Abstract

1. Introduction

Values of electron inelastic mean free paths (IMFPs) in solids are of fundamental physical importance for describing the inelastic scattering of electrons in different materials. The IMFPs also define the surface sensitivity of Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS) and are used in quantitative applications of these techniques.^[1,2]

We have previously calculated IMFPs for many solid materials over a wide energy range^{.[3,4,5,6,7,8]} Initially, we reported IMFPs for 50 eV to 2,000 eV electrons for 27 elemental solids^,[3,4] 15 inorganic compounds^,[5] and 14 organic compounds.^[6] These IMFPs were calculated from experimental optical data with the non-relativistic full Penn algorithm (FPA)^[9] for electron energies under 200 eV and the single-pole approximation or simple Penn algorithm (SPA)^[9] for higher energies. We analyzed these calculated IMFPs with the Bethe equation for inelastic scattering of electrons in matter^[10] to develop an IMFP predictive formula (designated TPP-2M).^[6] The TPP-2M equation could be used to estimate IMFPs in other materials, again for energies between 50 eV and 2,000 eV, the energy range of interest for many AES and XPS experiments.

In recent years, there has been growing interest in XPS and related experiments performed with X-rays of much higher energies for both scientific and industrial purposes. We then published IMFPs for 41 elemental solids for electron energies up to 30 keV^.[7] Since there is also a need for IMFPs in transmission electron microscopy (TEM), we also calculated IMFPs in 41 elemental solids for energies up to 200 keV with a relativistic version of the FPA.^[8] In addition, we developed a relativistic version of the TPP-2M equation that provides reasonable IMFP estimates for energies between 50 eV and 200 keV. The root-mean-square (RMS) deviation between the estimated IMFPs from the TPP-2M equation and the directly calculated values was 11.9 % for the group of 41 elemental solids.^[8] This RMS deviation was similar to that found (10.2 %) in a similar comparison for our original group of 27 elemental solids for the 50 eV to 2 keV energy range.^[6]

In this paper, we extend our previous calculations^[5] of IMFPs for 15 inorganic compounds and electron energies between 50 eV and 2,000 eV. We report here new calculations of IMFPs with the relativistic FPA for 42 inorganic compounds (AgBr, AgCl, AgI, Al₂O₃, AlAs, AlN, AlSb, cubic BN (c-BN), hexagonal BN (h-BN), CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF₂, MgO, NaCl, NbC_0.712, NbC_0.844, NbC_0.93, PbS, PbSe, PbTe, SiC, SiO₂, SnTe, TiC_0.7, TiC_0.95, VC_0.76, VC_0.86, Y₃Al₅O₁₂, ZnS, ZnSe, and ZnTe) for energies between 50 eV and 200 keV. These inorganic compounds were chosen because they had experimental ELF data below the bandgap energy and the errors in the two sum rules for evaluating ELFs were less than 10 %. Two inorganic compounds that we considered previously ^[5], LiF and Si₃N₄, were omitted in this study because their ELFs gave large errors in the sum rules discussed in Section 3.2. The relativistic TPP-2M formula^[8] must therefore be a more reliable means for determining IMFPs for these two compounds.

The IMFP calculations with the FPA require knowledge of the energy-loss functions (ELFs) of each material, typically for photon energies (or, equivalently, electron energy losses) between 1 eV and 200 keV. Our previous IMFP calculations were based on ELFs obtained from experimental optical data or from electron energy-loss spectroscopy (EELS). However, optical constants and optical ELFs as well as EELS ELFs are lacking for most semiconductor materials for energy losses larger than about 10 eV. We therefore calculated ELFs for the 27 compound semiconductors in our group of 42 inorganic compounds. We utilized the WIEN2k^[11] and FEFF^[12] codes to compute the imaginary part of the complex dielectric constant, typically for photon energies from several eV to 1 MeV, from which we calculated the real part and then the ELF. We note here that Werner et al.^[13] successfully used the WIEN2k code to calculate optical constants and ELFs for 16 elemental metals and one semimetal that agreed satisfactorily with corresponding data obtained from an analysis of reflection EELS data.

We also report comparisons of our calculated IMFPs for the 42 inorganic compounds with IMFPs calculated by Reich et al.^[14], Akkerman et al.^[15], Pandya et al.^[16], Kwei and Li^[17], Kwei et al.^[18], Boutboul et al.^[19], Ashley and Anderson^[20], Dapor and Miotello^[21, 22], and Dapor^[23], and with IMFPs measured by lakoubovskii et al.^[24], Egerton^[25, 26], Meltzman et al.^[27], Gurban et al.^[28], Pi et al.^[29], Krawczyk et al.^[30], Chung et al.^[31], Wang et al.^[32], MacCartney et al.^[33], Lee et al.^[34], and Bideux et al.^[35], Zommer et al.^[36], and Jung et al^.[37]. We also make comparisons between the measured IMFPs and IMFPs from the relativistic TPP-2M equation.

2. IMFP Calculations with the Relativistic Full Penn Algorithm

We utilized the relativistic FPA to calculate IMFPs for energies in the 50 eV to 200 keV range in order to be consistent with our previous IMFP calculations for elemental solids.^[8] The IMFPs were calculated at equal energy intervals on a logarithmic scale corresponding to increments of 10 % from 10 eV to 1 MeV. We will present IMFPs for energies between 10 eV and 50 eV and between 200 keV and 1 MeV in Figures but these results are shown only to illustrate trends.

The relativistic differential cross section (DCS) for inelastic scattering can be expressed as the sum of a longitudinal DCS and a transverse DCS.^[38] Since the transverse DCS can be neglected for electron energies less than about 0.5 MeV,^[39] the relativistic DCS can be written using Hartree atomic units (me=e=ℏ=1, where m_e is the electron rest mass, e is the elementary charge, and ℏ is the reduced Planck constant), as

where ω is the energy loss, c is the speed of light, Q is the recoil energy defined by^[40]

q is the momentum transfer, N is the number of molecules per unit volume, v is the electron velocity, and Im[−1/ε(Q,ω)] is the ELF expressed as a function of energy loss ω and recoil energy Q. Using Eqn (2) and dQ=q/[1+(Q/c2)]⋅dq, the ELF in Eqn (1) can be conveniently written as a function of ω and momentum transfer q.

We write the DCS as

where Im[−1/ε(q,ω)] is the ELF. We will use the Penn algorithm^[9] to evaluate Eqn (3). The dependence of the ELF on ω can be obtained from the measured ELF for each material (typically from optical experiments) while the dependence of the ELF on q can be obtained from the Lindhard model dielectric function.^[41] In this paper, we will show that we can also use calculated ELFs for materials for which there are no measured ELFs over key energy ranges.

The ELF in Eqn (3) can be expressed using the full Penn algorithm as

where εL denotes the Lindhard model dielectric function of a free-electron gas with plasmon energy ωp(=4πn−−−√), n is the electron density, g(ω_p) is a coefficient introduced to satisfy the condition Im[−1/ε(q=0,ω)]=Im[−1/ε(ω)], and Im[−1/ε(ω)] is the measured or calculated ELF. The coefficient g(ωp) is then given by

The ELF from the FPA in Eqn (4) can be described as the sum of two contributions, one associated with the plasmon pole and the other with single-electron excitations:

Details of the evaluations of the plasmon-pole and single-electron contributions were given in a previous paper ^[8].

We calculated IMFPs from Eqns (3) to (6) using the approach of Boutboul et al.^[19] to include the effect of the bandgap energy in semiconductors and insulators. The IMFP, λ, at electron energy T(>Eg+Ev), which is measured from the bottom of the valence band for semiconductors and insulators, can be expressed as ^[8]:

where T′=T−Eg and Eg is the bandgap energy. The integration domain D is determined from the maximum and minimum energy losses and the largest and smallest kinematically-allowed momentum transfers for a given energy T and ω:

where E_v is the width of the valence band for semiconductors and insulators, and q±T′(2+T′/c2)−−−−−−−−−−−√±(T′−ω)[2+T′−ω/c2]−−−−−−−−−−−−−−−−−−√.

Table 1 shows the material-property data used in our IMFP calculations and in our analyses of the ELFs and IMFPs. We show values of the molecular weight M, bulk density ρ (g cm⁻³), number of valence electrons per molecule (N_v), free-electron plasmon energy (E_p), bandgap energy (E_g), and valence-band width (E_v) for each compound. The bandgap energy is also an important material parameter in the use of our TPP-2M predictive equation for IMFPs ^[6]. We chose the median of the E_g values in the literature ^{[42,43,44,45,46,47,48]} for most of our compounds; however, only single values of E_g were available for NaCl ^[49] and Y₃Al₅O₁₂ ^[47]. The E_v values were obtained from the WIEN2k calculations for 27 compound semiconductors and values for the other eight insulators were estimated from the literature: Al₂O₃ ^[50, 51], KBr ^[52], KCl ^[52], MgF₂ ^[53], MgO ^[54], NaCl, ^[52], SiO₂ ^[55], Y₃Al₅O₁₂ ^[56]. For the niobium, titanium, and vanadium carbides, the Ev values correspond to the Fermi energies that were estimated from the literature ^[86, 87].

3. Optical energy-loss functions

We have shown that IMFPs can be calculated with the FPA and SPA from ELFs obtained from experimental optical data (i.e., optical ELFs) for many materials of interest.^[3–8] The ELF is thus the critical material-dependent parameter in our IMFP calculations. Ideally, optical ELFs should be determined from measured optical constants. In many cases, we utilized optical constants from Handbooks^[45–47] although we have occasionally used optical constants derived from analyses of electron energy-loss experiments.^[7] Our choices from sets of optical data were guided by evaluations of the ELFs using the two important sum rules described in the ‘Evaluations of energy-loss functions’ subsection.

Unfortunately, experimental optical constants are generally not available for many inorganic compounds, especially for the 20 eV to 50 eV energy range that is particularly important for the IMFP calculations. As an alternative, we have made first-principles calculations of optical constants using the WIEN2k^[11] and FEFF^[12] codes, as described in the next subsection. These calculations were made for 27 inorganic compounds (c-AgBr, c-AgCl, h– AgI, c– AlAs, h-AlN, c- AlSb, c– BN, h-BN, CdS, h-CdSe, c-CdTe, c– GaAs, h-GaN, c- GaP, c- GaSb, h– GaSe, c– InAs, c– InP, c– InSb, c– PbS, c– PbSe, c– PbTe, c– SiC, c– SnTe, c– ZnS, c– ZnS, and c– ZnTe). These compounds were selected because optical data for photon energies less than 0.1 eV were needed for evaluation of one of the sum rules and were mostly available in the literature.^{[44, 45, 46, 47]} Details of the first-principle calculations of ELFs will be published elsewhere.

Table 2 shows the sources of optical data used in our IMFP calculations. For 15 of the compounds (Al₂O₃, KBr, KCl, MgF₂, MgO, NaCl, NbC_0.712, NbC_0.844, NbC_0.93, SiO₂, TiC_0.7, TiC_0.95, VC_0.76, VC_0.86, and Y₃Al₅O₁₂), optical data from literature sources were available for photon energies up to at least 50 eV.

4. Results