In a milestone paper [F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)], Haldane elaborated a model of graphene within the time -reversal symmetry breaking is achieved by next -nearest -neighbors imaginary counterrotating hopping, hence conferring topological properties. In recent years, the time -reversal symmetry turned out to be broken also by light irradiation in so-called Floquet topological insulators (FTIs). On the other hand, Kane and Mele introduced a spin -orbit coupling (SOC) model [C. L. Kane et al., Phys. Rev. Lett. 95, 226801 (2005)] inspired by the Haldane's mechanism. In this paper, we present the topological properties of a FTI possessing SOC, using graphene as the playground. It was found that the interplay between sublattice subspace and the spin one triggers interesting topological phase transitions. Basically, in a FTI with SOC, two topological phases may be excited: charge quantum Hall effect (CQHE) and, respectively, spin quantum Hall effect (SQHE) phases. Also, it was demonstrated that the CQHE and SQHE coexistence is forbidden by the topology of the system. As well, it was identified a special driving regime of spin filter (SF), in which only one spin state is topological and, consequently, will be filtered in quantum transport.

In this paper, we present a theoretical perspective regarding the interaction of graphene with circularly polarized light and magnetic field, from the topological insulators point of view. We analyze how these two external fields affect the spectral, topological and transport properties of graphene and correlate the findings in order to explain in a fundamental and unified way the emerging topological phase transitions. In this respect, in the first step we introduce a model for interaction and charge transport. Then, based on the derived theory, we present numerical results aimed to explain the underlying processes which give graphene topological properties. The central point is represented by the time-reversal symmetry breaking which generates chiral edge states, namely electronic states localized at the edges of the system, having opposite velocity directions. We find that the light frequency, intensity and polarization state drastically influence the formation of the chiral edge states and their number. We correlate this effect with quantum Hall transport, analyzing the resulting transversal (Hall) resistance plateaus and their values. Moreover, if a supplementary magnetic field driving is applied, there emerge intricate topological phase transitions, characterized by introducing or removing specific Hall resistance plateaus.

In this paper we present an in-depth theoretical investigation of an electron scattering process on a graphene quantum dot (GQD) supposed to a perpendicular magnetic field. As it is well known, because of Klein tunneling, an electrostatic potential is not helpful to localize an electron inside a GQD. However, we report here that in the case of a magnetic driven GQD, there emerge scattering resonances characterized by trapped electronic states inside the dot for finite periods of time, otherwise known as quasi-bound states. Using a real space scattering analysis, we highlight in a very intuitive manner how the quasi-bound states are generated and, for a comprehensive investigation, we evaluate their corresponding lifetime (trapping time). As well, we explore the case of an electrostatically biased GQD. We show that this configuration may be very advantageous regarding the control of the trapping times which reach values much higher than in the unbiased case.

In this paper, we present a theoretical perspective concerning the scattering of electrons on a twisted light (TL) driven graphene quantum dot (GQD). Relatively recently discovered, TL is a novel type of electromagnetic field which carries a finite orbital angular momentum oriented on the propagation direction, besides its spin. This striking property of TL is due to its spatial structure. It is well known that the localization of electrons in a GQD is forbidden by the Klein tunneling, an effect that manifests by the perfect transmission of electrons through a potential barrier, regardless of its magnitude. Here we demonstrate that, for a suitable choice of the scattering regimes, there emerge scattering resonances characterized by trapping states of the incident electron inside the GQD for finite periods of time. The most interesting result is the prediction regarding the possibility to control the trapping times using a TL irradiation. Also, we mention that the investigation was performed for a frequency of the TL within the infrared spectrum.

Permanent localization of electrons inside a graphene quantum dot (GQD) is known to be forbidden, as a manifestation of Klein tunneling. However, an electron which scatters on a GQD may be transiently trapped inside and one known practice is the usage of magnetic field. These electronic states discussed here, called quasibound states, are scattering resonances typically characterized by a finite lifetime (trapping time). In this paper, we present a theoretical perspective concerning the opportunity to enhance the lifetime of quasibound states excited in a GQD placed in a uniform magnetic field, using circularly polarized light. Generally speaking, electron trapping inside GQDs is achievable for certain well-defined conditions, for instance, magnetic field intensity. We report here that the trapping time of an electron inside a GQD may be successfully enhanced by adjusting the light intensity while keeping the magnetic field constant.