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Received 4 Dec 2014 | Accepted 26 Mar 2015 | Published 11 May 2015

Ultrafast materials science promises optical control of physical properties of solids. Continuous-wave circularly polarized laser driving was predicted to induce a light-matter coupled state with an energy gap and a quantum Hall effect, coined Floquet topological insulator. Whereas the envisioned Floquet topological insulator requires high-frequency pumping to obtain well-separated Floquet bands, a follow-up question regards the creation of Floquet-like states in graphene with realistic low-frequency laser pulses. Here we predict that short optical pulses attainable in experiments can lead to local spectral gaps and novel pseudospin textures in graphene. Pump-probe photoemission spectroscopy can track these states by measuring sizeable energy gaps and Floquet band formation on femtosecond time scales. Analysing band crossings and pseudospin textures near the Dirac points, we identify new states with optically induced nontrivial changes of sublattice mixing that leads to Berry curvature corrections of electrical transport and magnetization.

DOI: 10.1038/ncomms8047

Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene

M.A. Sentef1,2, M. Claassen3, A.F. Kemper4, B. Moritz1,5, T. Oka6, J.K. Freericks7 & T.P. Devereaux1,3

1 Stanford Institute for Materials and Energy Sciences (SIMES), Stanford University and SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA. 2 HISKP University of Bonn, Bonn 53115, Germany. 3 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA.

4 Lawrence Berkeley National Lab, 1 Cyclotron Road, Berkeley, California 94720, USA. 5 Department of Physics and Astrophysics, University of North Dakota, Grand Forks, North Dakota 58202, USA. 6 Department of Applied Physics, University of Tokyo, Tokyo, 113-8656, Japan. 7 Department of Physics, Georgetown University, Washington, DC 20057, USA. Correspondence and requests for materials should be addressed to M.A.S.(email: mailto:[email protected]

Web End [email protected] ) or to T.P.D. (email: mailto:[email protected]

Web End [email protected] ).

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The ultrafast optical manipulation of materials by femtosecond laser pulses is rapidly becoming a major guiding theme in condensed matter physics13. 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 elds, 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 rst 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 bands1025, coined Floquet topological insulator (FTI).

The FTI concept is based on two things: rst, in the limit of continuous laser driving at frequency O, the temporal periodicity allows one to employ a repeated quasi-energy zone scheme with a temporal Brillouin zone of size O. Second, in the high-frequency limit, dened by O being larger than the electronic bandwidth, these repeated zones contain well-separated copies of the original electronic bands spaced by integer multiples nO, the so-called

Floquet sidebands. The effect of the laser on the original n 0 band

manifold is perturbative in 1/O. 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-dened 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/O perturbation theory is not applicable. In this regime, the overlap of different Floquet sidebands prevents a global topological classication 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-dened Floquet bands provided that a hierarchy of time scales is fullled between the duration of the pump-pulse, the duration of the probe-pulse, and the laser period: spump4sprobec2p /O. 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 eld strengths, we nd 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 classication scheme in terms of local gaps and Berry curvatures.

ResultsHaldane model. To set the stage for our results, we briey outline the basic ingredients for the low-energy physics of Haldanes equilibrium model. We start from two Dirac cones with effective Hamiltonian vD(qxsx#tz qysy#I). Here vD is the Dirac point

velocity, the Pauli matrices r label pseudospin arising from the graphene sublattices A and B, s labels the valley degree of free

dom corresponding to the Dirac cones around K and K0, 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 A sublattice character,

while a pseudospin pointing along the z (down) direction

indicates mainly B 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

Fq @qxP q

@qyP q

P q

26.

Pz( k) Pz( k)

[afii9797]

Trivial: 1/2 + 1/2 = 0

Nontrivial: + 1/2 + 1/2 = 1

E K

Figure 1 | Graphene with band gap. (a) Graphene band energy (E) versus momentum (k) dispersion with a band gap D at the Dirac points. (b) Trivial gap structure in Haldanes model: The pseudospin Pz(k), indicated by up and down arrows, points in the same direction at both Dirac points K and K0. 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 K0, leading to a nonzero Chern number 1 or 1. This gap structure appears if time-reversal

symmetry is broken and inversion symmetry is intact.

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Amax = 0.10 (lin.)

Amax = 0.20 (lin.)

Amax = 0.10

Amax = 0.315

0.5

[afii9853][eV]

0.5

1.5

Amax = 0.50

Amax = 0.72

Amax = 0.96

Amax = 1.00

0.5

1.5

0.2 0.1 0 0.1 0.2

0.2 0.1 0 0.1 0.2 0.2 0.1 0 0.1 0.2 0.2 0.1 0 0.1 0.2

(kK)x [1/aC-C]

Figure 2 | Transient angle-resolved photoemission spectrum. (ah) tr-ARPES spectral intensity of irradiated graphene at peak eld (delay time Dt 0)

versus binding energy o and momentum (k K)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 G K K0 direction. The colour scheme varies between 0 (white) and maximal (red) spectral intensity.

The pump laser eld is linearly polarized in panels (a) and (b), and circularly polarized in panels (ch). The pump-pulse eld 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) rst 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 eld strength is increased.

(kK)x [1/aC-C]

In Haldanes model, an effective mass term msz#tk leads to an energy gap D 2m at the Dirac points (Fig. 1a). Its relative

sign between K and K0 is determined by tk and depends on its origin: If the gap is induced by introducing a staggered sublattice potential breaking inversion symmetry, tk t0, implying that the

effective mass term has the same sign at K and K0, 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, tk tz, hence Pz points in opposite directions

(Fig. 1c).

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 eld modelled as a time-dependent vector potential A(t), which couples to the electrons via Peierls substitution. The relativistic magnetic component of the light eld is neglected. The pump-pulse has a temporal width spump

165 fs, photon frequency O 1.5 eV (laser period of

2.8 fs), with linear or circular light polarization, corresponding to a femtosecond pump-pulse. The eld strength is given by Amax,

which is measured in units of the inverse carboncarbon distance. For graphene, the conversion to the peak electric eld strength is Emax Amax 1,060 mV 1 for O 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 rst characterize the nonequilibrium band structures using tr-ARPES spectra. Fig. 2 shows the tr-ARPES spectra on a momentum cut along the G K K0 direction near K at peak

eld (Dt 0 fs). We rst perform a calculation using pump pulses

with linear polarization along the kx direction for two different eld 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 K0, which increases with increasing eld strength. This Dirac point shift is due to the nonlinearity of bands and does not happen for perfect Dirac cones.

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 nite driving eld A shows copies of the original bands shifted by integer multiples of O, the so-called Floquet sidebands. Energy gaps of n-th order in the eld open at avoided level crossings of sidebands which differ by n photon energies. For circular light, an energy gap of second order in the eld opens at the Dirac point. In our tr-ARPES simulation, for a moderate eld strength and1.5 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).

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[afii9821] = 1.5 eV

[afii9853][eV]

2 0 0.5 1 1.5 2

[afii9821] = 3.0 eV

[afii9821] = 4.5 eV

[afii9821] = 5.5 eV

0 0.5 1 1.5 2

5 0 0.5 1 1.5 2

6 0 0.5 1 1.5 2

Amax

Figure 3 | Gap opening and level crossings in Dirac point Floquet spectra. First two negative Floquet quasi-energies o at the Dirac points, shown below the Dirac point energy ( 0.5 eV) as a function of eld 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 eld strength for O 1.5 eV are indicated by violet

arrows. Black circles in panel (a) indicate the eld strengths used in Fig. 2ch. 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.

At larger eld 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 elds, 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 eld strength shown here (Amax 1.00), the bands are

almost at, indicating that the ac WannierStark limit is approached. This creation of at WannierStark bands in the strong driving limit impedes the continuous growth of the gap with increasing eld 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 rst two negative Floquet eigenvalues tracking the position of the rst two Floquet bands below ED. Fig. 3a shows the initial gap opening at O 1.5 eV, which is quadratic in the

eld 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 O 3.0 eV,

the eld range between the two Floquet band level crossings increases (Fig. 3b), then decreases at O 4.5 eV, and nally

vanishes for O 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 elds then arise from the nonlinearity of the bands further away from the Dirac points, which is specic to graphene.

Local pseudospin textures near Dirac points. We now turn to the discussion of local pseudospin content. Fig. 4ac present false color plots of the momentum-resolved pseudospin contents near the Dirac points for the driven system at O 1.5 eV. At small

eld before the rst 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 qxsx qysy 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 K0, which is consistent with breaking time-reversal symmetry.

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Px Py Pz (K) Pz (K)

(kK)x [1/aCC]

0.1

Amax = 0.10

Amax = 0.50

Amax = 1.00

0.1

(kK) y[1/a CC]

0.1

0.1 0 0.1

0.1 0 0.1 0.1 0 0.1

0.1 0 0.1

(kK)x [1/aCC]

1.5

A max

0.5

0 1 1.5 2 2.5 3 3.5 4 4.5 5

[afii9821] [eV]

Figure 4 | Field-induced changes in local pseudospin textures. (ac) Pseudospin textures of the components Px,y,z(k) near K and K0 at peak eld (delay time Dt 0) for O 1.5 eV, with varying maximal eld 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 S1 (c), with opposite pseudospin direction compared to the original S1 state. The Pz component changes sign between K and K0 since time-reversal symmetry is broken. (d) Phase diagram of local pseudospin windings near the Dirac points as a function of driving eld 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 eld is increased through the rst level crossing (Fig. 4b), the character of the local pseudospin textures changes. The effective mass term still changes sign between K and K0.

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-dened 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 eld30. In this analogy, we stress that the doubled winding in bilayer graphene persists even when the energy gap induced by the electric eld 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-eld limit. Also the effective perpendicular electric eld generated by the circularly polarized light pulse in our work points in opposite directions at K and K0, in contrast to the static electric eld 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 eld 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 ipped compared to the ones in Fig. 4a, and we coin this ipped state with single pseudospin winding S1.

Phase diagram at low-frequency driving. We are now in a position to discuss the phase diagram of local Berry curvatures,

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which follow from the pseudospin textures around the Dirac points, as a function of eld 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 eld strengths shown here. This is consistent with the fact that in the innite-frequency limit, only states with p-wave pseudospin textures corresponding to S1 and S1 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 O because different Floquet sidebands overlap if O 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 of1.5, 3.0 and 4.5 eV, respectively, and discussed in Supplementary Note 2.

DiscussionOur 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 sufciently 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 points3538.

The spectroscopic detection of pseudospin textures requires access to orbital band content. To this end, hexagonal structures with inequivalent orbitals on the A and B 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, articial 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

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 eld At Amaxps t sinOtex cosOtey

with a Gaussian

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 controlsthe nonequilibrium state and ensures a well-dened center frequency for the pump-pulse.

Model and time evolution. Our goal is to obtain the lesser Green function matrix

Gok; t; t0 in 2 2 orbital space (see below) with matrix elements

Goab k; t; t0

i ayktbk t0

D E

; 1

where ay

k bk

is a creation (annihilation) operator for a fermion at momentum k

in orbital a (b) A{a,b}. As shown below, the photocurrent and pseudospin contents are computed from these lesser Green functions.

Including the eld via Peierls substitution, the time-dependent Hamiltonian for

A and B sublattices with corresponding orbitals a and b reads

H t

k ay

k byk

0 g k At

g k A t

; 2

with the Hamiltonian matrix elements

gk V 2cos 3

p kx

cos

cos ky

i 2cos 3

p kx 2

2 sin

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 K0 given by 4p=3 3

;

2 sin ky

p ; 0 2:4184; 0

where momenta and the vector eld A(t) are measured in multiples of the inverse of the carboncarbon distance aCC8.

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

H k; t

0 g k A t

: 4 In the absence of a driving eld, this Hamiltonian is diagonalized by a rotation

Rk 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 Dt tpr tp 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 eld is an integer multiple of a reciprocal lattice vector.

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

U k; t; t0

T exp i Z

g k A t

shape function ps t exp t tp

for a pulse of width spump

centred around time tp, and photon frequency O. The phase shift of p/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 polarizedlight with vanishing y component. Throughout this work we use units wheree c 1. The electric eld is E(t) qA(t)/qt, and we neglect the relativistic

magnetic eld 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 p 1/sprobe27,28. The delay time of a pump peaked at tp and probe

peaked at tpr is given by Dt tpr tp.

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 eld 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 is the carboncarbon distance8. We choose a chemical

potential m 0.5 eV, which sets the Dirac point energy ED 0.5 eV relative to m.

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 eld has frequency O 1.5 eV, unless denoted otherwise, implying

oscillation periods of 2.58 fs. Its temporal width is spump 165 fs. We vary the peak

vector potential Amax 0.10 y 1.00 in units of aCC 1. This corresponds to peak

electric eld strengths Emax OAmax of 106 y 1,060 mV 1 for O 1.5 eV and

the graphene lattice parameters. The photoemission probe-pulse has a width sprobe

H k; t

d t

: 5

Since Uk; t at different times do not commute with each other, the time ordering

T in Uk; t; t0 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,

U k; t; t0

2=2s2

pump

exp iH k; t jdt=2

; 6

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where Nt,t is the number of ne time steps of size dt between t and t0, and the product is understood as time ordered with later times to the left.

The lesser Green function matrix results from

Go k; t; t0

U k; t; 0

o k; 0; 0

U k; t0; 0

; 7

Gok; 0; 0 Rk iN kRky; 8 where time 0 refers to an initial time where the system is in equilibrium before the pump-pulse is turned on, and N k is the time-independent diagonal matrix of

initial equilibrium band occupation with diagonal elements f E1k and f E2k

corresponding to Fermi function lling 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):

HF X

mO m; a

j i m; a

h j X

gm n k

m; A

j i n; B

h j h:c:

where gm n(k) are the Fourier series expansion coefcients of g(k A(t)):

gm nk

O 2p

Z 0 dt eim nOtg k At

ei k m n ei k

k m n

e i k

k m n

h i

Jm nA 11

Here Jn(A) is the Bessel function of the rst 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|r40.

Time-resolved ARPES formalism. The computation of the time-resolved photocurrent involves normalized Gaussian probe-pulse shape functions ssprobe(t) of width sprobe centred around time t. In the Hamiltonian gauge, the photocurrent (tr-ARPES intensity) at momentum k, binding energy o and pump-probe delay time Dt tpr tp is then obtained from27

I k; o; Dt

Im X

Z dt1Z dt2ss tpr t1

s tpr t2

eiot

t Goaa k; t1; t2

In the main text (Fig. 2) we show tr-ARPES spectra at peak eld strength with false colour plots of the tr-ARPES intensity I(k,o,Dt 0), that is, intensity variations as

a function of binding energy o 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 dened in equation (12) is computed from the lesser Green function in a xed gauge. We would like to point out that this quantity is not gauge-invariant. In fact, the general denition of a gauge-invariant photocurrent that fullls the positivity criterion I(k,o,Dt)Z0 for all k,o,Dt in the presence of a eld is an outstanding research problem43. The problem likely lies in the neglect of photoemission matrix elements, which can be momentum and eld dependent. The photocurrent according to equation (12) manifestly fullls 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 xed-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.

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8047

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Acknowledgements

We acknowledge helpful discussions with Patrick Kirchmann, Sri Raghu, Xiao-Liang Qi, Shou-Cheng Zhang, and Bruce Normand. This work was supported by the Department of Energy, Ofce of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract Nos. DE-AC02-76SF00515 (Stanford/SIMES), DE-FG02-08ER46542 (Georgetown) and DE-SC0007091 (for the collaboration). Computational resources were provided by the National Energy Research Scientic Computing Center supported by the Department of Energy, Ofce of Science, under ContractNo. DE- AC02-05CH11231. J.K.F. was also supported by the McDevitt bequest at Georgetown. A.F.K. was supported by the Laboratory Directed Research and

Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DE-AC02-05CH11231.

Author contributions

M.A.S. developed the computer code and performed the ARPES and pseudospin calculations. A.F.K. assisted in developing the computer code for the ARPES calculations. M.C. developed the computer code and performed the Floquet calculations, and T.O. assisted in developing the computer code for the Floquet calculations. M.A.S., M.C., A.F.K., B.M., J.K.F. and T.P.D. wrote the manuscript. T.P.D. and J.K.F. are responsible for project planning.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

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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).

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Title

Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene

Author

Sentef, Ma; Claassen, M; Kemper, Af; Moritz, B; Oka, T; Freericks, Jk; Devereaux, Tp

Pages

7047

Publication year

2015

Publication date

May 2015

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Nature Publishing Group

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20411723

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Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms8047

ProQuest document ID

1679882960

Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene

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