# Numerical Algorithms in III-V Semiconductor Heterostructures
## 1 Introduction  
Due to the increasing needs for high-quality nanostructures, several advanced growth techniques, e.g., molecular beam epitaxy (MBE), metal organic chemical vapor deposition (MOCVD), and Stranski-Krastanow growth [1], have been used to manufacture qualitative quantum structures constructed with dielectric materials. In polar dielectric crystals, phonon polaritons, which are the result of the coupling of optical phonons with an electromagnetic field (photon), are of crucial importance in low-dimensional structures (LDS) excitation processes. During recent decades, several theoretical and experimental results have been reported in research areas like surface phonon polaritons, polariton-electron interactions in semiconductor microcavities, phonon-polariton modes in superlattices, and polariton modes in ferroelectric/graphene heterostructure systems [2; 3; 4; 5; 6; 7], among others. The studying of phonon polaritons and their coupling to electrons within a semiconductor LDS (e.g., quantum wells and superlattices) play an important role to control and enhance the quantum efficiency of infrared (IR) detectors, quantum witches (QWs), and semiconductor lasing structures, among others [8]. Furthermore, numerical methods like finite elements [9; 10], direct diagonalization techniques [11], and integration methods are a few methods that have been employed to solve numerical problems in research related to phonon polariton processes. Many previous works (e.g., [3]) have used simple numerical methods to calculate the phonon polariton modes. This work suggests the application of artificial neural networks [12; 13] to estimate the phonon polariton modes. More specifically, for a quantum well structure, we have calculated the interface polariton frequencies as a function of the polariton in-plane wavevector in order to describe the reststrahlen frequency bands between the frequencies of the transverse and longitudinal optical phonons.  
Neural networks have been used in a variety of cases, such as problems from physics [14; 15; 16], as solutions of differential equations [17; 18], agriculture problems [19; 20], chemistry problems [21; 22; 23], problems related to economics [24; 25; 26], medicine problems [27; 28], etc. Artificial neural networks are usually formulated as functions \(N(\overrightarrow{x},\overrightarrow{w})\). Vector \(\overrightarrow{x}\) represents the input pattern and vector \(\overrightarrow{w}\) is called the weight vector. As suggested in [29], the neural network can be expressed as the following summation:  
\[N\big{(}\overrightarrow{x},\overrightarrow{w}\big{)}=\sum_{i=1}^{K}w_{(d+2)i-( d+1)}\sigma\Bigg{(}\sum_{j=1}^{d}x_{j}w_{(d+2)i-(d+1)+j}+w_{(d+2)i}\Bigg{)} \tag{1}\]  
Parameter \(K\) stands for the number of processing nodes, and parameter \(d\) represents the dimension of the input pattern. The used function \(\sigma(x)\) is known as sigmoid function in the relevant literature and is formulated as  
\[\sigma(x)=\frac{1}{1+\exp(-x)} \tag{2}\]  
Recently, artificial neural networks have been applied to some solid state physics problems, such as identification of quantum phase transitions [30], solving of the electronic Schrodinger equation [31], heat transfer problems [32], metal additive manufacturing [33], etc.  
The sections of this article are organized as follows: in Section 2, the objective problem and the methods used to tackle it are presented in detail; in Section 3, the experimental results are outlined, and, finally, in Section 4, some conclusions from the application of the optimization methods are discussed thoroughly.