Dynamic behaviour of a superconductor under time-dependent external excitation

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Dynamic behaviour of a superconductor under time-dependent external excitation

N. Perrin, C. Vanneste

To cite this version:

N. Perrin, C. Vanneste. Dynamic behaviour of a superconductor under time-dependent external excitation. Journal de Physique, 1987, 48 (8), pp.1311-1316. �10.1051/jphys:019870048080131100�.

�jpa-00210557�

Dynamic behaviour of a superconductor

under time-dependent external excitation

N. Perrin and C. Vanneste

Groupe ^de Physique des Solides de l’Ecole Normale Supérieure (*), ^{24, rue Lhomond,} 75231 Paris Cedex 05,

France

Laboratoire de Physique de la Matière Condensée (*), Université de Nice, ^Parc Valrose, 06034 Nice Cedex, France

(Reçu ^{le 9} janvier 1987, accepté le 14 avril 1987)

Résumé.

On analyse l’évolution temporelle ^d’un dispositif supraconducteur ^{soumis à une excitation} périodique, par ^un modèle simple ^{à deux} températures. On considère différentes sources d’excitation

(injection d’électrons ou de phonons, irradiation lumineuse) ^et différents mécanismes de couplage ^{au bain} thermique, ^{et on} compare les fréquences ^de réponse ^de quelques supraconducteurs caractérisés par leurs temps ^de relaxation intrinsèques. ^{On montre} que, selon la ^nature de la source d’excitation, ^des comportements

tout à fait inattendus peuvent ^être obtenus dans certaines conditions, des matériaux « lents » comme

l’aluminium et le zinc, pouvant ^{avoir une} réponse plus rapide que des matériaux ^« rapides ^{» comme le}

niobium.

Abstract.

The time evolution of a superconducting ^{device driven} by ^a periodic excitation is analysed through

a simple two-temperature model. Different excitation sources (electron ^or phonon injection, illumination) ^and coupling mechanisms to the thermal bath are considered, and the response frequencies ^{of some} superconduc-

tors characterized by their intrinsic relaxation times are compared. It is shown that, according ^{to the nature of}

the excitation _source, quite unexpected behaviours can be obtained under certain conditions, ^{« slow »} materials, like aluminium and zinc, being ^{able to exhibit a} shorter response than ^« fast » materials like niobium.

Classification

Physics ^Abstracts

74.30

1. Introduction.

During the last years, ^a large ^{number of studies have}

been devoted to nonequilibrium superconductors ⁱⁿ quasistationary conditions. The goal ^{of the} existing

theoretical descriptions ^{such as the} Rothwarf-Taylor equations [1], ^the g * model [2], ^{Parker’s T*}

model [3] ^{or more} complete ^theories which take the detailed spectrum ^{of the} nonequilibrium ^distribu-

tions of excitations into account [4-6], ^{is to describe}

the complex behaviour of the three main components of a superconductor, ^i.e. Cooper pairs, quasiparticles

and phonons. Some studies have also been devoted

to the non-stationary ^state [7-10] in connection with the problem ^{of the time behaviour of} superconduct- ing switching ^devices [11-15]. ^{This more} complex

case has been approached ^with simplifying assump- tions (linearized Rothwarf-Taylor equations [11], ^a

T* description ^{of the} quasiparticles [10]). ^{Even so,}

such approaches ^{are not} easy ^to carry ^out and the results obtained in specific ^{cases cannot be} simply extrapolated ^{to other} situations (various ^materials,

excitation sources and surroundings).

In order to compare ^the performances ^{of usual} superconducting ^{materials in different} environments,

the above difficulties have lead us to use a simple

model [7-9] ^{where the} nonequilibrium quasiparticle

and phonon populations ^{are described} by ^their

effective temperatures Te ^and Tp respectively. ^Con-

fidence is brought ^{in the} predictions ^{of this} simple

model by ^the good agreement with the results

previously ^{obtained with more} sophisticated descrip-

tions [10].

A superconductor ^driven by ^a periodic ^excitation

is considered here. Different sources (quasiparticle

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048080131100

1312

or phonon injection, ^and illumination) ^{and different} coupling mechanisms to the thermal bath are ex-

amined. Starting ^{from the usual} phonon coupling ^of

a thin film deposited ^{on an} insulating substrate, ^the

effect of increasing quasiparticle escape from the

sample ^is investigated. This mechanism has already

been considered by ^some physicists [11] ^{in order to} bypass ^the poorly ^{efficient former} phonon

mechanism. Classical results like phonon trapping ^or

bottleneck and also less known or intuitive results like better performances of slow materials under

some special conditions come out of this description.

2. Model calculation and discussion.

A superconductor ^driven by ^a periodic ^excitation P (t ) = P 0 (1 + cos co t ) ^can be described by ^the following equations :

where Te (Tp ) ^{is the} quasiparticle (phonon) ^effective

temperature ^{assumed to be} spatially uniform,

Ce(Cp) ^{is the} quasiparticle (phonon) specific ^heat

and _Te (Tp) ^{is the mean} energy-independent quasiparticle (phonon) electron-phonon ^collision

time [7].

The detailed balance of energy yields :

The nature of the external source is approximately

described by the factor f (0 = f 1 ) : f ^{= 1 for} quasiparticle injection, f

^{0 for} phonon injection,

and 0 f 1 ^{for a mixed source like il-}

lumination [7].

In equations (1) ^and (2), ^{the last terms} correspond

to the energy exchange between the sample ^{and the}

bath at the temperature Tb. ^The phonons ^and quasiparticles ^are characterized by the escape times Tg and Td, respectively. ^{In the usual case of a thin}

film deposited ^{on an} insulating substrate, ^{there is no} possible exchange between the quasiparticles ^and

the substrate (1/Ta = 0). However, ⁱⁿ particular

cases such as variable thickness bridges investigated by Chi et al. [11], ^{the term} (Ce/Td)(T,, - Tb ) ^{can be}

considered as a phenomenological description ^{of the} spatial ^{diffusion of the} quasiparticles ^{out of the} microbridge (Td may be ^as small as 10- 12 ^to 10-11 s [11]). This effect can be very efficient to

improve the response time of ^a nonequilibrium superconductor.

By introducing ^the complex amplitudes te ^and tp ^{of the Te and} Tp ac components respectively, ^the

solutions of equations (1) ^and (2) ^can be written :

where teo ^and tpo ^are the solutions for the low excitation frequency ^{limit w}

^{0. The} frequencies

w 1, CJ) 2 depend ^{on all the} relaxation times _Ti whereas

w3 and w 4 only depend ^{on the} relaxation times of the

phonons ^{and the} quasiparticles respectively, ^accord- ing ^{to :}

The last frequencies Cù 3 and _{w 4} both depend ^{on the}

external source parameter f

As the superconducting properties ^{are related to}

the quasiparticle population, ^{we are} mainly

interested in the time evolution described by ^the

solutions of equation (4). ^Before performing ^a

detailed comparison of these solutions for different

superconductors characterized by their relaxation times _Te and Tp, it is worth noticing ^that equation (4) predicts ^three types ^of solutions, according ^{to the} following possibilities for the relative values of Cù l’ w 2 and Cù 3 :

where _{wi 1} is the smaller of the frequencies w 1, w 2 satisfying equations (6) ^and (7). ^The correspond- ing ^{numerical solutions of} equation (4) ^are displayed

in figure 1 for the three limiting ^{cases with}

w1 &) 2 w 1 = W 3 (Fig. la), w 1 : W 3 (Fig.1b), ^and

w w 3’ ú) 2 W 3 (Fig. 1c). ^{In each case, after an}

initial plateau ^at the lowest excitation frequencies

where the superconductor ^{can follow the} time evolu- tion of the external source, the amplitude te I ^drops

as a function of w. This drop ^{occurs at a} characteristic

frequency ^{wich we} call below the response frequency

w r ; this characteristic frequency is either the higher frequency w2 (Fig.la), ^or the lower frequency

OJJ 1 (Fig. 1c), ^{or both} w 1 and _{w 2} (Fig.1b). ^{In the}

latter _case, we define the _response frequency

w r as the ^frequency corresponding ^to Itelteo I

1/ 1/2 ^as in the other two cases. However, ⁱⁿ spite ^of

this arbitrary ^{choice of} w r, it must be kept ^{in mind}

that the situation displayed ⁱⁿ figure Ib is charac-

terized by ^two frequencies.

, Fig. ^1.

Amplitude of the calculated electron tempera-

ture a.c. component ^{as a} function of the excitation

frequency (w ^{is in} arbitrary ^{units on a} logarithmic scale) ; a) ^{case 1 :} w W3(02; b) ^{case 2 :} w 1 w 3 w 2 ;

c) ^{case 3 :} w 1 w; i = 2,3.

Our aim is to consider some superconductors (Al, Sn, Pb, ...) characterized by ^their relaxation times T e and Tp ^{and to determine their behaviour under a} high frequency periodic excitation, ^{that is to estimate}

their response frequency Cù r. The quasiparticle ^and phonon relaxation times being energy and tempera-

ture dependent, the values of _Te and Tp ^{are chosen to} obtain a rather good approximation ^of the supercon- ductor behaviour ; ^{as we are} interested in situations where the superconducting properties (as the gap

parameter) vary appreciably ^{with the} quasiparticle

temperature, ^we choose Te and Tp values correspond- ing ^{to the} neighbourhood of the critical temperature ( T

^0.9 Tc). ^An advantage ^{of this} choice is that the relaxation times associated with quasiparticle ^recom-

bination and scattering processes roughly merge

near Tc [16]. ^{Moreover, the} phonon scattering ^and pair breaking ^{times are} nearly equal ^{at this} tempera-

ture. Therefore in this temperature range, it is reasonable to associate a single relaxation time with each type ^of excitation, Te for the quasiparticles ^and Tp for ^{the phonons.}

The times _Te and Tp ^are given in table I along ^with

the other constants characteristic of each material for the temperature ^{T == 0.9} Tc. The values of Te and zp have been chosen from [16] in which the temperature dependence ^{of the} quasiparticle ^and phonon relaxation times is given ^{as a} function of two

characteristic _{times To} and T 8h associated with each material. Those parameters ^{are of} the order of the

electron-phonon scattering times in the normal state at Tc ^{for the} electrons and the phonons respectively.

In table I, ^the quasiparticle ^and phonon specific

heats of each material are calculated from values of the critical temperature Tc, ^the Debye temperature

Td ^{and the} following ^{standard relations} [17-18] :

Table I.

Values of ie, Tp, Ce, Cp ^at temperature ^T

^0.9 Tc. ip ^and Le ^are estimated from [16] ^{with the choice} T - LghjO.15 ^and te ~ 0.4 To (except ^{the strong} coupling superconductors ^{Pb and} Hg [16]). ^{We have} verified ^that

the behaviours observed in figures 3, ^{4 and 5 are not} very sensitive ^to the choice of Le and Tp ^{that can be done accord-} ing ^to [16]. ^{Given the} uncertainties on the reported Debye temperature, ^extreme values of Td (and consequently of

Cp ^and CejCp) ^{are listed.}

1314

and :

where y T is the normal metal electron specific heat,

a=8.5 and b= 1.44.

Taking ^{account of the errors made in this evalua-}

tion of Te [16], _Tp [16], Ce [19-20] ^and Cp ^[21-23],

table I shows a reasonable agreement ^{between the} ratios T e/ T p ^and C e/ C p. ^{The position of each}

material in the plane (1/’te, 1 / T p) ^{is shown in}

figure ^2.

Fig. ^2.

Superconductors ^{in the} plane (1/,rp, 11’re) ^from

table I.

The example ^of tin is first considered in figure ³

where the response frequency wr is displayed ^{as a}

function of the factor f from 0 for phonon injection

Fig. ^3.

Response frequency w

of tin in units of

1/TS ^{as a} function of the factor f describing ^{the nature of}

the external source (from f

^{0 for} phonon ^{source to} f ^{=1 for} quasiparticle source). The different curves

correspond ^to different values of the quasiparticle escape time 7d : a) 1/,rd = 100/7-, ; b) 1/Td =10/TS ; c) 1/Td

1/Ts ; d) ’/Td

0.1/TS ; e) 1/Td

^0.

to 1 for electron injection. ^The value of the phonon

escape time has been fixed at T s =1 ^ns which is the order of magnitude ^{to be} expected ^{for a} good thermal coupling ^{between a 1 000 A thick film and}

an insulating substrate. The different curves corre-

spond ^to values of the quasiparticle escape time Td varying ^from 1/Ta =100/TS (fast quasiparticle

escape from the sample) ^to 1/’rd

⁰ (no quasiparti-

cle escape mechanism). Some features of this set of

curves can be emphasized.

First, without any noticeable quasiparticle escape

(1/Td

^0, Fig. ^{3d and} 3e), wr is close to, but slightly

smaller than the phonon escape frequency 1/TS and

does not depend ^{on the nature of} the external

source. This is a classical situation already ^examined

in past ^{efforts. The} independence ^{on the external}

source means that whatever the injected excitations, phonons ^or quasiparticles, ^{the energy can leave the} sample only ^{via the} phonons escaping ^{to the sub-}

strate or the helium bath. However, ^{it cannot be} straightforwardly deduced that the response time

W r ^{1 is} equal ^{to the} phonon escape time _TS. Actually,

as in tin the electron specific ^heat Ce ^is somewhat larger ^{than the} phonon specific ^heat Cp, ^{an excitation}

spends ^{more time as a} quasiparticle ^{rather than as a} phonon (Te> T p ^{with T e}

^{0.92 ns and} Tp = . 0.73 ns). Therefore, ^the efficiency ^{of the} energy escape via the phonons is reduced. In that _case, the response time is given by

This effect can be even more important ^{for materials} featuring ^a very large ^ratio Ce/Cp ^{like aluminium or}

zinc.

The second main result is the improvement ^{of the}

response time and the increasing sensitivity ^{to the}

nature of the excitation source when the quasiparti-

cle escape mechanism becomes important (Fig. ^3b

and 3a). ^This sensitivity results from the two compet-

ing energy escape mechanisms. ^{For a} very fast

quasiparticle escape, ^each quasiparticle-like ^excita-

tion leaves the sample very quickly. Therefore, very short _response times are predicted ^for quasiparticle

excitation sources. For phonon ^{sources, the fast} quasiparticle mechanism is slowed down by ^{the time} required ^{for a} phonon ^{to be} converted into a

quasiparticle. ^{A smaller} improvement ^{of the re-}

sponse time is then expected (Fig. 3) ^{for values of}

the factor f ^{closer to zero rather than to one.}

It is worth pointing ^{out that} the above results with

1 / T d

^{0 are in close} agreement ^{with those} previous- ly ^obtained by ^{a more} elaborated model [10]. ^We

can now consider some other typical superconduc-

tors : aluminium and niobium (Figs. ^{4 and 5} respect-

ively) in the scope of the twoitemperature ^model.

They ^{are both} characterized by their ratio

C e/ C p > 1 and their quasiparticle relaxation time

Fig. ^4.

Response frequency w

of aluminium as a

function of the factor f : a) 1/Td

100/Ts ; b)

Fig. ^5.

Response frequency (11 r of ^{niobium as a function}

of the factor f : a) 1/Td =100/TS ; b) 1/Td = 10/Ts ; c)

1 I- 1 j / -II j I I -,, j n

The same general features described for tin can be observed in both cases : the response time is im-

proved and becomes source dependent ^{when the} quasiparticle escape mechanism becomes _{very ef-} ficient. However, significant differences result from the rather distinct values of _Te and Tp associated with

those materials (Tab. I).

The response time of aluminium decreases very

rapidly ^with 1/Td (Fig. 4). ^This property ^{comes out}

from the very large ^ratio Ce/Cp associated with aluminium. Without quasiparticle escape mechanism, ^the decay ^{of the excess} quasiparticle population is controlled by ^the large ^time Te which

characterizes the difficulty ^{for the} quasiparticles ^to

transfer their _energy to the small-capacity phonon

reservoir and a large response time is obtained. As

soon as the quasiparticles ^{are allowed to} directly

leave the sample, ^a noticeable improvement ^{of the}

response time is observed from Cù; 1 = Cù 11 =

Te(1 ⁺ 7s/rp) ^{whatever f to wr 1}

W2 Td for

f ^{= 1 and} to Cùr

to, =e (1/7p ⁺ 1/rJ ^for f

^{0. It}

can be noticed that, despite larger ^{values of}

Te and Tp, the response time is ^as short for aluminium as for tin when the quasiparticle escape time _{T d} becomes smaller than the phonon escape time _{T S.}

In contrast with aluminium, ^{niobium is charac-} terized by very small values Te and Tp. Therefore, ^a

fast response ^can be expected from this material which has already ^been investigated for that purpose

[11, 15]. ^With only phonon escape from the sample ( 1 / T d = o ), comparison ^of figures 3, 4 and 5 shows that niobium is faster than aluminium but not faster than tin. This result is not surprising ^{as far as the}

response time is mainly ^limited by ^the phonon

escape time _TS (wr 1= w1 1= TS(1 ⁺ Te/Tp)). ^How-

ever, when the quasiparticle escape time _{T d} becomes very short, the response of niobium is different from the response of tin. A better improvement ^{is ob-}

served for phonon-like ^{sources than for} quasipartic-

le-like sources. Whatever the nature of the external source, the effect of the very small Te and Tp ^values

in niobium is to create a mixed cloud of quasiparti-

cles and phonons very quickly. Therefore, ^for pho-

non injection, ^{the fast} excitation of quasiparticles by

the phonons improves ^the efficiency ^{of the} quasipar-

ticle escape mechanism. On the other hand, ^for quasiparticle injection, ^{the effect of} the fast creation of phonons by ^the quasiparticles ^{is to} decrease the

efficiency ^{of the} quasiparticle escape mechanism.

Actually, ^{the fast} energy exchange ^between quasiparticles ^and phonons in this material leads to some trapping effect of the excess energy.

After the preceding ^results concerning tin,

aluminium and niobium, the behaviour of other materials can be deduced from their position ^{in the}

(1/Te, 1/Tp) ^{diagram (Fig. 2). Zinc, which is charac-}

terized by large relaxation times _Te and Tp, displays ^a dynamical ^{behaviour similar to} aluminium. Indium and thallium behave quite similarly ^{to tin, with}

Te/Tp 1 ^and wr 1- wi TS(l + Te/Tp). ^{With a}

phonon relaxation time T p smaller than that of tin,

tantalum exhibits intermediate properties ^between

tin and niobium. The last considered metals, ^lead

and mercury, ^are characterized by large phonon specific heats and Ce/Cp ^1. Therefore, ^these

materials tend to be rather insensitive to the fast

quasiparticle escape mechanism but in the ^case of

quasiparticle injection ( f =1 ) ^where w r = W 2

1 / Te ⁺ 1/ T d ; their response frequency ^is generally

determined either by ^the phonon relaxation times

1316

(wr = 1/, ^{s or} wr =1/TS + 1/?p ^{for f = 0, with}

1/Tj

^{0 and} 1/’rd =A ⁰ respectively), ^or by ^the quasiparticle relaxation time (wr 1- Te ^for f ^{= 1}

with 1 / Td

0). The above response frequencies ^for

lead are in good agreement ^{with the} experimental

results of Chi et al., ^obtained by ^a pulse technique,

either with thin films [12] ^or with variable thickness

microbridges [11], ^{where the} quasiparticle ^relaxation

time under phonon injection ^{is shown to be deter-}

mined respectively by Tg ^or by _Tp (1/Tp ^{+ 1/ T s =F} l/T p). Unfortunately, ^the quasiparticle injection

device has not yet been tested [11].

3. Conclusion.

The above results show that a two-temperature

model is quite ^{useful to} get ^a good qualitative insight

in the dynamic behaviour of an externally ^driven non-equilibrium superconductor. ^{In the} simplest

case of a uniform thin film from which excess energy

can only decay through phonon escape towards the

substrate, ^{several more or less} complex ^behaviours

can already be observed according ^{to the relative}

values of the times Te, Tp and TS, and the parameter f : ^the expected bottleneck due to the phonons

when the phonon escape time TS is long (niobium, lead, mercury) ^can be deduced from this model ; ^the trapping ^{of the} quasiparticles determines the re-

sponse frequency ^{when the} quasiparticle specific

heat is large (zinc, ^{aluminium and, to a} lesser extent,

tin). ^{In the more} complex ^{case where the} quasiparti-

cles are strongly coupled ^{to the} bath, ^{some unex-} pected ^{results are} obtained : for instance, ^under quasiparticle injection, « slow » materials like aluminium and zinc can exhibit a shorter response time than « fast » materials like niobium. Since the niobium relaxation times are small, ^{a rather} large

response frequency w, could be expected [11] ; however, ^the occurrence of a mixed cloud of

quasiparticles ^and phonons ^{and the} resulting trap-

ping ^{effect of the excess} energy prevent ^the efficiency

of the quasiparticle escape mechanism. Therefore,

the enhancement of _wr by ^the quasiparticle ^diffusion depends ^{both on the nature} of the excitation and on

the material, ^materials with small intrinsic relaxation times being rather insensitive to the ability ^{of a fast} quasiparticle escape mechanism.

Thus the superconductor ^{behaviour as a function}

of its intrinsic relaxation times and the external

conditions, ^{is well} pointed ^{out in such a} description

whereas in more elaborated models, ^such qualitative

behaviours first involve tedious numerical simula- tions and moreover can be obscured by ^the complexi-

ty ^{of the} calculations. Therefore, these results consti- tute a quite useful overview of the main supercon-

ducting material behaviours.

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