Fig. 1: Light-wave-controlled valley-selective bandgap modification.

a, The tailored intense light waveform, which resembles a trefoil structure on the lattice plane, is used to coherently manipulate the band structure of monolayer hBN. b, For such a two-dimensional system, the band structure shows hexagonal symmetry with a minimum bandgap at the K and K′ points (valleys) at the vertices of the hexagon. In monolayer hBN, the optical bandgap is Δ ≈ 5.9 eV. c, Upon interaction with a strong trefoil waveform, the couplings between the atoms are modified. The laser-dressed system is described by a Haldane-type tight-binding Hamiltonian, such that the next-nearest neighbour hoppings are complex and carry a phase difference of π between the two sublattices, with on-site energies Δ/2 and −Δ/2. d–g, As the field (and its associated vector potential shown with respect to the first Brillouin zone) rotates in space, the CNNN hoppings are modified, changing from purely real (d,f) to purely imaginary (e,g). In d and f, the waveform points in between K and K′ (that is, M), but with different orientations. In e and g, it points to K and K′, respectively. The latter lifts the degeneracy between the K and K′ valleys, reducing the bandgap in one of them and increasing it in the other. h, The effective bandgap oscillates as a function of the rotation of the trefoil light field, leading to different electron excitation dynamics between the valleys. i–l, Normalized electron populations in the conduction band corresponding to the trefoil field orientation shown in the inset of d–g. This is calculated using the code described in ref. ⁴⁴. The green hexagons delimit the first Brillouin zones, with the K and K′ points at the vertices, as indicated in the inset of d–g. The purple triangles indicate the orientation of the vector potential. The electron population is a maximum in the valley where the bandgap is lower during the laser–matter interaction. a.u., arbitrary units.