1. Introduction
Conceptually, a quantum dot (QD) [
1] is a “small” (less than the de Broglie wavelength), zero-dimensional (0D) nanostructure that confines carriers in all three directions in space [
2,
3,
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
Figure 1a. In particular, a (type I) semiconductor QD is a heterostructure [
6] made up of a small island (generally 10–20 nm in size) of a semiconductor material (“dot material”) embedded inside another with a higher bandgap (“barrier material” –BM–), which is able to confine both electrons and holes, as illustrated in
Figure 1b). This creates discrete energy levels for carriers, and modifies 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] technologies.
Figure 1c shows different SAQD growth modes. In the Stranski–Krastanow (SK) growth mode [
8], the deposition of the dot material starts with the formation of a two-dimensional, very thin wetting layer (WL), and when a critical amount of strained dot 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 consists of multilayer deposition with a fraction of a monolayer of dot material on the barrier matrix [
8]. SML-QDs have several advantages over SK-QDs, such as a smaller base diameter (5–10 nm), higher dot density (~
), and better control of QD size [
10,
11].
SAQD technologies help both research in Quantum Mechanics (QM) and manufacture novel devices. On the one hand, SAQD technologies 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], entanglement 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 interaction [
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,
25,
26,
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,
33,
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 represents 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) 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
. 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 Random 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
, 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
Section 2, 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.
The purpose of our work is to propose a particular class of RGG whose links have weights computed according to QM with the aim of studying electron transport in a system of disordered QDs like the one in
Figure 2a. It consists of a number of
N QDs—for instance, a layer of SAQDs [
6], which produce finite quantum confinement potentials (QCPs), randomly distributed in the physical metric space,
, characterized by the Euclidean distance,
.
In our model, any QD causing a QCP in
Figure 2a is represented by a node. To understand how links have been generated in
Figure 2b, it is convenient to have a look at
Figure 3. On its left side, we have represented, for illustrative purposes, two of these QDs, labeled
i and
j. For the sake of clarity, we have assumed that each QD has radius
. Their centers are separated by a Euclidean distance
. As shown in
Figure 3a, we have also represented an
axis that passes through the centers of both QDs. The
axis will assist us in clearly representing the associated concepts in
Figure 3b,c.
Figure 3b shows the corresponding finite QCP caused by a QD along
axis:
(inside the QD) and
(otherwise). Since the QCP is finite, there is a part of the electron wavefunction that spreads outside the QD, as qualitatively shown in
Figure 3c. Note that the QDs are close enough that the electron wavefunctions overlap. According to QM, the electron, is in both QDs with a probability amplitude as the one in
Figure 3c. The electron can tunnel from one QD to the other. We model this quantum phenomenon by forming a link between nodes
i and
j (
Figure 3d). We will show throughout the paper that the link weight
between two nodes
i and
j depends on the extent to which the electron wave-functions in both nodes overlap. Quite often, however, there are electron wave-functions in sufficiently remote QDs that do not overlap at all. Regarding this, we have represented in
Figure 3e two nodes that are so far apart that their corresponding wave-functions do not overlap (
Figure 3g). Thus, an electron that is in node
i at the initial time
will remain localized in that node even if
. We model this QM result through the absence of a link (
), as shown in
Figure 3h (not connected nodes).
Our main result is that the proposed RGG is able to capture inner properties of the complex quantum system: it predicts the system quantum state, its time evolution, and the emergence of quantum transport (QT) as the QD density increases. In fact, QT efficiency exhibits an abrupt change, from electron localization (no QT) to delocalization (QT emerges). This is an electron localization–delocalization transition that has also observed in [
60].
Our proposal could have potential application not only in improving the efficiency of QD-based optoelectronic devices (LEDs, solar cell, lasers, etc.) that make use of SAQD 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
Section 2,
Section 3 briefly introduces the QD system that we want to study, while
Section 4 explains our RGG proposal. The experimental work in
Section 5 allow for predicting inner features of the system such as the system quantum state, its time evolution, or the emergence of quantum transport.
Section 6 discusses potential applications, strengths, and weaknesses of the proposed method. Finally,
Section 7 completes the paper.
Appendix A 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,
73,
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
ℓ is small when compared to the total number of nodes
N in the network (
as
). The 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,
, 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
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.
3. The QD System
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 each nanostructure in the metric space
is determined by a position vector
. We consider a single electron (walker) freely tunneling among QDs (when allowed). An example of one-electron model is the tight-binding model [
89], which has been used for both lattice and random networks.
Aiming to later numerically illustrate the results of studying this system using CN concepts, we are going to assume a set of hypotheses about the QCF that the QDs produce. These hypotheses will allow us to tackle the problem by using some well known QM results on each individual nanostructure, so that we will be able to then focus on exploring the complete system as a RGG.
With this in mind, and for reasons that will be clearer later on, we first assume that the single band effective mass equation of electrons in the envelope approximation [
90] is a proper description of the dot and barrier bulk materials. This is because a QD size of 10 to 20 nm is much larger than the lattice constant of the material involved and, thus, it seems reasonable to consider that only the envelope part of the electron wave function is affected by the confinement potential. For the sake of clarity, we also assume that the QDs are identical (since in a closed system, energy is conserved, and the electron can only make transitions between QDs that have the same energy, that is, between QDs that have the same size [
91]) and spherical with radius
. The center of any QD
i is given by a position vector
in the metric space. We assume that its associated QCP is spherically symmetric (depending only on the radial co-ordinate
r), finite and “square”:
as shown in
Figure 4a.
The reason why we have made use of
is that the time-independent Schrödinger’s equation, which allows for computing both the electron wavefunction (
) and its energy (
),
can be solved analytically [
91,
92,
93].
in Equation (
2) is called the Hamiltonian operator and corresponds to the total energy of the system,
where
ℏ is the reduced Planck constant,
m the electron mass,
is the the Laplace operator [
94], and
is the energy potential operator.
Thus, according to QM, the electron energy in the QCF (
1) is quantized: it can only take discrete values [
91]. The number of “bound states” in this QCF depends on
(see [
91]): there is a range of values of
for which there is only one energy level. Now, if we assume, for simplicity, that the QD size,
, is so tiny that there is only one energy level,
, its associated wavefunction is a 1
s–orbital [
6,
93]. We have solved the problem for a single, isolated QD with
nm and
eV, typical in III-V semiconductors. There is only one bound state, whose energy is
eV. Its associated wave function
is a spherical 1
orbital. We have represented its squared modulus,
, in both Cartesian (
Figure 4b) and radial coordinates (
Figure 4c). The latter shows how
decreases very quickly as a function of the normalized radial coordinate,
.
With this idea in mind, the potential in the complete system is a function that varies from one QD to another, taking the
value inside each QD and zero in the space among QDs:
where
is the position vector locating 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
, as shown in
Figure 5a. 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
Section 1 when introducing our approach, we generate a link between two nodes (sites, kets),
and
, located at
and
, respectively, by computing to what extent their respective wave functions overlap. We compute the overlap between the wave functions
and
as the overlap integral [
94] over all the physical space
SNote that, because of symmetry, .
Using Expression (
5), we have computed the overlap integral for nodes whose centers are separated by a
normalized Euclidean distance
. The result appears in
Figure 6, where we have represented the overlap integral as a function of
. We have highlighted two situations in
Figure 6 for illustrative purposes.
The first one corresponds to the inset in which the centers of the nodes (sites)
i and
j are separated by a distance
for which
. Thus, we generate a link between nodes
i and
j whose weight is
. The second inset corresponds to two nodes whose centers are at a
for which
. A link is generated, but has a very small weight
. Note that the overlap integral
when
, and is
for
. That is, in our system, all those nodes whose centers are separated by a distance
(or
limit distance) are
not allowed to be linked. This is the case of nodes
and
in
Figure 5, where
.
Although it does not seem clear yet at this point,
defines a new
scale in the system,
. Note that our approach is a modified version of a RGG. As mentioned, an RGG is the simplest spatial network, consisting of randomly placing
N nodes in some metric space and connecting two nodes by a link if and only if their Euclidean distance is smaller than a given neighborhood radius,
r. In our case, this is distance is
. The novelty of our approach is that any link between nodes
i and
j is characterized by a weight that involves the overlap integral between kets
and
.
To advance in our model, it is necessary to introduce some essential 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
encodes the topology of a network, that is, whether or not there is a link (
or
) between any two pairs of nodes
i and
j. Sometimes, this binary information 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
[
41]. With our method, the
weighted adjacency matrix corresponding to the network represented in
Figure 5b is
which is symmetric (since
in this quantum systems), off-diagonal and non-negative. In particular,
since there is no overlap between the electron wave functions in kets
and
.
An interesting point in Expression (
7) is that its matrix elements have in QM the meaning of
probability amplitude,
, which is related to the
probability for an electron to be in
and
,
, as follows:
We can now generalize the idea from the toy system in
Figure 5 to the complete, complex, quantum system composed of
N QDs that are independently and uniformly distributed in the metric space
. The corresponding adjacency matrix is thus an
weighted adjacency matrix whose matrix elements,
, are the
probability amplitude for an electron in kets
and
,
Once we have defined our weighted adjacency matrix,
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
, where
is the set of nodes (
) and
is the set of links. We have specified the matrix
in the triplet
to emphasize the fact that the connections between the nodes are made using the
matrix and not, for example, a conventional adjacency matrix
(
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
quantifies connections that have physical meaning according to QM, and explicitly includes the spatial structure of the system (remember
Figure 6 and its associated, previous discussion). This is the key point that allows us to apply to
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
, whose elements
are the sum of weights of all links directly connecting node
i with others. In our particular system,
has physical meaning: using (
9),
is the sum of the probability amplitudes on ket
. We label it
to stress this physical meaning.
helps us obtain Laplacian matrices that will assist us in studying electron dynamics 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,
Note that the Laplacian matrix
computed using the weighted adjacency matrix
is different from the one used in other works [
72,
84,
95]. In these approaches, the matrix elements of
are assumed to be equal
. In our approach, the matrix elements take different values since they depend on the involved overlap integrals (or probability amplitudes,
) 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
[
68,
75,
77,
82,
84,
85,
96,
97,
98,
99,
100,
101].
The second one is the
normalized Laplacian matrix [
96],
, an Hermitian operator that, according to the way we have defined
in (
9), has matrix elements in the form
allows for generating the corresponding unitary CTQW [
96] of an electron on our graph
as
Note that, in the time evolution operator generated by
in (
12), the imaginary unit makes
be unitary [
75]. As in other CN approaches [
74,
76,
102,
103], we assume
so that time and energy can be treated as dimensionless. We will use
to study the temporal evolution of our quantum system.