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, inplane, effective mass and an effective period of the quantum wells. The inplane 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 inplane 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 bandtoband 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.