4.2 Inelastic Interactions of Electrons with Clad Cylindrical Systems
4.2.2.4 Limiting Cases
4.2.2.4 Limiting Cases
Taking ε2 =ε3 in Eq (4.27) or ε1 =ε2 in Eq (4.29), one obtains the same formulas of the stopping power as that derived in Eq. (4.9). Taking in Eq (4.29) or in Eq (4.27), one obtains the same formulas of the stopping power
as that derived in Eq. (4.10).
4.2.3 Differential Inverse Inelastic Mean Free Path
The stopping power is expressed in terms of the DIIMFP, μ( p), through
where p=I, II and III are for cases I, II and III respectively. Therefore, one obtains
the DIIMFPs as
( ) ∑
∞for an electron moving in medium 2, and
( ) ∑
∞for an electron moving in medium 3.
Taking ε2 =ε3 in Eq (4.31) or ε1 =ε2 in Eq (4.32), one obtains the same formulas of the DIIMFP as that derived in Eq. (4.12). Taking ε2 =ε3 in Eq (4.32) or in Eq (4.33), one obtains the same formulas of the DIIMFP as that derived in Eq. (4.13). Taking
2 1 =ε ε
ε
= ε
= ε
=
ε1 2 3 in Eqs. (4.31), (4.32) or (4.33), one obtains
the DIIMFP for an infinite solid as Eq. (4.14). Equation (4.14) may also be found by taking a→∞ in Eq. (4.31), a=0 and b→∞ in Eq. (4.32), or in Eq.
(4.33).
=0
= b a
Using Eqs. (4.31), (4.32) and (4.33), the DIIMFP for an electron moving parallel
to the axis of a Si cylindrical tube of inner radius and outer radius
is calculated. In these calculations, a sum-rule-constrained extended Drude dielectric function with dispersion (Kwei 2003) was applied. Figure 4.7 shows the results for the DIIMFP of a 500 eV electron traveling inside the Si tube, i.e.
, at several distances as a function of energy transfer. It is seen that the
DIIMFP is entirely contributed from surface excitations. The surface excitation peak (~ 12 eV) decreases in magnitude for decreasing
o
A
=15 a
o
A
=25 b
<a
ρ0 ρ0
ρ due to the weaker response by 0
the solid surface.
The DIIMFP of a 500 eV electron traveling inside the cylindrical shell of the Si tube, i.e. , is plotted in Fig. 4.8 as a function of energy transfer for several values of . Since now the electron travels inside the solid, the DIIMFP exhibits
b a<ρ0 <
ρ0
overlapping peaks due to the contributions from surface and volume excitations.
The relative contributions from surface and volume excitations depend on the location of the electron. As the electron moves along the midline between inner and outer surfaces (solid curve), volume excitations (the peak at ~17 eV) dominate. When the electron moves near the inner surface (dashed curve) or the outer surface (dotted curve), surface excitations (the peak at ~12 eV) become more prominent. In the case
of , for instance, the electron moves at away from and parallel to the cylindrical surface where the electron and the cylindrical axis are on opposite sides of
the surface. In the case of , on the other hand, the electron moves also at
away from and parallel to the cylindrical surface where the electron and the cylindrical axis are on the same side of the surface. For , the electron moves along the cylindrical surface bending away from it, leading to reduced surface
excitations and increased volume excitations. For , the electron moves along the cylindrical surface bending towards it, leading to enhanced surface excitations and decreased volume excitations.
o 0 =16A
ρ 1Ao
o 0 =24A ρ
o
A 1
o 0 =16A ρ
o 0 =24A ρ
Similar results on the DIIMFP of a 500 eV electron moving outside the Si tube, i.e. , for several are plotted in Fig. 4.9 as a function of energy transfer. It
is seen that in this case the DIIMFP is totally contributed from surface excitations, with its value decreasing for increasing electron distance from the surface. Figure
>b
ρ0 ρ0
4.10 shows the DIIMFP for electrons with various energies moving outside the Si tube
at . It is seen that the DIIMFP decreases with increasing electron energy.
This indicates that the contribution from surface excitations also decreases as electron velocity increases.
o 0 =26A ρ
Figure 4.11 shows the DIIMFP for a 500 eV electron moving at
outside a Si cylinder clad in a SiO
o case of , the DIIMFP exhibits a broad distribution contributed from surface excitations of SiO
o contributed from surface excitations of Si. For the Si cylinder clad in a SiO
o
A
=25 a
2 film of
thickness ) or ), the DIIMFP reveals the contributions from surface (SiO
o
2-vacuum) excitations and interface (Si-SiO2) excitations. For the case of a Si cylinder clad in a ( ) SiO
a 2 film, the DIIMFP approaches to that of the SiO2 cylinder wire. Another words, as the film thickness increases the DIIMFP gradually changes from a value of the Si cylindrical wire to that of the SiO2
wire. When the film thickness is greater than , nearly no contribution from interface excitations is found.
o
A 10
Fig. 4.7 Calculated DIIMFP for a 500 eV electron moving parallel to and at a
distance from the axis of a Si tube of inner radius and outer
radius .
o 0 <15A
ρ a=15Ao
o
A
=25 b
Fig. 4.8 Calculated DIIMFP for a 500 eV electron moving parallel to and at a
distance from the axis of a Si tube of inner radius and
outer radius .
o 0 o
A 25 A
15 <ρ < a=15Ao
o
A
=25 b
Fig. 4.9 Calculated DIIMFP for a 500 eV electron moving parallel to and at a
distance from the axis of a Si tube of inner radius and outer
radius .
o 0 >25A
ρ a=15Ao
o
A
=25 b
Fig. 4.10 Calculated DIIMFP for an electron moving parallel to and at a distance
from the axis of a Si tube of inner radius and outer radius
for several electron energies.
o 0 =26A
ρ a=15Ao
o
A
=25 b
Fig. 4.11 Calculated DIIMFP for a 500 eV electron moving parallel to and at a
distance from the axis of a Si cylinder clad in a SiO
o 0 =26A
ρ 2 film, having outer
radius and inner radius
o
A
=25
b a=0, 15, 22, 24 or . Results of
and correspond to the SiO
o
A
25 a =0
o
A 5
2 2 and the Si cylindrical wires.
CHAPTER 5 SUMMARY
In this dissertation, electronic excitations produced by the inelastic interaction between charged particles and solids were studied theoretically.
Electronic excitations are the important mechanism responsible for the energy loss of electrons in electron spectroscopies. The description of electronic excitations was based on the extended Drude model which characterized the dielectric response functions. Experimental data taken from the optical ellipsometry for small energy transfers and the electron energy-loss spectra for large energy transfers were used to obtain parameters in the model dielectric functions for semiconducting III-V
compounds. To assure the accuracy of the dielectric functions, sum-rules and critical-point energies are checked.
In the research on electronic excitations in planar systems, the inelastic response to a probe electron moving across the solid surface was determined. The angular and energy dependences of the SEP for electrons moving in vacuum and across the surface were analyzed. The SEP was fitted to a simple formula for the applications in
electron surface-sensitive spectroscopies. Moreover, a theoretical treatment was developed to account for the memory effect on the induced potential, stopping power,
DIIMFP, IMFP for a charged particle moving close and parallel to the surface of a solid. It was found that the consideration of memory effect was important for the calculation of inelastic interactions.
In the research on electronic excitations in cylindrical systems, analytic formulas were derived to deal with the DIIMFP and IIMFP for an electron moving parallel to the axis of a (clad) cylindrical structure based on the dielectric response theory. The dependences of the DIIMFP and IIMFP on the electron position and energy have been analyzed. All relevant inelastic interactions including volume, surface and interface excitations were considered. Information on electron inelastic interactions with cylindrical structures is essential in the applications of electron surface spectroscopies, involving nanowires and microcapillaries.
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