UNIVERSIDADE FEDERAL DE SÃO CARLOS
CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA

OSMAR DO NASCIMENTO SOUZA

Layers of planar hexagonal heterostructures modeled by quantum graphs

São Carlos - SP
August of 2022

UNIVERSIDADE FEDERAL DE SÃO CARLOS
CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA

OSMAR DO NASCIMENTO SOUZA

Layers of planar hexagonal heterostructures modeled by quantum graphs

Thesis submitted to PPGM/UFSCar as partial fulfillment of the requirements for
the degree of Doctor of Science.

Supervisor: Prof. Dr. César Rogerio de Oliveira

São Carlos - SP
August of 2022

To my sister Maria de Fátima (in memorian).

## Acknowledgment

I eternally thank God for all His Love, for the gift of life, for His infinite
Kindness and Mercy and for not giving up on me, and to Mary, the Mother of
God, for her “Yes”, and for her mother’s loving intercession.

I would like to say a huge thank you to my parents, Antônio e Luiza, for their
generosity in being open to life allowing the author, the 7o child, to come in
to this word; to my siblings Maria Ivani, Osvaldo, Maria Hilda, Maria de
Fátima, Maria Nazaré and José, and to my dears nephews Pedro, Laura, Verônica,
Nayele, Maria Luiza and Nádia. Anyway, i am very grateful to each and every
one of my family for the encouragement and, especially, for the prayers.

I also thank the teachers of the Federal University of São Carlos with whom I
had the opportunity to study each subject and, in particular, teacher César
Rogério de Oliveira, for the guindance, disposition, mathematical teachings
and,above all, for never backing down from difficulties of work and always
guide with patience, good humor and confidence. I extend my thanks to my
fellow professors of the Mathematics course at the Federal University of Mato
Grosso do Sul, Campus do Pantanal, for the encouragement of studies; to my big
friend John Adriano Silva Azeredo for his advices and spiritual directions; to
Winícius Jacinto Alves, Altair Santos de Oliveira Tosti, Giovana Alves,
Tallyta Ananda, Marcelo Augusto, Maicom Douglas Varella Costa, Luan Barbosa,
Thalyne Rocha, Ricardo Menotti, Gustavo Figueira and Lívia Massula, Nilton
Mendes and to the mineiros César and Priscila for bearing patiently my faults
and for the continuous friendship. I add to my thanks the Parish Street
Ministry of Nossa Senhora de Fátima for the learning in each pastoral meeting
and Opus Dei in Ribeirão Preto for the insights and direction; to Vinícius
Lourenço da Rocha for his disposition and help in the studies and, finally, to
friend Thales Fernando Vilamaior Paiva for the fellowship in the teaching
work, academic studies and for the daily conversations about Faith and others
virtues in the room 109 in the Department of Mathematics at UFSCar.

“There are people who want to know just to know, and that is curiosity;
others, to achieve fame, and that is vanity; others, to get rich with their
science, and that is a nasty business; others to be built, and that is
prudence; others, to built others, and that is charity.” - Saint Augustine

## Resumo

Iniciamos este trabalho revisando uma aplicação da teoria de grafos quânticos
periódicos para modelar monocamada de materiais hexagonais com parâmetros
$\delta_{a}$ e $\delta_{b}$ associados aos tipos de átomos distintos situados
em seus vértices. Verificamos que materiais dessa natureza possuem lacunas em
suas bandas espectrais e expressamos o tamanho dessa abertura em função desses
parâmetros. Nos Capítulos 2 e 3 estendemos essa modelagem para bicamadas
iguais, empilhadas no tipo $AA$ e $AA^{\prime}$, e no Capítulo 4 estudamos
heteroestruturas com duas camadas mistas e os “sanduíche” grafeno-nitreto de
boro hexagonal: folha de grafeno entre duas folhas de hBN, e hBN entre
grafenos. Em cada uma dessas configurações, usamos o operador de Schrödinger
com suas respectivas condições de contorno e introduzimos um parâmetro de
interação fraca $t_{0}$ entre as conexões de diferentes camadas. Analisamos
analiticamente a relação de dispersão obtida nesses modelos quanto à
existência de toques cônicos ou parabólicos e confirmamos, em modelos
rigorosos, resultados conhecidos na literatura física, a saber: bicamadas de
hBN não possuem cones de Dirac, porém no empilhamento $AA$ identificamos a
presença de toques parabólicos. No caso de bicamadas mistas, nosso estudo
permite concluir que a inclusão de uma camada de hBN sobre uma de grafeno pode
induzir um gap na folha de grafeno e expressamos a largura desse gap em função
dos parâmetros $t_{0}$ e $\delta_{a}$. No estudo dos “sanduíches”, hBN-
grafeno-hBN e grafeno-hBN-grafeno, para certos valores particulares dos
parâmetros, verificamos que a inclusão de uma folha de grafeno entre duas de
hBN não elimina o gap do hBN, mas induz uma redução na largura da lacuna
espectral em uma ordem de grandeza; por outro lado, no caso grafeno-hBN-
grafeno, o cone do grafeno na origem prevalece neste sanduíche, porém também
provocou lacunas nos outros cones de Dirac do grafeno. Tais resultados podem
ser justificados pelo fato de que, nessas heteroestruturas, os átomos de
carbono terem interagido com outros átomos inequivalentes de hBN, nitrogênio e
boro, provocando redução ou aumento das lacunas. Por fim, no último capítulo,
consideramos grafos quânticos hexagonais e adaptamos nossa proposta para
incluir um campo magnético na folha de hBN. Demonstramos que se o fluxo
magnético for constante na rede hexagonal e for múltiplo racional de $2\pi$,
então existirão valores desse fluxo de forma que, para certas condições de
contorno nos vértices (modelando o hBN), os toques cônicos na relação de
dispersão do operador deixarão de existir e garantimos a existência de
lacunas.

Palavras-chave: Grafos quânticos; Rede hexagonal; Nitreto de boro; Grafeno;
Heteroestruturas; Operador de Schrodinger; Campo magnético; Cones de Dirac.

## Abstract

We started this work by reviewing an application of periodic quantum graph
theory to model monolayer hexagonal materials with $\delta_{a}$ and
$\delta_{b}$ parameters associated with the different types of atoms located
in their vertices. We verified that materials of this nature have gaps in
their spectral bands and express the size of this opening according to these
parameters. In the Chapters 2 and 3, extend this modeling to equal bilayers,
stacked in type $AA$ and $AA^{\prime}$, and in the Chapters 4, we studied
heterostructures with two mixed layers and the “sandwich” hexagonal boron
graphene-nitride: a single graphene sheet between two layers of hBN, and a
single hBN sheet between two layers of graphene. In each of these
configurations, we use the Schrödinger operator with its respective boundary
conditions and we introduced a weak $t_{0}$ interaction parameter between the
connections of different layers. We analyzed initially the dispersion
relationship obtained in these models regarding the existence of conical or
parabolic touches and confirmed, in rigorous models, known results in the
physical literature, namely: hBN bilayers do not have Dirac cones, but, in
$AA$ stacking we identified the presence of parabolic touches. In the case of
mixed bilayers, our study allows us to conclude that the inclusion of an hBN
layer over a graphene layer can induce a gap in the graphene sheet and we
express the width of this gap according to the parameters $t_{0}$ and
$\delta_{a}$. In the study of “sandwiches”, hBN-graphene-hBN and graphene-hBN-
graphene, for certain particular values of the parameters, we found that the
inclusion of a single graphene sheet between two sheets of hBN does not
eliminate the gap of the hBN, but induces a reduction in the width of the
spectral gap in an order of magnitude; by on the other hand, in the case
graphene-hBN-graphene, the graphene cone at the origin prevails in this
sandwich, but it also caused gaps in the other Dirac cones of the graphene.
Such results can be justified by the fact that, in these heterostructures,
carbon atoms have interacted with other inequivalent hBN, nitrogen, and boron
atoms, causing a reduction or increase of the gaps. Finally, in the last
chapter, we consider hexagonal quantum graphs and we adapt our proposal to
include a magnetic field in the hBN sheet. We demonstrate that if the magnetic
flux is constant in the hexagonal network and is a rational multiple of
$2\pi$, then there will be values of this flux such that, for certain boundary
conditions at the vertices (modeling the hBN), the conical touches in the
operator scattering relation will cease to exist and we guarantee the
existence of gaps.

Keywords: Quantum graphs; Hexagonal lattice; Boron nitride; Graphene;
Heterostructures; Schrodinger operator; Magnetic field; Dirac cones.

###### Lista de Figuras

  1. 1 Graphene-graphite representation.
  2. 2 Van der Waals heterostructures. Comparison of Lego pieces with $2D$ materials, which can be stacked to form heterostructures linked by van der Waals forces (adapted from [18]).
  3. 3 Hexagonal bilayers - Side and top views, respectively, of the $AA$ and $AA^{\prime}$ stacks.
  4. 4 Representation of a hexagonal boron nitride sheet.
  5. 5 hBN structure seen as a 3D material: in each layer, boron and nitrogen atoms are linked by strong bonds, and these layers are also linked by van der Waals forces (Adapted from [23]).
  6. 1.1 Hexagonal network $G$ of hBN (the same as graphene): the fundamental domain $W$ and the generating vectors of the network $E_{1}$ and $E_{2}$.
  7. 1.2 $\alpha_{N}=\alpha_{B}=0.2$; Dirac cone at $F=0$ is formed when the parameters $\alpha_{N}$ and $\alpha_{B}$ coincide.
  8. 1.3 $\alpha_{N}\neq\alpha_{B}$: $\alpha_{N}=0.5,\alpha_{B}=0.2$; when the parameters $\alpha_{N}$ and $\alpha_{B}$ are distinct, gaps appear where there were Dirac cones
  9. 2.1 Side and top views, respectively, of material composed of two identical hexagonal layers aligned in type $AA$.
  10. 2.2 Material composed of two hexagonal layers stacked in the type $AA$.
  11. 2.3 Dirac cone if the atoms are equal; here $\alpha_{a}=\alpha_{b}=-1\,,t_{0}=0.3\,.$
  12. 2.4 Parabolic touch; here $\alpha_{a}=-1\,,\alpha_{b}=2t_{0}^{2}+\alpha_{a}\,,t_{0}=0.3\,.$
  13. 2.5 Gap of 0.589 in the dispersion relation at the origin for $\alpha_{a}=-1,\,\alpha_{b}=1\,,t_{0}=0.3\,.$
  14. 2.6 The curves intersect, but there are no parabolic or conic touches: $\alpha_{a}\\!=\\!-1,\alpha_{b}\\!=\\!-0.9,t_{0}\\!=\\!0.3\,.$
  15. 2.7 Dirac cone, $\alpha_{a}=0.25$ and $\alpha_{b}=0.09$ .
  16. 2.8 Parabolic Touch, $\alpha_{a}=0.25$ and $\alpha_{b}=-0.09$ .
  17. 2.9 Gaps: left with $\alpha_{a}=\alpha_{b}=-1$; on the right with $\alpha_{a}=1$ and $\alpha_{b}=-1$.
  18. 3.1 Stacking $AA^{\prime}$ of two hexagonal layers with 2 distinct atoms at the vertices, as in hBN. Side and top views, respectively.
  19. 3.2 Hexagonal bilayers with 2 distinct atoms at the vertices. Type-$A$ vertices are represented by red, and type-$B$ by blue.
  20. 3.3 Two Dirac cones when the atoms are equal; $\,\alpha_{a}=\alpha_{b}=-1\,,t_{0}=0.3$.
  21. 3.4 Gaps if atoms are distinct; $\alpha_{b}=1,\,\alpha_{a}=-1\,,t_{0}=0.3$.
  22. 4.1 A two-layer van der Waals heterostructure: side and top views, respectively.
  23. 4.2 Fundamental domain $W$ of a heterostructure bilayer formed by 8 edges and 4 vertices.
  24. 4.3 Gaps in the hBN-graphene dispersion relation; $t_{0}=0.3\,,\alpha_{a}=-1$.
  25. 4.4 Greater interaction between layers ($t_{0}=1)$ causes an increase in the gap; $\alpha_{a}=-1$.
  26. 4.5 The hBN-Graphene-hBN sandwich: side and top views, respectively.
  27. 4.6 Gaps at origin for $t_{0}=0.3$, $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=1\,,\alpha_{{}_{C}}=0$. The inclusion of a graphene sheet between two hBN sheets does not eliminate the gap at $F=0$, but causes a reduction in the gap width from $\approx 0.65$ in two hBN layers to $\approx 0.074$.
  28. 4.7 Gaps in $F\\!=\\!0$ from $\approx 0.00483$ (the smallest) for $t_{0}\\!=\\!0.3$, $\alpha_{{}_{B}}\\!=\\!-\alpha_{{}_{N}}\\!=\\!0.1\,,\alpha_{{}_{C}}\\!=\\!0$.
  29. 4.8 Gaps in $F=0$ of $\approx\\!0.00324$, $\approx\\!0.00647$ (smaller auxiliary figure on the side) and $\approx\\!0.04497$ for $t_{0}=0.3$, $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.01\,,\alpha_{{}_{C}}=0$.
  30. 4.9 Side and top views, respectively, of the Graphene-hBN-Graphene sandwich.
  31. 4.10 Presence of five iconic touches in the interaction of equivalent atoms, that is, $\alpha_{{}_{B}}=\alpha_{{}_{N}}=0=\alpha_{{}_{C}}$ , $t_{0}=0.3$.
  32. 4.11 $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=1$ and $t_{0}=0.3$. Gap formation in four of the five cones due to the interaction of carbon atoms with other inequivalents of hBN, the smallest of which is $\approx 0.07684$. A graphene cone always prevails in this type of sandwich for any values of the parameters $\alpha_{{}_{B}},\alpha_{{}_{N}}$.
  33. 4.12 $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.1$ and $t_{0}=0.3$. Prevalence of the graphene cone: red and green curves.
  34. 4.13 $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.01$ and $t_{0}=0.3$. The graphene cone is always maintained, while gaps are caused at other points, the smallest of which (smaller auxiliary figure on the side) is $\approx 0.00314$.
  35. 5.1 The fundamental domain $W_{\hexagon}$ and the generating vectors $E_{1}$ and $E_{2}$ of the ⎔ network. For $q=1$, the period of the ⎔ network coincides with that of the $H$ operator.
  36. 5.2 For $q=1$, the cone is maintained at $(\theta_{1},\theta_{2})\in[0,\pi)\times[-\pi,\pi)$ for $\alpha_{{}_{N}}=\alpha_{{}_{B}}$. Here $\alpha_{{}_{B}}=\alpha_{{}_{N}}=-1.$
  37. 5.3 Period of the operator $\Delta_{\beta}$ for $q=2$.
  38. 5.4 For $q=2$, the cone is maintained at $(\theta_{1},\theta_{2})\in[0,\frac{\pi}{2})\times[-\frac{\pi}{2},\frac{\pi}{2})$ for $\alpha_{{}_{N}}=\alpha_{{}_{B}}$, as expected, since in this case, $\alpha(v)$ is a constant.
  39. 5.5 Graphical representation of the roots $\eta^{\pm}_{\mp}$ with $\alpha_{{}_{B}}=1,\alpha_{{}_{N}}=-1$. For $q=2$, in $(\theta_{1},\theta_{2})\in[0,\frac{\pi}{2})\times[-\frac{\pi}{2},\frac{\pi}{2})$ the cone is not maintained for $\alpha_{{}_{N}}=-\alpha_{{}_{B}}$, that is, for non-constant $\alpha(v)$, the operator dispersion relation has an (open) gap.

###### Contents

  1. Introduction
  2. 1 Preliminary
    1. 1.1 Monocamadas
    2. 1.2 Floquet Theory
  3. 2 Structures of type $AA$
    1. 2.1 Structure with two identical leaves
    2. 2.2 Analysis of the dispersion relation
    3. 2.3 Analysis and conclusions
    4. 2.4 Appendix: the parameter $t_{0}$
  4. 3 Type structures $AA^{\prime}$
    1. 3.1 Structure with two identical leaves
    2. 3.2 Dispersion relationship analysis
    3. 3.3 Analysis and conclusions
  5. 4 Heterostructures
    1. 4.1 Distinct bilayers
      1. 4.1.1 Analysis of the dispersion relation.
    2. 4.2 Distinct trilayers
      1. 4.2.1 The hBN-Graphene-hBN case
      2. 4.2.2 The Graphene-hBN-Graphene case
    3. 4.3 Analysis and conclusions
  6. 5 Hexagonal monolayers with magnetic field
    1. 5.1 Magnetic potential
    2. 5.2 Graphene and boron nitride with magnetic field
    3. 5.3 Analysis and conclusions
  7. 6 Final considerations

## Introduction

Among the many features and motivations for studying graphene, is the fact
that it is a highly resistant material (more inclusive than diamond),
excellent thermal and electrical conductor, as well as a strong candidate to
replace silicon and revolutionize the technology industry, besides that it is
basis for others materials, like graphite (pencil tip) which is made up of
several layers of graphene (see Figure 1; basics references, including
historical aspects of this important material are [7, 22]. However, maybe the
most important property of the graphene is the presence of Dirac cones located
at a finite number of points in the Brillouin zone, called Dirac points, which
are of particular importance for the description of the eletronics properties
of this material. They are located exactly where the valence band and the
conduction band of graphene touch, and this touch occurs in a conical shape.

It is worth mentioning that the presence of a Dirac cone in the dispersions
relations indicate that, in the corresponding energy region, the behavior of
an electron in the material would be effectively described by a one-
dimensional Dirac operator, which would lead to relativistics behaviors of
that electron, justifying the great conduction of graphene and others
materials on what shuch cones are present. See, for example, [7, 22] and, for
a mathematical treatment of this phenomenonin a continuous model [15], which
was discussed in details by Fefferman and Weunstein [14] and, being
technically much more complicated than the graph model, it becomes more
difficult for generalizations the heterostructure and several layers (our main
objectivein this Thesis). The first study to predict the Dirac cones in
graphene is not recent, it was by Wallace [36], using tight-binding
approximations and published in 1947; however, the “two-dimensional”
experimental management of this material only occurred in 2004, which
resulkted in a Nobel Prize in Physics in 2010 for two authors, after they
isolated graphene experimentally, fact that boosted the growth in interest in
studying this material.

In summary, the single layer of graphene is a semimetal and, therefore, it
does not have an intrinsic gap and this limits the number of possible
applications. Thus, such material can´t be used alone as an transistor,
because the electricity will flow constantly, in other words, the electricty
flow cannot be turned off like other materials like silicon. The abscence of
gap between the conduction band and the valence band for the graphene, with
the presence Dirac points, leads to difficulties in using them in electronics
systems and transistors, because it “never turns off”. Therewith, some ways to
induce a gap in the single layer of graphene have been explored.

Figure 1: Graphene-graphite representation.

On the other hand, it is known that single layers of hexagonal boron nitride –
which in the course of that work we will denote as hBN – presents a great gap
between the valence and the conduction bands, characterized as an insulator.
In [19], the authors proposed to consider graphene-hexagonal boron nitride
stacking (the interaction between layers is by van der Walls) as a way to
induce gap in graphene. We mentioned here other very promising possibility,
two “twisted” graphene layers (see [35] and the references there cited there)
which includes others aspects of interests, but it will not be covered here.

From certain two-dimensional materials, differents heterostructures can be
created, which essentially consist in using a $2D$ material (in fact a sheet
with an atom thick) as a base and add another two-dimensional material on top
of it. The resulting stack represents an artificial material assembled in a
chosen sequence, as shown in Figure 2. Covalent (strong) occur between atoms
in the same layer, providing stability in the plane of $2D$ materials, while
weak forces between atoms in overlapping layers (van der Waals bonds) hold the
stack together. These staked materials are called van der Waals
heterostructures and the possibility of producing was demonstrated
experimentally and recently [16]. Throughout this work, when nothing is said,
we will only use the term heterostructures to refer to van der Waals
heterostructure.

Figure 2: Van der Waals heterostructures. Comparison of Lego pieces with $2D$
materials, which can be stacked to form heterostructures linked by van der
Waals forces (adapted from [18]).

Theoretical and numerical works indicate that the eletronic properties of
graphene can be modulated when considered these heterostructures formed by
graphene-hBN for certain stacking configurations (which can lead to graphene-
hBN devices with diferent mobilities [28]), so that it is possible predict the
opening of gaps in the eletronic estructure of graphene since in the new
structure graphene-hNB there are two types of atoms generating different
potencials, making the carbon atoms inequivalent, that is, such atoms begin to
interact with others atoms different from carbon and of different natures,
such as nitrogen and boron.

There are several ways to stack $n$-layers of $2D$ hexagonal-shaped materials,
like graphene or boron nitride, namely, $AB$ (also known as Bernal) and
$AB^{\prime}$ type stacking, however here we highlight the $AA$ and
$AA^{\prime}$ configurations, as shown in Figure 3, where two atoms from thge
sub-network $A$ and $B$ of the upper (layer labeled $A_{2}$ and $B_{2}$,
respectively) are points located on the vertices of the hexagon, which we will
say are vertices of type-$A$ and vertices of type-$B$, respectively.

Figure 3: Hexagonal bilayers - Side and top views, respectively, of the $AA$
and $AA^{\prime}$ stacks.

  * •

Stacking of type $AA$: all vertices are aligned so that a vertex of type-$A$
is located on a vertix of type-$A$; and on a vertex of type-$B$ there is one
of type-$B$.

  * •

Stacking of type $AA^{\prime}$: all vertices are ligned so that on a vertex of
type-$A$ there is one of type-$B$, and on a vertex of type-$B$ there is one of
type-$A$. 111note that for graphene bilayers, the stacking types $AA$ and
$AA^{\prime}$ coincide.

In [32] the spectral characterization and excistence of Dirac cones were
studied, using periodic quantum graph models, or even, through the Schrödinger
with Neumann vertex condition, a graphen sheet was modeled, as weel as two and
three layers of graphene stacked in the type $AB$ and also in $n$ graphene
sheets in the $AA$ stack. The study conclude that $n$-graphene layers have
Dirac cones if aligned in the $AA$ type and three sheets in the $AB$, but they
have only parabolic touches in bilayers $AB$.

In relation to others mathematical results in literature, quantum graph models
for graphene was also explored in [33, 17]. In the case of the $AA$ stack of
graphene, in any numbers of layers, the presence of Dirac cones was
mathematically demonstrated in [33, 17], in studies that focused in the
(ir)reducibility of the dispersion relations; adapting the elegants arguments
of symmetry of [4] another proof can be obtained, originally to the one-layer
continuous model of Fefferman and Weinstein mentioned above. We complete
reinforcing that even in the case $AA$ for graphene (with Neumann boundary
conditions), and including graphite, the cones were demonstrated in [32, 9],
but assuming that the interaction parameter between the layers, $t_{0}$, was
small enough. In the case of the $AB$ stack, the cones were mathematically
discussed [8] only in cases of three layers (with the existence cones,
regardless of the value of the interaction parameter) and two layers (absence
of cones and presence of parabolic touches).

With this in mind, our study proposal here is to consider quantum graph models
[5] (which simplify technicalities and result in several rigorous analytical
results) of heterostructures with bi- and tri-layers identical or mixed, such
as layers of boron nitride together with graphene. In particular, our work,
through a mathematical approach, in addition to contenplanting (within the
necessary simplifications) systems found in laboratories and in nature, can go
a step further and analyze the possible presence of Dirac cones in
heterostructures with two and three mixed layers not yet discussed in the
literature (as far as we know): graphene-hBN sandwiches formed by graphene
between two sheets of hexagonal boron nitride, and hBN between two sheets of
graphene.

Motived by structures of this nature we propose to analytically analyze the
existence or not of touches in the dispersions relations of operators
associated with various configurations. These touches can be characterized as
Dirac cones or parabolic touches. Intuitively, an Dirac cone222In chapter 1,
we present the precise definition of Dirac cone, see(1.2.1) is a point at
which two spectral bands touch each other linearly – at least in a first order
approximation – while a parabolic touch is a point at which two spectral bands
gently touch each other, in a parabolic way.

In [26], a layer of graphene was modeled using a periodic quantum Schrödinger
graph with Neumann vertex condition, that is, in terms of Robin condition
(1.2) with parameter $\delta_{v}=0$ for all vertices, while in [32, 8, 9] such
modeling is done considering $n$ sheets of graphene. Just like in [32], here
we will also use a more general boundery condition, that is, a constant that
is not necessarily zero $\delta_{v}$ at the vertices, to model the sime type
of atom (different types of atoms will be modeled by differents values of
$\delta_{v}$).

Figure 4 illustrates the two-dimensional hexagonal boron nitride with
hexagonal structure composed of nitrogen and boron atoms at its vertices.
Let’s say that the nitrogen atoms are located on the vertices of type-$A$,
while the boron atoms are at the vertices of type-$B$. Thus, we use different
values of the $\delta_{v}$ parameter in the Robin (1.2) condition as way of
distinguishing these atoms in the model: if $\delta_{v}$ is a vertice of
type-$A$ (or type-$B$) we choose
$\delta_{v}=\delta_{a}\quad\big{(}\mbox{or}\,\,\delta_{v}=\delta_{b}\big{)}$,
with $\delta_{a}\neq\delta_{b}$.

Figure 4: Representation of a hexagonal boron nitride sheet.

On the other hand, the hBN seen as a solid has a layered structure similar to
graphite333hexagonal boron nitride, due to its structure in layers, simillar
to graphite, is also called ”white graphite”, however it has some properties
entirely different from graphite; fact that arouses scientific interest in
studying it, including our own interest. and is described by sequencial layers
stacked in the form $AA^{\prime}$, with the layers located one above the
others, Figure 5.

Figure 5: hBN structure seen as a 3D material: in each layer, boron and
nitrogen atoms are linked by strong bonds, and these layers are also linked by
van der Waals forces (Adapted from [23]).

However, for the purposes of our work proposal, we will focus on the analysis
of configurations relevant to the study motivation: models of the form $AA$
and $AA^{\prime}$ for two sheets of hexagonal-shaped materials, such as hBN
and graphene, two and three distinct layers. With this in mind, in Chapter1,
we present in detail basic and necessary concepts that can be applied to model
an hBN sheet (and consequently graphene); in the Chapters 2 and 3, we cover
configurations of the type $AA$ and $AA^{\prime}$ for two identical $2D$
layers with two different types of atoms in the hexagonal net and, then, in
the Chapter4 we study these cases for two distinct sheets, the
heterostructures; in particular, in its last two sections, we attack the
central problem of this work: van der Walls heterostructures with three
distinct layers that, in particular, can be applied to the graphene-hBN
sandwich, which contributes to the current discussion regarding the production
of materials composed of graphene and hexagonal boron nitride, which tt is
still a point of debate and unresolved, with conflicting results in the
literature.

Before proceeding, we emphasize that this is a mathematical work with an
applied bias. With this in mind, we sometimes mix the nomenclature444For
example, when the term hexagonal boron nitride is mentioned in the text, we
are not necessarily referring to hBN, which is a real material, but rather a
simple modeling proposal to portray it from different areas, which we consider
natural.

##  Chapter
1
Preliminary

We begin this chapter with a quick abstract of the hexagonal structures which
discribe a single sheet of two-dimensional materials like graphene and boron
nitride, as already discussed in [26] (which was based on [1]) and [32],
respectively, is which concepts and preliminary results for the development of
this work are also presented.

###  1.1 Boron nitride and graphene monolayers

Quantum graphs are metric graphs with a self-adjoint operator on each edge;
they are one-dimensional singular varieties together with differential self-
adjoint operator defined on them. We then present, in this section, the
contruction of quantum graphos that can be proposed to represent a single
hexagonal layer of graphene and hBN materials. In particular, similar to the
work developed in [32, 10], we will focus, for the moment, on configurations
composed of a hexagonal boron nitride sheet. We highlight that in [32, 10] the
Dirac operator was used in the vertices, while here we will use Schrödinger
one.

Let $E_{1}=\left(\frac{3}{2},\frac{\sqrt{3}}{2}\right)$ and
$E_{2}=\left(0,\sqrt{3}\right)$ be vectors in $\mathbb{R}^{2}$ and the points
$A=(0,0)$, $B=(1,0)$. Consider the triangiular network
$r=\mathbb{Z}E_{1}\oplus\mathbb{Z}E_{2}$ and the subnets $r_{{}_{A}}=r+A$ and
$r_{{}_{B}}=r+B$. So the hexagonal network $G$ that discribe one boron nitride
sheet is

$G=r_{{}_{A}}\cup
r_{{}_{B}}=\Big{\\{}x\in\mathbb{R}^{2}:x=p_{1}E_{1}+p_{2}E_{2}+z,z\in
W\Big{\\}},$

being $p=(p_{1},p_{2})\,\in\mathbb{Z}^{2}$ and $W=\\{f,g,h,v_{1},v_{2}\\}$ the
fundamental network domain of $G$, which is the set where the two verticies
$v_{1},\,v_{2}$ and the three edges $f,g$ e $h$ are conveniently directed,
acordding to the Figuire 1.1. In general, a fundamental domain is the smaller
domain (not unique) where the entire network is recovered from translations
through the base vectores $E_{1}$ e $E_{2}$ (generators).

Figure 1.1: Hexagonal network $G$ of hBN (the same as graphene): the
fundamental domain $W$ and the generating vectors of the network $E_{1}$ and
$E_{2}$.

We emphasize that in all our modeling, we are always assuming that all adges,
both horizontal and vertical, has length $1$. In particular, for graphene it
is considered that the carbon atoms are located at the verticies of $G$ and
the covalents bonds are represented by adges of length $1$; while for boron
nitride, the verticies are interspersed with nitrogen atoms ($N$) anjd boron
atoms ($B$) – more precisely, a type of atom in the subnet $r_{B}$ –, and with
edges also assumed to be of length 1.

We will denote, respectively, by $\mathcal{E}(G)$ and $\mathcal{V}(G)$ the set
of edges and the set of verticies de $G$. We identify each edge with the
interval $[0,1]$, so that we induce from $[0,1]$ a metric on $G$ and then we
can consider $G$ embedded in $\mathbb{R}^{2}$. So,

  * •

we have a natural measure in $G$, denoted by $dx$;

  * •

we can defien functions in $G$, denoted by
$u=\\{u_{e}\\}_{e\,\in\,\mathcal{E}(G)}$;

  * •

we can differentiate and integrate functions in $G$.

The Hilbert space is also defined:

$L^{2}(G,{\mathbb{C}})=\bigoplus_{e\,\in\,\mathcal{E}(G)}L^{2}(e,{\mathbb{C}}),\quad\mbox{with}\quad
L^{2}(e,{\mathbb{C}})\approx L^{2}\big{(}[0,1],{\mathbb{C}}\big{)}.$

Note that

$u\in
L^{2}(G,{\mathbb{C}})\Leftrightarrow\|u\|^{2}_{{}_{L^{2}(G,{\mathbb{C}})}}{:=}\displaystyle\sum_{e\,\in\,E(G)}\\!\\!\|u_{e}\|^{2}_{{L^{2}(e,{\mathbb{C}})}}<\infty\,.$

therefore, in this work a quantum graph is a metric graph $G$ equipped with an
diferential operator $H$ acting on $L^{2}(G,{\mathbb{C}})$, and with
conditions of contour at the vertices so that $H$ is self-adjoint. In each
$e\in\mathcal{E}(G)$, the Schrödinger differential operator is considered,

$Hu_{e}=-\Delta u_{e}+q_{0}u_{e}\,,$ (1.1)

with $q_{0}:[0,1]\rightarrow\mathbb{R}$ pair function (in relation to the
center $x=\frac{1}{2}$) and continuous, that is,
$q_{0}(x)=q_{0}(1-x)\,\,\mbox{in}\,\,[0,1]$;

  1. 1.

${\mathrm{dom}}(H)\subset\mathcal{H}^{2}(G,{\mathbb{C}})$ ( Sobolev space in
$G$);

  2. 2.

$\displaystyle\sum_{e\,\in\,\mathcal{E}(G)}\|u_{e}\|^{2}_{{}_{\mathcal{H}^{2}(e,{\mathbb{C}})}}<\infty$;

  3. 3.

Robin condition at the vertex

$\\\ \left\\{\begin{array}[]{llll}(i)\quad\quad\quad
u_{e_{1}}(v)\,\,=\,\,u_{e_{2}}(v),\hskip
8.5359pt\forall\,\,e_{1},e_{2}\in\mathcal{E}_{v}(G)\,\,(\mbox{Continuity at
the vertex})\\\
(ii)\displaystyle\sum_{e\,\in\,\mathcal{E}_{v}(G)}\\!\\!u^{\prime}_{e}(v)=\delta_{v}u(v)\,,\hskip
28.45274pt(\mbox{Total flow})\end{array}\right.$ (1.2)

with $\delta_{v}$ a real number, $\mathcal{E}_{v}(G)$ the set of edges such
that $v$ is commom vertex, and $u^{\prime}_{e}(v)$ a derivative of $u_{e}$
along $e$ at the vertex $v$, which is is positive if $v$ is the initial vertex
of $e$, and negative otherwise. When the context is clear, we will omit the
subscript “$e$” to simplify the note.

###  1.2 Floquet Theory

For each quasimoment $\theta=(\theta_{1},\theta_{2})$ in the Brillouin zone
$B=[-\pi,\pi]^{2}$, there are Floquet operators, $H(\theta)=-\Delta+q_{0}$,
acting in the fundamental domain $W$, satisfying (1.2) and the cyclic
condition Floquet for periodic operators:

$u(x+p_{1}E_{1}+p_{2}E_{2})=e^{ip\cdot\theta}u(x)=e^{i(p_{1}\theta_{1}+p_{2}\theta_{2})}u(x),$
(1.3)

for all $p=(p_{1},p_{2})\in\mathbb{Z}^{2}$ and $x\in G$. It is known that
$H^{\theta}$ has a purely discrete spectrum [25],that is,
$\sigma(H(\theta))=\big{\\{}\lambda_{k}(\theta)\big{\\}}_{k\geq 1},$ com
$\lambda_{1}(\theta)\leq\lambda_{2}(\theta)\leq\cdots\leq\lambda_{k}(\theta)\leq\cdots\rightarrow\infty$,
when $k\to\infty$.

We call the union of the images of the functions
$\theta\mapsto\\{\lambda_{k}(\theta)\\}_{k\geq 1}$. From Floquet Theory [12,
24, 31], we know that the spectrum of the Schrödinger operator $H$ defined in
1.1, $\sigma(H)$, ís given by

$\displaystyle\sigma(H)=\bigcup_{\theta\,\in\,B}\sigma(H(\theta))\,.$ (1.4)

Therefore, by (1.4), to obtain $\sigma(H)$ just determine $\sigma(H(\theta))$,
that is, the spectrum problem of $H$ in the periodic graph is tranferred to
the spectrum of $H(\theta)$ in a fundamental domain, so we have to solve the
eigenvalue problem

$\displaystyle H(\theta)u=\lambda u,$ (1.5)

with $\lambda\in\mathbb{R},u\in{\mathrm{dom}}(H(\theta)),\theta\in B$, being
$u\in L^{2}(W)$ a function non-trivial that fulfills the (1.2) boundary
condition.

As each edge $e\in\mathcal{E}(G)$ is identified with the interval $[0,1]$, we
have the relations $v_{1}\sim\\!0$ and $v_{i}\sim\\!1$, being $v_{1},v_{i}$
vertices, for $i=2,3,4$, as shown in Figure 1.1. Therefore, we write the
continuity conditions at the vertex $(i)$ in (1.2) and the cyclic condition
(1.3) in the domain fundamental $W$, that is,

$\\\
\left\\{\begin{array}[]{llll}u_{a_{1}}(1)&=&u(v_{2})=u(v_{3}+E_{1})=e^{i\theta_{1}}u(v_{3})=e^{i\theta_{1}}u_{a_{2}}(1)\\\
u_{a_{1}}(1)&=&u(v_{2})=u(v_{4}+E_{2})=e^{i\theta_{2}}u(v_{4})=e^{i\theta_{2}}u_{a_{3}}(1)\end{array}.\right.$
(1.6)

To exemplify what has been exposed so far, we will briefly review the modeling
(proposed in our research group) of a boron nitride sheet, which consequently
also applies to a graphene sheet for particular parameter choices, so More
details can be found at [32, 26]. Using (1.6), we can rewrite the (1.2) and
cyclic Floquet conditions (1.3) in $W$ as

$\\\
\left\\{\begin{array}[]{llll}u_{a_{1}}(0)=u_{a_{1}}(0)=u_{a_{1}}(0)=:A_{1}\\\
u^{\prime}_{a_{1}}(0)+u^{\prime}_{a_{2}}(0)+u^{\prime}_{a_{3}}(0)=\delta_{a}A_{1}\\\
u_{a_{1}}(1)=e^{i\theta_{1}}u_{a_{2}}(1)=e^{i\theta_{2}}u_{a_{3}}(1)=:B_{1}\\\
-u^{\prime}_{a_{1}}(1)-e^{i\theta_{1}}u^{\prime}_{a_{2}}(1)-e^{i\theta_{2}}u^{\prime}_{a_{3}}(1)=\delta_{b}B_{1}\end{array}.\right.$
(1.7)

Consider the auxiliary operator Schrödinger–Dirichlet $H^{D}$ in $[0,1]$,

$H^{D}\\!\\!u=\frac{d^{2}u(x)}{dx^{2}}+q_{0}(x)u(x)\,,$ (1.8)

with Dirichlet boundary conditions $u(0)=u(1)=0$. Denoting by $\sigma(H^{D})$
its spectrum, it is shown that, for each, $\lambda\notin\sigma(H^{D})$, there
are functions 111We will omit the letter $\lambda$ in the notations
$\varphi_{\lambda,0},\varphi_{\lambda,1}$ when there is no need to emphasize
its dependence on $\lambda$. linearly independent,
$\varphi_{\lambda,0}:=\varphi_{{}_{0}},\varphi_{\lambda,1}:=\varphi_{{}_{1}}$
so that they solve the eigenvalue problem

$H(\theta)\varphi=\lambda\varphi,\,\qquad\lambda\,\in\mathbb{R},\varphi\neq
0,$ (1.9)
$\varphi_{{}_{0}}(0)=1=\varphi_{{}_{1}}(1)\,,\varphi_{{}_{1}}(0)=0=\varphi_{{}_{0}}(1)\,\,\mbox{e}\,\,\varphi^{\prime}_{{}_{1}}(x)=-\varphi^{\prime}_{{}_{0}}(1-x)\,,\,\forall\,\,x\in[0,1].$
(1.10)

Furthermore, on each edge $a_{i},\,i=1,2,3$ we obtain the representation

$\left\\{\begin{array}[]{lll}u_{a_{1}}(x)&=&A_{1}\varphi_{{}_{0}}(x)+B_{1}\varphi_{{}_{1}}(x)\\\
u_{a_{2}}(x)&=&A_{1}\varphi_{{}_{0}}(x)+e^{-i\theta_{1}}B_{1}\varphi_{{}_{1}}(x)\\\
u_{a_{3}}(x)&=&A_{1}\varphi_{{}_{0}}(x)+e^{-i\theta_{2}}B_{1}\varphi_{{}_{1}}(x)\end{array},\right.$
(1.11)

so that this linear combination fulfills the continuity conditions in (1.7)
and the eigenvalue problem (1.9). Combining (1.7) and (1.11), we obtain

$\left\\{\begin{array}[]{ll}(-3\eta-\alpha_{a})A_{1}+\bar{F}B_{1}=0\\\
FA_{1}+(-3\eta-\alpha_{b})B_{1}=0\end{array},\right.$ (1.12)

with $F=F(\theta):=1+e^{i\theta_{1}}+e^{i\theta_{2}}$, $\bar{F}$ its conjugate
complex,
$\eta:=\dfrac{\varphi^{\prime}_{{}_{1}}(1)}{\varphi^{\prime}_{{}_{1}}(0)}$ and
$\alpha_{i}:=\dfrac{\delta_{i}}{\varphi^{\prime}_{{}_{1}}(0)}\,,i=a,\,b$ (note
that $\varphi^{\prime}_{{}_{1}}(0)\neq 0$). Or, we can write (1.12) as
$M(\eta(\lambda),\theta)\\!\cdot X=\textbf{0}$, with

$M(\eta(\lambda),\theta)=\left(\begin{array}[]{cccc}-3\eta-\alpha_{a}&\bar{F}\\\
F&-3\eta-\alpha_{b}\end{array}\right),\,\quad
X=(A_{1},B_{1})^{T}\,,\,\textbf{0}=(0,0)^{T}\,.$

Furthermore,

$p(\eta):=\det(M(\eta(\lambda),\theta)))=9\eta^{2}+3(\alpha_{b}+\alpha_{a})\eta+\alpha_{b}\alpha_{a}-|F|^{2}$

ís a polynomial in $\eta(\lambda)$ whose roots are

$r^{\pm}(\theta)=\frac{-(\alpha_{b}+\alpha_{a})\pm\sqrt{(\alpha_{b}-\alpha_{a})^{2}+4|F|^{2}}}{6}\,.$
(1.13)

In other words, we see that there is $\theta\in B$ so that 222For simplicity,
when there is no confusion, we will just use the notation $\eta(\theta)$ or
$\eta(F)$ (since $F=F(\theta)$) instead of $\eta(\lambda)$. In fact, to
$\lambda\notin\sigma(H^{D})$, $\lambda\in\sigma(H)\Leftrightarrow$ exist
$\theta\in B$ such that $\eta(\lambda)$ is a function of $\theta$, that is,
$\eta(\lambda)=r(\theta)$, with $r(\theta)$ being a root of $P(\eta)=0$. This
fact gives the dispersion relation of $H$. $\det M(\eta(\lambda),\theta)=0$,
being $F=F(\theta)$ and, therefore, the representation (1.11) solves the
problem of eigenvalues (1.9) and, by (1.4), it follows that
$\lambda\in\sigma(H)$, according to the following proposition, whose
demonstration, can be found in [32], [25].

######  Proposition 1.2.1.

Let $\lambda\notin\,\sigma(H^{D})$. Then $\lambda\in\,\sigma(H)$ if, and only
if, there exists $\theta\,\in\,B$ such that

$\det(M(\eta(\lambda),\theta))=0\,.$

######  Observation 1.2.1.

For $\lambda\in\sigma(H^{D})$, it is possible to construct [26] (infinite)
eigenfunctions of $H$, each one supported on a single hexagon, satisfying the
continuity boundary conditions (together with Dirichlet); consequently,
$\lambda$ is an eigenvalue of $H$.

We want to interpret these roots in terms of the original potential $q_{0}(x)$
em $[0,1]$. To this end, we consider two auxiliary operators: the Schrödinger-
Dirichlet operator $H^{D}$ given by (1.8) and the Hill operator,
$H^{\mathrm{per}}$ in $\mathbb{R}$, which periodically extends $q_{0}(x)$,
given as in (1.1), for the entire line $\mathbb{R}$, with the resulting
periodic potential $q_{p}$:

$H^{\mathrm{per}}u(x)=\frac{d^{2}u(x)}{dx^{2}}+q_{p}(x)u(x)\,,\quad u\in
L^{2}(\mathbb{R}).$ (1.14)

We are interested in the spectral problem

$H^{\mathrm{per}}u=\lambda u\,.$ (1.15)

Let us then consider the monodromy matrix $M(\lambda)$ of $H^{\mathrm{per}}$
given by (see [6])

$\begin{pmatrix}\varphi(1)\\\
\varphi^{\prime}(1)\end{pmatrix}=M(\lambda)\begin{pmatrix}\varphi(0)\\\
\varphi^{\prime}(0)\end{pmatrix}\,,$ (1.16)

with $\varphi$ being a solution to the problem (1.15). The matrix $M(\lambda)$
shifts the solution $\big{(}\varphi(0)\,\,\varphi^{\prime}(0)\big{)}^{T}$ by
the period of $q_{{}_{p}}$, equal to 1 in this case. It is

$d(\lambda)={\mathrm{tr}}(M(\lambda))$

the discriminant of the Hill operator $H^{\mathrm{per}}$. Of the main results
and properties on the spectrum of $H^{\mathrm{per}}$ and $d(\lambda)$, we
highlight the relationship between the spectra of $H^{\mathrm{per}}$ and its
discriminant $d(\lambda)$:

$\sigma(H^{\mathrm{per}})=\\{\lambda\in\mathbb{R}:|d(\lambda)|\leq 2\\}\,.$
(1.17)

Furthermore, it is also known that $\sigma(H^{\mathrm{per}})$ is purely
absolutely continuous and that $\sigma(H^{\mathrm{per}})$ is the union of
intervals closed $B_{k}$,called $\sigma(H^{\mathrm{per}})$ bands, so that, for
$\lambda\in B_{k},d^{\prime}(\lambda)\neq 0$ and $d:B_{k}\rightarrow[-2,2]$ is
a homeomorphism, for each $k$ (see [26], Proposition 3.4, and also [6], [12],
[24], [27], [31]).

In particular, the roots $r^{\pm}$ in (1.13) are related to the discriminant
$d(\lambda)$ doof the periodic Hill operator $H^{\mathrm{per}}$, as shown in
the following results.

######  Lemma 1.2.1.

(Lemma 2.2.2 in [32]) Let $\lambda\notin\sigma(H^{{}^{D}})$ and $d(\lambda)$
be the discriminant of $H^{\mathrm{per}}$. So

$\eta(\lambda)=\frac{d(\lambda)}{2}\,,$ (1.18)

where $\eta(\lambda)$ is a root in (1.13).

######  Lemma 1.2.2.

(Lemma 2.2.3 in [32]) Let $\alpha_{b}=\alpha_{a}$. We have

  * (a)

If $\alpha_{a}\in[0,3]$, then $|r^{+}(\theta)|\leq
1\,,\,\,\forall\,\,\theta\,\in B_{d}$,

  * (b)

If $\alpha_{a}\in[-3,0]$, then $|r^{-}(\theta)|\leq
1\,,\,\,\forall\,\,\theta\,\in B_{d}$,

  * (c)

If $\alpha_{a}\in[-3,3]$, then $r^{+}(\theta_{D})=r^{-}(\theta_{D})\,$, with
$\theta_{D}=(\frac{2\pi}{3},-\frac{2\pi}{3})$, occur in $(-1,1)$. In
particular, the only case where $|r^{+}(\theta)|\leq 1$ and
$|r^{-}(\theta)|\leq 1$ occurs when $\alpha_{b}=\alpha_{a}=0$, in other words,
the standard Neumann vertex condition and the original modeling [26] to
represent the graphene material.

######  Lemma 1.2.3.

(Lemma 2.2.4 in [32]) Let $\alpha_{b}\neq\alpha_{a}$. Em (1.13), we have

  * (a)

If $\alpha_{b}\in[-3,\infty]$ and
$\frac{-3\alpha_{b}}{3+\alpha_{b}}\leq\alpha_{a}\leq 3$, then
$|r^{+}(\theta)|\leq 1\,,\,\,\forall\,\,\theta\,\in B_{d}$.

  * (b)

If $\alpha_{b}\in[-\infty,3]$ and
$-3\leq\alpha_{a}\leq\frac{-3\alpha_{b}}{3-\alpha_{b}}$, then
$|r^{-}(\theta)|\leq 1\,,\,\,\forall\,\,\theta\,\in B_{d}$.

######  Observation 1.2.2.

The Lemma 1.2.1 tells us that given $\lambda\notin\sigma(H^{D})$,
$\lambda\in\sigma(H)$ if, and only if, $r(\theta)=\frac{1}{2}d(\lambda)$ is a
solution of $\det(M(\eta(\lambda)))=0$. However, this information alone is not
enough to describe the $\sigma(H)$ spectrum. We saw that (1.17) presents a
complete characterization of $H^{\mathrm{per}}$ in terms of $d(\lambda)$, this
is, $\sigma(H^{\mathrm{per}})=\\{\lambda\in\mathbb{R}:|d(\lambda)|\leq 2\\}$
and, therefore, it is necessary to demand that $|r(\theta)|\leq 1$ in order to
relate the spectra $\sigma(H)$ and $\sigma(H^{\mathrm{per}})$. That said, we
see that the conditions in Lemmas 1.2.2 and 1.2.3 imply $|r(\theta)|\leq 1$
and, finally, the Spectral characterization of $H$ can be guaranteed.

These lemmas, together with the next theorems, further tell us that, under
certain conditions of $\alpha_{b}$ and $\alpha_{a}$, the dispersion relation
of $H$ has cone by Dirac; this is, a point $(\lambda,\theta_{D})$ at which two
spectral bands touch linearly, at least in a lower-order approximation:

$\lambda(\theta)-\lambda(\theta_{D})\approx\pm\gamma\\!\cdot\\!|\theta-\theta_{D}|\,,$

for some $\gamma>0$, as we present precisely below.

######  Definition 1.2.1.

( Dirac Cones )
A point $(\lambda,\theta_{D})$ in the dispersion relation, for
$\lambda\in\mathbb{R}$ and $\theta_{D}\in B$, is said  Dirac cone  if it
exists a constant $\gamma\neq 0$ such that

$\lambda(\theta)-\lambda(\theta_{D})+\mathcal{O}((\lambda(\theta)-\lambda(\theta_{D}))^{2})=\pm\\!\gamma\\!\cdot|\theta-\theta_{D}|+\mathcal{O}(|\theta-\theta_{D}|^{2}).$
(1.19)

In this case, $\theta_{D}$ is said to be a D-point and, in (1.19), the “–” and
“+” are signals for the bands of valence and conduction, respectively.

The description of the spectral structure of the $H$ operator in $G$ is given
by the following theorem.

######  Theorem 1.2.1.

(Theorem 3.6 in [26])

  * (a)

The continuous singular spectrum of $H$, $\sigma_{\mathrm{sc}}(H)$, is empty.

  * (b)

Except, possibly for a countable quantity of $\lambda$, the dispersion
relation of $H$ is given by

$d(\lambda)=\frac{-(\alpha_{b}+\alpha_{a})\pm\sqrt{(\alpha_{b}-\alpha_{a})^{2}+4|F|^{2}}}{3}\,,$
(1.20)

with $\alpha_{b}$ and $\alpha_{a}$ taken so that $|r^{\pm}(\theta)|\leq 1$,
for all $\theta\in B$ and $r^{\pm}(\theta)$ given in (1.13).

  * (c)

The absolutely continuous spectrum of $H$, $\sigma_{\mathrm{ac}}(H)$,
coincides, as a set, with $\sigma(H^{\mathrm{per}})$, that is, it has a "band-
gap" structure and

$\sigma_{\mathrm{ac}}(H)=\\{\lambda\in\mathbb{R}:|d(\lambda)|\leq 2\\}\,,$
(1.21)

being $d(\lambda)$ is the discriminant of $H^{per}$.

  * (d)

The pure point spectrum of $H$, $\sigma_{\mathrm{pp}}(H)$, coincides with
$\sigma({H^{D}})$.

######  Observation 1.2.3.

The countable set considered in item (b) of Theorem 1.2.1 is, in fact, formed
by the eigenvalues of the operator $H$ with Dirichlet boundary conditions.

######  Theorem 1.2.2.

(Theorem 2.3.1 em [32])

  * (a)

If $\alpha_{a}=\alpha_{b}$, the dispersion relation of $H$ has Dirac cones for
$\theta_{0}=\pm(\frac{2\pi}{3},-\frac{2\pi}{3})$ (Figure 1.2).

  * (b)

If $\alpha_{a}\neq\alpha_{b}$, the dispersion relation of $H$ does not have
Dirac cones (Figure 1.3).

The idea of proving Theorem 1.2.2 is interesting and will serve as a basis for
demonstrations throughout this work, in the cases of layers with two and three
sheets of boron nitride. Thus, we will present the idea of your demonstration,
omitting some technical accounts.

###### Proof.

Essentially, the argument consists of taking $\theta\in
B_{d}=\\{(\theta_{1},\theta_{2})\in B:\theta_{2}=-\theta_{1}\\}$ e
$\lambda\notin\sigma(H^{D})$ (then $d^{\prime}(\lambda)\neq 0$) and expanding
the functions $r^{\pm}$ around the point
$\theta_{D}=\pm(\frac{2\pi}{3},-\frac{2\pi}{3})$ and $d$ around
$\lambda(\theta_{0})$, with $\theta_{0}:=\pm\frac{2\pi}{3}$, thus obtaining

$r^{\pm}(\theta)-r^{\pm}(\theta_{D})=\pm\frac{\sqrt{3}}{3}\big{|}\theta_{1}-\theta_{0}\big{|}+\mathcal{O}\big{(}|\theta_{1}-\theta_{0}|^{2}\big{)},$
(1.22)

and

$d(\lambda(\theta))-d(\lambda(\theta_{D}))=d^{\prime}(\lambda(\theta_{D}))\big{(}\lambda(\theta)-\lambda(\theta_{D})\big{)}+\mathcal{O}\big{(}(\lambda(\theta)-\lambda(\theta_{D}))^{2}\big{)}\,.$
(1.23)

Combining these two expansions with the definition of a Dirac cone, we obtain

$\lambda(\theta)-\lambda(\theta_{D})+\mathcal{O}\big{(}(\lambda(\theta)-\lambda(\theta_{D}))^{2}\big{)}=\pm\frac{2\sqrt{3}}{3d^{\prime}(\lambda(\theta_{D}))}\big{|}\theta-\theta_{D}\big{|}+\mathcal{O}\big{(}|\theta-\theta_{D}|^{2}\big{)}\,,$
(1.24)

so that in (1.19) we obtain the non-zero constant,

$\gamma=\frac{2\sqrt{3}}{3d^{\prime}(\lambda(\theta_{D}))}\,.$ (1.25)

In the case where $\alpha_{b}\neq\alpha_{a}$, to demonstrate the non-existence
of Dirac cones, it is enough to note that, by the Lemmas 1.2.1 and 1.2.3, the
functions $r^{+}(\theta)$ and $r^{-}(\theta)$ never touch each other for
$\alpha_{b}\neq\alpha_{a}$. Furthermore, a simple calculation shows us that
the derivative of $r^{\pm}$ vanishes at $\theta_{D}$ and, consequently,
$r^{\pm}$ does not exhibit linear behavior around $\theta_{D}$. ∎

######  Observation 1.2.4.

In general, the following relation is valid between the constant $\gamma$,
obtained from the definition of Dirac Cones, and the constant
$\gamma_{{}_{D}}$, obtained from the Taylor expansion:

$\gamma=\dfrac{2\gamma_{{}_{D}}}{d^{\prime}(\lambda(\theta_{D}))}\,,$

desde que $d^{\prime}(\lambda(\theta_{D}))\neq 0$.

A consequence of this theorem can be obtained if we consider that, in a
hexagonal lattice, at the vertices of type-$A$ there is a nitrogen atom (N)
and at the vertices of type-$B$ there is a boron atom (B), or that is, if we
try to model boron nitride333We remember that here such modeling for hBN is a
very simplified proposal for this material.. Then, putting $\delta_{N}$ and
$\delta_{B}$ the parameters (in the boundary conditions) associated with N and
B, respectively, we have the following corollary.

######  Corollary 1.2.1.

The dispersion relation of the operator that models boron nitride does not
have Dirac cones if $\delta_{N}\neq\delta_{B}$ under vertex conditions. For
$\delta_{N}=\delta_{B}$ (that would be graphene), the relation has Dirac
cones.

Figure 1.2: $\alpha_{N}=\alpha_{B}=0.2$; Dirac cone at $F=0$ is formed when
the parameters $\alpha_{N}$ and $\alpha_{B}$ coincide. Figure 1.3:
$\alpha_{N}\neq\alpha_{B}$: $\alpha_{N}=0.5,\alpha_{B}=0.2$; when the
parameters $\alpha_{N}$ and $\alpha_{B}$ are distinct, gaps appear where there
were Dirac cones

Note that we can determine the width of the gap444We do not have clear
indications of what would be the values of the parameters $\delta_{N}$,
$\delta_{B}$ or $\delta_{C}$ that “best describe” the hBN and graphene, which
makes it difficult to make a more precise numerical comparison of the results
obtained in our model with those in the experimental physics literature. when
the atoms are distinct. In fact, for $\alpha_{b}=\alpha_{a}$ the conical touch
occurs when $\theta=\dfrac{2\pi}{3}$, that is, $F=0$. Thus, taking $F=0$ in
(1.13), we obtain that the gap is given by

$g(\alpha_{a},\alpha_{b})=\dfrac{|\alpha_{a}-\alpha_{b}|}{3}\,.$ (1.26)

In particular, for $\alpha_{b}=-\alpha_{a}=1$, we have a gap of width
${2}/{3}\approx 0.667$.

Throughout this work, we will make extensive use of this relationship to
justify the presence of Dirac cones in dispersion relations of two-and three-
layer stacks, including heterostructures. In general, to demonstrate that a
point $(\lambda,\theta)$ in the dispersion relation is a Dirac cone, we will
show that there is a constant $\gamma_{{}_{D}}\neq 0$ determined by the Taylor
expansion, as we saw in this example, without directly mentioning the (1.2.4)
relation above.

In the next chapter we extend the monolayer study, described here, to address
stacks composed of two identical $2D$ hexagonal layers, generalizing some
cases of graphene stacking described in [8, 9].

##  Chapter
2
Structures of type $AA$

Let us remember that the graphene bilayer consists of two vertically stacked
graphene sheets (with a certain type of stacking) that weakly interact with
each other, held together by van der Waals interactions. It is mainly found in
the so-called $AB$ [32] stacking, however there are other alternative stacking
configurations, in which one layer is translated or rotated by an angle in
relation to the other.

As described in [34], in these systems, in which layers of a material are
stacked in van der Waals structures, it is possible to exist regions that are
rich in $AB$ stacking and regions that mainly exhibit $AA$ stacking. In
particular, graphene bilayers aligned in the $AA$ type have attracted
theoretical interest from researchers because of experimental investigations
that show the presence of this type of stacking in certain samples and suggest
its experimental feasibility.

In this chapter we study $AA$ stackings with two equal layers in a hexagonal
shape in which we take two different types of atoms interspersed at the
vertices, of each hexagon, in each layer. We begin, in the first section, by
considering configurations of two identical sheets stacked so that each vertex
of type-$A$ on the lower sheet is overlapped and connected to a vertex of
type-$A$ on the upper sheet, and each vertex of type-$B$ of the lower layer is
superimposed and connected to a vertex of type-$B$ in the overlying sheet, as
shown in Figure 2.1. We will say that two atoms are equivalent if they are
identical (for example, both of type-$A$), otherwise we will say that they are
inequivalent.

Figure 2.1: Side and top views, respectively, of material composed of two
identical hexagonal layers aligned in type $AA$.

Notice in Figure 2.1 that each atom in a sheet interacts weakly with another
equivalent atom in the other layer. We emphasize that we do not aim here to
judge whether such a stacking model makes physical sense or describes any real
material found in nature. As it is a mathematical model, we are, a priori,
only interested in investigating the spectral properties and the possible
presence of Dirac cones for such a hypothetical material in the form of a van
der Waals heterostructure.

###  2.1 Structure with two identical leaves

In this model we consider two identical layers stacked in the form AA such
that an atom in the lower (upper) layer, in addition to interacting strongly
with three other distinct atoms (first neighbors) in the same layer, also
interacts weakly with another equivalent atom (i.e. similar atom) of the upper
(lower) sheet, as shown in Figure 2.2.

Figure 2.2: Material composed of two hexagonal layers stacked in the type
$AA$.

We will denote by $\bar{\mathcal{E}}(G^{AA})$ the set of edges in each leaf
and by $\hat{\mathcal{E}}(G^{AA})$ the set of edges between two consecutive
sheets; then we have the disjoint union

$\mathcal{E}(G^{AA})=\bar{\mathcal{E}}(G^{AA})\cup\hat{\mathcal{E}}(G^{AA}).$

For each vertex $v$ of the fundamental domain of $G^{AA}$, the Robin
conditions presented in Chapter 1 can be adapted and applied to two layers,
and become what we will call  modified Neumann vertex condition:

$\\\ \left\\{\begin{array}[]{ccll}(i)\quad u_{a_{1}}(v)\quad=\quad
u_{a_{2}}(v)\quad=\quad\dfrac{u_{f_{1}}(v)}{t_{0}}&=&\dfrac{u_{f_{2}}(v)}{t_{0}},\hskip
14.22636pt\forall\quad a_{1},a_{2},f_{1},f_{2}\in\mathcal{E}_{v}(G);\\\
(ii)\displaystyle\sum_{a\in\mathcal{E}_{v}(G)}\\!\\!u^{\prime}_{a}(v)\pm\\!\\!\displaystyle\sum_{f\in\hat{\mathcal{E}}_{v}(G)}t_{0}u^{\prime}_{f}(v)&=&\delta_{v}u(v)\end{array}\,,\right.$
(2.1)

where

  * •

$\delta_{v}$ a real number;

  * •

$u^{\prime}_{a}(v)$ the derivative of $u_{a}$ along the edge $a$ at the vertex
$v$;

  * •

$0<t_{0}\leq 1$ a (weak) interaction parameter between consecutive leaves;
sign $``+$ ” in $\pm u^{\prime}_{a}(v)$ if $v$ is a starting point of the
edge, sign $``-$ ” if it is the end point;

  * •

“ $a$ ” represents the edges in the leaves and $``\,f$ ” the edges between two
consecutive leaves.

In particular, in $(ii)$ of (2.1), when $\delta_{v}=0$ for any
$v\in\mathcal{V}(G)$, we have the Neumann vertex condition.

######  Observation 2.1.1.

We use a quantum graph model that, taken in its particular form, models
graphene, as well as hexagonal boron nitride, of two or three sheets with
$AA^{\prime}$ and $AA$ stacking. We consider the Schrödinger operator
described in (1.1) with the modified Neumann vertex condition, which requires
weighted continuity of functions and for this we consider the parameter
$t_{0}$, in the interval $(0,1]$, which also models the weak interaction
between consecutive leaves. The proposed idea is that the smaller $t_{0}$ is,
the weaker the interaction between such overlapping layers would be; this
comes from the following reasoning: from (2.1), the smaller $t_{0}$, the
smaller the value of $|u_{f}(v)|$ that connects consecutive layers and the
smaller the contribution of its derivative to the total flow in each vertex.
We emphasize that we do not consider the value $t_{0}=0$, as in this case we
have a singular limit and it is understood, intuitively, that the layers stop
interacting.

We also note that we could have considered two parameters $t_{a}$ and $t_{b}$,
instead of a single parameter $t_{0}$, to model the two weak interactions
between two consecutive hexagonal layers $\hexagon_{1}$ and $\hexagon_{2}$:
$t_{a}$ between atoms $A$ of the network $\hexagon_{1}$ with atoms $A$ of the
network $\hexagon_{2}$, and $t_{b}$ between atoms of type $B$. In this more
general case, as discussed in the Appendix 2.4, basically the results are
repeated, but with different values of the parameters.

On the other hand, although in $AA$ stacking it is pertinent to consider two
connection parameters $t_{a}$ and $t_{b}$ in the model, in $AA^{\prime}$
stacking this becomes unnecessary, as we will see in more detail in the
following chapter, because in $AA^{\prime}$ (according to our modeling
proposal) a single weak connection occurs: type $A$ atoms from the
$\hexagon_{1}$ network with type $B$ atoms from the $\hexagon_{2}$ network and
vice versa (we then use a single parameter $t_{0}:=t^{b}_{a}$).

In the $AA$ stacking, the fundamental domain for two layers is formed by six
horizontal edges, two vertical edges and the four initial and final vertices,
that is,

$W_{2}^{AA}=\\{a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},f_{1},f_{2},v_{11},v_{12},v_{21},v_{22}\\}\,.$

Let $H_{2}^{AA}$ be the Schrödinger operator associated with stacking of the
type $AA$ with two identical layers. As already mentioned in the previous
chapter and with analogous notation, we know that $H_{2}^{AA}(\theta)$has a
purely discrete spectrum [26], denoted by
$\sigma(H_{2}^{AA}(\theta))=\\{\lambda_{k}(\theta)\\}_{k\geq 1}$. The union of
the images of the functions $\theta\mapsto\\{\lambda_{k}(\theta)\\}$ is the
dispersion relation of $H_{2}^{AA}$, which determines its spectrum

$\sigma(H_{2}^{AA})=\bigcup_{\theta\in\mathcal{B}_{d}}\sigma(H_{2}^{AA}(\theta))\,.$
(2.2)

As we have already described in Chapter 1, from the Floquet cyclic condition,
we have

$u_{a_{k2}}(1)=e^{i\theta_{1}}u_{a_{k1}}(1)\quad\mbox{e}\quad
u_{a_{k3}}(1)=e^{i\theta_{2}}u_{a_{k1}}(1),\quad k=1,2\,,$ (2.3)

and with the conditions of continuity at the vertices,

$\left\\{\begin{array}[]{llll}u_{a_{11}}(0)&=&u_{a_{12}}(0)=u_{a_{13}}(0)=\dfrac{u_{f_{1}}(0)}{t_{0}}=:A_{1}\\\
u_{a_{11}}(1)&=&e^{i\theta_{1}}u_{a_{12}}(1)=e^{i\theta_{2}}u_{a_{13}}(1)=\dfrac{u_{f_{2}}(0)}{t_{0}}=:B_{1}\\\
u_{a_{21}}(0)&=&u_{a_{22}}(0)=u_{a_{23}}(0)=\dfrac{u_{f_{1}}(0)}{t_{0}}=:A_{2}\\\
u_{a_{21}}(1)&=&e^{i\theta_{1}}u_{a_{22}}(1)=e^{i\theta_{2}}u_{a_{23}}(1)=\dfrac{u_{f_{2}}(0)}{t_{0}}=:B_{2}\end{array},\right.$
(2.4)

By Robin condition (2.1), we have

$\left\\{\begin{array}[]{llll}u^{\prime}_{a_{11}}(0)+u^{\prime}_{a_{12}}(0)+u^{\prime}_{a_{13}}(0)+t_{0}u^{\prime}_{f_{1}}(0)&=&\delta_{a}A_{1}\\\
-u^{\prime}_{a_{11}}(1)-e^{i\theta_{1}}u^{\prime}_{a_{12}}(1)-e^{i\theta_{2}}u^{\prime}_{a_{13}}(1)-t_{0}u^{\prime}_{f_{2}}(1)&=&\delta_{b}B_{1}\\\
u^{\prime}_{a_{21}}(0)+u^{\prime}_{a_{22}}(0)+u^{\prime}_{a_{23}}(0)-t_{0}u^{\prime}_{f_{1}}(1)&=&\delta_{a}A_{2}\\\
-u^{\prime}_{a_{21}}(1)-e^{i\theta_{1}}u^{\prime}_{a_{22}}(1)-e^{i\theta_{2}}u^{\prime}_{a_{23}}(1)+t_{0}u^{\prime}_{f_{2}}(0)&=&\delta_{b}B_{2}\end{array}.\right.$
(2.5)

Let $\lambda\notin\sigma(H^{D})$. Then there are two linearly independent
solutions $\varphi_{0}=\varphi_{\lambda,0}$ and
$\varphi_{1}=\varphi_{\lambda,1}$ of the problem

$-\dfrac{d^{2}\varphi(x)}{dx^{2}}+q(x)\varphi(x)=\lambda\varphi(x)\,,$ (2.6)

such that

$\varphi_{0}(0)=1=\varphi_{1}(1)\,,\quad\varphi_{0}(1)=0=\varphi_{1}(0)\quad\mbox{e}\quad\varphi^{\prime}_{1}(x)=-\varphi^{\prime}_{0}(1-x)\,,\quad
x\in[0,1]\,.$ (2.7)

Since each edge of $W_{2}^{AA}$ is identified with the range $[0,1]$, we
define $\varphi_{\lambda,k}$ on each edge (with the same notation
$\varphi_{\lambda,k}$). Therefore, for each $\lambda\notin\sigma(H^{D})$, we
represent

$\left\\{\begin{array}[]{llll}u_{a_{k1}}&=&A_{k}\varphi_{\lambda,0}+B_{k}\varphi_{\lambda,1}\\\
u_{a_{k2}}&=&A_{k}\varphi_{\lambda,0}+e^{-i\theta_{1}}B_{k}\varphi_{\lambda,1}\\\
u_{a_{k3}}&=&A_{k}\varphi_{\lambda,0}+e^{-i\theta_{2}}B_{k}\varphi_{\lambda,1}\\\
u_{f_{1}}&=&t_{0}(A_{1}\varphi_{\lambda,0}+A_{2}\varphi_{\lambda,1})\\\
u_{f_{2}}&=&t_{0}(B_{2}\varphi_{\lambda,0}+B_{1}\varphi_{\lambda,1})\end{array},\quad
k=1,2\ \right..$ (2.8)

With this, we can verify that the continuity conditions and (2.6) are
satisfied. Now we have to check the “Total-Flow” condition in (2.5). Using the
representation of each function $u_{e}$ in (2.8), we have

$\left\\{\begin{array}[]{llll}(3+t_{0}^{2})\varphi^{\prime}_{\lambda,0}(0)A_{1}+\bar{F}(\theta)\varphi^{\prime}_{\lambda,1}(0)B_{1}+t_{0}^{2}\varphi^{\prime}_{\lambda,1}(0)A_{2}&=&\delta_{a}A_{1}\\\
F(\theta)\varphi^{\prime}_{\lambda,0}(1)A_{1}+(3+t_{0}^{2})\varphi^{\prime}_{\lambda,1}(1)B_{1}+t_{0}^{2}\varphi^{\prime}_{\lambda,0}(1)B_{2}&=&-\delta_{b}B_{1}\\\
-t_{0}^{2}\varphi^{\prime}_{\lambda,0}(1)A_{1}+\big{(}3\varphi^{\prime}_{\lambda,0}(0)-t_{0}^{2}\varphi^{\prime}_{\lambda,1}(1)\big{)}A_{2}+\bar{F}(\theta)\varphi^{\prime}_{\lambda,1}(0)B_{2}&=&\delta_{a}A_{2}\\\
-t_{0}^{2}\varphi^{\prime}_{\lambda,1}(0)B_{1}+\bar{F}(\theta)\varphi^{\prime}_{\lambda,0}(1)A_{2}+\big{(}3\varphi^{\prime}_{\lambda,1}(1)-t_{0}^{2}\varphi^{\prime}_{\lambda,0}(0)\big{)}B_{2}&=&-\delta_{b}B_{2}\end{array},\right.$
(2.9)

with $F(\theta):=1+e^{i\theta_{1}}+e^{i\theta_{2}}$ and $\bar{F}(\theta)$ its
complex conjugate. Like
$\varphi^{\prime}_{\lambda,0}(1)=-\varphi^{\prime}_{\lambda,1}(0)$,
$\varphi^{\prime}_{\lambda,0}(0)=-\varphi^{\prime}_{\lambda,1}(1)$ and, being
$\varphi^{\prime}_{\lambda,1}(0)\neq 0$, we put

$\eta(\lambda):=\dfrac{\varphi^{\prime}_{\lambda,1}(1)}{\varphi^{\prime}_{\lambda,1}(0)},$

multiplying the second and fourth lines by -1 in (2.9), we obtain the system

$M_{2}^{AA}(\eta(\lambda),\theta)X=0,\quad\mbox{com}\quad
X=[A_{1}\,\,B_{1}\,\,A_{2}\,\,B_{2}]^{T}$

and

$M_{2}^{AA}(\eta(\lambda),\theta)=\left(\begin{array}[]{cccc}-T\eta-\alpha_{a}&\bar{F}&t_{0}^{2}&0\\\
F&-T\eta-\alpha_{b}&0&t_{0}^{2}\\\ t_{0}^{2}&0&-T\eta-\alpha_{a}&\bar{F}\\\
0&t_{0}^{2}&F&-T\eta-\alpha_{b}\end{array}\right),$

where $T:=3+t_{0}^{2}$, $F=F(\theta)$, $\eta=\eta(\lambda)$ and
$\alpha_{k}:=\dfrac{\delta_{k}}{\varphi^{\prime}_{\lambda,1}(0)}$,
$k\in\\{a,b\\}$. Putting $P(\eta):=\det(M^{AA}_{2}(\eta(\lambda),\theta)$, we
have

$\begin{array}[]{lll}P(\eta)\\!\\!&\\!\\!=\\!\\!&\\!\\!T^{4}\eta^{4}\\!+\\!2(\alpha_{b}\\!+\\!\alpha_{a})T^{3}\eta^{3}\\!+\\!\big{[}(\alpha_{a}\\!+\\!\alpha_{b})^{2}\\!-\\!2(|F|^{2}\\!-\\!\alpha_{a}\alpha_{b}\\!+\\!t_{0}^{4})\big{]}T^{2}\eta^{2}\\\
&-&\\!\\!2(\alpha_{a}+\alpha_{b})\big{[}|F|^{2}\\!-\\!\alpha_{b}\alpha_{a}\\!+\\!t_{0}^{4}\big{]}T\eta+\big{[}|F|^{2}\\!-\\!\alpha_{b}\alpha_{a}\big{]}^{2}\\!+\\!t_{0}^{4}\big{[}t_{0}^{4}\\!-\\!\alpha^{2}_{n}-\alpha^{2}_{b}-2|F|^{2}\big{]}\\\
&=&\big{(}T^{2}\eta^{2}+(\alpha_{a}+\alpha_{b}-2t_{0}^{2})T\eta-|F|^{2}+t_{0}^{4}-(\alpha_{b}+\alpha_{a})t_{0}^{2}+\alpha_{b}\alpha_{a}\big{)}\cdot\\\
&&\big{(}T^{2}\eta^{2}+(\alpha_{a}+\alpha_{b}+2t_{0}^{2})T\eta-|F|^{2}+t_{0}^{4}+(\alpha_{a}+\alpha_{b})t_{0}^{2}+\alpha_{b}\alpha_{a}\big{)}\end{array}$

and it was possible to find its roots 111Here, the superscripts $\pm$ refer to
the first symbol, while the subscripts $\pm$ refer to the second., which are

$r^{\pm}_{\pm}(\theta)=-\dfrac{\alpha_{b}+\alpha_{a}\pm
2t_{0}^{2}\pm\sqrt{(\alpha_{a}-\alpha_{b})^{2}+4|F|^{2}}}{2T}\,.$ (2.10)

That is, there is $\theta\in\mathcal{B}$ such that
$\det\big{(}M^{AA}_{2}(\eta(\lambda),\theta)\big{)}=0$; therefore, the
representation (2.8) solves the problem of eigenvalues (2.6) and by
Proposition 1.2.1, we conclude that $\lambda\in\sigma(H_{2}^{AA})$.

###  2.2 Analysis of the dispersion relation

Let analyze the behavior of the (2.10) functions found in the previous
section, two-layer stacking. From now on, we study the Dirac cones for two
consecutive layers, making the values of $\gamma$ in each cone explicit. In
this case, in fact, for identical bi- or tri-layers it is enough to work with
the right side of (1.19), as the discussion for the left side repeats the
monolayer case made above in (1.23). We will check the presence of Dirac Cones
in this model, as well as parabolic touches. Through a direct calculation, we
see that the function

$|F(\theta)|^{2}=1+8\cos\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\cos\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\theta_{2}}{2}\right)$

has as its image the interval $[0,9]$, reaches a maximum at $(0,0)$ and
minimum at $\pm\left(\frac{2\pi}{3},-\frac{2\pi}{3}\right)$; Furthermore, the
functions $\pm|F|$ touch each other at $0$ only when
$\theta=\pm\left(\frac{2\pi}{3},-\frac{2\pi}{3}\right)$. Therefore, to analyze
the image of the functions $r^{\pm}_{\pm}$, simply take $\theta$ on the
diagonal $B_{d}$ of the Brillouin zone,

$B_{d}=\\{(\theta_{1},\theta_{2})\in[-\pi,\pi]^{2}:\theta_{2}=-\theta_{1}\\}\,$

and with that, we have $F(\theta)=1+2\cos(\theta_{1})$.

Initially, for $\alpha_{a}=\alpha_{b}$ in (2.10), we have

$r^{\pm}_{\pm}(\theta)=\dfrac{-\alpha_{b}\pm t_{0}^{2}\pm|F|}{3+t_{0}^{2}}\,.$
(2.11)

######  Lemma 2.2.1.

Under (2.11) conditions, we have for the images of the respective functions

${\mathrm{Img}}\big{(}r^{+}_{+}(\theta)\big{)}=\left[\frac{-\alpha_{b}+t_{0}^{2}}{3+t_{0}^{2}},\frac{-\alpha_{b}+t_{0}^{2}+3}{3+t_{0}^{2}}\right],\hskip
28.45274pt{\mathrm{Img}}\big{(}r^{+}_{-}(\theta)\big{)}=\left[\frac{-\alpha_{b}+t_{0}^{2}-3}{3+t_{0}^{2}},\frac{-\alpha_{b}+t_{0}^{2}}{3+t_{0}^{2}}\right],$
${\mathrm{Img}}\big{(}r^{-}_{+}(\theta)\big{)}=\left[\frac{-\alpha_{b}-t_{0}^{2}}{3+t_{0}^{2}},\frac{-\alpha_{b}-t_{0}^{2}+3}{3+t_{0}^{2}}\right],\hskip
28.45274pt{\mathrm{Img}}\big{(}r^{-}_{-}(\theta)\big{)}=\left[\frac{-\alpha_{b}-t_{0}^{2}-3}{3+t_{0}^{2}},\frac{-\alpha_{b}-t_{0}^{2}}{3+t_{0}^{2}}\right]\,.$

Consequently:

  1. 1.

$\min\left(r^{+}_{+}(\theta)\right)\\!\\!=\\!\\!\dfrac{-\alpha_{b}+t_{0}^{2}}{3+t_{0}^{2}}$
and
$\max\left(r^{+}_{+}(\theta)\right)\\!\\!=\\!\\!\dfrac{-\alpha_{b}+t_{0}^{2}+3}{3+t_{0}^{2}}$
reached at $\theta=\pm\big{(}\\!-\frac{2\pi}{3},\frac{2\pi}{3}\big{)}$ and
$\theta\\!=\\!(0,0)$, respectively.

  2. 2.

$\min\left(r^{+}_{-}(\theta)\right)=\dfrac{-\alpha_{b}+t_{0}^{2}-3}{3+t_{0}^{2}}$
and
$\max\left(r^{+}_{-}(\theta)\right)=\dfrac{-\alpha_{b}+t_{0}^{2}}{3+t_{0}^{2}}$
reached at $\theta=(0,0)$ e
$\theta=\pm\big{(}\\!-\frac{2\pi}{3},\frac{2\pi}{3}\big{)}$, respectively;

  3. 3.

$\min\left(r^{-}_{+}(\theta)\right)\\!\\!=\\!\\!\dfrac{-\alpha_{b}-t_{0}^{2}}{3+t_{0}^{2}}$
e
$\max\left(r^{-}_{+}(\theta)\right)\\!\\!=\\!\\!\dfrac{-\alpha_{b}-t_{0}^{2}+3}{3+t_{0}^{2}}$
reached at $\theta=\pm\big{(}\\!-\frac{2\pi}{3},\frac{2\pi}{3}\big{)}$ and
$\theta=(0,0)$, respectively.

  4. 4.

$\min\left(r^{-}_{-}(\theta)\right)=\dfrac{-\alpha_{b}-t_{0}^{2}-3}{3+t_{0}^{2}}$
and
$\max\left(r^{-}_{-}(\theta)\right)=\dfrac{-\alpha_{b}-t_{0}^{2}}{3+t_{0}^{2}}$
reached at $\theta=(0,0)$ and
$\theta=\pm\big{(}\\!-\frac{2\pi}{3},\frac{2\pi}{3}\big{)}$, respectively.

Finally, we arrive at the following theorem, which establishes results
regarding the dispersion relation of $H^{AA}_{2}$.

######  Theorem 2.2.1.

(Conclusion on the dispersion relationship of $H^{AA}_{2}$) Fix
$t_{0}\in(0,1)$ and $\alpha_{b}\in\mathbb{R}$ such that
$|r^{\pm}_{\pm}(\theta)|\leq 1$ in (2.10) and let
$\alpha_{a}=(1-u)\cdot\big{(}\\!\\!-2t_{0}^{2}+\alpha_{b}\big{)}+u\cdot\big{(}2t_{0}^{2}+\alpha_{b}\big{)}$.
The dispersion relation of the $H^{AA}_{2}$ operator is given by

$r^{\pm}_{\pm}(\theta)=\dfrac{\pm\sqrt{|F|^{2}+t_{0}^{4}(1-2u)^{2}}\pm
t_{0}^{2}+t_{0}^{2}(1-2u)-\alpha_{b}}{T}\,,$

and we conclude that

  1. (a)

For $\alpha_{a}=\alpha_{b}$, there are Dirac cones (Figure 2.3).

  2. (b)

For $\alpha_{a}=-2t_{0}^{2}+\alpha_{b}$ or $\alpha_{a}=2t_{0}^{2}+\alpha_{b}$,
there are parabolic touches (Figure 2.4).

  3. (c)

For $\alpha_{a}\notin[-2t_{0}^{2}+\alpha_{b},2t_{0}^{2}+\alpha_{b}]$, the
dispersion relation has gaps (Figure 2.5).

  4. (d)

For $0<u<1$ and $\alpha_{a}\neq\alpha_{b}$, the curves only intersect, but
there are no parabolic or conic touches (Figure 2.6).

We will only give an idea of the demonstration here.

###### Proof.

Let take
$\theta\\!\in\\!B_{d}\\!=\\!\big{\\{}(\theta_{1},-\theta_{1}):\theta_{1}\\!\in\\![-\pi,\pi]\big{\\}}$.
For $\alpha_{a}=\alpha_{b}$, we will apply the Taylor expansion to the
function $r^{\pm}_{+}$ in 2.11 at the point
$\theta_{D}=\pm(\frac{2\pi}{3},-\frac{2\pi}{3})$. Note that, putting
$\theta_{0}:=\pm\frac{2\pi}{3}$, we have

$F(\theta)=1+2\cos(\theta_{1})=\pm\sqrt{3}(\theta_{1}-\theta_{0})+\mathcal{O}\big{(}(\theta_{1}-\theta_{0})^{2}\big{)}\,.$

Since $T:=3+t_{0}^{2}$, we have

$r^{\pm}_{+}(\theta):=\dfrac{-\alpha_{b}\pm
t_{0}^{2}+|F(\theta)|}{T}=\dfrac{-\alpha_{b}\pm
t_{0}^{2}+\sqrt{3}|\theta_{1}-\theta_{0}|}{T}+\mathcal{O}\big{(}|\theta_{1}-\theta_{0}|^{2}\big{)}$

and

$r^{\pm}_{+}(\theta_{D}):=\dfrac{-\alpha_{b}\pm t_{0}^{2}}{T}\,.$

Therefore,

$r^{\pm}_{+}(\theta)-r^{\pm}_{+}(\theta_{D})=\pm\dfrac{\sqrt{3}}{T}|\theta_{1}-\theta_{0}|+\mathcal{O}\big{(}|\theta_{1}-\theta_{0}|^{2}\big{)}\,.$

Similarly, for $r^{\pm}_{-}$, we obtain

$r^{\pm}_{-}(\theta)-r^{\pm}_{-}(\theta_{D})=\mp\dfrac{\sqrt{3}}{T}|\theta_{1}-\theta_{0}|+\mathcal{O}\big{(}|\theta_{1}-\theta_{0}|^{2}\big{)}\,.$

For $\alpha_{a}=2t_{0}^{2}+\alpha_{b}$ (or
$\alpha_{a}=-2t_{0}^{2}+\alpha_{b})$, we see that the functions $r^{+}_{-}$
and $r^{-}_{+}$ touch when $\theta_{p}=\pm(\frac{2\pi}{3},-\frac{2\pi}{3})$,
or even, $r^{+}_{-}(\theta_{p})=r^{-}_{+}(\theta_{p})=0$ so that there is the
parabolic touch $(\theta_{p},0)$. In fact, in 2.10, taking $\theta\in B_{d}$,
$F=F(\theta)=1+\cos(\theta)$ and putting $\alpha_{a}=\pm
2t_{0}^{2}+\alpha_{b}$, we get

$r^{\pm}_{\mp}(\theta)=-\dfrac{-\alpha_{b}-t_{0}^{2}\pm
t_{0}^{2}\mp\sqrt{t_{0}^{4}+|F|^{2}}}{3+t_{0}^{2}}\,.$

If $\theta\neq\theta_{p}$, a direct calculation shows us that
$\dfrac{dr^{\pm}_{\mp}}{d\theta}$ cancels out in $\pm\frac{2\pi}{3}$.
Furthermore, to obtain the behavior of the functions $r^{\pm}_{\mp}$ around
$\pm\frac{2\pi}{3}$, we apply the relation $(1+x)^{\frac{1}{2}}\approx
1+\frac{x}{2}-\frac{x^{2}}{8}$, for small $x$, in the expression
$\sqrt{t_{0}^{4}+|F|^{2}}$ and we conclude that $r^{\pm}_{\mp}$ has parabolic
behavior around $\pm\frac{2\pi}{3}$, that is, next of $F=0$.

For the other cases, in particular, when
$\alpha_{a}\notin[-2t_{0}^{2}+\alpha_{b},2t_{0}^{2}+\alpha_{b}]$, a simple
calculation shows us that the functions $r^{\pm}_{\pm}$ do not touch each
other regardless of the values of $\theta$, for any $\theta\in B$. ∎

Below are graphic images of some cases obtained from the theorem above.

Figure 2.3: Dirac cone if the atoms are equal; here
$\alpha_{a}=\alpha_{b}=-1\,,t_{0}=0.3\,.$ Figure 2.4: Parabolic touch; here
$\alpha_{a}=-1\,,\alpha_{b}=2t_{0}^{2}+\alpha_{a}\,,t_{0}=0.3\,.$ Figure 2.5:
Gap of 0.589 in the dispersion relation at the origin for
$\alpha_{a}=-1,\,\alpha_{b}=1\,,t_{0}=0.3\,.$ Figure 2.6: The curves
intersect, but there are no parabolic or conic touches:
$\alpha_{a}\\!=\\!-1,\alpha_{b}\\!=\\!-0.9,t_{0}\\!=\\!0.3\,.$

We summarize these results in the following section.

###  2.3 Analysis and conclusions

We saw in this chapter that in a van der Waals heterostructure, stacked in the
$AA$ type with two equal hexagonal layers, in which two types of atoms are
considered interspersed at the vertices of each hexagon, the following
situations basically occur: (a) if the atoms are identical
($\alpha_{a}=\alpha_{b}$), the model has a Dirac cone; (b) has parabolic
touches if $\alpha_{b}=\pm\,2t_{0}^{2}\,+\,\alpha_{a}$; (c) has gaps
otherwise; (d) has no conical or parabolic touches for
$\alpha_{b}\neq\alpha_{a}$ in the interval
$(-2t_{0}^{2}\,+\,\alpha_{a},2t_{0}^{2}\,+\,\alpha_{a})$. Note also that for
(only formally) $t_{0}=0$ (in which case there is no interaction between the
layers and the bilayers are reduced to the monolayer), we recover the results
for the monolayer seen in the previous chapter. In particular, according to
our model, heterostructures of two sheets of boron nitride stacked in the $AA$
type do not have Dirac cones, but can have parabolic touches.

###  2.4 Appendix: the parameter $t_{0}$

Let us remember that, in the previous section, we used different values of the
$\delta_{v}$ parameter in the Robin (1.2) condition as a way of distinguishing
the atoms in the model: if $\delta_{v}$ is associated with a vertex of
type-$A$ (or type-$B$), we choose $\delta_{v}=\delta_{a}$
$\big{(}\mbox{ou}\,\,\delta_{v}=\delta_{b}\big{)}$, with
$\delta_{a}\neq\delta_{b}$ if A$\neq$B. Furthermore, in stacking $AA$ we admit
the same parameter $t_{0}$ simulating the weak connections between vertices of
consecutive layers, that is, both to vertices of type-$A$ and to those of
type-$B$, we associate a single parameter $t_{0}$. Let’s now generalize this
modeling: for bonds between type A atoms we will associate an interaction
parameter $t_{a}$, and for bonds between type B atoms, we will associate
$t_{b}$.

In this case, the Floquet cyclic condition and the continuity conditions at
the vertices will be given by

$\left\\{\begin{array}[]{llll}u_{a_{11}}(0)&=&u_{a_{12}}(0)=u_{a_{13}}(0)=\dfrac{u_{f_{1}}(0)}{t_{a}}=:A_{1}\\\
u_{a_{11}}(1)&=&e^{i\theta_{1}}u_{a_{12}}(1)=e^{i\theta_{2}}u_{a_{13}}(1)=\dfrac{u_{f_{2}}(1)}{t_{b}}=:B_{1}\\\
u_{a_{21}}(0)&=&u_{a_{22}}(0)=u_{a_{23}}(0)=\dfrac{u_{f_{1}}(1)}{t_{a}}=:A_{2}\\\
u_{a_{21}}(1)&=&e^{i\theta_{1}}u_{a_{22}}(1)=e^{i\theta_{2}}u_{a_{23}}(1)=\dfrac{u_{f_{2}}(0)}{t_{b}}=:B_{2}\end{array}.\right.$
(2.12)

Again, introducing the parameters $\delta_{a},\delta_{b}$ to the respective
vertices type-$A$ and type-$B$, using the Robin condition (2.1), we obtain the
Total Flow equations

$\left\\{\begin{array}[]{llll}u^{\prime}_{a_{11}}(0)+u^{\prime}_{a_{12}}(0)+u^{\prime}_{a_{13}}(0)+t_{a}u^{\prime}_{f_{1}}(0)&=&\delta_{a}A_{1}\\\
-u^{\prime}_{a_{11}}(1)-e^{i\theta_{1}}u^{\prime}_{a_{12}}(1)-e^{i\theta_{2}}u^{\prime}_{a_{13}}(1)-t_{b}u^{\prime}_{f_{2}}(1)&=&\delta_{b}B_{1}\\\
u^{\prime}_{a_{21}}(0)+u^{\prime}_{a_{22}}(0)+u^{\prime}_{a_{23}}(0)-t_{a}u^{\prime}_{f_{1}}(1)&=&\delta_{a}A_{2}\\\
-u^{\prime}_{a_{21}}(1)-e^{i\theta_{1}}u^{\prime}_{a_{22}}(1)-e^{i\theta_{2}}u^{\prime}_{a_{23}}(1)+t_{b}u^{\prime}_{f_{2}}(0)&=&\delta_{b}B_{2}\end{array}.\right.$
(2.13)

Similarly, for each $\lambda\notin\sigma(H^{D})$, we represent

$\left\\{\begin{array}[]{llll}u_{a_{k1}}&=&A_{k}\varphi_{0}+B_{k}\varphi_{1}\\\
u_{a_{k2}}&=&A_{k}\varphi_{0}+e^{-i\theta_{1}}B_{k}\varphi_{1}\\\
u_{a_{k3}}&=&A_{k}\varphi_{0}+e^{-i\theta_{2}}B_{k}\varphi_{1}\\\
u_{f_{1}}&=&t_{a}(A_{1}\varphi_{0}+A_{2}\varphi_{1})\\\
u_{f_{2}}&=&t_{b}(B_{2}\varphi_{0}+B_{1}\varphi_{1})\end{array},\quad k=1,2\
\right.\,.$ (2.14)

After some calculations, we obtain the system

$M_{ab}^{AA}(\eta(\lambda),\theta)X=0,\quad\mbox{com}\quad
X=[A_{1}\,\,B_{1}\,\,A_{2}\,\,B_{2}]^{T}$

and

$M_{ab}^{AA}(\eta(\lambda),\theta)=\left(\begin{array}[]{cccc}-T_{a}\eta-\alpha_{a}&\bar{F}&t_{a}^{2}&0\\\
F&-T_{b}\eta-\alpha_{b}&0&t_{b}^{2}\\\
t_{a}^{2}&0&-T_{a}\eta-\alpha_{a}&\bar{F}\\\
0&t_{b}^{2}&F&-T_{b}\eta-\alpha_{b}\end{array}\right),$

where $T_{a}:=3+t_{a}^{2},T_{b}:=3+t_{b}^{2},F=F(\theta)$,
$\eta=\eta(\lambda)$ e
$\alpha_{k}:=\dfrac{\delta_{k}}{\varphi^{\prime}_{1}(0)}$, $k\in\\{a,b\\}$.
Putting $P(\eta)\\!:=\det(M_{ab}^{AA}(\eta(\lambda),\theta))$, we have

$\displaystyle P(\eta)\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\big{[}T_{a}T_{b}\eta^{2}+\big{(}(\alpha_{a}-t_{a}^{2})T_{b}+(\alpha_{b}-t_{b}^{2})T_{a}\big{)}\eta+t_{a}^{2}t_{b}^{2}-\alpha_{a}t_{b}^{2}-\alpha_{b}t_{a}^{2}+\alpha_{a}\alpha_{b}-|F|^{2}\big{]}$
$\displaystyle\cdot\\!$
$\displaystyle\\!\big{[}T_{a}T_{b}\eta^{2}+\big{(}(\alpha_{a}+t_{a}^{2})T_{b}+(\alpha_{b}+t_{b}^{2})T_{a}\big{)}\eta+t_{a}^{2}t_{b}^{2}+\alpha_{a}t_{b}^{2}+\alpha_{b}t_{a}^{2}+\alpha_{a}\alpha_{b}-|F|^{2}\big{]}.$

In particular, note that for $t_{a}=t_{b}$, we recover the polynomial (2.1)
and the relation (2.10), and the results above coincide with those obtained
previously, as predicted: there will be a cone if $\alpha_{a}=\alpha_{b}$, and
there will be a gap if $\alpha_{a}\neq\alpha_{b}$.

Suppose then $t_{a}\neq t_{b}$. Let us analyze two particular cases,
initially.

Case $\alpha_{a}=t_{a}^{2}$ and $\alpha_{b}=t_{b}^{2}$. We state that a Dirac
cone occurs. In fact, in this case, we obtain the polynomial

$\displaystyle
P(\eta)\\!=\\!\big{(}T_{a}T_{\eta}^{2}-|F|^{2}\big{)}\big{(}T_{a}T_{b}\eta^{2}+2(T_{a}t_{b}^{2}+T_{b}t_{a}^{2})\eta+4t_{a}^{2}t_{b}^{2}-|F|^{2}\big{)}$

whose roots are

$\displaystyle r^{\pm}(F)\\!=\\!\pm\dfrac{|F|}{T_{a}T_{b}}\quad\mbox{and}\quad
r_{\mp}(F)\\!=\\!\mp\dfrac{\sqrt{(T_{b}t_{a}^{2}-T_{a}t_{b}^{2})^{2}+4|F|^{2}}}{T_{a}T_{b}}-\dfrac{T_{a}t_{b}^{2}+T_{b}t_{a}^{2}}{T_{a}T_{b}}$

Clearly, we see that $r^{\pm}$ has conical behavior, whose touch occurs for
$F=0$, that is, at $r^{+}(0)=0=r^{-}(0)$, while in $r_{\mp}$ there are no
touches, since $t_{a}\neq t_{b}$. An analogous result is obtained for
$\alpha_{a}=-t_{a}^{2}$ and $\alpha_{b}=-t_{b}^{2}$.

Case $\alpha_{a}=t_{a}^{2}$ and $\alpha_{b}=-t_{b}^{2}$. We claim that there
is a parabolic touch. In effect, here, we obtain the polynomial

$\displaystyle
P(\eta)\\!=\\!\big{(}T_{a}T_{b}\eta^{2}+2T_{b}t_{a}^{2}\eta-|F|^{2}\big{)}\big{(}T_{a}T_{b}\eta^{2}-2T_{a}t_{b}^{2}\eta-|F|^{2})\big{)}$

and its roots are

$\displaystyle
r^{\pm}(F)\\!=\\!\pm\dfrac{\sqrt{T_{a}^{2}t_{b}^{4}+T_{a}T_{b}|F|^{2}}}{T_{a}T_{b}}-\dfrac{T_{a}t_{b}^{2}}{T_{a}T_{b}}\quad\mbox{and}\quad
r_{\mp}(F)\\!=\\!\mp\dfrac{\sqrt{T_{a}^{2}t_{b}^{4}+T_{a}T_{b}|F|^{2}}}{T_{a}T_{b}}+\dfrac{T_{a}t_{b}^{2}}{T_{a}T_{b}}\,.$

With this, we see that $r^{\pm}$ and $r_{\mp}$ have parabolic behavior, whose
touch occurs at $r^{+}(0)=r_{-}(0)$. Similarly, we obtain a similar result if
$\alpha_{a}=-t_{a}^{2}$ and $\alpha_{b}=t_{b}^{2}$.

In general, the roots of $P$ are given by

$\displaystyle r^{\pm}(F)\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\pm\dfrac{\sqrt{(\alpha_{a}-t_{a}^{2})^{2}T_{b}^{2}+(\alpha_{b}-t_{b}^{2})^{2}T_{a}^{2}-2T_{a}T_{b}\big{[}(\alpha_{a}-t_{a}^{2})(\alpha_{b}-t_{b}^{2})-2|F|^{2}\big{]}}}{2T_{a}T_{b}}$
$\displaystyle-$
$\displaystyle\dfrac{(\alpha_{a}-t_{a}^{2})T_{b}+(\alpha_{b}-t_{b}^{2})T_{a}}{2T_{a}T_{b}}$

and

$\displaystyle r_{\mp}(F)\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\mp\dfrac{\sqrt{(\alpha_{a}+t_{a}^{2})^{2}T_{b}^{2}+(\alpha_{b}+t_{b}^{2})^{2}T_{a}^{2}-2T_{a}T_{b}\big{[}(\alpha_{a}+t_{a}^{2})(\alpha_{b}+t_{b}^{2})-2|F|^{2}\big{]}}}{2T_{a}T_{b}}$
$\displaystyle-$
$\displaystyle\dfrac{(\alpha_{a}+t_{a}^{2})T_{b}+(\alpha_{b}+t_{b}^{2})T_{a}}{2T_{a}T_{b}}$

Through a direct calculation and graphical representation of the dispersion
relation, with distinct $t_{a},t_{b}\in(0,1]$, analyzing the functions
$r^{\pm}$ and $r^{\mp}$ above, we get the result below.

######  Theorem 2.4.1.

Let $t_{a},t_{b}\in(0,1]$ be distinct, weak interaction parameters between
type A and type B atoms in different layers, respectively, of two hexagonal
lattices. Consider the dispersion relation associated with the type structure
$AA$. Then

  * (a)

If $\alpha_{a}=t_{a}^{2}$ (or $\alpha_{a}=-t_{a}^{2}$) and
$\alpha_{b}=t_{b}^{2}$ (or $\alpha_{b}=-t_{b}^{2}$), conical touch will occur
in $F=0$ (Figure 2.7).

  * (b)

If $\alpha_{a}\\!=\\!t_{a}^{2}$ (or $\alpha_{a}\\!=\\!-t_{a}^{2}$) and
$\alpha_{b}\\!=\\!-t_{b}^{2}$ (or $\alpha_{b}=t_{b}^{2}$), there will be a
parabolic touch $F\\!=\\!0$ (Figure 2.8).

  * (c)

If $|\alpha_{a}|\neq t_{a}^{2}$ or $|\alpha_{b}|\neq t_{b}^{2}$, conical and
parabolic touches no longer exist and the dispersion relation will have gaps
(Figure 2.9).

Figure 2.7: Dirac cone, $\alpha_{a}=0.25$ and $\alpha_{b}=0.09$ . Figure 2.8:
Parabolic Touch, $\alpha_{a}=0.25$ and $\alpha_{b}=-0.09$ . Figure 2.9: Gaps:
left with $\alpha_{a}=\alpha_{b}=-1$; on the right with $\alpha_{a}=1$ and
$\alpha_{b}=-1$.

######  Observation 2.4.1.

In stacking $AA$ it is pertinent to consider two connection parameters $t_{a}$
and $t_{b}$ in the model, as explained above, since in this case two weak
interactions occur between the hexagonal layers $\hexagon_{1}$ and
$\hexagon_{2}$. However, in the $AA^{\prime}$ stacking, as we will see in more
detail in the following chapter, a single weak bond occurs: atoms A from the
network $\hexagon_{1}$ with atoms B from the network $\hexagon_{2}$ and vice
versa (we then use a single parameter $t^{b}_{a}=:t_{0}$).

Note that if we took the parameters $\alpha_{a}$ and $\alpha_{b}$ equal, such
modeling with two different parameters would be reduced to the model with a
single $t_{0}$.

A comparison of these results with the $AA^{\prime}$ stacking case appears in
Observation 3.3.1.

######  Observation 2.4.2.

$AA$ stacking for hBN layers appears to be rare in nature; thus, the wealth of
behaviors described here for this case were not found in the physical
literature, remaining as a proposal for future experiments.

##  Chapter
3
Type structures $AA^{\prime}$

Let us remember that hexagonal boron nitride is composed of alternating boron
$(B)$ and nitrogenene $(N)$ atoms arranged in a structure similar to graphene,
however, with strong connections in a plane and a weak van der Waals force
between the layers.

In [13] these layers are arranged in the $AA^{\prime}$ stacking with $B$ atoms
placed directly above the $N$ atoms. Therefore, similarly to the previous
chapter, boron nitride bilayers can be considered, in which the nitrogen atoms
interact with the boron atoms (and vice versa) both in the interaction the
strong one (represented by the horizontal edges) as well as the van der Waals
one (represented by the vertical edges), as shown in Figure 3.1.

Figure 3.1: Stacking $AA^{\prime}$ of two hexagonal layers with 2 distinct
atoms at the vertices, as in hBN. Side and top views, respectively.

However, in this chapter we will study stacks formed by two layers of the same
material with a hexagonal shape, modeled by a quantum graph, in this case
$AA^{\prime}$. Clearly, in the case of bilayers, when one of them only has the
same type of atom at the vertices, for example in the case of graphene, the
configurations $AA$ and $AA^{\prime}$ coincide.

###  3.1 Structure with two identical leaves

To apply our model – in which quantum graph theory, Floquet theory, modified
Neumann conditions are used – to this stacking, we will consider that the
layers are arranged so that a type-A vertex (red color) of a leaf is connected
to a type-B vertex (blue color) on the overlying leaf and vice versa,
illustrated in Figure 3.2. To vertices of type-$A$ (type-$B$) we associate a
contour parameter $\delta_{a}$ ( $\delta_{b}$ ) in the condition flux; again,
this is the proposal to differentiate the types of atoms.

Figure 3.2: Hexagonal bilayers with 2 distinct atoms at the vertices. Type-$A$
vertices are represented by red, and type-$B$ by blue.

As in the previous chapter, here we consider the fundamental domain given by

$W_{2}^{AA^{\prime}}=\\{a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},f_{1},f_{2},v_{11},v_{12},v_{21},v_{22}\\}\,.$

Let us denote by $H_{2}^{AA^{\prime}}$ the Schrödinger operator, associated
with $AA^{\prime}$ type stacking. with two layers. The spectrum of each
operator $H_{2}^{AA^{\prime}}(\theta)$ is purely discrete [26], which we
denote $\sigma(H_{2}^{AA^{\prime}}(\theta))=\\{\lambda_{k}(\theta)\\}_{k\geq
1}$; furthermore, the union of the images of the functions
$\theta\mapsto\\{\lambda_{k}(\theta)\\}$ (he Dispersion Relation of
$H_{2}^{AA^{\prime}}$) determines its spectrum

$\sigma(H_{2}^{AA^{\prime}})=\bigcup_{\theta\in\mathcal{B}_{d}}\sigma(H_{2}^{AA^{\prime}}\\!(\theta)).$
(3.1)

With this, our objective now is to determine the spectrum of
$H_{2}^{\\!AA^{\prime}}\\!(\theta)$. To do so, let us remember that the cyclic
Floquet conditions, together with the continuity conditions at the vertices
(2.1) applied to the fundamental domain $W_{2}^{AA^{\prime}}$, we get

$\left\\{\begin{array}[]{lll}u_{a_{11}}(0)&=&u_{a_{12}}(0)=u_{a_{13}}(0)=\dfrac{u_{f_{1}}(0)}{t_{0}}=:A_{1}\\\
u_{a_{11}}(1)&=&e^{i\theta_{1}}u_{a_{12}}(1)=e^{i\theta_{2}}u_{a_{13}}(1)=\dfrac{u_{f_{2}}(1)}{t_{0}}=:B_{1}\\\
u_{a_{21}}(0)&=&u_{a_{22}}(0)=u_{a_{23}}(0)=\dfrac{u_{f_{2}}(0)}{t_{0}}=:A_{2}\\\
u_{a_{21}}(1)&=&e^{i\theta_{1}}u_{a_{22}}(1)=e^{i\theta_{2}}u_{a_{23}}(1)=\dfrac{u_{f_{1}}(1)}{t_{0}}=:B_{2}\end{array}\,.\right.$
(3.2)

Applying the modified Neumann vertex condition (2.1), we obtain the equations
of

$\left\\{\begin{array}[]{llllll}u^{\prime}_{11}(0)+u^{\prime}_{12}(0)+u^{\prime}_{13}(0)+t_{0}u^{\prime}_{f_{1}}(0)&=&\delta_{a}A_{1}\\\
-u^{\prime}_{11}(1)-e^{i\theta_{1}}u^{\prime}_{12}(1)-^{i\theta_{2}}u^{\prime}_{13}(1)-t_{0}u^{\prime}_{f_{2}}(1)&=&\delta_{b}B_{1}\\\
u^{\prime}_{21}(0)+u^{\prime}_{22}(0)+u^{\prime}_{23}(0)+t_{0}u^{\prime}_{f_{1}}(0)&=&\delta_{a}A_{2}\\\
-u^{\prime}_{21}(1)-e^{i\theta_{1}}u^{\prime}_{22}(1)-e^{i\theta_{2}}u^{\prime}_{23}(1)-t_{0}u^{\prime}_{f_{2}}(1)&=&\delta_{b}B_{2}\end{array}\right.\,.$
(3.3)

Let $\lambda\notin\sigma(H^{D})$. Then there are two linearly independent
solutions $\varphi_{0}=\varphi_{\lambda,0}$ and
$\varphi_{1}=\varphi_{\lambda,1}$ of the problem

$-\dfrac{d^{2}\varphi(x)}{dx^{2}}+q(x)\varphi(x)=\lambda\varphi(x)\,,$ (3.4)

such that

$\left\\{\begin{array}[]{ll}\varphi_{0}(0)=1=\varphi_{1}(1)\\\
\varphi_{0}(1)=0=\varphi_{1}(0)\end{array}\right.\,\,\mbox{and}\quad\varphi^{\prime}_{1}(x)=-\varphi^{\prime}_{0}(1-x)\,\,,x\in[0,1]\,.$
(3.5)

As every edge of $W_{2}^{AA^{\prime}}$ is identified with the range $[0,1]$,
we define $\varphi_{\lambda,k}$ on each edge (with the same notation
$\varphi_{\lambda,k}$). Then, for each $\lambda\notin\sigma(H^{D})$, we write

$\left\\{\begin{array}[]{llllll}u_{a_{k1}}&=&A_{k}\varphi_{0}+B_{k}\varphi_{1}\\\
u_{a_{k2}}&=&A_{k}\varphi_{0}+B_{k}\varphi_{1}e^{-i\theta_{1}}\\\
u_{a_{k3}}&=&A_{k}\varphi_{0}+B_{k}\varphi_{1}e^{-i\theta_{2}}\\\
u_{f_{1}}&=&t_{0}(A_{1}\varphi_{0}+B_{2}\varphi_{1})\\\
u_{f_{2}}&=&t_{0}(A_{2}\varphi_{0}+B_{1}\varphi_{1})\\\ \end{array},\quad
k=1,2\right.\,\,,$ (3.6)

we can verify that the continuity conditions in (2.1) are satisfied. Now we
are left check Robin’s condition (3.4). Using the representation of each
function $u_{e}$ in (3.6) with the total flow equations in (3.3), we have

$\left\\{\begin{array}[]{llllll}(3+t_{0}^{2})\varphi^{\prime}_{0}(0)A_{1}+\bar{F}\varphi^{\prime}_{1}(0)B_{1}+t_{0}^{2}\varphi^{\prime}_{1}(0)A_{2}&=&\delta_{a}A_{1}\\\
F\varphi^{\prime}_{0}(1)A_{1}+(3+t_{0}^{2})\varphi^{\prime}_{1}(1)B_{1}+t_{0}^{2}\varphi^{\prime}_{0}(1)B_{2}&=&-\delta_{b}B_{1}\\\
t_{0}^{2}\varphi^{\prime}_{1}(0)B_{1}+(3\varphi^{\prime}_{0}(0)-t_{0}^{2}\varphi^{\prime}_{0}(0))A_{2}+F\varphi^{\prime}_{1}(0)B_{2}&=&\delta_{a}A_{2}\\\
t_{0}^{2}\varphi^{\prime}_{0}(1)A_{1}+\bar{F}\varphi^{\prime}_{0}(1)A_{2}+(3\varphi^{\prime}_{1}(1)+t_{0}^{2}\varphi^{\prime}_{1}(1))B_{2}&=&-\delta_{b}B_{2}\par\end{array},\right.$
(3.7)

with $F=F(\theta):=1+e^{i\theta_{1}}+e^{i\theta_{2}}$ and $\bar{F}(\theta)$
the conjugate complex of $F(\theta)$. As
$\varphi^{\prime}_{\lambda,0}(1)=-\varphi^{\prime}_{\lambda,1}(0)$,
$\varphi^{\prime}_{\lambda,0}(0)=-\varphi^{\prime}_{\lambda,1}(1)$ and, where
$\varphi^{\prime}_{\lambda,1}(0)\neq 0$, putting

$\eta=\eta(\lambda):=\dfrac{\varphi^{\prime}_{\lambda,1}(1)}{\varphi^{\prime}_{\lambda,1}(0)},$

$T:=3+t_{0}^{2}$,
$\alpha_{k}:=\dfrac{\delta_{k}}{\varphi^{\prime}_{\lambda,1}(0)}$, with
$k\in\\{a,b\\}$, and multiplying the even rows by -1 in (3.7), we obtain the
following equivalent system

$M_{2}^{AA^{\prime}}\\!(\eta(\lambda),\theta)\cdot X=0,\quad\mbox{with}\quad
X:=[A_{1}\,\,B_{1}\,\,A_{2}\,\,B_{2}]^{T}$

and

$M^{AA^{\prime}}_{2}(\eta(\lambda),\theta):=\left(\begin{array}[]{cccccc}-T\eta-\alpha_{a}&\bar{F}&0&t_{0}^{2}\\\
F&-T\eta-\alpha_{b}&t_{0}^{2}&0\\\ 0&t_{0}^{2}&-T\eta-\alpha_{a}&\bar{F}\\\
t_{0}^{2}&0&F&-T\eta-\alpha_{b}\end{array}\right).$

The determinant of the matrix $M^{AA^{\prime}}_{2}(\eta(\lambda),\theta)$ is
given by

$\begin{array}[]{lll}\det(M^{\\!AA^{\prime}}_{2}(\eta(\lambda),\theta)&\\!\\!=\\!\\!&\Big{(}T^{2}\eta^{2}+(\alpha_{a}+\alpha_{b})T\eta+\alpha_{a}\alpha_{b}-F^{2}-2t_{0}^{2}\mathcal{Re}(F)-t_{0}^{4}\Big{)}\\\
&\cdot\\!\\!\\!&\\!\\!\\!\Big{(}T^{2}\eta^{2}+(\alpha_{a}+\alpha_{b})T\eta+\alpha_{a}\alpha_{b}-F^{2}+2t_{0}^{2}\mathcal{Re}(F)-t_{0}^{4}\Big{)}\end{array}\,,$

where $\Re(F)=1+\cos(\theta_{1})+\cos(\theta_{2})$ the real part of the
complex number $F=F(\theta_{1},\theta_{2})$. The roots of
$\det(M^{\\!AA^{\prime}}_{2})$ are

$r^{\pm}(F)=\frac{-\alpha_{a}-\alpha_{b}\pm\sqrt{4F^{2}-8t_{0}^{2}\Re(F)+4t_{0}^{4}+(\alpha_{a}-\alpha_{{}_{b}})^{2}}}{2T}$
(3.8)

and

$r_{\mp}(F)=\frac{-\alpha_{a}-\alpha_{b}\mp\sqrt{4F^{2}+8t_{0}^{2}\Re(F)+4t_{0}^{4}+(\alpha_{a}-\alpha_{b})^{2}}}{2T}\,.$
(3.9)

With this, we see that there is $\theta\in\mathcal{B}$ such that
$\det(M_{2}^{\\!AA^{\prime}}(\eta(\lambda),\theta)=0$; that is, the
representation (3.6) solves the eigenvalue problem (3.4) and, therefore, by
the Proposition 1.2.1 and [25], we conclude
$\lambda\in\sigma(H_{2}^{\\!AA^{\prime}})$.

In the following section, we will analyze these functions for the existence of
conical or parabolic touches in the operator dispersion relation
$H^{\\!AA^{\prime}}_{2}$. Note that we have abandoned the notation $r(\theta)$
in which the roots explicitly depend on $\theta$, and we started writing the
roots with the new notation $r(F)$ in which dependence on $\theta$ is
implicit, since $F=F(\theta)$. This change facilitates the analysis of the
dispersion relation, as well as the construction of its graphs, as we will see
later.

###  3.2 Dispersion relationship analysis

In this section we analyze the roots found in piles of the type $AA^{\prime}$
of two consecutive sheets. In a similar way to Chapter 1, from now on, when
nothing is said, we will always consider $\theta\in
B_{d}\\!=\\!\\{(\theta_{1},\theta_{2})\\!\in\\![-\pi,\pi]^{2}\\!:\\!\theta_{2}\\!=\\!-\theta_{1}\\}$,
that is, we restrict the quasimoment $\theta$ to the diagonal of the Brillouin
zone $B$. With this, $F$ becomes real, that is,and the roots in (3.8) and
(3.9) can be written in the form

$r^{\pm}_{\mp}(F)=\frac{-\alpha_{a}-\alpha_{b}\pm\sqrt{4(F\mp
t_{0}^{2})^{2}+(\alpha_{a}-\alpha_{b})^{2}}}{2(3+t_{0}^{2})}\,\,.$ (3.10)

Let us remember that the case $\alpha_{b}=\alpha_{a}=0$ reduces to what is
discussed in [32], with two sheets of graphene. So, let us initially assume
that $\alpha_{b}=\alpha_{a}\neq 0$; then in (3.10), we get

$r^{\pm}_{\mp}(F)=\dfrac{-\alpha_{b}\pm\left|F\mp
t_{0}^{2}\right|}{3+t_{0}^{2}}\,,$ (3.11)

with $t_{0}\in(0,1)$ fixed.

With this, we see that the analysis remains analogous to the study carried out
in Chapter 2, in case of stacking $AA$, just as the results and demonstrations
are easily adapted from the Lemma 2.2.1 and of the Theorem 2.2.1. We then
reach the following conclusion about the $AA^{\prime}$ type stacking with two
atoms alternating at the vertices.

######  Theorem 3.2.1.

(Operator $H^{AA^{\prime}}_{2}$ dispersion relation) Consider $t_{0}\in(0,1)$
fixed and $r^{\pm}_{\pm}$ as in (3.10). Then

  1. (a)

If $\alpha_{a}=\alpha_{b}\neq 0$, the operator dispersion relation
$H^{AA^{\prime}}_{2}$ always has Dirac cones in 111That is, when
$\theta_{D}=\pm\left(\arccos\big{(}\frac{t_{0}^{2}-1}{2}\big{)},-\arccos\big{(}\frac{t_{0}^{2}-1}{2}\big{)}\right)$,
there is $F_{D}:=F(\theta_{D})=\pm\left(t_{0}^{2},-t_{0}^{2}\right)$.
$F_{D}=\pm\left(t_{0}^{2},-t_{0}^{2}\right)$ (Figure 3.3).

  2. (b)

For $\alpha_{a}\neq\alpha_{b}$, the dispersion relation of
$H^{AA^{\prime}}_{2}$ it has no touch, that is, it always has gaps (Figure
3.4).

Figure 3.3: Two Dirac cones when the atoms are equal;
$\,\alpha_{a}=\alpha_{b}=-1\,,t_{0}=0.3$. Figure 3.4: Gaps if atoms are
distinct; $\alpha_{b}=1,\,\alpha_{a}=-1\,,t_{0}=0.3$.

In the following section, we summarize these results.

###  3.3 Analysis and conclusions

In the calculations and analyzes carried out so far, any material formed by
hexagons with two atoms interspersed at the vertices was considered and
modeled with a quantum graph. However, item (b) of the theorem above tells us
that if, in particular, we apply these results to boron nitride, since the
boron and nitrogen atoms are distinct, we obtain the following conclusion:

If $\alpha_{a}$ and $\alpha_{b}$ represent the boundary conditions associated
with nitrogen and boron, respectively, we conclude that the dispersion
relation of the $hBN$ operator, in $AA^{\prime}$ stacking, has no touch, or
even that hexagonal boron nitride (bilayer) always has gaps.

This is in agreement with the results found in the experimental physics
literature [13]; A material that has gaps between its valence bands is said to
be an insulator. In particular, boron nitride is an insulator, which is a
well-known fact, corroborating the above results for two layers in this
configuration.

######  Observation 3.3.1.

There is a small and interesting distinction between the results presented by
$AA$ and $AA^{\prime}$ models when considering $hBN$ bilayers.

  1. 1.

The $AA^{\prime}$ model eliminates the parabolic touches that occur in
stacking $AA$.

  2. 2.

If $\alpha_{b}=\alpha_{a}$, the $AA^{\prime}$ model horizontally displaces the
Dirac cones of $AA$ by $t_{0}^{2}$.

  3. 3.

While in $AA$ stacking we found a gap of 0.589, in $AA^{\prime}$ the gap
(where the parabolic touch occurred) is 0.64987 and (where the conical touch
occurred) is 0.64725, as shown in Figures 2.5 and 3.4. Finally, we note that
we did not find experimental studies in the physical literature considering
two layers of hBN stacked in the type $AA$, a fact that we makes it impossible
to make more precise analyzes and comparisons of our study with others.

Note also that, although $AA$ stacking makes sense because it is a
mathematical model, such a structure does not describe the hexagonal boron
nitride material seen as a solid composed of n-layers, as an essential
characteristic of this material is that the nitrogen interacts with the boron
atom and vice versa, both in the strong interaction in the same layer and in
the van der Walls interaction, between the sheets [13], [19]. However,
modeling with $AA^{\prime}$ stacking presents a better proposal for describing
hBN, as each boron atom interacts with a nitrogen atom and vice versa, whether
in a weak or strong interaction, so that the property of the dispersion
relationship of a monolayer in having holes is preserved in type $AA^{\prime}$
stacking for bilayers, a fact corroborated by [13].

######  Observation 3.3.2.

An analogous calculation (not presented here) considering three sheets, that
is, adding another layer of the same material to the $AA^{\prime}$ stack of
two sheets, shows us that the dispersion relation obtained always has gaps for
$\alpha_{b}=-\alpha_{a}$.

In the next chapter, we will address stacks with two and three distinct two-
dimensional layers, that is, cases of heterostructures.

##  Chapter
4
Heterostructures: boron nitride and graphene

Now we will use quantum graphs to model systems formed by some combined layers
of graphene and boron nitride, particularly the one studied in [19], that is,
some heterostructures with two and three layers. In the aforementioned
article, the authors considered a heterostructure with a graphene layer on (a
block of) boron nitride, so that the entire analysis and conclusion are
carried out considering only the interaction between the graphene layer and
the boron nitride layer; see Figure 4.1.

Figure 4.1: A two-layer van der Waals heterostructure: side and top views,
respectively.

###  4.1 Distinct bilayers

Initially, we will stick to the two-layer heterostructure: on the bottom
sheet, we fix a layer formed by hexagons with two atoms (mathematically
represented by the parameters $\delta_{a}$ and $\delta_{b}$ in the Robin (2.1)
boundary conditions ) interspersed at the vertices of each hexagon; in the
upper one, we take a hexagonal sheet with an atom ($\delta_{{}_{C}}$) at each
vertex. In particular, when convenient, we will also use the constants
$\delta_{{}_{B}},\delta_{{}_{N}}$ and $\delta_{{}_{C}}$ to indicate the
boundary condition attributed to the vertices, modeling with boron, nitrogen
and carbon atoms respectively.

Figure 4.2: Fundamental domain $W$ of a heterostructure bilayer formed by 8
edges and 4 vertices.

As in the previous chapter, we considered the fundamental domain given by

$W=\\{a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},f_{1},f_{2},v_{11},v_{12},v_{21},v_{22}\\}\,.$

In a similar way to the previous chapters, we apply Floquet cyclic conditions
and continuity conditions to the fundamental domain $W$ and obtain

$\left\\{\begin{array}[]{lll}u_{a_{11}}(0)&=&u_{a_{12}}(0)=u_{a_{13}}(0)=\dfrac{u_{f_{1}}(0)}{t_{0}}=:A_{1}\\\
u_{a_{11}}(1)&=&e^{i\theta_{1}}u_{a_{12}}(1)=e^{i\theta_{2}}u_{a_{13}}(1)=\dfrac{u_{f_{2}}(1)}{t_{0}}=:B_{1}\\\
u_{a_{21}}(0)&=&u_{a_{22}}(0)=u_{a_{23}}(0)=\dfrac{u_{f_{2}}(0)}{t_{0}}=:A_{2}\\\
u_{a_{21}}(1)&=&e^{i\theta_{1}}u_{a_{22}}(1)=e^{i\theta_{2}}u_{a_{23}}(1)=\dfrac{u_{f_{1}}(1)}{t_{0}}=:B_{2}\end{array}\,.\right.$
(4.1)

Applying (ii) of the modified Neumann Vertex Condition (2.1), we obtain the
total flow equations at each vertex of the fundamental domain:

$\left\\{\begin{array}[]{llllll}u^{\prime}_{11}(0)+u^{\prime}_{12}(0)+u^{\prime}_{13}(0)+t_{0}u^{\prime}_{f_{1}}(0)&=&\delta_{a}A_{1}\\\
-u^{\prime}_{11}(1)-e^{i\theta_{1}}u^{\prime}_{12}(1)-e^{i\theta_{2}}u^{\prime}_{13}(1)-t_{0}u^{\prime}_{f_{2}}(1)&=&\delta_{b}B_{1}\\\
u^{\prime}_{21}(0)+e^{i\theta_{1}}u^{\prime}_{22}(0)+e^{i\theta_{2}}u^{\prime}_{23}(0)+t_{0}u^{\prime}_{f_{1}}(0)&=&\delta_{c}A_{2}\\\
-u^{\prime}_{21}(1)-u^{\prime}_{22}(1)-u^{\prime}_{23}(1)-t_{0}u^{\prime}_{f_{2}}(1)&=&\delta_{c}B_{2}\end{array}\right.\,.$
(4.2)

We also consider, for $\lambda\notin\sigma(H^{D})$, the functions
$\varphi_{0}=\varphi_{\lambda,0}$, $\varphi_{1}=\varphi_{\lambda,1}$ and we
write the linear combination

$\left\\{\begin{array}[]{llllll}u_{a_{k1}}&=&A_{k}\varphi_{\lambda,0}+B_{k}\varphi_{\lambda,1}\\\
u_{a_{k2}}&=&A_{k}\varphi_{\lambda,0}+e^{-i\theta_{1}}B_{k}\varphi_{\lambda,1}\\\
u_{a_{k3}}&=&A_{k}\varphi_{\lambda,0}+e^{-i\theta_{2}}B_{k}\varphi_{\lambda,1}\\\
u_{f_{1}}&=&t_{0}(A_{1}\varphi_{\lambda,0}+B_{2}\varphi_{\lambda,1})\\\
u_{f_{2}}&=&t_{0}(A_{2}\varphi_{\lambda,0}+B_{1}\varphi_{\lambda,1})\end{array},\quad
k=1,2\right.\,\,;$ (4.3)

with the same notations as in previous chapters and the appropriate
substitutions, we obtain the system

$M^{ab}_{c}(\theta)X=0,\quad\mbox{com}\quad
X=[A_{1}\,\,B_{1}\,\,A_{2}\,\,B_{2}]^{T}$

with

$M^{ab}_{c}(\theta)=\left(\begin{array}[]{cccccc}-T_{1}\eta-\alpha_{a}&\bar{F}&0&t_{0}^{2}\\\
F&-T_{1}\eta-\alpha_{b}&t_{0}^{2}&0\\\
0&t_{0}^{2}&-T_{1}\eta-\alpha_{c}&\bar{F}\\\
t_{0}^{2}&0&F&-T_{1}\eta-\alpha_{c}\end{array}\right)\,.$

Let’s assume that $\alpha_{c}=\alpha_{C}=0$, as admitted in [9]. Having done
this, the determinant $\det(M^{ab}_{c}(\theta))=:P(\eta)$ is given by

$\begin{array}[]{lll}\\!P(\eta)\\!=\\!\eta^{4}T^{4}\\!+\\!(\\!\alpha_{a}\\!+\\!\alpha_{b})T^{3}\eta^{3}\\!+\\!\left(\\!\alpha_{a}\alpha_{b}\\!-\\!2F^{2}\\!-\\!2t_{0}^{4}\right)T^{2}\eta^{2}\\!-\\!(\\!\alpha_{a}\\!+\\!\alpha_{b})(\\!F^{2}\\!+\\!t^{4})T\eta\\!+\\!(\\!F^{2}\\!-\\!t_{0}^{4})^{2}\\!-\\!\alpha_{a}\alpha_{b}F^{2}\end{array}$

In the subsequent section, we will analyze the behavior of the roots of $P$.
Although we do not make it explicit, we highlight that when analyzing the
functions $\eta$ as a dispersion relation $\lambda$ of the operator, in fact,
we are considering the observation (1.2.2) from Chapter 1, in which we saw
that it is necessary to require that $|\eta(\lambda)|\leq 1$ in order to
relate the spectra $\sigma(H)$ and $\sigma(H^{\mathrm{per}})$. That said, we
will always be considering, even if implicitly, only the cases in which
$r(F(\theta))\leq 1$ in our analysis of the dispersion relation.

####  4.1.1 Analysis of the dispersion relation.

Initially, we highlight that since $P$ is a polynomial of degree 4, there is
enormous difficulty in displaying the general form of the zeros of $P$, so
that the general expression of its roots, that is, with arbitrary values of
$t_{0}$, $\alpha_{a}$ and $\alpha_{b},F$ are extremely extensive and
intractable. However, considering the case111Such a choice is partially
justifiable, since it has some physical motivation, because thinking about the
parameters $\alpha$s as electronegativity, we would have
$\alpha_{{}_{B}}<\alpha_{{}_{C}}<\alpha_{{}_{N}}$, and with the choice
$\alpha_{{}_{C}}=0$, then $\alpha_{{}_{N}}>0$ and
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}$ becomes reasonable.
$\alpha_{b}=-\alpha_{a}\neq 0$, the roots of $P(\eta)$ become simpler and are
given by

$r^{\pm}(F)=\pm\dfrac{\sqrt{2F^{2}+2t_{0}^{2}+\alpha_{a}^{2}+\sqrt{16t_{0}^{4}F^{2}+4\alpha_{a}^{2}t_{0}^{4}+\alpha_{a}^{4}}}}{\sqrt{2}(3+t_{0}^{2})}$
(4.4)

and

$r_{\mp}(F)=\mp\dfrac{\sqrt{2F^{2}+2t_{0}^{2}+\alpha_{a}^{2}-\sqrt{16t_{0}^{4}F^{2}+4\alpha_{a}^{2}t_{0}^{4}+\alpha_{a}^{4}}}}{\sqrt{2}(3+t_{0}^{2})}\,.$
(4.5)

Figure 4.3: Gaps in the hBN-graphene dispersion relation;
$t_{0}=0.3\,,\alpha_{a}=-1$.

Note that, as shown in Figure 4.3, no conical ringing occurs at the zero
energy level, so a spectral gap is induced by the inclusion of the boron
nitride layer on top of the graphene layer. This is in complete agreement with
the results found in laboratory experiments and published in the literature,
namely, in [13] and [19]. In this type of stacking with bilayers of different
materials, graphene carbon atoms, which until then only interacted with other
equivalent carbon atoms (and always generated conical touches at the origin),
began to have weak bonds with inequivalent atoms (boron and nitrogen) from the
upper layer. This inequivalence of the bonds caused the rupture of the Dirac
cones and the appearance of gaps at the origin. Furthermore, in (4.5) we can
determine the width of the smallest gap (that is, at the origin) formed as a
function of the parameters $t_{0}$ and $\alpha_{a}$, which we will denote by
$g$, and given by

$g(\alpha_{a},t_{0})=\big{|}r_{+}(0)-r_{-}(0)\big{|}=\sqrt{2}\,\frac{\sqrt{2\,t_{0}^{4}+\alpha_{a}^{2}-\sqrt{4\,\alpha_{a}^{2}\,t_{0}^{4}+\alpha_{a}^{4}}}}{3+t_{0}^{2}}\,.$
(4.6)

In particular, for $t_{0}=0.3$ and $\alpha_{a}=-1$, we obtain, at the origin,
a gap of at least $g(-1,0.3)\approx 0.0052$, illustrated in Figure 4.3. In
general, for each fixed $\alpha_{a}$, when $t_{0}\to 0^{+}$, the gap decreases
so that $\displaystyle\lim_{t_{0}\to 0^{+}}g(\alpha_{a},t_{0})=0$; and as
$t_{0}\to 1^{-}$, the gap between the valence and conduction bands increases,
as shown in figure 4.4, that is,

$g(\alpha_{a},1)=\dfrac{\sqrt{2}}{4}\sqrt{2+\alpha_{a}^{2}-\sqrt{\alpha_{a}^{4}+4\alpha_{a}^{2}}}\,.$

Figure 4.4: Greater interaction between layers ($t_{0}=1)$ causes an increase
in the gap; $\alpha_{a}=-1$.

From a physical point of view, increasing $t_{0}$ means increasing interaction
between layers. In particular, as mentioned at the beginning of the work, when
$t_{0}=1$ the weak and strong interactions coincide.

Note that, in (4.4), for any nonzero values of $t_{0},\alpha_{a},\alpha_{b}$,
$r^{+}$ is always positive and $r^{-}$ is always negative, so that the two
never meet. null and therefore do not intersect, as all parameters
$t_{0},\alpha_{a},\alpha_{b}$ are not null.

######  Theorem 4.1.1 (Conclusion on hBN-graphene stacking).

Let $\alpha_{{}_{N}},\alpha_{{}_{B}}$ be distinct and non-null, and
$0<t_{0}\leq 1$ be fixed. So the dispersion relation obtained from hBN-
graphene stacking always has holes at $F=0$ for
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}$.

In the next section, we extend the case discussed in this one and consider two
new cases of physical interest: the “sandwiches” formed with these two
hexagonal materials. In both cases, we will apply the modified Neumann
conditions, the cyclic Floquet conditions and the equations of linear
combinations of the solutions, which are easily adapted and applied to the
case of structures with three layers (although with much longer calculations),
so To avoid repetition, we will omit the calculations and display only the
most essential details for the analysis.

Let us initially remember that, seeking to bring our model closer to reality,
it is reasonable to choose certain values for the parameters associated with
the atoms in boron nitride, which we will denote by $\alpha_{{}_{N}}$ for the
nitrogen atom and $\alpha_{{}_{B}}$ for boron.

###  4.2 Distinct trilayers

As briefly presented in the Introduction, the configuration formed by three
distinct layers, so that the middle layer differs from the two similar outer
layers, has aroused interest as it is a promising proposal that would solve
the problem of graphene not being able to be used as a type of transistor

We address in the two subsections 222We highlight here that, although the
stacking is of three layers, in our model we consider interactions only
between consecutive layers. following two types of “sandwiches”: graphene
sheet between hBN, and hBN between graphene.

####  4.2.1 The hBN-Graphene-hBN case

In this subsection we consider the “sandwich” formed by a monolayer of
graphene between two sheets of hexagonal boron nitride, as illustrated in
Figure 4.5.

Figure 4.5: The hBN-Graphene-hBN sandwich: side and top views, respectively.

There is great interest in this type of configuration, as according to the
current wide discussion, the aim is to know what new properties will be
obtained by including a layer of graphene (an excellent conductor) between two
layers of boron nitride (an excellent insulating). In other words,
mathematically we will analyze the behavior of the dispersion relation
obtained from this configuration regarding the presence of Dirac cones,
parabolic touches or gaps between the valence and conduction bands. Let us
remember that denoting by $\alpha_{{}_{N}},\alpha_{{}_{B}}$ and
$\alpha_{{}_{C}}$ the constants associated with the boundary conditions of
nitrogen, boron and carbon, respectively, the matrix $M_{{BNGBN}}$ associated
with this model, can be obtained in an entirely analogous way to those in the
previous chapters (although in a more laborious way), and is given by

$M_{{BNGBN}}\\!\\!=\\!\\!\left(\\!\\!\\!\\!\begin{array}[]{cccccc}-T_{1}\eta-\alpha_{{}_{N}}&\bar{F}&0&t_{0}^{2}&0&0\\\
F&-T_{1}\eta-\alpha_{{}_{B}}&t_{0}^{2}&0&0&0\\\
0&t_{0}^{2}&-T_{2}\eta-\alpha_{{}_{C}}&\bar{F}&0&t_{0}^{2}\\\
t_{0}^{2}&0&F&-T_{2}\eta-\alpha_{{}_{C}}&t_{0}^{2}&0\\\
0&0&0&t_{0}^{2}&-T_{1}\eta-\alpha_{{}_{N}}&\bar{F}\\\
0&0&t_{0}^{2}&0&F&-T_{1}\eta-\alpha_{{}_{B}}\end{array}\\!\\!\\!\\!\right)$

with $T_{1}:=3+t_{0}^{2}$ e $T_{2}:=3+2t_{0}^{2}$.

For simplicity, taking the constant associated with the carbon atom as null,
that is, $\alpha_{{}_{C}}=0$, as admitted in [9], we obtain
$\det(M_{{BN.G.BN}})(\eta)=:P(\eta)=P_{2}(\eta)P_{N}(\eta)$, where

$P_{2}(\eta)=(3+t_{0}^{2})\eta^{2}+(\alpha_{{}_{N}}+\alpha_{{}_{B}})(3+t_{0}^{2})\eta+\alpha_{{}_{N}}\alpha_{{}_{B}}-F^{2}$

and $P_{N}$ a polynomial of degree 4 in $\eta$. The roots of $P_{2}$ are given
by

$r_{\pm}(F)=\dfrac{-\alpha_{{}_{B}}-\alpha_{{}_{N}}\pm\sqrt{(\alpha_{{}_{B}}-\alpha_{{}_{N}})^{2}+4F^{2}}}{2(3+t_{0}^{2})}\,,$
(4.7)

which clearly have gaps for any distinct values of $\alpha_{{}_{B}}$ and
$\alpha_{{}_{N}}$, but have a Dirac cone 333The case
$\alpha_{{}_{N}}=\alpha_{{}_{B}}$ does not represents the hexagonal boron
nitride material, since in hBN we assume that the atoms are distinct. when
$\alpha_{{}_{B}}=\alpha_{{}_{N}}$ (see Figures 4.6, 4.7 and 4.8). As already
highlighted in Chapter 3, Observation 3.3.1, in configurations like $AA$ or
$AA^{\prime}$, with two or three sheets of boron nitride, there are always
gaps for $\alpha_{{}_{B}}=-\alpha_{{}_{N}}\neq 0$, however the inclusion of a
graphene sheet between two boron nitride sheets, although it did not eliminate
the gap, induced a reduction in the width of the spectral gap of $\approx
0.65$ to $\approx 0.07$, that is, an order of magnitude; certainly this is the
main property of this configuration.

The roots of $P_{N}$ are given by extremely long expressions so that their
general form becomes intractable. However, taking
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}$, we were able to carry out some analyzes so
that we obtained important and interesting results, that is

$P_{N}(\eta)=A(t_{0},F,\alpha_{{}_{N}})\eta^{4}+B(t_{0},F,\alpha_{{}_{N}})\eta^{2}+C(t_{0},F,\alpha_{{}_{N}})\,,$

being

$A(t_{0},F,\alpha_{{}_{N}})=(3+t_{0}^{2})^{2}(3+2t_{0}^{2})^{2}\,,$
$B(t_{0},F,\alpha_{{}_{N}})=-\left[(5t_{0}^{4}+18t_{0}^{2}+18)F^{2}+8t_{0}^{8}+36t_{0}^{6}+(4\alpha_{{}_{N}}+36)t_{0}^{4}+12\alpha_{{}_{N}}^{2}t_{0}^{2}+9\alpha_{{}_{N}}^{2}\right]$

and

$C(t_{0},F,\alpha_{{}_{N}})=-(F^{2}-2t_{0}^{4})^{2}-\alpha_{{}_{N}}^{2}F^{2}\,,$

so that, in this case, the roots of $P_{N}$ are given by

$r^{\mp}_{\pm}(F)=\mp\sqrt{\dfrac{-B(t_{0},F,\alpha_{{}_{N}})\pm\sqrt{B^{2}(t_{0},F,\alpha_{{}_{N}})-4A(t_{0},F,\alpha_{{}_{N}})C(t_{0},F,\alpha_{{}_{N}})}}{2A(t_{0},F,\alpha_{{}_{N}})}}$
(4.8)

Figure 4.6: Gaps at origin for $t_{0}=0.3$,
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}=1\,,\alpha_{{}_{C}}=0$. The inclusion of a
graphene sheet between two hBN sheets does not eliminate the gap at $F=0$, but
causes a reduction in the gap width from $\approx 0.65$ in two hBN layers to
$\approx 0.074$. Figure 4.7: Gaps in $F\\!=\\!0$ from $\approx 0.00483$ (the
smallest) for $t_{0}\\!=\\!0.3$,
$\alpha_{{}_{B}}\\!=\\!-\alpha_{{}_{N}}\\!=\\!0.1\,,\alpha_{{}_{C}}\\!=\\!0$.
Figure 4.8: Gaps in $F=0$ of $\approx\\!0.00324$, $\approx\\!0.00647$ (smaller
auxiliary figure on the side) and $\approx\\!0.04497$ for $t_{0}=0.3$,
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.01\,,\alpha_{{}_{C}}=0$.

######  Observation 4.2.1.

Although not presented here, we analyzed stacks of the $AA$ or $AA^{\prime}$
type, with two or three boron nitride sheets, and found that there are always
gaps for $\alpha_{{}_{B}}=-\alpha_{{}_{N}}\neq 0$. Let us now note that,
despite the particularities in the values $\alpha_{{}_{B}}$ and
$\alpha_{{}_{N}}$, our model allows us to infer some important and interesting
conclusions; namely, the inclusion of a graphene sheet between two hBN sheets
does not induce conical or parabolic touches, that is, it does not eliminate
the gap at $F=0$, however, it causes a reduction of an order of magnitude in
the width of the opening at two layers in this graph model of hexagonal boron
nitride.

####  4.2.2 The Graphene-hBN-Graphene case

Let us now consider another type of “sandwich”, which also arouses physical
interest and is formed by a hexagonal boron nitride sheet between two layers
of graphene, as shown in Figure 4.9.

Figure 4.9: Side and top views, respectively, of the Graphene-hBN-Graphene
sandwich.

In this stacking mode, the aim is to analyze the disturbance caused by the
inclusion of a boron nitride monolayer, an insulating material whose
dispersion relationship always has gaps, between graphene bilayers 444Remember
that the dispersion relationship obtained in stacking type $AA$ (or
$AA^{\prime}$) of three layers of graphene always has conical touches [32]. –
which we already know is an excellent conductor and whose dispersion relation
has Dirac cones.

The matrix $M_{{G.BN.G}}$ obtained from this stacking is given by

$M_{{G.BN.G}}(\eta)\\!=\\!\left(\begin{array}[]{cccccc}\\!\\!\\!-T_{1}\eta-\alpha_{{}_{C}}\\!&\\!\bar{F}\\!&\\!0&t_{0}^{2}\\!&\\!0\\!&\\!0\\\
F\\!&\\!-T_{1}\eta-\alpha_{{}_{C}}\\!&\\!t_{0}^{2}\\!&\\!0\\!&\\!0\\!&\\!0\\\
0\\!&\\!t_{0}^{2}\\!&\\!-T_{2}\eta-\alpha_{{}_{N}}\\!&\\!\bar{F}\\!&\\!0\\!&\\!t_{0}^{2}\\\
t_{0}^{2}\\!&\\!0\\!&\\!F\\!&\\!-T_{2}\eta-\alpha_{{}_{B}}\\!&\\!t_{0}^{2}\\!&\\!0\\\
0\\!&\\!0\\!&\\!0\\!&\\!t_{0}^{2}\\!&\\!-T_{1}\eta-\alpha_{{}_{C}}\\!&\\!\bar{F}\\\
0\\!&\\!0\\!&\\!t_{0}^{2}\\!&\\!0\\!&\\!F\\!&\\!-T_{1}\eta-\alpha_{{}_{C}}\\!\\!\\!\end{array}\right)$

with $T_{1}:=3+t_{0}^{2}$ e $T_{2}:=3+2t_{0}^{2}$.

Again we consider $\alpha_{{}_{C}}=0$, we obtain
$\det(M_{{G.BN.G}})(\eta)=:P(\eta)=P_{2}(\eta)P_{N}(\eta)$, being

$P_{2}(\eta)=(3+t_{0}^{2})\eta^{2}-F^{2}$

and $P_{N}$ a polynomial of degree 4 in $\eta$. The roots of $P_{2}$ are given
by

$r_{\pm}(F)=\pm\dfrac{|F|}{3+t_{0}^{2}}$ (4.9)

which clearly always have Dirac cones at the origin regardless of the values
of $\alpha_{{}_{B}}$ and $\alpha_{{}_{N}}$. Remember that configurations of
two and three graphene layers always have conical touches at the origin when
stacked in the $AA$ (or $AA^{\prime}$) form, but the inclusion of an hBN sheet
between two graphene sheets induced gaps in four of the five Dirac cones (see
below).

As in the previous subsection, the roots of $P_{N}$ are very long and its
general form becomes intractable. Again let us take
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}$, and we obtain

$P_{N}(\eta)=A(t_{0})\eta^{4}+B(t_{0},F,\alpha_{{}_{N}})\eta^{2}+C(t_{0},F,\alpha_{{}_{N}})\,,$

being

$A(t_{0})=(3+t_{0}^{2})^{2}(3+2t_{0}^{2})^{2}\,,$
$B(t_{0},\alpha_{{}_{N}},F)=-\left[(5t_{0}^{4}+18t_{0}^{2}+18)F^{2}+8t_{0}^{8}+36t_{0}^{6}+(\alpha_{{}_{N}}+36)t_{0}^{4}+6\alpha_{{}_{N}}^{2}t_{0}^{2}+9\alpha_{{}_{N}}^{2}\right]$

and

$C(t_{0},\alpha_{{}_{N}},F)=(F^{2}-2t_{0}^{4})^{2}+\alpha_{{}_{N}}^{2}F^{2}\,,$

so that, for $\alpha_{{}_{B}}=-\alpha_{{}_{N}}$, the roots of $P_{N}$ are
given by

$r^{\mp}_{\pm}(F)=\mp\sqrt{\dfrac{-B(t_{0},F,\alpha_{{}_{N}})\pm\sqrt{B^{2}(t_{0},F,\alpha_{{}_{N}})-4A(t_{0},F,\alpha_{{}_{N}})C(t_{0},F,\alpha_{{}_{N}})}}{2A(t_{0},F,\alpha_{{}_{N}})}}\,.$
(4.10)

Independently of the values of $\alpha_{{}_{B}}$ and $\alpha_{{}_{N}}$, two
roots in the graphene-hBN-graphene case always cause cones; more precisely,
the graphene cone prevails in this sandwich.

Figure 4.10: Presence of five iconic touches in the interaction of equivalent
atoms, that is, $\alpha_{{}_{B}}=\alpha_{{}_{N}}=0=\alpha_{{}_{C}}$ ,
$t_{0}=0.3$.  Figure 4.11: $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=1$ and
$t_{0}=0.3$. Gap formation in four of the five cones due to the interaction of
carbon atoms with other inequivalents of hBN, the smallest of which is
$\approx 0.07684$. A graphene cone always prevails in this type of sandwich
for any values of the parameters $\alpha_{{}_{B}},\alpha_{{}_{N}}$. Figure
4.12: $\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.1$ and $t_{0}=0.3$. Prevalence of
the graphene cone: red and green curves. Figure 4.13:
$\alpha_{{}_{B}}=-\alpha_{{}_{N}}=0.01$ and $t_{0}=0.3$. The graphene cone is
always maintained, while gaps are caused at other points, the smallest of
which (smaller auxiliary figure on the side) is $\approx 0.00314$.

######  Observation 4.2.2.

As in the previous subsection, we highlight an interesting fact here: the
inclusion of hBN monolayer between graphene bilayers (which always have
conical touches) induced gaps in some graphene Dirac points (bilayers),
however a Dirac cone at the origin is always kept in this sandwich, regardless
of the non-zero values of $\alpha_{{}_{B}}$ and $\alpha_{{}_{N}}$.

###  4.3 Analysis and conclusions

As mentioned in the Introduction of this work, the authors Giovannetti et al.
[19] carried out numerical simulations, using “ab initio” density functional
calculations, considering bilayers in the form of van der Walls
heterostructures: one made of boron nitride and the other made of graphene.
The energies were found as a function of the distance between the graphene
sheet and the hBN sheet, as well as gaps between the conduction and valence
bands (i.e., without Dirac points); it is interpreted that the boron nitride
sheets induced this gap in the graphene. The authors conclude that the
appearance of these gaps offers the potential to assemble electronic systems
based on these configurations.

Furthermore, as demonstrated in [9], and inferred as a particular case of two
or three layers of boron nitride analyzed here, two or three layers of
graphene stacked in the form $AA$ do not have gaps, so that in $F=0$ (that is,
in
$(\theta_{1},\theta_{2})=\big{(}\frac{2\pi}{3},\frac{-2\pi}{3}\big{)}\Big{)}$
there are always conical touches. However, in our study, in addition to the
presence of Dirac cones in the heterostructure of graphene-hBN bilayers, we
also identified conical touches or gaps in trilayers arranged in a sandwich
shape: graphene between boron nitride, and boron nitride between graphene.
More precisely, our work allows us to conclude that the inclusion of a boron
nitride layer between two graphene layers induces a gap at some Dirac points,
although a conical touch is always maintained in this stacking, regardless of
the parameters $\alpha_{a}$ and $\alpha_{b}$. This allows us to conclude that
in any sandwich with a hexagonal material between two sheets of graphene there
will always remain at least one Dirac cone. On the other hand, the inclusion
of a graphene sheet between two boron nitride sheets induced a reduction in
the gaps formed by hBN bilayers.

##  Chapter
5
Hexagonal monolayers with magnetic field

The electronic properties of graphene with a magnetic field have also
attracted strong interest. In the mathematical articles [2, 3], quantum graphs
are considered as a model of graphene with a magnetic field and a complete
analysis of the spectrum is provided, for any constant magnetic flux. The work
[3] demonstrates that, considering graphene, if the magnetic flux $\phi$ is a
rational multiple of $2\pi$, then the continuous spectrum is absolutely
continuous and, complemented by [2], showing that there are Dirac cones in
these cases; and if the flux $\phi$ is an irrational multiple of $2\pi$, the
spectrum is a Cantor set with zero Lebesgue measure [3]. At the vertices, we
always start from the Kirchhoff boundary conditions (i.e., $\alpha=0$ at all
vertices).

We will use quantum graphs in a proposal to model hexagonal boron nitride
under the effect of a magnetic field, through the Schrödinger operator, and
demonstrate that if the magnetic flux $\phi$ is constant in the network and of
the form $\phi=2\pi\dfrac{p}{q}$ (rational multiple of $2\pi$), then there are
integer values $p$ and $q$ such that, for certain boundary conditions at the
vertices (starting from the Robin condition with different parameter values,
our proposal to model boron nitride), the conic touches in the operator
dispersion relation cease to exist so that we guarantee the existence of gaps.

First, however, we present some necessary definitions and results; we will
start in the following section with a very succinct review of known results,
and more details can be found in [2, 3, 29, 30].

###  5.1 Magnetic potential

Consider a graph $G$. For an edge $e\in\mathcal{E}$ we denote by
$i(e)\in\mathcal{V}$ its initial vertex, and $t(e)\in\mathcal{V}$ its terminal
vertex. Let us remember that each edge $e$ of $G$ can be identified with a
copy of the segment $[0,1]$, such that 0 is identified with the vertex $i(e)$,
1 with the vertex $t(e)$, as constructed in Chapter 1.

In this context, we will approach models of boron nitride layers (and
consequently graphene) through quantum graphs considering a magnetic field in
the modeling through the Schrödinger operator; on each edge $e$ , let us take
$a_{e}\in C^{1}[0,1]$ and associate, to each $e$, the operator

$L_{e}:=(i\partial+a_{e})^{2}+U\,;$

we define the maximal operator $(g_{e})\mapsto(L_{e}g_{e})$ in the functions
$g=(g_{e})\in\bigoplus_{e}{\mathcal{H}}^{2}[0,1]$, so that, introducing the
boundary conditions at the vertices, we obtain a self-adjoint operator:

$\left\\{\begin{array}[]{ll}g_{a}(v)=g_{b}(v)=:g(v)\,,\forall\,\,a,b\in\mathcal{E}_{v}\,;\\\
\displaystyle\sum_{e;i(e)=v}\\!\\!\big{(}g^{\prime}_{e}(0)-ia_{e}(0)g_{e}(0)\big{)}-\\!\\!\sum_{e;t(e)=v}\\!\\!\big{(}g^{\prime}_{e}(1)-ia_{e}(1)g_{e}(1)\big{)}=\alpha(v)g(v)\,,\,\,v\in\mathcal{V}\end{array},\right.$
(5.1)

where $\alpha(v)\in\mathbb{R}$ is a parameter and $\mathcal{E}_{v}$ is the set
of edges that contain $v$.

More generally, now consider the 1-differential form
$A(\textbf{x})=A_{1}(x_{1},x_{2})dx_{1}+A_{2}(x_{1},x_{2})dx_{2}$ and the
scalar potential $A_{e}\in C^{\infty}(e)$ on the edge $e\in\mathcal{E}(G)$
obtained by calculating the form $A$ in the graph along the vector field
generated by edge $[e]\in\mathcal{E}(G)$:

$A_{e}(\textbf{x})=A(\textbf{x})\cdot([e]_{1}\partial_{1}+[e]_{2}\partial_{2})\,.$

The integrated vector potentials are defined by

$\beta_{e}:=\int_{e}A_{e}(\textbf{x})d\textbf{x}\,,\,\,e\in\mathcal{E}(G)\,.$

For the hexagonal lattice, define the magnetic flux by
$\phi=\int_{\hexagon}dA$ and assume that it is always constant in the hexagons
$\hexagon$ of the lattice, that is,

$\beta_{f,\gamma_{1}\gamma_{2}}-\beta_{h,\gamma_{1}\gamma_{2}+1}+\beta_{g,\gamma_{1}\gamma_{2}+1}-\beta_{f,\gamma_{1}-1\gamma_{2}+1}+\beta_{h,\gamma_{1}-1,\gamma_{2}+1}-\beta_{g,\gamma_{1}\gamma_{2}}=\phi\,,\quad\forall\,\,\gamma\in\mathbb{Z}^{2}\,\,.$

We introduce the discrete Hilbert space

$l^{2}(G)=\Big{\\{}f:\mathcal{V}\to\mathbb{C};\|f\|^{2}=\sum_{v\in\mathcal{V}}\operatorname{deg}(v)|f(v)|^{2}<\infty\Big{\\}}\,,$

with the inner product given by

${\left\langle
f,g\right\rangle}_{G}=\sum_{v\in\mathcal{V}}\operatorname{deg}(v)\bar{f}(v)g(v)\,,$

where $\operatorname{deg}(v)=\operatorname{outdeg}(v)+\operatorname{indeg}(v)$
the degree of the vertex $v$, that is, the sum of the number of edges leaving
$v$, $\operatorname{outdeg}(v)$, and the number of edges arriving to $v$,
$\operatorname{indeg}(v)$.

The discrete Laplacian operator $\Delta$ is defined in $l^{2}(G)$

$\Delta
f(v)=\frac{1}{\operatorname{deg}(v)}\\!\left(\\!\sum_{e\,;\,i(e)=v}\\!\\!\\!f(t(e))+\\!\\!\sum_{e\,;\,t(e)=v}\\!\\!\\!f(i(e))\right)\,.$
(5.2)

Let’s now start building the continuous version of the Laplacian in $G$. The
magnetic Schrödinger operator in such a structure can be defined as follows.

Consider the Hilbert space
$\mathcal{H}=\bigoplus_{e\in\mathcal{E}}{\mathcal{H}}_{e}$, with
${\mathcal{H}}_{e}=L^{2}[0,1]$, consisting of functions
$f=(f_{e}),\,f_{e}\in{\mathcal{H}}_{e}$, and define in $\mathcal{H}$ the
operator $H,H(f_{e})=(-f^{\prime\prime}_{e})$ satisfying the boundary
conditions at the vertices

$\left\\{\begin{array}[]{ll}f_{a}(v)=f_{b}(v)\,,\forall\,\,a,b\in\mathcal{E}_{v}\\\
\displaystyle\sum_{e;\,i(e)=v}\\!\\!f^{\prime}_{e}(0)-\\!\\!\sum_{e;\,t(e)=v}\\!\\!f^{\prime}_{e}(1)=0\end{array}.\right.$
(5.3)

The operator $H$ is self-adjoint and its spectrum is closely related to the
spectrum of $\Delta$: denoting by
$\sigma_{D}=\\{(\pi\,n)^{2}:n\in\mathbb{N}\\}$ and, as previously, by
$\sigma(\Delta)$ the spectrum of the operator $\Delta$, we have the relation

$\sigma(H)\backslash\sigma_{D}=\\{z\notin\sigma_{D}:\cos(\sqrt{z})\in\sigma(\Delta)\\}.$
(5.4)

Let $\Omega$ be a Borel set in $\mathbb{R}$ and $A$ be a self-adjoint
operator; denote by $A_{\Omega}$ the part of $A$ in $\Omega$, that is,
$A_{\Omega}=A1_{\Omega}(A)$ considered as an operator on the image
$1_{\Omega}(A)$. Here $1_{\Omega}(A)$ ís the spectral projection of $A$ onto
$\Omega$. In [29], Pankrashkin demonstrates the following result:

######  Proposition 5.1.1.

(Prop. 1, page 641, [29]) For any interval
$J\subset\mathbb{R}\backslash\sigma_{D}$, the operator $H_{J}$ ís unitarily
equivalent to the operator $\mu^{-1}(\Delta_{\mu(J)})$, with
$\mu(z)=\cos(\sqrt{z})$.

Let us now take a real potential $U\in L^{2}[0,1]$ and fix
$\alpha:\mathcal{V}\to\mathbb{R}$. We will say that the symmetry condition is
satisfied if one of the following properties is fulfilled:
$(a)\,\operatorname{indeg}(v)=\operatorname{outdeg}(v),\,\forall\,\,v\in\mathcal{V}$
; $(b)$ $U$ is even, that is, $U(x)=U(1-x)$.

Denote by $H$ the self-adjoint operator in
$\mathcal{H}=\displaystyle\bigoplus_{e\in\mathcal{E}}L^{2}[0,1]$ as

$H(f_{e})=-f^{\prime\prime}_{e}+Uf_{e}\,,$ (5.5)

in the functions
$f=\displaystyle(f_{e})\in\bigoplus_{e\in\mathcal{E}}\mathcal{H}^{2}[0,1]$
($\mathcal{H}^{2}[0,1]$ and the usual Sobolev space in the interval)
fulfilling the boundary conditions

$\left\\{\begin{array}[]{ll}f_{a}(v)=f_{b}(v)\,,\forall\,\,a,b\in\mathcal{E}_{v}\\\
\displaystyle\sum_{e;\,i(e)=v}\\!\\!f^{\prime}_{e}(0)-\\!\\!\sum_{e;\,t(e)=v}\\!\\!f^{\prime}_{e}(1)=\alpha(v)f(v)\,,v\in\mathcal{V}\,.\end{array}\right.$
(5.6)

Therefore, remembering that $\sigma_{D}$ denotes the spectrum of the operator
$f\mapsto-f^{\prime\prime}+Uf$ in $[0,1]$ with Dirichlet boundary conditions,
the Proposition 5.1.1 is a simple corollary of the following result.

######  Theorem 5.1.1.

(Teor. 17, p. 652, [29]) Suppose $H$ defined as above and the potential $U$
even, and that $\alpha(v)=\alpha\operatorname{deg}(v)$, for some
$\alpha\in\mathbb{R}$.

Then, for any interval $J\subset\mathbb{R}\backslash\sigma_{D}$, the operator
$H_{J}$ is unitarily equivalent to
$\mu^{-1}_{\alpha}(\Delta_{\mu_{\alpha}(J)})$, with $\Delta$ being the
operator in $l^{2}(G)$ given by (5.2) and
$\mu_{\alpha}(z)=c(1;z)+\alpha\,s(1;z)$, with $c(1;z),s(1;z)$ fundamental
solutions of the differential equation $-y^{\prime\prime}(t)+U(t)y(t)=zy(t)$
that meet the boundary conditions $s(0;z)=c^{\prime}(0;z)=0$ and
$s^{\prime}(0;z)=c(0;z)=1$.

Note that, in particular, for $U=0$ and $\alpha=0$, we have
$\mu_{0}(z)=\cos(\sqrt{z})$, which gives the Proposition 5.1.1.

These results can be extended to the case in which a magnetic field is
considered, as per the construction in [29], briefly presented below. Consider
now the discrete magnetic Laplacian defined by

$\Delta_{\beta}f(v)=\frac{1}{\operatorname{deg}(v)}\\!\left(\sum_{e\,;\,i(e)=v}\\!\\!\\!e^{-i\beta_{e}}f(t(e))+\\!\\!\\!\sum_{e\,;\,t(e)=v}\\!\\!\\!e^{i\beta_{e}}f(i(e))\right)\,,$
(5.7)

where $\beta:\mathcal{E}\to\mathbb{R}$ is an arbitrary function, which is
self-adjoint in $l^{2}(G)$ and limited, since
$\left\|\Delta_{\beta}\right\|\leq 1.$ Indeed,

$\begin{array}[]{lll}\left\|\Delta_{\beta}f\right\|^{2}_{G}&=&\displaystyle\sum_{v\in\mathcal{V}}\dfrac{1}{\operatorname{deg}(v)}\left|\sum_{e:\,i(e)=v}\\!\\!\\!e^{-i\beta_{e}}f(t(e))+\\!\\!\sum_{e\,;\,t(e)=v}\\!\\!\\!e^{i\beta_{e}}f(i(e))\right|^{2}\\\
&\leq&\displaystyle\sum_{v\in\mathcal{V}}\left(\sum_{e:\,i(e)=v}\\!\\!\\!e^{-i\beta_{e}}|f(t(e))|^{2}+\\!\\!\sum_{e\,;\,t(e)=v}\\!\\!\\!e^{i\beta_{e}}|f(i(e))|^{2}\right)\\\
&=&\displaystyle\sum_{w\in\mathcal{V}}\operatorname{deg}(w)|f(w)|^{2}\\\
&=&\left\|f\right\|^{2}_{G}\,.\end{array}\,.$

Taking a magnetic potential $a_{e}\in C^{1}[0,1]$ in each $e\in\mathcal{E}$,
we associate to each $e$, the operator $\tilde{H}_{e}=(i\partial+a_{e})^{2}+U$
and we define the Schrödinger operator $(g_{e})\mapsto(\tilde{H}_{e}g_{e})$ in
functions $g=(g_{e})\in\bigoplus_{e\,\in\,\mathcal{E}}{\mathcal{H}}^{2}[0,1]$
which models this type of system with a magnetic field Using integration by
parts, we see that $\tilde{H}_{e}$ does not is symmetric, and it is necessary
to introduce boundary conditions at the vertices to obtain a self-adjoint
operator. The standard boundary conditions for these magnetic operators, which
make them self-adjoint, are:

$\left\\{\begin{array}[]{ll}g_{a}(0)=g_{b}(1)=:g(v)\,,\forall\,\,a,b\in\mathcal{E}_{v}\,;\\\
\displaystyle\sum_{e;i(e)=v}\\!\\!\big{(}g^{\prime}_{e}(0)-ia_{e}(0)g_{e}(0)\big{)}-\\!\\!\sum_{e;t(e)=v}\\!\\!\big{(}g^{\prime}_{e}(1)-ia_{e}(1)g_{e}(1)\big{)}=\alpha(v)g(v)\,,\,\,v\in\mathcal{V}\end{array},\right.$
(5.8)

with $\alpha(v)\in\mathbb{R}$ being a parameter.

Applying the gauge transformation $\displaystyle
g_{e}(t)=e^{i\int^{t}_{0}a_{e}(s)ds}f_{e}(t)$ and introducing the parameter
$\displaystyle\beta(e):=\int^{1}_{0}a_{e}(s)ds$, we remove the magnetic
potential $a_{e}$ from $\tilde{H}_{e}$ so that $\tilde{H}_{e}$ is unitarily
equivalent to operator $-f^{\prime\prime}_{e}+Uf_{e}$:

$g_{e}^{-1}\left((i\partial+a_{e})^{2}+U\right)g_{e}=-f^{\prime\prime}_{e}+Uf_{e}\,,$
(5.9)

and now the magnetic potential appears in the new boundary conditions:

$\left\\{\begin{array}[]{ll}f_{a}(0)=e^{i\beta_{e}}f_{b}(1)=:f(v)\,,\forall\,\,a,b\in\mathcal{E}_{v}\,;\\\
f^{\prime}(v)=\displaystyle\sum_{e;i(e)=v}f^{\prime}_{e}(0)-\sum_{e;t(e)=v}e^{\beta_{e}}f^{\prime}_{e}(1)=\alpha(v)f(v)\,,\,\,v\in\mathcal{V}\end{array}\,.\right.$
(5.10)