The article information
 Yang Sun, Huan Xu, Bo Da, Shifeng Mao, Zejun Ding
 孙阳, 徐欢, 达博, 毛世峰, 丁泽军
 Calculations of EnergyLoss Function for 26 Materials
 26种材料能量损失函数的计算研究
 Chinese Journal of Chemical Physics, 2016, 29(6): 663670
 化学物理学报, 2016, 29(6): 663670
 http://dx.doi.org/10.1063/16740068/29/cjcp1605110

Article history
 Received on: May 16, 2016
 Accepted on: May 23, 2016
When electrons transport in a solid, the inelastic scattering is a fundamental process. Therefore, for quantitative surface chemical analysis by surface electron spectroscopy techniques, such as Xray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES), the physical modeling of electron inelastic scattering is essential. The electron inelastic interaction with a sample is closely related to the energyloss function which determines the probability of inelastic scattering event, the energyloss distribution and the scattering angular distribution [1]. Energyloss function, defined as Im[1/
In this work, we first briefly describe the DrudeLindhard model of energyloss function and the Monte Carlo method for REELS simulation. Then we present the fitting calculations of energyloss function for 26 materials based on the experimentally measured optical data. We validate the fitting results with oscillator strength sum rule and perfectscreeningsum rule, respectively. For the application of the fitting results, one example is shown by using the fitted energyloss function to simulate REELS spectrum and to compare with experimental measurement.
Ⅱ. THEORY A. Energy loss functionThe dielectric functional theory [11] is employed to describe the electron inelastic interaction inbulk materials by:
$\varepsilon (q, \omega ) = 1 +\frac{{{\omega _\textrm{p}}^{\rm{2}} }}{{{\omega _q}^{\rm{2}}  {\omega _\textrm{p}}^{\rm{2}}  \omega \left( {\omega + i\gamma } \right)}}$  (1) 
where
${\omega _q}^{\rm{2}} = {\omega _\textrm{g}}^{\rm{2}} + {\omega _\textrm{p}}^{\rm{2}} + \beta ^2 q^2 + \frac{q^4}{4}$  (2) 
where the energy
Therefore, the bulk energyloss function can be decomposed into Nterms of DrudeLindhard model energyloss function:
${\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( {q, \omega } \right)}}} \right] = \sum\limits_{i = 1}^N {a_i {\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( {q, \omega ;\omega _{\textrm{p}i}, \gamma _i } \right)}}} \right]}$  (3) 
where the
${\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( \omega \right)}}} \right] = \sum\limits_{i = 1}^N {a_i {\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( {0, \omega ;\omega _{\textrm{p}i}, \gamma _i } \right)}}} \right]}$  (4) 
This implies the core electron excitations in a solid are also treated as free electrons in the spirit of a statistical model [12], and described by a DrudeLindhard dielectric function for a plasmon pole with finite damping:
$\begin{align} & {{\varepsilon }_{i}}=\varepsilon (q, \omega ;{{\omega }_{\text{p}i}}, {{\gamma }_{i}})= \\ & 1+\frac{{{\omega }_{\text{p}i}}^{2}}{{{\beta }^{2}}{{q}^{2}}+{{q}^{4}}/4\omega \left( \omega +i{{\gamma }_{i}} \right)} \\ \end{align}$  (5) 
while the constant
Thus, we can fit the optical data in the optical limit (q=0) with Nterm analytic form equation:
${\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( \omega \right)}}} \right] = \sum\limits_{i = 1}^N {a_i \frac{{{\omega _{\textrm{p}i}}^2 \gamma _i \omega }}{{\left( {\omega ^2  {\omega _{\textrm{p}i}}^2 } \right)^2 + \left( {\gamma _i \omega } \right)^2 }}}$  (6) 
Then according to Ritchie and Howie's scheme [2], we make an extrapolation from optical limit to other momentum transfers, and derive Im
We validate our fitting results with two widely used sum rules, the oscillator strength sum rule and the perfectscreening sum rule, which are limiting form of the KramersKronig integral [13]. The fsum rule is given by:
$Z_{{\rm{eff}}} {\rm{ = }}\frac{{\rm{2}}}{{\pi {\Omega _\textrm{p}}^2 }}\int_0^{\omega _{\max } } {\omega {\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( \omega \right)}}} \right]{\rm{d}}\omega }$  (7) 
where
The KramersKronig relations lead to perfectscreeningsum rule given by:
$P_{{\rm{eff}}} = \frac{2}{\pi }\int_0^{\omega _{\max } } {\frac{1}{\omega }} {\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( \omega \right)}}} \right]{\rm{d}}\omega + {\mathop{\rm Re}\nolimits} \left[{\frac{1}{{\varepsilon \left( 0 \right)}}} \right]$  (8) 
For conductors, Re[
$P_{{\rm{eff}}} = \frac{2}{\pi }\int_0^{\omega _{\max } } {\frac{1}{\omega }} {\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( \omega \right)}}} \right]{\rm{d}}\omega + \frac{1}{{n^2 \left( 0 \right)}}$  (9) 
where
To simulate electron scattering in a solid, we need to deal with electron elastic and inelastic scattering process. Mott's crosssection [16] is employed for the treatment of electron elastic scattering,
$\frac{\text{d}\sigma }{\text{d}\Omega }={{\left f\left( \vartheta \right) \right}^{2}}+{{\left g\left( \vartheta \right) \right}^{2}}$  (10) 
$\begin{align} & f(\vartheta )=\frac{1}{2ik}\sum\limits_{l=0}^{\infty }{[}(l+1)({{\text{e}}^{2i{{\delta }_{l}}^{+}}}1)+ \\ & l\left( {{\text{e}}^{2i{{\delta }_{l}}^{}}}1 \right)]{{P}_{l}}\left( \cos \vartheta \right) \\ \end{align}$  (11) 
$g\left( \vartheta \right)=\frac{1}{2ik}\sum\limits_{l=1}^{\infty }{\left( {{\text{e}}^{2i{{\delta }_{l}}^{+}}}+{{\text{e}}^{2i\delta _{l}^{}}} \right)}P_{l}^{1}\left( \cos \vartheta \right)$  (12) 
where
The differential inverse inelastic mean free path (DIIMFP) is represented in the dielectric theory as:
$\frac{{{\rm{d}}^2 \lambda _{\textrm{in}}^{  1} }}{{{\rm{d}}\left( {\hbar \omega } \right){\rm{d}}q}} = \frac{1}{{\pi a_0 E}}{\mathop{\rm Im}\nolimits} \left[{\frac{{1}}{{\varepsilon \left( {q, \omega } \right)}}} \right]\frac{1}{q}$  (13) 
where
We have also considered the surface electronic excitation. The surface dielectric function
$\frac{1}{{{\varepsilon }_{s}}\left( {{\mathbf{q}}_{}}, \omega \right)}2=\sum\limits_{i=1}^{N}{{{a}_{i}}\left[\frac{1}{{{\varepsilon }_{\text{s}i}}\left( {{\mathbf{q}}_{}}, \omega \right)}2 \right]}$  (14) 
$\frac{1}{{{\varepsilon }_{\text{s}i}}\left( {{\mathbf{q}}_{}}, \omega \right)}=1+\frac{{{q}_{}}}{\pi }\int_{\infty }^{\infty }{\frac{\text{d}{{q}_{\bot }}}{{{q}^{2}}{{\varepsilon }_{i}}\left( \boldsymbol{q}, \omega \right)}}$  (15) 
Based on the theory of specular surface reflection model [18, 19], we have presented the surface response function and the electron selfenergy previously [5, 6, 7]. By using a vanishing surface potential and a fastelectron approximation, the randomphaseapproximation selfenergy of a system which is inhomogeneous in the zdirection is obtained in terms of the bulk dielectric function of solid, for the case of an electron moving toward the surface from vacuum side [20] and from the solid side [6], and for the case of an electron in the vacuum and in the solid, respectively, as follows:
$\Sigma \left( {z\left \omega \right.} \right) = \left\{ \begin{gathered} {\Sigma _1}\left( {z\left \omega \right.} \right),\;\;\left( {z > {\text{0}},0.2\;\;{\nu _ \bot } < {\text{0}}} \right) \hfill \\ {\Sigma _{\text{1}}}\left( {z\left \omega \right.} \right) + {\Sigma _{\text{2}}}\left( {z\left \omega \right.} \right),\;\;\left( {z > {\text{0}},\;\;{\nu _ \bot } > {\text{0}}} \right) \hfill \\ {\Sigma _b}\left( \omega \right) + {\Sigma _i}\left( {z\left \omega \right.} \right) + {\Sigma _s}\left( {z\left \omega \right.} \right) + \hfill \\ {\Sigma _{i  s}}\left( {z\left \omega \right.} \right),\;\;\left( {z < {\text{0}},\;\;{\nu _ \bot } < {\text{0}}} \right) \hfill \\ {\Sigma _b}\left( \omega \right) + {\Sigma _i}\left( {z\left \omega \right.} \right) + {\Sigma _s}\left( {z\left \omega \right.} \right), \hfill \\ \left( {z < {\text{0}},\;\;{\nu _ \bot } > {\text{0}}} \right) \hfill \\ \end{gathered} \right.$  (16) 
where
$\sigma \left( {\omega \left {E, \alpha, z} \right.} \right) =  \frac{2}{\nu }{\rm{Im}}\left\{ {\sum {\left( {\omega \left {\alpha, z} \right.} \right)} } \right\}$  (17) 
where α is the angle between the velocity vector and surface normal.
With the bulkand surfaceinelastic crosssection, the DIIMFP, and the Mott's elastic crosssection, we have performed Monte Carlo simulation to calculate the REELS spectrum for the considered materials. By using a Monte Carlo method [4], the flight length s between successive individual scattering events is sampled from an exponential probability distribution:
$f\left( s \right)\text{=}\sigma \left( s \right)\text{exp}\left\{ \int_{0}^{s}{\sigma \left( s' \right)\text{d}s'} \right\}$  (18) 
where the total crosssection
As the interband transitions happen at the low to intermediate energy region and innershell ionization locates at high energy region, a wide photon energy range is necessary to fit a complete optical energyloss function. The energy range has been shown in Table Ⅰ. For most materials, we take the data from 10^{1} eV to 10^{5} eV. But some are in a smaller range due to the lack of experimental data, especially for semiconductors. The experimental data are mostly compiled in Palik's handbooks [21]. As some of the optical data are measured by different groups under different conditions, we validate the data by using the sum rules mentioned above. The selected references of each material at various energy ranges are listed in detail in Table Ⅰ. We also applied a linear interpolation for the serious missing data as shown in Table Ⅰ. To give an accurate fitting of these selected data, we have to use a sufficiently large number of DrudeLindhard terms. It is obvious that the set of parameters is not unique. We also allow a negative value of
Figure 1 shows the fitting results of bulk energyloss function Im[
Figure 2 shows the sumrule validation for Al, Si, and SiO_{2}. Eq.(9) is used for Al, while Eq.(10) for semiconductors Si and SiO_{2}. It also shows a good agreement between the fitting results and experimental data. As the maximum values of energy loss in each figures are above 10^{4} eV, the energy range is large enough to include various excitations from all innershells. Note that the phonon excitation mentioned above makes a remarkable contribution in the perfectscreeningsum rule for silicon oxide in Fig. 2(f). The sumrule validations of all the fitted material are shown in Table Ⅱ. The theoretically ideal
With the fitted energy loss function, we can simulate REELS spectrum with a Monte Carlo method [4, 8, 23, 24]. Here we compare the simulated REELS spectrum with an experimental measurement [25, 26] for silver. The REELS spectra were measured with a cylindrical mirror analyzer (CMA) equipped with a coaxial electron gun. In this system, signal electron current was measured with a Faraday cup. The measurement was performed at a primary energy of 500 eV and for normal incidence of electrons. CMA only detects those electrons emitted from sample surface into a solid corn by the angular aperture from 36.3° to 48.3°. Hence, the Monte Carlo simulation has been performed to exactly match the experiment by counting only those reflected electrons that are detected by the CMA. Figure 3 shows the comparison between the simulation and the experiment. The inelastic scattering peaks have been normalized to the elastic peak for both simulated and experimental data. Two peaks at 4.0 and 7.4 eV correspond to surface and bulk excitation, respectively. One can see that the simulation based on the fitted energy loss function well captured both excitation phenomena and obtained a good agreement with experimental measurement.
Ⅳ. CONCLUSIONIn summary, we have performed a systematic fitting procedure of energyloss function for 26 materials based on the dielectric functional theory by using a finite number of DrudeLindhard terms. We have also evaluated experimental data and fitting data with fsum rule and perfectscreeningsum rule. It shows that the fitting procedure is accurate and the error of sum rules is mostly caused by the experimental data. The optical data are then extended to the qdependent energyloss function by assuming a plasmon dispersion. Thus, the surface excitation can also be described by using the derived surface dielectric function. To verify the present fitting data, we have performed a Monte Carlo simulation of the REELS spectrum for silver, and compared it with an experimental measurement. The line shapes of the simulated and experimental spectra agree with each other reasonably well for both surface and bulk features. Our fitting parameter database for energy loss function can be used in various problems of inelastic scattering in electron transport process in solids for application to surface chemical analysis by surface electron spectroscopy.
Ⅴ. ACKNOWLEDGMENTSThis work is supported by the National Natural Science Foundation of China (No.11274288 and No.11574289).
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