Edge magnetization in chiral graphene nanoribbons and quantum anomalous Hall effect interfaces in graphene

Table of contents

1 Introduction

2 Theoretical Background
2.1 Crystal Structure: Direct and Reciprocal Lattices
2.2 Band Structure
2.3 Dirac Fermions p.10
2.4 Graphene Nanoribbons: basic aspects
2.5 Magnetic behaviour: Stoner mechanism and Lieb’s theorem

3 Chiral Graphene Nanoribbons
3.1 Geometry of chiral GNR
3.2 Model Hamiltonian
3.3 Non-interacting case (U = 0)

4 Edge Magnetization in Graphene Nanoribbons
4.1 Nanoribbons with zigzag edges
4.2 Magnetization in GNR with chiral edges
4.3 LDOS in chiral graphene nanoribbons

5 Quantum Anomalous Hall Effect in Graphene
5.1 Hall effect
5.2 Model for the QAHE in Graphene
5.2.1 Exchange Field: Zeeman-like term
5.2.2 Spin-Orbit Coupling
5.3 Quantum Anomalous Hall effect in a constant exchange field
5.4 Quantum anomalous Hall effect in a periodic exchange field

6 Conclusion

Appendix A - Berry phase and Berry connection

Appendix B - Spin-orbit and Zeeman interaction
B.1 Spin-Orbit coupling
B.2 The Zeeman effect p.73

References

Abstract

This thesis is composed of two theoretical studies related to properties of edge states in nanostructures of graphene monolayers. In the first one, we analyze the magnetic properties of chiral graphene nanoribbons. Chiral edges corresponds to a symmetry prop­erty whose mirror image cannot be superposed on to the original one, inversely of an achiral [1]. There are only two cases of achiral nanoribbon: armchair or zigzag edges. Chiral graphene nanoribbons, as well as those with zigzag edges, have localized states that favour to edge magnetization. In our analysis we use the tight-binging (TB) model with an electron-electron Hubbard mean-field interaction term. We show that only the standard tight-binding model with nearest-neighbor hopping is not sufficient to describe the low-energy and magnetic properties of graphene nanoribbons, i.e., the inclusion of next-nearest-neighbor hopping terms is necessary for an accurate modeling. We compare the results from our model with recent data from scanning tunneling spectroscopy and propose a new interpretation for the peaks experimentally observed in the local density of states.

The second subject of this thesis corresponds to a study in progress of conducting states from the quantum anomalous Hall effect (QAHE) in graphene in the presence of a periodic exchange field and a Rashba spin-orbit interaction. We call interfaces the point where two systems meet, in this case, a system with positive sign in the exchange field interaction (EF) meet the analogous system with negative sign in the EF. The conducting states appear at the point of meet: gapless interfaces states. To this end, we analyze the formation of gapless states in the change of the sign from exchange field parameter in graphene with Rashba spin-orbit interaction. While the system in QAHE with a constant sign in exchange field has energy gap separating the highest occupied electronic band from the lowest empty band our results shows that the possibility of tunable exchange field sign creates conducting interfaces states, remaining a bulk energy gap.

Resumo

Esta tese e composta de dois estudos relacionados a estados de borda em nanoestru- turas de monocamadas grafeno. No primeiro estudo analisamos as propriedades magneticas de nanofitas quirais de grafeno. A quiralidade das bordas e uma propriedade de simetría onde a imagem espelhada nao pode ser sobreposta a original, ao contrario do que acontece em bordas aquirais [1]. Existe dois casos de nanofitas aquirais: com bordas “poltrona” ou “ziguezague”. As nanofitas quirais, assim como as de bordas ziguezague, apresentam estados localizados que favorecem a magnetizacao. Em nossa analise usamos o modelo de ligaçôes fortes (TB, do ingles “tight-binding”) com um termo de interaçao eletron-eletron do tipo Hubbard de campo medio. Mostramos que somente a inclusao de integrais de transferencia entre vizinhos mais próximos no modelo TB nao e suficiente para descrever propriedades de baixas energias e magneticas nanofitas de grafeno, i.e., a inclusao de inte­grais de transferencia entre segundos vizinhos e necessaria para uma modelagem realística das propriedades eletrónicas. Comparamos nossos calculos com resultados recentes de mi- croscopia de varredura por tunelamento e propomos uma nova interpretacao para picos observados experimentalmente na densidade local de estados.

O segundo trabalho e um estudo em andamento sobre estados condutores do efeito Hall quantico anomalo (EHQA) em grafeno na presenca de um campo de troca periódico e interacao Rashba spin-orbita . Nos nomeamos de interfaces o ponto de encontro de dois sistemas, neste caso, um sistema com sinal positivo de campo de troca (CT) encontra um sistema anóalogo com sinal negativo de CT. Estados condutores aparecem no ponto de encontro: estados de interfaces condutores. Para tal, nos analisamos a formaçao de estados sem lacuna de energia na mudança de sinal do parametro campo de troca. Enquanto o sistema em EHQA com sinal constante no campo de troca tem uma lacuna separando estados de valencia e conducao nossos resultados mostram que a possibilidade de ajustar o sinal do CT cria estados de interfaces condutores, mantendo uma lacuna de energia no volume.

List of Figures

1 Example for set vectors in Honeycomb lattice: aí and a2 are basis (or primitive) vectors of the direct lattice, and Si are next-nearest vectors, with i = 1, 2 and 3. (Left) Representation for Eq. (2.2) and (2.4). (Right) Representation for Eq. (2.5) and (2.6)

2 (a) Honeycomb lattice: a1 and a2, Eq. (2.2), are basis (primitive) vectors of the direct lattice, a0 is the lattice constant and acc = ао/л/3 is the bond length. (b) Reciprocal lattice: b1 and b2 are the basis vectors of the reciprocal lattice, the high-symmetry k-points K and K' are located at the corners of the Brillouin zone Σ. For the crystalline orientation of Eq. (2.5) the direct and reciprocal lattices above are rotated by 30° (Figure adapted from Ref. [15])

3 Band structure of honeycomb lattice by tight-binding with nearest-neighbors for kx = 2n/3acc

4 Ilustration of the chiral angle θ and chiral vector C in a honeycomb lattice. Armchair and Zigzag edges are represented by dashed lines. Adapted from Ref. [23]

5 (a) Armchair ribbon, where θ = 30° with respect to the zigzag direction (b) Zigzag GNR defined by cutting the graphene sheet along the direction of the basis vector a1. Adapted from Fig. [22]

6 (a) Relation between the BZ of bulk graphene and armchair nanoribbons by zone folding. The hexagonal BZ of graphene is mapped onto the shaded blue rectangle as the BZ of armchair nanoribbons and this phase space rise of the boundary condition of the armchair GNR Ref. [22]. For a width of N = 5 dimers the cutting line r = 4, of Eq. (2.27) passes through the Dirac point. (b) Energy band structure of armchair strip in correspondence to cutting line of BZ. Figure adapted from Ref. [22]. . .

7 (a) Unlike the armchair graphene nanoribbon, slicing the Brillouin zone of graphene does not show relation with zigzag-ribbon because the kx depends on ky and N. The cutting lines for zigzag GNR of N=5. (c) Energy band structure of zigzag strip for width of N chains, which in the flat band for ky > |kc| correspond to localized states. Figure adapted from Ref. [22]

8 The probability distribution for the highest-energy valence band with spin-down as function of the transversal site and crystal momentum ky of a zigzag graphene ribbon. The width is w = 32 sites, or 16 zigzag chains, without electron-electron interaction (U = 0) and considering a tight binding model with nearest-neighbour hopping only

9 Representation of a chiral nanoribbon with sublattices A (B) in red (blue). The chiral vector is Ch = (5,1) and the width is character­ized by W = (-4,8). The sites with dangling bonds at the GNR edges are indicated by circles p.22

10 Evolution of the band structure obtained by tight-binding calculations of the graphene ribbons for several chiralities obtain from our model Hamiltonian for w = 12, U = 0 and ť = 0, Eq. 3.11. The chiral angles 0c for each GNR configuration (n,m) follows: 0c(1, 0) = 0°, 0c(5,1) «

8.9°, 0c(4,1) « 10.9°, 0c(3,1) « 14.9°, 0c(2,1) « 19.1°, 0c(3, 2) « 23.4°, 0c(4, 3) « 25.3° and 0c(1,1) = 30° p.29

11 Schematic band structures of “chiral” GNRs with (S, 0) after folding the band structure of a GNR (1,0) with zigzag edges Stimes. The shaded areas represent the band continuum of states. The degeneracies of the flat bands, namely, g = 2P and g = 2(P + 1), are put in correspondence with their respective edges of the Brillouin zone at the charge neutrality point. Adapted from Ref. [48]

12 Band structure (left column) and corresponding density of states (right column) of a zigzag graphene nanoribbon of N = 24 for ť = 0 (upper row) and ť/t = 0.1 (lower row). The solid lines stand for the case of U/t = 1, while the dashed ones for U = 0

13 Edge magnetization Μχ of zigzag graphene nanoribbons as a function of their width N. Inset: Band gaps Δ0 and Δι versus N. In both cases U/t = 1.0 and ť/t = 0.0,0.1 and 0.2

14 Edge magnetization Μχ/Δχ as a function of the chemical potential μ/Δχ for zigzag nanoribbons with different GNR widths N for (a) ť/t = 0.0 and (b) ť/t = 0.1. In both cases U/t = 1

15 Electronic band structure for GNRs of width w = 12 and U/t = 1, with chiralities (a) (2,1) and (c) (3,1). Corresponding density of states for the (b) (2,1) and (d) (3,1) chiralities. The solid lines stand for the case of ť/t = 0.1, while the dashed ones for ť/t = 0.0. The energy gaps Δ0 and Δχ are only indicated for the ť/t = 0.1 case

16 Gap Δχ as a function of the GNR width w for different next-nearest- neighbor hopping parameters ť/t. Here we use U/t = 1.0

17 Edge magnetization Μ/Δχ as a function of the chemical potential μ/Δχ for chiral nanoribbons of different widths w for (a) ť/t = 0.0 and (b) ť/t = 0.1. In both cases U/t = 1

18 (a) Band structures of a (8, 1) chiral graphene nanoribbon of w = 12

and (b) edge magnetization M/a0 as a function of the chemical potential μ/t. The dashed (red) lines stand for the case of ť/t = 0.0 and the solid (blue) ones for ť/t = 0.1

19 Local density of states for a (3,1) chiral graphene nanoribbon for (a) ť/t = 0 and (b) ť/t = 0.1. Here, w =12 and U/t = 1

20 Local density of states of a (8,1) chiral graphene nanoribbon for (a) ť/t = 0 and (b) ť/t = 0.1. Here, w =12 and U/t = 1

21 (a) Scanning tunnelling microscopy (STM) of the terminal edge of an (8,1) GNR, Ref. [8]. (b) dI/dV spectra obtained at different positions (as marked) of the GNR edge shown in (a) along a line perpendicular to the GNR edge. (c) Shows a higher resolution dI/dV spectrum for the edge of a (5, 2). (d) The LDOS results by our model of chiral ribbon (8,1) as shown previously in Fig. 20 with blue (purple) star highlighter in e3/t = 0.04 (e2/t = 0.02)

22 Local density of states (logarithmic scale) and edge magnetization (linear scale) M of a graphene nanoribbon with chirality (8,1) for (a) ť/ŕ = 0 and (b) ť/ŕ = 0.1. Here, w =12 and U/t = 1

23 (a) Negative carriers flow with a drift velocity vd opposite in direction to the conventional current density J = Jxeb (b) The same for the case of positive charge carriers. For the same J and external magnetic field B = Bxe2, the resulting Lorentz force F = Fze3 is equal in (a) and (b). The Hall field EH = Eee3 differs in orientation and the charge density at the edge differs in sign. Edited from Ref [64]

24 Band structure of a zigzag graphene nanoribbon of width w = 32 and exchange field parameter λζ = 0.1t

25 Top: Probability distribution |Tk|(n)|2 as a function of the momentum ky and site position for zigzag nanoribbon of width w = 32, with the band represented by the red circles in Fig. 24. Bottom: The same as above for |Tkf(n)|2 for the band corresponding in red solid line in Fig. 24.

26 Band structure of bulk graphene in the presence of a Rashba spin-orbit interaction with a coupling parameter tSо = 0.4t (tso = 0.1t) in blue (red).

27 Band structure of zigzag graphene ribbon of width w = 32 and Rashba SOC parameter tSO = 0.1t

28 Expectation value of the z projection of the spin operator, (Sz), as a function of position across the ribbon and crystal momentum ky. Top: tSO = 0.1t, Middle: tSO = 0.2t, Bottom: tSO = 0.3t

29 Evolution of the band structures of bulk graphene for kx = 2n/(3acc) and reciprocal lattice vectors given by Eq. (2.8) and (2.9): (Black) Only hopping, tSO = λζ = 0, the same as Fig. 2. Here the spins states are degenerate; (Blue) Only with the Zeeman-like term on λζ = 0.4t. The spin-up (spin-down) states have an upwards (downwards) energy shift, proportional to λζ; (Green) Rashba SOC parameter tSO = 0.1t and λζ = 0; (Red) with both interactions tSO = 0.1t and λζ = 0.4t. In this last case there is a bulk gap, but this gap is non-trivial (see text)

30 Band structure of zigzag graphene ribbons of width 100 sites in the prim­ itive unit cell (PUC) with tSO = 0.0471t and λζ = 0.1885t. The Fermi energy Ef = 0.0125t indicated in red corresponds to four different edge states, namely, (a), (b), (c), and (d)

31 Top: Band structure of zigzag graphene ribbons of width 100 sites in the PUC with tso = 0.0471t and λζ = 0.1885t. The Fermi level Ef = 0.0125t in red corresponds to four different edge states (a), (b), (c), and (d). The probability density |Ψ|2 across the width for the four edge states are shown: (a), (c) states are localized at the left boundary, the opposite for (b) and (d) states

32 The fitting band structure of graphene and graphene with boundary con­dition of width diameter = 300acc (Ch(100,100)) for: 1) tSO = λζ = 0 shows degenerate spins states; 2) Only exchange field λζ = 0.4t the spin-up (spin-down) states are shifted upwards (downwards) in energy; 3) Only Rashba SOC tSO = 0.1t; 4) Both interactions tSO = 0.1t and λζ = 0.4t showing a non trivial bulk gap (see text)

33 The band structures of graphene with boundary condition under periodic exchange field λΖ0) = 0.0471t and tSO = 0.1885t. The Fermi level Ef = 0.0115t gives eight different “edge” states. The widths are w = 100 and w = 200

34 The probability density |Ψ|2 across the width for the eight interfaces states in an graphene with boundary condition are shown. The width is w = 100 sites in the PUC with Rashba parameter tSO = 0.0471t and periodic exchange field λΖ0 = 0.1885t . The Fermi level is Ef = 0.0115t. The vertical black lines represent the sites where there is sign change in λρ

35 The probability density |Ψ|2 across the width for the eight interfaces states in an graphene with boundary condition are shown. The width is w = 200 sites in the PUC with Rashba parameter tSO = 0.0471t and periodic exchange field λΖ0 = 0.1885t . The Fermi level is Ef = 0.0115t. The vertical black lines represent the sites where there is a sign change in λρ

36 Sketch showing the direction of the “interfaces” modes propagation (indi­ cated by arrows) in the (a) quantum anomalous Hall (QAH) with positive exchange field; (b) QAH with negative exchange field

37 Left: Normal Zeeman effect in levels l = 2 and l = 1, both with S = 0. Before external magnetic field is possible only one transition is possible, in the presence of the field each level split in Δm = 0, ±1 and the nine transitions is permitted in agree selection rule [85]. There are three length of lines which correspond three different frequencies. Right: Anomalous Zeeman effect. The shift in the energy levels depends on spin S, orbital L and total angular momenta J. The spectroscopic notation is 2S+1XJ, X is S, P ... corresponding to l values. The transitions between levels follow selection rule. Figure adapted from Ref. [63]

1 Introduction

The synthesis of graphene opened new directions in the research of two-dimensional (2D) systems [2]. Having a single-atom thickness, graphene systems are the thinnest possible films one can produce. Graphene monolayers show remarkable mechanical prop­erties and very peculiar electronic ones. More recently, the class of graphene-like two­dimensional materials has been broadened substantially with the synthesis of transition metal dichalcogenides, germanene, silicene and phosphorene [3]. Graphene and these new 2D materials provide a broad range of challenging problems for basic and applied research.

The motivations for the strong scientific interest in graphene are many. We highlight some of the most important. From the perspective of applications, several properties of graphene have possible technological potential. Since graphene is practically transparent, substantially stronger than steel, it is a flexible conductor [4], just to name some of its qualities, this material has a high potential for innovation. For instance, graphene transistors are predicted to be substantially faster than those made out of silicon today [5], which makes this very attractive in view that silicon-based electronics are reaching their miniaturization limits. There is an expectation to have graphene-based products in the market within few years in flexible electronics and composites. Those are discussed at length in technological roadmaps by, for instance, the Graphene Flagship [6].

In addition to these promising technological applications, one of the major motivations for fundamental research that is worth emphasizing is the relativistic behavior of the electrons in graphene. The effective Hamiltonian describing the behavior of the low- energy electrons in graphene monolayer is formally identical to the Dirac equation for massless fermions. The main experimental and theoretical efforts have been focused in trying to understand the consequences of the linear spectrum [7] associated with Dirac equation. For instance, one interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox [2], an effect so far not observed in particle physics.

The pristine graphene is a non-magnetic material, but for many nanosized graphitic systems there are theoretical scenarios for the emergence of edge magnetism [8]. Mag­netic graphene nanostructures constitute a particularly promising new path to explore applications in the field of spintronics at the nanometre scale [9]. The success in the miniaturization of graphene-based electronic devices demands elucidation of the effect of edges on the properties of graphene nanostructures.

In this thesis, we study the edge magnetization and the local density of states of chiral graphene nanoribbons using a π-orbital Hubbard model in the mean-field approximation [10]. We show that the inclusion of a realistic next-nearest-neighbor hopping term in the tight-binding Hamiltonian changes the graphene nanoribbons band structure significantly and affects its magnetic properties. We study the behavior of the edge magnetization upon departing from half filling as a function of the nanoribbon chirality and width. We find that the edge magnetization depends very weakly in the nanoribbon width, regardless of chirality as long as the ribbon is sufficiently wide. We compare our results to recent scanning tunneling microscopy experiments [8] reporting signatures of magnetic ordering in chiral nanoribbons and provide a new interpretation for the observed peaks in the local density of states, that does not depend on the antiferromagnetic interedge interaction.

We also explore the unusual electronic properties of graphene to study the possibility of observing conducting states through the anomalous quantum Hall effect. For that, one needs spin-orbit interactions and the presence of an exchange field, another manifestation of the magnetic properties of graphene. We review the recent proposal for the realization of the QAHE [11] with gapless edge states in graphene nanoribbons. The first result of our project on about QAHE, in progress, shows the formation of gapless interface states in graphene sheet based on a model system with periodic exchange field.

This thesis is structured as follows: In Chapter 2 we introduce the basic elements of our theoretical analysis. There, we discuss the crystal and electronic structure of bulk graphene, as well as those of graphene nanoribbons. We conclude the chapter with a general discussion of the magnetic properties of graphene with edges and defects.

Chapter 3 is devoted to discuss the geometrical and simple band structure properties of chiral graphene nanoribbons. We introduce the model Hubbard mean-field Hamiltonian used in the analysis of the magnetic properties of GNRs, justifying the parameters choice in view of DFT calculations.

In Chapter 4 we discuss the edge magnetism properties of graphene nanoribbons with zigzag and chiral edges. Particular attention is dedicated to the local density of states (LDOS) and the formation of local magnetic moments, making connection to recent scanning tunneling microscopy and spectroscopy (STM/STS) experiments [8] mentioned above.

In Chapter 5 we switch subjects and study the transport properties of graphene sys­tems in the presence of spin-orbit interaction and exchange fields. In this setting we shows the effect of each interaction in graphene and nanoribbon. We start reviewing the main aspect of the classical Hall and anomalous Hall effects. We then presents the topologi­cal theory for the standard quantum Hall effect. The manifestation of conducting states in graphene sheet is performed by graphene with boundary condition with two areas of different signs in the exchange field parameter.

We conclude presenting a summary of our findings in Chapter 6.

2 Theoretical Background

The central purpose of this chapter is to present the basic structural and electronic properties of graphene bulk and graphene nanoribbons. We begin by describing the crys­tal structure of pristine graphene and shows the main features of its electronic struc­ture within the nearest-neighbor tight-binding approximation. Next, we address the confinement effects that appear when one considers graphene nanoribbons, a quasi one­dimensional system. Finally, the effect of electron-electron interactions and the emergence of magnetic properties in graphene is present in general terms .

2.1 Crystal Structure: Direct and Reciprocal Lat­tices

The honeycomb crystal structure can be seen like an overlap of two triangular sub­lattices A and B with the unitary cell containing one site of each sublattice. All sites of the two dimensional triangular Bravais lattice can be expressed by

illustration not visible in this excerpt

where (n¿, m¿) are integers that specify the position of the site i belonging to the sublattice A. A possible choice for the primitive vectors ai and a2, particular to triangular lattices, is:

where a0 = 2.46 A is the lattice parameter and acc = а0Д/3 = 1.42 Â is the carbon­carbon bond length.

The vectors a1 and a2 are represented in Fig. 1(left), its primitive unit cell and first Brillouin Zone in Fig. 2. The sites of sublattice B, placed at RB, are shifted by a vector

δi in relation the sites in A:

illustration not visible in this excerpt

where ái is one of the nearest-neighbor vectors

That is, the separation between nearest-neighbor atoms belonging to different sublattices is represented by the vectors á¿. The Hamiltonian with the basis 2.2 and 2.4 have the same eigenvalues as the one obtained by rotating the axes in real space by π/б, the difference is a phase in the eigenstates [12].

Another possible choice of vectors set, rotating the axes in real space by π/6:

In the Section 2.2 we employ the representation for Eq. (2.5) and (2.6) in the derivation of the electronic band structure.

The structural relations are identical, but not the orientational relations because the array of points looks the same when viewed from adjacent points only if the arrangement is rotated through 180° each time one moves between nearest-neighbors points [13]. Since two-dimensional honeycomb lattice cannot be described by combination of primitive vec­tors, as in Eq. (2.1), it is not a Bravais lattice^[1]. However, the 2D hexagonal lattice can be mapped in a triangular (Bravais lattice) which in each point consists of the sites A and B. A and B.

The translational invariance or the periodic array of the ions on the microscopic level (crucial property of a crystal [13]), makes it very convenient to use Fourier Analysis and work in a Fourier space associated with the crystal, known as reciprocal lattice. The reciprocal lattice vectors b can be found as usually by demanding that they satisfy [14]

illustration not visible in this excerpt

The system of linear equations with Eq. (2.2) leads to

illustration not visible in this excerpt

with the vectors b1 and b2 one defines the lattice in the k-space, which have triangular form, and the Wigner — Seitz cell2 of the reciprocal lattice, also known as Brillouin Zone, has an hexagonal shape.

2.2 Band Structure

Let us consider the simplest model for the electronic structure of graphene, namely, the tight-binding model. In this picture, bands formed from the sp^[2] hybridized orbitals are assumed to be filled and inert (chemical bond), and mobile carriers move in the x — y plane by hopping between the pz orbitals of the carbon atoms.

The tight-binding Hamiltonian with one orbital and nearest-neighbor hopping taken into account is

illustration not visible in this excerpt

where a*(RA) and ασ(RA) are, respectively, the creation and annihilation operators of electrons with spin projection σ at the site R¿ of the sublattice A, Eq. (2.1). Analogous for operators b*. The symbols (i,j) indicate sums over nearest neighbor. Fourier transforming

illustration not visible in this excerpt

This behavior characterizes pristine graphene as a zero-gap semiconductor. As we show in the following, the dispersion relation around K and K' is linear, more specifically, Ek has conical shape. K and K' are called Dirac points. The electronic band structure with two Dirac points is shows in Fig. 3 and in the Section 5.3 we will see the effect of the Rashba spin-orbit coupling and exchange field interaction on the dispersion relation.

2.3 Dirac Fermions

Let us linearize the Hamiltonian Eq. (2.15) for values of k near the inequivalent Dirac points, K and K'. We introduce the substitution k ^ ±K + k, with |k| ^ |K|, to write

illustration not visible in this excerpt

One can write the matrix elements of Eq. (2.23) in exponential form as a complex number z = kx — iky = |z|e-iö, with 0k = arg(z) = arctan(ky/kx). Accordingly, the wave function of Eq. (2.23) in momentum space, for the momentum around K, has the form

where the ± signs refer to the positive and negative energy eigenvalues of Eq. (2.24), respectively. A similar Hamiltonian and eigenstates can be derived for the K' point. We recommend Ref. [2] to review interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials, the so-called Klein tunneling.

2.4 Graphene Nanoribbons: basic aspects

Let us review how the bulk electronic properties of graphene are modified by in­troducing edges that constrain the electronic dynamics. The simplest case is that of nanoribbons, which have been extensively studied both experimentally [16, 17, 18] and theoretically [19, 20, 21, 22].

Graphene nanoribbons (GNR) can be obtained by cutting a graphene sheet along of two parallel straight lines. As we will discuss, the crystalline direction of the edges play an important role in the electronic structure of a GNR. The edges are characterized by a chiral angle θ, see Fig. 4. The angles θ = 0° and θ = 30° correspond to the high-symmetry directions known as zigzag and armchair, respectively. In this section we discuss these particular cases, leaving the analysis of the general chiral case, 0° < θ < 30°, to the next chapter.

illustration not visible in this excerpt

Figure 4: Ilustration of the chiral angle θ and chiral vector C in a honeycomb lattice. Armchair and Zigzag edges are represented by dashed lines. Adapted from Ref. [23]

Figure 5 illustrates the lattice structure of armchair and zigzag GNRs. It shows the primitive unit cell (PUC) for both cases and defines the notation used in this study.

illustration not visible in this excerpt

Figure 5: (a) Armchair ribbon, where θ = 30° with respect to the zigzag direction (b) Zigzag Gevild, defined by cutting the graphene sheet along the direction oil the basis vector aí. Adapted from Fig;. [22].

The width of both GNRs is conventionally defined by [122]

illustration not visible in this excerpt

2 cc ° (2.26) 3NNacc + acc ξ Wz for zigzag ribbons.

For armchair GNRs, N stands for the number of dimer (two c arbon ¡sites) lines, parallel to the y axis (define in Fig- 5). fror zigzag GNRs, N is time number of zigzag chains across the ribbons transversal direction. The armchair nanoribbon shown in Fig. 5a lias x-axis reflection lymmetry. This symmetry breaks down by adding ([or removing) an A — B dimer line to (or Crom) the unit cell. Ins contrast, the zigzag GNRs shown in Fig. 5b does not have reflection symmetry. This symmetry is obtained by adding °or removing) zigzag chain to (or from) the unit cell. The ribbons with reflection symmetry corresponds to odd N for armchair and even N in case o° zigzag strip.

Let us now discuss the effects of confinement in the electronic structure. The electronic states of the low-dimensional system can be considered as a subset of the eigenvalues of the original bulk material. The restriction is that the wave vector components in the nanoscale directions can only take discrete values to maintain an integer number of wave function nodes. That is, the wave vector components become quantized. The number of quantized states is equal to the number of unit cells of the bulk material in the PUC of the low-dimensional structure [24]. This reasoning allows one to infer the main features of the band structure of graphene carbon nanotubes and armchair nanoribbon from the band structure graphene, except for the special case of zigzag GNRs [21, 24].

The method of constructing 1D electronic bands by “slicing” the 2D dispersion re­lations is known as the zone-folding scheme [24]. The analytical solution of the nearest- neighbor tight-binding model for armchair nanoribbons shows that energy bands can be obtained by slicing the band structure of graphene, making the transverse wavenumber discrete, in accordance with the edge boundary condition [21, 22]. For armchair GNRs, the results is very simple. The discrete kx value, in unit of primitive vector, depend only on the width N, namely [22]

illustration not visible in this excerpt

This construction is illustrated in Fig. 6. The value of N determines if there is a slice that intersects a 2D Dirac point. The kx component of the point K is situated at 2n/3a0. Hence, Eq. (2.27) indicates that for N = 3i — 1 (i E positive integers) the armchair graphene ribbon has a Dirac point and, hence, shows a metallic behaviour. For the other cases, where N = 3i — 1, armchair ribbons are semiconductors with a gap that depends on their width.

The discretization of the wave number kx imposed by the edge boundary condition in zigzag ribbons does not follow a simple formula, as in the case of armchair nanoribbons. One can show [22, 25] that for E > 0, the energy spectrum of the zigzag GNR is composed by N extended (non-localized) states in the interval — kc < ky < kc, and N — 1 states in kc < |ky| < π, where kc = 2 arccos ^ 1 l+i/v). By examining the probability amplitude of the electronic wave functions one finds that these electronic states corresponding to the almost flat band in kc < |ky | < π are localized along the zigzag edges.

For zigzag graphene nanoribbons, the transverse wave number kx depends not only on N, but also on the longitudinal wave number ky. Figure 7a shows the values of kx (equivalent to p in Ref. [22]) versus the longitudinal wave number ky. The transverse wave

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Figure 6: (a) Relation between the BZ of bulk graphene and armchair nanoribbons by zone folding. The hexagonal BZ of graphene is mapped onto the shaded blue rectangle as the BZ of armchair nanoribbons and this phase space rise of the boundary condition of the armchair GNR Ref. [22]. For a width of fV = 5 dimers the cutting line r = 4, of Eq. (2.27) passes through the Dirac point, (b) Energy band structure of armchair strip in correspondence to cutting line of BZ. Figure adapted from Ref. [22]. number is complex for fcc < |kj < π. The imaginary part η of kx is shown in Fig. 7b. Here, Fig. 7c, the low energy features of the energy spectrum are not directly related to the band structure of bulk graphene, i.e., no such flat band is expected by projection of the 2D dispersion relation for zigzag edges [20, 22, 21].

Figure 8 shows the wave function probability distribution |Tk(i)|2 for the highest- energy valence band as a function of kyalong the Brillouin zone and the site Iransverse position of a zigzag ribbon of width w = 32 sites (or N = 16 zigzag chains). One can observe that the wave functions arehighly localized ait the edge s(tes i = 1 and i = 32 for states with ky = ±7r/a0. The probability amplitude decreases as one; moves away from the domain where the band is flat, —kc < ky < kc. These results do not depend on

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Figure 7: (a) Unlike the armchair graphene nanoribbon, slicing the Brillouin zone of graphene does ^o^ show relationwifh zigzag-ribbon because the kx depends on ky and N. The cutting lines for zigzag GNR of N=5. (c) Energy band structure of zigzag strip for width of N chains, which in the flat band Sor ky > |kc| correspond to localized states. Figure adapted Crom ReC. [22].

spin-projection, which means that, in the absence of electron-electron interactions, the system is not spin-polar[zed.

The localized states at the edges or zigzag GNRs and the corresponding sharp peak of the density of states (DOS) near the charge neutrality point, lead to very interesting phenomena, when one takes electron-electron interactione into account;. Thore phenomena are absent in armchair GNRs, but — as we discuss in tine next chapter — are manifest in chiral ribbons.

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Figure 8: The probability distribution for the highest-energy valence band with spin-down as function of the transversal site and crystal momentum ky of a zigzag graphene ribbon. The width is w = 32 sites, or 16 zigzag chains, without electron-electron interaction (U = 0) and considering a tight binding model with nearest-neighbour hopping only.

2.5 Magnetic behaviour: Stoner mechanism and Lieb’s theorem

The magnetic moment of a free atom has three principal sources: the intrinsic mag­netic momentum (spin) of the electron; their orbital angular momentum around the nu­cleus; and the change in the orbital moment induced by an applied magnetic field [26]. These elements are sufficient to built a crude model to understand the some magnetic properties of insulators. This is certainly not the case in graphene. Next we enunciate the main types of magnetic behavior in solids, making the parallel with graphene.

- Diamagnetism is the state where the induced moment in the system is opposite to the applied field, as a result, the substance experiences a repulsion by the external magnetic field, with a negative magnetic susceptibility. The repulsive force is a property of all materials and makes a weak contribution to the material in a magnetic field, but is negligible in presence of the other form of magnetism [13]. In graphene the electrons are delocalized and the diamagnetism properties are captured by the Landau susceptibility, which is particularly small close to the charge neutrality point due to the vanishing density of states.

- Paramagnetism: In this state the material has no inherent magnetization, but when is subject to an external field it develops magnetization which is aligned with the field, a positive susceptibility is the consequence. This state corresponds to situ­ations where the unpaired intrinsic magnetic moments (spin) tends to be oriented in the same direction as the external magnetic field [26]. In delocalized electronic system this regime is described by the Pauli susceptibility, which is in graphene.

- Ferromagnetism is a state in which the system have a non-vanishing magnetic mo­ment even in the absence of a magnetic field, below a critical temperature Tc (Curie temperature), taking into account that the thermal motion of the atoms can pre­vent a complete orientation of spins, i.e., the temperatures influences in the degree of magnetization. To explain this magnetic ordering is necessary to go well beyond the independent electron approximation [13]. To a first approximation, interactions like the magnetic dipole-dipole one and the spin-orbit coupling are superseded by electrostatic effects, which leads one to retain only ordinary Coulomb interactions. If there were no magnetic coupling, in the absence of a field the individual magnetic moments would be thermally disordered, pointing in random directions, and could not sum to a net moment for the bulk [26].

The first insights of ferromagnetism are due to P. Weiss who in 1907 put forward the hypotheses that below characteristic temperature Tc a ferromagnet is composed of small spontaneously magnetized regions called domains and each region is spon­taneously magnetized because a very strong “molecular field”, of unknown origin to Weiss [27]. The source of the Weiss field was demonstrated by Heisenberg by ac­counting for quantum-mechanical exchange effects. The exchange interactions are nothing but electrostatic interaction energies and the Pauli exclusion principle [13]. We will return to this issue on chapter 5, when discussing the quantum anomalous Hall effect.

There are other types of magnetic states, like ferrimagnetism, superdiamagnetism, and superparamagnetism that are not so relevant for graphene. We will not address these phenomena in this thesis.

So far we have considered only non-interacting model to describe the electronic struc- ture of graphene. Several studies established that a minimal model for interaction in such systems is the Hubbard model [28]. The effect of a finite U in materials with d—orbitals leads to magnetism. However for the p—orbitals of carbon, the formation of local magnetic moments is still controversial with elusive experimental evidences.

The situation changes for zigzag graphene nanoribbons. In this case, there is a very strong enhancement of the local density of states (LDOS) at the edges for E = 0, corre­sponding to the flat bands discussed in the previous section.

For U = 0, such localized states give rise to a competition between the kinetic and Coulomb energy, favoring spin polarization. This mechanism of magnetization of band electrons is nicely discussed in textbooks [29]. The occurrence of spontaneous magne­tization is predicted by the Stoner criterion, namely υρ(ε) > 1 where ρ is the density of states. This naive estimate relies on a mean-field theory for electrons with a smooth (featureless) DOS with a Hubbard interaction. The presence of flat bands in zigzag edge GNRs modifies the Stoner criterion in a surprising way as we discuss in the forthcoming chapters.

Let us recall that the localization of states in zigzag GNRs is different from the standard localization discussed in disordered systems. Here the states are localized only in the transverse direction, while in the longitudinal one they are extended. This is why the theory of band magnetism is applicable in zigzag GNRs.

Let us conclude with some remarks on a theorem due to Lieb [30] that serves as a guide to interpret the magnetic properties of GNRs. In the seminal paper, Lieb proved an important property of the ground state of the half-filled Hubbard model. In the repulsive case of Hubbard U (U > 0), for a bipartite system with a non-degenarate ground state the total spin is Stot = |NA — Nb |/2, where NA (NB) is the number of the sites in the sublattice A (B). The definition of the bipartite system is a lattice which can be decomposed two sublattices and the electronic hopping is allowed only between different sublattices, i.e., first-nearest-neighbor. An example of the bipartite system is honeycomb lattice with nearest-neighbor hopping terms, that corresponds the the graphene bulk Hamiltonian discussed earlier in the Chapter. Half-filling, in turn, means that each atom provides one electron to the system [30].

It should be noted, however, that the knowledge of the total spin of the ground states of a bipartite system at half-filling does not necessarily determine its properties [31]. For example, zigzag nanoribbons where the two sublattices have the same number of sites Na = NB have a ground states with Stot = 0, but exhibit magnetic ordering of edge states. Hence, one can still have spin polarization at the edges, but opposite edges have to show an antiferromagnetic configuration.

A realistic description of the electronic properties of graphene is certainly different from the next-nearest-neighbor tight-binding model with a Hubbard interaction term. To begin with, the graphene spectrum show features that are consistent with a small next-nearest-neighbor hopping term. All that indicates that the Lieb's theorem does not necessarily apply to zigzag graphene nanoribbons. We will show in the next chapter that, nevertheless, it provides an excellent approximation.

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3 Chiral Graphene Nanoribbons

Graphene nanoribbons (GNRs) with regular edges cut along an orientation other than the armchair or the zigzag are called chiral. Given the contrast between the low lying electronic properties of armchair and zigzag GNRs, the natural question to ask is what to expect for the chiral ones. The focus of this study is on the modifications in the local density of states (LDOS) due to the edges, and its implication in physical phenomena, like the emergence of magnetism [9].

Chiral graphene nanoribbons (GNRs) are strips of graphene, see Sec. 2.4, with edges that follow a combination of zigzag and armchair high-symmetry orientations. The study of edges with low-symmetry orientation is essential for the understanding of the basic physics of graphene nanoribbons, giving a more realistic description of its properties. In particular, GNRs synthesized by chemically unzipping carbon nanotubes show chiral edges.

One of the predictions of interest is the emergence of magnetism in graphene nanos­tructures [32]. The prediction of localized states at the edges of graphene nanostructures [20] are believed to give rise to edge magnetization [19, 33, 34] with potential applica­tions in spintronics [35] and has attracted a lot of theoretical and experimental attention. Electronic structure calculations indicate that graphene nanoribbons (GNRs), depending on the crystallographic orientation of their edges, exhibit a ferromagnetic spin alignment along the edges and an antiferromagnetic interedge ordering [9, 19, 35, 36, 37]. Several experiments report evidences of edge states [8, 38, 39, 40], but direct observations of edge magnetization in graphene remains rather elusive.

The synthesis of GNRs was pioneered by lithographic patterning [16, 17, 18]. This technique produces rough edges, that give rise to short range scattering, detrimental to the electronic mobility [41] and to the formation of local magnetic moments [42]. More recently, by chemically unzipping carbon nanotubes, it became possible to obtain GNRs with very smooth edges [43, 44, 45]. In general, the latter are chiral, that is, their edges do not follow neither the zigzag nor the armchair high-symmetry orientations. The local density of states (LDOS) of ultrasmooth edge chiral GNRs was recently investigated using scanning tunneling microscopy/spectroscopy (STM/STS) [8]. The obtained STS spectra are the first direct experimental evidence of localized edge states in chiral GNRs. These results are the experimental motivation for these works.

The theoretical studies of the electronic properties of graphene nanoribbons with arbitrary edges date back to the pioneering works in the field [20], where it has been established that, for sufficiently wide ribbons, there is always an enhancement of the density of states (DOS) due to dispersionless zero-energy edge states, except for GNRs with armchair terminations [46]. In chiral GNRs, as in the zigzag case, electron-electron interactions split the zero-energy bands and give rise to edge magnetization [9, 47]. While the Hubbard mean-field calculations indicate that local magnetization appears whenever the non-interacting DOS in enhanced [9], density functional theory (DFT) calculations point to a sharp suppression of the edge magnetization for chiralities close to the armchair orientation [47].

In this chapter we discuss efficient ways to characterize the geometry of chiral GNRs. Next we present the model Hamiltonian we use to describe the electronic structure of ribbons, with electron-electron interaction. Finally, we present a qualitative discussion of what to expect from interaction and their interplay with edge effects.

3.1 Geometry of chiral GNR

The definition for primitive unit cell (PUC) of a chiral GNR adopted is in terms of their widths and the crystallographic direction of their edges [9, 46]. The longitudinal orientation in the strip is characterized by the translation (or chiral) vector Ch, see Fig. 9, defined as

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where n and m are integers, whereas a1 and a2 are the lattice unit vectors. The length of the translation vector is a = a0Vm2 + mn + n2, where a0 ~ 0.246 nm is the graphene lattice constant. For later convenience, we write Ch in terms of its projection on the zigzag and armchair directions

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In general, Ch does not provide a unique characterization of the edges, since GNRs with the same translation vector can have a different number of edge atoms Ne and dangling bonds N4 per unit cell. The constraint that neither Ne nor N4 can be smaller than m + n [46], is used to define “minimal edge” GNRs [46, 48], where Ne = N4 = m + n. In this case, Ch describes unambiguously the nanoribbon edges. For the sake of simplicity, in this work we consider only minimal edge chiral GNRs.

The GNR orientation is also often specified by the chiral angle 9C, defined in terms of the of product between vectors

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Due to the symmetry of the honeycomb lattice, 0° < 9C < 30° accounts for all possible crystallographic directions. The high-symmetry cases are those where the GNR edges correspond to the zigzag and armchair directions, that is, 0C = 0° with a translation vector (/?., 0) and 9C = 30° with (/?., ??,), respectively. GNRs whose edges are neither armchair nor zigzag, are called chiral.

The GNR width is conveniently characterized by vector [9]

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where w is an integer. The vector W, represented in the Fig. 9, is parallel to the armchair lattice orientation . The width of chiral (and zigzag) GNR is IT = y/Hwao cos9c. Defining the width this way facilitates the computational implementation of the sites for indexation. It is enough to know the coordinates of the atoms of the “lower” edge, see Fig. 9, to determine the positions of the other site according to a arithmetic progression with an increment (—1,2).

The edge sites are determined as follows: We start with a reference site, placed at the sublattice A, and draw the chiral vector Ch with its origin at (0, 0). The ж-axis is parallel to Ch, while the y-axis is perpendicular, see Fig. 9. Next, we eliminate all A sites with abscissas smaller than that of the line defined by chiral vector. Finally, we store the positions of all A lattices indicated by red in the Fig. 9. Once the “lower” edge sites are determined the unit cell can be constructed in a simple way: All sites are placed in lines parallel to the vector (—w, 2w) and are indexed using the coordinates of the edge plus integer multiples (-1, 2). In this way we construct an array with the positions of the N/2 sites of the sublattice A, labelled by (nA,mA). In the Cartesian coordinates, the site (па,ша) corresponds to the position

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The set {Ra} corresponds to the sublattice A basis vectors of the chiral GNR primitive unit cell.

The labelling is completed with the indexing of the sublattice B in a similar manner. In coordinate space, B lattice basis vectors of the chiral graphene nanoribbons primitive unit cell read

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where δ1 is the nearest-neighbor vector, Eq. (2.4), that translates a site A to a site B at the left neighboring row, see Fig. 1.

With all the basis vectors defined, it is a simple task to determine the lattice nearest- neighbor sites, that satisfy ||Ra — Rb|| = acc, as well as the next-nearest neighbors pairs, where for A one requires ||Ra — RAH = a0.

3.2 Model Hamiltonian

Let us now present the model Hamiltonian we use to study the low-energy electronic and magnetic properties of chiral GNRs. The ground state magnetic ordering driven by electron-electron interaction has been extensively studied for GNRs with zigzag edges by a number of methods [22]. Band structures calculated by density functional theory with local spin density approximation (DFT-LSDA) [35, 49] show remarkable agreement with those obtained from a tight-binding model with a Hubbard term in the mean-field approximation [50]. Further studies treating the e-e interaction beyond mean-field, like Hartree-Fock with configuration interactions [51, 52], and quantum Monte Carlo [53, 54, 55] confirm that the mean-field approximation provides a good description of the magnetic ground state properties of zigzag GNRs.

In this study we use the Hubbard mean-field approximation to compute the elec­tronic and magnetic properties of chiral GNRs. As discussed, this simple model leads to results that agree with more sophisticated methods. Moreover, it allows to assess the ground states properties of chiral GNRs with large primitive unit cells at a very modest computational cost.

The tight-binding Hamiltonian with a Hubbard interaction term reads where α|σ and αίσ are, respectively, the creation and annihilation operators of electrons with spin projection σ at the site i, while ηί , σ = aj σαί σ is the number operator. The sym­bols (- · - ) and ((· · · )) indicate sums over nearest-neighbor and next-nearest-neighbor lat­tice sites, respectively. Note that this model Hamiltonian is different from that presented in chapter 2, which describes the main low-energy electronic features of bulk graphene.

The magnitude of the on-site Coulomb energy U in graphene systems is under current debate in the literature [28]. We adjust the model parameters t, ť and U to reproduce the band structure and the local magnetization obtained by DFT-LSDA calculations for narrow nanoribbons [50]. In our study we do not consider edge reconstructions [56] and assume that the dangling bond of undercoordinated edge atoms are passivated by hydrogen atoms, that have a similar electronegativity to the carbon ones.

One can obtain the mean-field approximation [57] of the model Hamiltonian given by Eq. (3.7) by introducing

where (ui,a·) is the average electron occupation of site i. Hence,

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where we assume that the fluctuations 5ui,σ are small and thus neglect the term containingöui>(T squared. Usingöui>(7 = ui,σ — (u^*), we write

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are independent of m due to translational invariance, namely, (uia-) = (uml,a-) = (ulo-). For sites within the same PUC, tll/,k represents the nearest and the next-nearest hopping integrals, t and ť. For neighboring sites at different PUCs, the hopping terms acquires the phase e±ika.

The occupations (ul,a-) are obtained self-consistently. The problem is defined with the help of the eigenenergies {ε^,σ} and eigenfunctions {φ^,σ(l)} of И11/,ka, defined by where ν is the band index. The transformations that diagonalizes Ик, defined by Eq. (3.13), are

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The matrix elements of the above unitary transformations are associated to the tight- binding wave functions, namely

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We define, as standard, (η^,σ) = (c^ σckνσ). For a given wave number k, the vth state occupation follows the Fermi distribution at zero temperature, namely, (nkv,σ) = Θ(μ — ekv,o-), where Θ is the Heaviside step function and μ is the chemical potential. The probability amplitudes φ^,σ (l) allow one to calculate the l site occupation for a fixed k, using

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Finally, the occupation appearing in Eq. (3.14) is obtained by integrating (nkl^) over the Brillouin zone, namely,

where VBZ is the “volume” of the Brillouin zone.

The ground state energy per unit cell, E0, is a sum of the occupied self-consistent single-particle state energies minus a standard term accounting for double counting the on-site Coulomb interaction energy, namely

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The lth site magnetization (in units of μΒ/2) is defined as

For chiral nanoribbons, one is frequently interested in the average edge magnetization, conveniently defined as the magnetic moment per edge unit length

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where a is the length of the chiral translation vector, a = \Ch\, and the sum runs over the sites of the sublattice (A or B) with the largest number of dangling bonds along the chiral GNR edge, see Fig. 9.

The local density of states (LDOS) reads

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where δΓ is the Dirac δ-function broadened over an energy range Γ, taken to be much smaller then the typical energy separation between bands. In turn, the density of states (DOS) is given by

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In the forthcoming chapter, we use this theory to study the LDOS and the formation of local magnetic moment in chiral GNRs. Before we proceed it is useful to review the band structure non-interacting electrons in chiral GNRs that has been extensively studied in the literature [9, 48].

3.3 Non-interacting case (U = 0)

Here, we discuss the main features of the low energy states of chiral graphene nanorib­bons in the absence of electron-electron interactions. We compute the band structures using the model Hamiltonian defined by Eq. (3.11) taking U = 0. In this case the disper­sion relations depends only on the chiral angle 0c and on the GNR width w.

Let us first take ť = 0. Figure 10 shows the band structure for a number of chiral GNRs upon the change of chirality from zigzag (Bc = 0°) to armchair (Bc = 30°) at fixed nanoribbon width (w = 12). We obtain these results using a Fortran code we have developed following the prescription described at the beginning of the chapter to construct the PUC. This program constitutes the basis for the self-consistent mean-field calculations of next chapter. Our results, Fig 10 reproduces those presented in [9].

As discussed in section 2.4, the high-symmetry zigzag nanoribbon exhibits a flat band at the charge neutrality point (E = 0), that spans an interval between kc < | ky | < π of the 1D Brillouin zone (BZ), with kc = 2 arccos ^2i+i/v)· Depending on their width, armchair GNRs are either metallic or semiconductor with no electronic states localized at their edges·

In Fig. 10, one observes that for all chiralities the flat band at E = 0, characteristic of zigzag GNR is modified, but is still present, except for the armchair geometry. This can be understood as follows: Using a continuous rotation of the graphene band structure, it was shown [46] that in the infinite-width limit the average edge density of states at e = 0 due to a dispersionless band corresponding to edge states is

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in the absence of electron-electron interaction. The density p0 is largest for zigzag nanorib­bons and vanishes for the armchair ones. For chiral GNRs, 0 < 0c < 300, p0 shows a nearly linear dependence on 0c. As discussed in Section 2.5, the large enhancement of p0 at the charge neutrality point is key to explain the edge magnetization in GNRs in terms of the Stoner mechanism. As discussed in Ref. [9], while the edge magnetization M is propor­tional to p0, the band gap Δ0 (for ť = 0) is related to U/t.

Alternatively, it is possible to predict the degeneracy of the zero energy bands of GNRs with arbitrary chiralities without performing any band structure calculation by following the band folding prescription put forward by Chico and collaborators [48].

The construction is based on two pivotal observations. First, as already noticed in Ref. [46], in the “minimal edges” addressed here, the edges of chiral GNRs can be viewed as a combination of zigzag and armchair termination. The armchair ones do not contribute to the zero energy flat bands. This suggests that it is convenient to write the chiral vectors, Eq. 3.2, in terms of zigzag and armchair projection, namely,

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where [S, m] indicate the zizag and armchair components of |Ch|.

The second key observation is that the zero-energy state degeneracy of a chiral [S, m] GNR is the same that of a [S, 0] GNR.

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Figure 10: Evolution of the band structure obtained by tight-binding calculations of the graphene ribbons for several chiralities obtain from our model Hamiltonian for w =12, U = 0 and ť = 0, Eq. 3.11. The chiral angles 0c for each GNR configuration (n,m) follows: 0c(1,0) = 0°, 0c(5,1) « 8.9°, 0c(4,1) « 10.9°, 0c(3,1) « 14.9°, 0c(2,1) « 19.1°,

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These elements lead to the prescription proposed in Ref. [48]. S is conveniently written as S = I + 3P, where I = 1, 2, 3 and P = 0,1, 2, · · · . For I = 1 and 2 the spectrum has a Dirac-like point at k ~ 2π/3, while for I = 3 the Dirac point moves to k = 0. For S > 3 the zero-energy states extend over the whole Brillouin zone.

The degeneracy g of the zero-energy dispersionless band is either g = 2P or g = 2(P + 1), depending on I. Figure 11 summarizes the prescription to determine g: It is simple to show that in the limit of S ^ 1 the folding rule leads to the p0 given by the Eq. 3.26.

Let us now discuss the cases presented in Fig. 10 to show how the prescription works. The chiral GNRs with S = 1 (I = 1, P = 0), corresponding to (2,1) = [1,1], (3, 2) = [1, 2] and (4, 3) = [1,3], are similar to the zigzag one, in the low energy, since that flat bands are determined by zigzag components, [S, 0]. The flat band is originated from merging a valence with a conduction band, giving g = 2 for k > kc and g = 0 for k < kc. This is in agreement with Fig. 11 (left panel). As m is increased, |kc| increases and the flat region corresponds to a smaller fraction of the Brillouin zone. The (5,1) G NR has 5 = 4 (/ = 1, P = 1). It differs from the S' = 1 by the fact that its band structure shows a flat band along the whole Brillouin zone, with degeneracy g = 2 close to ka = 0 and g = 4 at ka = π/2. The (3,1) GNR is a example of / = 2, central panel of Fig. 11, and the (4,1) GNR has 1 = 3 following the prediction of Fig. 11 (right panel).

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Figure 11: Schematic band structures of “chiral” GNRs with (S', 0) after folding the band structure o“ a GNR (1,0) with zigzag edges Stimes. The shaded areas represent title band continuum of states. The degentracies od the flat bands, namely, g = 2P and g = 2(P+1), are put in sorrtspondenct with their respective ed.es ot the Brillouin zone e“ filie chaage neutrality point. Adapted trom Red. [48].

The concepts disarased here are not direc“(y applicable for the case of ť = 0. However, despite the modifications in the dispersion relation, for realistic values of ť = 0 the localized states in the edges are preserved, leading to the formation of magnetic moment, as analysed in the next chapter.

4 Edge Magnetization in Graphene Nanoribbons

In this chapter we study the edge magnetization and the local density of states (LDOS) of chiral graphene nanoribbons using the π-orbital Hubbard model in the mean-field approximation introduced in section 3.2. We show that the inclusion of a realistic next- nearest-neighbor hopping term in the tight-binding Hamiltonian changes the graphene nanoribbons band structure significantly and affects its magnetic properties. We study the behavior of the edge magnetization upon departing from half filling as a function of the nanoribbon chirality and width. We find that the edge magnetization depends very weakly on the nanoribbon width, regardless of chirality as long as the ribbon is sufficiently wide. We compare our results to recent scanning tunneling microscopy (STS) experiments reporting signatures of magnetic ordering in chiral nanoribbons [8] and provide a new interpretation for the observed peaks in the local density of states. Our interpretation, in distinction to other works [9], does not depend on the antiferromagnetic inter-edge interaction.

4.1 Nanoribbons with zigzag edges

The origin of the ferromagnetism along the GNR edge is due the competition between kinetic and Coulomb energy [37]. Ignoring the later altogether, we already discussed in Chapter 2 that the zigzag terminated GNRs are zero band-gap semiconductors with an almost flat dispersion close to the charge neutrality point. The latter gives rise to a remarkably sharp peak in the density of states. Early theoretical studies [20], already showed the zigzag edges host localized states. The large density of states combined with the corresponding Coulomb energy results in a Stoner instability that leads to a ferromagnetic ordered state along the edge.

We begin by presenting the band structure and magnetic properties of zigzag GNRs.

Part of this material can be found in the literature (see, for instance, Refs. [22, 32] for a review), but the analysis we present serves as an important guide for the subsequent discussion of the chiral GNRs results.

As qualitatively discussed above, for U = 0 and ť = 0, the dispersionless edge modes enhance dramatically the LDOS at the GNR charge neutrality point [20]. When U = 0, due the Stoner mechanism, the large LDOS at the GNR edges give rise to a local magnetization and the electronic band structure shows a gap around e = 0 for μ = 0 [20]. See Fig. 12a. The ground state shows a parallel spin alignment along each edge and antiferromagnetic inter-edge order. This is consistent with the Lieb’s theorem [30] which asserts that the ground state of the Hubbard model of a bipartite lattice with nearest-neighbor hopping has spin S = 0.

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Ab initio calculations of zigzag GNRs band structures do not show particle-hole sym­metry [35, 49]. The Hamiltonian (3.11) successfully reproduces the DFT band structure dispersion relations in the vicinity of the charge neutrality point at the expense of tak­ing ť = 0 [50]. In this case, Lieb’s theorem [30] is no longer applicable and the natural question to ask is how robust is the ground state antiferromagnetic phase.

This issue is partially understood by a closer analysis of the localized edge states as a function of k. For U = 0, the lowest energy |e| modes become dispersionless at k > 2π/3α, see Fig.12a and the Section 2.4. As k increases the states become increasingly localized at the GNR edges. The consideration of NNN shifts the Fermi level 0.3ť [58], we shift so as put the Ef to charge neutrality point. This behaviour persists when U = 0. Accordingly, one spots two characteristic gaps in the GNRs dispersion relations, see Fig. 12. The band gap, Δ0, occurs at the vicinity of the edge localization transition. The gap Δι at k = π/a is more relevant to the analysis of the system magnetic properties. That k-point corresponds to the most localized states along the GNR edges, which dominate the Stoner magnetization criterion.

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Figure 13: Edge magnetization Mi of zigzag graphene nanoribbons as a function of their width N. Inset: Band gaps Δ0 and Δι versus N. In both cases U/ť = 1.0 and ť/ť = 0.0, 0.1 and 0.2.

Based on this argument, one expects Δι and the edge magnetization Mi to be related. This is indeed observed in Fig. 13, that shows Δ0, Δι, and Mi as a function of the zigzag GNR width, here conveniently expressed in terms of N the number of zigzag chains crossing the ribbon transversal direction, namely, W = (λ/3Ν/2 + 1Д/3)а0. While Δ0 decreases with increasing GNR width, Δι and Mi show a weak N dependence. We stress that neither Δι nor Mi show a significant dependence on ť.

Before proceeding, it is worth to notice that Figs. 12b and 12d anticipate some impor­tant features we discuss below, in the analysis of the STS spectra of chiral GNRs. While in the particle-hole symmetric case (t = 0) the lowest energy |ε| peaks in the density of states can be clearly associated with spin-polarized states, this is not true for t = 0. In Fig. 12d, the peak at t/t ~ -0.19 corresponds to the van Hove singularity of a band unaffected by turning on the interaction U, while the peak at t/t ~ -0.22 is the one related to a magnetic state.

Let us now study the behavior of the edge magnetization M1 away from the charge neutrality point, or more precisily, as a function of the chemical potential |μ|. For μ values close to Δι/2 the states with opposite spin orientation with respect to the ground state start to be occupied and the edge magnetization to be suppressed. This is nicely illustrated in Fig. 14, that shows that Mi vanishes for |μ| > Δι/2. Our calculations also indicate that Δ1 slowly decreases with increasing doping (not shown).

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Figure 14 shows that the edge magnetization M1 scaled by Δ1 as a function of the chemical potential μ shows a universal-like behavior for N ^ 1. For ť = 0, Μ1/Δ1 is an even function of μ/Δ1. For very narrow GNRs (N < 20), Μ1/Δ1 versus μ shows a maximum value, corresponding to a plateaux of a width of the order of μ/Δ0. With increasing width, A0 decreases and so does the plateaux width. Figure 14a shows that, already for N > 30, the edge magnetization no longer depends on N. The results are qualitatively similar for the more realistic case of ť = 0 [50], as illustrated by Fig. 14b.

A lot of attention has been devoted to the study of the competition between the anti- and ferromagnetic phases in GNR [59, 60]. Within the Hubbard mean field (for ť = 0) approximation, Jung and collaborators [37] studied the antiferromagnetic inter­edge superexchange interaction to estimate the energy difference ΔΕ = Ε^Μ — E^™ between the antiferromagnetic ground state and the energetically lowest ferromagnetic configuration. A good fit to the numerical calculations is

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where α = 0.245 and C = 38.9 for ť = 0, while α = 0.198 and C = 45.9 for ť/t = 0.1. These values are smaller than the values reported in Ref. [36], but in line with those of Ref. [37] for ť = 0 and U = 2.0 eV. Note that ΔΕ becomes comparable with kBT at room temperature for N > 10. Hence, for most experimental GNR samples currently available, where N ^ 1, one expects inter-edge interaction to be negligible.

For doped systems, the Lieb’s theorem [30] does not apply and ground state phases other than the antiferromagnetic one are allowed. Some authors [59, 60] have extensively studied the ť = 0 case and found a very rich phase diagram for zigzag GNRs of 10 < N < 30. In our calculations, as long as ť = 0, we only observe antiferromagnetic ground states for the whole range of doping |μ| < Δι/2.

A more interesting question to ask is: What is the ground state configuration for ť = 0? In this case, the ferromagnetic ground state at half-filling is no longer forbidden by the Lieb’s theorem. For realistic values of ť/t [50], our calculations only lead to (interedge) antiferromagnetic ground states at the charge neutrality point, but we find other phases very close in energy. We conclude that, except for very narrow GNRs, the experimental assessment of this phase diagram is very daunting. For this reason, we focus our study on the spin alignment along a single edge.

4.2 Magnetization in GNR with chiral edges

As discussed above, for zigzag GNRs the next-nearest-neighbor hopping term modifies and reduces the edge LDOS at the charge neutrality point, but surprisingly it does not change the magnitude of edge magnetization. Is the scenario the same for chiral GNRs and how robust the edge magnetization in this case? These are the issues we address in this section.

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Figure 15: Electronic band structure for GNRs of width w =12 and U/t = 1, with chiralities (a) (2,1) and (c) (3,1). Corresponding density of states for the (b) (2,1) and (d) (3,1) chiralities. The solid lines stand for the case of ť/t = 0.1, while the dashed ones for ť/t = 0.0. The energy gaps Δ0 and Δ! are only indicated for the ť/t = 0.1 case.

In Fig. 15 we show the band structures obtained using the model Hamiltonian of Eq. (3.11) for the chiralities (a) (2,1) with 0c = 19.1o and (b) (3,1) with 0c = 13.9o. Here, we take U/t =1 and consider the cases of ť = 0 and ť/t = 0.1, both at half-filling. As before, ť = 0 breaks the particle-hole symmetry. As a result, the band gap Δ0 goes to zero in most cases, even for very narrow nanoribbons. In distinction to the zigzag case, the k-point corresponding the maximally localized states at the edges depends on the chirality, namely, ka = π (Fig. 15a) for the chirality (2,1) and k = 0 (Fig. 15c) for (3,1).

Accordingly, we define Δ! as the energy gap around c ~ 0 calculated at the k point, corresponding to maximally localized edge states. Figure 16 shows Δ! as a function of the GNR width w for the chiralities (3,1) and (2,1). We find that Δι is independent of

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Figure 16: Gap Δι as a function of the GNR width w for different next-nearest-neighbor hopping parameters ť/ŕ. Here we use U/t = 1.0.

ť, within the parameter range that fits the DFT calculations [50]. Hence, the numerical results indicate that ť = 0 does not significantly change the edge localized states.

Figure 16 also shows that Δι increases with w for very narrow nanoribbons and becomes almost independent of the GNR width for w > 10. Other chiralities show a similar behavior (not show here). These observations make possible to relate our findings based on calculations for GNRs of w < 20 to ribbons with experimentally realistic sizes, where w « 20 · · · 50 [8].

We now turn to the analysis of the edge magnetization as a function of doping (or chemical potential μ). Figure 17 shows the magnetic moment per edge unit length M versus the chemical potential μ, scaled by Δι. (We use U/t = 1.0.) We find that, for sufficiently wide GNRs (w > 10), the magnetization M/Δi as a function of μ/Δι becomes independent of w. This behavior is obtained for both the ť = 0 and ť = 0 cases. Figure 17 indicates that M is not a smooth function of μ. The reason is that the inter-edge antiferromagnetic phase is no longer always the ground state of GNRs away from half filling. The phase diagram is very rich, but M does not change appreciably. For this reason we did not pursue this line of investigation.

The chiralities we address above show a strong resemblance with GNRs with zigzag edges. We find that by increasing 0c, for sufficiently wide GNRs, the edge magnetization M decreases almost linearly with 0c and, as expected, vanishes for armchair terminations. The other limit is more interesting. For U = 0, by decreasing 0c, one increases S and the degeneracy of the zero-energy states modes. For U = 0, these states split and give raise to a complicated band structure around half-filling, as illustrated in Fig. 18a for the (8,1)

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Figure 17: Edge magnetization M/Αι as a function of the chemical potential μ/Δι for chiral nanoribbons of different widths w for (a) t'/t = 0.0 and (b) ť/ŕ = 0.1. In both cases U/t = 1.

chirality. In the notation introduced in the previous chapter, S = 7,P = 2, and I = 1. The corresponding edge magnetization M as a function of μ is shown in Fig. 18b. The latter clearly indicates that the nearly dispersionless modes of Fig. 18a are the ones that contribute most to M.

4.3 LDOS in chiral graphene nanoribbons

As already pointed out in the introduction, the current experimental evidence for edge magnetization in GNRs is indirect: The local density of states measured by scanning tunneling spectroscopy (STS) in graphene nanoribbons is claimed to show a behaviour consistent with the theory for a variety of chiralities [9, 8].

The STS data main features are the following [8]: When the tip is placed at the GNR edge the measured spectra display two clear peaks close the charge neutrality point. As the tip is moved away from the edge, the peak amplitudes are quickly suppressed. By

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Figure 18: (a) Band structures of a (8, 1) chiral graphene nanoribbon of w =12 and (b) edge magnetization M/a0 as a function of the chemical potential μ/t. The dashed (red) lines stand for the case of ť/t = 0.0 and the solid (blue) ones for ť/t = 0.1.

moving the tip parallel to the edge, the peak amplitudes show modulations, with a period consistent with the size of the translation vector, a = |Ch| [8]. In general, the peak heights show a large asymmetry that remains unexplained.

The experimental peak spacing has been associated with Δ0 [9, 8]. For that, it is necessary to take U = 0.5t, a value somewhat smaller than the conventional one [50], based on the argument of screening effects due to the metallic substrate. It is argued that the opening of an inelastic phonon scattering channel at |e| =65 meV makes hard to observe higher energy peaks in the DOS. In what follows, we discuss how the nnn hopping term changes this picture.

In Fig. 19 we present the local density of states (LDOS) as a function of energy t/t for a GNR of chirality (3,1). The LDOS is calculated along the edge (referred as y = 0) and inside the ribbon along the longitudinal orientation (y = 3a0, in red). Figures 19a and 19b correspond to the ť/t = 0 and ť/t = 0.1 cases, respectively. We use U = t. The LDOS decreases exponentially with increasing y, indicating that the peaks in the LDOS correspond to edge states. Note that, for the case of ť/t = 0.1, that best reproduces DFT band structure calculations, the LDOS peak amplitudes become asymmetric and the peak spacing can be understood in terms of Δ0 and Δι, defined in Figs. 15c and d.

The GNR chirality (8,1) is experimentally analyzed in detailed in Ref. [8] (see, for example, Fig. 2c therein). Its corresponding low-energy band structure is more complex than that of the (3,1) case.

Figure 20 shows the LDOS of a GNR with chirality (8,1). From Fig. 18 we infer that the states that contribute most to the edge magnetization are those corresponding to the dispersionless bands. For ť = 0, the latter are located at é[/t ~ ±0.08 and e^/t ~ ±0.12. Accordingly, Fig. 20a shows that the LDOS peaks at the energies e^'/t and é^/t are the ones that are most localized at the GNR edges. For the more realistic parametrization corresponding to ť/t = 0.1, the states that predominantly drive the magnetization M (see Fig. 18b), are the flat bands at e\/t & —0.18, e|/t « 0.02, and e\/t & 0.04. The

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LDOS peak at tb4/t — 0.05 corresponds to a van Hove singularity of an ordinary band, that is, a band whose states are not localized at the GNR edges. Hence, the band gap Δ0 involves localized and delocalized states.

The comparative between experimental data Ref. [8] and the results from our model is shown in Fig. 21. The nanoribbon can be synthesized by unzipping of carbon nanotubes Fig. 21a and shows low symmetry along the edges, i.e. they are chiral. The measures by scanning tunneling spectroscopy (STS) shows dl/dV values in Fig. 21b, the peaks are more higher as we approach the edge , and finally goes to zero when the gold substract is found. This is evidence of localized edge states in chiral as theoretically predicted. The dI/dV spectra obtained at different positions (as marked) near the edge of the (8,1) GNR pictured in Fig. 21a. As explained in the previous paragraph the LDOS with energy in ¿b/t ~ 0.02 (with purple star), and /t ~ 0.04 (with blue star) Fig. 21d are the ones that are most localized at the GNR edges and display similar behavior with Fig. 21c, an dI/dV

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Figure 21: (a) Scanning tunnelling microscopy (STM) of the terminal edge of an (8,1) GNR, Ref. [8]. (b) dl/dV spectra obtained at different positions (as marked) of the

GNR edge shown in (a) along a line perpendicular to the GNR edge. (c) Shows a higher resolution dl/dV spectrum for the edge of a (5, 2). (d) The LDOS results by our model of chiral ribbon (8,1) as shown previously in Fig. 20 with blue (purple) star highlighter in вз/t = 0.04 (e2/t = 0.02).

for chiral edge (5,1) and is the inset of the measures by scanning tunneling spectroscopy shown in Fig. 21b for chiral edge (8,1).

Let us now examine the local magnetization along the edges. In Fig. 22 we select values of t/t corresponding to representative sharp peaks of Fig. 20 and plot the corresponding LDOS and edge magnetization M as a function of x, the position oriented along the GNR edge. The case t/t = 0 corresponds to Fig. 22a and ť/t = 0.1 to Fig. 22b. These figures indicate that the edge magnetization M and LDOS corresponding to the dispersionless states at t/t = 0.11 (for t/t = 0) and t/t = 0.04 (for t/t = 0.1) display a very similar behavior with x. This observation gives further support to the discussion of the previous paragraph, corroborating the picture that, for S ^ 1, the splitted dispersionless states dominate the edge magnetization.

In summary, our calculations indicate that the interpretation of STM/STS data is very different for ť = 0 as compared with the more realistic parametrization, where t/t = 0.1. For the (8,1) chirality we show that next-nearest-neighbor with t/t = 0.1 leads to an

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Figure 22: Local density of states (logarithmic scale) and edge magnetization (linear scale) M of a graphene nanoribbon with chirality (8,1) for (a) t'/t = 0 and (b) t'/t = 0.1. Here, w = 12 and U/t = 1.

energy peak spacing Δ = — e\ æ 20meV and peaks hight asymmetry that is consistent with the experimental results Fig. 21b, with Δβχρ = 23.8 ± 3.2meV, reported in Ref. [8]. These observations suggest that further experimental input is needed to establish a clear theoretical picture.

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5 Quantum Anomalous Hall Effect in Graphene

In this chapter we change the emphasis on the emergence of magnetic properties in GNRs to put forward a proposal to observe Quantum Anomalous Hall Effect (QAHE) in graphene by combining spin-orbit and exchange field interactions. This study was developed under the supervision of Prof. Luis Brey (Madrid) during my 9 month visit to Spain in the framework of a “Doutorado Sanduiche” supported by CAPES.

The chapter is structured as follows: We begin with the standard Hall effect and the anomalous Hall effect. We then review the main features of the Quantum Hall Effect (QHE) and present the topological approach to calculate its conductivity [61]. Next, we introduce the general model Hamiltonian to study the QAHE and review the results obtained by MacDonald and collaborators [62]. Finally, we present our model and results.

5.1 Hall effect

Let us begin with a short historical note. In 1879, E. H. Hall observed that electrical currents in conductors placed in a magnetic field develops a voltage across the sample, in a direction perpendicular to the current and the magnetic field. This voltage is known as Hall voltage (VH). Vh is found proportional to the magnetic field and to the current density [13].

This effect can be explained in simple terms by a classical analysis of the dynamics of the charges carriers under a magnetic field perpendicular to the conductor and an in­plane electric field E [13]. The electrons are accelerated by the applied electric field and deflected by a Lorentz force. As a result, there is an accumulation of charges at the edges of the conductor. The accumulated charges create a transverse electric field, the Hall field, that opposes further accumulation of carriers and counteracts the force of the magnetic field. Figure 23 illustrates the forces acting on the charge carries of a bulk conductor, indicating the Hall field and the charge accumulation at the conductor edges.

The Hall effect has several applications. The most standard are in instruments, called Hall probes, which are used to measure magnetic field through the Hall voltage, and to determine the sign of the charge carriers in a conductor [63]. The direction of the conventional current is arbitrarily defined as the same one as that of the positive charge carriers flow.

Assuming an electron mean scattering time τ, the Drude model [13] allows to calculate the magnetoresistance of a Hall bar device in the high temperature (classical) regime. For a conductor where the charge carriers have an effective mass m* and a sheet concentration ns, the conductivity tensor reads [65] where wc is the cyclotron frequency, wc = |e|Bz/m* and σ0 is the Drude conductivity in the absence of a magnetic field. The above resuls are obtained for E = Exe3 and B = Bze3. Alternatively, one can express the transport properties in terms of the resistivity, namely The transverse component pxy is called Hall resistivity, it is independent of t and linear in B. The longitudinal component pxx is just the inverse Drude conductivity σ0 and does not depend on B.

The Hall effect in a ferromagnetic sample is frequently much larger than the one in ordinary materials and has a different nature depending on the magnetization of the material. For that reason it is called Anomalous Hall Effect (AHE) [66]. The total Hall resistivity is the sum of the normal and the anomalous contributions, namely,

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where B — μ0(Η + M). While the first term is the normal Hall resistivity discussed above, with a linear B dependence, the second one pAN grows with the magnetization M of the material. Hence, one needs a spin-orbit interaction to convert the direction of magnetization to a preferred direction of the electron motion. There are two main theories to explain the AHE: The extrinsic one that requires (non-magnetic) impurities to produce the spin-orbit coupling SOC and the intrinsic scenario, proposed by Karplus and Luttinger [67]. The latter can be related to the Berry curvature and to topological aspects [68]. This is the scenario we explain in our analysis of the quantum anomalous Hall effect (QAHE) in the next section. Before doing that let us discuss the topological aspects of the normal QHE.

The Quantum Hall Effect (QHE) was first observed in 1980 [61], almost 100 years after of the classical Hall effect. In two dimensions the spectrum of electronic systems at high magnetic fields is strongly degenerated, characterized by Landau levels. At sufficiently low temperatures, much smaller than the Landau level spacing, the transport properties show remarkable features: The Hall resistivity shows well pronounced plateaus, namely

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where N is an integer number and Rk — h/e2 is the Klitzing constant (or resistance quantum) measured with a relative accuracy of 10-8. This effect can be explained by the Landauer-Buttiker picture by considering edge states [65]. Thouless, Kohmoto, Nightin­gale and Nijs (TKNN) proposed an alternatively explanation based on a topological theory [61]. This is one of the approaches we adopt in our analysis of the QAHE.

The theory developed by TKNN uses the Kubo formula to show that the Hall conduc­tivity is related to the Berry curvature of the Bloch states. The TKNN theory was later cast in terms of topological elements [68] to the form that it currently known [69]. Here, we present a simple semiclassical derivation of the TKNN formula in order to introduce the main elements of the theory [68].

Let us consider a two-dimensional electron gas (2DEG) under a strong perpendicular magnetic field. The system is driven out-of-equilibrium by a weak in-plane electric field that we treat as a perturbation. We choose a gauge where E = —dAE/dt, AE = -Et. The model Hamiltonian [70] reads

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where B = Vx A0 and V¡at is the lattice potential. The above Hamiltonian has a standard quadratic dispersion, but since the main ingredients one is interested in are of topological nature, the results are applicable to systems with linear dispersion relation [Bernevig2013], such as graphene.

Let us change representation and use a k-dependent Hamiltonian, H = e-ibrHeibr, and introduce periodic Bloch wave functions unk(r), eigenstates of H|unk(r)) = Enk|unk(r)), to calculate the electron group velocity vn(k) and the Hall conductivity. We assume that the eigenvalues and eigenfunctions of the problem

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in the absence of the external electric field are known. Note that the external magnetic field breaks lattice symmetry. Hence, to solve the Schrödinger equation one needs to use the so-called magnetic translation symmetry (we recommend Ref. [71] and references therein). The electric field is treated as perturbation. In first order perturbation theory (see Appendix of Ref. [71])

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where the crystal momentum k(t) = k° — ^ encodes the effect of the external electric field. For notational convenience we introduce the abbreviations: |n) = |un k(t) ) and en = E°o k. The velocity operators are defined by v = Г = [H, r] |. In the k-representation it reads v(k) = e-ibr[H, r]|eik^r = [d/d(hk),H\ = dH(hk)/d(kk). Hence, the average group velocity in the nth-band is

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The first term is the group velocity in the absence of the perturbation caused by the external electric field. It does not contribute to the current since it does not contain any non-equilibrium information. As shown above, [д/д(hk),H] = дН/дk, hence one can write

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with 6αβΎ the Levi Civita symbol. Qn is the Berry curvature, see Appendix A, in momen­tum space. For a 2DEG and, in general, for two-dimensional systems, Qn = Hnz.

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The Hall conductivity reads

TKNN showed [61] that the integral of the Berry curvature over the Brillouin zone divided by 2π is always an integer number Cn. This integer number, called Chern number, characterizes the topological nature of Bloch states in the two-dimensional Brillouin zone.

5.2 Model for the QAHE in Graphene

In the quantum anomalous Hall effect the Hall conductance is quantized for bulk insulators in the absence of an external perpendicular field and is related to a bulk topo­logical number [61, 68]. Unlike the quantum spin Hall effect, proposed by Kane and Mele [72], the QAHE does not require the conservation of the time-reversal symmetry. On the contrary, the presence of internal exchange field plays a key role. However, in distinction to the quantum Hall effect, its nature is not purely orbital due to external magnetic field [62].

The QAHE in graphene can arise due to two key elements: a Rashba spin-orbit and exchange fields. Next we discuss the origin of both separately.

5.2.1 Exchange Field: Zeeman-like term

In this Section we present and justify the simple model we employ to account for exchange fields. In the context of graphene, the model is motivated by extrinsic effects such as using a ferromagnetic insulating substrate or by depositing magnetic dopants or adding defects to the graphene sheet.

The exchange interaction may be thought to give rise to an effective field called the exchange field, molecular field or Weiss field [73]. This can be understood by taking the mean-field of the Heisenberg model (based on the exchange interaction between local electrons) [13], namely

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where Si is the spin operator on the jth site and Jij is a coupling constant related to the exchange contribution to the microscopic electronic Hamiltonian [74]. If we assume that the coupling is J and acts only on nearest neighbor sites, it is a simple exercise to obtain a mean-field model: One considers the effect of all other spins on a single atom of the lattice

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where z is the coordination number. Hence, one can define an effective field BW as

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where g is the Lande factor and μΒ is the Bohr magneton. As a result the mean-field

Hamiltonian on a single site i reads

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For simplicity, let us assume Bw is oriented along the z axis, Bw = BWez. We then obtain the exchange field for a single atom

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The above Hamiltonian has the form of the Zeeman term that describes the splitting in several components of the spectral lines emitted by atoms in the presence of an external static magnetic field. In the Appendix B.2 we shall take a brief historical note on the development of the understanding of the theory of the Zeeman effect. We point out, based in the literature [73], the distinction between the effect on electronic level splittings of a “real” external magnetic field and the exchange field.

In the case of interest, Sz can be represented by the Pauli spin matrix σζ. Hence, the Hamiltonian defined by Eq. (5.19) leads to a splitting of the energy bands: The states with spin f are shifted in energy by λ, while the ones with Sz =X are shifted by -λ. In general, the splitting in a two dimensional sample depends on the three components of the field. However, considering only the z-component, we model the exchange field contribution as where a =f, X stands for the spin orientation along the z-direction.

The effect of the Zeeman-like exchange interaction in the band structure of zigzag nanoribbons is represented in Fig. 24. Note the overall effect is a constant energy shift of the spin up and spin down bands.

In distinction to the last chapter, where the Hubbard mean-field Hamiltonian favored an antiferromagnetic ground state, here the Zeeman-like interaction tends to align the spins along the same orientation. (Note that since in the study of the QAHE we are interested in bulk properties, here we neglect the effect of electron-electron interactions.)

Let us discuss the spin properties of the states near the charge neutrality point. Figure 25 shows the spin projected wave function probability distribution of the band indicated by red in Fig. 24. For |ky | > 0.7π/α, corresponding to a dispersionless behavior, the states are localized close to the edges of the GNR with a strong spin polarization X, as shown in the upper panel of Figure 25. At |ky| ~ 0.7π/α, there is a band crossing and the states

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corresponding to |ky | < 0.7π/α have α =/ polarization, as illustrated by the bottom panel of Figure 25

5.2.2 Spin-Orbit Coupling

The spin-orbit coupling (SOC) [75, 73] consists of the interaction between the electron spin magnetic dipole moment and the internal magnetic field related to the electron orbital angular momentum. In Appendix B we present a historical note where we discuss the SOC effect in atomic physics and the Rashba SOC in solid state.

In graphene, Kane and Mele in a seminal paper [72] have proposed that intrinsic spin­orbit can give rise to spin-polarized edge states in a new phase of matter, the quantum spin Hall phase [72]. Unfortunately, realistic calculations found that the intrinsic SOC in graphene is too weak to make possible a physical realization of the effect [11]. Several authors have shown [11, 62] that the quantum spin Hall effect can still be produced in graphene by extrinsic Rashba spin-orbit coupling, that is found to be two-three orders of magnitude larger than the estimated intrinsic SOC [76].

The microscopic Hamiltonian describing the Rashba spin-orbit coupling in graphene was introduced by Ref. [72] and deduced by tight binding model Ref. [77] where both the π and σ bands of graphene were considered with an intra-atomic spin-orbit coupling tSO

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in the presence of a perpendicular electric field Ez. The Rashba SO term reads

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where aj (bj,ß) creates (annihilates) an electron with spin projection α(β) at the i—th (j—th) site of the sublattice A (B), σ are the Pauli matrices and the vector Uj [78] is where E is the effective electric field on the graphene sheet, acc is the carbon-carbon distance, and δ] = RA — RB is a vector between the two adjacent sites (i,j), see Eq. (2.1) and (2.3).

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where the strength of the spin-orbit coupling is denoted by tSO.

To include the spin degrees of freedom and the Rashba term one doubles the basis dimension of Eq. (2.10), namely, (a^, Щ, a¿, b¿). The full Hamiltonian in matrix form reads

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where 0o(k) is the hopping, defined in Eq. (2.16), and the Rashba matrix elements are Those matrices elements are almost the same as the ones reported in Ref. [78]. The only difference is in a sign definition of the wave-vectors. Equation (5.24) can be diagonalized analytically, but unlike the Zeeman Hamiltonian which can be separated in matrix blocks, the presence of a Rashba interaction mixes the blocks, hybridizing the bands. The band structure of bulk graphene in the presence of a Rashba term, as the one described above, is shown in Fig. 26.

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The Rashba spin-orbit interaction strongly affects the dispersion relation near the two Dirac points. It does not open a gap, but depending on the magnitude of the coupling parameter the Rashba term causes a crossing between the valence and the conductance bands. This is illustrated in Fig. 26 for the Rashba parameter tSO = 0.4t [78]. Away from the charge neutrality point, the Rashba Spin-Orbit (RSO) leads to a trigonal warping in the energy bands in monolayer graphene graphene sheets, as well as, to the coupling between different layers in bilayer graphene [76].

In graphene zigzag nanoribbons the Rashba spin-orbit interaction produces an average magnetization perpendicular to the nanoribbon plane, suggesting that the RSO coupling could be used to produce spin-polarized currents [78]. Our numerical calculations confirm this statement.

The band structure of zigzag graphene nanoribons with a Rashba SO look very similar to that of pristine zigzag GNR, as it is shown in Fig. 27.

The effect of extrinsic spin-orbit effect in zigzag nanoribbons is better noticed in the local spin polarization, as illustrated in Fig. 28. There we show the expectation value of the spin operator Sz as a function of the transversal site position. The (Sz) is defined as (Sz(ky, n)) = |фку,t(n)|2 — |фку4(n)|2 for the wave function of the highest valence band of a ribbon with w = 32 and tSO = 0.1t. The RSO interaction produces a spin polarization on the edge states of the zigzag ribbon in agreement with previous theories [78]. Our results suggest that at the values of wave vector π/α and —π/α, for some tSO parameters, the system tends to a ferromagnetic configuration, besides the population of the sites of “bulk”.

5.3 Quantum Anomalous Hall effect in a constant ex­change field

Let us now discuss how the combination of the Rashba spin-orbit coupling term and the exchange field, given respectively by Eqs. (5.23) and (5.20), generate the quantum anomalous Hall effect in bulk graphene. This theory presented in this subsection follows the proposal put forward in Ref. [62]. Our numerical calculations reproduce their results (which consists in an appropriate check for our codes).

The full tight-binding Hamiltonian is given in second quantization by where cja(cia) creates (annihilates) an electron with spin projection a at the ith site, σ are the Pauli matrices, Sij is defined in Eq. (5.22). Finally, tSO and λ* are the Rashba spin-orbit coupling and the exchange field parameters, respectively.

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Figuro 28: Expectation value of the z projection of the spin operator. (SZ}. as a (unction of position across the ribbon and crystal momentum ky. Top: tSо = 0.1t, Middle: tSO = 0.2t, Bottom: tSO = 0.3t.

Alternatively, one can express the above Hamiltonian in matrix form as

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which in φ0 is the electronic hopping, Eq. (2.16), λζ is the Zeeman energy, and φ± are the Rashba terms.

The evolution of the dispersion for graphene with Rashba and exchange field is shown in Fig. 29.

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reciprocal lattice vectors given by Eq. (2.8) and (2.9): (Black) Only hopping, tSO = λζ = 0, the same as Fig. 2. Here the spins states are degenerate; (Blue) Only with the Zeeman-like term on λζ = 0.4t. The spin-up (spin-down) states have an upwards (downwards) energy shift, proportional to λζ; (Green) Rashba SOC parameter tSO = 0.1t and λζ = 0; (Red) with both interactions tSO = 0.1t and λζ = 0.4t. In this last case there is a bulk gap, but this gap is non-trivial (see text).

Niu and collaborators [62] analyzed the topological properties of the bulk Bloch states in situations where the Rashba SOC and the exchange field are combined by calculating where the summation is taken over all occupied bands below the bulk gap, un = En/h, and vx(y) is the velocity operator, previously discussed. This integral has been evaluated numerically [62] and analytically in the long wave length limit [11] to obtain where sgn(Az) is the sign of the exchange field parameter. The Chern number C =2 corresponds to the number of gapless chiral edge states along an edge of the 2D system, a nanoribbon has four in total.

We have reproduced the analytical calculation of the Chern number found in Ref. [11], but not the numerical one reported in Ref. [11, 62].

Let is now switch to nanoribbons. Similar to the quantum Hall regime, the analysis of the edge states provides an alternative way of explaining the QAHE. Figure 30 shows the band structure of a zigzag nanoribbon of 100 sites across the transversal direction, calculated using Eq. (5.27) The parameters used here are tSо = 0.0471t and Az = 0.1885t. For a given Fermi energy in the “bulk” gap, there are four edge states labelled as (a), (b), (c) and (d). This is in agreement with Ref. [62].

In Fig. 31 we plot the probability density profile of the edge-state wave functions \фп,ку (i)\2 at E = 0.0125t, associated to the four different edge states channels labelled as (a), (b), (c), and (d).

From the group velocity v(k) = ^fk^- Ref. [74] obtained by inspecting Fig. 30, one finds that electrons in the states (b) and (d) propagate along the same direction +y (increasing y), while in (a) and (c) they propagate along — y (decreasing y). This leads to a quantized Hall conductance axy = 2e2/h.

From the density profiles of Fig. 31 one finds that the states (a) and (c) [(b) and (d)] are localized along the left [right] edge. This observations agree with those for ribbons with 800 sites in the PUC, presented Ref. [11]. In this case the edge states are topologically distinct from the helical edge states of the quantum spin Hall effect, where opposite spins

illustration not visible in this excerpt

Figure 30: Band structure of zigzag graphene ribbons of width 100 sites in the primitive unit cell (PUC) with tSо = 0.0471t and λζ = 0.1885t. The Fermi energy EF = 0.0125t indicated in red corresponds to four different edge states, namely, (a), (b), (c), and (d). propagate in opposite directions along the same boundary [62].

5.4 Quantum anomalous Hall effect in a periodic ex­change field

The Hall conductance of the QAH effect depends on the exchange field parameter sign, Eq. (5.30). A natural question to ask is how the topological states change with parameter along of the system. The answer can be obtained from a systematic study of the wave functions, by Rashba spin-orbit coupling under a periodic exchange field.

We consider a graphene monolayer with an exchange field coupling that changes sign periodically along the system. To model such system we impose periodic boundary condi­tions. In way, graphene can be seen as a nanotube with a very large radius (that is, with­out curvatures effects). In our study we consider the graphene with boundary condition (graphene-BC), that correspond a large armchair carbon nanotubes (ACNT) without the

illustration not visible in this excerpt

Figure 31: Top: Band structure of zigzag graphene ribbons of width 100 sites in the PUC with tSO = 0.0471t and λζ = 0.1885t. The Fermi level EF = 0.0125t in red corresponds to four different edge states (a), (b), (c), and (d). The probability density |Ψ|2 across the width for the four edge states are shown: (a), (c) states are localized at the left boundary, the opposite for (b) and (d) states.

curvature effects. The radius of the generic nanotube is a0Vn2 + m2 + nm/(2π), which is the length of chiral vector Eq. (3.1)1 divided by 2π. The graphene-BC of width w = 100 (w = 200) carbon atoms along the circumference has ~ 13.6 nm (æ 27.1 nm) of diameter. Then, a graphene with boundary condition represents our large unit cell with translational vector in ky and with kx = 0.

In Fig. 32 we compare the evolution of the band structures of bulk graphene (in red) with an graphene-BC of width w = 100 (in black). In both cases we consider a Rashba SOC and a constant exchange field.

The dispersion relations of the graphene in red, Fig. 32, is centered on Dirac points K and K'. The structure bands comparing graphene and graphene-BC reveals that the behavior of the bands is radius—independent, i.e., for different values of width w the

1The chiral vector in nanoribbons characterizes the edge of nanoribbon and corresponds the same translational (longitudinal) vector. A nanotube is in turn characterized by the transversal section that is the chiral vector.

illustration not visible in this excerpt

Figure 32: The fitting band structure of graphene and graphene with boundary condition of width diameter = 300acc (Ch(100,100)) for: 1) tSο = Az = 0 shows degenerate spins states; 2) Only exchange field Az = 0.4t the spin-up (spin-down) states are shifted upwards (downwards) in energy; 3) Only Rashba SOC tSO = 0.1t; 4) Both interactions tSO = 0.1t and Az = 0.4t showing a non trivial bulk gap (see text). graphene with boundary condition bands follows the graphene bulk ones. For kx one obtain the same characteristic band structure for grapheme (in red). The behavior for bands in kx direction for our primitive unit cell (PUC) is performed by a fixed ky and a translational vector in kx between the PUC with 100 sites. The results follows the graphene bulk ones in kx direction (not shown).

The periodicity in the exchange field is defined as follows:

where Ap(i) defined by Eq. (5.31) is the new parameter in the exchange field Hamiltonian of Eq. (5.19). Here x(i) gives the location of the site i in the x-axis and acc the carbon­carbon distance. This means that the exchange field at the site i = 1(i = w/2 + 1) corresponds to a coupling value Ap = aZ0) (Ap = — aZ0) ). The acc is included to fulfill the periodic boundary conditions.

The periodic exchange field creates a system in which there are two zones with distinct signs in the interaction. The dispersion relations for two width of graphene with boundary condition under periodic exchange field aZ0) = 0.0471t and constant tSO = 0.1885t are

illustration not visible in this excerpt

Figure 33: The band structures of graphene with boundary condition under periodic exchange field λΖ0) = 0.0471t and tSо = 0.1885t. The Fermi level EF = 0.01157 gives eight different “edge” states. The widths are w = 100 and w = 200.

shown in Fig. 33. The effect in the change of the sign along the transversal direction creates conduction states as the ones seen in the nanoribbons of the previous section, but the number of bands is doubled, as seen for fixed energy 0.0115t.

In Figs. 34 and 35 we plot the probability density profile of the wave functions |фп,ку(i)|2 for E = 0.0115t, associated to the eight “interfaces” states channels.

From the group velocity v(k) = dE(k)/dk obtained by inspecting Fig. 33, one finds that electrons in the states (2), (4), (6) and (8) propagate along the same direction +y (increasing y), while in (1), (3), (5) and (7)they propagate along — y (decreasing y). This leads to a quantized Hall conductance axy = 2e2/h.

From the density profiles of Fig. 34 (Fig. 35) one finds that the states are localized along the sites where λρ changes sign. This condition corresponds to the sites i = 26 and 76 (i = 51 and 151) for the graphene-BC of width w = 100 (w = 200).

illustration not visible in this excerpt

Figure 34: The probability density |Ψ|2 across the width for the eight interfaces states in an graphene with boundary condition are shown. The width is w = 100 sites in the PUC with Rashba parameter tSO = 0.0471t and periodic exchange field λΖ0) = 0.1885t . The Fermi level is EF = 0.0115t. The vertical black lines represent the sites where there is sign change in λρ.

Figure 36 represents the current flows in graphene with quantum anomalous Hall effect with periodic exchange field λρ. The regions indicated in brown, (a) , is connected by

illustration not visible in this excerpt

Figure 35: The probability density |Ψ|2 across the width for the eight interfaces states in an graphene with boundary condition are shown. The width is w = 200 sites in the PUC with Rashba parameter tSO = 0.0471t and periodic exchange field λΖ0) = 0.1885t . The Fermi level is EF = 0.0115t. The vertical black lines represent the sites where there is a sign change in λρ. the periodic boundary conditions and have a positive exchange field; while the region (b) corresponds to a negative one. Eight states are localized along the sites where λρ changes sign. The blue (red) lines correspond to the right (left) “interfaces” channel states and the black line represents the periodic boundary connecting the region (a). The change of the sign in the exchange field acts as boundary edge, generating the conditions for the appearance of the QAHE without the need of confinement by cutting real edges. As we saw in the Hall conductivity, Eq. (5.30), the brown (light blue) region has axy of 2e2/h (—2e2/h). For the 2e2/h case (brown color) one has in the top (defined as (a)) region) two left moving interfaces channels (in red) and right moving two channels at the bottom.

The conclusion in the second subject, in progress, of this thesis is the formation of conducting states from the quantum anomalous Hall effect (QAHE) in the presence of a periodic exchange field corresponds: gapless interfaces states.

illustration not visible in this excerpt

Figure 36: Sketch showing the direction of the “interfaces” modes propagation (indicated by arrows) in the (a) quantum anomalous Hall (QAH) with positive exchange field; (b) QAH with negative exchange field.

6 Conclusion

This thesis is composed of two theoretical studies related to properties of edge states in nanostructures of graphene monolayers.

In the first one, we studied the electronic band structure, the local density of states, and the edge magnetization of chiral graphene nanoribbons using a π-orbital Hubbard model in the mean-field approximation. We show that the inclusion of a next-nearest- neighbor hopping term ť in the tight-binding Hamiltonian, necessary for a realistic mod­eling of the electronic properties of GNRs, changes its band structure significantly: While ť = 0 has little effect on the average magnitude of the edge magnetization at the charge neutrality point, the nnn hoping term largely modifies the behaviour of M as a function of doping. We believe that these observations call for a more realistic analysis of the spin wave excitations in GNRs [79, 80, 81].

The most notable effect of a ť non-zero is on the density of states. Our study indicates that the interpretation of STM/STS data is very different for ť = 0 as compared with the more realistic parametrization, where ť/t = 0.1. In the latter and for the (8,1) chirality, the energy peak spacing δ = — tbb and the peak hight asymmetry is consistent with the results reported in Ref. [8]. However, in analogy to the discussion of Δι, we do not expect δ to depend on the width of the GNR w. We believe that further experimental LDOS studies of GNR of a fixed chirality and different widths can be of great help for the understanding of the edge magnetization in GNRs.

The second subject of this thesis corresponds to the formation of conducting interfaces channels in graphene due to a periodic exchange field and a Rashba spin-orbit coupling term. Our study is inspired by Ref. [11], where the authors propose that graphene with a Rashba spin-orbit and an exchange field gives rise to a quantum anomalous Hall effect with Hall conductivity axy = [Sign(Xz)]2eb/h. Here the sign of axy depends on the sign of the exchange field coupling. The factor 2 in the conductivity corresponds to the Chern number and implies that each sample edge must have 2 edge states.

Our proposal is to study graphene systems where the exchange field change sign as a function of site position. Our proposal is that there is formation of an “interfaces” channel in regions where the exchange field changes of sign. For a Hall conductivity with +2e2/h and —2e2/h, one must have 4 interfaces channels in the vicinity of the region where the sign changes.

The system in QAHE with a constant sign in exchange field has energy gap separating the highest occupied electronic band from the lowest empty band, our results shows that the possibility of tunable exchange field sign creates conducting interfaces states, remaining a bulk energy gap. We can conclude, in this subject in progress, that the change in the sign in the exchange field parameter in graphene with Rashba coupling enables the formation of interfaces systems of Hall conductivity +2e2/h and —2e2/h in graphene.

APPENDIX A - Berry phase and Berry

In this Appendix we review the theory of Berry phase and Berry connection, a key element for the topological analysis of the Hall conductivity in Chapter 5.

Let us assume the Hamiltonian of the quantum system of interest has a time-dependence that can be described by a “slow” variable R(t), that it H(t) = H(R(t). Here R de­scribes a parameter space over which the Hamiltonian can be changed. The corresponding Schrodinger equation reads

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The eigenvalue problem is conveniently solved using the instantaneous basis expansion given by

illustration not visible in this excerpt

where an(t) is called dynamical phase and reads

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Let is now insert the instantaneous basis expansion into the Schrodinger equation to calculate the expansion coefficients an(t), namely

that gives

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For simplicity, we denote the derivatives of an(t) and \n(R(t))) by an and \n), respectively. By taking the inner product of the terms in the above equation with an arbitrary (m\ state, we obtain

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By going back to the Schrödinger equation, H\n) = En\n), with implicit time dependence H = H (R(t)) and En ξ En (R(t)), we write

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where H = dH/dt and En = dEn/dt. Hence one obtains

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Substituting back into Eq. (A.7)

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So far this formal derivation is exact. Let us now choose an initial condition

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where an(0) = 1 and am(0) = 0 for any m = n. Now let us consider an adiabatic evolution, for which \an(t)\ = 1 and \am(t)\ = 0 for all times. Hence, the initial state evolves as

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By projecting the terms of Eq. (A.6) on the state (n\ one writes

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Since \an(t)\ = 1, one can write it in the form an(t) = ei7n(i), where

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As a result, \Ψ^)) becomes

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Let us write Eq. (A.14) in a more convenient form by using df (x(t))/dt = Vx/(x)-x(t), namely, 7n(t) = i(n(R(í))|VRn(R(t))> · R(t) ξ A(t) · R(t) where the real vector field (sometimes viewec as a gauge potential)

illustration not visible in this excerpt

is called Berry connection.

For situations where R(t) evolves periodically (or not) around a close curve C, the net change of the phase γη (t) is

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Hence the net change in phase depends on the close path C in parameter space, but not on the rate of change in R. This the famous Berry phase, also called geometrical phase. Finally, by using the Stoke’s theorem, one transforms the line integral in a surface integral

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is a gauge invariant (sometimes viewed as a gauge field) is called the Berry curvature. This quantity (in momentum space) is key for the topological theory of the quantum Hall effect.

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APPENDIX B - Spin-orbit and Zeeman

In this Appendix we shall take a brief historical note on the development of the understanding of the spin-orbit coupling (SOC) term and the theory of the Zeeman effect. We point out, based in the literature [73], the distinction between the effect on electronic level splittings of a “real” external magnetic field and the exchange field.

B.1 Spin-Orbit coupling

We present an historical note, where we discuss the SOC effect in atomic physics.

The spin-orbit coupling was first heuristically introduced in atomic physics, to explain the origin of the splitting of spectral lines. The semiclassical reasoning to justify the SOC goes as follows:

In the electron reference the nucleus revolves around the electron giving rise to a current loop of Z positive charges. The Biot-Savart law for a circular loop, relates the electronic orbital motion with an induced magnetic field, namely

illustration not visible in this excerpt

2R where μ0 is the magnetic permeability of the vacuum, R the radius of revolution and I is the current given I = Z = Z= Z= 2пд2т , where L is the orbital angular moment of the electron (for the rest frame of the nucleus). Therefore the magnetic field created by the atomic core is

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The magnetic field B exerts a torque on the electron spin to align μ3 with direction of the field. The energy associated with the work performed by this torque the so called spin­orbit interaction between internal magnetic field Eq. B.2 and spin magnetic momentum

illustration not visible in this excerpt

We recall that the quantum mechanical solution of the Coulomb problem gives (-R3 к -¿¡η? where a0 is the Bohr radius and n is the principal quantum number. As a result, one learns that ESO scales with Z4. Hence, the SOC interaction is weak for light elements and becomes much more important for heavy elements, especially for the inner shells [82]. The result (B.3) needs a correction because it uses a non-inertial frame. Since the electron is accelerated by the Coulomb interaction between electron and nucleus, a non-inertial frame, and the correction is known as Thomas precession [75, 73]. The results of the correction in Eq. (B.3), which in can be rewritten in terms of electric potential multiplied by a factor 1/2.

The modern derivation of the SO interaction is based on the Dirac equation. By writing the Dirac Hamiltonian as an expansion in power of v/c, the leading term is just the Schrodinger equation and, among the relativistic correction, one finds the SOC. one of the first by successes of the Dirac theory as to obtain the semiclassical SOC discussed above with the correct prefactor to reproduce the experiments.

The full quantum mechanical derivation of the SOC is obtained when the relativistic Dirac equation is evaluated up to order in the non-relativistic limit, as done in standard quantum mechanical textbooks [83].

The SOC reads (in Gaussian units)

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For a Coulomb potential V(r) = (Ze2/r)f, one recovers Hso = (a/r)dN/drS · L, the SOC presented in introductory Quantum Mechanics lectures. The expression (B.4) is more general and suitable for applications in condensed matter. In this case V(r) is the lattice potential, due to electron-ion interactions. Equation (B.4) is obtained by considering an electrostatic potential φ(r) = V(r)/e in the Dirac equation. The latter, can be intrinsic in which the material inherits a strong SO interaction from its atomic constituents or extrinsic, that is, generated by an effective external electric field E = —Уф(r), which allows to tune the SO interaction.

Equation (B.3) suggest that the SOC can split a spin degenerate state into levels with spin parallel and antiparallel to the orbital motion. This is indeed the case in atomic physics. In solids, such symmetry can be forbidden, depending on the crystal symmetry.

To understand this statement, let us recall that time-reversal symmetry preserves the Kramers degeneracy [83]. Hence, at any point of the Brillouin zone,

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One the other hand, if the crystal has inversion symmetry (the symmetry operationr ^ —r, does not change the crystal lattice), one has

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For systems with both time-reversal and inversion symmetry (like those with crystal structure fcc, hcp, etc), one obtains the condition

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which means that the bulk energy states are spin-degenerate.

Surfaces break the crystal thrr-dimensional inversion symmetry. This allows for

illustration not visible in this excerpt

where k| is the wave vector parallel to the surface. Hence, the SOC allows for spin­splitting in the absence of magnetic fields. This effect is described be the Rashba SOC, as we discuss below.

In the last years several applications have been proposed to control/ engineer the SO interaction in condensed matter to manipulate the electron spin [84]. Those constitute the basis for spintronics.

B.2 The Zeeman effect

In 1896 Pieter Zeeman observed that, in the present of an external static magnetic field, the spectral lines emitted by atoms split in several components. The phenomenon is called Zeeman effect. The change in the frequency of a spectral lines implies in energy displacements of the states involved in transition and are a function of magnetic field. Historically, one distinguishes between the normal and anomalous cases. The “normal” Zeeman effect can be explained by classical physics [63]. The “anomalous” case requires an intrinsic spin, which was still unknown in the early 20th century. We begin with a general explanation of effect of magnetic field in the magnetic moment, then we will discuss about normal and anomalous Zeeman cases.

Let us describe the classical picture: A particle of mass M and charge q in circular motion has a magnetic moment proportional^[3] to L given by μ = qL/2M. A system with a magnetic moment in a external magnetic field is subject to in a torque τ = μ x B, which tends to turn the μ into the direction of the magnetic field B. The work on the system is W = ƒ τ άθ = ƒ μ x B άθ = —μΒ cos θ, which means that as one increases the angle θ, the work becomes positive and the system gaisn energy. So, the energy associated with a dipole μ in a field Β is given by

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By further exploring semiclassical considerations, let us note that:The orbital angular momentum is quantized, namely, |L| = \Jl(l + 1)h where l is the orbital quantum number. Then, the magnetic moment associated to the orbital motion for electron in an atom is

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For B pointing in the z direction, the expectation value of Hz is

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since mih are the allowed values of the z component of the angular momentum. The Eq. (B.11) can be writes in terms of the quantity μΒ = 2m ~ 5-79 x 10-5 eV/Tesla, known as Bohr magneton. The minus signal in Eq. (B.11) is the negative charge of the electron.

The conclusion from Eq. (B.10) is that the magnetic moment aligned with magnetic field have the smaller energy and a system with different orientations in the magnetic moment μ (refeq:mu-orbital) will show different energies. Since the energy of the states depends upon which of the possible orientations it assumes in the field result is splitting in the spectrum for each orientation of the magnetic moment, see Figure 37. The possible transitions follows the selection rule Δml = 0, ±1 [63, 85].

For anomalous Zeeman the spin must be considered by at least one of states involved in the transition as illustrated in Fig. 37. The sodium atom has a ground state elec­tronic configuration [(1s2) (2s2) (2p6)]3s1 ξ [Ne]3s1. The inner filled shells indicated by square brackets have zero spin and angular momentum so that the magnetic properties are determined by by electron unpaired in 3s1 [73]. The emission spectrum of Na atoms recorded by Zeeman in the presence of a magnetic field that is perpendicular shows a different number of frequencies in the spectral lines. While the normal Zeeman exhibits three frequencies corresponding corresponding to the transitions permitted by selection rule. Notice that without spin will only be these energies for any initial and final values of l. When considerate the intrinsic moment S in general there are more than three different transition energies due to the fact that the upper and lower states are split by different amounts [63].

As discussed before the orbital 3s has l = 0 and its magnetic properties are therefore entirely determined by the spin alone, whereas the 3p state occupied by one electron and quantum numbers l = 1 and s = 1/2. The total angular momentum is J = L + S, while the magnetic moment (B.11) in addiction to orbital part has the spin according

illustration not visible in this excerpt

where the factor gs is the g-factor, gs ~ 2 for electron. To preserve symmetry with equations normally introduces gi factor in μι [85]. Each energy level is split into 2j + 1 levels, for the possible values of mj. Unlike the normal Zeeman, the consideration of spin leads more than three different transition energies, see Fig. 37.

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^[1] The primitive cell or primitive unit cell (PUC) is the volume of space that, when translated, by primitive vectors, fills all the space without either overlapping itself or leaving voids, and must contain precisely one lattice point. The PUC do not have a unique shape, but the most common such choice is the Wigner — Seitz cell. Translations of the primitive cell may possess common surface points; the non-overlapping condition is only intended to prohibit overlapping regions of nonzero volume [13].

^[2] Wigner — Seitz cell is a primitive cell spanning the entire direct space

^[3] In the classical physics a particle in circular motion of radius r and linear moment p has angular momentum defined by L = r x p. If the particle is charged arise a magnetic field, in the axis of the movement, and a vector pointing in the direction along its axis, namely magnetic moment and defined by current times area of a circle performed by charge, m = IA).