Absorption of quantum wells¶
For modelling the optical properties of QWs we use the method described by S. Chuang (). 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 ():
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.
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. ():
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 ().
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.