Doping and gap size dependence of high-harmonic generation in graphene: Importance of consistent formulation of light-matter coupling

High-harmonic generation (HHG) in solids is a fundamental nonlinear phenomenon, which can be efficiently controlled by modifying system parameters such as doping level and temperature. To correctly predict the dependence of HHG on these parameters, consistent theoretical formulation of the light-matter coupling is crucial. Recently, contributions to the current that are often missing in the HHG analysis based on the semiconductor Bloch equations have been pointed out [Wilhelm et al., Phys. Rev. B 103, 125419 (2021)]. In this paper, by systematically analyzing the doping and gap-size dependence of HHG in gapped graphene, we discuss the practical impact of such terms. In particular, we focus on the role of the current $J_{\mathrm{ra}}^{\left(2\right)}$, which originates from the change of the intraband dipole via interband transition. When the gap is small and the system is close to half filling, intraband and interband currents mostly cancel, thus suppressing the HHG signal—an important property that is broken when neglecting $J_{\mathrm{ra}}^{\left(2\right)}$. Furthermore, without $J_{\mathrm{ra}}^{\left(2\right)}$, the doping and gap-size dependence of HHG becomes qualitatively different from the full evaluation. Our results demonstrate the importance of the consistent expression of the current to study the parameter dependence of HHG for the small gap systems.

Figure 1

Summary of the relation between different representations discussed in this paper.

Figure 2

Tight-binding model on the two-dimensional honeycomb lattice. Blue circles indicate the $A$ sublattice, while red circles indicate the $B$ sublattice.

Figure 3

(a), (b) HHG spectra $I_{\mathrm{HHG}}={\left|\omega J_X\left(\omega \right)\right|}^2$ of gapped graphene for indicated values of chemical potential and field strength. (c), (d) The intensity of the peaks of the HHG spectra $I_{\mathrm{peaks}}\left(n\right)$ as a function of chemical potential. (e), (f) The intensity of the fifth harmonic peak of $I_{\mathrm{HHG},\mathit{k}}\left(\omega \right)={\left|\omega J_{\mathit{k},X}\left(\omega \right)\right|}^2$ for indicated values of field strength around the Dirac point [$\mathit{K}=(K_x,K_y)$]. The dashed circles indicate the Fermi surface for $\mu =-0.4$, while the dot-dashed lines indicate the edge of the Brillouin zone of the graphene. In all cases, we set $t_{\mathrm{hop}}=3, m=0.001, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$. These results are obtained from the analysis of the Dirac models.

Figure 4

(a)–(d) HHG spectra of gapped graphene for indicated values of chemical potential and field strength. We compare the contributions from different types of currents; $I_{\mathrm{HHG}}={\left|\omega J_X\left(\omega \right)\right|}^2, I_{\mathrm{er}}={\left|\omega J_{\mathrm{er},X}\left(\omega \right)\right|}^2, I_{\mathrm{ra}}={\left|\omega J_{\mathrm{ra},X}\left(\omega \right)\right|}^2, I_{\mathrm{ra}}^{\left(1\right)}={\left|\omega J_{\mathrm{ra},X}^{\left(1\right)}\left(\omega \right)\right|}^2$. (e), (f) The intensity of the peaks of the HHG spectra $I_{\mathrm{peaks}}^{\prime }\left(n\right)$, which is evaluated from $I_{\mathrm{HHG}}^{\prime }\left(\omega \right)={\left|\omega (J_{\mathrm{er},X}\left(\omega \right)+J_{\mathrm{ra},X}^{\left(1\right)}\left(\omega \right))\right|}^2$ (i.e., without $J_{\mathrm{ra}}^{\left(2\right)}$), are shown with open markers, as a function of chemical potential. The filled markers indicate the results from $I_{\mathrm{HHG}}\left(\omega \right)$ shown in Figs. 3 and 3. In all cases, we use $t_{\mathrm{hop}}=3, m=0.001, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$. These results are obtained from the analysis of the Dirac models.

Figure 5

(a)–(f) HHG spectra of gapped graphene for indicated values of the gap ($2m$) and field strength. We compare contributions from different types of currents: $I_{\mathrm{HHG}}={\left|\omega J_X\left(\omega \right)\right|}^2, I_{\mathrm{er}}={\left|\omega J_{\mathrm{er},X}\left(\omega \right)\right|}^2, I_{\mathrm{ra}}={\left|\omega J_{\mathrm{ra},X}\left(\omega \right)\right|}^2, I_{\mathrm{ra}}^{\left(1\right)}={\left|\omega J_{\mathrm{ra},X}^{\left(1\right)}\left(\omega \right)\right|}^2$. In all cases, we use $t_{\mathrm{hop}}=3, \mu =0, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$. These results are obtained from the analysis of the Dirac models.

Figure 6

The intensity of the peaks in the HHG spectra $I_{\mathrm{peaks}}\left(n\right)$, which is evaluated from $I_{\mathrm{HHG}}\left(\omega \right)$, and $I_{\mathrm{peaks}}^{\prime }$, which is evaluated from $I_{\mathrm{HHG}}^{\prime }\left(\omega \right)={\left|\omega (J_{\mathrm{er},X}\left(\omega \right)+J_{\mathrm{ra},X}^{\left(1\right)}\left(\omega \right))\right|}^2$ (i.e., without $J_{\mathrm{ra}}^{\left(2\right)}$), as a function of the mass term $m$. In all cases, we use $t_{\mathrm{hop}}=3, \mu =0, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$. These results are obtained from the analysis of the Dirac models.

Figure 7

HHG spectra of gapped graphene for indicated values of the gap ($2m$) and field strength. We compare contributions from different types of currents; $I_{\mathrm{HHG}}={\left|\omega J_X\left(\omega \right)\right|}^2, {\tilde{I}}_{\mathrm{HHG}}={\left|\omega (J_{\mathrm{ra},X}\left(\omega \right)-i\omega P_{\mathrm{er},X}\left(\omega \right))\right|}^2, I_{\mathrm{er}}={\left|\omega J_{\mathrm{er},X}\left(\omega \right)\right|}^2$ and ${\tilde{I}}_{\mathrm{er}}={\left|{\omega }^2{P}_{\mathrm{er},X}\left(\omega \right)\right|}^2$. In all cases, we use $t_{\mathrm{hop}}=3, \mu =0, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$. These results are obtained from the analysis of the Dirac models.

Figure 8

Comparison of HHG spectra polarized along the $X$ direction evaluated with the tight-binding model or with the Dirac model for gapped graphene. In all cases, we use $t_{\mathrm{hop}}=3, \mu =0, T_1=150$, and $T_2=30$. The parameters of the electric field are $t_0=280,\sigma =40$, and $\mathrm{\Omega }=0.26$.