Adjustments to Energy Spectra of Quasi-elastically Backscattered Electrons

There are two experimental factors that affect the height and the width in the energy distribution of the electron elastic peak detected by a spectrometer (Tóth et al.

1998; Gergely et al. 2001; Sulyok et al. 2001). One is the energy distribution of the primary electron beam emitted from an electron gun. The other is the energy resolution of the analyzer of a spectrometer. To simulate the experimental results of elastic peak electron spectra, these two effects were considered in our MC algorithm for the energy distribution of backscattered electrons.

We assumed that both the energy distribution of the electron beam and the resolution function are Gaussian distributions. Let G ,;₀ be the Gaussian function of a variable  with the mean value  and the standard deviation0 .

Considering the Gaussian energy distribution of the primary electron beam, the energy distribution of quasi-elastically backscattered electrons before the measuring of the spectrometer can be expressed as

 ^{E I E ω E} ^G





h











 , (4.13)

where E₀ is the mean energy of the primary electron beam and  is the standard_g energy deviation of the energy distribution of the electron beam. By taking the further consideration of the energy resolution of the spectrometer, the energy distribution becomes

  ^{E h E'G E E'} ^dE'

H ^







 , (4.14)

where  is the standard energy deviation of the resolution function. Using Eqs._r (4.13) and (4.14), the simulated quasi-elastic electron peak spectra is comparable with experimental data.

In Fig. 4.4, we plot the energy distributions of elastic peaks for electron beams normally incident into Si with 5000 eV mean energy and two different FWHMs, 0.3 eV and 0.5 eV. It can be seen that the spectra are broadened Gaussian peaks. As compared to the primary electron beams, the width broadenings in the two Gaussian peaks are arisen from the recoil width broadening displayed in Fig. 4.1. In addition, the FWHM of the resulting spectrum is wider when  of the electron beamE_g

increases. Here E_g is the FWHM of the energy distribution of the electron beam.

A similar plot is shown in Fig. 4.5 for Au. The FWHM of the resulting spectrum for Au is obviously narrower than that for Si. This is because the recoil width broadening is much smaller for Au. Figures 4.6 and 4.7 show the energy distributions of the elastic peaks for Si and Au, respectively, using different spectrometer resolutions but the same Gaussian distributed normally incident electron beam with 0.3 eV FWHM and 5000 eV mean energy. The spectrometer function was assumed to be a Gaussian distribution with two different FWHMs, 0.3 eV and 0.5 eV. The results without the consideration of the energy resolution of the spectrometer are also included for comparison. It is found that all the spectra depend strongly on the spectrometer functions. The larger E_s , the FWHM of the spectrometer function, results in the larger E_t , the FWHM of the total spectrum.

Fig. 4.4 A plot of the MC results on the energy spectra of electrons quasi-elastically backscattered from Si for normally incident electrons of 5000eV mean energy and 0.3 eV (solid curve) and 0.5 eV (dashed curv) FWHM.

Fig. 4.5 A plot of the MC results on the energy spectra of electrons quasi-elastically backscattered from Au for normally incident electrons of 5000eV mean energy and 0.3 eV (solid curve) and 0.5 eV (dashed curv) FWHM.

Fig. 4.6 A plot of the MC results on the energy spectra of electrons quasi-elastically backscattered from Si for normally incident electrons of 5000eV mean energy and 0.3 eV FWHM. The spectrometer resolutions are 0 eV (solid curve), 0.3 eV (dashed curve) and 0.5 eV (dotted curve).

Fig. 4.7 A plot of the MC results on the energy spectra of electrons quasi-elastically backscattered from Au for normally incident electrons of 5000eV mean energy and 0.3 eV FWHM. The spectrometer resolutions are 0 eV (solid curve), 0.3 eV (dashed curve) and 0.5 eV (dotted curve).

It is seen that E_t of the present calculated spectra are in good agreement with the approximation (Tóth et al. 1998; Gergely et al. 2001)

2 2 2

s g r

t E E E

E    

 , (4.15)

where E_r is the FWHM of the recoil width broadening. For our case of 5000 eV normally incident electrons, E_r is 0.18 eV for Si and 0.0048 eV for Au. E_r is small in comparison with E_g and E_s used in our work. Therefore, all the spectra are almost Gaussian distributions so that the widths of the quasi-elastic elastic peaks are dominated by E_g and E_s. Figure 4.8 shows our calculated spectra without and with the consideration of the thermal corrections in the recoil losses.

The experimental results (Varga et al. 2001) are included for comparison. Here the electron beam with 5000 eV mean energy and E_g 0.4eV was obliquely incident into Si. The incident angle was 50⁰. The acceptance angles were between 0⁰and 3⁰. The energy resolution of the spectrometer E_s was 0.28 eV. The spectrum without the thermal effect is a Gaussian distribution with its maximum at 4999.69 eV and a FWHM of 0.5 eV (solid curve). For such narrow acceptance angles, the broadening is expected to be completely dominated by E_g and E_s . The spectrum including the thermal effect was calculated by applying the single scattering model (Varga et al. 2001). In this model, it was assumed that the recoil energy losses followed a Gaussian distribution with its maximum at the most probable recoil energy loss for atoms at rest. Here the thermal Gaussian function with its maximum at

Fig. 4.8 A plot of the MC simulation results on the energy spectra of electrons quasi-elastically backscattered from Si for normally incident electrons of 5000 eV mean energy and 0.4 eV FWHM. Here electrons are incident at an angle 50^o; acceptance angles are between 0^oand 3^o; the spectrometer resolution is 0.28 eV. The thermal effect of atoms is neglected (solid curve) and considered (dash-dot) by applying the single scattering model (Varga et al. 2001) with a FWHM of ET= 0.3 eV at the room temperature. For comparison, the experimental data (dashed curve) (Varga et al. 2001) are also included.

The resulting spectrum with the thermal effect (dash-dot curve) is a Gaussian function with a FWHM of 0.58 eV. Then we can find that this FWHM, E, is in good agreement with E E_T²E_g²E_s² . However, the peak position is associated with the most probable recoil energy loss for the vibrating atoms, not for atoms at rest.

In addition, the thermal effect in multiple scatterings shifts the peak position and widens the spectra. Thus, this simple model is not enough to describe the shift and the broadening. Owning to the experimental data (Varga et al. 2001) were in arbitrary units, the experimental data shown in the figure were adjusted in magnitude to match our expectation of absolute values.

4.1.4 Formula for Energy Spectra Contributed by