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Reflection Electron Energy Loss Spectra beyond the optical limit
Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Reflection Electron Energy Loss Spectra beyond the optical limit Lucia Calliari a,⇑, Maurizio Dapor a, Giovanni Garberoglio a, Sergey Fanchenko b a European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK) and Trento Institute for Fundamental Physics and Applications (TIFPA-INFN), via Sommarive 18, I-38123 Trento, Italy b National Research Center ‘‘Kurchatov Institute’’, Kurchatov Square 1, 123182 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 25 June 2014 Received in revised form 9 October 2014 Accepted 29 November 2014 Available online xxxx Keywords: Inelastic scattering Electron energy loss Monte Carlo Plasmon energy dispersion Plasmon damping dispersion
a b s t r a c t A position-dependent Inverse Inelastic Mean Free Path (IIMFP), calculated according to the Chen–Kwei theory of inelastic scattering, is used to obtain Reflection Electron Energy Loss Spectra (REELS) by a Monte Carlo approach. The basic ingredient of the theory is the energy-loss and momentum-transfer dependent dielectric function, for which we use a plasmon-pole approximation to the Lindhard dielectric function. The dependence on the momentum transfer enters by assuming a proper dispersion relation for the energy loss. Experiments reveal however that, beside energy loss, also plasmon damping disperses with momentum transfer. By comparing measured and calculated REEL spectra, we explore the role of damping dispersion for a quasi-free-electron material like silicon. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Within the dielectric framework, all information on electronic excitations initiated by external perturbations (e.g. photons or electrons) is included in the material dielectric function eðx; qÞ x and momentum hq transferred to which depends on energy h target electrons. Calculating the dielectric function from first principles is however not feasible in most instances, so that extensive use is made of semi-empirical models which exploit the direct connection existing between the bulk Energy Loss Function (ELF), Im eð1 x;qÞ , and measurable properties of materials, in particular optical properties. The latter probe the energy dependence of electron excitations at zero momentum transfer and make the basis of so-called ‘‘optical-data’’ models. However, to describe electronic excitations (e.g. electron-induced) where a non-negligible momentum is transferred to the material, one needs to go beyond the ‘‘optical limit’’. Extension algorithms have been proposed [1–3] to extrapolate ‘‘optical-data’’ models into the non-zero momentum transfer region. The extension is performed by introducing suitable dispersion relations for all or some of the parameters entering the dielectric function, essentially plasmon energy and damping. Among the proposed models, we have recently considered [4,5] a plasmon-pole approximation to the Lindhard dielectric function where extension to non-zero momentum transfer was ⇑ Corresponding author. Tel.: +39 0461 314483. E-mail address: [email protected] (L. Calliari).
obtained by accounting for plasmon energy dispersion. Such a dielectric function was used within the Chen–Kwei theory [6,7] to calculate REEL spectra by a Monte Carlo approach. Examining several prescriptions for the plasmon energy dispersion, we concluded that inclusion of a proper dispersion relation is crucial for a satisfactory agreement between measured and calculated REEL spectra for a quasi-free electron material like silicon [5]. No dispersion was introduced however for plasmon damping, though momentum-resolved measurements on nearly-free electron materials (i.e. exhibiting well-defined plasmon peaks) show that damping increases with momentum transfer [8–10]. More specifically, two regimes, separated by so-called critical momentum x qc v Fp (xp = plasma frequency, vF = Fermi velocity), are observed, with damping increasing at a definitely higher rate above the critical momentum where single electron excitations become kinematically allowed. In the present paper, we examine if and how damping dispersion affects REEL spectra. Once more, we will take Si as a casestudy and results will be evaluated by comparing calculated to measured spectra. 2. Experimental Experimental REEL spectra are measured on a silicon wafer which was sputter-cleaned in Ultra High Vacuum by 4 keV Ar+ ions. Sample cleanliness was checked by measuring Auger spectra within a PHI545 instrument equipped with a double-pass cylindrical mirror analyser (CMA) and a coaxial electron gun.
http://dx.doi.org/10.1016/j.nimb.2014.11.106 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106
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REEL spectra are excited by 2 keV electrons within the same instrument. The electron analyser was operated in the constant energy resolution mode. The analyser energy resolution was 0.6 eV, as measured on the Pd Fermi edge of a HeI (hm = 21.2 eV) excited valence band photoemission spectrum, leading to a measured FWHM of 0.9 eV for the Zero Loss Peak (ZLP). Once acquired, REEL spectra are corrected for the (previously measured) energy dependence (E0.9) of the analyser transmission function. Details of the CMA geometry are discussed elsewhere [5]. 3. Theory of inelastic scattering A diagram illustrating the basic event of a REEL experiment is shown in Fig. 1. An inelastic scattering event, characterized by scattering angles h (polar) and u (azimuthal), is supposed to take place while the electron enters from the vacuum into the solid.
eðx; hÞ ¼ 1
2
h2 x2 b2E h2 þ hE
2 x2p h 2
2 E h2 þ h2E h x2g þ ihx hc0 þ 2Eðh2 þ h2E Þ
x
ð1Þ
x x x þ ixcq 2 q
hx
E is the energy of incident electrons and hE ¼ 2Ep is the so-called characteristic angle of scattering. Using the dielectric function (2), we calculate the positiondependent Differential Inverse Inelastic Mean Free Path (DIIMFP) according to the Chen–Kwei theory [6,7]:
DIIMFPðE; x; a; zÞ ¼
2 p
2
ð2Þ
2
The angle of surface crossing, a, is the angle between the electron trajectory and the normal to the surface. The distance from the surface is measured on the z-axis, perpendicular to the surface and pointing towards the vacuum. We describe the material response using a plasmon-pole approximation to the Lindhard dielectric function:
eðq; xÞ ¼ 1
x
the measured spectrum, taking into account that the ratio xpg is constrained by 2 the value of the static dielectric function x eð0; 0Þ ¼ 1 þ xp2g ; 11 12 for Si [12,13]. We chose hxp ¼ 15:8 eV and hxg ¼ 4:9 eV [5] (notice that the average energy gap of silicon, as reported in Ref. [14], is 4.8 eV, in very good agreement with our estimate). For the damping parameter, we assume a quadratic dis2 2 persion hcq ¼ hc0 þ d h mq , formally introduced by adding an imagi c0 ¼ 3:2eV. Though nary part to b, i.e. b ¼ 65 EF i hxd. We take h the prescription does not account for the strong damping dispersion above the critical momentum, a quadratic momentum-dependence is nonetheless measured for plasmon widths up to the critical momentum [8–10]. The phenomenological parameter d thus accounts for the increasing probability of plasmon decay as the momentum transfer increases below the critical momentum. Parameter d should range between 0 and 1 [9]. In line with our previous papers, we write the dielectric function as a function of the polar angle of scattering h:
2 g
1 a0 p2 2E
Z
hB
dh 0
Z 2p 0
du
h h2 þ h2E
f ðz; a; h; u; EÞ ð3Þ
where hxp is the plasmon energy ( h = Planck constant divided by 2p) and hxg is a parameter representing the average energy that electrons have to overcome to take part in collective excitations. We therefore expect it to be larger than the minimum energy gap 2 4 2 of the target. The q-dependence enters via x2q ¼ b qm þ h4mq2 (m is the electron mass) and cq . The former term accounts for plasmon energy dispersion, the latter for damping dispersion. For b, characterizing energy dispersion, we take b ¼ 65 EF (EF = Fermi energy) [5], i.e. the value obtained within the Random Phase Approximation xp and (RPA) [11]. Parameters h hxg are chosen by comparison with
where a0 is the Bohr radius. Function f ðz; a; h; u; EÞ, dictated by the dielectric function eðx; hÞ, depends on the distance from the surface, z, on the angle of surface crossing, a, on the energy of electrons, E, and on the scattering angles h and u. Integration over h extends qffiffiffiffiffi up to the Bethe ridge angle hB ¼ hEx [4,5]. Explicit functional forms of f ðz; a; h; u; EÞ for all the relevant cases (ingoing and outgoing electrons, either in the vacuum or in the solid) are found elsewhere [5] together with details on the calculations. As an example, we report here the expression for f ðz; a; h; u; EÞ relative to incoming electrons in the solid (i.e. the case illustrated in Fig. 1 and considered in the examples below). With obvious notation:
sol f in ðz; a; h; uÞ ¼ 2 cosðqz zÞ eQ jzj
eðQ ; xÞ eðQ ; xÞ 1 1 1þ Im eðq; xÞ eðQ ; xÞ þ 1 eðq; xÞ eðz; Q ; xÞ
z
vacuum
ð4Þ
α
where q, qz and Q are, respectively, the modulus of the momentum transfer and its components perpendicular and parallel to the solid surface. In terms of the scattering angles:
solid θ
ϕ Fig. 1. Diagram illustrating an inelastic scattering event. An electron enters the solid from the vacuum, crossing the surface at an angle a. Inelastic scattering, characterized by scattering angles h (polar) and u (azimuthal), takes place in the solid.
q¼
mv h
qz ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2E þ h2
mv ðhE cos a þ h sin a cos uÞ h
ð5Þ ð6Þ
Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mv 2 2 Q¼ ðhE sin a h cos a cos uÞ þ ðh sin uÞ h
ð7Þ
with m the electron mass. Functions eðQ ; xÞ and eðz; Q ; xÞ are defined as:
1 Q ¼ eðQ ; xÞ p
Z
þ1
1
1 Q ¼ eðz; Q ; xÞ p
Z
0
dqz 1 2 2 02 2 ; x Q þ q02 e Q þ q z z
þ1
1
0
ð8Þ
0
eiqz z dqz 1 2 02 2 ; x ðQ 2 þ q02 Þ e Q þ q z z
ð9Þ
4. Monte Carlo While a detailed treatment of the elastic scattering, requiring numerical quantum mechanical calculations, is described in [15], here we briefly remember a few basic points. We use the Hartree–Fock atomic potential and, to account for solid-state related changes in the outer electronic orbitals, the muffin-tin model is used (enabling the atomic potential to be affected by nearestneighbor atoms). Elastic scattering is treated by calculating the phase shifts. Since the asymptotic behavior of the wave function at a large radial coordinate is known, the phase shifts can be computed by solving the Dirac’s equation for a central electrostatic field up to a large radius for which the atomic potential can be safely ignored (Mott cross section: Relativistic partial wave expansion method). Exchange effects (the incident electron can be captured by an atom with emission of a new electron, the two electrons being indistinguishable) are taken into account. Concerning inelastic scattering, the position dependent DIIMFP, calculated for a distance up to 7 nm from the surface both into the solid and into the vacuum, is used to obtain the position-dependent cumulative probability for energy loss, Pcum, and the position dependent IIMFP:
R hx DIIMFPðE; x; a; zÞdx 0 Pcum ðE; x; a; zÞ ¼ R hx max DIIMFPðE; x; a; zÞdx 0 IIMFPðE; zÞ ¼
1 pa0 E
Z
ð10Þ
define the step-length between successive scattering events. Details on the overall procedure are given in Ref. [5]. Once the step-length is defined, the elastic or inelastic nature of the scattering event, the polar angle for elastic scattering and the energy loss for inelastic scattering are all sampled, according to the usual Monte Carlo recipes, via random numbers from the relevant cumulative probabilities [15]. Inelastic events are assumed to change only the electron energy and not the electron direction. Monte Carlo simulations were performed by generating 128 million trajectories, with electrons starting at 10 nm from the surface and crossing the surface at an angle of 30°. The full threedimensional electron trajectory is generated. Relations to calculate the trajectory direction in the laboratory frame after each scattering event are taken from Ref. [16]. Electron trajectories are followed until the energy loss is larger than 100 eV, because this is the energy loss range probed experimentally, or until the electron exits the surface and it reaches a distance of 10 nm from it. At this point, its outgoing angle aout is tested for detection, i.e. whether it satisfies the condition for being detected by the analyser.
5. Results and discussion Fig. 2 shows the damping parameter as a function of the polar angle of scattering h over a range which includes the critical angle qffiffiffiffiffi hxp hc pffiffiffiffiffiffi and the Bethe ridge angle hB ¼ hEx, the latter calculated 2
EEF
x ¼ 16 eV. The dashed, horizontal line reprefor an energy loss h sents the non-dispersive case, d ¼ 0. Continuous and dotted lines cðhÞ ¼ hc0 þ d2Eðh2 þ h2E Þ, with refer to the dispersive case, h d ¼ 0:5 and d ¼ 1, respectively. The effect of damping dispersion on the Energy Loss Function (ELF) is shown in Fig. 3 where the bulk ELF ¼ Im eð1 x;hÞ is plotted as a function of the energy loss for several scattering angles within the 0 hB ð16 eVÞ range. The upper panel refers to d ¼ 0, the intermediate panel to d ¼ 0:5 and the lower panel to d ¼ 1. For d ¼ 0, the ELF moves to higher energy loss and it decreases in intensity as the scattering angle increases. For d–0, broadening of the ELF is additionally observed. It might be worth mentioning that experimental proof of such a broadening of the ELF as the
hxmax
DIIMFPðE; xÞdx
ð11Þ
0
δ =0
4
where hxmax ¼ 100 eV corresponds to an energy loss range wide enough to account for all processes relevant for determining the IMFP. Functions (10) and (11) are used, respectively, to sample the energy loss associated with inelastic scattering events and to
2 0
45
4
ELF
40
hγ/2π (eV)
35
δ =0.510
20
30
δ =1 10
20
30
10
20
30
2
30 25
0
20
4
15
2
10
θ
5 0 0
θ
c
B
0 0.02
0.04
0.06
θ (rad)
0.08
0.1
Fig. 2. Damping hc vs. polar angle of scattering h. Dashed line: d = 0; continuous line: d = 0.5; dotted line: d = 1. The critical angle, hc, and the Bethe ridge angle, hB (16 eV), are marked by arrows.
0
40
ENERGY LOSS (eV) Fig. 3. Bulk ELF as a function of the energy loss and for the scattering angle h increasing from 0 to 0.08 rad in steps of 0.02 rad. Each panel refers to a different choice for d in the damping dispersion.
Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106
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Fig. 5 compares Monte Carlo spectra (continuous curves) to the measured spectrum (dotted curve). Spectra refer to 2 keV electrons and they are all normalized to a common area of the ZLP. On moving from the upper to the lowest panel, spectra are calculated for parameter d increasing from d ¼ 0 to d ¼ 1. Hardly any change can be observed in the spectra (not shown here) as long as d < 0:1 0:2. For greater d, spectral changes mimic changes observed for the DIIMFP in Fig. 4. In particular, intensity shifts towards low energy loss and the surface plasmon gains in intensity over the bulk one. Though one could question the assumed momentum-dependence for damping, it is nonetheless clear that:
z=0nm
0.02
DIIMFP (eV nm-1)
0.01 0 0.02
10
20
30z=-0.5nm
10
20
30z=-1nm
0.01 0 0.02
Neglecting damping dispersion has no significant effect on calculated spectra as long as d < 0:2. Calculated spectra are not brought into better agreement with measured spectra by accounting for damping dispersion; on the contrary, they deviate more from the experiment.
0.01 0
0
10
20
30
40
50
ENERGY LOSS (eV) Fig. 4. DIIMFP calculated for 2 keV ingoing electrons in the solid at variable distance z from the surface. In each panel, different curves correspond to different d values: continuous: d ¼ 0; dashed: d ¼ 0:5; dotted: d ¼ 1.
δ =0
0.1
INTENSITY (eV -1)
0.05 0 0.1
10
20
30
40 δ =0.5 50
10
20
30
40 δ =1 50
10
20
30
40
0.05 0 0.1 0.05 0
0
50
60
ENERGY LOSS (eV) Fig. 5. Comparison between experimental (dotted) and Monte Carlo (continuous) spectra. All spectra are normalized to a common area of the zero loss peak. Spectra in different panels are calculated for different values of the damping dispersion parameter d.
momentum transfer increases is found not only in solids [8–10], but also, for instance, in liquid water [17]. Fig. 4 shows the DIIMFP calculated at variable depth, z, within the solid for ingoing electrons of energy 2 keV. The distance from the surface increases on moving from the upper to the lowest panel. In each panel, continuous, dashed and dotted lines refer to calculations performed, respectively, with d ¼ 0, d ¼ 0:5 and d ¼ 1 in the damping dispersion relation. Accounting for damping dispersion results in a decrease in intensity at the high energy loss tail of the DIIMFP and, at the same time, in a shift of intensity from the bulk to the surface plasmon. In any case, the marked effects observed on the momentum-resolved ELF in Fig. 3 turn into less striking effects on the DIIMFP. We remember that the latter is obtained, according to Eq. (3), by integrating function f ðz; a; h; u; EÞ over the whole momentum transfer range after multiplication by a factor which disfavours the contribution of high momenta.
Alternative reasons should therefore be considered to explain the still existent disagreement between calculated and measured spectra. According to the upper panel of Fig. 5, calculations tend to underestimate the line-width (or, which is the same, overestimate the lifetime) of surface plasmons, suggesting that not all the relevant plasmon decay mechanisms are taken into account. Semi-empirical models (like the plasmon-pole approximation to the dielectric function) assume plasmons to be excited in a freeelectron–gas where the only decay channel is the creation of electron–hole pairs. For this decay to take place, energy and momentum conservation requires that the momentum transfer exceeds a critical momentum. Plasmons are damped however even at zero momentum transfer, the required momentum being supplied by the lattice. To cope with this observation, semi-empirical models incorporate the role of the lattice in a phenomenological way via the damping parameter c which is thus intended to account for damping due to several lattice effects in the bulk, such as interband transitions and scattering of electrons with phonons, lattice defects or impurities. Beside these damping mechanisms however, it has long been recognized [9,18–22] that physical boundaries represent themselves an additional source for damping. Plasmon line-widths broaden in fact on moving from crystalline to polycrystalline materials or from bulk materials to small clusters, and broadening increases with decreasing size of crystallites or clusters. If scattering at surfaces and interfaces provides an additional decay channel for plasmons, damping should depend on space in the surface region in the same way as it is taken to depend on crystal or cluster size [19–21]. However, not only is the mere presence of a boundary crucial in this context, but also are features of the boundary itself. For instance, the crucial role of surface roughness with respect to plasmon excitation and decay has long been known [23] and its effect on REEL spectra was demonstrated [24,25]. Recently, roughness was included in the calculation of REEL spectra [26], showing that good agreement could be obtained with experiments. 6. Conclusions Within the dielectric response framework, REEL spectra were synthesized for a nearly-free electron material like Si. The Chen– Kwei theory of inelastic scattering was used to obtain the position-dependent IMFP needed to account for surface excitations. The main ingredient of the theory is the energy-loss and momentum-transfer dependent dielectric function, for which a plasmonpole approximation to the Lindhard dielectric function was used. Dependence on the momentum transfer was incorporated by accounting for dispersion in the energy and damping of plasmon excitations. While the effects of energy dispersion on REEL spectra
Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106
L. Calliari et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
were characterized previously, here we examined the effects of a quadratic dependence of damping on the momentum transfer. Results were evaluated by comparing spectra calculated by the Monte Carlo method to measured spectra. We found that REEL spectra are unaffected if damping dispersion is low. However, as damping dispersion becomes stronger calculated spectra deviate more and more from the measurements, a suggestion that the dispersion strength is overestimated. We thus concluded that, while energy dispersion is crucial to satisfactorily model REEL spectra, damping dispersion can safely be neglected. Our calculated spectra still fail at reproducing the width of surface plasmons. We discussed possible reasons for such disagreement, leaving to future work an investigation of this point. Acknowledgements This work was supported by NATO (Belgium) through Collaborative Linkage Grant CBP.EAP.CLG.984158 and by Istituto Nazionale di Fisica Nucleare (INFN, Italy) through the ‘Supercalcolo’ agreement with FBK. Computer simulations were performed on the KORE cluster at Fondazione Bruno Kessler. References [1] R.H. Ritchie, A. Howie, Phil. Mag. 36 (1977) 463–481. [2] Z.-J. Ding, R. Shimizu, Scanning 18 (1996) 92–113. [3] D. Emfietzoglou, I. Abril, R. Garcia-Molina, I.D. Petsalakis, H. Nikjoo, I. Kyriakou, A. Pathak, Nucl. Instr. Meth. B 266 (2008) 1154–1161.
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[4] L. Calliari, S. Fanchenko, Surf. Interface Anal. 44 (2012) 1104–1109. [5] L. Calliari, M. Dapor, G. Garberoglio, S. Fanchenko, Surf. Interface Anal. 46 (2014) 340–349. [6] Y.F. Chen, C.M. Kwei, Surf. Sci. 364 (1996) 131–140. [7] Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589 (2005) 67–76. [8] P. Zacharias, Z. Physik 256 (1972) 92–96. [9] T. Kloos, Z. Physik 265 (1973) 225–238. [10] R.V. Baltz, Spectroscopy and dynamics of collective excitations in solids, in: B. Di Bartolo, S. Kyrkos (Eds.), NATO ASI Series, vol. 356, Plenum Press, New York, 1997, pp. 303–338. [11] P. Nozieres, D. Pines, Phys. Rev. 113 (1959) 1254–1267. [12] W.C. Dunlap, R.L. Watters, Phys. Rev. 92 (1953) 1396–1397. [13] J.P. Walter, M.L. Cohen, Phys. Rev. B 2 (1970) 1821–1826. [14] J.C. Phillips, Bonds and Bands in Semiconductors, Academic Press Inc., New York, 1973. p. 116. [15] M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems, Springer-Verlag, Berlin, 2003. [16] R. Shimizu, Z.-J. Ding, Rep. Prog. Phys. 55 (1992) 487–531. [17] N. Watanabe, H. Hayashi, Y. Udagawa, Bull. Chem. Soc. Jpn. 70 (1997) 719–726. [18] V. Krishan, R.H. Ritchie, Phys. Rev. Lett. 24 (1970) 1117–1119. [19] P. Apell, D.R. Penn, Phys. Rev. Lett. 50 (1983) 1316–1319. [20] M. Mitome, Y. Yamazaki, H. Takagi, T. Nakagiri, J. Appl. Phys. 72 (1992) 812– 814. }vel, S. Fritz, A. Hilger, U. Kreibig, M. Vollmer, Phys. Rev. B 48 (1993) [21] H. Ho 18178–18188. [22] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, Springer, New York, 2011. [23] E.A. Stern, Phys. Rev. Lett. 19 (1967) 1321–1324. [24] B. Strawbridge, R.K. Singh, C. Beach, S. Mahajan, N. Newman, J. Vac. Sci. Technol., A 24 (2006) 1776–1781. [25] B. Strawbridge, N. Cernetic, J. Chapley, R.K. Singh, S. Mahajan, N. Newman, J. Vac. Sci. Technol., A 29 (2011) 041602 (1–5). [26] B. Da, S.F. Mao, G.H. Zhang, X.P. Wang, Z.J. Ding, J. Appl. Phys. 112 (2012) 034310 (1–9).
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Reflection Electron Energy Loss Spectra beyond the optical limit Lucia Calliari a,⇑, Maurizio Dapor a, Giovanni Garberoglio a, Sergey Fanchenko b a European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK) and Trento Institute for Fundamental Physics and Applications (TIFPA-INFN), via Sommarive 18, I-38123 Trento, Italy b National Research Center ‘‘Kurchatov Institute’’, Kurchatov Square 1, 123182 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 25 June 2014 Received in revised form 9 October 2014 Accepted 29 November 2014 Available online xxxx Keywords: Inelastic scattering Electron energy loss Monte Carlo Plasmon energy dispersion Plasmon damping dispersion
a b s t r a c t A position-dependent Inverse Inelastic Mean Free Path (IIMFP), calculated according to the Chen–Kwei theory of inelastic scattering, is used to obtain Reflection Electron Energy Loss Spectra (REELS) by a Monte Carlo approach. The basic ingredient of the theory is the energy-loss and momentum-transfer dependent dielectric function, for which we use a plasmon-pole approximation to the Lindhard dielectric function. The dependence on the momentum transfer enters by assuming a proper dispersion relation for the energy loss. Experiments reveal however that, beside energy loss, also plasmon damping disperses with momentum transfer. By comparing measured and calculated REEL spectra, we explore the role of damping dispersion for a quasi-free-electron material like silicon. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Within the dielectric framework, all information on electronic excitations initiated by external perturbations (e.g. photons or electrons) is included in the material dielectric function eðx; qÞ x and momentum hq transferred to which depends on energy h target electrons. Calculating the dielectric function from first principles is however not feasible in most instances, so that extensive use is made of semi-empirical models which exploit the direct connection existing between the bulk Energy Loss Function (ELF), Im eð1 x;qÞ , and measurable properties of materials, in particular optical properties. The latter probe the energy dependence of electron excitations at zero momentum transfer and make the basis of so-called ‘‘optical-data’’ models. However, to describe electronic excitations (e.g. electron-induced) where a non-negligible momentum is transferred to the material, one needs to go beyond the ‘‘optical limit’’. Extension algorithms have been proposed [1–3] to extrapolate ‘‘optical-data’’ models into the non-zero momentum transfer region. The extension is performed by introducing suitable dispersion relations for all or some of the parameters entering the dielectric function, essentially plasmon energy and damping. Among the proposed models, we have recently considered [4,5] a plasmon-pole approximation to the Lindhard dielectric function where extension to non-zero momentum transfer was ⇑ Corresponding author. Tel.: +39 0461 314483. E-mail address: [email protected] (L. Calliari).
obtained by accounting for plasmon energy dispersion. Such a dielectric function was used within the Chen–Kwei theory [6,7] to calculate REEL spectra by a Monte Carlo approach. Examining several prescriptions for the plasmon energy dispersion, we concluded that inclusion of a proper dispersion relation is crucial for a satisfactory agreement between measured and calculated REEL spectra for a quasi-free electron material like silicon [5]. No dispersion was introduced however for plasmon damping, though momentum-resolved measurements on nearly-free electron materials (i.e. exhibiting well-defined plasmon peaks) show that damping increases with momentum transfer [8–10]. More specifically, two regimes, separated by so-called critical momentum x qc v Fp (xp = plasma frequency, vF = Fermi velocity), are observed, with damping increasing at a definitely higher rate above the critical momentum where single electron excitations become kinematically allowed. In the present paper, we examine if and how damping dispersion affects REEL spectra. Once more, we will take Si as a casestudy and results will be evaluated by comparing calculated to measured spectra. 2. Experimental Experimental REEL spectra are measured on a silicon wafer which was sputter-cleaned in Ultra High Vacuum by 4 keV Ar+ ions. Sample cleanliness was checked by measuring Auger spectra within a PHI545 instrument equipped with a double-pass cylindrical mirror analyser (CMA) and a coaxial electron gun.
http://dx.doi.org/10.1016/j.nimb.2014.11.106 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106
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L. Calliari et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
REEL spectra are excited by 2 keV electrons within the same instrument. The electron analyser was operated in the constant energy resolution mode. The analyser energy resolution was 0.6 eV, as measured on the Pd Fermi edge of a HeI (hm = 21.2 eV) excited valence band photoemission spectrum, leading to a measured FWHM of 0.9 eV for the Zero Loss Peak (ZLP). Once acquired, REEL spectra are corrected for the (previously measured) energy dependence (E0.9) of the analyser transmission function. Details of the CMA geometry are discussed elsewhere [5]. 3. Theory of inelastic scattering A diagram illustrating the basic event of a REEL experiment is shown in Fig. 1. An inelastic scattering event, characterized by scattering angles h (polar) and u (azimuthal), is supposed to take place while the electron enters from the vacuum into the solid.
eðx; hÞ ¼ 1
2
h2 x2 b2E h2 þ hE
2 x2p h 2
2 E h2 þ h2E h x2g þ ihx hc0 þ 2Eðh2 þ h2E Þ
x
ð1Þ
x x x þ ixcq 2 q
hx
E is the energy of incident electrons and hE ¼ 2Ep is the so-called characteristic angle of scattering. Using the dielectric function (2), we calculate the positiondependent Differential Inverse Inelastic Mean Free Path (DIIMFP) according to the Chen–Kwei theory [6,7]:
DIIMFPðE; x; a; zÞ ¼
2 p
2
ð2Þ
2
The angle of surface crossing, a, is the angle between the electron trajectory and the normal to the surface. The distance from the surface is measured on the z-axis, perpendicular to the surface and pointing towards the vacuum. We describe the material response using a plasmon-pole approximation to the Lindhard dielectric function:
eðq; xÞ ¼ 1
x
the measured spectrum, taking into account that the ratio xpg is constrained by 2 the value of the static dielectric function x eð0; 0Þ ¼ 1 þ xp2g ; 11 12 for Si [12,13]. We chose hxp ¼ 15:8 eV and hxg ¼ 4:9 eV [5] (notice that the average energy gap of silicon, as reported in Ref. [14], is 4.8 eV, in very good agreement with our estimate). For the damping parameter, we assume a quadratic dis2 2 persion hcq ¼ hc0 þ d h mq , formally introduced by adding an imagi c0 ¼ 3:2eV. Though nary part to b, i.e. b ¼ 65 EF i hxd. We take h the prescription does not account for the strong damping dispersion above the critical momentum, a quadratic momentum-dependence is nonetheless measured for plasmon widths up to the critical momentum [8–10]. The phenomenological parameter d thus accounts for the increasing probability of plasmon decay as the momentum transfer increases below the critical momentum. Parameter d should range between 0 and 1 [9]. In line with our previous papers, we write the dielectric function as a function of the polar angle of scattering h:
2 g
1 a0 p2 2E
Z
hB
dh 0
Z 2p 0
du
h h2 þ h2E
f ðz; a; h; u; EÞ ð3Þ
where hxp is the plasmon energy ( h = Planck constant divided by 2p) and hxg is a parameter representing the average energy that electrons have to overcome to take part in collective excitations. We therefore expect it to be larger than the minimum energy gap 2 4 2 of the target. The q-dependence enters via x2q ¼ b qm þ h4mq2 (m is the electron mass) and cq . The former term accounts for plasmon energy dispersion, the latter for damping dispersion. For b, characterizing energy dispersion, we take b ¼ 65 EF (EF = Fermi energy) [5], i.e. the value obtained within the Random Phase Approximation xp and (RPA) [11]. Parameters h hxg are chosen by comparison with
where a0 is the Bohr radius. Function f ðz; a; h; u; EÞ, dictated by the dielectric function eðx; hÞ, depends on the distance from the surface, z, on the angle of surface crossing, a, on the energy of electrons, E, and on the scattering angles h and u. Integration over h extends qffiffiffiffiffi up to the Bethe ridge angle hB ¼ hEx [4,5]. Explicit functional forms of f ðz; a; h; u; EÞ for all the relevant cases (ingoing and outgoing electrons, either in the vacuum or in the solid) are found elsewhere [5] together with details on the calculations. As an example, we report here the expression for f ðz; a; h; u; EÞ relative to incoming electrons in the solid (i.e. the case illustrated in Fig. 1 and considered in the examples below). With obvious notation:
sol f in ðz; a; h; uÞ ¼ 2 cosðqz zÞ eQ jzj
eðQ ; xÞ eðQ ; xÞ 1 1 1þ Im eðq; xÞ eðQ ; xÞ þ 1 eðq; xÞ eðz; Q ; xÞ
z
vacuum
ð4Þ
α
where q, qz and Q are, respectively, the modulus of the momentum transfer and its components perpendicular and parallel to the solid surface. In terms of the scattering angles:
solid θ
ϕ Fig. 1. Diagram illustrating an inelastic scattering event. An electron enters the solid from the vacuum, crossing the surface at an angle a. Inelastic scattering, characterized by scattering angles h (polar) and u (azimuthal), takes place in the solid.
q¼
mv h
qz ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2E þ h2
mv ðhE cos a þ h sin a cos uÞ h
ð5Þ ð6Þ
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L. Calliari et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mv 2 2 Q¼ ðhE sin a h cos a cos uÞ þ ðh sin uÞ h
ð7Þ
with m the electron mass. Functions eðQ ; xÞ and eðz; Q ; xÞ are defined as:
1 Q ¼ eðQ ; xÞ p
Z
þ1
1
1 Q ¼ eðz; Q ; xÞ p
Z
0
dqz 1 2 2 02 2 ; x Q þ q02 e Q þ q z z
þ1
1
0
ð8Þ
0
eiqz z dqz 1 2 02 2 ; x ðQ 2 þ q02 Þ e Q þ q z z
ð9Þ
4. Monte Carlo While a detailed treatment of the elastic scattering, requiring numerical quantum mechanical calculations, is described in [15], here we briefly remember a few basic points. We use the Hartree–Fock atomic potential and, to account for solid-state related changes in the outer electronic orbitals, the muffin-tin model is used (enabling the atomic potential to be affected by nearestneighbor atoms). Elastic scattering is treated by calculating the phase shifts. Since the asymptotic behavior of the wave function at a large radial coordinate is known, the phase shifts can be computed by solving the Dirac’s equation for a central electrostatic field up to a large radius for which the atomic potential can be safely ignored (Mott cross section: Relativistic partial wave expansion method). Exchange effects (the incident electron can be captured by an atom with emission of a new electron, the two electrons being indistinguishable) are taken into account. Concerning inelastic scattering, the position dependent DIIMFP, calculated for a distance up to 7 nm from the surface both into the solid and into the vacuum, is used to obtain the position-dependent cumulative probability for energy loss, Pcum, and the position dependent IIMFP:
R hx DIIMFPðE; x; a; zÞdx 0 Pcum ðE; x; a; zÞ ¼ R hx max DIIMFPðE; x; a; zÞdx 0 IIMFPðE; zÞ ¼
1 pa0 E
Z
ð10Þ
define the step-length between successive scattering events. Details on the overall procedure are given in Ref. [5]. Once the step-length is defined, the elastic or inelastic nature of the scattering event, the polar angle for elastic scattering and the energy loss for inelastic scattering are all sampled, according to the usual Monte Carlo recipes, via random numbers from the relevant cumulative probabilities [15]. Inelastic events are assumed to change only the electron energy and not the electron direction. Monte Carlo simulations were performed by generating 128 million trajectories, with electrons starting at 10 nm from the surface and crossing the surface at an angle of 30°. The full threedimensional electron trajectory is generated. Relations to calculate the trajectory direction in the laboratory frame after each scattering event are taken from Ref. [16]. Electron trajectories are followed until the energy loss is larger than 100 eV, because this is the energy loss range probed experimentally, or until the electron exits the surface and it reaches a distance of 10 nm from it. At this point, its outgoing angle aout is tested for detection, i.e. whether it satisfies the condition for being detected by the analyser.
5. Results and discussion Fig. 2 shows the damping parameter as a function of the polar angle of scattering h over a range which includes the critical angle qffiffiffiffiffi hxp hc pffiffiffiffiffiffi and the Bethe ridge angle hB ¼ hEx, the latter calculated 2
EEF
x ¼ 16 eV. The dashed, horizontal line reprefor an energy loss h sents the non-dispersive case, d ¼ 0. Continuous and dotted lines cðhÞ ¼ hc0 þ d2Eðh2 þ h2E Þ, with refer to the dispersive case, h d ¼ 0:5 and d ¼ 1, respectively. The effect of damping dispersion on the Energy Loss Function (ELF) is shown in Fig. 3 where the bulk ELF ¼ Im eð1 x;hÞ is plotted as a function of the energy loss for several scattering angles within the 0 hB ð16 eVÞ range. The upper panel refers to d ¼ 0, the intermediate panel to d ¼ 0:5 and the lower panel to d ¼ 1. For d ¼ 0, the ELF moves to higher energy loss and it decreases in intensity as the scattering angle increases. For d–0, broadening of the ELF is additionally observed. It might be worth mentioning that experimental proof of such a broadening of the ELF as the
hxmax
DIIMFPðE; xÞdx
ð11Þ
0
δ =0
4
where hxmax ¼ 100 eV corresponds to an energy loss range wide enough to account for all processes relevant for determining the IMFP. Functions (10) and (11) are used, respectively, to sample the energy loss associated with inelastic scattering events and to
2 0
45
4
ELF
40
hγ/2π (eV)
35
δ =0.510
20
30
δ =1 10
20
30
10
20
30
2
30 25
0
20
4
15
2
10
θ
5 0 0
θ
c
B
0 0.02
0.04
0.06
θ (rad)
0.08
0.1
Fig. 2. Damping hc vs. polar angle of scattering h. Dashed line: d = 0; continuous line: d = 0.5; dotted line: d = 1. The critical angle, hc, and the Bethe ridge angle, hB (16 eV), are marked by arrows.
0
40
ENERGY LOSS (eV) Fig. 3. Bulk ELF as a function of the energy loss and for the scattering angle h increasing from 0 to 0.08 rad in steps of 0.02 rad. Each panel refers to a different choice for d in the damping dispersion.
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L. Calliari et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
Fig. 5 compares Monte Carlo spectra (continuous curves) to the measured spectrum (dotted curve). Spectra refer to 2 keV electrons and they are all normalized to a common area of the ZLP. On moving from the upper to the lowest panel, spectra are calculated for parameter d increasing from d ¼ 0 to d ¼ 1. Hardly any change can be observed in the spectra (not shown here) as long as d < 0:1 0:2. For greater d, spectral changes mimic changes observed for the DIIMFP in Fig. 4. In particular, intensity shifts towards low energy loss and the surface plasmon gains in intensity over the bulk one. Though one could question the assumed momentum-dependence for damping, it is nonetheless clear that:
z=0nm
0.02
DIIMFP (eV nm-1)
0.01 0 0.02
10
20
30z=-0.5nm
10
20
30z=-1nm
0.01 0 0.02
Neglecting damping dispersion has no significant effect on calculated spectra as long as d < 0:2. Calculated spectra are not brought into better agreement with measured spectra by accounting for damping dispersion; on the contrary, they deviate more from the experiment.
0.01 0
0
10
20
30
40
50
ENERGY LOSS (eV) Fig. 4. DIIMFP calculated for 2 keV ingoing electrons in the solid at variable distance z from the surface. In each panel, different curves correspond to different d values: continuous: d ¼ 0; dashed: d ¼ 0:5; dotted: d ¼ 1.
δ =0
0.1
INTENSITY (eV -1)
0.05 0 0.1
10
20
30
40 δ =0.5 50
10
20
30
40 δ =1 50
10
20
30
40
0.05 0 0.1 0.05 0
0
50
60
ENERGY LOSS (eV) Fig. 5. Comparison between experimental (dotted) and Monte Carlo (continuous) spectra. All spectra are normalized to a common area of the zero loss peak. Spectra in different panels are calculated for different values of the damping dispersion parameter d.
momentum transfer increases is found not only in solids [8–10], but also, for instance, in liquid water [17]. Fig. 4 shows the DIIMFP calculated at variable depth, z, within the solid for ingoing electrons of energy 2 keV. The distance from the surface increases on moving from the upper to the lowest panel. In each panel, continuous, dashed and dotted lines refer to calculations performed, respectively, with d ¼ 0, d ¼ 0:5 and d ¼ 1 in the damping dispersion relation. Accounting for damping dispersion results in a decrease in intensity at the high energy loss tail of the DIIMFP and, at the same time, in a shift of intensity from the bulk to the surface plasmon. In any case, the marked effects observed on the momentum-resolved ELF in Fig. 3 turn into less striking effects on the DIIMFP. We remember that the latter is obtained, according to Eq. (3), by integrating function f ðz; a; h; u; EÞ over the whole momentum transfer range after multiplication by a factor which disfavours the contribution of high momenta.
Alternative reasons should therefore be considered to explain the still existent disagreement between calculated and measured spectra. According to the upper panel of Fig. 5, calculations tend to underestimate the line-width (or, which is the same, overestimate the lifetime) of surface plasmons, suggesting that not all the relevant plasmon decay mechanisms are taken into account. Semi-empirical models (like the plasmon-pole approximation to the dielectric function) assume plasmons to be excited in a freeelectron–gas where the only decay channel is the creation of electron–hole pairs. For this decay to take place, energy and momentum conservation requires that the momentum transfer exceeds a critical momentum. Plasmons are damped however even at zero momentum transfer, the required momentum being supplied by the lattice. To cope with this observation, semi-empirical models incorporate the role of the lattice in a phenomenological way via the damping parameter c which is thus intended to account for damping due to several lattice effects in the bulk, such as interband transitions and scattering of electrons with phonons, lattice defects or impurities. Beside these damping mechanisms however, it has long been recognized [9,18–22] that physical boundaries represent themselves an additional source for damping. Plasmon line-widths broaden in fact on moving from crystalline to polycrystalline materials or from bulk materials to small clusters, and broadening increases with decreasing size of crystallites or clusters. If scattering at surfaces and interfaces provides an additional decay channel for plasmons, damping should depend on space in the surface region in the same way as it is taken to depend on crystal or cluster size [19–21]. However, not only is the mere presence of a boundary crucial in this context, but also are features of the boundary itself. For instance, the crucial role of surface roughness with respect to plasmon excitation and decay has long been known [23] and its effect on REEL spectra was demonstrated [24,25]. Recently, roughness was included in the calculation of REEL spectra [26], showing that good agreement could be obtained with experiments. 6. Conclusions Within the dielectric response framework, REEL spectra were synthesized for a nearly-free electron material like Si. The Chen– Kwei theory of inelastic scattering was used to obtain the position-dependent IMFP needed to account for surface excitations. The main ingredient of the theory is the energy-loss and momentum-transfer dependent dielectric function, for which a plasmonpole approximation to the Lindhard dielectric function was used. Dependence on the momentum transfer was incorporated by accounting for dispersion in the energy and damping of plasmon excitations. While the effects of energy dispersion on REEL spectra
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L. Calliari et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
were characterized previously, here we examined the effects of a quadratic dependence of damping on the momentum transfer. Results were evaluated by comparing spectra calculated by the Monte Carlo method to measured spectra. We found that REEL spectra are unaffected if damping dispersion is low. However, as damping dispersion becomes stronger calculated spectra deviate more and more from the measurements, a suggestion that the dispersion strength is overestimated. We thus concluded that, while energy dispersion is crucial to satisfactorily model REEL spectra, damping dispersion can safely be neglected. Our calculated spectra still fail at reproducing the width of surface plasmons. We discussed possible reasons for such disagreement, leaving to future work an investigation of this point. Acknowledgements This work was supported by NATO (Belgium) through Collaborative Linkage Grant CBP.EAP.CLG.984158 and by Istituto Nazionale di Fisica Nucleare (INFN, Italy) through the ‘Supercalcolo’ agreement with FBK. Computer simulations were performed on the KORE cluster at Fondazione Bruno Kessler. References [1] R.H. Ritchie, A. Howie, Phil. Mag. 36 (1977) 463–481. [2] Z.-J. Ding, R. Shimizu, Scanning 18 (1996) 92–113. [3] D. Emfietzoglou, I. Abril, R. Garcia-Molina, I.D. Petsalakis, H. Nikjoo, I. Kyriakou, A. Pathak, Nucl. Instr. Meth. B 266 (2008) 1154–1161.
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Please cite this article in press as: L. Calliari et al., Reflection Electron Energy Loss Spectra beyond the optical limit, Nucl. Instr. Meth. B (2014), http:// dx.doi.org/10.1016/j.nimb.2014.11.106