**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

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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 nanorib‐ bon [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 predict‐ ed 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 *BS* leads to Landau quantization and edge states that circulate in opposite directions [47, 48], and the strain creates graphene pseudo‐ magnetic fields. Then, without breaking the TRS, the strain may induce a gap in the bulk and edges without helical gap. Thus, strain, EX and SOC can be used as a versatile tool for control of topological phase transitions [32]. These facts motivated us to propose ways in which the spin‐orbit coupling, uniaxial mechanical strain and exchange (instead of an external magnet‐ ic field) to be used to carry out phase transitions in graphene nanoribbons [49].

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 nanorib‐ bons. Finally, we show systematically the strain‐engineered QPT from the QSH to QAH states.
