Linear Photogalvanic Effect

• Linear photogalvanic effect is the generation of a steady-state dc current in non-centrosymmetric media under linearly polarized illumination via second-order nonlinear processes.
• It involves both ballistic and shift current contributions, characterized by a third-rank tensor that reflects carrier scattering and quantum geometric effects.
• Applications range from 2D materials and topological insulators to bilayer graphene, making LPGE a vital probe of symmetry breaking and carrier dynamics.

The linear photogalvanic effect (LPGE) denotes the generation of a steady-state direct current (dc) in a non-centrosymmetric medium under irradiation by linearly polarized light. The current arises from second-order nonlinear processes that depend sensitively on crystal symmetry, carrier band structure, scattering dynamics, and quantum geometric properties of the electronic states. LPGE provides a powerful probe of symmetry breaking and carrier physics across diverse classes of materials, ranging from two-dimensional electron gases over liquid helium, group-V monolayers, quantum wells, and topological insulators to Weyl semimetals and bilayer graphene.

1. Fundamental Principles and Tensor Formalism

In a material lacking inversion symmetry, the dc current density $j_i$ generated under monochromatic illumination $E_j(\omega)$ is described by the third-rank photogalvanic tensor $\chi_{ijk}$ (Wittmann et al., 2010, Zhang et al., 2023, Yu et al., 3 Jan 2026): $$j_i = \sum_{j,k}\chi_{ijk}E_j(\omega)E_k^*(\omega),$$ where $\chi_{ijk}$ encodes the symmetry constraints and microscopic mechanisms governing the effect. For linearly polarized light, $E_jE_k^*$ reduces to $E_jE_k$, so the response depends quadratically on the electric field amplitude and is parameterized by material- and geometry-specific tensor components $\chi_{ijk}$, which can be explicitly calculated via Kubo-type expressions and nonlinear perturbation theory (Zhang et al., 2023, Yu et al., 3 Jan 2026).

The LPGE tensor is nonzero only in crystals or nanostructures with broken inversion symmetry, often dictated by structural inversion asymmetry (e.g., quantum wells, surfaces of topological insulators) or externally induced asymmetry (e.g., gate voltage in bilayer graphene) (Zheng et al., 2023). Its detailed structure is determined by the point-group symmetry (e.g., $C_{2v}$, $C_{3v}$, $C_1$) and the orientation of the polarization vector relative to the symmetry axes.

2. Microscopic Mechanisms: Ballistic vs. Shift Contributions

Microscopically, the LPGE current in a noncentrosymmetric system is typically decomposed into two principal contributions (Leppenen et al., 2022, Budkin et al., 2024, Yu et al., 3 Jan 2026):

• Ballistic (Injection) Current: Originates from asymmetric carrier distributions in momentum space generated by interference between direct optical absorption and disorder- or phonon-assisted scattering. This term requires electron-hole asymmetry or $k$-linear anisotropy in the band structure and is sensitive to momentum relaxation time $\tau$. Its amplitude depends on both inversion asymmetry (parameter $Q$ in Dirac-like models) and on the quasi-momentum mean free path $\ell$, and dominates when scattering is weak, often scaling as $1/\ell$ (Budkin et al., 2024).
• Shift Current: Results from a real-space shift of the center of charge during each optical excitation event, quantified by the shift vector $R_{mn}(k)$ (a quantum-geometric quantity related to Berry connection differences and phase gradients of dipole matrix elements). Its strength depends on the interband dipole structure and quantum geometric properties, is largely intensity-independent in the linear regime, and scales with the Bohr radius $a_B$ in standard semiconductors (Leppenen et al., 2022, Yu et al., 3 Jan 2026, Budkin et al., 2024).

In many experimentally relevant situations, the ballistic term overwhelms the shift term—their ratio is generally $a_B/\ell$ ($a_B$ is the Bohr radius, $\ell$ is the quasi-momentum mean free path) (Budkin et al., 2024).

3. Symmetry, Polarization, and Frequency Dependencies

The LPGE exhibits rich dependencies on crystal symmetry, incidence angle, and the polarization plane orientation (Wittmann et al., 2010, Plank et al., 2017, Plank et al., 2016):

• Tensor Component Selection: Only those $\chi_{ijk}$ allowed by the crystal’s point group are nonzero. For instance, in $C_{2v}$, only $\chi_{xxz}$, $\chi_{yyz}$ are present, resulting in currents under oblique incidence with the electric field having both in-plane and out-of-plane components (Plank et al., 2016, Wittmann et al., 2010).
• Polarization Angle Dependence: The current varies sinusoidally with polarization azimuth $\alpha$, typically following $\sin 2\alpha$ and $\cos 2\alpha$ harmonics. For trigonal ($C_{3v}$) surfaces, the LPGE current can exhibit the characteristic $\cos 2\phi$ dependence (Plank et al., 2017, Leppenen et al., 2022).
• Frequency Response: In the Drude regime (intraband absorption), the LPGE amplitude scales as $1/[1+(\omega\tau)^2]$; in the interband regime, resonances can appear due to direct transitions between subbands or between surface and bulk states (e.g., in topological insulators, at $\hbar\omega\sim E_c-E_F$) (Plank et al., 2017, Entin et al., 2013, Magarill et al., 2015).

The combined symmetry and polarization dependences enable experimental separation of LPGE from other nonlinear effects such as the circular photogalvanic effect (CPGE).

4. Quantum Geometric and Scattering Effects

Advances in quantum geometric theory have established that LPGE responses fundamentally involve quantities such as the shift vector (Berry connection differences and phase gradients), Berry curvature, and the quantum metric (Yu et al., 3 Jan 2026, Zhang et al., 2023, Zheng et al., 2023). Ab initio calculations in bulk ferroelectrics like BaTiO$_3$ reveal that phonon-mediated scattering can substantially enhance the shift current, in some cases overshadowing the naïve excitation-only result (Yu et al., 3 Jan 2026). The phonon-shift current is determined by real-space displacements during electron-phonon scattering, which at low phonon momentum couple directly to the Berry curvature.

A summary table of relevant microscopic contributions is:

Mechanism	Origin	Dominance Criterion
Ballistic current	Asymmetric distribution (scattering + absorption)	High mean free path ($\ell\gg a_B$)
Shift current	Quantum geometric real-space shift	$\ell\sim a_B$
Phonon-shift current	Electron-phonon scattering mediated	Strong e-ph coupling

In real devices, the balance between these terms is sensitive to detailed band structure, carrier scattering parameters, and quantum geometric features.

5. Experimental Realizations and Application Domains

2D Electron Gases on Liquid Helium

In systems such as electrons floating over liquid helium surfaces, LPGE originates from asymmetric quantum wells formed by the image potential and static electric field, with microwave-induced transitions between quantized subbands and ripplon scattering yielding pronounced resonance features (Entin et al., 2013, Magarill et al., 2015). The LPGE current in these cases shows a Lorentzian dependence on the detuning from subband resonance, with strength set by the intersubband dipole matrix element $z_{12}$ and width determined by friction (e.g., electron–ripplon interaction).

Group-V Monolayers: Charge/Spin Photocurrent

First-principles NEGF-DFT calculations in group-V monolayers (As, Sb, Bi) reveal highly anisotropic LPGE, where the armchair direction supports a pure charge current and the zigzag direction admits a pure spin current, tunable via the polarization angle and buckling height (Zhang et al., 2023). The symmetry ($C_{2v}$) dictates which tensor components are nonzero, and strong buckling enables record LPGE coefficients.

Topological Insulators

In three-dimensional topological insulators (TIs) such as Bi$_2$Te$_3$, LPGE is robust at room temperature over a wide spectral range. At low frequencies (THz), LPGE arises via Drude-like intraband absorption in Dirac surface states. At higher photon energies, resonant enhancement occurs due to surface-to-bulk transitions mediated by shift and ballistic mechanisms; nonlinear dependence on intensity allows determination of Fermi level and energy relaxation times (Plank et al., 2017, Leppenen et al., 2022, Danilov et al., 2021). In the nonlinear regime (bleaching/saturation), the ballistic mechanism dominates and shows higher-order harmonics (Leppenen et al., 2022, Danilov et al., 2021).

Bilayer Graphene: Gate Control

Application of a gate voltage to centrosymmetric bilayer graphene breaks inversion symmetry, yielding sizable LPGE (shift current) in AB-stacked bilayers. The response is sharply peaked near band-gap and van Hove singularity energies and can be tuned via gate voltage, chemical potential, and polarization orientation; expected device currents are in the milliampere range for micrometer-scale samples under high illumination (Zheng et al., 2023).

Weyl Semimetals: Surface Engineering

In Weyl semimetals, LPGE is sensitive to surface boundary conditions that control the connectivity and orientation of Fermi arc states. Analytical formulas show that by tuning surface angles (boundary spinors), the magnitude and direction of LPGE can be switched or reversed, exploiting surface state engineering rather than bulk symmetry (Steiner et al., 2021).

6. Advanced Theoretical Frameworks and Computational Approaches

Recent ab initio quantum kinetic theories incorporate density matrix dynamics and all bosonic scattering channels to calculate LPGE in realistic materials, capturing both transient and steady-state regimes (Yu et al., 3 Jan 2026). These frameworks connect LPGE responses to fundamental quantum geometric quantities and resolve discrepancies with experimental measurements by including, e.g., phonon-induced shift currents in ferroelectrics.

Moreover, semiclassical Boltzmann approaches complemented by Berry phase formalism clarify the distinction between LPGE (dominated by skew scattering and side jump mechanisms) and CPGE (Berry curvature driven), providing compact expressions for the response tensor in noncentrosymmetric conductors (0904.1917).

7. Summary and Outlook

The linear photogalvanic effect is a universal manifestation of second-order nonlinear optical transport allowed by broken inversion symmetry, combining ballistic and quantum-geometric shift mechanisms. Its precise characterization—enabled by modern theoretical development and first-principles computation—offers direct insight into carrier dynamics, quantum geometry, and device-relevant properties in platforms ranging from quantum wells and 2D materials to topological and strongly correlated systems. The LPGE is particularly rich in materials with strong spin-orbit coupling, engineered symmetry breaking, and complex surface states, facilitating both high-efficiency optoelectronic detection and fundamental exploration of quantum response phenomena.

Key references: (Entin et al., 2013, Leppenen et al., 2022, Plank et al., 2017, Zheng et al., 2023, Zhang et al., 2023, Yu et al., 3 Jan 2026, Plank et al., 2016, 0904.1917, Budkin et al., 2024, Steiner et al., 2021, Wittmann et al., 2010, Danilov et al., 2021, Magarill et al., 2015).