Ultrafast optical currents in gapped graphene

We study theoretically the interaction of ultrashort optical pulses with gapped graphene. Such strong pulse results in finite conduction band population and corresponding electric current both during and after the pulse. Since gapped graphene has broken inversion symmetry, it has an axial symmetry about the $y$-axis but not about the $x$-axis. We show that, in this case, if the linear pulse is polarized along the $x$-axis, the rectified electric current is generated in the $y$ direction. At the same time, the conduction band population distribution in the reciprocal space is symmetric about the $x$-axis. Thus, the rectified current in gapped graphene has inter-band origin, while the intra-band contribution to the rectified current is zero.


I. INTRODUCTION
The availability of ultrashort laser pulses with the duration of a few femtoseconds provides effective tools to manipulate and study the electron dynamics in solids at ultrafast time scale with high temporal resolution 1-21 . Among solids two dimensional (2D) crystalline materials exhibits unique properties due to the confinement of electron dynamics to a plane 22 . Graphene, a layer of carbon atoms with the thickness of one atom, is well known 2D material with fascinating properties. Graphene has a honeycomb crystal structure made of two sublattices, A and B -see Fig. 1(a) 23,24 . Having two Dirac points, K and K at the edges of the Brillouin zone -see Fig.  1(b), makes graphene a suitable platform to study the dynamics of massless Dirac fermions [22][23][24][25] . In graphene, both time reversal and inversion symmetries are conserved. However, there is a broad class of semiconductors with honeycomb crystal structure where two sublattices are made of two different atoms, and the inversion symmetry is broken, which results in a finite bandgap at the K and K points 26,27 . One of such materials is a monolayer of transition metal dichalcogenides (TMDCs) that has a direct bandgap with nonzero Berry curvature around the K and K valleys. Gapped graphene, which has broken inversion symmetry, has topological properties similar to TMDC monolayer. Namely, the Berry curvature in gapped graphene is extended over the finite region near the K and K points. Such broadening of the Berry curvature, which can be tuned by the bandgap, results in nontrivial topological properties of gapped graphene 20,26,28,29 . One of such properties is recently predicted topological resonance, which produces finite valley polarization in transition metal dichalcogenides and gapped graphene 19 .
In this article, we study the ultrafast nonlinear electron dynamics in gapped graphene. The dynamics is induced by a single cycle ultrafast linearly polarized pulse. Although the linear pulse does not produce any residual valley polarization, it results in electric current, the magnitude and the direction of which can be controlled by the bandgap. Gapped graphene, considered in the present article, is a model of direct bandgap semiconduc-tors with honeycomb lattice structures. Opening of the bandgap in graphene can be achieved by several methods, for example, by placing graphene on Boron Nitride (BN) or silicon carbide (SiC) substrate 30,31 .

II. MODEL AND MAIN EQUATIONS
In the presence of an applied ultrafast optical pulse, F(t), with the duration of less than 5 fs, the electron dynamics is coherent. This assumption is valid since the electron scatering time in 2D materials is longer than 10 fs [32][33][34][35][36][37] . To find the coherent electron dynamics in gapped graphene we solve time-dependent Schrödinger equation (TDSE) with the Hamiltonian where e is an electron charge, and H 0 is the nearest neighbor tight binding Hamiltonian of gapped graphene 38 , ( Here ∆ is the bandgap, γ = −3.03 eV is the hopping integral, and where a = 2.46Å is a lattice constant. The eigenenergies of the tight-binding Hamiltonian, H 0 , can be found as follows where c and v stand for the conduction band (CB) and the valence band (VB), respectively. Figure 1(c) shows the calculated energy dispersion from Eqs. (5) and (6) for the bandgap of ∆ = 0.5 eV.
The coherent electron dynamics in solids has two major components: intraband and interband dynamics. The intraband dynamics is governed by the Bloch acceleration theorem The solution of this equation has the following form where q is the initial crystal wavevector of an electron in the first Brillouin zone. The corresponding wave functions, which are the solutions of Schrödinger equation (1) within a single band α, i.e., without interband coupling, are the Houston functions 39 , where α = v, c stand for the VB and CB, respectively, Ψ (α) k are Bloch-band eigenstates in the absence of the external field, E α (k) are the eigenenergies, and the dynamic phase, φ Here is the intraband Berry connection. The expressions for the intraband Berry connec-tions, A αα = (A αα x , A αα y ), can be found from the tightbinding Hamiltonian as follows A αα y (k) = The interband electron dynamics is described by TDSE (1). The solution of TDSE can be expanded in the basis of Houston functions Φ where β αq (t) are expansion coefficients, which satisfies the following system of coupled differential equations where the wave function (vector of state) B q (t) and Hamiltonian H (q, t) are defined as where Here A cv (q) is a matrix element of the non-Abelian Berry connection [41][42][43] , which has the following expression where The ultrafast field drives electric current, J(t) = {J x (t), J y (t)}. The current has both interband and intraband contributions, The intraband current is proportional to the group velocity and has the following form where v is the group velocity (intraband velocity) and g s = 2 is the spin degeneracy. The group velocities can be found from Eqs. (5) The interband current is given by the following expression

III. RESULTS AND DISCUSSION
In gapped graphene, sublattices A and B are unequivalent, which results in broken inversion symmetry. The gapped graphene is symmetric with respect to the y-axis, but there is no symmetry with respect to the x-axis, see Fig. 1. Thus, if the linear optical pulse is polarized along the y-axis, then the CB population distribution in the reciprocal space is symmetric with respect to the y-axis and the electric current is generated only along the yaxis, and not along the x-axis. But if the pulse is polarized along the x-axis, the current is expected to flow both along the x and y directions. Below we consider only this case, i.e., we assume that the optical pulse is polarized along the x-axis.
We consider a linearly x-polarized ultrafast optical pulse that is applied normally on the gapped graphene monolayer and has the following waveform where F 0 is the amplitude of the pulse, u = t/τ , and τ = 1 fs. We assume that the pulse is polarized along the x-axis. It should be mentioned that the x-axis is not the axis of symmetry of the gapped graphene, while the y-axis is the axis of symmetry.
In the presence of the pulse, we solve the TDSE assuming that the VB is initially occupied and the CB is empty. The electron dynamics in the field of the pulse is highly nonlinear and is characterized by redistribution of electrons between the valence and the conduction bands. After the pulse, there is a nonzero residual electron population, N  44, where interaction of a linear optical pulse with pristine graphene has been studied. For gapped graphene, the interference pattern becomes smeared, see Fig. 2. This is because the interband coupling is determined by non-Abelian Berry connection, the distribution of which is broadened with increasing the bandgap. At the same time, the separation between the fringes is inversely proportional to the nonlocality distance and, thus, does not depend on the bandgap 44 . Another interesting property of the CB population distribution is that it is symmetric with respect to both x and y axes. This is nontrivital property since the the x-axis is not the axis of symmetry of the system. The CB population distribution is shown in Fig. 3(a)-(d) at different moments of time. It illustrates the formation of the interference-induced hot spots in the CB population distribution. At all moments of time the CB population distribution is symmetric with respect to the x axis. Initially, at −2.5 fs ≤ t ≤ −0.7 fs, the applied field is negative so the electrons are accelerated to the right. Since the interband coupling is strong near the K and K points only, the CB population within this time interval is large on left side of the Dirac points, see Fig.  3

(a).
For time interval −0.7 fs ≤ t ≤ 0 fs, the field is positive and the electrons move to the left and pass the Dirac points the second time, which results in interference fringes or hot spots on the left sides of the valleys as shown in Fig.3(b). The field remains positive for 0 fs ≤ t ≤ 0.7 fs and now the electrons from the right side of the Dirac points pass the region near the K or K points, which results in large CB population on the right side of the K and K points, see Fig.3(c). The field changes its sign at 0.7 fs ≤ t ≤ 2.5 fs. Then the electrons from the right side of the Dirac points pass through the region of large interband coupling the second time, which produce hot spots of CB population on the right side of the K and K points. The electron CB distributions shown in Figs. 2 and 3 could be observed by the time resolve angle-resolved photoelectron spectroscopy (tr-ARPES) 45,46 .
Redistribution of electrons between the VB and CB during the pulse generates an electric current. For the pulse polarized along the x-axis, which is not the axis of symmetry for the gapped graphene, both the longitudinal current, i.e., the current in the x direction, and the transverse current, i.e., the current in the y direction, are generated. Such currents are shown in Fig. 4 for different values of the bandgap, ∆. For zero bandgap, i.e., for pristine graphene, the transverse current is zero. The transverse current increases with the bandgap. The electric current, generated during the pulse, has two contributions: intraband and interband. The intraband current is completely determined by the electron density distributions in the CB and VB. It can be also considered as a measure of asymmetry of such distributions. Such the CB population distribution is symmetric with respect to the y-axis both during the pulse and after the pulse, the intraband transverse current, J y , is zero. Thus, the transverse current for gapped graphene is determined by the interband contribution only. As the results, the trans- verse current as a function of time is oscillating with the frequency that depends on the bandgap, see Fig. 4(a). At the same time, the longitudinal current is almost unidirectional with small oscillations, see Fig. 4

(b).
Since the bandgap determines the strength of the asymmetry of the system, we expect that the magnitude of the transverse current increases with the bandgap, which is shown in Fig. 4(a). For the longitudinal current, there is a different tendency. The longitudinal current first increases with ∆ and then at large bandgaps, ∆ ∼ 2eV , decreases. Such suppression of the longitudinal current at large values of ∆ is due to the specific dependence of the interband dipole matrix elements (non-Abelian Berry connection) on the bandgap. At small bandgaps, the interband dipole matrix element is strongly localized near the K and K points. With increasing the bandgap, the dipole matrix element becomes delocalized and nonzero at large part of the Brillouin zone, where the maximum of the dipole matrix element decreases with the bandgap keeping the net dipole matrix element, i.e., the integral of the dipole matrix element over the whole Brillouin zone, constant. As a results the total CB population near the K or K points decreases with ∆, which finally results in suppression of the longitudinal current.
In Fig. 5 the longitudinal and transverse currents are shown for different field amplitudes. As expected, with increasing the field amplitude, the magnitudes of both currents increase. The frequency of oscillations of the transverse current also shows the dependence on the magnitude of the pulse, while the longitudinal current is almost unidirectional.
The direction of the current is determined by the direction of the field maximum. For the field profile (31), the field maximum is pointing in the positive direction of the x-axis. If we change the direction of the field maximum to the negative one, i.e., it is pointing in the negative direction of the x-axis, then the longitudinal current, J x , changes its sign, while the transverse current, J y , remains the same. The transverse current changes its sign if we change the signs of the on-site energies of sublattices A and B, i.e., change the sign of parameter ∆ in Hamiltonian (3).
The generated electric current during the pulse results in the transfer of an electric charge through the system. Such transferred charge can be calculated as For the pulse polarized along the x-axis, the charge is transferred in both x and y directions. In Fig. 6 the transferred charge is shown as a function of the pulse amplitude for different values of the bandgap. As expected, for zero bandgap, there is no charge transfer in the transverse direction, Q y = 0. As a function of the field amplitude, the transverse transferred charge shows oscillations, which is due to oscillations in the transverse current as a function of time. The longitudinal transferred charge, Q x , monotonically increases with the field amplitude and has weak dependence on the bandgap. At large bandgap, ∆ ∼ 2eV , transferred charge Q x becomes smaller, which is related to suppression of the CB population and correspondingly the longitudinal electric current at large ∆. Changing the direction of the applied field and applying it in -x direction changes the sign of the longitudinal current however, it does not have any effect on the normal current. This current only changes the sign if we change the on-site energies of different sublattices.

IV. CONCLUSION
In pristine graphene, which has an inversion symmetry, there are two axes of symmetry, say x and y. If an external linear pulse is polarized along these two directions, then it will produce CB population distribution that is symmetric with respect to the axis of polarization of the pulse. The pulse will also generate an electric current and the corresponding transferred charge along the direction of polarization only, but not in the transverse direction.
For gapped graphene, the inversion symmetry is broken. In this case there is only one axis of symmetry, say the y-axis. If the linear pulse is polarized along the x axis, then since this axis is not the axis of symmetry, the electric current is generated in both x and y directions. The transverse current does not depends on the direction of the field maximum, while the longitudinal current changes its sign when the direction of the maximum is reversed. At the same time, for the same polarization of the pulse, i.e., along the x-axis, similar to pristine graphene, the CB population distribution is symmetric with respect to the x-axis both during the pulse and after the pulse. It means that the electron dynamics above (k y > 0) and below (k y < 0) the K (K ) point is exactly the same, which results in symmetric CB population distribution. Although the electron dynamics depends on the geometric phase, which is different above and below the K (K ) point, this phase is exactly canceled by the phase of the interband dipole matrix element (non-Abelian Berry connection). This is the property of the two-band model of gapped graphene which will be discussed somewhere else. If more bands are included into the model, then there will be no cancellation of the geometric phase and the net (topological) phase, which is the sum of the geometric phase and the phase of the interband dipole coupling, will be nonzero. The topological phase has different time dependence above and below the K (K ), which results in topological resonance. The topological resonance occurs due to a partial cancellation of the dynamic phase by the topological phase. Such partial cancellation is different above and below the K (K ) point, which finally results in different CB populations and asymmetric CB population distribution. Such small asymmetry of CB population will introduce small intraband contribution to the transverse current.