# Numerical Algorithms in III-V Semiconductor Heterostructures
## 2 Materials and Methods
### Theory  
In a semiconductor structure, the electron-polariton Hamiltonian can be described by the following formula:  
\[H=H_{0}+H_{free}+H_{int} \tag{3}\]  
where the unperturbed electron Hamiltonian is approximated (effective mass approximation) to  
\[H_{0}=\frac{p^{2}}{2m^{*}}+V_{CB} \tag{4}\]  
with \(V_{CB}\) the conduction band profile, \(p\) the electron momentum, and \(m_{*}\) the electron effective mass. The free field Hamiltonian has the form  
\[H_{free}=\frac{e_{o}}{2}\sum_{\omega}\biggl{(}\int\frac{\partial(\omega e( \omega))}{\partial\omega}E^{2}+cB^{2}\biggr{)} \tag{5}\]  
where \(e(\omega)\) is frequency-dependent, \(e_{o}\) is the permittivity of free space, and \(c\) is the velocity of light in vacuum. The electric and the magnetic field, related to the polariton, are respectively denoted by \(E\) and \(B\). By ignoring higher-order processes, the electrons interact with polaritons via the interaction Hamiltonian  
\[H_{int}=\frac{e}{m^{*}}\mathbf{A}\cdot\mathbf{p} \tag{6}\]  
where \(\mathbf{A}\) is the vector potential that describes the polaritons. Let us consider a double heterostructure constructed with GaAs/AlAs with a well width of \(d\). The dielectric functions to describe the interface Fuchs-Kliewer (FK) polaritons in the heterostructure are provided by [6]  
\[\epsilon_{i}=\epsilon_{\infty,j}\frac{\omega^{2}-\omega_{L,i}^{2}}{\omega^{2}- \omega_{T,i}^{2}} \tag{7}\]  
where \(\epsilon_{\infty,j}\) is the high-frequency dielectric constant, \(\omega_{L,i}\) and \(\omega_{T,i}\) are the zone center LO and TO optical frequencies of the \(i\)-th material. The symmetric and the antisymmetric interface mode dispersion relations are, respectively, provided by the following equations  
\[\frac{\epsilon_{2}(\omega)q_{1}}{\epsilon_{1}(\omega)q_{2}}=-coth\bigg{(}\frac {q_{2}d}{2}\bigg{)} \tag{8}\]  
\[\frac{\epsilon_{2}(\omega)q_{1}}{\epsilon_{1}(\omega)q_{2}}=-tanh\bigg{(}\frac {q_{2}d}{2}\bigg{)} \tag{9}\]  
Wavevectors \(q_{i}\) and the in-plane wavevector \(q_{||}\) are provided by  
\[q_{i}^{2}=q_{||}^{2}-\omega^{2}\epsilon_{i}(\omega)/c^{2} \tag{10}\]  
Hence, by combining Equations (8) and (9), the following optimization problem can be formulated:  
\[\min\sum_{q_{||}=t_{0}}^{t_{1}}\Bigg{(}\bigg{(}\frac{\epsilon_{2}(\omega_{1})q _{1}}{\epsilon_{1}(\omega_{1})q_{2}}+coth\bigg{(}\frac{q_{2}d}{2}\bigg{)} \bigg{)}^{2}+\bigg{(}\frac{\epsilon_{2}(\omega_{2})q_{1}}{\epsilon_{1}(\omega _{2})q_{2}}+tanh\bigg{(}\frac{q_{2}d}{2}\bigg{)}\bigg{)}^{2}\Bigg{)} \tag{11}\]  
Equation (11) should be minimized with respect to independent variables \(\omega_{1},~{}\omega_{2}\), and parameter \(q_{||}\) varies from \(t_{0}\) to \(t_{1}\). In the current implementation, the artificial neural networks \(N_{\text{I}}\big{(}q_{||},\overrightarrow{\omega}\big{)}\), \(N_{\text{I}}\big{(}q_{||},\overrightarrow{\omega}\big{)}\)were used in place of variables \(\omega_{1},~{}\omega_{2}\). Using Equation (1) with \(d=1\), the final form of used neural network is  
\[N\big{(}q_{||},\overrightarrow{\omega}\big{)}=\sum_{i=1}^{K}w_{3i-2}\sigma \Big{(}q_{||}w_{3i-1}+w_{3i}\Big{)} \tag{12}\]  
where \(K\) denotes the number of weights for the neural network. Hence, the optimization problem of Equation (11) is transformed to the following one:  
\[H_{0}=\frac{p^{2}}{2m^{*}}+V_{CB} \tag{4}\]0  
For experimental purposes, the interval \([t_{0},t_{1}]\) is divided into \(NP\) equidistant points, forming the set \(X=[x_{0}=t_{0},x_{1},\ldots,x_{NP}=t_{1}]\), and hence the following equation will be minimized:  
\[H_{0}=\frac{p^{2}}{2m^{*}}+V_{CB} \tag{4}\]1  
In the following subsection, two methods will be analyzed that were used to optimize Equation (14) with respect to weight vectors \(\overrightarrow{w_{1}},~{}\overrightarrow{w_{2}}\).