Review Article Electrodynamics of Metallic...

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2013 Article ID 104379 25 pageshttpdxdoiorg1011552013104379

Review ArticleElectrodynamics of Metallic Superconductors

M Dressel

1 Physikalisches Institut Universitat Stuttgart Pfaffenwaldring 57 70550 Stuttgart Germany

Correspondence should be addressed to M Dressel dresselpi1physikuni-stuttgartde

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

Academic Editor Victor V Moshchalkov

Copyright copy 2013 M Dressel This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The theoretical and experimental aspects of the microwave terahertz and infrared properties of superconductors are discussedElectrodynamics can provide information about the superconducting condensate as well as about the quasiparticles The aim isto understand the frequency dependence of the complex conductivity the change with temperature and time and its dependenceon material parameters We confine ourselves to conventional metallic superconductors in particular Nb and related nitrides andreview the seminal papers but also highlight latest developments and recent experimental achievementsThe possibility to producewell-defined thin films of metallic superconductors that can be tuned in their properties allows the exploration of fundamentalissues such as the superconductor-insulator transition furthermore it provides the basis for the development of novel and advancedapplications for instance superconducting single-photon detectors

1 Introduction

11 Historical Developments The history of superconductiv-ity in the 1950s is one of the best examples of how theinterplay of experiment and theory advances science towardsa profound understanding After having moved to Urbanain 1951 and even more intense after having received theNobel Prize for the development of the transistor in 1956John Bardeen was pressing to solve the longstanding problemof superconductivity During these years he was very wellaware of the experimental facts and recent developments butalso triggered experimental investigations and pushed certainmeasurements

The presence of an energy gap in the density of states ofa superconductor (and thus in the excitation spectrum) canbe inferred from the temperature dependence of the specificheat that had been measured already in Heike KamerlinghOnnesrsquo laboratory in Leiden [1] however it was Bardeen whoseriously discussed this idea inmore detail in 1955 [2 3] basedon considerations proposed by Welker [4] years earlier Heassumed ldquothat in the superconducting state a finite energy gapΔ asymp 119896

119861119879119888is required to excite electrons from the surface of

the Fermi seardquo Cooper developed this concept further whenhe introduced the electron pairs in the following year [5]

In the seminal BCS paper of 1957 [6] Bardeen Cooper andSchrieffer finally derived the fundamental ratio 2Δ119896

119861119879119888=

35 and the temperature dependence of the energy gap Intheir calculation of the matrix elements they explicitly referto the ongoing NMR relaxation-rate experiments by Hebeland Slichter [7 8] ultrasound-attenuation measurements byMorse and Bohm [9] and microwave-absorption experi-ments by the Tinkham group [10ndash12]

Concerning the high-frequency absorption in an articlesubmitted 1955 [13] Buckingham suggested that (for consis-tency reasons the notation is transformed)

ldquoa gap of width 2Δ(0) in the electron energy spec-trum in the ground state its width 2Δ at finitetemperatures will decrease eventually vanishingat the transition temperature119879

119888The normal state

is regarded as one in which the energy spectrumis without a gap and is independent of tempera-ture the system representing the superconductingstate becoming identical with the normal one for119879 gt 119879

119888 The electromagnetic properties of such a

system will depend strongly on the size of the gapbut at high frequencies and near the transition

2 Advances in Condensed Matter Physics

temperature when ℏ1205962Δ is large will approachthose of the normal state (without gap)rdquo

Looking at the experimental results of Blevins et al whomeasured the surface resistance of tin in the millimeter-wave range [14] he suggested some scaling of frequency andtransition temperature close to119879

119888Thiswas further developed

by Tinkham [11] based on his far-infrared transmissionmea-surements on thin lead films [10] where an anomalously hightransmission was observed (cf Figure 1(b)) He suggestedtwo contributions (superconducting and normal electrons)becoming relevant in the limiting cases of low and highfrequencies

ldquoWe consider a model of a superconductor atabsolute zero inwhich a gap ofwidthℏ120596

119892asymp 3119896119861119879119888

appears in the spectrum of one-electron energystates [sdot sdot sdot ] The lossy conductivity 120590

1will be zero

for photon energies ℏ120596 lt ℏ120596119892 For 120596 gt 120596

119892 1205901

will rise gradually with120596 as an increasing fractionof the states originally within ℏ120596 of the occupiedstates below the Fermi level are still available forexcitation If the gap merely excluded the stateswithin ℏ120596

119892leaving all else unchanged this simple

picture would suggest a rise as 1205901= 120590119899(1minus120596

119892120596)

The experimental rise seems to be faster cuttingoff at least as fast as (1 minus 120596

2

1198921205962) This might

correspond to the states displaced from the gapjust ldquopiling uprdquo on either side so that the changeaverages to zero more rapidly for 120596 ≫ 120596

119892rdquo

These ideas are further developed and extended in laterpapers [12 15] where the size of the superconducting energygap and the shape 120590

1(120596) at 120596

119892were 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 thesituation on the superconducting energy gap in a review [17]Previously they had published several thorough microwaveand millimeter-wave results on superconducting aluminumand tin [18ndash20] and several nice papers followed [21ndash23] Atthe same time Tinkham gave an overview on the work of hisgroup [24]While Biondi andGarfunkel alwaysmeasured thetemperature dependence for certain frequencies Tinkhamand his collaborators focussed at the spectral properties atfixed temperatures This allowed him to make the sum-ruleargument introduced in the previous quote [11] which wasfinalized by Ferrell et al in 1959 [25 26] In Figure 1 thefrequency dependence of the electrodynamical properties ofa superconductor at 119879 = 0 is sketched Upon decreasing thefrequency the real part of the conductivity120590

1(120596)120590

119899gradually

vanishes at the gap frequency120596119892= 2Δℏ while the imaginary

part 1205902(120596)120590

119899diverges as 120596 rarr 0 The maximum in the

relative transmission Tr119904Tr119899is observed right at the gap

(with values well above the unity) before it goes to zero for120596 = 0 There is also a maximum of the absorption above thesuperconducting gap but no absorption takes place below thegap 2Δ It poses a real challenge to measure the reflectivityof a metallic superconductor since it approaches unity at 120596

119892

but in general it is well above 99 already in the normal state(Figure 1(c))

The problem remaining was the determination of thesuperconducting conductivity that is the actual frequencyand temperature dependence of 120590

1(120596) and 120590

2(120596) individually

without the assumption of any model or Kramers-Kronigrelation Figure 2 displays temperature and frequency-dependent conductivity 120590

1120590119899as it is calculated fromMattis-

Bardeen theory 1205901120590119899features a minimum at ℏ1205962Δ(119879) =

1 which shifts to lower frequencies and is smeared out as119879 approaches 119879

119888from below Also seen is the enormous

enhancement of the conductivity 1205901(120596)120590

119899for low frequen-

cies observed right below the superconducting gap it becameknown as the coherence peakDespite continuous efforts overthe years [30ndash32] only in 1968 Palmer and Tinkham reachedthis goal of measuring the real and imaginary parts of theelectrodynamic response independently [33] simultaneousmeasurements of the far-infrared transmission through andreflection off thin films of lead made it possible to calculatethe conductivity unambiguously The status is summarizedin several reviews [34 35] The precise measurement ofthe temperature-dependent conductivity right below thesuperconducting transition however was achieved only aquarter of a century later in the group of Gruner [36 37]Previous efforts by the Cambridge group of Pippard andothers were less convincing [38ndash40]The so-called coherencepeak known from the NMR experiments of Hebel andSlichter [7 8] is also seen in the microwave conductivityof lead niobium and eventually even in alumimum [29]where for the first time the complete frequency evolutionof the coherence peak was measured exactly fifty yearsafter it was calculated Still ultrasound absorption remainsa desideratum in the exploration of the coherence effectsa peak is predicted above 2Δ but no frequency-dependentultrasound absorption measurements on superconductorshave been carried out in the vicinity of the energy gap

Even today Tinkhamrsquos book [41ndash43] gives the best intro-duction to the electrodymical properties of conventionalsuperconductors More detailed descriptions can be found byRickayzen [44] and Parks [45] or more recent textbooks [46]

12 Recent Developments Having understood the bulk prop-erties of metallic superconductors fairly well there are stillmany open questions when going to the vortex latticevery thin films nanostructures inhomogeneous materialsgranular superconductors highly disordered films and soforth which are subject of todayrsquos research efforts Thin filmsof metallic superconductors are of both technological andacademic interests

As we will see the properties of interacting electronsystems are governed by the electronic density of states[47] which changes upon reducing spatial dimensionsThusdimensional reduction from three dimensions to two dimen-sions (3D rarr 2D) may affect optical electronic and ther-modynamic properties of the system In superconductivitystudying quasi-two-dimensional thin-film systems turns outto be a particularly fruitful field and leads to remarkable find-ings such as insulators featuring a superconducting gap [48]

Advances in Condensed Matter Physics 3

00

05

10

15

20

0 1 2 3 4 5

Con

duct

ivity

Frequency ℏ1205962Δ

1205902120590n

1205901120590n

(a)

0 1 2 3 4 500

05

15

10

20

25

Tran

smiss

ion

Abso

rptio

n


AsAn

TrsTrn

(b)

0 1 2 3 4 50997

0998

0999

1000

Refle

ctiv

ity


RsRn 120590dc = 104 (Ωcm)minus1

(c)

Figure 1 Sketch of the electrodynamic properties of a superconductor at 119879 = 0 (a) in the upper panel the frequency dependence of the realand imaginary parts of the conductivity 120590

1and 120590

2is plotted normalized to the normal state behavior 120590

119899 which basically can be assumed to

be constant in the spectral range of interest where 120596 ≪ Γwith Γ the quasiparticle scattering rate (b)The second panel exhibits the frequencydependence of the relative transmission Tr

119904Tr119899and absorption119860

119904119860119899 (c)The optical reflectivity of a superconductor 119877

119904is compared to the

reflectivity of a normal metal 119877119899 assuming a dc conductivity of 104 (Ωcm)minus1

1205901120590

n

4

2

002

0406

0810

12 0002

0406

08

TTc

ℏ1205962Δ(0)

Figure 2 Frequency and temperature dependence of the real partof the conductivity 120590

1120590119899calculated according the BCS theory [27

28] with the ratio of the coherence length to the mean free path120587120585(0)2ℓ = 10 The pronounced maximum for low frequencies ata temperature slightly below 119879

119888is the coherence peak The kink at

ℏ120596 = 2Δ(0) corresponds to the superconducting gap that decreaseswith temperature and vanishes for 119879 rarr 119879

119888(after [29])

pseudogap in conventional 119904-wave superconductors [49]or extremely strong-coupling superconductivity in Kondolattices [50] NbN has recently gained attention as a modelsystem for the superconductor-insulator transition a topicintensively investigated in thin films of Bi InO or TiNtunneling studies [51] reveal a gapped density of states slightlyabove 119879

119888in highly disordered films as it has been found

for TiN films [49] This peculiar state resembles the well-established pseudogap in high-119879

119888cuprate superconductors

[52 53] In case of NbN however 119879119888is about one order of

magnitude higher compared to TiNThus measurements onNbN in the vicinity of the superconductor-insulator transi-tion (SIT) are much less demanding [54 55] Consequentlyresults obtained for NbN might serve as a key to understandthe still puzzling pseudogap state in high-119879

119888cuprates

Besides pure academic interest thin-film superconduc-tors play a key role in many state-of-the-art applications suchas superconducting quantum interference devices (SQUID)thermal and electrical switches using Josephson junctions ormicrowave resonators [56] Contrary to nonmetallic systemssuch as MgB

2or cuprates thin and ultrathin films of NbN

and TaN can be produced of very high quality for exampleby reactive dc-magneton sputtering [57 58] They are ofparticular interest due to their high critical temperature that


makes them suited for novel approaches to build single-photon detectors for example [57 59ndash63]

2 Theory

21 Mattis-BardeenTheory Based on the BCS theory [6] thedescription of the electrodynamic properties of a supercon-ductor was given by Mattis and Bardeen in their seminalpaper [27] Miller applied these considerations on the surfaceimpedance [64] In parallel Abrikosov and coworkers [65 66]also described the electromagnetic absorption of a super-conductor including the coherence factors and the nonlocalrelationship between vector potential and induced currents

The celebrated Mattis-Bardeen equations read

1205901 (120596 119879)

120590119899

=2

ℏ120596int

infin

Δ

[119891 (E) minus 119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

+1

ℏ120596int

minusΔ

Δminusℏ120596

[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(1)

1205902 (120596 119879)

120590119899

=1

ℏ120596int

Δ

Δminusℏ120596minusΔ

[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(Δ2 minusE2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(2)

where for ℏ120596 gt 2Δ the lower limit of the integral in (2)becomesminusΔ Here119891(E 119879) = (expE119896

119861119879 minus 1)

minus1 is the usualFermi-Dirac distribution function The expression of 120590

1120590119899

describes the response of the normal carriers that is thedissipative part The first term of (1) represents the effects ofthermally excited quasiparticles which vanishes for 119879 = 0The second term mainly accounts for the contribution ofphoton-excited quasiparticles since it requires the breakingof a Cooper pair it is basically zero for ℏ120596 lt 2Δ(119879) How-ever there is a small contribution possible of quasiparticlesthermally excited across the gap for 119879 ≲ 119879

119888

The sketch in Figure 3 illustrates the electronic densityof states for a normal metal (panel a) a superconductorat finite temperatures (panel b) and the superconductorat 119879 = 0 (panel c) Single-particle excitations only takeplace from occupied states (dark colors) to empty states(light colors) In the case of a metal (119879 gt 119879

119888) this is

possible for any finite photon energy ℏ120596 leading to a Druderesponse limited only by the scattering rate 1120591 When thesuperconducting gap opens in the density of states for 119879 lt

119879119888 the states from the gap region pile up above and below

plusmnΔ(119879) making a large number of excitations possible Wehave to distinguish excitations across the gap (solid arrows)for ℏ120596 gt 2Δ(119879) from those above (and correspondinglybelow) the gap between occupied and unoccupied states due

to the finite-temperature Fermi distribution 119891(E 119879) Thelatter ones have no lower-energy limit and dominate the low-frequency response since the joint density of states is largestfor ℏ120596 rarr 0 These thermally activated contributions tothe optical conductivity [first term in (1)] are shown by reddashed lines in Figure 4 They are actually observed as thecoherence peak in the temperature dependent conductivityright below 119879

119888(see Section 31 below) These quasiparticle

excitations are limited by the occupation given by the Fermidistribution The across-gap excitations on the other handalways have a clear-cut lower-frequency limit 120596

119892determined

by the gap 2Δℏ this is the energy required to break aCooper pair In Figure 4 these excitations (second termin (1)) are plotted as blue dashed lines Due to the case-2coherence factor describing the quasiparticle excitations in asuperconductor the conductivity only smoothly increases for120596 gt 2Δℏ despite the divergency in the density of states rightabove and below the gap [46] For 119879 = 0 these are the onlyquasiparticle contributions to 120590

1(120596) The superconducting

condensate is observed as a 120575-peak at 120596 = 0 and dominatesthe imaginary part 120590

2(120596) A generalized Mattis-Bardeen

theory which accounts for quasiparticle scattering is derivedby Zimmermann et al [28]The inset of Figure 4 displays howa finite scattering rate modifies 120590

1(120596) and 120590

2(120596)

An interesting aspect was elucidated by Chang andScalapino [67ndash69] who found an enhancement of the super-conducting energy gap and a change in the quasiparticledensity of states upon microwave irradiation in addition to aredistribution of quasiparticles by themicrowaves Eventuallyit extends to the field of nonequilibrium superconductivitythat has been a widely studied field for half a century [70 71]

Already from the London equations a relation betweenthe imaginary part of the conductivity and the Londonpenetration depth is obtained [46]

1205902 (120596) =

1198731198902

119898120596=

1198882

41205871205822

119871120596 (3)

where the London penetration 120582119871

= (11989811988824120587119873

1199041198902)12 is

determined by the superconducting charge carrier mass anddensity119898 and119873

119904 Through the Kramers-Kronig relation the

real part of the conductivity is then given by

1205901 (120596) =

120587

2

1198731199041198902

119898120575 120596 = 0 =

1198882

81205822

119871

120575 120596 = 0 (4)

where 120575120596 = 0 denote a delta peak centered at 120596 = 0Figures 5 and 6 summarize the frequency and temper-

ature dependence of 1205901and 120590

2as derived from (1) and (2)

assuming the finite scattering effects 120587120585(0)ℓ = 10 discussedin [28 73ndash77] Although the density of states diverges atplusmnΔ the conductivity 120590

1(120596) does not show a divergency

but a smooth increase which follows approximately thedependence 120590

1(120596)120590

119899prop 1minus(ℏ120596119896

119861119879119888)minus165 The temperature

dependent conductivity 1205901(119879) shows a peak just below 119879

119888

at a low frequency the maximum occurs around 08119879119888and

slightly shifts up with frequency It is called the coherencepeak because it does not just reflect the enhanced density ofstates but is also an expression of the quantum mechanical


Ener

gy

EF = 0

Single-particle density of states

T gt Tc

(a)

Ener

gy

EF = 0

Δ(T) + ℏ120596

Δ(T)

minusΔ(T)

minusΔ(T) minus ℏ120596


0 lt T lt Tc

(b)

Ener

gy EF = 0

minusΔ(T = 0)

Δ(T = 0)


T = 0

(c)

Figure 3 The density of electronic states in the close vicinity of the Fermi energy 119864119865 (a) For a normal metal the density of states is basically

constant The dark colored area indicates the occupied states according to the Fermi-Dirac statistic at finite temperature (b) In the case of asuperconductor an energy gap opens around 119864

119865 it grows continuously as the temperature is reduced below 119879

119888 The dotted arrow indicates

possible excitations of the occupied states above the gap [first term in (1)] leading to a quasiparticle peak at120596 = 0 For the electronic excitationsshown by the solid arrow a minimum energy of 2Δ is required their contribution is captured by the second term in (1)The dark shaded areaup to |Δ(119879) + ℏ120596| indicates states that can contribute to the conductivity by absorption of photons of arbitrary energy ℏ120596 (c) The full size ofthe superconducting energy gap is given by 2Δ

0for 119879 = 0 No quasiparticle peak is present leading to absorption only above 120596

119892= 2Δℏ The

states removed from the gap area are pilled up below and above the gap leading to a 119864radic1198642 minus Δ20divergency

factor relevant for these excitations The so-called coherencefactor 119865(EkEk1015840) describes the scattering of a quasiparticlefrom a state k with energyEk to a state k1015840 = k+qwith energyEk1015840 = Ek + ℏ120596 upon absorption of a photon with energy ℏ120596and momentum q If summed over all k values it reads [41ndash43 46]

119865 (ΔEE1015840) =

1

2(1 +

Δ2

EE1015840) (5)

Only for energies below the gap 2Δ this factor is appreciable119865 asymp 1 for ℏ120596 ≪ 2Δ For ℏ120596 ge 2Δ the coherencefactors are reversed and 119865 vanishes in the present case Forlarge energies the coherence effects become negligible sinceEE1015840 ≫ 2Δ and 119865 asymp 12 Hence the coherence peak isseen as a maximum in 120590

1(119879) in the low-frequency limit

it becomes smaller with increasing frequency and shifts tohigher temperaturesThe height of the peak has the followingfrequency dependence

(1205901

120590119899

)

maxsim log 2Δ (0)

ℏ120596 (6)

The peak disappears completely for ℏ120596 ge Δ2 (well below2Δ) At 119879 = 0 and 120596 lt 2Δℏ the complex part of theconductivity120590

2120590119899describes the response of the Cooper pairs

and is related to the gap parameter through the expression

1205902 (119879)

120590119899

asymp120587Δ (119879)

ℏ120596tanhΔ (119879)

2119896119861119879 asymp lim119879rarr0

120587Δ (0)

ℏ120596 (7)


Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)

Decreasing T 09 TcT = 08Tc

1120591 = 40 times 2Δ(0)

T = 08Tc

Frequency ℏ1205962Δ(0)0 1

Frequency ℏ1205962Δ(0)

00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc

0 1Frequency ℏ1205962Δ(0)

0

0

0

1 2 3

1Frequency ℏ1205962Δ(0)


Increasing 1120591

09 Tc

Total responsePhoton-activatedThermally activated

Im (120590

sc) =

1205902

01 Tc

(b)

Figure 4 (a) Real and (b) imaginary parts of the complex optical conductivity (solid line) of a superconductor as a function of frequencybased on generalized Mattis-Bardeen theory [27] for 119879 = 08119879

119888and a finite scattering rate [28] Dashed lines disentangle the contributions of

both the thermal electrons [first term in (1) and dashed arrows in Figure 3(b)] and photon-activated electrons [second term in (1) and solidarrow in Figure 3(b)] to the conductivity Insets of (a) and (b) show 120590

1and 120590

2for different temperatures (119879119879

119888= 09 08 07 06 05 04

03 02 and 01) and scattering rates (1120591 = 40 20 4 2 08 04 and 004 times 2Δ(0)ℏ) respectively (Used parameters 2Δ(0)ℎ119888 = 25 cmminus1119879119888= 10K and 120590dc = 1066Ω

minus1 cmminus1) (after [72])

000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2

Frequency 1205962Δℏ

T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)

Figure 5 Frequency dependence of the conductivity of a superconductor at different temperatures119879 relative to the transition temperature119879119888

(a)The real part 1205901(120596)120590

119899exhibits a sharp drop at the superconducting gap 2Δ and vanishes below for 119879 = 0 For finite temperatures the gap

gradually moves to lower frequencies in addition a finite conductivity emerges at low frequencies that increases considerably as 119879 rarr 119879119888 1205901

even exceeds the normal state conductivity in a narrow frequency range right below 119879119888 (b) The imaginary part of the conductivity 120590

2(120596)120590

119899

diverges for low temperatures as 1205902(120596) prop 120596

minus1 The kink of 1205901(120596) at 120596 = 2Δℏ also shows up in the imaginary part 120590

2(120596) [46]

22 Scattering Effects Length Scales and Sum Rule TheEquations (1) and (2) were derived by Mattis and Bardeen[27] assuming the infinite scattering time 120591 rarr infin In1967 Nam showed [76 77] however that these equationsare more general and also valid for 120591 rarr 0 He alsoextended the application towards the strong coupling regime2Δ119896119861119879119888

gt 353 Nevertheless in order to describe theelectrodynamics of superconductors different length scales

have to be considered as their relative magnitude determinesthe nature of the superconducting state and also the responseto electromagnetic fields The first length scale is the Londonpenetration depth

120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07

00 02 04 06 08 10Temperature TTc

(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)

Figure 6 Temperature dependence of the conductivity of a superconductor at different frequencies120596with respect to the gap frequency 2Δℏ(a) For low frequencies the conductivity first rises as 119879 decreases below 119879

119888and exhibits a pronounced maximum before it drops to zero for

119879 rarr 0 This so-called coherence peak becomes smaller as the frequency increases eventually 1205901(120596)120590

119899drops monotonously to zero (b)

The imaginary part 1205902(120596)120590

119899increases continuously with decreasing temperature In the static limit it resembles the development of the

superconducting gap Δ(119879) [46]

determined through the plasma frequency 120596119901

=

(41205871198731198902119898)12 by the parameters of the metallic state

that is the density of charge carriers 119873 the carrier mass 119898and the electronic charge minus119890 The second length scale is thecorrelation length

1205850=ℏV119865

120587Δ(9)

describing crudely speaking the spatial extension of theCooper pairs This is the length scale over which the wavefunction can be regarded as rigid and cannot respond toa spatially varying electromagnetic field This parameteris determined through the energy gap Δ which is linkedto critical temperature by the coupling parameter 120572 =

2Δ119896119861119879119888 The third length scale is the mean free path of the

uncondensed electrons

ℓ = V119865120591 (10)

set by the impurities and lattice imperfections at low temper-atures V

119865is the Fermi velocity 120591 is the time between two

scattering events For typical metals in the superconductingstate the three length scales are of the same order ofmagnitude and depending on the relativemagnitude varioustypes of superconductors are observed these are referred toas the various limits The local limit is where ℓ ≪ 120585 120582more commonly it is referred to as the dirty limit defined byℓ120585 rarr 0 In the opposite so-called clean limit ℓ120585 rarr infinit is necessary to distinguish the following two cases thePippard or anomalous limit defined by the inequality 120582 ≪

120585 ℓ (type I superconductor) and the London limit for which120585 ≪ 120582 ℓ (type II superconductor) Figure 7 summarizesthe relation between these different length scales and therespective limiting cases

The relative magnitude of themean free path with respectto the coherence length has important consequences on the

penetration depth and this can be described using spectralweight arguments The mean free path is given by ℓ asymp

V119865120591 while for the coherence length we get 120585 asymp V

119865Δ and

consequently in the clean limit 1120591 lt Δℏ while 1120591 gt Δℏ

is the so-called dirty limit In the former case the width of theDrude response in the metallic state just above 119879

119888 is smaller

than the frequency corresponding to the gap 2Δℏ and theopposite is true in the dirty limit

The spectral weight associated with the excitations isconserved by going from the normal to the broken symmetrystates While we have to integrate the real part of the normal-state conductivity spectrum 120590

119899

1(120596) in the superconducting

phase there are two contributions one from the condensateof the Cooper pairs 120590coll

1(120596) and one from the single-particle

excitations 120590sp1(120596) thus

int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)

assuming that all the normal carriers condense The argu-ments for superconductors were advanced by Ferrell et al[25 26] who noted that the area 119860 which has been removedfrom the integral upon going through the superconductingtransition

int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)

is redistributed to give the spectral weight of the collectivemode with 120590

1(120596 = 0) = 119860120575120596 By comparison of the

following expression with the Cooper pair response derivedin (4)

1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit

Clean limit Dirty limit

Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)

Figure 7 Schematic representation of the local London and Pippard limits in the parameter space given by the three length scales thecoherence length 120585

0 the London penetration depth 120582

119871 and the mean free path ℓ In the local regime ℓ is smaller than the distance over which

the electric field changes ℓ lt 120585(0) lt 120582 When ℓ120585(0) rarr 0 the superconductor is in the so-called dirty limit The opposite situation is inwhich ℓ120585(0) rarr infin is the clean limit in which nonlocal effects are important and we have to consider Pippardrsquos treatment This regime canbe subdivided if we also consider the third parameter 120582 The case 120585(0) gt 120582 is the Pippard or anomalous regime which is the regime of type Isuperconductors 120585(0) lt 120582 is the London regime the regime of the type II superconductors [46]

we see that the missing spectral weight 119860 is connected to thepenetration depth 120582 by

120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)

This relation is expected to hold also at finite temperaturesand also for various values of the mean free path andcoherence length In the limit of long relaxation time 120591 thatis for ℏ120591 ≪ Δ all the Drude spectral weight of the normalcarriers collapses in the collective mode giving

1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)

and through (4) leading to the London penetration depth120582119871 With increasing 1120591 moving towards the dirty limit

the spectral weight is progressively reduced resulting in anincrease of the penetration depth In the limit 1120591 ≫ 2Δℏthe missing spectral weight is approximately given by 119860 asymp

1205901(2Δℏ) = 2119873119890

2120591Δℏ119898 which can be written as

119860 =1198882

2120587

120591Δ

ℏ (16)

Then the relation between the area 119860 and the penetrationdepth 120582(ℓ)mdashwhich now is mean free path dependent as

ℓ = V119865120591mdashis given by 1205822(ℓ) = 1205822

119871ℏ120587(4120591Δ) which leads to the

approximate expression

120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)

3 Experimental Results

31 Coherence Peak In order to determine the tempera-ture dependence of the complex conductivity at frequencieswell below the superconducting gap Klein et al [37] haveperformed precise measurements of the complex surfaceimpedance of Nb using cavity perturbation methods [78ndash80] A bulk Nb sample of 999 purity was used to replacethe endplate of a 60GHz cylindrical copper cavity Anabrupt drop of the bandwidth was observed upon coolingthrough the 93 K transition temperature In Figure 8(a) thetemperature dependence of the surface resistance 119877

119904and

surface reactance 119883119904normalized to the normal state value

(defined as the value when the temperature of the sample isslightly above 119879

119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where

1198850is the free space impedance) corresponding to 120590

119899(119879 =

10K) = 085 sdot 106(Ω cm)minus1 Below 119879

119888 the surface resistance

119877119904(119879) drops rapidly with decreasing temperature as shown

in the logarithmic plot of Figure 8(a) The length scale forthe surface reactance is determined by the skin depth in thenormal state 120575 = 2270 A119883

119904(119879) saturates at low temperatures

yielding a penetration depth 120582(0) = 440 A Using the


measured values of 119877119904and 119883

119904 the temperature dependence

of the complex conductivity was evaluated and displayedin Figure 8(b) The coherence peak can be seen as a widepeak below119879

119888The solid curves represent theMattis-Bardeen

results [(1) and (2)] while the dashed line corresponds to theEliashberg theory for strong coupling [37]

Only recently broadband microwave measurements onAl films succeeded to probe the coherence peak of a super-conductor (as shown in Figure 2) both as a function oftemperature and frequency Steinberg et al [29] employeda broadband microwave Corbino spectrometer [81] wherebasically a coaxial microwave line is terminated by thesample in this case a 550 A aluminum film on a sapphiresubstrate Using a vector network analyzer the complexreflection coefficient is measured in the spectral range from45MHz to 60GHz allowing the calculation of the real andimaginary parts of the conductivity after also the calibrationmeasurements have been performed with high precisionThemethod is most sensitive when the film impedance is closeto the waveguide impedance of 50Ω For thicker or thinnerfilms different geometries are possible [82] The specimenis cooled down to 11 K and even lower temperatures areavailable [83] Figure 9(a) shows data (taken at differentfrequencies on a film with thickness 500 A and a roomtemperature resistivity of 87 sdot 10minus5Ω cm) of the temperaturedependence of the real part of the conductivity120590

1is governed

by thermally excited charge carriers above and below the gaptheir density of states diverges at the gap edge as depictedin Figure 3 Right below 119879

119888 Δ(119879) is small both the thermal

energy and the photon energy ℏ120596 are sufficient to break upCooper pairs The imaginary part 120590

2(120596)120590

119899displayed in the

lower frame basically describes the temperature dependenceof energy gap in full agreement with (7)

In accord with theory the coherence peak is seen asa maximum in 120590

1(119879) at approximately 08119879

119888in the low-

frequency limit it becomes smaller with increasing frequencyand shifts to higher temperatures in according to (6) this isclearly observed in Figure 9(a) Above 28 GHz [correspond-ing to 02 sdot Δ(0)] the peak is completely suppressed and forhigher frequencies 120590

1(119879)monotonously drops below 119879

119888

32 Energy Gap In order to explore the energy gap in asuperconductor the most straight-forward experiment is atransmissionmeasurement throughmetallic films in the rele-vant frequency range around 120596

119892 and to vary the temperature

from above to below 119879119888 as first has been done by Glover III

and Tinkham on lead films in 1956 [10] In Figure 10(a) thetransmission through a NbN film at different temperatures asindicated normalized to the transmission in themetallic statejust above 119879

119888= 135K [84] With decreasing temperature

a maximum in Tr119904Tr119899is observed that becomes more

pronounced and shifts to higher frequencies As sketched inFigure 1(b) it marks the superconducting energy gap 2Δ inthis case 2Δℎ119888 = 35 cmminus1

The effect of the scattering rate on the transmissionthrough superconducting NbN films has been exploredexperimentally [86] and analyzed on the basis of Leplaersquostheory [73] that describes strong-coupling materials By

reducing the film thickness from 100 to 20 nm the criticaltemperature 119879

119888is lowered from 17 to 14K but more impor-

tantly the energy gap 2Δ(0)ℎ119888 is reduced from 52 cmminus1 to39 cmminus1 Also the transmission maximum becomes signifi-cantly lower Recently it was shown [87 88] that when thesuperconducting state is suppressed by a magnetic field thistransmissionmaximum broadens and shifts to lower temper-atures This is in agreement with the fact that the maximumin transmission corresponds to the energy gap Using thetransmission of 135 cmminus1 gas laser the transmission through a80 nmNbN film on a Si substrate was measured as a functionof temperature for different external fields (Figure 11) Withhigher frequencies the transmission peak first becomesmorepronounced and shifts to lower temperatures however iteventually vanishes at 120596

119892asymp 2Δℏ When a magnetic field

is applied parallel to the propagation direction (Faradaygeometry) the electric field vector is perpendicular to theexternal magnetic field For the Voigt geometry the externalfield is chosen parallel to the electric field vectorThe differentbehavior reflects the vortex dynamics in this constraintgeometry determined by the thin superconducting film Thehigh-frequency response of type-II superconductors in themixed state was extensively studied by Coffey and Clem[89 90]

The absorption of electromagnetic radiation by a super-conductor was directly investigated by measuring the tem-perature increase due to the deposited energy (bolometricabsorption) A sputtered thin film of NbN on a siliconsubstrate (039mm) was mounted at the bottom of a highlyreflecting copper light cone supported by four strands of25 120583mdiameter gold wire of about 15mm longThis gives thebolometer a thermal response time constant of about 30ms at108 KThe temperature is measured by a small doped siliconthermometer glued to the back of the silicon substrate Theresistance of this thermometer is exponentially decreasingwith increasing temperature [85] In Figure 10(b) the relativepower absorption spectra are plotted for a 250 nmNbNsuperconducting film (119879

119888= 133K) on a 039mm silicon

substrate The curves are taken at 53 K and 108 K and eachnormalized to a normal state spectrum taken at 119879 = 14KWith decreasing frequency the absorption increases andexhibits a maximum around the superconducting energygap before it rapidly drops to zero the behavior is morepronounced for lower temperatures On the same thin filmreflectionmeasurements have been performed using Fourier-transform spectrometer with a resolution of 05 cmminus1 Fig-ure 10(c) shows the reflectivity of theNbNfilm on Si substrateabove and below 119879

119888 The reflectivity in the normal state is

relatively low (due to the particular film thickness) makingthe superconductivity behavior very evident While in thenormal state at 155 K the reflectivity levels off around 119877 asymp

093 a pronounced dip is observed in the superconductingstate before the reflectivity rapidly rises to 10 plusmn 001 forfrequencies below 40 cmminus1 clearly indicating the super-conducting gap frequency In any case the high-resolutionreflection spectra exhibit pronounced interference fringesdue to multireflection within the Si substrate In particularone can see in the inset that the phase of the fringes at 53 K


0 025 050 075 100Temperature TTc

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)

Figure 8 (a) Real component 119877119904(119879) (referring to the logarithmic scale on the left axis) and imaginary component 119883

119904(119879) (right axis) of

the surface impedance of niobium normalized to the normal state surface resistance 119877119899measured at 60GHz as a function of temperature

(b) Temperature dependence of the components of the conductivity 1205901(119879) and 120590

2(119879) in niobium calculated from the results of the surface

impedance shown in (a) Note the enhancement of 1205901(119879) just below 119879

119888= 92K The full lines are calculated using the MattisndashBardeen

formalism [(1) and (2)] in the local limit and the dashed lines follow from the Eliashberg theory of a strong coupling superconductor (after[37 46])

4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)

Figure 9 Temperature dependence of the (a) real and (b) imaginary parts of the conductivity of aluminum normalized to the normal stateconductivity 120590

119899for selected frequencies The solid lines are calculated by the BCS theory using ℓ(120587120585) = 003 [28]


0

1

2

3

50 K85 K105 K

118 K125 K

0 50 100 150Frequency (cmminus1)

Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K

Frequency (cmminus1)

(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

ExperimentMatthis-Bardeen-theory

(d)

Figure 10 (a) Ratio of the infrared transmission through a NbN film in the superconducting state and the normal state for severaltemperatures the critical temperature of this film is 119879

119888= 135K (after [84]) (b) The bolometric absorption spectra of a superconducting

NbN film at different temperatures below 119879119888= 133K ratioed with a 14K spectrum above 119879

119888 The curves have been scaled individually

to agree at 110 cmminus1 (for the purpose of comparison) and overall to give a 53 K spectrum which equals 10 at large wave numbers Theresolution is 2 cmminus1 (c) Optical reflectivity of a 250 nm NbN film on silicon (119889 = 039mm) above and below the superconducting transitiontemperature The resolution was 05 cmminus1 below 100 cmminus1 and 2 cmminus1 above The inset gives the low-frequency part on a magnified scale inorder to demonstrate the shift of the Fabry-Perot resonance pattern At high frequencies the reflection maxima are in phase but for lowfrequency the reflectivity in the superconducting state exhibits minima where maxima are observed for 119879 gt 119879

119888 (d) Real and imaginary parts

of the conductivity at 119879 = 53K normalized to the normal state value as extracted from the reflection data shown above The dashed linescorrespond to the calculations based on Mattis-Bardeen theory yielding an energy gap 2Δℎ119888 = 345 cmminus1 (after [85])


NbN-theoretical calculation

NbN-experimental data

8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05

2 4 6 8 10 12Temperature (K)


Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)

Figure 11 Temperature-dependent normal-incidence transmission of 135 cmminus1 radiation through a 80 nm NbN film on a Si substrate of025mm thicknessThe orientation is either in Faraday geometry (119861

0perp 119864) shown in the left column or in Voigt geometry (119861

0|| 119864) as shown

in the right column The calculations are done using a phenomenological effective medium approach (after [87])


changes with respect to those at 155 K near 46 cmminus1 Above46 cmminus1 the fringes are in phase while below they are 180∘out of phase From the lower panel (Figure 10(d)) we see thatthis is the frequencywhere the real and imaginary parts of theconductivities are equal at 53 KThemetallic reflection turnsinto a dielectric reflection governed by the out-of-phaseresponse of the Cooper pairs leading to a sign change in thereflection coefficient Using Fresnelrsquos equations in order todescribe the complete systemof film and substrate the Fabry-Perot pattern allows the calculation of the real and imaginaryparts of the conductivity as displayed in Figure 10(c) NoKramers-Kronig transformation is used here and thus noextrapolation or other model is required The results agreevery well with the theoretical predictions by Mattis andBardeen [27] except slightly above 120596

119892 The superconducting

gap is larger than expected from weak-coupling mean-fieldtheory 2Δ(0)119896

119861119879119888= 39 this is in agreement with the strong

coupling nature of the Nb compoundsWhile the previous experiments have been performed

utilizing Fourier-transform spectrometers Gorshunov et alemployed a THz spectrometer on the basis of tunablecoherent monochromatic backward wave oscillators [93ndash95] to measure 115 A thin NbN films on sapphire substrate[91 92] They demonstrated that for frequencies below 2Δ

the error bars are too large to draw conclusions aboutthe superconducting state and in-gap states for instanceHowever if the substrate is covered with a superconductingfilm on both sides (double-sided film) the Fabry-Perotresonances get much sharper (increase in quality factor)and the conductivity can be evaluated with a much largerprecision allowing even to observe the coherence peakright below the transition temperature The raw data if theTHz transmission are plotted in Figures 12(a) and 12(b) forsingle and double sided films the corresponding conductivityspectra are plotted in panel (c) for two different frequenciesclose to and well below the superconducting gap 2Δ

33 Complex Conductivity Very similar results have beenachieved by Nuss et al [96] using THz time-domain spec-troscopy The amplitude transmitted through a thin Nb film(119879119888= 625K) is plotted in Figure 13 as a function of time Both

the amplitude and phase of the transients change as the filmbecomes superconducting In accordwith the observations ofGlover III and Tinkham [10 12] and others the transmissionin the superconducting state is higher than in the normalstate Since time-domain spectroscopy actually probes theelectric field intensity and the shift in time the real andimaginary parts of the conductivity can be calculated directlyas 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 oneto measure both the change in transmission and phaseshifts when the electromagnetic radiation passes through ametallic and superconducting film This method was inten-sively explored by Gorshunov and collaborators [93ndash95]Figure 14 displays the results of a transmission experiment[97] performed on a thin niobium film (thickness 150 A) ona 045mm thick sapphire substrate using a Mach-Zehnder

interferometer The substrate acts as a Fabry-Perot resonatordue to multireflection leading to pronounced interferencefringes in the spectra this effect enhances the interaction ofthe radiation with the film considerably and thus improvesthe sensitivity and accuracy In Figures 14(a) and 14(b) thetransmission Tr

119865and relative phase shifts 120601] respectively

through this composite sample are shown as a function offrequency As the temperature decreases below the supercon-ducting transition 119879

119888= 83K the transmitted power and

phase shifts are modified significantly Since the propertiesof the dielectric substrate do not vary in this range offrequency and temperature the changes observed are due tothe electrodynamic properties of the superconductor [97]When cooling from 119879 = 9 to 6K the overall transmissionincreases for frequencies above 15 cmminus1 while it drops forlower frequencies The inset illustrates this behavior nicelyThis corresponds to the theoretical predictions (Figure 1(b))and previous observations The maximum in the transmis-sion corresponds to the superconducting energy gap 2ΔSimilar changes are observed in the relative phase shift(Figure 14(b)) where a strong decrease is found for 120596 lt 120596

119892

The energy gap is evaluated to 2Δ(0)ℎ119888 = 24 cmminus1 Thedata can be used to directly calculate the real and imaginaryparts of the complex conductivity without using any modelor assumption In Figure 15 the real and imaginary partsof the conductivity of a Nb film are plotted as a functionof frequency covering an extremely large range Since thedata are not normalized the Drude roll-off becomes obviousabove 100 cmminus1 placing the material into the dirty limit Inall cases it seems to be difficult to actually measure the zero-conductivity well below the gap for very low temperaturesIn many cases some residual absorption seems to be presentwhich can either be intrinsic due to some in-gap absorptionor due to inhomogeneities impurities surface effects and soforth

Thin films of NbN and TaN are perfectly suited forsuperconducting nanowire single-photon detectors [57 59]which are built by thin and narrow superconducting stripestypically in a meander-like geometry The device is drivenwith a dc bias current slightly below the critical currentvalue Absorption of a photon locally reduces the criticalcurrent below the applied dc bias current consequently thesuperconducting state breaks down locally this generatesa normal-conducting spot (hot spot) and increases theresistance The recovery time is in the picosecond range [63]Since their performance in the far-infrared strongly dependson the optical properties of the device in this spectral rangethe energy gap 2Δ and the development of 120590

2(120596) the infor-

mation of THz and infrared investigations are highly desiredIn particular it is important to study the influence of substratefilm thickness processing parameters structuring and soforth In Figure 16 the frequency dependence of the real andimaginary parts of the conductivity120590

1(120596) and120590

2(120596) is plotted

for selected temperatures [72 98] The NbN film is depositedon a 10mm times 10mm R-plane-cut birefringent sapphiresubstrate using reactive dc-magnetron sputtering which isa well-established and powerful way of film deposition withhigh demands concerning the performance of single-photon


001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF

Energy ℏ120596 (meV)

Frequency n (cmminus1)

(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1

29 cmminus11205901(103Ωminus1cm

minus1)

(c)

Figure 12 (a) Transmission spectra Tr119865(]) of an asymmetrical Fabry-Perot resonator formed by a NbN film (119889

2= 115 A) on a sapphire

substrate (043mm) measured at two temperatures above and below critical temperature 119879119888asymp 12K (b) Transmission spectra Tr

119865(]) of a

plane parallel sapphire plate (041mm) covered with NbN films on both sides (1198892= 250 A and 119889

4= 60 A) again measured at 300K and 5K

The solid lines represent the theoretical fit (c) For the latter case the temperature dependence of the optical conductivity 1205901(119879) of NbN film

is calculated for two frequencies 8 cmminus1 and 29 cmminus1 (after [91 92])

detectors [57 58] Typical films have a thickness of 119889 = 39 to47 A Since the thickness 119889 is similar to the superconductingcoherence length 120585 and much smaller than the penetrationdepth 120582 the films are considered ultrathin that is quasi-two dimensional from the superconducting point of viewHowever the electron mean free path ℓ is about one order ofmagnitude smaller than 119889 which justifies the 3D expressionsused in the analysis and places the material in the dirtylimit (Figure 7) Depending on the deposition current 119868

119889

while sputtering the critical temperature is lowered in thecase of NbN from 122 K (119868

119889= 100mA) to 85 K (200mA)

[58 59 72 98 99] The degree of disorder can be describedby the Ioffe-Regel parameter 119896

119865ℓ (with 119896

119865the Fermi wave

vector and ℓ the mean free path) ranging from 45 to 55 thatis in the present case these films are still weakly disordered

Figure 17(a) displays the temperature dependent energygap 2Δ(119879) of the sample with lowest disorder and highest119879119888 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 BCSexpression For 03119879

119888lt 119879 lt 08119879

119888the measured energy

gap is slightly above the BCS curve and for 08119879119888lt 119879 lt

119879119888below Upon decreasing 119879

119888 the energy gap (at 42 K) is

also reduced from 45 to 33meV compare Figure 17(b)Consequently 2Δ(119879) depends on the amount of disorder andis the greatest for low-disorder films There are no traces orindications of superconductivity seen in the optical behaviorof the normal state among the higher disordered samplesas it is observed in tunneling studies [51] At 42 K 2Δ(119879)is reduced as 119879

119888decreases (Figure 17(b)) ratio 2Δ(0)119896

119861119879119888

indicates that upon increasing disorder (decreasing 119879119888)

the ratio 2Δ(0)119896119861119879119888which is first reduced from about

41 indicating strong-coupling superconductivity exhibits aminimum and becomes larger again (about 45) towards thehighest-disorder sample studied

34 Dynamics The question of how long it takes to breakor recombine a Copper pair was considered from early


0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)

Figure 13 (a) THz transients transmitted through a thin niobium film (119879119888= 625K) on quartz in the normal state (dashed red line) and the

superconducting state (solid blue line) (b) Real and imaginary parts of the normalized complex conductivity 1205901(120596) and 120590

2(120596) of Nb in the

superconducting state evaluated from the above data (after [96])The dashed black line indicates the predictions by the BCS theory using theMattis-Bardeen equations (1) and (2) The energy gap is given by 2Δ(0) = 38119896

119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2

1 10 100Temperature T (K)

TrFTr

ansm

issio

n TrF

6 K

9 K

100 20 30Frequency (cmminus1)

57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)

Figure 14 Raw data of transmission Tr(120596) and relative phase shift 120601(])] spectra of the Nb film on the sapphire substrate The criticaltemperature is 119879

119888= 83K The phase shift is divided by the frequency ] = 1205962120587119888 for the better perception The dots represent the measured

values the red lines for 9K are the fits with the Drude model The blue lines correspond to the fit using the Mattis-Bardeen equation In theinsets the temperature dependence of Tr and 120601] is displayed for two particular frequencies corresponding to the first and last transmissionmaxima in measured spectral range (after [97])

on [100 101] and already linked to the possibility to usesuperconductors as quantum detectors in the microwave andTHz range Soon it became clear that not only radiativeprocesses are important (emission of photons) but theemission of phonons dominates [102ndash104] From early studiesof the nonequilibrium conditions of superconductors [101]it became clear that the recombination of two quasiparticlesto a Cooper pair requires the emission of phonons withenergy above 2Δ Rothwarf and Taylor [104] pointed out thatreabsorption of high-energy phonons leads to the so-calledphonon bottleneck the recovery of the superconducting

state is not governed by the bare recombination of twoquasiparticles into the condensate the return to equilibriumis rather limited by the phonon escape rate It took some effortbefore the development of suitable light sources made theexperimental investigations possible

The first time Testardi [105] was able to break Cooperpairs by short laser light pulses (547 nm 6ndash40 120583s)The exper-iments are challenging due to the short penetration lengthof visible light into Pb metal and the heating of the sampleFederici et al explored the dynamics of the optical responseby looking at the photoinduced Cooper-pair breaking in lead


3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb

Frequency (cmminus1)10 100

(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)

Figure 15 Frequency dependence of the real and imaginary parts of the conductivity 1205901(120596) and 120590

2(120596) in niobium at various temperatures

119879 above and below the critical temperature 119879119888= 831K The transmission through a 150 A thick film on sapphire was measured by a Mach-

Zehnder interferometer the stars were obtained by reflection measurements (after [97]) The lines are calculated using the theory of Mattisand Bardeen The inset shows the temperature dependence of the penetration depth evaluated from the 120590

2(119879) at the lowest frequency

6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K

2 6 10 14 18 22 26 30 34 38Frequency (cmminus1)

(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)

Figure 16 (a) Real part and (b) imaginary part of the complex conductivity versus frequency for 42 K (dots) and 7K (diamonds) well below119879119888= 96K and 20K (squares) in the normal state for both main axes of a NbN film Solid black and blue lines are predictions calculated

directly from raw data fits for both axes individually The kink in 1205901signaling the energy gap is resolved clearly The error bar shown is

representative for all data (after [98])

[106] 100 fs light pulses (630 nm 1 120583Jcm2) lead to an abruptdecrease of the transmission maximum within about 1 ps ittakes several nanoseconds before the superconducting stateis recoveredThis implies that the superconducting gap closesin less than one picosecond after the optical excitation Kaindlet al showed [107] that the imaginary part of the opticalconductivity in NbN films abruptly changes and then relaxesin an exponential way (Figure 18(c)) The recovery time 120591 =580 ps does not depend on the on intensity 119860

0of the light

pulse

A more detailed picture is drawn by Demsar and col-laborators who succeeded to perform a detailed study onthe superconducting state relaxation phenomena in NbNfilms both the Cooper-pair breaking and the recovery ofthe superconducting state depend on the temperature andenergy density [108] In Figure 18(d) the temporal evolutionof the superconducting gap is shown after the sample isexcitated by a femtosecond pulse The photoinduced changeof the gap 120575Δ (normalized to the equilibrium energy gap atzero temperature) is plotted in Figure 19(a) for various laser


4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis

BCS (+ axis)BCS (minus axis)

Temperature (K)

(a)

8 9 10 11 12Critical temperature Tc (K)

44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)

Figure 17 (a) Energy gap 2Δ versus temperature of a NbN sample for both main axes of the substrate Solid lines are the fits according toBCS theory Results on both main axes yield very similar values of 2Δ(119879) The error bar shown is representative for all data (b) 2Δ at 42 Kand (c) ratios 2Δ(0)119896

119861119879119888versus 119879

119888of several NbN films deposited with different parameters yielding different transition temperatures (after

[72])

intensities It can be seen that both the Cooper-pair breakingand the recovery of superconductivity depend on intensity119860The long timescale of the Cooper pair breaking implies thatphotoexcited hot electrons and holes create a high density ofhigh-frequency phonons which subsequently break Cooperpairs this process reaches some equilibrium between quasi-particle population and high-frequency phonon population[108] It is seen from Figure 19(b) that the maximum inducedchange first increases linearly with excitation intensity (notethe logarithmic abscissa) but then saturates when supercon-ductivity is suppressed 120575ΔΔ

0= 1 minus expminus119860119860 sat where

the temperature dependence of 119860 sat(119879) plotted in the insetbasically follows the superconducting density Δ

2(119879) The

superconducting recovery time 120591rec depends on temperature119879 and on intensity119860 as shown in Figure 19(c)These findingssupport the phenomenologicalmodel of Rothwarf and Taylor[104] on the phonon bottleneck

Matsunaga and Shimano [109] investigated the non-linear dynamics of superconducting state in a NbN filmby using intense THz pulses to pump the system whichis then probed by time-domain spectroscopy in the samefrequency range After the intense pulse superconductivityis rapidly switched off with the duration of the THz pulseIn contrast to the previous experiments with visible orinfrared pump lasers the THz pulse resonantly excites thelow-energy quasiparticles without the process of phononemission from hot electrons The rapid time evolution afterthe THz pulse is basically independent of intensity implying

direct photoinjection of quasiparticles Figure 20 shows theconductivity spectra obtained at 20 ps after the pump withvarious intensities With increasing pump intensity 120590

1(120596)

below the gap energy significantly increases and exceedsthe normal state value (dashed line) At the gap energy ofapproximately 5meV the conductivity 120590

1(120596) remains always

below the metallic behavior This cannot be explained bythe increase of the effective temperature and implies a highdensity of quasiparticles For their analysis they apply aneffective medium approach of superconducting and normalstate phases The conclusion is a spatially inhomogeneoussuppression of the superconducting state on a length scale of1 120583m [110 111]

4 Outlook

The effect of disorder on superconductivity is an ongoingintriguing topic of research that has puzzled physicists fordecades It was argued by Anderson in a seminal paper[112] in 1959 that 119904-wave superconductivity is not affectedby the presence of mild nonmagnetic disorder persistingeven in polycrystalline or amorphous materials Experimen-tally however it was found that superconductivity can bedestroyed by a sufficiently large degree of disorder [113ndash118]Once superconductivity is destroyed the system undergoes atransition to an insulating state in what has been named thesuperconductor-insulator transition (SIT)


6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)

0 500Time delay (ps)

1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)

Figure 18 Temperature-dependence of the (a) real and (b) imaginary parts of the conductivity spectra obtained on a 15 nm NbN filmon MgO substrate (119879

119888= 154K) The solid lines are fits with the Mattis-Bardeen equations [27 28] The inset of panel (a) shows the

temperature dependence of the energy gap 2Δ extracted from fits to 120590(120596) (symbols) overlaying the normalized transmissivity ratio TRThe BCS temperature dependence of Δ is shown by the dashed line The inset to panel (b) displays the transmitted THz transients throughthe NbN film below and above 119879

119888 the arrow denotes the time 1199051015840 = 119905

0with maximum change in the transmitted electric field (after [108])

(c) The recovery of the imaginary part of the superconducting conductivity for different intensities can be described by a single exponentialdependence with 120591 = 580 ps [107] (d) Time evolution of normalized transmissivity ratio (TR) recorded at 119860 = 22mJcm3 together withΔ(119905) extracted from 120590

1(120596 119905119889) fit with the BCS theory [27 28] (after [108])


01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r

ec(10minus2psminus1)

120591 rec

(ps)

(c)

Figure 19 (a) The relative change in gap 120575ΔΔ0recorded at 43 K for various intensities A in mJcm3 The dashed lines are fits to the data

with (1) (b) The dependence of 120575ΔΔ0on intensity A at 43 K The dashed line is a fit to the simple saturation model the solid line is the

linear fit Inset the temperature dependence of the saturation energy density 119860 sat compared to the 119879-dependence of Δ2 (dashed line) Panel(c) shows the 119879-dependence of the superconducting recovery time 120591rec for several intensities119860 (in mJcm 3) Inset 120591minus1rec(119860) recorded at 43 KFor low intensities 119860 the relaxation rate increases linearly with 119860 (dashed line) (after [108])


0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)

0 kVcm10 kVcm14 kVcm23 kVcm

27 kVcm42 kVcm67 kVcm

(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation

Figure 20 The spectra of the (a) real and (b) imaginary parts of the optical conductivity for various THz excitation intensities at 4 K Forreference 120590

1(120596) in the normal metal state at 16 K is plotted as a dotted line Panel (c) and (d) exhibit the calculated real and imaginary parts

of the optical conductivity spectra respectively using an effective medium approachThe inset is a schematic image of the inhomogeneouslysuppressed superconductivity (after [109])

Interest in this field has been revived recently with theexperimental observations of a number of dramatic featuresnear the superconductor-insulator transition in InO TiNand NbN such as nonmonotonic temperature dependenceof the resistivity simple activated temperature dependenceon the insulating side large peak in the magnetoresistancepeculiar 119868-119881 characteristics and existence of an energy gap attemperatures above119879

119888[119ndash130] Another reason for renewed

interest in this field is that the superconductor-insulatortransition may be a very basic realization of a quantum phasetransition that is a phase transition which occurs at 119879 = 0 as

a function of some nonthermal parameters (pressure dopingmagnetic field etc) and is driven by quantum instead ofthermal fluctuations

The nature of the superconductor-insulator transitionand especially that of the insulating phase is still underdebate Recently several indications for the presence ofsuperconductivity in the insulating state have been reported[131ndash133] and a number of possible explanations have beenput forward One suggestion relates the phenomena to theinhomogeneous 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 ofsuperconducting material embedded in an insulating matrixIn this picture the transition occurs when the isolatedsuperconducting islands are able to Josephson-couple andallow the many-body wave function to percolate throughoutthe entire system There are basic arguments [136 137]and theoretical considerations [138 139] that actually antic-ipate that inherent inhomogeneity has to occur near thesuperconductor-insulator transition even when the systemis structurally uniform and compatible with the underlyingdisorder (which near the SIT of a typical superconductor isfairly strong) Therefore some inhomogeneity is expected tobe present near the superconductor-insulator transition in allsystems where the transition temperature does not drop tozero before a substantial disorder has set in (which in 2D isequivalent to the SIT occurring when the sheet resistance 119877

◻

is of the order of the quantum resistance ℏ41198902)There are quite a few systems that seem to exhibit

transport properties that are consistent with this expectationAn example is the situation in amorphous InO films one ofthe systems that showed the peculiar features alluded aboveStructural study reveals morphological inhomogeneities onscales of order 100 A Transport studies however exposedscale dependences up to few microns [136 137] Anothercandidate is films of NbN that exhibit higher 119879

119888and thus are

suitable for optical experiments in the THz rangeA second picture invokes the existence of uncorrelated

preformed electron pairs which do not constitute a conden-sate [129 130 140 141] but are characterized by an energygap that is associated with the pair binding energy Othermodels adapt concepts from both pictures [142] and suggestthat above 119879

119888and on the insulating site the film is composed

of small superconducting islands that are uncorrelated andare too small to sustain bulk superconductivity Hence thesituation is unclear and further experimental information isrequired

Acknowledgments

The authours thank A Frydman B Gorshunov E Heinze UPracht M Scheffler and D Sherman for the fruitful collabo-ration and useful discussionsThe work was supported by theDeutsche Forschungsgemeinschaft (DFG)

References

[1] F London Superfluids vol 1 ofMacroscopicTheory of Supercon-ductivity John Wiley amp Sons New York NY USA 1950

[2] J Bardeen ldquoTheory of the Meissner effect in superconductorsrdquoPhysical Review vol 97 no 6 pp 1724ndash1725 1955

[3] J Bardeen ldquoTheory of Supercondgctivityrdquo in Handbuch derPhysik vol 15 pp 274ndash369 Springer Berlin Germany 1956

[4] H Welker ldquoSupraleitung und magnetische Austauschwechsel-wirkungrdquo Zeitschrift fur Physik vol 114 no 9-10 pp 525ndash5511939

[5] L N Cooper ldquoBound electron pairs in a degenerate fermi gasrdquoPhysical Review vol 104 no 4 pp 1189ndash1190 1956

[6] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 no 5 pp 1175ndash1204 1957

[7] L C Hebel and C P Slichter ldquoNuclear relaxation in supercon-ducting aluminumrdquo Physical Review vol 107 no 3 p 901 1957

[8] L CHebel andC P Slichter ldquoNuclear spin relaxation in normaland superconducting aluminumrdquo Physical Review vol 113 no6 pp 1504ndash1519 1959

[9] R W Morse and H V Bohm ldquoSuperconducting energy gapfrom ultrasonic attenuation measurementsrdquo Physical Reviewvol 108 no 4 pp 1094ndash1096 1957

[10] R E Glover III and M Tinkham ldquoTransmission of super-conducting films at millimeter-microwave and far infraredfrequenciesrdquo Physical Review vol 104 no 3 pp 844ndash845 1956

[11] M Tinkham ldquoEnergy gap interpretation of experiments oninfrared transmission through superconducting filmsrdquo PhysicalReview vol 104 no 3 pp 845ndash846 1956

[12] R E Glover III and M Tinkham ldquoConductivity of super-conducting films for photon energies between 03 and 40kTcrdquoPhysical Review vol 108 no 2 pp 243ndash256 1957

[13] M J Buckingham ldquoVery high frequency absorption in super-conductorsrdquo Physical Review vol 101 no 4 pp 1431ndash1432 1956

[14] G S Blevins W Gordy and W M Fairbank ldquoSuperconductiv-ity atmillimeter wave frequenciesrdquo Physical Review vol 100 no4 pp 1215ndash1216 1955

[15] M Tinkham and R E Glover III ldquoSuperconducting energy gapinferences from thin-film transmission datardquo Physical Reviewvol 110 no 3 pp 778ndash779 1958

[16] A T Forrester ldquoSuperconducting energy gap inferences fromthin-film transmission datardquo Physical Review vol 110 no 3 pp776ndash778 1958

[17] M A Biondi A T Forrester M P Garfunkel and C BSatterthwaite ldquoExperimental evidence for an energy gap insuperconductorsrdquo Reviews of Modern Physics vol 30 no 4 pp1109ndash1136 1958

[18] M A Biondi M P Garfunkel and A OMcCoubrey ldquoMillime-ter wave absorption in superconducting aluminumrdquo PhysicalReview vol 101 no 4 pp 1427ndash1429 1956

[19] M A Biondi M P Garfunkel and A O McCoubreyldquoMicrowave measurements of the energy gap in superconduct-ing aluminumrdquo Physical Review vol 108 no 2 pp 495ndash4971957

[20] M A Biondi A T Forrester and M P Garfunkel ldquoMillimeterwave studies of superconducting tinrdquo Physical Review vol 108no 2 pp 497ndash498 1957

[21] M A Biondi and M P Garfunkel ldquoMeasurement of thetemperature variation of the energy gap in superconductingaluminumrdquo Physical Review Letters vol 2 no 4 pp 143ndash1451959

[22] M A Biondi andM P Garfunkel ldquoMillimeter wave absorptionin superconducting aluminum I Temperature dependence ofthe energy gaprdquo Physical Review vol 116 no 4 pp 853ndash8611959

[23] M A Biondi andM P Garfunkel ldquoMillimeter wave absorptionin superconducting aluminum II Calculation of the skindepthrdquo Physical Review vol 116 no 4 pp 862ndash867 1959

[24] M Tinkham ldquoAbsorption of electromagnetic radiation insuperconductorsrdquo Physica vol 24 supplement 1 pp S35ndashS411958

[25] R A Ferrell and R E Glover III ldquoConductivity of supercon-ducting films a sum rulerdquo Physical Review vol 109 no 4 pp1398ndash1399 1958


[26] M Tinkham and R A Ferrell ldquoDetermination of the supercon-ducting skin depth from the energy gap and sum rulerdquo PhysicalReview Letters vol 2 no 8 pp 331ndash333 1959

[27] D C Mattis and J Bardeen ldquoTheory of the anomalous skineffect in normal and superconducting metalsrdquo Physical Reviewvol 111 no 2 pp 412ndash417 1958

[28] W Zimmermann E H Brandt M Bauer E Seider and LGenzel ldquoOptical conductivity of BCS superconductors witharbitrary purityrdquo Physica C vol 183 no 1ndash3 pp 99ndash104 1991

[29] K Steinberg M Scheffler and M Dressel ldquoQuasiparticleresponse of superconducting aluminum to electromagneticradiationrdquo Physical Review B vol 77 no 21 Article ID 2145172008

[30] D M Ginsberg P L Richards and M Tinkham ldquoApparentstructure on the far infrared energy gap in superconducting leadand mercuryrdquo Physical Review Letters vol 3 no 7 pp 337ndash3381959

[31] D M Ginsberg and M Tinkham ldquoFar infrared transmissionthrough superconducting filmsrdquo Physical Review vol 118 no 4pp 990ndash1000 1960

[32] P L Richards and M Tinkham ldquoFar-infrared energy gapmeasurements in bulk superconducting in Sn Hg Ta V Pband Nbrdquo Physical Review vol 119 no 2 pp 575ndash590 1960

[33] L H Palmer andM Tinkham ldquoFar-infrared absorption in thinsuperconducting lead filmsrdquo Physical Review vol 165 no 2 pp588ndash595 1968

[34] M Tinkham ldquoThe electromagnetic properties of superconduc-torsrdquo Reviews of Modern Physics vol 46 no 4 pp 587ndash5961974

[35] PWyder ldquoFar-infrared and superconductivityrdquo Infrared Physicsvol 16 no 1-2 pp 243ndash251 1976

[36] K Holczer O Klein and G Gruner ldquoObservation of theconductivity coherence peak in superconducting Pbrdquo Solid StateCommunications vol 78 no 10 pp 875ndash877 1991

[37] O Klein E J Nicol K Holczer and G Gruner ldquoConductivitycoherence factors in the conventional superconductors Nb andPbrdquo Physical Review B vol 50 no 9 pp 6307ndash6316 1994

[38] A B Pippard ldquoMetallic conduction at high frequencies and lowtemperaturesrdquo Advances in Electronics and Electron Physics vol6 pp 1ndash45 1954

[39] A B Pippard ldquoThe anomalous skin effect in anisotropicmetalsrdquoProceedings of the Royal Society of London A vol 224 no 1157pp 273ndash282 1954

[40] J R Waldram ldquoSurface impedance of superconductorsrdquoAdvances in Physics vol 13 no 49 pp 1ndash88 1964

[41] M Tinkham Introduction to Superconductivity Mc-Graw HillNew York NY USA 1st edition 1975

[42] M Tinkham Introduction to Superconductivity Mc-Graw HillNew York NY USA 2nd edition 1996

[43] M Tinkham Introduction to Superconductivity Dover Publica-tion Mineola NY USA 2nd edition 2004

[44] G Rickayzen Theory of Superconductivity Interscience Pub-lishers New York NY USA 1965

[45] R D Parks Ed Superconductivity Vol I and II Marcel DekkerNew York NY USA 1969

[46] MDressel andGGrunerElectrodynamics of Solids CambridgeUniversity Press Cambridge UK 2002

[47] D N Basov R D Averitt D van der Marel M Dressel andK Haule ldquoElectrodynamics of correlated electron materialsrdquoReviews of Modern Physics vol 83 no 2 pp 471ndash541 2011

[48] D Sherman G Kopnov D Shahar and A Frydman ldquoMea-surement of a superconducting energy gap in a homogeneouslyamorphous insulatorrdquo Physical Review Letters vol 108 no 17Article ID 177006 2012

[49] B Sacepe C Chapelier T I Baturina V M Vinokur M RBaklanov and M Sanquer ldquoPseudogap in a thin film of aconventional superconductorrdquo Nature Communications vol 1no 9 article 140 2010

[50] Y Mizukami H Shishido T Shibauchi et al ldquoExtremelystrong-coupling superconductivity in artificial two-dimen-sional Kondo latticesrdquoNature Physics vol 7 no 11 pp 849ndash8532011

[51] S P Chockalingam M Chand A Kamlapure et al ldquoTunnelingstudies in a homogeneously disordered s-wave superconductorNbNrdquo Physical Review B vol 79 no 9 Article ID 094509 2009

[52] T Timusk and B Statt ldquoThe pseudogap in high-temperaturesuperconductors an experimental surveyrdquo Reports on Progressin Physics vol 62 no 1 pp 61ndash122 1999

[53] N D Basov and T Timusk ldquoElectrodynamics of highmdashT119888

superconductorsrdquo Reviews of Modern Physics vol 77 no 2 pp721ndash729 2005

[54] U S PrachtM SchefflerMDressel D F Kalok C Strunk andT I Baturina ldquoDirect observation of the superconducting gapin a thin film of titanium nitride using terahertz spectroscopyrdquoPhysical Review B vol 86 no 18 Article ID 184503 2012

[55] E F C Driessen P C J J Coumou R R Tromp P J deVisser and TM Klapwijk ldquoStrongly disordered tin and nbtin s-wave superconductors probed by microwave electrodynamicsrdquoPhysical Review Letters vol 109 no 10 Article ID 107003 2012

[56] M R Vissers M P Weides J S Kline M Sandberg and DP Pappas ldquoIdentifying capacitive and inductive loss in lumpedelement superconducting hybrid titanium nitridealuminumresonatorsrdquo Applied Physics Letters vol 101 Article ID 0226012012

[57] D Henrich S Dorner M Hofherr et al ldquoBroadening ofhotspot response spectrum of superconducting NbN nanowiresinglephoton detector with reduced nitrogen contentrdquo Journalof Applied Physics vol 112 no 7 Article ID 074511 2012

[58] K Ilrsquoin M Hofherr D Rall et al ldquoUltra-thin TaN films forsuperconducting nanowire single-photon detectorsrdquo Journal ofLow Temperature Physics vol 167 pp 809ndash814 2011

[59] A Engel A Schilling K Ilin and M Siegel ldquoDependence ofcount rate on magnetic field in superconducting thin-film TaNsingle-photon detectorsrdquo Physical Review B vol 86 Article ID140506 2012

[60] A J Kerman E A Dauler W E Keicher et al ldquoKinetic-inductance-limited reset time of superconducting nanowirephoton countersrdquo Applied Physics Letters vol 88 no 11 ArticleID 111116 2006

[61] G GolrsquoTsman O Okunev G Chulkova et al ldquoFabricationand properties of an ultrafast NbN hot-electron single-photondetectorrdquo IEEE Transactions on Applied Superconductivity vol11 no 1 pp 574ndash577 2001

[62] J Zhang W Slysz A Verevkin et al ldquoResponse time charac-terization of NbN superconducting single-photon detectorsrdquoIEEE Transactions on Applied Superconductivity vol 13 no 2pp 180ndash183 2003

[63] C M Natarajan M G Tanner and R H Hadfields ldquoSuper-conducting nanowire single-photon detectors physics andapplicationsrdquo Superconductor Science and Technology vol 25no 6 Article ID 063001 2012


[64] P B Miller ldquoSurface impedance of superconductorsrdquo PhysicalReview vol 118 no 4 pp 928ndash934 1960

[65] A A Abrikosov L P Gorkov and I M Khalatnikov ldquoASuperconductor in a high frequency fieldrdquo Soviet Physics JETPvol 8 no 1 pp 182ndash189 1959

[66] A A Abrikosov L P Gorkov and I M Khalatnikov ldquoAnalysisof experimental data relating to the surface impedance ofsuperconductorsrdquo Soviet Physics JETP vol 10 no 1 pp 132ndash134 1960

[67] J-J Chang and D J Scalapino ldquoGap enhancement in supercon-ducting thin films due tomicrowave irradiationrdquo Journal of LowTemperature Physics vol 29 no 5-6 pp 477ndash485 1977

[68] J-J Chang and D J Scalapino ldquoKinetic-equation approach tononequilibrium superconductivityrdquo Physical Review B vol 15no 5 pp 2651ndash2670 1977

[69] J-J Chang and D J Scalapino ldquoNonequilibrium superconduc-tivityrdquo Journal of Low Temperature Physics vol 31 no 1-2 pp1ndash32 1978

[70] D N Langenberg and A L Larkin Eds NonequilibriumSuperconductivity North-Holland Amserdam The Nether-lands 1986

[71] N KopninTheory of Nonequilibrium Superconductivity OxfordUniversity Press Oxford UK 2009

[72] U S Pracht E Heintze C Clauss et al ldquoElectrodynamics of thesuperconducting state in ultra-thin films at THz frequenciesrdquoIEEE Transactions on Terahertz Science and Technology vol 3no 3 pp 269ndash280 2013

[73] L Leplae ldquoDerivation of an expression for the conductivityof superconductors in terms of the normal-state conductivityrdquoPhysical Review B vol 27 no 3 pp 1911ndash1912 1983

[74] J-J Chang and D J Scalapino ldquoElectromagnetic response oflayered superconductorsrdquo Physical Review B vol 40 no 7 pp4299ndash4305 1989

[75] A J Berlinsky C Kallin G Rose and A-C Shi ldquoTwo-fluidinterpretation of the conductivity of clean BCS superconduc-torsrdquo Physical Review B vol 48 no 6 pp 4074ndash4079 1993

[76] S B Nam ldquoTheory of electromagnetic properties of supercon-ducting and normal systems Irdquo Physical Review vol 156 no 2pp 470ndash486 1967

[77] S B Nam ldquoTheory of electromagnetic properties of strong-coupling and impure superconductors IIrdquo Physical Review vol156 no 2 pp 487ndash493 1967

[78] O Klein S Donovan M Dressel and G Gruner ldquoMicrowavecavity perturbation technique part I principlesrdquo InternationalJournal of Infrared and Millimeter Waves vol 14 no 12 pp2423ndash2457 1993

[79] S Donovan O Klein M Dressel K Holczer and G GrunerldquoMicrowave cavity perturbation technique part II experimen-tal schemerdquo International Journal of Infrared and MillimeterWaves vol 14 no 12 pp 2459ndash2487 1993

[80] M Dressel O Klein S Donovan and G Gruner ldquoMicrowavecavity perturbation technique part III applicationsrdquo Interna-tional Journal of Infrared and Millimeter Waves vol 14 no 12pp 2489ndash2517 1993

[81] M Scheffler and M Dressel ldquoBroadband microwave spec-troscopy in Corbino geometry for temperatures down to 17 KrdquoReview of Scientific Instruments vol 76 no 7 Article ID 0747022005

[82] M Scheffler S Kilic and M Dressel ldquoStrip-shaped samplesin a microwave Corbino spectrometerrdquo Review of ScientificInstruments vol 78 no 8 Article ID 086106 2007

[83] K Steinberg M Scheffler and M Dressel ldquoBroadbandmicrowave spectroscopy in Corbino geometry at 3He temper-aturesrdquo Review of Scientific Instruments vol 83 no 2 ArticleID 024704 2012

[84] R P S M Lobo J D LaVeigne D H Reitze et al ldquoPho-toinduced time-resolved electrodynamics of superconductingmetals and alloysrdquo Physical Review B vol 72 no 2 Article ID024510 2005

[85] K E Kornelsen M Dressel J E Eldridge M J Brett and KL Westra ldquoFar-infrared optical absorption and reflectivity of asuperconducting NbN filmrdquo Physical Review B vol 44 no 21pp 11882ndash11887 1991

[86] D Karecki R E Pena and S Perkowitz ldquoFar-infrared transmis-sion of superconducting homogeneous NbN films scatteringtime effectsrdquo Physical Review B vol 25 no 3 pp 1565ndash15711982

[87] M Sindler R Tesar J Kolacek L Skrbek and Z SimSa ldquoFar-infrared transmission of a superconducting NbN filmrdquo PhysicalReview B vol 81 no 18 Article ID 184529 2010

[88] R Tesar M Sindler K Ilin J Kolacek M Siegel and L SkrbekldquoTerahertz thermal spectroscopy of a NbN superconductorrdquoPhysical Review B vol 84 no 13 Article ID 132506 2011

[89] M W Coffey and J R Clem ldquoTheory of high-frequency linearresponse of isotropic type-II superconductors in the mixedstaterdquo Physical Review B vol 46 no 18 pp 11757ndash11764 1992

[90] M W Coffey and J R Clem ldquoTheory of microwave transmis-sion and reflection in type-II superconductors in the mixedstaterdquo Physical Review B vol 48 no 1 pp 342ndash350 1993

[91] B P Gorshunov G V Kozlov A A Volkov et al ldquoMeasurementof electrodynamic parameters of superconducting films in thefar-infrared and submillimeter frequency rangesrdquo InternationalJournal of Infrared and Millimeter Waves vol 14 no 3 pp 683ndash702 1993

[92] B P Gorshunov I V Fedorov G V Kozlov A A Volkov andAD Semenov ldquoDynamic conductivity and the coherence peak inthe submillimeter spectra of superconducting NbN filmsrdquo SolidState Communications vol 87 no 1 pp 17ndash21 1993

[93] B Gorshunov A Volkov I Spektor et al ldquoTerahertz BWO-spectrosopyrdquo International Journal of Infrared and MillimeterWaves vol 26 no 9 pp 1217ndash1240 2005

[94] B P Gorshunov A A Volkov A S Prokhorov et al ldquoTerahetzBWO spectroscopy of conductors and superconductorsrdquoQuan-tum Electronics vol 37 no 10 pp 916ndash923 2007

[95] M Dressel N Drichko B Gorshunov and A Pimenov ldquoTHzspectroscopy of superconductorsrdquo IEEE Journal on SelectedTopics in Quantum Electronics vol 14 no 2 pp 399ndash406 2008

[96] M C Nuss K W Goossen J P Gordon P M MankiewichM L OrsquoMalley and M Bhushan ldquoTerahertz time-domainmeasurement of the conductivity and superconducting bandgap in niobiumrdquo Journal of Applied Physics vol 70 no 4 pp2238ndash2241 1991

[97] A V Pronin M Dressel A Pimenov A Loidl I V Roshchinand L H Greene ldquoDirect observation of the superconductingenergy gap developing in the conductivity spectra of niobiumrdquoPhysical Review B vol 57 no 22 pp 14416ndash14421 1998

[98] U Pracht M Scheffler M Dressel K S Ilin and MSiegelunpublished

[99] A Semenov B Gunther U Bottger et al ldquoOptical and transportproperties of ultrathin NbN films and nanostructuresrdquo PhysicalReview B vol 80 no 5 Article ID 054510 2009


[100] E Burstein D N Langenberg and B N Taylor ldquoSuperconduc-tors as quantum detectors for microwave and sub-millimeter-wave radiationrdquo Physical Review Letters vol 6 no 3 pp 92ndash941961

[101] D M Ginsberg ldquoUpper limit for quasi-particle recombinationtime in a superconductorrdquo Physical Review Letters vol 8 no 5pp 204ndash207 1962

[102] J R Schrieffer and D M Ginsberg ldquoCalculation of thequasiparticle recombination time in a superconductorrdquo PhysicalReview Letters vol 8 no 5 pp 207ndash208 1962

[103] A Rothwarf and M Cohen ldquoRate of capture of electronsinjected into superconducting leadrdquo Physical Review vol 130no 4 pp 1401ndash1405 1963

[104] A Rothwarf and B N Taylor ldquoMeasurement of recombinationlifetimes in superconductorsrdquo Physical Review Letters vol 19no 1 pp 27ndash30 1967

[105] L R Testardi ldquoDestruction of superconductivity by laser lightrdquoPhysical Review B vol 4 no 7 pp 2189ndash2196 1971

[106] J F Federici B I Greene P N Saeta D R Dykaar FSharifi and R C Dynes ldquoDirect picosecond measurement ofphotoinduced Cooper-pair breaking in leadrdquo Physical ReviewB vol 46 no 17 pp 11153ndash11156 1992

[107] Experiments on NbN films by R Kaindl M Carnahan and DChemla at the Advanced Light Source of the Lawrance BerkeleyNational Laboratory Figure reproduced from N Gedik THzspectroscopy of quasiparticle dynamics in complex materialshttpsportalslacstanfordedusitesconf publicTHz 201209presentationsgedik Frontiers of THz Sciencepdf

[108] M Beck M Klammer S Lang et al ldquoEnergy-gap dynamicsof superconducting NbN thin films studied by time-resolvedterahertz spectroscopyrdquo Physical Review Letters vol 107 no 17Article ID 177007 2011

[109] R Matsunaga and R Shimano ldquoNonequilibrium BCS statedynamics induced by intense terahertz pulses in a supercon-ducting NbN filmrdquo Physical Review Letters vol 109 no 18Article ID 187002 1 page 2012

[110] J-J Chang andD J Scalapino ldquoNew instability in superconduc-tors under external dynamic pair breakingrdquo Physical Review Bvol 10 no 9 pp 4047ndash4049 1974

[111] D J Scalapino and B A Huberman ldquoOnset of an inhomoge-neous state in a nonequilibrium superconducting filmrdquo PhysicalReview Letters vol 39 no 21 pp 1365ndash1368 1977

[112] P W Anderson ldquoTheory of dirty superconductorsrdquo Journal ofPhysics and Chemistry of Solids vol 11 no 1-2 pp 26ndash30 1959

[113] M Strongin R S Thompson O F Kammerer and J ECrow ldquoDestruction of superconductivity in disordered near-monolayer filmsrdquo Physical Review B vol 1 no 3 pp 1078ndash10911970

[114] R C Dynes A E White J M Graybeal and J P GarnoldquoBreakdown of eliashberg theory for two-dimensional super-conductivity in the presence of disorderrdquo Physical ReviewLetters vol 57 no 17 pp 2195ndash2198 1986

[115] D B Haviland Y Liu and AM Goldman ldquoOnset of supercon-ductivity in the two-dimensional limitrdquo Physical Review Lettersvol 62 no 18 pp 2180ndash2183 1989

[116] J M Valles Jr R C Dynes and J P Garno ldquoElectrontunneling determination of the order-parameter amplitude atthe superconductor-insulator transition in 2Drdquo Physical ReviewLetters vol 69 no 24 pp 3567ndash3570 1992

[117] A Yazdani and A Kapitulnik ldquoSuperconducting-insulatingtransition in two-dimensional a-MoGe thin filmsrdquo PhysicalReview Letters vol 74 no 15 pp 3037ndash3040 1995

[118] A M Goldman and N Markovic ldquoSuperconductor-insulatortransitions in the two-dimensional limitrdquo Physics Today vol 51no 11 pp 39ndash44 1998

[119] M A Paalanen A F Hebard and R R Ruel ldquoLow-temperatureinsulating phases of uniformly disordered two-dimensionalsuperconductorsrdquo Physical Review Letters vol 69 no 10 pp1604ndash1607 1992

[120] V F Gantmakher M V Golubkov V T Dolgopolov G ETsydynzhapov and A A Shashkin ldquoDestruction of localizedelectron pairs above themagnetic-field-driven superconductor-insulator transition in amorphous In-O filmsrdquo JETP Letters vol68 no 4 pp 363ndash369 1998

[121] V Y Butko and P W Adams ldquoQuantum metallicity in a two-dimensional insulatorrdquo Nature vol 409 no 6817 pp 161ndash1642001

[122] V F Gantmakher S N Ermolov G E Tsydynzhapov AA Zhukov and T I Baturina ldquoSuppression of 2D supercon-ductivity by the magnetic field quantum corrections vs thesuperconductor-insulator transitionrdquo JETP Letters vol 77 no8 pp 424ndash428 2003

[123] N Hadacek M Sanquer and J-C Villegier ldquoDouble reentrantsuperconductor-insulator transition in thin TiN filmsrdquo PhysicalReview B vol 69 no 2 Article ID 024505 2004

[124] G Sambandamurthy L W Engel A Johansson and D Sha-har ldquoSuperconductivity-Related Insulating Behaviorrdquo PhysicalReview Letters vol 92 no 10 Article ID 107005 2004

[125] G Sambandamurthy LW Engel A Johansson E Peled andDShahar ldquoExperimental evidence for a collective insulating statein two-dimensional superconductorsrdquo Physical Review Lettersvol 94 no 1 Article ID 017003 2005

[126] M Steiner and A Kapitulnik ldquoSuperconductivity in the insu-lating phase above the field-tuned superconductor-insulatortransition in disordered indiumoxide filmsrdquoPhysica C vol 422no 1-2 pp 16ndash26 2005

[127] T I Baturina J Bentner C Strunk M R Baklanov andA Satta ldquoFrom quantum corrections to magnetic-field-tunedsuperconductor-insulator quantum phase transition in TiNfilmsrdquo Physica B vol 359-361 pp 500ndash502 2005

[128] T I Baturina C Strunk M R Baklanov and A Satta ldquoQuan-tum metallicity on the high-field side of the superconductor-insulator transitionrdquo Physical Review Letters vol 98 no 12Article ID 127003 2007

[129] B Sacepe C Chapelier T I Baturina V M Vinokur M RBaklanov andM Sanquer ldquoDisorder-induced inhomogeneitiesof the superconducting state close to the superconductor-insulator transitionrdquo Physical Review Letters vol 101 no 15Article ID 157006 2008

[130] B Sacepe T Dubouchet C Chapelier et al ldquoLocalizationof preformed Cooper pairs in disordered superconductorsrdquoNature Physics vol 7 no 3 pp 239ndash244 2011

[131] S M Hollen H Q Nguyen E Rudisaile et al ldquoCooper-pair insulator phase in superconducting amorphous Bi filmsinduced by nanometer-scale thickness variationsrdquo PhysicalReview B vol 84 no 6 Article ID 064528 2011

[132] V M Vinokur T I Baturina M V Fistul A Y MironovM R Baklanov and C Strunk ldquoSuperinsulator and quantumsynchronizationrdquo Nature vol 452 no 7187 pp 613ndash615 2008

[133] S Poran E Shimshoni and A Frydman ldquoDisorder-inducedsuperconducting ratchet effect in nanowiresrdquo Physical ReviewB vol 84 no 1 Article ID 014529 2011


[134] Y Dubi Y Meir and Y Avishai ldquoTheory of the magnetoresis-tance of disordered superconducting filmsrdquo Physical Review Bvol 73 no 5 Article ID 054509 2006

[135] Y Dubi Y Meir and Y Avishai ldquoNature of the superconductor-insulator transition in disordered superconductorsrdquoNature vol449 no 7164 pp 876ndash880 2007

[136] D Kowal andZOvadyahu ldquoDisorder induced granularity in anamorphous superconductorrdquo Solid State Communications vol90 no 12 pp 783ndash786 1994

[137] D Kowal and Z Ovadyahu ldquoScale dependent superconductor-insulator transitionrdquo Physica C vol 468 no 4 pp 322ndash3252008

[138] N Trivedi R T Scalettar and M Randeria ldquoSuperconductor-insulator transition in a disordered electronic systemrdquo PhysicalReview B vol 54 no 6 pp R3756ndashR3759 1996

[139] Y Imry M Strongin and C C Homes ldquoAn inhomogeneousJosephson phase in thin-film and high-T

119888superconductorsrdquo

Physica C vol 468 no 4 pp 288ndash293 2008[140] M V Feigeirsquoman L B Ioffe V E Kravtsov and E A

Yuzbashyan ldquoEigenfunction fractality and pseudogap statenear the superconductor-insulator transitionrdquo Physical ReviewLetters vol 98 no 2 Article ID 027001 2007

[141] M V Feigelrsquoman L B Ioffe V E Kravtsov and E CuevasldquoFractal superconductivity near localization thresholdrdquo Annalsof Physics vol 325 no 7 pp 1390ndash1478 2010

[142] K Bouadim Y L Loh M Randeria and N Trivedi ldquoSingle-and two-particle energy gaps across the disorder-drivensuperconductor-insulator transitionrdquo Nature Physics vol 7 no11 pp 884ndash889 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


temperature when ℏ1205962Δ is large will approachthose of the normal state (without gap)rdquo

Looking at the experimental results of Blevins et al whomeasured the surface resistance of tin in the millimeter-wave range [14] he suggested some scaling of frequency andtransition temperature close to119879

119888Thiswas further developed

by Tinkham [11] based on his far-infrared transmissionmea-surements on thin lead films [10] where an anomalously hightransmission was observed (cf Figure 1(b)) He suggestedtwo contributions (superconducting and normal electrons)becoming relevant in the limiting cases of low and highfrequencies

ldquoWe consider a model of a superconductor atabsolute zero inwhich a gap ofwidthℏ120596

119892asymp 3119896119861119879119888

appears in the spectrum of one-electron energystates [sdot sdot sdot ] The lossy conductivity 120590

1will be zero

for photon energies ℏ120596 lt ℏ120596119892 For 120596 gt 120596

119892 1205901

will rise gradually with120596 as an increasing fractionof the states originally within ℏ120596 of the occupiedstates below the Fermi level are still available forexcitation If the gap merely excluded the stateswithin ℏ120596

119892leaving all else unchanged this simple

picture would suggest a rise as 1205901= 120590119899(1minus120596

119892120596)

The experimental rise seems to be faster cuttingoff at least as fast as (1 minus 120596

2

1198921205962) This might

correspond to the states displaced from the gapjust ldquopiling uprdquo on either side so that the changeaverages to zero more rapidly for 120596 ≫ 120596

119892rdquo

These ideas are further developed and extended in laterpapers [12 15] where the size of the superconducting energygap and the shape 120590

1(120596) at 120596

119892were 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 thesituation on the superconducting energy gap in a review [17]Previously they had published several thorough microwaveand millimeter-wave results on superconducting aluminumand tin [18ndash20] and several nice papers followed [21ndash23] Atthe same time Tinkham gave an overview on the work of hisgroup [24]While Biondi andGarfunkel alwaysmeasured thetemperature dependence for certain frequencies Tinkhamand his collaborators focussed at the spectral properties atfixed temperatures This allowed him to make the sum-ruleargument introduced in the previous quote [11] which wasfinalized by Ferrell et al in 1959 [25 26] In Figure 1 thefrequency dependence of the electrodynamical properties ofa superconductor at 119879 = 0 is sketched Upon decreasing thefrequency the real part of the conductivity120590

1(120596)120590

119899gradually

vanishes at the gap frequency120596119892= 2Δℏ while the imaginary

part 1205902(120596)120590

119899diverges as 120596 rarr 0 The maximum in the

relative transmission Tr119904Tr119899is observed right at the gap

(with values well above the unity) before it goes to zero for120596 = 0 There is also a maximum of the absorption above thesuperconducting gap but no absorption takes place below thegap 2Δ It poses a real challenge to measure the reflectivityof a metallic superconductor since it approaches unity at 120596

119892

but in general it is well above 99 already in the normal state(Figure 1(c))

The problem remaining was the determination of thesuperconducting conductivity that is the actual frequencyand temperature dependence of 120590

1(120596) and 120590

2(120596) individually

without the assumption of any model or Kramers-Kronigrelation Figure 2 displays temperature and frequency-dependent conductivity 120590

1120590119899as it is calculated fromMattis-

Bardeen theory 1205901120590119899features a minimum at ℏ1205962Δ(119879) =

1 which shifts to lower frequencies and is smeared out as119879 approaches 119879

119888from below Also seen is the enormous

enhancement of the conductivity 1205901(120596)120590

119899for low frequen-

cies observed right below the superconducting gap it becameknown as the coherence peakDespite continuous efforts overthe years [30ndash32] only in 1968 Palmer and Tinkham reachedthis goal of measuring the real and imaginary parts of theelectrodynamic response independently [33] simultaneousmeasurements of the far-infrared transmission through andreflection off thin films of lead made it possible to calculatethe conductivity unambiguously The status is summarizedin several reviews [34 35] The precise measurement ofthe temperature-dependent conductivity right below thesuperconducting transition however was achieved only aquarter of a century later in the group of Gruner [36 37]Previous efforts by the Cambridge group of Pippard andothers were less convincing [38ndash40]The so-called coherencepeak known from the NMR experiments of Hebel andSlichter [7 8] is also seen in the microwave conductivityof lead niobium and eventually even in alumimum [29]where for the first time the complete frequency evolutionof the coherence peak was measured exactly fifty yearsafter it was calculated Still ultrasound absorption remainsa desideratum in the exploration of the coherence effectsa peak is predicted above 2Δ but no frequency-dependentultrasound absorption measurements on superconductorshave been carried out in the vicinity of the energy gap

Even today Tinkhamrsquos book [41ndash43] gives the best intro-duction to the electrodymical properties of conventionalsuperconductors More detailed descriptions can be found byRickayzen [44] and Parks [45] or more recent textbooks [46]

12 Recent Developments Having understood the bulk prop-erties of metallic superconductors fairly well there are stillmany open questions when going to the vortex latticevery thin films nanostructures inhomogeneous materialsgranular superconductors highly disordered films and soforth which are subject of todayrsquos research efforts Thin filmsof metallic superconductors are of both technological andacademic interests

As we will see the properties of interacting electronsystems are governed by the electronic density of states[47] which changes upon reducing spatial dimensionsThusdimensional reduction from three dimensions to two dimen-sions (3D rarr 2D) may affect optical electronic and ther-modynamic properties of the system In superconductivitystudying quasi-two-dimensional thin-film systems turns outto be a particularly fruitful field and leads to remarkable find-ings such as insulators featuring a superconducting gap [48]


00

05

10

15

20

0 1 2 3 4 5

Con

duct

ivity


1205902120590n

1205901120590n

(a)

0 1 2 3 4 500

05

15

10

20

25

Tran

smiss

ion

Abso

rptio

n


AsAn

TrsTrn

(b)

0 1 2 3 4 50997

0998

0999

1000

Refle

ctiv

ity



(c)


1and 120590








1205901120590

n

4

2

002

0406

0810

12 0002

0406

08

TTc

ℏ1205962Δ(0)






119888(after [29])







119888cuprates






2 Theory



1205901 (120596 119879)

120590119899

=2

ℏ120596int

infin

Δ

[119891 (E) minus 119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

+1

ℏ120596int

minusΔ

Δminusℏ120596

[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(1)

1205902 (120596 119879)

120590119899

=1

ℏ120596int

Δ


[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(Δ2 minusE2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(2)


119861119879 minus 1)


1120590119899


119888


119888) this is







119892determined






1(120596) and 120590

2(120596)



1205902 (120596) =

1198731198902

119898120596=

1198882

41205871205822

119871120596 (3)


= (11989811988824120587119873

1199041198902)12 is




1205901 (120596) =

120587

2

1198731199041198902

119898120575 120596 = 0 =

1198882

81205822

119871

120575 120596 = 0 (4)







1(120596)120590

119899prop 1minus(ℏ120596119896



119888




Ener

gy

EF = 0


T gt Tc

(a)

Ener

gy

EF = 0

Δ(T) + ℏ120596

Δ(T)

minusΔ(T)



0 lt T lt Tc

(b)

Ener

gy EF = 0

minusΔ(T = 0)

Δ(T = 0)


T = 0

(c)







119892= 2Δℏ The



119865 (ΔEE1015840) =

1

2(1 +

Δ2

EE1015840) (5)




(1205901

120590119899

)

maxsim log 2Δ (0)

ℏ120596 (6)




1205902 (119879)

120590119899

asymp120587Δ (119879)

ℏ120596tanhΔ (119879)

2119896119861119879 asymp lim119879rarr0

120587Δ (0)

ℏ120596 (7)


Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)


1120591 = 40 times 2Δ(0)

T = 08Tc



00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc


0

0

0

1 2 3



Increasing 1120591

09 Tc


Im (120590

sc) =

1205902

01 Tc

(b)




1and 120590


119888= 09 08 07 06 05 04



000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2


T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)


(a)The real part 1205901(120596)120590




2(120596)120590

119899



2(120596) [46]




120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

05

10

15

20

0 1 2 3 4 5

Con

duct

ivity


1205902120590n

1205901120590n

(a)

0 1 2 3 4 500

05

15

10

20

25

Tran

smiss

ion

Abso

rptio

n


AsAn

TrsTrn

(b)

0 1 2 3 4 50997

0998

0999

1000

Refle

ctiv

ity



(c)


1and 120590








1205901120590

n

4

2

002

0406

0810

12 0002

0406

08

TTc

ℏ1205962Δ(0)






119888(after [29])







119888cuprates






2 Theory



1205901 (120596 119879)

120590119899

=2

ℏ120596int

infin

Δ

[119891 (E) minus 119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

+1

ℏ120596int

minusΔ

Δminusℏ120596

[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(1)

1205902 (120596 119879)

120590119899

=1

ℏ120596int

Δ


[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(Δ2 minusE2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(2)


119861119879 minus 1)


1120590119899


119888


119888) this is







119892determined






1(120596) and 120590

2(120596)



1205902 (120596) =

1198731198902

119898120596=

1198882

41205871205822

119871120596 (3)


= (11989811988824120587119873

1199041198902)12 is




1205901 (120596) =

120587

2

1198731199041198902

119898120575 120596 = 0 =

1198882

81205822

119871

120575 120596 = 0 (4)







1(120596)120590

119899prop 1minus(ℏ120596119896



119888




Ener

gy

EF = 0


T gt Tc

(a)

Ener

gy

EF = 0

Δ(T) + ℏ120596

Δ(T)

minusΔ(T)



0 lt T lt Tc

(b)

Ener

gy EF = 0

minusΔ(T = 0)

Δ(T = 0)


T = 0

(c)







119892= 2Δℏ The



119865 (ΔEE1015840) =

1

2(1 +

Δ2

EE1015840) (5)




(1205901

120590119899

)

maxsim log 2Δ (0)

ℏ120596 (6)




1205902 (119879)

120590119899

asymp120587Δ (119879)

ℏ120596tanhΔ (119879)

2119896119861119879 asymp lim119879rarr0

120587Δ (0)

ℏ120596 (7)


Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)


1120591 = 40 times 2Δ(0)

T = 08Tc



00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc


0

0

0

1 2 3



Increasing 1120591

09 Tc


Im (120590

sc) =

1205902

01 Tc

(b)




1and 120590


119888= 09 08 07 06 05 04



000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2


T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)


(a)The real part 1205901(120596)120590




2(120596)120590

119899



2(120596) [46]




120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2 Theory



1205901 (120596 119879)

120590119899

=2

ℏ120596int

infin

Δ

[119891 (E) minus 119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

+1

ℏ120596int

minusΔ

Δminusℏ120596

[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(E2 minus Δ2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(1)

1205902 (120596 119879)

120590119899

=1

ℏ120596int

Δ


[1 minus 2119891 (E + ℏ120596)] (E2 + Δ2 + ℏ120596E)

(Δ2 minusE2)12[(E + ℏ120596)

2minus Δ2]

12119889E

(2)


119861119879 minus 1)


1120590119899


119888


119888) this is







119892determined






1(120596) and 120590

2(120596)



1205902 (120596) =

1198731198902

119898120596=

1198882

41205871205822

119871120596 (3)


= (11989811988824120587119873

1199041198902)12 is




1205901 (120596) =

120587

2

1198731199041198902

119898120575 120596 = 0 =

1198882

81205822

119871

120575 120596 = 0 (4)







1(120596)120590

119899prop 1minus(ℏ120596119896



119888




Ener

gy

EF = 0


T gt Tc

(a)

Ener

gy

EF = 0

Δ(T) + ℏ120596

Δ(T)

minusΔ(T)



0 lt T lt Tc

(b)

Ener

gy EF = 0

minusΔ(T = 0)

Δ(T = 0)


T = 0

(c)







119892= 2Δℏ The



119865 (ΔEE1015840) =

1

2(1 +

Δ2

EE1015840) (5)




(1205901

120590119899

)

maxsim log 2Δ (0)

ℏ120596 (6)




1205902 (119879)

120590119899

asymp120587Δ (119879)

ℏ120596tanhΔ (119879)

2119896119861119879 asymp lim119879rarr0

120587Δ (0)

ℏ120596 (7)


Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)


1120591 = 40 times 2Δ(0)

T = 08Tc



00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc


0

0

0

1 2 3



Increasing 1120591

09 Tc


Im (120590

sc) =

1205902

01 Tc

(b)




1and 120590


119888= 09 08 07 06 05 04



000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2


T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)


(a)The real part 1205901(120596)120590




2(120596)120590

119899



2(120596) [46]




120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Ener

gy

EF = 0


T gt Tc

(a)

Ener

gy

EF = 0

Δ(T) + ℏ120596

Δ(T)

minusΔ(T)



0 lt T lt Tc

(b)

Ener

gy EF = 0

minusΔ(T = 0)

Δ(T = 0)


T = 0

(c)







119892= 2Δℏ The



119865 (ΔEE1015840) =

1

2(1 +

Δ2

EE1015840) (5)




(1205901

120590119899

)

maxsim log 2Δ (0)

ℏ120596 (6)




1205902 (119879)

120590119899

asymp120587Δ (119879)

ℏ120596tanhΔ (119879)

2119896119861119879 asymp lim119879rarr0

120587Δ (0)

ℏ120596 (7)


Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)


1120591 = 40 times 2Δ(0)

T = 08Tc



00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc


0

0

0

1 2 3



Increasing 1120591

09 Tc


Im (120590

sc) =

1205902

01 Tc

(b)




1and 120590


119888= 09 08 07 06 05 04



000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2


T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)


(a)The real part 1205901(120596)120590




2(120596)120590

119899



2(120596) [46]




120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re (120590

sc) =

1205901

1205901

0 0 1

1120591 = 40 times 2Δ(0)


1120591 = 40 times 2Δ(0)

T = 08Tc



00 1 2 3


Increasing 1120591

120590dc

01Tc

(a)

1205902

0

1120591 = 40 times 2Δ(0)

Decreasing T

T = 08Tc


0

0

0

1 2 3



Increasing 1120591

09 Tc


Im (120590

sc) =

1205902

01 Tc

(b)




1and 120590


119888= 09 08 07 06 05 04



000 1 2 3

1205901(120596)120590n

15

10

05

06

03T = 0 Case 2


T = 09Tc

(a)

1205902(120596)120590n

2

1

00 1 2 3

T = 0

06


T = 09Tc

(b)


(a)The real part 1205901(120596)120590




2(120596)120590

119899



2(120596) [46]




120582119871= (

1198981198882

41205871198731198902)

12

=119888

120596119901

(8)


1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1205901(T)120590n

2

1

0

Case 2 ℏ1205962Δ = 0005

005

02

07


(a)

1205902(T)120590n

100

10

1

01

ℏ1205962Δ = 0005

005

02

07


(b)









=



1205850=ℏV119865

120587Δ(9)




ℓ = V119865120591 (10)








119865Δ and



119888 is smaller



119899





int

infin

minusinfin

[120590coll1

+ 120590sp1] 119889120596 = int

infin

minusinfin

120590119899

1119889120596 =

120587

2

1198731198902

119898(11)


int

infin

0+

[120590119899

1minus 120590119904

1] 119889120596 = 119860 (12)




1205901 (120596) =

1198882

81205822

119871

120575 120596 = 0 (13)


Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Anomalous regime

Pipp

ard

limit

Lond

on li

mit


Local regime

985747 gt 1205850 gt 120582L

985747 gt 120582L gt 1205850

120582L gt 985747 gt 1205850

120582L gt 1205850 gt 985747

1205850 gt 985747 gt 120582L

1205850 gt 120582L gt 985747

120582L = 1205850

985747=120585 0

985747=120582 Llo

g(1205850120582

L)

log(1205850985747)

log(120582L 985747)






120582 =119888

radic8119860or 119860 =

1198882

8120582 (14)


1205901 (120596 = 0) =

1205871198731199041198902

2119898120575 120596 = 0 (15)



1205901(2Δℏ) = 2119873119890


119860 =1198882

2120587

120591Δ

ℏ (16)





120582 (ℓ) asymp 120582119871radic1205850

ℓ (17)



119904and



119888 ie sim10 K) 119877

119899= (00538Ω)119885

0(where


119899(119879 =














1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










1is governed




2(120596)120590





119888in the low-



119888






















101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





101

100

10minus1

10minus2

10minus3

10minus4

10minus5

Surfa

ce re

sista

nceR

sRn

Surfa

ce re

acta

nceX

sRn

100

050

075

025

0

RsRnXsRn

Nb

(a)

0

20

15

10

05

0

0 025 050 075 100

20

15

10

5

0

Con

duct

ivity1205901120590n

1205901120590n

Con

duct

ivity1205902120590n

1205902120590n

Temperature TTc

(b)


119904(119879) (right axis) of







4

3

2

1

0

1205901120590n

Al film C

d = 50nm

985747 = 18nm

Tc = 19K

3 GHz20 GHz25 GHz

29 GHz32 GHz

T (K)10 15 20

(a)

20

10

0

1205902120590n

2 GHz5 GHz

8 GHz10 GHz

T (K)10 15 20

(b)




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

1

2

3

50 K85 K105 K

118 K125 K


Tran

smiss

ion

Tr(T

)Tr

(T=14

K)

(a)

0

05

1

15

Abso

rptio

nA(T)A

(T=14

K) T = 108K

T = 53K

NbN on SiTc = 133K


(b)

0 50 100 150

09

08

10

08

09

10

30 40 50 60

Refle

ctiv

ityR

T = 155K

T = 53K


(c)

0

05

1

15

2


Con

duct

ivity120590120590n

1205901120590n

1205902120590n


(d)











8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






8 T6 T4 T2 T1 T0 T

15

10

00

05

15

10

00

05



Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)



8 T6 T4 T2 T1 T0 T



Sample Radiation

Magneticfield

Faraday geometry

E

B

(a)



8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T

00

15

10

00

05

15

10

05





8 T

10 T

6 T

4 T2 T

0 T

8 T

10 T

6 T

4 T

2 T

0 T



Sample Radiation

Magneticfield

E

B

Voigt geometry

Tran

smiss

ion

(Tr sc

Tr n

)Tr

ansm

issio

n (T

r scT

r n)

(b)



0|| 119864) as shown


















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



















119892





1(120596) and120590




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




001

01

10

0 10 20 30 40

NbN on sapphire

0 1 2 3 4

300 K

5 KTran

smiss

ion

TrF



(a)

002

00

004

006

008

01

0 10 20 30 40

300 K

5 K

Tran

smiss

ion

TrF

0 1 2 3 4Energy ℏ120596 (meV)


(b)

2 10 100 5000

5

10

Temperature T (K)

8 cmminus1


minus1)

(c)




119865(]) of a






119889


119889= 100mA) to 85 K (200mA)


119865ℓ (with 119896






119888lt 119879 lt 08119879







119861119879119888






0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 5 10 15 20

Nb

Am

plitu

de (a

u)

minus15

minus1

minus05

0

05

147 K

7 K

Time (ps)

(a)

0

00

05

1

15

05 10

1 2 3 4

Frequency f (THz)


Con

duct

ivity120590120590n

1205901120590n

1205902120590n

T = 47K

(b)





119861119879119888

Nb filmon sapphire

10minus5

10minus4

10minus3

10minus2

10minus4

10minus3

10minus2


TrFTr

ansm

issio

n TrF

6 K

9 K


57 cmminus1

288 cmminus1

(a)

04100 20 30

06

08

06

08


57 cmminus1

288 cmminus1

6 K

9 K120601

(rad

cmminus1)

Phas

e shi

ftfre

quen

cy (r

adc

mminus1)


(b)








3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




3

2

1

0

1205901

(105Ωminus1cm

minus1)

Tc = 831K

Data Fit9 K7 K45 K

Nb


(a)

8

4

0

10

05

000 4 8

1205902

(105Ωminus1cm

minus1)

T (K)


1205822(0)1205822(T)

57 cmminus1

BCS

10 100

(b)






6000

5000

4000

3000

2000

1000

0

1205901(Ω

minus1cm

minus1)

20 K

7 K

42 K


(a)

0

2800024000200001600012000

80004000

1205902(Ω

minus1cm

minus1)

7 K

42 K

0∘

90∘BCS (0∘)BCS (90∘)

NbN 570Tc = 96K


(b)





0of the light

pulse



4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4

3

2

1

00 2 4 6 8 10 12

302520151050

NbNTc = 122K2Δ

(meV

)

2 Δ(c

mminus1)

+ axisminus axis


Temperature (K)

(a)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

(b)


44403632M

ean

ener

gy g

ap(4

2 K

) (m

eV) 45

423936 2Δ

(0)kBTc

Weak-coupling BCS

(c)


119861119879119888versus 119879


[72])




2(119879) The




1(120596)




4 Outlook



6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




6 9 12 150

1

2

3

0 1TR4

3

2

1

0

2Δ(T

Hz)

T (K)

0 1 2 3 4 (THz)

1205901(120596)(104Ωminus1cm

minus1)

(a)

430 K1000 K1300 K

1400 K1475 K1600 K

0 1 2 3 4

8

6

4

2

0

(THz)

minus1 0 1

5

0

minus5

16 K

4 K

t998400 (ps)

t0

Etr

(au

)

1205902(120596)(104Ωminus1cm

minus1)

(b)


1000

minus08

minus06

minus04

minus02

00

Δ1205902120590n

03 A0

10 A0

20 A0

30 A0

(c)

0 1TR

1

2

3

1 10 100Time (ps)

(T

Hz)

(d)









01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




01

001

120575ΔΔ

0

td (ps)

355

141

071

035

014

0 20 50 100

(a)

120575ΔΔ

0

A (mJcm3)01 1 10 100

0

1

Asa

t(m

Jcmminus3)

20

00 5 10 15

T (K)

T = 43K

(b)

355141071 71

1106

400

200

A (mJcm3)0 10

0

0

1

5 10 15T (K)

T = 43K

1120591 r


120591 rec

(ps)

(c)





0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

1

2

3

0 2 4 6 8 10Energy (meV)

16 K

T = 4K

1205901

(104Ωminus1cm

minus1)



(a) experiment

00 2 4 6 8 10

1

2

3

4

5

Energy (meV)

T = 4K

1205902

(104Ωminus1cm

minus1)



(b) experiment

0 2 4 6 8 10Energy (meV)

0

1

2

3

1205901

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(c) calculation

0 2 4 6 8 10Energy (meV)

0

1

2

3

4

5

1205902

(104Ωminus1cm

minus1)

0f01f02f04f

06f07f08f

(d) calculation











◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





◻



119888and thus are





Acknowledgments


References



























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Review Article Electrodynamics of Metallic...

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