Theory of the electron relaxation in metals excited by an ultrashort optical pump

# Theory of the electron relaxation in metals excited by an ultrashort optical pump

V. V. Baranov and V.V. Kabanov Department for Complex Matter, Jozef Stefan Institute, 1001 Ljubljana, Slovenia
###### Abstract

The theory of the electron relaxation in simple metals excited by an ultrashort optical pump is developed on the basis of the solution of the linearized Boltzmann kinetic equation. The kinetic equation includes both the electron-electron and the electron-phonon collision integrals and assumes that Fermi liquid theory is applicable for the description of a simple metal. The widely used two temperature model follows from the theory as the limiting case when the thermalization due to the electron-electron collisions is fast with respect to the electron-phonon relaxation. It is demonstrated that the energy relaxation has two consecutive processes. The first and most important step describes the emission of phonons by the photo-excited electrons. It leads to the relaxation of 90% of the energy before the electrons become thermalized among themselves. The second step describes electron-phonon thermalization and may be described by the two temperature model. The second stage is difficult to observe experimentally because it involves the transfer of only a small amount of energy from electrons. Thus the theory explains why the divergence of the relaxation time at low temperatures has never been observed experimentally.

###### pacs:
71.38.-k, 74.40.+k, 72.15.Jf, 74.72.-h, 74.25.Fy

## I Introduction

Investigations of ultrafast nonequilibrium dynamics in metals, superconductors and other strongly correlated systems after excitation by an ultrashort laser pulse have attracted a lot of attention during the last couple of decades. The particular interest to this field of research is related to the possibility to obtain unique information on the strength of electron electron (e-e) and of electron-phonon (e-ph) interactions in metals and superconductors.

Up to now detailed experimental data on relaxation processes are available for metals brorson ; schoenlein ; elsayed ; groeneveld1 ; brorson1 , high temperature superconductors eesley ; han ; chekalin ; albrecht ; stevens ; demsar ; kabanov ; christoph , and pnictide superconductors mertelj ; rettig using standard optical pump optical probe technique. Recently a comprehensive analysis of experimental data on optical-pump broad-band probe in high temperature superconductors has been performed gianetti . Most of the data are analyzed in the framework of the so-called two temperature model (TTM) kaganov ; anisimov and e-ph coupling constants were obtained by using P.B. Allen’s theory allen , which relates the e-ph relaxation time with the second moment, , of the Eliashberg function eliashberg2 ; eliashberg4 . The basic assumption of the model is that electrons and phonons are in a thermal quasi-equilibrium (QE) at two different time-dependent temperatures and , respectively. This assumption is correct if the e-e thermalization occurs on a much shorter timescale than the e-ph relaxation. Indeed the QE electron distribution function, characterized by the nonequilibrium electronic temperature , nullifies the electron-electron collision integral in the Boltzmann kinetic equation (BKE) and the remaining e-ph collision integral leads to the thermalization between electrons and phonons. However this approach has severe problems. The e-ph thermalization is in the low limit and in the high limitallen ; kabanovAlex , where is the Debye frequency, and is equilibrium temperature. As it was demonstrated in Refs.groeneveld1 ; kabanovAlex ; Fann the e-e thermalization rate , where is the Fermi energy, is much smaller than the electron-phonon thermalization rate in the temperature range where most of the experiments are performed . Therefore the main assumption of the TTM is not justified.

Experimentally it was demonstrated that the electron distribution function of a laser-heated metal is a nonthermal distribution on the time scale of the e-ph relaxation time groeneveld1 . It was shown that in Ag and Au the intensity dependence of the pump-probe signal as well as the temperature dependence of the relaxation time are inconsistent with the TTM groeneveld1 . More convincing arguments against the TTM were obtained by the direct measurements of the electron distribution function using time-resolved photoemission spectroscopy Fann ; lisowski ; perfetti . The interpretation of the photoemission spectra in terms of the distribution function of electrons assumes that the matrix element involved in the photoemission experiments and the density of electronic states near are smooth functions of energy Hufner . In the case of ordinary metals this assumption is usually correct. In some metals additional bands appear in the spectra, but usually these bands have relatively large excitation energy and do not influence the determination of the energy dependence of the distribution function (see Fig.5 in Ref.lisowski for example). In the measurements Fann ; lisowski ; perfetti the transient electron distribution function is not thermal and has high energy tails which survive till the thermalization occurs Fann . Moreover in gold at about 400 fs after excitation about 30% of the pump energy is already in the phonon subsystemFann , while the thermalization is observed only after 1ps. Therefore, the transfer of energy from electrons to phonons occurs much faster than the electronic thermalization. A similar effect is observed in Ru lisowski . It is estimated that 100 fs after the pump about 20% of quasiparticles are in the high energy tails of the distribution function. To account for this effect it was suggested that the electron distribution function can be represented as a sum of thermal and nonthermal parts Fann ; lisowski . It was also suggested to approximate the nonthermal distribution function by a Fermi-Dirac function with reduced amplitude and a nonphysical auxiliary temperature lisowski . We argue that such a decomposition is unphysical and cannot describe the electron energy relaxation. Instead the distribution function should be determined as a solution of the complete set of BKEs.

The accurate comparison of the TTM with experimental data on Ag and Au is presented in Ref. groeneveld1 . It was experimentally demonstrated that the relaxation time decreases with temperature and does not show any pump intensity dependence contrary to the TTM predictions. In order to account for the discrepancy between the TTM and experimental data the nonthermal electron model was introduced. This model represents the BKE for a nonequilibrium electron distribution function with electron-electron and electron-phonon collision integrals, while assuming that phonons are at equilibrium. Numerical integration of the BKEs allowed to reproduce experimental observations of the pump-probe experiments on Ag and Au. Since experimental data do not demonstrate any nonlinear effects as a function of the pump intensity the integro-differential BKE for the electrons was reduced to differential form kabanovAlex . The reduced equation is integrable and has an analytic solution. As a result it was shown again that the relaxation time should increase at low temperatures kabanovAlex , contrary to the experimental results groeneveld1 .

Recent analysis of the pump-probe experiments on the superconducting MgB jure and LaSrCuO primoz has demonstrated that the superconducting order parameter is reduced due to the nonequilibrium phonons generated by the photoexcited carriers. It indicates that on the sub-picosecond scale the reabsorption of the nonequilibrium phonons with the creation of low energy electron-hole pairs is an important process which influence the energy relaxation in metals and superconductors. Note that within the nonthermal electron model, used in Refs. groeneveld1 ; kabanovAlex , this mechanism of relaxation is absent, because phonons are considered to be in equilibrium.

In this paper we develop the theory where both nonthermal electron and phonon distribution functions are obtained by solving of linearized BKE. The e-e collisions are described on the basis of the theory, developed in Ref.kabanovAlex , which explicitly accounts for the conservation of energy. In the second paragraph we derive general linearized BKE’s. Then the high temperature limit of the BKEs is derived for the Eliashberg function , which is valid for disordered simple metals, where electrons interact with the acoustic phonons. By the direct simulations of the linearized BKEs with the different Eliashberg functions (Debye model for the acoustic phonons) and (Einstein model for the optical phonons) we demonstrate that the derived Fokker-Planck equations describe well the energy relaxation in the high and low temperature range. It was also demonstrated that the energy relaxation is not sensitive to the particular form of the Eliashberg function and is determined by the second moment of . Note that recent results on the time-resolved angle-resolved photoemission spectroscopy (tr-ARPES) indicate that the return to equilibrium of electronic excitations is determined by the momentum and energy dependent equilibrium self-energydevereaux . Therefore the particular form of the Eliashberg function may be resolved in the tr-ARPES experiments. The accurate solution of the BKE, presented in the fourth paragraph, leads to a distribution function which is very similar to that observed in the time-resolved photoemission experimentsFann ; lisowski ; perfetti . Then we show that the energy transfer from the photoexcited electrons takes place on a much shorter time scale than e-e and e-ph thermalization. Therefore the main experimentally observed process is determined by the emission of phonons by the photoexcited electrons and not by the e-e or e-ph thermalization as assumed in the TTM

## Ii Main equations

To derive linearized BKEs, describing both electron and phonon distribution functions we start from the general BKE which takes both e-ph and e-e collision integrals into account. According to Ref.stat_fiz2 the applicability of the BKE in metals is restricted by two inequalities. 1. . Here is the Fermi momentum and is the characteristic size of inhomogeneity of the distribution function. In our case is restricted by the penetration depths of light or the thickness of the film. Both of them are larger than interatomic distance and therefore this condition is fulfilled. 2. . Here is the characteristic timescale of the changes of the distribution function. Therefore the application of the BKE for the description of the pump-probe experiments in metals is justified on the timescale fs, well below the time resolution of all known experimental data discussed here. This quasi-classical description neglects any effects of quantum coherence which are not important in the case of simple metals. Therefore any effects related to the quantum coherence and dephasing are out of scope of this paper. The BKE for electrons reads:

 ˙fζ=Ie−e+Ie−ph, (1)

where is the electron distribution function, averaged over the surface of constant energy:

 fζ=1N(ζ)∑kδ(ζk−ζ)fk, (2)

Note that this averaging is justified if the distribution function depends only on the electron energy and does not depend on the direction of the momentum . For excitation by a spatially uniform fast optical pulse it is a reasonable assumption. If the pulse did not penetrate the sample fully, the drift and the field terms should be included in the equation. In that case the distribution function will be dependent on the direction of and the expansion defined by Eq.(2) is not justified. In that case more accurate expansions should be appliedSmithJensen . Here is the density of electronic states per spin, is the effective mass of electron, and is the electron energy counted from . The density of states is a very weak function of energy and we assume that it is constant. The e-e collision integral has the form:

 Ie−e=∫∫∫dζ′dϵdϵ′K(ζ,ζ′,ϵ,ϵ′)δ(ζ+ϵ−ζ′−ϵ′)× (3) [fζ′fϵ′(1−fζ)(1−fϵ)−fζfϵ(1−fζ′)(1−fϵ′)] (4)

with the kernel defined as:

 K(ζ,ζ′,ϵ,ϵ′)=2πℏN(ζ)∑k,p,qV2c(q)δ(ζk−ζ)× (5) δ(ζp−ϵ)δ(ζk+q−ζ′)δ(ζp−q−ϵ′). (6)

Here is the Fourier component of the effective e-e potential. Since we consider the relaxation of nonequilibrium electron-hole excitations with energies less than we neglect all energy dependence of the kernel , where is the Coulomb pseudopotentialkabanovAlex . The electron-phonon collision integral reads:

 Ie−ph = 2π∫dω∫dζ′Q(ω,ζ,ζ′)× (8) {δ(ζ−ζ′−ℏω)[(fζ′−fζ)Nω−fζ(1−fζ′)] + δ(ζ−ζ′+ℏω)[(fζ′−fζ)Nω+fζ′(1−fζ)]}. (9)

Here is the phonon distribution function averaged over the surface of constant frequency:

 Nω=1D(ω)∑qδ(ωq−ω)Nq, (10)

with the density of phonon states in the Debye approximation. This averaging is also possible if the excitation pulse is spatially homogeneous and gradient terms may be omitted in the kinetic equation for phonons. The kernel of the e-ph interaction is defined as:

 Q(ω,ζ,ζ′) = 1ℏN(ζ)∑k,qM2(q)δ(ζk−q−ζ′) (12) δ(ζk−ζ)δ(ωq−ω).

Here is the matrix element of e-ph interaction. Because the characteristic energy of electron-hole excitations are much less than the Fermi energy we neglect the dependence of on . As a result the e-ph collision integral is expressed in terms of the Eliashberg functionallen ; kabanovAlex :

 Q(ω,ζ,ζ′)≈Q(ω,0,0)≡α2F(ω) (13)

The kinetic equation for the phonon distribution function readsfiz_kin :

 ˙Nω = 4π∫∞−∞dζ′∫∞−∞dζQph(ω,ζ,ζ′)[fζ(1−fζ′) (14) (1+Nω)−fζ′(1−fζ)Nω]δ(ζ′−ζ+ℏω).

Note that here we neglect the anharmonic scattering of phonons which may be important at high temperatures. Here we assume that the relaxation in the phonon subsystem is described by the inelastic phonon-electron scattering. In general the anharmonic effects may lead to an additional temperature dependence of the relaxation time at high temperatures. The phonon-electron kernel is expressed in terms of the Eliashberg function:

 Qph(ω,0,0)=Q(ω,0,0)N(0)D(ω)=ω2Dβ2ℏα2F(ω)ω2, (15)

here

 β=2N(0)ℏωD/9∼ℏωD/EF≪1. (16)

Since in most of the experiments the pump-probe response is a linear function of the pump intensity groeneveld1 ; christoph we linearize the kinetic equations. The electron and phonon distribution function have the form:

 fζ=f0ζ+ϕ(ζ,t), (17)
 Nω=N0ω+η(ω,t), (18)

where and are small nonequilibrium corrections to the equilibrium distribution functions of electrons and phonons , respectively.

In order to simplify the following calculations we introduce the dimensionless electron energy

 ξ=ζ/kBT (19)

and dimensionless phonon frequency

 ν=ℏω/kBT. (20)

Therefore the functions and .

Let us consider first the linearized e-e collision integral. The linearized BKE was derived in Ref.kabanovAlex and reduced to a differential form applying Fourier transform over energy (see also Ref.SmithJensen , where a similar equation is derived for the e-e collision integral). This form of the collision integral is very useful if we consider e-e collisions only. Consideration of the e-ph interaction in the high temperature limit leads to the differential form of the collision integral as a function of energy . Therefore it is more convenient to rewrite the e-e collision integral as a function of energy . Indeed, the linearized e-e collision integral has the form (Eq.(7) in Ref.kabanovAlex ):

 Ie−e=1π2τe[−ϕ(ξ,t)(π2+ξ2)+ (21) 3∫∞−∞dξ′ϕ(ξ′,t)Φ(ξ,ξ′)], (22)

where is the e-e thermalization time, and the kernel . At large therefore we can simplify the e-e collision integral:

 Ie−e=1π2τe[−ϕ(ξ,t)(π2+ξ2)+ (23) 3∫∞−∞dξ′ϕ(ξ′,t)Φ1(ξ−ξ′)+6(E1(∞)tanh(ξ2)− (24) E1(ξ)−ξ(E0(∞)−E0(ξ)))], (25)

where the kernel , and . represents the dimensionless density () and energy () of nonequilibrium electrons with the energy less than . This form of the e-e collision integral Eq.(25) is very easy to treat numerically since the integral part represents a convolution of the distribution function and the kernel .

Let us turn back to the e-ph collision integrals. Substituting Eqs. (17,18) to Eqs.(1, 14) yields:

 ˙ϕ(ξ,t)= F1[ϕ]+F2[η]; (26) ˙η(ν,t)= F3[ϕ]+F4[η]. (27)

Here the expressions for to have the following form:

 F1[ϕ] = −ϕ(ξ,t)τ1(ξ)+2πkBTℏcosh(ξ/2)∫∞−∞dξ′sign(ξ−ξ′) ×α2F(kBT|ξ−ξ′|ℏ)cosh(ξ′/2)2sinhξ−ξ′2ϕ(ξ′,t) +Ie−e, F2[η] = 2πkBTℏ∫∞0dνα2F(kBTνℏ) ×sinh2(ν/2)tanh(ξ/2)cosh(ξ+ν2)cosh(ξ−ν2)η(ν,t), F3[ϕ] = 12τ2(ν)ν∫∞−∞dξ′ϕ(ξ′,t)sinh(ξ′)cosh(ν+ξ′2)cosh(ν−ξ′2), F4[η] = −η(ν,t)τ2(ν).

The energy dependent electron-phonon relaxation rate is defined as:

 τ1(ξ)−1=2πkBTℏ∫∞0dνα2F(kBTνℏ) ×⎡⎢⎣1sinh(ν2)cosh(ν2)+sinh2(ξ/2)tanh(ν/2)cosh(ν+ξ2)cosh(ν−ξ2)⎤⎥⎦, (28)

The frequency dependent relaxation rate of the phonons due to the collisions with electrons has the form:

 τ2(ν)−1=2πℏω2Dβ kBTα2F(kBTνℏ)ν. (29)

Note that equation (27) for the phonon distribution function has an analytical solution:

 η(ν,t)=12τ2(ν)ν∫t0dt′exp((t′−t)/τ2(ν)) ∫∞−∞dξ′ϕ(ξ′,t′)sinh(ξ′)cosh(ν+ξ′2)cosh(ν−ξ′2), (30)

Substituting it into equation (26) leads to a single integro-differential equation for the distribution function of electrons . This equation allows only numerical analysis and is therefore not very practical from the point of view of the analysis of experimental data. On the other hand some assumption about the Eliashberg function and applying the high-temperature expansion leads to more simple equation. An important simplification is obtained if we consider the model of a disordered metal with strong phonon damping. The Eliashberg function in that limit has the following formpoor1 :

 α2F(ω)={λω2ωD,ω<ωD0,ω>ωD (31)

This form of the Eliashberg function leads to a frequency independent phonon-electron relaxation rate , where the electron-phonon coupling is defined as:

 λ=2∫∞0dωα2F(ω)ω. (32)

The above equations are valid for any temperature . Since most of the time-resolved photoemission experiments in metals are performed at temperatures above 100K in the next section we consider the high temperature limit of the kinetic equation Ref.kabanovAlex .

## Iii High-temperature limit

In the high temperature limit the BKEs (26,27) can be further simplified. Let us first consider equation (27) for the phonon distribution function. In the high-temperature limit in the expression for we can neglect under the integral over . As the result the equation for has the form:

 ˙η(ν,t)=−η(ν,t)τ2+2I(t)ντ2, (33)

where the function is defined by the equation:

 I(t)=∫∞0dξtanh(ξ/2)ϕ(ξ,t). (34)

This function has a very simple meaning. It describes the rate at which electrons are losing their energy to phonons (Eq.(36) in Ref.kabanovAlex ). Equation (33) defines the frequency dependence of the nonequilibrium phonon distribution function. Indeed, if we substitute

 η(ν,t)=2p(t)/ν (35)

to Eq.(33) we obtain an ordinary differential equation for the function :

 ˙p(t)=−(p(t)−I(t))τ2. (36)

It means that at high temperatures, where , phonons are always described by the quasiequilibrium distribution function and the function describes the time evolution of the nonequilibrium phonon temperature. Substituting the phonon distribution function Eq.(35) to the equation for leads to the generalized Fokker-Planck equation for the nonequilibrium electron distribution function :

 γ−1˙ϕ(ξ,t)=∂∂ξ[tanh(ξ2)ϕ(ξ,t)+∂∂ξϕ(ξ,t)]+ p(t)sinh(ξ/2)2cosh3(ξ/2)+γ−1Ie−e, (37)

where

 γ=πℏλ⟨ω2⟩kBT (38)

is the e-ph relaxation rate, and the e-e collision integral is given by Eq.(25). The detailed derivation of the differential form for is presented in Refs.kabanovAlex ; gusevWright .

Note that equations (36,37) are derived for the case of a disordered metal with electrons interacting with acoustical phonons. In the Appendix we present the results of the numerical simulations of equations (26,27) with three different types of Eliashberg functions at high and low temperatures, here is the Debye temperature. We demonstrate that equations (36,37) describe the relaxation of the photo-excited electrons with the accuracy more than 10% in the whole temperature range provided that is constant for different Eliashberg functions. It proves that Eqs.(36,37) are useful because they are insensitive to the approximations and assumptions that were used. It justifies their applicability for the description of the energy relaxation of optically excited electrons in a large variety of ordinary metals in the whole temperature range.

Equations (36,37) describe the relaxation of both phonon and electron distribution functions after perturbation. These equations represent the generalization of the TTM. The TTM may be derived from these equations in the limit when e-e relaxation is much faster than e-ph relaxation, i.e. when . It is easy to check that the equilibrium distribution functions, corresponding to the simultaneous increase of the temperature of electrons and phonons , represent a solution of Eqs. (36,37). Moreover, it is easy to check that the energies accumulated in the phonon and electron systems are proportional to the phonon and electron specific heats. Indeed, the energy accumulated in the phonon system is . The energy in the electronic system is . Therefore, the energies in the phonon and electron systems are exactly proportional to their specific heats.

From Eqs.(36,37) we can define two dimensionless parameters. The first parameter

 κ1=γτ2=ℏλ⟨ω2⟩λωDkBTβ∼EF/kBT≫1. (39)

The parameter describes the relative time of generation of the electron-hole pairs at low energy by the hot phonons which are created by electrons at large energies. This parameter is explicitly related to the electronic and phonon specific heats: . The second parameter

 κ2=γτe=4ℏ2λ⟨ω2⟩EFμ2cπ2(kBT)3≫1. (40)

It describes the relative time of the electron thermalization due to e-e collision. The ratio does not contain small parameter. The order of magnitude of these dimensionless parameters for ordinary metals follows from the ratio of specific heats of electrons and phonons: . Note that parameter is temperature dependent.

## Iv Results

In this section we discuss the results of the high temperature limit of the theory and demonstrate TTM results obtained from the Eqs.(36,37) in the limit of the fast e-e relaxation. We also show that the process of emission of phonons dominates the relaxation in the low temperature limit as well. We assume that excitation of electrons with an ultrashort laser pulse creates at a broad nonequilibrium distribution of photoexcited electrons. The width of the distribution is of the order of the light frequency and is much larger than the Debye frequency . The initial distribution function after the pump pulse is approximated by the formula

 ϕ(ξ,0)=ξΩ3exp(−ξ2/Ω2). (41)

Here the dimensionless excitation frequency which is defined as frequency of light measured in units of . This formula preserves the energy per pulse for different frequencies of excitation .

Note that this formula is different from the standard distribution of quasiparticles created by an optical pulseelesin . It has a characteristic energy which is of the order of the frequency of excitation. This function allows the existence of quasiparticles with the energy higher than , but it is decreasing very quick at . In the Appendix we present the results of numerical simulations assuming different initial distribution functions. The simulations show that irrespective of the particular choice of , the distribution function becomes independent on initial conditions on the time scale . Therefore the particular choice of does not influence the process of energy relaxation of photo-excited electrons at the time scale longer than .

In Fig.1 the time evolution of the electron distribution function is presented in the absence of the e-ph interaction. The evolution is characterized by the fast reduction of the high energy part of the distribution function and fast increase of the distribution function at small energies . As it follows from Fig.1 the high energy tails of the distribution functions disappear at the time scale determined by . This is consistent with the previous analysis of the e-e relaxation kabanovAlex . The time dependence of the density of nonequilibrium electrons, presented in Fig.1 (inset), shows that electron thermalization occurs on the time scale of irrespective of the photon energy . The difference is important only on a very short time scale for relatively low frequency and is due to the fact that the initial density of nonequilibrium electrons depends on the pump frequency , provided that the energy per pulse is constant. Here

 n(t)=∫∞0dξϕ(ξ,t) (42)

is the dimensionless density of nonequilibrium electrons. The dimensionless density is actually equal to the measured density of nonequilibrium electrons expressed in units of . Note that the scaling arguments presented in Ref.tas predict . Our calculations do not support this behaviour. However with the increase of the dependence of becomes more and more similar to .

In order to demonstrate the relation of the TTM to Eqs.(36,37) we consider the unphysical limit . This limit corresponds to the case of the fast electron thermalization with respect to e-ph relaxation. The time evolution of the nonequilibrium distribution function in this limit is plotted in Fig.2. As it follows from Fig.2 at the time scale the high energy part of the nonequilibrium distribution function disappears and . Here is the change of the electron temperature. This function nullifies the e-e collision integral and further evolution of the distribution function is described by the time dependence of as it follows from Fig.2. The dependence of the distribution function at is always the same and the evolution with time is described by the time dependent pre-factor. Note that theory is linear in the excitation intensity and therefore , therefore the time evolution of is described by the linearized TTM. Note that for the case, presented in Fig.2 the dimensionless energy density transferred from electrons to phonons during the thermalization time is much smaller than the energy absorbed by electrons . The dimensionless energy accumulated by nonequilibrium electrons is defined as:

 E(t)=∫∞0dξξϕ(ξ,t). (43)

Therefore at can be evaluated using the electronic specific heat . Then thermalization between electrons and phonons occurs on the time scale in accordance with the TTM. The distribution function at corresponds to , where is found from the equation: . Note that the measure for electronic temperature in that case is not the width of the distribution function but the value of at the maximum which is approximately equal to . The most important property of this limit is the absence of the high-energy tails of the distribution function, which directly follows from the fact that .

Now let us consider the more realistic case . These parameters roughly correspond to the case of metals like AuFann or Rulisowski . The time evolution of the distribution function is presented in Fig.3. At the timescale of the high energy tail of the distribution function disappears and the electron distribution function can be approximated by the quasiequilibrium distribution function characterized by the electronic temperature . The electronic temperature is not defined by the conservation of energy, because by the time of ”thermalization” about 90% of the energy has gone to phonons (See inset of Fig.3). The dimensionless energy accumulated by nonequilibrium electrons is defined in Eq.(43). This is consistent with the time resolved photoemission data on AuFann and Rulisowski . According to Ref.lisowski the estimated electronic temperature at the peak is much less than electronic temperature estimated from TTM . The final stage of the relaxation in that case can be characterized by the slight decrease of the electronic temperature to its equilibrium value and can be described by the TTM. Note, however that the final stage of relaxation is very difficult to observe experimentally, because it involves the transfer of a very small amount of energy (less than 10% of the pump energy) from electrons to phonons. The biggest changes in the nonequilibrium distribution function take place during the initial stage of relaxation where the TTM is not applicable.

From these calculations the following qualitative picture of the relaxation of the photoexcited electrons emerges. The pump pulse creates a broad distribution of electron-hole pairs with large excitation energy. The high energy electrons relax to the low energy scale due to e-e collisions. It happens on the time scale . The photoexited electron-hole pairs emit phonons immediately after excitation. The emission rate is temperature independent and is not affected by the Fermi distribution function, because the average energy of nonequilibrium electrons is large in comparison with the phonon frequency and therefore the factor in the emission probability can be replaced by 1. In the Appendix it is shown that the Fokker-Planck equation describes well energy relaxation in the low temperature limit as well. (see Fig. 8, 9 and 10 in the Appendix). The absence of the divergence of the relaxation time at low temperatures for different e-e collision times is presented in Fig. 4. The relaxation time, defined as the time when half of the energy is transferred from electrons to phonons, is weakly temperature dependent at high temperature and temperature independent at low temperatures. Note that electron-electron collisions have an important effect on energy relaxation. When the electron-electron collision rate increases, the high-energy excitations are decaying with the creation of more low energy electron-hole pairs. The energy relaxation on the other hand is proportional to the number of nonequilibrium electrons. Therefore when the e-e collision rate increases the energy relaxation rate of the nonequilibrium electrons increases as well. This is demonstrated in Fig.4.

When the average energy of nonequilibrium electrons, because of emission of phonons, is reduced to the scale the factor becomes important leading to the strong slowing down of the relaxation, indicating the second stage of relaxation. Since most of the energy is transferred to the phonon subsystem before the time when the width of the nonequilibrium distribution function becomes of the order of , the final and the longest stage of thermalization is difficult to observe. In order to demonstrate that we multiply Eq.(37) by and integrate over from to with the following result:

 ˙E(t)=−γ(I(t)−p(t)) (44)

is defined in Eq.(43). The e-e collision integral conserves electron energy and therefore does not contribute to Eq.(44). The first term in this equation describes the loss of the energy due to generation of phonons. The second term describes the reabsorption of phonons with the generation of low energy electron-hole pairs and slows down the relaxation. If we can neglect the second term in this equation. When the distribution function is broad and has tails at , in the integral Eq.(34) can be replaced to 1 and therefore , where is defined in Eq.(42). Therefore Eq.(44) indicates that energy loss by electrons is proportional to the nonequilibrium electron density. In order to illustrate these arguments we plot in Fig.5 the time dependence of the electron energy for the case of large in comparison with the approximate formula:

 E(t)=E(0)−γ∫t0dt′n(t′). (45)

As it follows from Fig.5, this approximation describes well the time evolution of the energy until when the high energy tails in the nonequilibrium distribution function disappear.

Note that the slope of the energy relaxation curve at is the same in both cases (see inset of Fig.5) because the number of photo-excited electrons is the same at . The difference between the two cases is due to the e-e collisions. Since for the case of the quasiparticle multiplication is much faster the energy relaxation rate is larger. This is demonstrated in Fig.6. The characteristic maximum in the quasiparticle density for the case of is approximately 2 times larger than for the case of . Therefore the relaxation rate is determined by the nonequilibrium quasiparticle density at relatively high energy . The quasiparticle density strongly depends on . Therefore analysis of the energy relaxation after the ultrashort pump pulse allows one to obtain not only information about the e-ph interaction constant, but also about and as a consequence about the Coulomb pseudopotential .

The physical meaning of Eq.(44) is very simple. Every non-equilibrium electron emits phonons with the temperature independent rate . The formula for the emission rate is valid in the low temperature limit as well. Indeed, if we consider the zero temperature limit in Eq.(9) and calculate the phonon emission rate we obtain the same expression for as in high temperature limit (for details of calculations see Ref.beyer ). Since the emission rate for non-equilibrium electrons is much larger than the thermalization rate in both low temperature and high temperature limits, we conclude that low temperature divergence of the relaxation time will not be observed experimentally. Most of the energy will be transferred to the phonon subsystem before low temperature relaxation processes become important. This is confirmed by the comparison of the results of calculations of the temperature dependent energy relaxation time using the Focker-Planck equation (36,37) with the results obtained from the kinetic equations (26,27) with different Eliashberg functions (Fig.10).

Note that the theory predicts the dependence of the relaxation rate on the pump frequency (Fig.7). Indeed, preservation of the energy per pulse leads to the increase of the number of photoexcited electrons . If the pump frequency is large enough, e-e collisions lead to the quick disappearance of quasiparticles with the energy higher than the threshold energy. The threshold energy is defined as the energy where the life-time of a nonequilibrium electron due to e-e collisions becomes comparable with the life-time due to e-ph collisions. Therefore excitation with the light frequency higher than this threshold energy does not lead to the pump frequency dependence. The situation is different when the pump frequency is below the threshold energy. In that case the e-e collisions are not important and the relaxation rate is governed by the number of photoexcited electrons , leading to the increase of the relaxation rate. This behaviour is demonstrated in Fig.7.

Experimental measurements of time-resolved photoemission in metalsFann ; lisowski provide direct information about the time dependent nonequilibrium distribution function (see Fig.7 in lisowski ). This can be directly compared with the results of our calculations. One of the most striking resemblance between our theory and the experiments is the existence of the high energy tails in the nonequilibrium distribution function when a substantial amount of the energy is already transferred to the phonon subsystem (Fig.3). Very often in the experimental analysis these high energy tails are interpreted as the nonequilibrium electron temperature . Usually the results of the measurements of the nonequilibrium distribution function are plotted on a logarithmic scale as a function of the energy and the slope of defines the nonequilibrium electron temperature lisowski ; perfetti . In this analysis a large part of the nonthermal electrons is accounted for thermal. We suggest a different way of defining the electron temperature. In Fig.7 of Ref.lisowski the nonequilibrium part of the electron distribution function is plotted for different time delays after pump. From this graph it is clear that at any time delay and therefore the condition for the linearization of the BKEs is fulfilled. If is described by the nonequilibrium temperature then . The shape of in Fig.7 of Ref.lisowski is similar to that described by this formula. The function has a maximum at which is about 1. Therefore the electronic temperature can be defined directly from the maximum of in Fig.7lisowski where is the experimental value of the maximum of . Therefore after 100fs contrary to the estimate from the slope of the distribution function .

Another important consequence of the time-resolved photoemission experiments is the possibility to evaluate the electron energy and the number of the photoexcited electrons and holes. According to our theoretical results the maximum in energy and maximum in the density of photoexcited electrons should be shifted in time with respect to each other (Figs.5,6). If the pump pulse is much shorter than the energy has its maximum immediately after pump pulse, because the photo-excited electrons cannot emit any phonon during the short pulse. The high energy photoexcited electrons reduce the energy due to e-e collisions leading to an increase in the number of nonequilibrium electrons. The electron-hole recombination is slow at the short time scale because the number of nonequilibrium electrons within the energy interval is small when the pump frequency is large . It means that the number of photoexcited electrons increases immediately after the pump. The maximum in the nonequilibrium electron density occurs when the process of electron multiplication due to e-e collisions is compensated by the electron-hole recombination with the emission of phonons. The time when the density of the nonequilibrium electrons has its maximum indicates the end of the first stage of electron thermalization. After that the energy relaxation rate decreases. The difference in the positions of maxima can be clearly seen in Figs. 8 and 9 of Ref.lisowski .

Note that Eq.(44) allows the evaluation of the electron-phonon coupling constant. If we evaluate the decay rate of the electron energy experimentally and divide it with on a short time scale after the maximum of the curve this ratio should be constant and is exactly equal to . Note that the determination of the coupling constant from the single color pump-probe measurements only is problematic. The energy relaxation rate depends on the e-e and e-ph coupling constants as demonstrated in Fig.4 and 5 and requires the knowledge of both and dependences. Therefore, independent measurements of the nonequilibrium density are necessary to evaluate the electron-phonon coupling constant .

## V Conclusion

We developed the theory of the electron relaxation in metals excited by an ultrashort optical pump. The theory is based on the solution of the linearized BKE which includes the electron-electron and the electron-phonon collision integrals. The well known two temperature model represents the limiting case of the theory when thermalization due to the electron-electron collisions is fast with respect to the electron-phonon relaxation.

We demonstrated that for realistic parameters the energy transfer from electrons to phonons occurs on a timescale which is much faster than the electron thermalization. The high energy tails in the electron distribution function disappear when most of the energy is already transferred to phonons as it is observed in the time-resolved photoemission experiments Fann ; lisowski . The reabsorption of nonequilibrium phonons slows down the relaxation.

We demonstrate that the relaxation of the photo-excited electrons occurs in two steps. The first and the most important step represents the emission of phonons by the nonequilibrium electrons. The rate of electron energy loss at that stage is proportional to the density of nonequilibrium electrons. The temperature dependence of the relaxation at this stage is significantly different from the predictions of the two temperature model. The density of nonequilibrium electrons is strongly influenced by the electron-electron collisions. It makes the relaxation time dependent on the pump frequency if the pump frequency is smaller than some threshold frequency. It also allows the evaluation of the electron-phonon coupling constant from time-resolved photoemission data.

The second stage of the relaxation describes the electron-phonon thermalization. This stage may be described approximately by the two temperature model. Since it involves a small energy transfer (about 10% or less) from electrons to phonons it is difficult to observe experimentally. Our theory explains one of the most severe problems of the theory why the divergence of the relaxation time at low temperatures was never observed experimentally.

## Vi Acknowledgments

We are grateful to Jure Demsar, Dragan Mihailovic, Christoph Gadermaier, Tomaz Mertelj, Alexander Balanov, Laurenz Rettig and Uwe Bovensiepen for sharing with us their insight into non-equilibrium phenomena in metals and superconductors.

## Vii Appendix

In order to demonstrate that Eqs. (36,37) are independent of the approximations that were made, we perform extensive numerical simulations of equations (26,27) in the limit of large with three different Eliashberg functions: the linear function of phonon frequency Eq.(31), the Debye model

 α2F(ω)=⎧⎨⎩λω2ω2D,ω<ωD0,ω>ωD (46)

and the Einstein model

 α2F(ω)=λω0δ(ω−ω0). (47)

Since the e-ph collision integral for all these cases is similar to the e-e collision integral it is very easy to treat numerically. The evolution of the nonequilibrium distribution function at low temperature is presented in Fig.8.

It is easy to see that in the whole range of time the electron distribution function calculated with Eqs.(26,27) with the Eliashberg function (31) is almost undistinguishable from the distribution function obtained by the solution of the Fokker-Planck equation (36,37). In Fig.9 we present the electron energy as a function of time calculated on the basis of Eqs.(26,27) for different functions defined by Eqs.(31,46, and 47) in comparison with the solution of the Fokker-Planck equation. Here we keep constant for different Eliashberg functions. Again the energy relaxation is almost the same for different cases in the high temperature region and differs not more than by 10% at low temperatures . Note that the difference between different models increases when almost all energy is transferred to phonons and electron thermalization takes place. It is because the Fokker-Planck equation overestimates the thermalization rate in the low temperature range. On the other hand the temperature dependence of the thermalization rate for different types of Eliashberg functions is different. In the case of the Eliashberg function defined by Eq.(31) the thermalization time is proportional to , in the Debye model and in the case of Einstein model related to the e-ph interaction with optical phonons it is an exponential function of temperature .

Therefore we can conclude that energy relaxation of the photo-excited electrons is well described by the Fokker-Planck equation (37) in the whole temperature region. The relaxation does not depend on the particular form of the Eliashberg function and is defined by the second moment of the Eliashberg function . The energy relaxation is not exponential, therefore we define the relaxation time as the time at which half of the absorbed energy is transferred from electrons to phonons.

The temperature dependence of the energy relaxation time is plotted in Fig.10 for three different Eliashberg functions and for the Fokker-Plank equation. As it is clearly seen from this graph, there is marginal difference between these cases at temperatures . Therefore the relaxation time is independent of the particular form of the Eliashberg function. It is determined by the second moment of the Eliashberg function in this temperature range. It is almost independent of the particular form of the function and is not sensitive to whether the acoustic or optical phonons dominate the the relaxation, provided that is constant. Note, that in the case of tr-ARPES experiments, where the the relaxation time may be momentum dependent some of the features of the Eliashberg function may be resolveddevereaux .

In the low temperature region there is a very small increase of the relaxation time for the Debye (2% at ) and for the Einstein(5% at ) models for the Eliashberg function. As it was mentioned in the introduction the two temperature model becomes valid at very low temperatures. Since the e-ph relaxation time increases exponentially with decreasing of temperature in the case of the interaction with optical phonons, the range of temperatures where the two-temperature model is efficient may be relatively broad . In the case of interaction with the acoustic phonons this range is more narrow. Therefore if the relaxation is dominated by the interaction with the optical phonons it may be possible to observe the increase of the relaxation time at low temperatures. From an experimental point of view this is very unlikely because the interaction with the acoustic phonons will dominate the relaxation at low temperatures.

In order to show that the results are independent of the initial distribution function of the photo-excited electrons we present the results of the time evolution of the distribution function for different initial conditions. In Fig.11 we plot the time evolution of the distribution function for compared with the case where . The results clearly demonstrate that the essential difference between these two cases survives only on the time scale less than when the transfer of energy from nonequilibrium electrons to phonons is negligible. Therefore the particular choice of the initial distribution function is not important. The only important parameter is the characteristic energy scale of this distribution, which in both cases is described by the characteristic frequency which is related to the pump frequency.

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