14.4 Energy-Dependent 3D-SPB
285
14.4.2.2 Inelastic Electron Dissipation
The energy dependence of the inelastic potential, ImV(E), represents all the dissipative processes that are dominated by phonon and single-electron excitation at
energies below that for plasma excitation. This means that the single-electron excitation occurs in the electron-occupied space, described by ρ(r), and with any incident
energy E greater than the work function φ.
As remarked by Pendry and Alldredge [55], no contribution to ImV(E) from a
particular loss mechanism can come about until the incident electron has enough
energy to excite this mechanism. Plasma excitation needs energies around 15 eV
below E F [56] in metals with a free-electron conduction band. The contribution of
plasma excitation to ImV(E) does not come into play below this value. Even in nonfree-electron metals there is a more general cluster of excitations usually at around the
equivalent free-electron plasmon energy. Below the plasmon energy, single electrons
can still be excited from the conduction band, though they produce a smaller ImV(E)
features than plasma excitation.
In metals, single electron can be excited by an incident electron that has any energy
greater than the work function, (E ≥ φ), but in insulators the situation is different. On
the other hand, phonon excitation and photon excitation may occur at energies that
are much lower than the E F . Therefore, at energies between the work function and
the plasma excitation threshold, single electron excitation is dominant, which adds
“humps” to the inelastic damping showing the DOS features in this energy regime.
14.4.2.3 Amplitude and Phase Shift of Diffracted Beams
The z-directional integration of the ImV(z, E) and the ReV(z) corresponds, respectively, to the amplitude loss A and the phase change of the reflected electron
beams that are described with plane waves:
ϕ = Aexp(−ik · r + ), and,
A ∝
−∞
a
ϕImV (z, E)ϕ
∗ dz,
∝
−∞
a
ϕReV (z)ϕ
∗ dz
(14.3)
where k is the plane wave vector. Integration starts from a certain point, a, inside the
crystal, to infinitely far away of the surface. The parameter a varies depending on
the penetration depth of the incident beams. This constraint provides a leeway for
the mathematical expressions of the specific forms of ImV(z), ImV(E), and ReV(z).
Therefore, the exact values of the strongly correlated parameters are much less important than the integration is in the physics, see Eq. (14.2). The correlation among the
285
14.4.2.2 Inelastic Electron Dissipation
The energy dependence of the inelastic potential, ImV(E), represents all the dissipative processes that are dominated by phonon and single-electron excitation at
energies below that for plasma excitation. This means that the single-electron excitation occurs in the electron-occupied space, described by ρ(r), and with any incident
energy E greater than the work function φ.
As remarked by Pendry and Alldredge [55], no contribution to ImV(E) from a
particular loss mechanism can come about until the incident electron has enough
energy to excite this mechanism. Plasma excitation needs energies around 15 eV
below E F [56] in metals with a free-electron conduction band. The contribution of
plasma excitation to ImV(E) does not come into play below this value. Even in nonfree-electron metals there is a more general cluster of excitations usually at around the
equivalent free-electron plasmon energy. Below the plasmon energy, single electrons
can still be excited from the conduction band, though they produce a smaller ImV(E)
features than plasma excitation.
In metals, single electron can be excited by an incident electron that has any energy
greater than the work function, (E ≥ φ), but in insulators the situation is different. On
the other hand, phonon excitation and photon excitation may occur at energies that
are much lower than the E F . Therefore, at energies between the work function and
the plasma excitation threshold, single electron excitation is dominant, which adds
“humps” to the inelastic damping showing the DOS features in this energy regime.
14.4.2.3 Amplitude and Phase Shift of Diffracted Beams
The z-directional integration of the ImV(z, E) and the ReV(z) corresponds, respectively, to the amplitude loss A and the phase change of the reflected electron
beams that are described with plane waves:
ϕ = Aexp(−ik · r + ), and,
A ∝
−∞
a
ϕImV (z, E)ϕ
∗ dz,
∝
−∞
a
ϕReV (z)ϕ
∗ dz
(14.3)
where k is the plane wave vector. Integration starts from a certain point, a, inside the
crystal, to infinitely far away of the surface. The parameter a varies depending on
the penetration depth of the incident beams. This constraint provides a leeway for
the mathematical expressions of the specific forms of ImV(z), ImV(E), and ReV(z).
Therefore, the exact values of the strongly correlated parameters are much less important than the integration is in the physics, see Eq. (14.2). The correlation among the