Shift Current in Noncentrosymmetric Materials

• Shift current is a second-order nonlinear optical response where light induces a directional real-space shift of electrons in noncentrosymmetric crystals.
• It is governed by the Berry-phase formalism and characterized by the shift vector, which quantifies the displacement during interband optical transitions.
• Ultrafast time-domain studies reveal its rapid onset (10–20 fs) and decay, highlighting its potential for high-speed photodetection applications.

Shift current is an intrinsic dc photocurrent generated under illumination in a noncentrosymmetric material as part of the bulk photovoltaic effect (BPVE). Unlike junction-based photovoltaics, it does not require a $p$-$n$ interface or an external dc bias. In the modern formulation, it is a second-order nonlinear optical response governed by the Berry connection of Bloch states and by the shift vector that encodes the real-space displacement of charge during an interband optical transition (Young et al., 2012, Fei et al., 2018).

1. Definition and physical interpretation

In its conventional form, shift current appears when monochromatic light excites electrons from valence to conduction bands in a crystal lacking inversion symmetry. The essential point is not post-excitation drift, but the fact that the optical transition itself carries a directed real-space displacement of the electronic wave packet. Because the underlying crystal potential is noncentrosymmetric, these transition-induced displacements do not cancel over the Brillouin zone, and a dc current results (Fei et al., 2018, Young et al., 2012).

This mechanism is distinct from conventional photoconductive or $p$-$n$-junction current. In those cases, light creates carriers and a built-in or external field separates them. In shift current, the displacement is intrinsic to the optical transition. Ferroelectrics are natural hosts because spontaneous polarization implies broken inversion symmetry, but the effect is not simply proportional to polarization and does not require ferroelectricity per se; the necessary condition is absence of inversion symmetry (Young et al., 2012).

The modern geometric picture identifies the shift current with Berry-phase structure. In one standard notation, the shift vector is written as

$$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$

where $\phi(m,n,k)$ is the phase of the interband optical matrix element and $A_Q$ is the Berry connection. This combination is gauge invariant even though its two terms are separately gauge dependent (Fei et al., 2018).

Ordinary shift current vanishes in centrosymmetric crystals because inversion forces cancellation between symmetry-related transitions. A notable extension is that finite photon momentum can activate a geometric photon-drag version in centrosymmetric materials. In that case the leading response is governed by a “shift current dipole,” and the resulting current can be purely transverse because of the pseudovector transformation of the shift vector under mirror symmetry (Shi et al., 2020).

2. Response theory and geometric formalism

The perturbative description is a second-order conductivity. A standard form is

$J_q = \sigma_q^{rs} E_r E_s,$

or, equivalently for monochromatic light,

$$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$

In the first-principles formulations used for ferroelectrics and related BPVE materials, the conductivity can be written as a Brillouin-zone integral of a transition-intensity factor multiplied by the shift vector (Young et al., 2012, Fei et al., 2018).

One common decomposition is

$$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$

where $n$0 contains the optical transition matrix elements and the energy-conserving $n$1-function, while $n$2 is the gauge-invariant real-space shift accompanying the transition. This form makes explicit that a large shift current requires both strong optical weight and a large geometric displacement; neither ingredient alone is sufficient (Young et al., 2012).

The same structure survives in alternative gauges and basis sets. A general Gaussian-basis prescription has been developed in both length and velocity gauges, with explicit treatment of nonorthogonal overlap matrices, analytical $n$3-derivatives of $n$4 and $n$5, and symmetry folding to the irreducible Brillouin zone. That framework was designed to preserve the full tensor symmetry of $n$6 under general light polarization and, in magnetic cases, under magnetic point-group symmetry (García-Blázquez et al., 2023).

This formalism also clarifies a common misconception. The shift vector is not itself the observable current; the current is the transition-by-transition product of shift vector and optical intensity, integrated over all $n$7. This is why aggregate quantities such as an integrated shift vector can be qualitatively useful while remaining insufficient to predict the sign or magnitude of the measured conductivity (Young et al., 2012).

3. Time-domain dynamics and microscopic current decomposition

The standard perturbative formulation is frequency resolved and stationary. Real-time dynamics were missing until explicit time-domain studies became available. In monolayer WS$n$8, real-time propagation time-dependent density functional theory showed that the shift current is established within $n$9–$p$0 fs after illumination begins and decays within a few tens of femtoseconds after the field is turned off. At $p$1 eV, the saturation times are about $p$2 fs at $p$3 K, $p$4 fs at $p$5 K, and $p$6 fs at $p$7 K, while the corresponding total-current decay times are $p$8, $p$9, and $n$0 fs (He et al., 2022).

That same study decomposed the current into off-diagonal and diagonal parts by projecting the time-dependent states onto ground-state Kohn–Sham orbitals. For WS$n$1, the diagonal contribution dominates: at $n$2 eV the total current is $n$3 mA, split into $n$4 mA off-diagonal and $n$5 mA diagonal, while at $n$6 eV the total is $n$7 mA with $n$8 mA off-diagonal and $n$9 mA diagonal. The diagonal channel therefore accounts for about $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$0 of the current at $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$1 eV and $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$2 at $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$3 eV. The smaller off-diagonal term is oscillatory and remains within $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$4–$$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$5 mA over the simulated $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$6 fs window (He et al., 2022).

Experimentally, ultrafast time-domain evidence was obtained in ferroelectric SbSI by terahertz emission spectroscopy. Above-gap excitation at $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$7 eV produces a THz peak delayed by about $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$8 fs and a long tail extending to roughly $$R_Q(m,n,k)= -\frac{\partial\phi(m,n,k)}{\partial k_Q}-(A_Q(n,n,k)-A_Q(m,m,k)),$$9 ps, whereas below-gap excitation at $\phi(m,n,k)$0 eV yields an intraband optical-rectification waveform confined to the pump-pulse timescale. The reconstructed current is consistent with a prompt shift followed by a sub-picosecond relaxation with $\phi(m,n,k)$1–$\phi(m,n,k)$2 ps, and the associated charge-shift distance is estimated as about $\phi(m,n,k)$3 Å (Sotome et al., 2018).

These time-domain studies establish two distinct points. First, shift current is genuinely ultrafast: it appears on the primary photoexcitation timescale, not on the slower drift or diffusion timescale. Second, the measured waveform generally contains both the instantaneous geometric excitation and subsequent relaxation dynamics. This suggests that device-relevant performance depends not only on the static conductivity tensor but also on how the geometric current is converted into a transient or steady extracted signal.

4. Symmetry, materials, and representative platforms

Shift current is strongly tensorial. Crystal symmetry determines which conductivity components vanish, which survive, and how they transform under light polarization. In monolayer WS$\phi(m,n,k)$4, mirror symmetry $\phi(m,n,k)$5 forces the current along $\phi(m,n,k)$6 to vanish, so $\phi(m,n,k)$7 and $\phi(m,n,k)$8 are zero while $\phi(m,n,k)$9 is symmetry allowed. In elemental phosphorene-like ferroelectrics As, Sb, and Bi, mirror symmetry $A_Q$0 allows only $A_Q$1, $A_Q$2, and $A_Q$3, with current constrained to the polar $A_Q$4 axis (He et al., 2022, Qian et al., 2022).

System	Representative shift-current result	arXiv
BaTiO$A_Q$5, PbTiO$A_Q$6	First-principles BPVE; response is not simply set by ferroelectric polarization	(Young et al., 2012)
SbSI	THz-emission observation of above-gap shift current; $A_Q$7 Å shift distance and $A_Q$8–$A_Q$9 ps relaxation	(Sotome et al., 2018)
Monolayer WS$J_q = \sigma_q^{rs} E_r E_s,$0	rt-TDDFT shows $J_q = \sigma_q^{rs} E_r E_s,$1–$J_q = \sigma_q^{rs} E_r E_s,$2 fs buildup and tens-of-fs decay in the allowed $J_q = \sigma_q^{rs} E_r E_s,$3 channel	(He et al., 2022)
Bulk BaTiO$J_q = \sigma_q^{rs} E_r E_s,$4 vs monolayer SnSe	GW and excitons strongly renormalize bulk response, but thin-film SnSe is much less affected near the gap	(Fei et al., 2018)
Twisted bilayer graphene	giant $J_q = \sigma_q^{rs} E_r E_s,$5–$J_q = \sigma_q^{rs} E_r E_s,$6 THz BPVE interpreted as momentum shift current	(Kaplan et al., 2021)
Monolayer As, Sb, Bi	very large 2D responses; reliable spectra require beyond-GGA treatment and SOC	(Qian et al., 2022)

Material engineering now extends beyond static noncentrosymmetric crystals. In multilayer $J_q = \sigma_q^{rs} E_r E_s,$7-MoS$J_q = \sigma_q^{rs} E_r E_s,$8, twisting breaks untwisted crystal symmetry and activates additional tensor components forbidden in the untwisted structures. In the calculations reported for twisted bilayers, $J_q = \sigma_q^{rs} E_r E_s,$9 is zero in the untwisted limit but grows strongly with angle, and the corresponding shift distance $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$0 reaches about $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$1 Å for the largest angle studied (Bagaglini et al., 15 Jun 2026).

A related design principle appears in alternating-angle twisted multilayer graphene. There, enhancement does not come only from a single direct transition but from virtual multiband processes through nearby flat bands. The paper links this to large Fubini–Study metric and increased Wannier spread, and identifies new twist-angle optima for shift current that do not coincide with the usual magic-angle condition (Chen et al., 2024).

5. Many-body corrections and extensions beyond the conventional Bloch-electron picture

Independent-particle DFT is often qualitatively useful but quantitatively incomplete. In bulk BaTiO$$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$2, $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$3 increases the direct band gap from $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$4 eV to $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$5 eV and enlarges the valence and conduction bandwidths by $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$6 and $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$7, respectively, which shifts and stretches the shift-current spectrum to higher frequency and reduces its magnitude. Excitons, treated through the Bethe–Salpeter equation, strongly reshape absorption within about $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$8 eV above the band gap and reduce the near-edge shift current. In monolayer SnSe, by contrast, the physical thickness $$j_Q(\omega)=\sum_s \sigma_{ssQ}(0,\omega,-\omega)E_s(\omega)E_s(-\omega).$$9 Å is much smaller than the optical penetration depth; for $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$0 eV light, $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$1 Å, so the measured current becomes largely insensitive to the excitonic renormalization of $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$2 (Fei et al., 2018).

The conventional Bloch-band picture is also not exhaustive. In a one-dimensional noncentrosymmetric Anderson insulator, dc shift current survives under global illumination if electron-phonon relaxation is present throughout the bulk. The effect remains robust even when the disorder scale exceeds the clean bandwidth, $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$3. By contrast, if relaxation occurs only via electrodes, or if excitation is local and far from contacts, the photocurrent decays exponentially with distance or system size (Ishizuka et al., 2020).

The mechanism has further generalizations. A multiferroic electromagnon can generate a dc charge photocurrent of shift-current type, with the analytical result

$$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$4

showing resonance at the electromagnon frequency $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$5 (Morimoto et al., 2019). Ordered ferrimagnetic and antiferromagnetic insulators support a bosonic analogue: a dc spin current induced by linearly polarized light, essentially independent of Gilbert damping, which the paper explicitly interprets as a shift current of magnons (Ishizuka et al., 2019).

There are also transport analogues in which the relevant transition is not optical. A thermal nonequilibrium between electrons and phonons in a noncentrosymmetric quantum well can generate a dc current

$$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$6

where $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$7 is the gauge-invariant coordinate shift accompanying inelastic electron-phonon scattering (Budkin et al., 2019). An analogous nonlinear heat response can be written in terms of the same shift vector that appears in electric shift current, but with an extra weighting factor $$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$8, so the shift heat current becomes directly tunable through chemical potential (Onishi et al., 2022).

6. Device physics, misconceptions, and unresolved questions

Several misconceptions recur in the literature. The first is that large spontaneous polarization should automatically imply large shift current. First-principles comparisons contradict this. PbTiO$$\sigma_q^{rs}(\omega)=e\sum_{n',n''}\int d\mathbf{k}\, I^{rs}(n',n'',\mathbf{k};\omega)\,R_q(n',n'',\mathbf{k}),$$9 has more than double the polarization of BaTiO$n$00, yet their photocurrent susceptibilities are of similar magnitude; more generally, strong response correlates better with asymmetric, delocalized, covalent states along the current direction than with polarization alone (Young et al., 2012).

A second misconception is that any zero-bias THz emission above or below a gap is equivalent to shift current. In SbSI, below-gap intraband optical rectification also emits THz radiation, but it integrates to zero and does not produce net transported charge; only the above-gap interband process shows the delayed, relaxing waveform associated with genuine shift current (Sotome et al., 2018).

A third misconception is that the intrinsic geometric current fully determines device performance. The zero-bias shift current itself is geometric, but the current–voltage characteristic and shot noise depend on the diagonal velocity difference $n$01 and the relaxation time $n$02. In the transport theory of shift-current photovoltaics,

$n$03

with $n$04, while the shot noise scales as $n$05. This suggests that one can, in principle, maximize the geometric current while suppressing noise and load dependence by designing bands with small $n$06. The same analysis identifies Landau levels in noncentrosymmetric two-dimensional systems as especially promising because $n$07 there (Morimoto et al., 2018).

On the application side, shift current has been linked to ultrafast photodetection because of the $n$08–$n$09 fs turn-on and few-tens-of-femtoseconds turn-off demonstrated in monolayer WS$n$10 (He et al., 2022). It has also been invoked to interpret anomalous photovoltaic responses in a ferroelectric double perovskite, where photovoltage up to about $n$11 V was reported for a band gap of about $n$12 eV. That study tied the effect to Cs$n$13 displacement and a calculated peak shift current of about $n$14 at $n$15 eV, but it also discussed light-induced ion migration, so the intrinsic electronic and slower ionic contributions were not fully disentangled (Wei et al., 3 Sep 2025).

More speculative proposals now use shift current outside conventional optoelectronics. One example is broadband axion-dark-matter detection in TaAs, where a strong experimental electric field mixes with the axion-induced electric field through the second-order shift-current tensor, downconverting the signal to an accessible difference frequency while exploiting the non-dissipative character of the response (Kondo et al., 22 May 2025).

The open questions are correspondingly clear. Microscopic relaxation channels after the initial geometric excitation are not yet fully resolved; in WS$n$16, the ultrafast current relaxation was speculated to be “probably due to electron-electron scattering,” but not derived from a complete scattering theory (He et al., 2022). Excitonic, quasiparticle, and other many-body effects remain decisive for quantitative predictions, especially in low-dimensional materials (Fei et al., 2018, Qian et al., 2022). In moiré and flat-band systems, the most useful microscopic descriptors may extend beyond the quantum geometric tensor itself, as argued for momentum shift current in twisted bilayer graphene and for multiband virtual-transition enhancement in twisted multilayer graphene (Kaplan et al., 2021, Chen et al., 2024). These developments suggest that shift current is best understood not as a single material coefficient, but as a broad geometric transport phenomenon whose quantitative realization depends on symmetry, band topology, optical matrix elements, nonequilibrium dynamics, and sample geometry.