Introduction

The study of lightwave-driven electronic dynamics occurring on sub-optical-cycle time scales in condensed matter and nanosystems is a fascinating frontier of attosecond science originally developed in atoms and molecules1. Adapting attosecond metrology techniques2 to observe and control the fastest electronic dynamics in the plethora of known solids and novel quantum materials3 is very promising for studying correlated electronic dynamics (e.g., excitonic effects, screening) on atomic length and time scales, thereby potentially impacting future technologies such as emerging petahertz electronic signal processing2,4 or strong-field optoelectronics5,6.

The nonlinear process of high-order harmonic generation (HHG) in gases is one of the cornerstones of attosecond science and is well understood by the semiclassical three-step model1. In solids, nonperturbative HHG up to 25th harmonic order without irreversible damage was first reported in ref. 7. This work triggered extensive research activities aimed at unraveling the microscopic interband and intraband dynamics underlying HHG from crystals (for a comprehensive overview, see ref. 8), thereby extending attoscience techniques to solids. The prevailing strong-field dynamics were successfully identified in specific cases, even if a global picture has not yet emerged. Other works demonstrated isolated attosecond extreme ultraviolet (XUV) pulses emitted from thin SiO2 films9, or investigated HHG from two-dimensional (2D) materials such as graphene10, 2D transition-metal dichalcogenides10,11, and monolayer hexagonal boron nitride12.

Elucidating the complex microscopic electronic dynamics producing HHG without making a priori severe assumptions poses a challenge for theory. Indeed, the theory must capture at the same time the transitions between discrete electronic bands, and the ultrafast motion of electrons within the bands; two mechanisms usually decoupled in the description of either optical properties or transport in semiconductors and insulators. An effective way to account for the full interacting many-body electronic dynamics and real crystal structure is using ab-initio time-dependent density functional theory (TDDFT) simulations13,14,15. Some of us recently used this theoretical framework to reveal how the microscopic mechanisms governing HHG in solids depend on the ellipticity of the driving field and the underlying band structure15. That work predicted that different harmonics react differently to the driver ellipticity, as they can either originate mainly from intraband contributions or from coupled interband and intraband dynamics14.

The symmetry properties of the light-matter interaction Hamiltonian distinguishes HHG in crystals from atoms and molecules, with major ramifications for the selection rules of different harmonics and their polarization states. HHG from atoms driven by propeller-shaped bichromatic waveforms produces circular harmonics16 (for convenience, we often use the terminology linear/circular/elliptical instead of linearly/circularly/elliptically polarized to refer to the polarization state). In molecules, both the point group and the driving field determine the symmetries of the coupled light-matter system. Consequently, depending on the molecular symmetries and the molecule’s orientation with respect to the light polarization direction, elliptical high-order harmonics can be produced by linear17 or elliptical driver pulses18. For bichromatic bicircular driver fields, circular harmonics with alternating helicities can be obtained, provided that the molecule’s symmetries are compatible with that of the driving field19. In crystals, several recent works studied the high-harmonic response on driver pulse ellipticity ε, which can strongly differ qualitatively from the atomic and molecular cases. Whereas earlier work20 looked exclusively at the harmonic yield, later research also investigated the polarization states and selection rules of the higher harmonics from various solids of different crystal symmetries15,2). The pulses are focused onto the sample with a 25-cm CaF2 lens, resulting in a 1/e2 focus diameter of 2w0 = 95 μm. After 50 cm of propagation, an iris is used to spatially suppress the otherwise very strong third harmonic. A curved UV-enhanced Al mirror is used to direct the output light to an Ocean Optics UV-VIS HR4000 spectrometer with a slit width of 10 μm. To determine the ellipticities and major axes of the generated harmonics, a Rochon polarizer is placed between sample and iris and rotated in total by 360°, measuring a spectrum every 18°. To detect the helicity of the circular harmonics, a tunable zero-order QWP (from Alphalas) is placed between sample and polarizer. For post-processing the polarizer scans, the harmonic intensities are fitted with a sin-square curve offset from zero, the ellipticity calculated as \(|\varepsilon _{{\mathrm{HH}}}| = \sqrt {I_{{\mathrm{min}}}/I_{{\mathrm{max}}}}\) and the major-axis rotation as ϕHH = arctan(Iy/Ix). The driving-intensity scan in Supplementary Figure 3 is performed employing reflective neutral-density filters.

Ab-initio TDDFT simulations of high-harmonic generation in solids

Within the framework of TDDFT, the evolution of the wavefunctions and the evaluation of the time-dependent current are computed by propagating the Kohn–Sham equations

$${\mathrm{i}}\hbar \frac{{\partial \psi _{n,{\mathbf{k}}}({\mathbf{r}},t)}}{{\partial t}} = H_{{\mathrm{KS}}}({\mathbf{r}},t)\psi _{n,{\mathbf{k}}}({\mathbf{r}},t),$$
(1)

where ψn,k is a Bloch state, n a band index, k a point in the first Brillouin zone (BZ), and HKS is the Kohn–Sham Hamiltonian given by

$$H_{{\mathrm{KS}}}({\mathbf{r}},t) = \frac{1}{{2m}}\left( { - {\mathrm{i}}\hbar \nabla + \frac{e}{c}{\mathbf{A}}(t)} \right)^2 + v_{{\mathrm{ext}}}({\mathbf{r}},t) + v_{{\mathrm{H}}}({\mathbf{r}},t) + v_{{\mathrm{xc}}}({\mathbf{r}},t).$$
(2)

The different terms correspond to the kinetic energy, the ionic potential, the Hartree potential, that describes the classical electron–electron interaction, and exchange-correlation potential, which contains all the correlations and nontrivial interactions between the electrons. The latter needs to be approximated in practice34.

We perform the calculations using the Octopus code35, employing the TB0936 meta generalized gradient approximation (MGGA) functional to approximate the exchange-correlation potential using the adiabatic approximation. To ensure the stability of our time-propagation, we solved the time-dependent Kohn–Sham equations self-consistently at every time step using the enforced time-reversal symmetry propagator37. The c-value entering in the TB09 functional is recomputed at each time step using the gauge-invariant kinetic energy density. We employ norm-conserving pseudo-potentials. We emphasize that within TB09 MGGA, the experimental bandgap of common semiconductors and insulators is well reproduced38, which is an important improvement over the local-density approximation (LDA) used in refs. 14,15, permitting direct comparison between experiment and theory. As shown in ref. 14 for adiabatic LDA (ALDA), local-field effects and dynamical correlations (at the level of the ALDA functional) do not seem to affect the HHG spectra of Si. The excitonic effects in Si mainly come from the long-range part of the exchange-correlation potential39, i.e., a renormalization of the Hartree term (which does not play any role in HHG from Si14); therefore, excitonic effects are not expected to modify the HHG spectra of materials such as Si. We note that this is not necessarily true for all materials, in particular materials with strongly localized excitons, for which bound states will form in the bandgap, or in strongly correlated materials23,24.

All calculations for bulk Si are performed using the primitive cell of bulk Si, using a real-space spacing of 0.484 atomic units, corresponding to 15 points along each primitive axis. We consider a laser pulse of 50-fs FWHM duration with a sin-square envelope and a carrier wavelength λ of 2.08 μm, corresponding to 0.60 eV carrier photon energy. We employ an optimized 36 × 36 × 36 grid shifted four times to sample the BZ, and we use the intensity corresponding to the experimental intensity, using the value for the optical index n of ~3.4 for computing the intensity in matter. The four shifts of the k-point grid are (in reduced coordinates) (0.5, 0.5, 0.5), (0.5, 0.0, 0.0), (0.0, 0.5, 0.0), (0.0, 0.0, 0.5). We use the experimental lattice constant a leading to a MGGA bandgap (direct) of silicon of 3.09 eV. In all our calculations, we assume a CEP of ϕ = 0.

We compute the total electronic current j(r, t) from the time-evolved wavefunctions, the HHG spectrum is then directly given by

$${\mathrm{HHG}}(\omega) = \left| {{\mathrm{FT}}\left( {\frac{\partial }{{\partial t}}{\int} d^3{\mathbf{r}}\,{\mathbf{j}}({\mathbf{r}},t)} \right)} \right|^2,$$
(3)

where FT denotes the Fourier transform.

Supplementary Figure 5 shows a comparison of a computed HHG spectrum to a corresponding experimental spectrum. Note that, as mentioned above, in our experiments we only detect harmonics up to HH9 due to the spectrometer used (Ocean Optics UV-VIS HR4000). Our TDDFT calculations predict that harmonics up to HH19 in the XUV spectral region are generated for our experimental conditions.