Electron inelastic mean free paths in condensed matter down to a few electronvolts
aa r X i v : . [ c ond - m a t . o t h e r] J u l Electron inelastic mean free paths in condensed matter down to a few electronvolts
Pablo de Vera ∗ and Rafael Garcia-Molina Departamento de F´ısica – Centro de Investigaci´on en ´Optica y Nanof´ısica,Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, E-30100 Murcia, Spain
A method is reported for a simple, yet reliable, calculation of electron inelastic mean free pathsin condensed phase insulating and conducting materials, from the very low energies of hot electronsup to the high energies characteristic of electron beams. Through a detailed consideration of theenergy transferred by the projectile in individual and collective electronic excitations, as well asionizations, together with the inclusion of higher order corrections to the results provided by thedielectric formalism, inelastic mean free paths are calculated for water, aluminum, gold and copper inexcellent agreement with the available experimental data, even at the elusive very low energy region.These results are important due to the crucial role played by low energy electrons in radiobiology(owing to their relevant effects in biodamage), and also in order to assess the not yet elucidateddisagreement between older and recent measurements of low energy electron mean free paths inmetals (which are relevant for low energy electron transport and effects in nanostructured devices).
Electrons interacting (in a broad range of energies)with condensed matter appear either as direct projectiles,as the result of target ionization by external radiation,or as charge carriers in electronic devices. The preciseknowledge of the transport of these electrons throughmatter, which depends on the electron energy and themedium characteristics, provides very useful informationfor controlling material properties (through modificationor analysis) [1, 2], improving the yield of nanoelectronicdevices that relies on charge mobility [3, 4], or to gainknowledge on energy conversion, catalysis at surfaces andnanomaterials [5, 6] or oncological studies at the molec-ular level [7, 8], all of them depending on the transportand intereactions of electrons. To properly understandand model all these phenomena, the knowledge of theaverage distance between inelastic collisions (i.e., the in-elastic mean free path, IMFP) is of paramount relevance.The dielectric framework to treat the interaction ofcharged particles with matter, which dates back to Fermi[9], relies on the first Born approximation (FBA). Cur-rently, it represents a reliable, yet simple, method to cal-culate electronic mean free paths [10] and, even, ioniza-tion cross sections in condensed matter [11, 12]. Gen-eral consensus between models and experimental data isfound for electron energies &
200 eV [13, 14], with dis-crepancies appearing at the lower energies ( .
50 eV).This raises some debate [15–17] due to the influence ofthe maximum energy transferred by the electron wheninteracting with the target [18–20]. Also, due to pos-sible corrections to the FBA for low energy (sometimesreferred as “hot”) electrons [21], so relevant in nanoelec-tronics [4], catalysis [6] or biomolecular damage [7, 8].In this letter we discuss the different role played by(both individual and collective) excitations, as well asionizations, induced by electrons in insulators and con-ductors, as they have rather different types of electronicexcitations. This is illustrated for representative mate-rials (water, aluminum, gold and copper) relevant forradiological and nanoelectronic applications. Water is a molecular material where collective electronic excita-tions are unlikely [22], thus inelastic interactions resultin ionized (i.e., free) or excited (i.e., bound) electrons.In metals, both collective (e.g., plasmons) and individ-ual electronic excitations are possible, the latter directlyleading to free electrons in the conduction band. Whilethe excitation spectrum of aluminum is dominated by astrong plasmon, gold and copper present complex exci-tation spectra, where both individual and collective ex-citations can coexist.In this work, we will show that a proper considerationof the excitation spectrum of the target, together withthe inclusion of higher order corrections to the FBA, al-low the calculation of IMFP in excellent agreement withthe available experimental data for a wide energy range,down to a few electronvolts. In particular, for these ma-terials (especially for metals) there is an extensive set ofexperimental data over a wide energy range which willserve as a benchmark for our model. For the case of cop-per, there is also a discrepancy between the low energyIMFP derived from modern experimental techniques [23]and older measurements [24–26]. Our analysis will pro-vide clues in order to elucidate these discrepancies.Within the dielectric formalism (i.e., FBA), the inelas-tic mean free path λ e of an electron of kinetic energy T ,mass m and charge e is [10, 27]: λ − ( T ) = e mπ ~ T n X i =1 Z E + ,i E − ,i d E Z k + k − d kk f ex ( k, T ) × F ( E − E th ,i )Im (cid:20) − ǫ ( k, E ) (cid:21) i , (1)where E = ~ ω and ~ k are, respectively, the energy andmomentum transferred in an inelastic collision. The tar-get electronic excitation spectrum is represented by itsEnergy-Loss Function (ELF), Im h − ǫ ( k,E ) i = P i F ( E − E th ,i )Im h − ǫ ( k,E ) i i , with F ( E − E th ,i ) and E th ,i beinga smooth step function and threshold energies, respec-tively. Although the contributions i are commonly usedto reproduce the measured ELF of materials, they can begiven a physical meaning, i.e., the excitation of differentelectronic levels, as it will be explained later on. Theindistinguishability between the incident and target elec-trons (when applicable) is introduced through the Born-Ochkur exchange factor f ex ( k, T ) [28].In practice, Eq.(1) is evaluated by means of extendedoptical data models [29], where the ELF is usually fittedto experimental data in the optical limit ( k = 0), and ap-propriately extended to finite momentum transfers (i.e.,over the whole Bethe surface). In this work, we use theMELF-GOS method to cover the whole ( k, E )-space [29].The IMFP calculation depends, as seen in Eq.(1), onthe integration limits on momentum and energy trans-fers. The former, obtained by energy and momentumconservation, are ~ k ± = √ m (cid:16) √ T ± √ T − E (cid:17) . Thelatter, generally considered independent of the electronicexcitation ( E ± ,i = E ± ) are determined not only by en-ergy conservation, but also by the Pauli exclusion prin-ciple and electron indistinguishability. On the one hand,the primary electron cannot fall into an occupied level ofthe target. Therefore, for metals, E + = T − E F , where E F is the Fermi energy. On the other hand, when aprimary electron creates a free secondary electron (pre-viously bound with energy B i ), the maximum energytransfer occurs when both have the same final energy, so E + = ( T + B i ) /
2. On this basis, Bourke and Chantler [19]discussed whether the maximum energy transfer (in thecase of metals) should be ∼ T / T − E F . Theyconcluded that the former is a too constraining limit, ar-guing that collective excitations (e.g., plasmons, distin-guishable from the primary electron), represent the mainexcitation channel in metals. Considering the differentnature of each inelastic interaction will lead to maximumenergy transfers that depend on the specific electronicexcitation ( E ± ,i = E ± ), as explained in what follows.For metals, both individual and collective electronicexcitations (e.g., plasmons) are possible. Every individ-ual transition will promote an electron to the partiallyfilled conduction band (through which the primary elec-tron is moving), so E indiv+ ,i = min [( T + B i ) / , T − E F ].In turn, a collective excitation is distinguishable from theprimary electron, thus E collect+ ,i = T − E F and f ex = 0.For insulators, two possible types of excitations willbe considered: an electron transition to a localized dis-crete energy level (an “excitation”), or to the conductionband (an “ionization”). As shown in Refs. [11, 12] forwater and other molecular materials, the introduction ofa mean binding energy B for the outer shell electronsallows the distinction between both types of excitations.When E < B , an electron is excited to a bound state,while the primary electron moves freely in the conduc-tion band; the latter can then lose all its energy, and E excit+ ,i = min (cid:2) T, B (cid:3) . If
E > B , an electron is ionized;
Exp. [31] Total (this work) Contributions (this work) Ref. [32] E L F ( k = , E ) E (eV)Plasmon
FIG. 1: (Color online) Energy-loss function of gold, at k =0. Symbols depict experimental data [31] while solid linesrepresent the present model, together with the correspondingparameterization by Kwei [32] (dashed lines) for comparison. then both primary and secondary electrons move in theconduction band, so E ioniz+ ,i = ( T + B ) /
2. All these criteriawill be assessed later on, when comparing the calculatedIMFP with experimental data.Once the criteria for the maximum energy transfer foreach inelastic interaction type are established, it is nec-essary to know when they apply. For this purpose, theexcitation spectrum of the target (i.e., its ELF) must beexamined for the different excitation types. Liquid wa-ter, an insulating material, has an ELF characterized by asingle broad peak at ∼
20 eV [30], where excitations andionizations can be separated by means of its mean bind-ing energy B [11, 12]. The spectrum of aluminum [31] isdominated by a sharp and intense plasmon excitation at ∼
15 eV [1]; thus, aluminum will be used as an exampleof a material where collective excitations dominate. Fi-nally, gold and copper present complex excitation spectrawhere individual and collective excitations coexist.Figure 1 contains the optical ( k = 0) ELF of gold (sym-bols) [31]. The peak around 25 eV is considered to be theplasmon resonance [1]. The rest of peaks will be assignedto the excitation of the different bands. A parameteriza-tion made by Kwei et al. [32] is shown by dashed lines.In the first part of our discussion, all excitations, exceptthe plasmon, will be regarded as individual ones.We have parameterized the ELF in terms of Mermin-type (MELF) contributions [33], introduced in Eq.(1).Threshold energies for each level have been taken fromthe literature [34, 35], while fitting of the optical ELF hasbeen constrained to respect as much as possible the num-ber of electrons expected in each energy level, accordingto its electronic configuration, by evaluating individual f -sum rules [36]. Our parameterization of the differentcontributions (thin solid lines in Fig. 1) corresponds rea-
10 100 10000.000.020.040.060.080.10
Ionization:Calculation: FBA Higher orderExperiments:[37][38][39]
T (eV)
Ionization
Excitation:Calculation: FBA Higher orderExperiments:[40] (one level, scaled, x2.59)[41] (six levels)[42] (six levels)[43] (two levels, scaled, x1.43) I n v e r s e m ean f r ee pa t h ( ¯ - ) Excitation
FIG. 2: (Color online) Electron inverse IMFP in water due tothe processes of ionization and excitation. See the main textfor the meaning of symbols and lines. sonably well to that by Kwei et al. [32] (dashed lines).The thick line, representing the sum of all the contribu-tions, agrees very well with the experimental data [31].Our analysis of the electron IMFP in the selected ma-terials starts with liquid water, an insulating materialwhere both electronic transitions to bound levels (excita-tions) and to the conduction band (ionizations) are pos-sible. Figure 2 depicts the inverse IMFP correspondingto these cases. Dashed lines represent our calculations,using B = 12 . & ab initio calcu-lations [5]. Red dotted and gray dot-dashed lines cor-respond, respectively, to calculations using Eq.(1) with Calculations: FBA, E indiv+ = T / 2, with exchange FBA, E collect+ = T - E F , with exchange Higher order, E collect+ = T - E F , w/o exchange Ab initio calculations, Ref. [5]Experiments: [26] [46] [47] [48] [49] [50] M ean f r ee pa t h ( ¯ ) T - E F (eV) FIG. 3: (Color online) Electron IMFP in Al. See the maintext for the meaning of symbols, lines and the shaded area. E indiv+ = T / E collect+ = T − E F [19] (collective excitations dom-inate); exchange is included in both cases. Finally, theblack line uses E collect+ = T − E F , but excludes electronicexchange, which should not be considered for plasmonexcitations due to distinguishability from the primaryelectron. The latter condition yields the best agreementwith most of the experimental data in the whole energyrange, which validates the maximun energy transfer forcollective excitations [19], as it will be used in the follow-ing. In this case, the calculation including higher ordercorrections by means of the Coulomb-field perturbationterm is practically identical to the black solid line.Gold is a material with a complex excitation spectrum(Fig. 1) where both individual and collective excitationscoexist. Figure 4(a) shows the calculated (lines) and ex-perimental (symbols) [26, 46, 54–57] electron IMFP ingold. Thin solid lines represent contributions from dif-ferent excitations, evidencing their relevance at differentelectron energies. The red dotted and blue dashed linescorrespond, respectively, to calculations where E indiv+ = T / E collect+ = T − E F (all regarded as collec-tive, exchange not included). Clearly, the former over-estimates the IMFP around its minimum, although itsbehavior at low and high energies is reasonable. Besidesthis, the latter also reproduces the minimum of the IMFPat ∼
70 eV. This manifests the importance of the collec-tive excitations, as pointed out in Ref. [19]. However,this calculation seems to differ from the experiments atlower energies and, particularly, does not reproduce thestructure of the experimental IMFP for energies . FBA, E indiv+ = T / 2, with exchange FBA, E collect+ = T - E F , w/o exchange FBA, individual d excitation [46] [54] [55] [56] (b) M ean f r ee pa t h ( ¯ ) Plasmon
5d 5p (a) Experiments: [14] [23] [24] [25] [26] [57]Calculations: Ab initio calculations (e-e + e-ph) Ref. [5] Higher order, collective d excitation Higher order, coll. d excit. + derived e-ph
T - E F (eV) FIG. 4: (Color online) Electron IMFP in (a) Au and (b)Cu. See the main text for the meaning of symbols, lines andshaded area. two more considerations. First, higher order correctionscan be added as in the previous cases. Second, the ex-citation of the 5d electrons, with onset around 20 eV,was suggested to be an atomic giant resonance [51, 52].As this is a collective atomic excitation [53], it has to betreated as such, and its associated IMFP evolves fromthe blue solid thin line to the blue dashed thin line inFig.4(a). All the previous ingredients are incorporatedin the calculations shown by the black solid thick line.It is striking its excellent agreement with the experimen-tal data over the entire energy range, from high energiesdown to 3–5 eV, and particularly around 5 – 100 eV,where the structure of the experimental IMFP is verywell reproduced.Finally, we apply the previously detailed methodologyto calculate the IMFP in copper, where low energy datahave recently been obtained through XAFS experiments[23]. These measurements, shown in Fig. 4(b) by an or-ange line (with symbols and error bars), are in conflictwith older measurements [24–26]. Our full calculation(black solid line) reproduces very well the older experi-mental data [14, 24–26, 57] and is close to the ab initio results from Ref. [5] down to ∼ ab initio calculations by theelectron-phonon interaction, quite strong in certain crys-tallographic orientations, resulting in a significant disper-sion of their results, which spans from the older to theXAFS-derived data. Therefore, it is plausible that un-accounting for electron-phonon interaction in the XAFSexperiment could affect the derived IMFP, λ XAFS .To deeper investigate this point, we have obtainedthe difference λ − = λ − − λ − , where λ e is ourcalculated electronic IMFP. We have fitted λ diff to theasymptotic form of the electron-acoustic phonon meanfree path [58, 59]. The blue double-dotted-dashed curvein Fig. 4(b) corresponds to a total IMFP calculated as λ = (cid:0) λ − + λ − (cid:1) − , which perfectly agrees with boththe high energy experimental IMFP [14, 57] and theXAFS-derived IMFP [23] down to 20 eV. This indicatesthat electron-phonon interaction (among other possibleprocesses) could have affected the interpretation of theXAFS experiments [23]. This plausible explanation shedslight on the disagreement between the older and theXAFS measurements at low energies.In conclusion, we have analyzed the role played by thedifferent excitations (collective or individual), as well asionizations, in the maximum energy transferred in theinelastic interactions of an electron moving through ei-ther conducting (aluminum, gold and copper) or insu-lating (liquid water) media appearing in nanostructureddevices and biological environments. The discussion andresults presented in this work highlight the importanceof a proper description of the material excitation spec-trum (through its Energy-Loss Function) for an accuratecalculation of the electron inelastic mean free path. Also,the need for higher order corrections to the dielectric for-malism to obtain accurate inelastic mean free path atthe lower energies is remarked. The calculated IMFP forthese materials are in excellent agreement with the ex-perimental data in practically the entire energy range,covering from the low energies of hot electrons [26] upto the high energies typical in electron beams [1]. Ourresults also help to elucidate the discrepancy between re-cent [23] and older [24–26] measured electron IMFP incopper at low energies, which could be due to the unac-counted electron-phonon interaction in the former. Thepresented results are of great relevance for understandingthe dynamics of electron transport in condensed matter.The authors are indebted to Prof. Isabel Abril forenlighting discussions and constant support while devel-oping this work. Financial support was provided by theSpanish Ministerio de Econom´ıa y Competitividad andthe European Regional Development Fund (Project No.FIS2014-58849-P), as well as by the Fundaci´on S´eneca(Project No. 19907/GERM/15). ∗ Corresponding author: [email protected][1] R. F. Egerton,
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