10.2 Band–Band Recombination
309
10.2.6 Lasing
The net rate for stimulated emission and absorption is
r st (E) − r abs (E) =
1 − exp
E − (F n − F p )
kT
(10.22)
×
E+E V
E C
dE e B D e (E e ) f e (E e ) D h (E e − E) f h (E e − E) N ph (E) .
The net rate at photon energy E = is only larger than zero (i.e. dominating stimulated emission)
when
F n − F p > E ≥ E g .
(10.23)
When the difference of the quasi-Fermi levels is larger than the band gap, the carrier population is
inverted, i.e. close to the band edges the conduction-band states are more strongly populated with
electrons than the valence-band states, as shown in Fig. 10.4. An incoming optical wave of energy E
will then be net amplified by stimulated emission. Equation (10.23) is also called the thermodynamic
laser condition. We note that lasing requires further conditions as discussed in Sect. 23.4.
10.3 Exciton Recombination
10.3.1 Free Excitons
The observation of free-excitons is limited for semiconductors with a small exciton binding energies
(such as in GaAs) to low temperatures. However, for large exciton binding energy, recombination from
free-excitons is observed even at room temperature, as shown in Fig. 10.5 for ZnO.
Fig. 10.4 Charge-carrier
distribution during
inversion, necessary for
lasing. Shaded areas are
populated with electrons. A
stimulated transition
between an electron and a
hole is indicated
D(E)
E V
E C
E
F n
F p
309
10.2.6 Lasing
The net rate for stimulated emission and absorption is
r st (E) − r abs (E) =
1 − exp
E − (F n − F p )
kT
(10.22)
×
E+E V
E C
dE e B D e (E e ) f e (E e ) D h (E e − E) f h (E e − E) N ph (E) .
The net rate at photon energy E = is only larger than zero (i.e. dominating stimulated emission)
when
F n − F p > E ≥ E g .
(10.23)
When the difference of the quasi-Fermi levels is larger than the band gap, the carrier population is
inverted, i.e. close to the band edges the conduction-band states are more strongly populated with
electrons than the valence-band states, as shown in Fig. 10.4. An incoming optical wave of energy E
will then be net amplified by stimulated emission. Equation (10.23) is also called the thermodynamic
laser condition. We note that lasing requires further conditions as discussed in Sect. 23.4.
10.3 Exciton Recombination
10.3.1 Free Excitons
The observation of free-excitons is limited for semiconductors with a small exciton binding energies
(such as in GaAs) to low temperatures. However, for large exciton binding energy, recombination from
free-excitons is observed even at room temperature, as shown in Fig. 10.5 for ZnO.
Fig. 10.4 Charge-carrier
distribution during
inversion, necessary for
lasing. Shaded areas are
populated with electrons. A
stimulated transition
between an electron and a
hole is indicated
D(E)
E V
E C
E
F n
F p