qw_absorption.rst

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:

$$\begin{aligned} \label{eq:QW_abs2} \begin{split} \alpha_0(E) & = C_0(E) \sum_{n,m} |I_{hm}^{en}|^2 | \hat{e} \cdot \vec{p} |^2 \rho_{rmn}^{2D} \\\ & \times \left[ H(E-E^{en} + E_{hm}) + F_{nm}(E) \right] \end{split} \end{aligned}$$

where |I_hm^en|² is the overlap integral between the holes in level m and the electrons in level n; H is a step function, H(x) = 1 for x > 0, 0 and 0 for x < 0, ρ_rmn^2D is the 2D joint density of states, C₀ a proportionality constant dependent on the energy, and F the excitonic contribution, which will be discussed later.

$$\begin{aligned} \begin{aligned} \label{eq:qw_abs} C_0 (E) & = \frac{\pi q^2 \hbar }{n_r c \epsilon_0 m_0^2 E} \\\ \rho_r^{2D} &= \frac{m_{rmn}^*}{\pi \hbar L}\end{aligned} \end{aligned}$$

Here, n_r is the refractive index of the material, m_rmn = m_enm_hm/(m_en + m_hm) the reduced, in-plane, effective mass and L 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):

$$\begin{aligned} \label{eq:in_plane} m_{\perp} = \int_{0}^{L} m(z) | \psi(z) |^2\end{aligned}$$

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], |ê ⋅ p⃗|² is the momentum matrix element, which depends on the polarization of the light and on the Kane’s energy E_p, specific to each material and determined experimentally. For band edge absorption, where k = 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
c − hh	3/2Mb2	0
c − lh	1/2Mb2	2Mb2

Table: Momentum matrix elements for transitions in QWs. M_b² = m₀E_p/6 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 F_nm. This term is a Lorenzian (or Gaussian) defined by an energy E_nmx, j and oscillator strength f_ex, j. It is zero except for m = n ≡ j where it is given by Klipstein et al. (³):

$$\begin{aligned} \begin{aligned} F_{nm} &= f_{ex, j} \mathcal{L}(E - E_{nmx, j}, \sigma) \\\ E_{nmx, j} &= E^{en} - E_{hm} - \frac{R}{(j-\nu)^2} \\\ f_{ex, j} &= \frac{2R}{(j-\nu)^3} \\\ R &= \frac{m_r q^4}{2 ( 4\pi \epsilon_r \epsilon_0)^2 \hbar^2 }\end{aligned} \end{aligned}$$

Here, ν is a constant with a value between 0 and 0.5 and σ is the width of the Lorentzian, both often adjusted to fit some experimental data. In Solcore, they have default values of ν = 0.15 and σ = 6 meV. R is the exciton Rydberg energy (⁴).

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.

solcore.absorption_calculator.absorption_QW

References

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1. Chuang, S.L.: Physics of Optoelectronic Devices. Wiley- Interscience, New York (1995)↩

2. Barnham, K., Vvedensky, D. (eds.): Low-Dimensional Semi- conductor Structures: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2001)↩

3. Klipstein, P.C., Apsley, N.: A theory for the electroreflectance spec- tra of quantum well structures. J. Phys. C Solid State Phys. 19(32), 6461–6478 (2000)↩

4. Chuang, S.L.: Physics of Optoelectronic Devices. Wiley- Interscience, New York (1995)↩