Magnon Valley Degree of Freedom

• Magnon valley degree of freedom refers to the presence of inequivalent energy valleys in magnets, enabling valley-selective magnon transport via symmetry breaking.
• It leverages topological properties such as Berry curvature and valley Chern numbers to yield protected edge states and observable valley Hall effects.
• Applications include low-power, dissipationless magnonic devices where material engineering enables robust information processing and valley manipulation.

The magnon valley degree of freedom originates from the presence of inequivalent energy extrema (“valleys”) in the Brillouin zone of crystalline magnets, typically at high-symmetry points such as K and K′ in honeycomb or kagome lattices. In analogy to valleytronics in electronic systems, control over magnon occupation in these valleys—by symmetry breaking, substrate engineering, or interfacial manipulation—enables the realization and manipulation of additional pseudospin-like channels in magnonic systems. Valley-polarized magnon phases exhibit nontrivial band topology, Berry curvature, and allow for valley-selective transport phenomena, including the magnon valley Hall effect, protected chiral edge modes, and tunable interface states. The magnon valley degree of freedom is now central to proposals for dissipationless, low-power information processing and non-charge-based magnonic devices.

1. Theoretical Framework for Magnon Valleys

The magnon valley degree of freedom is realized in lattices hosting multiple band extrema at inequivalent points, such as the K and K′ points in 2D hexagonal lattices (e.g., CrI₃, honeycomb bilayers, kagome magnets). The essential ingredients are:

• Spin Hamiltonian and Sublattice/Layer Asymmetry: Broken inversion symmetry is paramount. For example, in monolayer CrI₃, a two-sublattice Heisenberg model with unequal on-site anisotropy $D_\alpha$ on sublattices $a, b$ yields valley-contrasting magnon gaps (Hidalgo-Sacoto et al., 2020). In AB-stacked honeycomb bilayers, electrostatic doping introduces a layer-dependent onsite potential $U$, breaking inversion and enabling valley-resolved physics (Ghader, 2020).
• Effective Low-Energy Dirac Theory: Around the K, K′ points, the magnon Hamiltonian can be expanded to a Dirac form:

$$H_{\text{eff}}^{\pm K}(q) = v_F (q_x\tau_x + q_y\tau_y) + M_\eta\tau_z$$

where $M_\eta$ is a valley-dependent mass term, $v_F$ the magnon “velocity,” and $\tau_i$ Pauli matrices in sublattice or pseudospin space (Xing et al., 2020).

• Berry Curvature and Topological Indices: The magnon bands near valley points host concentrated Berry curvature. Each valley can carry opposite “valley Chern numbers” of $\pm1$ for a gapped Dirac magnon band, despite the total Chern number vanishing in the full Brillouin zone (Hidalgo-Sacoto et al., 2020, Ghader, 2020). In magnetic systems with Dzyaloshinskii-Moriya interaction (DMI), true quantum anomalous Hall phases with nonzero band Chern number are also possible (Xing et al., 2020).

2. Valley Hall and Anomalous Transport Phenomena

Manipulating the valley degree of freedom leads to unique magnonic transport phenomena:

• Magnon Valley Hall Effect (MVHE): When Berry curvature is sharply peaked with opposite signs in different valleys, a thermal gradient drives a transverse magnon flow with valley polarization. The associated magnon Hall conductivity is

$$\kappa_{xy}(T) = -\frac{k_B^2\,T}{\hbar^2}\sum_n \int_{BZ} [c_2(n_B(\epsilon_{n,k})) - \pi^2/3] \Omega_n^z(k)\,d^2k$$

where $n_B$ is the Bose occupancy and $c_2$ a monotonous function (Hidalgo-Sacoto et al., 2020, Xing et al., 2020).

• Single- and Multi-Valley Hall Regimes: In staggered kagome ferromagnets, inversion symmetry breaking opens gaps at K, K′ (yielding pure valley Hall responses with valley-resolved edge states), while DMI breaks valley symmetry and produces a net anomalous magnon Hall effect (Xing et al., 2020).
• Edge and Interface States: Valley-contrasting topology manifests as topologically protected magnon edge or domain-wall states. At domain walls or layer-stacking boundaries with opposite valley mass terms, chiral valley-polarized magnon modes become localized at the interface, giving rise to velocity–valley locking (Ghader, 2020).

3. Material Realizations and Symmetry Engineering

Several materials systems demonstrate the practical realization and control of magnon valleys:

Material System	Valley Gap Mechanism	Key Tunable Parameters	Notable Effects
CrI₃	vdW-substrate	Sublattice-anisotropy, Δ	$D_a-D_b$ (substrate-induced)
Bilayer honeycomb FM	ED (inversion), DMI	$U$ (doping), DMI $D$	Valley Chern phases, domain-wall modes (Ghader, 2020)
Kagome ferromagnet	Exch. staggering, DMI	$J_1\neq J_2$, D	Valley/single-edge Hall, phase transitions (Xing et al., 2020)
Monolayer graphene QH	Valley Zeeman, anisotropy	$E_V$ (substrate), $u_z,u_{xy}$ (field)	Magnon transmission controlled by valley DOF (De et al., 2024)

For instance, in CrI₃-based heterostructures, substrates such as MoTe₂ induce large sublattice-anisotropy and thus sizable valley-dependent magnon gaps, while pristine CrI₃ is valley-degenerate (Hidalgo-Sacoto et al., 2020). In bilayer honeycomb ferromagnets, electrostatic doping breaks inversion, generating Berry curvature of opposite sign at each valley, realized without DMI (Ghader, 2020).

4. Device Physics and Valley Manipulation

The valley degree of freedom enables novel magnonic device architectures:

• Valley Encoding: Magnons can be regarded as carrying a pseudospin—“valley up” (+K) or “valley down” (–K). Selective excitation and detection can encode information in the valley channel, orthogonal to spin-based encoding (Hidalgo-Sacoto et al., 2020).
• Domain Wall Waveguides: At interfaces where valley mass Δ changes sign, one traps counterpropagating, valley-polarized magnon modes—realizing protected magnonic interconnects (Hidalgo-Sacoto et al., 2020, Ghader, 2020).
• Valley Filters and Valves: Locally gating the substrate or layer structure enables regions that permit only one valley to propagate, acting as valley-selective magnon filters (Hidalgo-Sacoto et al., 2020).
• Single-Edge Transport: Differential valley-dependent localization lengths afford the realization of edge-selective magnon flow: at appropriate frequencies, heat/magnon current is localized on one edge only (Xing et al., 2020).
• Magnon Transmission Control in Graphene: In gate-controlled QH junctions, the nature (spin- vs valley-rotated) of the interface determines whether high-energy magnon transmission across $\nu=1|-1|1$ junctions is perfect or suppressed. Tuning the strength of substrate-induced valley Zeeman $E_V$ versus exchange-driven anisotropy using the external field $B_\perp$ induces phase transitions in the valley pseudospin texture at the interface (De et al., 2024).

5. Topological Invariants and Phase Transitions

Magnon valley phases are characterized by topological indices, tunable via material and external parameters:

• Chern and Valley Chern Numbers: For bulk bands,

$$C_{\text{valley}}^{\pm} = \frac{1}{2\pi}\int_{|k-K^{\pm}|<\Lambda} \Omega(k)\,d^2k = \pm1$$

with $\sum_{\text{valleys}} C_{\text{valley}} = 0$ (Hidalgo-Sacoto et al., 2020, Ghader, 2020, Xing et al., 2020).

• Competition of Symmetry Breakings: In bilayer honeycomb FMs, DMI ($D$) and electrostatic doping ($U$) compete: $U=0$ and $D\neq0$ yield nontrivial Chern number but no valley Hall; $D=0$, $U\neq0$ gives quantized valley Chern number with protected domain-wall modes (Ghader, 2020).
• Topological Phase Transitions: In staggered kagome lattices, tuning the ratio $|M_D|/|M_s|$ transitions the system between trivial and magnon TI phases; the boundary is $D = \pm (\Delta J)/\sqrt{3}$ (Xing et al., 2020). In graphene QH systems, the critical field $B_c$ at which interface rotation transitions from spin to valley character is set by $B_c u_{\text{eff}}^0 \simeq E_V$ (De et al., 2024).

6. Detection, Experimental Probes, and Outlook

Detection and utilization of the magnon valley degree of freedom are at the forefront of current research:

• Experimental Detection: Valley-polarized magnon currents are accessed via thermal Hall measurements (nonzero $\kappa_{xy}$), spatially resolved Brillouin light scattering, NV-center magnetometry, and Kerr microscopy (Hidalgo-Sacoto et al., 2020, Xing et al., 2020). In graphene QH devices, the frequency and field dependence of magnon transmission across controlled interfaces directly probes valley pseudospin textures and determines microscopic anisotropy parameters (De et al., 2024).
• Device Prospects: Magnon valleytronics is poised to enable low-dissipation logic elements, Y-junctions, ring interferometers exploiting valley phase, and robust interconnects based on protected valley-polarized edge states (Hidalgo-Sacoto et al., 2020, Xing et al., 2020).
• Broader Implications: The magnon valley degree of freedom bridges magnonics, spintronics, and valleytronics, offering a new channel for encoding, transport, and control in insulating magnets without electronic charge motion.

The systematic manipulation and harnessing of the magnon valley degree of freedom expand the toolkit for topological information transfer and processing in magnetic and quantum materials, with ongoing developments in both theoretical models and experimental realization (Hidalgo-Sacoto et al., 2020, Ghader, 2020, Xing et al., 2020, De et al., 2024).