Ferrovalley–Ferroelectric Coupling

• Ferrovalley–ferroelectric coupling is a phenomenon where switchable ferroelectric polarization modulates valley polarization, making the valley degree of freedom electrically addressable.
• The mechanism involves polarization-induced spin–orbit coupling and symmetry breaking that reversibly alters Berry curvature, optical selection rules, and valley splitting.
• Various strategies, including charge ordering, sliding ferroelectricity, and multiferroic interactions, enable tunable valley responses with implications for spintronics and topological devices.

Ferrovalley-ferroelectric coupling denotes a class of symmetry-governed phenomena in which a switchable ferroelectric polarization generates, modulates, or reverses spontaneous valley polarization, so that the valley degree of freedom becomes electrically addressable rather than being controlled only by magnetic-field-driven spin reversal. Within the recent literature, the terminology includes “ferroelectrovalley coupling,” “ferroelectric-ferrovalley (FE-FV) coupling,” and, in some bilayer and oxide contexts, “ferroelectric-valley coupling,” but the common content is a nonvolatile linkage between polar order and inequivalent band extrema at time-reversal-related valleys (Tong et al., 2016, Bhardwaj et al., 23 May 2025, Zhao et al., 2024). The subject emerged from the earlier ferrovalley concept—defined as spontaneous valley polarization analogous to ferroelectric and ferromagnetic order—yet has developed into a broader multiferroic framework in which inversion breaking, spin–orbit coupling (SOC), layer pseudospin, magnetic order, charge ordering, and interlayer sliding can all participate in electrically switchable valley physics (Tong et al., 2016, Yamauchi et al., 2015).

1. Conceptual framework and nomenclature

The modern discussion begins with the definition of a ferrovalley material as a system with spontaneous, intrinsic polarization of carriers into one valley over the other, without requiring an external field, optical pumping, or doping to sustain it (Tong et al., 2016). In that formulation, the microscopic origin is the coexistence of SOC and intrinsic exchange interaction, exemplified by monolayer $2H$-VSe$_2$, and the principal signatures are chirality-dependent optical gaps, Berry-curvature asymmetry, and anomalous valley Hall response. That work explicitly did not derive a direct microscopic ferrovalley–ferroelectric coupling; possible coupling to ferroelectricity was described only as a qualitative outlook (Tong et al., 2016).

Subsequent literature shifted the emphasis from spontaneous valley order alone to electrically switchable valley order. In oxide heterostructures, ferroelectricity was proposed as a non-volatile handle to manipulate valley-contrasting spin polarization, first in BiAlO$_3$/BiIrO$_3$, where ferroelectric offcentering converts a hidden layer-resolved spin texture into a net valley-dependent spin polarization (Yamauchi et al., 2015). A related oxide programme showed that ferroelectric distortion can simultaneously induce spin-valley coupling and drive a topological transition in [111]-grown honeycomb bilayers (Yamauchi et al., 2016). More recent two-dimensional work introduced the explicit language of “ferroelectrovalley” and “FE-FV coupling,” in which ferroelectric polarization itself is the control variable for valley switching (Zhao et al., 2024, Bhardwaj et al., 23 May 2025).

A key conceptual distinction now separates at least three regimes. In conventional ferrovalley systems, valley polarization is intrinsic and usually tied to broken time-reversal symmetry. In ferroelectrovalley systems, valley polarity is reversed by ferroelectric switching, often without reversing spin orientation. In multiferroic FE-valley systems, polarization, magnetism, layer character, and valley polarization can be mutually entangled, so that electric and magnetic switching may become functionally equivalent for the low-energy valley state (Shen et al., 27 Jan 2026).

2. Symmetry principles and microscopic mechanisms

The most explicit direct mechanism for ferroelectrovalley coupling is the polarization-induced SOC term proposed for a $1T$-phase rare-earth halide. In the presence of an electric dipole moment $\vec d$, the SOC is written as

$$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$

and, when the spin is locked out of plane and the dipole is in plane, this reduces to

$H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$

Because the term is odd in $\vec k$, an in-plane polarization breaks the valley degeneracy once inversion symmetry is lost. If $d_x\neq 0$ and $_2$0, the splitting occurs along $_2$1; if $_2$2 and $_2$3, the splitting occurs along $_2$4. Reversing $_2$5 by $_2$6 reverses the sign of the SOC term and therefore switches the valley polarization (Bhardwaj et al., 23 May 2025). In this formulation, valley polarization is a secondary effect of ferroelectric polarization rather than a pre-existing property of an intrinsically acentric lattice.

Oxide heterostructures realize a different but closely related mechanism. For the nonpolar BiAlO$_2$7/BiIrO$_2$8 bilayer, the effective model is

$_2$9

while ferroelectric offcentering $_3$0 adds

$_3$1

The coefficients $_3$2, $_3$3, $_3$4, and $_3$5 are all proportional to $_3$6, so they reverse sign when polarization is reversed. The associated valley spin polarization is

$_3$7

This makes the dependence on ferroelectric distortion explicit: when $_3$8, the net valley spin polarization vanishes even though hidden layer-resolved spin polarization can remain (Yamauchi et al., 2015).

A further oxide generalization couples ferroelectricity, spin–valley physics, and topology. In a [111]-grown honeycomb bilayer, the centrosymmetric low-energy Hamiltonian

$_3$9

acquires a polar-distortion term $_3$0 with

$_3$1

Here the ferroelectric order parameter directly controls the valley-dependent spin texture, while the same structural distortion can drive a quantum spin-Hall to trivial-insulator transition (Yamauchi et al., 2016).

Not all coupling mechanisms are SOC terms linear in polarization. In ferroelectrovalley kagome lattices, the crucial symmetry statement is that $_3$2 rotation can replace time reversal for operating valley index. In that setting, ferroelectric switching is equivalent to a structural transformation that swaps the $_3$3 valleys (Zhao et al., 2024). In triangular non-collinear magnets with in-plane magnetization, the decisive condition is different: mirror symmetry is required for remarkable valley polarization, whereas time-reversal–mirror joint symmetry must be excluded; electric-field control proceeds through ferroelectricity-induced changes in magnetic chirality or offset angle rather than through a simple scalar dipole term (Liu et al., 2024).

3. Charge ordering, polar distortions, and single-layer realizations

The clearest direct FE-FV realization by charge ordering is the $_3$4 Gd-substituted $_3$5-EuCl$_3$6 monolayer. Pristine $_3$7-EuCl$_3$8 is inversion symmetric and valley degenerate, whereas the $_3$9 monolayer lacks inversion symmetry and exhibits a valley splitting of about $1T$0 meV; the $1T$1 phase is lower in energy than the $1T$2 phase by $1T$3 meV/formula unit, and the exfoliation energy of monolayer EuCl$1T$4 is $1T$5 meV/$1T$6 (Bhardwaj et al., 23 May 2025). Gd substitution in a $1T$7 supercell introduces an extra electron and produces bond-centered charge ordering (BCO), which shifts the Gd atoms away from the centers of Eu hexagons and breaks the parent $1T$8 symmetry. Two ordered structures appear: an AFE phase with space group $1T$9, where alternate 1D chains shift in opposite directions and inversion symmetry is retained, and an FE phase with space group $\vec d$0, where all Gd atoms shift in the same direction and inversion symmetry is lost. The formation energy is $\vec d$1 eV under Cl-rich conditions; the FE structure has a spontaneous polarization of $\vec d$2 pC/m; the AFE phase is lower in energy than the FE phase by $\vec d$3 meV per formula unit of Eu$\vec d$4GdCl$\vec d$5; and the FE-to-AFE barrier from SS-NEB is $\vec d$6 meV per formula unit. Both phases are dynamically stable. In the unstrained FE phase the valence-band maximum remains at $\vec d$7, but under $\vec d$8 biaxial tensile strain the VBM shifts toward $\vec d$9, the valley splitting becomes about $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$0 meV, and the Berry curvature peaks are approximately $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$1 and $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$2 along the two time-reversed paths. Polarization reversal interchanges the valley ordering and swaps the Berry-curvature distribution (Bhardwaj et al., 23 May 2025).

The same paper extends the mechanism to ferroelectric $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$3, where the extra electron from Li intercalation induces site-centered charge ordering, orbital ordering, and an asymmetric Jahn–Teller distortion at alternate Cr sites. The resulting polar structure has a valley splitting of about $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$4 meV between $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$5 and $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$6, again switchable by reversing the in-plane polarization (Bhardwaj et al., 23 May 2025).

Other monolayer and quasi-monolayer systems realize related but not identical versions of the coupling. In orthorhombic GeSe, the in-plane ferroelectric states $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$7 and $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$8 lift the degeneracy between valleys $$H_{soc} = \lambda(\vec{d}\times \vec{k})\cdot \vec{S},$$9 and $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$0, with spontaneous polarization $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$1. In the $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$2 state, the global band gap is at $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$3 with about $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$4 eV, while the $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$5 gap is about $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$6 eV; the situation reverses in the $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$7 state. The coupling is optically distinctive because $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$8 and $H_{soc}=\lambda_z(d_xk_y-d_yk_x)S_z.$9 couple to $\vec k$0- and $\vec k$1-polarized light, respectively, yielding electrically tunable linear dichroism rather than the circular dichroism familiar from hexagonal ferrovalleys (Shen et al., 2017).

In single-layer Ti$\vec k$2Br$\vec k$3, ferroelectricity is tied to a breathing kagome distortion of Ti$\vec k$4 trimers. The monolayer has spontaneous polarization $\vec k$5 pC/m along $\vec k$6, two degenerate ferroelectric states related by reversing the octahedral distortion, and a CI-NEB switching barrier of $\vec k$7 meV/atom. The isolated top valence band is separated from lower bands by about $\vec k$8 meV and shows a valley splitting of about $\vec k$9 meV; reversing the ferroelectric phase reverses which of $d_x\neq 0$0 is higher in energy (Zhao et al., 2024).

Single-layer W$d_x\neq 0$1Cl$d_x\neq 0$2 demonstrates that electrically reversible valley polarization need not require out-of-plane magnetization. In the $d_x\neq 0$3 breathing-kagome structure, a Y-type antiferromagnetic state with in-plane non-collinear order is lowest in energy, the magnetocrystalline anisotropy energy is $d_x\neq 0$4 meV per unit cell, and the valley splitting reaches about $d_x\neq 0$5 meV at $d_x\neq 0$6. Reversing chirality under the same offset angle changes the valley polarization from $d_x\neq 0$7 meV to $d_x\neq 0$8 meV, and the switching pathway is mediated by the multiferroic linkage between ferroelectricity and magnetic texture (Liu et al., 2024).

4. Sliding ferroelectricity and bilayer multiferroics

A second major design route is sliding ferroelectricity. In bilayer GdI$d_x\neq 0$9, AA stacking has $_2$00 symmetry and a mirror plane about the $_2$01-plane, so no out-of-plane dipole is allowed. Sliding to AB or BA breaks the $_2$02-mirror symmetry, lowers the symmetry to $_2$03, and generates opposite out-of-plane ferroelectric states with vacuum-level discontinuities $_2$04 meV and polarization $_2$05 (Xun et al., 2024). The ferrovalley response is coupled to a stacking-driven magnetic transition: monolayer GdI$_2$06 already has valley splitting $_2$07 meV, but in the bilayer the AA structure is AFM with no net valley polarization, whereas AB and BA are FM and exhibit valley polarization of approximately $_2$08 meV with opposite sign. Berry curvature reflects the reversal: AA gives $_2$09 and $_2$10, AB gives $_2$11 and $_2$12, and BA gives $_2$13 and $_2$14. The interlayer magnetic physics is modeled by

$_2$15

with stacking-dependent super-superexchange through I-$_2$16 states (Xun et al., 2024). This is therefore a symmetry-driven and magnetism-mediated FE-FV coupling rather than a purely electrostatic one.

Bilayer MnPTe$_2$17 provides a more explicitly antiferromagnetic realization. AB and BA stackings are degenerate ferroelectric states related by interlayer sliding, with a spontaneous out-of-plane polarization of about $_2$18 pC/m and an energy barrier of $_2$19 meV (Jiang et al., 28 Jul 2025). The effective four-band spinful $_2$20 description introduces a layer electrostatic potential $_2$21 generated by the dipole, and the spontaneous valley polarization is quantified as

$_2$22

Polarization reversal simultaneously inverts layer-resolved valley polarization and altermagnetic spin splitting. In AB stacking, the $_2$23-valley conduction-band minimum is about $_2$24 meV lower than the $_2$25-valley minimum, while the $_2$26-valley valence-band maximum is about $_2$27 meV above the $_2$28-valley maximum; sliding to BA reverses both splittings while keeping their magnitudes nearly unchanged (Jiang et al., 28 Jul 2025).

Bilayer VS$_2$29 realizes a dual-switch version of the same general theme. Interlayer sliding between AA-1 and AA-2 produces opposite out-of-plane ferroelectric states with polarization magnitude $_2$30 C/m, a small sliding barrier of $_2$31 meV/f.u., and a layer-dependent built-in electrostatic potential of about $_2$32 meV (Shen et al., 27 Jan 2026). The bilayer remains interlayer AFM for all six stackings examined, but the FE-AFM states display a valley polarization

$_2$33

of $_2$34 meV or $_2$35 meV, depending on polarization and spin configuration. The Berry curvature is approximately $_2$36 Bohr$_2$37 at $_2$38 and $_2$39 Bohr$_2$40 at $_2$41 in one state, with signs reversed in the switched state. A notable feature is that ferroelectric reversal and a magnetic-field-induced spin-flip transition are functionally equivalent in modulating valley, layer, and spin indices (Shen et al., 27 Jan 2026).

5. Experimental signatures, optical responses, and switching phenomenology

Berry curvature is the central transport observable across the field. In the original ferrovalley framework, unequal Berry-curvature magnitudes at inequivalent valleys underlie the anomalous valley Hall effect, directly analogous to anomalous Hall transport in ferromagnets (Tong et al., 2016). FE-coupled systems preserve this logic but add nonvolatile electrical reversibility. In Gd-substituted EuCl$_2$42, polarization switching swaps the Berry-curvature pattern between the two time-reversed valley paths (Bhardwaj et al., 23 May 2025). In sliding ferroelectrics such as bilayer GdI$_2$43, MnPTe$_2$44, and VS$_2$45, polarization reversal changes not only the sign of the valley splitting but also the layer selectivity and the Hall response because the Berry curvature is simultaneously valley-contrasting and layer-sensitive (Xun et al., 2024, Jiang et al., 28 Jul 2025, Shen et al., 27 Jan 2026).

Optical selection rules depend strongly on lattice class and coupling mechanism. Orthorhombic GeSe does not follow the circular dichroism paradigm of hexagonal valley systems; instead, the ferrovalley state is manifested as linearly polarized optical selectivity, enabling an electrically tunable polarizer in which the emitted radiation is $_2$46-polarized in the $_2$47 state and $_2$48-polarized in the $_2$49 state (Shen et al., 2017). By contrast, oxide honeycomb bilayers preserve the familiar valley-contrasting spin and circular optical selection rules. In ferroelectric BiAlO$_2$50/BiIrO$_2$51, the valence-band maximum lies at $_2$52 and all valence bands at $_2$53 are spin polarized, with $_2$54; reversing the ferroelectric polarization reverses the sign of the dominant valley-contrasting $_2$55 (Yamauchi et al., 2015). In the topological oxide heterostructure, the polar distortion preserves valley-dependent spin polarization while reversing the Berry curvature and the handedness of the fundamental circular dichroic transition across the topological phase transition (Yamauchi et al., 2016).

Switching phenomenology is not uniform across material classes. A common misconception is that electrical switching necessarily removes magnetism from the problem. In the in-plane-polarized EuCl$_2$56-based mechanism, the SOC-induced valley splitting requires spins aligned along the out-of-plane $_2$57-axis; a magnetic field may therefore still be needed to maintain spin orientation, even though a changing magnetic field is not needed to switch the valleys (Bhardwaj et al., 23 May 2025). A related caveat applies to $_2$58, where the spin easy axis lies in plane parallel or antiparallel to the polar axis, so an external magnetic field is needed to bring the spin out of plane if one seeks the same SOC-induced splitting (Bhardwaj et al., 23 May 2025). By contrast, in bilayer VS$_2$59, electric and magnetic switching are explicitly described as functionally equivalent routes to the same valley-layer-spin state (Shen et al., 27 Jan 2026).

6. Scope, distinctions, and unresolved boundaries

Several boundaries of the concept are now clear. First, not every ferrovalley material is ferroelectrically coupled. Monolayer $_2$60-VSe$_2$61 remains the archetype of spontaneous ferrovalley order from SOC plus intrinsic exchange, but it does not furnish a direct FE-FV coupling theory (Tong et al., 2016). Similarly, the $_2$62-NbX$_2$63/TaX$_2$64 family displays large intrinsic valley splitting and anomalous valley Hall response, but no spontaneous electric dipole moment, ferroelectric switching, or coupling Hamiltonian between ferroelectric order and valley order is established there (Islam et al., 24 Aug 2025).

Second, broken inversion symmetry alone is not equivalent to ferroelectric controllability. In BiAlO$_2$65/BiIrO$_2$66, the paraelectric $_2$67 reference structure is globally nonpolar yet already acentric and hosts hidden layer-resolved spin polarization; only the ferroelectric distortion lowering the symmetry to $_2$68 makes the net valley-contrasting spin polarization electrically switchable (Yamauchi et al., 2015). In the EuCl$_2$69 mechanism, the crucial point is even stronger: inversion symmetry breaking is not built into the lattice from the outset but is induced by the polarization or charge-ordering pattern itself, so the valley polarization is genuinely secondary to ferroelectric order (Bhardwaj et al., 23 May 2025).

Third, the coupling can be direct, symmetry-driven, or magnetism-mediated. The in-plane-dipole SOC term in Gd-substituted EuCl$_2$70 is a direct polarization-induced valley-splitting mechanism (Bhardwaj et al., 23 May 2025). Sliding ferroelectricity in bilayer GdI$_2$71 is primarily symmetry-driven and magnetism-mediated because sliding changes both polarization and interlayer exchange, which then determines valley polarization (Xun et al., 2024). Bilayer MnPTe$_2$72 and VS$_2$73 occupy an intermediate multiferroic regime in which polarization, layer asymmetry, AFM order, and valley response are inseparable at the band-edge level (Jiang et al., 28 Jul 2025, Shen et al., 27 Jan 2026).

Finally, adjacent literature extends the logic beyond conventional $_2$74 valley splitting. Fractional quantum multiferroics based on fractional quantum ferroelectricity and altermagnetism reverse momentum-dependent spin splitting by electric polarization switching, but the controlled quantity there is altermagnetic spin splitting rather than a conventional valley index. This places such systems next to, rather than within, the strict ferrovalley-ferroelectric category (Dong et al., 19 Oct 2025).

Taken together, the current literature defines a coherent design space for electrically programmable valley physics. Charge-order-driven in-plane polarization, sliding ferroelectricity, breathing kagome distortions, orthorhombic ferroelectricity, and multiferroic control of non-collinear chirality all provide routes by which ferroelectric order can create or reverse valley polarization. This suggests that ferrovalley-ferroelectric coupling is best understood not as a single mechanism but as a family of symmetry-linked strategies for converting the valley index into a nonvolatile electrically writable state (Bhardwaj et al., 23 May 2025, Zhao et al., 2024, Xun et al., 2024, Liu et al., 2024).