Page 274 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Photon
E a) b) c)
2
1 photon 1 photon
E 2 photons
1

Figure 7.10. Electron-hole recombination. a) Non-radiative (without photon emission)
recombination. The energy is dissipated into other degrees of freedom, e.g., phonons. b)
Spontaneous emission of a photon. c) Emission is stimulated by an incoming photon that
is in phase with the stimulation.

The energy levels of the respective states are E 1 and E 2 , which means
⁄
that the electromagnetic wave at ω = ( E – E ) — couples resonantly
0
1
2
to the electronic system. We insert this into the equation of motion for the
superposition coefficients as given from (3.74) in first order perturbation
theory to give
i –
c˙ t() = ----V c t()exp ( i – ω t)
1 12 2 0
—
(7.82)
i –
c˙ t() = ----V c t()exp ( iω t)
2 21 1 0
—
Rotating Remember that V has a harmonic time dependence through the elec-
mn
Wave tric field vector with frequency , near the resonance frequency ω 0 . Let
ω
approxi- us assume that c t() and c t() change very slowly on time scales where
mation 1 2
exp ( − i + ω +( ω )t) is changing. Integrating (7.82) over time will yield a
0
vanishing contribution of the exponential term since it will perform many
oscillations in the integration period that cancel each other. Therefore, we
are led to neglect the terms with the summed frequencies ω +( ω ) in the
0
exponents compared to those with the frequency difference ωω ) .
(
–
0
This so called rotating wave approximation yields for the transition
dipole matrix element the new form V = µexp ( i – ωt)E ⁄ 2 , where
12 0
µ is the projection of the dipole matrix element µµ µµ along the electric
12
Semiconductors for Micro and Nanosystem Technology 271