Hot electron effect in high-order harmonic generation from graphene driven by elliptically polarized light

Due to the development of intense ultrashort pulse lasers, it has become possible to apply high electric fields to materials without damaging them, and this also led to significant progress in research on high-order harmonic generation (HHG) in solids.^1–7 HHG is important for applications such as attosecond pulse generation and light sources in the extreme ultraviolet region.⁸ The spectrum of the emitted high-order harmonics (HHs) is influenced by the band structure of the investigated material,^9–13 and it has been shown that the sources of HH emission are nonlinear currents produced by interband and intraband transitions.^14–20 HHG can be controlled via these two types of transitions by using nanocrystals (to manipulate the intraband transitions) and also by modifying the polarization of the excitation pulse or by using two-color excitation.^{17,19,21–23}

A characteristic feature of the HH emission from solids is a nonmonotonic behavior with respect to the ellipticity of the excitation light.^17,24–27 This behavior differs from the behavior of HHG in gases, where the HH intensity decreases monotonically with increasing ellipticity.^28,29 Graphene is one of the materials in which an enhancement of the emission intensity can be observed;²⁵ under elliptically polarized strong laser excitation, a new polarization component (orthogonal to the HH emission under linearly polarized light) with a higher intensity can be generated. It has been proposed that the gapless band structure of graphene causes this enhancement.²⁵ However, there is also another theoretical interpretation of this enhancement in graphene, in which the role of coupling between interband and intraband transitions was pointed out.¹⁸ In this latter study, the k-space distribution of interband- and intraband-transition strengths was calculated, and it was concluded that the HH intensities are enhanced in the case of elliptically polarized light because of the following characteristic of the k-space distributions: the distribution of the interband excitation induced by one polarization component of the driving field and the distribution of the intraband transitions induced by the orthogonal polarization component of the driving field are similar. More recently, an experiment on a field-effect transistor in the weak excitation regime has been performed.²⁰ One peculiar feature of this ellipticity dependence is that the y-component of the seventh-order harmonic intensity obtained using elliptically polarized MIR light exceeds that obtained using linearly polarized MIR light. However, so far, the details of the enhancement of the seventh-order harmonic achieved by using elliptically polarized MIR light have not been investigated. In addition, in contrast to static gate voltages, pulsed excitation with visible photons or also terahertz (THz) photons allows us to control the thermal distribution of carriers at higher speeds and allows us to obtain complementary information about the HHG enhancement mechanism.^30–38 Therefore, studies that show how THz-induced changes in the thermal distribution affect HHG from graphene can be also important for applications.

In this study, we investigated HHG in graphene induced by either elliptically or linearly polarized MIR light with and without additional THz irradiation. The used THz pulse acts as an external electric field that breaks the inversion symmetry and thus leads to the generation of even-order harmonics. Simultaneously, the seventh-order harmonic intensities decrease due to the suppression of electronic transitions, which is attributed to a THz-induced change in the thermal distribution of electrons. We provide a simple model that explains how the different quantum pathways for HHG are affected by the THz irradiation.

Figure 1(a) is a schematic of the experimental setup used to measure HH spectra of graphene in the case of excitation with linearly polarized MIR pulses. The THz pulses and the MIR pulses are focused at the same spot on the sample, and the THz beam can be blocked using a shutter. We used two types of monolayer graphene sheets transferred to quartz substrates: one purchased from ACS Material and another grown on Cu(111)/sapphire by chemical vapor deposition (CVD).³⁹ Both showed similar results. The used graphene sheets were about 10 × 10 mm² in size and were excited near the center, thus removing the need to consider sample-edge effects in our analysis.⁴⁰ The MIR pulses (pulse duration: 80 fs, photon energy ℏω₀: 0.28 eV, 4.5 µm) were generated by using a multi-stage optical parametric amplifier (OPerA, Coherent) and 800-nm pulses from a Ti:sapphire laser (repetition rate: 1 kHz). The output of the optical parametric amplifier was focused on the sample surface using a lens, and the excitation power density was adjusted to a sufficiently low value to avoid sample damage (the maximum excitation power density of the MIR pulse was 21 GW/cm², corresponding to 4.0 MV/cm in terms of the electric field). We also investigated the HHG process driven by elliptically polarized MIR light. To control the polarization direction and ellipticity of the MIR pulse, we placed a λ/2 waveplate and a λ/4 waveplate in the MIR beam path. The angle between the x-axis and the polarization direction of the MIR pulse is denoted by θ. The HH spectra were recorded using a spectrometer (SpectraPro, Teledyne Princeton Instruments) and a charge-coupled-device (CCD) camera (PIXIS, Teledyne Princeton Instruments).

The square of the time-domain waveform of the electric field of the THz pulse, E_THz, is shown by the gray shaded region in Fig. 1(c). The maximum electric-field amplitude of the THz pulse in air (E_THz,max) was 0.4 MV/cm, which was estimated by electro-optic (EO) sampling at the sample position.⁴¹ In all experiments, the THz pulse was polarized along the direction of the x-axis. The delay time t between the THz pulse train and the MIR pulse train was controlled using a delay stage. The time origin is defined as the time when the electric field of the THz pulse reaches its maximum at the sample position, and t is the time when the peak of the MIR pulse envelope reaches the sample (when the MIR pulse reaches the sample after the THz-pulse peak, the delay time t is positive). All experiments were performed at room temperature.

First, we present the HH spectra obtained using linearly polarized MIR light at θ = 0° with and without the modulation by the THz pulses. Figure 1(b) shows the HH spectra for t = 0. The data in the energy region where the CCD camera has a low quantum yield (<1.2 eV) are not shown. The error bar for the photon energy is about 30 meV. In the case of no modulation (the red data), only odd-order harmonics are observed, which reflects the inversion symmetry of the graphene lattice structure. When the THz field is added (the blue data), the odd-order harmonic intensities are reduced by about 50% and the sixth-order harmonic appears. Figure 1(c) presents the changes that occur in the harmonic intensities when the delay time t is varied. The solid curves are the integrated peak intensities, and the gray shaded area shows the THz field intensity for comparison. The intensity reduction of the odd-order harmonics roughly correlates with the THz electric field amplitude, but the effect is not symmetric in time. Focusing on the seventh-order harmonic, we can confirm that the intensity reaches a minimum at t = 200 fs, and then, it gradually recovers over a relatively long period of about 2 ps. The low intensities in this time range are clearly not due to the instantaneous electric field of the THz pulse. Instead, they are due to an effect of a certain population that grows during THz excitation. Since the recovery time is on the same time scale as the cooling of a thermal electron distribution in graphene,³⁴ these data suggest that the THz-induced intensity reduction is related to the onset of the thermalized electron distribution (a brief discussion of the change of the onset with the electron temperature is provided in the supplementary material). On the other hand, the delay time dependence of the sixth-order harmonic intensity follows the instantaneous THz field intensity.

In Fig. 1(d), we show three curves for t = 0. In the case of the odd orders, we plotted the ratio of the harmonic intensity obtained with modulation to that obtained without modulation as a function of θ. The θ dependence of the sixth-order harmonic is normalized to its maximum. The sixth-order harmonic is strongest when E_MIR is parallel to E_THz. This result indicates that the even-order generation is enhanced by the component of the THz field that has the same direction as the driving field, which suggests the following θ dependence of the nonlinear polarization in the perturbative regime: $P6ω∝χ7EMIR6ETHz⁡cos⁡θ$ [χ⁽⁷⁾ is the seventh-order susceptibility].^42,43 This predicted dependence roughly reproduces the experimentally observed behavior. On the other hand, Fig. 1(d) evidences that the THz-induced intensity reductions of the odd-order harmonics are independent of θ. This fact supports the above finding that the THz-induced odd-order intensity reduction is not an electric field effect, but an effect of the onset of hot electrons.

The MIR intensity dependence of HHG in graphene in Fig. 1(e) shows a non-perturbative behavior (we were able to measure a clear dependence for the fifth- and sixth-order harmonics). Based on Ref. 42, the above argument regarding the θ dependence of the even harmonics also holds for the non-perturbative case. Figure 1(f) clarifies that the crystal angle (φ) dependence of the sixth-order harmonic is rather isotropic (it does not show features related to the sixfold rotational symmetry of graphene). We consider that this is a result of the isotropic nature of the Dirac cone of graphene.

The data in Fig. 2 are used to explain the effect of the THz pulse on the HH spectra of graphene in the case of elliptically polarized MIR light. It is known that, in graphene (and also several other materials), the intensities of the HHs can be enhanced by using elliptically polarized MIR light instead of linearly polarized MIR light.²⁵ To further clarify the mechanism of this phenomenon, we changed the ellipticity of the MIR pulse by rotating the λ/4 waveplate in the MIR beam path and measured the intensity of seventh-order harmonic as a function of ellipticity. Figure 2(a) shows the experimental setup. We define the amplitude of the x-component of the driving field as E_x and that of the y-component as E_y. The ellipticity is defined as ε = E_y/E_x. For each ε value, the λ/2 waveplate in front of the λ/4 waveplate was adjusted in such a way that the long axis of the polarization ellipse is parallel to the x-axis. The excitation power density of the MIR pulse was 21 GW/cm² for all ε values. We also installed a wire-grid (WG) polarizer in front of the spectrometer to measure the ε dependence for both polarization components of the emitted HHs (the corresponding integrated peak intensities are denoted by I_x and I_y). The data shown in Fig. 2(b) are the intensities normalized to the value of I_x for ε = 0. The results obtained without the modulation by the THz pulses are shown in the left-hand side of Fig. 2(b). Here, I_x (the red data) shows an overall monotonically decreasing behavior with increasing MIR ellipticity. On the other hand, I_y (the blue data) exhibits a peak near ε = 0.3 and I_y is even stronger than I_x for ε = 0. The results obtained with additional modulation by the THz pulses [Fig. 2(b), right-hand side] reveal a significantly smaller (normalized) peak of I_y near ε = 0.3. Since the absolute intensities of the odd-order harmonics are reduced by THz excitation as shown in Fig. 1(b), this result for the normalized I_y values indicates that the absolute THz-induced intensity reduction in I_y for ε ≈ 0.3 is larger than the absolute THz-induced intensity reduction in I_x for ε = 0.

Note that the behavior observed without modulation is not restricted to the seventh-order; the sixth-order harmonic behaves similar, as shown in the supplementary material. The ellipticity dependence of the fifth-order harmonic is also shown in the supplementary material.

To clarify the mechanism of the THz-induced odd-order intensity reduction, we measured the THz power density dependence of the seventh-order harmonic intensity for ε = 0.38. We chose ε = 0.38, because the x-component of the seventh-order harmonic is almost zero for this value [Fig. 2(b)]. The MIR excitation power density was fixed at 21 GW/cm², and the maximum electric-field amplitude of the THz pulse was varied from 0 to 0.4 MV/cm. Figure 2(c) shows that the seventh-order harmonic intensity remains constant up to about E_THz,max = 0.027 MV/cm, and then, it monotonically decreases as the THz field amplitude is increased. Note that such a pulse with E_THz,max = 0.027 MV/cm is considered to result in an electron temperature of about 900 K.⁴⁴ 

It is known that the coupling between interband and intraband transitions is important for HHG in graphene.^18,20,45 The HHs generated through a combination of both types of transitions are considered to be a sum of harmonics generated via various quantum pathways, as shown in Fig. 3(a). A theoretical study has shown that similar transition-strength distributions in k-space are obtained for interband transitions and intraband transitions if the two types of transitions are induced by mutually orthogonal components of the electric field.¹⁸ For example, the intraband transitions induced by the x-component of the electric field and the interband transitions induced by the y-component of the electric field are strong in the region of the Brillouin zone defined by k_y ≈ 0 and k_x values around zero (excluding k_x = 0). Hence, under such an excitation condition, strong intraband and interband transitions can be induced in the same k-space region. The good overlap of these particular distributions of different types of transitions results in more efficient coupling and hence also in enhanced HHG. This interpretation suggests that the contribution of each quantum pathway in HHG depends on the ellipticity ε. In the following, we discuss how hot electrons in graphene can suppress these transitions and thereby reduce the odd-order harmonic intensities.

First, we confirm that the odd-order intensity reduction can be explained by a change in the thermal distribution of electrons. For this, we investigate how the ellipticity dependence of the seventh-order harmonic intensity is related to the electron temperature T_e. The theoretical model is briefly introduced in the supplementary material, and the details can be found in the literature.^18,46–51 Furthermore, conductivity measurements on graphene have shown that T_e increases up to 7000 K using THz pulses with E_THz,max = 0.12 MV/cm.⁴⁴ Therefore, we calculated the ellipticity dependence for T_e = 300 and 7000 K. Figure 3(b) shows the results: while I_x decreases monotonically with increasing MIR ellipticity, I_y has a peak near ε = 0.3. The peak of I_y near ε = 0.3 is even stronger than I_x at ε = 0. The comparison of the I_y-results for T_e = 300 and 7000 K shows that the peak of I_y is significantly smaller in the latter case. This is the same trend as that of the experimental results, indicating that the origin of the change in the ellipticity dependence is related to the thermal distribution of electrons. Based on the same calculation procedure, we also calculated the electron temperature dependence of the seventh-order harmonic intensity for ε = 0.38. The result is shown in Fig. 3(c), which qualitatively reproduces the experimentally observed THz power density dependence in Fig. 2(c), including the electron temperature at which the intensity begins to decrease (≈900 K). These results suggest that the THz-induced odd-order intensity reduction is due to a change in the thermal distribution of electrons.

The effect of incoherent carriers (e.g., thermally distributed electrons) on HHG can also be inferred from the following results: the suppression of interband transitions by Pauli blocking was discussed in a study on HHG in ZnO,⁵² and two studies on HHG in MoS₂ and WSe₂ discussed the intraband scattering of electrons (electron–electron scattering).^53,54 Note that our calculation procedure does not take into account the change in the scattering rate that occurs in the case of a temperature increase. On the other hand, our calculation reproduces the experimentally observed trend. Therefore, the THz-induced reduction of the odd-order harmonic intensities can be attributed to Pauli blocking of interband transitions. This result is consistent with a previous report on HHG in graphene, where the interband transitions were suppressed by controlling the Fermi energy.

Finally, we discuss the origin of the larger intensity reduction of odd-order harmonics in the case of elliptically polarized light. The Fermi energy μ of CVD-grown graphene is 0.07 eV above the Dirac point,⁴⁴ and the energy of the interband transition from the valence band to the Fermi level (2μ = 0.14 eV) is smaller than the MIR photon energy (0.28 eV). If we consider a fixed electron temperature T_e, then Eq. (1) implies that the occupation probability of a conduction-band state closer to the Dirac point is larger, and thus, the blocking of interband transitions to such states is more pronounced. Among the various quantum pathways shown in Fig. 3(a), the HHG via interband transitions to these low-energy states is more suppressed by the thermal distribution than the HHG via interband transitions to the high-energy states. Furthermore, under elliptically polarized MIR excitation, the THz-induced intensity reduction is larger than that under linearly polarized MIR excitation. This implies that the low-energy states play a rather important role in the HHG process under elliptically polarized MIR excitation. This means that the quantum pathways including interband transitions to low-energy states are the origin of the enhancement of HHG that can be achieved in graphene by using elliptically polarized MIR light.

In the following, we explain why processes involving low-order interband transitions make a relatively large contribution to the total HHG in the case of elliptically polarized light: the HHG process in graphene can be interpreted in terms of adding quantum pathways as illustrated in Fig. 3(a). The intraband transitions can be categorized into those originating from the x-component of the driving field, E_x, and those originating from E_y. If we consider several quantum pathways for the same harmonic order n, the probability of each quantum pathway is different: for a quantum pathway with a low interband-transition order, the number of possible combinations of E_x- and E_y-derived intraband transitions is larger than that of a quantum pathway with a higher interband-transition order. We consider that these combinations lead to the result that the contribution from quantum pathways with low-order interband transitions in the case of elliptically polarized excitation is larger than that in the case of linearly polarized excitation, where only E_x-derived intraband transitions are present.

We have observed HHG in graphene using rather complex excitation conditions: HHG was driven by either linearly or elliptically polarized MIR light, and the electron distribution in graphene was modified by THz pulses. We found that the THz pulses affect the odd and even orders differently: they suppress electronic transitions for odd orders by modulating the thermal electron distribution, and for even order, they act as an external electric field that breaks the inversion symmetry. In this work, a similar enhancement in the case of elliptically polarized excitation was also observed for the sixth-order harmonic. In the case of elliptically polarized MIR light, the quantum pathways including interband transitions to energy states that are closer to the Dirac point are more important for HHG than in the case of linearly polarized MIR light, and these pathways are the origin of the HHG enhancement achieved by using elliptically polarized light instead of linearly polarized light. These quantum pathways are strongly suppressed by increasing the temperature of the thermal electron distribution, which leads to an efficient THz-induced suppression of the HHG enhancement. This study shows how interband- and intraband-transitions in graphene can be controlled on ultrafast time scales.

The supplementary material contains supplementary figures used in the discussion in this article and the details of the microscopic theory of HHG in solids.

Part of this study was supported by JSPS KAKENHI (Grant Nos. JP19H05465, JP21H01842, JP21H05232, and JP21H05233). This work was supported by the International Collaborative Research Program of Institute for Chemical Research, Kyoto University (Grant No. 2024-19).

The authors have no conflicts to disclose.

Kotaro Nakagawa: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Wenwen Mao: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Writing – review & editing (equal). Shunsuke A. Sato: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Hiroki Ago: Funding acquisition (equal); Investigation (equal); Resources (equal); Writing – review & editing (equal). Angel Rubio: Formal analysis (equal); Investigation (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Yoshihiko Kanemitsu: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Hideki Hirori: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.