# Optical Absorption in N-Dimensional Colloidal Quantum Dot Arrays: Influence of Stoichiometry and Applications in Intermediate Band Solar Cells

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## Abstract

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## 1. Introduction

**k·p**approaches, [3] and have therefore neglected atomistic details. The importance of considering such detail will be evident when we show the effects on both electronic structure and optical properties of inverting the stoichiometry of a CQD by interchanging anions and cations. We will also analyse the implications, for their application in IBSCs, of assembling CQDs with anion- and cation-rich surfaces.

## 2. Theoretical Framework

## 3. Results and Discussion

#### 3.1. Electronic Structure

#### 3.2. Absorption Coefficient

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CB | Conduction Band |

CBM | Conduction Band Minimum |

CQD | Colloidal Quantum Dot |

EQD | Epitaxial Quantum Dot |

IB | Intermediate Band |

IBSC | Intermediate Band Solar Cell |

QD | Quantum Dot |

QDSC | Quantum Dot Solar Cell |

VB | Valence Band |

VBM | Valence Band Maximum |

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**Figure 1.**Schematic representation of the simulated CQD arrays in (

**a**) 1D, (

**b**) 2D and (

**c**) 3D. For the sake of clarity, a single quantum dot is highlighted in the 3D structure.

**Figure 2.**Miniband structure for 1D arrays of (

**a**) InP, (

**b**) InAs, and (

**c**) InSb QDs with anion-rich (configuration A, green lines) and cation-rich (configuration B, red lines) surfaces. The energies on the vertical axis are relative to the vacuum level; the horizontal axis sweeps the Brillouin zone, q stands for $\pi $/2${R}_{1}$, i.e., the limits of the Brillouin zone.

**Figure 3.**Miniband structure for 2D arrays of (

**a**) InP, (

**b**) InAs, and (

**c**) InSb QDs with anion-rich (configuration A, green lines) and cation-rich (configuration B, red lines) surfaces. The energies on the vertical axis are relative to the vacuum level; the horizontal axis sweeps the Brillouin zone. The inset in (

**a**) shows a schematic view of the path along the Brillouin zone together with the indexing of its major points.

**Figure 4.**Miniband structure for 3D arrays of (

**a**) InP, (

**b**) InAs, and (

**c**) InSb QDs with anion-rich (configuration A, green lines) and cation-rich (configuration B, red lines) surfaces. The energies on the vertical axis are relative to the vacuum level; the horizontal axis sweeps the Brillouin zone. The inset in (

**a**) shows a schematic view of the path along the Brillouin zone together with the indexing of its major points.

**Figure 5.**Absorption coefficient calculated at 300 K for 1D (

**a**,

**b**), 2D (

**c**,

**d**) and 3D (

**e**,

**f**) arrays of InP dots with radius $R=11.9$ Å and either anion-rich (configuration A, left panels) or cation-rich (configuration B, right panels) surfaces.

**Figure 6.**Absorption coefficient calculated at 300 K for 1D (

**a**,

**b**), 2D (

**c**,

**d**) and 3D (

**e**,

**f**) arrays of InAs dots with radius $R=12.2$ Å and either anion-rich (configuration A, left panels) or cation-rich (configuration B, right panels) surfaces.

**Figure 7.**Absorption coefficient calculated at 300 K for 1D (

**a**,

**b**), 2D (

**c**,

**d**) and 3D (

**e**,

**f**) arrays of InSb dots with radius $R=13.1$ Å and either anion-rich (configuration A, left panels) or cation-rich (configuration B, right panels) surfaces.

**Figure 8.**High symmetry points contribution to the CB states of InP, InAs and InSb QDs with anion-rich surfaces.

**Figure 9.**High symmetry points contribution to the CB states of InP, InAs and InSb QDs with cation-rich surfaces.

**Table 1.**Overlap integrals between neighbouring QDs for CBM states. The integral on the left quantifies the confinement of the wavefunction in the QD, whereas the integral on the right is related to the miniband width.

Overlap Integrals | |||
---|---|---|---|

$\int {\psi}^{*}(\overrightarrow{r})\psi (\overrightarrow{r}-\overrightarrow{R})d\overrightarrow{r}$ (adim.) | $\int {\psi}^{*}(\overrightarrow{r})V(\overrightarrow{r})\psi (\overrightarrow{r}-\overrightarrow{R})d\overrightarrow{r}$ (eV) | ||

InP | 0.0123 | −0.0479 | |

Configuration A | InAs | 0.0141 | −0.0571 |

InSb | 0.0146 | −0.0599 | |

InP | 0.0032 | −0.0076 | |

Configuration B | InAs | 0.0066 | −0.0171 |

InSb | 0.0058 | −0.0125 |

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**MDPI and ACS Style**

Hahn, R.V.H.; Rodríguez-Bolívar, S.; Rodosthenous, P.; Skibinsky-Gitlin, E.S.; Califano, M.; Gómez-Campos, F.M.
Optical Absorption in N-Dimensional Colloidal Quantum Dot Arrays: Influence of Stoichiometry and Applications in Intermediate Band Solar Cells. *Nanomaterials* **2022**, *12*, 3387.
https://doi.org/10.3390/nano12193387

**AMA Style**

Hahn RVH, Rodríguez-Bolívar S, Rodosthenous P, Skibinsky-Gitlin ES, Califano M, Gómez-Campos FM.
Optical Absorption in N-Dimensional Colloidal Quantum Dot Arrays: Influence of Stoichiometry and Applications in Intermediate Band Solar Cells. *Nanomaterials*. 2022; 12(19):3387.
https://doi.org/10.3390/nano12193387

**Chicago/Turabian Style**

Hahn, Rebeca V. H., Salvador Rodríguez-Bolívar, Panagiotis Rodosthenous, Erik S. Skibinsky-Gitlin, Marco Califano, and Francisco M. Gómez-Campos.
2022. "Optical Absorption in N-Dimensional Colloidal Quantum Dot Arrays: Influence of Stoichiometry and Applications in Intermediate Band Solar Cells" *Nanomaterials* 12, no. 19: 3387.
https://doi.org/10.3390/nano12193387