Calculation of low-energy-electron lifetimes

The effect of the density of states on the lifetime of low-energy electrons in metals has been studied using a golden rule approach. Simple approximations to the real density of states of metals have been used allowing analytical results, which show that the free-electron scaling ( E2EF) 22 of the lifetime is affected above the onset ofd-electron contributions. Hence, in noble metals a scaling with ( E2EF2vd) , wherevd is the energy distance from the Fermi energy to the top of the d band, appears once d electrons can be excited. In ferromagnetic Co, the ratio between the lifetimes of majority and minority spin electrons is found energy independent below the threshold for the excitation of majority d electrons while this ratio increases with energy above that point. @S0163-1829 ~99!09327-3#


I. INTRODUCTION
The transient behavior of excited electrons at or near metal surfaces is fundamental to many important surface processes, such as chemical reactions, transport, surfacephase transitions, molecule-surface interactions, as well as in technological applications of electronic materials. 1,2][5][6][7][8][9][10] These kind of experiments, although their interpretation is still not settled, indicate that a description more accurate than the one given by the free-electron approach 11 is necessary.Such lifetime calculations have a long story. 12Quinn 13 also introduced the effect of a polarizable background with a static dielectric constant ⑀ 0 showing that the lifetimes increased as ͱ⑀ 0 since the electron-electron interaction is reduced due to the additional screening.Krolikowsky and Spicer 14 used experimental density of states ͑DOS͒ to calculate the qualitative behavior of the lifetime, which was compared to the pioneering work by Kanter 15 who measured lifetimes in Ag, Al, and Au in transmission experiments for electrons about 10 eV above the Fermi energy.There are many studies related to the calculation of the electronic mean-free path (l) both for ions and electrons, whose magnitude is directly related to the lifetime by the velocity (v), lϭv.
Using the self-energy formalism, Shelton 16 calculated the inelastic mean-free path of electrons in jellium for energies from 10 to 500 eV above the Fermi energy.In the same formalism, Penn 17 introduced the effects of exchange and correlation and calculated the mean-free path for different metals and insulators.Tung, Ashley, and Ritchie 18 used a statistical approach to calculate the mean-free path as an average of the free-electron mean-free paths in each point of the unit cell in the energy range between 1 and 10 5 eV, and studied the effect of core polarizability, damping, and localfield corrections on the mean-free path in Aluminum between 10 and 50 eV. 19With a similar approach as in Ref. 18, Penn 20 used optical response functions to calculate the meanfree path within a dielectric formalism from 5 up to 10 4 eV.Ashley 21 proposed a prescription for constructing a dielectric-response function including dispersion effects, from the optical one, and calculated the mean-free path and the energy loss for electrons between 40 and 10 4 eV in noble metals.
In ferromagnets, Penn, Apell, and Girvin 22 made a study of the secondary electron cascades, which allows to obtain information about the lifetimes from the polarization of the emitted secondaries.Several experiments in ultrathin magnetic layers have confirmed the spin dependence of the lifetime in these materials ͑see e.g., Refs.23-25͒.Siegman 26 showed that d electrons should play an important role in understanding the mean-free path in transition metals.Passek and coworkers 27 measured spin-dependent lifetimes of image states in Fe͓110͔ and analyzed the data by calculating the convolution of first-principles DOS.Recently, Drouhin 28 developed a detailed mathematical approach to the problem of spin-dependent lifetimes in ferromagnetic materials in terms of the density of states.
In this paper, we try to delineate the effect of the density of states ͑DOS͒ on the lifetime of low-energy electrons (E ϪE F Ͻ4 eV).Usually the approximation of assuming a constant DOS is made but in the case of transitions metals, with d bands near or even at the Fermi energy, one expects that such an assumption will not provide a good description.The well-known scaling of the lifetime as (EϪE F ) Ϫ2 is a direct consequence of assuming a smooth DOS in the energy region of the involved electronic transitions, that allows a Taylor expansion around the Fermi energy.This expansion will only be valid very close to the Fermi energy.We have thus found that the mentioned scaling property of the lifetime is affected by the inclusion of a more realistic DOS once the energy of the electron is larger than the interband excitation energy.
In order to model the d bands, we have used simple descriptions that allow analytic expressions for the lifetime in the approximation of constant matrix elements.They give the same characteristic features as calculations based on full first-principles DOS.Within this model we are able to describe in a reasonable way the available experimental data.
In the next section, we use our approach to study the behavior of the lifetime in noble metals ͑Cu and Ag͒.In Sec.III, we generalize the model to a ferromagnet ͑Co͒ in which a spin-dependent lifetime is expected.In Sec.IV, we analyze the available experimental data in terms of our approach.Finally, in Sec.V we discuss the major conclusions from our study.

II. LIFETIME IN NOBLE METALS
Consider an electron with momentum p and energy E above the Fermi energy.The interaction with the electrons in the solid will make, after some typical time , this electron to decay to a lower energy state of momentum pЈ and energy EЈ, exciting one of the electrons in the solid from a state k to kЈ ͑See Fig. 1͒.From a perturbative approach one can write this lifetime ͑inverse of the probability per unit time͒ using Fermi's golden rule as done in Ref. 22 arriving at the expression where ϵEϪEЈ is the energy transfer and being f (E) the Fermi-Dirac occupation function and (E) the DOS for energy E and spin .¯is the opposite spin to .M () is the matrix element of the screened electronic interaction that connects the initial and final states.To arrive at Eq. ͑1͒ we have made an angular average 29,30 and the interference between direct and exchange terms has been neglected since they appear to be negligible when exchange effects are also taken into account in the interaction. 31 a nonmagnetic material ϭ ¯so the spin subindex can be dropped in Eq. ͑1͒ and

͑3͒
where now is the spin-independent density of states.We will in our study make the approximation of constant matrix elements, because we are mainly interested in the direct effect of the DOS.One can suppose that, in the small energy range we are interested in ͑0-4 eV above E F ), matrix elements will remain fairly constant while in the same energy range the DOS could change about one order of magnitude.This constant matrix approximation has been widely used in the analysis of photoemission data with excellent results, see for example the pioneering work by Berglund and Spicer 29 where experimental photoemission results in copper and silver were successfully explained using constant matrix elements.Kane 30 calculated the matrix elements in the case of silicon obtaining that they were quite insensitive to energy changes ͑less than 30% when going from 5 to 8 eV͒.In the study of secondary electron cascades in ferromagnets, although energy-dependent matrix elements were necessary to obtain full numerical agreement with the experiment, the general trend was successfully explained using constant matrix elements. 22Also in ferromagnets, after removing from the scattering amplitudes for Stoner processes the contribution of the DOS, a relatively small energy dependence is found. 32nce the matrix element has been extracted out of the integral, only the convolution integral of the DOS corresponding to the electrons involved in the deexcitation process has to be calculated.From Eq. ͑3͒ taking a constant value (E)ϭ s for the DOS and assuming constant matrix elements, one arrives immediately at the familiar (EϪE F ) Ϫ2 scaling of the lifetime, an expression that was first obtained in Ref. 29.By comparison with the well-known Quinn's result, 11 valid for free electrons in the low-energy limit (EϷE F ), the matrix element is given by The strength of the interaction decreases with increasing density due to the enhancement of the screening, thus increasing the lifetimes.It can be shown that in free-electron-like metals ϰn 5/6 in the high-density limit.
In order to describe noble metals, a more detailed description of the density of states will be necessary.We will then distinguish between s and d electrons, describing them in different ways.Noble metals are characterized by a quite flat sp band and large d bands lying below the Fermi energy.The distance from E F to the d band is about 2 eV in the case of Cu and near 4 eV in the case of Ag.It is not difficult to use first-principles calculations of band structure to perform the integral in Eq. ͑3͒ but, in order to understand the qualitative behavior, we have used simple descriptions for the DOS, which allow to obtain analytical results.In this way, we describe the DOS of noble metals using boxes with different shapes.We use a square box of height s for the sp electrons, extended over the whole energy range and another one for the d electrons, of height d , between E b and E t ͓see Fig. 2͑a͔͒.Thus, we introduce In Eq. ͑7͒ (x) is the unit step function.Although the matrix elements are taken as constants we distinguish them by the participating electrons because one would expect that the overlap between different wave functions will be different, and hence yield different matrix elements. 33͉M l f s f l i s i ͉ 2 is the matrix element of when the primary electron goes from band l i to l f and the secondary electron goes from band s i to s f .We suppose that ͉M kl i j ͉ 2 ϭ͉M i j kl ͉ 2 ϭ͉M lk ji ͉ 2 due to timereversal symmetry and indistinguishable electrons, respectively.
When (EϪE F )Ͻ t , being t ϵ(E F ϪE t ) the energy distance from the top of the d band to the Fermi energy, no d electrons can be excited and the expression shown in Eq. ͑4͒ is reproduced, , part of the d electrons can be excited and the lifetime is given by

͑9͒
Due to the new channel for the decaying provided by the d electrons, an additional contribution appears, which has not the standard (EϪE F ) Ϫ2 behavior predicted by the Fermi Liquid's theory in the case of low energies.This new term scales with (EϪE F Ϫ t ) 2 and results in a sudden increase of the probability ͑decrease of the lifetime͒ as soon as it becomes possible to excite d electrons.Note also that the selectron contribution scaling with (EϪE F ) Ϫ2 enters with a factor ͓( s / d )Ӷ1͔ thus being smaller than the d-electron contribution only considering DOS.As a result, and supposing as a first approximation that matrix elements are of the same order, once d electrons can take part in the deexcitation process, they dominate it while the s-electron contribution becomes a small correction.Finally for energies large enough for all d electrons to be excited, namely (E ϪE F )Ͼ b , the term due to excitation of d electrons becomes a constant and only the s electron part keeps the energy dependence.
One can also use a triangular DOS for the d electrons ͓see Fig. 2͑a͔͒ d Following the same procedure, below the d-electron excitation threshold (EϪE F Ͻ t ) the result in Eq. ͑8͒ remains unchanged.If part of the d electrons can be excited

͑11͒
Although the result is in this case more complicated, there is still a term related with the excitation of d electrons that has a energy dependence different from the (EϪE F ) 2 behavior.Finally, when all d electrons can be excited ( b ϽE ϪE F ) the d term becomes again a constant.Keep in mind that when using triangles, in order to keep the number of electrons constant, d in Eqs.͑10͒ and ͑11͒ is twice d entering in Eqs.͑7͒-͑9͒.
These analytic results have been compared with numeric calculations of the convolution integrals where firstprinciples density of states has been used.In order to compare both approaches, when evaluating the analytic expressions we will use for s the value given by first-principles calculation at the Fermi energy. 34To determine d we first calculate the total contribution of the d electrons by integration of the d part of the first-principles density of states and then we divide it by the d band width.( s / d ) is approximately 0.07 in Cu and 0.09 in Ag, being small parameters in both cases.In spite of the huge simplification made in the DOS, the agreement found between our simplified model and the first-principles evaluation is very good, supporting the validity of our approach.A triangle gives the best agreement in the case of copper while a square box is more suitable for silver.These shapes are intuitively suggested by the appearance of the first-principles DOS.

III. LIFETIME IN FERROMAGNETIC METALS: COBALT
Ferromagnetic metals are characterized by different band structures for each spin state.This implies different DOS for each spin and so one expects different lifetimes too.Regarding the way of introducing this spin-dependent DOS, one will have to go back to Eq. ͑1͒ in which the spin label appears explicitly.The DOS in these materials are characterized by completely filled d bands for the majority spin state ͑↑͒, which means that d ↑ bands are below the Fermi energy.Nevertheless, minority d bands are partially filled, crossing the Fermi energy.This implies that there exist d ↓ states above the Fermi energy.One expects that these empty d ↓ states will yield an increase of the available phase space for the decaying process leading to shorter lifetimes for spindown electrons. 22n the same way as we did for noble metals, we will describe the majority electrons by a constant value for the s ↑ and a box below Fermi energy for the d ↑ .In the case of minority spins the d box will cross the Fermi energy ͓See Fig. 2͑b͔͒.Then we have where For simplicity, it has been assumed that s ↑ ϭ s ↓ ϭ s and d ↑ ϭ d ↓ ϭ d ; this means that the whole bulk polarization comes from the differences in E b ↓↑ and E t ↓↑ .Introducing these expressions into Eq.͑1͒, it is possible to obtain analytic results in the whole energy range but the expression becomes more and more involved while increasing the energy and so we will restrict ourselves to the lowest energy regions described below.In the same way as happened in the case of noble metals, the model using boxes is able to reproduce all the main features of a full first-principles calculation. 35n order to confront later the experimental data, 10 we are going to choose the , respectively the energy distances to the top of the majority and minority d bands, because this is the case for Co ( ↑ Ӎ0.6 eV, ↓ Ӎ1.3 eV). 35With this selection of parameters, the first energy range is (EϪE F )Ͻ ↑ .In this region, the primary electron cannot excite any majority d electrons.Nevertheless, minority d electrons will be excited for any energy.Equation ͑1͒ for ↑ then gives where the fact that a majority electron above the Fermi energy is always an s electron has been taken into account.
Only the matrix elements entering with the lowest power in ( s / d ) have been considered because in Co ( s / d )Ӎ0.2.Notice that this approximation does not affect the energy dependence of the lifetime, only the prefactor of this dependence.In the case of minority electrons both spins are possible in this energy range, but we will consider that the electron above the Fermi level will be a d ↓ electron because d is much bigger than s .Thus, we obtain, In this energy range the ratio between lifetimes is constant and its value obviously depends both on the DOS and on the matrix elements.To lowest order in ( s / d ) we find Notice that this ratio remains energy independent if higher order terms in ( s / d ) are included in Eqs.͑13͒ and ͑14͒.
For Co this result extends from the Fermi energy up to ↑ Ӎ0.6 eV.
The next energy region is characterized by ↑ Ͻ(E ϪE F )Ͻ ↓ , where both minority and majority d electrons will be excited.In this case, As can be seen, similar to the case of copper, there appear new terms that do not scale as (EϪE F ) 2 .These new terms change the constant ratio between ↑ and ↓ , which will now depend on energy.Due to the fact that the new term entering in ↓ has a smaller power in ( s / d ) than the one in ↑ , the ratio ( ↑ / ↓ ) is expected to increase with increasing energy.Above this energy range (EϪE F Ͼ1.3 eV) the expressions are too complicated to be of any use to spell out.

IV. ANALYSIS OF THE EXPERIMENTAL DATA
Information on excited-electron lifetime has been obtained from the measure of linewidths in photoemission and inverse photoemission experiments.A recent compilation of these experiments 36 suggests a linear dependence Ϫ1 ϭ␣•(EϪE F ) with a universal value for ␣.Ballistic electron emission microscopy 37 ͑BEEM͒ seems also to be able to al-low the determination of the mean-free path of very lowenergy electrons.The lifetimes obtained for Au by BEEM ͑Ref.38͒ can be well described by Quinn's approach while in the case of palladium an unexplained linear dependence with (EϪE F ) is obtained. 39This linear dependence can be explained in materials where a large density of unoccupied states exists just above the Fermi energy, for instance in transition metals, 22 but the quadratic behavior should be retrieved for energies low enough.
Nor linewidth analysis, where approximations for the elastic width are needed to separate the inelastic linewidth, neither BEEM, where the lifetime enters as one of the fitting parameters, provide a direct measure of the lifetime of the excited states.TR-2PPE does it.In a TR-2PPE experiment an electron is excited above the Fermi level but below the work function using an ultrashort laser pulse.A second pulse extracts the electron above the vacuum level and it can be detected.Measuring the final energy, the energy corresponding to the intermediate excited state can be determined simply by subtracting the photon energy.Varying the delay between the two laser pulses the time evolution of the population of this state, and hence its lifetime, can be directly monitored.
][5][6][7][8][9][10] There exist problems related to the interpretation of the data and there is presently a lack of agreement between the results of different groups.Different processes could be the origin of this disagreement.First of all, the deexcitation of the primary electrons ͑and their corresponding holes͒ excited by the laser beam gives rise to a series of secondary cascade processes that would affect the measured lifetime by a delayed repopulation of the states under measure.This kind of process would mainly affect the states up to about 1 eV above the Fermi energy.When the energy of the laser is enough to excite electrons from the deeper d bands the corresponding d holes are localized and presumably have lifetimes longer than those of the free states that could be the reason for some surprising results. 8,9Transport effects would also affect lifetimes by taking the excited electrons out from the experimental region.This transport effects would depend on the quality of the sample and on its orientation and could explain the observed discrepancies and the face-dependent lifetimes observed by some groups. 6,7Most probably, the discrepancies reflect different experimental approaches ͑like different pulse-width or different strategies of data analysis͒ and/or a considerably different quality of the used samples.See Ref. 40 for a recent discussion on this question.Thus, even if TR-2PPE experiments do not measure the intrinsic lifetime of the excited state but a superposition of the different processes involved in the photoexcitation event, we think that an analysis of the experimental data in terms of our approach could be useful to check the consistency of both experimental and theoretical results.
6][7][8] We have fitted the experimental data to the expression obtained for a triangular density of states of the d band, which gives the best agreement with the first-principles calculation, where t ϭ1.9 eV and WӍ4.1 eV from band-structure calculations. 34a 0 and a 1 are directly related to the matrix elements ͓c.f.Eqs.͑8͒ and ͑11͔͒, The experimental lifetimes are reasonably well described by Eq. ͑18͒, which supports the validity of the constant matrix approximation that has been used.The results from our fittings are summarized in Table I.As can be seen the values for a 0 look rather consistent but there is a large spread in the values for a 1 .However, it is in the region where a 1 plays a role where the biggest experimental errors are expected because lifetimes are much shorter and closer to the experimental resolution.Regarding the ratio between the matrix elements for Cu, it varies between 1.1Ͻ ͉M ss ss ͉ ͉M ss sd ͉ Ͻ2.8.

͑20͒
One would expect, supposing the screening is the same for all electrons, that this ratio should in fact be greater than unity due to a bigger overlap between the s-wave functions. 33or silver, we have two sources 4,5 and due to the large d band threshold there is no experimental information of the behavior above that threshold.We cannot, therefore, obtain information about a 1 .The available experimental data has been fitted using the expression

͑21͒
The a 0 values obtained are shown in Table I.Since Cu an Ag have very similar s values, the difference in the a 0 values has to be related to the matrix elements and the screening.To first order, we expect the bare matrix elements to be rather similar but the screening is expected to be less efficient in Ag than in Cu ͑in Ag, d bands are deeper and s is smaller͒ and hence a 0 (Ag) should be larger than a 0 (Cu) as found ͑see Table I͒.Aeschlimann et al. 10 have measured spin-dependent lifetimes in Co in the energy range we have studied.We have analyzed both spin states with the expression from the previous section.Figure 3 shows the fitting of the available experimental data for both spin states together with the ratio of the lifetimes.The model seems to be able to provide a good description of the experiment, which supports the validity of the approximations being used.To lowest order in ( s / d ), This means that for energies where no d ↑ electrons can be excited the lifetimes will be dominated by the excitation of d ↓ electrons giving a contribution scaling with (EϪE F ) 2 .
When excitation of d ↑ electrons becomes possible there appears a correction, which scales with (EϪE F Ϫ ↑ ) 2 .Again to lowest order in ( s / d ), Notice that, in contrast to the case of noble metals, a 1 has a higher power in ( s / d ) than a 0 and hence, the correction is supposed to be small.The best-fitting parameters are shown in Table II.It is interesting to notice that a 0 ↑↓ (Co) is about one order of magnitude larger than a 0 (Cu) and a 0 (Ag).This is a consequence of the existence of empty minority d states above the Fermi energy, which increases the available phase space with respect to the case of noble metals.The absence of these states in the case of majority electrons is also the key to understand why a 1 ↑ is about one order of magnitude smaller than the other parameters.The same argument would predict a larger value for a 0 ↓ .This is probably not found due to a small overlap between the localized d states. 33inally, from the different ratios between the a 1,2 ↑,↓ the ratios between the different matrix elements can be obtained, showing that ͉M dd dd ͉Ͻ͉M sd sd ͉Ͻ͉M ss sd ͉Ͻ͉M ds dd ͉. ͑25͒ Except for the case ͉M ds dd ͉ the matrix elements follow the intuitive trend that the more d electrons the smaller the matrix elements should be, which is supported by previous results. 33

V. SUMMARY
We have developed a model that introduces the effect of the real density of states in the calculation of the lifetime of low-energy electrons in transition metals, in a simple way.With the assumption of constant matrix elements and using them as fitting parameters, different sets of experimental data can be accounted for.The values obtained for the matrix elements presently differ between different experiments on the order of 40% and show a qualitative agreement with theoretical estimations.Mechanisms affecting the measurements such as transport, cascades or surface effects can be the reason for this disagreement.On the other hand, energydependent matrix elements and interference terms should also be taken into account, but this is beyond the present initial analysis.

FIG. 1 .
FIG. 1. Scheme for the deexcitation process in a scattering picture.

FIG. 2 .
FIG. 2. ͑a͒ Box and triangle description for the DOS in noble metals.Observe that d triangle ϭ2 d box in order to keep the same number of electrons.͑b͒ Box model for the spin dependent DOS in Co.We will assume that d and s are spin independent, therefore, the ferromagnetic behavior will be a consequence of the positions relative to the Fermi energy of the d boxes.

FIG. 3 .
FIG.3.Spin-dependent lifetimes in Cobalt ͑top͒.Open triangles are experimental data for majority spin electrons while full triangles correspond to minority spin electrons ͑Ref.10͒.The solid lines are the results of our model applied to both sets of data.The ratio between the spin-dependent lifetimes ͑bottom͒ is also compared with our results.

TABLE I .
Fitting parameters for the different experimental sources.a 0 accounts for the term with usual (EϪE F ) 2 scaling and a 1 for the one scaling with (EϪE F Ϫ t ) 2 ͓see Eqs.͑18͒ and ͑21͔͒.Units are ͓fs Ϫ1 eV Ϫ2 ͔.

TABLE II .
Fitting parameters for Co. a 0 accounts for the term with usual (EϪE F ) 2 scaling and a 1 for the one scaling with (E ϪE F Ϫ ↑ ) 2 for both majority ͑↑͒ and minority ͑↓͒ spin states ͓see Eq. ͑22͔͒.Units are ͓fs Ϫ1 eV Ϫ2 ͔.