Quantum geometry-driven photogalvanic responses in semi-Dirac systems

Abstract

The photogalvanic effect (PGE), a fundamental nonlinear optical phenomenon in non-centrosymmetric materials, generates direct photocurrent under polarized light. Using quantum kinetic theory within the relaxation-time approximation, we theoretically investigate the PGE as a probe of quantum geometry in anisotropic type-I and type-II semi-Dirac (SD) systems, characterized by distinct electronic structures. We systematically analyse various microscopic contributions to the PGE conductivity, including injection, shift, resonance, higher-order pole, and anomalous terms, and emphasize their connections to different quantum geometric quantities, namely, Berry curvature, quantum metric, and metric connection. By studying the frequency and chemical-potential dependence of the PGE conductivity in SD systems, we find that the optical conductivities in the type-II case are significantly enhanced relative to those in type-I. For the circular PGE (CPGE), Berry-curvature-driven contributions remain qualitatively similar in both phases, whereas the linear PGE (LPGE) displays clear qualitative differences. In particular, the $xxx$ component of the shift conductivity in the type-II phase reverses sign upon tuning the perturbation parameter $δ$, providing a direct signature of the Lifshitz transition. In contrast, other components remain sign-invariant, as in type-I SD systems. These combined CPGE and LPGE signatures provide an unambiguous distinction between the two SD phases. The predicted effects, realizable in TiO$_2$/VO$_2$ heterostructures, establish PGE as a sensitive probe of quantum geometry with potential applications in polarization-selective photodetection, optical rectification, and next-generation optoelectronic devices.

Figures (8)

• Figure 1: Schematic illustration of type-I and type-II semi-Dirac (SD) band structures and their corresponding photogalvanic responses. (a,b) Low-energy band dispersions for type-I and type-II SD systems for different values of the perturbation parameter $\delta$. For $\delta>0$, the type-I system hosts two Dirac nodes that merge at $\delta=0$ to form an SD node and become gapped for $\delta<0$. In contrast, the type-II system hosts three Dirac nodes for $\delta>0$, which merge at $\delta=0$ at the Brillouin-zone center, and the node moves away from the zone center without opening a gap for $\delta>0$. (c) Schematic of photogalvanic responses in a SD sample. Circularly polarized (CP) light probes the Berry curvature $\Omega_{+-}$, producing the circular photogalvanic effect (CPGE), whereas linearly polarized (LP) light probes the symplectic connection $\tilde{\Gamma}_{+-}$, which governs the shift current underlying the linear photogalvanic effect (LPGE). The bottom panels show representative frequency-dependent responses: the CPGE magnitude is markedly enhanced in the type-II phase relative to type-I, while the $xxx$ component of the shift conductivity changes sign across the three phases of type-II SD ($\delta>0,\delta=0,\delta<0,$), while no such sign change occurs in the type-I case. The vertical dashed line corresponds to twice the chemical potential.
• Figure 2: Circular photogalvanic conductivities are plotted as functions of photon frequency for several values of the chemical potential $\mu$. Panels (a–d) show the injection ($\sigma^{xyx}_{\text{Inj}}$), anomalous ($\sigma^{xyx}_{\text{Anm}}$), resonance ($\sigma^{xyx}_{\text{Res}}$), and higher-order-pole ($\sigma^{xyx}_{\text{HOP}}$) contributions for the type-I SD system, while panels (e–h) display the corresponding components for the type-II SD system. Injection and anomalous responses display a single prominent peak, while the resonance and higher-order-pole terms reveal two peaks. The model parameters used are ($\alpha$, $\hbar v_F$, $\Delta$, $T$, $\tau$) = ($0.75$ eV Å$^2$, $0.65$ eV Å, $0.05$ eV, 25 K, 0.05 ps) for both systems, with $\beta = 3.5$ eV Å$^2$ for the type-II SD system.
• Figure 3: Dependence of circular photogalvanic conductivity on the photon frequency as the perturbation parameter $\delta$ is varied. The (a) injection ($\sigma^{xyx}_{\text{Inj}}$), (b) anomalous ($\sigma^{xyx}_{\text{Anm}}$), (c) resonance ($\sigma^{xyx}_{\text{Res}}$), and (d) higher-order-pole ($\sigma^{xyx}_{\text{HOP}}$) conductivities for the type-I semi-Dirac system are plotted. Panels (e)–(h) present the corresponding $\sigma^{yyx}$ components for the type-II semi-Dirac system. Notably, type-II displays systematically larger amplitudes for all contributions due to its additional Dirac node. Peak positions remain largely insensitive to $\delta$, while amplitudes adjust smoothly. The model parameters are the same as in Fig. \ref{['fig:mu_var']}.
• Figure 4: Chemical potential ($\mu$) dependence of different components of LPGE shift conductivity for type-I semi-Dirac system as functions of photon frequency. The photon frequency dependence of (a) $\sigma^{yyx}_{Sh}$, (b) $\sigma^{xxy}_{Sh}$, and $\sigma^{xyx}_{Sh}$-components of shift conductivity is shown for multiple values of $\mu$. The components exhibit identical qualitative behaviour: a single negative peak whose position shifts to higher frequencies with increasing $\mu$. The model parameters used are ($\alpha$, $\hbar v_F$, $\Delta$, $T$, $\tau$) = ($0.75$ eV Å$^2$, $0.65$ eV Å, $0.05$ eV, 25 K, 0.1 ps).
• Figure 5: Perturbation parameter ($\delta$) dependence of LPGE shift current components for the type-I system is plotted for (a) $\sigma^{yyx}_{Sh}$, (b) $\sigma^{xxy}_{Sh}$, and (c) $\sigma^{xyx}_{Sh}$. Different components display identical spectral behaviour, with a single negative peak whose amplitude varies smoothly with $\delta$, while the peak position remains essentially unchanged. This reflects the fact that $\delta$ alters the nodal geometry and phase-space weighting without modifying the characteristic interband transition energy. The model parameters are the same as in Fig. \ref{['fig:type1_mu_var_shift']}.