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

^{1}

^{2}

^{*}

## Abstract

**:**

## 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. |

## References

- Cotta, M.A. Quantum Dots and Their Applications: What Lies Ahead? ACS Appl. Nano Mater.
**2020**, 3, 4920–4924. [Google Scholar] [CrossRef] - Harrison, P.; Valavanis, A. Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures; John Wiley & Sons: Hoboken, NJ, USA, 2016. [Google Scholar]
- Sengupta, S.; Chakrabarti, S. Structural, Optical and Spectral Behaviour of InAs-based Quantum Dot Heterostructures: Applications for High-performance Infrared Photodetectors; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Nowozin, T. Self-Organized Quantum Dots for Memories: Electronic Properties and Carrier Dynamics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Cipriano, L.A.; Di Liberto, G.; Tosoni, S.; Pacchioni, G. Quantum confinement in group III–V semiconductor 2D nanostructures. Nanoscale
**2020**, 12, 17494–17501. [Google Scholar] [CrossRef] - Bimberg, D.; Grundmann, M.; Ledentsov, N.N. Quantum Dot Heterostructures; John Wiley & Sons: Hoboken, NJ, USA, 1999. [Google Scholar]
- Wang, Z.M. Self-Assembled Quantum Dots; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007; Volume 1. [Google Scholar]
- Kim, J.S.; Shin, J.C.; Kim, J.O.; Noh, S.K.; Lee, S.J.; Krishna, S. Photoluminescence study of InAs/InGaAs sub-monolayer quantum dot infrared photodetectors with various numbers of multiple stack layers. J. Lumin.
**2019**, 207, 512–519. [Google Scholar] - Leon, R.; Lobo, C.; Chin, T.; Woodall, J.; Fafard, S.; Ruvimov, S.; Liliental-Weber, Z.; Stevens Kalceff, M. Self-forming InAs/GaP quantum dots by direct island growth. Appl. Phys. Lett.
**1998**, 72, 1356–1358. [Google Scholar] [CrossRef] - Germann, T.; Strittmatter, A.; Pohl, J.; Pohl, U.; Bimberg, D.; Rautiainen, J.; Guina, M.; Okhotnikov, O. High-power semiconductor disk laser based on In As/Ga As submonolayer quantum dots. Appl. Phys. Lett.
**2008**, 92, 101123. [Google Scholar] [CrossRef] - Xu, Z.; Birkedal, D.; Hvam, J.M.; Zhao, Z.; Liu, Y.; Yang, K.; Kanjilal, A.; Sadowski, J. Structure and optical anisotropy of vertically correlated submonolayer InAs/GaAs quantum dots. Appl. Phys. Lett.
**2003**, 82, 3859–3861. [Google Scholar] [CrossRef][Green Version] - Kiyama, H.; Korsch, A.; Nagai, N.; Kanai, Y.; Matsumoto, K.; Hirakawa, K.; Oiwa, A. Single-electron charge sensing in self-assembled quantum dots. Sci. Rep.
**2018**, 8, 1–6. [Google Scholar] [CrossRef][Green Version] - Gao, W.; Fallahi, P.; Togan, E.; Miguel-Sánchez, J.; Imamoglu, A. Observation of entanglement between a quantum dot spin and a single photon. Nature
**2012**, 491, 426–430. [Google Scholar] [CrossRef] - De Greve, K.; Yu, L.; McMahon, P.L.; Pelc, J.S.; Natarajan, C.M.; Kim, N.Y.; Abe, E.; Maier, S.; Schneider, C.; Kamp, M.; et al. Quantum-dot spin–photon entanglement via frequency downconversion to telecom wavelength. Nature
**2012**, 491, 421–425. [Google Scholar] [CrossRef] - Michler, P.; Kiraz, A.; Becher, C.; Schoenfeld, W.; Petroff, P.; Zhang, L.; Hu, E.; Imamoglu, A. A quantum dot single-photon turnstile device. Science
**2000**, 290, 2282–2285. [Google Scholar] [CrossRef][Green Version] - Press, D.; Ladd, T.D.; Zhang, B.; Yamamoto, Y. Complete quantum control of a single quantum dot spin using ultrafast optical pulses. Nature
**2008**, 456, 218–221. [Google Scholar] [CrossRef] [PubMed] - Deacon, R.S.; Oiwa, A.; Sailer, J.; Baba, S.; Kanai, Y.; Shibata, K.; Hirakawa, K.; Tarucha, S. Cooper pair splitting in parallel quantum dot Josephson junctions. Nat. Commun.
**2015**, 6, 1–7. [Google Scholar] - Hamaya, K.; Masubuchi, S.; Kawamura, M.; Machida, T.; Jung, M.; Shibata, K.; Hirakawa, K.; Taniyama, T.; Ishida, S.; Arakawa, Y. Spin transport through a single self-assembled InAs quantum dot with ferromagnetic leads. Appl. Phys. Lett.
**2007**, 90, 053108. [Google Scholar] [CrossRef][Green Version] - Kanai, Y.; Deacon, R.; Takahashi, S.; Oiwa, A.; Yoshida, K.; Shibata, K.; Hirakawa, K.; Tokura, Y.; Tarucha, S. Electrically tuned spin–orbit interaction in an InAs self-assembled quantum dot. Nat. Nanotechnol.
**2011**, 6, 511–516. [Google Scholar] [CrossRef] - Takahashi, S.; Deacon, R.; Oiwa, A.; Shibata, K.; Hirakawa, K.; Tarucha, S. Electrically tunable three-dimensional g-factor anisotropy in single InAs self-assembled quantum dots. Phys. Rev. B
**2013**, 87, 161302. [Google Scholar] [CrossRef][Green Version] - Kanai, Y.; Deacon, R.; Oiwa, A.; Yoshida, K.; Shibata, K.; Hirakawa, K.; Tarucha, S. Electrical control of Kondo effect and superconducting transport in a side-gated InAs quantum dot Josephson junction. Phys. Rev. B
**2010**, 82, 054512. [Google Scholar] [CrossRef][Green Version] - Qi, H.; Wang, S.; Jiang, X.; Fang, Y.; Wang, A.; Shen, H.; Du, Z. Research progress and challenges of blue light-emitting diodes based on II–VI semiconductor quantum dots. J. Mater. Chem. C
**2020**, 8, 10160–10173. [Google Scholar] [CrossRef] - Lv, Z.; Wang, Y.; Chen, J.; Wang, J.; Zhou, Y.; Han, S.T. Semiconductor quantum dots for memories and neuromorphic computing systems. Chem. Rev.
**2020**, 120, 3941–4006. [Google Scholar] [CrossRef] - Chen, J.; Du, W.; Shi, J.; Li, M.; Wang, Y.; Zhang, Q.; Liu, X. Perovskite quantum dot lasers. InfoMat
**2020**, 2, 170–183. [Google Scholar] [CrossRef][Green Version] - Geiregat, P.; Van Thourhout, D.; Hens, Z. A bright future for colloidal quantum dot lasers. NPG Asia Mater.
**2019**, 11, 1–8. [Google Scholar] [CrossRef][Green Version] - Norman, J.C.; Jung, D.; Zhang, Z.; Wan, Y.; Liu, S.; Shang, C.; Herrick, R.W.; Chow, W.W.; Gossard, A.C.; Bowers, J.E. A review of high-performance quantum dot lasers on silicon. IEEE J. Quantum Electron.
**2019**, 55, 1–11. [Google Scholar] [CrossRef] - Bimberg, D.; Kirstaedter, N.; Ledentsov, N.; Alferov, Z.I.; Kop’Ev, P.; Ustinov, V. InGaAs-GaAs quantum-dot lasers. IEEE J. Sel. Top. Quantum Electron.
**1997**, 3, 196–205. [Google Scholar] [CrossRef] - Vichi, S.; Bietti, S.; Khalili, A.; Costanzo, M.; Cappelluti, F.; Esposito, L.; Somaschini, C.; Fedorov, A.; Tsukamoto, S.; Rauter, P.; et al. Droplet epitaxy quantum dot based infrared photodetectors. Nanotechnology
**2020**, 31, 245203. [Google Scholar] [CrossRef] - Ren, A.; Yuan, L.; Xu, H.; Wu, J.; Wang, Z. Recent progress of III–V quantum dot infrared photodetectors on silicon. J. Mater. Chem. C
**2019**, 7, 14441–14453. [Google Scholar] [CrossRef] - Sogabe, T.; Shen, Q.; Yamaguchi, K. Recent progress on quantum dot solar cells: A review. J. Photonics Energy
**2016**, 6, 040901. [Google Scholar] [CrossRef] - Grundmann, M.; Stier, O.; Bimberg, D. InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure. Phys. Rev. B
**1995**, 52, 11969. [Google Scholar] [CrossRef] - Barabási, A.L. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep.
**2006**, 424, 175–308. [Google Scholar] [CrossRef] - Strogatz, S.H. Exploring complex networks. Nature
**2001**, 410, 268–276. [Google Scholar] [CrossRef][Green Version] - Cuadra, L.; Salcedo-Sanz, S.; Del Ser, J.; Jiménez-Fernández, S.; Geem, Z.W. A critical review of robustness in power grids using complex networks concepts. Energies
**2015**, 8, 9211–9265. [Google Scholar] [CrossRef][Green Version] - Cuadra, L.; Pino, M.D.; Nieto-Borge, J.C.; Salcedo-Sanz, S. Optimizing the structure of distribution smart grids with renewable generation against abnormal conditions: A complex networks approach with evolutionary algorithms. Energies
**2017**, 10, 1097. [Google Scholar] [CrossRef][Green Version] - Doyle, J.C.; Alderson, D.L.; Li, L.; Low, S.; Roughan, M.; Shalunov, S.; Tanaka, R.; Willinger, W. The “robust yet fragile” nature of the Internet. Proc. Natl. Acad. Sci. USA
**2005**, 102, 14497–14502. [Google Scholar] [CrossRef][Green Version] - Chimal-Eguía, J.C.; Castillo-Montiel, E.; Paez-Hernández, R.T. Properties of the vascular networks in malignant tumors. Entropy
**2020**, 22, 166. [Google Scholar] [CrossRef][Green Version] - Braun, P.; Gingras, A.C. History of protein–protein interactions: From egg-white to complex networks. Proteomics
**2012**, 12, 1478–1498. [Google Scholar] [CrossRef] - Guimera, R.; Amaral, L.A.N. Functional cartography of complex metabolic networks. Nature
**2005**, 433, 895–900. [Google Scholar] [CrossRef][Green Version] - Newman, M. Networks; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47. [Google Scholar] [CrossRef][Green Version] - Montoya, J.M.; Solé, R.V. Small world patterns in food webs. J. Theor. Biol.
**2002**, 214, 405–412. [Google Scholar] [CrossRef][Green Version] - Pond, T.; Magsarjav, S.; South, T.; Mitchell, L.; Bagrow, J.P. Complex contagion features without social reinforcement in a model of social information flow. Entropy
**2020**, 22, 265. [Google Scholar] [CrossRef][Green Version] - Iannelli, F.; Koher, A.; Brockmann, D.; Hövel, P.; Sokolov, I.M. Effective distances for epidemics spreading on complex networks. Phys. Rev. E
**2017**, 95, 012313. [Google Scholar] [CrossRef][Green Version] - Liu, W.; Liu, C.; Yang, Z.; Liu, X.; Zhang, Y.; Wei, Z. Modeling the propagation of mobile malware on complex networks. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 37, 249–264. [Google Scholar] [CrossRef] - Ding, L.; Liu, S.Y.; Yang, Q.; Xu, X.K. Uncovering the Dependence of Cascading Failures on Network Topology by Constructing Null Models. Entropy
**2019**, 21, 1119. [Google Scholar] [CrossRef][Green Version] - Dobson, I.; Carreras, B.A.; Lynch, V.E.; Newman, D.E. Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization. Chaos Interdiscip. J. Nonlinear Sci.
**2007**, 17, 026103. [Google Scholar] [CrossRef] - Fu, X.; Li, W. Cascading failures of wireless sensor networks. In Proceedings of the 11th IEEE International Conference on Networking, Sensing and Control, Miami, FL, USA, 7–9 April 2014; pp. 631–636. [Google Scholar]
- Cui, L.; Kumara, S.; Albert, R. Complex networks: An engineering view. Circuits Syst. Mag. IEEE
**2010**, 10, 10–25. [Google Scholar] [CrossRef] - Crucitti, P.; Latora, V.; Marchiori, M. Model for cascading failures in complex networks. Phys. Rev. E
**2004**, 69, 045104. [Google Scholar] [CrossRef][Green Version] - Kinney, R.; Crucitti, P.; Albert, R.; Latora, V. Modeling cascading failures in the North American power grid. Eur. Phys. J. Condens. Matter Complex Syst.
**2005**, 46, 101–107. [Google Scholar] [CrossRef] - Barthélemy, M. Spatial networks. Phys. Rep.
**2011**, 499, 1–101. [Google Scholar] [CrossRef][Green Version] - Bashan, A.; Berezin, Y.; Buldyrev, S.V.; Havlin, S. The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys.
**2013**, 9, 667–672. [Google Scholar] - Barthelemy, M. Morphogenesis of spatial networks; Springer: New York, NY, USA, 2018. [Google Scholar]
- Zhao, J.; Li, D.; Sanhedrai, H.; Cohen, R.; Havlin, S. Spatio-temporal propagation of cascading overload failures in spatially embedded networks. Nat. Commun.
**2016**, 7, 1–6. [Google Scholar] [CrossRef][Green Version] - Penrose, M. Random Geometric Graphs; Oxford University Press: Cambridge, UK, 2003; Volume 5. [Google Scholar]
- Kenniche, H.; Ravelomananana, V. Random Geometric Graphs as model of wireless sensor networks. In Proceedings of the 2010 The 2nd International Conference on Computer and Automation Engineering (ICCAE), Singapore, 26–28 February 2010; Volume 4, pp. 103–107. [Google Scholar]
- Nemeth, G.; Vattay, G. Giant clusters in random ad hoc networks. Phys. Rev. E
**2003**, 67, 036110. [Google Scholar] [CrossRef][Green Version] - Gong, L.; Tong, P. von Neumann entropy and localization-delocalization transition of electron states in quantum small-world networks. Phys. Rev. E
**2006**, 74, 056103. [Google Scholar] [CrossRef][Green Version] - Umeyama, T.; Igarashi, K.; Sasada, D.; Tamai, Y.; Ishida, K.; Koganezawa, T.; Ohtani, S.; Tanaka, K.; Ohkita, H.; Imahori, H. Efficient light-harvesting, energy migration, and charge transfer by nanographene-based nonfullerene small-molecule acceptors exhibiting unusually long excited-state lifetime in the film state. Chem. Sci.
**2020**, 11, 3250–3257. [Google Scholar] [CrossRef][Green Version] - Biamonte, J.; Faccin, M.; De Domenico, M. Complex networks from classical to quantum. Commun. Phys.
**2019**, 2, 1–10. [Google Scholar] [CrossRef][Green Version] - Sánchez-Burillo, E.; Duch, J.; Gómez-Gardenes, J.; Zueco, D. Quantum navigation and ranking in complex networks. Sci. Rep.
**2012**, 2, 605. [Google Scholar] [CrossRef][Green Version] - Venegas-Andraca, S.E. Quantum walks: A comprehensive review. Quantum Inf. Process.
**2012**, 11, 1015–1106. [Google Scholar] [CrossRef] - Masuda, N.; Porter, M.A.; Lambiotte, R. Random walks and diffusion on networks. Phys. Rep.
**2017**, 716, 1–58. [Google Scholar] [CrossRef] - Susskind, L.; Friedman, A. Quantum Mechanics: The Theoretical Minimum; Penguin Books: London, UK, 2015. [Google Scholar]
- Ritter, S.; Nölleke, C.; Hahn, C.; Reiserer, A.; Neuzner, A.; Uphoff, M.; Mücke, M.; Figueroa, E.; Bochmann, J.; Rempe, G. An elementary quantum network of single atoms in optical cavities. Nature
**2012**, 484, 195. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Dolgushev, M.; Galiceanu, M. Complex quantum networks: From universal breakdown to optimal transport. Phys. Rev. E
**2016**, 93, 022304. [Google Scholar] [CrossRef][Green Version] - Faccin, M.; Migdał, P.; Johnson, T.H.; Bergholm, V.; Biamonte, J.D. Community detection in quantum complex networks. Phys. Rev. X
**2014**, 4, 041012. [Google Scholar] [CrossRef][Green Version] - Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys.
**2008**, 129, 11B603. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Blumen, A. Efficiency of quantum and classical transport on graphs. Phys. Rev. E
**2006**, 73, 066117. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Blumen, A. Continuous-time quantum walks: Models for coherent transport on complex networks. Phys. Rep.
**2011**, 502, 37–87. [Google Scholar] [CrossRef][Green Version] - Darázs, Z.; Kiss, T. Pólya number of the continuous-time quantum walks. Phys. Rev. A
**2010**, 81, 062319. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Volta, A.; Blumen, A. Asymmetries in symmetric quantum walks on two-dimensional networks. Phys. Rev. A
**2005**, 72, 042334. [Google Scholar] [CrossRef][Green Version] - Agliari, E.; Blumen, A.; Mülken, O. Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. Math. Theor.
**2008**, 41, 445301. [Google Scholar] [CrossRef] - Mülken, O.; Bierbaum, V.; Blumen, A. Coherent exciton transport in dendrimers and continuous-time quantum walks. J. Chem. Phys.
**2006**, 124, 124905. [Google Scholar] [CrossRef][Green Version] - Agliari, E.; Blumen, A.; Muelken, O. Quantum-walk approach to searching on fractal structures. Phys. Rev. A
**2010**, 82, 012305. [Google Scholar] [CrossRef][Green Version] - Blumen, A.; Bierbaum, V.; Mülken, O. Coherent dynamics on hierarchical systems. Phys. Stat. Mech. Its Appl.
**2006**, 371, 10–15. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Blumen, A. Slow transport by continuous time quantum walks. Phys. Rev. E
**2005**, 71, 016101. [Google Scholar] [CrossRef][Green Version] - Xu, X.P.; Li, W.; Liu, F. Coherent transport on Apollonian networks and continuous-time quantum walks. Phys. Rev. E
**2008**, 78, 052103. [Google Scholar] [CrossRef][Green Version] - Xu, X.; Liu, F. Coherent exciton transport on scale-free networks. New J. Phys.
**2008**, 10, 123012. [Google Scholar] [CrossRef] - Mülken, O.; Pernice, V.; Blumen, A. Quantum transport on small-world networks: A continuous-time quantum walk approach. Phys. Rev. E
**2007**, 76, 051125. [Google Scholar] [CrossRef][Green Version] - Salimi, S. Continuous-time quantum walks on star graphs. Ann. Phys.
**2009**, 324, 1185–1193. [Google Scholar] [CrossRef][Green Version] - Anishchenko, A.; Blumen, A.; Mülken, O. Enhancing the spreading of quantum walks on star graphs by additional bonds. Quantum Inf. Process.
**2012**, 11, 1273–1286. [Google Scholar] [CrossRef][Green Version] - Kulvelis, N.; Dolgushev, M.; Mülken, O. Universality at breakdown of quantum transport on complex networks. Phys. Rev. Lett.
**2015**, 115, 120602. [Google Scholar] [CrossRef][Green Version] - Watts, D.J.; Strogatz, S.H. Collective dynamics of “small-world” networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] - Caldarelli, G.; Vespignani, A. Large Scale Structure and Dynamics of Complex Networks: From Information Technology to Finance and Natural Science; World Scientific: Singapore, 2007; Volume 2. [Google Scholar]
- Newman, M.E.; Watts, D.J. Renormalization group analysis of the small-world network model. Phys. Lett. A
**1999**, 263, 341–346. [Google Scholar] [CrossRef][Green Version] - Martinez-Mendoza, A.; Alcazar-López, A.; Méndez-Bermúdez, J. Scattering and transport properties of tight-binding random networks. Phys. Rev. E
**2013**, 88, 012126. [Google Scholar] [CrossRef][Green Version] - Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Mandl, F. Quantum Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1992. [Google Scholar]
- Galindo, A.; Pascual, P. Quantum Mechanics I; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Ballentine, L.E. Quantum Mechanics: A Modern Development; World Scientific Publishing Company: Singapore, 2014. [Google Scholar]
- Cohen-Tannoudji, C.; Diu, B.; Laloe, F.; Dui, B. Quantum Mechanics; Wiley-Interscience: Hoboken, NJ, USA, 2006. [Google Scholar]
- Farhi, E.; Gutmann, S. Quantum computation and decision trees. Phys. Rev. A
**1998**, 58, 915. [Google Scholar] [CrossRef][Green Version] - Faccin, M.; Johnson, T.; Biamonte, J.; Kais, S.; Migdał, P. Degree distribution in quantum walks on complex networks. Phys. Rev. X
**2013**, 3, 041007. [Google Scholar] [CrossRef][Green Version] - Mülken, O.; Blumen, A. From Continuous-Time Random Walks to Continuous-Time Quantum Walks: Disordered Networks. In Nonlinear Phenomena in Complex Systems: From Nano to Macro Scale; Springer: Berlin/Heidelberg, Germany, 2014; pp. 189–197. [Google Scholar]
- Darázs, Z.; Anishchenko, A.; Kiss, T.; Blumen, A.; Mülken, O. Transport properties of continuous-time quantum walks on Sierpinski fractals. Phys. Rev. E
**2014**, 90, 032113. [Google Scholar] [CrossRef][Green Version] - Anishchenko, A.; Blumen, A.; Muelken, O. Geometrical aspects of quantum walks on random two-dimensional structures. Phys. Rev. E
**2013**, 88, 062126. [Google Scholar] [CrossRef][Green Version] - Ray, R.K. Solving Quantum Random Walker Using Steepest Entropy Ascent Ansatz: A Pathway Towards Typicality. arXiv
**2019**, arXiv:1907.04548. [Google Scholar] - Gualtieri, V.; Benedetti, C.; Paris, M.G. Quantum-classical dynamical distance and quantumness of quantum walks. Phys. Rev. A
**2020**, 102, 012201. [Google Scholar] [CrossRef] - Mülken, O.; Blumen, A. Spacetime structures of continuous-time quantum walks. Phys. Rev. E
**2005**, 71, 036128. [Google Scholar] [CrossRef][Green Version] - Stevanovic, D. Applications of graph spectra in quantum physics. In Selected Topics in Applications of Graph Spectra; Mathematical Institute of the Serbian Academy of Sciences and Arts: Belgrade, Serbia, 2011; pp. 85–111. [Google Scholar]
- Lee, D.; Kahng, B.; Cho, Y.; Goh, K.I.; Lee, D.S. Recent advances of percolation theory in complex networks. J. Korean Phys. Soc.
**2018**, 73, 152–164. [Google Scholar] [CrossRef][Green Version] - Luque, A.; Martí, A.; Stanley, C. Understanding intermediate-band solar cells. Nat. Photonics
**2012**, 6, 146–152. [Google Scholar] [CrossRef][Green Version] - Luque, A.; Martí, A. Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels. Phys. Rev. Lett.
**1997**, 78, 5014. [Google Scholar] [CrossRef] - Shockley, W.; Queisser, H.J. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys.
**1961**, 32, 510–519. [Google Scholar] [CrossRef] - Luque, A.; Linares, P.; Antolín, E.; Ramiro, I.; Farmer, C.; Hernández, E.; Tobías, I.; Stanley, C.; Martí, A. Understanding the operation of quantum dot intermediate band solar cells. J. Appl. Phys.
**2012**, 111, 044502. [Google Scholar] [CrossRef][Green Version] - López, E.; Datas, A.; Ramiro, I.; Linares, P.; Antolín, E.; Artacho, I.; Martí, A.; Luque, A.; Shoji, Y.; Sogabe, T.; et al. Demonstration of the operation principles of intermediate band solar cells at room temperature. Sol. Energy Mater. Sol. Cells
**2016**, 149, 15–18. [Google Scholar] [CrossRef][Green Version] - Luque, A.; Martí, A.; López, N.; Antolín, E.; Cánovas, E.; Stanley, C.; Farmer, C.; Caballero, L.; Cuadra, L.; Balenzategui, J. Experimental analysis of the quasi-Fermi level split in quantum dot intermediate-band solar cells. Appl. Phys. Lett.
**2005**, 87, 083505. [Google Scholar] [CrossRef] - Martí, A.; Antolín, E.; Stanley, C.; Farmer, C.; López, N.; Díaz, P.; Cánovas, E.; Linares, P.; Luque, A. Production of photocurrent due to intermediate-to-conduction-band transitions: A demonstration of a key operating principle of the intermediate-band solar cell. Phys. Rev. Lett.
**2006**, 97, 247701. [Google Scholar] [CrossRef] - Datas, A.; López, E.; Ramiro, I.; Antolín, E.; Martí, A.; Luque, A.; Tamaki, R.; Shoji, Y.; Sogabe, T.; Okada, Y. Intermediate band solar cell with extreme broadband spectrum quantum efficiency. Phys. Rev. Lett.
**2015**, 114, 157701. [Google Scholar] [CrossRef][Green Version] - Wełna, M.; Żelazna, K.; Létoublon, A.; Cornet, C.; Kudrawiec, R. Stability of the intermediate band energy position upon temperature changes in GaNP and GaNPAs. Sol. Energy Mater. Sol. Cells
**2019**, 196, 131–137. [Google Scholar] [CrossRef] - Jiang, J.; Zhou, W.; Xue, Y.; Ning, H.; Liang, X.; Zhou, W.; Guo, J.; Huang, D. Intermediate band insertion by group-IIIA elements alloying in a low cost solar cell absorber CuYSe2: A first-principles study. Phys. Lett. A
**2019**, 383, 1972–1976. [Google Scholar] [CrossRef] - Ramiro, I.; Martí, A.; Antolín, E.; Luque, A. Review of experimental results related to the operation of intermediate band solar cells. IEEE J. Photovoltaics
**2014**, 4, 736–748. [Google Scholar] [CrossRef][Green Version] - Martí, A.; Luque, A. Fundamentals of intermediate band solar cells. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 209–228. [Google Scholar]
- Marrón, D.F. Thin-Film Technology in Intermediate Band Solar Cells. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 277–307. [Google Scholar]
- Foxon, C.T.; Novikov, S.V.; Campion, R.P. InGaN Technology for IBSC Applications. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 309–319. [Google Scholar]
- Olea, J.; Pastor, D.; Luque, M.T.; Mártil, I.; Díaz, G.G. Ion implant technology for intermediate band solar cells. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 321–346. [Google Scholar]
- Antolín, E.; Martí, A.; Olea, J.; Pastor, D.; González-Díaz, G.; Mártil, I.; Luque, A. Lifetime recovery in ultrahighly titanium-doped silicon for the implementation of an intermediate band material. Appl. Phys. Lett.
**2009**, 94, 042115. [Google Scholar] [CrossRef][Green Version] - Persans, P.D.; Berry, N.E.; Recht, D.; Hutchinson, D.; Peterson, H.; Clark, J.; Charnvanichborikarn, S.; Williams, J.S.; DiFranzo, A.; Aziz, M.J.; et al. Photocarrier lifetime and transport in silicon supersaturated with sulfur. Appl. Phys. Lett.
**2012**, 101, 111105. [Google Scholar] [CrossRef][Green Version] - Luque, A.; Martí, A.; Antolín, E.; Tablero, C. Intermediate bands versus levels in non-radiative recombination. Phys. Condens. Matter
**2006**, 382, 320–327. [Google Scholar] [CrossRef] - Mott, N. Metal-insulator transition. Rev. Mod. Phys.
**1968**, 40, 677. [Google Scholar] [CrossRef] - Mott, N.F.; Davis, E.A. Electronic Processes in Non-Crystalline Materials; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Scholes, G.D.; Fleming, G.R.; Olaya-Castro, A.; Van Grondelle, R. Lessons from nature about solar light harvesting. Nat. Chem.
**2011**, 3, 763–774. [Google Scholar] [CrossRef] - Kundu, S.; Patra, A. Nanoscale strategies for light harvesting. Chem. Rev.
**2017**, 117, 712–757. [Google Scholar] [CrossRef][Green Version] - Sarovar, M.; Ishizaki, A.; Fleming, G.R.; Whaley, K.B. Quantum entanglement in photosynthetic light-harvesting complexes. Nat. Phys.
**2010**, 6, 462–467. [Google Scholar] [CrossRef] - Nalbach, P.; Thorwart, M. Enhanced quantum efficiency of light-harvesting in a biomolecular quantum “steam engine”. Proc. Natl. Acad. Sci. USA
**2013**, 110, 2693–2694. [Google Scholar] [CrossRef][Green Version] - Jiang, Y.; McNeill, J. Light-harvesting and amplified energy transfer in conjugated polymer nanoparticles. Chem. Rev.
**2017**, 117, 838–859. [Google Scholar] [CrossRef] - Yang, P.Y.; Cao, J. Steady-state analysis of light-harvesting energy transfer driven by incoherent light: From dimers to networks. J. Phys. Chem. Lett.
**2020**, 11, 7204–7211. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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