---
title: Ferrovalley Phase in 2D Materials
url: https://www.emergentmind.com/topics/ferrovalley-phase
type: topic
---

# Ferrovalley Phase in 2D Materials

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 \(\Delta E_v \equiv E_K-E_{K'}\) [2111.07110]. 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 [1706.06374] [2304.12002] [2409.04739] [2508.15909].

## 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 [2111.07110]. 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 \(\Delta_v=E_K-E_{K'}\) [2409.04739].

The symmetry logic is recurrent across the literature. One formulation states that a nonzero valley polarization requires both \(\mathcal{T}\) and \(\mathcal{P}\) to be spontaneously broken, whereas ordinary valley materials may break \(\mathcal{P}\) but preserve \(\mathcal{T}\), and conventional antiferromagnets often preserve a combined \(\mathcal{P}\mathcal{T}\) symmetry and therefore show no net valley polarization [2411.05518]. In monolayer GdI\(_2\), the condition is phrased equivalently as the simultaneous breaking of time-reversal symmetry and inversion or horizontal mirror symmetry, yielding \(\Delta E_v\neq 0\) [2403.01070].

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 \(P_s\) rather than magnetism, so the ferrovalley order is ferroelectric-driven and electrically switchable [1706.06374]. 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 [2304.12002].

## 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 \(3d\) levels together with the breaking of inversion and the horizontal mirror \(\sigma_h\) by the Cl–Fe–F geometry. The low-energy valley Bloch states are approximately linear combinations of \(d_{x^2-y^2}\mp i d_{xy}\), carrying \(L_z=\pm 2\hbar\), or pure \(d_{z^2}\), carrying \(L_z=0\). SOC then produces an effective first-order term \(H_{\mathrm{SOC}}^0=\alpha L_z S_z\), with \(\Delta E_v \approx 4\alpha S_z\) when the valley is dominated by the \((d_{x^2-y^2},d_{xy})\) manifold, whereas \(d_{z^2}\)-dominated valleys show \(\Delta E_v\approx 0\) to first order [2111.07110]. The same orbital logic reappears in 2H-OsBr\(_2\), where projection onto \(|\psi_v^\tau\rangle=(d_{xy}+i\tau d_{x^2-y^2})\otimes|\downarrow\rangle\) yields \(\Delta E_v \simeq 4\alpha\) in the ferromagnetic \(z\)-axis ground state [2502.18190].

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\(_2\) the instability appears when \(U>U_c\) with \(U_c\simeq 2.38\) eV for \(U'=0\), and more generally \(U>U_c+1.4U'\) [1712.07157].

A distinct nonmagnetic mechanism arises from ferroelectric distortion. In monolayer GeSe, the paraelectric phase has four degenerate valleys \(\pm V_x\) and \(\pm V_y\), but ferroelectric distortion along \(x\) lowers the symmetry and yields \(E(V_x)\neq E(V_y)\), with a valley order parameter \(\Delta E_v \equiv E(V_y)-E(V_x)\propto P_s\). Microscopically, \(V_x\) is built from \(p_x\)-dominated conduction and valence states, whereas \(V_y\) is \(p_y\)-dominated, so the ferroelectric displacement changes their relative energies [1706.06374].

Several recent works identify symmetry-lowering by sliding as a general route. In BSFV, lateral translation breaks \(C_{3z}\) or \(C_{4z}\) symmetries that connect valleys, inducing \(E(\boldsymbol{k}_i)\neq E(\boldsymbol{k}_j)\) without SOC-driven magnetic splitting [2304.12002]. In stacked bilayer altermagnets such as Fe\(_2\)MX\(_4\), interlayer sliding breaks the symmetries that exchange \(X\) and \(Y\), producing a valley polarization \(VP\equiv \Delta_Y-\Delta_X\neq 0\), captured analytically by \(VP=2\sigma t'[\cos(\pi A_y)-\cos(\pi A_x)]\) [2410.03155].

The requirement of out-of-plane magnetization is not absolute. In altermagnetic semiconductors Nb\(_2\)Se\(_2\)O and Nb\(_2\)SeTeO, in-plane magnetization plus SOC generates valley polarization through magnetovalley coupling, with \(\Delta E_v \propto \lambda^2/E_g\) in perturbation theory [2411.05518]. In single-layer W\(_3\)Cl\(_8\), in-plane ferro-valleytricity is linked to Y-type \(120^\circ\) 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 [2409.04739].

## 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 \(C=0\). In Janus FeClF under perpendicular magnetic anisotropy, increasing \(U\) drives the sequence
\[
\mathrm{FV}\rightarrow \mathrm{HVM}\rightarrow \mathrm{QAH}\rightarrow \mathrm{HVM}\rightarrow \mathrm{FV},
\]
where the initial and final FV states are trivial insulators, the HVM states are half-valley-metals, and the intermediate QAH state has \(C=1\) and a chiral edge state [2111.07110]. The corresponding Chern number is defined by
\[
C=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega(\mathbf{k})\,d^2k.
\]

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 \(100\%\) valley polarized [2111.07110]. In Janus VSiGeN\(_4\), the VQAH state at zero strain is separated from ferrovalley semiconductor regimes by HVM critical points at \(\varepsilon_c{}_1=-0.32\%\) and \(\varepsilon_c{}_2=+0.10\%\), where the metallic valley hosts a Dirac cone with Fermi velocity \(\sim 3.6\times 10^5\) m/s [2206.00892].

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\(_2\), the FVI phase occurs for \(1.75\) eV \(<U<1.95\) eV and again for \(U>1.95\) eV. In that regime the Berry curvatures at \(K\) and \(-K\) have opposite signs, with valley Chern numbers \(C_K=+1\) and \(C_{-K}=-1\), so \(C=0\) but the valley Hall conductivity is quantized as \(\sigma_{xy}^v=2e^2/h\) [2203.05225].

Multiple valley-related topological phase transitions have been reported in several Janus systems. In VSiXN\(_4\) monolayers, built-in electric field and strain induce the sequence valley semiconductor \(\rightarrow\) valley-half-semimetal \(\rightarrow\) valley quantum anomalous Hall insulator \(\rightarrow\) valley-half-semimetal \(\rightarrow\) valley semiconductor (or valley-metal), with the mechanism identified as band inversion between \(d_{xy}/d_{x^2-y^2}\) and \(d_{z^2}\) orbitals at \(K\) and \(K'\) [2305.13670]. In FeO\(_2\)SiGeN\(_2\), correlation strength similarly drives a ferrovalley \(\rightarrow\) half-valley-metal \(\rightarrow\) QAVH \(\rightarrow\) half-valley-metal \(\rightarrow\) ferrovalley sequence, with critical values \(U_1\approx 1.78\) eV and \(U_2\approx 1.92\) eV at zero strain [2410.15786].

A recurrent topological signature is sign-reversible Berry curvature. FeClF, OsBr\(_2\), and VSiGeN\(_4\) all exhibit Berry-curvature sign changes as \(U\) or strain drives sequential valley band inversions, so the sign pattern of \(\Omega\) at the valleys can reverse even when the total phase remains topologically trivial [2111.07110] [2203.05225] [2206.00892].

## 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\(_4\), VSiXN\(_4\), XYZH | Spontaneous \(K/K'\) splitting; correlation- or strain-driven FV/HVM/QAH transitions [2111.07110] [2206.00892] [2305.13670] [2502.21135] |
| Rare-earth and halide monolayers | 2H-RI\(_2\), 2H-OsBr\(_2\), GdI\(_2\) | Valley splittings from \(15\) to \(143\) meV in RI\(_2\); \(175.49\) meV in 2H-OsBr\(_2\); \(155.5\) meV in monolayer GdI\(_2\) [2206.14626] [2502.18190] [2403.01070] |
| Ferroelectric and stacking-driven systems | GeSe, RhCl\(_3\), InI, bilayer OsBr\(_2\), bilayer GdI\(_2\) | Ferroelectric-driven valley order, BSFV without broken \(\mathcal{T}\), sliding-switchable valley polarization [1706.06374] [2304.12002] [2502.18190] [2403.01070] |
| Altermagnetic and correlated systems | Nb\(_2\)Se\(_2\)O, Nb\(_2\)SeTeO, Fe\(_2\)MX\(_4\), V\(_2\)OSSe, rhombohedral hexalayer graphene | Piezovalley and magnetovalley coupling, sliding-induced ferrovalley, antiferromagnetic half-metallicity, orbital ferrovalley pocket near zero field [2411.05518] [2410.03155] [2509.20687] [2508.15909] |

Among ferromagnetic monolayers, several quantitative benchmarks are notable. FeClF at \(U=2.5\) eV exhibits a conduction-band valley splitting of about \(109\) meV [2111.07110]. In Janus XYZH monolayers, HSE06+\(U\)+SOC gives \(\Delta E=46.3\) meV for ScBrSH, \(51.4\) meV for YBrSH, and \(52.9\) meV for LaBrSH, with valley splittings across the family of about \(40\)–\(75\) meV [2502.21135]. The 2H-RI\(_2\) series spans \(15\)–\(143\) meV, with GdI\(_2\) reaching the largest reported splitting of \(143\) meV in that family [2206.14626].

Bilayer and multilayer platforms extend the phenomenon beyond monolayer ferromagnets. Bilayer GdI\(_2\) has \(\Delta E_v=0\) in AA stacking, \(\Delta E_v\approx +158\) meV in AB stacking, and \(\Delta E_v\approx -158\) meV in BA stacking, with interlayer sliding simultaneously controlling ferroelectric polarization, interlayer magnetism, and valley polarization [2403.01070]. In bilayer OsBr\(_2\), certain slid stackings develop out-of-plane ferroelectric polarization and switchable valley polarization, while the 1T bilayer exhibits tri-state valley polarization at three \(R\)-point valleys [2502.18190]. In rhombohedral hexalayer graphene, a ferrovalley state occupies a small triangular pocket around zero externally applied field in the \(n\)–\(E\) phase diagram and is diagnosed by butterfly-shaped hysteresis in \(R_{xy}(E)\) [2508.15909].

## 5. Control parameters and switching pathways

Magnetization reversal is the most direct switching channel in magnetic ferrovalley systems. In FeClF, reversing \(S_z\rightarrow -S_z\) flips the sign of \(\Delta E_v\) and the carrier spin polarization [2111.07110]. The same reversal of valley sign upon flipping the magnetization is reported for OsBr\(_2\), Janus XYZH monolayers, VSiGeN\(_4\), and FeO\(_2\)SiGeN\(_2\) [2203.05225] [2502.21135] [2206.00892] [2410.15786].

Electronic correlation \(U\) is a major internal tuning parameter in Fe- and Os-based systems. In FeClF, the orbital character of conduction and valence valleys swaps as \(U\) is tuned: for \(U\lesssim 1.4\) eV the ferrovalley splitting resides in the valence band, whereas for \(U\gtrsim 1.5\) eV it resides in the conduction band [2111.07110]. In OsBr\(_2\), \(U\) drives three gap closures and associated band inversions between \(d_{xy}/d_{x^2-y^2}\) and \(d_{z^2}\) states [2203.05225]. This suggests that ferrovalley order in correlated \(d\)-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 \(\varepsilon_{c1}\approx 5.25\%\), a QAH state for \(5.25\%<\varepsilon<5.48\%\), and a return to \(C=0\) beyond \(\varepsilon_{c2}\approx 5.48\%\) [2502.21135]. In VSiGeN\(_4\), small compressive or tensile strain moves the system between VQAH, HVM, and ferrovalley semiconductor states [2206.00892]. In rare-earth iodides, biaxial strain tunes \(\Delta E_v\) by altering the \(d\)-band dispersion and crystal-field splitting [2206.14626]. In altermagnets Nb\(_2\)Se\(_2\)O and Nb\(_2\)SeTeO, uniaxial strain generates a piezovalley effect even without SOC, with \(\Delta E_v(\varepsilon)=D\varepsilon\) and values up to \(264.08\) meV in Nb\(_2\)Se\(_2\)O at \(\varepsilon_x=-3.44\%\) [2411.05518].

Electric-field control is explicit in ferroelectric or multiferroic realizations. In GeSe, electric fields switch the in-plane ferroelectric polarization and thereby select whether \(V_x\) or \(V_y\) is lower in energy [1706.06374]. In W\(_3\)Cl\(_8\), an external \(E_z\) couples to \(P_z\), modulates the Cl-layer buckling and Rashba SOC, and can reverse the sign of \(\Delta_v\) through ferroelectric switching and the associated change in vector-spin chirality [2409.04739]. In rhombohedral hexalayer graphene, small perpendicular electric fields of order \(\pm 6\) mV nm\(^{-1}\) switch the sign of the ferrovalley order parameter and orbital magnetization inside the triangular ferrovalley pocket [2508.15909].

Interlayer sliding provides a structurally nonvolatile control route. RhCl\(_3\) and InI bilayers exhibit stacking-dependent valley splittings of \(39\) meV and \(326\) meV, respectively, with switching barriers of order \(13\) meV/f.u. in RhCl\(_3\) and \(25\) meV/f.u. in InI [2304.12002]. Bilayer GdI\(_2\) and bilayer OsBr\(_2\) 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 [2403.01070] [2502.18190] [2410.03155].

Practical stability depends on anisotropy and magnetic ordering scales. In FeClF, the magnetocrystalline anisotropy energy is \(MAE=E_{\mathrm{in\ plane}}-E_{\mathrm{out\ of\ plane}}\), and only the \(U\) window with \(MAE>0\) supports perpendicular magnetic anisotropy and the full FV \(\rightarrow\) HVM \(\rightarrow\) QAH sequence; the nearest-neighbor exchange \(J\) and the Curie temperature fall sharply with increasing \(U\), from \(J\approx 25.3\) meV and \(T_C\approx 311\) K at \(U=1.5\) eV to \(J\approx 4.4\) meV and \(T_C\approx 63\) K at \(U=2.5\) eV [2111.07110].

## 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 \(\sigma^{AH}\approx (e^2/\hbar)\Omega_0 n_{\mathrm{hole}}\) with \(\Omega_0\simeq 20\) \(\text{\AA}^2\) from a two-band \(k\cdot p\) model [1712.07157]. In Janus XYZH monolayers, Berry-curvature peaks of opposite sign occur at \(K\) and \(K'\), with ScBrSH giving \(\Omega(K)\simeq +37.6\) \(\text{\AA}^2\) and \(\Omega(K')\simeq -32.5\) \(\text{\AA}^2\), leading to an anomalous valley Hall effect [2502.21135].

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 \(C=0\), even though they can possess strong valley splitting and sizable Berry curvature. FeClF has trivial FV phases flanking a narrow \(C=1\) QAH window [2111.07110], and OsBr\(_2\) exhibits an FVI with quantized valley Hall response but zero total Chern number [2203.05225].

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 \(0.53\) eV in the \(P_x\) phase, symmetry enforces \(\Omega\approx 0\) at all valleys, so there is no intrinsic anomalous valley Hall effect [1706.06374].

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\(_4\), strain-driven sign reversal of Berry curvature flips which valley couples to \(\sigma^+\) or \(\sigma^-\) light [2206.00892]. In orthorhombic GeSe, by contrast, the optical selection rule is linear rather than circular: \(V_x\) couples exclusively to \(x\)-polarized light and \(V_y\) to \(y\)-polarized light, enabling an electrically tunable polarizer [1706.06374]. In stacked bilayer altermagnets Fe\(_2\)MX\(_4\), the reported linear optical dichroism is independent of SOC and is tied to the symmetry distinction between the \(X\) and \(Y\) valleys [2410.03155].

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\(_3\)Cl\(_8\), piezovalley and magnetovalley coupling in altermagnets, and an orbital-magnetic ferrovalley state in rhombohedral hexalayer graphene [1706.06374] [2304.12002] [2409.04739] [2411.05518] [2508.15909].

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.

Source: https://www.emergentmind.com/topics/ferrovalley-phase