## Abstract

In this chapter, we discuss the new classes of matter, such as the quantum spin Hall (QSH) and quantum anomalous Hall (QAH) states, that have been theoretically predicted and experimentally observed in graphene and beyond graphene systems. We further demonstrate how to manipulate these states using mechanical strain, internal exchange field, and spin‐orbit couplings (SOC). Spin‐charge transport in strained graphene nanoribbons is also discussed assuming the system in the QAH phase, exploring the prospects of topological devices with dissipationless edge currents. A remarkable zero‐field topological quantum phase transition between the time‐reversal‐symmetry‐broken QSH and quantum anomalous Hall states is predicted, which was previously thought to take place only in the presence of external magnetic field. In our proposal, we show as the intrinsic SOC is tuned, how it is possible to two different helicity edge states located in the opposite edges of the graphene nanoribbons exchange their locations. Our results indicate that the strain‐induced pseudomagnetic field could be coupled to the spin degrees of freedom through the SOC responsible for the stability of a QSH state. The controllability of this zero‐field phase transition with strength and direction of the strain is also explored as additional phase‐tuning parameter. Our results present prospect of strain, electric and magnetic manipulation of the QSH, and QAH effect in these novel two‐dimensional (2D) materials.

### Keywords

- Graphene
- graphene nanoribbon
- quantum spin Hall
- quantum anomalous Hall
- topological insulator
- 2D materials
- strain

## 1. Introduction

Starting from the work by Landau and Peierl’s work [1, 2], two‐dimensional (2D) materials were regarded as theoretical structures, thermodynamically unstable to be obtained in laboratory. This is because of fusion temperature decreases as function of thickness of thin films, causing the material to segregate in islands or decomposing in typically thicknesses of tens of atomic layers [3, 4]. In 1947, Wallace [5] demonstrated the electronic properties of what became the first theoretical work predicting the one‐atom thick of carbon atoms. Past 57 years, his theoretical predictions were experimentally synthesized by Novoselov et al., which now is widely known as graphene. In 2004, Geim and Novoselov [6, 7] created by mechanical exfoliation an one‐atom thick layer made of graphite, the so‐called graphene. Due to their well‐succeeded experiment, many other techniques have been developed to grow graphene on several possible substrate materials such as on hydrogenated silicon carbide, copper, cobalt, and gold [8–16].

Before 2004, graphite systems were also widely studied [5, 17, 18], and their electronic properties used to theoretically describe other materials based on carbon, such as fullerene [19] and carbon nanotubes [20]. These chemical elements have attracted much attention because of their exotic electronic and mechanical properties, such as high tensile strength and, in the case of nanotubes, tunable electronic structure according to chirality, radius, and high thermal conductivity. A new type of derivative of graphene arose after 2004: the graphene nanoribbon [21], in which some electronic properties of the graphene were modified and could be controlled. These properties depends on the type of crop that was carried out on graphene and can be simpler cuts, called zigzag and armchair or being modeled in a specific way, such as triangles to form quantum dots [21–25] or even with Z formats [26].

The interest in two‐dimensional materials started from the nineteenth century, mainly for its electronic transport properties after the discovery of the Hall effect. In 1988, Haldane predicted that another type of Hall effect, called anomalous quantum Hall effect, could be observed in a two‐dimensional crystal with hexagonal lattice [27]. Recently, the new classes of matter, such as quantum Hall effect (QHE) [28, 29], quantum anomalous Hall (QAH) effect [30–32], and quantum spin Hall (QSH) effect [33, 34], have been discovered or predicted in the graphene, as well as other 2D materials such as topological insulators [35–37], HgTe‐CdTe quantum wells [38, 39], silicene [40], two‐dimensional germanium [40], and transition metal dichalcogenides [41]. Among these new classes of matter, the QSH and QAH states possess topologically protected edge states at the boundary, where the electron backscattering is forbidden, offering a potential application to electronic devices to transport current without dissipation [24, 42]. However, the QSH state and QAH are very different states of matter. The quantum spin Hall is characterized by a gap completely insulating the bulk, and their edge states are helical with no gap, wherein opposite spins propagate in opposite directions on each side of the sample and are protected by time reversal symmetry (TRS) [27, 33, 36, 38–40, 43, 44]. In the case of quantum anomalous Hall, chiral edge states takes place, also without gap, where one spin channel is suppressed because of the TRS break [35, 37, 45]. Therefore, to observe topological phase transitions (QPT) between quantum spin Hall and quantum anomalous Hall states, it is necessary to apply a condition, which might break the TRS [28]. An external magnetic field is a potential solution but from applicability point of view, an internal exchange field (EX) which takes the main spin band to be completely filled while the minority spin band becomes empty, becomes an more attractive alternative [31, 32, 35, 46]. As it is known, a pseudo‐magnetic field induced by strain

In this chapter, firstly, we make a brief description on tight‐binding model. Then, we report energy band structure of the graphene and the individual effects of the intrinsic SOC, the Rashba SOC, and the EX. After that, we present the effects of applied uniaxial strain on both electronic structure and transport property of the graphene. Then, we demonstrate the effects of strain on single‐particle energy and quantum transport property of graphene nanoribbons. Finally, we show systematically the strain‐engineered QPT from the QSH to QAH states.

## 2. Electronic structure and transport properties of graphene

### 2.1. Electronic structure of graphene tight‐binding approach

An isolated atom has its own electronic levels ranging and depending on its fundamental characteristics. When two or more atoms are approximate to each other, their electronic levels are recombined to obtain a new structure for the system as a whole. And the periodic clustering of atoms in a structure is meant by the crystal lattice. In the case of an insulating material, superposition of the wave functions of the valence electrons in the crystal lattice atoms is low. In the case of a conductive material, such superposition of the wave function of the electrons is large and acquires great mobility through the solid. Semiconductor materials have an electronic distribution that is not very well located, because there is not a too strong electrical attraction between electrons and protons on the atomic nucleus but low overlap between the valence electrons from neighboring atoms are observed. The tight‐binding method is useful in those cases [50]. Thus, one can assume that the lattice Hamiltonian

In order to apply tight‐binding method to graphene, we begin with the wave function of an electron on its lattice as a linear combination of atomic orbitals of A and B sites, the two distinct atoms in graphene unit cell

where

where

Lattice Hamiltonian are made up of two terms: on‐site

The first one,

with

where

and B‐site distance vector can be obtained in a similar way. Here,

To find energy equations for our system, we solve Schrödinger’s equation using Eqs. (1) and (5). This will give

In Figure 2, valence and conduction bands are shown in profile: with

Quantum anomalous Hall and quantum spin Hall effect can be induced in graphene without a magnetic field if we consider Rashba and intrinsic spin‐orbit coupling and also exchange field [53, 54]. Intrinsic spin‐orbit coupling is weak on graphene [55–57], however, graphene is easily affected by disturbance at low energies and the effects due to spin‐orbit coupling should become relevant at low temperatures [33]. Although it is challenging the experimental envision, this type of coupling can be controlled with graphene deposition on other materials. The exchange interaction that occurs between electron spins can be observed in graphene with stabilization of a ferromagnetic phase, when it has a low doping [53, 58].

The intrinsic spin‐orbit interaction evolves the next nearest‐neighbors and is written as

being

and the negative of right side for

Rashba spin‐orbit coupling can be induced in graphene with application of an external electric field perpendicular to the sheet plane [60], interaction of carbon atoms with a substrate [43] or by curving the sheet [61–63]. Its Hamiltonian reads

where the summation

with

Using electron wave function, Eq. (1) and Hamiltonian part 11, we find four spin nondegenerated energy equations:

which, together with energy Eqs. (7) and (8), allow one to plot energy levels in Figure 5. As one can note, the Rashba SOC lifts the spin degeneracy, breaking the SU(2) symmetry. However, due to time‐reversal symmetry, we still have

Calculations of * ab initio*have recently shown that graphene doped with Fe on its surface may have intrinsic ferromagnetism [30]. This interaction arises when the change in the spin of an electron changes the electrostatic repulsion between electrons near it. Its Hamiltonian includes the coupling of orbital motion and the spin of the electrons with the exchange field. In this paper, to simplify the calculations and not lose generality, only the portion of spin will be considered. Now, the Hamiltonian is

where

the same for

### 2.2. Electronic structure of strained graphene

Deformation can naturally be observed when graphene is grown on top of other materials, because of distinct atomic arrangements between the atoms of graphene and the substrate. The application of an external tension on graphene sheet or nanoribbons can change its electronic properties, as with the nanotubes [64–68]. Some calculations [69] and experiments [70] have shown that these deformations can reach about 20% of the initial interatomic distance without permanently deform the graphene.

Strain is calculated in graphene using the strain matrix defined in [52]:

where

The transformation of an atom position from (* x*,

*) to the new position (*y

x

^{′},

y

^{′}) after the strain application will be

Or, in vectorial form

where

Hopping term

With these modifications, energy equations for strained graphene are calculated in the same way as in Section 2.1 and gives

with summation made over the nearest‐neighbors. For

where

now, with new distances

and nearly the same for other expressions, with

In Eq. (31),

In Figure 8, we show the effects of the combination of intrinsic SOC and uniaxial strain applied along

where

and we defined

In Figure 9, we can still observe the broken spin‐degeneracy, but compared with the case where the uniaxial strain is absent, one can note that the effect of strain is to renormalize the Rashba SOC and shifts the Dirac point relative to the original one.

### 2.3. Quantum anomalous Hall effect in strained graphene

In this section, we discuss the prospects of external manipulation of the quantum anomalous Hall effect (QAHE) in graphene by strains [73–76]. We present here our results of the microscopic study of the QAHE in graphene under uniaxial strains [32]. For this purpose, we have theoretically explored the dependence of electronic structure, topological and transport properties upon the orientation and modulus of uniaxial strain, in the presence of Rashba, Intrinsic SO, and an exchange field interaction [32].

To identify the topological properties of the Dirac gap and study the origin of QAHE, we have calculated the Berry curvature of the * n*th bands

where * n*th band and

where the summation is taken over all the occupied states below the Fermi level, and the integration is carried out over the whole first Brillouin zone.

Since the Berry curvatures are highly peaked around the Dirac points

It is interesting to mention that in the above integral, a momentum cutoff is set around each valley for which the Chern number calculation is guaranteed to converge.

As known, intrinsic spin‐orbit (ISO) interaction respects the crystal symmetries and does not couple states of opposite spins. But it opens up a topologically nontrivial bulk band gap at zero magnetic field [43]. This bulk band gap hosts two counter‐propagating edge modes per edge in the graphene nanoribbon, with opposite spins: this topological phase is known as the QSH phase and may be regarded as two opposite QH phases (i.e., each spin performs the QH effect, with opposite chirality) [27]. Therefore, the Chern number must vanish in a system with TRS. In contrast, the Rashba term explicitly violates the

Let us now calculate the Hall conductivity of the strained graphene considering both Rashba SOC and ISO. Figure 10(a) and (b) shows the Hall conductance for

The distinct behaviors observed along different strain directions for the QAHE phase transition can be explained by the competition of the Rashba SOC and ISO in the bulk band gap‐closing phenomena for a given critical exchange field

## 3. Electronic structure and transport properties of graphene nanoribbon

A graphene nanoribbon is defined as a graphene sheet in which one of its dimensions is narrow and the other approximately infinite. The unique properties arising due to the reduced dimensions become very important because shape of the edges and width of nanoribbon defines its electronic structure. The main nanoribbons classification is based on the edge design, which can be armchair, zigzag, chiral, and bearded nanoribbons depending on the edge terminations [82]. We will focus on the electronic dispersion of only two types: armchair and zigzag.

### 3.1. Electronic structure of graphene nanoribbon

The electron wave function in a armchair nanoribbon is

where

is a vector between the atom and its neighbors in the next unit cell with same type of site. In armchair nanoribbons, the unit cell

And similar vectors could be find for B sites. Then, Hamiltonian of armchair nanoribbon is

where the summation

Zigzag nanoribbon has unit cell

Now, the energies are

and energy dispersion for a

* Electronic structure of strained graphene nanoribbon*‐For the case of strained graphene nanoribbons, we need to replace the strain‐invariant hopping integrals by the strain‐dependent ones [52, 83] as described in Section 2.2. Many interesting properties are observed in the optical conductivity [72] and electronic transport [84, 85] when the uniaxial strain is considering in the graphene nanoribbons. In the next section, we discuss in detail the effects of uniaxial strain in the spin‐charge electronic transport and QAH effect.

### 3.2. Transport properties of strained graphene nanoribbon

Here, we aim to analyze the electronic transport control in GNR with different terminations in the QAH phase by means of uniform strain deformations [86]. The electronic transport can be performed using a two‐terminal device akin to a field electron transistor (FET). QAH phase can be determined experimentally, by spin‐resolved density of states, that can be accessed by spatially scanning tunneling microscope (STM) or by scanning tunneling spectroscopy (STS) [87–89]. To calculate the spin‐resolved conductance, we have implemented the standard surface Green’s function approach [90, 91]. The GNR device is divided into three regions: left lead, central conductor, and right lead. The uniaxial strain is applied to either the longitudinal (

where

where the trace runs through the lattice sites at the central conductor,

To study the conductance characteristics in the presence of both Rashba SOC and exchange field [86], we set the parameters

In the QAH phase, however, there is a weak scattering between forward and backward movers, leading to a low‐dissipation spin transport. At low energy, this interesting strain‐controllable behavior of conducting channel suppression might be efficiently used to filter electrical current of desired spins, in spin filtering devices [86]. In Figure 15(d), we show the total conductance, which is nearly robust against strains, specially close to the charge neutrality point, where the deviations due to strain are quite small. In contrast, the conductance of AGNR shows a drastic modification as one can notice in Figure 15(e)–(h), with the development of a transport gap, which is insensitive to the electron spin that is injected and collected in the device. However, this induced transport gap is dependent upon the direction of the applied strain, with a larger conduction suppression along

Another remarkable phenomenon is the oscillatory dependence of the spin components of * precesses*as it propagates in the presence of the Rashba field, acquiring a net phase that is proportional to

### 3.3. Quantum phase transitions in strained graphene nanoribbon

Quantum spin Hall and quantum anomalous Hall (QAH) states have topologically protected edge states, where the electron back scattering is forbidden, making these systems good candidates for electronic devices with dissipationless electronic transport [33, 35, 38, 41]. The potential possibility to explore the different Quantum Hall phases in strained graphene has motivated us to study the strain‐related physics at zero magnetic field in graphene nanoribbons [49].

If the mirror symmetry about the graphene plane is preserved, then the intrinsic SOC which opens gaps around Dirac points is the only allowed spin‐dependent term in the Hamiltonian. Otherwise, if the mirror symmetry is broken, then a Rashba term is allowed, which mixes spin‐up and spin‐down states around the band crossing points. Besides, Rashba SOC pushes the valence band up and the conduction band down, reducing the bulk gap. Following Reference [45], we present our results for the ZGNR in Figure 17, which shows the effects of intrinsic‐ and Rashba‐SOCs and EX upon the energy band of the ZGNR [49]. Notice in Figure 17(a) that the interplay between intrinsic‐ and Rashba‐SOCs partially lifts the degeneracies of both bulk‐ and edge‐state, breaks particle‐hole symmetry and pushes the valence band up. In turn, the presence of the EX breaks the TRS and lifts the Kramer’s degeneracy of electron spin, pushing the spin‐up (spin‐down) bands upward (downward), as shown in Figure 17(b). In strong contrast with Figure 17(b), the presence of Rashba SOC and EX induces coupling between edge and bulk states, which significantly modifies the group velocity of edge states, as shown in Figure 17(c). The combined effects of intrinsic, Rashba SOCs and EX are shown in Figure 17(d), which are in agreement with results reported in Reference [45] (see for instance Figure 2). Notice that the Fermi level enters into the valence band and the energies of some edge modes are smaller than the valence band maximum.

The intrinsic SOC can be strongly enhanced by impurity (adatom) coverage on the surface of graphene, which produces strong lattice distortions [59]. In this context, one may ask how the quantum phase transition in a graphene ribbon changes as the intrinsic SOC is tuned [49]. Following the discussion of Reference [45], the effects of strain fields are shown in Figure 18 (with a similar representation to the one introduced in Reference [45]) with parameters W=48,

To understand the QPT and show intuitively how it takes place [49], we follow Reference [45] and introduce the average value of the position

To seek the controllable topological QPTs induced either by strain (EX), or intrinsic SOC, or any of their combinations [49], the phase diagrams in which the phase is characterized by the difference in the average value of position

The underlying physics of the strain tuned phase diagram is as follows. It is well established that uniaxial mechanical strain does not break the sublattice symmetry, but rather deforms the Brillouin zone, such as, the Dirac cones located in graphene at points * this is not true*for graphene with SOC. Since SOC couples the spin and the momentum degrees of freedom of the carriers,

## 4. Conclusion

In summary, we have performed a systematic investigation of the effects of uniaxial strains, exchange field, staggered sublattice potential, and SOC on the electronic and transport properties of graphene and graphene nanoribbons. We have employed the tight‐binding approximation, and Green’s function formalism in order to fully describe the electronic and transport properties of these interesting nanostructures.

Using an effective low energy approximation, we were able to describe the Berry curvature and the associated Chern numbers for different orientation and uniaxial strain strength, as function of exchange field interaction. The QSH–QAH phase transition associated to the tunability of Chern number for the bulk graphene displays an interesting behavior according to specific directions of strains: an increase in the critical exchange field

Our results demonstrated in this Chapter offer the prospect to efficiently manipulate the electronic structure, transport properties, and consequently the QAHE by strain engineering of the graphene. We also envision that our work can be extended to other layered materials (for instance, transition metal dichalcogenides), with a great potential application on novel electronic devices with the focus on dissipationless charge current.