increases for frequencies above 15 [cm.sup.-1] while it drops for lower frequencies. The inset illustrates this behavior nicely. This corresponds to the theoretical predictions (Figure 1(b)) and previous observations. The maximum in the transmission corresponds to the superconducting energy gap 2[DELTA]. Similar changes are observed in the relative phase shift (Figure 14(b)), where a strong decrease is found for [omega] < [[omega].sub.g].

The energy gap is evaluated to 2[DELTA](0)//hc = 24 [cm.sup.-1] .The data can be used to directly calculate the real and imaginary parts of the complex conductivity without using any model or assumption. In Figure 15 the real and imaginary parts of the conductivity of a Nb film are plotted as a function of frequency covering an extremely large range. Since the data are not normalized, the Druderoll-off becomes obvious above 100 [cm.sup.-1], placing the material into the dirty limit. In all cases it seems to be difficult to actually measure the zero conductivity well below the gap for very low temperatures. In many cases some residual absorption seems to be present which can either be intrinsic due to some in-gap absorption, or due to inhomogeneities, impurities, surface effects, and so forth.

Thin films of NbN and TaN are perfectly suited for superconducting nanowire single-photon detectors [57, 59] which are built by thin and narrow superconducting stripes typically in a meander-like geometry. The device is driven with a dc bias currents lightly below the critical current value. Absorption of a photon locally reduces the critical current below the applied dc bias current; consequently the superconducting state breaks down locally: this generates a normal-conducting spot (hot spot) and increases the resistance. The recovery time is in the picosecond range [63]. Since their performance in the far-infrared strongly depends on the optical properties of the device in this spectral range, the energy gap 2[DELTA], and the development of [[sigma].sub.2]([omega]),the information of THz and infrared investigations are highly desired. In particular it is important to study the influence of substrate, film thickness, processing parameters, structuring, and so forth. In Figure 16 the frequency dependence of the real and imaginary parts of the conductivity [[sigma].sub.1]([omega]) and [[sigma].sub.2]([omega]) is plotted for selected temperatures [72, 98]. The NbN film is deposited on a 10 mm x 10 mm R-plane-cut birefringent sapphire substrate using reactive dc-magnetron sputtering, which is a well-established and powerful way of film deposition with high demands concerning the performance of single-photon detectors [57, 58]. Typical films have a thickness of d = 39 to 47 [Angstrom]. Since the thickness d is similar to the superconducting coherence length [xi] and much smaller than the penetration depth [lambda], the films are considered ultrathin, that is, quasi-two dimensional, from the superconducting point of view. However, the electron mean free path l is about one order of magnitude smaller than d, which justifies the 3D expressions used in the analysis and places the material in the dirty limit (Figure 7). Depending on the deposition current [I.sub.d] while sputtering, the critical temperature is lowered in the case of NbN from 12.2 K ([I.sub.d] = 100 mA) to 8.5 K (200 mA) [58, 59, 72, 98, 99]. The degree of disorder can be described by the Ioffe-Regel parameter [K.sub.Fl], (with [K.sub.F] the Fermi wave vector and l the mean free path) ranging from 4.5 to 5.5; that is, in the present case these films are still weakly disordered.

Figure 17(a) displays the temperature dependent energy gap 2[DELTA](T) of the sample with lowest disorder and highest [T.sub.c]. Filled and open symbols refer to different main axes of the substrate and coincide within the range of error. Data for each main axis is fitted separately with the BCS expression. For 0.3 [T.sub.c] < T < 0.8[T.sub.c] the measured energy gap is slightly above the BCS curve and for 0.8[T.sub.c] < T < [T.sub.c] below. Upon decreasing [T.sub.c], the energy gap (at 4.2 K) is also reduced from 4.5 to 3.3 meV; compare Figure 17(b). Consequently, 2[DELTA](T) depends on the amount of disorder and is the greatest for low-disorder films. There are no traces or indications of superconductivity seen in the optical behavior of the normal state among the higher disordered samples as it is observed in tunneling studies [51]. At 4.2 K, 2[DELTA](T) is reduced as [T.sub.c] decreases (Figure 17(b)); ratio 2[DELTA](0)/[K.sub.B][T.sub.c] indicates that upon increasing disorder (decreasing [T.sub.c]), the ratio 2[DELTA](0)/[K.sub.B][T.sub.c] which is first reduced from about 4.1, indicating strong-coupling superconductivity, exhibits a minimum and becomes larger again (about 4.5) towards the highest-disorder sample studied.

3.4. Dynamics. The question of how long it takes to break or recombine a Copper pair was considered from early on [100, 101] and already linked to the possibility to use superconductors as quantum detectors in the microwave and THz range. Soon it became clear that not only radiative processes are important (emission of photons), but the emission of phonons dominates [102-104]. From early studies of the nonequilibrium conditions of superconductors [101] it became clear that the recombination of two quasiparticles to a Cooper pair requires the emission of phonons with energy above 2[DELTA].Rothwarf and Taylor [104] pointed out that reabsorption of high-energy phonons leads to the so-called phonon bottleneck: the recovery of the superconducting state is not governed by the bare recombination of two quasiparticles into the condensate; the return to equilibrium is rather limited by the phonon escape rate. It took some effort before the development of suitable light sources made the experimental investigations possible.

The first time Testardi [105] was able to break Cooper pairs by short laser light pulses (547 nm, 6-40 [micro]s). The experiments are challenging due to the short penetration length of visible light into Pb metal and the heating of the sample. Federici et al. explored the dynamics of the optical response by looking at the photoinduced Cooper-pair breaking in lead [106]. 100 fs light pulses (630 nm, 1 [micro]J/[cm.sup.2]) lead to an abrupt decrease of the transmission maximum within about 1 ps; it takes several nanoseconds before the superconducting state is recovered. This implies that the superconducting gap closes in less than one picosecond after the optical excitation. Kaindl et al. showed [107] that the imaginary part of the optical conductivity in NbN films abruptly changes and then relaxes in an exponential way (Figure 18(c)). The recovery time [tau] = 580 ps does not depend on the on intensity [A.sub.0] of the light pulse.

A more detailed picture is drawn by Demsar and collaborators who succeeded to perform a detailed study on the superconducting state relaxation phenomena in NbN films: both the Cooper-pair breaking and the recovery of the superconducting state depend on the temperature and energy density [108]. In Figure 18(d) the temporal evolution of the superconducting gap is shown after the sample is excitated by a femtosecond pulse. The photo induced change of the gap [delta][DELTA] (normalized to the equilibrium energy gap at zero temperature) is plotted in Figure 19(a) for various laser intensities. It can be seen that both the Cooper-pair breaking and the recovery of superconductivity depend on intensity A. The long timescale of the Cooper pair breaking implies that photoexcited hot electrons and holes create a high density of high-frequency phonons, which subsequently break Cooper pairs; this process reaches some equilibrium between quasiparticle population and high-frequency phonon population [108]. It is seen from Figure 19(b) that the maximum induced change first increases linearly with excitation intensity (note the logarithmic abscissa) but then saturates when superconductivity is suppressed: [delta][DELTA]/[[DELTA].sub.0] = 1 - exp{-A/[A.sub.sat]}, where the temperature dependence of [A.sub.sat](T), plotted in the inset, basically follows the superconducting density [[DELTA].sup.2] (T).The superconducting recovery time [[tau].sub.rec] depends on temperature T and on intensity A, as shown in Figure 19(c).These findings support the phenomenological model of Rothwarf and Taylor [104] on the phonon bottleneck.

Matsunaga and Shimano [109] investigated the nonlinear dynamics of superconducting state in a NbN film by using intense THz pulses to pump the system which is then probed by time-domain spectroscopy in the same frequency range. After the intense pulse, superconductivity is rapidly switched off with the duration of the THz pulse. In contrast to the previous experiments with visible or infrared pump lasers, the THz pulse resonantly excites the low-energy quasiparticles without the process of phonon emission from hot electrons. The rapid time evolution after the THz pulse is basically independent of intensity implying direct photoinjection of quasiparticles. Figure 20 shows the conductivity spectra obtained at 20 ps after the pump with various intensities. With increasing pump intensity [[sigma].sub.1]([omega]) below the gap energy significantly increases and exceeds the normal state value (dashed line). At the gap energy of approximately 5 meV the conductivity [[sigma].sub.1]([omega]) remains always below the metallic behavior. This cannot be explained by the increase of the effective temperature and implies a high density of quasiparticles. For their analysis they apply an effective medium approach of superconducting and normal state phases. The conclusion is a spatially inhomogeneous suppression of the superconducting state on a length scale of 1 [micro]m [110, 111].

4. Outlook

The effect of disorder on superconductivity is an ongoing intriguing topic of research that has puzzled physicists for decades. It was argued by Anderson in a seminal paper [112] in 1959 that s-wave superconductivity is not affected by the presence of mild nonmagnetic disorder, persisting even in polycrystalline or amorphous materials. Experimentally, however, it was found that superconductivity can be destroyed by a sufficiently large degree of disorder [113-118]. Once superconductivity is destroyed, the system undergoes a transition to an insulating state in what has been named the superconductor-insulator transition (SIT).

Interest in this field has been revived recently with the experimental observations of a number of dramatic features near the superconductor-insulator transition in InO, TiN and NbN, such as nonmonotonic temperature dependence of the resistivity, simple activated temperature dependence on the insulating side, large peak in the magnetoresistance, peculiar I-V characteristics, and existence of an energy gap at temperatures above [T.sub.c] [119-130]. Another reason for renewed interest in this field is that the superconductor-insulator transition may be a very basic realization of a quantum phase transition, that is, a phase transition which occurs at T = 0 as a function of some nonthermal parameters (pressure, doping, magnetic field, etc.) and is driven by quantum instead of thermal fluctuations.

The nature of the superconductor-insulator transition and especially that of the insulating phase is still under debate. Recently several indications for the presence of superconductivity in the insulating state have been reported [131-133], and a number of possible explanations have been put forward. One suggestion relates the phenomena to the inhomogeneous nature of the system under study [134, 135]; near the superconductor-insulator transition on both sides of the transition the system is composed of regions of superconducting material embedded in an insulating matrix. In this picture, the transition occurs when the isolated superconducting islands are able to Josephson-couple and allow the many-body wave function to percolate throughout the entire system. There are basic arguments [136, 137] and theoretical considerations [138, 139] that actually anticipate that inherent inhomogeneity has to occur near the superconductor-insulator transition even when the system is structurally uniform and compatible with the underlying disorder (which, near the SIT of a typical superconductor, is fairly strong). Therefore, some inhomogeneity is expected to be present near the superconductor-insulator transition in all systems where the transition temperature does not drop to zero before a substantial disorder has set in (which in 2D is equivalent to the SIT occurring when the sheet resistance R[] is of the order of the quantum resistance [??]/4[e.sup.2]).

There are quite a few systems that seem to exhibit transport properties that are consistent with this expectation. An example is the situation in amorphous InO ilms, one of the systems that showed the peculiar features alluded above. Structural study reveals morphological inhomogeneities on scales of order 100 [Angstrom]. Transport studies, however, exposed scale dependences up to few microns [136, 137]. Another candidate is films of NbN that exhibit higher [T.sub.c] and thus are suitable for optical experiments in the THz range.

A second picture invokes the existence of uncorrelated preformed electron pairs which do not constitute a condensate [129, 130, 140, 141] but are characterized by an energy gap that is associated with the pair binding energy. Other models adapt concepts from both pictures [142] and suggest that above [T.sub.c] and on the insulating site the film is composed of small superconducting islands that are uncorrelated and are too small to sustain bulk superconductivity. Hence, the situation is unclear, and further experimental information is required.


The authours thankA. Frydman, B. Gorshunov, E. Heinze, U. Pracht, M. Scheffler, and D. Sherman for the fruitful collaboration and useful discussions. The work was supported by the Deutsche Forschungsgemeinschaft (DFG).


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(1.) Physikalisches Institut, Universitat Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany

Correspondence should be addressed to M. Dressel;

Received 11 April 2013; Revised 23 May 2013; Accepted 9 June 2013

Academic Editor: Victor V. Moshchalkov
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Date:Jan 1, 2013
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