378
12 Heterostructures
Fig. 12.41 a
Recombination spectra
(solid lines, scaled to same
height) of a model quantum
well for different
temperatures as labeled and
complete thermalization,
dashed (dash-dotted) line
is unperturbed
(inhomogeneously
broadened by σ = 20 meV)
shape of the QW
absorption edge. The
energy scale is relative to
the energy position of the
unperturbed QW
absorption edge at E 0 . b
Theoretical energy position
and c linewidth of exciton
recombination from a
model disordered quantum
well. The high-temperature
limits are marked by
arrows. Parts (b, c) adapted
from [1204]
(a)
(b)
(c)
probability distribution p(δ E) ∝ exp[−(δ E)
2
/2σ
2
)] with δ E being the deviation from the QW band
edge δ E = E − E 0 . The resulting lineshape is given by the convolution of the Gaussian with the
unperturbed absorption spectrum yielding an error-function-like spectrum
6 as shown in Fig. 12.41a.
For complete thermalization the level population is given by the Boltzmann function. The recombination spectrum is given by the product of the absorption spectrum (or JDOS) and the Boltzmann
function. It is (red-) shifted with respect to E 0 by about
7
) = −
σ
2
kT
= γ(T ) kT .
(12.11)
This shift between emission and absorption is also called the Stokes shift.
Within their lifetime, limited at least by radiative recombination, the excitons are typically unable
to reach the energy position required by the Boltzmann function, but only a local minimum. Thus, their
thermalization may be incomplete due to insufficient lateral diffusion. This effect is particularly important at low temperatures when thermal emission into adjacent deeper potential minima is suppressed. In
this case, the red-shift is smaller than expected from (12.11). A numerical simulation [1204] yields such
behavior of the energy position of the recombination line as shown in Fig. 12.41b. Simultaneously, the
width of the recombination spectrum also exhibits a minimum (Fig. 12.41c). These findings are in agreement with experiments [1206, 1207]. An analytical model for temperature dependent exciton localization in the presence of disorder has been given in [1208], yielding a value 0 ≤ γ ≤ γ 0 = (σ/kT )
2
in (12.11).
6 The error function is defined as erf(x) = (2/
√ π)
z
0 exp −t 2 dt.
7 Formula (12.11) is exact for the product of a Gaussian and the Boltzmann function.
12 Heterostructures
Fig. 12.41 a
Recombination spectra
(solid lines, scaled to same
height) of a model quantum
well for different
temperatures as labeled and
complete thermalization,
dashed (dash-dotted) line
is unperturbed
(inhomogeneously
broadened by σ = 20 meV)
shape of the QW
absorption edge. The
energy scale is relative to
the energy position of the
unperturbed QW
absorption edge at E 0 . b
Theoretical energy position
and c linewidth of exciton
recombination from a
model disordered quantum
well. The high-temperature
limits are marked by
arrows. Parts (b, c) adapted
from [1204]
(a)
(b)
(c)
probability distribution p(δ E) ∝ exp[−(δ E)
2
/2σ
2
)] with δ E being the deviation from the QW band
edge δ E = E − E 0 . The resulting lineshape is given by the convolution of the Gaussian with the
unperturbed absorption spectrum yielding an error-function-like spectrum
6 as shown in Fig. 12.41a.
For complete thermalization the level population is given by the Boltzmann function. The recombination spectrum is given by the product of the absorption spectrum (or JDOS) and the Boltzmann
function. It is (red-) shifted with respect to E 0 by about
7
) = −
σ
2
kT
= γ(T ) kT .
(12.11)
This shift between emission and absorption is also called the Stokes shift.
Within their lifetime, limited at least by radiative recombination, the excitons are typically unable
to reach the energy position required by the Boltzmann function, but only a local minimum. Thus, their
thermalization may be incomplete due to insufficient lateral diffusion. This effect is particularly important at low temperatures when thermal emission into adjacent deeper potential minima is suppressed. In
this case, the red-shift is smaller than expected from (12.11). A numerical simulation [1204] yields such
behavior of the energy position of the recombination line as shown in Fig. 12.41b. Simultaneously, the
width of the recombination spectrum also exhibits a minimum (Fig. 12.41c). These findings are in agreement with experiments [1206, 1207]. An analytical model for temperature dependent exciton localization in the presence of disorder has been given in [1208], yielding a value 0 ≤ γ ≤ γ 0 = (σ/kT )
2
in (12.11).
6 The error function is defined as erf(x) = (2/
√ π)
z
0 exp −t 2 dt.
7 Formula (12.11) is exact for the product of a Gaussian and the Boltzmann function.