9.5 Dielectric Function due to Optical Phonons
265
Fig. 9.6 Dielectric
function according to
(9.24) with = 3 and
= 2 (without
damping). Grey area
denotes the region of
negative
4
2
0
3
1
2
3
0
1
2
Frequency (
)
/ T
( )
(0)
1
The (long-wavelength) TO-phonon does not create a long-range electric field. Using ∇ · D = 0 and
(9.23) and looking at the longitudinal fields, we have
0 E = b 12 u + b 22 E .
(9.28)
This can be rewritten as
E = − ω LO
M r
0
1
(∞)
−
1
(0)
u ∝ −u .
(9.29)
The (long-wavelength) LO-phonon thus creates a long-range electric field acting against the ion displacement and represents an additional restoring force; this is consistent with the fact that ω LO > ω TO .
9.6 Electron–Photon Interaction
The absorption process within the band structure is quantum mechanically described by the coupling
of electrons and photons. The process is described with time-dependent perturbation theory. If H em is
the perturbation operator (electromagnetic field), the transition probability per time w fi for electrons
from (unperturbed) state ‘i’ (initial) to state ‘f’ (final) is given (with certain approximations) by Fermi’s
golden rule
w fi ( =
2π
H
fi
2 δ(E f − E i − ,
(9.30)
where is the photon energy, E i (E f ) is the energy of the initial (final) state. H
fi is the matrix element
H
fi =
f
H
i
,
(9.31)
where i ( f ) are the wavefunctions of the unperturbed initial (final) state.
265
Fig. 9.6 Dielectric
function according to
(9.24) with = 3 and
= 2 (without
damping). Grey area
denotes the region of
negative
4
2
0
3
1
2
3
0
1
2
Frequency (
)
/ T
( )
(0)
1
The (long-wavelength) TO-phonon does not create a long-range electric field. Using ∇ · D = 0 and
(9.23) and looking at the longitudinal fields, we have
0 E = b 12 u + b 22 E .
(9.28)
This can be rewritten as
E = − ω LO
M r
0
1
(∞)
−
1
(0)
u ∝ −u .
(9.29)
The (long-wavelength) LO-phonon thus creates a long-range electric field acting against the ion displacement and represents an additional restoring force; this is consistent with the fact that ω LO > ω TO .
9.6 Electron–Photon Interaction
The absorption process within the band structure is quantum mechanically described by the coupling
of electrons and photons. The process is described with time-dependent perturbation theory. If H em is
the perturbation operator (electromagnetic field), the transition probability per time w fi for electrons
from (unperturbed) state ‘i’ (initial) to state ‘f’ (final) is given (with certain approximations) by Fermi’s
golden rule
w fi ( =
2π
H
fi
2 δ(E f − E i − ,
(9.30)
where is the photon energy, E i (E f ) is the energy of the initial (final) state. H
fi is the matrix element
H
fi =
f
H
i
,
(9.31)
where i ( f ) are the wavefunctions of the unperturbed initial (final) state.