nanomaterials Article Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints 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 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 Citation: Cuadra, L.; Nieto-Borge, in QD clusters and make the wave function better extended. As a potential application, we provide J.C. Approaching Disordered recommendations for improving QD intermediate-band solar cells. Quantum Dot Systems by Complex Networks with Spatial and Keywords: quantum dot; disordered system of quantum dots; complex networks; spatial networks; Physical-Based Constraints. quantum transport; quantum dot intermediate-band solar cells Nanomaterials 2021, 11, 2056. https://doi.org/10.3390/ nano11082056 1. Introduction Academic Editors: Maria E. Dávila Low-dimensional nanomaterials are systems that confine quantum particles in at and Iván Mora-Seró least one of the Euclidean dimensions (D). As illustrated in Figure1a, quantum wells, quantum wires and quantum dots (QDs) confine particles in 2-D, 1-D and 0-D, respec- Received: 1 July 2021 tively. Specifically, a QD [1,2] is a “tiny” (shorter than the de Broglie wavelength) 0-D Accepted: 3 August 2021 Published: 12 August 2021 nanostructure that confines carriers in all three directions in space [1,3,4], mimicking an “artificial atom”. QDs lead to a delta-like density of states (DOS) [1], which is very different Publisher’s Note: MDPI stays neutral from those corresponding to the other low-dimensional structures [1] (Figure1b). Put simply, a semiconductor QD is a heterostructure [5] formed by a small piece ( 10–15 nm with regard to jurisdictional claims in ∼ published maps and institutional affil- in size) of a semiconductor material (“dot material” (DM)) embedded inside another with iations. a higher bandgap (“barrier material” (BM)). This leads to the formation of confinement potentials (CPs) that confine particles, as illustrated in Figure1c. This creates discrete or bound energy levels for carriers and modifies both the electronic and optical properties [3] when compared to those of both bulk and other nano-materials [1]. All these properties would not be useful if there were no techniques to manufacture Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. a high density of QDs. The most successful methods are the self-assembled QD (SAQD) This article is an open access article growth technologies [6] illustrated in Figure1d. In the Stranski–Krastanow (SK) mode [7], distributed under the terms and the deposition of the DM starts with the formation of a very thin 2-D wetting layer, and conditions of the Creative Commons when a critical amount of strained dot material has been deposited, the formation of Attribution (CC BY) license (https:// (usually) pyramidal QDs occurs by relaxing strain. Conversely, sub-monolayer (SML)- creativecommons.org/licenses/by/ QDs [8] can be formed as disks or spherical QDs [7]. SML-QDs exhibit some advantages 11 2 4.0/). over SK-QDs, such as a smaller diameters (5–10 nm), a higher dot density ( 5 10 cm ), ∼ × − Nanomaterials 2021, 11, 2056. https://doi.org/10.3390/nano11082056 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, theyeven to 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 that they electronsare 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, itfrom is also suggesting said that QW a QDs for is aim zero-dimensionalplementing thequasi-Ferm IBSC. Ani level,electron EFC optically. A quasi-Fermsim ilarpum discussionpedi level,from EcanFC. Abe simappliedilar discussionto the hole can sub-bands. be applied to the hole sub-bands. structure (0D). This statement requires somthee groundcomme ntsstate in in order the wellto clarify (actually what the “0D” fiTherefore,rst sub-band) as onlyup to two the quasi-FermsecondTherefore, sub-bandi levelsas only (or exist two in quasi-Ferm a QW-cell,i levels(EFC andexist E inFV , awhich QW-cell, (EFC and EFV, which physically means. It is known that theeven reduction to the incontinuum dimensionality in the fromCB) woulda bulkdescribe easily the relax carrier to theconcentration firstdescribe sub-band. inthe th Thuscarriere electron concentration and hole sub-bands),in the electron the QWand -cellhole sub-bands), the QW-cell semiconductor to a thin layer confines electrelectronsons (or in holes)the QW to sub-bandsthis layer. andThis in causes thbehavese CB continuum like a conventional are describedbehaves single-gap by likethe sam acell conventionale whose effective single-gap gap correspondscell whose effectiveto the gap corresponds to the outstanding changes in some of the carrier properties. This is the case, for example, of a Nanomaterials 2021quasi-Ferm, 11, 2056 i level, EFC. A similar discussionseparation can bebetween applied the to groundtheseparation hole electron sub-bands. between and hole the sub-bands.ground electron Although and holethe MQW sub-bands. cell2 of 26Although the MQW cell QW, which currently is the best knownTherefore, and more as used only low-dim two quasi-Fermensional istructure levelshas exist the inpotential a QW -cell,advantage (EFChas andof the increasingE FVpotential, which its advantage ISC by theof absorptionincreasing itsof sub-bandgapISC by the absorption of sub-bandgap [Sh99]. But more exciting modifications describein electronic the carrier and optical concentration properties in canthe photons,beelectron this and is, hole nevertheless, sub-bands),photons, at thethe expenses thisQW is,-cell nevertheless, of reducing atits the VOC expenses. of reducing its VOC. achieved by a further reduction in the dimebehavesnsionality like of athe conventional carrierand surroundings better single-gap control from cell a of whose QD effective size [8 ,9gap]. corresponds SAQD technologies to the are crucial for implementing two-dimensional (2D) QW to a one-dimensionalseparation (1D) between quantum devicesthe wire, ground and such electron finally, as QD-based ato nd a hole sub-bands. light-emitting Although diodes the MQW (LEDs) cell [10], QD-lasers [11–14], QD-infrared zero-dimensional QD. This process of reduchas ingthe dimpotentialensionality advantagephotodetectors has artisticallyof increasing [been7, 15its , 16ISC], by QD-solar the absorption cells [of17 sub-bandgap], or QD-memories [4,18]. A key point for all these devices is that the position of carrier level(s) can be tuned by controlling the dot represented in Fig. 4.10. photons, this is, nevertheless, at the expenses of reducing its VOC. size [1], this being achieved by modifying the growth conditions [5,8,9,19]. DOS DOS DOS Quantum well Quantum wire Q-Well Q-Wire QD constant QD L. CUADRA, DEVELOPMENT OF INTERMEDIATE BAND SOLAR CELLS (2-D) (1-D) E E E (0-D) (a) (b) Wetting layer QD Fig. 4.11. Electron density ofFig. states, 4.11. N Electron(E): (a) indensity a bulk of sem states,iconductor, N(E): (a) (b) in in a abulk quantum semiconductor, (b) in a quantum Fig. 4.10. Representation of the three different kinds of barrierlow-dim ensional structures: CB CB quantum well (two-dimensional heterostructure), quantum wire (one-dimensional) andwell, (c) in a quantum wire,well, and (d)(c) in a QDEquantume, [2Le00 wire,]. and (d) in a QD [Le00]. QD material BM quantum dot (zero-dimensional). (BM) intra-band IB transition CB-CP Stranski-Krastanov E Ee,1 EI 2R I QD EG Fig. 4.11.