# Optoelectronic Properties of a Cylindrical Core/Shell Nanowire: Effect of Quantum Confinement and Magnetic Field

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## Abstract

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## 1. Introduction

## 2. Background Theory

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- For ${E}_{e\rho}<{V}_{0}$$$f\left({\rho}_{e}\right)=\left(\right)open="\{"\; close>\begin{array}{c}{A}_{1}{I}_{m}\left(\right)open="("\; close=")">\sqrt{\frac{{V}_{0}-{E}_{e\rho}}{\mu}}{\rho}_{e},\phantom{\rule{62.59596pt}{0ex}}\mathrm{if}\phantom{\rule{14.22636pt}{0ex}}0{\rho}_{e}a\end{array}\\ {B}_{1}{J}_{m}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}{\rho}_{e}+{Y}_{m}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}{\rho}_{e}& ,\phantom{\rule{14.22636pt}{0ex}}\mathrm{if}\phantom{\rule{14.22636pt}{0ex}}a{\rho}_{e}b.$$
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- For ${E}_{e\rho}>{V}_{0}$$$f\left({\rho}_{e}\right)=\left(\right)open="\{"\; close>\begin{array}{c}{A}_{2}{J}_{m}\left(\right)open="("\; close=")">\sqrt{\frac{{E}_{e\rho}-{V}_{0}}{\mu}}{\rho}_{e},\phantom{\rule{62.59596pt}{0ex}}\mathrm{if}\phantom{\rule{14.22636pt}{0ex}}0{\rho}_{e}a\end{array}\\ {B}_{2}{J}_{m}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}{\rho}_{e}+{Y}_{m}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}{\rho}_{e}& ,\phantom{\rule{14.22636pt}{0ex}}\mathrm{if}\phantom{\rule{14.22636pt}{0ex}}a{\rho}_{e}b.$$

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- For ${E}_{e\rho}<{V}_{0}$$${A}_{1}\sqrt{\mu \left(\right)open="("\; close=")">\frac{{V}_{0}}{{E}_{e\rho}}-1}=-{B}_{1}{J}_{1}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}a$$
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- For ${E}_{e\rho}>{V}_{0}$$${A}_{2}\sqrt{\mu \left(\right)open="("\; close=")">1-\frac{{V}_{0}}{{E}_{e\rho}}}={B}_{2}{J}_{1}\left(\right)open="("\; close=")">\sqrt{{E}_{e\rho}}a$$

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the cylindrical core/shell nanowire and the corresponding conduction band structure. The core is taken to be Al${}_{x}$Ga${}_{1-x}$As material with controllable radius a, surrounded by GaAs material-based shell with radius b. we assume that the structure is under the effect of an external magnetic field $\overrightarrow{B}$ along the z-axis.

**Figure 2.**Electron ground state energy of Al${}_{x}$Ga${}_{1-x}$As/GaAs core/shell NW with $x=0.1$ calculating via variational method (solid line) according to Equation (10) and via FEM implemented in COMSOL Multiphysics (dashed line). The FEM is used by modeling the Schrödinger equation ${H}_{0e}\left(B\right){\psi}_{e}\left({\rho}_{e}\right)={E}_{0e}\left(B\right){\psi}_{e}\left({\rho}_{e}\right)$ with Partial Differential Equations model. The results of the two methods are sufficiently consistent. The core radius was varied by the ratio ($a/b$) and three cases of the applied magnetic field are considered: $\gamma =1,1.5$ and 2.

**Figure 3.**Electron probability density in Al${}_{x}$Ga${}_{1-x}$As/GaAs core/shell NW along the radial direction $\rho $ at zero magnetic field ($\gamma =0$). Three core radius values are considered $a=1,1.2$ and $1.4{a}_{D}^{*}$ less than the threshold radius ${a}_{0}$. Each linear representation corresponds to the planar representation, which is the xy-plane cross-section of the electron density distribution at the middle of the cylinder.

**Figure 4.**Electron probability density in Al${}_{x}$Ga${}_{1-x}$As/GaAs core/shell NW along the radial direction $\rho $ at zero magnetic field ($\gamma =0$). Three core radius values are considered $a=1.7,1.8$ and $1.9{a}_{D}^{*}$ greater than the threshold radius ${a}_{0}$. Each linear representation corresponds to the planar representation, which is the xy-plane cross-section of the electron density distribution at the middle of the cylinder.

**Figure 5.**Variation of the electron ground state energy inside Al${}_{x}$Ga${}_{1-x}$As/GaAs-based core/shell NW, as a function of the magnetic field for different values of the core radius: from 1 by 0.1 to 1.5 ${a}_{D}^{*}$. The pace of evolution is semi-parabolic, the magnetic field has a remarkable effect only from an intensity $\gamma \ge 1$.

**Figure 6.**The calculated impurity binding energy as a function of the $a/b$ ratio. The dimensionless measure $\gamma $ of the magnetic field strength is varied from 1 by 0.5 to 2. The donor impurity is located at the core center along the z-axis.

**Figure 7.**The probability density distribution of the electron inside Al${}_{x}$Ga${}_{1-x}$As/GaAs-based core/shell NW with shell radius $b=2{a}_{D}^{*}$ and height $h=20{a}_{D}^{*}$, for different values of the core radius: from $a=1{a}_{D}^{*}$ by 0.2 to 1.8${a}_{D}^{*}$. The square of the wave function describes the ground state level of (

**a**) single electron and (

**b**) electron-donor impurity system. The representation is the $\rho $z-plane cross-section with 2D axi-symmetric model.

**Figure 8.**On-center donor impurity-related photoionization cross-section coefficient in cylindrical Al${}_{x}$Ga${}_{1-x}$As/GaAs-based core/shell NW as a function of the incident photon energy at zero magnetic field. The core radius varies from $1.2{a}_{D}^{*}$ by 0.2 to 1.8${a}_{D}^{*}$.

**Figure 9.**On-center donor impurity related photoionization cross-section coefficient in cylindrical Al${}_{x}$Ga${}_{1-x}$As/GaAs-based core/shell NW as a function of the incident photon energy for two typical core radiuses $a=1.2{a}_{D}^{*}$ and 1.8${a}_{D}^{*}$. We take the following magnetic field strength: $\gamma $ = 0, 1, 1.5 and 2.

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**MDPI and ACS Style**

El-Yadri, M.; El Hamdaoui, J.; Aghoutane, N.; Pérez, L.M.; Baskoutas, S.; Laroze, D.; Díaz, P.; Feddi, E.M.
Optoelectronic Properties of a Cylindrical Core/Shell Nanowire: Effect of Quantum Confinement and Magnetic Field. *Nanomaterials* **2023**, *13*, 1334.
https://doi.org/10.3390/nano13081334

**AMA Style**

El-Yadri M, El Hamdaoui J, Aghoutane N, Pérez LM, Baskoutas S, Laroze D, Díaz P, Feddi EM.
Optoelectronic Properties of a Cylindrical Core/Shell Nanowire: Effect of Quantum Confinement and Magnetic Field. *Nanomaterials*. 2023; 13(8):1334.
https://doi.org/10.3390/nano13081334

**Chicago/Turabian Style**

El-Yadri, Mohamed, Jawad El Hamdaoui, Noreddine Aghoutane, Laura M. Pérez, Sotirios Baskoutas, David Laroze, Pablo Díaz, and El Mustapha Feddi.
2023. "Optoelectronic Properties of a Cylindrical Core/Shell Nanowire: Effect of Quantum Confinement and Magnetic Field" *Nanomaterials* 13, no. 8: 1334.
https://doi.org/10.3390/nano13081334