Advances in Condensed Matter Physics

Volume 2018, Article ID 5703197, 10 pages

https://doi.org/10.1155/2018/5703197

## Time Evolution of Floquet States in Graphene

^{1}Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/A, I-41125 Modena, Italy^{2}CNR-Institute of NanoSciences-S3, Italy

Correspondence should be addressed to F. Manghi; ti.erominu@ihgnam.acnarf

Received 29 January 2018; Revised 18 April 2018; Accepted 20 June 2018; Published 2 August 2018

Academic Editor: Sergio E. Ulloa

Copyright © 2018 F. Manghi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on a solution of the Floquet Hamiltonian we have studied the time evolution of electronic states in graphene nanoribbons driven out of equilibrium by time-dependent electromagnetic fields in different regimes of intensity, polarization, and frequency. We show that the time-dependent band structure contains many unconventional features that are not captured by considering the Floquet eigenvalues alone. By analyzing the evolution in time of the state population we have identified regimes for the emergence of time-dependent edge states responsible for charge oscillations across the ribbon.

If a time-periodic field is applied to electrons in a periodic lattice the Bloch theorem can be applied twice, both in space and in time. This is the essence of Floquet-Bloch theory [1–3] that has recently attracted a renewed interest for its ability to describe topological phases in driven quantum systems [4–7]. The discovery that circularly polarized light may induce nontrivial topological behaviour in materials that would be standard in static conditions [8–11] has opened the way to the realization of the so-called Floquet topological insulators, where a topological phase may be engineered and manipulated by tunable controls such as polarization, periodicity, and amplitude of the external perturbation.

When the field is applied for a sufficiently long time (pulse duration much larger than the field oscillation period) electrons reach a nonequilibrium steady state characterized by a periodic time-dependence of the wave functions and, consequently, of the expectation values of any observable [12, 13]. In this paper we focus on this time-dependence, looking for the time evolution of some relevant quantities such as energy and charge density. How these characteristics affect the time behaviour of these observables will be our focus. We will consider the prototypical case of graphene that under the influence of circularly polarized light exhibits in its Floquet band structure the distinctive characteristics of a 2D Chern insulator, namely, a gap in 2D and linear dispersive edge states in 1D [9, 11, 14, 15]. These Floquet edge states are topologically protected and responsible for a quantized Hall conductance in the absence of a magnetic field [16–18], a remarkable realization of the so-called “quantum Hall systems without Landau levels” originally proposed by Haldane [19].

Under a periodic drive, the nonequilibrium steady states, solutions of the time-dependent Schrödinger equationevolve in time aswhere is periodic in time and , the Floquet quasi-energies, are the eigenvalues of an effective Hamiltonian , the so-called Floquet Hamiltonian:Here is the full Hamiltonian of the driven systemwith being the static Hamiltonian and being the external periodic driving. The factorization in (2) is exact and represents the temporal analogue of the Bloch theorem. In the following we will consider states described by (2) characterized by a single wave vector and of a given Floquet quasi-energy . With being time-periodic it can be expressed as a Fourier series:where in turn can be expanded on a complete set, for instance, on a localized basis:with being a site index, being the number of sites in the unit cell, and being the localized orbitals. In practice the Fourier expansion is truncated to include a finite number of modes, up to a sufficiently large whose value depends obviously on . This allows formulating the eigenvalue problem in (3) in a standard matrix form whose eigenvalues turn out to be replicas of the static band structure with gaps opening at their crossing points [1, 11].

The field-free Hamiltonian of graphene is described in the tight-binding scheme with a single hopping parameter eV between nearest neighbor sites, reproducing the well-known Dirac-like valence and conduction bands [20]. In the presence of the oscillating field described by the vector potential , the hopping between neighboring sites is modified according to Peierls’ substitution [21, 22]:

We are interested in the effects of reduced dimensionality, namely, on the gapless edge states that arise in graphene ribbons; we chose a zig-zag terminated ribbon 50 atoms wide (Figure 1). We consider two frequency values ( eV, eV) representative of the intermediate and large frequency regime (, ). We study also the effect of different amplitudes of the external vector potential ( and in units of the inverse carbon-carbon distance [7]). In Figure 2 we compare the Floquet quasi-energies obtained for the honeycomb lattice in 2D and 1D exposed to a circularly polarized field .