Topological Valley Transport

• Topological valley transport is defined by valley-contrasting Berry curvature and half-integer valley Chern numbers that enable robust, one-way edge modes immune to backscattering.
• It utilizes engineered domain walls and symmetry-breaking in materials like graphene and TMDs to create chiral, valley-polarized conduction channels with quantized conductance.
• The principles extend across electronics, photonics, phononics, and magnonics, offering pathways for low-dissipation on-chip routing and reconfigurable quantum circuits.

Topological valley transport refers to a class of dissipationless, robust transport phenomena in which the carrier's propagation direction is locked to its valley degree of freedom—an internal quantum number associated with inequivalent extrema (valleys) in the Brillouin zone—by engineered band topology. This paradigm spans electronics, photonics, phononics, and magnonics, relying on Berry curvature, valley Chern number, and symmetry-breaking-induced gaps to produce one-way edge or channel states that are immune to backscattering from smooth disorder and geometry, provided valley mixing is suppressed. The key elements include valley-contrasting Berry curvature, domain wall/channel engineering between bulk regions of differing valley topology, and the emergence of protected, valley-polarized 1D/2D modes. This article provides a comprehensive, technical account of valley transport rooted in recent theoretical, computational, and experimental literature.

1. Theoretical Foundations: Valley–Chern Numbers, Dirac Gaps, and Berry Curvature

Topological valley transport is universally underpinned by Dirac-like band crossings at K, K′ valleys in systems with hexagonal symmetry (graphene, TMDs, photonic and phononic hexagonals). Breaking inversion (and/or mirror) symmetry via mass terms in an effective two-band Dirac Hamiltonian induces valley-contrasting Berry curvature and valley Chern numbers. For a generic 2D Dirac model near valley τ=±1,

$$H_\tau(\mathbf{k}) = v_D(\tau\sigma_x k_x + \sigma_y k_y) + m\,\sigma_z$$

the Berry curvature for the lower band is

$$\Omega_\tau(\mathbf{k}) = -\tau\,\frac{m\,v_D^2}{2(v_D^2|\mathbf{k}|^2 + m^2)^{3/2}}$$

and the valley Chern number is

$C_v(\tau) = \frac{\tau}{2}\,\mathrm{sgn}(m)$

per Dirac cone. Thus, breaking inversion symmetry gaps the Dirac points, generating Berry curvature “hot spots” at K, K′, and assigning half-integer valley Chern numbers (±½) to each valley. Across an interface/domain wall where m changes sign, the total change in C_v per valley is unity, guaranteeing by bulk–boundary correspondence the emergence of a chiral kink (interface) mode at each valley (Lensky et al., 2014, Chen et al., 2020, Lu et al., 2017).

In multilayer systems (AB/BA-stacked bilayer graphene, TMD multilayers, designer plasmonic and photonic crystals), more complex four-band or multi-band models can be reduced to effective two-band Dirac forms at low energies, recovering the same valley-based topology, with details determined by interlayer coupling, stacking order, and local fields (Sui et al., 2015, Pan et al., 16 Nov 2025, Wu et al., 2018).

2. Elementary Manifestations: Valley Hall Effect, Edge States, and Channels

The valley Hall effect (VHE) arises when an applied electric field generates transverse valley currents due to the Berry curvature’s anomalous velocity. In a gapped Dirac material,

$j^v_y = \sigma_{xy}^v E_x$

with

$$\sigma_{xy}^v = \frac{e^2}{\hbar}\sum_\tau \int \frac{d^2k}{2\pi} f(E_\pm(\mathbf{k}))\,\Omega_\tau(\mathbf{k})$$

which remains quantized (per valley) so long as the Fermi level lies in the gap: $\sigma_{xy}^v=\pm(e^2/2h)$ (Lensky et al., 2014).

At domain walls/interfaces with $\Delta C_v = 1$, Jackiw–Rebbi theory guarantees a localized, valley-polarized chiral mode with dispersion $E(k_{||})=v_D \tau k_{||}$ traversing the bulk gap, exponentially confined with penetration depth $\xi=v_D/|m|$ (Chen et al., 2020, Lu et al., 2017). In twisted or sliding bilayer graphene, periodic AB/BA domains separated by AA′ walls generate a network of 1D kink channels hosting valley-locked, topologically protected, ballistic conduction, with experimental quantization of conductance as $G = h/(4e^2)$ per spin per domain wall (Mania et al., 2019, Pan et al., 16 Nov 2025, Rickhaus et al., 2018, Bal et al., 2022). For generic valley Chern transitions, the total number of edge or channel modes is determined by the Chern mismatch per valley (Chen et al., 2020, Bal et al., 2022).

In classical wave systems—photonic, acoustic (phononic), magnonic—the same topology arises, with edge states seen as unidirectional interface modes between domains of opposite valley Chern number. These have been verified experimentally for microwave, THz, and sonic systems, often visualized directly via field-mapping (Chen et al., 2020, Wu et al., 2018, Lu et al., 2017, Zhai et al., 2020, Zhou et al., 19 Sep 2024).

3. Robustness, Nonlocal Response, and Device-Level Metrics

Topologically protected valley transport possesses immunity against backscattering, provided disorder or bends do not couple the well-separated K and K′ valleys (which would require large-momentum transfer). The suppression of backscattering and intervalley scattering underlies nearly dissipationless 1D conduction, as seen in the saturation of two-terminal resistance to quantized values in folded/sliding/twisted BLG (Mania et al., 2019, Pan et al., 16 Nov 2025, Rickhaus et al., 2018).

For 2D “bulk” valley Hall effect, the nonlocal response is quantified via nonlocal resistance $R_{NL}$. For small valley Hall angles $$\theta_v = \arctan(\sigma_{xy}^v / \sigma_{xx})\ll1$$, $R_{NL} \propto (R_L)^3$, where $R_L$ is the local, longitudinal resistance; for large $\theta_v$, $R_{NL}$ saturates and becomes insensitive to $\sigma_{xx}$, permitting robust valley currents over micron scales at room temperature (Beconcini et al., 2016, Sui et al., 2015, Wu et al., 2018). Valley diffusion lengths of $0.5$–$1$ μm have been extracted for monolayer MoS$_2$ and BLG, with valley transport observed up to 300 K (Wu et al., 2018, Sui et al., 2015).

In photonic and acoustic systems, robustness is established by high transmission through sharp bends and disorder, absence of back-reflection and crosstalk at channel intersections, and mode purity under k-space decomposition. Channel confinement, propagation loss, and bandwidth metrics (e.g., $\Gamma_B$, bandwidth up to $1.68$ GHz, loss $\lesssim0.1$ dB per period) have been reported in TVLWs (Chen et al., 2020, He et al., 2018, Lu et al., 2017, Zhou et al., 19 Sep 2024).

4. Extensions, Synthetic Dimensions, Modal Character, and Controversies

Recent work demonstrates that topological valley transport can transcend the specific values of valley Chern number, being dictated largely by the presence of Dirac points and “momentum-hotspots” at K/K′. Even when inversion symmetry is weakly broken or restored (C$_v\to0$), the edge state remains immune to 120$^\circ$ or 150$^\circ$ bends due to k-space matching, a modal character effect (Liu et al., 2023). Thus, valley locking and bending immunity are not always strictly tied to topological invariants but can persist for non-topological edge states with strong valley localization in momentum space.

For metacrystals and synthetic systems, the rigorous topological classification of the valley Hall phase is subtle: true topology (in the homotopical sense) requires (i) the gap to be much narrower than bandwidth, so that Berry curvature is valley-localized and half-integer quantization exact; (ii) inter-valley separation much larger than the width of the wavefunction in k-space (Fan et al., 2022). With large gaps or strong disorder, the valley Chern is only approximate, and robust interface transport must be argued from synthetic Weyl point constructions or momentum-matching rules (Fan et al., 2022, Xu et al., 2019).

Beyond pure edge states, “bulk-by-boundary” mechanisms allow for boundary-induced chiral bulk states (CABSs) in completely topologically trivial media, enforcing valley-polarized transport over an inert bulk, mimicking the valley Hall effect without a nonzero bulk Chern number (Wang et al., 2022).

5. Material Platforms and Experimental Realizations

Electronic systems:

• Bilayer Graphene (BLG): Gate-tunable topological valley Hall effect and kink transport at domain walls and engineered junctions (Sui et al., 2015, Mania et al., 2019, Pan et al., 16 Nov 2025).
• Twisted BLG: Moiré superlattice pattern creates a network of AB/BA domain walls hosting topological QVH channels, with quantized conductance and robust coherence under perpendicular magnetic fields (Rickhaus et al., 2018, Bal et al., 2022).
• Monolayer/Trilayer MoS$_2$: Intrinsic VHE demonstrated by nonlocal response scaling with local resistance cubed, surviving at room T over micron scale (Wu et al., 2018).

Photonics:

• Microwave TVLWs: Hexagonal photonic crystals with tunable width and robust channel intersections, energy concentrators, topological cavities (Chen et al., 2020).
• SOI slabs: VPCs with valley-dependent edge states, routing via microdisk generators, CMOS-compatible, sub-μm-scale (He et al., 2018).
• Designer surface plasmon crystals: Layer pseudospin degree of freedom enables new edge/channel functionalities (Wu et al., 2018).
• THz Kagome structures: Ultrafast valley Hall phase transitions, Z-shaped domain wall transport, positive/negative refraction (Zhou et al., 19 Sep 2024).

Phononics / Acoustics:

• Sonic crystals: First experimental observation of valley Hall states of sound, direct field mapping, reflection immunity at sharp corners (Lu et al., 2017).
• Boundary-induced CABS: Valley Hall-like transport realized without bulk topology (Wang et al., 2022).
• Synthetic Weyl metacrystals: Quantized valley transport via 3D synthetic (spatial+structural parameter) topology (Fan et al., 2022).

Magnonics:

• Bilayer CrBr$_3$ and similar: Magnon valley Hall currents, valley Seebeck effect through symmetry-tuned band structure (Zhai et al., 2020).

Nonlinear/Active Systems:

• Graphene metasurfaces: All-optical switching of valley transport via Kerr effect (Wang et al., 2023).

6. Applications, Device Implications, and Outlook

Valley-resolved, topologically protected channels serve as robust on-chip waveguides, logic interconnects, delay lines, optical routers, isolators, multiplexers, and topological lasers or masers. The channel width-tunability and modal flexibility of TVLWs and their analogs facilitate interfacing with conventional circuit elements, refractive steering, and compact logic designs (Chen et al., 2020, He et al., 2018, Zhou et al., 19 Sep 2024). Photonic routers exploit valley-spin or layer locking to direct signals to selected ports, with demonstrated high directionality and isolation (Lai et al., 2017).

In electronics, long valley diffusion lengths, room-temperature persistence, and ballistic quantization in multi-μm channels challenge conventional limits of interconnect scaling and offer paths toward all-electric low-dissipation logic and memory (Sui et al., 2015, Pan et al., 16 Nov 2025, Wu et al., 2018). Engineering with sliding or twisting provides continuous topological reconfiguration, inexpensively implementable in 2D materials.

Acoustic and magnonic platforms present unique prospects for non-electronic information carriers in robust, reconfigurable circuits, including thermoelectric detection of valley Seebeck effects (Zhai et al., 2020). The “mode-hotspot” design strategy extends robust bending immunity to topologically trivial systems, conceivably broadening the design space for low-scattering channels in arbitrary geometries (Liu et al., 2023).

Rigorous index theory, spectral numerics, and synthetic topological tracking now allow for predictive modeling and bulk-edge-junction correspondences even in complex multi-junction channel networks (Bal et al., 2022).

Valley-based topological transport—spanning quantum, classical, active, and reconfigurable systems—is thus a robust, versatile paradigm for guided, dissipationless channeling of information, with material-dependent performance set by intervalley scattering length, gap size, and the fidelity of symmetry-breaking mechanisms. The ongoing integration of synthetic dimensions, modal engineering, and topological control promises further applications in quantum information processing, on-chip wave manipulation, and tunable valleytronic circuits.