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 in relax the to CB) the would first 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 QW CB continuumsub-bands andare describedin the CB by continuum 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-Ferm sim ilarpum discussionpedi level,from E canFC. Abe simappliedilar discussion to the hole can sub-bands. be applied to the hole sub-bands. structure (0D). This statement requiresthe ground some comstatem inents the in well order (actually to clarify the whatfiTherefore,rst sub-band)“0D” as onlyup to two the quasi-FermsecondTherefore, sub-bandi levels as only(or exist two in quasi-Ferm a QW-cell,i levels (EFC andexist E inFV , a which QW-cell, (EFC and EFV, which physically means. It is known thateven the to reduction the4. I continuumM4.P LEMIMP LEMENT inEAT NT dimIONAT inIONOFe nsionality TOF theHE T INTH E CB) EINTRMEEDIATRM wouldEDIAT fromE BANDdescribeE BAND aSOLAR easily bulkSOLAR CELL the relaxCELL BY carrier M BY EANS toMEANS OF the concentration QUANTOF firstQUANTUdescribeM DOTU sub-band.M DOTS S in the th Thuscarriere electron concentration and hole sub-bands), in the electron the QWand -cell hole sub-bands), the QW-cell semiconductor to a thin layer confineselectrons electr inons the (or QW holes) sub-bands to this and layer. in Thisthbehavese CB causes continuum like a conventional are describedbehaves single-gap by likethe sam a cell conventionale whose effective single-gap gap corresponds cell whose effective to the gap corresponds to the outstanding changes in some of the quasi-Fermcarrier propei rties.level, ThisEFC .is A the sim case,ilar for discussion examseparationple, can of a be between applied the to ground theseparation hole electron sub-bands. between and hole the sub-bands. ground electron Although and holethe MQW sub-bands. cell Although the MQW cell sizesize is sois sosm small allthat that carriers carriers are are alm almosto comst completelypletely localised localised within within the the dot dot and and discrete discrete QW, which currently is the best Therefore, known and as m onlyore usedtwo quasi-Ferm low-dimensionali levelshas structure exist the in potential a QW -cell, advantage (EFChas and of the increasingE FV potential, which its advantage ISC by the of absorption increasing itsof sub-bandgapISC by the absorption of sub-bandgap levelslevels are are form formed.ed This. This is justis just one one of ofthe the reasons reasons that that com compelspels us usto topropose propose the the use use of of [Sh99]. But more exciting modificationsdescribe in the electronic carrier concentration and optical properties in thephotons, electron can be this and is, hole nevertheless, sub-bands),photons, at the the expenses this QW is,-cell nevertheless, of reducing atits the VOC expenses. of reducing its VOC. achieved by a further reduction in thebehaves dimens likeionalityquantum aquantum conventional of dots:the dots: carrier the the single-gapability surroundings ability to to create cell create fromwhose a discretea adiscrete effective energy energy gap level level corresponds within within the the tofo rbidden theforbidden gap gap of of two-dimensional (2D) QW to a one-dimseparationensional betweenthe the (1D)barrier barrier thequantum sem groundsemiconductori conductorwire, electron and (i.e., finally,(i.e., a nd our our hole “interm to “interm sub-bands.a edieatediate level”) Although level”) and and thecontrol control MQW its itsenergycell energy [Ma00]. [Ma00]. zero-dimensional QD. This processhas of the reduc potentialingSinceSince dim the advantagee thensionality density density of of hasofstates increasing states artistically (DOS) (DOS) itsof 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. The The IB IBwould would ideally ideally derive derive from from thethe interm intermediateediate electron electron bound bound state state in inthe the ordered ordered superlattice superlattice of ofdots dots [Ma01b [Ma01b] as] as DOSQ-Well DOS DOS Quantumillustratedillustrated inwell inFig. Fig. 4.2 Quantum4.2 (c). (c). By By doping wiredoping the the QD-array QD-array at ata ratea rate of ofone one donorQ-Wire donor im impuritypurity per per QD QD,QD, the the IB IBm amy apotentiallyy potentially be behalf half filled filled with with electrons electronsconstant [Ma01a [Ma01a, Ma01b]., Ma01b]. (2-D) QD

(1-D) E E E (0-D) EI EI EI (a) Fig. 4.10. Representation of the three different kinds of low-dimensionalbarrier Fig. structures: 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 material Wetting layer QD quantum well (two-dimensional heterostructure), quantumQD wire (one-dimensional)well, (c) and in a quantum wire,well, and (d)(c) in a QDquantum [Le00 wire,]. and (d) in a QD [Le00]. (BM) quantum dot (zero-dimensional). BM

E 2RQD Stranski-Krastanov

Fig. 4.11. Electron densityCB of states, N(E): (a) in a bulk semiconductor, (b) in a quantum well, (c) in a quantum wire, and (d) in a QD [Le00 ]. QD 2 NQD On the contrary, the QD DOSOn the is contrary,discreteBM , isthe a QD delta-function DOS is discrete [Su99a,, is Su99a, a delta-function [Su99a, Su99a, Ue EI ⇥ electron As86, Ba93] and, as said inAs86 advance,, Ba93] there and, are as true said sub-gaps in advance, with there zero are density true sub-gapsof states with zero density of states level Volmer–Weber In this context, the term “dimensionality” refers to the number of degreesbetween of the intermediate-DOS levelbetween in the the dot interm andediate the continuum level in the in dotthe andCB and the continuumthe VB in the CB and the VB hole level freedom in the electron momentum. For instance, as shown in Fig. 4.10, an electron in a QD quantum well is confined in only one directionOn Uthe (theh contrary, growth direction) the QD DOS while is it discreteis free to, is a delta-function [Su99a, Su99a,BM SML move in the other two. There are thus two degrees of freedomVB in its motion. A further As86, Ba93] and, as said in advance, there are true sub-gaps with zeroDOS density of states 102 102 Sub-monolayer (SML) between the intermTypeediate I levelQD in the dot and the continuum in the CB and the VB

100 (b) (c) Fig.Fig. 4.2. 4.2. (a) (a) Illustration Illustration of of a asphe sphericalrical quantum quantum dot dot of ofradius radius R DR andD and the the three- three- dimdimensionalensional confinem confinementent energy energy potentials, potentials, U e Uande and U hU, forh, for electrons electrons and and holes holes (not (not to to ( ) Conceptual representation of nano-structures and their corresponding density of states Figurescale).scale). 1. They aThey have have been been represented represented102 along along an anim imaginaryaginary line line that that crosses crosses its itscentre. centre. EI EI (DOS)representsrepresents for quantum the the electron electron wells, energy quantum energy level. level. wires (b) (b) Density and Density quantum of of states states dots; (DOS), (DOS), (b) confinementg, g in, in the the system system potentials for electrons QD/barrier.QD/barrier. N N QDis theis the dot dot density. density. (c) (c) Band Band structure structure derived derived from from an anordered ordered array array of of (Ue) and holesQD (Uh) in a type I QD (or simply, QD), and its corresponding density of states; (c) different thesethese identical identical dots dots (From (From [Cu03c, [Cu03c, Cu04a]). Cu04a]). classes of growth in self-assembled quantum dots. See the main text for details.

SAQD technologies help both research in Quantum Mechanics (QM) and manufacture novel devices. On the one hand, SAQD87 technologies87 assists in manufacturing high quality QDs for studying QM and exploring novel effects in electronics, photonics, and spintronics. For instance, SAQD-based devices help achieve single-electron charge sensing [12], entan- glement between spins and photons [13,14], single-photon sources [15], or single-spin [16], and help also the control of Cooper pair splitting [17], spin transport [18], spin–orbit in- teraction [19], g-factor [20], and Kondo effect [21]. On the other hand, SAQD technologies allow for manufacturing high density of QDs, which are crucial for implementing opto- electronic devices such as QD-based light-emitting diodes (LEDs) [22], QD-memories [4,23], QD-lasers [24–27], QD-infrared photodetectors [8,28,29], and QD-solar cells [30]. A key point in these devices is that the position of carrier level(s) can be tuned by controlling the dot size [2], and, this, by modifying the growth conditions [6,10,11,31]. In this work, we model a system formed by QDs as a special graph in the effort of exploring electron transport. Our approach could be considered as belonging to Complex Networks (CN) Science. CN have become a multidisciplinary research field [32–34] for studying systems with a huge amount of components that interact with each other. These range from artificial systems (such as power grids [35,36], or the Internet [37]) to natural systems (vascular networks [38], protein interactions [39], or metabolic networks [40]). More examples showing the feasibility of CN to study many other systems can be found in [32,41,42] and the references therein. All these very different systems have in common that all of them can be described in terms of a graph [32]: a set of entities called nodes (or vertices) that are connected to each other by means of links (or edges). A node repre- sents an entity (generator/load in a power grid [36], or a species in an ecosystem [43]), which is connected with others (linked by electrical cables in power grids, or related by trophic relationships in ecosystems). This way, any system can be encoded as a graph, a mathematical abstraction of the relationships (links) between the constituent units (nodes) Nanomaterials 2021, 11, 375 3 of 22

of the system. In broad sense, CN science models not only the structure (topology), but also some dynamic phenomena such as information spreading [44], epidemic processes (both biological [45], and artificial viruses [46]) or cascading failures [47,48]. These are very common in large engineered networks: wireless sensor networks [49], Internet [50], power grids [35,51,52], or transportation networks [53]. Some of these CN, such as transportation networks, are networks in which the nodes are spatially embedded [54] or constrained to locations in a physical space with a metric. For many practical applications, the space is the two-dimensional space and the metric is the Euclidean distance dE. This geometric constraint usually makes the probability of having a link between two nodes decrease with the Euclidean distance [53]. This particular subset of networks are called spatial networks (SN) [53,55] or spatially embedded CN [56]. A particular class of SN are Ran- dom Geometric Graphs (RGGs) [57] in which N nodes are uniformly distributed over the unit square, while the link between two nodes i and j is formed if the Euclidean distance dE i, j < r, a given model parameter. RGGs have been successfully used to model wireless sensor networks [58] and ad hoc networks [59] in which r is related to the range of the radio( ) devices. The previous paragraph has shown that CN science has been applied to a broad variety of macroscopic, “classical” (non-quantum) systems. As will be shown in our review of the current state of the art in Section2, CN science has also been used to study quantum nanosystems, although to a much lesser extent and not with the RGG approach that we will show throughout this paper. VersionThe June purpose 25, 2020 submitted of our work to Entropy is to propose a particular class of RGG whose links have 3 of 6 Version June 25, 2020 submitted to Entropy 3 of 6 weights computed according to QM with the aim of studying electron transport in a system of disordered QDs like the one in Figure2a. It consists of a number of N QDs—for in- stance, a layerNode of SAQDs [6], which produce finite quantum confinementLink potentials (QCPs), Node 2 Link randomly distributed in the physical metric space, R , characterized by the Euclidean distance, dE.

Node Link

(a) (a) (b)(b)

FigureFigure 1. This 1. isThis a figure, is a figure, Schemes Schemes follow follow the same the same formatting. formatting. If there If there are multipleare multiple panels, panels, they they should should be listedbe listed as: (a) as: Description (a) Description of what of what is contained is contained in the in first the first panel. panel. (b) Description (b) Description of what of what is contained is contained

in thein second the second panel. panel. Figures Figures should should be2 placed be placed in the in main the main text text near near to the to thefirst first time time they they are are cited. cited. A A Figure 2. (a) Random distribution of QDs in R . Each QD, represented by a node, causes a quantum captioncaption on a single on a single line should line should be centered. be centered. 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 65 2.65 Results2. Results overlap. See the main text for details.

66 66 This sectionThis section may maybe divided be divided by subheadings. by subheadings. It should It should provide provide a concise a concise and and precise precise description description

67 In our model, any QD causing a QCP in Figure2a is represented by a node. To 67 of theof experimental the experimental results, results, their their interpretation interpretation as well as well as the as theexperimental experimental conclusions conclusions that that can can be be understand how links have been generated in Figure2b, it is convenient to have a 68 drawn. 68 drawn. look at Figure3. On its left side, we have represented, for illustrative purposes, two

of69 these QDs,This labeled sectioni mayand j be. For divided the sake by of subheadings. clarity, we have It should assumed provide that each a concise QD and precise 69 This section may be divided by subheadings. It should provide a concise and precise 70 description of the experimental results, their interpretation as well as the experimental 70 description of the experimental results, their interpretation as well as the experimental 71 conclusions that can be drawn. 71 conclusions that can be drawn.

72 2.1. Subsection 72 2.1. Subsection

73 2.1.1. Subsubsection 73 2.1.1. Subsubsection

74 Bulleted lists look like this: 74 Bulleted lists look like this:

75 First bullet 75 First• bullet • 76 Second bullet 76 Second• bullet • 77 Third bullet 77 Third• bullet • 78 Numbered lists can be added as follows: 78 Numbered lists can be added as follows:

79 1. First item 79 1.80 First2. itemSecond item 80 2.81 Second3. Third item item 81 3. Third item 82 The text continues here. 82 The text continues here.

83 2.2. Figures, Tables and Schemes 83 2.2. Figures, Tables and Schemes 84 All figures and tables should be cited in the main text as Figure 1, Table 1, etc.

84 All figures and tables should be cited in the main text as Figure 1, Table 1, etc. Nanomaterials 2021, 11, 375 4 of 22

Version January 18, 2021 submitted to NanomaterialsVersion January 18, 2021 submitted to Nanomaterials 4 of 24 4 of 24

has radius RQD. Their centers are separated by a Euclidean distance dE i, j . As shown in Figure3a, we have also represented an ξ axis that passes through the centers of both 82 dE i, j .QDs. As shown The ξ axis in will Figure assist823 usa,d inweE clearlyi, havej . As representing also shown represented in the Figure associated an3a,x axis conceptswe have that( in) passes also Figure represented3 throughb,c. the an centersx axis that passes through the centers 83 of bothFigure QDs.3bx showsaxis will the correspondingassist83 usof in both clearly finite QDs. QCP representingx causedaxis will by aassist the QD associated along us inξ clearlyaxis: conceptsV representingξ = −V inC Figure the3b associated and c. concepts in Figure 3b and c. (inside the QD) and V ξ = 0 (otherwise). Since the QCP is finite, there is a part of the 84 Figure( ) 3b shows the corresponding84 Figure( finite) 3b QCPshows caused the corresponding by a QD along finitex axis: QCPV causedx = V byC (inside a QD along the x axis: V x = VC (inside the electron wavefunction that spreads outside the QD, as qualitatively shown in( Figure) 3c. 85 QD) and V x 0 (otherwise).85 SinceQD) andthe QCPV x is finite0 (otherwise)., there is a Since part of the the QCP electron is finite wavefunction, there is a part that of the electron wavefunction that Note that= the QDs are close( ) enough that the electron= wavefunctions overlap. According to ( ) ( ) 86 spreadsQM, outside the electron, the QD,a, as isqualitatively in86 bothspreads QDs with outside shown a probability in the Figure QD, amplitude as3c. qualitatively Note as that the one the shown in QDs Figure are in3 Figurec. close enough3c. Note that that the QDs are close enough that 87 the electronThe( electron wavefunctions) can tunnel overlap. from87 the one electronAccording QD to the( wavefunctions) other. to QM, We the model electron, overlap. this quantuma According, is in phenomenonboth QDs to QM, with the a electron, probabilitya, is in both QDs with a probability by forming a link between nodes i and j (Figure3d). We will show throughout the paper 88 amplitude as the one in Figure88 3amplitudec. The electron as the can one tunnel in Figure from3c. one The QD electron to the other. can tunnel We model from onethis QD to the other. We model this that the link weight wij between two nodes i and j depends on the extent to which the 89 quantumelectronphenomenon wave-functionsby forming89 in bothquantum a nodes link between overlap.phenomenon Quite nodes often,byi and forming however,j (Figure a link there3d). between are We electron will nodes showi throughoutand j (Figure 3d). We will show throughout 90 the paperwave-functions that the link in sufficiently weight90 wtheij remotebetween paper QDs that twothat the do nodes link not overlap weighti and atj wdepends all.ij between Regarding on thetwo this, extent nodeswe toi and whichj depends the on the extent to which the

91 electronhave wave-functions represented in in Figure both91 3 nodeseelectrontwo overlap.nodes wave-functions that Quite are so often, far in apart both however, that nodes their there overlap. corresponding are electron Quite often, wave-functions however, there are electron wave-functions wave-functions do not overlap (Figure3g). Thus, an electron that is in node i at the initial 92 in sufficiently remote QDs that92 doinnot sufficientlyoverlap atremote all. Regarding QDs that this,do not weoverlap have represented at all. Regarding in Figure this,3e we have represented in Figure 3e time t = 0 will remain localized in that node even if t → ∞. We model this QM result 93 two nodes that are so far apart93 thattwo nodestheir corresponding that are so far wave-functions apart that their docorrespondingnot overlap (Figurewave-functions3g). do not overlap (Figure 3g). through the absence of a link (wij = 0), as shown in Figure3h (not connected nodes). 94 Thus, an electron that is in node94 Thus,i at the an initial electron time thatt = is0 inwill node remaini at the localized initial time in thatt = node0 will even remain if localized in that node even if 95 t ô. We model thisd QME(i, j result95) t throughô. We the modelabsence thisof QM a link resultdE( (i,w jij through) = 0), as the shownabsence in Figureof a link3h (w (notij = 0), as shown in Figure 3h (not 96 connected nodes). 2RQD 96 connected nodes). 2RQD i j ⇠ i j ⇠

(ad)E(i, j) dE(i,(e j))dE(i, j) dE(i, j) 0 2RQCP ⇠ 0 2R2RQCPQCP ⇠ 2RQCP

VC i j VC ii j j i j (b) (a) (a)(f) (e) (e) probability amplitude0 00 0

VC VCVC VC overlap ⇠ ⇠ (c) (b) (b)(g) (f) (f) weighted probability probability no link amplitude link amplitude i j i wij =0 j wij > 0 disconnected nodes overlap (d) overlap (h) (c) (c) (g) (g) weighted weighted Figure 3. (a) Two of the QDs as those in Figure2a. We consider QDs with radius RQD and whose link link no link no link centers are separated by a Euclidean distance d i, j . ξ represents an axis that passes through the i j E ii wjij =0 j i wij =0 j center of both QDs. Along the ξ-axis, we have represented in (b) the corresponding QCP (−V inside w > 0 w disconnected> 0 nodes C disconnected nodes the node, 0 otherwise) andij the electron probability( amplitudes) (c).ij An electron can be in both QDs (d) (d) (h) (h) with the represented probability amplitude. (d) We model this with a link whose weight wij is given by the overlap. (e) Opposite case in which the QDs are so far apart (f) that there is no overlap (g) and ( ) Two of the QDs that create QCPs( ) as Two those of thein Figure QDs that2a. createWe have QCPs considered as those QDs in Figure with 2a. We have considered QDs with Figurelink formation 3. a is not allowed (h). Figure 3. a radius RQD and whose centers are separatedradius RQD byand a Euclidean whose centers distance ared separatedE i, j . x represents by a Euclidean an axis distance that dE i, j . x represents an axis that passes throughOur main the result center is of that both the QDs. proposedpasses Along through RGGx-axis, is the able we center have to capture represented of both inner QDs. properties in Along (b) thex-axis, corresponding of the we have represented QCP in (b) the corresponding QCP complex quantum system: it predicts the system quantum state, its time evolution,( ) and the ( ) (VC inside the node, 0 otherwise) and(V theC inside electron thewave-functions node, 0 otherwise) (c). and As wave-functions the electron wave-functions overlap, an (c). As wave-functions overlap, an emergence of quantum transport (QT) as the QD density increases. In fact, QT efficiency electron can be in both QDs with theelectron represented can be probability in both QDs amplitude. with the (representedd) We model probability this with a amplitude. link (d) We model this with a link exhibits an abrupt change, from electron localization (no QT) to delocalization (QT emerges). whoseThis weight is an electronwij is given localization–delocalization by the overlap.whose (f weight) The transition rightmostwij is given that panel has by thealso represents overlap. observed the (f in) opposite The [60]. rightmost case in panel which represents the opposite case in which the nano-structuresOur proposal are could so far have apart potentialthe that nano-structures the application wave functions not are only so do in far not improving apart overlap that the ( theg), efficiency andwave link functions formation do not is overlap (g), and link formation is notof allowed QD-based (g). optoelectronic devicesnot (LEDs, allowed solar (g cell,). lasers, etc.) that make use of SAQD

97 Our main result is that97 the proposedOur main RGG result is able is that to capture the proposed inner properties RGG is able of the to capture complex inner properties of the complex

98 quantum system: it predicts98 thequantum system system:quantum it state, predicts its time the system evolution, quantum and the state, emergence its time of evolution, and the emergence of

99 quantum transport (QT) as99 thequantum QD density transport increases. (QT) Inas fact,the QD QT density efficiency increases. exhibits Inan fact, abrupt QT efficiency exhibits an abrupt

100 change, from electron localization100 change, (no fromQT) to electron delocalization localization (QT (no emerges). QT) to delocalizationThis is an electron (QT emerges). This is an electron

101 localization-delocalization transition101 localization-delocalization that has also observed transition in [60]. that has also observed in [60]. Nanomaterials 2021, 11, 375 5 of 22

layers but also in single, huge macromolecules (light-harvesting molecules [61]) to study the quantum transport (energy, charge) between specific areas of their structures. The rest of this paper has been structured as follows. After reviewing of the current state of the art in Section2, Section3 briefly introduces the QD system that we want to study, while Section4 explains our RGG proposal. The experimental work in Section5 allow for predicting inner features of the system such as the system quantum state, its time evolution, or the emergence of quantum transport. Section6 discusses potential applica- tions, strengths, and weaknesses of the proposed method. Finally, Section7 completes the paper. AppendixA lists the symbols used in this work.

2. Current State of the Art There are basically two approaches that combine CN and QM concepts [62]. The first one applies concepts inspired by QMs to better study CN. For instance, Ref. [63] proposes a way to navigate complex, classical (non-quantum) networks (such as, the Internet) based on quantum walks [64] (the quantum mechanical counterpart of classical random walks [65]). More examples belonging to this first framework can be found in [62] and in the references therein. The second approach, in which the present article can be included, is based on applying CN concepts to explore nanosystems, which are governed by the laws of QM [66] and not by those of classical physics. A representative instance is the system studied in [67]: any atom trapped in a cavity is represented by a node, while the photon that the two atoms (nodes) exchange is encoded by a link between them. This and other papers have in common the fact of studying quantum properties on networks using quantum walks. This is because the quantum dynamics of a discrete system can be re-expressed and interpreted as a single-particle quantum walk [68,69]. This is the reason why quantum walks have been used to study the transport of energy through biological complexes involved in light harvesting in photosynthesis [70]. Quantum walks have also been used to explore transport in systems described by means of CN with different topologies [71,72]. Specifically, continuous-time quantum walks (CTQW)—a class of quantum walks on continuous time and discrete space [64]—have been used extensively to study quantum transport (QT) on CN [72], and will also be used in our work. There are several works that have studied QT over regular lattices [72–74], branched structures [75,76] (including dendrimers [76]), fractal patterns [77], Husimi cacti [78], Cayley trees [79], Apollonian networks [80], scale-free networks [81], small-world (SW) networks [82] and start graphs [83,84], leading to the conclusion that QT differs from its classical counterpart. Having a quantitative measure of the efficiency of QT in a CN has been found to be important for practical and comparative purposes. In this regard, Ref. [85] has recently found bounds that allow for measuring the global transport efficiency of CN, defined by the time-averaged return probability of the quantum walker. QT efficiency can undergo abrupt changes, and can have transitions from localization (no QT) to delocalization (QT appears). In this respect, the authors of [60] have studied localization–delocalization transition of electron states in SW networks. The SW feature is interesting because it makes it easy to navigate a network since SW networks exhibit a relatively short path between any pair of nodes [86,87]: the mean topological distance or average path length ` is small when compared to the total number of nodes N in the network (` = ln N as N → ∞). The O usual technique of rewiring [86] or adding links [88] in macroscopic, non-quantum CN to create SW networks have also been extended to quantum system( [82,84) ] to enhance QT. In [82], SW networks have been generated from a one-dimensional ring of N nodes by randomly introducing B additional links between them. The quantum particle dynamics has been modeled by CTQWs, computing the averaged transition probability to reach any node of the network from the initially excited one. Finally, the strategy of adding new links have been explored in star networks with the aim of enhancing the efficiency of quantum walks to navigate the network [84]. Please note that all of these key works have focused their research from the viewpoint of the topological properties. In particular, the topological (geodesic) distance between two nodes i and j, d i, j , is the length of the

( ) Nanomaterials 2021, 0,5 6 of 22

research from the viewpoint of the topological properties. In particular, the topological (geodesic) distance between two nodes i and j, d i, j , 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 [50]. The distance between two( nodes) i and j that are directly linked is d i, j = 1, regardless of where they are located in physical space. We propose in the next paragraph to use the Euclidean distance for reasons that will be clearer later on. ( ) Nanomaterials 2021, 11, 375 3. The QD System 6 of 22 Let us consider a microscopic physical system, which is closed, and made of a set of N QDs that are randomly distributed in a metric space as shown in Figure 2a. The position of shortest path (geodesic path) between them, that is, the minimum number of links when 2 each nanostructure in the metric space R is determined by a position vector r. We consider going from one node to the other [50]. The distance between two nodes and that are a single electron (walker)i freelyj tunneling among QDs (when allowed). An example of , ,5 directly linked is d i, j 1, regardless of where6 they of 22 are located in physical space. We Nanomaterials 2021 0 = one-electron model is the tight-binding model [89], which has been used for both lattice propose in the next paragraph to use the Euclidean distance for reasons that will be clearer and random networks. later on. ( ) Aiming to later numerically illustrate the results of studying this system using CN research from the viewpoint of the topological properties. In particular, the topological 3. The QD System concepts, we are going to assume a set of hypotheses about the QCF that the QDs produce. (geodesic) distance between two nodes i and j, d i, j , is the length of the shortest path These hypotheses will allow us to tackle the problem by using some well known QM (geodesic path) between them, that is, the minimumLet us consider number a microscopic of links when physical going system, from onewhich is closed, and made of a set of N QDs that are randomly distributed in a metric spaceresults as shown on in each Figure individual2a. The position nanostructure, of so that we will be able to then focus on exploring node to the other [50]. The distance between two( nodes) i and j that are directly linked is each nanostructure in the metric space 2 is determinedthe complete by a position system vector asr. a We RGG. consider d i, j = 1, regardless of where they are located in physical space. We proposeR in the next a single electron (walker) freely tunneling among QDsWith (when this allowed). in mind, An and example for reasons of that will be clearer later on, we first assume that paragraph to use the Euclidean distance for reasons that will be clearer later on. one-electron model is the tight-binding model [89],the which single has band been effectiveused for both mass lattice equation of electrons in the envelope approximation [90] ( ) 3. The QD System and random networks. is a proper description of the dot and barrier bulk materials. This is because a QD size of Aiming to later numerically illustrate the results10 to of 20 studying nm is much this system larger using than CN the lattice constant of the material involved and, thus, it Let us consider a microscopic physical system, which is closed, and made of a set of N concepts, we are going to assume a set of hypothesesseems about reasonable the QCF that to the consider QDs produce. that only the envelope part of the electron wave function QDs that are randomly distributed in a metric space as shown in Figure 2a. The position of These hypotheses will allow us to tackle the problemis affected by using by some the confinement well known potential. QM For the sake of clarity, we also assume that the each nanostructure in the metric space results2 is determined on each individual by a position nanostructure, vector r. so We that consider we will be able to then focus on exploring R QDs are identical (since in a closed system, energy is conserved, and the electron can only a single electron (walker) freely tunnelingthe complete among systemQDs (when as a RGG. allowed). An example of make transitions between QDs that have the same energy, that is, between QDs that have one-electron model is the tight-binding modelWith [ this89], in which mind, has and been for reasons used for that both will lattice be clearer later on, we first assume that the same size [91]) and spherical with radius R . The center of any QD i is given by a and random networks. the single band effective mass equation of electrons in the envelope approximation [90] QD position vector r in the metric space. We assume that its associated QCP is spherically Aiming to later numerically illustrateis a proper the results description of studying of the dot this and system barrier using bulk materials. CN This is becausei a QD size of 10 to 20 nm is much larger than the lattice constantsymmetric of the material (depending involved only and, on thus, the it radial co-ordinate r), finite and “square”: concepts, we are going to assume a set of hypotheses about the QCF that the QDs produce. seems reasonable to consider that only the envelope part of the electron wave function These hypotheses will allow us to tackle the problem by using some well known QM is affected by the confinement potential. For the sake of clarity, we also assume that the VC , if r < RQD results on each individual nanostructure, so that we will be able to then focus on exploring UC r = , (1) QDs are identical (since in a closed system, energy is conserved, and the electron can only 0 , if r > RQD the complete system as a RGG. make transitions between QDs that have the same energy, that is, between QDs that have as shown in Figure 4a. ( ) v With this in mind, and for reasonsthe that same will size be [ clearer91]) and later spherical on, we with first radius assumeRQD that. The center of any QD i is given by a the single band effective mass equationposition of electrons vector inr in the the envelope metric space. approximation We assume [ that90] its associated QCP is spherically i U is a proper description of the dot and barriersymmetric bulk (depending materials. only This on is the because radial a co-ordinate QD size ofr),C finiteRQCP and “square”: 2 2 10 to 20 nm is much larger than the lattice constant of the material involved and, thus, it QD 0 r QD | | −VC , if r < RQD | | seems reasonable to consider that only the envelope part of the electronUC r = wave function , (1) 0 , if r > RQD is affected by the confinement potential. For the sake of clarity, we also assume that the EQD 20 as shown in Figure4a. QDs are identical (since in a closed system, energy is conserved, and the( ) electronv can only make transitions between QDs that have the same energy, that is, between QDs that have –20 y(nm) the same size [91]) and spherical with radiusUC RQD. The center of any QD i is given by��� a 2 2 QD x(nm) position vector ri in the metric space. We assume that its associated QCP is spherically��� QD ��� |VC | –20 0 r 20 symmetric (depending only on the radial co-ordinate r), finite and| “square”:| ��� ��� r/RQCP EQD 20 ��� V , if r R (a) (b) (c) C < QD ��� UC r = ,��� (1) 0 , if r > RQD –20 Figure 4. (a) Quantum confinement potential. VC is the depth of the potential well while RQD is y(nm) ��� ��� as shown in Figure 4a. ( ) v x(nm) the radius of the QD producing UC. EQD is the energy of the bound state; (b) squared modulus of VC –20 2 2 20 the corresponding� � wave� function,� �� yQD�� , in Cartesian coordinates; (c) yQD as a function of the U normalized radial coordinate, r R . C RQCP 2 QD (a) (b) (c) ∂ ∂ ∂ ∂ 2 QD 0 r QD | | The reason why we have/ made use of UC is that the time-independent Schrödinger’s | | Figure 4. (a) Quantum confinement potential. VC is the depth of the potential well while RQD is equation, which allows for computing both the electron wavefunction (yQD) and its en- E the radius of the QD producing UC. EQD is the energy of the bound state; (b) squared modulus of QD 20 2 ergy (E ), 2 the corresponding wave function, ψQD , in Cartesian coordinates;QD (c) ψQD as a function of the normalized radial coordinate, r RQD. –20 ∣ ∣ ∣ ∣ y(nm) x(nm) The reason why we have/ made use of UC is that the time-independent Schrödinger’s VC –20equation, which allows for computing both the electron wavefunction (ψQD) and its en- 20 ergy (EQD), r/RQCP (a) (b) (c)

Figure 4. (a) Quantum confinement potential. VC is the depth of the potential well while RQCP is the radius of the nanostructure producing UC. EQD is the energy of the bound state; (b) squared modulus 2 2 of the corresponding wave function, yQD , in Cartesian coordinates; (c) yQD as a function of the normalized radial coordinate, r RQCP. ∂ ∂ ∂ ∂ The reason why we have/ made use of UC is that the time-independent Schrödinger’s equation, which allows for computing both the electron wavefunction (yQD) and its en- ergy (EQD), Nanomaterials 2021, 11, 375 7 of 22

HsψQD = EQD ⋅ ψQD, (2)

can be solved analytically [91–93]. Hs in Equation (2) is called the Hamiltonian operator and corresponds to the total energy of the system,

2 h¯ 2 H − ▽ +V (3) s ≡ 2m s where h¯ is the reduced Planck constant, m the electron mass, ▽2 is the the Laplace opera- tor [94], and Vs is the energy potential operator. Thus, according to QM, the electron energy in the QCF (1) is quantized: it can only take 2 discrete values [91]. The number of “bound states” in this QCF depends on VC ⋅ 2RQCP 2 (see [91]): there is a range of values of VC ⋅ 2RQCP for which there is only one energy level. Now, if we assume, for simplicity, that the QD size, 2RQCP, is so tiny that( there) is only one energy level, EQD, its associated wavefunction( ) is a 1 s–orbital [6,93]. We have solved the problem for a single, isolated QD with RQCP = 10 nm and VC = 0.47 eV, typical in III-V semiconductors. There is only one bound state, whose energy is EQD = 0.4 eV. Its associated wave function ψQD is a spherical 1s−orbital. We have represented its squared 2 modulus, ψQD , in both Cartesian (Figure4b) and radial coordinates (Figure4c). The 2 latter shows how ψQD r decreases very quickly as a function of the normalized radial coordinate,∣ r RQCP∣ . With this idea∣ in mind,( )∣ the potential in the complete system is a function that varies from one QD/ to another, taking the −VC value inside each QD and zero in the space among QDs:

−V , inside a QD at r V V r C (4) ≡ = 0 , outside a QD , where r is the position vector locating( ) v each QD.

4. Modeling the QD System as a Spatial Network: The Proposed Model Aiming at generating the network associated with the proposed system, we represent any QD i by means of a node, and we label this node using its ket i , as shown in Figure5a. Our next step is to generate the links in a way that makes physical sense according to QM, and also takes into account that the nodes are spatially embedded.∣ ⟩ As outlined in Section1 when introducing our approach, we generate a link between two nodes (sites, kets), i and j , located at ri and rj, respectively, by computing to what extent their respective wave

functions overlap. We compute the overlap between the wave functions ψQDi and ψ∣QD⟩ j as ∣the⟩ overlap integral [94] over all the physical space S

∗ ψQDj ψQDi dr = j i = wij. (5) ES Note that, because of symmetry, j i = i j ⟨. ∣ ⟩ Using Expression (5), we have computed the overlap integral for nodes whose centers are separated by a normalized⟨ Euclidean∣ ⟩ ⟨ ∣ ⟩ distance dE RQCP. The result appears in Figure6, where we have represented the overlap integral as a function of dE RQCP. We have highlighted two situations in Figure6 for illustrative/ purposes. / Version January 18, 2021 submitted to Nanomaterials 8 of 24 Version January 18, 2021 submitted to Nanomaterials 8 of 24

Node w12 = 1 2 Node | w12 = 1w223 = 2 3 | | w23 = 2 3 2 2 | | 2 | 2 r2 | | 1 3 r2 1 3 | r1 1 | 3w13 = 1 3 =01 | 3 Version January 18, 2021 submitted to Nanomaterials rr1 | |8 of 24 | w13 = 1 3 =0 | | 3 No link | | r3 No link (a) (b) (a) (b) Node w12 = 1 2 Figure 5. (a) System with| three QDs. Vector r isw the23 position= 2 3 vector in the metric space. We represent Figure 5. (a) System with three QDs. | Vector r is the position vector in the metric space. We represent 2 each QD as a node, and we label it with the ket notation i .(b) Methodology to form weighted links each QD as a node,2 and we label it with the ket notation i .(b) Methodology to form weighted links | according to QMs. Each weight wij is the| overlap integral between yQDi and yQDj (or, equivalently, r according to QMs. Each weight w is the overlap integral between y and y (or, equivalently, 2 i j in Dirac’s notation). ∂ijã QDi QDj 1 3 1 3 ∂ ã | r1 | | wi j13in= Dirac’s1 3 notation).=0 | r3 Ö ∂ ã | Ö ∂ ã No link

(a) (b)

Nanomaterials 2021, 11, 375 8 of 22 Figure 5. (a) System with three QDs. Vector r is the position vector in the metric space.w We0.65 represent ij w 0.65 each QD as a node, and we label it with the ket notation i .(b) Methodology to form weighted links ij i j , according to QMs. Each weight wij is the overlap integral between yQDi and yQDj (or, equivalently, i j Node w12 = 1 , 2 i j in Dirac’s notation). ∂ ã dE(i,h j)=3| i RQCP w23 = 2 3 2 d2E(i, j)=3Rh QCP| i wgh 0.05 | i g | i wgh 0.05 Ö ∂ ã r2 g h Overlap 3 3 h 1 1 dE(g, h) = 15RQCP r1 |i Overlap w13 = 1 3 =0 | i | i | i h | i dE(g, h) = 15RQCP r3 No link (a) (b) Figure 5. (a) System with three QDs. Vector r is the positiond vectorE/R inQCP the metric space. We represent wij 0.65 dE/RQCP each QD as a node, and we label it with the ket notation i ;(b) methodology to form weighted links Figure 6. Overlap between the electron wave functions in two QDs as a function of the distance

accordingi to QMs.j Each weight wFigureij is the 6. overlapOverlap integral between between theψQDi electronand ψQDj wave(or, functions equivalently, in two QDs as a function of the distance , (between dot centers) normalized by the radius∣ ⟩ of the QD, dE RQD. The insets aim to graphically i j in Dirac’s notation). (between dot centers) normalized by the radius of the QD, dE RQD. The insets aim to graphically dE(i,illustrate j)=3 howRQCP increasing the separation between the nodes implies a longer link with much less weight. illustrate how increasing the separation between/ the nodes implies a longer link with much less weight. ⟨ ∣ ⟩ / ��� wwgh 0.065.05 g ij ⇡ h Overlap ��� d (g, h) =j 15R 206 The firstE one correspondsi QCP to the inset in which the centers of the nodes (sites) i and j are separated 206 The first one corresponds to the inset in which the centers of the nodes (sites) i and j are separated 207 by��� a distance dE(i,i, jj)=3= 3RQCPRQD for which i j = wij ⌅ 0.65. Thus we generate a link between nodes 207 by a distance dE i, j = 3 RQD for which i j = wij ⌅ 0.65. Thus we generate a link between nodes 208 i and j whose weight is wwij gh= 0.650..05 The second inset corresponds to two nodes whose centers are at ��� 208 Overlap gi and j whose⇡ weighth is wij = 0.65. The second inset corresponds to two nodes whose centers are at 209 a dE g, h = 15(RQD) for whichwgh =g hÖ ∂=ã wgf ⌅ 0.05. A link is generated, but has a very small ��� 209 a ddEE(g,g, h) =15 R(QCPRQD) for which wgh = g hÖ ∂=ã wgf ⌅ 0.05. A link is generated, but has a very small 210 weight dwEgh /R⌅ 0.05QCP. Note that the overlap integral wij 0 when dE i, j > 25 RQD, and is wij = 0 for 210 weight wgh ⌅ 0.05. Note that the overlap integral wij 0 when dE i, j > 25 RQD, and is wij = 0 for 211 dE ���i,(j ')30 RQD. That is, in our system,Ö ∂ allã those nodes whose centers are separated by a distance Overlap between the electron wave functions211 indE twoi,(j QDs')30 as R aQD function. That is, of in the our distance system,Ö ∂ allã those nodes whose centers are separated by a distance Figure 6. 212 dE ' 30� RQD� = dE��,Lim (or��limit��distance)�� are��not allowed to be linked.( This) is the case of nodes 1 and 212 dE ' 30 RQD = dE,Lim (or limit distance) are not allowed to be linked.( This) is the case of nodes 1 and (between dot centers) normalized by213 the3 ( radiusin) Figure of the5, where QD, d1E 3RQD= w.13 The= 0. insets aim to graphically 213 3 (in) Figure 5, where 1 3 = w13 = 0. illustrate how increasing the separation214 betweenAlthough the nodes it does implies not seem a longer clear link yet with at this much point, less weight. ∂ ã Figure 6. Overlap between the electron wave functions in two QDs as a function of the distance ∂ ã ∂ ã 214 Ö ∂/Althoughã it does not seem clear yet at this point, (between dot centers) normalized∂ ã by the radius of theÖ ∂ QD,ã dE RQD. The insets aim to graphi- cally illustrate how increasing the separation between thedE,Lim nodes d impliesS, a longer link with much (6) d , d , (6) less weight. / E Lim S 215 defines a new scale in the system, dS. Note that our approach is a modified version of a RGG. As 206 The first one corresponds to the inset in which the centers215 defines of the a new nodesscale (sites)in thei and system,j ared separatedS. Note that our approach is a modified version of a RGG. As 216 mentioned,The first one a RGG corresponds is the simplest to the inset spatial in which network, the centers consisting of the innodes randomly (sites) i placingand j N nodes in some 207 by a distance d i, j 3 R for which i j w 0.65216 mentioned,. Thus we agenerate RGG is the a link simplest between spatial nodes network, consisting in randomly placing N nodes in some E = QD 217 aremetric separated space= byij and a⌅ distance connectingdE i, j two= 3 ⋅ nodesRQD for by which a link ifi j and= w onlyij ≈ 0.65 if their. Thus, Euclidean we generate distance is smaller than a link between nodes217i andmetricj whose space weight and connecting is w 0.65 two. The nodes second by inset a link corresponds if and only to if their Euclidean distance is smaller than 208 i and j whose weight is wij = 0.65. The218 seconda given insetneighborhood corresponds radius, tor. two In our nodes caseij = this whose is distance centers is aredE,Lim at. The novelty of our approach is two nodes whose centers218 a are given( at a) neighborhooddE g, h = 15 ⋅ R radius,QD ⟨for∣ ⟩ whichr. In ourwgh case= g thish = isw distanceg f ≈ 0.05. is dE,Lim. The novelty of our approach is 209 a dE g, h = 15 (RQD) for which wgh = g hÖ ∂=ã wgf ⌅ 0.05. A link is generated, but has a very small A link is generated, but has a very small weight wgh ≈ 0.05. Note that the overlap integral 210 weight w 0.05. Note that the overlap integral w 0 when d i, j 25 R , and is w 0 for gh ⌅ wij → 0 when dEiji, j > 25 ⋅ RQD, andE ( is w>ij) = 0 for dQDE i, j ≥ 30 ⋅ RQDij =. That⟨ ∣ is,⟩ in our system, 211 dE i,(j ')30 RQD. That is, in our system,allÖ ∂ those allã those nodes nodeswhose centers whose are centers separated are by separated a distance dE by≥ a30 distance⋅ RQD = dE,Lim (or limit distance) are not( allowed) to be linked. This is the case of( nodes) 1 and 3 in Figure5, where 212 dE ' 30 RQD = dE,Lim (or limit distance) are not allowed to be linked.( This) is the case of nodes 1 and 1 3 w 0. ( ) = 13 = 213 3 in Figure 5, where 1 3 = w13 = 0. Although it does not seem clear yet at this point, ∣ ⟩ ∣ ⟩ 214 Although it does not seem clear yet⟨ ∣ at⟩ this point, ∂ ã ∂ ã Ö ∂ ã dE,Lim ≡ dS, (6)

definesd aE, newLim scaledSin, the system, dS. Note that our approach is a modified (6) version of a RGG. As mentioned, an RGG is the simplest spatial network, consisting of randomly 215 defines a new scale in the system, dS.placing Note thatN nodes our in approach some metric is space a modified and connecting version two of nodes a RGG. by a link As if and only if their Euclidean distance is smaller than a given neighborhood radius, r. In our case, this is 216 mentioned, a RGG is the simplest spatial network, consisting in randomly placing N nodes in some distance is dE,Lim. The novelty of our approach is that any link between nodes i and j is 217 metric space and connecting two nodescharacterized by a link if by and a weight only if that their involves Euclidean the overlap distance integral is between smaller kets thani and j . 218 a given neighborhood radius, r. In our caseTo this advance is distance in our model, is dE, itLim is. necessary The novelty to introduce of our some approach essential is concepts. The first one arises from the very interaction between nodes. When two nodes∣ ⟩ are∣ directly⟩ connected by a link, they are then said to be “adjacent” or neighboring. The adjacency matrix A encodes the topology of a network, that is, whether or not there is a link (aij = 1 or aij = 0) between any two pairs of nodes i and j. Sometimes, this binary information Nanomaterials 2021, 11, 375 9 of 22

encoding whether or not a node is connected to another is not enough, and it is necessary to quantify the “importance” of any link (the strength of a tie between two users in a social network, or the flow of electricity between two nodes in a power grid [35]) by assigning to each link a “weight”. In that case, the matrix that encodes the connections is called weighted adjacency matrix W [41]. With our method, the weighted adjacency matrix corresponding to the network represented in Figure5b is

0 1 2 1 3 0 1 2 0 2 1 0 2 3 = 2 1 0 2 3 , (7) ⎛ 3 1 ⟨3∣2⟩⟨ 0∣ ⟩⎞ ⎛ 0 ⟨3∣2⟩ 0 ⎞ ⎜⟨ ∣ ⟩ ⟨ ∣ ⟩⎟ ⎜⟨ ∣ ⟩ ⟨ ∣ ⟩⎟ which is symmetric (since i j = j i in this quantum systems), off-diagonal and non- ⎝⟨ ∣ ⟩⟨ ∣ ⟩ ⎠ ⎝ ⟨ ∣ ⟩ ⎠ negative. In particular, 1 3 = 3 1 = 0 since there is no overlap between the electron wave functions in kets 1 and⟨ ∣ ⟩3 . ⟨ ∣ ⟩ An interesting point⟨ ∣ in⟩ Expression⟨ ∣ ⟩ (7) is that its matrix elements have in QM the meaning of probability amplitude∣ ⟩ ∣, ⟩i j , which is related to the probability for an electron to be in i and j , Pi↝j, as follows: ⟨ ∣ ⟩ ∗ 2 ∣ ⟩ ∣ ⟩ j i j i = j i . (8)

We can now generalize the idea⟨ ∣ ⟩ from⟨ ∣ ⟩ the∣ ⟨ toy∣ ⟩system ∣ in Figure5 to the complete, complex, quantum system composed of N QDs that are independently and uniformly 2 distributed in the metric space R . The corresponding adjacency matrix is thus an N × N weighted adjacency matrix WPA whose matrix elements, i j , are the probability amplitude for an electron in kets i and j , ⟨ ∣ ⟩ ∣ ⟩ ∣ ⟩ 0 , if i = i WPA ij = (9) i j , if i ≠ j

Once we have defined our( weighted) adjacencyv matrix, WPA in (9) and interpreted its meaning in QM, we now have enough knowledge⟨ ∣ ⟩ to represent the system as a network by using the undirected, weighted graph ≡ , , WPA , where is the set of nodes G G N L N (card = N) and is the set of links. We have specified the matrix WPA in the triplet N L ≡ , , WPA to emphasize the fact that the( connections) between the nodes are made G G N L using( the) WPA matrix and not, for example, a conventional adjacency matrix A (aij = 1 if i and j (are directly) linked; 0 otherwise), which would result in different results. Note that, because of the way we have generated the links, the weighted adjacency matrix WPA quantifies connections that have physical meaning according to QM, and explicitly includes the spatial structure of the system (remember Figure6 and its associated, previous discussion). This is the key point that allows us to apply to WPA techniques that are well known in network science. For instance, in addition to the weighted adjacency matrix, it is common to use the diagonal degree matrix D, whose elements Di are the sum of weights of all links directly connecting node i with others. In our particular system, D has physical meaning: using (9), D W i j s is the sum of the i = ∑i≠j PA i = ∑i≠j ≡ APi probability amplitudes on ket i . We label it sAPi to stress this physical meaning. WPA helps us obtain Laplacian matrices( that will) assist⟨ us∣ ⟩ in studying electron dy- namics using CTQW, quantum∣ ⟩ walks that are continuous in time and discrete on space. See [64] for a very illustrative discussion on CTQW and their use in the simulation of quantum systems. The first type of Laplacian matrix, the (combinatorial) Laplacian, or simply, Laplacian matrix,

L = D − WPA. (10)

Note that the Laplacian matrix L computed using the weighted adjacency matrix WPA is different from the one used in other works [72,84,95]. In these approaches, the matrix elements of L are assumed to be equal γij ≡ γ = 1. In our approach, the matrix elements Nanomaterials 2021, 11, 375 10 of 22

take different values since they depend on the involved overlap integrals (or probability amplitudes, 0 ≤ wij < 1) and, as shown throughout the paper, they play a natural role in the probability for an electron to tunnel from one node to another. The Laplacian acts as a node to node transition matrix. The Hamiltonian of the CTQW can be written as H = L [68,75,77,82,84,85,96–101]. −1 2 −1 2 The second one is the normalized Laplacian matrix [96], LN = D LD , an Hermi- tian operator that, according to the way we have defined WPA in (9),/ has matrix/ elements in the form 1 , if i = j L i j (11) N ij = − , if i j sAPi⋅ sAPj ≠ ⟨ ∣ ⟩ Ó Ô LN allows for generating( the) correspondingw unitary CTQW [96] of an electron on our graph ≡ , , WPA as G G N L

−iLN⋅t ( ) UsLN t = e (12) Note that, in the time evolution operator generated by L in (12), the imaginary unit ( ) N makes UsLN be unitary [75]. As in other CN approaches [74,76,102,103], we assume h¯ ≡ 1 so that time and energy can be treated as dimensionless. We will use UsLN t to study the temporal evolution of our quantum system. ( ) 5. Experimental Work: Simulations 5.1. Network Parameters The spatial constraints stated by the overlap integral have effects on CN parameters such as degree distribution, clustering, and average shortest path, defined as: • The degree distribution of a network captures the probability P k that a randomly chosen node exhibits “degree” k (= number of links). P k and its mean value k (mean degree) are very useful since it quantifies to what extent( nodes) are heteroge- neous 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 > k 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], charac- −γ terized by a degree distribution P k ∼ k . 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 ki links that connects it to ki other nodes. The clustering coefficient of node i is defined as the ratio between the number Mi of links that actually exist between these ki nodes and the maximum possible number of links, that is, Ci = 2Mi ki ki − 1 . The clustering coefficient of the whole network is: 1 / ( = ) Ci. (13) C N = i • The average shortest path length, `,⟨ 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:

1 ` = dij (14) N N − 1 = i≠j

( ) Nanomaterials 2021, 11, 375 11 of 22

where dij is the distance between node i and node j. The length of the shortest path between two nodes i and j in a network is the minimum number of links for going from node i to j. Its average is computed over all possible pairs of nodes. When ` is small when compared to the “network size” (number of nodes, N), the small-world property arises. Intuitively, this means that any pair of nodes are relatively “close”. Mathematically, this means that the average shortest path scales logarithmically with the network size [32]: ` ∼ ln N. The fact that an electron cannot tunnel from one QD to another that is at a distance longer than dE,Lim leads to inhibiting the existence of long-range links to connect hubs (that is, shortcuts are not allowed). The increasing tendency to establish connections in the neighborhood (d < dE,Lim) leads to a high clustering coefficient ( ≈ 0.57). C 5.2. The Network Has a Percolation Transition as the Dot Density Increases⟨ ⟩ Consider the density of QDs as ρQD = N A, where A = 1 is the area of a unit square. When ρQD is very small (and, so does k , k being the number of links per node), the nodes are so far apart that there is no overlap between/ the wave functions and, as a consequence, there is no links: k → 0 and there are⟨ N⟩ isolated nodes. As ρQD increases (and, so does k ), the QDs are closer and closer and electron wavefunctions in⟨ some⟩ of them start to overlap, causing the formation of links among some of them. As a consequence, small⟨ clusters⟩ begin to appear. They are disconnected from each other. These isolated subnetworks are called components and all of them have similar size. However, there is a value of k for which one of the clusters becomes dominant and begins to grow to the detriment of the others. This cluster is called a giant component (GC). The fraction or normalized size of⟨ the⟩ giant component with respect to the total number of nodes is quantified as SGC = NGC N. Figure7 shows SGC of our network as a function of the average node degree k . Note that, for k < 3.4, there is no giant component. There are several disconnected networks, many/ of then being trees. The point k = 3.4 in Figure7 seems to be a critical point⟨ ⟩ at which SGC has⟨ ⟩ an abrupt transition. For 3.4 < k < 10, there is a single giant component with small clusters. For k ≥ 10, there is⟨ only⟩ a single giant component, that is, the network is connected. ⟨ ⟩ This formation of a GC on a macroscopic scale⟨ is⟩ an example of percolation transi- tion [104]. The fraction of nodes belonging to the giant cluster becomes the order parameter of the percolation transition. Following [104], we have denoted it as m k . Ref. [104] explores the most recent advances of percolation theory in CN, and studies the order parameter m for continuous, explosive, discontinuous, and hybrid percolation(⟨ ⟩) transitions. According to [104], our network exhibits a Hybrid Percolation Transition (HPT) since it has properties of both second-order (critical phenomena) and first-order (abrupt jump of the order parameter) phase transitions at the same transition point, k C, in our case. In our case, m k ≡ SGC k fulfills ⟨ ⟩ (⟨ ⟩) (⟨ ⟩) 0 , if k < k C m k = βm (15) m0 + r ⋅ k − k C , if k ≥ k C ⟨ ⟩ ⟨ ⟩ where m0 and r are constants,(⟨ ⟩) w and βm is the critical exponent of the order parameter. For (⟨ ⟩ ⟨ ⟩ ) 0.099 ⟨ ⟩ ⟨ ⟩ k C ≥ 3.4, m k = 0.0492937 + 0.788699 k − 3.4 .

⟨ ⟩ (⟨ ⟩) (⟨ ⟩ ) Nanomaterials 2021, 11, 375 12 of 22

���

��� 0.0492937 + 0.788699( k 3.4)0.099 ��� h i ���

���

Giant Component Size Component Giant ��� � � �� �� k k =2.7 k h=5i.0 k = 10 h i h i h i

SGC =0.20 SGC =0.88 SGC =1.0 Figure 7. Normalized size of the giant component with respect to the total number of nodes, SGC = NGC N, as a function of the average node degree k . See the main text for further details.

5.3. Studying/ the Emergence of Electron Transport ⟨ ⟩ We characterize the network’s transport efficiency by using the average return probability α t , defined as [85] N 1 2 α t j U, t j , (16) ( ) = N = s LN j=1 ( ) ∣ ⟨ ∣ ( ) ∣ ⟩ ∣ where the operator UsLN t , stated in Equation (12), is the unitary time evolution operator governing the evolution of the probability amplitudes. Please note that, as shown in a num- ber of papers [68,75,77,82( ,)84,85,96–101], the Hamiltonian of the network is the Laplacian matrix (also called Connectivity matrix in some contexts). High values of α t suggest inefficient transport since the quantum particle tends to remain at the initial node [85]. On the contrary, α t ≪ 1 means that the electron, localized at the initial node in t(=)0, tends to be delocalized, with different probability components on each node. ( ) With this concepts in mind, we define the quantum transport efficiency as

ηQT t = 1 − α t . (17)

Figure8 shows the quantum transport( ) efficiency( ) ηQT stated by (17) as a function of the average node degree k , k being the number of links per node. Note that ηQT = 0 for k < 10 while there is and abrupt transition at k = 10 so that ηQT ≈ 1 for k > 10. ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ Nanomaterials 2021, 11, 375 13 of 22

���

���

���

���

���

��� Quantum Transport Efficiency Transport Quantum � � �� �� �� k (1) h i k =2.7 k =5.0 k = 10 h i h i h i

(2)

SGC =0.20 SGC =0.88 SGC =1.0 inset 1 inset 2 inset 3

Figure 8. Quantum transport efficiency as a function of k . Insets “1”, “2” and “3” shows net- work connectivity at different values of k . SGC = NGC N is the fraction (normalized size) of the giant component. ⟨ ⟩ ⟨ ⟩ / The results shown in Figure8 has been computed with t = 500. Note that time and energy can be treated as dimensionless when assuming ¯h ≡ 1, as mentioned before [74,76,102,103]. To explain this result, we have studied what happens to the connectivity of the network when the density of QDs per unit area (A), ρQD = N A, increases. When ρQD is very small (and, so does k ), the nodes are so far apart that there is no overlap between wavefunctions and, as a consequence, there is no links: k →/0, and there are N isolated nodes. As ρQD increases (and⟨ ⟩ so does k ), the QDs are closer and closer and the electron wavefunctions in some of them start to overlap, causing the⟨ ⟩ formation of links among some of them. This is the case of the two nodes⟨ ⟩ labeled “(1)” in Figure8. Small clusters begin to appear too (cluster labeled “(2)”). Note that they are disconnected from each other. These isolated subnetworks are called components and many of them have a similar size. In any component, the electron can tunnel among the involved QDs. This is just the situation in the below, leftmost inset, called “inset 1”. However, as k continues to grow, one of the clusters becomes dominant and begins to grow more and more as forming links to other smaller clusters. This cluster is called a giant component (GC).⟨ ⟩ The fraction or normalized size of the giant component with respect to the total number of nodes is quantified as SGC = NGC N. In “inset 2”, with k = 5, we can see that there are small clusters along with a single GC whose size is SGC = 0.88. Just when k increases up to k = 10, one single/ GC (SGC = 1) appears,⟨ ⟩ which has captured all the other small components, making the complete network be connected. The electron can⟨ thus⟩ tunnel from any⟨ ⟩ node to any other node. When the network is connected, an electron always has at least one path to pass from any node j to another. As a consequence, α t = 0 in Equation (16) since the electron is no longer confined at node

( ) N ) is the 33 19 of 24 all 2 slits. N (market in inset 3 ]), but much more , and there are j 0 61 then the QTE in ( } ã ) that has captured all the ) since the electron is no N k 1 , Ö 32 = ⇧ GC . S 1, 2, { in Eq. ( " j 0 æ = 0 connected ) t = ( j ) t à ( j j P à ), the QCN are closer and closer and the electron j , ã } P k Ö ), the nodes are so far apart that there is no overlap N ã , k

Ö ⇧

hr bullet Third • 470

1, 2, , we can see that there are small clusters along with a single

eodbullet Second { • 5 469 , there is only one single GC (

)

" = is bullet First • 10 468 14 ) is ã j wave function whose components are distributed (although with k = 15

æ Ö 15 of 21

ã ultdlsslo iethis: like look lists Bulleted 467 k Ö in Eq. ( Entropy

0 Version August 25, 2020 submitted to Entropy 19 of 24

...Subsubsection 8.1.1. = 466 , our system is conceptually equivalent to an experiment with wave function extended ) t m (in non-technical language: the electron has links by which it can “escape”), ( increases (and, so does j

j 0.88.

à

j 1. Subsection 8.1. 465 427 When rQCN is very small (and, so does k ), the nodes are so far apart that there is no overlap P = continues to grow, one of the clusters becomes dominant and begins to grow , extended = } 428 between the wave functions and, as a consequence, there is no links: k 0, and there are N QCN ã

N } k r j

, GC ocuin htcnb drawn. be can that conclusions Ö ã 464 429 Ö isolated nodes. As rQCN increases (and, so does k ), the QCN are closer and closer and the electron is very small (and, so does S à ⇧ . The electron can thus tunnel j

of a connected network with 430 wavefunctions in some of them start to overlap, causing the formation of linksÖ ã among some of them. ecito fteeprmna eut,teritrrtto swl steexperimental the as well as interpretation their results, experimental the of description 463 ã P n

) appears, which has captured all 1, 2, . In “inset 2”, with ∂

1 Ö ã { 431 As a consequence, small clusters begin to appear. They are disconnected from each other. These

hsscinmyb iie ysbedns tsol rvd ocs n precise and QCN concise a provide should It subheadings. by divided be may section This N 462 = " 1 r 432 j isolated subnetworks are called components and many of them have similar size. Some of them are / connected { , 0.000201637, 0.0188286, 0.000133976, GC æ

6 ã

S 433

trees.This is just the situation in the below, leftmost inset, called “inset 1”.

GC Nanomaterials 2021, 11, 375 14 of 22 drawn. 461 y 10 ∂ ã

N 434 min However, as k continues to grow, one of the clusters becomes dominant and begins to grow , we can see that there are small clusters along

n

n 5 fteeprmna eut,teritrrtto swl steeprmna ocuin htcnbe can that conclusions experimental the as well as interpretation their results, experimental the of 460 First bullet Second bullet Third bullet When However, as Just when increases up to A quantum walker can thus hop from any node to anyWhen other the node. network In is QM connected, the an electron electron passes always has at least 1 path to pass from any node In fact, from the point of view of QM, unless additional measurements are made at the nodes, ∂ Ö This section may be divided by subheadings. It should provide a concise and precise description This section may bedescription divided of by the subheadings. experimentalconclusions results, that It their can should be interpretation drawn. provide as a well concise as and the precise experimental Bulleted lists look like this: then the the quantum transport efficiency ( =

= 435 more and more and linking to others. This cluster is called a giant component (GC). The fraction or =

}

ã hsscinmyb iie ysbedns tsol rvd ocs n rcs description precise and concise a provide should It subheadings. by divided be may section This N 459 k 436 Ö ã

, and can now behave as an normalized size of the giant component with respect to the total number of nodes is quantified as ,

Ö j, and can now behave as an extended wave function whose components are distributed , on each of the kets to another. As a consequence, j GC QT ã j ⇧

S h between the wave functionsisolated and, nodes. as As a consequence, there is no links: isolated subnetworks are called components andtrees.This many is of just them the have situation similar in size. the Some below, of leftmost them inset, are called “insetmore 1”. and more and linking tonormalized others. size This of cluster the is called giant a component giant with component respect (GC). to The the fractionGC or total whose number size of is nodes is quantified as other small components, making the complete network be through all intermediate nodes longer confined at node and can now behave as an different amplitudes) between different nodes. As nodes of the network. This iscomplex. equivalent If to we the are famous interested two-slit inin experiment knowing Figure (see if ) [ the to electron a passes final from node an initial node 6. Discussion 7. Summary and conclusions 8. Results of the experimental results, their interpretationdrawn. as well as the experimental conclusions that can be 8.1. Subsection 8.1.1. Subsubsection • • • y Version August 25, 2020 submitted to As a consequence, small clusters begin to appear. They are disconnected from each other. These wavefunctions in some of them start to overlap, causing the formation of links among some of them. we cannot know which path the electron has chosen. Being strict, the electron goes through (although with different437 probabilityS N components)N. In “inset between 2”, different with k nodes5, (see we Figure can see9). that there are small clusters along with a single

∂ GC = GC =

n .Results 8. 458 Ö As α t 0, then the quantum transport efficiency (17) is ηQT t 1 − α t 1. , one single GC ( ≈ 438 GC whose size is 0.88. = ≈ 1, 2, S GC = 2 {

10 Figure9 is an example of the electron probability components, n ψ , on each of 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 439461 462 463 /464 465 466 467 Ö ã 468 469 470 " Just when increases up to k = 10, there is only one single GC (SGC = 1) that has captured all the = j

0.88. the kets( ) n of a network with N = 100 nodes. In this example,( ) the electron( ) was localized ã

æ 440 k other small components, making the complete network be .

= connected .Smayadconclusions and Summary 7. Ö 457 in node 7 at t 0 (initial state), and after applying the unitary evolution∣ ⟨ ∣ ⟩ ∣ operator (12)

0 = 441 Ö ã

GC A quantum walker can thus hop from any node to any other node. In QM the electron passes = for t = 500∣ ⟩, it has evolved to a extended wave function whose probability components S Nanomaterials ) 2 to another. As a consequence, . In “inset 2”, with

t n ψ ∣ are⟩ distributed442 overthrough the N 100 all intermediate nodes. nodes ( j = j

N

ã ∂

/ Ö

à .Discussion 6. j 456 443 When the network is connected, an electron always has at least 1 path to pass from any node

P

GC

1. ∣ ⟨ ∣ ⟩ ∣ ã N ∂ 444 j to another. As a consequence, æj " 1, 2, ⇧, N , Pjàj t = 0 in Eq. (32) since the electron is no =

=

} increases up to j n ). As y 445 455 ���� longer confined at node j (in non-technical language: the electron has links by which it can “escape”), 9 ã à j GC

k { } ( ) S

Ö 446

P and can now behave as an extended wave function whose components are distributed (although with n 454 Electron probability amplitudes,

2 ���� 447 different amplitudes) between different nodes. As P t 0 æj 1, 2, ⇧, N then the QTE in (33) is

1 jàj = "

2 N m slits. 2 with experiment an to equivalent conceptually is system our , nFgr oafia node final a to ) Figure in ã∂ { 453 ∂ y

100 nodes. 448 P

∂ hQT = min 1 jàj = 1.

= QD n j min mre nist3 inset in (market node initial an from passes electron the if knowing in interested are we If complex.

452 ����

y 449 ( ) { }

N In fact, from the point of view of QM, unless additional measurements are made at the nodes, Just when Figure 9. ∂ ∂Ö 0.000172176, 0.00582029, 0.00387988, 0.00106017, 0.000970012, 0.0000719148, 0.00413783, =

) u uhmore much but ]), 61 [ (see experiment two-slit famous the to equivalent is This network. the of nodes 451 450 we cannot{ know which} path the electron has chosen. Being strict, the electron goes through all the QT

h is quantified as the other small components, making thefrom complete any network node be to any other node. When the network issince connected, the an electron electron is always no has longer at confined least at 1 node nodes (see Figure 0.000454207, 0.0768675, 0.00242522, 0.0298138, 0.00921407, 0.0000565646,0.0938865, 0.000102786, 0.00010824, 0.0000761972, 0.0027705, 0.0000545876, 0.000735455,0.0000908889, 0.000247052, 0.000190905, 0.00886395, 0.00274959, 0.00197876,0.000360844, 0.00202883, 0.00247567, 0.003812, 0.0193588,0.00103512, 0.0197223, 0.000150275, 0.0000203756, 0.0000862307, 0.0300932, 0.0294795, 0.0000281, 0.0203633,0.0000315912, 0.000148517, 0.00025177, 0.0172012, 0.00847614, 0.0000518012, 0.00144027, 0.000285581, 0.00469129, 0.00814919, 0.000261477, 0.000582733,0.0032206, 0.0147341, 0.00371945, 0.0336232, 0.00980407, 0.000807111,0.000115002, 0.000121867, 0.0447992, 0.0001218, 0.0459497,0.0156714, 0.00431284, 0.0137177, 0.0105552, 8.72451 0.0022036, 0.000168711, 0.00168941, 0.000250133, 0.0166772,0.000493777, 0.00534086, 0.00114926, 0.0344186, 0.0928017, 0.00393284, 0.00341317, 0.00876622,0.0000476467, 0.0000574528, 0.000893781, 0.0117769, 0.0451846, 0.0178166, 0.0034605, 0.0901622,0.000129045, 0.000244178, 0.00373535, 0.0000291204, 0.000142244, 0.00499453, 0.0118115 0.00037752, Version January 17, 2021 submitted to path to pass from any node (GC). The fraction or normalized size of the giant component with respect to the total number of nodes with a single GC whose size is whose components are distributed (although with different probability amplitudes) between different ����

451 all the ecno nwwihpt h lcrnhscoe.Bigsrc,teeeto osthrough goes electron the strict, Being chosen. has electron the path which know cannot we nodes of the network. This is equivalent to the famous two-slit experiment (see [61]), but much more 450 } {

345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 452364 complex.365 366 367 368 If369 we370 are interested in knowing if the electron passes from an initial node j (market in inset 3

nfc,fo h on fve fQ,uls diinlmaueet r aea h nodes, the at made are measurements additional unless QM, of view of point the from fact, In 449 } { )

( ����

� �� 453 in�� Figure )�� to a final�� node m,��� our system is conceptually equivalent to an experiment with N 2 slits.

= = j j QT à 1. 1 min

h P

448 454 n

" = j j à N j t is ) 33 ( in QTE the then , 2, 1, 0 As nodes. different between amplitudes) different

P

⇧ æ 447 Figure 9. Electron probability455 components,n y n ψ 2, on each of the kets n of a connected network

∂ ã extended aefnto hs opnnsaedsrbtd(lhuhwith (although distributed are components whose function wave an as behave now can and 446 ) ( } { with N = 100 nodes.

456 6. DiscussionÖ ∂ ∣ã ⟨ ∣ ⟩ ∣ ∣ ⟩

j i o-ehia agae h lcrnhslnsb hc tcn“escape”), can it which by links has electron the language: non-technical (in node at confined longer

445 For the sake of clarity, Table1 lists the corresponding numerical values represented in

= " j j Figure9. Please note that the sum of the probability components in the whole system is à t N j j

ic h lcrni no is electron the since ) 32 ( Eq. in 0 , , 2, 1, consequence, a As another. to

P ⇧ æ 444 N 100 2 457 7. Summary and conclusions

= n ψ 1.

∑n=1 = hntentoki once,a lcrnawy a tlat1pt ops rmaynode any from pass to path 1 least at has always electron an connected, is network the When 443

Table 1. List of the electron probability components, n ψ 2, on each of the kets n of a connected hog l nemdaenodes intermediate all through ∣ ⟨ ∣ ⟩ ∣ 458 442 8. Results

network with N = 100 nodes. The sum of the probability components in the whole system is

unu akrcntu o rmaynd oayohrnd.I Mteeeto passes electron the QM In node. other any to node any from hop thus can walker quantum A N 100 2

441 ã Ö =

n ψ 1. 459 This section∣ ⟨ may∣ ⟩ ∣ be divided by subheadings.∣ ⟩ It should provide a concise and precise description

∑n=1 =

connected . te ml opnns aigtecmlt ewr be network complete the making components, small other 460 440 of the experimental results, their interpretation as well as the experimental conclusions that can be

0.000172176∣ ⟨ ∣ ⟩ ∣ 0.00582029 0.00387988 0.00106017 0.000970012

461

= =

GC drawn. S k 0.0000719148 0.00413783 0.000454207 0.0768675 0.00242522

hthscpue l the all captured has that ) 1 ( GC single one only is there , 10 to up increases when Just

439 ã Ö

/ 0.0298138 0.00921407 0.0000565646 0.000102786 0.0000761972

= GC

S 0.88. is size whose GC

438 0.0938865 0.00010824462 This 0.0027705section may be 0.0000545876 divided by subheadings. 0.000735455 It should provide a concise and precise

0.000247052 0.000190905 0.00274959 0.0000908889 0.00886395

= = GC

GC 463 k N N S

ecnseta hr r ml lsesaogwt single a with along clusters small are there that see can we , 5 n“ne ” with 2”, “inset In . description of the experimental results, their interpretation as well as the experimental 437 0.00197876 0.00202883 0.003812 0.0197223 0.0300932

0.000148517 0.000360844464 conclusions 0.00247567 that can 0.0193588be drawn. 0.0000203756

omlzdsz ftegatcmoetwt epc otettlnme fndsi unie as quantified is nodes of number total the to respect with component giant the of size normalized 436 ã Ö 0.0000281 0.0172012 0.00814919 0.0147341 0.00103512

0.000150275 0.0000862307 0.0294795 0.0203633 0.00025177 oeadmr n ikn oohr.Ti lse scle in opnn G) h rcinor fraction The (GC). component giant a called is cluster This others. to linking and more and more

435 465 8.1. Subsection

0.0000518012 0.00469129 0.0000315912 0.00847614 0.00144027

k otne ogo,oeo h lsesbcmsdmnn n eist grow to begins and dominant becomes clusters the of one grow, to continues as However, 434 0.000285581 0.000261477 0.000582733 0.00371945 0.0336232

0.0032206 0.00980407466 8.1.1. Subsubsection 0.000807111 0.000121867 0.0459497

re.hsi uttestaini h eo,lfms ne,cle ist1”. “inset called inset, leftmost below, the in situation the just is trees.This 433 0.0105552 0.00168941 0.00534086 0.000115002 0.0447992

−

0.0001218 0.00431284467 Bulleted8.72451 lists× 10 look6 like0.000201637 this: 0.0188286 sltdsbewrsaecle opnnsadmn fte aesmlrsz.Sm fte are them of Some size. similar have them of many and components called are subnetworks isolated

432 0.000133976 0.0156714 0.0137177 0.0022036 0.000168711

sacneune ml lsesbgnt per hyaedsonce rmec te.These other. each from disconnected are They appear. to begin clusters small consequence, a As 468 431 ã Ö 0.000250133 0.0166772• First 0.00114926 bullet 0.0344186 0.00393284

0.000493777 0.0928017 0.00341317 0.00876622 0.000893781

469 • Second bullet

aeucin nsm fte tr ooelp asn h omto flnsaogsm fthem. of some among links of formation the causing overlap, to start them of some in wavefunctions 430 ã Ö 0.0451846 0.0901622 0.0000291204 0.0000476467 0.0000574528

0.0117769 0.0178166470 • Third 0.0034605 bullet 0.000244178 0.000142244

QCN k ,teQNaecoe n lsradteelectron the and closer and closer are QCN the ), does so (and, increases As nodes. isolated r 429 ã

Ö 0.00037752 0.000129045 0.00373535 0.00499453 0.0118115

N k n hr are there and , 0 links: no is there consequence, a as and, functions wave the between 428

QCN k ,tendsaes a pr htteei ooverlap no is there that apart far so are nodes the ), does so (and, small very is When r 427

Entropy 9o 24 of 19 to submitted 2020 25, August Version Nanomaterials 2021, 11, 375 15 of 22

6. Potential Applications, Strengths, and Weaknesses of the Proposed Method 6.1. Prospective Applications To the best of our knowledge, there are two possible groups of applications: interme- diate band (IB) materials and light-harvesting materials.

6.1.1. Intermediate Band Materials The first potential application of the proposed method could be found within the field of IB materials to put into practice the concept of intermediate band solar cell (IBSC) [105]. This solar cell has an isolated IB within what, otherwise, would be the semiconductor gap, EG. The IBSC was proposed by Luque and Martí in [106] and exhibits a limiting efficiency of 63.2%, much higher than the Shockley–Queisser (SQ) limit [107] of the single- gap solar cell (40.7%). Unlike other sub-bandgap absorbing proposals, the IBSC concept surpasses the SQ limit by means of: (1) increasing the photogenerated current via the two-step absorption of sub-bangap photons via an IB located within the gap EG; and (2) without degrading the cell output voltage. Several technological approaches have been proposed aiming to obtain IB materials. The first one, the QD approach has led to the quantum dot intermediate band solar cell (QD-IBSC) [108], the first device on which it has been possible to experimentally prove the two concepts the IBSC is based on [109–112]. A sufficiently dense array of QDs with a single bound state leads to an IB material. Regarding this, a possible future application of our method could consist in exploring the electron conductivity in the IB that arises from the bound states within the QDs. A second feasible strategy for obtaining IB materials is based on semiconductor bulk materials containing a high density of adequate deep-level impu- rities and those corresponding to materials that “naturally” have an IB (theoretically predicted by ab-initio methods) [113–119]. Regarding the use of deep-level impurities, the proposed method could potentially be used to study when a sufficiently large density of impurities produces a transition from localized states (in the impurities) to states that are extended to the whole volume of the host semiconductor. It could be interesting to compare with experimental data [120,121] and theoretical models [122]. Putting it simply, deep centers transform into a band when their density is sufficiently high (Mott transition) [123,124], and, as a consequence, the electron wave-function becomes extended instead of localized. Regarding this, the experimental test of the theoretical model stated in [122] have been recently proved in silicon implanted with Ti [120] and in silicon supersaturated with sulfur [121].

6.1.2. Light-Harvesting Materials In the field of light-harvesting materials [125], QDs have been found to be promising light-harvesting materials because of their size-, shape-, and composition-dependent elec- tronic properties, and exciton generation after photoexcitation [126]. The key challenge is to model and design nanoscale materials with tailored properties for light harvesting work. Regarding this, our proposal could be used to explore the transport of energy (ex- citons) in light-harvesting QDs [126], light-harvesting molecules [125,127,128], polymer nanoparticles [129], dimers, and networks [130].

6.2. Strengths and Weaknesses The proposed method has a set of strengths and weaknesses. Among the strengths, the method allows for obtaining more realistic simulations of nanostructures than others found in the literature, where the fact that the tunneling probability decreases exponentially with the distance is not considered. The method is also generalizable to nanostructures other than QDs, such as impurities in semiconductors, light-harvesting molecules, etc. Another advantage is that it allows obtaining an approximation of the electron behavior in systems with a high number of components per area unit (whether they are QDs or impurities). The main weakness of the proposed method is that it is a first approach in the sense that identical quantum dots have been assumed. However, self-assembled quantum dot growth Nanomaterials 2021, 11, 375 16 of 22

methods produce ensembles of quantum dots that have size dispersion, which leads to a dispersion in bound levels from one QD to another. The probability of electron hopping from a QD to another with the same energy level must be considered when computing the weight of links. Additionally, it is necessary to consider not only the electron system but also the phonon system, which provides/absorbs the energy when an electron hops from one bound state in a QD to another with different energy. Because of its complexity, we leave this refinement for future work.

7. Summary and Conclusions This paper has proposed the use of a special class of Random Geometric Graphs (RGG) to model systems formed by N disordered quantum dots (QDs). While discerning what a node is seems easy (QD ≡ node), what requires a bit more care and physical intuition is determining how the links between QDs are formed in such a way that they have meaning. Specifically, in the network model that we have proposed, the most novel aspect is the link formation mechanism between nodes (≡ QDs): any link between two nodes i and j is formed if and only if the corresponding electron wave function at such nodes have non-zero overlap. Any link has thus a weight wij that is the real number (0 ≤ wij < 1) corresponding to the electron overlap integral (also called probability amplitude (PA)). The aforementioned link formation mechanism leads to a N × N weighted adjacency matrix WPA whose matrix elements are the probability amplitudes for a single electron in the involved nodes. The corresponding Laplacian matrix L, which assists in computing continuous time quantum walks (CTQW) on the associated network, is different from the one used in other works [72,84,95]. In these approaches, the matrix elements of L are assumed to be equal γij ≡ γ = 1. In our approach, the matrix elements take different values since they depend on the involved overlap integrals (or probability amplitudes, 0 ≤ wij < 1) and, as shown throughout the paper, they play a natural role in the probability for an electron to tunnel from one node to another. Regarding this, our main results point out: 1. 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. 2. 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: L.C. has worked on the conceptualization, methodology, software, investiga- tion, validation, writing—original draft preparation and writing—review and editing. J.C.N.-B. has worked on the conceptualization, methodology, and writing—review and editing. All authors have read and agreed to the published version of the manuscript. Funding: This research has been partially funded by the Ministerio de Economía y Competitividad of the Spanish Government (Grant No. TIN2017-85887-C2-2-P). Conflicts of Interest: The authors declare no conflict of interest. Nanomaterials 2021, 11, 375 17 of 22

Abbreviations The following abbreviations are used in this manuscript:

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 A Adjacency matrix of a graph . G aij Element of the adjacency matrix A α t Average return probability Mean clustering coefficient of a network. C D( ) Node degree matrix: diag k1, ⋯, kN . It is the diagonal matrix formed from ⟨ ⟩ the nodes degrees. dE i, j Euclidean distance between( any pair) of nodes i and j in a network. dij 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. dE,Lim dE,Lim ≡ dS Euclidean distance limit beyond which there is no link formation. EQD Discrete electron energy in a quantum dot (QD). ηQT Quantum transport efficiency. Graph ≡ , , WPA , where is the set of nodes (card = N), is G G G N L N N L the set of links, and WPA is weighted adjacency matrix that emerges from our method to link( formation.) ( ) Hs Hamiltonian operator corresponding to the total energy of a quantum system. H Hamiltonian in matrix form. h Planck constant. h¯ Reduced Planck constant. i Ket vector in the Hilbert space . It corresponds to the electron wave function H in nanostructure (≡ site ≡ node ≡ ket) i. ∣i⟩ Bra vector in the dual space corresponding to the ket i ∈ H k Average node degree. ⟨ki∣ Degree of a node i. It is the number of links connecting∣ ⟩ i to any other node. ⟨` ⟩ Average path length of a network. It is the mean value of distances between any pair of nodes in the network. Set of links (edges) of a network (graph). L L Laplacian matrix of a graph . G Nanomaterials 2021, 11, 375 18 of 22

−1 2 −1 2 LN Normalized Laplacian matrix, LN = D LD . m Electron mass. / / M Size of a graph . It is the number of links in the set . G L N Order of a graph = , . It is the number of nodes in set , that is, the G N L N cardinality of set : N = ≡ card . N N N Set of nodes (or vertices)( of) a graph. N ▽2 Laplace operator. ∣ ∣ ( ) Pj↝k Probability for an electron to evolve between kets j and k in the time interval t. P k Probability density function giving the probability that∣ ⟩ a randomly∣ ⟩ selected node has k links. ψ( ) Ket or vector state in Dirac notation corresponding to the wave function ψ. RQD Radius of the quantum dot. ∣ψQD⟩ Electron wavefunction in a quantum dot. SGC SGC = NGC N normalized size of the giant component (GC) with respect to the total number of nodes N. s Sum of the probability amplitudes on ket i , s W i j . APi / APi ≡ ∑i≠j PA i = ∑i≠j Vs Potential energy operator. −VC Depth of confinement potential. ∣ ⟩ ( ) ⟨ ∣ ⟩ UC r Confining, spherical (depending only on the radial co-ordinate r), finite, and “square” potential energy. ( ) L UsLN t Time evolution operator generated by the normalized Laplacian matrix N. wij Weight of the link between node i and j. We define it as the overlap inte- ( ) gral between the electron wave functions in kets i and j or the probability amplitude i j . WPA weighted adjacency matrix whose elements are quantum probability ampli- tudes. ⟨ ∣ ⟩

References 1. Cotta, M.A. Quantum Dots and Their Applications: What Lies Ahead? ACS Appl. Nano Mater. 2020, 3, 4920–4924. [CrossRef] 2. Harrison, P.; Valavanis, A. Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures; John Wiley & Sons: Hoboken, NJ, USA, 2016. 3. 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. 4. Nowozin, T. Self-Organized Quantum Dots for Memories: Electronic Properties and Carrier Dynamics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. 5. Cipriano, L.A.; Di Liberto, G.; Tosoni, S.; Pacchioni, G. Quantum confinement in group III–V semiconductor 2D nanostructures. Nanoscale 2020, 12, 17494–17501. [CrossRef] 6. Bimberg, D.; Grundmann, M.; Ledentsov, N.N. Quantum Dot Heterostructures; John Wiley & Sons: Hoboken, NJ, USA, 1999. 7. Wang, Z.M. Self-Assembled Quantum Dots; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007; Volume 1. 8. 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. 9. 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. [CrossRef] 10. 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. [CrossRef] 11. 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. [CrossRef] 12. 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. [CrossRef] 13. 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. [CrossRef] 14. 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. [CrossRef] Nanomaterials 2021, 11, 375 19 of 22

15. 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. [CrossRef] 16. 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. [CrossRef][PubMed] 17. 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. 18. 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. [CrossRef] 19. 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. [CrossRef][PubMed] 20. 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. [CrossRef] 21. 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. [CrossRef] 22. 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. [CrossRef] 23. 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. [CrossRef] 24. Chen, J.; Du, W.; Shi, J.; Li, M.; Wang, Y.; Zhang, Q.; Liu, X. Perovskite quantum dot lasers. InfoMat 2020, 2, 170–183. [CrossRef] 25. Geiregat, P.; Van Thourhout, D.; Hens, Z. A bright future for colloidal quantum dot lasers. NPG Asia Mater. 2019, 11, 1–8. [CrossRef] 26. 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. [CrossRef] 27. 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. [CrossRef] 28. 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. [CrossRef][PubMed] 29. 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. [CrossRef] 30. Sogabe, T.; Shen, Q.; Yamaguchi, K. Recent progress on quantum dot solar cells: A review. J. Photonics Energy 2016, 6, 040901. [CrossRef] 31. Grundmann, M.; Stier, O.; Bimberg, D. InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure. Phys. Rev. B 1995, 52, 11969. [CrossRef] 32. Barabási, A.L. Network Science; Cambridge University Press: Cambridge, UK, 2016. 33. Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep. 2006, 424, 175–308. [CrossRef] 34. Strogatz, S.H. Exploring complex networks. Nature 2001, 410, 268–276. [CrossRef] 35. 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. [CrossRef] 36. 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. [CrossRef] 37. 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. [CrossRef][PubMed] 38. Chimal-Eguía, J.C.; Castillo-Montiel, E.; Paez-Hernández, R.T. Properties of the vascular networks in malignant tumors. Entropy 2020, 22, 166. [CrossRef][PubMed] 39. Braun, P.; Gingras, A.C. History of protein–protein interactions: From egg-white to complex networks. Proteomics 2012, 12, 1478–1498. [CrossRef][PubMed] 40. Guimera, R.; Amaral, L.A.N. Functional cartography of complex metabolic networks. Nature 2005, 433, 895–900. [CrossRef] [PubMed] 41. Newman, M. Networks; Oxford University Press: Oxford, UK, 2018. 42. Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys. 2002, 74, 47. [CrossRef] 43. Montoya, J.M.; Solé, R.V. Small world patterns in food webs. J. Theor. Biol. 2002, 214, 405–412. [CrossRef][PubMed] 44. 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. [CrossRef] 45. 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. [CrossRef][PubMed] Nanomaterials 2021, 11, 375 20 of 22

46. 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. [CrossRef] 47. 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. [CrossRef] 48. 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. [CrossRef][PubMed] 49. 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. 50. Cui, L.; Kumara, S.; Albert, R. Complex networks: An engineering view. Circuits Syst. Mag. IEEE 2010, 10, 10–25. [CrossRef] 51. Crucitti, P.; Latora, V.; Marchiori, M. Model for cascading failures in complex networks. Phys. Rev. E 2004, 69, 045104. [CrossRef] [PubMed] 52. 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. [CrossRef] 53. Barthélemy, M. Spatial networks. Phys. Rep. 2011, 499, 1–101. [CrossRef] 54. Bashan, A.; Berezin, Y.; Buldyrev, S.V.; Havlin, S. The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys. 2013, 9, 667–672. 55. Barthelemy, M. Morphogenesis of spatial networks; Springer: New York, NY, USA, 2018. 56. 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. [CrossRef] 57. Penrose, M. Random Geometric Graphs; Oxford University Press: Cambridge, UK, 2003; Volume 5. 58. 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. 59. Nemeth, G.; Vattay, G. Giant clusters in random ad hoc networks. Phys. Rev. E 2003, 67, 036110. [CrossRef] 60. 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. [CrossRef] 61. 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. [CrossRef] 62. Biamonte, J.; Faccin, M.; De Domenico, M. Complex networks from classical to quantum. Commun. Phys. 2019, 2, 1–10. [CrossRef] 63. Sánchez-Burillo, E.; Duch, J.; Gómez-Gardenes, J.; Zueco, D. Quantum navigation and ranking in complex networks. Sci. Rep. 2012, 2, 605. [CrossRef][PubMed] 64. Venegas-Andraca, S.E. Quantum walks: A comprehensive review. Quantum Inf. Process. 2012, 11, 1015–1106. [CrossRef] 65. Masuda, N.; Porter, M.A.; Lambiotte, R. Random walks and diffusion on networks. Phys. Rep. 2017, 716, 1–58. [CrossRef] 66. Susskind, L.; Friedman, A. Quantum Mechanics: The Theoretical Minimum; Penguin Books: London, UK, 2015. 67. 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. [CrossRef] 68. Mülken, O.; Dolgushev, M.; Galiceanu, M. Complex quantum networks: From universal breakdown to optimal transport. Phys. Rev. E 2016, 93, 022304. [CrossRef] 69. Faccin, M.; Migdał, P.; Johnson, T.H.; Bergholm, V.; Biamonte, J.D. Community detection in quantum complex networks. Phys. Rev. X 2014, 4, 041012. [CrossRef] 70. Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 2008, 129, 11B603. [CrossRef] 71. Mülken, O.; Blumen, A. Efficiency of quantum and classical transport on graphs. Phys. Rev. E 2006, 73, 066117. [CrossRef] 72. Mülken, O.; Blumen, A. Continuous-time quantum walks: Models for coherent transport on complex networks. Phys. Rep. 2011, 502, 37–87. [CrossRef] 73. Darázs, Z.; Kiss, T. Pólya number of the continuous-time quantum walks. Phys. Rev. A 2010, 81, 062319. [CrossRef] 74. Mülken, O.; Volta, A.; Blumen, A. Asymmetries in symmetric quantum walks on two-dimensional networks. Phys. Rev. A 2005, 72, 042334. [CrossRef] 75. Agliari, E.; Blumen, A.; Mülken, O. Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. Math. Theor. 2008, 41, 445301. [CrossRef] 76. Mülken, O.; Bierbaum, V.; Blumen, A. Coherent exciton transport in dendrimers and continuous-time quantum walks. J. Chem. Phys. 2006, 124, 124905. [CrossRef][PubMed] 77. Agliari, E.; Blumen, A.; Muelken, O. Quantum-walk approach to searching on fractal structures. Phys. Rev. A 2010, 82, 012305. [CrossRef] 78. Blumen, A.; Bierbaum, V.; Mülken, O. Coherent dynamics on hierarchical systems. Phys. Stat. Mech. Its Appl. 2006, 371, 10–15. [CrossRef] 79. Mülken, O.; Blumen, A. Slow transport by continuous time quantum walks. Phys. Rev. E 2005, 71, 016101. [CrossRef] Nanomaterials 2021, 11, 375 21 of 22

80. Xu, X.P.; Li, W.; Liu, F. Coherent transport on Apollonian networks and continuous-time quantum walks. Phys. Rev. E 2008, 78, 052103. [CrossRef] 81. Xu, X.; Liu, F. Coherent exciton transport on scale-free networks. New J. Phys. 2008, 10, 123012. [CrossRef] 82. Mülken, O.; Pernice, V.; Blumen, A. Quantum transport on small-world networks: A continuous-time quantum walk approach. Phys. Rev. E 2007, 76, 051125. [CrossRef] 83. Salimi, S. Continuous-time quantum walks on star graphs. Ann. Phys. 2009, 324, 1185–1193. [CrossRef] 84. 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. [CrossRef] 85. Kulvelis, N.; Dolgushev, M.; Mülken, O. Universality at breakdown of quantum transport on complex networks. Phys. Rev. Lett. 2015, 115, 120602. [CrossRef][PubMed] 86. Watts, D.J.; Strogatz, S.H. Collective dynamics of “small-world” networks. Nature 1998, 393, 440–442. [CrossRef][PubMed] 87. 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. 88. Newman, M.E.; Watts, D.J. Renormalization group analysis of the small-world network model. Phys. Lett. A 1999, 263, 341–346. [CrossRef] 89. Martinez-Mendoza, A.; Alcazar-López, A.; Méndez-Bermúdez, J. Scattering and transport properties of tight-binding random networks. Phys. Rev. E 2013, 88, 012126. [CrossRef] 90. Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, UK, 1997. 91. Mandl, F. Quantum Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1992. 92. Galindo, A.; Pascual, P. Quantum Mechanics I; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. 93. Ballentine, L.E. Quantum Mechanics: A Modern Development; World Scientific Publishing Company: Singapore, 2014. 94. Cohen-Tannoudji, C.; Diu, B.; Laloe, F.; Dui, B. Quantum Mechanics; Wiley-Interscience: Hoboken, NJ, USA, 2006. 95. Farhi, E.; Gutmann, S. Quantum computation and decision trees. Phys. Rev. A 1998, 58, 915. [CrossRef] 96. Faccin, M.; Johnson, T.; Biamonte, J.; Kais, S.; Migdał, P. Degree distribution in quantum walks on complex networks. Phys. Rev. X 2013, 3, 041007. [CrossRef] 97. 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. 98. 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. [CrossRef] 99. Anishchenko, A.; Blumen, A.; Muelken, O. Geometrical aspects of quantum walks on random two-dimensional structures. Phys. Rev. E 2013, 88, 062126. [CrossRef] 100. Ray, R.K. Solving Quantum Random Walker Using Steepest Entropy Ascent Ansatz: A Pathway Towards Typicality. arXiv 2019, arXiv:1907.04548. 101. Gualtieri, V.; Benedetti, C.; Paris, M.G. Quantum-classical dynamical distance and quantumness of quantum walks. Phys. Rev. A 2020, 102, 012201. [CrossRef] 102. Mülken, O.; Blumen, A. Spacetime structures of continuous-time quantum walks. Phys. Rev. E 2005, 71, 036128. [CrossRef] 103. 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. 104. 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. [CrossRef] 105. Luque, A.; Martí, A.; Stanley, C. Understanding intermediate-band solar cells. Nat. Photonics 2012, 6, 146–152. [CrossRef] 106. Luque, A.; Martí, A. Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels. Phys. Rev. Lett. 1997, 78, 5014. [CrossRef] 107. Shockley, W.; Queisser, H.J. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys. 1961, 32, 510–519. [CrossRef] 108. 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. [CrossRef] 109. 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. [CrossRef] 110. 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. [CrossRef] 111. 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. [CrossRef][PubMed] 112. 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. [CrossRef][PubMed] Nanomaterials 2021, 11, 375 22 of 22

113. Wełna, M.; Zelazna,˙ 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. [CrossRef] 114. 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. [CrossRef] 115. 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. [CrossRef] 116. Martí, A.; Luque, A. Fundamentals of intermediate band solar cells. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 209–228. 117. Marrón, D.F. Thin-Film Technology in Intermediate Band Solar Cells: Advanced Concepts for Chalcopyrite Solar Cells. In Next, Generation of Photovoltaics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 277–307. 118. 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. 119. 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. 120. 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. [CrossRef] 121. 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. [CrossRef] 122. Luque, A.; Martí, A.; Antolín, E.; Tablero, C. Intermediate bands versus levels in non-radiative recombination. Phys. Condens. Matter 2006, 382, 320–327. [CrossRef] 123. Mott, N. Metal-insulator transition. Rev. Mod. Phys. 1968, 40, 677. [CrossRef] 124. Mott, N.F.; Davis, E.A. Electronic Processes in Non-Crystalline Materials; Oxford University Press: Oxford, UK, 2012. 125. 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. [CrossRef] 126. Kundu, S.; Patra, A. Nanoscale strategies for light harvesting. Chem. Rev. 2017, 117, 712–757. [CrossRef][PubMed] 127. Sarovar, M.; Ishizaki, A.; Fleming, G.R.; Whaley, K.B. Quantum entanglement in photosynthetic light-harvesting complexes. Nat. Phys. 2010, 6, 462–467. [CrossRef] 128. 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. [CrossRef][PubMed] 129. Jiang, Y.; McNeill, J. Light-harvesting and amplified energy transfer in conjugated polymer nanoparticles. Chem. Rev. 2017, 117, 838–859. [CrossRef][PubMed] 130. 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. [CrossRef][PubMed]