Valleytronics: Fundamentals & Device Applications

• Valleytronics is a field of condensed matter physics that uses distinct energy valleys in 2D materials as information carriers, analogous to spin in spintronics.
• It leverages both global valley polarization and local valley magnetic moments to enable functionalities in memory, logic circuits, and quantum operations.
• Recent advances show that spatial inhomogeneities and edge-induced symmetry breaking in materials like graphene can create innovative valleytronic device architectures.

Valleytronics is a field of condensed matter physics and device engineering that exploits the "valley" degree of freedom—the occupation of inequivalent energy extrema in a material's electronic band structure—as an information carrier, analogous to spin in spintronics. The valleys, which typically appear at high-symmetry points such as K and K′ in the Brillouin zone of 2D hexagonal crystals, act as a binary pseudospin. Manipulating and reading valley occupation enables new paradigms for nonvolatile memory, logic, optoelectronics, and quantum information processing. Recent research has extended valleytronics beyond global valley polarization, revealing local valley moments, device architectures, and new platforms including symmetric materials.

1. Fundamentals of the Valley Degree of Freedom

In materials with multiple energy extrema (valleys) in their band structure, each electronic state can be labeled by a valley index (e.g., τ = ±1 for K and K′). In two-dimensional hexagonal lattices such as monolayer graphene or transition metal dichalcogenides (TMDCs), the valleys occur at the Brillouin zone corners and are protected from mixing by large momentum separation. The valley index emerges as a robust internal quantum number, with physical manifestations in orbital magnetization, Berry curvature, and optical selection rules.

A central quantitative object is the total valley magnetic moment per unit area for state distribution $f_n(k)$ in band $n$: $$\mu_{v} = \frac{e}{\hbar} \sum_{n} \int \frac{d^2k}{(2\pi)^2} f_n(k)\,\Omega_n(k)$$ where $$\Omega_n(k) = i\langle \partial_{k_x}u_{n,k}|\partial_{k_y}u_{n,k}\rangle - (x\leftrightarrow y)$$ is the Berry curvature. In crystals with global inversion symmetry, $\Omega_n(k) = -\Omega_n(-k)$ and $f_n(k)=f_n(-k)$, so $\mu_{v}=0$ globally (Huang et al., 2023).

Valley indices underlie phenomena such as valley Hall effect, valley-selective circular dichroism in TMDCs (Cao et al., 2011), linearly polarized valley selection in group-IV monochalcogenides (Rodin et al., 2015, Dien et al., 16 Jan 2024), and gate/electric field control of valley occupation in layered and bulk materials (Hossain et al., 2020, Gindl et al., 18 Nov 2024).

2. Local Valley Magnetic Moments and Spatially Resolved Valleytronics

Conventional valleytronics assumes that only materials with broken inversion symmetry (e.g., monolayer MoS₂ (Cao et al., 2011)) exhibit nonvanishing global valley magnetic moments. However, recent theoretical work has generalized the concept to spatially inhomogeneous systems, introducing the local valley magnetic moment $m_v(r)$ (Huang et al., 2023). This field:

• Captures space-resolved contributions to magnetization, operationally defined via the local linear response to a probe field,

$$m_v(r) = -\left. \frac{\delta E_{Zeeman}^{valley}}{\delta B_z^{probe}(r)} \right|_{B^{probe}\rightarrow 0}$$

• In Dirac-type materials (e.g., graphene), for a state of valley τ and energy E,

$m_v(r) = -\tau \frac{\rho_A(r) - \rho_B(r)}{2E}$

where $\rho_{A/B}(r) = |\psi_{A/B}(r)|^2$ are sublattice densities.

Critically, in inversion-symmetric but inhomogeneous environments (e.g., a zigzag nanoribbon of graphene), $m_v(r)$ may be sizable locally even though its spatial integral vanishes:

• At the nanoribbon edge terminated by sublattice A, $m_v(y)>0$; at B-terminated edge, $m_v(y)<0$. The profile $m_v(y)$ is locally large and antisymmetric ($m_v(y)=-m_v(-y)$).
• Global inversion and time reversal require the system's spatial average $\int m_v(r) d^2r = 0$, but do not prevent device functionalities based on nonzero $m_v(r)$.

Thus, spatially resolved tactics—using local symmetry breaking or inhomogeneity—enable “local valleytronics” in a broader range of materials, including gapless monolayer graphene.

3. Symmetry, Inhomogeneity, and the Relaxation of Constraints

Global valleytronic indicators such as $\mu_v$ are governed by the interplay between inversion (Inv) and time-reversal (TR) symmetry:

• Inv: $k\rightarrow -k$, $\mu_v\rightarrow \mu_v$
• TR: $\tau\rightarrow -\tau$, $\mu_v\rightarrow -\mu_v$
• Combined: $\mu_v=0$

However, local $m_v(r)$ can remain finite even when the above symmetries are preserved globally, provided there is inhomogeneity or edge-induced sublattice imbalance. This key point permits extension of valleytronic principles to previously excluded materials.

For example, in an inversion-symmetric system with different terminations or inhomogeneities (e.g., graphene nanoribbons), strong local pseudospin imbalance at the boundaries leads to pronounced local valley moments. The spatial profile is dictated by edge sublattice asymmetry and underlying quantum state properties. Under external fields, these local moments can give rise to significant effective valley-dependent energy shifts and response fields, even as the globally integrated response remains symmetry-forbidden.

4. Coupling Local Valley Moments to External Fields: Device Implications

The spatially dependent $m_v(r)$ couples linearly and locally to external magnetic and electric fields, enabling device-scale valley manipulation:

• Magnetic field coupling (local Zeeman effect):

$H_{Zeeman}^{valley} = -\int m_v(r) B_z(r) d^2r$

In a nanoribbon, applying a step-like $B_z(y)$ to part of the width induces a valley-Zeeman splitting localized to a subregion, without generating a global valley magnetization.

• Electric field (valley–orbit interaction):

For a state of momentum $k_x$ and transverse electric field $\epsilon_y(y)$,

$$\Delta E_{valley-orbit} = \int k_x\,\epsilon_y(y)\,m_v(y) dy$$

An antisymmetric field (e.g., sawtooth gate) shifts K- and K′-edge subbands differentially (valley Rashba effect), potentially opening indirect energy gaps.

These interactions open the possibility of constructing devices with spatially precise, valley-selective functionality, such as local valley filters, waveguides, or quantum-dot islands embedded within otherwise symmetric materials.

Table: Coupling of Local Valley Moment to External Fields

Field Type	Coupling Hamiltonian	Device Effect
Magnetic $B_z(r)$	$-\int m_v(r) B_z(r)\,d^2 r$	Local valley Zeeman shift
Electric $\epsilon_y(r)$	$\int k_x\,\epsilon_y(y)\,m_v(y)\,dy$	Valley–orbit splitting, selective band edge shifts

5. Device Architectures and Experimental Probes

By exploiting local valley magnetic moments in inhomogeneous or edge-engineered materials, a wider range of valleytronic device architectures becomes possible. Key outcomes include:

• Material flexibility: Inversion symmetry is no longer a strict constraint. Gapless monolayer graphene, with robust and reproducible growth methods, becomes a practical candidate for valleytronic devices by leveraging edge or spatial inhomogeneity for local $m_v(r)$ generation.
• Spatial control: Nanopatterned magnetic gates or in-plane gate fields ($\epsilon_y(r)$) enable programmable landscape engineering for valley-selective functionalities, such as local filters, valves, or waveguides.
• Readout strategies:
  • Local valley Zeeman shifts can be detected using spatially resolved magneto-optical methods or local transport spectroscopies.
  • Valley–orbit-induced energy differences can be probed via scanning tunneling spectroscopy, which is sensitive to spatially varying sublattice and valley configurations.

Device approaches therefore become compatible with high-density integration, scalable architectures, and multi-functional valleytronic circuits, even without the need for globally broken symmetry or gapped band structures.

6. Broader Context and Outlook

The expansion of valleytronics to the local scale, and to materials with otherwise forbidden global valley responses, alters fundamental device design principles. By marrying the microscopic physics of local valley magnetic moments with nanoscale patterning of fields and materials, it becomes possible to realize memory, logic, and quantum operations in regimes previously considered inapplicable.

These advances dramatically broaden the range of candidate materials and device topologies for next-generation valleytronic systems, provide new functionalities for information encoding and manipulation, and open new avenues for quantum valley pseudospintronics, including valley-based qubits and ultrafast, subwavelength valley logic—paving the way toward highly flexible, scalable information technologies (Huang et al., 2023).