Ultrafast nonequilibrium dynamics of electrons in metals

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Laboratoire d'Acoustique de l'Université du Maine UMR CNRS 6613, Av. O. Messiaen, 72085 Le Mans Cedex 9, France. Oliver B. .... electron-hole distribution extending by photon energy h ei- ..... 1.1 in Eq. 3.24. IV. ..... JETP 4, 173 1957.
PHYSICAL REVIEW B

VOLUME 57, NUMBER 5

1 FEBRUARY 1998-I

Ultrafast nonequilibrium dynamics of electrons in metals Vitalyi E. Gusev* Laboratoire d’Acoustique de l’Universite´ du Maine UMR CNRS 6613, Av. O. Messiaen, 72085 Le Mans Cedex 9, France

Oliver B. Wright† Department of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060, Japan ~Received 11 July 1997! Approximate analytic solutions for the athermal relaxation and diffusion of carriers in free-electron metals 2/3 excited by ultrashort optical pulses are derived. The energy-loss time of the electrons is shown to be ; t 1/3 0 tS , where t 0 and t S are characteristic electron-electron and electron-phonon scattering times. By evaluating the average excess energy per excitation we are able to establish an approximate criterion for the time of transition from an athermal electron distribution to a thermalized distribution. We thus demonstrate how one can calculate the evolution of the total energy and number densities of the nonequilibrium electrons at a given point in the metal, linking the athermal to the thermalized regime by interpolation. Quantitative agreement of the theory with some recent experiments on hot electrons in metals is good. @S0163-1829~98!05006-1#

Rapid advances in time resolution have been ushered in by the use of ultrashort duration optical pulses. These have allowed femtosecond physical processes to be effectively frozen at a chosen moment for detailed perusal. Solid state physics in particular has been transformed by new findings on these time scales because the fundamental properties of metals and semiconductors are to a large extent controlled by electrons and phonons, excitations that can scatter after a few femtoseconds. Such short interaction times were implicit in the first microscopic theories of the electrical resistance of metals.1 It is only recently, however, that the subpicosecond nonequilibrium dynamics of carriers could be probed directly. An understanding of this dynamics is vital if progress in circuit switching speed and electronic devices is to be continued. Metals have come under intense scrutiny not only because of their ubiquitous presence in electronics, but also because nearly free-electron metals exist to provide ideal test cases for theories of ultrafast carrier relaxation and transport. In contrast to the situation in undoped semiconductors, photoexcited electrons in metals are mainly scattered by the equilibrium part of the electron distribution, and the electronelectron (e-e) scattering integral can be linearized. Hot carriers in the noble metals have, in particular, been the theme of an ongoing series of ultrafast optical experiments involving photoemission,2 reflectance or transmittance,3–5 and coherent phonon generation.6–8 Having one atom per unit cell, no optical phonons are present to complicate the electronphonon (e-p) dynamics in these metals. Theoretical treatment is difficult because of simultaneous e-e and e-p scattering, which are of commensurate importance. Although abundant contributions have been made in this field over ten years,3 there is still no general approach to immediately determine, for example, the rate of energy loss of the electrons for a given metal or their thermalization time. This is the case even for the simple geometry of films homogeneously excited by light pulses. This has hampered efforts to obtain an overall picture of the ultrafast relaxation processes in different metals and, moreover, has prevented a 0163-1829/98/57~5!/2878~11!/$15.00

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link being made between fundamental microscopic parameters and the key characteristic times governing the physics of the relaxation. Basic questions, such as how these times scale with the e-e and e-p scattering strengths, therefore remain unanswered. The purpose of the present paper is to provide an analytical framework for understanding and predicting the athermal electron relaxation and transport in metals, and thus to go some way towards addressing these problems. A gas of electrons interacting through screened Coulomb repulsion should be treated as a Fermi liquid consisting of a distribution of electron and hole excitations, or quasiparticles, above the ground state. Laser photons absorbed in the region of the sample surface perturb this distribution by producing such excitations out of the initial Fermi-Dirac distribution. Symmetry ensures that the hole distribution mirrors that of the electrons, and hereafter we refer to the quasiparticles above the Fermi energy as electrons for brevity. The evolution in space and time of the perturbed distribution depends on the ratio of the sample thickness d to the optical absorption depth z ~z ;10 nm in metals in the visible or near visible!: for d/ z !1, conditions are approximately spatially homogeneous provided that no electrons diffuse out the film; for d/ z @1 the electrons can diffuse freely into the bulk of the sample. In both cases we assume that the illuminated region of the sample is large enough for any gradients in the lateral directions to be negligible in the analysis. ~In practice, this entails the use of a laser spot size much larger than d for the spatially homogeneous case, or much larger than the electron diffusion length for the diffusive case. Even with diffraction limited spot sizes, ;1 m m, obtainable in the optical region, this turns out to be a reasonable assumption.! The electron distribution function n can be described by a Boltzmann equation. In this paper we formulate this equation without recourse to the relaxation time approximation.9 In the spatially homogeneous case ~d/ z !1, for a free standing film or one with an insulating substrate!,

S D S D

]n ]n 5 ]t ]t 2878

1

e-e

]n ]t

.

~1!

e-p

© 1998 The American Physical Society

57

ULTRAFAST NONEQUILIBRIUM DYNAMICS OF . . .

Approximate solutions to this equation and its generalization to diffusive conditions in one dimension ~d/ z @1, with illumination on one surface! will be found to predict the relaxation of the electrons, and the characteristic times for energy loss t E and thermalization to a Fermi-Dirac distribution t T will be derived. For times greater than t T we shall exploit a simpler ‘‘two-temperature’’ model,10 in which the electrons and lattice are respectively described by FermiDirac and Bose-Einstein distributions. In Sec. I we reformulate the two-temperature model in terms of the total energy of the electrons to facilitate its analytical solution in the linear approximation. In Sec. II we make use of a linearized form of the Boltzmann equation in the spatially homogeneous case, and find approximate solutions for the total energy and electron number for times up to the thermalization time t T . For later times these quantities are determined using analytical solutions of the twotemperature model. In Sec. III we extend the treatment to the diffusion of electrons in a one-dimensional geometry in which the surface of a bulk sample is irradiated. The solution for the total energy distribution in space is found and compared with experiment for the case of gold. Finally, in Sec. IV, we discuss the extension of the analysis to different metals and its general utility in the interpretation of experiments involving hot athermal electron distributions in metals.

2879

F G S F GD F G F

] 1 ] k0 ] 1 2 T g T 2e 5 ]t 2 ]z Tl ]z 2 e 5D e

2g ~ T e 2T l ! 1S ~ z,t !

G

T 2e 2T 2l ]2 1 2 g T 2g 1S ~ z,t ! , e ]z2 2 T e 1T l

where D e 5 k e /C e is the electron diffusivity. Therefore,

]Ee ] 2E e 5D e 2g ]t ]z2

Ag F A 2

E e 2E 0 E e 1 AE 0

G

1S ~ z,t ! .

Substituting E5E e 2E 0 gives E ]E ] 2E 5D e 2 2 1S ~ z,t ! , ]t ]z t e~ E !

~4!

where

t e ~ E ! 5 ~ C e /2g !~ 11 @ 11E/E 0 # 1/2! is the e- p energy exchange time, typically ;1 ps. For small excursions in T e , that is assuming T e 2T l !T l , one may write t e (E)5C e /g5 g T l /g[ t e , and the equation becomes linear. We shall deal in this paper with this linear approximation. Equation ~4! has the advantage of directly revealing the role of the e-p energy exchange time, the characteristic time for the electron energy to be lost to the lattice in this model.

I. TWO-TEMPERATURE MODEL

The two-temperature model ~TTM!, initially proposed by Kaganov, Tanatarov, and Lifshitz,10 has been widely used for interpretation of hot electron relaxation and diffusion in metals. After excitation with an ultrashort optical pulse the electrons are described by a Fermi-Dirac distribution at temperature T e and the lattice by a Bose-Einstein distribution at temperature T l . Although the model was initially formulated using the Debye theory to describe the phonon spectrum, it is straightforward to generalize the model to arbitrary phonon spectra. Allowing for the possibility of diffusion in one dimension, the general expression of the model is11 C e~ T e !

S

D

]Te ] ]Te 5 k e~ T e ! 2g ~ T e 2T l ! 1S ~ z,t ! , ]t ]z ]z

Cl

]Tl 5g ~ T e 2T l ! , ]t

~2! ~3!

where C e 5 g T e and C l are the specific heats of the electrons and the lattice, S(z,t) is the optical source term, and k e 5 k 0 T e /T l ~k 0 the equilibrium value! is the electronic thermal conductivity.12 The e- p coupling constant g drives the return to T e 5T l . For the purposes of comparison with the solution of the Boltzmann equation it is convenient to rearrange these equations in terms of the electron excess energy density E 5 g (T 2e 2T 2l )/25E e 2E 0 when changes in T l are relatively small compared to those in T e . This is a reasonable assumption since C e !C l . We first express the differential equations in terms of T 2e , treating T l as a constant:12

II. SOLUTIONS FOR THE SPATIALLY HOMOGENEOUS CASE

We first consider the athermal regime (t, t T ). A flat electron-hole distribution extending by photon energy h n either side of the Fermi level ~at « F ! is created by phonon assisted excitation, assuming k B T l !h n !« F and a laser pulse with a duration, typically ;100 fs, small compared to the electron energy loss time t E . 7 The initial occupation probability is taken as n 0 5N V /h n D(« F ), where D(«) is the electron density of states per unit volume and N V is the photoexcited electron density ~5F/dh n for d/ z !1, F the absorbed fluence!. We ignore the smoothing effect of the finite laser pulse duration, and therefore take the initial condition to be n « 5n 0 @ 12 u («2h n ) # , u the unit step function. We shall often refer to gold, a free electron metal near « F 55.5 eV. ~Only intraband transitions need be considered if h n ,2.4 eV.! For spatially homogeneous conditions, we shall work in the regime in which, typically, n 0 ;331024 . The relatively low value of n justifies the use of a linearized theory. For gold this value of n 0 corresponds to F ;3 m J cm22 if d;10 nm and h n 52 eV, for example. The e-e scattering integral has been evaluated in compact form:7

S D ]n ]t

52 e-e



t 0 ~ « 2F /« 2 !

1

6

t 0 « 2F

E

`

«

~ « 8 2« ! n « ~ « 8 ! d« 8 ,

~5!

where n « is the nonequilibrium component of n @] n/ ] t 5 ] n « / ] t in Eq. ~1!#. The first term represents the decay of electrons of excess energy « by the overwhelmingly dominant scattering from ground state electrons below « F . The lifetime

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VITALYI E. GUSEV AND OLIVER B. WRIGHT

t ee ~ « ! 5 t 0 « 2F /« 2 , ~for «@k B T l ;0.025 eV at 300 K!, increases towards «50 because of increasing restrictions from quasimomentum and energy conservation combined with the Pauli exclusion principle. Assuming one free electron per atom, t 0 can be simply estimated as ;d 0 / n F ~'0.3 fs in gold!, where d 0 is the interatomic spacing and n F the Fermi velocity.13 The second term in Eq. ~5! arises from new electrons being created at level « in the scattering process. As in the case with all the equations in this paper, electron-hole symmetry is assumed, and Eq. ~5! applies to energies «.0. ~This equation can be derived by expanding the general form of the e-e scattering integral to first order in deviations in n.! The e-p scattering integral is given by13

S D ]n ]t

2p 5 \ e- p

E

E

d 3q u M kk8 u 2 ~ 2p !3

3 @ N q~ n k8 2n k! 2n k~ 12n k8 !# d ~ « k2« k8 2\ v q! , where N q is the phonon distribution function, n k is the electron distribution function, and M kk8 is the e- p matrix element. Assuming isotropic n k , n k8 and quasielastic scattering ~«@\ v q , the phonon energy!, we expand n k8 5n «6\ v q to leading powers in \ v q , neglecting \ v q in the d functions:

S D ]n ]t

F

G

]n ] 2n 1k B T l 2 . ' p \l ^ v & ~ 122n ! ]« ]« e- p 2

~6!

We have assumed N q'k B T l /\ v q@1, since T. u D ~the Debye temperature '165 K in gold!, and used the notation of Allen14 to relate the e-p spectral function to the e-p coupling parameter l and the mean square of the phonon frequency ^ v 2 & , where g53 g \l ^ v 2 & / p k B . When n5n FD (T e ), a Fermi-Dirac distribution, Eq. ~6! yields the TTM energy exchange rate g(T e 2T l ). We deal here with a strongly athermal distribution. Linearization relative to the cold Fermi sea leaves only the first term ~the second is only appreciable at «;k B T l !, arising from spontaneous phonon emission:

S D ]n ]t

' p \l ^ v 2 & e-p

energy loss rate q˙ is defined with respect to «50. ~Including the effect of the mirror distribution for «,0 gives a total rate of twice this amount.! The electron energy loss rate is related to the characteristic time for spontaneous phonon emission t e p 51/p l ^ v & , 14 through the expression q˙ 5\ v ¯ / t ep , where v ¯ 5 ^ v 2 & / ^ v & is the characteristic frequency of the emitted phonons ~v ¯ ; ^ v & ; v D , the Debye frequency!. For later use we define

t e p ~ « ! 5«/ p \l ^ v 2 & , the characteristic time for a single electron to lose excess energy « @so t e p [ t e p (\ v ¯ );100 fs in gold#. The solution of the Boltzmann equation in the absence of e-e scattering is easily found. With the initial condition n « 5n 0 @ 12 u ~ «2h n !# ,

d 3q u M kk8 u 2 ~ 2p !3

3 @ N q~ n k8 2n k! 1n k8 ~ 12n k!# d ~ « k2« k8 1\ v q! 2p 1 \

57

]n« . ]«

~7!

Net energy loss at an excess energy « only occurs for ] n « / ] «Þ0 because the rate of loss of electrons with energy « is then different from the rate of their replenishment from the decay of electrons of higher energy. The electron energy loss rate for a single electron is given by q˙ 5 p \l ^ v 2 & . ~An approximate, classical form is used in Ref. 7.! This can be understood by taking n5 d («2« 0 )/D(« F ), « 0 a constant energy, and evaluating * «( ] n/ ] t) e-p D(« F )d«. The electron

the constant energy loss rate for each electron leads to a steady shift of this flat distribution function towards «50: n « 5n 0 @ 12 u ~ «1q˙ t2h n !# . In this case, all the excess energy is lost after a time t 5 t e p (h n ) ~;20 ps in gold with h n 52 eV!. If the e-p scattering term in the Boltzmann equation is set to zero and the e-e scattering term retained, Eqs. ~1! and ~5! can be solved exactly using Laplace transform methods with the same initial condition to give ~see the Appendix! 3 « Ap˜t ~ 2t˜11 !˜ 2

˜ 2˜ ˜ n ~˜ « ,t˜! 5 ~ 113t˜! e 2 ˜« t 23t˜ ˜ «e2 t 1

3 @ erf~ ˜ « A˜ t ! 2erf~ A˜t !#

~8!

˜ 2˜ ˜Ap˜t erfc~ ˜ « A˜t !# , '3t˜ @ e 2« t2«

~9!

˜ « 5«/h n , and ˜t 5t(h n ) 2 / t 0 « 2F where ˜ n 5n « /n 0 , 5t/ t ee (h n ) @;t/(2 fs) at h n 52 eV in gold#. Evolution with increasing electron number ever closer to the Fermi level occurs while the total energy remains constant. After a few collisions, ˜t @1, the scaling form, Eq. ~9!, applies. This scaling behavior has been discussed elsewhere.7 The electron number density N in this scaling regime can be calculated from Eq. ~9! using the definition N'

E

`

0

n « ~ « ! D ~ « F ! d«5N V

E

`

0

˜ ˜, n ~˜ « ! d«

where N V 5D(« F )h n n 0 , to give N 3 Ap 5 NV 4

At

t ee ~ h n !

.

~10!

This result was derived by Tas and Maris.7 The increase in N proportional to At becomes invalid for occupation numbers ;1, imposing the fluence-dependent limit t, t L ; t ee ~ h n ! /3n 0 on the use of this linearized theory when d/ z !1. @This expression for t L is obtained from Eq. ~9!.# For gold with h n 52 eV and n 0 5331024 , t L ;10 ps. With both e-e and e-p scattering terms present, we have not been able to solve for n for the complete problem @Eqs.

57

ULTRAFAST NONEQUILIBRIUM DYNAMICS OF . . .

~1!, ~5!, and ~7!#. Instead, we use an approximate method to estimate the nonequilibrium electron number (N) and energy (E) densities. To the lowest order of approximation, solutions can be obtained by taking N from the solution to the Boltzmann equation without the e-p scattering term, and then evaluating E from the exact relation

]E 522q˙ N, ]t

~11!

where E'2

E

`

0

n « ~ « ! «D ~ « F ! d«52E V

E

`

0

˜ ˜. n ~˜ « !˜ « d«

Equation ~11! is a consequence of each electron losing energy at the constant rate q˙ . Although this is a rough approach, we shall see that it leads to a result that is similar to that obtained with a more sophisticated treatment. The total electron energy density can be obtained using Eqs. ~10! and ~11!:

S D

E t 512 EV tE

3/2

~12!

,

where

t E5

S D

1 « Fg 1/3 1/3 2/3 1/3 t 0 t s '0.31t 0 p k Bg

2/3

,

~13!

and E V (5N V h n 5F/d) is the initial value of E. The characteristic time

t S 5 t ep ~ « F ! 5« F / p \l ^ v 2 & depends on the e-p scattering strength ~;60 ps in gold!. The constant 1/p 1/3 in Eq. ~13! was chosen to make E50 at t 5 t E in Eq. ~12!. Clearly, t E represents a characteristic time for electron energy loss ~t E ;0.8 ps for gold!. The exact choice of constant is a matter of convenience; it is sufficient that a significant amount of energy should be lost by the time t5 t E . Equation ~12! does not hold at times up to t; t E , but rather is valid when (t/ t E ) 3/2 is small (!1). Moreover, the derivation applies for t@ t ee (h n ), since it is based on Eq. ~9!. The remarkably simple Eq. (13) reveals explicitly how the e-e and e-p scattering strengths govern the electron energy release. The intriguing 31 and 23 powers arise because dE/dt}2N/ t S and, to first order, N} A(t/ t 0 ), leading to 1/2 t 3/2 E } t S t 0 . That t E is independent of h n is a direct consequence of the approximate scaling form of Eq. ~9!. The pivotal role played by t E can be appreciated by calculating approximate forms for the distribution-averaged scattering rates ^ 2( ]˜ n / ] t) e-p & and ^ 2( ]˜ n / ] t) e-e & , using Eqs. ~5!, ~7!, and ~9! @where n / ] t) & ^ 2( ]˜ 5 * `0 ( ] n/ ] t)d«/ * `0 n « d«]. Quantities ^ t ep & and ^ t ee & , the distribution averaged e-p and e-e scattering times, can be defined as the reciprocals of these scattering rates:

^ t ep & 5

p t 4 E

At

^ t ee & 52t.

E

t

,

2881

The latter time, strikingly, is independent of material parameters. Initially, when t; ^ t ee & ; t ee (h n ), e-e scattering is dominant: @where m ^ t e p & ; t e p (h n )5 t ee (h n )/ m 5 t ee (h n )/ t e p (h n )!1 is a dimensionless parameter#. The relative influence of e-p scattering increases until, at t ;0.5t E , ^ t e p & 5 ^ t ee & ; t E . We may also write t E 5 t e p (h n )( m / p ) 1/3, demonstrating directly the enhancement of the energy release owing to the increase in electron number from e-e scattering @m ;1024 , ( m / p ) 1/3;0.03, and t e p (h n );20 ps for h n 52 eV in gold#. This enhancement can be appreciated by comparing Eqs. ~10! and ~12! with the corresponding results for the case in which there is e-p scattering but no e-e scattering: N q˙ t t 512 512 , NV hn t e p~ h n !

S

E q˙ t 5 12 EV hn

D S 2

5 12

t t e p~ h n !

D

2

.

In order to find the solution for E and N in the presence of both scattering mechanisms to the next order of approximation, one must consider in more detail how e-p scattering affects the distribution function. Its introduction has two effects: ~i! it displaces the distribution at a uniform rate towards the Fermi level causing an effective recombination of electrons with holes at «50; 15 ~ii! the resulting energy loss slows the increase in electron number. Corrections due to these processes can be estimated by using the method of moments. We outline the method here, leaving the details to the Appendix. In dimensionless form Eqs. ~1!, ~5!, and ~7! reduce to

]˜ n ˜ 2˜ 52« n 16 ]˜t

E

`

«

˜ !˜ ˜ 81 m « 8 2« n ~˜ « 8 ! d« ~˜

]˜ n ]˜ « 8

~14!

˜21). To solve for the evowith initial condition ˜ n 512 u (« lution of the energy density we introduce moments of the electron distribution function M m5

E

`

0

˜ ˜, « m˜ n ~˜ « ! d«

~15!

where m is an integer. Recursion relations are obtained between integrals M m using Eq. ~14!. Solving for M m to first order in m by algebraic manipulation, we derive M 0 and M 1 , and hence N and E: N 3 Ap 5 NV 4

S DG S D S D

At

F

t t 122 b h n t ! ~ ee E

E t 512 EV tE

3/2

1b

t tE

3/2

,

~16!

3

,

~17!

where b 5(113 p 2 /16)/2p '0.45 @b 51/p in the absence of effect ~ii!#. These equations, still consistent with Eq. ~11!, reduce to Eqs. ~10! and ~12! when t is small. Equations ~16! and ~17! hold for ^ « & @k B T l , but eventually, at a time we define by t5 t T , the average energy per excitation ^«& becomes equal to that of a thermal ~Fermi-

VITALYI E. GUSEV AND OLIVER B. WRIGHT

2882

57

Dirac! distribution. For such a distribution function E 5 p 2 k 2 D(« F )(T 2e 2T 2l )/6 and N5D(« F )k(T e 2T l )ln2, giving

^ « & 5E/2N5 p 2 k B ~ T l 1T e ! /6 ln2. Provided that T e 2T l !T l , the approximation used in the present treatment, ^ « & 5E/2N5 p 2 k B T l /6 ln2;k B T l . Just after the arrival of the laser pulse the initial athermal distribution is characterized by ^ « & 5h n /2@k B T l . The increase in electron number due to e-e scattering and phonon emission lead to the decrease in ^«&, until ^ « & ;k B T l . However, Eqs. ~5! and ~7! are no longer valid for ^ « & ;k B T l , in which case temperature-dependent corrections to these equations become appreciable. Under such conditions the stimulated emission and absorption of phonons, described by the term in Eq. ~6! proportional to k B T l ] 2 n/ ] « 2 , is no longer negligible. The evolution of n « («), now described by an equation containing a term in ] 2 n « / ] « 2 , becomes diffusive in energy space, resulting in a significant rounding of n « («) for 0,« ,k B T l . As a simplifying assumption we shall assume that this rounded distribution n « («) can be described by a FermiDirac function at some temperature T e (t) for t. t T . This assumption implies that the linearized TTM in the spatially homogeneous case can be used for t. t T . This corresponds to the use of Eq. ~4! with D e 50 and S(z,t)50. In order to have an analytical expression for all times, we also make use of the approximate solutions ~16! and ~17! of the linearized Boltzmann equation up to t5 t T . Taking the total energy from Eq. ~17! as the initial condition for the solution of the linearized TTM for t. t T , one obtains

F S D S DG

E tT 5 12 EV tE

3/2

tT 1b tE

3

e 2 ~ t2 t T ! / t e ,

~18!

for times t. t T . For the TTM with T e 2T!T l , N is related to E through N5 ~ 3 ln2! E/ ~ p 2 k B T l ! , as discussed above. The small overheating of the electrons at t5 t T is given in this approximation by T e ( t T )2T l 5E( t T )/ g T l and for subsequent times by E(t)/ g T l . With estimates given below for t T and t E appropriate to gold, the condition T e ( t T )2T l !T l is equivalent to F!10 m J cm22 for d;10 nm. Stimulated phonon processes, whose effects are appreciable for t* t T , are controlled not by the spontaneously emitted phonons but by the interaction with the thermalized phonon bath at the lattice temperature T l . This is because the total energy of the phonon bath characterized by a Bose-Einstein distribution with temperature T l is assumed to be significantly larger than the energy of the emitted nonequilibrium phonons. This assumption is valid because DT l !T l for all times. ~In fact the assumption T e 2T l !T l imposes more stringent conditions on the fluence than does DT l !T l .! Reflectivity and photoemission data in gold confirm this relaxation to a Fermi-Dirac distribution, and give t T ;0.5 ps. 2,4 A rapid reshaping of the distribution at t; t T can be justified by estimates based on Eq. ~6! for the time, ;100 fs (! t T ), needed for stimulated phonon processes to effect the required smoothing of n « («) up to «;k B T l . 16

FIG. 1. Calculated evolution of ~a! nonequilibrium electron number and ~b! energy, under homogeneous conditions after excitation with an ultrashort optical pulse. Solid curves: interpolation model for t T 50.5 ps and t 0 50.33 fs, appropriate to gold; dashed curves: for t T 50.25 ps and t 0 50.22 fs; dotted curves: twotemperature model.

The condition for thermalization is obtained by equating the expression for E/2N for the two models at t5 t T , under the approximation T e 2T l !T l :

S D

12 u 3/21 bu 3 p 7/3 t S 1/2 3/2 5 u ~ 122 bu ! 4 ln2 t 0

1/3

k BT l , «F

~19!

where u 5 t T / t E , and use has been made of Eq. ~13!. Expressing u as a function of t 0 using Eq. ~13!, Eq. ~19! can be converted to a quadratic equation in t 0 1/2, with coefficients depending on t T and t S . Therefore, given values for t T and t S it is possible to solve analytically for t 0 . ~Conversely, by a numerical approach it would also be possible to evaluate t T given t 0 and t S .! Taking t T '0.5 ps from reflectivity measurements in the linear response regime,4 and using reasonable literature estimates of k (5317 W m21 K21), C e (52.0 4 23 21 310 J m K ), and t e (51 ps) for gold at 300 K ~g 5C e / t e 52031015 W m23 K21, t S 565 ps!,4,6 we derive t 0 50.33 fs and t E 50.75 ps. ~Only the larger root of the quadratic equation for t 1/2 0 is physically significant, giving the correct limiting behavior when t T →0.! Hot electron and low-temperature resistivity experiments yield similar t 0 , from 0.3 to ;1 fs. 2–5,17,18 Theoretical calculations based on the random phase approximation give t 0 '7.5/v p '0.55 fs for a free electron gas ~v p the plasma frequency!,19 whereas those based on the e-e scattering contribution to resistivity give t 0 '0.3 fs, 14,20 the latter estimate including the effects of the exchange interaction, umklapp processes, and band structure in general. The predicted variations of N and E in the spatially homogeneous case are shown for gold in Figs. 1~a! and 1~b! for our ‘‘interpolation model’’ that combines the Boltzmann equation and TTM ~see solid curves! and for the TTM used

ULTRAFAST NONEQUILIBRIUM DYNAMICS OF . . .

57

2883

t ee ~ k B T l ! t 0 « 3F h5 5 !140 t e p~ k BT l ! t S ~ k BT l ! 3

FIG. 2. Calculated evolution of the rate of energy deposition to the lattice under homogeneous conditions after excitation with an ultrashort optical pulse. Solid curves: interpolation model for t T 50.5 ps and t 0 50.33 fs, appropriate to gold; dashed curves: for t T 50.25 ps and t 0 50.22 fs; dotted curves: two-temperature model.

alone ~equivalent to setting t T 50—see dotted curves!. The interpolation model predicts a slower rate of energy loss at short times, consistent with the initially lower concentration of phonon emitting electrons. Only this model accounts for the increase in electron number, giving a peak in N, and hence also in dE/dt, at ;0.4t E ~, t T for t 0 50.33 fs!. The time t E thus gives a measure of the time when the sign of dN/dt changes. The electron number increases up to a maximum corresponding to ;10 times the number present just after excitation. @Equation ~16! applies for t@ t ee (h n ). This is the reason why N/N V 50 rather than 1 in Fig. 1~a! at t 50.# The peak in dN/dt described by Eq. ~16! arises when the effects of e-p scattering on the distribution function are correctly accounted for, and is not predicted with the simple treatment of Eq. ~10!. The curves for t T 50.25 ps ~t 0 50.22 fs and t E 50.66 ps—dashed lines! demonstrate the acceleration of thermalization with increasing e-e scattering rate. Figure 2 shows the corresponding normalized curves for dE L /dt52dE/dt, the rate of energy deposition to the lattice, where E L 5E V 2E is the energy lost to the lattice. The discontinuity at t5 t T gives an idea of the maximum error, typically ;20%, involved in our approximate treatment. This error arises from the extrapolations involved when using the two theories in the region t; t T . The value of ( t T / t E ) 3/2 for t T 50.5 ps is '0.5, indicating that our solution of the Boltzmann equation is at its limit of applicability at t5 t T . The overheating T e ( t T )2T l should be small compared to T l in the linearized version of the TTM. For the example of gold with F;3 m J cm22 and d;10 nm, T e ( t T ) 2T l '90 K. The energy deposition to the lattice in the thermalized stage of the relaxation process is ;1.4 times that in the nonthermalized stage. One can obtain a simple criterion for application of the analysis to different metals from the approximate relation

t T'

t 0 « 2F t ee ~ k B T l ! 5 . 40 40~ k B T l ! 2

giving ;0.4 ps for gold. This can be derived by neglecting the time-dependent terms in square brackets in Eqs. ~16! and ~17! @or by keeping only 1/u 1/2 on the left-hand side of Eq. ~19!#. Equations ~16! and ~17! are valid up to t5 t T if ( t T / t E ) 3/2!1, or, equivalently, if

~using the above relation for t T !. This condition, independent of h n , is valid for gold ( h 540). However, with estimates appropriate for a simple polyvalent metal such as Al, for example, one finds h ;3500. Thermalization for t T , t E is impossible, and nearly all energy is lost before ^«& becomes of the same order as k B T l . A more precise formulation of the scattering integrals should then be used for t* t E . Nevertheless, Eqs. ~16! and ~17! do apply at shorter times when ^ « & @k B T l : for Al one can derive t E '0.2 ps from estimated values of t 0 ~0.3 fs!, t s ~8 ps!, and t e ~0.06 ps!.7 For spatially homogeneous excitation we predict a peak in dE L /dt at ;0.1 ps, of the same order as that deduced numerically in Ref. 7. Noble metals Ag and Cu are probably in the intermediate regime with h nearer to the critical value of 140, but they should still be close in behavior to gold in line with their similar band structures and common grouping. Since h }1/T 3l , higher T l favors the use of the interpolation model. Direct comparison with experiment is possible in the diffusive case, and this is discussed in the next section. III. DIFFUSIVE CONDITIONS

The geometry we consider here is the case of the illumination of a bulk sample (d/ z @1) on one surface. The initial photoexcited electron density is now given by N V 5Fe 2z/ z /( z h n ). The essential difference compared to the spatially homogeneous case is that energy from the photoexcited region can now be lost not only by e-p scattering, but also by diffusion of the hot athermal electron distribution out of this region. For a laser pulse duration in the region of ;100 fs! t E , it is sufficient to include D e ] 2 n « / ] z 2 on the right-hand side of Eq. ~1!.7 Although ballistic transport plays a role at short times,3,21 this formulation is valid after the randomization of the hot electron momentum through e-e umklapp processes or e-p collisions. For «5h n 52 eV, e-e scattering is dominant in gold in determining this randomization time ; t ee (h n )/D;7 fs, where the dimensionless ratio D'0.35 arises since normal e-e scattering processes do not degrade the electron current.17 The corresponding e-p scattering time, the Drude relaxation time ~;25 fs at 300 K! is the limiting factor for lower « since t ee («)}1/« 2 . For t @25 fs, this Drude time determines D e ~independent of «! since most electrons will have relaxed to lower energies. Considerable simplification is achieved by noticing that the diffusive case can be reduced to the same algebraic problem as the homogeneous case: if the equations are expressed in terms of n 8« , where n « 5n « 8 z exp@ 2z 2 /4D e t # / A~ p D e t ! 5n 8« c ~ z,t ! , the Boltzmann equation with diffusion reduces to the form of Eq. ~1! for t@ t D 5 z 2 /D e ~in gold, z '15 nm and t D '15 fs!. Moreover, the initial condition in the Boltzmann equation for diffusion n « 5n 0 @ 12 u ~ «2h n !# exp~ 2z/ z ! ,

VITALYI E. GUSEV AND OLIVER B. WRIGHT

2884

57

also reduces to the homogeneous case n 8« 5n 0 @ 12 u (« 2h n ) # . In fact this reduction is exact without any restriction on t with

c ~ z,t ! 5exp~ T 2 !@ exp~ a z ! erfc~ T1Z ! 1exp~ 2 a z ! erfc~ T2Z !# /2, where a 51/z , T5 a A(D e t) and Z5z/2A(D e t). However, the simpler form for c (z,t) is sufficient in the present case ~for which t@ t D !. This simpler form corresponds to the approximation of the absorption of light at the surface of the sample. The boundary condition for diffusion is zero particle flux across the illuminated surface. Similar simplifications also apply to the TTM with diffusion, which reduces to the TTM without the diffusion term by use of the substitution T(z,t)5T(t) c (z,t). The existence of electron diffusion leads to an enhanced rate of cooling of the electron gas and to less stringent conditions for linearity. For example, even with the higher fluence F;50 m J cm22 ~N V ;1020 cm23 and n 0 '331023 ! the linearized theory turns out to be a valid approximation. As in the spatially homogeneous case, the use of the linearized theory becomes invalid for an occupation number ;1. This imposes the restriction t, t L8 5 p t 2L / t D . For the fluence F;50 m J cm22 in gold, t L8 '16 ps. The solutions to the Boltzmann equation for N(z,t) and E(z,t) in the diffusive case are thus obtained by multiplying Eqs. ~16! and ~17! by c (z,t). In this case the increase N } At, as in the homogeneous case, is balanced by the 1/At term in c, and N(z50,t) is independent of time during the regime of application of these equations. The expression for t T is the same as in the spatially homogeneous case because the ratio E/N is unchanged. The solution for E in the TTM including diffusion can be found in a similar way by multiplying Eq. ~18! by c (z,t). These variations determine the energy deposition rate to the lattice dE L /dt given by 2q˙ N(z,t) for t, t T and, from Eq. ~4!, by E(z,t)/ t E for t. t T . The normalized curves for the calculated energy lost to the lattice E L (t) and final energy deposition profiles E L (z,t5`) are shown in Figs. 3~a! and 3~b! for the example of gold. Compared to the initial optical absorption @dashed line in Fig. 3~c!#, the electrons diffuse about 10 times deeper (;150 nm). The integrated profiles * `z E L (z 8 ,t5`)dz 8 , shown in Fig. 3~c!, can be measured in coherent phonon experiments.6,8,22 ~Reflectivity and photoemission, essentially surface sensitive techniques, are not suited to depth profiling in the diffusive limit.! The temporal shapes of the echoes in such phonon experiments give a snapshot of the spatial integrated heat deposition profiles provided that the electron diffusion length is large compared to the optical absorption depth, valid for the case of gold. The predictions of both models, similar because t E ; t e , are in good agreement with the observed integrated profile for gold,6 as shown for the interpolation model in Fig. 4.23 The data is the average of both sides of the first echo in Ref. 6. Linearized models apply here because t T ! t L8 and T e 2T l &80 K!T l , and since t e (E)/ t e &1.1 in Eq. ~3!.24

FIG. 3. ~a! Calculated evolution of the energy deposited to the lattice under diffusive conditions for gold. ~b! Final energy deposition profiles in the bulk ~at t5`!. ~c! Normalized integral of the final energy deposition profile; dashed curve: initial optical absorption profile for h n 52 eV; solid curves: interpolation model; dotted curves: two-temperature model. IV. DISCUSSION AND CONCLUSIONS

In order to obtain analytical solutions we have been forced to make several approximations. What we have done in the case of gold is to effectively join two asymptotic solutions, one valid at short times and one ~the TTM! valid at long times. At intermediate times, near what we have termed the thermalization time t T , the analytical solutions are least

FIG. 4. Normalized integral of the final energy deposition to the lattice for gold as a function of depth (z/ z ) for the interpolation model compared to experimental results from coherent phonon experiments ~Ref. 6!.

57

ULTRAFAST NONEQUILIBRIUM DYNAMICS OF . . .

accurate. Thermalization is not a sudden process, but in fact occurs over a time interval ;k B T l t S /2« F . 16 Our definition of t T as the time for which the average energy per excitation becomes equal to that of a Fermi-Dirac distribution is the natural choice. This does not mean to say that at t5 t T the electron distribution is exactly described by a Fermi-Dirac distribution, but rather that this should approximately be the case. We have been able to find an analytical relation between t T , t 0 , and t S . However, it turns out that the quadratic equation for t 1/2 0 in terms of t T and t S , derived from Eqs. ~13! and ~19!, only has a real solution when t T / t e ,0.533.25 The parameters that we selected from the literature for the example of gold lead to t T / t e 50.5, and it is therefore possible to obtain t 0 . However, the value of t e depends on the value assumed for g, the e- p coupling constant, for which a wide range of estimates exist in the literature. The good agreement we found for the deposited energy profile with the results of coherent phonon experiments is in part due to the relative insensitivity of this profile to the value of t 0 . This is evident in Fig. 3~c! because the results for the TTM model used alone ~effectively for t 0 50! are in good agreement with those for t 0 50.33 fs. As shown previously,6,22 the linearized TTM predicts a deposited energy profile extending to a distance ; A( k e /g) into the bulk. Even in the presence of an athermal electron distribution, this result therefore remains a reasonable approximation for gold in the linear regime ~that is, at fluences for which the characteristic electron diffusion time is less than t L8 !. It is also possible that our estimate of k e is too large owing to the presence of grain boundaries and defects in the thin film samples of gold used in the coherent phonon experiments.26 In view of these uncertainties one should not be too hasty in claiming an excellent correlation with experiment until a wider range of measurements, for example on single-crystal gold films, are available. For any metal, there should always come a time t T when the conditions for effective thermalization to an overheated Fermi-Dirac distribution are met. The solutions we present are not precise, being valid for times & t E . However, we have shown how to determine whether these conditions are fulfilled on such time scales. At low excess energies « ( ^ « & ;k B T l ) corrections to the e-e and e- p scattering integrals render our treatment invalid. It is unlikely that an analytical solution can be found for t T in the case when t T * t E . In conclusion, we have obtained analytic solutions for the ultrafast athermal relaxation and diffusion of hot electrons in metals. We derive the electron energy loss time, allowing its scaling with e-e and e-p scattering strengths to be understood. This time is shown to govern the crossover between dominant e-e and dominant e-p phases of the relaxation. The thermalization time can also be derived for metals in which it is shorter than the electron-energy-loss time. In such metals this thermalization time is shown to be largely determined by the strength of the e-e interaction. Coupling the athermal stage with a thermalized stage by interpolation, we solve for both homogeneous and diffusive conditions, finding for the latter good agreement with the significant penetration of hot electrons observed in gold. Our approach may also prove useful for the interpretation of other hot electron

2885

experiments in metals above the Debye temperature. We hope that this analytical treatment of athermal electron relaxation in metals, supplying the link between critical relaxation times and fundamental microscopic parameters, will help stimulate further work in characterizing and elucidating the intricate interplay between the scattering mechanisms at work on ultrashort time scales. APPENDIX

Using the dimensionless variables ˜ « 5«/h n , ˜t 2 2 5t(h n ) / t 0 « F and the function ˜ n 5n « /n 0 , Eqs. ~1! and ~5!, in the absence of e-p scattering, take the form

]˜ n ˜ 2˜ 52« n 16 ˜ ]t

E

`

˜«

˜ !˜ ˜ 8, « 8 2« n ~˜ « 8 ! d« ~˜

~A1!

with initial condition ˜ n ~˜t 50 ! 512 u ~ ˜ « 21 ! ,

˜ « >0.

~A2!

The Laplace transform over time nˆ 5

E

`

˜

˜ n e 2s t dt˜

0

reduces Eqs. ~A1! and ~A2! to the equation

E

« 2 1s ! nˆ 56 ~˜

`

˜«

˜ ! nˆ ~ ˜ ˜ 8 112 u ~ ˜ « 8 2« « 8 ! d« « 21 ! . ~˜

The latter equation is transformed by two successive differentiations into a differential equation « 2 1s ! ~˜

] 2 nˆ ] nˆ ˜ 24nˆ 52 d 8 ~ ˜ « 21 ! . 2 14« ˜ ]« ]˜ «

~A3!

where d 8 refers to the derivative of the delta function. Since the right-hand side of Eq. ~A3! is active only at the boundary ˜51), Eq. ~3! can be simplified of the electron distribution (« to the homogeneous form « 2 1s ! ~˜

] 2 nˆ ] nˆ ˜ 24nˆ 50, 2 14« ˜ ]« ]˜ «

~A4!

provided that the appropriate boundary conditions are applied. From Eqs. ~A2! and ~A1! it follows that

]˜ n ˜ ]t

U

˜, 52n

˜

˜ n ~˜ « 51 ! 52e 2 t .

˜ « 51

˜,s) is One boundary condition for the Laplace transform nˆ (« therefore nˆ ~ ˜ « ,s ! 51/~ 11s ! .

~A5!

From Eqs. ~A1! and ~A2! it also follows that

]˜ n ]˜t

U

56 ˜ « 50

E

`

0

˜ ˜ 8 53, « 8˜ n ~˜ « 8 ! d«

because the energy of nonequilibrium carriers is conserved in the absence of e-p scattering. This leads to

2886

VITALYI E. GUSEV AND OLIVER B. WRIGHT

˜ n ~˜ « 50 ! 5113t˜

F

~A6!

~A7!

The exact solution of Eqs. ~A4!, ~A5!, and ~A7! is nˆ 5

F

1 1 ~ s13 ! 11s

3

E

1

`

E

G

dj ˜ « 2 ~ s13 ! j 2 ~ s1 j 2 ! 2

1

`

dj . j ~ s 1 j 2˜ «2!2 2

~A8!

2

It is not necessary to evaluate the integrals in Eq. ~A8! because the functions in this equation can be easily transformed by the inverse Laplace transform. The final result is Eq. ~8!. In order to take into account the weak influence of the e-p scattering on the evolution of the distribution function, it is possible in principle to solve Eq. ~14! with m !1 by a method involving successive approximations ~that is, by a perturbative approach!, using the solution in Eq. ~8! as a zeroth order approximation. However, we found it more suitable to apply the iterative procedure not to Eq. ~14! but to the set of equations for the moments of the distribution function @Eq. ~15!#. That is, we have concentrated on finding the electron number density and energy density rather than the distribution function itself. The zeroth and first moments in Eq. ~15! are proportional to N and E, respectively ~M 0 5N/N V and M 1 5E/2E V !. Applying the integral operation in Eq. ~15! to Eq. ~14! and to the initial condition ~A2!, one can derive the infinite set of equations

] M m8 ~ m21 !~ m14 ! 8 2 m mM m21 8 , 52 M m12 ˜ ]t ~ m11 !~ m12 !

8 5M ~m0 ! 1 m M ~m1 ! , Mm

~A10!

From Eq. ~A9! it immediately follows that, in particular,

]M1 52 m M 0 , ]˜t

] M ~m0 ! ~ m21 !~ m14 ! ~ 0 ! 52 M m12 , ˜ ]t ~ m11 !~ m12 !

8 2m M m 5M m

FE

˜t 0

G

˜ n ~˜ « 50,t˜ ! dt˜ d m,0 .

~A12!

In this approximation the third term on the right-hand side of Eq. ~A9! contributes only to the evolution of M 0 . This contribution can be evaluated with the use of Eq. ~A6!:

M ~m0 ! ~˜t 50 ! 5

1

, m11 ~A16!

and

] M ~m1 ! ~ m21 !~ m14 ! ~ 1 ! 0! 52 M m12 2mM ~m21 , ]˜t ~ m11 !~ m12 ! M ~m1 ! ~˜t 50 ! 50.

~A17!

We demonstrate the method of solution of these sets of equations for the case of the odd moments in Eq. ~A16! (m 52k21, k51,2,3, . . . ). Applying the Laplace transform in the time variable, one obtains ˆ ~ 0 ! 1 ~ k21 !~ 2k13 ! M ˆ ~0! 5 1 . sM 2k21 2k11 k ~ 2k11 ! 2k

~A11!

equivalent to Eq. ~11!. This demonstrates that once the zeroth moment is found, M 1 can be easily obtained by integration of Eq. ~A11!. As we are looking for an approximate solution that accounts for contributions not smaller than first order in the small parameter m !1, the last term on the right-hand side of Eq. ~A9! can be eliminated by the following substitution:

~A15!

where the corresponding sets of equations and initial condi(1) tions for M (0) m and M m are

~A9!

1 , m50,1,2 . . . . m11

~A14!

it follows that there could be other corrections of the same order in m because of the changes in the e-e scattering caused by the influence of phonon emission on the distribution function @referred to as effect ~ii! in Sec. II#. We have solved Eq. ~A14! using a method involving successive approximations, looking for a solution in the form

with initial conditions M m ~˜t 50 ! 5

~A13!

where the latter, scaling form applies for ˜t @1. This correction is caused by an effective carrier recombination @referred to as effect ~i! in Sec. II#. This corresponds to the loss of nonequilibrium electrons by phonon emission as they cross the Fermi level, resulting in a motion of the electron distribution as a whole towards the Fermi level at a constant velocity in energy space. This was taken into account in Ref. 7. However, from the equations

]Mm ~ m21 !~ m14 ! 52 M m12 2 m mM m21 ]˜t ~ m11 !~ m12 ! 2 m˜ n ~˜ « 50,t˜! d m,0 ,

G

3 3 M 0 5M 80 2 m ˜t 1 ˜t 2 'M 80 2 m ˜t 2 , 2 2

and to a second boundary condition, ˜ n ~˜ « 50,s ! 51/s13/s 2 .

57

This can be rewritten in the form R k11 2R k 5

2k11 , 2k ~ k21 !~ 2s ! k

~A18!

with R k5

2k11 ˆ ~0! . M ~ k21 !~ 2s ! k21 2k21

Assuming that R k →0 when k→` ~an assumption that can be checked after the solution is obtained!, one can eliminate all the unknowns in Eq. ~A18! except R k by summation of all the terms in Eq. ~A18!, starting at the fixed value k and finishing at `:

ULTRAFAST NONEQUILIBRIUM DYNAMICS OF . . .

57 `

R k 52

( l5k

method of solution to Eq. ~A17!, one can derive corrections for the even moments. In particular, one finds

~ 2l11 ! . 2l ~ l21 !~ 2s ! l

`

Or, in terms of the moments, ˆ ~ 0 ! ~ s ! 52 ~ k21 !~ 2s ! M 2k21 2k11

ˆ ~ 1 ! ~ s ! 52 M 0 k21

`

( l5k

~ 2l11 ! . 2l ~ l21 !~ 2s ! l ~A19!

`

( l50

~ 21 ! l11 ~ l11 ! . ~ 2l21 !~ 2l11 !~ s ! l11

After an inverse Laplace transform, one obtains

FA S D

G

1 3 Ap˜t ˜ erf~ A˜t ! 1e 2 t ' . 4 6t˜ 4 ~A20! ˜ The latter approximation holds for t @1, and leads to Eq. ~10! @obtained before by the use of Eq. ~9!#. Applying the same M ~00 ! ~˜t ! 5

3

p˜t 11

( l50

l ~ l21 ! ˆ ~ 0 ! ~ s ! . ~A21! M ~ 2l21 !~ 2s ! l11 2l21

By substituting Eq. ~A19! into Eq. ~A21! and performing the inverse Laplace transform, one obtains, in the limit ˜t @1,

A similar procedure for the even moments in Eq. ~A16! leads, in particular, to ˆ ~ 0 !~ s ! 5 M 0

2887

M ~01 ! ~˜t ! 5

F

G

3 3p2 2 ˜t , 12 4 16

~A22!

where the terms growing more slowly in time have been neglected. Therefore, the contribution to M 0 (t˜) from ˜ m M (1) 0 (t ) is of the same order of magnitude as that from the effective carrier recombination. Combining Eqs. ~A13!, ~A15!, ~A20!, and ~A22!, we find M 0 ~˜t ! 5

F

S

D G

3 Ap˜t 1 3 p 2 3/2 ˜t 12 11 , 4 16 Ap

which is equivalent to Eq. ~16!. Equation ~17! then follows from Eq. ~A11!. M. Kaveh and N. Wiser, Adv. Phys. 33, 257 ~1984!. C. R. Crowell and S. M. Sze, in Physics of Thin Films, edited by G. Hass and R. E. Thun ~Academic, New York, 1967!, p. 325. 19 J. J. Quinn and R. A. Ferrell, Phys. Rev. 112, 812 ~1958!. 20 In order to derive t 0 from parameters appropriate to lowtemperature resistivity measurements, it is convenient to work with the formula for the e-e scattering rate in the low-energy limit. Using the e-e scattering integral and taking account of the Fermi-Dirac distribution at a finite temperature T ~rather than using the approximation T50!, we obtain t 21 ee 2 2 5 t 21 0 ( p k B T/« F ) @ 11(«/ p k B T) # , a formula respecting the required electron-hole symmetry. We shall compare this formula with that for the frequency-dependent relaxation rate in optical absorption experiments @Eq. ~16.2! in Ref. 17#, 21 (\ v /2p k B T) 2 # 5 b ee (2 p k B T) 2 @ t 21 opt ( v ) # ee 5 t 0,ee (T) @ 1 1 21 2 1b ee (\ v ) , where t 0,ee is the dc limit for the scattering rate in optical experiments, and b ee is a temperature-independent parameter. Because only umklapp processes contribute to the optical absorption, t 21 0,ee should be equated with the zero « limit of t 21 ee D, where D is the dimensionless factor discussed in 21 ( p k B T/« F ) 2 Sec. III. One therefore obtains t 21 0,ee 5D t 0 2 2 5b ee (2 p k B T) , giving t 0 5D/(4b ee « F ). Furthermore, b ee can be related to the coefficient A ee arising in the e-e scattering contribution to low-temperature resistivity r ee (T)5A ee T 2 , through b ee 5 v 2p A ee /(16p 3 k 2B ). Taking theoretically calculated values of A ee for gold from Ref. 17 gives t 0 '0.28 fs. ~Experimental values of b ee from high-temperature optical data for gold give t 0 '0.29 fs.! 21 S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Phys. Rev. Lett. 59, 1962 ~1987!. 22 O. B. Wright and V. E. Gusev, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 329 ~1995!. 23 A single adjustable parameter was used to match the amplitude of the data to that of the theoretical prediction. 24 T e (t5 t T ,z)2T l 5 p 5/2Fk B c (z, t T ) A( t T / t 0 )/(« F g z ln 2), maximum at z50.

*On leave from International Laser Center, Moscow State Univer-

17

sity, 117899 Moscow, Russian Federation. † Also with PRESTO, Japan Science and Technology Corporation ~JST!, Kawaguchi 332, Japan. 1 P. Drude, Ann. Phys. ~Leipzig! 1, 566 ~1900!. 2 W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, Phys. Rev. B 46, 13 592 ~1992!. 3 See, e.g., C. Suarez, W. E. Bron, and T. Juhasz, Phys. Rev. Lett. 75, 4536 ~1995! and references therein. 4 C.-K. Sun, F. Vallee, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, Phys. Rev. B 50, 15 337 ~1994!. 5 R. H. M. Groenveld, R. Sprik, and A. Lagendijk, Phys. Rev. B 51, 11 433 ~1995!. 6 O. B. Wright, Phys. Rev. B 49, 9985 ~1994!. 7 See G. Tas and H. J. Maris, Phys. Rev. B 49, 15 046 ~1994!. The finite width of the optical pulse spectrum is ignored. 8 O. B. Wright and K. Kawashima, Phys. Rev. Lett. 69, 1668 ~1992!. 9 O. B. Wright and V. E. Gusev, Physica B 219&220, 770 ~1996!. 10 Originally due to M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Sov. Phys. JETP 4, 173 ~1957!. 11 S. I. Anisimov, B. L. Kapliovich, and T. L. Perel’man, Sov. Phys. JETP 39, 375 ~1974!. 12 P. B. Corkum, F. Brunel, N. K. Sherman, and T. Srinivasan-Rao, Phys. Rev. Lett. 61, 2886 ~1988!. 13 E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Vol. 10 of Course on Theoretical Physics ~Pergamon, Oxford, 1981!. 14 P. B. Allen, Phys. Rev. Lett. 59, 1460 ~1987!; Phys. Rev. B 3, 305 ~1971!. 15 One can account for effect ~i! by the modified distribution func˜1 m˜t ,t˜). tion ˜ n (« 16 Introduction of the term in ] 2 n/ ] « 2 implies that the distribution function evolves according to the diffusion equation in energy space. Variations in n on an energy scale ;k B T l are smoothed in a time ;(k B T l ) 2 /2D where D5k B T l « F / t S is the effective diffusion coefficient. This smoothing time is ;k B T l t S /2« F ;100 fs for gold at 300 K.

18

2888 25

VITALYI E. GUSEV AND OLIVER B. WRIGHT

Application of Eq. ~19! leads to the additional restriction t T / t E ,0.61 in order to avoid a second solution for which E/2N has the same value for a time smaller than t T . We have in the present paper extrapolated the solution to a value of t T / t E

57

slightly greater than this limiting value of 0.61 ~which in fact corresponds to t T '0.48 ps for the case of gold.! 26 T. Juhasz, H. E. Elsayed-Ali, G. O. Smith, C. Suarez, and W. E. Bron, Phys. Rev. B 48, 15 488 ~1993!.