# Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links

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## Abstract

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## 1. Introduction

## 2. Current State of the Art

## 3. The QD System

## 4. Modeling the QD System as a Spatial Network: The Proposed Model

## 5. Experimental Work: Simulations

#### 5.1. Network Parameters

- The degree distribution of a network captures the probability $P\left(k\right)$ that a randomly chosen node exhibits “degree” k (= number of links). $P\left(k\right)$ and its mean value $\langle k\rangle $ (mean degree) are very useful since it quantifies to what extent nodes are heterogeneous with respect to their connectivity. In fact, many real-world networks exhibit broad, heterogeneous degree distributions. In a degree-heterogeneous network, the probability to find a node with $k>\langle k\rangle $ decreases slower than exponentially, leading to the existence of a non-negligible number of nodes with very high degrees. A key feature of such degree distributions is the so-called scale-free behavior [32], characterized by a degree distribution $P\left(k\right)\sim {k}^{-\gamma}$. This means that most of the nodes have very few links, while only a few nodes have a large percentage of all links. These most connected nodes are called “hubs”.
- The clustering or transitivity [32] quantify the probability that two neighbors of a given node i are connected. This concept is clear in social networks: the fact that usually “the friend of a friend is a friend” leads to high clustering coefficient. The “clustering coefficient” is a local property capturing “the density” of triangles in the graph, that is, two nodes that both are connected to a third node are also directly connected to each other. A node i in the network has ${k}_{i}$ links that connects it to ${k}_{i}$ other nodes. The clustering coefficient of node i is defined as the ratio between the number ${M}_{i}$ of links that actually exist between these ${k}_{i}$ nodes and the maximum possible number of links, that is, ${C}_{i}=2{M}_{i}/{k}_{i}({k}_{i}-1)$. The clustering coefficient of the whole network is:$$\langle \mathcal{C}\rangle =\frac{1}{N}\sum _{i}{C}_{i}.$$
- The average shortest path length, ℓ, quantifies the extent to which a node is accessible from any other [32]. The average path length of a network is the average value of distances between any pair of nodes in the network:$$\ell =\frac{1}{N(N-1)}\sum _{i\ne j}{d}_{ij}$$

#### 5.2. The Network Has a Percolation Transition as the Dot Density Increases

#### 5.3. Studying the Emergence of Electron Transport

## 6. Potential Applications, Strengths, and Weaknesses of the Proposed Method

#### 6.1. Prospective Applications

#### 6.1.1. Intermediate Band Materials

#### 6.1.2. Light-Harvesting Materials

#### 6.2. Strengths and Weaknesses

## 7. Summary and Conclusions

- The spatial network generated by the proposed model prohibits the existence of shortcuts between distant nodes because of the impossibility of the electron tunneling between two very distant QDs. This leads, as expected, to high clustering coefficient and makes it impossible for the network to be small-world.
- The proposed network is also able to capture the inner properties of the QD system: it predicts the system quantum state, its time evolution, and the emergence of quantum transport (QT) as the mean node degree increases (or, equivalently, when the QD increases). In fact, QT efficiency exhibits an abrupt change, from electron localization (no QT) to delocalization (QT emerges), which has also been observed in [60], although with a different approach.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

0D | Zero-dimensional |

1D | One-dimensional |

2D | Two-dimensional |

CN | Complex Networks |

CTQW | Continuous-Time Quantum Walks |

GC | Giant Component |

IB | Intermediate Band |

IBSC | Intermediate Band Solar Cell |

QCP | Quantum Confinement Potential |

QD | Quantum Dot |

QD-IBSC | Quantum Dot Intermediate Band Solar Cell |

QM | Quantum Mechanics |

QT | Quantum Transport |

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 |

VW | Volmer–Weber |

WL | Wetting layer |

## Appendix A

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

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

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

$\langle \mathcal{C}\rangle $ | Mean clustering coefficient of a network. |

$\mathbf{D}$ | Node degree matrix: 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. |

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

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

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

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

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

h | Planck constant. |

ℏ | Reduced Planck constant. |

$|i\rangle $ | Ket vector in the Hilbert space $\mathcal{H}$. It corresponds to the electron wave function in nanostructure (≡ site ≡ node ≡ ket) i. |

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

$\langle k\rangle $ | Average node degree. |

${k}_{i}$ | Degree of a node i. It is the number of links connecting i to any other node. |

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 $\mathcal{G}$. |

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

m | 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. |

${\mathsf{P}}_{j\rightsquigarrow k}$ | Probability for an electron to evolve between kets $|j\rangle $ and $|k\rangle $ in the time interval t. |

$P\left(k\right)$ | 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. |

${S}_{GC}$ | ${S}_{GC}={N}_{GC}/N$ normalized size of the giant component (GC) with respect to the total number of nodes N. |

${s}_{A{P}_{i}}$ | Sum of the probability amplitudes on ket $|i\rangle $, ${s}_{A{P}_{i}}\equiv $ ${\sum}_{i\ne j}{\left({\mathbf{W}}_{PA}\right)}_{i}={\sum}_{i\ne j}\langle i|j\rangle $. |

$\widehat{V}$ | 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 normalized Laplacian matrix ${\mathcal{L}}_{N}$. |

${w}_{ij}$ | Weight of the link between node i and j. We define it as the overlap integral between the electron wave functions in kets i and j or the probability amplitude $\langle i|j\rangle $. |

${\mathbf{W}}_{PA}$ | weighted adjacency matrix whose elements are quantum probability amplitudes. |

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**Figure 1.**(

**a**) Conceptual representation of nano-structures and their corresponding density of states (DOS) for quantum wells, quantum wires and quantum dots; (

**b**) confinement potentials for electrons (${U}_{e}$) and holes (${U}_{h}$) in a type I QD (or simply, QD), and its corresponding density of states; (

**c**) different classes of growth in self-assembled quantum dots. See the main text for details.

**Figure 2.**(

**a**) Random distribution of QDs in ${\mathbb{R}}^{2}$. Each QD, represented by a node, causes a quantum confinement potential. (

**b**) A link between two nodes is allowed if and only if the electron wave-functions at such nodes have non-zero overlap. Any link has a weight that is quantified by such overlap. See the main text for details.

**Figure 3.**(

**a**) Two of the QDs as those in Figure 2a. We consider QDs with radius ${R}_{QD}$ and whose centers are separated by a Euclidean distance ${d}_{E}(i,j)$. $\xi $ represents an axis that passes through the center of both QDs. Along the $\xi $-axis, we have represented in (

**b**) the corresponding QCP ($-{V}_{C}$ inside the node, 0 otherwise) and the electron probability amplitudes (

**c**). An electron can be in both QDs with the represented probability amplitude. (

**d**) We model this with a link whose weight ${w}_{ij}$ is given by the overlap. (

**e**) Opposite case in which the QDs are so far apart (

**f**) that there is no overlap (

**g**) and link formation is not allowed (

**h**).

**Figure 4.**(

**a**) Quantum confinement potential. ${V}_{C}$ is the depth of the potential well while ${R}_{QD}$ is the radius of the QD producing ${U}_{C}$. ${E}_{QD}$ is the energy of the bound state; (

**b**) squared modulus of the corresponding wave function, $|{\psi}_{QD}{|}^{2}$, in Cartesian coordinates; (

**c**) $|{\psi}_{QD}{|}^{2}$ as a function of the normalized radial coordinate, $r/{R}_{QD}$.

**Figure 5.**(

**a**) System with three QDs. Vector $\mathbf{r}$ is the position vector in the metric space. We represent each QD as a node, and we label it with the ket notation $|i\rangle $; (

**b**) methodology to form weighted links according to QMs. Each weight ${w}_{ij}$ is the overlap integral between ${\psi}_{QDi}$ and ${\psi}_{QDj}$ (or, equivalently, $\langle i|j\rangle $ in Dirac’s notation).

**Figure 6.**Overlap between the electron wave functions in two QDs as a function of the distance (between dot centers) normalized by the radius of the QD, ${d}_{E}/{R}_{QD}$. The insets aim to graphically illustrate how increasing the separation between the nodes implies a longer link with much less weight.

**Figure 7.**Normalized size of the giant component with respect to the total number of nodes, ${S}_{GC}={N}_{GC}/N$, as a function of the average node degree $\langle k\rangle $. See the main text for further details.

**Figure 8.**Quantum transport efficiency as a function of $\langle k\rangle $. Insets “1”, “2” and “3” shows network connectivity at different values of $\langle k\rangle $. ${S}_{GC}={N}_{GC}/N$ is the fraction (normalized size) of the giant component.

**Figure 9.**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.

**Table 1.**List of the 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. The sum of the probability components in the whole system is ${\sum}_{n=1}^{N=100}{\left|\langle n|\psi \rangle \right|}^{2}=1$.

0.000172176 | 0.00582029 | 0.00387988 | 0.00106017 | 0.000970012 |

0.0000719148 | 0.00413783 | 0.000454207 | 0.0768675 | 0.00242522 |

0.0298138 | 0.00921407 | 0.0000565646 | 0.000102786 | 0.0000761972 |

0.0938865 | 0.00010824 | 0.0027705 | 0.0000545876 | 0.000735455 |

0.000247052 | 0.000190905 | 0.00274959 | 0.0000908889 | 0.00886395 |

0.00197876 | 0.00202883 | 0.003812 | 0.0197223 | 0.0300932 |

0.000148517 | 0.000360844 | 0.00247567 | 0.0193588 | 0.0000203756 |

0.0000281 | 0.0172012 | 0.00814919 | 0.0147341 | 0.00103512 |

0.000150275 | 0.0000862307 | 0.0294795 | 0.0203633 | 0.00025177 |

0.0000518012 | 0.00469129 | 0.0000315912 | 0.00847614 | 0.00144027 |

0.000285581 | 0.000261477 | 0.000582733 | 0.00371945 | 0.0336232 |

0.0032206 | 0.00980407 | 0.000807111 | 0.000121867 | 0.0459497 |

0.0105552 | 0.00168941 | 0.00534086 | 0.000115002 | 0.0447992 |

0.0001218 | 0.00431284 | $8.72451\times {10}^{-6}$ | 0.000201637 | 0.0188286 |

0.000133976 | 0.0156714 | 0.0137177 | 0.0022036 | 0.000168711 |

0.000250133 | 0.0166772 | 0.00114926 | 0.0344186 | 0.00393284 |

0.000493777 | 0.0928017 | 0.00341317 | 0.00876622 | 0.000893781 |

0.0451846 | 0.0901622 | 0.0000291204 | 0.0000476467 | 0.0000574528 |

0.0117769 | 0.0178166 | 0.0034605 | 0.000244178 | 0.000142244 |

0.00037752 | 0.000129045 | 0.00373535 | 0.00499453 | 0.0118115 |

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**MDPI and ACS Style**

Cuadra, L.; Nieto-Borge, J.C.
Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links. *Nanomaterials* **2021**, *11*, 375.
https://doi.org/10.3390/nano11020375

**AMA Style**

Cuadra L, Nieto-Borge JC.
Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links. *Nanomaterials*. 2021; 11(2):375.
https://doi.org/10.3390/nano11020375

**Chicago/Turabian Style**

Cuadra, Lucas, and José Carlos Nieto-Borge.
2021. "Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links" *Nanomaterials* 11, no. 2: 375.
https://doi.org/10.3390/nano11020375