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

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Modeling Quantum Dot Systems As Random Geometric Graphs with Probability Amplitude-Based Weighted Links nanomaterials Article Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links Lucas Cuadra 1,2,* and José Carlos Nieto-Borge 2 1 Department of Signal Processing and Communications, University of Alcalá, 28801 Alcalá de Henares, Spain 2 Department of Physics and Mathematics, University of Alcalá, 28801 Alcalá de Henares, Spain; [email protected] * Correspondence: [email protected] Abstract: This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected. Keywords: quantum dot; disorder array of quantum dots; probability amplitude; complex networks; spatial network; Random Geometric Graphs; quantum transport Citation: Cuadra, L.; Nieto-Borge, J.C. Modeling Quantum Dot Systems as Random Geometric Graphs with 1. Introduction Probability Amplitude-Based Weighted Links. Nanomaterials 2021, Conceptually, a quantum dot (QD) [1] is a “small” (less than the de Broglie wave- 11, 375. https://doi.org/10.3390/ length), zero-dimensional (0D) nanostructure that confines carriers in all three directions nano11020375 in space [2–4], and exhibits a delta density of states (DOS), unlike those of quantum wells (2D nanostructure [5]) and quantum wires (1D) [2], as shown in Figure1a. In particular, a Academic Editor: Juan P. Martínez (type I) semiconductor QD is a heterostructure [6] made up of a small island (generally 10– Pastor 20 nm in size) of a semiconductor material (“dot material”) embedded inside another with a Received: 21 December 2020 higher bandgap (“barrier material” –BM–), which is able to confine both electrons and holes, Accepted: 27 January 2021 as illustrated in Figure1b). This creates discrete energy levels for carriers, and modifies Published: 2 February 2021 both its electronic and optical properties [3]. The problem of manufacturing high densities of high-quality QDs is successfully addressed by using self-assembled QD (SAQD) [7] Publisher’s Note: MDPI stays neu- technologies. Figure1c shows different SAQD growth modes. In the Stranski–Krastanow tral with regard to jurisdictional clai- (SK) growth mode [8], the deposition of the dot material starts with the formation of a ms in published maps and institutio- two-dimensional, very thin wetting layer (WL), and when a critical amount of strained dot nal affiliations. material has been deposited, the formation of pyramidal QDs occurs to relax strain. In the Volmer–Weber (VW) mode, the QDs grow directly on the bare substrate [9]. Sub-monolayer (SML)-QDs [10] can be formed as a disk or as a spherical QD because the growth method Copyright: © 2021 by the authors. Li- consists of multilayer deposition with a fraction of a monolayer of dot material on the censee MDPI, Basel, Switzerland. barrier matrix [8]. SML-QDs have several advantages over SK-QDs, such as a smaller 11 −2 This article is an open access article base diameter (5–10 nm), higher dot density (∼5 ×10 cm ), and better control of QD distributed under the terms and con- size [10,11]. ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Nanomaterials 2021, 11, 375. https://doi.org/10.3390/nano11020375 https://www.mdpi.com/journal/nanomaterials L. CUADRA, DEVELOPMENT OF INTERMEL.DIAT CUADRAE BAND, D SEVELOPMOLAR CELLSENT OF INTERMEDIATE BAND SOLAR CELLS L. CUADRA, DEVELOPMENT OF INTERMEDIATE BAND SOLAR CELLS from suggesting QWs for imfromplem suggestingenting the QW IBSC.s for An im electronplementing optically the IBSC. pum pedAn fromelectron optically pumped from L. CUADRA, DEVELOPMENT OF INTERMEDIATE BAND SOLAR CtheELLS ground state in the wellthe (actually ground thestate fi rstin sub-band)the well (actually up to the the second first sub-band) sub-band up(or to the second sub-band (or From this perspective, since QDs are able to create discrete energy levels,even tothey the continuum ineven the CB)to the would continuum easily inrelax the toCB) the wouldfirst sub-band.easily relax Thus to the first sub-band. Thus mimic some of the properties of the atoms, and sometimes, even it is said thatelectrons they are in the QW sub-bandselectrons and inin thethe QWCB continuumsub-bands andare describedin the CB bycontinuum the same are described by the same “artificial atoms” [We91]. In this context,from suggesting it is also said QW thats for a imQDplem is aenting zero-dimensional thequasi-Ferm IBSC. An i level,electron EFC optically. A quasi-Fermsim ilarpum discussionpedi level,from EcanFC. 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This processhas of the reduc potentialingSinceSince dim theadvantagee thensionality density density ofof hasofstates increasing states artistically (DOS) (DOS) its of beenofIcarriersSC carriers by theconf conf absorptioninedined in ina QD aof QD issub-bandgap ideallyis ideally a delta a delta function function Nanomaterials 2021, 11, 375 2 of 22 represented in Fig. 4.10. photons, this[Fig. is,[Fig. nevertheless, 4.2 4.2 (b)], (b)], the the atinterm theinterm expensesediateediate state ofstate reducingin inthe the dot itsdot isV OCisseparated . separated from from the the CB CB and and from from the the VBVB by bym eansmeans of ofsub-gaps sub-gaps of ofzero zero density density of ofstates. states. 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