Circular Photogalvanic Effect in Quantum Materials

• Circular Photogalvanic Effect (CPGE) is a nonlinear optical process where circularly polarized light induces a dc photocurrent in noncentrosymmetric materials.
• Quantized CPGE currents in Weyl and multifold semimetals link the photocurrent response to topological invariants, such as the Chern number.
• Experimental tuning via geometry, strain, and gating makes CPGE a promising tool for applications in spintronics, valleytronics, and topological photonics.

The circular photogalvanic effect (CPGE) is a nonlinear, second-order optical response in which circularly polarized light incident on a noncentrosymmetric material induces a dc photocurrent whose sign reverses with the helicity of the light. This effect is fundamentally rooted in symmetry-breaking, spin–orbit interaction, and Berry phase geometry, and is realized in a broad range of quantum materials, including semiconductors, metals, Weyl and multifold semimetals, Rashba interfaces, van der Waals heterostructures, and antiferromagnetic insulators. CPGE is notable for manifestations such as quantization tied to topological invariants, spin-to-charge conversion, electrical tunability, and dissipationless spin or valley currents.

1. Theoretical Principles and Tensor Formalism

The CPGE is most generally expressed as a dc photocurrent density

$$j_i(\omega) = \sum_{j} \beta_{ij}(\omega)\; [E(\omega) \times E^*(\omega)]_j,$$

where $\beta_{ij}(\omega)$ is the CPGE response tensor (a rank-2 pseudotensor, antisymmetric under index interchange), $E(\omega)$ is the incident optical field, and $[E(\omega) \times E^*(\omega)]_j$ is proportional to the light helicity and defines a "circularity pseudovector." The CPGE tensor originates from the second-order (in $E$) nonlinear optical response; it only acquires nonzero values when inversion symmetry is broken.

A general microscopic expression for the injection current contribution to the CPGE, in the language of multiband Bloch theory, is given by (Juan et al., 2016, Pal et al., 2023, Yahyavi et al., 2023, Le et al., 2021)

$$\beta_{ij}(\omega) = \frac{\pi e^3}{\hbar^2 V} \sum_{\mathbf{k},n,m} f_{nm}(\mathbf{k})\, \Delta_{nm}^i(\mathbf{k})\, \varepsilon_{jkl}\, r_{nm}^k(\mathbf{k})\, r_{mn}^l(\mathbf{k})\, \delta(\hbar\omega - E_{mn}(\mathbf{k})),$$

where $f_{nm}$ is the difference of Fermi occupations, $\Delta_{nm}^i$ the group velocity difference, $r_{nm}^k$ interband Berry connections, and $\varepsilon_{jkl}$ the Levi–Civita symbol. For two-band systems, this reduces to Berry-curvature-based expressions that directly reveal the topological nature of the response.

2. Symmetry, Quantization, and Topological Origin

The presence and structure of the CPGE tensor are strictly dictated by crystal symmetry. It vanishes in centrosymmetric (inversion-symmetric) materials and, in noncentrosymmetric lattices, its allowed components are determined by the material’s point group. For example, in chiral point groups (such as the cubic $T$ or cyclic groups $C_3$, $C_4$, $C_6$), only certain tensor elements are nonzero (Le et al., 2021).

A striking property is the "quantized" CPGE that appears in Weyl and multifold semimetals lacking inversion and mirror symmetries. In such systems, the trace of the CPGE tensor is directly proportional to the net Chern number ($C$) enclosed by the resonant optical transitions:

$$\mathrm{Tr}\,\beta(\omega) = i\,\pi\,\frac{e^3}{h^2}\,C \quad (2|\varepsilon_L| < \hbar\omega < 2|\varepsilon_R|),$$

where $\varepsilon_{L/R}$ are the energies of the left/right-handed Weyl nodes (Juan et al., 2016, Pal et al., 2023, Le et al., 2021, Le et al., 2020). This quantization is topologically protected in the noninteracting and clean limit, immune to disorder and material parameters but sensitive to many-body interactions (Avdoshkin et al., 2019, Mandal, 2020). Experimental observation of quantized CPGE is possible in compounds such as RhSi, CoSi, or SrSi$_2$ under suitable gating and doping (Le et al., 2020, Ni et al., 2020, Le et al., 2021).

3. Microscopic Mechanisms in Representative Material Classes

• Weyl and Multifold Semimetals: In these, CPGE reflects Berry curvature monopoles at band crossings. For an isolated Weyl node, helicity-selective excitation leads to an asymmetric carrier population in momentum space—a net current proportional to monopole charge (Sun et al., 2016, Juan et al., 2016, Ni et al., 2020). Interplay of node tilting and quadratic warping yields rich tensor structure and additional, "transverse" CPGE signals sensitive to Lorentz-violating effects (Yahyavi et al., 2023).
• Rashba Systems and Interfaces: In materials with strong Rashba spin–orbit interaction, such as Cu/Bi bilayers or Tl–Pb monolayers, CPGE arises from helicity-selective pumping between spin-split subbands. The resulting current is controlled by the Rashba parameter, incidence geometry, and selection rules (Hirose et al., 2018, Taniuchi et al., 2023). For tunneling-induced interfacial Rashba splitting, the effect is maximized at oblique incidence and vanishes for in-plane excitation.
• Gyrotropic and Wurtzite Semiconductors: In noncentrosymmetric semiconductors with gyrotropic point groups, the CPGE can be formulated using semiclassical Boltzmann theory, where Berry curvature, side-jump, and skew-scattering mechanisms are all manifest; these semiclassical and quantum mechanical approaches yield quantitatively identical results in high-frequency regimes (Golub et al., 2020).
• Strain-Engineered and Heterostructure Systems: In materials like strained Cd_xHg_{1–x}Te, CPGE can be induced by symmetry reduction due to strain, resulting in Drude-type absorption pathways interfering in a helicity-sensitive manner (Hubmann et al., 2019). In van der Waals heterostructures (e.g., MoS$_2$/WSe$_2$), CPGE is generated through valley-selective optical selection rules, built-in electric fields, and relaxation-time asymmetries between electron and holes, yielding electrically tunable and gate-dependent responses (Rasmita et al., 2019).
• Third-Order and Electrically Switchable CPGE: In Rashba systems under static applied electric field, a third-order "magnetic shift current" can be induced, associated with Rashba–Edelstein spin-polarization, exhibiting orders-of-magnitude enhancement and electrical switchability of the CPGE (Zhu et al., 2024).
• Spin-CPGE and AFM Insulators: In PT-symmetric antiferromagnets, a pure spin CPGE (spin-polarized current without net charge flow) is symmetry-allowed, robust even when conventional charge CPGE is forbidden. This spin CPGE is insensitive to SOC strength and may achieve high efficiency at room temperature (Fei et al., 2021).

4. Experimental Realization and Signatures

CPGE manifests as a helicity-dependent dc current or voltage, detected under circularly polarized illumination. The spatial, polarization, and angular dependence of CPGE is prescribed by the symmetry-restricted tensor structure. Experimentally, tuning geometry (angle of incidence, polarization configuration) and device structure (contacts, electrodes, gating) enables the isolation of the CPGE from linear and polarization-independent photoresponses (Hirose et al., 2018, Ni et al., 2020, Rasmita et al., 2019, Zhu et al., 2024).

Analysis often proceeds by fitting the photocurrent (or photo-voltage) as a function of a polarization-control angle (e.g., quarter-wave plate) using

$$I(\alpha) = C \sin[2(\alpha+\alpha_0)] + L_1\sin[4(\alpha+\alpha_0)] + L_2\cos[4(\alpha+\alpha_0)] + D,$$

where the $C$-term encodes the CPGE, which flips sign for circular polarization reversal (Hirose et al., 2018, Taniuchi et al., 2023).

Giant and quantized CPGE signals have been established in topological materials (e.g., CoSi $\sim 550\;\mu A/V^2$ at 0.4 eV (Ni et al., 2020), RhSi and PtAl up to $4\beta_0$ (Le et al., 2020)), as well as in Rashba monolayers, where even atomic-thin films generate CPGE of comparable magnitude to bulk semiconductors (Taniuchi et al., 2023). Spin-CPGE magnitudes in AFM insulators like CrI$_3$ are of order tens–hundred $\mu$A/V$^2$ (Fei et al., 2021).

5. Corrections, Limitations, and Advanced Effects

• Role of Disorder and Interactions: The quantized CPGE is robust against disorder (which only controls steady-state saturation and time scales) but is generally not topologically protected against electron–electron interactions. First-order Hubbard and Coulomb interactions produce corrections of order $\alpha^* \sim e^2/(\hbar v_F\varepsilon)$, which can reach tens of percent, spoiling exact quantization (Avdoshkin et al., 2019, Mandal, 2020).
• Finite Frequency Windows: Quantized plateaus only appear for photon energies where only one Weyl or multifold node is optically accessible (all others Pauli-blocked), leashed by node energy separation and thermal smearing. Band-structure details, such as nodal energy offsets or hybridization with trivial bands, can significantly restrict plateau windows (Le et al., 2021, Le et al., 2020).
• Spatially Dispersive and Nonlocal CPGE: In systems lacking in-plane CPGE by symmetry, spatially nonuniform light beams (nonzero wavevector $q$) can give rise to a "spatially dispersive" CPGE (s-CPGE), generating circulating photocurrents whose direction and magnitude are controlled by beam profile and helicity (Ji et al., 2018). In perovskites, CPGE can be highly nonlocal, with spin diffusion lengths up to tens of microns, relevant for spintronics (Zhu et al., 2024).
• Tunability and Control: CPGE is tunable via static electric and magnetic fields, gating, doping, interfacial engineering, structural phase, and twist angle (e.g., in twisted bilayer systems, a layer-polarized CPGE resonance is angle-selectable) (Rasmita et al., 2019, Gao et al., 2019, Zhu et al., 2024).
• Ultrathin and Monolayer Systems: Monolayer Rashba alloys, and bilayer Xene (e.g., germanene) heterostructures with inversion symmetry broken at the interface, exhibit layer-selective CPGE—the signal is maximized when the active 2D sheet is isolated from substrate hybridization, providing efficient spin-to-charge conversion (Nishijima et al., 2024, Taniuchi et al., 2023).

6. Applications and Outlook

CPGE serves as a direct experimental probe of Berry phase topology, Chern number, and symmetry-breaking in quantum materials, providing a unique diagnostic for topological semimetal states (Ni et al., 2020, Juan et al., 2016, Pal et al., 2023). Quantized and electrically tunable CPGE currents offer opportunities for robust, contactless mid-IR and THz detectors, helicity-sensitive optoelectronics, valleytronics, spin–photon interfaces, and low-dissipation spintronic and valleytronic devices (Rasmita et al., 2019, Zhu et al., 2024, Fei et al., 2021).

Device concepts include:

• Topological photodetectors exploiting quantized CPGE plateaus.
• All-optical spin-injectors and logic elements based on Rashba or Dirac excitations.
• Electrically switchable helicity detectors and opto-spintronics in perovskites and 2D Xenes.
• Pure spin-current sources in PT-symmetric antiferromagnets.

The CPGE thus epitomizes the intersection of nonlinear optics, symmetry, and topology, enabling the exploration of new regimes of quantum photonics and topologically protected optoelectronic phenomena (Juan et al., 2016, Le et al., 2020, Le et al., 2021, Yahyavi et al., 2023, Nishijima et al., 2024).