Ultrafast topological phenomena in gapped graphene
UUltrafast topological phenomena in gapped graphene
S. Azar Oliaei Motlagh, Vadym Apalkov, and Mark I. Stockman
Center for Nano-Optics (CeNO) and Department of Physics and Astronomy,Georgia State University, Atlanta, Georgia 30303, USA (Dated: December 24, 2018)In the model of a gapped graphene, we have shown how the recently predicted topological res-onances are solely related to the presence of an energy band gap at the K and K (cid:48) points of theBrillouin zone. In the field of a strong single-oscillation chiral (circularly-polarized) optical pulse, thetopological resonance causes the valley-selective population of the conduction band. This populationdistribution represents a chiral texture in the reciprocal space that is structured with respect to thepulse separatrix as has earlier been predicted for transition metal dichalcogenides. As the band gapis switched off, this chirality gradually disappears replaced by a achiral distribution characteristicof graphene. I. INTRODUCTION
Two dimensional (2D) materials with hexagonalsymmetry – graphene, silicene, transition metaldichalcogenides (TMDC’s), hexagonal boron nitride (h-BN), etc. – possess nontrivial topological properties inthe reciprocal space . We aim at study of nonlinear be-havior of such materials in strong ultrafast (one or a fewoptical oscillations) laser pulse fields. This behavior waspredicted to be fundamentally different for graphene andTMDC’s .In graphene, linearly-polarized pulses cause appear-ance of interference fringes in the reciprocal space, whichare due to transitions in the vicinity of the K (or, K (cid:48) )point that occur twice per cycle; the corresponding tran-sition amplitudes interfere causing the fringes . Fora circular polarized pulses, there is no effect of pulsehandedness for a single optical oscillation. For two ormore optical oscillations there is weak preferential pop-ulation of one of the K or K (cid:48) valley and and charac-teristic forks of the electron interferogram indicating ef-fect of the Berry phase. The electron distribution forboth linear and circular-polarized pulses are asymmetric,which causes currents that have recently been observedexperimentally .In a contrast, for TMDC’s there is a strong prefer-ential population of the valley that corresponds by itschirality to the handedness of the circularly polarizedexcitation pulse (called the valley polarization). Thereis also texturing of the reciprocal space with respect tothe separatrix [see below Eq. (9) for definition]. Namely,the preferentially populated valley is populated outsideof the separatrix while the low-population valley is popu-lated inside the separatrix. These phenomena, which wecalled the topological resonance , are due to the interfer-ence of the topological (Berry) phase and the dynamicphase of the polarization oscillation.In this article, we explore a model of gapped graphene where the center symmetry is removed by introducingsublattice-specific oncite energies ± ∆ – see below Eq. (3);this opens up a band gap of 2∆ at the K and K (cid:48) points.We show that, as the band gap increases, the distributionof the carriers in the reciprocal space gradually changes from that characteristic of graphene with a low valleypolarization to a dramatically different texture charac-teristic of TMDC’s with a high valley polarization. Notethat experimentally the band gap in graphene can beopen, in particular, by growing it on a SiC substrate .Graphene, a two dimensional (2D) layer of carbonatoms with a honeycomb symmetry, possesses uniquephysical properties: gapless Dirac-fermion spectrum atthe K and K (cid:48) points, nonzero Berry curvature concen-trated at these Dirac points corresponding to the ± π Berry phase, tunable carrier density and plasmonic prop-erties, unusual magnetic properties including the quan-tum Hall effect at room temperatures, etc. .To elucidate high-field ultrafast behavior of graphene,we have theoretically studied its behavior for linear-polarized and chiral (circularly-polarized) few-oscillationoptical pulses, which cause population transfer from thevalence band (VB) to the conduction band (CB). Wefound that linearly-polarized pulses caused appearanceof interference fringes in the CB population to the quan-tum interference of two passages of electrons by the Diracpoints where VB → CB transitions occurred . This quan-tum interference also caused field-induced currents andtotal charge transfer per pulse (optical rectification) pre-dicted in Ref. 3. However, as expected, the linear pulsewas “blind” to the valley chirality: the CB populationdistribution in the K and K (cid:48) valleys were exactly thesame as protected by the time reversal ( T ) symmetry.The chiral single-oscillation optical pulses also did notproduce any significant valley-specific CB population orinterference fringes, which was explained by the fact thatan electron experiences only a single passage in the vicin-ity of a Dirac point. For a pulse with a few (two or more)optical oscillations, the quantum pathways of the pas-sages by a Dirac point for different optical cycles wouldinterfere causing the appearance of pronounced chiral in-terference patterns . These were different for the K and K (cid:48) valleys depending on whether the pulse chirality isthe same or opposite to that of the corresponding valley.These chiral structures contained characteristic “forks”revealing vortices of the Berry connection. The result-ing electron distributions in the reciprocal space wereasymmetric, which obviously would lead to electron cur- a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec rents. Experimentally, such electric currents in grapheneinduced by both linearly- and circularly-polarized opticalpulses were observed in Ref. 7.We have also studied another class of hexagonal-symmetry 2D systems – transition metal dichalcogenides(TMDC’s) such as MoS and WS . In a dramatic con-trast to graphene, the TMDC’s showed very strong val-ley selectivity (preference to valley chirality) even for asingle-oscillation circularly-polarized optical pulse wherethe valley polarization was (cid:38) where the band gap can be opened and continuouslytuned controlled by an on-site energy ∆ parameter. Thismaterial is subjected to a strong-field single-oscillationoptical pulse. We aim to investigate the transition froman almost achiral strong-field population of the grapheneCB with no signs of the topological resonances and a lowvalley polarization to a highly-chiral and valley-selectivepopulation of the CB’s in the TMDC’s, which does ex-hibit a pronounced topological resonance and a high val-ley polarization.The gapped graphene is described by a two-bandmodel for direct band gap 2D semiconductors with the K and the K (cid:48) valleys mimicking the bilayer graphene, h-BN,or TMDC’s. We use a two band tight-binding Hamil-tonian of graphene where we additionally introduce anadjustable gap through different on-site energies, ∆ and − ∆, for the two sublattices, A and B. This difference ofthe on-site energies causes the breakdown of the inver-sion symmetry and opens up the band gaps of 2∆ at the K and the K (cid:48) points whose equality is protected by the T -inversion symmetry. II. MODEL AND MAIN EQUATIONS
The electron-collision relaxation times in graphene and2D materials are on the order or significantly longer than10 fs. . Therefore, for an ultrashort optical pulse withthe duration of less than 10 fs, we assume that the elec-tron dynamics in the field of the pulse is coherent andthe electron collision effects are negligible. With the de-scribed assumption, the electron dynamics is describedby the time-dependent Schr¨odinger equation (TDSE),
FIG. 1. (Color online) (a) Hexagonal lattice structure ofgraphene with sublattices A and B. (b) The first Brillouinzone of the reciprocal lattice of graphene with two valleys K and K (cid:48) . (c) Energy dispersion is shown as a function ofcrystal momentum for gapped graphene (1 eV) which has the following form i (cid:126) d Ψ dt = H ( t )Ψ (1)with Hamiltonian H ( t ) = H − e F ( t ) r , (2)where F ( t ) is the pulse’s electric field, e is electron charge,and H is the nearest neighbor tight binding Hamiltonianfor gapped graphene, H = (cid:18) ∆ γf ( k ) γf ∗ ( k ) − ∆ (cid:19) , (3)2∆ is the aforementioned finite gap between the CB andthe VB, γ = − .
03 is hopping integral, and f ( k ) = exp (cid:16) i ak y √ (cid:17) + 2 exp (cid:16) − i ak y √ (cid:17) cos (cid:16) ak x (cid:17) , (4)where a = 2 .
46 ˚A is lattice constant. The energies of CBand VB can be found from the above Hamiltonian, H ,as the following expressions E c ( k ) = + (cid:113) γ | f ( k ) | + ∆ ,E v ( k ) = − (cid:113) γ | f ( k ) | + ∆ , (5)where c and v stand for the CB and VB, respectively.This energy dispersion is shown in Fig. 1(c).Below we assume that the VB is fully occupied andthe CB is empty. We will be applying intense fields withamplitude F (cid:38) . / ˚A. At such intensities, the numberof photons, N p , per pulse within the minimum coherencearea of ∼ λ , where λ ∼ µ m is wavelength, N p ∼ cτ p λ F π (cid:126) ¯ ω ∼ × , (6)where c is speed of light; we assume realistic parameters: τ p ∼ (cid:126) ¯ ω ∼ F ( t ) as a classical elec-tric field keeping the quantum-mechanical description forthe solid. This is a usual semi-classical approach in high-field optics – see, e.g., Refs. 26–28. Note that quantizedoptical fields are used for much lower intensities .Such a full quantum mechanical approach is not neededfor the fields of the amplitude we consider. We solvethe Schr¨odinger equation in the truncated basis of Hous-ton functions (10) numerically without further approx-imations. Our pulse is just a single optical oscillation;therefore field F ( t ) is not periodic, and its effect cannotbe described as band gap modification as in Refs. 29 and31. However, the dynamic Stark effect and other field-dressing effects during the pulse are indeed fully takeninto account by our solution.In solids, the applied electric field generates both theintraband and interband electron dynamics. The intra-band dynamics is determined by the Bloch accelerationtheorem for time evolution of the crystal momentum, k , (cid:126) d k dt = e F ( t ) . (7)From this, for an electron with an initial crystal momen-tum q , time-dependent crystal momentum k ( q , t ) is ex-pressed as k ( q , t ) = q + e (cid:126) (cid:90) t −∞ F ( t (cid:48) ) dt (cid:48) . (8)Related to Bloch trajectories (8), we also define theseparatrix as a set of initial points q for which electrontrajectories pass precisely through the corresponding K or K (cid:48) points . Its parametric equation is q ( t ) = K − k (0 , t ) , or q ( t ) = K (cid:48) − k (0 , t ) , (9)where t ∈ ( −∞ , ∞ ) is a parameter.The corresponding wave functions, which are solutionsof Schr¨odinger equation (1) within a single band α , i.e.,without interband coupling, are the well-known Houstonfunctions ,Φ ( H ) α q ( r , t ) = Ψ ( α ) k ( q ,t ) ( r ) e − i (cid:126) (cid:82) t −∞ dt E α [ k ( q ,t )] , (10)where α = v, c for the VB and CB, correspondingly, andΨ ( α ) k are Bloch-band eigenfunctions in the absence of thepulse field, and E α ( k ) are the corresponding eigenener-gies.The interband electron dynamics is determined by thesolution of the TDSE (1). Such a solution can be ex-panded in the basis of the Houston functions Φ ( H ) α q ( r , t ),Ψ q ( r , t ) = (cid:88) α = c,v β α q ( t )Φ ( H ) α q ( r , t ) , (11)where β α q ( t ) are expansion coefficients. Let us introduce the following quantities D cv ( q , t ) = A cv [ k ( q , t )] exp (cid:16) iφ (d) cv ( q , t ) (cid:17) , (12) φ (d) cv ( q , t ) =1 (cid:126) (cid:90) t −∞ dt (cid:48) ( E c [ k ( q , t (cid:48) )] − E v [ k ( q , t (cid:48) )]) , (13) A cv ( q ) = (cid:28) Ψ ( c ) q | i ∂∂ q | Ψ ( v ) q (cid:29) . (14)Here Ψ (v) q and Ψ (c) q are periodic Bloch functions, i.e.,eigenfunctions of the Hamiltonian without an opticalfield; A cv ( q ) is a matrix element of the well-knownnon-Abelian Berry connection , and φ (d) cv ( q , t ) is thedynamic phase; the trajectory in the reciprocal space, k ( q , t ), is given by the Bloch theorem (8). Note that theinterband dipole matrix element, which determines op-tical transitions between the VB and the CB at crystalmomentum q , is D cv ( q ) = e A cv ( q ).The non-Abelian Berry connection matrix elementscan be found analytically as A cvx ( k ) = N (cid:32) − a | f ( k ) | (cid:33)(cid:32) sin ak x a √ k y i ∆ E c (cid:32) cos a √ k y ak x ak x (cid:33)(cid:33) (15) A cvy ( k ) = N (cid:32) a √ | f ( k ) | (cid:33)(cid:32) − − cos a √ k y ak x ak x − i E c sin a √ k y ak x (cid:33) (16)where N = | γf ( k ) | (cid:113) ∆ + | γf ( k ) | . (17)In these terms, we introduce Schr¨odinger equation inthe interaction representation in the adiabatic basis ofthe Houston functions as i (cid:126) ∂B q ( t ) ∂t = H (cid:48) ( q , t ) B q ( t ) , (18)where wave function (vector of state) B q ( t ) and Hamil-tonian H (cid:48) ( q , t ) are defined as B q ( t ) = (cid:20) β c q ( t ) β v q ( t ) (cid:21) , (19) H (cid:48) ( q , t ) = − e F ( t ) ˆ A ( q , t ) , (20)ˆ A ( q , t ) = (cid:20) D cv ( q , t ) D vc ( q , t ) 0 (cid:21) . (21)Schr¨odinger equation (18) defines a solution for dy-namics of the system, whose accuracy is limited by the FIG. 2. (Color online) F x and F y components of right handedcircularly polarized field. size of the basis set (i.e., truncation of the Hilbert space ofthe crystal). In particular, it contains such phenomenonas band gap opening in the field of a circularly-polarizedpulse. A formal general solution of this equation can bepresented in terms of the evolution operator, ˆ S ( q , t ), as B q ( t ) = ˆ S ( q , t ) B q ( −∞ ) , (22)ˆ S ( q , t ) = ˆ T exp (cid:20) i (cid:90) t −∞ ˆ A ( q , t (cid:48) ) d k ( t (cid:48) ) (cid:21) , (23)where ˆ T is the well-known time-ordering operator , andthe integral is affected along the Bloch trajectory [Eq.(8)]: d k ( t ) = e (cid:126) F ( t ) dt .The total charge current, J ( t ) = { J x ( t ) , J y ( t ) } , gener-ated during the pulse is the summation of the interbandand intraband currents, and it is determined by the fol-lowing expression J j ( t ) = ea (cid:88) q (cid:88) α ,α = v,c β ∗ α q ( t ) V α α j ( k ( q , t )) β α q ( t ) , (24)where j = x, y , a =2.46 ˚A is lattice constant, and V α α j ( k ) are matrix elements of the velocity operator,ˆ V j = 1 (cid:126) ∂H ∂k j . (25)For the known eigenstates, the intraband velocities are FIG. 3. (Color online) Residual CB population N (res)CB ( k ) forgraphene with adjustable bandgap in the extended zone pic-ture. The white solid line shows the boundary of the firstBrillouin zone with K, K (cid:48) -points indicated. The amplitudeof the optical field with the right handed polarization is 0.5V˚A − . The band gap is 0 (a), 0.2 eV (b), 0.8 eV (c), and 1.6eV (c).FIG. 4. (Color online) Residual CB population N (res)CB ( k ) forgraphene with adjustable bandgap in the extended zone pic-ture. The white solid line shows the boundary of the firstBrillouin zone with K, K (cid:48) -points indicated. The amplitude ofthe optical field with the left handed polarization is 0.5 V˚A − .The band gap is 0 (a), 0.2 eV (b), 0.8 eV (c), and 1.6 eV (c). FIG. 5. (Color online) Topological phase φ (T)cv ( q , t ) as a func-tion of time for gapped graphene in the field of left-handedoptical pulse with the amplitude of 0.5 V˚A − . The topolog-ical phase is calculated along the electron trajectory in thereciprocal space for (a) initial q point outside of the sepratrixand (b) initial q point inside of the sepratrix. Inset: solidblack line illustrates the separatrix for K (cid:48) valley, while thered line in panel (a) and the blue line in panel (b) show thecorresponding electron trajectories. FIG. 6. (Color online) Residual CB population N (res)CB ( k ) forgraphene with 1.6 eV bandgap in the extended zone picture.The white solid line shows the first Brillouin zone boundarywith K, K (cid:48) -points indicated. The amplitude of the opticalfield with right handed polarization is 0.5 V˚A − . Residualpopulation, N (res)CB ↑ ( k ), for (a) field amplitude is equal to 0.1V˚A − , (b) field amplitude is equal to 0.4 V˚A − , (c) field am-plitude is equal to 0.7 V˚A − , and (d) field amplitude is equalto 1 V˚A − . calculated as following V cc x ( k ) = − V vv x ( k ) = − aγ (cid:126) (cid:112) | γf ( k ) | + ∆ × sin ak x (cid:16) cos √ ak y ak x (cid:17) (26) V cc y ( k ) = − V vv y ( k ) = −√ aγ (cid:126) (cid:112) | γf ( k ) | + ∆ × sin √ ak y ak x . (27)The interband velocities can be expressed in terms ofthe non-Abelian Berry connection matrix elements as thefollowing V cv x = i (cid:126) A cvx ( E c − E v ) ,V cv y = i (cid:126) A cvy ( E c − E v ) . (28) III. RESULTS AND DISCUSSIONA. Circularly polarized pulse
Consider the ultrafast valley polarization induced bya circularly polarized pulse in the gapped graphene.We apply an ultrafast circularly-polarized optical pulse F =( F x , F y ), which is parametrized as the following F x = F (1 − u ) e − u (29) F y = ± F ue − u (30)Here, ± determines the handedness: the upper sign isfor the right-handed circular polarization, and the loweris for the left-handed circular polarization, which is T -reversed with respect to the first one; F is the ampli-tude of the optical oscillation, and u = t/τ , where τ is acharacteristic half duration of the optical oscillation (incalculations, we choose τ = 1 fs). The x and y compo-nents of F ( t ) [Eqs. (29) and (30)] for the right-handedcircularly polarized pulse are displayed in Fig. 2.By using the theory described above in Sec. II, wesolve TDSE (18) numerically with the initial condition( β c q , β v q ) = (0 , − , the CBpopulation after the pulse ends, known as residual CBpopulation, N (res)CB ( q ) = | β c q ( t = ∞ ) | , is shown in Figs3(a)-3(d) for different values of the band gap. In the caseof graphene, when the band gap is zero [Fig. 3(a)], theoptical pulse populates the CB along the correspondingseparatrix but does not produce any appreciable interfer-ence fringes or hot spots.For the case of graphene (the band gap is zero), thedistributions of the residual CB population in the K and K (cid:48) valleys in Fig. 3(a) are very close to each other. How-ever, there are some small differences, especially visibleinside the separatrix, which are mirror images of eachother due to the reflection ( P y ) symmetry of the lattice.In the reciprocal space near the K valley, with an in-crease of the band gap, the area inside the separatrix getless populated in comparison to the area outside of sepa-ratrix [Figs. 3(b)-(d)]. The opposite happens for the K (cid:48) valley where the majority of the population is inside ofseparatrix. For the T -reversed (left handed) pulse, thedistributions shown in Fig. 4 are T reversed (or, center-reflected) images of the distributions in Fig. 3; in partic-ular, the K and K (cid:48) valleys are exchanged places.As we can see from comparison of the cases of a differ-ent band gap [different panels in Figs. 3(a)-(d) or in Figs.4(a)-(d)], we conclude that with an increase of the bandgap, the K and K (cid:48) valleys become increasingly populateddifferently (valley polarization); simultaneously asymme-try of the population with respect to the separatrix ap-pears: the major (dominating) population occurs outsideof the separatrix while the minor population is inside. Note that the separatrix is a topological object: it di-vides the reciprocal space into two distinct regions: anypulse-field-induced electron Bloch trajectory, which origi-nates inside the separatrix, encircles the K (or, K (cid:48) ) pointthat is the center of the topological (Berry) curvature. Tothe opposite, a Bloch trajectory, which originates fromthe outside of the separatrix, does not encircle the K (or, K (cid:48) ) point. This difference causes an effect of thetopological resonance , which we briefly explain below.The fundamental evolution operator (23) can berewritten in the formˆ S ( q , t ) = ˆ T exp (cid:20) i (cid:90) t −∞ ˆ A (cid:107) ( q , t (cid:48) ) dk ( t ) (cid:21) , (31)where a longitudinal component of the non-Abelian Berry connection is defined as ˆ A (cid:107) ( q , t ) =ˆ A ( q , t ) F ( t ) /F ( t ), and dk ( t ) = e (cid:126) F ( t ) dt . Explicitly,matrix ˆ A (cid:107) ( q , t ) has the formˆ A (cid:107) ( q , t ) = (cid:34) D (cv) (cid:107) ( q , t ) D (cv) ∗(cid:107) ( q , t ) 0 (cid:35) , (32)where D (cv) (cid:107) ( q , t ) = (cid:12)(cid:12)(cid:12) A (cv) (cid:107) ( k ( q , t ) (cid:12)(cid:12)(cid:12) exp (cid:104) iφ (tot)cv ( q , t ) (cid:105) , (33)and the total phase, φ (tot)cv , is a sum of the dynamic andtopological phases, φ (tot)cv ( q , t ) = φ (d)cv ( q , t ) + φ (T)cv ( q , t ) . (34)Here, the topological phase is defined as φ (T)cv ( q , t ) =arg (cid:2) A (cid:107) ( q , t ) (cid:3) . This phase is the nontrival phase thatthe interband coupling amplitude acquires as a functionof time. For two classes of trajectories, which correspondto points q outside and inside of the separatrix, the topo-logical phase behaves completely differently. This phaseis displayed in Fig. 5 for point q outside (a) and inside(b) of the separatrix. The results are shown for K (cid:48) -valley.For the K -valley the corresponding phases have oppositesigns. As we see, if point q is outside of the separa-trix, the topological phase deceases with time near the K (cid:48) point ( t ≈
0) with the total change of ≈ − π . Thistotal change is almost independent on the band gap, 2∆.If point q is inside of the separatrix, the topological phaseas a function of time increases near the K (cid:48) point with themagnitude of the local increase that strongly depends onthe band gap. For zero band gap the topological phaseremains constant, while with increasing of the band gapthe magnitude of the local change of the topological phasenear the K (cid:48) point monotonically increases. Thus thetopological phase for gapped grapheneAs we see from Eq. (33), the interband electron dy-namics is determined by the total phase φ (tot)cv , which isa sum of the dynamic and topological phases. While thedynamic phase monotonically increases with time irre-spective of the position of point q (inside or outside ofthe separatrix), the dependence of the topological phaseon time is different for points q inside and outside ofthe separatrix. As a result the interference of the dy-namic and topological phases results in either a signifi-cant change of the total phase along the Bloch trajectory,which leads to small CB population, or mutual cancella-tion of the dynamic and topological phases, which resultsin coherent accumulation of the CB excitation amplitudeand enhancement of CB population. This is a topologi-cal resonance effect. For the right handed polarized pulseand for K (cid:48) valley, see Fig. 4, the topological resonanceoccurs for q points outside of the separatrix, while for K valley the topological resonance occurs for q point out-side of the separatrix. WIth increasing the band gap thetopological resonance becomes more pronounced. Notethat the conventional resonance can also be described ascancellation between the dynamic phase 2∆ t/ (cid:126) (where2∆ is excitation energy) and the field phase − ωt , whichoccurs for ω ≈ / (cid:126) .For a case of left-handed pulse illustrated in Fig. 5,the topological resonance occurs for crystal momentum q inside the separatrix for the K -point and outside of theseparatrix for the K (cid:48) -point. B. Linearly polarized pulse
A unique feature of circularly polarized pulse is thatan electron trajectory passes through a given point inthe reciprocal space only once. As a result the topo-logical resonance becomes well pronounced, which mani-fests itself in large valley polarization and clear asymme-try between electron residual CB populations inside andoutside of the separatrix. For linearly polarized pulse anelectron passes through each given point in the reciprocalspace twice. In this case manifestation of the topologicalresonance in the residual CB population is suppressed,while the features of the topological resonance are visibleduring the pulse in both CB population and generatedelectric current.Here we consider interaction of a linearly polarizedpulse with gapped graphene. The pulse is polarized along x axis and has the following profile F x = F (1 − u ) e − u , (35)where F is the amplitude of the pulse, u = t/τ , and τ = 1 fs. Similar to a circularly polarized pulse, weassume that initially the valence band is occupied andthe conduction band is empty.The residual CB population distribution in the recip-rocal space is shown in Fig. 7 for different values of theband gap. The hot spots are clearly visible in the popu-lation distribution. They are due to double passages byelectrons of the region near the K ( K (cid:48) ) point during thepulse, which finally results in the corresponding interfer-ence pattern. Such hot spots were discussed in Ref. ,where interaction of a linear optical pulse with graphene FIG. 7. (Color online) Residual CB population N (res)CB ( k ) ofgapped graphene with different bandgaps in the extendedzone picture. The solid white line shows the first Brillouinzone boundary with K, K (cid:48) -points indicated. The amplitudeof the optical field is 1 V˚A − . has been studied. For gapped graphene, the interferencepattern becomes smeared, see Fig. 7, which is due tobroadening of the interband dipole matrix element (non-Abelian Berry connection) for large band gaps.For gappless graphene, the CB population distributionis symmetric with respect to both x and y -axes, see Fig.7(a). For gapped graphene, the CB population distribu-tion is centrosymmetric only without any axial symme-tries. This is a manifestation of topological resonance forlinearly polarized pulse. In this case, the population dis-tribution is also chiral. Such chirality results in non-zeroresidual valley current in y direction, while the y compo-nent of the charge current vanishes after the pulse.The topological resonance is more clearly visible in thetime evolution of the CB population distribution, whichis shown in Fig. 8 for gapped graphene with the band gapof 1.6 eV that is similar to the band gap of MoS mono-layer. The amplitude of the pulse is 1 V / ˚A. At t = − . K and K (cid:48) points. This difference is due to topologicalresonance. Indeed, at t < − . k x axis andpassing through the K or K (cid:48) point only once. Then thecondition of the topological resonance is satisfied for q points above the K point and for q points below the K (cid:48) point. At t = 0, electronsThe distribution of the CB population during the pulseis shown in Fig. 8 for gapped graphene with bandgap 1.6eV (similar to MoS monolayer). The amplitude of the FIG. 8. (Color online) CB population N CB ( k ) as a functionof initial lattice vector for gapped graphene with bandgap 1.6eV in the extended zone picture at different moment of time.The white solid line shows the first Brillouin zone boundarywith K, K (cid:48) -points indicated. The applied pulse in linearlypolarized in x direction and its amplitude is 1 V˚A − . applied pulse is 1 V˚A − . Here this distribution is notsymmetric respect to x axis which is dramatically differ-ent from the case of graphene. Initially, for t ≤ − . − . < t ≤ . . < t ≤ FIG. 9. Longitudinal current density, J x , in gapped grapheneas a function of time for different bandgaps, 0 eV, 1 eV, and2 eV. F = 1V˚A − to create a bandgap, one sublattice, A, gets higher on-site energy respect to the other sublattice, B, so electronsmove from the sublattice with higher energy to the sub-lattice with lower energy see Fig. 10 (b). This Hall cur-rent is an addition to the longitudinal current generatedin the direction of the applied pulse (see Fig. 9). Apply-ing the pulse in opposite direction does not change thedirection of the Hall current ( J y ) which is determined bythe on-site energies of sublattices A and B. By increasingthe bandgap modified by the on-site energies, the am-plitude of the Hall current increases shown in from Fig.10(a). This unbalanced current causes the net transferredcharge which can be measured experimentally.In addition to the charge current and the Hall current,each valley generates a valley current shown in Fig.11 .Total valley current, J ( T ) , characterized by the followingexpression J ( T ) α = J ( K ) α − J ( K (cid:48) ) α (36)where α shows the direction of the current. IV. CONCLUSION
We demonstrated that a fundamentally fastest valleypolarization could be induced in gapped graphene by asingle oscillation circularly polarized pulse. This effectis similar to TMDC where the circular pulse populatesone valley significantly respect to the other depending onthe polarization of the pulse. We also showed the effectof bandgap on the valley polarization. The existence ofthe bandgap is necessary to have a valley polarization
FIG. 10. (a) Hall current density, J y , in gapped graphene asa function of time for different bandgaps, 0 eV, 1 eV, and 2eV. The amplitude of the applied field is F = 1V˚A − . (b)The lattice structure of graphene with two sublattices, A andB, is shown here. Where A has higher on-site energy respectto B it causes the electron motion from A to B and createsHall current in positive direction normal to the applied field. since it causes a gradual accumulation of the topologicalphase along the Bloch k − space electron trajectory, whichis necessary to compensate the gradually accumulatingdynamic phase.Also, we predicted that the distribution of the CBpopulation in the reciprocal space induced by the ap-plied linear pulse is chiral. This electron distribution canbe observed by time resolve angle-resolved photoelectronspectroscopy (tr-ARPES). The linear pulse generates alongitudinal current in the direction of the field and aphotovoltaic Hall current normal to the field in gappedgraphene. This Hall current is generated in the absenceof a magnetic field by a linearly polarized pulse. Whileapplying pulse in the opposite direction changes the di-rection of the longitudinal current it does not change thedirection of the Hall current which is only affected bythe on-site energies of different sublattices. Additionally,the unbalanced profiles of currents generate transferredcharges in the direction of the applied field and the direc-tion normal to the applied field. In addition to the chargecurrent, there is a nonzero net valley current which canbe measured experimentally.The predicted ultrafast valley polarization has the po-tential to be used in ultrafast quantum memory devicesfor quantum information processing. K. S. Novoselov, A. Mishchenko, A. Carvalho, andA. H. C. Neto, “2d materials and van der waals het-erostructures,” Science , 461–1–11 (2016). D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effectson electronic properties,” Rev. Mod. Phys. , 1959–2007(2010). H. K. Kelardeh, V. Apalkov, and M. I. Stockman,“Graphene in ultrafast and superstrong laser fields,” Phys.Rev. B , 045439–1–8 (2015). H. K. Kelardeh, V. Apalkov, and M. I. Stockman, “Ultra-fast field control of symmetry, reciprocity, and reversibil-ity in buckled graphene-like materials,” Phys. Rev. B ,045413–1–9 (2015). H. K. Kelardeh, V. Apalkov, and M. I. Stockman, “At-tosecond strong-field interferometry in graphene: Chi-rality, singularity, and Berry phase,” Phys. Rev. B ,155434–1–7 (2016). S. A. Oliaei Motlagh, J.-S. Wu, V. Apalkov, and M. I.Stockman, “Femtosecond valley polarization and topolog-ical resonances in transition metal dichalcogenides,” Phys.Rev. B , 081406(R)–1–6 (2018). T. Higuchi, C. Heide, K. Ullmann, H. B. Weber, andP. Hommelhoff, “Light-field-driven currents in graphene,”Nature , 224–228 (2017). Thomas G. Pedersen, Antti-Pekka Jauho, and KjeldPedersen, “Optical response and excitons in gappedgraphene,” Phys. Rev. B , 113406 (2009). D. Jariwala, A. Srivastava, and P. M. Ajayan, “Graphenesynthesis and band gap opening,” J. Nanosci. Nanotechno. , 6621–6641 (2011). M. S. Nevius, M. Conrad, F. Wang, A. Celis, M. N. Nair,A. Taleb-Ibrahimi, A. Tejeda, and E. H. Conrad, “Semi-conducting graphene from highly ordered substrate inter-actions,” Phys. Rev. Lett. , 136802 (2015). C. L. Kane and E. J. Mele, “Quantum spin hall effect ingraphene,” Phys. Rev. Lett. , 226801 (2005). Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim,“Experimental observation of the quantum hall effect andBerry’s phase in graphene,” Nature , 201–204 (2005). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, andA. A. Firsov, “Two-dimensional gas of massless Diracfermions in graphene,” Nature , 197–200 (2005). A. K. Geim and K. S. Novoselov, “The rise of graphene,”Nat. Mater. , 183–191 (2007). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, “The electronic propertiesof graphene,” Rev. Mod. Phys. , 109–162 (2009). A. N. Grigorenko, M. Polini, and K. S. Novoselov,“Graphene plasmonics,” Nat. Phot. , 749–758 (2012). K. S. Novoselov, V. I. Falko, L. Colombo, P. R. Gellert,M. G. Schwab, and K. Kim, “A roadmap for graphene,”Nature , 192–200 (2012). M. Titov, R. V. Gorbachev, B. N. Narozhny, T. Tu-dorovskiy, M. Schutt, P. M. Ostrovsky, I. V. Gornyi, A. D.Mirlin, M. I. Katsnelson, K. S. Novoselov, A. K. Geim,and L. A. Ponomarenko, “Giant magnetodrag in grapheneat charge neutrality,” Phys. Rev. Lett. , 166601–1–5 FIG. 11. Valley current density, (a) J x and (b) J y in gappedgraphene (1 eV bandgap) as a function of time for F =1V˚A − (2013). Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi,“Proximity-induced ferromagnetism in graphene revealedby the anomalous hall effect,” Phys. Rev. Lett. , 016603(2015). E. H. Hwang and S. Das Sarma, “Single-particle relaxationtime versus transport scattering time in a two-dimensionalgraphene layer,” Phys. Rev. B , 195412–1–6 (2008). M. Breusing, S. Kuehn, T. Winzer, E. Malic, F. Milde,N. Severin, J. P. Rabe, C. Ropers, A. Knorr, and T. El- saesser, “Ultrafast nonequilibrium carrier dynamics in asingle graphene layer,” Physical Review B , 153410(2011). Ermin Malic, Torben Winzer, Evgeny Bobkin, and An-dreas Knorr, “Microscopic theory of absorption and ultra-fast many-particle kinetics in graphene,” Phys. Rev. B ,205406 (2011). D. Brida, A. Tomadin, C. Manzoni, Y. J. Kim, A. Lom-bardo, S. Milana, R. R. Nair, K. S. Novoselov, A. C. Fer-rari, G. Cerullo, and M. Polini, “Ultrafast collinear scat-tering and carrier multiplication in graphene,” Nat Com-mun , 1987–1–9 (2013). I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. Turcu,E. Springate, A. Stohr, A. Kohler, U. Starke, and A. Cav-alleri, “Snapshots of non-equilibrium Dirac carrier distri-butions in graphene,” Nat. Mater. , 1119–24 (2013). Andrea Tomadin, Daniele Brida, Giulio Cerullo, Andrea C.Ferrari, and Marco Polini, “Nonequilibrium dynamics ofphotoexcited electrons in graphene: Collinear scattering,auger processes, and the impact of screening,” Phys. Rev.B , 035430 (2013). P. B. Corkum and F. Krausz, “Attosecond science,” Nat.Phys. , 381 – 387 (2007). F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod.Phys. , 163–234 (2009). Ferenc Krausz and Mark I. Stockman, “Attosecond metrol-ogy: from electron capture to future signal processing,”Nat Photon , 205–213 (2014). O. V. Kibis, “Metal-insulator transition in graphene in-duced by circularly polarized photons,” Phys. Rev. B ,165433–1–5 (2010). K. Kristinsson, O. V. Kibis, S. Morina, and I. A. Shelykh,“Control of electronic transport in graphene by electro-magnetic dressing,” Sci. Rep. , 20082–1–7 (2016). M. Claassen, C. J. Jia, B. Moritz, and T. P. Dev-ereaux, “All-optical materials design of chiral edge modesin transition-metal dichalcogenides,” Nat. Commun. ,13074–1–8 (2016). F. Bloch, “ ¨Uber die Quantenmechanik der Elektronen inKristallgittern,” Z. Phys. A , 555–600 (1929). W. V. Houston, “Acceleration of electrons in a crystal lat-tice,” Phys. Rev. , 184–186 (1940). F. Wilczek and A. Zee, “Appearance of gauge structurein simple dynamical systems,” Phys. Rev. Lett. , 2111–2114 (1984). F. Yang and R. B. Liu, “Nonlinear optical response inducedby non-Abelian Berry curvature in time-reversal-invariantinsulators,” Phys. Rev. B , 245205 (2014). A. A. Abrikosov, L. P. Gorkov, and I. E. Dzialoshinskii,