Phonon Spectrum in Photoexcited Superconducting Films

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Phonon Spectrum in Photoexcited Superconducting Films

N. Perrin

To cite this version:

N. Perrin. Phonon Spectrum in Photoexcited Superconducting Films. Journal de Physique, 1979, 40

(11), pp.1089-1092. �10.1051/jphys:0197900400110108900�. �jpa-00209196�

Phonon Spectrum ⁱⁿ Photoexcited Superconducting ^Films

N. Perrin

Groupe ^de Physique des Solides (*), Ecole Normale Supérieure, 24,

Lhomond, 75231 Paris 05, France

(Reçu ^{le 18 mai} 1979, révisé le 3 juillet 1979, accepté ^le 23 juillet 1979)

Résumé.

Nous considérons le système couplé ^de quasiparticules, paires ^et phonons ^{dans un film} supraconduc-

teur porté ^hors d’équilibre par irradiation optique. ^Nous étudions la forme de la distribution de phonons ^en sup- posant que la distribution de quasiparticules ^{est une} fonction de Fermi Dirac à la température effective T*. Nous montrons que la distribution de phonons ^est différente à la fois d’une distribution d’équilibre ^{à T* et d’une dis-}

tribution d’équilibre ^{à la} température ^{ambiante T.}

Abstract.

We consider the coupled quasiparticle-pair-phonon system ^{in a} superconducting film driven out of equilibrium by optical irradiation. The form of the phonon distribution is studied, assuming ^a Fermi Dirac function at an effective temperature T* for the quasiparticles. It is shown to deviate from both an equilibrium

distribution at T* and at the ambient temperature ^T.

Classification

Physics ^Abstracts

74.30 - 74.20

During ^{the last} years, the properties of supra- conductors driven out of equilibrium ^states by ^an

external perturbation (injection ^of electrons, phonons

or photons) ^{have been} largely investigated [1]. ^Several

theoretical models have been proposed ^to explain

the results of experiments ^{such as} irradiation of thin films by optical photons [2-5]. ^{Owen and} Scalapino [2]

and Chang ^and Scalapino [3] proposed ^the ,u* ^model

based on the assumption ^{of a} recombination bot- tleneck. In this model, ^the system ^of quasiparticles

and Cooper pairs ^{remains in thermal} equilibrium ^at

the lattice temperature, ^{but not in chemical} equili-

brium. A first order phase transition to the normal

state is predicted, ^{but has not been} directly ^observ-

ed [6-9]. ^However, this model leads to a good agree- ment with photoexcitation experiments ^{for a small}

excess quasiparticle ^number [10, 11 and ^{with tunnel} injection experiments ^for quasiparticles ^of energy

slightly ^{above the} gap [12]. ^Parker [5] proposed ^an

alternative model, the so-called T* model in which it is assumed that the quasiparticles ^{are in both}

thermal and chemical equilibrium ^{at an effective}

temperature ^{T *} greater than the ambient tempe-

rature T. The phonons of energy greater ^{than twice} the energy gap d ^are described by ^a thermal distri- bution at T*, ^{while the} phonons of energy less than

(*) Laboratoire associé

Centre National de la Recherche

Scientifique.

2 L1 remain unperturbed by ^the optical ^radiation,

and are therefore described by ^a thermal distribution at the ambient temperature ^T.

In this _paper, we are interested in the distribution functions of both quasiparticles (q.p.) ^and phonons

in a superconducting film driven out of equilibrium by optical irradiation, ^under spatially ^uniform steady

state conditions, ^and particularly ^{in the} energy

phonon distribution. The crucial importance ^of

recombination phonons ^{in the} coupled quasi particle- pair-phonon system ^{out of} equilibrium ^{has been} pointed ^{out first} by Rothwarf and Taylor [13] ⁱⁿ

tunnel junction experiments. The behaviour of these

phonons ^which may have a probability ^of being

reabsorbed in a pair breaking process higher ^{than the} probability ^of escaping ^{from the} sample (phonon trapping effect), ^{as well as} the behaviour of the relaxation phonons ^emitted through ^the decay ^of high energy q.p. created by optical energy ^are essential in the interpretation ^of non-equilibrium experi-

ments [14, 15]. ^{Still further, Parker} [5] ^{has shown}

from the Rothwarf-Taylor equations [13] ^that phonons

of _energy greater ^{than 2 L1 are even more out of}

equilibrium ^{than the excess} q.p. under conditions where tes » zB- Les being ^the phonon escape time, and tB 1 ^{1 the mean rate at which} phonons ^create

q.p. More recently, Chang ^and Scalapino [16] ^have

shown that the phonon distribution is perturbed ^as

well as the _q.p. distribution in the linear regime,

i.e. in the case of weak external drives.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197900400110108900

1090

We report ^{here some} early ^{results of a} study ^{of the} steady ^{state of a} superconducting film under either weak or intense irradiation by ^laser light [14]. ^{A full}

calculation of the coupled q.p.-pair-phonon ^system

would require taking ^{into account the} spatial depen-

dence of the distribution functions with _appro-

priate boundary conditions at the film-helium bath

or substrate interface and, for thick enough films, of the non-uniform light absorption ^{in the film.}

We assume the distribution functions spatially ^uni-

form in order to formulate a simple tractable model.

Furthermore, ^{we take for the} q.p. energy distribution

one which is identical in form to the BCS distribution,

i.e. a Fermi distribution at the _q.p. steady ^state tempe-

rature T*, ^which depends ^{on the} injected energy.

This choice is supported by ^recent experiments

where the _q.p. distribution was observed directly by measuring ^the absorption of acoustic waves [17].

But, ⁱⁿ opposition ^to the Parker model, ^the phonons

of energy greater ^{than 2 d are not assumed to be} in thermal equilibrium ^{at the} q.p. temperature ^T*.

Neither are the phonons ^of energy 2 L1 assumed to be in equilibrium ^{at the ambient} temperature. ^The

phonon distribution is determined from the phonon

Boltzmann equation.

The assumption ^{of a} Fermi function with a chemical

potential y

^{0, for the} q.p. energy distribution

implies that the time for recombination is short

compared ^{to the time for the excess} quasiparticles

to thermalize with respect ^{to the} lattice, ^{which is} indeed the case for a quasiparticle ^{created at a} high

energy level, [18, ^19, 5] ^{at least for} temperatures T* > ^0.5 T,, [19]. The thermalization of the _q.p. at

the effective temperature ^{T* takes} place ^{on a} very short time scale [14]. ^{Further, we assume that the} superconducting ^{film is} uniformly illuminated by ^the

laser light and that the optical energy is initially

absorbed by the excited _q.p. [10, 14] ^{and then} partly

converted into phonon energy through ^{the electron-} phonon interaction. Thus, ⁱⁿ writing ^{down our} phonon ^Boltzmann equation, we assume that the

phonon distribution is not perturbed directly by ^the

laser light [20]. Only phonon-electron interaction is

taken into account and the particles ^interact only

with longitudinal phonons ^{in normal} processes.

In the steady state, ^the phonon distribution Nq ^is given by :

where (oNq/ot)es ^and (oNq/ot)c ^{are the rates of change}

of the phonon distribution due respectively ^to phonons escaping from the film and ^to relaxation _processes

including quasiparticles. ^{The time} ’tes required ^for

the phonons ^to thermalize to the bath temperature ^T is of the order of ildlu, ^where ~ is an acoustic mismatch

parameter, d is the film thickness and u is the longitu-

dinal sound velocity. ^{We can write :}

thus neglecting ^{the energy} dependence ^{of the} escape time. N(T ) ^{is the} phonon equilibrium distribution at the temperature ^T.

The rate of change (oNq/ot)c ^{of the phonon distri-}

bution Nq ^{due to} scattering by q.p., ^to recombination of excited _q.p. and the inverse _process, pair breaking by phonons, ^are derived from the Fermi golden

rule [21]. ^The electron-phonon matrix element Vq

is obtained from Ziman [22] ^{and is} given by :

for a particle transition from state k to state k’ with k’ = k - q and bNq

^{+ 1} for the emission of a

phonon, and, ^k’

^{k +} q, bNq

^{1 for the absorp-}

tion of a phonon of angular frequency ^co and of wave vector q (0)

uq in the Debye approximation).

V is the volume and _PD is the density ^{of the} crystal.

EF ^is the Fermi potential of the metal. Then, ^we

obtain :

where (oNq/ot)sc ^and (oNq/ot)R ^{are the rates of change} of Nq ^due respectively ^to phonon scattering by ^excited

q.p., and to recombination of excited quasiparticles

and pair breaking by phonons ; f (E) ^{is the Fermi}

function at T* : f(E) = [exp(E/k. ^{T*) +} 1] - 1 ;

d is the energy gap ^at T*, p(E) ^is the ratio of the

density ^{of states in the} superconducting phase ^{to the}

normal phase p(E)

E/(E2 - ⁴ 2)1/2 when E >,A;

C is a constant :

The effective excited quasiparticles temperature ^T*

is determined by ^the following energy balance _equa- tion :

P is the absorbed light power.

The solution of the coupled equations (1) ^and (7) along ^with equations (2) ^to (6) ^{is carried out nume-} rically ^{on an} IBM-370-168 for a typical super-

conducting film (Sn : EF =10.3 3 eV ; d (0) = 0.561 meV ;

pp = 7 g cm - 3 ; u = 3.32 x 103 ms -1; m = m°, the elec- tron mass). ^{We assume the ambient} temperature ^is T

2.5 K. The _escape phonon ^{time is} ’tes = 0.3 ^ns, for a film thickness d

103 Â and

^{10. We assume}

an effective temperature ^T*

^{3.052 K} > 0.5 T,, (T,

^{3.722 K)} and derive the phonon distribution from equations (1-2) ^and (4-6). Equation (7) ^then gives P = 2.9 x 101 ° W m- 3 and an absorbed opti-

cal flux P,

^{P x d}

^{0.29 W cm-2.} This value of

P. ^{seems small} compared ^{to the} experimental energy fluxes [6, 7, 10, 11, 14] ^{but it must} be borne in mind that we only ^consider longitudinal phonons ^{and that} taking ^{into account the} transverse phonons ^would yield ^a largely increased absorbed _power due to the

large density ^of transverse phonons.

The main difficulty ^{lies in} the calculation of the

integrals ⁱⁿ equations (5) ^and (6) ^{since the} q.p. density

of states (oc p(E)) ^{has an} inverse square ^root diver- gence ^at E

d . A cut-off energy EMAX

^{8 d has}

been chosen in equation (5).

In figure ^{1, is shown the excess phonon spectra}

w2(Nq - N(T)). ^A discontinuity ^{in this} spectra ^is observed at hw

2 J. This is expected ^{from the}

recombination of quasiparticle ^{of low} energy, since the q.p. density ^{of states} is very large ^for energies

E - d : the high energy excitations created by ^the

laser light decay by emitting phonons [18], ^thus giving

lower _energy quasiparticles ^which themselves will recombine and emit phonons ^of energy > ^{2 d . For} phonon energies ^{1ïw 2 J} (region I), ^the phonon

distribution shows a deviation from the equilibrium

distribution at T. The maximum of the excess phonon

spectra ^{in this} region ^{occurs for a} frequency

wo

1.83 d/h whereas the equilibrium phonon ^den-

Fig. ^1.

^Excess phonon spectra w2(Nq - N(T )) ^{(full line) for}

ambient temperature ^T

^{2.5 K and}

^effective quasiparticle temperature ^T*

^{3.052 K. The} escape time _tes

0.3

and the

phonon lifetime in pair breaking process TpB(2 ⁰³⁹⁴⁾

^0.15

^The

dashed

represents ^{the function} (J)2(N(T*) - Nq), ^{i.e. the} departure ^{from the thermal} equilibrium ^{at T* of the} phonon spectra.

sity ^W2 N(T) ^{has a maximum for} nmm

^0.9,A (the corresponding maximum for a thermal distribution

at T * occurs at 1iw

1.09 A). This results from the q dependence ^{of the} phonon relaxation on excited

quasiparticles. ^In region ^II (hw > 2 0394 ) ^the phonon density ^deviates appreciably ^{from its} equilibrium

value until energies 1iw ’" 7 d . However, contrarily

to the region ^{1 where the} departure of°the phonon

distribution from equilibrium ^{is due to} relaxation of

phonons ^on quasiparticles, ^{in this} region, ^{the deviation}

from equilibrium essentially results from both _q.p.

recombination and phonon trapping ^effect. Indeed,

in our calculation, ^the phonon ^{lifetime in} pair breaking

process ! PB for the energy hv = 2 .1 is of the same order of magnitude ^than Ter, (iPB(2 A)= ^{0.15 ns N 0.5} Tes).

We have also ^{drawn in figure 1 (dashed curve) the}

function co2(N(T *) - Nq). ^{It is} shown that the pho-

nons are not exactly ^described by ^an equilibrium ^distri-

bution at T* [5] ⁱⁿ region ^II (hw > 2 0394 ) though ^their

temperature ^{is much nearer to T* than to T. In} region I, ^the phonon distribution is far from an

equilibrium distribution at the effective quasiparticle temperature ^{T*. These two} later results justify ^the

use of the T* Parker model as a first approximation

for the phonon distribution [5].

1092

We have shown in figure ^{2, the excess} quasiparticle

distribution ôf

f (E, T*) - f (E, T ) corresponding

to the phonon results shown in figure ^1.

Fig. ^2.

^Excess quasiparticle distribution bf = f (E, T*) - f (E, T) for

ambient temperature ^T

^{2.5 K and}

^effective quasipar-

ticle temperature ^T*

^{3.052 K.}

Though ^{our work is} different from that of Chang

et al. [16] essentially ^{in the} following points : ^{the Fermi}

distribution at T* of the _{q.p. ;} no phonon injection

rate ; ^a q.p. driving ^term corresponding ^{to the} light

power P absorbed by ^the q.p. ; ^use of the electron-

phonon matrix element V,, (equation ^{(3)), our results}

agree qualitatively with the results obtained by Chang

et al. in the linear regime [16]. ^{This is} particularly interesting ; first, ^{because our choice of T* to cha-}

racterize the _q.p. distribution is justified a posteriori, if necessary ; then, ^our simple approach might ^be

more easily ^{used for} comparisons ^to experiments ^and

for calculations of the transport coefficients.

In summary, ^we have shown that the phonon

distribution is perturbed whatever the frequency hco 5 2 L1, ^{when the} superconducting film is irradiated

by optical light and therefore weakly ^or largely

driven out of equilibrium. ^{Its detailed form} depends

on both the phonon escape time and the _q.p. phonon

interaction. However, ^the phonon distribution can be characterized by ^three temperatures : first, ^a tempe-

rature 7B I slightly higher ^{than T in} region 1 ; then,

a temperature ^{T* for} high energy phonons ^of region II, and for phonons ^of energy 2 J hco 4 J a tempe-

rature T2. intermediate between T and T*.

The introduction of a Fermi _q.p. distribution at an

effective temperature T*, ^{which seems to be a} good approximation ^{in all cases} [23], ^{enables us to consider} large external drive and to obtain rather easily quan- titative results. Information concerning ^{the mean rate}

at which phonons ^create quasi particles ^{and the}

recombination coefficient [13] ^can be deduced from these calculations as well as the values of the phonon

temperatures characterizing the distribution. They

are beyond ^the scope of this _paper.

Acknowledgments.

^We thank Dr. Pannetier for

helpful discussions.

References

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the 14th International Conference

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