Introduction

The inelastic interaction of a fast electron with matter results in excitation of the internal degrees of freedom of the sample, or, in other words, in a change of the target state. An internal motion can be regarded as either a single particle or a quasiparticle characterized by its momentum and energy. In the course of inelastic scattering the energy and momentum of the electron are transferred to the internal motion, that is, excitation or ionization of discrete atomic levels and generation of plasmons and electron-hole pairs take place.

There are various approaches to description of inelastic scattering process. Perhaps the best approach should use dielectric function which, as well known, characterizes specific excitation processes in a sample (Pines, 1964)[14]. The dielectric function \varepsilon(q,\omega) can provide a detailed information about energy loss cross-section and scattering-angular distribution for inelastic scattering of electrons. These values have been calculated first with a help of Lindhard dielectric function (Ganachaud and Cailler, 1979; Cailler and Ganachaud, 1990)[10,3], describing the plasmon excitation and electron-hole pair production. Unfortunately, the Lindhard dielectric function of a free-electron gas in random phase approximation (Fetter and Walecka, 1971)[9] is only valid for a very restricted class of materials (so-called free electron metals) being hardly applicable to other materials, such as transition and noble metals, for which optical dielectric data have complicated structure due to interband transitions. Theoretical calculation of the q-dependent dielectric function \varepsilon(q,\omega) is difficult and it has been evaluated numerically (Walter and Cohen, 1972; Sramek and Cohen, 1972; Nizolli, 1978)[19,16,12] only for selected q's and the first few reciprocal lattice vectors, using a realistic band structure data for some simple metals and semiconductors (Sturm, 1982)[17]. Therefore, it is impossible to use the dielectric function derived from band structure calculation for the evaluating the results of inelastic scattering of electrons.

A way out of the situation is to use optical dielectric data available experimentally from optical methods and electron energy loss spectroscopy (Egerton, 1986)[8]. According to Penn (1987)[13], the q-dependent electron energy loss function related to \varepsilon(q,\omega) by

F(q,\omega)={\rm Im}\left(\frac{1}{\varepsilon(q,\omega)} \right)

can be obtained by extrapolating the optical dielectric function \varepsilon(0,\omega) into the region of finite wavelengths (q > 0). This approach was developed by Ding and Shimizu (1989)[5] and Ashley (1988)[1] to calculate the energy loss and scattering-angular distribution in electron inelastic scattering. It has been shown that the method yields the Bethe stopping powers at high energies and the calculated electron mean free paths fit experimental data in a wide energy region for a set of materials. The differential inelastic mean free paths (IMFP) obtained from dielectric data have been successfully used in calculation of X-ray depth profiles (Ding and Wu, 1993)[7], the background in Auger electron spectroscopy (Ding et. al, 1994)[6], and reflected-electron energy loss spectrum (Tokesi et al., 1995)[18].

In this approach, the electron energy loss function can be treated as being composed of multi-modes of a localized plasmon. The expansion coefficients are determined from experimental optical data. The q-dependence is introduced through plasmon dispersion relation. This relation can be chosen in various forms. The most simple one,

\omega_p(q)=\omega_p+\hbar q^2/2m,

where \omega_p is the plasmon energy, was proposed by several authors (Ritchie and Howie, 1977; Kwei and Tung, 1986; Ashley, 1991)[15,11,2] and was used by Ding (1990)[4] for IMFP calculations. Ding showed that the differences in IMFP's and stopping power, caused by the choice of dispersion relation, are insignificant; however, the true secondary electron yield is somewhat sensitive to this choice.

General relations

Data on differential and total probabilities of inelastic scattering of fast electrons in solids per unit length (differential and total inverse inelastic mean free paths) calculated according to the dielectric formulation from optical data are presented in this archive.

The differential probability of inelastic scattering per unit length d W(E, Q)/d Q depends on the composition of a solid, initial energy of electron E and energy loss Q. The total probability of inelastic scattering per unit length (inverse inelastic mean free path)

W(E)=\int\limits_0^{Q_{\rm max}}\frac{{\rm d} W(E, Q)}{{\rm d}Q} {\rm d} Q

depends on the composition of the solid and the initial energy of electron E. Here Qmax is the maximum energy loss for a given solid at given E. If Fermi level lies inside the gap and the energy of the bottom edge of the conduction band counted from the Fermi level is EF, then Qmax = E–EF else Qmax = E.

The following data are also included in the archive. The first set represents the function

R(q,E)=\int_0^{Q_{\rm max}}\frac{{\rm d} W(E, Q)}{{\rm d} Q} \frac{{\rm d} Q}{W(E)}

which gives the probability of inelastic interaction with energy loss below qQmax.

The second set is connected with the representation of the inverse dependence q(R, E) which is widely used in Monte-Carlo simulations of electron transport in solids. To approximate this dependence at fixed E, the 5-parametric analytical function

F(a,b,g,p,m,R)= \left{ \begin{array}{lc} F_1(a,b,g,p,R) & m=0 \\ F_2(a,b,g,p,m,R) & m \ne 0 \\ \end{array} \right.

has been proposed. The parameters a, b, g, p and m take into account the influence of E on the form of R-dependence of q. The function F1(R) (hereinafter the above parameters are omitted in the argument string) is defined as

F1(R) = 1–Y(1–R)g (1–Y(1–R)),

where

Y(R)=R \left( a+\frac{1-a}{1-p} \frac{1-(1-r^b)^p-pR^b}{R^b} \right).

The function F2(R) is given by

F_2(R)=\frac{Z(R)}{Z(1)},

where

Z(R) = (G(R)+H(R)–H(0)) arctan(5000 c R),

and the dependences G(R), H(R) and parameter c are specified, in their turn, by the formulae

G(R)=\left(1+ \left( \frac{cR}{g} \right)^m \right)^{-p/m}b(cR)^p,

H(R)=\left( \left| {1-g} \right|^{-|m|} + \left( {1-cR} \right)^{-|m|} \right)^{1/|m|},

c=\frac{1}{1+a}.

The relative error of approximation is described by the expression

\delta=\int_0^1 \frac{|F(R)-q(E,R)|}{q(E,R)} {\rm d} R.

The data presented in the archive are intended to be used for material diagnostics. This fact determines the choice of the energy range for which the calculations were performed. The initial energies of fast electrons were taken from some eV up to a value of 30 keV. The energy loss Q varies from 0 up to Qmax. All the energies are measured from the Fermi level. The values of d W(E, Q)/d Q and W(E) are given in (cm  eV)–1 and cm–1, respectively.

The results of computation are presented in this archive in numerical (in downloadable data files) and graphic forms. In Data files section the file-naming convention and the file structure are described in detail. It will be helpful to read Content and use of the archive before viewing and retrieving the data.

References

  1. Ashley J. C. Interaction of low-energy electrons with condensed matter: Stopping powers and inelastic mean free paths from optical data, J. Electron Spectrosc. Related Phenomena, 46, 199–214 (1988).
  2. Ashley J. C. Energy loss probabilities for electrons, positrons and protons in condensed matter, J. Appl. Phys., 69, 674–678 (1991).
  3. Cailler M. and Ganachaud J. P. Secondary electron emission from solids II. Theoretical description, In Fundamental Electron and Ion Beam Interactions with solids for Microscopy, Microanalysis and Microlithography (Eds. Schou J., Kruit P., Newbury D.E.), Scanning Microscopy Supplement 4, Scanning Microscopy International, 81–110, Chicago, 1990.
  4. Ding Z.-J. Fundamental studies on the interactions of kV electrons with solids for applications to electron spectroscopies, PHD Thesis, Osaka University, Japan, 1990.
  5. Ding Z.-J. and Shimizu R. Inelastic collision of kV electrons in solids, Surf. Sci., 222, 313–331 (1989).
  6. Ding Z.-J., Shimizu R. and Goto K. Background formation in the low energy region in Auger electron spectroscopy, J. Appl. Phys., 76, 1187–1195 (1994).
  7. Ding Z.-J., Wu Z. Q. A Comparison of Monte Carlo simulation of electron scattering and X-ray production in solids, J. Phys., D, 507–516 (1993)
  8. Egerton R. F. Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum, New York, 1986.
  9. Fetter A. L. and Walecka J. D. Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971.
  10. Ganachaund J. P. and Cailler M. A Monte Carlo calculation of secondary electron emission of normal metals, Surf. Sci., 83, 498–530 (1979).
  11. Kwei C. M. and Tung C. J. Stopping power of semiconducting III-V compounds for low-energy electrons, J. Phys. D, 19, 255–263 (1986).
  12. Nizzoli F. A model calculation of the dielectric function in trigonal Se and Te with local-field corrections included, J. Phys. C, 11, 673–683 (1978).
  13. Penn D. R. Electron mean free path calculations using a model dielectric function, Phys. Rev. B, 35, 482–486 (1987).
  14. Pines D. Elementary Excitation in Solids, Benjamin, New York, 1964.
  15. Ritchie R. H. and Howie A. Electron excitation and the optical potential in electron microscopy, Phyl. Mag., 36, 463–481 (1977).
  16. Sramek S. J. and Cohen M. L. Frequency and wave-vector-dependent dielectric function for Ge, GaAs and ZnSe, Phys. Rev. B, 6, 3800–3804 (1972).
  17. Sturm K. Electron energy loss in simple metals and semiconductors, Adv. Phys, 31, 1–64 (1982).
  18. Tokesi K., Nemethy A., Kover L., Varga D. and Mukoyama T. Modeling of electron scattering in thin manganese films on silicon by Monte Carlo methods, Surf. Sci., 1995.
  19. Walter J. P. and Cohen M. L. Frequency- and wave-vector-dependent dielectric function for Silicon, Phys. Rev. B, 5, 3101–3110 (1972).

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