Topological edge-state contributions to high-order harmonic generation in finite flakes

Edge states play a major role in the electron dynamics of topological insulators as they are the only conducting part in such materials. In this work we consider the Haldane model for a two-dimensional topological insulator, subjected to an intense laser field. We compare the numerically simulated high-harmonic generation (HHG) in the bulk of the Haldane model to HHG in corresponding finite flakes with edge states present, and explain the differences. In particular, peaks for energies below the bulk band gap appear in the spectra for the finite flakes. The positions of these peaks show a strong dependence on the size of the flakes, which can be explained using the dispersion relation for the edge states.

Figure 1

Sketch of a finite Haldanite flake of size $N=4$ (number of hexagons on each edge). There are two sublattice sites: A (open circles) and B (filled circles). The hopping between nearest neighbors (amplitude $t_1\in \mathbb{R}$) is indicated by solid lines connecting the sites, the distance between two sites being $a$. The next-nearest neighbor hopping is only highlighted for one example hexagon by the dotted arrows. The hopping amplitude along the arrows is $i{t}_2$ and $-i{t}_2$ in the opposite direction ($t_2\in \mathbb{R}$). The polarization direction of the laser field is indicated by the double-headed arrow. The edge of the flake is marked by the red line. Its length, the circumference, is denoted $s\left(N\right)$ in the text.

Figure 2

(a) Energies of a finite flake ($N=10$) plotted in ascending order. The edge states (ES) are highlighted in red and without a shaded background. The states of the valence band (VB) and conduction band (CB) are indicated by a shaded background. The dotted, horizontal line indicates the energy of the highest occupied state ($E_{299}$). (b) Color-coded Fourier transform of the edge states. The slope of the solid line is determined by the maxima for the highest occupied ($i=299$) and lowest unoccupied ($i=300$) state. (c) Probability density of the highest occupied state ($E_{299}$). The radii of the circles centered at the lattice sites are proportional to the probability density. The localization very close to the edge sites is clearly visible.

Figure 3

Group velocity $v_g$ calculated with (19) as a function of the flake size $N$.

Figure 4

Comparison between the spectra for the finite ($N=7$) and the bulk ($50\times 50$ points in $k$ space) system in (a) parallel and (b) perpendicular polarization direction to the external laser field. The shaded areas indicate energies below the band gap of the bulk system. The spectrum of the bulk is shifted in yield such that the fundamental harmonic in parallel direction to the external field has a yield equal to 1. The spectrum of the finite flake is shifted in yield such that the plateaus (nonshaded area) from both systems have a comparable yield in parallel polarization direction.

Figure 5

Harmonic spectra in parallel polarization direction for different flake sizes $N$. The red bullets (blue crosses) indicate the first (second) peak that shifts continuously to smaller harmonic orders as the flake size increases. For $N=[2,3,4]$ the second peak could not be unambiguously identified. The spectra are multiplied by a factor of ${10}^{12(N-1)}$ for better visibility. The dotted, vertical line indicates harmonic order three.

Figure 6

(a) Hexagonal flake ($N=7$) with indicated corners (blue numbers). (b) Zoom into one edge (between two corners) with indication of segments containing three sites each. (c) and (d) Frequency-filtered current, spatially and temporally resolved, for (c) ${\omega }_1=2.4{\omega }_0$ and (d) ${\omega }_2=11.3{\omega }_0$. The white vertical lines in (c) and (d) indicate the positions of the corners of the flake. The diagonal black line segments correspond to a velocity $v=0.5976$, i.e., the group velocity determined by the band structure for $N=7$. These line segments are plotted for each edge between two corners separately. The horizontal lines in (c) and (d) depict different times $t_l^i$ (with $i=1$ for the first peak, $i=2$ for the second one, $l=0,1,2,3$) which are sorted in an ascending order $t_0^i<t_1^i<t_2^i<t_3^i$. (e) and (f) Direction of the spatially integrated current (frequency filtered) over the whole edge of the flake depicted as arrow for the first $i=1$ (e) and second $i=2$ (f) peak and for different times $t_l^i$.

Figure 7

Data points for the inverse peak positions $1/{\omega }_i$ for the first ($i=1$) and second ($i=2$) peak as a function of the flake size $N$. The red solid line ($i=1$) corresponds to the theoretical angular frequency ${\omega }_1^{\mathrm{th}}$ determined from the group velocity of the edge states. The solid blue line ($i=2$) corresponds to five times that group velocity.