High-harmonic generation from few-layer hexagonal boron nitride: Evolution from monolayer to bulk response

Two-dimensional materials offer a versatile platform to study high-harmonic generation (HHG), encompassing as limiting cases bulklike and atomiclike harmonic generation [Tancogne-Dejean and Rubio, Sci. Adv. 4, eaao5207 (2018)]. Understanding the high-harmonic response of few-layer semiconducting systems is important and might open up possible technological applications. Using extensive first-principles calculations within a time-dependent density functional theory framework, we show how the in-plane and out-of-plane nonlinear nonperturbative responses of two-dimensional materials evolve from the monolayer to the bulk. We illustrate this phenomenon for the case of multilayer hexagonal BN layered systems. Whereas the in-plane HHG is found not to be strongly altered by the stacking of the layers, we found that the out-of-plane response is strongly affected by the number of layers considered. This is explained by the interplay between the induced electric field, resulting from the electron-electron interaction, and the interlayer delocalization of the wave functions contributing most to the HHG signal. The gliding of a bilayer is also found to affect the high-harmonic emission. Our results will have important ramifications for the experimental study of monolayer and few-layer two-dimensional materials beyond the case of hexagonal BN studied here as the results we found are generic and applicable to all two-dimensional semiconducting multilayer systems.

Figure 1

(a) Side view and (b) top view of a four-layer hBN slab with ${\mathrm{AA}}^{\prime }$ stacking as explained in the text. (c) Another possible stacking, referred to as the AD stacking. Decomposition of the harmonic emission into (d) parallel and (e) perpendicular contributions, defined with respect to the laser polarization direction. This latter is defined by the angle $\alpha$ with respect to the crystallographic direction $\overline{K\mathrm{\Gamma }}$.

Figure 2

(a) Normalized HHG spectra for one- to six-layer slabs and for the bulk. The laser polarization is taken along the $\overline{K\mathrm{\Gamma }}$ crystallographic direction. (b) Comparison between the bulk HHG spectrum (black curve) and the spectrum obtained from the innermost layers of a six-layer slab, as explained in the text. (c) Parallel and (d) perpendicular contributions to the HHG spectrum of the monolayer versus the in-plane polarization angle, as defined in Fig. 1. (e) and (f) Same as (c) and (d), but for the bilayer case.

Figure 3

(a) HHG normalized spectra for one- to six-layer slabs and the bulk one. The intensity in matter used for the bulk calculation is $6.85\times {10}^{12}\mathrm{W}{\mathrm{cm}}^{-2}$, as explained in the text. (b) Harmonic yield integrated between 43 and 60 eV, normalized to the one of the monolayer, versus the number of layers. The shaded areas in (a) indicate the low-energy and high-energy regions; see the text for details.

Figure 4

Induced electric field along the out-of-plane direction, averaged in the plane (blue curve). This induced electric field can be split into a surface contribution, which gives a uniform induced electric field, and a bulk contribution, which, on average, gives no induced electric field. The bulk induced field (red curve) is shifted to the value of the depolarization field (dashed line), as explained in the text.

Figure 5

External electric field (red curve), surface-induced electric field (blue curve), and total electric field (green solid curve) for the six-layer hBN slab driven by an out-of-plane electric field; see the text for details. The green dashed curve shows the estimated total field from the Fresnel transmission coefficients, using the value of the bulk refractive index for a field polarized along the optical axis ($n=2.25$ [35]).

Figure 6

(a)–(f) Square modulus of the six highest occupied wave functions at the $\mathbf{k}$-point $K$ for the six-layer hBN slab. The value of the isosurfaces is taken here to be 0.08.

Figure 7

Effect of the out-of-plane phase on the HHG spectra of the bulk hBN and a six-layer slab of hBN. (a) HHG spectra for the bulk and six-layer slab (black and red curves) and HHG spectra computed using Eq. (5), i.e., without the out-of-plane phase (gray and purple dashed curves). (b) Same as Fig 3, but using Eq. (5) to compute the harmonic spectra.

Figure 8

Time-frequency analysis of the HHG from (a) a monolayer of hBN to (f) the six-layer slab. A time window of 0.25 fs was used to compute the Gabor transforms. The black arrows in (b)–(e) show a secondary structure which cannot be directly described by the three-step model. These structures appear as soon as two or more layers are stacked. The spectra become noisier as the number of layers stacked increases because of the spatial interferences, as explained in the text.

Figure 9

(a) Atomiclike HHG spectra of the monolayer (blue curve), the bilayer with ${\mathrm{AA}}^{\prime }$ stacking (red curve), and the bilayer with AD stacking (orange curve). The left panel corresponds to the low-order harmonics, whereas the right panel shows harmonics close to the energy cutoff. Anisotropy of the harmonic emission for an in-plane polarized laser for (b) the parallel and (c) perpendicular contributions for the bilayer with AD stacking.