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

Abstract

This paper focuses on modeling a disordered system of quantum dots (QDs) by using complex networks with spatial and physical-based constraints. The first constraint is that, although QDs (=nodes) are randomly distributed in a metric space, they have to fulfill the condition that there is a minimum inter-dot distance that cannot be violated (to minimize electron localization). The second constraint arises from our process of weighted link formation, which is consistent with the laws of quantum physics and statistics: it not only takes into account the overlap integrals but also Boltzmann factors to include the fact that an electron can hop from one QD to another with a different energy level. Boltzmann factors and coherence naturally arise from the Lindblad master equation. The weighted adjacency matrix leads to a Laplacian matrix and a time evolution operator that allows the computation of the electron probability distribution and quantum transport efficiency. The results suggest that there is an optimal inter-dot distance that helps reduce electron localization in QD clusters and make the wave function better extended. As a potential application, we provide recommendations for improving QD intermediate-band solar cells.

Persistence in complex systems

“…The existence of a threshold that separates two different macroscopic phases (absence/ persistence of a disease) is an instance of percolation transitions on complex networks [214]. An illustrative example of percolation during the growth of networks is the emergence of electron transport in some networks that model systems of disordered quantum dots (QDs) [348,349]. In this approach, a QD −which confines most of the wavefunction inside it− is encoded by a node, while electron hopping between two QDs (nodes) is represented by a link.…”

“…In this approach, a QD −which confines most of the wavefunction inside it− is encoded by a node, while electron hopping between two QDs (nodes) is represented by a link. Electron hopping is related to the overlapping of the electron wavefunctions in both nodes [348,349]. In the case in which the dot density is very small, these are disconnected because the wavefunctions do not overlap.…”

Organic Disordered Semiconductors as Networks Embedded in Space and Energy

“…As suggested, the most striking idea of small-world networks characterized by high local clustering and short average shortest path between any two nodes is that even though they can be made up of a huge number of interacting nodes, they nonetheless greatly enhance the exchange of information [ 81 ] (social networks [ 82 ], human brain [ 83 , 84 ]); matter (electrons in quantum dot systems [ 54 , 55 ], sap in vascular networks in plants [ 85 ]); or energy (in power grids [ 61 , 62 ]) between the involved nodes. However, as discussed in [ 95 ], it is necessary to quantify the shortness of ℓ is and the height of .…”

“…All of them have in common the fact that they consist of a large number of interacting elements that can be represented using a network (or, mathematically, a graph) [ 53 ], that is, a collection of “nodes” (or “vertices”) attached by “links” (or “edges”). Simply put, a node represents an interacting element of a system that is connected to others by means of a relationship (human networks) or by the exchange of particles ([ 54 , 55 ] in nanostructures), energy (in electric grids [ 56 ]) or information (communication networks [ 57 ]). Thanks to this versatility, NS allows for understanding the structure and behavior of systems showing very different natures [ 44 , 52 , 58 , 59 ], involving both artificial systems (blockchain [ 60 ], electric grids [ 56 , 61 , 62 ], the Internet [ 63 ], transport networks [ 64 ]), natural systems (the emergence of interstellar molecular complexity [ 65 ], complex Earth systems [ 66 ], the human brain [ 67 ], ecosystems [ 68 ], vascular networks [ 69 ], and metabolic networks [ 70 ].…”

“…Section 3 contains our simulation work. This suggests that, at room temperature, the network exhibits a strong propensity for a small-network nature, a beneficial property which, as will be demonstrated later on, has been found to enhance the exchange of information [ 81 ] (social networks [ 82 ], human brain [ 83 , 84 ]); matter (electrons in quantum dot systems [ 54 , 55 ] sap in vascular networks in plants [ 85 ]); and energy (in power grids [ 61 , 62 ]) in the field of network science. The results suggest that there could be a parallelism between the well-known dependence of carrier mobility on temperature and the potential emergence of the small-world property with increasing temperature.…”