23.4 Lasers
663
(a)
(b)
(c)
(d)
Fig. 23.34 Gain in the two-band model for GaAs. a Electron and hole concentrations at T = 300 K as a function of
the quasi-Fermi energies counted relative to the band edges, i.e. F n − E C and E V − F p . b Difference of quasi-Fermi
levels as a function of carrier concentration (n = p) for GaAs at two different temperatures. c Gain spectra according to
(23.12) for n = 2 × 10 18 and T = 300 K (solid line), increased carrier density n = 2.1 × 10 18 and T = 300 K (dashed
line), higher temperature n = 2 × 10 18 and T = 314 K (dash-dotted line) and same difference of the quasi-Fermi levels
as for the solid line, n = 2.1 × 10 18 and T = 314 K (dotted line). d Maximum gain (solid line) and gain at a particular
energy (dashed line, for photon energy E g + 26.2 meV for which the gain is maximal for n = 2 × 10 18 and T = 300 K,
see solid line in part c)
with E
= ω − E g − E. The gain is positive for those photon energies for which light is amplified
and negative for those that are absorbed. In Fig. 23.34a, the electron and hole concentrations are shown
for GaAs as a function of the quasi-Fermi energies. In Fig. 23.34b, the difference of the quasi-Fermi
energies is shown as a function of the carrier density (for neutrality n = p). The gain spectrum
is shown in Fig. 23.34c for a simple two-band model.
18 For a more elaborate discussion of such
matters we refer to [1979]. In the case of inversion, the gain is positive for energies between E g
and F n − F p . At = F n − F p , the gain is zero (transparency) and for larger energies negative
(positive absorption coefficient). The agreement of experimental gain spectra of quantum wells with
theoretical considerations, including carrier collision effects at the level of quantum kinetic theory in
the Markovian limit, is very good (Fig. 23.35a) [1980].
For a given fixed energy, the gain increases with increasing pumping and increasing carrier density n
(Fig. 23.34d). For very small density, it is given as g(n → 0) = −α. The gain rises around transparency
approximately linearly with the pumping intensity. At transparency carrier density n tr , the gain is zero.
Therefore, the relation g(n) can be approximated as (linear gain model)
18 One electron and one hole band are considered; the heavy and light hole bands are taken into account via the mass
according to (6.73).
663
(a)
(b)
(c)
(d)
Fig. 23.34 Gain in the two-band model for GaAs. a Electron and hole concentrations at T = 300 K as a function of
the quasi-Fermi energies counted relative to the band edges, i.e. F n − E C and E V − F p . b Difference of quasi-Fermi
levels as a function of carrier concentration (n = p) for GaAs at two different temperatures. c Gain spectra according to
(23.12) for n = 2 × 10 18 and T = 300 K (solid line), increased carrier density n = 2.1 × 10 18 and T = 300 K (dashed
line), higher temperature n = 2 × 10 18 and T = 314 K (dash-dotted line) and same difference of the quasi-Fermi levels
as for the solid line, n = 2.1 × 10 18 and T = 314 K (dotted line). d Maximum gain (solid line) and gain at a particular
energy (dashed line, for photon energy E g + 26.2 meV for which the gain is maximal for n = 2 × 10 18 and T = 300 K,
see solid line in part c)
with E
= ω − E g − E. The gain is positive for those photon energies for which light is amplified
and negative for those that are absorbed. In Fig. 23.34a, the electron and hole concentrations are shown
for GaAs as a function of the quasi-Fermi energies. In Fig. 23.34b, the difference of the quasi-Fermi
energies is shown as a function of the carrier density (for neutrality n = p). The gain spectrum
is shown in Fig. 23.34c for a simple two-band model.
18 For a more elaborate discussion of such
matters we refer to [1979]. In the case of inversion, the gain is positive for energies between E g
and F n − F p . At = F n − F p , the gain is zero (transparency) and for larger energies negative
(positive absorption coefficient). The agreement of experimental gain spectra of quantum wells with
theoretical considerations, including carrier collision effects at the level of quantum kinetic theory in
the Markovian limit, is very good (Fig. 23.35a) [1980].
For a given fixed energy, the gain increases with increasing pumping and increasing carrier density n
(Fig. 23.34d). For very small density, it is given as g(n → 0) = −α. The gain rises around transparency
approximately linearly with the pumping intensity. At transparency carrier density n tr , the gain is zero.
Therefore, the relation g(n) can be approximated as (linear gain model)
18 One electron and one hole band are considered; the heavy and light hole bands are taken into account via the mass
according to (6.73).