ELECTRON QUASI-ELASTIC PEAK
4.3 Elastic Reflection Coefficient for a Compound
4.3.2 Using Elastic Reflection Coefficient to Determine Electron Inelastic Mean Free Path for a Compound
n
j
j e j
j e j
a j a
Pe
1 , ,
, (4.22)
Applying Eq. (4.22), the j th element is collided by the electron in each elastic scattering event if
k
j k
j
j P R
j
Pe e
1 4 1
1
, for k = 1, 2, 3, …, n. (4.23)
Eq. (4.23) can be used to determine the element which is collided by the electron.
Then the EDCS of the element is taken to find the polar scattering angle after the scattering through the help of Eqs. (4.1) and (4.2). In addition, the average EMFP can be obtained from the individual element EMFPs in the compound according to Eq.
(2.6). Note that we used the relativistic model with free-atom potentials depicted in chapter 2 to calculate the total elastic cross sections and the average EMFPs for compounds. Then the elastic peak spectra of electrons backscattered from a compound can be carried out using the modified MC algorithm. Also, the elastic reflection coefficient for electrons backscattered from a compound can be obtained.
4.3.2 Using Elastic Reflection Coefficient to Determine Electron Inelastic Mean Free Path for a Compound
The method to extract the IMFP from the elastic peak electron spectra is based on the calculation of the elastic reflection coefficient. To determine the IMFP from
coefficient for a compound sample relative to a Ni reference. In this section, the SEPs and various values of depth-independent IMFP are first applied in the modified MC algorithm. Then the calibration curves can be obtained. Besides, using the extended Drude dielectric function, the calculated depth-dependent IMFPs and SEPs were applied to determine the electron elastic reflection coefficient for a compound by means of the modified MC simulations. Subsequently, an effective IMFP for the compound can be extracted by the intersection of the calibration curve and the calculated elastic reflection coefficient.
Figure 4.11 shows the intensity ratio of electrons backscattered from GaAs to those from Ni for a 50oincident angle and 0o~ 90oemission angles (solid circles) and for a 0o(normally) incident angle and 36.3o ~ 48.3oemission angles (solid triangles).
Corresponding experimental data (open circles and triangles) (Krawczyk et al. 1998;
Zommer et al. 1998) are plotted for comparisons. The differences between present results and experimental data are due to the lack of information on experimental configurations.
Figure 4.12 shows the same intensity ratio as a function of depth-independent electron IMFP, or the calibration curve, calculated using the same simulation configurations as those applied to Fig. 4.11. The solid and dashed curves are, respectively, for a 50o incident angle and 0o ~ 90o emission angles and for a 0o incident angle and 36.3o ~ 48.3o emission angles. The symbols are the corresponding results obtained by MC simulations using the depth-dependent electron IMFPs and SEPs. From the intersection of the calibration curve and the simulated intensity ratio, the effective IMFP may be determined. These results of the effective IMFP are plotted in Fig. 4.13 as a function of electron energy for a 50oincident angle and 0o ~ 90o emission angles (solid circles) and for a 0o incident angle and 36.3o ~ 48.3oemission angles (solid triangles). These results are also compared with
Fig. 4.11 The intensity ratio of electrons backscattered from GaAs to those from Ni for a 50o incident angle and 0o - 90o emission angles (solid circles) and for a 0o (normally) incident angle and 36.3o - 48.3o emission angles (solid triangles).
Experimental data of Krawczyk et al. (1998) (open circles) and Zommer et al. (1998) (open triangles) are plotted for comparison.
Fig. 4.12 The intensity ratio of electrons backscattered from GaAs to those from Ni calculated using the depth-independent electron IMFP (abscissa). The solid and dashed curves are, respectively, for a 50o incident angle and 0o- 90oemission angles and for a 0o incident angle and 36.3o - 48.3o emission angles. The symbols are the corresponding results obtained by MC simulations using the depth-dependent electron IMFPs and SEPs.
Fig. 4.13 The effective IMFP as a function of electron energy for a 50o incident angle and 0o- 90oemission angles (solid circles) and for a 0oincident angle and 36.3o - 48.3o emission angles (solid triangles). Also plotted are experimental data of Krawczyk et al. (1998) (open circles) and Zommer et al. (open triangles), and calculated IMFPs for volume excitations (solid curve).
experimental data (open circles and triangles) (Krawczyk et al. 1998; Zommer et al.
1998), and with calculated IMFPs for volume excitations (solid curve). The effective IMFPs determined here are in good agreement with electron IMFPs for volume excitations. This indicates that a depth-independent IMFP is approximately valid due to the compensation of volume and surface excitations inside the solid (Chen and Kwei 1996; Kwei et al. 1998b). Since parameters in the extended Drude dielectric function were fitted mainly for the response of valence electrons (Chen 2002), the lack of inner shell responses caused the deviation between calculated effective IMFPs and experimental values at high energies.
CHAPTER 5 SUMMARY
In this thesis, the elastic and inelastic interactions between electrons and solids were studied theoretically.
For elastic interaction, the non-relativistic and the relativistic approximations with free-atom and solid-atom potentials to determine EDCS and EMFP were described and compared. The calculated results showed that the relativistic elastic-scattering model is more realistic, especially for the solids composed of heavy atoms. In addition, the derivation of the electron total elastic cross sections and total EMFPs for compounds was further made based on elastic cross sections and based on elastic mean free paths of individual elements. These two methods also can determine the probability of each element in a compound being bumped by an electron. Besides, the recoil effect was also discussed for the energy loss of the electron after elastic scattering. It was found that the recoil effect is significant for atoms of low and intermediate atomic numbers.
In the research about inelastic interaction, a new model was constructed based on the dielectric response theory. Formulas were derived to calculate the stopping power, DIIMFP, IMFP and SEP for electrons escaping from and incident into solids.
It was found that the DIIMFP, IMFP and SEP for electrons near surfaces are dependent of the electron’s moving direction and position. These results arise from the number of the surface excitations varying with the electron’s distance to the solid surface. Deep inside the solid, the DIIMFP and IMFP reduced to the values for
conservations of energy and momentum. Moreover, the inelastic-scattering model with the inclusion of retardation effect was constructed for electrons moving parallel to solid surfaces. The calculated results for high-speed electrons revealed that the stopping power and DIIMFP with retardation effect were lower than the stopping power and DIIMFP without retardation effect.
Based on the theories on the elastic and inelastic interactions of electrons with solids, we further established a MC method to simulate the energy distribution of elastic peaks of electrons backscattered from solid surfaces. The energy distribution is due to the Rutherford-type recoil energy losses occurring in elastic scatterings.
The simulated results showed that the recoil energy loss caused the peak energy shift and width broadening in the ideal energy spectrum of an electron elastic peak.
However, the ideal energy spectra did not agree with the experimental data well.
Adjustment of the contributions from spectrometer energy resolution, the energy distribution of the primary electrons and the thermal motion of atoms were included.
It was found that the width broadenings of the energy spectra of elastic peaks were mainly contributed by the spectrometer energy resolution, the energy distribution of the primary electrons and the thermal effect. Regarding the applied model of the thermal effect, it was also discussed that the recoil energy loss should be centered at the most probable value of this loss for vibrating atoms rather than for atoms at rest, and the thermal effect due to multiple elastic scatterings could shift the peak position and widen the peak. More information on both phenomena should be given to study the energy distribution of electron elastic peaks. Besides, the method to extract effective IMFPs from electron elastic peaks was discussed. We obtained the effective IMFPs for GaAs using our MC method. The obtained effective IMFPs were in agreement with the calculated IMFPs for volume excitation due to the compensation of surface excitation and volume excitation inside the solid.
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簡 歷
姓 名：黎 裕 群
性 別：男
出生日期：民國 66 年 10 月 12 日 出 生 地：台灣省花蓮縣
學 歷：
國立交通大學電子工程學系畢業（85 年 6 月～89 年 2 月）
國立交通大學電子研究所碩士班（89 年 2 月～90 年 2 月）
國立交通大學電子研究所博士班（90 年 2 月入學）
論文名稱：準彈性反射電子模擬與分析
The Simulation and Analysis of Quasi-elastically Backscattered Electrons