# Modeling of Quantum Dots with the Finite Element Method

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

**.**

## 3. Results and Discussion

#### 3.1. Acuracy and Computational Time

#### 3.2. Electronic Energy and Wavefunctions of Semiconductor QDs

#### 3.3. Properties of Semiconductor Core/Shell QDs

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The mesh regions used for the calculations: (

**a**) mesh for a rectangular QD, (

**b**) mesh for a spherical QD, (

**c**) mesh for a conical QD, (

**d**) mesh for a cylindrical QD, (

**e**) mesh for an ellipsoid QD, (

**f**) mesh for an ellipsoid of revolution (spheroid), (

**g**) mesh for a nanotadpole, (

**i**) mesh for a nanostar, (

**j**) mesh for the quantum ring.

**Figure 2.**Piecewise Woods–Saxon potential plotted for the following parameters: ${R}_{core}=1\mathrm{nm}$, ${R}_{shell}=1.3\mathrm{nm}$, ${a}_{core}=0.03$.

**Figure 3.**The deviation of the analytically obtained electron energy value from the numerical values for the first 26 states in (

**a**) rectangular, (

**b**) spherical, and (

**c**) cylindrical QDs. The comparison was carried out for the $MaxCellMeasure=0.01$.

**Figure 4.**The dependence of the electron energy on the $\mathrm{MaxCellMeasure}$ parameter for (

**a**) rectangular QDs, (

**b**) spherical QDs, (

**c**) cylindrical QDs, (

**d**) ellipsoidal QDs, (

**e**) spheroidal QDs, (

**f**) QRs, (

**g**) conical QDs, (

**h**) nanotadpole QDs, (

**i**) nanostars.

**Figure 5.**The dependence of computation time of the first 10 states on the $\mathrm{MaxCellMeasure}$ parameter for (

**a**) rectangular QDs, (

**b**) spherical QDs, (

**c**) cylindrical QDs, (

**d**) ellipsoidal QDs, (

**e**) spheroidal QDs, (

**f**) QRs, (

**g**) conical QDs, (

**h**) nanotadpole QDs, (

**i**) nanostars.

**Figure 6.**The values of energy for the electron’s ground state and the first twenty-five excited states for (

**a**) rectangular QDs, (

**b**) spherical QDs, (

**c**) cylindrical QDs, (

**d**) ellipsoidal QDs, (

**e**) spheroidal QDs, (

**f**) QRs, (

**g**) conical QDs, (

**h**) nanotadpole QDs, (

**i**) nanostars.

**Figure 7.**The probability density of the ground state and first three excited states of an electron confined in a spherical GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 8.**The probability density of the ground state and first three excited states of an electron confined in a rectangular GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 9.**The probability density of the ground state and first three excited states of an electron confined in a cylindrical GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 10.**The probability density of the ground state and first three excited states of an electron confined in an ellipsoidal GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 11.**The probability density of the ground state and first three excited states of an electron confined in a spheroidal GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 12.**The probability density of the ground state and first three excited states of an electron confined in a conical GaAs QD. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 13.**The probability density of the ground state and first three excited states of an electron confined in a conical GaAs nanotadpole. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 14.**The probability density of the ground state and first three excited states of an electron confined in a GaAs nanostar. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 15.**The probability density of the ground state and first three excited states of an electron confined in a GaAs QR. The mesh domain used for the calculation is overlaid on the probability density.

**Figure 16.**The segmented blue line is the probability density of the particle’s ground state localized in the core region, the solid red line is the ground state energy of the particle depending on the inner core size for an electron (

**a**) and a hole (

**b**). In (

**a**), you can see three regions: the region that has a red pattern represents the core radii where the localization probability of the hole in the core is less than $50\%$. The second region, represented by the red and blue pattern, shows the core radii where the hole’s localization probability in the core is larger than $50\%$ but the electron’s localization probability is lower than $50\%$. The third region does not have a pattern, and shows the core radii where both the hole’s and the electron’s localization probability in the core are larger than $50\%$.

The Structure | Values of Geometrical Parameters (nm) |
---|---|

Rectangular QD | ${L}_{x}=40,\hspace{0.17em}\hspace{0.17em}{L}_{y}=20,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{L}_{z}=20$ |

Spherical QD | $R=10$ |

Cylindrical QD | $R=5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}H=30$ |

Ellipsoidal QD | ${a}_{x}=\hspace{0.17em}30,\hspace{0.17em}\hspace{0.17em}{b}_{y}=10,\hspace{0.17em}{c}_{z}=\hspace{0.17em}20,\hspace{0.17em}\hspace{0.17em}$ |

Spheroidal QD | $a=\hspace{0.17em}30,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}b=\hspace{0.17em}20,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}c=20$ |

QR | ${R}_{inner}=10,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_{outer}=20$ |

Conical QD | $R=12,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}H=20$ |

Nanotadpole | ${R}_{head}=12,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_{tail}=6,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{H}_{tail}=20$ |

Nanostar | ${R}_{sphere}=10,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_{cone}=10,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{H}_{cone}=7.5$ |

Part Name | Model |
---|---|

CPU | AMD Ryzen Thresadripper 3990X |

RAM | Kingston 4 × 32 GB 3600 MHz |

GPU | Double NVIDIA GeForce RTX 3090 |

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## Share and Cite

**MDPI and ACS Style**

Mantashian, G.A.; Mantashyan, P.A.; Hayrapetyan, D.B.
Modeling of Quantum Dots with the Finite Element Method. *Computation* **2023**, *11*, 5.
https://doi.org/10.3390/computation11010005

**AMA Style**

Mantashian GA, Mantashyan PA, Hayrapetyan DB.
Modeling of Quantum Dots with the Finite Element Method. *Computation*. 2023; 11(1):5.
https://doi.org/10.3390/computation11010005

**Chicago/Turabian Style**

Mantashian, G.A., P.A. Mantashyan, and D.B. Hayrapetyan.
2023. "Modeling of Quantum Dots with the Finite Element Method" *Computation* 11, no. 1: 5.
https://doi.org/10.3390/computation11010005