Excitonic Shift Current in Noncentrosymmetric Materials

• Excitonic shift current is a nonlinear dc photocurrent generated by resonant exciton transitions in inversion-asymmetric crystals, characterized by strong electron–hole interactions and quantum geometric effects.
• It exhibits distinct resonant enhancements and symmetry-driven selection rules that differentiate its behavior from conventional independent-particle photocurrents.
• Practical modeling employs GW+BSE, Wannier interpolation, and real-time Green’s function techniques to accurately predict its nonlinear optoelectronic response in diverse materials.

Excitonic shift current is the nonlinear dc photocurrent generated by resonant excitation of bound electron-hole pairs (excitons) in noncentrosymmetric crystals under optical illumination. This phenomenon generalizes the canonical bulk photovoltaic "shift current" mechanism—originally formulated for independent-particle interband transitions—to regimes where many-body electron-hole correlations dominate the optical response. The excitonic shift current is governed by the quantum geometric properties of excitonic wavefunctions and manifests strongly near exciton resonances, exhibiting amplitude, selection rules, and symmetry constraints distinct from those found in the independent-particle framework. Robust in both three-dimensional and low-dimensional materials, the excitonic shift current underlies ultrafast nonlinear photogalvanic effects in bulk semiconductors, two-dimensional materials, and nanostructures, and is central to the understanding and engineering of next-generation photovoltaic, terahertz emission, and nonlinear optoelectronic devices.

1. Fundamentals and Formal Theory

The shift current is the dc component of the second-order nonlinear optical response in noncentrosymmetric crystals. In the length gauge, it arises from real-space displacements—the "shift vector"—experienced by electron-hole pairs upon photoexcitation. For independent electrons, the shift current density is given by

$$J^a = \sum_{b,c} \sigma^{abc}(0;\omega,-\omega)\, E_b(\omega)\, E_c(-\omega)$$

where $\sigma^{abc}$ is the shift-current conductivity tensor (Podzimski et al., 2017, Morimoto et al., 2015, Lai et al., 3 Feb 2024). Microscopically, the shift vector for interband transitions is

$$R_{cv}(\mathbf{k}) = \nabla_{\mathbf{k}}\, \mathrm{Arg}[v_{cv}(\mathbf{k})] + A_c(\mathbf{k}) - A_v(\mathbf{k})$$

with $v_{cv}$ the velocity or dipole matrix element and $A_{c/v}$ the respective Berry connections.

When the electron–hole Coulomb interaction is included, the optical response is dominated by excitonic bound states rather than free particle transitions. The many-body formalism requires a projection onto the excitonic eigenstates of the Bethe–Salpeter Hamiltonian (BSE). The excitonic shift-vector $R_S$ for exciton $S$ at crystal momentum $\mathbf{k}$ generalizes the above expression: $$R_S(\mathbf{k}) = \nabla_{\mathbf{k}} \mathrm{Arg}\,[\Psi_{S,\mathbf{k}}] + A_c(\mathbf{k}) - A_v(\mathbf{k})$$ where $\Psi_{S,\mathbf{k}}$ is the exciton wavefunction. The quadratic (second-order) shift current at resonance becomes

$$J_{\rm shift} = 2\pi e \int \frac{d^d k}{(2\pi)^d} |M_S(\mathbf{k})|^2 R_S(\mathbf{k})\, \frac{\Gamma}{\left[(\hbar\omega - E_S)\right]^2 + \Gamma^2} |E(\omega)|^2$$

with $M_S$ the excitonic optical matrix element, $E_S$ the exciton energy, and $\Gamma$ a phenomenological linewidth (Morimoto et al., 2015, Podzimski et al., 2017).

2. Quantum Geometry, Symmetry, and the Many-Body Shift Vector

The quantum geometric interpretation of the excitonic shift vector reveals fundamental distinctions from the free-particle scenario. The many-body shift vector is defined as

$$R^{\alpha}_{S0} = \langle S| \hat{R}^\alpha |S\rangle - \langle 0 | \hat{R}^\alpha |0\rangle$$

where $\hat{R}^\alpha$ is the position operator and $|S\rangle$ is the exciton state (Lai et al., 3 Feb 2024). This quantity is gauge-invariant and physical: it measures the shift of the electronic charge centroid under excitation into the bound exciton.

A sum-rule relates momentum and position matrix elements: $$\sum_{S' \neq 0} r^a_{0S'} p^b_{S'S} - \sum_{S'\neq S} p^b_{0S'} r^a_{S'S} = R^a_{S0} p^b_{0S}$$ allowing practical evaluation via the exciton envelopes and their k-space overlap.

Crucially, the excitonic shift vector transforms intrinsically under the crystal point group and is independent of light polarization (Yang et al., 9 Jul 2025). In non-polar, noncentrosymmetric crystals, this enforces a vanishing shift vector for vertical (Q=0) transitions—i.e., excitonic shift current is forbidden by symmetry even if free-particle shift current persists. Conversely, in polar classes, a nonzero shift vector is symmetry-allowed and can be engineered for device purposes (Yang et al., 9 Jul 2025).

3. Computational Methodologies and Model Approaches

Accurate prediction of excitonic shift currents demands realistic modeling of both bandstructure and many-body interactions:

• Wannier Interpolation and GW+BSE: Realistic tight-binding models are constructed from DFT, using Wannier90. GW corrections are applied to open quasiparticle gaps and adjust band dispersions. The Bethe–Salpeter equation (BSE) is then solved (often in the Tamm–Dancoff approximation) to obtain exciton eigenstates $A^{S}_{v c \mathbf{k}}$, which encode both real-space and momentum-space structure (Esteve-Paredes et al., 20 Jun 2024, Huang et al., 2023, Lai et al., 3 Feb 2024).
• Shift current tensor evaluation: Once the exciton states and their envelopes are known, the nonlinear conductivity is computed:

$$\sigma^{abc}_{\text{exc}}(\omega) = \frac{\pi e^3}{\hbar^2} \sum_S \int \frac{d\mathbf{k}}{(2\pi)^d} X^b_S(\mathbf{k}) X^{c*}_S(\mathbf{k}) R^a_S(\mathbf{k}) \delta(\hbar\omega - E_S)$$

where $X^b_S$ is the exciton dipole (Huang et al., 2023).

• Screening and Dimensionality: The electron–hole kernel incorporates Rytova–Keldysh 2D screening in monolayers, or a screened Coulomb potential for bulk. The dielectric environment and monolayer thickness are critical for exciton binding (Quintela et al., 2023).
• Green’s-function and Real-time Propagation: Non-equilibrium approaches (e.g., Keldysh contour Green's function) can capture ultrafast dynamics and strong-field effects (Chan et al., 2019, Mao et al., 19 Jun 2025).
• Symmetry and Selection Rules: Tensor structure (e.g., availability of out-of-plane vs. in-plane elements, dependence on buckling) directly follows from point group analysis and the structure of the exciton wavefunctions (Quintela et al., 2023).

4. Material Case Studies and Numerical Results

Bulk GaAs

Numerical solutions of 14-band k·p semiconductor Bloch equations show that including the electron-hole Coulomb attraction yields a sharp excitonic peak in both absorption and shift current. The excitonic shift current peak is enhanced by ~30% over the non-interacting case, but the net microscopic displacement per excitation (the shift vector) at resonance is reduced (d_shift ≈ 89 pm vs. 200 pm without Coulomb) due to electron–hole localization (Podzimski et al., 2017).

BN Nanotubes and Monolayer BN

GW+BSE calculations yield a giant in-gap excitonic shift-current peak attributed to the A exciton. The peak shift current exceeds that for the free-carrier continuum by factors of 3–10, and achieves values up to ≈25–30 μA/V²—nearly an order of magnitude above leading ferroelectrics. Current direction is robust to tube chirality, in contradistinction to model π-band predictions (Huang et al., 2023).

Monolayer MoS₂, GeS, and Janus TMDs

In monolayer GeS, excitonic enhancements drive the shift current peak at subgap frequencies to ≈40–50 μA/V² (an order of magnitude above the IPA value and at frequencies far below the GW bandgap) (Chan et al., 2019). In MoS₂ and Janus TMDs, both ab initio and real-time approaches reveal pronounced shift-current resonances associated with both "bright" (A/1s) and "dark" (2p, C excitons) states: for instance, C-exciton peaks in Janus TMDs reach ≈0.15 A V⁻² m⁻¹, two orders of magnitude above other features, owing to spatial charge separation across different chalcogen sublayers (Mao et al., 19 Jun 2025, Esteve-Paredes et al., 20 Jun 2024).

BaTiO₃ (Bulk Ferroelectrics) vs. SnSe (2D)

In bulk BaTiO₃, inclusion of excitonic renormalization reduces the calculated photocurrent magnitude by ~30% and redshifts features near threshold, highlighting the more prominent suppression of shift current in 3D systems with weak screening (Fei et al., 2018). In 2D SnSe, the exciton binding energy is small and geometric factors conspire to minimize excitonic effects, keeping the shift current amplitude close to the IPA prediction.

5. Physical Insights, Symmetry, and Device Implications

• Resonance and Enhancement: Excitonic poles in σ(ω) boost the shift current by factors as large as 10–100 over the independent-particle background, especially in low-dimensional materials with strong Coulomb binding and high k-space overlap among nearly-degenerate excitons (Lai et al., 3 Feb 2024, Huang et al., 2023, Chan et al., 2019).
• Symmetry Constraints: The polarization independence and transformation properties of the excitonic shift vector allow for strict symmetry-based selection: in nonpolar but noncentrosymmetric point groups, excitonic shift current is forbidden for Q=0 transitions, regardless of the oscillator strength (Yang et al., 9 Jul 2025). In polar systems, large resonant currents are possible and can be tuned via external fields, strain, or chemical modification.
• Topology and Quantum Geometry: The shift vector and associated Berry curvature of excitons reflect deeper topological structure. The shift current couples directly to Berry-phase-derived properties of the underlying bands and the quantum geometry inherited by the interacting excitonic manifold (Paiva et al., 19 Aug 2024).
• Device Outlook: Excitonic shift currents are accessible in ultrafast measurements (e.g., THz emission in bulk MoS₂ at cryogenic temperatures) (Dhakar et al., 11 Nov 2025), and promise direct photovoltaic effects unmediated by junctions or conventional diffusive dynamics. Material engineering targeting polar symmetry, reduced screening, and density of optically active, nearly-degenerate excitons leads to giant, spectrally tunable nonlinear photocurrents (Lai et al., 3 Feb 2024, Huang et al., 2023).

6. Practical Considerations and Experimental Realizations

• Disorder, Sample Quality, and Linewidth: Narrower excitonic linewidths (γ_S < 10 meV) yield sharper and higher shift-current peaks. Clean samples maximize both the amplitude and discernibility of excitonic features (Esteve-Paredes et al., 20 Jun 2024).
• Thickness and Screening: In 2D, strong dielectric screening suppresses exciton binding and can reduce excitonic modulation of the shift current—advantageous for applications where independent-particle behavior is preferred (Fei et al., 2018).
• Competing Many-Body Phases: At high exciton densities, the formation of electron–hole liquids suppresses the coherent excitonic shift current, as observed for fluence-dependent THz emission in bulk MoS₂ (Dhakar et al., 11 Nov 2025).
• Temperature and Phonon Effects: Exciton–phonon interaction broadens peaks and diminishes resonant shift currents at elevated temperatures (Lai et al., 3 Feb 2024).

7. Outlook and Design Principles

The design of materials exhibiting strong excitonic shift current is optimized by:

1. Maximizing exciton binding (tight real-space localization, broad k-space support).
2. Engineering polar symmetry, either by elemental selection, strain, or stacking.
3. Controlling the density and energetic alignment of optically active, nearly-degenerate exciton states.
4. Minimizing disorder and exciton–phonon coupling for sharper, higher-resonance currents.
5. Utilizing low-dimensional, van der Waals systems for enhanced Coulomb correlations and symmetry tuning (Lai et al., 3 Feb 2024, Mao et al., 19 Jun 2025, Huang et al., 2023).

Excitonic shift current phenomena thus encode rich geometric and many-body effects, offer practical routes to nonlinear optoelectronic functionalities, and provide sensitive probes of symmetry and quantum geometry in solids.