# Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Theoretical Framework

#### 3.1. The Density Operator

#### 3.2. Electron Dynamics of the Open Quantum System S

## 4. Approaching the QD System by a Network with Spatial and Physical-Based Constraints

#### 4.1. A Single QD

#### 4.2. The Quantum System S

#### 4.3. Generating the Network Associated to S System

## 5. Simulation Work

#### 5.1. Methodology

#### 5.2. Testing the Weak Overlap Hypothesis

#### 5.3. Influence of the Minimum Inter-Dot Distance on Quantum Transport

## 6. A Prospective Application

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations and Nomenclature

## Abbreviations

0D | Zero-dimensional |

1D | One-dimensional |

2D | Two-dimensional |

ARP | Average Return Probability |

CB | Conduction Band |

CN | Complex Networks |

CP | Confinement Potential |

CTQW | Continuous-Time Quantum Walks |

DM | Dot Material |

GS | Ground State |

IB | Intermediate Band |

IBSC | Intermediate-Band Solar Cell |

LCAO | Linear Combination of Atomic Integrals |

LME | Lindblad Master Equation |

QD | Quantum Dot |

QD-IBSC | Quantum Dot Intermediate-Band Solar Cell |

QM | Quantum Mechanics |

QT | Quantum Transport |

QW | Quantum Walk |

RGG | Random Geometric Graph |

RN | Random Network |

SAQDs | Self-Assembled Quantum Dots |

SML-QDs | Sub-Monolayer Quantum Dots |

SN | Spatial Network |

SK | Stranski–Krastanow |

SW | Small World |

TB | Tight-binding |

VB | Valence band |

VW | Volmer–Weber |

WL | Wetting layer |

## Nomenclature

$\U0001d7d9$ | Identity operator in Quantum Mechanics. |

$\mathbf{A}$ | Adjacency matrix of a graph $\mathcal{G}$. |

${a}_{ij}$ | Element of the adjacency matrix $\mathbf{A}$. |

$\overline{\alpha}\left(t\right)$ | Average return probability. |

$\mathbf{D}$ | Node degree matrix: $\mathrm{diag}({k}_{1},\cdots ,{k}_{N})$. It is the diagonal matrix formed from the nodes degrees. |

${d}_{E}(i,j)$ | Euclidean distance between any pair of nodes i and j in a network. |

${d}_{ij}$ | Distance between two nodes i and j. It is the length of the shortest path (geodesic path) between them, that is, the minimum number of links when going from one node to the other. |

${d}_{E,Lim}$ | ${d}_{E,Lim}\equiv {d}_{S}$ Euclidean distance limit beyond which there is no link formation. |

${E}_{e}$ | Energy level of a confined electron in a quantum dot. |

${E}_{h}$ | Energy level of a confined hole in a quantum dot. |

${E}_{QD}$ | Discrete electron energy in a quantum dot (QD). |

${\eta}_{QT}$ | Quantum transport efficiency. |

$\mathcal{G}$ | Graph $\mathcal{G}\equiv \mathcal{G}(\mathcal{N},\mathcal{L},\mathbf{W})$, where $\mathcal{N}$ is the set of nodes ($\mathrm{card}\left(\mathcal{N}\right)=N$), $\mathcal{L}$ is the set of links, and $\mathbf{W}$ is weighted adjacency matrix that emerges from our method to link formation. |

${\mathcal{F}}_{B,ij}$ | Boltzmann factor. |

$\widehat{H}$ | Hamiltonian operator corresponding to the total energy of a quantum system. |

$\mathbf{H}$ | Hamiltonian in matrix form. |

ℏ | Reduced Planck constant. |

$\mathcal{H}$ | Hilbert space. |

${\mathcal{H}}_{eq}$ | Equilibrium Hilbert subspace. |

$|i\rangle $ | Ket vector in the Hilbert space $\mathcal{H}$. |

$\langle i|$ | Bra vector in the dual space corresponding to the ket $|i\rangle $ $\in \mathcal{H}$ |

ℓ | Average path length of a network. It is the mean value of distances between any pair of nodes in the network. |

$\mathcal{L}$ | Set of links (edges) of a network (graph). |

$\mathbf{L}$ | Laplacian matrix of a graph $\mathcal{G}$. |

${\mathcal{L}}_{N}$ | Normalizad Laplacian matrix, ${\mathcal{L}}_{N}=$ ${\mathbf{D}}^{-1/2}\mathbf{L}{\mathbf{D}}^{-1/2}$. |

${m}_{e}$ | Electron mass. |

M | Size of a graph $\mathcal{G}$. It is the number of links in the set $\mathcal{L}$. |

N | Order of a graph $\mathcal{G}=(\mathcal{N},\mathcal{L})$. It is the number of nodes in set $\mathcal{N}$, that is the cardinality of set $\mathcal{N}$: $N=\left|\mathcal{N}\right|\equiv \mathrm{card}\left(\mathcal{N}\right)$. |

$\mathcal{N}$ | Set of nodes (or vertices) of a graph. |

${\u25bf}^{2}$ | Laplace operator. |

$P\left(k\right)$ | Probability density function giving the probability that a randomly selected node has k links. |

$|\psi \rangle $ | Ket or vector state in Dirac notation corresponding to the wave function $\psi $. |

${R}_{QD}$ | Radius of the quantum dot. |

${\psi}_{QD}$ | Electron wavefunction in a quantum dot. |

${\tau}_{B}$ | Boltzmann time. |

$\widehat{V}$ | Potential energy operator. |

$-{V}_{C}$ | Depth of confinement potential. |

${U}_{C}\left(r\right)$ | Confining, spherical (depending only on the radial co-ordinate r), finite, and square potential energy. |

${\widehat{U}}_{{\mathcal{L}}_{N}}\left(t\right)$ | Time evolution operator generated by the normalizad Laplacian matrix ${\mathcal{L}}_{N}$. |

${w}_{ij}$ | Weight of the link between node i and j. |

$\mathbf{W}$ | weighted adjacency matrix. |

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**Figure 1.**(

**a**) Illustration of low-dimensional nanostructures. (

**b**) Corresponding density of states (DOS). (

**c**) Simplified quantum dot (QD). (

**d**) Different classes of growth in self-assembled QDs. (

**e**) Three electron gases in an intermediate-band solar cell. (

**f**) QD and energy levels. (

**g**) Distribution of QDs (=nodes). (

**h**) Allowed electron hopping situation. (

**i**,

**j**) Forbidden electron hopping cases. (

**k**) Link generation in the spacial node distribution in (

**g**) according to the processes illustrated in (

**h**,

**i**). See the main text for further details.

**Figure 2.**(

**a**) Position of the electron energy level (eV) below the CB (${E}_{C}=0$ is assumed to be as the energy reference origin) as a function of the quantum dot radius ${R}_{QD}$. (

**b**) Square modulus of the electron wave function for ${R}_{QD}=8$ nm.

**Figure 3.**(

**a**) Mean value of the overlap as a function of the normalized inter-dot distance ${d}_{E,ij}/{R}_{QD}$ in the case in which ${r}_{min}=20\phantom{\rule{4pt}{0ex}}{R}_{QD}$. (

**b**) Mean value of the overlap as a function of ${d}_{E,ij}/{R}_{QD}$ in the case in which ${r}_{min}=40\phantom{\rule{4pt}{0ex}}{R}_{QD}$.

**Figure 4.**(

**a**) Mean value (over 50 networks) of the quantum transport efficiency (QTE) as a function of the minimum inter-dot distance normalized by ${R}_{QD}$, ${r}_{min}/{R}_{QD}$. (

**b**) QD density $(\times {10}^{10}$ cm${}^{-2})$ as a function of ${r}_{min}/{R}_{QD}$.

**Figure 5.**(

**a**) Electron probability components, ${\left|\langle n|\psi \rangle \right|}^{2}$, on each of the kets $|n\rangle $ of a connected network with $N=100$ nodes. (

**a**) Probability components for networks with ${r}_{min}=20{R}_{QD}$. (

**b**) Probability components for networks with ${r}_{min}=60{R}_{QD}$.

**Figure 6.**Average return probability (ARP) as a function of time (adimensional). As we have assumed $\hslash \equiv 1$, then time and energy can be treated as dimensionless. Each value has been obtained as the mean value of the ARP over 50 networks with $N=100$ nodes each. (

**a**) ARP for an ensemble of networks with ${r}_{min}=20{R}_{QD}$. (

**b**) ARP for an ensemble of networks with ${r}_{min}=60\phantom{\rule{4pt}{0ex}}{R}_{QD}$.

**Figure 7.**Gradient of electron wave function, $\widehat{\mathit{e}}\xb7\u25bd|{\psi}_{i}\rangle $, in the dot. Note that $\widehat{\mathit{e}}\xb7\u25bd|{\psi}_{i}\rangle \to 0$ when approaching the dot center.

**Figure 8.**Gradient of electron wave function, $\widehat{\mathit{e}}\xb7\u25bd|{\psi}_{i}\rangle $, in two cases. (

**a**) Gradient of an initial state in which the probability components are very unbalanced. Its overlap with an extended final function on the continuum is expected to be very small. (

**b**) Gradient of an initial state in which the probability components are unevenly distributed. Its overlap with an extended final function on the continuum is expected to be greater than in the case (

**a**).

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**MDPI and ACS Style**

Cuadra, L.; Nieto-Borge, J.C.
Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints. *Nanomaterials* **2021**, *11*, 2056.
https://doi.org/10.3390/nano11082056

**AMA Style**

Cuadra L, Nieto-Borge JC.
Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints. *Nanomaterials*. 2021; 11(8):2056.
https://doi.org/10.3390/nano11082056

**Chicago/Turabian Style**

Cuadra, Lucas, and José Carlos Nieto-Borge.
2021. "Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints" *Nanomaterials* 11, no. 8: 2056.
https://doi.org/10.3390/nano11082056