The ultrafast optical manipulation of materials by femtosecond laser pulses is rapidly becoming a major guiding theme in condensed matter physics1,2,3. At the same time, the quest for novel topological states of matter triggered enormous research activity since the discovery of topological insulators4. Merging both of these vibrant fields, a recent work reported the coupling of short laser pulses to surface Dirac fermions in the topological insulator Bi2Se3 (ref. 5). This work demonstrated the creation of Floquet-like sidebands during irradiation as well as the opening of a small band gap at the surface state Dirac point for circular light polarization.

Time-reversal symmetry protects massless Dirac fermions on the surface of topological insulators6,7 and, in combination with inversion symmetry, also in graphene in the absence of spin-orbit coupling8. In a milestone paper, Haldane envisioned that breaking either or both of these symmetries would open a gap at the Dirac points in graphene, allowing one to tune between a trivial insulator and a Chern insulator9. While equilibrium band gap engineering has become a major theme since the first synthesis of monolayer graphene, it was only recently proposed that circularly polarized, high-frequency laser light could turn trivial equilibrium bands into topological nonequilibrium Floquet bands10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, coined Floquet topological insulator (FTI).

The FTI concept is based on two things: first, in the limit of continuous laser driving at frequency Ω, the temporal periodicity allows one to employ a repeated quasi-energy zone scheme with a temporal Brillouin zone of size Ω. Second, in the high-frequency limit, defined by Ω being larger than the electronic bandwidth, these repeated zones contain well-separated copies of the original electronic bands spaced by integer multiples , the so-called Floquet sidebands. The effect of the laser on the original n=0 band manifold is perturbative in 1/Ω. If the laser is circularly polarized, time-reversal symmetry is broken and an energy gap opens at the Dirac points in graphene due to the fact that photon emission and absorption processes do not commute. The FTI concept then follows from an exact mapping of driven graphene to the Haldane model, leading to a well-defined nonzero Chern number.

Whereas this envisioned high-frequency strong pumping limit that is required for nontrivial topological states is currently experimentally unattainable, a natural follow-up question regards the engineering of local spectral gaps in realistic pump-probe experiments. This leads to the question on which time scales the quasi-steady Floquet regime can be reached when continuous-wave driving is replaced by a short laser pulse. Moreover, it requires the investigation of the low-frequency regime, in which 1/Ω perturbation theory is not applicable. In this regime, the overlap of different Floquet sidebands prevents a global topological classification of states.

In this work, we address these problems by simulating the real-time development of single-particle energy gaps in graphene coupled to short laser pulses, using realistic parameters for time-resolved, angle-resolved photoemission spectroscopy (tr-ARPES). We show that the tr-ARPES band structure shows well-defined Floquet bands provided that a hierarchy of time scales is fulfilled between the duration of the pump-pulse, the duration of the probe-pulse, and the laser period: σpump>σprobe2π/Ω. We predict the opening of a Dirac point gap and the formation of Floquet sidebands that form on femtosecond time scales. An important difference to the high-frequency limit arises from the overlap of Floquet sidebands. At frequency-dependent critical driving field strengths, we find a sequence of level crossings and energy gap closings at the Dirac points. The analysis of snapshots of pseudospin textures near the Dirac points allows us to identify optically induced nontrivial changes of sublattice mixing at these level crossing points, that manifest themselves in Berry curvature corrections of electrical transport and magnetization. Even though a global topology cannot be assigned to the low-frequency driven states, we show that the analysis of level crossings and energy gap closings leads to a classification scheme in terms of local gaps and Berry curvatures.


Haldane model

To set the stage for our results, we briefly outline the basic ingredients for the low-energy physics of Haldane’s equilibrium model. We start from two Dirac cones with effective Hamiltonian vD(qxσxτz+qyσyI). Here vD is the Dirac point velocity, the Pauli matrices σ label pseudospin arising from the graphene sublattices and , τ labels the valley degree of freedom corresponding to the Dirac cones around K and K′, and I is the 2 × 2 identity matrix. Electron spin can be neglected in the absence of spin-orbit coupling. Momentum q is measured from the respective Dirac points. The pseudospin content P(q) essentially measures orbital band content (see Supplementary Note 1). For instance, a pseudospin pointing along the +z (up) direction means that the band is predominantly of sublattice character, while a pseudospin pointing along the −z (down) direction indicates mainly sublattice character. Together with the winding of the Px and Py in-plane pseudospin components around the Dirac points, Pz determines the local Berry curvature 26.

In Haldane’s model, an effective mass term zτκ leads to an energy gap Δ=2m at the Dirac points (Fig. 1a). Its relative sign between K and K′ is determined by τκ and depends on its origin: If the gap is induced by introducing a staggered sublattice potential breaking inversion symmetry, τκ=τ0, implying that the effective mass term has the same sign at K and K′, and the out-of-plane pseudospin component Pz is the same at both Dirac points (Fig. 1b). By contrast, if the gap originates from breaking time-reversal symmetry, τκ=τz, hence Pz points in opposite directions (Fig. 1c).

Figure 1: Graphene with band gap.
figure 1

(a) Graphene band energy (E) versus momentum (k) dispersion with a band gap Δ at the Dirac points. (b) Trivial gap structure in Haldane’s model: The pseudospin Pz(k), indicated by up and down arrows, points in the same direction at both Dirac points K and K′. The winding number contributions cancel. This gap structure appears if inversion symmetry is broken, and time-reversal symmetry is intact. (c) Nontrivial gap structure: Pz(k) points in opposite directions at K and K′, leading to a nonzero Chern number +1 or −1. This gap structure appears if time-reversal symmetry is broken and inversion symmetry is intact.

Time-resolved photoemission spectra of driven graphene

We now come to the discussion of our nonequilibrium results. We start from the minimal honeycomb-lattice tight-binding model of graphene. We drive this system by coupling to a time-dependent, spatially homogeneous electric field modelled as a time-dependent vector potential A(t), which couples to the electrons via Peierls substitution. The relativistic magnetic component of the light field is neglected. The pump-pulse has a temporal width σpump=165 fs, photon frequency Ω=1.5 eV (laser period of 2.8 fs), with linear or circular light polarization, corresponding to a femtosecond pump-pulse. The field strength is given by Amax, which is measured in units of the inverse carbon–carbon distance. For graphene, the conversion to the peak electric field strength is Emax=Amax × 1,060 mV Å−1 for Ω=1.5 eV. We track the time- and momentum-resolved single-particle spectrum of the pump-driven electrons using a short 26 fs probe-pulse that emits photoelectrons and thereby generates a photocurrent27,28, as measured experimentally with tr-ARPES (see Methods).

We first characterize the nonequilibrium band structures using tr-ARPES spectra. Fig. 2 shows the tr-ARPES spectra on a momentum cut along the Γ−KK′ direction near K at peak field (Δt=0 fs). We first perform a calculation using pump pulses with linear polarization along the kx direction for two different field strengths (Fig. 2a,b). One can see the formation of Floquet sidebands, but since the pump preserves time-reversal symmetry, the spectrum remains gapless at the Dirac point energy. The main effect of the linearly polarized pump is a shift of the Dirac point location from K towards K′, which increases with increasing field strength. This Dirac point shift is due to the nonlinearity of bands and does not happen for perfect Dirac cones.

Figure 2: Transient angle-resolved photoemission spectrum.
figure 2

(ah) tr-ARPES spectral intensity of irradiated graphene at peak field (delay time Δt=0) versus binding energy ω and momentum (kK)x near the Dirac K point (vertical dashed lines) with Dirac point energy at −0.5 eV (horizontal dashed lines) on a momentum cut along the Γ−KK′ direction. The colour scheme varies between 0 (white) and maximal (red) spectral intensity. The pump laser field is linearly polarized in panels (a) and (b), and circularly polarized in panels (ch). The pump-pulse field strength Amax is varied between panels, with Amax values and laser polarizations as indicated. The driving frequency is 1.5 eV for all panels. The solid curves show the corresponding quasi-static Floquet band structures. The linearly polarized laser in panels (a) and (b) does not induce a band gap, whereas the circularly polarized laser in panels (ch) first induces a band gap at the Dirac point and then leads to a sequence of level crossings (panels (d) and (f)) and gap closing (g) as the field strength is increased.

Floquet spectrum and level crossings

Next, we turn to circular light polarization, thereby breaking time-reversal symmetry. In Floquet theory, the quasi-static eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of Ω, the so-called Floquet sidebands. Energy gaps of n-th order in the field open at avoided level crossings of sidebands which differ by n photon energies. For circular light, an energy gap of second order in the field opens at the Dirac point. In our tr-ARPES simulation, for a moderate field strength and 1.5 eV photons, an energy gap exceeding 100 meV at K is induced, accompanied by avoided level crossing gaps nearby (Fig. 2c). Due to the aforementioned hierarchy of time scales, we observe an excellent agreement of tr-ARPES spectra and the quasi-static Floquet band structure obtained by diagonalizing the Floquet Hamiltonian involving large numbers of sidebands (solid lines, see Supplementary Material).

At larger field strength (Fig. 2d), the Floquet bands move closer to each other and cross. They separate again and the Dirac point energy gap decreases (Fig. 2e). At even larger fields, there is another crossing between Floquet bands (Fig. 2f) before the Dirac point gap closes (Fig. 2g) and eventually reopens (Fig. 2h). At the largest field strength shown here (Amax=1.00), the bands are almost flat, indicating that the ac Wannier–Stark limit is approached. This creation of flat Wannier–Stark bands in the strong driving limit impedes the continuous growth of the gap with increasing field strength.

To analyse the pump photon frequency dependence of the Dirac point level crossings and gap closing in more detail, we show in Fig. 3 the first two negative Floquet eigenvalues tracking the position of the first two Floquet bands below ED. Fig. 3a shows the initial gap opening at Ω=1.5 eV, which is quadratic in the field at small Amax, followed by two level crossings between the two Floquet bands indicated by two arrows. The gap at the Dirac point is then closed, indicated by the third arrow. For Ω=3.0 eV, the field range between the two Floquet band level crossings increases (Fig. 3b), then decreases at Ω=4.5 eV, and finally vanishes for Ω=5.5 eV. The initial quadratic gap opening is the same for all photon frequencies due to the linearity of the graphene bands near the Dirac points. The differences between different photon frequencies at larger fields then arise from the nonlinearity of the bands further away from the Dirac points, which is specific to graphene.

Figure 3: Gap opening and level crossings in Dirac point Floquet spectra.
figure 3

First two negative Floquet quasi-energies ω at the Dirac points, shown below the Dirac point energy (−0.5 eV) as a function of field strength Amax for increasing driving frequencies varied between panels (ad), as indicated. The level crossings between the two Floquet bands are indicated by green arrows, the gap closings at higher field strength for Ω=1.5 eV are indicated by violet arrows. Black circles in panel (a) indicate the field strengths used in Fig. 2c–h. The sequence of level crossings and gap closings implies nontrivial changes in the pseudospin contents due to changes in the sublattice mixing of orbital band contents.

Local pseudospin textures near Dirac points

We now turn to the discussion of local pseudospin content. Fig. 4a–c present false color plots of the momentum-resolved pseudospin contents near the Dirac points for the driven system at Ω=1.5 eV. At small field before the first level crossing (Fig. 4a), the Px and Py components have two sign changes along a path around K, as expected for weakly driven graphene. This nodal structure is directly related to the qxσx+qyσy term in the effective low-energy Hamiltonian introduced above, which shows that the Px component transforms like qx and the Py component transforms like qy. We coin this state S1, with one nodal line and therefore a single pseudospin winding in the vicinity of the Dirac points. Importantly, the Pz component changes sign between K and K′, which is consistent with breaking time-reversal symmetry.

Figure 4: Field-induced changes in local pseudospin textures.
figure 4

(ac) Pseudospin textures of the components Px,y,z(k) near K and K′ at peak field (delay time Δt=0) for Ω=1.5 eV, with varying maximal field strengths Amax in panels (ac) as indicated. The colour scheme varies between pseudospin component being −1 (red), 0 (white) and +1 (blue). The pseudospin winding changes from p-wave in state S1 (a) to fourfold nodal lines in S2 (b) to p-wave in S'1 (c), with opposite pseudospin direction compared to the original S1 state. The Pz component changes sign between K and K′ since time-reversal symmetry is broken. (d) Phase diagram of local pseudospin windings near the Dirac points as a function of driving field strength and frequency. The phase boundaries are obtained from the Dirac point level crossings and gap closings. Black circles indicate the parameter values used in panels (ac).

When the field is increased through the first level crossing (Fig. 4b), the character of the local pseudospin textures changes. The effective mass term still changes sign between K and K′. Remarkably, the Px and Py components double their winding number, changing sign four times along a path around K. We call this state S2, since it has two nodal lines in the vicinity of the Dirac point, which cross at the Dirac point. Although the corresponding ARPES spectrum for the same parameters (Fig. 2f) has only little spectral weight near K in the Floquet bands close to ED, the pseudospin texture is well-defined at all momenta considered here. Also, other sidebands have higher spectral weight, and each of the Floquet sidebands carries the same pseudospin information.

A similar texture is also obtained for Bernal stacked bilayer graphene29 in a perpendicular electric field30. In this analogy, we stress that the doubled winding in bilayer graphene persists even when the energy gap induced by the electric field goes to zero. By contrast, in our case the doubled winding state S2 vanishes when the Floquet sidebands cross, and gives way to a single winding state S1 in the low-field limit. Also the effective perpendicular electric field generated by the circularly polarized light pulse in our work points in opposite directions at K and K′, in contrast to the static electric field applied in ref. 30, which corresponds to the Haldane mass term. In any case, the observation of state S2 suggests the possibility to dynamically engineer effective models with higher pseudospin winding numbers, similarly to higher-order spin-orbital textures in topological insulators31.

When the field is increased further through the second level crossing and the gap closing, one obtains the pseudospin textures shown in Fig. 4c. Here all the pseudospin components are flipped compared to the ones in Fig. 4a, and we coin this flipped state with single pseudospin winding S'1.

Phase diagram at low-frequency driving

We are now in a position to discuss the phase diagram of local Berry curvatures, which follow from the pseudospin textures around the Dirac points, as a function of field strength and driving frequency. Fig. 4d shows the positions of the Floquet band level crossings indicating the transition to a pseudospin texture with a doubled number of nodal lines, as well as the Dirac energy gap closing leading to a state with inverted Pz component. There is an upper frequency limit for the former state in the range of field strengths shown here. This is consistent with the fact that in the infinite-frequency limit, only states with p-wave pseudospin textures corresponding to S1 and S'1 were found, which can be understood from the exact mapping to the static Haldane model in this limit32.

On one hand, the characterization of nonequilibrium states in terms of local pseudospin textures is restricted to momenta near the Dirac points by Floquet sideband level crossings. Such level crossings generically appear at low driving frequency Ω because different Floquet sidebands overlap if Ω is smaller than the electronic bandwidth. On the other hand, sideband level crossings at the Dirac point are the root cause of the appearance of the exotic pseudospin textures in Fig. 4b. The low-frequency behaviour of driven graphene is therefore more complicated, but also contains new states that are absent in the high-frequency limit. The complete evolution of the tr-ARPES spectra and pseudospin textures across the boundaries of the S2 state is shown in the Supplementary Figs 1, 2 and 3 for driving frequencies of 1.5, 3.0 and 4.5 eV, respectively, and discussed in Supplementary Note 2.


Our combined results show that band gaps induced by breaking time-reversal symmetry in graphene are within reach under realistic experimental conditions. In particular, the achievable energy resolution for probe-photon energies which are sufficiently high to reach the Dirac points33,34 should allow for the detection of photoemission gap sizes exceeding 100 meV. The change in pseudospin texture near critical driving offers the exciting opportunity of optical manipulation of local Berry curvatures near Dirac points on ultrafast time scales. Moreover, the combination of broken inversion symmetry and broken time-reversal symmetry opens up the possibility of controlling the valley degree of freedom and inducing different energy gaps at the two Dirac points35,36,37,38.

The spectroscopic detection of pseudospin textures requires access to orbital band content. To this end, hexagonal structures with inequivalent orbitals on the and sublattices having different photoemission probe-energy cross-sections could be examined. A candidate material for this purpose is hexagonal boron nitride. The demonstration of pseudospin imbalance at the two Dirac points by circularly polarized light in boron nitride would be intriguing. Alternatively, artificial hexagonal lattices with sublattice potentials have already been demonstrated with cold atoms39. Thus the proposed pseudospin textures could in principle also be realized in driven ultracold quantum gases40,41.


Methods summary

The simulations presented here start from a minimal tight-binding model of spinless electrons with nearest-neighbour hopping on the honeycomb lattice8,42. The pump-pulse drives the electrons via minimal coupling to a gauge field , with a Gaussian shape function for a pulse of width σpump centred around time tp, and photon frequency Ω. The phase shift of π/2 between the x and y components, represented by the unit vectors ex and ey, describes circular light polarization. For comparison we also study linearly polarized light with vanishing y component. Throughout this work we use units where e==c=1. The electric field is E(t)=−∂A(t)/∂t, and we neglect the relativistic magnetic field of the laser pulse. This means that the electronic spin degree of freedom maintains its full degeneracy, and it is therefore not explicitly included in our calculations. In particular, both the photoemission spectra and the pseudospin textures are identical for both physical spin species.

The tr-ARPES is computed from the trace of the nonequilibrium lesser Green function by a postprocessing step involving the probe-laser-pulse shape sW(t) with time resolution W, which leads to an effective tr-ARPES energy resolution 1/σprobe27,28. The delay time of a pump peaked at tp and probe peaked at tpr is given by Δttprtp.

Simulations are performed for an initial equilibrium sample temperature T=116 K. We typically use 500,000−1,500,000 time steps for the computation of the time evolution operator, depending on the field parameters. This corresponds to a maximal time-step size of 0.0016, fs. Green functions for tr-ARPES measurements are sampled on a grid with 5,000−15,000 real-time steps and a maximal step size of 0.16 fs. The Dirac point velocity is given by vD=4.2 eV aC−C, where aC−C=1.42 Å is the carbon–carbon distance8. We choose a chemical potential μ=0.5 eV, which sets the Dirac point energy ED=−0.5 eV relative to μ. This choice is motivated by the fact that typical graphene samples on substrates are doped, and that states both below and above the Dirac point energy are occupied in the initial equilibrium states and therefore nicely visible in the tr-ARPES spectra.

The simulation parameters for the pump-probe setup are as follows: The pump laser field has frequency Ω=1.5 eV, unless denoted otherwise, implying oscillation periods of 2.58 fs. Its temporal width is σpump=165 fs. We vary the peak vector potential Amax=0.10 … 1.00 in units of aC–C−1. This corresponds to peak electric field strengths Emax=ΩAmax of 106 … 1,060 mV Å−1 for Ω=1.5 eV and the graphene lattice parameters. The photoemission probe-pulse has a width σprobe=26 fs. This choice of parameters is motivated by the hierarchy of time scales in the system: The oscillation period for the pump laser light, the temporal width of the probe-pulse, which controls the time and energy resolution for the tr-ARPES signal, and the temporal width of the pump-pulse, which controls the nonequilibrium state and ensures a well-defined center frequency for the pump-pulse.

Model and time evolution

Our goal is to obtain the lesser Green function matrix in 2 × 2 orbital space (see below) with matrix elements

where is a creation (annihilation) operator for a fermion at momentum k in orbital α (β) {a,b}. As shown below, the photocurrent and pseudospin contents are computed from these lesser Green functions.

Including the field via Peierls substitution, the time-dependent Hamiltonian for and sublattices with corresponding orbitals a and b reads

with the Hamiltonian matrix elements

where V=2.8 eV is the nearest-neighbour hopping matrix element matching the graphene bandwidth and Dirac point velocity. In equilibrium, the Hamiltonian has two Dirac points at momenta K and K′ given by , where momenta and the vector field A(t) are measured in multiples of the inverse of the carbon–carbon distance aC–C8.

It is convenient to define the Hamiltonian matrix for momentum k in orbital basis,

In the absence of a driving field, this Hamiltonian is diagonalized by a rotation at t=0, which is a time before the pump-pulse is turned on. Note that t=0 is used here as a notation for the earliest real time we consider, not to be confused with zero delay time Δttprtp=0, which refers to the time where the Gaussian pump-pulse envelope is maximal. For later times, the given rotation does not diagonalize the Hamiltonian except for accidental cases where the gauge field is an integer multiple of a reciprocal lattice vector.

The computation of double-time propagators requires the evaluation of the time evolution operators

Since at different times do not commute with each other, the time ordering in is taken into account by discretization of the real-time axis and multiplication of the resulting time-step evolution operators. We then obtain the time evolution operator as 2 × 2 matrices in band basis,

where Nt,t is the number of fine time steps of size δt between t and t′, and the product is understood as time ordered with later times to the left.

The lesser Green function matrix results from

where time 0 refers to an initial time where the system is in equilibrium before the pump-pulse is turned on, and is the time-independent diagonal matrix of initial equilibrium band occupation with diagonal elements and corresponding to Fermi function filling for the two energy eigenvalues.

Floquet spectra

The Floquet spectra shown in Figs 2 and 3 are calculated from the Floquet Hamiltonian corresponding to equation (3):

where gmn(k) are the Fourier series expansion coefficients of g(kA(t)):

Here Jn(A) is the Bessel function of the first kind. As the pump frequencies considered in this work are small with respect to the electronic bandwidth, the corresponding spectrum must be evaluated numerically via truncation of the full Floquet Hamiltonian. In practice, we achieve convergence for |m|≤40.

Time-resolved ARPES formalism

The computation of the time-resolved photocurrent involves normalized Gaussian probe-pulse shape functions sσprobe(t) of width σprobe centred around time t. In the Hamiltonian gauge, the photocurrent (tr-ARPES intensity) at momentum k, binding energy ω and pump-probe delay time Δttprtp is then obtained from27

In the main text (Fig. 2) we show tr-ARPES spectra at peak field strength with false colour plots of the tr-ARPES intensity I(k,ωt=0), that is, intensity variations as a function of binding energy ω along selected momentum cuts k. The location of energy bands E(k) can be obtained from the maxima in the ARPES intensity as a function of binding energy at constant momentum, the so-called energy distribution curves. As seen in Fig. 2 of the main text, these bands are in excellent agreement with quasi-static Floquet bands, whose calculation is described below.

The photocurrent as defined in equation (12) is computed from the lesser Green function in a fixed gauge. We would like to point out that this quantity is not gauge-invariant. In fact, the general definition of a gauge-invariant photocurrent that fulfills the positivity criterion I(k,ωt)≥0 for all k,ωt in the presence of a field is an outstanding research problem43. The problem likely lies in the neglect of photoemission matrix elements, which can be momentum and field dependent. The photocurrent according to equation (12) manifestly fulfills the positivity criterion. In addition, it also matches the Floquet band structure, as shown in the present work. The Floquet band structure is not gauge-invariant either. Importantly, general conclusions drawn from the analysis of Floquet sidebands, level crossings and gap closings are valid even in a fixed-gauge calculation. This is due to the fact that the time-resolved, momentum-integrated photoemission spectrum (tr-PES) is always manifestly gauge-invariant and positive. The tr-PES signal is obtained by integrating the tr-ARPES spectrum in any gauge over all momenta. Hence, conclusions about the presence or absence of energy gaps can be drawn even in a gauge-variant formalism.

Additional information

How to cite this article: Sentef, M.A. et al. Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene. Nat. Commun. 6:7047 doi: 10.1038/ncomms8047 (2015).