Local Valley Magnetic Moments in Dirac Materials

Updated 11 March 2026

Local valley magnetic moments are spatially resolved, valley-indexed dipoles arising from Berry curvature and symmetry breaking in multi-valley materials such as honeycomb and Dirac systems.
They are tunable via gating, magnetic fields, or optical excitation, providing control over valleytronics, magnetoelectric transport, and quantum device functionalities.
Detectable through magnetotransport, optical probes, and scanning magnetometry, these moments reveal key insights into symmetry-driven electronic properties in advanced materials.

A local valley magnetic moment is a spatially resolved, valley-indexed magnetic dipole moment associated with the self-rotation and Berry curvature of electronic Bloch states in solids with multiple valleys, such as honeycomb and Dirac materials. Originating from the electronic structure, symmetry breaking, and (in some systems) the presence of local or adsorbed magnetic moments, local valley magnetic moments play a central role in the fields of valleytronics, magnetoelectric transport, and quantum information devices. Their definition generalizes the global, valley-integrated moment to a local quantity with possible spatial variation even in inversion-symmetric, inhomogeneous environments.

1. Microscopic Theory and Definitions

The orbital magnetic moment of a crystalline Bloch state in band $n$ at momentum $\mathbf{k}$ is given by

$\mathbf{m}_n(\mathbf{k}) = -\frac{e}{2\hbar} \,\mathrm{Im} \left\langle \nabla_{\mathbf{k}} u_{n\mathbf{k}} \left| \times \left[H(\mathbf{k})-E_n(\mathbf{k})\right] \right| \nabla_{\mathbf{k}} u_{n\mathbf{k}} \right\rangle \,,$

where $|u_{n\mathbf{k}}\rangle$ is the cell-periodic part of the Bloch state, $H(\mathbf{k})$ the Bloch Hamiltonian, and $E_n(\mathbf{k})$ its eigenvalue (Chen et al., 2016, Song et al., 2014, Mu et al., 21 Mar 2025). For two-band Dirac systems, one finds a direct relation $m_n(\mathbf{k}) = (e/\hbar) E_n(\mathbf{k}) \Omega_n(\mathbf{k})$ with Berry curvature $\Omega_n(\mathbf{k})$ .

A local valley magnetic moment generalizes this to real space, defined as

$m_v(\mathbf{r}) = \frac{\hbar}{2E} \left[ |\psi_A(\mathbf{r})|^2 - |\psi_B(\mathbf{r})|^2 \right]$

in Dirac models, where $|\psi_A|^2$ , $|\psi_B|^2$ are the sublattice-resolved probability densities and $E$ is the state energy (Huang et al., 2023). The global valley moment is recovered by integrating $m_v(\mathbf{r})$ over all space.

The valley degree, indexed by $\tau = \pm 1$ , enters both the intrinsic moment and the Berry curvature: $m_\tau(\mathbf{k}) = \tau \frac{e \hbar v_F^2 \Delta}{2\left(\Delta^2 + \hbar^2 v_F^2 k^2\right)} \,,\qquad \Omega_\tau(\mathbf{k}) = -\tau \frac{\hbar^2 v_F^2 \Delta}{2\left(\Delta^2 + \hbar^2 v_F^2 k^2\right)^{3/2}}$ for massive Dirac systems (Cai et al., 2013, Grujić et al., 2014).

2. Symmetry, Origin, and Control of Valley Moments

Time-reversal symmetry ( $\mathcal{T}$ ) links valleys and enforces $m_{n}(\mathbf{k}) = -m_{n}(-\mathbf{k})$ ; inversion symmetry ( $\mathcal{P}$ ) gives $m_{n}(\mathbf{k}) = m_{n}(-\mathbf{k})$ . Only when inversion symmetry is broken does a nonzero valley-contrasting magnetic moment arise (Wu et al., 2012, Song et al., 2014).

Intrinsic moments: Emergent in systems with broken inversion symmetry—either structurally (as in monolayer TMDs) or by external gating (bilayer MoS₂, rhombohedral graphene multilayers).
Extrinsic/local moments: Can be introduced by spatially nonuniform adsorption of transition-metal atoms (creating site-localized moments that act as Zeeman fields), by interaction-induced symmetry breaking, or by spatial edge-state inhomogeneity in ribbons and nanostructures (Chen et al., 2016, Huang et al., 2023).

In homogeneous, inversion-symmetric systems, the total valley moment vanishes, but inhomogeneous structures (e.g. nanoribbons, gated domains) support finite $m_v(\mathbf{r})$ with spatial antisymmetry (Huang et al., 2023).

The direction, magnitude, and sign of local valley moments are continuously tunable via electrostatic gates (modulating inversion symmetry), application of an external magnetic field (rotating local moments, tuning Zeeman splitting), or by optical excitation and elliptical pumping (selectively populating valleys—see below) (Wu et al., 2012, Chen et al., 2016, Song et al., 2014).

3. Manifestations in Materials and Device Contexts

Transition Metal Dichalcogenides and Graphene Multilayers

TMDs: Monolayer MoS₂ or WSe₂ exhibit strong spin-orbit and valley-contrasting moments (valley g-factors up to $\sim-4$ for exciton transitions), directly measured via valley-resolved photoluminescence under magnetic fields (Aivazian et al., 2014, Chen et al., 2016, Song et al., 2014).
Rhombohedral graphene multilayers: DFT predicts giant valley moments ( $m_{\max} \sim 30\,\mu_B$ at the band edge), which are gate-tunable by perpendicular electric fields and responsible for nanometer-scale cycloidal current loops (Mu et al., 21 Mar 2025). This leads to large, valley-resolved orbital Hall effects and strongly field-dependent valley polarization.
Trilayer graphene (ABA stacking): Localized quantum-dot states have valley g-factors exceeding 1000, with $m_z$ up to $800\,\mu_B$ per state, controlled by gating and magnetic fields. Sublattice-resolved STM can detect valley Zeeman splittings directly (Ge et al., 2021).

Valley Order and Correlated Systems

In moiré Mott insulators and spin-valley models, local valley magnetic moments appear as generalized SU(4) "pseudospins" that may exhibit spontaneous valley order (ferromagnetic, antiferromagnetic, or liquid-like) depending on microscopic exchange couplings. Functional RG studies show the interplay of SU(4) symmetry, XXZ anisotropy, and valley ordering, with pronounced peaks in valley-valley susceptibilities indicating the emergence of valley magnetic order at critical interaction strengths (Gresista et al., 2022).

Local Moments and Defects

Spatial inhomogeneity, including line defects or adsorbed local moments in graphene, can imprint spin- and valley-dependent scattering. Local moments adjacent to defects can create moderate spin polarizations without significantly degrading valley filtering, while the local valley moments themselves remain robust due to underlying symmetry (Gunlycke et al., 2012, Huang et al., 2023).

4. Detection, Measurement, and Tunable Responses

Local valley moments yield observable signatures in:

Magnetotransport: Asymmetric Landau levels, giant orbital magnetic susceptibility, negative magnetoresistance in gapped valley systems (e.g. ionic-liquid gated graphene), and valley-dependent shifts in quantum oscillations (Cai et al., 2013, Cai et al., 2011, Zhou et al., 2019, Iwasaki et al., 2018).
Optical probes: Valley-resolved circular dichroism in photoluminescence or absorption; tunable through pump polarization, gating (e.g. MoS₂, WSe₂), and external fields (Aivazian et al., 2014, Wu et al., 2012, Song et al., 2014).
Orbital/Valley Hall effect: The transverse anomalous current of orbital moment (without net charge current) as a function of valley polarization is a distinctive probe, especially under in-plane electric fields and optical pumping (Song et al., 2014, Mu et al., 21 Mar 2025).
Local scanning magnetometry: SQUID or NV-center probes can image real-space patterns of $m_v(\mathbf{r})$ in designed nanostructures (Grujić et al., 2014, Huang et al., 2023).

Table: Typical Valley Moments in Selected Systems

System	Max $m_\tau(0)$ ( $\mu_B$ )	Modulation/Control
Monolayer MoS₂ (adsorbed TM)	3–6 [Mn, Fe, etc.]	TM type, magnetic field, gating (Chen et al., 2016)
Monolayer WSe₂	$g_v\sim-4$ ( $\sim2\,\mu_B$ )	Magneto-PL, gating (Aivazian et al., 2014)
Rhombohedral graphene (5L)	$\sim 30$	Gate field, valley, $B_z$ (Mu et al., 21 Mar 2025)
ABA trilayer graphene	$400$–$800$	Gate, $B_z$ ; local STM (Ge et al., 2021)
Graphene/hBN (gap $30\,\mathrm{meV}$ )	$\sim200$	Substrate, gate (Grujić et al., 2014)

5. Tunability and Device Applications

Gating and Symmetry Control: Electric fields in bilayers or multilayers (e.g. MoS₂, rhombohedral graphene) permit continuous switching of both valley moments and Berry curvature, enabling in situ valley-state manipulation (Wu et al., 2012, Mu et al., 21 Mar 2025). The strength and sign of local valley moments can be continuously tuned, and the effect is absent in centrosymmetric pristine stacked systems.

Magnetic Field: The valley splitting $\Delta E_v$ in the band edge is linear in $B_z$ and proportional to the valley moment; when local or proximity-induced moments are present, valley splitting can be enhanced and sign-reversed by rotating the magnetization vector, demonstrated in TM-adsorbed TMDs (Chen et al., 2016, Iwasaki et al., 2018). In rare-earth Weyl semimetals, external $B$ tunes valley populations via $f$ - $d$ exchange fields (Kapon et al., 2022).

Optical Excitation: Population imbalances and hence valley moments are efficiently generated via elliptical or circular pumping; the degree of polarization determines the chemical potential shifts $\mu_\tau$ and thus the local valley magnetization (Song et al., 2014). Orbital-moment Hall effects can be induced purely by tuning optical parameters.

Local Valve and Qubit Architectures: Patterned local gates or magnetic domains in nanoribbons exploit spatially varying $m_v(\mathbf{r})$ for valley filtering, waveguiding, and logic. Valley qubits and reconfigurable logic gates can be realized by dynamically controlling these local moments using electric or magnetic fields (Huang et al., 2023).

6. Extensions: Multipolar Moments and Beyond

Recent theory has generalized the notion of local valley magnetic moment to magnetic multipole moments. In systems with broken inversion or time-reversal symmetry (e.g. black phosphorus under $E_z$ ), higher-order moments such as valley-contrasting octupoles can be induced by applied currents, generating edge or corner accumulation of multipole densities—effectively translating valley physics into spatially patterned higher-rank magnetic textures (Tahir et al., 2022).

This opens avenues for "valley-magnetoelectric multipole" devices, where local valley and orbital multipole moments are exploited for nontrivial transport and memory functionalities, robust to the absence of global inversion symmetry breaking (Tahir et al., 2022, Huang et al., 2023).

References:

(Cai et al., 2011, Gunlycke et al., 2012, Wu et al., 2012, Cai et al., 2013, Aivazian et al., 2014, Song et al., 2014, Grujić et al., 2014, Chen et al., 2016, Iwasaki et al., 2018, Zhou et al., 2019, Ge et al., 2021, Gresista et al., 2022, Kapon et al., 2022, Tahir et al., 2022, Huang et al., 2023, Mu et al., 21 Mar 2025).