Absorption of quantum wells¶
For modelling the optical properties of QWs we use the method described by S. Chuang ([1]). The absorption coefficient at thermal equilibrium in a QW is given by:
where is the overlap integral between the holes
in level
and the electrons in level
;
is a
step function,
= 1 for
, 0 and 0 for
,
is the 2D joint density of states,
a proportionality constant dependent on the energy, and
the excitonic contribution, which will be discussed later.
Here, is the refractive index of the material,
the reduced,
in-plane, effective mass and
an effective period of the
quantum wells. The in-plane effective mass of each type of carriers is
calculated for each level, accounting for the spread of the wavefunction
into the barriers as ([2]):
This in-plane effective mass is also used to calculate the local density
of states shown in Figure [fig:qw]b. In Eq. [eq:QW_abs2],
is the momentum matrix element,
which depends on the polarization of the light and on the Kane’s energy
, specific to each material and determined experimentally.
For band edge absorption, where
= 0, the matrix elements for
the absorption of TE and TM polarized light for the transitions
involving the conduction band and the heavy and light holes bands are
given in Table [tab:matrix_elements]. As can be deduced from this
table, transitions involving heavy holes cannot absorb TM polarised
light.
TE |
TM |
|
---|---|---|
0 |
||
Table: Momentum matrix elements for transitions in QWs.
is the bulk matrix element.
In addition to the band-to-band transitions, QWs usually have strong
excitonic absorption, included in Eq. [eq:qw_abs] in the term
. This term is a Lorenzian (or Gaussian) defined by an
energy
and oscillator strength
. It
is zero except for
where it is given by Klipstein
et al. ([3]):
Here, is a constant with a value between 0 and 0.5 and
is the width of the Lorentzian, both often adjusted to
fit some experimental data. In Solcore, they have default values of
= 0.15 and
= 6 meV.
is the exciton
Rydberg energy ([1]).
Fig. [fig:QW_absorption] shows the absorption coefficient of a range of InGaAs/GaAsP QWs with a GaAs interlayer and different In content. Higher indium content increases the depth of the well, allowing the absorption of less energetic light and more transitions.
