Ultrafast dynamics of nonequilibrium electrons in metals under femtosecond laser irradiation

PHYSICAL REVIEW B, VOLUME 65, 214303

Ultrafast dynamics of nonequilibrium electrons in metals under femtosecond laser irradiation

B. Rethfeld*Institut fur Laser- und Plasmaphysik, Universita¨t Essen, D-45117 Essen, Germany

A. KaiserMax-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnitzer Straße 38, D-01187 Dresden, Germany

M. Vicanek and G. SimonInstitut fur Theoretische Physik, TU Braunschweig, D-38106 Braunschweig, Germany

~Received 5 December 2001; revised manuscript received 21 March 2002; published 22 May 2002!

Irradiation of a metal with an ultrashort laser pulse leads to a disturbance of the free-electron gas out ofthermal equilibrium. We investigate theoretically the transient evolution of the distribution function of theelectron gas in a metal during and after irradiation with a subpicosecond laser pulse of moderate intensity. Weconsider absorption by inverse bremsstrahlung, electron-electron thermalization, and electron-phonon cou-pling. Each interaction process is described by a full Boltzmann collision integral without using any relaxation-time approach. Our model is free of phenomenological parameters. We solve numerically a system of time- andenergy-dependent integro-differential equations. For the case of irradiation of aluminum, the results show thetransient excitation and relaxation of the free-electron gas as well as the energy exchange between electronsand phonons. We find that laser absorption by free electrons in a metal is well described by a plasmalikeabsorption term. We obtain a good agreement of calculated absorption characteristics with values experimen-tally found. For laser excitations near damage threshold, we find that the energy exchange between electronsand lattice can be described with the two-temperature model, in spite of the nonequilibrium distributionfunction of the electron gas. In contrast, the nonequilibrium distribution leads at low excitations to a delayedcooling of the electron gas. The cooling time of laser-heated electron gas depends thus on excitation parametersand may be longer than the characteristic relaxation time of a Fermi-distributed electron gas depending oninternal energy only. We propose a definition of the thermalization time as the time after which the collectivebehavior of laser-excited electrons equals the thermalized limit.

DOI: 10.1103/PhysRevB.65.214303 PACS number~s!: 63.20.Kr, 05.20.Dd, 42.50.Ct

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I. INTRODUCTION

Nonequilibrium dynamics of the electron gas in metirradiated by ultrashort laser pulses has been an area otense research during the last couple of decades. Withadvent of femtosecond laser pulses, direct experimental sies of fundamental processes such as electron-electrontering and electron-phonon interaction in metals havecome possible.1–14

Nonequilibrium between electrons and phonons is imptant already on a picosecond time scale. In a metal, first,electrons absorb energy from the laser while the latticemains cold. On a femtosecond time scale the energy istributed among the free electrons by electron-electron cosions leading to the thermalization of the electron gas. Tenergy exchange between electrons and the lattice iserned by electron-phonon collisions. Though the electrphonon collision timetep may be as short as the electroelectron collision timetee,15 the energy transfer from the hoelectrons to the lattice will last much longer than the thmalization of the electron gas due to the large mass difence of electrons and phonons; typically a few tens of piseconds. This picture was widely verified experimeally.1–4,16,17It is described by the classical two-temperatumodel,18,19 assuming that electron distributions and phondistributions can be characterized in terms of electron teperatureTe and lattice temperatureTp , respectively. Thenthe energy exchange is given by

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where ce and cp are the specific heats of electrons aphonons, respectively. The energy exchange ratea betweenelectrons and the lattice is related to the electron-phocoupling constant.20,21 The validity range of the two-temperature model is obviously limited to times longer ththe electron-electron collision timetee, lying in the femto-second range. However, as will be discussed below, theredoubts about its validity on time scales up to 1 ps.

In metals electron-electron scattering acts on femtosectime scales. This result was obtained by probing the lifetiof laser-excited electrons in two-photon photoemissexperiments.9,10,22 As soon as an electron is excited to aenergy« above Fermi energy«Fermi, it will decay after atime tee}(«2«Fermi)

22.23,24 However, the distribution func-tion of electrons is in nonequilibrium for a much longer timbecause secondary electrons are created. Thus, melectron-electron collisions are needed to establish finalFermi distribution. It was shown experimentally that thelectron distribution function, after excitation with a femtsecond laser pulse, is not in thermal equilibrium for a fe100 fs up to the picosecond regime.5,7 Theoretical calcula-tions confirm a nonequilibrium of the electron gas on thtime scale.15,25Thus, the description of the electron gas wia temperatureTe as done in the two-temperature model~1! isquestionable for time scales below 1 ps.

©2002 The American Physical Society03-1

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B. RETHFELD, A. KAISER, M. VICANEK, AND G. SIMON PHYSICAL REVIEW B65 214303

When investigating nonequilibrium dynamics of electrgases in metals, one fundamental question is the mutuafluence of electron-electron interaction and electron-phointeraction. For low excitations, experiments on gold asilver have shown a delay of electron-phonon relaxatcompared with the prediction of the two-temperature modThis effect was attributed to a nonthermalized electgas.11–14

Theoretical studies are essential to model the dynamica laser-excited electron gas and to understand the coquence of a nonthermalized electron gas in metals. Wheand when a distribution function may be assumed to be tmalized, strongly depends on the features one wishes toserve. When studying electron emission, for example, ononly interested in high-energy electrons. Then it is sufficito focus on the high-energy tail of the distribution functioand on the lifetime of one electron, which is the timeelectron stays in the high-energy region. Since the electelectron collision time depends on the electron energy,lifetime is much shorter than the collision time of an electrin the low-energy region. However, low-energy electronsessential when investigating the time needed to establiFermi distribution, which is important for the collective bhavior of the free-electron gas. The transient evolution ofelectron distribution in metals after laser excitation is thusfundamental interest for a wide field of research.

In this work we investigate the nonequilibrium dynamiof electrons in metals irradiated with a laser pulse of modate intensity, where the electron gas is significantly disturbut no lattice damage occurs. We use a time- and enedependent kinetic description, applying Boltzmann collisiintegrals explicitly without any relaxation-time approximtion, thus, without preanticipating any feature of the systeway of relaxation to equilibrium. In other investigations pulished so far7,12,15,25,26,14different approaches are usedmodel the energy absorption, the electron-phonon intetion, and the electron-electron interaction While the lattemostly calculated using a detailed collision term,7,27

electron-phonon collisions are usually described byrelaxation-time approach. For modeling the energy absotion strength, different phenomenological collision ratesapplied. Now we present, to the best of our knowledge,first calculation of the temporal evolution of the electrdistribution function in laser-excited metal representing eprocess by a detailed collision integral. Neither relaxatiotime approaches nor phenomenological parameters areplied in our calculations. Therefore, since we do not maany preassumptions about the behavior of the electronsare able to observe any unexpected behavior of the nonelibrium electron gas. Our results show the transient evolutof the distribution function of the laser-excited electron gTo interpret the calculated dynamics of the distribution funtion of the laser-excited electron gas we focus on the conquence of the nonequilibrium found. According to our rsults, after strong excitation even a nonequilibrium electgas maybehaveas if it were in a Fermi distribution, andtherefore, its energy exchange with lattice may be describy Eq. ~1!. However, if the electron gas is only slightly excited, the energy exchange between electrons and lattic

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delayed compared with Eq.~1! and the two-temperaturemodel fails. We thus confirm the experimental resultsRefs. 11–14. Our detailed microscopical model gives a cexplanation for the delayed energy exchange betweenweakly excited electron gas and the lattice. Finally, we ppose a definition of the thermalization time of the lasexcited electron gas after which its collective behavior equthe behavior of a Fermi-distributed electron gas.

In the following section we introduce the applied modand the collision integrals used for the numerical calculatiSection III explains the numerical algorithm and the adaption of the model to a specific material. As an example,calculate the distribution functions in the free-electron-limetal aluminum. The resulting electron dynamics is psented in Sec. IV, where the evolution of the nonequilibriudistribution function is shown. We compare the calculatenergy absorption with known absorption characteristThe energy exchange between electrons and lattice is intigated for high and low excitations. We determine the thmalization time of a laser-excited electron gas as a funcof excitation strength.

II. THEORETICAL MODEL

We assume a perfect, homogeneous, and isotropic mrial. The laser shall have a moderate intensity that is suciently large to disturb the free-electron gas significantly bis not so high that lattice damages may occur. We negenergy transport as well as spatial variation of pulse intsity. This can be interpreted as the irradiation of a thin filmmetal. Corresponding to our assumption of homogeneityisotropy we take an average over all polarization directioof the incident laser light. We are aware that these assutions are simplifications of the real situation, however, ointention is to reveal some universal features of thetransientexcitation and relaxation dynamics of electrons and phonin a metal. Therefore, we focus on the time-dependenthavior of the distribution functions, believing that our resuare fairly general. With our simplifications, the model is sptially independent and the distribution functions depend oon time and the modulus of the momentum or, equivalenon time and energy.

In order to consider photon absorption by inverse bremstrahlung, electron-electron interaction, and electron-phointeraction, one collision integral is used for each listed pcess. None of these complete collision integrals will beproximated by a relaxation-time approach. Moreover, aphenomenological parameters are avoided in our calculatThus, even the dynamics of a highly nonequilibrium electrdistribution is described by our model. The phonon gas sbe affected only by the electron gas directly. Therefore,ergy absorption by the lattice directly from the laser as was phonon-phonon collisions are neglected.

In total this yields to a system of Boltzmann’s equatiofor the distribution function of the electron gas,f (k), and ofthe phonons gas,g(q), which read

] f ~k!

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ULTRAFAST DYNAMICS OF NONEQUILIBRIUM . . . PHYSICAL REVIEW B65 214303

]g~q!

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. ~3!

Here, the distribution functionsf (k) andg(q) depend on themodulus of the wave vectors of electrons,k and phonons,q,respectively. Before irradiation,f (k) is assumed to be aFermi-Dirac distribution andg(q) shall be a Bose-Einsteindistribution, both at room temperature.

In the following we will have a closer look at the consiered interaction processes and their collision terms occurin Eqs.~2! and ~3!.

A. Electron-electron collisions

Electron-electron collisions lead to an energy relaxatwithin the electron gas called thermalization. Through tprocess the absorbed energy is distributed among theelectrons so that the electron gas tends towards a theequilibrium and thus to a Fermi distribution.

The collision term that describes the electron-electronteraction is given by7,27

] f ~k!

]t Uel-el

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\ (k1

(k2

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2 f ~k! f ~k2!$12 f ~k3!%$12 f ~k1!%#, ~4!

whereDk5k12k25k2k3 is the exchanged momentum anthe wave vectork3 results from momentum conservatio«(k) is the energy of an electron with the modulus of twave vectork. The matrix elementMee is derived from ascreened Coulomb potentialMee}(Dk21ksc

2 )21, whereksc

is the static screening length. This screening length resents an important parameter for the electron-electron inaction and is calculated at each time step for the curdistribution functionf (k) according to Ref. 28,

ksc2 5

e2me

p2\2«0E

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Here,me is the effective mass of a free electron in the coduction band. Using Eq.~5!, the screening lengthksc is con-sistently calculated even for a highly nonequilibrium electrgas.

B. Electron-phonon collisions

The main effect of electron-phonon collisions is to tranfer energy from the laser-heated electron gas to the coldtice. By emission of phonons the electron gas cools doSince the maximum phonon energy is small compared tokinetic electron energy, one electron-phonon collisichanges the energy of the electron gas only slightly. Thmany electron-phonon collisions are necessary to decrthe amount of kinetic energy stored in the electron gasaddition, electron-phonon interactions have an equilibrateffect on the electron gas. Phonon emission and absorpoccur in such a way that the electron gas tends to establFermi distribution~at phonon temperature!.

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The collision term for the electron-phonon interactioreads21,29

] f ~k!

]t Uel-phon

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\ (q

uMel-phon~q!u2$d„«~k1!2«~k!

2«ph~q!…@ f ~k1!$12 f ~k!%$g~q!11%2 f ~k!

3$12 f ~k1!%g~q!#1d„«~k2!2«~k!

1«ph~q!…@ f ~k2!$12 f ~k!%g~q!

2 f ~k!$12 f ~k2!%$g~q!11%#%, ~6!

wherek15k1q and k25k2q are the wave vectors of aelectron with resulting or initial wave vectork before or afteremission or absorption of a phonon with wave vectorq re-spectively.«ph(q) is the energy of a phonon with the modulus of wave vectorq.

In metals, not only the interaction of one electron with tlattice deformation has to be taken into account but alsoscreening of the lattice deformation by free electrons30,31andthe interaction of that electron with the screening electron24

This is considered by the electron-phonon matrix elemMel-phon(q) given in Ref. 32 withuMel-phon(q)u2}«ph(q)(q2

1ksc2 )21. The electron screening lengthksc is given above

by Eq. ~5!.

C. Energy absorption

Energy absorption from the laser occurs in a metal maithrough free electrons. In the classical picture, a free elecoscillates in the electromagnetic field of the laser and absoenergy only when it is changing its momentum parallel tooscillation direction. This can happen through a third cosion partner, which disturbs the oscillation of the electroThe kinetic oscillation energy isekin&5e2EL

2/(4mevL2) on

average, wherevL is the angular frequency of the laser ligandEL is the amplitude of its electric field. In our case, wconsider laser irradiation with parameters of such magnitthat the average kinetic energy of an oscillating elect^ekin& is much less than the photon energy\vL . Therefore,the absorption has to be described in quanta of photrather than by classical absorption.33 Analogously to the clas-sical picture, a third collision partner is also needed inquantum-mechanical one in order to ensure energy andmentum conservation.

In the literature, two collision integrals for absorption anemission of photons by free electrons can be found. Theone was given by E´pshtein.34 He used second quantizationderive a collision term that describes the absorption of ptons by free electrons in the conduction band of an insulawhen colliding with phonons. For this case, photon absotion mediated by electron-phonon collisions leads to goagreement of our calculations with experiments.35 Seely andHarris36 derived such a collision term of photon absorptifor inverse bremsstrahlung in a plasma, which differs frothe expression of E´pshtein only by the mediating matrix eement of the three-particle interaction. In metals the largcontribution to absorption is mediated by electron-icollisions,37,38 as in a plasma. Therefore, we apply the e

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pression of Seely and Harris to model the photon absorpby free electrons. With laser light of frequencyvL and am-plitude of electric fieldEL , this electron-photon-ion collisionterm reads

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\ (Dk

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2«~k!1 l\vL…. ~7!

HereDk is the exchanged momentum, andk85k1Dk isthe resulting or initial wave vector of the colliding electro

For the electron-ion interaction we assume againscreened Coulomb potential like for electron-electron cosions, which yields the same matrix element, see Sec. IThe probability of absorption~or emission! of l photons isgiven by the square of the Bessel functionJl

2 . The productEL•Dk in the argument of the Bessel function only allowabsorption of a photon if the change of the electron wavectorDk has a componentparallel to the electric laser fieldEL . This is analogous to the classical description. Becawe do not consider a particular polarization of the laser ligan average is made over all directions ofEL . In our calcu-lations electric laser field amplitudesEL of such magnitudeare assumed so that the argument of the Bessel functionJl

2 isalways small compared to unity. Therefore, the probabilitymultiphoton processes withu l u.1 is much smaller than thaof one-photon processes. Preliminary calculations hshown that multiphoton processes can be neglected wthe considered range of laser parameters.

At first glance, the consideration of ions as wellphonons in our model may appear surprising. However,a consistent description since in our model metal electrare considered asfree electrons, such as those in a plasmrather than Bloch electrons. Therefore, the interactionelectrons with fixed ions has to be included separately. Tcan be understood when looking at the Hamiltonian ofmetal,

H5(j

\2

2mekj

21(j ,a

V~r j2Ra!1Hee1Hi i , ~8!

where the first term describes the energy of the free etrons, the second term denotes the interaction of electronpositions r j with ions at positionsRa , and the third andfourth terms denote the electron-electron interaction andion-ion interaction, respectively. The second term is usuexpanded, leading to a term describing the interactionelectrons with fixed ions and the interaction of electrons wthe potential caused by the displacementdRa of ions fromtheir equilibrium positionsRa,0 :

H5(j

\2kj2

2me1(

j ,aV~r j2Ra,0!2(

j ,adRa¹V~r j2Ra,0!

1Hee1Hi i . ~9!

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The first two terms of Eq.~9! are usually combined to Blochelectrons, while the third term describes the electron-phointeraction. In our work, however, we describe the electroas free electrons, thus we consider only the first term of E~9!. The second term of Eq.~9!, i.e. the interaction of elec-trons with fixed ions, should therefore be considered seately. This separation does not affect the standard electphonon interaction, described by the third term in Eq.~9!.Note that collisions of electrons withfixedions change solelythe momentumof free electrons, while theenergyof a freeelectron changes in collisions with ionvibrations, which arerepresented by phonons. Thus, in our model, electroncollisions mediate the energy absorption of laser light by felectrons, while energy transfer between the electron gaslattice occurs through electron-phonon interaction.

We consider only one dispersion relation of free electrothus only intraband absorption mechanisms are possiblepolyvalent materials, also interband absorption betwenearly parallel bands may play a role.39–41 In contrast tointraband absorption, interband absorption occurs withmomentum transfer to the absorbing electron. This absotion mechanism, associated with bound electrons ratherwith free electrons,39 is neglected in our model.

D. Equation for phonons

The phonon distribution function is assumed to chanonly due to phonon-electron collisions. Direct absorptionthe laser energy is neglected. For simplicity, we consionly one phonon mode and do not take umklapp procesinto account. Thus, phonon-phonon collisions are neglecas well. Phonon-electron interaction leads to heating ofphonon gas, analogously to the cooling of the electron gaelectron-phonon collisions, see Sec. II B.

The phonon-electron collision term is given by

]g~q!

]t Uphon-el

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\uMphon-el~q!u2(

kd„«~k1!2«~k!

2«ph~q!…@ f ~k1!$11g~q!%$12 f ~k!%

2g~q! f ~k!$12 f ~k1!%# ~10!

with k15k1q. The matrix elementMphon-el is the same asthe matrix elementMel-phon, described in Sec. II B.

III. NUMERICAL SOLUTION

In this section we give an overview of how and wiwhich assumptions the equation system with the above induced collision terms is numerically solved. First the cosion sums are transformed into collision integrals usingcommon approaches, so we are dealing with one sixfoldtegral and three threefold integrals. Due to the assumedropy, the integrals can be analytically reduced to a sintwofold integral and three one-fold integrals, respectivewhich depend only on the modulus of the electron and pnon wave vectors.27,35,42

The energy dispersion of electrons is approximated bparabola,«(k)5\2k2/2me , with the effective electron masme . For phonons, Debye’s dispersion relation is assum

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«ph(q)5vsq, wherevs is the speed of sound. Then, the sytem of nonlinear integro-differential equations~2! and ~3!and the collision integrals can be rewritten dependingtime and energy, using the distribution functionf „«(k),t…andg„«ph(q),t….

In order to solve the equation system numerically, discrelectron energies and phonon energies are considered, reing in a system of about 250 fully coupled, nonlinear ornary differential equations. The integration over time is doby applying Runge-Kutta integration of fifth order with atomatic step-size control.43 With this procedure, we are ablto follow the time evolution of the distribution functionf „«(k),t… andg„«ph(q),t… and observe their changes dueexcitation of the electron gas by the laser beam, thermaltion of the electron gas by electron-electron collisions, aenergy exchange between the electron gas and phonondue to electron-phonon collisions.

While in metals the assumption of Debye’s dispersionlation for phonons is not really a restriction, the parabodispersion for the electrons is. However, in aluminumelectrons dispersion relation resembles rather well a felectron-like parabolic dispersion relation; thereforechose aluminum for our calculations. Other dispersion retions could also be included, but this would make the callations more demanding. Gold is often chosen for simcalculations, however, in gold there ared-band electronsabout 2.5 eV below the Fermi edge of the frees electrons,requiring much more complicated calculations: Thesed elec-trons cannot be neglected even for lasers with\vL,2.5 eV, since during irradiation free states occur belthe Fermi edge~see Sec. IV A! and, therefore, thed electronsmay also be strongly excited.

For the calculation of the time evolution of the distribtion functions in a given material, here aluminum, seveparameters have to be provided. These are the free-eledensityne518.031028 m23, the wave number at Fermi energy forTe50 K, kF,05(3p2ne)

1/351.74731010 m21, theion density ni5ne/3, and Debye’s wave numberqDebye5(6p2ni)

1/351.52631010 m21.44,45 The effective electronmassme and the speed of soundvs are needed for the dispersion relations. We calculated the internal energy of etrons and phonons for different temperaturesTe andTp , re-spectively, and compared the resulting heat capacitiesexperimental values.42 This leads to an effective electromass ofme51.45me,free for aluminum, as also derived inRef. 46. The phonon heat capacity turned out to be breproduced when applying the sound speed of longitudphonons,vs5vs, long56260 m/s.45 Note that all these parameters describe theundisturbedcrystal. No parameters describing the investigated dynamics of the electron gasphonon gas or the interaction with the laser beam are induced. Neither do any phenomenological parameters oparameters occur in our calculations.

IV. RESULTS FOR ALUMINUM

We assume a laser pulse of constant intensity with dution tL5100 fs and vacuum wavelengthl5630 nm, cor-responding to a photon energy of\vL51.97 eV

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50.245«Fermi. Electrons and phonons are assumed to betially in thermal equilibrium at 300 K.

A. Change of electron distribution function

Figure 1 shows the transient behavior of the occupatnumber of free electrons for an electric laser field of amptude EL51.43108 V/m, corresponding to an intensity oI L573109 W/cm2 and an absorbed fluence ofFabs50.7 mJ/cm2. A function F, which is defined as

F„f ~«!…ª2 lnS 1

f ~«!21D , ~11!

is shown as a function of electron energy«/«Fermi,0. Thisfunction increases monotonically with increasing occupatnumberf («) and is particularly suitable to visualize the peturbation of the electron gas. In thermal equilibrium, whelectrons obey a Fermi-Dirac distribution,F(«) equals(«Fermi,02«)/(kBTe). In this case,F(«) is a linear functionwith a slope proportional to the inverse electron temperat1/Te . Thus, a deviation of the electron gas from thermequilibrium is directly reflected in a deviation ofF(«) froma straight line. Figure 1~a! shows a strong perturbation of thelectron gas immediately after the beginning of irradiatioIn comparison with the straight solid line representing tinitial Fermi distribution at 300 K, the absorption of photonlead to a steplike distribution function: Electrons belo

FIG. 1. Distribution function of free electrons in aluminium~a!during and~b! after irradiation. The quantityF( f ) defined by Eq.~11! is shown as a function of electron energy. A laser pulse of 1fs duration with constant intensity was assumed with a photonergy of\vL51.97 eV50.245«Fermi and an electric-field amplitudeEL51.43108 V/m. ~b! shows a section of about«Fermi6\vL ofthe energy scale.

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Fermi energy absorb photons, leading to an increase ofoccupation number of electrons with energies up to\vLabove Fermi energy. An excited electron may absorb ather photon, leading to an increase of the occupation numfor energies up to 2\vL above Fermi energy. The occupationumber of electrons below Fermi energy decreases atsame rate, reproducing the Fermi edge in steps of\vL . Asimilar steplike electron distribution function was found fenergies above the Fermi energy in Refs. 47 and 48, incing also one step below the Fermi energy. Experimentallyfirst plateau of excited electrons with energies up to«Fermi1\vL was observed in gold in Refs. 5 and 6, and theorcally reproduced in Refs. 15 and 25. Our logarithmlike pof the functionF(«) has the advantage that not only excitelectrons above Fermi energy but also the resulting ‘‘holbelow Fermi energy are easily observed. In Fig. 1~a! theevolution of the electron distribution function during irradition is shown. The clear steplike structure at the beginningthe pulse is washed out quickly due to electron-electronlisions, which act towards thermal equilibrium within thelectron gas and smoothen the functionF(«). Figure 1~b!shows the completion of electron thermalization after irradtion for energies of about6\vL around the Fermi edge. Astraight line F(«), corresponding to a Fermi distributiof Fermi(«), is reached about 200 fs after irradiation has endits slope, however, is still much smaller than those forFermi distribution at 300 K. Thus the electron gas is heasignificantly. Later on, the effect of electron-phonon interation is also visible. The cooling of the hot electron gas leato an increasing slope ofF(«), corresponding to lower electron temperatures.

B. Change of phonon distribution function

Figure 2 shows the change of the distribution functionthe phonons due to electron-phonon interaction. A functG, defined as G„g(«ph)…ª2 ln@111/g(«ph)#, is shown,which is a linear function with a slope proportional to 1/Tp inthermal equilibrium, when phonons obey a Bose-Einstdistribution.

FIG. 2. Distribution function of phonons for the same laser prameters as in Fig. 1. The functionG(g)52 ln(111/g) of the dis-tribution function g is shown in dependence on phonon enerwhere the maximum phonon energy«ph, max5\qDebye.

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fn

n

The phonon gas is nearly unaffected during the puAfter irradiation ends, the occupation number and thusinherent energy of the phonon gas increases. Since the melement of phonon-electron interaction vanishes for«ph→0,the phonon distribution function becomes slightly equilirium shifted at later times.

In the following we focus on electron and phonon behaior during and directly after irradiation, where the distubance of phonons from thermal equilibrium canneglected.

C. Absorption characteristics

The internal energy within the electron gas,ue(t), or pho-non gas,up(t), is calculated by integrating over the corrsponding distribution functionsf (k,t) and g(q,t), respec-tively. Due to the assumption of isotropy, this reads for telectronsue(t)52/2p2*0

` f (k,t)\2k2/(2me)k2dk.

The gain of additional internal energy due to laser exction in the electron gas at timet, due(t), is given by thedifference betweenue(t) and the initial energyue(2`).Since we assumed the free electrons to be initially in therequilibrium at a temperature of 300 K, for the absorbedergy of the electron gas,due(t) follows

due~ t !ªue~ t !2ue~2`!

52

2p2E0

`

@ f ~k,t !2 f Fermi,300 K~k!#\2k2

2mek2dk. ~12!

Here, f Fermi,300 Kis the Fermi-Dirac distribution at room temperature~300 K!. Analogously, the gain of absorbed enerper volume by phononsdup(t)ªup(t)2up(2`) is calcu-lated keeping in mind that initially the phonons are in BosEinstein distribution at a temperature of 300 K. The sumboth increments of internal energy,du(t)5ue(t)1up(t),gives the total energy absorbed from the laser. Checkwhetherdu(t) remains constant after irradiation can be usto verify the numerical stability of our calculations.

Figure 3 shows the transient behavior of energy increfor the electron gas, the phonon gas, and the total abso

-

,

FIG. 3. Transient energy increase of electron gas,due , and ofphonon gas,dup , respectively, and total absorbed energydu. Thelaser pulse was assumed with the same parameters as in Fig.

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energy, respectively, for the same laser parameters as in1. The absorbed energydu(t) increases linearly during irradiation with constant intensity, as expected. After irradiatdu remains constant at a valueduendªdu(`), in this caseduend57.13108 J/m3.

We applied a wide range of electric laser field amplitudEL , respective of laser intensitiesI L . For all laser intensitiesconsidered up to damage threshold, the absorbed energduincreases linearly during irradiation with constant intensthus the energy increase can be expressed asdu/dt5duend/tL . As shown in Fig. 4, the absorbed energy afirradiationduend is proportional to the square of the electrlaser fieldEL

2 , and hence proportional to the laser intensI L . Since, compared with spatially dependent absorpcharacteristics, the equationdI/dz52du/dt and thus forour casedI/dz}I holds, the absorption characteristics in ocalculations correspond to the usual exponential absorpprofile I (z)5I L exp(2mz). The absorption strength in oucalculations can thus be compared with literature by extring the corresponding absorption constantm52I (z)21dI/dzuz , which for z50 in our case equalsm5I L

21duend/tL . The intensityI L inside the material is givenby the density of the energy flux of an electromagnetic wain metal, I L5Ae0 /m0nEL

2 ,49 wheren is the real part of thecomplex refractive index. In aluminumn equals 1.36 forl5630 nm.50 From Fig. 3, we find that the calculated absortion strength corresponds to an absorption constant ofmnum51.013108 m21. The experimental value for aluminum fol5630 nm ismLit51.523108 m21.50 However, in alumi-num absorption is to a large extent caused by interbandcesses, not considered in our model. At the applied phoenergy, intraband absorption provides about the sixth pathe total absorption.40,41 Thus, our calculated absorptiostrength appears to be larger than the expected intrabansorption strength. We also performed calculations withhigher photon energy atl5350 nm, where interband absorption is less pronounced. In this case, the calculatedsorption strength ismnum51.13108 m21,42 in good com-parison with the literature value ofmLit51.543108 m21.50

Thus, keeping in mind that no phenomenological paraeter entered our calculation, the calculated absorp

FIG. 4. Total absorbed energyduend as a function of electriclaser fieldEL . A laser pulse of 100 fs duration,l5630 nm, andconstant electric field was assumed.

21430

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strength compares well with literature. The comparison cfirms the validity of our applied absorption term developin Sec. II C. Note that we did not consider spatial variatiof the electric laser field in our calculations but used knoexperimental spatial absorption characteristics only for coparison with our calculations. The comparison refers toqualitative dependence of absorption on laser intensity anthe quantitative absorption strength.

D. Energy relaxation between electrons and phonons

In Fig. 3 the energy increase in the electron gasdue(t)and the energy increase in the phonon gasdup are showntogether with the total absorbed energydu during and afterirradiation with a laser pulse specified in Fig. 1. The eneincrease in the electron gas during irradiation follows ttotal absorbed energy with a slight decrease due to entransfer to the phonon gas during the pulse. After irradiathas ended, energy transfer from the electron gas to thenon gas continues. The electron gas loses energy at therate as the phonon gas is heated up. This process, causelectron-phonon collisions, continues until both systehave the same temperature. Because of the large differin heat capacity, this corresponds to a much lower interenergy of the electron gasdue than of the phonon gasdup .

When aluminum is irradiated by a 100-fs laser pulse wa constant electric-field amplitude ofEL51.43108 V/m ~in-tensity I L573109 W/cm2), the electron gas absorbs aamount of energy ofdue56.73108 J/m3. If this electrongas was to thermalize without energy loss, i.e., with consinternal energy, it would end in a Fermi-Dirac distributiona temperature ofTe53215 K. An interesting question ishow the cooling behavior of the laser-excited electron gdiffers from that of a Fermi-distributed electron gas with tsame internal energy. The cooling of both, the laser-excelectron gas as well as the corresponding Fermi-distribuelectron gas, is shown in Fig. 5~a!. According to this figure,the energy transfer rates from both kinds of electron gasthe phonon gas are essentially the same. This implies thathese relatively high laser intensities the energy exchabetween free electrons and phonons is determined byinternal energydue only, which can be characterized bytemperatureTe , and can thus be described by the twtemperature model~1!. The electron-phonon coupling constanta in Eq. ~1! can be extracted from Fig. 5~a!. We find avalue ofa531031015 J/K s m3 for aluminum.

In the case of irradiation with a smaller intensity, e.g.,a 100-fs laser pulse with an electric laser field amplitudeEL543107 V/m ~laser intensity ofI L55.83108 W/cm2),the two-temperature model does not hold, see Fig 5~b!. Inthis case, the electron gas is excited more weakly than in5~a!; it has absorbed an energy ofdue55.53107 J/m3. AFermi distribution atTe5960 K has the same internal energy. Figure 5~b! shows that the initial cooling rate of thlaser-heated electron gas is substantially lower than the cing rate of the corresponding Fermi-distributed electron gSuch delay of energy transfer to the lattice for a laser pturbed electron gas compared with the two-temperatmodel was observed experimentally at low temperatures

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for very low intensities.11–14 Thus, for weak excitation, thecooling rate of the laser-heated electron gas is determineits particular electron distribution function in nonequilibriuand not only by the internal energy as in the case of stronexcitation.

In order to explain this different behavior, we have plottin Figs. 6~a! and 6~b! the above-mentioned electron distribtions near the Fermi edge at a time when the laser pulsejust ended (t50 in Fig. 5!. The functionsF of the distribu-tion functions given by Eq.~11! are shown near the Fermedge. In Fig. 6~a! F is shown for an electron gas after lasexcitation with a field ofEL51.43108 V/m, leading to anabsorbed energy ofdue56.73108 J/m2. The functionF ofa Fermi-distributed electron gas withTe53215 K, havingthe same internal energy, and withTe5300 K are shown aswell. The same was done in Fig. 6~b! for the electron gasexcited by a laser pulse withEL543107 V/m, whichcauses an energy increase ofdue55.53107 J/m3. Also F ofthe corresponding Fermi distribution withTe5960 K andF( f Fermi, 300 K) are plotted in Fig. 6~b!.

At t50 the hot electron gas interacts with a phonon gof about Tp.300 K. Thus, at this moment the electrophonon interaction acts on the electron gas in such a waythe electron gas tries to establish a Fermi distribution300 K. To this end, electrons above Fermi energy have to

FIG. 5. Transient internal energy of laser excited electron~solid lines! for excitation with a laser-pulse of 100 fs duration aa constant electric-field amplitude of~a! EL51.43108 V/m ~inten-sity I L573109 W/cm2) and ~b! EL543107 V/m ~intensity I L

55.83108 W/cm2). The dashed-dotted lines show the cooling ocorresponding Fermi-distributed electron gas, which has the sinternal energy at the timet50.

21430

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transferred to states below Fermi energy. However, duethe small phonon energy, the phonons are able to act onelectrons only in a small region around the Fermi edge. Tmaximum phonon energy in the case of aluminum is ab0.008«Fermi.

In Fig. 6~b! the distribution functions in the case of lowexcitation are shown. In the region of one maximum phonenergy around Fermi energy~shaded area!, the laser-excitedelectron gas shows nearly no deviation from the Fermi dtribution at 300 K. In contrast, the Fermi distribution witTe5940 K of a thermalized electron gas with the sameternal energy differs strongly from the Fermi distribution300 K. Thus, after low exitation, the phonon cooling of thnonequilibrium electron gas is much less efficient thancooling of a corresponding thermalized Fermi-distributelectron gas. Therefore, the cooling rates are far from bethe same, which explains the different slopes of energy deat t50 in Fig. 5~b!.

In contrast, for high excitations as shown in Fig. 6~a!, thedistribution function of the laser-excited electron gas de

s

e

FIG. 6. Distribution function of laser-excited electron gas~solidlines! for excitation with a laser pulse of 100 fs duration andconstant electric-field amplitude of~a! EL51.43108 V/m ~inten-sity I L573109 W/cm2) and ~b! EL543107 V/m ~intensity I L

55.83108 W/cm2). The quantityF( f ), defined by Eq.~11! isshown as a function of electron energy. The dashed-dotted lshow the corresponding Fermi-distributed electron gas, whichthe same internal energy as the laser-excited electron gas. Theted lines show the Fermi distribution for an electron gas at rotemperature. A region of one maximum phonon energy aroundFermi edge is shaded.

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ates strongly from the Fermi distribution at 300 K in thregion of one maximum phonon energy. In this case the cing of the laser-excited electron gas by phonon emissionearly as efficient as the cooling of the Fermi-distributelectron gas. Though also here the distribution functionthe laser-excited electron gas differs from the correspondFermi distribution, i.e., the electron gas has a highly nequilibrium distribution function, our calculations show thits cooling behavioris nearly the same as the behavior othermalized electron gas.

E. Relaxation time and thermalization time for electrons

The so-called relaxation time is a characteristic timethe energy exchange between the electron gas and phgas. Yet, when speaking about relaxation timet rel , equilib-rium states and only those transitions that run through elibrium states are considered. In this case,t rel is a generalfeature of the heated electron gas, which depends only oninternal energy or temperature of the electron gas. Incase the time for energy exchange depends strongly on pcharacteristics and thus on the history of the electron gTherefore, we prefer to speak about an ‘‘initial cooling timtcool,0, which characterizes the energy decrease directly athe pulse. We extract it from our calculations through tslope of the energy decrease of the electron gas after irration has ended, see Fig. 7.

Figure 8 shows the cooling timestcool,0of the electron gasin aluminum as a function of the internal energy. To demostrate the dependence of the cooling timetcool,0 on the his-tory of the electron gas, we assumed excitations with a lapulse of 20 fs and 100 fs duration, respectively, and a cstant intensity. The internal energy was adjusted by usdifferent laser intensities. Furthermore, the relaxation tit rel of a corresponding Fermi-distributed electron gas of

FIG. 7. Determination of the cooling timetcool,0 and the ther-malization timet therm. The example is shown for the case of irrdiation with a 100-fs laser pulse with an electric-field amplitudeEL563107 V/m. The cooling time is extracted through the sloof energy decrease~dashed line! directly after excitation (t50,point O). The thermalization time is found by shifting the coolincurve of the Fermi-distributed electron gas keeping constant en~dotted line! until from a certain pointT onwards, both coolingcurves coincide.

21430

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same internal energy is shown. We see that the energy trfer between the laser-excited electron gas and lattice islayed for small excitations and short pulse lengths. Thependence on excitation strength was discussed in Sec. IVthe dependence on laser-pulse duration is evident, sincelonger pulses electron-electron interaction already workswards a thermalized electron gas during the pulse.

Although an excited electron gas in equilibrium andnonequilibrium acts differently in detail, its macroscopicbe-havior, in particular, the cooling behavior, is not distinguisable after a specific time. In Figs. 5~b! and 7 we see that atime proceeds, the rate of electron cooling~slope of thecurve! of the laser-excited electron gas becomes equal tocooling rate of the Fermi-distributed electron gas, compaat the same energy. This provides an opportunity to definnew value to characterize the electron gas. For this, wetroduce the thermalization timet therm, which gives the timewhen an electron gas in nonequilibrium exchanges enewith a cold lattice at the same rate as a Fermi-distribuelectron gas, which has the same internal energy. Figushows graphically how the thermalization timet therm is ex-tracted from our calculations. Note that this isnot the timeafter which a Fermi distribution is established but the timewhich the laser-excited electron gasbehavesas a Fermi-distributed electron gas. The behavior of the laser-excelectron gas may be the same as the behavior of the Fedistributed electron gasbeforean equilibrium is reached inthe laser-excited electrons.

Figure 9 shows the thermalization time of the lasexcited electron gas depending on internal energy after idiation with a laser pulse of constant intensity and a duratof 20 fs and 100 fs, respectively. The thermalization timt therm after an excitation of 100 fs duration is about 80shorter than thet therm for a 20-fs excitation because ithis case the electron gas is already thermalizing durirradiation.

V. CONCLUSIONS

We investigated irradiation of metals with a femtosecolaser pulse using a kinetic description. The effects of parti

f

gy

FIG. 8. Cooling timetcool,0 of nonequilibrium electron gas afteexcitation with a laser pulse of constant intensity and duration100 fs ~solid line! and 20 fs~dashed line! depending on internaenergy, together with the relaxation timet rel of the correspondingFermi-distributed electron gas~dashed-dotted line!.

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lar collision processes on the electron distribution functwere studied. In our model we consider absorption byverse bremsstrahlung, electron-electron thermalization,electron-phonon coupling. In contrast to other kineapproaches7,12,15,25,26,14we explicitly calculate the transienfree-electron distribution by microscopic collision integraavoiding any phenomenological or averaging parameterswithout using any relaxation-time approach. In this way,are able to follow the transient evolution of a laser-excitelectron gas even when it is in a highly nonequilibrium stawithout anticipating any feature of its behavior.

FIG. 9. Thermalization timet therm of nonequilibrium electrongas after excitation with a laser pulse of constant intensityduration of 100 fs~solid line! and 20 fs~dashed line! depending oninternal energy.t therm is defined as the time when the cooling ratethe laser-excited electron gas equals the cooling rate of a cosponding Fermi-distributed electron gas, see Fig. 7.

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. J

. B

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We solve a system of time- and energy-dependent Bomann equations numerically for the free-electron-like mealuminum. The results show in detail the transient excitatand thermalization of the free-electron gas as well asenergy exchange between electrons and phonons. Theabsorbed energy was calculated for different laser intensand compares well with absorption characteristics knofrom literature. For excitations near damage threshold,found that the nonequilibrium electron gas does not affthe electron-phonon interaction and the energy exchangebe described by the two-temperature model. In contrast,lower excitations, the energy transfer from electron gaslattice is influenced by the nonequilibrium of the laseexcited electron gas. In this case, the cooling of the elecgas is delayed compared to a thermalized electron gas osame internal energy, which is in agreement with earlierperimental results.11–14 The reason is that the distributiofunction of a weakly excited electron gas shows only a vslight deviation from an electron gas at room temperaturethe region around the Fermi edge. The calculated eleccooling time thus depends on excitation parameters andbe longer than the characteristic relaxation time of a Ferdistributed electron gas, which depends on internal eneonly. To characterize this, we defined an electron thermaltion time as the time after which the collective behaviorthe laser-excited electron gas is the same as the behaviocorresponding Fermi-distributed electron gas. Note that athe thermalization time depends on laser parameters.

We are grateful to S.I. Anisimov, K. Sokolowski-Tintenand D. von der Linde for valuable suggestions and helpdiscussions. This work was supported in part by the Desche Forschungsgemeinschaft under Grant No. Si 158/1

d

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Documents

Ultrafast dynamics of nonequilibrium electrons in metals under femtosecond laser irradiation