Absorption of quantum wells =========================== For modelling the optical properties of QWs we use the method described by S. Chuang ([#Ref18]_). The absorption coefficient at thermal equilibrium in a QW is given by: .. math:: \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} where :math:|I_{hm}^{en}|^2 is the overlap integral between the holes in level :math:m and the electrons in level :math:n; :math:H is a step function, :math:H(x) = 1 for :math:x>0, 0 and 0 for :math:x<0, :math:\rho_{rmn}^{2D} is the 2D joint density of states, :math:C_0 a proportionality constant dependent on the energy, and :math:F the excitonic contribution, which will be discussed later. .. math:: \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} Here, :math:n_r is the refractive index of the material, :math:m_{rmn} = m_{en} m_{hm} / (m_{en} + m_{hm}) the reduced, in-plane, effective mass and :math: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 ([#Ref19]_): .. math:: \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], :math:| \hat{e} \cdot \vec{p} |^2 is the momentum matrix element, which depends on the polarization of the light and on the Kane’s energy :math:E_p, specific to each material and determined experimentally. For band edge absorption, where :math: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 | +================+=====================+===================+ | :math:c-hh | :math:3/2 M_b^2 | 0 | +----------------+---------------------+-------------------+ | :math:c-lh | :math:1/2 M_b^2 | :math:2 M_b^2 | +----------------+---------------------+-------------------+ Table: Momentum matrix elements for transitions in QWs. :math:M_b^2=m_0 E_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 :math:F_{nm}. This term is a Lorenzian (or Gaussian) defined by an energy :math:E_{nmx, j} and oscillator strength :math:f_{ex, j}. It is zero except for :math:m=n \equiv j where it is given by Klipstein et al. ([#Ref20]_): .. math:: \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} Here, :math:\nu is a constant with a value between 0 and 0.5 and :math:\sigma is the width of the Lorentzian, both often adjusted to fit some experimental data. In Solcore, they have default values of :math:\nu = 0.15 and :math:\sigma = 6 meV. :math:R is the exciton Rydberg energy ([#Ref18]_). 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. .. image:: qw_absorption.png :align: center .. automodule:: solcore.absorption_calculator.absorption_QW :members: References ---------- .. [#Ref18] Chuang, S.L.: Physics of Optoelectronic Devices. Wiley- Interscience, New York (1995) .. [#Ref19] Barnham, K., Vvedensky, D. (eds.): Low-Dimensional Semi- conductor Structures: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2001) .. [#Ref20] 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)