Lightwave topology for strong-field valleytronics
LLightwave topology for strong-field valleytronics ´A. Jim´enez-Gal´an , R. E. F. Silva , , O. Smirnova , , & M. Ivanov , , Max-Born-Institute, Berlin, Germany. Department of Theoretical Condensed Matter Physics, Universidad Aut ´onoma de Madrid, Spain Technische Universit¨at Berlin, Berlin, Germany. Department of Physics, Humboldt University, Berlin, Germany. Blackett Laboratory, Imperial College London, London, United Kingdom.
Modern light generation technology offers extraordinary capabilities for sculptinglight pulses, with full control over individual electric field oscillations within each laser cy-cle . These capabilities are at the core of lightwave electronics – the dream of ultrafastlightwave control over electron dynamics in solids, on a few-cycle to sub-cycle timescale,aiming at information processing at tera-Hertz to peta-Hertz rates. Here we show a robustand general approach to ultrafast, valley-selective electron excitations in two-dimensionalmaterials , by controlling the sub-cycle structure of non-resonant driving fields at a few-femtosecond timescale. Bringing the frequency-domain concept of topological Floquet sys-tems
10, 11 to the few-femtosecond time domain, we develop a transparent control mechanismin real space and an all-optical, non-element-specific method to coherently write, manipulateand read selective valley excitations using fields carried in a wide range of frequencies, ontimescales orders of magnitude shorter than valley lifetime, crucial for implementation ofvalleytronic devices . a r X i v : . [ phy s i c s . op ti c s ] F e b wo-dimensional graphene-like systems with broken inversion symmetry, such as mono-layer hexagonal boron nitride (hBN) or transition metal dichalcogenides (TMDs), are candidatesfor next generation quantum materials due to their high carrier mobility and, especially, to theirvalley degree of freedom , with potential applications in quantum information processing. Valleysare local minima in the crystal band structure corresponding to different crystal momenta; in 2Dhexagonal lattices they are located at the K and K ′ = − K points of the Brillouin zone (Fig. 1a).Selective excitation of K or K ′ can be achieved using weak circularly polarized field resonant withthe direct band gap of the material : it couples to either K or K ′ depending on light’s helicity(the optical valley selection rule
14, 15 ).However, such weak fields pose a challenge for switching the generated excitations at theultrafast time-scales, desirable due to short valley lifetimes ( ∼ − fsec for excitons andelectrons, respectively). A major step towards meeting this challenge has been made recently :switching of the population between the K and K ′ valleys was achieved using the combinationof a resonant pump pulse, which populated the desired valley, and a strong terahertz pulse, whichmoved the excited population within the Brillouin zone by controlling the THz field strength .The feasibility of applying strong non-resonant fields to bulk dielectrics
7, 17–19 and ultrathintransition methal dichalcogenide (TMD) films without material damage opens major new op-portunities in valleytronics. Our approach capitalizes on them. In contrast to previous work ,we require neither resonant light, nor the precise tuning of the strength of the control field. In-stead, we use far off-resonant light to modify the topological properties of the system by inducing2aldane-type complex-valued second neighbour hoppings in a topologically-trivial lattice. Thisis done by using the bicircular light field composed of counter-rotating fundamental and its sec-ond harmonic. We find that rotating the Lissajous figure drawn by the electric field vector of suchpulse (Fig. 1b), relative to the lattice (Fig. 1c), controls the magnitude and the phase of the com-plex light-induced second-neighbour hoppings, thus controlling the cycle-averaged band structure(Fig. 1d) and the Berry curvature in each valley. Exponential sensitivity of multi-photon excitationto the effective bandgap naturally leads to selective excitation in the valley where the bandgap isreduced. Thus, valley selection is achieved by tailoring the symmetry of the Lissajous figure to thelattice and controlling its orientation.Using light to control topological properties of solids has led to the concept of topologicalFloquet lattices
10, 11, 22 . In this context, in addition to providing a real-space description of theeffect, our key results are as follows. First, we find that strong low-frequency circularly polarizedfields show opposite valley polarization than those in the weak-field, one-photon resonant regime,as a consequence of light-induced streaking of the excited electrons. Second, we initialize andmanipulate valley polarization on a few femtosecond time scale in a way that remains consistent fora broad range of frequencies and field intensities, and independent of the specifics of the material.We give two examples, hexagonal boron nitride (hBN) and MoS . Third, using an additionallinearly polarized probe pulse, we map the valley pseudospin onto the polarization of its harmonics,providing an all-optical measure of the valley asymmetry. Finally, we show numerical evidence ofa topological phase transition induced by non-resonant, tailored light, occurring at specific valuesof intensity and wavelength of the driving field, just as predicted by our analytical model.3 a) (b) K K KGK'K' K' K + = ω ω (c) -6 E n e r g y ( e V ) -4-20246 φ = - π /2 φ = 0 φ = π M G -M K'G G φφ = - π /2 φ = 0 φ = π (d) Figure 1:
Light-induced modification of the band structure with tailored field . (a) 2D hexago-nal lattice with broken inversion symmetry in real space (left; red and green represent two differentatoms) and reciprocal space, with valleys K and K ′ (right). (b) The trefoil Lissajous figure gener-ated by the field has the symmetry of the sub-lattice and can be rotated by changing the two-colorphase ϕ . (c) Depending on the field orientation (grey trefoil), the two atomic sites are addresseddifferently. For ϕ = − π / , the field interacts with both atoms equally and the bands show valley-degeneracy (d, black solid line). For ϕ = , the field interacts with the two types of atoms differently(note how the two atoms inside the trefoil are now not interchangeable, irrespective of where thefield is placed in the lattice). This lifts the valley degeneracy (d, red dashed line). For ϕ = π , thesituation is reversed (d, blue dashed dotted line).4onsider first strong circularly polarized fields with frequencies well below the band gapenergy. In Fig. 2a,b,c we show the electron populations in the p z conduction band of hBN afterapplying a strong ( I = TW/cm ) circularly polarized field with three different frequencies ω andthe same helicity. The same observable can be obtained, e.g., by angularly-resolved photoemis-sion spectroscopy (ARPES). The most excited valley switches as we transition from the highestfrequency ( Fig. 2a) to the lowest frequency ( Fig. 2c). All panels switch K for K ′ when the helicityof the laser is reversed (See Supplementary Note 1).This switch in valley polarization is a consequence of streaking: the rotation of the fieldselects excitation at the K valley, as in the one-photon case, while the large magnitude of thevector potential ( A = √ I / ω ≃ . a.u.) displaces the electron population towards the K ′ valley.Thus, one can control valley polarization using non-resonant, low-frequency fields and controllingthe field helicity. However, just like in one-photon resonant fields, the result is material specificand depends crucially on the relation between the band gap and the field frequency (Fig. 2a,b,c).A robust approach is offered by a field widely used for controlling strong-field processes inatoms
2, 23, 24 , which combines circularly polarized fundamental ω with its counter-rotating secondharmonic: F L = ˆ x [− F cos ( ωt ) + F cos ( ωt + ϕ )] + ˆ y [ F sin ( ωt ) + F sin ( ωt + ϕ )] , (1)where F and F are the field strengths of the fundamental and second harmonic respectively, ϕ is the sub-cycle phase delay between the two drivers. During one cycle, the field draws thetrefoil shown in Fig 1b, which fits ideally the geometry of two triangular sub-lattices and can break5 .50.0-0.5 0.00.0150.03 0.00.0150.03 0.00.0150.00.020.04 0.00.010.02 0.00.01-0.5 0.0 0.5 k y k x KK' KK K'K'KK' KK K'K' KK' KK K'K' KK' KK K'K'KK' KK K'K'KK' KK K'K' -0.5 0.0 0.50.50.0-0.5 k y -0.5 0.0 0.5 k x -0.5 0.0 0.5 k x -0.5 0.0 0.5-0.5 0.0 0.50.50.0-0.5 0.50.0-0.50.50.0-0.50.50.0-0.5 (a) (b) (c)(f)(e)(d) Decreasing frequency Figure 2:
Selective valley excitation in strong fields . (a-c) Populations in the first Brillouin zone(red dashed hexagon) of the conduction band of monolayer hBN (band gap of ∆ = . eV) afterapplying a strong, circularly polarized pulse with frequencies: (a) ω = . eV, (b) ω = . eV and(c) ω = . eV. All other parameters (helicity, peak field strength F L = . a.u. and duration τ = fs) remain fixed. Valley selection changes in the low-frequency regime; all panels switch K for K ′ when the helicity is reversed (not shown). Panels (d-f) show the same for a bicircularfield with the same F L , τ and fundamental frequency ω as in the corresponding panels (a-c) above.The most populated valley now remains the same; reversing the overall helicity of the fields doesnot switch K to K ′ (not shown), indicating new robust mechanism for valley polarization.6he symmetry between these two identical sub -lattices in graphene inducing charge oscillationsbetween them . Its orientation relative to the sublattices is controlled by ϕ .Fig. 2d,e,f shows the valley polarization in hBN at fixed ϕ , for three different frequencies.In contrast to (a-c), now the most populated valley remains robust as we transition from the few-photon to the deep multiphoton regime (Fig. 2d,e,f). It is selected by the orientation of the trefoilrelative to the lattice. The feasibility of changing the trefoil orientation during the pulse implies theability to switch the valley pseudospin on the fly. Reversing the helicity of the two drivers ( ω and ω ) does not switch the valley polarization (see Supplementary Note 1), suggesting a new, robustmechanism for valley selection.As shown in the Methods section, energy conserving processes involving both ̵ hω and ̵ hω photons, such as absorption of one ̵ hω photon and re-emission of two ̵ hω photons, lead to complexsecond-neighbour hopping t . Its amplitude and phase are controlled by the field strengths andthe two-color phase ϕ . A non-zero imaginary part lifts the valley degeneracy. For moderatelystrong fields (see Methods), the cycle-averaged, the imaginary component of laser-induced secondneighbour hopping is I { t } ∼ J ( √ a F ω ) J ( √ a F ω ) cos ϕ, (2)where a is the lattice constant and J n is the Bessel function of the first kind of order n . Thus,the sub-cycle control over the field geometry, ϕ , controls the topological properties of the dressedsystem. The associated modification of the band structure leads to the valley asymmetry, controlledby the two-color delay ϕ in a way that is not material specific.7o illustrate this, we consider monolayer hBN (see Methods for numerical details) and use abicircular pulse with fundamental frequency ω = . eV ( λ = nm), duration of τ = fs,intensity ratio of I ( ω )/ I ( ω ) = , and maximum peak intensity of I = TW/cm (lower than itspredicted damage threshold ). We change the two-color phase ϕ to control the orientation of thefield with respect to the lattice. The valley population asymmetry is calculated as A = ± ( f n,K − f n,K ′ )/( f n,K + f n,K ′ ) , where f n,K ( f n,K ′ ) is obtained by integrating the electron population insidethe black dashed circles encircling K ( K ′ ) in (a-c,e-g), and the + ( − ) sign is used for hBN (MoS ),due to their opposite Berry curvatures.When ϕ = − π / , both sub-lattices are addressed equally, no valley asymmetry is present inthe cycle-averaged band structures (Fig. 3a). We find that K and K ′ valleys of the conductionband are nearly equally excited. In contrast, when ϕ = , the cycle-averaged band structure showsa strong valley asymmetry (Fig. 3b). This reflects in the valley populations, which show a 60%contrast (Fig. 3d). The situation is reversed for ϕ = π , the populations switch to the opposite valley(Fig. 3c). The sense of rotation of the pulse also contributes to the valley asymmetry due to theorbital propensity rule, but its effect is relatively weak. It manifests in small valley polarization for ϕ = ± π / , and in that ϕ = and ϕ = π are not exact opposites (compare Figs. 3b,c).To confirm that the mechanism is general, we performed calculations in MoS for the samelaser frequencies, which allows us to maintain the same timescale of valley control. Due to thesmaller band gap of MoS , the intensity peak of the total field was kept at I = . TW/cm ,well below its damage threshold . Fig. 3(e)-(g) shows valley excitation after the pulse in the8 (b) (c) -404G K G K' G E n e r g y ( e V ) -404G K G K' G E n e r g y ( e V ) (f) (g) KK' KK' k x k x k x k x /2 /2/2 3 /2 (rad) V a ll e y a s y mm e t r y ( % ) H a ll c o n d u c t i v i t y ( a . u . ) integrated pop.helicity H3 xy Berry curvature -660 k x k y /2 (rad) V a ll e y a s y mm e t r y ( % ) xy integrated pop.helicity H3 H a ll c o n d u c t i v i t y ( a . u . ) -60600 Berry curvature k x k y (d) (h) -0.50.00.5-0.40.40 k y k y (a) -404G K G K' G E n e r g y ( e V ) KK' k x -0.4 0.0 0.4 k x -0.5 0.0 0.5 (e) KK' KK' KK'
Figure 3:
Strong-field manipulation and optical reading of valley polarization . Electron popu-lations in the lowest conduction bands of hBN (a-c) and spin-integrated MoS (e-g), after applyinga bicircular field with λ = µ m: (a,e) ϕ = − π / , (b,f) ϕ = , and (c,g) ϕ = π , sketches of fieldorientations relative to the hexagonal cell are also shown. The insets in panels (a)-(c) show thecycle-averaged band structures for hBN (see Methods) vs ϕ . (d,h) Valley population asymmetry(dotted solid blue curve), helicity of the third harmonic of the probe (faint blue curve) and anoma-lous Hall conductivity (faint red curve), as a function of ϕ in (d) hBN and (h) MoS . The probe iscarried at ω for hBN and ω for MoS ; ω is the fundamental frequency of the bicircular pulse.9pin-integrated lowest conduction band of MoS (see Supplementary Note 2 for spin-resolvedresults), illustrating similar control as for hBN, but with lower values of the valley polarization.Higher values of valley polarization can be obtained by increasing the field strength, while totalpopulations can be controlled with the pulse duration.We now turn to optical reading of the valley pseudospin. Since the Berry curvature is op-posite at the K and K ′ valleys, when an in-plane electric field is applied, the carriers generatea current perpendicular to the electric field (anomalous current), with opposite direction at eachvalley. For equal valley populations, the anomalous current will thus cancel, leading to a zeroanomalous Hall conductivity (AHC), σ xy = − e ̵ h ∑ n ∫ BZ d k ( π ) f n ( k ) Ω n,z ( k ) , (3)where f n and Ω n are the population and Berry curvature of the n -th band. If the valley populationsare not equal, the perpendicular currents originating from K and K ′ do not compensate each other,leading to a non-zero AHC . This so-called valley Hall effect has been demonstrated for MoS monolayers by measuring the direction of the transverse Hall voltage .Alternatively, the direction of the anomalous current can be retrieved all-optically from thehelicity of the harmonics of a linearly polarized probe . To this end, we apply a probe fieldwith frequency ω or ω , linearly polarized along the G - M direction of the lattice, and monitorthe harmonic response at a 3N multiple of the probe frequency. This guarantees a background-free measurement without interference from harmonics generated by the bi-circular field, since thelatter does not generate N ω harmonics due to symmetry. The probe pulse arrives after the end of10he bicircular pump.The component of the optical response polarized orthogonal to the probe polarization un-dergoes a phase jump of π as the population switches from one valley to another (see Methods),leading to a rotation in the helicity of the probe response, Fig. 3(d,h). The helicity is calculated as h = ( I ⟳ − I ⟲ )/( I ⟳ + I ⟲ ) , where I ⟳ ( I ⟲ ) is the component of the harmonic intensity rotatingclockwise (anticlockwise).When the dynamics occur mainly in one conduction band, as in monolayer hBN, the helicityof the optical response reads the AHC and, consequently, the valley polarization (Fig. 3d). In thecase of MoS (Fig. 3h), the helicity follows the valley polarization qualitatively, but deviates fromthe AHC due to the influence of higher bands.Our approximate analytical model for an effective t Eq.(2) predicts a topological phasetransition. To explore this prediction, we monitor the time-dependent anomalous Hall conductivity(AHC) in hBN during the laser pulse. Fig. 4 shows the gated time-dependent AHC defined as ¯ σ xy ( t ) = ∫ ∞−∞ σ xy ( t ′ ) G ( t ′ − t ) dt ′ , (4)where σ xy ( t ′ ) is the instantaneous AHC, defined by Eq. 3 using instantaneous electron populations f n ( k , t ) . The instantaneous AHC is averaged over several laser cycles using the gate function G ( t ′ − t ) = e − ( t ′ − t ) / T , T = × π / ω , to produce ¯ σ xy ( t ) .Note that σ xy ( t ) uses field-free Berry curvature Ω ( k ) and field-free bands, for which f n ( k , t ) are computed. Strikingly, we find that ¯ σ xy ( t ) changes sign when the laser field strength and wave-11 H a ll c o n d u c t i v i t y ( a . u . ) (pred. topo. trans.) fi eld peak (a)
20 10 0 10 20time (optical cycle)0.0250.0200.0150.0100.0050.0000.0050.010 2.0 m2.5 m3.0 m3.5 m4.0 m4.5 m5.0 m5.5 m6.0 m (pred. topo. trans.) fi eld peak (b) H a ll c o n d u c t i v i t y ( a . u . ) Figure 4:
Light-induced topological phase transition . (a,b) Gated time-dependent anomalousHall conductivity (AHC) in hBN, for bi-circular field with (a) different total electric field amplitudefor fixed carrier λ = µ m, and (b) different λ for fixed total intensity I tot = TW/cm . The pulseduration is 20 cycles (FWHM), I ( ω )/ I ( ω ) = , two-color delay ϕ = . The grey lines at the totalintensity I tot = . TW/cm (a) and λ = µ m (b) correspond to the analytical prediction of thetopological phase transition. It coincides with the change of sign of the gated AHC close to thepeak of the field (warm color lines).length reach the parameter regime for the topological phase transition predicted by the model, bothas a function of intensity and frequency (Fig. 4a,b). The sign change in σ xy ( t ) manifests the keyfeature of the topological phase transition, that is, the band gap closing and the associated changein the cycle-averaged anomalous Hall conductivity of the dressed system.Thus, intense low-frequency fields offer unusual and robust routes for controlling electronicresponse in 2D materials at PHz rates. Fields with polarization states tailored to the geometry ofthe lattice provide the most opportunities. Controlling orientation of the field polarization relative12o the lattice controls the topological properties of the trivial dielectric by inducing and controllingcomplex Haldane-type second-neighbor couplings. This enables robust valley selection for multi-photon excitation. The new rules for valley selective excitation could also enable spin selectivityin materials with strong spin-orbit interaction, exploiting exponential sensitivity of strong-fieldexcitation to the bandgap in the way similar to spin-polarization in strong-field ionization of atoms
29, 30 . Our work thus lays the grounds for a new regime of valleytronics, spintronics, and light-induced topology.
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