Relaxation of the resistive superconducting state in

boron-doped diamond films

A. Kardakova, A. Shishkin, A. Semenov, G. N. Goltsman, S. Ryabchun, T. M.

Klapwijk, J. Bousquet, D. Eon, B. Sacepe, Th. Klein, et al.

To cite this version:

A. Kardakova, A. Shishkin, A. Semenov, G. N. Goltsman, S. Ryabchun, et al.. Relaxationof the resistive superconducting state in boron-doped diamond films. Physical Review B :Condensed matter and materials physics, American Physical Society, 2016, 93 (6), pp.064506.<10.1103/PhysRevB.93.064506>. <hal-01319638>

HAL Id: hal-01319638

https://hal.archives-ouvertes.fr/hal-01319638

Submitted on 21 May 2016

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Relaxation of the resistive superconducting state in boron-doped diamond films

A. Kardakova∗ and A. ShishkinPhysics Department, Moscow State Pedagogical University, Russia

A. SemenovPhysics Department, Moscow State Pedagogical University, Russia and

Moscow Institute of Physics and Technology, Russia

G.N. Goltsman† and S. RyabchunPhysics Department, Moscow State Pedagogical University, Russia and

National Research University Higher School of Economics, Russia

T.M. Klapwijk‡

Physics Department, Moscow State Pedagogical University, RussiaKavli Institute of Nanoscience, Delft University of Technology, The Netherlands and

Donostia International Physics Center, Donostia-San Sebastian, Spain

J. Bousquet, D. Eon, B. Sacepe, Th. Klein, and E. BustarretInstitut Neel, CNRS, Grenoble, France andUniversite Grenoble Alpes, Grenoble, France

(Dated: January 12, 2016)

We report a study of the relaxation time of the restoration of the resistive superconducting statein single crystalline boron-doped diamond using amplitude-modulated absorption of (sub-) THzradiation (AMAR). The films grown on an insulating diamond substrate have a low carrier densityof about 2.5 × 1021 cm−3 and a critical temperature of about 2 K. By changing the modulationfrequency we find a high-frequency roll-off which we associate with the characteristic time of energyrelaxation between the electron and the phonon systems or the relaxation time for nonequilibriumsuperconductivity. Our main result is that the electron-phonon scattering time varies clearly asT−2, over the accessible temperature range of 1.7 to 2.2 K. In addition, we find, upon approachingthe critical temperature Tc, evidence for an increasing relaxation time on both sides of Tc.

Usage:

PACS numbers: 74.70.Wz, 72.10.-d, 74.62.En, 74.25.N , 74.40.Gh

Structure:

I. INTRODUCTION

Electron-phonon (e-ph) scattering is well-understoodin clean bulk normal metals, and is part of standard text-books on solid-state physics. Within that framework,it is also embedded in the theory of superconductivity,as well as in that of non-equilibrium superconductivity,where the energy relaxation rate is a crucial parameter1,for example in applications where hot-electron effects areexploited. In practice, one uses normal-metal or super-conducting thin films, in which there is a high degree ofimpurity scattering dominating the resistivity. Thin filmswith impurity scattering have been extensively studied inthe field of weak localization, where the energy relaxationprovides is one of the contributions to the single parti-cle phase-coherence length. Quite generally the impurityscattering is considered to be elastic, only changing thedirection of momentum, and derivable from the observedresistivity at temperatures above Tc. This common ap-proach neglects the fact that inelastic electron-phononscattering is strongly affected by the presence of impu-rities, or strong disorder. Moreover, it overlooks that

in many cases of practical interest impurity scattering ispredicted to be not elastic but also inelastic, i.e. con-tributing to the energy relaxation and phase-breaking.

In particular, various interference effects modify thetemperature dependence of the relaxation rate. In im-pure metals, where the electronic mean free path is lessthan the wavelength of a thermal phonon, the electron-phonon interaction is found to be suppressed in compar-ison to the clean case2–5, and at low temperatures therelaxation rate evolves from the standard T 3 dependencein pure metals to T 4 in impure metals. In this case,the theory assumes that the impurities and defects vi-brate in phase with the host atoms of the lattice (theso-called complete drag of impurities). More recently, itwas pointed out that in the case of even a small differ-ence in the vibrations of the electron scatterers and thehost atoms an enhancement of the electron-phonon inter-action is expected6. Such disorder-enhanced relaxation,with a T 2 dependence, was reported for normal metals inmeasurements of the phase-breaking length in a varietyof metallic alloys, and summarized by Lin and Bird7. Analternative method to study the inelastic aspects of impu-

2

rity scattering is by heating the electrons and measuringdirectly the temperature difference between the electronbath and the phonon bath8. The applicability of this ex-perimental method to real materials depends strongly onthe compatibility with the fabrication technology.

We studied superconducting boron-doped diamondfilms grown on diamond substrates. An important ex-perimental advantage compared to many previous thin-film studies is that there is no acoustic mismatch betweenthe phonons the film and the substrate, and the unifiedphonon bath should be in equilibrium at the bath tem-perature. Superconductivity in diamond was first foundby Ekimov et al.9 in polycrystalline material and by Bus-tarret et al.10 in single-crystal thin films. Boron dopesinto a shallow acceptor level close to the top of the va-lence band that is separated from the conduction bandof diamond by Eg ≈ 5.5 eV. At low boron concentrationsnB ≈ 1017 − 1019 cm−3 the material is semiconducting.If the doping concentration nB exceeds the critical value(> 1020 cm−3), the system passes through an insulator-to-metal transition and shows metallic behaviour11. At aboron concentration of ≥ 5× 1020 cm−3 superconductiv-ity is observed, with an increase in the critical tempera-ture Tc with increasing carrier concentration. Crystallinediamond is therefore an attractive model system for thestudy of the impurity scattering at low temperatures6.The boron dopants provide charge carriers but also playthe main role for impurity scattering12. Superconduc-tivity has been attributed to the optical phonons aris-ing from the B-C stretching mode12 and these phononsare therefore direcly related to the acceptor atoms. Tak-ing into account the absence of a Kapitza resistance forphonons, the CVD-grown boron-doped diamond filmsform a unique system to study electron-phonon inter-action processes. We use amplitude-modulated absorp-tion of (sub-) THz radiation (AMAR), introduced byGershenzon et al.13, suitable for the study of electron-phonon processes in various thin films, provided that theybecome superconducting at an experimentally accessibletransition temperature.

Our main result is that the relaxation of the resistivesuperconducting state of boron-doped diamond is con-trolled by an electron-phonon inelastic scattering rate,which varies as T 2. In addition we find that on bothsides of Tc the observed relaxation time increases, sug-gesting a divergent behavior upon approaching Tc.

II. DETERMINATION OF THEENERGY-RELAXATION TIME IN THIN

SUPERCONDUCTING FILMS

The method introduced by Gershenzon et al.13

(AMAR) allows to measure the energy relaxation ratebetween electrons and phonons. A superconducting filmis brought into a regime where the film is in a supercon-ducting resistive state (cf. Fig. 1).

A small dc current IDC is applied and the voltage VDC

VDC (mV)

I DC (μ

A)

0 2 3 4 5

20

40

60

RL

Tcr = 2.2 K, B = 0

2.185 K2.205 K2.215 K2.240 K

δV

RL

PRF

t tδV

normal state

FIG. 1. Current-voltage curves at different temperatures nearthe critical temperature T r

c in zero magnetic field. Under theRF power an operation point (I = 20 µA and V = 1 mV)shifts along the equivalent load resistance line that producesthe voltage signal δV . The inset shows the equivalent circuitfor determining an output signal of the superconducting filmupon the absorption of the amplitude-modulated radiation.

is monitored, while an amplitude-modulated signal froma submillimeter source is directed at the film. The mod-ulation frequency for the amplitude is ωm. The absorbedradiation power causes an increase of the electron tem-perature Te (cf. Fig. 2), which leads to an increase ofthe film resistance δR followed by a voltage signal pro-portional to the bias current δV = IδR. It is found thatthis method provides a frequency-dependent roll-off (cf.Fig. 3), which is taken as the measure of the energy-relaxation rate. Application of a perpendicular magneticfield brings the film into a resistive state at various tem-peratures, which makes it possible to measure the energy-relaxation rate as a function of temperature, althoughover a limited range of temperatures. The technique hasbeen applied to various materials, usually providing dif-ferent temperature dependences of the relaxation rate,for example T 2 in Nb film13, T 1.6 in NbN14 and T 3 inTiN15 and in NbC16.

Inelastic relaxation times are important in non-equilibrium superconductivity, in hot-electron effects innormal metals and in magnetoresistance due to quantum-interference processes. In the latter, the normal-metalphase-breaking time τφ is limiting the phase coherenceof elastically scattered electron waves. It is usually as-sumed to be limited by inelastic electron-phonon scatter-ing and at lower temperatures by electron-elecron scat-tering. Gershenzon et al.13 have made a comparisonof results based on weak-localization experiments withthose obtained with the AMAR method. The outcome,for strongly disordered niobium films with the elasticmean free path in the 1-nm range and a Tc of 3.2 to8.5 K, is that the results are comparable in the tem-

3

perature range from 10 to 20 K. At lower temperaturesthe weak-localization results differ quite strongly, both inthe temperature dependence and in the absolute value,which is attributed to electron-electron scattering as thelimiting process for weak localization. It supports theassumption that an elevated electron temperature canbe assigned to the electron system in comparison tothe phonon-temperature. Although the complexities ofelectron-electron scattering, in particular in relation toits material-dependence, have meanwhile become muchmore detailed 17–21, we assume that the basic premise ofthe existence of an electron temperature is justified.

A boron-doped diamond film is considered a thermo-dynamic system composed of two interacting subsystems:electrons and phonons which are coupled via electron-phonon interaction22. If the film is exposed to amplitude-modulated radiation, the temperature of the electronsubsystem will change accordingly. The amplitude of thetemperature response will depend on the modulation fre-quency and the time constant of the electron subsystem.This time constant will be a function of the electronicspecific heat and the heat conductance between the elec-trons and the phonons in the film. Due to the absence ofa film-substrate interface for phonons between the doped-layer and the substrate we assume that the escape time ofnon-equilibrium phonons into the substrate is very short,considerably less than the resistance-relaxation times wefind.

The resistance relaxation time is determined from 3-dBroll-off of the frequency dependence of the amplitude ofthe output voltage δV (ωm). Since the diamond film hasno Kapitza resistance, the phonons can be treated as aheat bath in equilibrium with the cryogenic environment.Then the dynamics of the film can be described by asingle heat-balance equation:

CedTedt

= −G(Te − Tb) + PDC + PRF , (1)

with Ce the heat capacity of the electrons, Te the electrontemperature, Tb the phonon-bath temperature, G theheat conductance from the electrons to the phonon-bath,PDC = I2R the Joule power dissipated in the film, andPRF the absorbed radiation-power. Eq.(1) is valid undertwo assumptions. The first one is that the Joule heatingand the RF drive are sufficiently weak so that the depar-ture of Te from Tb is small in the sense |Te − Tb| � Tb.This regime is achieved when the distance L between thecontact pads is larger than the thermal diffusion lengthLdiff =

√Dτe−ph (with D the electronic diffusion coef-

ficient and τe−ph the electron-phonon interaction time).As will be shown below, the condition Ldiff � L is sat-isfied in our case. In the experiments the radiation powerwas modulated, so that PRF (t) = P0 + P1exp(iωmt).This allows us to use the results of the lumped-circuitmodel for hot-electron bolometers given by Karasik andElantiev23 for the power of the response signal generatedby the film:

metal/SC filminsulating substrate

p+ diamond epilayerdiamond substrate

electron-phononinteraction

(a) (b)

ħω

Phonons

Te

TphPhonons in film

Phonons in substrate

f(E)ElectronsQuasi-particles/ Cooper-pairsVortices/ Fluctuations

ħω

f(E)Te

ElectronsQuasi-particles/ Cooper-pairsVortices/ Fluctuations

Tph

Tph

electron-phononinteraction

phonon transport

FIG. 2. Coupling between the thermodynamic subsystems inthe case of (a) a thin metal or superconducting (SC) film onan insulating substrate; (b) a boron-doped diamond film ona diamond substrate, illustrating, in comparison to (a) theabsence of a Kapitza resistance. The electron reservoir underillumination can be described by the Fermi-Dirac distributionfunction f(E) with an effective electron temperature Te ex-ceeding the phonon temperature Tph. In practice, the metalfilm is in a resistive superconducting state which is due tovortices, phase and/or amplitude fluctuations. It is assumedthat the electron temperature Te, increased by DC power andby RF power, controls the changes of the resistivity of thesuperconducting state.

Pout(ωm) =P0

1 + (ωmτB)2, (2)

with P0 the power for low modulation frequency and

τB =τe−ph1 + α

,with α =I2

G

∂R

∂Te

R−RLRT +RL

. (3)

In Eq. (3), R = V/I, is the Ohmic resistance at theoperating point, RL is the equivalent load resistance de-termined by the read-out electronics and the bias circuit,RT = R+I(∂R/∂I). By plotting Pout(ωm) we determinethe time constant τB . Choosing the operating point sothat the current is as small as possible, while the re-sponse power is still measurable, allows minimizing theparameter α. Physically, in this regime the Joule heat-ing is minimal and is assumed not to slow down the en-ergy relaxation process of the electron system. Keepingthe current low also reduces non-thermal effects such asvortex-creation or enhanced phase-slip rates, which arenot included in Eq.(1).

An equivalent electrical circuit for analysing theresponse of the superconducting film to amplitude-modulated radiation is presented in the inset of Fig. 1.

4

As a radiation source, we use a backward-wave oscillator(BWO) with a carrier frequency of 350 GHz. The BWOpower is amplitude-modulated at frequencies of 10 to2000 kHz. The response voltage from the film, δV (ωm),and the frequency ωm are measured with a spectrum an-alyzer. To determine the temperature dependence of theresistance-relaxation time we varied the bath tempera-ture and applied a magnetic field to get into a usableresistive superconducting state by the creation of vor-tices.

A crucial ingredient of this AMAR method is the ex-ploitation of the resistive superconducting state. Theobserved relaxation is in essence the restoration of theresistive superconducting state after exposure to radia-tion with a frequency higher than the energy gap of thesuperconductor. For the results presented here we use,in practice, two types of resistive superconducting states.

Case A: the resistive transition of a superconduct-ing film in zero magnetic field. Above the mean fieldcritical temperature Tc, the resistive transition is deter-mined by amplitude fluctuations of the order parameter,and known as Aslamazov-Larkin24 and Maki-Thompsoncontributions25,26. Below Tc it is, for one-dimensional su-perconductors, determined by thermally activated phaseslip events. For a 2-dimensional film the situation ismore complex. There exists a well-defined regime forfilms with a high resistance per square, where the emer-gence of resistivity is controlled by the theory of theBerezinskii-Kosterlitz-Thouless (BKT) model, which fo-cuses on the macroscopic phase-fluctuations. In this the-ory a superconducting film upon approaching Tc will firstpass another critical temperature, TBKT , where vortex-antivortex pairs unbind, providing free vortices. Thesefree vortices will move under the influence of a transportcurrent and will therefore provide a voltage across thesuperconductor and makes it appear resistive. The tem-perature dependence of the resistivity, the exponentialrise at the resistive transition, is due to the increasingpresence of free vortices. The relevant temperature isthe electron temperature, since it controls the supercon-ducting properties including the density of free vortices.As shown by Kamlapure et al.27 a detailed analysis ofthe resistive superconducting properties in terms of theBKT theory can be made, for example for NbN, providedfinite-size effects of the films are properly taken into ac-count. For the diamond films studied here, with a not toohigh sheet resistance, the expected TBKT is close to Tc,which rules out such an analysis based only on the phase.In the regime where TBKT is close to Tc, the emergence ofresistance is then not exclusively controlled by the vortexdensity occurring in the BKT theory, but includes bothphase and amplitude fluctuations of the order parame-ter. Therefore, the observed resistance is most likely dueto the interplay of the time-dependent phase differencesand non-equilibrium conversion currents, as was studiedexperimentally by Carlson and Goldman28–30. Becauseof this complexity a quantitative description of the emer-gence of resistance in a 2-dimensional superconducting

film cannot be based on a sharply delimited conceptualframework31. It is known that for uniform systems in thelimit of ∆ � kT the relaxation of non-equilibrium stateinduced by radiation should be called longitudinal non-equilibrium32. Its relaxation is controlled by the electrontemperature Te, by the mean-field critical temperatureTc and the inelastic relaxation rate τE . In many cases τEis the elecron-phonon time, which itself is temperaturedependent. In addition there is the temperature depen-dence related to the restoration of the superconductingstate, which is dependent on (1− Te/Tc)−1/2.

In practice a dc bias is also used to move the resistivetransition to lower temperatures, which allows a range oftemperatures close to Tc to be accessed. The shift of theresistive transition is due to the fact that a transport cur-rent also contributes to the creation of extra free vorticesin the BKT theory33.

Case B: the resistive state of the superconductor isreached by applying a perpendicular magnetic field,which creates vortices, with their flux in the directionof the applied magnetic field above the field Bc1. Witha current applied these vortices move under the Lorentzforce, provided the force exceeds the pinning force. Thisflux-flow regime including the breakdown of collectiveflux-pinning was studied recently for NbN films by Lin etal.34. Case B enables carrying out measurements over alarger range of temperatures. By choosing a bath temper-ature Tb and adjusting the magnetic field close to Bc2 atthat temperature, we access the temperature-dependentresistive superconducting state. By appying a low bias-current we choose a resistive state where a measurablevoltage response exists. Also in this case the resistive su-perconducting state is controlled by the electron temper-ature. In contrast to Case A the superconducting stateis in principle in the regime ∆ � kTc. In this way itis possible to measure a voltage from the resistive super-conductor, caused by the modulation of the electron tem-perature as a function of applied modulation frequencyat different bath temperatures. The main assumption isthat the changes of the resistive superconducting state,caused by flux flow, with absorbed power are due to arise in electron temperature and do not contain any cor-rections due to the fact that a magnetic-field-induced re-sisitive superconducting state is used. In other words, thefact that the resisitivity is due to flux-flow processes andby the density of vortices is not effecting the observationsin a significant way. The only significant parameter is theeffective electron temperature for a given B-field and thecurrent, in comparison with the phonon-bath tempera-ture. In addition, it is experimentally verified that theobserved response does not depend on the level of themicrowave power (linear regime in power).

With the assumptions stated above the temporal re-sponse of the resistive superconducting state, which weobserve in the experiment, serves in all cases as a measureof the temporal response of the electron temperature. Forthe regime of the time constants that we find this seemslike a justified assumption. However, since we are ob-

5

TABLE I. Parameters of the films

Sample d R� ρ T rc D α p

(nm) (Ohm) (µOhm×cm) (K) (cm2s−1) (µs×Kp)N1 300 50 1500 2.245 1.38±0.04 1.91 1.88±0.05N2 70 220 1540 2.195 1.30±0.02 1.92 2.06±0.05

serving the resistivity of the superconducting state therestoration of the superconducting state adds in princi-ple an additional temperature dependence around Tc.

III. SAMPLES

Two p+ epilayers of diamond were grown in a home-made vertical silica tube reactor35 by Microwave Plasma-enhanced Chemical Vapor Deposition (MPCVD) on 0.3×3 × 3 mm3 (001)-oriented type Ib diamond substrates,on top of a 500-nm-thick non-intentionally doped bufferlayer. The growth was carried out at 880 ◦C in a gasmixture of H2, CH4, and B2H6. The total pressure was33 torr, i.e. 44 hPa, and the total gas flow was 100 sccm.The molar methane-to-hydrogen ratio was 3.5 %, andthe boron-to-carbon molar ratio in the introduced gasmixture was 0.25 % for sample N1 and 0.33 % for sam-ple N2. The other difference between the two sampleswas the duration of the growth, leading to thicknesses dof 300 and 70 nm for samples N1 and N2, respectively.Four parallel silver-paste contacts drawn across the wholesample were used to measure the sheet resistance R� ofthe film at 300 K, and the thickness was deduced fromspectroscopic ellipsometry measurements36 performed insitu37. This allowed the determination of the resistivityρ = R�d, which was very similar for both samples. Thecritical temperature T rc was determined as the tempera-ture of the mid-point of the resistive transition where thesample’s resistance is 50 % of RN (the RN is the normalstate resistanse above the transition). We also measuredthe electron diffusion constant D from the temperaturedependence of the second critical magnetic field Hc2 as

D = −1.28kBc

e

(dHc2

dT

)−1∣∣∣∣∣T=Tc

(4)

The results are summarized in Table I.

IV. THREE REGIMES

A typical experimental result is shown in Fig. 3 forone of the samples (sample N2). This set of data is takenfollowing the Case-B method described above. The bathtemperature is set between 1.7 K and 2.11 K. A perpen-dicular magnetic field is applied until a resistive stateis reached, providing the resistive state shown in Fig. 1.The output voltage, δV (ωm), as a function of modulation

frequency is shown in Fig. 3. With increasing frequencywe find for each bath temperature a clear roll-off. We ap-ply a least-square fit to the measured data using Eq. (2),which leads to a characteristic relaxation time, shown inthe inset. Similar curves are obtained for measurementsusing the Case-A method. All the results are put togetherin the inset of Fig. 3 as a function of the normalized crit-ical temperature T rc , determined from the mid-point ofthe transition in the absence of a magnetic field whichwe call Tc of the film, as listed in the Table I.

We identify three different regimes in the response:

• Regime I: At temperatures 0.75 Tc < T < 0.95 Tc,the data for both samples are shown in Fig. 4. Weobserve, for both samples, a very similar trend withτ = αT−p, with α and p used as fitting param-eters. The values of α and p obtained from theleast-square fit are listed in Table I.In both casesthe value of p is very close to 2. The values of τ(T )run from 400 to 700 ns over the temperature range1.7 K to 2.2 K.

• Regime II: At temperatures 0.95 Tc < T < 0.99 Tc,the relaxation time increases in a divergent man-ner upon approaching Tc (inset of Fig. 3). Thisis reminiscent of data reported before by Gershen-zon et al.38 for dirty niobium and interpreted asthe observation of the relaxation of the supercon-ducting order parameter, the so-called longitudi-nal relaxation time known from non-equilibriumsuperconductivity32.

• Regime III: At even higher temperatures of0.99 Tc < T < 1.02 Tc the relaxation time de-creases with temperature. In this regime the resis-tive superconducting state is close to the normal-state resistance. It should be considered as withinthe range where the superconducting state emergesout of the normal state due to time-dependentGinzburg-Landau fluctuations i.e. within the re-sistive transition.

These three regimes represent in our view three differ-ent physical processes. We consider the fact that RegimeII and III have been measured according to the Case Amethod and Regime I with the Case B method an im-portant distinction of which the significance is to be ad-dressed further. In Case A we are in a regime where manyprocesses are entangled and where one can safely statethat ∆ � T ∼ Tc. In Case B there is a well developedenergy gap ∆ outside the regime where the vortex-cores

6

T/Tcr

1.70 K1.78 K1.80 K1.90 K2.06 K2.11 K

100 1000Frequency (kHz)

Out

put s

igna

l (dB

m)

0

-3

-6

-9

-120.8 0.9 1.0

0.4

0.6

Rel

axat

ion

time

(μs)

FIG. 3. The frequency dependence of the sample responseat different bath temperatures Tb. The experimental data aremeasured at a temperature in the middle of the superconduct-ing transition (where ∂R/∂T = max) at the same bias current.The temperature of the resistive transition shifts when a mag-netic field normal to the film plane is applied. The data ofeach curve were normalized to 0 dB for convenience. The solidlines are a least-square fit with Eq. 2. The fit standard error ofthe roll-off frequency does not exceed 10 %. The inset showsthe energy relaxation time vs. the normalized temperature(T/T r

c ), where the critical temperature T rc is the temperature

of the middle point of the resistive transition. The experimen-tal results correspond to both types of measurements: case A(red triangles) and case B (black squares).

are located, but the magnetic field is close to Bc2. There-fore the resistive superconducting state is controlled bya complex inhomogeneous nonequilibrium process. Therestoration of the resistive state occurs in a spatially dis-tributed way with, on a microscopic level, scattering andrecombination processes known from non-equilibrium su-perconductivity, as well as diffusion processes.

V. REGIME I

The observed resistance relaxation time indicates howfast the resistance changes with a modulation of the in-put power. As argued above we interpret this time asthe energy-relaxation time between the electron- and thephonon-system. In previous experiments, such as forTiN15, we find that it obeys a power law with the ex-ponent p = 3. Here we find clearly the exponent p = 2(Fig. 4).

A straightforward explanation for p = 2 might bethe dimensionality. A phonon system is two-dimensionalwhen λT � d, with λT the wavelength of the thermalphonons and d the film thickness. This is definitely notour case because the wavelength of the thermal phononλT = (~ul)/(kBT ) is ≈ 60 nm, which is less than thethicknesses of both our samples. Besides, since our super-conducting layer is grown on diamond the perfect acous-tic match between the boron-doped diamond epilayer and

Temperature (K)

Ener

gy re

laxa

tion

time

(μs)

1.8 2.0 2.2

0.8

0.6

0.4

Sample N2Sample N1

τe-ph ~ T-2

τe-ph ~ T-3

FIG. 4. The temperature dependence of the energy-relaxationtime at low temperatures far from Tc (0.75T r

c < T < 0.95T rc ).

The full lines represent the theoretical estimate for τe−ph(T )according to Eq. (6). For comparison the dashed line repre-sents a T−3 dependence.

the diamond substrate makes it unreasonable to think interms of 2-dimensional phonons.

In the case of a 3-dimensional phonon system sub-stantial modifications in the electron-phonon interactiondue to electron-impurity scattering have been developed.They depend on the polarization mode, transverse or lon-gitudinal phonons, and on the effect of disorder. Further-more, the samples we study are in the dirty limit in thesense l � λT , which is already achieved at T ≈ 2.2 Kwith l/λT = 0.2.

In disordered metals the electron-phonon interactionis non-local with a characteristic size of the interactionregion about equal to λT . In the diffusive limit, when l�λT , the theory predicts, in the presence of strong elasticscattering, a weakened electron phonon interaction3–5:

τeph =1

9.1

10

3π4βt

(pFut)3

pF l

1

(kBT )4, (5)

with βt = (2εF /3)2(N0/(2ρmu2t )) the coupling constant,

pF and εF the Fermi momentum and Fermi-energy, N0

the density of states at the Fermi energy and ρm themass density (these parameters are listed in Table II).The coefficient 9.1 results from averaging over all elec-tron states contributing to τeph

16. This theoretical modelassumes that all the impurity scattererers for electrons,vibrate in phase with the host atoms. Experimentally,the T−4 behaviour of τeph has been predominately ob-served in elemental thin films, such as Cu8, Au8, Hf39

and Ti39, mostly at very low temperatures, below hun-dreds of mK. Nevertheless, the T−4 dependence was alsofound in disordered amorphous InO films40 and heavilydoped silicon41 at low temperatures. Obviously, the re-sults found in our diamond films are not in agreementwith this model for electron-phonon interaction in thepresence of impurity scattering.

7

TABLE II. Calculated parameters of the films for evaluation of τe−ph(T )

Sample nB εF kF l N0 b(cm−3) (eV) (cm−1) (nm) (eV−1µm−3)

N1 3×1021 1.5 4.5×107 0.41 3.1×109 0.066N2 3×1021 1.5 4.5×107 0.39 3.1×109 0.073

A T−2 dependence, found from weak localization ex-periments, has been reported for alloys, polycrystallinefilms and metallic glasses such as CuZrAl42, TiAl43,TiAlSn44, AuPd45, VAl alloys46, CuCr47, ZrSn, Au-doped In2O3−x films48 and in Mn-doped Al films49. Suchdisorder-enhanced relaxation, with a T−2 dependence, ispredicted by a recent model of Sergeev and Mitin (SM) ofscattering of electrons by static impurities such as heavyimpurity atoms, the defects and grain boundaries6:

τeph =1

1.6

1

b

(pF l)(pFut)

3π2βt

1

(kBT )2, (6)

where the coefficient b (bmax = 0.25)50 describes thedifference in the vibration of the scatterers and thehost atoms. We apply this theoretical prediction toour data as follows. The carrier density is given bynB = 3 × 1021 cm−3 from the experimental dependenceof the critical temperature as a function of the boronconcentration11. The Fermi wave vector and the elastic-scattering length are determined from kF =

3√

3π2nand l = ((3π2)1/3~)/(3e2ρπ2n2/3) within the Drude-Sommerfeld model51. The effective carrier mass followsfrom m∗ = (pF l)/(3D) ≈ 0.5me, where me is the elec-tron mass. The density of states at the Fermi level isestimated from the experimental values of the resistiv-ity ρ and the electron diffusion constant D through theexpression N0 = 1/(e2ρD). The mass density ρm wastaken for diamond with the value ρm = 3.5 g/cm3. Thesound velocities for the longitudinal mode ul = 16× 105

cm/s and for the transverse mode ut = 9.7 × 105 cm/sare estimated from the phonon dispersion relations us-ing Giustino et al.52. The calculated dependencies areshown in Fig. 4, using only the parameter b as a fittingparameter (see Table II).

Since the elastic mean free path l of electrons is com-parable with average distance between the boron atoms,we assume that the carriers are scattered predominantlyat sites of boron atoms. However, the mass differencebetween the boron and carbon is only about 10%, whichitself is not a sufficient condition for the applicability ofthe SM model. Therefore in the case of boron-dopeddiamond one should consider as scatterers also clustersof boron atoms (dimers, trimers and etc.)53, but furtherstudies are needed to identify the exact nature of thescatterers.

For completeness we point out that a T−2 dependenceof the relaxation time is also predicted for semiconduc-tors for the case of low screening54. The e-ph interaction(through the deformation potentials) in semiconductors

is of different nature than those for metals55. This inter-action in semiconductors has a different dependence ondisorder and on the electronic concentration. However,because of the relatively high boron concentration (of or-der 1021 cm−3), our diamond samples are in the strongscreening limit (with the screening length κ−1 ≈ 1.5 A,where κ2 = 4πe2N0), and hence the theory for e-ph inter-action in a semiconductor is not applicable to this case.

It should also be stressed that the time τe−ph, whichis measured with the AMAR technique is by definitionthe time of relaxation of the electron temperature due toelectron-phonon interaction. It differs from another char-acteristic electron-phonon time - the time of the relax-ation of the distribution function - by a numerical factorand is several times shorter. The reason for this differenceis the following. The rate of relaxation for a quasipar-ticle depends on its energy and increases with it. Thusthe total energy of the thermal distribution of quasipar-ticles (and correspondingly the temperature), which isdetermined mainly by quasiparticles with the highest en-ergies, relaxes faster than the number of quasiparticles.The value of the numerical factor depends on the particu-lar form of the electron-phonon collision integral and hasnot been calculated for the most general case56. Becausethe time of the relaxation of the distribution function,or of the quasiparticle number, is also often referred toas the electron-phonon time, one should bear in mindthe numerical difference between it and the time of therelaxation of the electron temperature. But, the temper-ature dependences for both are the same. The formulas(see above) we will use to fit the experimental data givethe time of the relaxation of the quasiparticle number,not the time of the energy relaxation; but because of anuncertainty in the numerical coefficient in these formulasand of an unknown factor between the two times, we willneglect this difference.

VI. REGIME II

At the temperatures in the range 0.95Tc < T < 0.99Tc,the relaxation time is found to increase sharply (Fig. 3).In this temperature range the photons of the THz sourcescatter quasi-particles to energies well above the super-conducting energy gap at the given temperature. Theresistive superconducting state is expected to relax backto the equilibrium state on a time scale of the order ofthe so-called longitudinal relaxation time, Eq. (7).

Since the longitudinal relaxation time is inversely pro-portional to the energy gap, we plot the data, Fig. 5,

8

as the inverse square of the relaxation time vs. temper-ature. We obtain straight lines suggesting that we areindeed observing the longitudinal relaxation time, whichdiverges as (Tc/(Tc − T ))1/2, although the extrapolatedvalue goes to a Tc, which we called TLc which is slightlydifferent from T rc . This longitudinal relaxation time isgiven by:

τL ≈ 3.7τEkBTc/∆ (7)

with ∆(T ≈ Tc) ≈ 3.1kBTc(1 − T/Tc)1/2, where Tc is

the critical temperature, i.e. the temperature at whichthe gap is completely suppressed, and τE is the energy-relaxation or inelastic-scattering time for an electron atthe Fermi surface. In this case the critical temperature Tcis determined as the temperature TLc at which the valueof the order parameter approaches zero. The values ofTLc for both samples are almost identical to the values ofT rc determined from the resistive transition.

The time τE is the characteristic time for the non-equilibrium distribution function to relax to the Fermifunction. In the standard analysis of, for example Kaplanet al.1, this τE is related to electron-phonon interactionsas measured in energy-dependence of the superconduct-ing energy gap in a tunneling experiment. Hence, the in-elastic scattering rate is coupled to the electron-phononinteraction responsible for superconductivity. However,in general, two processes may be responsible for inelas-tic scattering: electron-electron interaction and electron-phonon interaction. The faster of the two will dominate.The estimated values of τE at TLc are τE ≈ 52 ns for the70-nm sample (with TLc = 2.2 K) and τE ≈ 72 ns forthe 300-nm sample (with TLc = 2.24 K). While compar-ing the inferred values of τE to the characteristic time ofelectron-phonon interaction, one should remember thatτe−ph measured in Regime I is the time of the energyrelaxation and should be several times shorter than thetime of the relaxation of the distribution function in thesame process. Thus τE is more than order of magnitudeless than the time of relaxation of the distribution func-tion due to electron-phonon interaction.

In a pure metal the relaxation time due to electron-electron interaction follows, for states near the Fermi sur-face, an inverse quadratic temperature dependence:

τee ∝~εF

(kBT )2(8)

The values of τee in the clean case are of the order of15 ns at Tc = 2.25 K. In dirty metals with a short elec-tronic mean free path the electron-electron interaction isenhanced compared to the clean case. The actual relationdepends on the dimensionality, which in turn depends onthe ratio of the film thickness to the characteristic lengthLT =

√~D/(kBT ) called the thermal diffusion length. It

defines the length scale over which electrons loose coher-ence as a result of the thermal smearing of their energy57.

TcL

= 2.24 K Res

ista

nce

(Ohm

)

0

5

10

2.10 2.20 2.30 2.400

5

10

15

20

τ-2×1

012 (µ

s-2)

Temperature (K)

Sample N1

τE = 72 ns

Temperature (K)

Res

ista

nce

(Ohm

)Sample N2

τ-2×1

012 (µ

s-2)

2.10 2.15 2.20 2.25 2.300

50

100

0

5

10

τE = 52 nsTc

L = 2.20 K

FIG. 5. The inverse square of the relaxation time (τ−2) andthe film resistance as function of the temperature. The dashedline corresponds to the longitudinal relaxation time, whichdiverges as (TL

c /(TLc −T ))1/2, where TL

c is determined as thetemperature at which the superconducting gap is completelysuppressed and τ−2 = 0. The values of TL

c , indicated in thelegend, are close to the temperature T r

c determined from theresistive transition.

Since for our samples LT ≈ 25 nm, both samples arein three-dimensional regime for electron-electron interac-tion.

In 3-D dirty metals, for an electron at the Fermi sur-face, the electron-electron scattering rate58,59 is given by

1

τee= c

1

kF l

(kBT )3/2

εF√~τ

(9)

where c = (3√

3π)/16ζ(3/2)(√

8 − 1) ∼= 2.75, and τ =l2/3D is the elastic scattering time. For Tc = 2.25 K weobtain τee ≈ 0.2 ns. This value is considerably less thanthe experimentally determined values of τE . Thereforewe beleive that the electron-electron interaction does notplay a role in the interpretation of the data in this regime.

We conclude this Section by emphasizing that the re-laxation time of 500 ns found in Regime I is different fromthe relaxation time of 50 to 70 ns found in Regime II.We note however, that the unprocessed relaxation timesmeasured in Case A and Case B are quantitatively at the

9

same level. Therefore we assume that our identificationof the data obtained in Case B as representing the bareτE and the ones of Case A the longitudinal relaxationrate is too simplified can not be used too strongly forthe absolute value. In reality the restoration of the re-sistive state in Case B involves an inhomogeneous statewith uni-polar vortices and elsewhere a well-developedenergy-gap, although close to Bc2. In Case A we dealwith a system very close to Tc also inhomogeneous and inthe limit where ∆� kBT and where multiflux-quantumdomains may exist with opposite polarity. The restora-tion of the resistive superconducting state in the time-domain involves a complex process, which may influ-ence the absolute values. We believe however, that wecan safely attribute significance to the observed T−2 and(Tc/(Tc − T ))1/2-dependences in comparison with othersuperconducting materials.

VII. REGIME III

Finally, at higher temperature in the range (0.99Tc <T < 1.02Tc) the resistance-relaxation time falls withtemperature. The decrease of τ corresponds to thetemperature region of the conventional resistive transi-tion, where thermally activated processes generate vor-tices/phase slips, as well as amplitude fluctuations, whichgradually merge towards the regime of superconductingfluctuations out of the normal state. Above the super-conducting mean field Tc the fluctuations can be de-scribed by the Aslamazov-Larkin theory using the time-dependent Ginzburg-Landau (TDGL )equations32.

According to this time-dependent Ginzburg-Landautheory the characteristic time is controlled by

τ0 =π~

8kB(T − Tc), (10)

which is a measure how quickly a temporary existenceof superconducting order gets restored to the normalstate. As shown in Fig. 6, for descending temperaturesupon approaching Tc the lifetime of superconducting or-der gets extended in order to become ’infinitely’ long.The temperature located at the peak between RegimeII and Regime III, is denoted as T pc . The values of T pcdiffer from the values of T rc and TLc within 1% - 3%.This difference in temperatures may be due to the T rchaving been measured with minor disturbance, whereasTLc and T pc both represent bias-conditions with a sizablemeasurement current. The temperature T pc can be con-sidered as the superconducting mean field Tc above whichthe fluctuations of the order parameter dominate. Theobserved temperature dependence τ(T ) is in agreementwith such a scenario (Fig. 6), but the data are too limitedto conclude that this is indeed what we observe. RegimeIII has not been reported before with this experimen-tal method. In comparison with previous measurements,where the spontaneous thermal fluctuations of the order

T - TcP

0 0.1

Rel

axat

ion

time

(µs) Tc

P = 2.174 K

-0.10.3

0.4

0.5Sample N1

Rel

axat

ion

time

(µs)

T - TcP

0 0.05-0.05

0.3

0.4

0.5

0.6 Sample N2Tc

P = 2.175 K

FIG. 6. The relaxation time vs. T − T pc , where the tempera-

ture T pc correspond to a temperature at the peak separating

Regime II and Regime III. On both sides of the temperatureT pc the observed relaxation time increases, suggesting a diver-

gent behaviour upon approaching T pc . For Sample N1, since

the last 3 points were measured in the limit where the film isnormal we left them out of the discussion.

parameter were measured with a tunnel junction 60,61,the advantage of our method is a direct restoration ofthe electron system after a disturbance and a possibilityto maintain phonons in equilibrium. However, more de-tailed measurements are needed to resolve the situationmore accurately. Nevertheless, it is to be expected thaton both sides of the mean-field critical temperature Tcwe will have a divergent slowing-down of the restorationof fluctuations.

VIII. CONCLUSIONS

In conclusion, we have been able to study supercon-ducting boron-doped diamond films by using the methodof amplitude-modulation of the absorbed THz radiation.By changing the frequency of the modulation we find dif-

10

ferent regimes with different values and different temper-ature dependences of the energy-relaxation time. Theslow energy-relaxation at low temperatures is governedby a T−2-dependence with a value of 0.7 µs at T = 1.7K. At temperatures closer to Tc we identify the longitu-dinal non-equilibrium time, in the narrow temperaturerange (0.95Tc < T < 0.99Tc). The associated inelastic-scattering time differs by an order of magnitude from theenergy relaxation time found at lower temperatures. Weargue that we cannot assign a conclusive interpretationto the differences in the absolute value.

Blase et al.12 have pointed out that the superconduc-tivity in boron-doped diamond may be intimately relatedto the contribution of the stretching bond of the C-atomsto the B-atoms. It implies that the electron-phonon inter-action leading to superconductivity is intimately relatedto the presence of the B-atoms. The results presentedhere suggest that it is worth analyzing in more depththe time-dependence of the non-equilibrium processes by

combining the insights from Blase et al. with insightsfrom theories like the Sergeev-Mitin theory6.

ACKNOWLEDGMENTS

This work was supported by the Ministry of Educa-tion and Science of the Russian Federation, contract No.14.B25.31.0007. The study was also implemented in theframework of the Basic Research Program at the Na-tional Research University Higher School of Economics(HSE) in 2015. T.M.K. also acknowledges the financialsupport from the European Research Council Advancedgrant No. 339306 (METIQUM). Alexander Semenov ac-knowledges the financial support of the Russian Founda-tion for Basic Research Grant No. 15-52-10044 and theGrant of the President of the Russian Federation No.MK-6184.2014.2 and MSPU Grant No.3.2575.2014/K.We thank A. Sergeev for helpful correspondence.

∗ kardakova@rplab.ru† goltsman@rplab.ru‡ t.m.klapwijk@tudelft.nl1 S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang,

S. Jafarey, and D. J. Scalapino, Phys. Rev. B 14, 4854(1976).

2 A. B. Pippard, Philos. Mag. 46, 1104 (1955).3 A. Schmid, Z. Physik 259, 421 (1973).4 B. Keck and A. Schmid, J. of Low Temp. Phys. 24, 611

(1976).5 M. Y. Reizer and A. Sergeev, Zh. Eksp. Teor. Fiz. 90, 1056

(1986), [Sov. Phys. JETP 63, 616 (1986)].6 A. Sergeev and V. Mitin, Phys. Rev. B 61, 6041 (2000).7 J. J. Lin and J. P. Bird, J. Phys.: Condens. Matter 14,

R501 (2002).8 J. T. Karvonen, L. J. Taskinen, and I. J. Maasilta, Phys.

Rev. B 72, 012302 (2005).9 E. A. Ekimov, V. A. Sidorov, E. D. Bauer, N. N. Mel‘nik,

N. J. Curro, J. D. Thompson, and S. M. Stishov, Nature428, 542 (2004).

10 E. Bustarret, J. Kacmarcik, C. Marcenat, E. Gheeraert,C. Cytermann, J. Marcus, and T. Klein, Phys. Rev. Lett.93, 237005 (2004).

11 T. Klein, P. Achatz, J. Kacmarcik, C. Marcenat,F. Gustafsson, J. Marcus, E. Bustarret, J. Pernot,F. Omnes, B. E. Sernelius, C. Persson, A. F. da Silva,and C. Cytermann, Phys. Rev. B 75, 165313 (2007).

12 X. Blase, C. Adessi, and D. Connetable, Phys. Rev. Lett.93, 237004 (2004).

13 E. M. Gershenzon, M. Gershenzon, G. N. Gol’tsman, A. M.Lyul‘kin, A. D. Semenov, and A. V. Sergeev, Zh. Eksp.Teor. Fiz. 97, 901 (1990), [Sov. Phys. JETP 70, 505(1990)].

14 Y. P. Gousev, G. N. Gol’tsman, A. D. Semenov, E. M.Gershenzon, R. S. Nebosis, M. A. Heusinger, and K. F.Renk, J. of Appl. Phys. 75, 3695 (1994).

15 A. Kardakova, M. Finkel, D. Morozov, V. Kovalyuk, P. An,C. Dunscombe, M. Tarkhov, P. Mauskopf, T. M. Klapwijk,and G. N. Goltsman, Appl. Phys. Lett. 103, 252602 (2013).

16 K. S. Il’in, N. G. Ptitsina, A. V. Sergeev, G. N. Gol’tsman,E. M. Gershenzon, B. S. Karasik, E. V. Pechen, and S. I.Krasnosvobodtsev, Phys. Rev. B 57, 15623 (1998).

17 H. Pothier, S. Gueron, N. O. Birge, D. Esteve, and M. H.Devoret, Phys. Rev. Lett. 79, 3490 (1997).

18 A. B. Gougam, F. Pierre, H. Pothier, D. Esteve, and N. O.Birge, J. of Low Temp. Phys. 118, 447 (2000).

19 F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Es-teve, and N. O. Birge, Phys. Rev. B 68, 085413 (2003).

20 A. Anthore, F. Pierre, H. Pothier, and D. Esteve, Phys.Rev. Lett. 90, 076806 (2003).

21 B. Huard, A. Anthore, N. O. Birge, H. Pothier, and D. Es-teve, Phys. Rev. Lett. 95, 036802 (2005).

22 F. C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B49, 5942 (1994).

23 B. S. Karasik and A. I. Elantiev, Appl. Phys. Lett. 68, 853(1996).

24 L. G. Aslamazov and A. I. Larkin, Phys. Tverd. Tela 10,1104 (1968), [Sov. Phys. Solid State 10, 875 (1986)]; Phys.Lett. A 26, 238 (1968).

25 K. Maki, Prog. Theor. Phys. 39, 897 (1968).26 R. S. Thompson, Phys. Rev. B 1, 327 (1970).27 A. Kamlapure, M. Mondal, M. Chand, A. Mishra, J. Je-

sudasan, V. Bagwe, L. Benfatto, V. Tripathi, and P. Ray-chaudhuri, Appl. Phys. Lett. 96, 072509 (2010).

28 R. V. Carlson and A. M. Goldman, J. of Low Temp. Phys.25, 67 (1976).

29 R. V. Carlson and A. M. Goldman, Phys. Rev. Lett. 34,11 (1975).

30 R. V. Carlson and A. M. Goldman, Phys. Rev. Lett. 31,880 (1973).

31 E. J. Konig, A. Levchenko, I. V. Protopopov, I. V. Gornyi,I. S. Burmistrov, and A. D. Mirlin, “Berezinskii-kosterlitz-thouless transition in homogeneously disordered supercon-ducting films,” e-print arXiv:cond-mat/1509.01474.

32 M. Tinkham, Introduction to superconductivity, 2nd ed.(McGraw-Hill, Inc., 1996).

33 B. I. Halperin and D. R. Nelson, J. of Low Temp. Phys.36, 599 (1979).

11

34 S.-Z. Lin, O. Ayala-Valenzuela, R. D. McDonald, L. N.Bulaevskii, T. G. Holesinger, F. Ronning, N. R. Weisse-Bernstein, T. L. Williamson, A. H. Mueller, M. A. Hoff-bauer, M. W. Rabin, and M. J. Graf, Phys. Rev. B 87,184507 (2013).

35 P. Achatz, F. Omnes, L. Ortega, C. Marcenat, J. Vacık,V. Hnatowicz, U. Koster, F. Jomard, and E. Bustarret,Diamond and Related Materials 19, 814 (2010).

36 J. Bousquet, G.Chicot, D. Eon, and E. Bustarret, Appl.Phys. Lett. 104, 021905 (2014).

37 A. Fiori, J. Bousquet, D. Eon, F. Omnes, E. Bellet-Amalric, and E. Bustarret, Appl. Phys. Lett. 105, 081109(2014).

38 E. M. Gershenzon, M. E. Gershenzon, G. N. Gol‘tsman,A. D. Semenov, and A. V. Sergeev, Zh. Eksp. Teor. Fiz.86, 758 (1984), [Sov. Phys. JETP 59, 442 (1984)].

39 M. E. Gershenson, D. Gong, T. Sato, B. S. Karasik, andA. V. Sergeev, Appl. Phys. Lett. 79, 2049 (2001).

40 M. Ovadia, B. Sacepe, and D.Shahar, Phys. Rev. Lett.102, 176802 (2009).

41 P. Kivinen, A. Savin, M. Zgirski, P. Torma, J. Pekola,M. Prunnila, and J. Ahopelto, Journal of Applied Physics94, 3201 (2003).

42 L. Li, S. T. Lin, C. Dong, and J. J. Lin, Phys. Rev. B 74,172201 (2006).

43 J. F. DiTusa, K. Lin, M. Park, M. S. Isaacson, and J. M.Parpia, Phys. Rev. Lett. 68, 1156 (1992).

44 J. J. Lin and C. Y. Wu, Europhys. Lett. 29, 141 (1995).45 S. Y. Hsu, P. J. Sheng, and J. J. Lin, Phys. Rev. B 60,

3940 (1999).46 Y. L. Zhong and J. J. Lin, Phys. Rev. Lett. 80, 588 (1998).

47 A. K. Meikap, Y. Y. Chen, and J. J. Lin, Phys. Rev. B69, 212202 (2004).

48 R. Ceder, O. Agam, and Z. Ovadyahu, Phys. Rev. B 72,245104 (2005).

49 J. M. Underwood, P. J. Lowell, G. C. ONeil, and J. N.Ullom, Phys. Rev. Lett. 107, 255504 (2011).

50 B. S. Karasik, A. Sergeev, and D. E. Prober, IEEE Trans-actions on THz Science and Technology 1, 97 (2012).

51 N. W. Ashcroft and N. D. Mermin, Solid State Physics,college ed. (Saunders College Publishing, Harcourt CollegePublishers, 1976).

52 F. Guistino, J. R. Yates, I. Souza, M. L. Cohen, and S. G.Louie, Phys. Rev. Lett. 98, 047005 (2007).

53 E. Bourgeois, E. Bustarret, P. Achatz, F. Omnes, andX. Blase, Phys. Rev. B 74, 094509 (2006).

54 A. Sergeev, M. Y. Reizer, and V. Mitin, Phys. Rev. Lett.94, 136602 (2005).

55 V. F. Gantmakher and Y. B. Levinson, Carrier Scatter-ing in Metals and Semiconductors (North-Holland Ams-terdam, 1987).

56 For the conventional case of electron-phonon interaction inclean limit, it is 4.5, see in P. B. Allen, Phys. Rev. Lett.59, 1460 (1987).

57 B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev.Lett. 44, 1288 (1980).

58 A. Schmid, Z. Physik 271, 251 (1974).59 J. Rammer, Quantum transport theory, Frontiers in Physics

Series No. 99 (Westview Press, 2004).60 J. T. Anderson, R. V. Carlson, and A. M. Goldman, J. of

Low Temp. Phys. 8, 29 (1972).61 W. J. Skocpol and M. Tinkham, Rep. Prog. Phys. 38, 1049

(1975).