Ferrovalley Phase in 2D Materials

• Ferrovalley phase is a symmetry-broken electronic state where inequivalent valleys exhibit spontaneous energy splitting due to intrinsic ferromagnetism, spin–orbit coupling, or ferroelectric distortion.
• It is observed in diverse 2D platforms including Janus ferromagnets, orthorhombic ferroelectrics, and stacking-controlled systems, with measurable valley splitting and Berry curvature signatures.
• Control parameters such as magnetic anisotropy, strain, sliding, and electronic correlations enable tuning between different topological phases and valley-polarized states.

The ferrovalley phase is a symmetry-broken electronic state in which inequivalent valleys in momentum space acquire a spontaneous energy splitting, so that the valley degree of freedom becomes intrinsically polarized without continuous external bias. In the conventional formulation for two-dimensional crystals with valleys at $K$ and $K'$, this requires simultaneous breaking of time-reversal symmetry and inversion or mirror symmetry, typically through the coexistence of ferromagnetism and spin–orbit coupling (SOC), and is quantified by a built-in valley splitting $\Delta E_v \equiv E_K-E_{K'}$ (Guo et al., 2021). Subsequent work has broadened the concept to include ferroelectric-driven valley polarization in orthorhombic lattices, stacking-induced nonvolatile valley polarization without broken time-reversal symmetry, in-plane-magnetization-driven ferro-valleytricity, and correlated semimetallic realizations with orbital magnetization (Shen et al., 2017, Yu et al., 2023, Liu et al., 2024, Deng et al., 21 Aug 2025).

1. Definition, symmetry content, and variants

In the standard hexagonal setting, “valley” denotes inequivalent extrema at the Brillouin-zone corners $K$ and $K'$. A ferrovalley material exhibits a spontaneous splitting of these valleys analogous to ferromagnetic spin splitting, but acting on the valley index. In Janus FeClF, this is described as a spontaneous, symmetry-broken lifting of the degeneracy between $K$ and $K'$, driven by intrinsic ferromagnetism and SOC; the resulting state carries both a net magnetic moment and a built-in valley polarization (Guo et al., 2021). A closely related formulation defines ferro-valleytricity as the long-range-ordered phase characterized by spontaneous valley polarization, with valley polarization measured by $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$ and valley splitting by $\Delta_v=E_K-E_{K'}$ (Liu et al., 2024).

The symmetry logic is recurrent across the literature. One formulation states that a nonzero valley polarization requires both $\mathcal{T}$ and $K'$0 to be spontaneously broken, whereas ordinary valley materials may break $K'$1 but preserve $K'$2, and conventional antiferromagnets often preserve a combined $K'$3 symmetry and therefore show no net valley polarization (Xie et al., 2024). In monolayer GdI$K'$4, the condition is phrased equivalently as the simultaneous breaking of time-reversal symmetry and inversion or horizontal mirror symmetry, yielding $K'$5 (Xun et al., 2024).

The term has also acquired broader usages. In orthorhombic group-IV monochalcogenides such as monolayer GeSe, the valley splitting is induced by in-plane ferroelectric polarization $K'$6 rather than magnetism, so the ferrovalley order is ferroelectric-driven and electrically switchable (Shen et al., 2017). In bilayer stacking ferrovalley (BSFV), spontaneous valley polarization is produced by bilayer stacking that breaks the crystalline point symmetries connecting valleys while preserving time-reversal symmetry; in that usage, the “ferro” character refers to nonvolatile spontaneous valley polarization rather than intrinsic magnetization (Yu et al., 2023).

2. Microscopic mechanisms

A central microscopic mechanism is the interplay of exchange splitting, SOC, and orbital angular momentum. In Janus FeClF, the effect originates from exchange-split Fe $K'$7 levels together with the breaking of inversion and the horizontal mirror $K'$8 by the Cl–Fe–F geometry. The low-energy valley Bloch states are approximately linear combinations of $K'$9, carrying $\Delta E_v \equiv E_K-E_{K'}$0, or pure $\Delta E_v \equiv E_K-E_{K'}$1, carrying $\Delta E_v \equiv E_K-E_{K'}$2. SOC then produces an effective first-order term $\Delta E_v \equiv E_K-E_{K'}$3, with $\Delta E_v \equiv E_K-E_{K'}$4 when the valley is dominated by the $\Delta E_v \equiv E_K-E_{K'}$5 manifold, whereas $\Delta E_v \equiv E_K-E_{K'}$6-dominated valleys show $\Delta E_v \equiv E_K-E_{K'}$7 to first order (Guo et al., 2021). The same orbital logic reappears in 2H-OsBr$\Delta E_v \equiv E_K-E_{K'}$8, where projection onto $\Delta E_v \equiv E_K-E_{K'}$9 yields $K$0 in the ferromagnetic $K$1-axis ground state (Wu et al., 25 Feb 2025).

An itinerant mechanism was established for hole-doped monolayer transition-metal dichalcogenides. There the combination of a multi-orbital tight-binding band structure, strong intrinsic SOC, and moderate to large on-site Coulomb repulsion produces an itinerant ferromagnetic state in which all doped holes reside in a single spin-split valley. The exchange field lowers one spin–valley sector and raises the other, so spin polarization and valley polarization occur simultaneously; for WS$K$2 the instability appears when $K$3 with $K$4 eV for $K$5, and more generally $K$6 (Braz et al., 2017).

A distinct nonmagnetic mechanism arises from ferroelectric distortion. In monolayer GeSe, the paraelectric phase has four degenerate valleys $K$7 and $K$8, but ferroelectric distortion along $K$9 lowers the symmetry and yields $K'$0, with a valley order parameter $K'$1. Microscopically, $K'$2 is built from $K'$3-dominated conduction and valence states, whereas $K'$4 is $K'$5-dominated, so the ferroelectric displacement changes their relative energies (Shen et al., 2017).

Several recent works identify symmetry-lowering by sliding as a general route. In BSFV, lateral translation breaks $K'$6 or $K'$7 symmetries that connect valleys, inducing $K'$8 without SOC-driven magnetic splitting (Yu et al., 2023). In stacked bilayer altermagnets such as Fe$K'$9MX$K$0, interlayer sliding breaks the symmetries that exchange $K$1 and $K$2, producing a valley polarization $K$3, captured analytically by $K$4 (Li et al., 2024).

The requirement of out-of-plane magnetization is not absolute. In altermagnetic semiconductors Nb$K$5Se$K$6O and Nb$K$7SeTeO, in-plane magnetization plus SOC generates valley polarization through magnetovalley coupling, with $K$8 in perturbation theory (Xie et al., 2024). In single-layer W$K$9Cl$K'$0, in-plane ferro-valleytricity is linked to Y-type $K'$1 non-collinear magnetism on a triangular lattice, where large valley splitting requires broken out-of-plane mirror symmetry and the absence of time-reversal–mirror joint symmetry (Liu et al., 2024).

3. Phase taxonomy and topological structure

The ferrovalley phase is not a single topological category. In the simplest case it is a trivial insulator with spontaneous valley splitting and total Chern number $K'$2. In Janus FeClF under perpendicular magnetic anisotropy, increasing $K'$3 drives the sequence

$K'$4

where the initial and final FV states are trivial insulators, the HVM states are half-valley-metals, and the intermediate QAH state has $K'$5 and a chiral edge state (Guo et al., 2021). The corresponding Chern number is defined by

$K'$6

The half-valley-metal is the phase boundary most frequently reported in valley-related topological transitions. In FeClF, the gap first closes at one valley and then at the opposite valley; in the HVM regime only one valley remains gapped, and the carriers are intrinsically $K'$7 valley polarized (Guo et al., 2021). In Janus VSiGeN$K'$8, the VQAH state at zero strain is separated from ferrovalley semiconductor regimes by HVM critical points at $K'$9 and $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$0, where the metallic valley hosts a Dirac cone with Fermi velocity $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$1 m/s (Liu et al., 2022).

A second recurring topological pattern is the ferrovalley insulator (FVI), in which both valleys remain gapped and the total Chern number vanishes, but the valley-resolved topology is nontrivial. In monolayer OsBr$P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$2, the FVI phase occurs for $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$3 eV $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$4 eV and again for $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$5 eV. In that regime the Berry curvatures at $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$6 and $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$7 have opposite signs, with valley Chern numbers $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$8 and $P_v \equiv (n_K-n_{K'})/(n_K+n_{K'})$9, so $\Delta_v=E_K-E_{K'}$0 but the valley Hall conductivity is quantized as $\Delta_v=E_K-E_{K'}$1 (Guo et al., 2022).

Multiple valley-related topological phase transitions have been reported in several Janus systems. In VSiXN$\Delta_v=E_K-E_{K'}$2 monolayers, built-in electric field and strain induce the sequence valley semiconductor $\Delta_v=E_K-E_{K'}$3 valley-half-semimetal $\Delta_v=E_K-E_{K'}$4 valley quantum anomalous Hall insulator $\Delta_v=E_K-E_{K'}$5 valley-half-semimetal $\Delta_v=E_K-E_{K'}$6 valley semiconductor (or valley-metal), with the mechanism identified as band inversion between $\Delta_v=E_K-E_{K'}$7 and $\Delta_v=E_K-E_{K'}$8 orbitals at $\Delta_v=E_K-E_{K'}$9 and $\mathcal{T}$0 (Li et al., 2023). In FeO$\mathcal{T}$1SiGeN$\mathcal{T}$2, correlation strength similarly drives a ferrovalley $\mathcal{T}$3 half-valley-metal $\mathcal{T}$4 QAVH $\mathcal{T}$5 half-valley-metal $\mathcal{T}$6 ferrovalley sequence, with critical values $\mathcal{T}$7 eV and $\mathcal{T}$8 eV at zero strain (Tian et al., 2024).

A recurrent topological signature is sign-reversible Berry curvature. FeClF, OsBr$\mathcal{T}$9, and VSiGeN$K'$00 all exhibit Berry-curvature sign changes as $K'$01 or strain drives sequential valley band inversions, so the sign pattern of $K'$02 at the valleys can reverse even when the total phase remains topologically trivial (Guo et al., 2021, Guo et al., 2022, Liu et al., 2022).

4. Representative material platforms

The reported material space for ferrovalley physics is diverse, spanning ferromagnetic semiconductors, ferroelectrics, altermagnets, and correlated graphene systems.

Platform	Representative systems	Reported ferrovalley features
Janus and hexagonal ferromagnets	FeClF, VSiGeN$K'$03, VSiXN$K'$04, XYZH	Spontaneous $K'$05 splitting; correlation- or strain-driven FV/HVM/QAH transitions (Guo et al., 2021, Liu et al., 2022, Li et al., 2023, Tian et al., 28 Feb 2025)
Rare-earth and halide monolayers	2H-RI$K'$06, 2H-OsBr$K'$07, GdI$K'$08	Valley splittings from $K'$09 to $K'$10 meV in RI$K'$11; $K'$12 meV in 2H-OsBr$K'$13; $K'$14 meV in monolayer GdI$K'$15 (Sharan et al., 2022, Wu et al., 25 Feb 2025, Xun et al., 2024)
Ferroelectric and stacking-driven systems	GeSe, RhCl$K'$16, InI, bilayer OsBr$K'$17, bilayer GdI$K'$18	Ferroelectric-driven valley order, BSFV without broken $K'$19, sliding-switchable valley polarization (Shen et al., 2017, Yu et al., 2023, Wu et al., 25 Feb 2025, Xun et al., 2024)
Altermagnetic and correlated systems	Nb$K'$20Se$K'$21O, Nb$K'$22SeTeO, Fe$K'$23MX$K'$24, V$K'$25OSSe, rhombohedral hexalayer graphene	Piezovalley and magnetovalley coupling, sliding-induced ferrovalley, antiferromagnetic half-metallicity, orbital ferrovalley pocket near zero field (Xie et al., 2024, Li et al., 2024, Zhang, 25 Sep 2025, Deng et al., 21 Aug 2025)

Among ferromagnetic monolayers, several quantitative benchmarks are notable. FeClF at $K'$26 eV exhibits a conduction-band valley splitting of about $K'$27 meV (Guo et al., 2021). In Janus XYZH monolayers, HSE06+$K'$28+SOC gives $K'$29 meV for ScBrSH, $K'$30 meV for YBrSH, and $K'$31 meV for LaBrSH, with valley splittings across the family of about $K'$32–$K'$33 meV (Tian et al., 28 Feb 2025). The 2H-RI$K'$34 series spans $K'$35–$K'$36 meV, with GdI$K'$37 reaching the largest reported splitting of $K'$38 meV in that family (Sharan et al., 2022).

Bilayer and multilayer platforms extend the phenomenon beyond monolayer ferromagnets. Bilayer GdI$K'$39 has $K'$40 in AA stacking, $K'$41 meV in AB stacking, and $K'$42 meV in BA stacking, with interlayer sliding simultaneously controlling ferroelectric polarization, interlayer magnetism, and valley polarization (Xun et al., 2024). In bilayer OsBr$K'$43, certain slid stackings develop out-of-plane ferroelectric polarization and switchable valley polarization, while the 1T bilayer exhibits tri-state valley polarization at three $K'$44-point valleys (Wu et al., 25 Feb 2025). In rhombohedral hexalayer graphene, a ferrovalley state occupies a small triangular pocket around zero externally applied field in the $K'$45–$K'$46 phase diagram and is diagnosed by butterfly-shaped hysteresis in $K'$47 (Deng et al., 21 Aug 2025).

5. Control parameters and switching pathways

Magnetization reversal is the most direct switching channel in magnetic ferrovalley systems. In FeClF, reversing $K'$48 flips the sign of $K'$49 and the carrier spin polarization (Guo et al., 2021). The same reversal of valley sign upon flipping the magnetization is reported for OsBr$K'$50, Janus XYZH monolayers, VSiGeN$K'$51, and FeO$K'$52SiGeN$K'$53 (Guo et al., 2022, Tian et al., 28 Feb 2025, Liu et al., 2022, Tian et al., 2024).

Electronic correlation $K'$54 is a major internal tuning parameter in Fe- and Os-based systems. In FeClF, the orbital character of conduction and valence valleys swaps as $K'$55 is tuned: for $K'$56 eV the ferrovalley splitting resides in the valence band, whereas for $K'$57 eV it resides in the conduction band (Guo et al., 2021). In OsBr$K'$58, $K'$59 drives three gap closures and associated band inversions between $K'$60 and $K'$61 states (Guo et al., 2022). This suggests that ferrovalley order in correlated $K'$62-electron monolayers is often controlled as much by orbital reordering as by the absolute magnitude of SOC.

Strain is a second pervasive tuning knob. In Janus XYZH monolayers, biaxial tensile strain reduces the FV band gap and drives a half-valley-metal at $K'$63, a QAH state for $K'$64, and a return to $K'$65 beyond $K'$66 (Tian et al., 28 Feb 2025). In VSiGeN$K'$67, small compressive or tensile strain moves the system between VQAH, HVM, and ferrovalley semiconductor states (Liu et al., 2022). In rare-earth iodides, biaxial strain tunes $K'$68 by altering the $K'$69-band dispersion and crystal-field splitting (Sharan et al., 2022). In altermagnets Nb$K'$70Se$K'$71O and Nb$K'$72SeTeO, uniaxial strain generates a piezovalley effect even without SOC, with $K'$73 and values up to $K'$74 meV in Nb$K'$75Se$K'$76O at $K'$77 (Xie et al., 2024).

Electric-field control is explicit in ferroelectric or multiferroic realizations. In GeSe, electric fields switch the in-plane ferroelectric polarization and thereby select whether $K'$78 or $K'$79 is lower in energy (Shen et al., 2017). In W$K'$80Cl$K'$81, an external $K'$82 couples to $K'$83, modulates the Cl-layer buckling and Rashba SOC, and can reverse the sign of $K'$84 through ferroelectric switching and the associated change in vector-spin chirality (Liu et al., 2024). In rhombohedral hexalayer graphene, small perpendicular electric fields of order $K'$85 mV nm$K'$86 switch the sign of the ferrovalley order parameter and orbital magnetization inside the triangular ferrovalley pocket (Deng et al., 21 Aug 2025).

Interlayer sliding provides a structurally nonvolatile control route. RhCl$K'$87 and InI bilayers exhibit stacking-dependent valley splittings of $K'$88 meV and $K'$89 meV, respectively, with switching barriers of order $K'$90 meV/f.u. in RhCl$K'$91 and $K'$92 meV/f.u. in InI (Yu et al., 2023). Bilayer GdI$K'$93 and bilayer OsBr$K'$94 couple sliding to ferroelectricity and valley polarization, while sliding bilayer altermagnets use the sliding vector itself to switch the sign or magnitude of the valley polarization (Xun et al., 2024, Wu et al., 25 Feb 2025, Li et al., 2024).

Practical stability depends on anisotropy and magnetic ordering scales. In FeClF, the magnetocrystalline anisotropy energy is $K'$95, and only the $K'$96 window with $K'$97 supports perpendicular magnetic anisotropy and the full FV $K'$98 HVM $K'$99 QAH sequence; the nearest-neighbor exchange $\Delta E_v \equiv E_K-E_{K'}$00 and the Curie temperature fall sharply with increasing $\Delta E_v \equiv E_K-E_{K'}$01, from $\Delta E_v \equiv E_K-E_{K'}$02 meV and $\Delta E_v \equiv E_K-E_{K'}$03 K at $\Delta E_v \equiv E_K-E_{K'}$04 eV to $\Delta E_v \equiv E_K-E_{K'}$05 meV and $\Delta E_v \equiv E_K-E_{K'}$06 K at $\Delta E_v \equiv E_K-E_{K'}$07 eV (Guo et al., 2021).

6. Experimental signatures, responses, and recurring misconceptions

The principal transport signature is an anomalous Hall-type response derived from Berry curvature. In hole-doped TMDs, the ferrovalley phase yields anomalous Hall, anomalous spin-Hall, and spin-valley Hall conductivities, and for small hole doping $\Delta E_v \equiv E_K-E_{K'}$08 with $\Delta E_v \equiv E_K-E_{K'}$09 $\Delta E_v \equiv E_K-E_{K'}$10 from a two-band $\Delta E_v \equiv E_K-E_{K'}$11 model (Braz et al., 2017). In Janus XYZH monolayers, Berry-curvature peaks of opposite sign occur at $\Delta E_v \equiv E_K-E_{K'}$12 and $\Delta E_v \equiv E_K-E_{K'}$13, with ScBrSH giving $\Delta E_v \equiv E_K-E_{K'}$14 $\Delta E_v \equiv E_K-E_{K'}$15 and $\Delta E_v \equiv E_K-E_{K'}$16 $\Delta E_v \equiv E_K-E_{K'}$17, leading to an anomalous valley Hall effect (Tian et al., 28 Feb 2025).

A first misconception is that ferrovalley order necessarily implies a quantum anomalous Hall state. The literature does not support this. Trivial FV and FVI phases have $\Delta E_v \equiv E_K-E_{K'}$18, even though they can possess strong valley splitting and sizable Berry curvature. FeClF has trivial FV phases flanking a narrow $\Delta E_v \equiv E_K-E_{K'}$19 QAH window (Guo et al., 2021), and OsBr$\Delta E_v \equiv E_K-E_{K'}$20 exhibits an FVI with quantized valley Hall response but zero total Chern number (Guo et al., 2022).

A second misconception is that valley splitting always implies a nonzero anomalous valley Hall effect. GeSe provides a counterexample: despite a large ferroelectric-driven valley splitting of about $\Delta E_v \equiv E_K-E_{K'}$21 eV in the $\Delta E_v \equiv E_K-E_{K'}$22 phase, symmetry enforces $\Delta E_v \equiv E_K-E_{K'}$23 at all valleys, so there is no intrinsic anomalous valley Hall effect (Shen et al., 2017).

Optical signatures depend strongly on lattice class. In hexagonal systems, valley-selective circular dichroism follows from opposite orbital angular momentum at the two valleys; in VSiGeN$\Delta E_v \equiv E_K-E_{K'}$24, strain-driven sign reversal of Berry curvature flips which valley couples to $\Delta E_v \equiv E_K-E_{K'}$25 or $\Delta E_v \equiv E_K-E_{K'}$26 light (Liu et al., 2022). In orthorhombic GeSe, by contrast, the optical selection rule is linear rather than circular: $\Delta E_v \equiv E_K-E_{K'}$27 couples exclusively to $\Delta E_v \equiv E_K-E_{K'}$28-polarized light and $\Delta E_v \equiv E_K-E_{K'}$29 to $\Delta E_v \equiv E_K-E_{K'}$30-polarized light, enabling an electrically tunable polarizer (Shen et al., 2017). In stacked bilayer altermagnets Fe$\Delta E_v \equiv E_K-E_{K'}$31MX$\Delta E_v \equiv E_K-E_{K'}$32, the reported linear optical dichroism is independent of SOC and is tied to the symmetry distinction between the $\Delta E_v \equiv E_K-E_{K'}$33 and $\Delta E_v \equiv E_K-E_{K'}$34 valleys (Li et al., 2024).

A third misconception is that ferrovalley physics is restricted to out-of-plane ferromagnetic monolayers. Canonical cases do rely on perpendicular magnetization and SOC, but the broader record now includes ferroelectric-driven orthorhombic ferrovalley order, BSFV without broken time-reversal symmetry, in-plane non-collinear ferro-valleytricity in W$\Delta E_v \equiv E_K-E_{K'}$35Cl$\Delta E_v \equiv E_K-E_{K'}$36, piezovalley and magnetovalley coupling in altermagnets, and an orbital-magnetic ferrovalley state in rhombohedral hexalayer graphene (Shen et al., 2017, Yu et al., 2023, Liu et al., 2024, Xie et al., 2024, Deng et al., 21 Aug 2025).

Across these formulations, one recurring design principle is the selective lifting of valley degeneracy by a symmetry-breaking field that couples differently to the relevant valley orbital manifolds. In magnetic Janus monolayers that field is exchange plus SOC; in ferroelectrics it is polar distortion; in stacked bilayers it is sliding-induced symmetry lowering; and in correlated graphene it is a self-consistent valley-polarization order parameter coupled to Berry-curvature-driven orbital magnetization. This suggests that “ferrovalley phase” is best understood not as a single mechanism, but as a family of equilibrium states with spontaneous valley asymmetry and material-specific couplings among spin, orbital, lattice, layer, and topological degrees of freedom.