---
title: Second-Order Shift Photocurrent
url: https://www.emergentmind.com/topics/second-order-shift-photocurrent
type: topic
---

# Second-Order Shift Photocurrent

Second-order shift photocurrent refers to a nonlinear DC photoresponse in noncentrosymmetric and, in magnetic variants, magnetically ordered or engineered crystalline systems. It is governed by the geometry and phase structure of the electronic wavefunctions, encapsulated in the "shift vector," and is a foundational contribution to the bulk photovoltaic effect—a rectification of light into electrical current without p–n junctions. Second-order shift currents may occur under continuous-wave monochromatic light, pulsed excitation, or more complex tailored optical fields. Their tensorial structure, frequency dependence, and quantum-geometric origin make them central objects in nonlinear optics, quantum materials engineering, and optoelectronic device research.

## 1. Fundamental Theory and Mathematical Formulation

The second-order shift current arises as the leading DC contribution to the photoconductivity in a noncentrosymmetric crystal under irradiation by an electromagnetic field. The general tensor relation for the induced current density is
\[
J^a(0) = \sum_{b,c} \sigma_{a b c}(0; \omega, -\omega) E_b(\omega) E_c(-\omega)
\]
where $E_b(\omega)$ is the complex electric field at frequency $\omega$, and $\sigma_{a b c}(0; \omega, -\omega)$ is the second-order shift-current conductivity tensor [2406.14215, 1904.12813].

Within perturbation theory, in the independent-particle (IP) approximation, this tensor has the prototypical "shift vector" form:
\[
\sigma_{a b c}^{\rm IP}(0; \omega, -\omega) = \frac{\pi e^3}{\hbar^2} \int_{\rm BZ} \frac{d^d k}{(2\pi)^d} \sum_{n,m} (f_n - f_m) r^b_{nm}(k) r^c_{mn}(k) R^a_{nm}(k) \delta[\omega_{mn}(k) - \omega]
\]
with
- $r^b_{nm}(k) = \langle u_{n k} | i \partial_{k_b} | u_{m k} \rangle$ the interband velocity (Berry connection) matrix element,
- $R^a_{nm}(k) = \partial_{k_a} {\rm arg} [r^b_{nm}(k)] - [A^a_{nn}(k) - A^a_{mm}(k)]$ the shift vector,
- $A^a_{nn}(k) = i \langle u_{n k}| \partial_{k_a} u_{n k} \rangle$ the Berry connection,
- $f_n$ Fermi occupation, and $\omega_{mn} = (E_m - E_n)/\hbar$ [2302.03090, 2310.06168, 2409.05571].

The shift vector measures the real-space displacement of the carrier wavepacket during an interband transition and is strictly gauge invariant. Shift current persists only in systems lacking inversion symmetry; time-reversal symmetry and magnetic order further classify allowed tensor components [2006.13573, 1607.03888].

## 2. Quantum Geometry, Symmetry, and the Role of the Shift Vector

The shift current intimately reflects the quantum geometry of the Bloch bands. The shift vector $R^a_{nm}(k)$ encodes the change in the intracell coordinate after photoexcitation and is linked to interband phase structure—a quantum geometric property. The magnitude and sign of the shift current are set by
- The joint density of states at resonance $\omega_{mn}(k) = \omega$,
- The product $|r^b_{nm}|^2$ (optical transition probability),
- The local shift vector $R^a_{nm}(k)$, which is typically large near band extrema and van Hove singularities, or where the phase of $r^b_{nm}$ varies rapidly in $k$-space [2107.09090].

Symmetry dictates tensorial selection rules:
- **Noncentrosymmetry**: $\sigma_{a b c}$ nonzero only if inversion symmetry is broken [2310.06168, 2302.03090].
- **Time-reversal symmetry**: Pure charge shift current is allowed under linear polarization in T-symmetric systems, while magnetic shift currents (weighted by Berry curvature) require broken T [2006.13573].
- **Crystal Point Group**: For instance, in 2H-MoS$_2$ (D$_{3h}$), only specific in-plane components are nonzero; in TaIrTe$_4$ (Pmn2$_1$), only five components survive [2302.03090, 2506.16067].

## 3. Exciton-Enhanced Shift Currents and Many-Body Effects

Electron–hole interactions strongly modify the shift current response, particularly in 2D and low-dimensional systems with large exciton binding energies. Many-body treatments employ either time-dependent nonequilibrium Green's functions or Bethe-Salpeter equation (BSE) frameworks:
- The shift-current tensor is obtained from real-time propagation of the interacting density matrix under an external field,
- The "sum-over-excitons" approach yields
\[
\sigma_{a b c}^{\rm exc}(0; \omega, -\omega) \approx \frac{\pi e^3}{m_e V_x} \sum_{m, n} P^a_m Q^b_{mn} P^c_n \frac{1}{E_m} \frac{1}{E_n - \hbar \omega - i\eta}
\]
with $P^a_n$ the optical dipole from ground to exciton $n$, $Q^b_{mn}$ the inter-exciton position matrix element, and $E_n$ the exciton energy [1904.12813, 2406.14215].

Excitonic resonances enable shift currents at subgap photon energies ($\hbar\omega < E_{\rm gap}$), resulting in distinctive in-gap peaks in $\sigma_{a b c}(\omega)$, with amplitudes far exceeding the IP value and displaying strong dependence on exciton oscillator strength and linewidth. In monolayer GeS, such effects yield shift current enhancements by up to 20× over the non-interacting theory and produce responsivities on par with Si photodiodes, but in atomically thin layers [1904.12813]. Similar mechanisms underpin strong shift current responses in Janus TMDs at the C-exciton resonance, with the electron and hole localized on different atomic layers, maximizing the shift vector [2506.16067].

The Coulomb attraction further enhances the shift current through the Sommerfeld factor, amplifying both absorption and shift current by the same multiplicative factor $Z(\omega)$ [2409.05571].

## 4. Real-Time and First-Principles Methodologies

First-principles computation of the shift current employs several complementary approaches:
- **Perturbative length-gauge Berry connection formalism**: Band structures and Berry connections from DFT, MLWF interpolation, and dense $k$-grid integration [2310.06168, 2302.03090].
- **Real-time propagation**: Direct evaluation of the current from time-dependent Kohn–Sham (TDDFT, TD-aGW) or Peierls-substituted Hamiltonians, allowing for arbitrary light fields (pulse, CW, two-color, etc.), the inclusion of ultrafast carrier relaxation, and many-body screening [2207.03772, 2506.16067].
- **Many-body BSE frameworks**: Explicit exciton basis for the nonlinear optical response, producing a matrix expression for $\sigma_{a b c}^{\rm exc}$ and accounting for dark-bright coupling mechanisms [2406.14215, 1904.12813].
- **Nonequilibrium Green's Function (TD-NEGF)**: Extends to strongly pulsed regimes, nonperturbative intensities, and quantum transport in devices, capturing two-photon (intensity-squared) shift currents and superballistic carrier propagation [1803.04404].

## 5. Material Systems, Dynamical and Device Aspects

Second-order shift current effects have been extensively studied and confirmed in:
- **2D semiconductors**: MoS$_2$, WS$_2$, GeS, Janus TMDs—where shift currents can be ultrafast ($\sim$10–20 fs response), large in magnitude (up to $10^{-3}$ A/V$^2$), and strongly tunable via gating, stacking, or strain [2207.03772, 2506.16067, 1904.12813].
- **Topological quantum materials**: TaIrTe$_4$, TaAs, Bi$_2$X$_3$ TIs (Dirac surface states)—featuring large nonlinear response and, for TIs, possible realization of shift spin kernels under magnetic perturbation [2302.03090, 1607.03888].
- **Twisted bilayer graphene**: Sharp dependence of $\sigma(\omega)$ on twist angle, doping, and interaction-driven band renormalization; shift current spectroscopy directly probes quantum geometry and filling-related topological transitions [2107.09090].

Gating, external fields, or dynamical-symmetry-breaking multicolor pulses enable dynamically switchable and sign-reversible shift currents ("bulk electro-photovoltaic effect") [2304.13977, 2410.07767]. Application to ultrafast photodetectors and optoelectronics stems from the fs-scale current dynamics in high-mobility 2D materials [2207.03772].

Antisymmetric, phase-controlled shift current responses are realized via temporally shaped or phase-mismatched pulse pairs, introducing new classes of ultrafast coherent control schemes [1209.1473].

## 6. Competing Mechanisms and Magnetic/Multiferroic Generalizations

In real materials, shift current competes with other second-order (and higher) mechanisms:
- **Ballistic (asymmetric population) current**: Dominant in clean systems with long carrier mean-free paths ($j^{\rm sh}/j^{\rm bal} \sim a_B/\ell$) [2409.05571].
- **Injection current (CPGE)**: Associated with circularly polarized light, often dominant in ferroelectric and topological systems [2310.06168].
- **Magnetic shift current (MSC)**: Emerges in $\mathcal{PT}$-symmetric topological magnets, combines real-space shift with interband Berry curvature, yielding circular dichroism-driven nonlinear responses; allows for electrical, magnetic, and optical switching [2006.13573].

Device-relevant parameters such as open-circuit voltage can exceed the band gap, enabling photoresponse beyond the Shockley–Queisser limit; practical $V_{\rm oc}$, however, is limited by sample conductivity and geometry [1904.12813].

## 7. Outlook: Tunability, Control, and Applications

Second-order shift photocurrents offer avenues for:
- Broadband, junction-free photovoltaic devices exceeding traditional efficiency limits [1904.12813, 2506.16067].
- Ultrafast, field- and polarization-tunable photodetectors and nonlinear optical switches [2207.03772, 2304.13977].
- All-optical and optoelectronic control of ferroic, topological, and phase-transition phenomena (photostrictive effects, structural switching) [2310.06168].
- Quantum metrology and material probing: Spectroscopy of $\sigma(\omega)$ traces band geometry, electron-electron interaction effects, Berry curvature, and emergent topology [2107.09090, 2302.03090].

Recently developed experimental and computational methods—real-time propagation, ultrafast pump-probe, and complex pulse-shaping—continue to broaden the accessible regimes for shift current studies, including decoherence times, pulse envelope effects, two-photon/THz ultrafast processes, and hybrid magnetic, topological, or correlated phases [1209.1473, 1803.04404, 2410.07767].

---

**Key arXiv References**:  
[1904.12813], [2310.06168], [2409.05571], [2506.16067], [2406.14215], [2107.09090], [2304.13977], [2302.03090], [2207.03772], [1607.03888], [1803.04404], [1209.1473], [2006.13573], [2410.07767]

Source: https://www.emergentmind.com/topics/second-order-shift-photocurrent