2.13. Excitons

The exciton states are computed using the approach from [ChuangOpto1995]. The explanation below only covers the most important aspects of the model, for detailed derivation please refer to the [ChuangOpto1995] book. The exciton computation is only available for 1D systems.

2.13.1. Model

Assuming the effective mass approximation for the electron and hole, the Schrödinger equation for the exciton can be written as:

(2.13.1.1)\[(H_e(\bar{r}_e) + H_h(\bar{r}_h) - \frac{e^2}{4 \pi \varepsilon |\bar{r}_e - \bar{r}_h|}) \Phi(r_e,r_h) = E\Phi(r_e,r_h)\]

where \(H_e\) and \(H_e\) are the Hamiltonians for the electron and hole, respectively, \(\bar{r}_e\) and \(\bar{r}_h\) are the coordinates of the electron and hole, respectively, \(\varepsilon\) is the dielectric constant, and \(E\) is the energy of the exciton.

The wave function of the exciton, formed by electron \(n\) and hole \(m\), will be calculated in the form:

(2.13.1.2)\[\Phi(r_e,r_h) = exp(i\bar{K_t}\bar{R_t})F(\rho,z_e,z_h) = exp(i\bar{K_t}\bar{R_t}) \phi_{nm}(\rho)f_n(z_e)g_m(z_h)\]

where \(\bar{K_t}\) is the in-plane wave vector of the exciton, \(\bar{R_t}\) is the in-plane coordinate of the exciton, \(F(\rho,z_e,z_h)\) is the exciton envelope function, \(f_n(z_e)\) and \(g_m(z_h)\) are the single particle envelope wave functions of electron and hole in the growth direction. Then, the equation for the unknown \(\phi_{nm}(\rho)\) is given by:

(2.13.1.3)\[\left ( -\frac{\hbar^2}{2m_r} \nabla^{2}_{\rho} - V_{nm}(\rho) \right ) \phi_{nm}(\rho) = E_{binding} \phi_{nm}(\rho)\]

where \(m_r\) is the reduced mass of the exciton, \(E_{binding}\) is the binding energy of the exciton, and \(V_{nm}(\rho)\) is expressed as:

(2.13.1.4)\[V_{nm} = \int dz_e|f_n(z_e)|^2 \int dz_h |g_m(z_h)|^2 \frac{e^2}{4 \pi \varepsilon_s (\rho^2 + |z_e-z_h|^2)}\]

The solution of the equation for \(\phi_{nm}(\rho)\) can be found variationally by minimizing the binding energy of the exciton. The form of the solution is assumed to be similar to 1S state of 2D hydrogen atom:

(2.13.1.5)\[\phi (\rho) = \sqrt{\frac{2}{\pi}}\frac{1}{\lambda} exp(-\rho/\lambda)\]

where \(\lambda\) is the variational parameter, which has an interpretation of exciton inplane Bohr radius. The variational parameter \(\lambda\) is determined by minimizing the binding energy of the exciton from equation (2.13.1.3).

2.13.2. Averaging model parameters

The model depends on dielectric constant \(\varepsilon\), effective masses of the electron and hole \(m_e\) and \(m_h\), which are not constant in heterostructures. If not given in the input file, the volume averaged values of these parameters are used. For effective masses, density weighted average is also possible.

2.13.3. Excitons in multiband Hamiltonians

The computation of the exciton in the case of 8-band \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian is complicated by the fact, that the electron and hole Hamltonians are no longer separable. In that case the equations derived from effective mass Hamiltonians are used, using wave functions computed with the 8-band Hamiltonian. As the effective masses are not longer parameters of the Hamiltonian, the effective masses used are computed from the parameters for the 8-band Hamiltonian: \(L, M, N, E_P, S, E_{gap}\). The same approach is used for 6-band \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian, where the effective masses are computed from the parameters \(L, M, N\).