---
title: Local Valley Magnetic Moments in Dirac Materials
url: https://www.emergentmind.com/topics/local-valley-magnetic-moments
type: topic
---

# Local Valley Magnetic Moments in Dirac 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 [1609.00880][1408.0685][2503.16761]. 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 [2309.00091]. 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 [1309.3814][1408.1200].

## 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 [1208.6069][1408.0685].

- **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 [1609.00880][2309.00091].

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 [2309.00091].

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) [1208.6069][1609.00880][1408.0685].

## 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 [1407.2645][1609.00880][1408.0685].
- **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 [2503.16761]. 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 [2104.01783].

### 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 [2202.05029].

### 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 [1205.3923][2309.00091].

## 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 [1309.3814][1103.5851][1907.04992][1802.01060].
- **Optical probes**: Valley-resolved circular dichroism in photoluminescence or absorption; tunable through pump polarization, gating (e.g. MoS₂, WSe₂), and external fields [1407.2645][1208.6069][1408.0685].
- **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 [1408.0685][2503.16761].
- **Local scanning magnetometry**: SQUID or NV-center probes can image real-space patterns of $m_v(\mathbf{r})$ in designed nanostructures [1408.1200][2309.00091].

### 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 [1609.00880]      |
| Monolayer WSe₂           | $g_v\sim-4$ ($\sim2\,\mu_B$) | Magneto-PL, gating [1407.2645]  |
| Rhombohedral graphene (5L)       | $\sim 30$             | Gate field, valley, $B_z$ [2503.16761] |
| ABA trilayer graphene    | $400$–$800$            | Gate, $B_z$; local STM [2104.01783]  |
| Graphene/hBN (gap $30\,\mathrm{meV}$) | $\sim200$          | Substrate, gate [1408.1200]   |

## 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 [1208.6069][2503.16761]. 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 [1609.00880][1802.01060]. In rare-earth Weyl semimetals, external $B$ tunes valley populations via $f$-$d$ exchange fields [2203.01612].

**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 [1408.0685]. 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 [2309.00091].

## 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 [2210.15753].

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 [2210.15753][2309.00091].

---

**References:**  
[1103.5851], [1205.3923], [1208.6069], [1309.3814], [1407.2645], [1408.0685], [1408.1200], [1609.00880], [1802.01060], [1907.04992], [2104.01783], [2202.05029], [2203.01612], [2210.15753], [2309.00091], [2503.16761].

Source: https://www.emergentmind.com/topics/local-valley-magnetic-moments