1. Introduction

High-harmonic generation (HHG) is nowadays a well established technique to generate highly coherent extreme-ultraviolet/x-ray radiation [1–7]. The phenomenon stems from the non-perturbative interaction of intense laser light with matter. For atomic and molecular targets, HHG can be understood in semiclassical terms, according to the so-called three-step model [8]. In this point of view, harmonics are radiated when a tunnel-ionized electron returns to the parent ion, where it recombines releasing its kinetic energy in the form of high-frequency radiation. This later step is only efficient if the electron’s trajectory returns to the parent ion, which is a condition naturally found in linearly-polarized driving fields. For elliptically-polarized drivers, the electrons are pulled away from the parent ion, and the efficiency of the HHG process decreases accordingly [9]. It is worth noticing also that the electron may not recombine at the first encounter with the parent ion, therefore the harmonics can be emitted at subsequent higher-order recollisions [10–12]. In such case, however, the radiated photon energies fall well below the maximum photon energy of the first recollisions (the so-called harmonic cutoff frequency). Moreover, since high-order recollisions imply electron’s excursion times longer than the driver’s cycle, they are associated to lower harmonic conversion efficiencies, resulting from the progressive spread of the electron’s wavepacket in the continuum [13–15].

Solid targets provide richer scenarios for HHG [6,16–21]. In this case, the irradiation geometry defines the HHG mechanism. For grazing incidences, i.e. when the polarization plane of the driving electric field is perpendicular to the target’s surface, HHG follows from the recombination of electrons detached from the target [22], in close resemblance to the ionization mechanism in atomic and molecular cases. For solid targets, however, the crystal periodicity leaves a signal in the detached electron’s wavefunction, in form of Talbot modulations, which in turn are reflected in the HHG spectrum. This phenomenon gives ground to the recent proposal of ultrafast Talbot spectroscopy at the nanometer scale [23]. On the other hand, for the case of normal incidence or bulk solid targets, the HHG mechanism follows from an adapted semiclassical three-step model [24]. According to this view, the electron’s tunnel ionization step is replaced by tunnel excitation to the conduction band, and the consequent creation of a hole. Harmonics result from the electron-hole recombination at times when the electron and hole trajectories overlap, i.e. the so called perfect recollisions (ehPR). As a main characteristic, the harmonic cutoff frequency corresponds to the maximum energy gap during the electron/hole excursion [24–28].

Despite of these fundamental similarities, HHG in solids turns out to be far more complex than in atoms and molecules. On the one side, the crystal band’s dispersion is not quadratic and thus, the dynamics of the electron in the conduction band is not as trivial as that of an electron ionized in free space. In addition, Berry curvatures introduce a deflection in the electron and hole trajectories. Therefore, while in gas systems the electron trajectories are universally defined, in solids they depend on the target species. On the other side, the details of the excitation of the electron to the conduction band are also affected by the band structure. For instance, it has been demonstrated that HHG from elliptically-polarized drivers results from electrons tunnel-excited at points surrounding the minimum band gap, rather than at the exact point [29]. In 2D semimetals enclosing Dirac points, the excitation details are more involved, as tunneling is replaced by the non-adiabatic crossing near these singular points [30]. This simple variation in the first step leads to substantial differences in the HHG process, as electron-hole excitation is no longer linked to the maxima of the driving field’s amplitude. In other low dimensional targets, as carbon nanotubes, HHG follows from a similar non-adiabatic excitation mechanism near the Van Hove singularities [31].

In the general case, the complex dynamics in crystals does not only affect the maximum photon’s energy but also the polarization and phase properties of the emitted harmonics. In particular, the band geometry around the symmetry points can lead to a highly non-linear anisotropic response, resulting in the generation of harmonics with elliptical polarization even when driven by linearly-polarized drivers [22,32–41]. In addition, the convoluted electronic dynamics during HHG in solids leads to high harmonics with complex phase properties. As a result, accounting for the spatial phase distribution in HHG from a macroscopic target [26,42] has allowed to achieve a better understanding of the harmonic spectrum properties.

It has been recently shown that although the modified semiclassical three-step model accounts for the main features of the HHG mechanism in solids, non-classical pathways can be also relevant in systems with large Berry curvatures [43] or when HHG is driven by elliptically polarized fields [44]. In these cases, harmonics are emitted during electron-hole imperfect recollisions (ehIR), i.e. when the electron-hole wave packets overlap even though their centers are spatially separated by distances one order of magnitude larger than the lattice constant [43].

In this article, we demonstrate that ehIR’s have a relevant role also for HHG in solids with vanishing Berry curvature and driven by linearly-polarized laser fields. In particular, we show the enhanced role of high-order ehIR at times larger than the laser cycle. The excellent agreement between our semiclassical and quantum calculations demonstrates that ehIR must be considered to obtain a better understanding of the complex phase properties of HHG in solids.

The article is organized as follows. In Section 2, we present the details of the computation of the nonlinear response, where we use two models: the resolution of the time dependent Schrödinger equation within the first neighbor tight-binding approximation, and a semiclassical saddle-point approach which allow us to interpret our results. In Section 3, we present and discuss our findings, where we identify the role of ehPR and ehIR for several cases of HHG in single-layer graphene driven by a linearly-polarized laser field. Finally we present our conclusions in Section 4.

2. Theoretical description of HHG in single-layer graphene

We consider the emission of high-order harmonics in single-layer graphene irradiated by a short mid-infrared pulse, linearly polarized along the graphene sheet plane. To this end, we compute the nonlinear response of graphene following the method described in [30], and we interpret our results in terms of classical electron-hole trajectories in real space. The dynamics of the interaction of a single layer graphene sheet with an ultrashort laser pulse will be described in the first neighbor tight-binding approximation [45]. We express the electron wavefuntion in terms of the graphene’s eigenstates, $\Phi _{\mathbf k}^\pm ({\mathbf r})$,

The evolution of $\Psi ( {\mathbf r}, t)$ is solved in the reciprocal space’s moving frame $\hbar \boldsymbol {\kappa }_t= \hbar {\mathbf k}-q_e {\mathbf A}(t)/c$, where ${\mathbf A}(t)$ is the vector potential of the electromagnetic field, $c$ the speed of light, and $q_e$ the electron’s charge. The numerical instabilities associated with the singularities at the Dirac points can be circumvented by using the change of variables,

Finally, by solving Eqs. (6) and (7), we calculate the harmonic emission from the dipole acceleration, ${\mathbf a}(t)=\frac {d^2}{dt^2}{\mathbf d}(t)$ as [30],

3. Results and discussion

We consider a six optical cycles (o.c.) driving field with a $\sin ^2$ envelope, which corresponds to 2.2 o.c., 22 fs, of full width at half maximum (FWHM) in intensity (see inset in Fig. 1(a)). The field is assumed linearly polarized along the $\Gamma$-K direction. The laser pulse is centered at 3 $\mu$m wavelength, with a peak intensity of 5 $\times$ 10$^{11}$ W/$cm^2$, and it is aimed perpendicularly to the graphene’s layer. Note that the pulse duration is shorter that the typical electron thermalization time [46]. The resulting harmonic spectrum and its time-frequency analysis are presented in Figs. 1(a) and 1(b), respectively. The time-frequency representation has been obtained scanning over the HHG spectral amplitudes with a Gaussian mask of 4$\omega _0$ in FWHM, and subsequently performing the Fourier transform of each sample. As a reference, Figs. 1(c) and 1(d) show the same information for the case of HHG in a hydrogen atom. HHG in hydrogen is calculated from the solution of the three-dimensional time dependent Schrödinger equation, using a 0.72 $\mu$m driving field of peak intensity of 1.6 $\times$ $10^{14}$ W/$cm^2$, parameters that have been adjusted to produce a HHG spectrum with similar cutoff frequency as the graphene case.

 

Fig. 1. Harmonic spectrum (a) and time-frequency analysis (b) of the high order harmonic emission from single-layer graphene. The inset in (a) plots the driving electric field, a 3 $\mu$m, six-cycles sin$^2$ pulse (22 fs FWHM) linearly polarized in the graphene plane along the $\Gamma$-K direction, with peak intensity of 5$\times$10$^{11}$ W/cm$^2$. Panels (c) and (d) plot the same information from a hydrogen atom target, irradiated with a linearly-polarized 0.72 $\mu$m, six-cycles sin$^2$ pulse, with peak intensity of 1.6$\times$10$^{14}$ W/cm$^2$. Inset in (b) is the normalized color map of the electron excitation in the conduction band. The time at which this snapshot is taken is depicted in the inset of panel (a), at 2.75 cycles after the beginning of the laser pulse.

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The comparison of both spectra reveals that the emission of harmonics in graphene presents a temporal structure far more complex than that in atoms. In the atomic case (Fig. 1(d)) the high-order harmonics result from two well-resolved emission bursts per half-cycle, the so-called short and long trajectories, which present a positive and negative slope structure in the time-frequency representation, respectively [47–51]. In contrast, in HHG from graphene, the first step —the creation of the electron-hole pair— is connected with the Dirac points: electron-hole pair excitation follows from the non-adiabatic crossing near the singular points (see the inset of Fig. 1(b)) and, therefore, pairs are created continuously instead of at the driving field’s maxima, as it is the general case for bulk crystals, atoms and molecules. As a result, the number of interfering trajectories playing a relevant role in HHG in graphene is substantially larger, and not necessarily regularly spaced in time [30], as it can be observed in Fig. 1(b). Note that, once the driving field interaction with the graphene layer ends, the time-frequency picture reveals that there is still emission from the excited sample, a behavior that has been reported in previous works [36]. In the following we shall see that, even though the semiclassical trajectories leading to ehPR successfully account for the main spectral features in Figs. 1(a) and 1 (b), the harmonic signal can only be fully explained considering also the role of ehIR.

We present in Fig. 2 the results of the SPAM calculations for the graphene case presented in Fig. 1. The background in Figs. 2(a) and 2(b) are replicas of Fig. 1(b). We analyse the temporal distribution of electron-hole recombination energies from ehPR (Fig. 2(a)) and ehIR (Fig. 2(b)). The plotted ehIR energies correspond to the cases in which the electron and hole are 10 unit cells (24 Å, pink lines) and 70 unit cells (172 Å, grey lines) apart, at the instant of recombination. Lighter tones are used for trajectories initiated near the K Dirac points, and darker tones for those initiated near the K$'$ points (see the inset in Fig. 2(c)). Note that, while the temporal distribution of radiated frequencies from ehPR explains reasonably the fundamental features of the HHG spectrum in (Fig. 2(a)), it becomes also clear that ehPR do not provide a full explanation of the harmonic contributions at intermediate times and, specially, at the end of the driving pulse.

 

Fig. 2. Results of the harmonic emission corresponding to trajectories leading to ehPR —red, panel (a)— and ehIR —pink and grey, panel (b)—. The electron-hole pairs are created at points K (light colors) and K$'$ (dark colors) of the graphene’s Brillouin zone. Panels (a) and (b) show the energy of the electron-hole pair at recombination obtained with the semiclassical SPAM calculations against the background corresponding to the time-frequency analysis shown in Fig. 1 (b). In (a) we plot the frequency of the harmonic emission corresponding to ehPR, i.e. when the electron and hole classical trajectories intersect, and in (b) the same results for ehIR, where the electron-hole distances at the recombination time are 24 Å$\,$ (pink) and 172 Å$\,$ (grey). We have added a long recombination distance of 492 Å$\,$ (black) in order to recover the harmonic emission at the rear part of the pulse. Panel (c) shows the distribution of energies radiated from ehIR as a function of the pair recombination (horizontal axis) and excitation (vertical axis) times. The inset shows the geometry of graphene’s Brillouin zone. Labels $\alpha _i$ and $\beta _i$ correspond to two pairs of sampled ehIR trajectories recombining at 24 Å$\,$ and 172 Å, respectively, with excitation times of $\sim$1.9 o.c. ($i$=1), and $\sim$2.9 o.c. ($i$=2).

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To simplify the discussion of the results from ehIR in Fig. 2(b), we shall focus our attention to the highest harmonic orders, $q>15$, and to the trajectories originated at the Dirac point K (lighter lines). The discussion of the emission for pairs originated near points K$'$ (darker lines) is analogous, due to the crystal’s inversion symmetry.

Figure 2(b) points out that the emissions from ehIR’s for increasingly longer recombination distances are progressively delayed. This is consistent with the increasingly larger excursion times needed for the electron-hole pair to reach longer interparticle distances. This aspect is reproduced in Fig. 2(c), where we show the energy of the electron-hole pair at any potential recombination time, as a function of the pair recombination (horizontal axis) and excitation (vertical axis) times. Labels $\alpha _i$ and $\beta _i$ correspond to two pairs of sampled ehIR trajectories recombining at distances of 24 Å$\,$ and 172 Å, respectively, with different excitation times: $\sim$1.9 o.c. for $i$=1, and $\sim$2.9 o.c. for $i$=2. To complete our analysis, we have also included in Fig. 2(b) the trajectory contributions of ehIR that recombine at longer distances (200 unit cells, 492 Å, black lines), which present a relevant contribution at the rear part of the harmonic emission. Thus, our analysis unequivocally demonstrates the relevance of ehIR in HHG in graphene. The harmonic contributions that are not reproduced by ehPR in Fig. 2(a) are mostly fulfilled by the ehIR sampled in Fig. 2(b), whereas the nature of ehIR is clearly identified in Fig. 2(c). We note that a larger sample of ehIR born at different excitation times, and with other recombination distances, completely fulfills the harmonic emmission not reproduced by ehPR in Fig. 2(a).

It is important to point out that the excursion time, i.e. the time lapse between excitation and emission times, increases with the recombination distance. In Ref. [29] ehIR are understood from non-classical pathways, since while the mean position of the electron and hole wavefunctions are distant, dispersion leads to the overlap of the wavefunctions. Threfore, ehIR at larger distances require a larger wavepacket dispersion and, thus, a larger excursion time. Results in Fig. 2(c) show excursion times of the order of 1.5 cycles for ehIR recombining at 172 Å$\,$($\beta _i$).

Note that the wavepacket dispersion rate is nearly six times larger than the one reported for hBN in [29]. This enhanced dispersion suggests that the excited electron-hole wavepackets are more localized in our case, therefore the dispersion rate is larger. We interpret this as consequence of the different excitation mechanisms for each species. In graphene, the non-adiabatic electron-hole excitation results in a broad wavepacket in the reciprocal space, around the Dirac points. In contrast, tunnel excitation, as is the case of hBN, is very sensitive to the band gap, therefore resulting in narrower wavepackets in the reciprocal space around the gap minima.

Finally, in Fig. 3 we show a comparative analysis of the role of ehPR against ehIR for different driving peak intensities. According to the current understanding of HHG in solids, the frequency of the harmonic radiation is given by the energy gap at the time of recombination of the electron-hole pair. Since the trajectories quiver in the reciprocal space with an amplitude $(q_e {\mathbf A} /c)$, the harmonic cutoff frequency is expected at the maximum gap energy during this excursion. For the two cases represented in Fig. 3, the maximum gap is reached at the turning point $(2q_e {\mathbf A_0} /c)$, ${\mathbf A_0}$ being the amplitude of the vector potential. This maximum quivering for each case is represented in the energy band representation in the insets of Fig. 3. According to this, the prediction for the harmonic cutoff frequency is the 15th order ($\simeq$ 6 eV) for a peak intensity of 1.3 $\times$ 10$^{11}$ (Fig. 3(a)), and the 43rd order ($\simeq$ 18 eV) for a peak intensity of 2 $\times$ 10$^{12}$ (Fig. 3(b)), which agrees with the corresponding time-frequency analyses. Figure 3 also shows the prediction of the harmonic emission from the semiclassical SPAM for ehPR (red) and ehIR at distances of 86 Å$\,$(green) and 172 Å$\,$(grey). As a main observation, we can see that in all cases, ehIR are necessary to reproduce the harmonic emission, specially at longer times. Note that the quiver trajectory of electrons and holes in the reciprocal space is the same for ehPR and ehIR. However, by relaxing the constraint of zero distance at recombination, ehIR harvest the full potential harmonic emission during the quivering, making it possible the emission of the maximum cutoff frequency in situations excluded by the ehPR’s overlapping constrain.

 

Fig. 3. Time-frequency analysis of the harmonic emission corresponding to two different peak intensities, $1.3 \times 10^{11}$ W/$cm^2$ (a) and $2.0 \times 10^{12}$ W/$cm^2$ (b), with the same length duration and central wavelength as in the previous figures. For the scan of the spectrum we have used a Gaussian mask of 2.5 $\omega _0$ in (a), and 4 $\omega _0$ in (b). In both panels we plot the frequency of the harmonic emission corresponding to ehPR (red) and to ehIR recombining at 86 Å$\,$ (green) for (a) and at 172 Å$\,$ (grey) for (b).

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4. Conclusion

We have presented quantum calculations and classical analysis of HHG in single-layer graphene. A comparative analysis with the results of HHG from atoms reveals the extraordinary complex nature of the process in the solid layer. The main trends of the characteristics of this complex emission can be associated with the particular mechanism of HHG in graphene, in which the electron-hole pair excitation is not associated with the maxima of the driving field amplitude. However, our study demonstrates that still relevant spectral features are connected with non-classical recollisions —the so-called imperfect recollisions— where the electron-hole pair can recombine even when the classical trajectories do no overlap. We have found that these imperfect recollisions typically contribute to the harmonic emission at increasing delayed times for longer recombination distances. These time delays, which are connected with the spreading of the electron and hole wavefunctions to the recombination distance, are significantly larger for graphene. We interpret this result as the consequence of the linear dispersion relation near the Dirac points.