Valley-Hall Photonic Crystals

• Valley-Hall Photonic Crystals are photonic structures that leverage inequivalent Brillouin-zone corners and inversion-symmetry breaking to produce valley-polarized, backscattering-resistant edge modes.
• These crystals employ unit-cell engineering to induce a controlled Dirac gap, yielding opposite Berry curvature and half-integer valley Chern numbers that underpin efficient wave routing and slow-light phenomena.
• Implemented across diverse material platforms—from silicon slabs to microwave ceramics—VPhCs enable applications such as topological routing, chiral excitation, lasing, and biosensing with high transmission fidelity.

Valley-Hall photonic crystals (VPhCs) are photonic crystals in which the inequivalent Brillouin-zone corners $K$ and $K'$ act as a binary valley degree of freedom, and inversion-symmetry breaking gaps the Dirac degeneracy while preserving time-reversal symmetry. In the standard realization, this produces valley-contrasting Berry curvature localized near $K$ and $K'$, opposite valley Chern numbers, and domain-wall edge states that are valley polarized and strongly resistant to backscattering when intervalley mixing is weak. Across microwave, terahertz, and telecommunication implementations, VPhCs have become a central platform for topological routing, chiral excitation, slow light, lasing, nonlinear frequency conversion, and related higher-order or hybrid topological effects (Chen et al., 2016, Shalaev et al., 2017, Chen et al., 2018).

1. Topological framework and valley invariants

The canonical description of a VPhC begins with a hexagonal, honeycomb-like, or triangular-derived lattice whose bulk bands host Dirac cones at $K$ and $K'$. Breaking inversion symmetry opens a gap and yields a massive Dirac description near each valley. In a standard notation used repeatedly in the literature, the effective Hamiltonian is written as

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

or equivalently with $m$ in place of $\Delta$, where $\tau=\pm1$ labels $K'$0, $K'$1 is the Dirac velocity, and the mass term is controlled by the inversion-breaking asymmetry of the unit cell. When the mass vanishes, the spectrum is gapless; when it is nonzero, a valley gap opens with opposite Berry-curvature sign in the two valleys (Chen et al., 2016, Chen et al., 2018).

For a Bloch band with eigenstate $K'$2, the Berry curvature is

$K'$3

and in the massive Dirac approximation it is sharply peaked near the valleys. Integrating over a single valley region yields a half-integer valley Chern number, typically $K'$4, so the total Chern number remains zero by time-reversal symmetry while the valley contrast is nonzero. This is the formal basis for the photonic analogue of the quantum valley Hall effect and for the existence of kink states at interfaces where the mass changes sign (Chen et al., 2016, Chen et al., 2020, Zhang et al., 21 May 2025).

At a domain wall between two VPhCs with opposite mass signs, the valley-projected topological index changes by $K'$5 per valley, and bulk–edge correspondence predicts one edge mode per valley with opposite group velocities. In the usual zigzag-oriented setting, the projected $K'$6 and $K'$7 wavevectors remain well separated, so backscattering requires large momentum transfer and intervalley mixing is strongly suppressed. This is the central protection mechanism in VPhCs, but it is conditional rather than absolute: it depends on preserving valley separation in momentum space and on avoiding interface geometries or defects that efficiently couple the two valleys (Chen et al., 2016, Noh et al., 2017).

2. Unit-cell engineering, valley chirality, and excitation selectivity

The most common route to a VPhC is geometric asymmetry within a hexagonal or honeycomb unit cell. Representative examples include silicon rods of diameters $K'$8 and $K'$9 in air, ceramic rods of diameters $K$0 and $K$1 in a microwave parallel-plate system, unequal triangular air holes in silicon slabs, breathing Kagome lattices, and deformed SSH-type triangular-pillar cells. In each case, the relevant perturbation breaks inversion symmetry, gaps the Dirac point, and reverses sign when the sublattice asymmetry is swapped (Chen et al., 2016, Chen et al., 2018, Shalaev et al., 2017, Yu et al., 2022, Gong et al., 2020).

A distinctive property of many VPhCs is that the valley eigenfields carry phase vortices. In the all-dielectric photonic valley crystal of Chen, Chen, and Dong, the out-of-plane field $K$2 exhibits opposite orbital angular momentum (OAM) chirality at $K$3 and $K$4: at the lower band edge near $K$5, the $K$6 state is left-handed and the $K$7 state is right-handed. A circular $K$8 line source with azimuthal phase winding,

$K$9

selectively excites the matching valley, with $K'$0 coupling to $K'$1 and $K'$2 coupling to $K'$3 (Chen et al., 2016).

Microwave experiments later reproduced the same selection rule with a three-monopole phased array. There, a left-hand circular polarization vortex $K'$4 excites $K'$5, while a right-hand circular polarization vortex $K'$6 excites $K'$7. By continuously varying the antenna phases, the bulk splitting ratio was tuned smoothly from approximately $K'$8 to $K'$9, establishing source-phase control as a direct way to steer valley-polarized power flow without structural reconfiguration (Chen et al., 2018).

The phase-vortex picture extends beyond the simplest $K$0 OAM case. In a $K$1 tri-sublattice metallic-rod photonic crystal, the lower valley gap supports valley eigenfields with OAM $K$2, while the upper valley gap supports OAM $K$3, and both coexist with a pseudospin topological gap at $K$4. In inverse-designed structures, the same chirality has been recast in terms of transverse spin angular momentum (TSAM), with the design objective explicitly targeting arbitrary pseudospin states at $K$5 in two separated bandgaps (Chen et al., 2020, Sato et al., 28 Mar 2025).

Not all valley-selective phenomena require globally nonzero Berry curvature. A $K$6-symmetric triangular photonic crystal with globally vanishing Berry curvature near the valleys was shown to possess a uniform distribution of opposite phase vortices, giving a location-defined “local valley Hall effect” in which valley selectivity depends on the source position and OAM chirality rather than on a bulk valley topological phase. This suggests that phase-vortex structure and valley-addressable transport can survive outside the standard inversion-breaking paradigm (Bisharat et al., 2023).

3. Domain walls, transport regimes, and the limits of robustness

The operational signature of a VPhC is the domain-wall edge state. In the 2016 all-dielectric valley crystal, a zigzag interface between domains with $K$7 supports one positive-velocity $K$8 mode and one negative-velocity $K$9 mode. A Z-shaped domain-wall waveguide with two sharp $K'$0 corners exhibited broadband, nearly unity transmission over $K'$1 to $K'$2, corresponding to about $K'$3 fractional bandwidth, without corner morphological optimization (Chen et al., 2016).

Experiments in the microwave regime established the same transport picture in a distinct platform. A domain wall between two valley photonic crystals with opposite mass signs showed high $K'$4 transmission for both straight and Z-shaped channels, and field mapping at $K'$5 confirmed guidance along the interface without observable backscattering at the bend. The same work also showed that a boundary between a VPhC and a perfect electric conductor can host valley-dependent edge states whose dispersion evolves from gapless to gapped flat bands as the nearest boundary rod diameter is increased from $K'$6 to $K'$7, thereby tuning the high-transmission frequency window (Chen et al., 2018).

At telecommunication wavelengths, silicon implementations made the bend robustness directly relevant to integrated photonics. A suspended silicon membrane with unequal triangular holes showed a bulk bandgap in $K'$8–$K'$9, while straight and trapezoidal domain walls maintained high transmittance through the same window; later CMOS-compatible SOI devices with silica overcladding showed nearly equivalent transmission for straight and Z-shaped valley waveguides over approximately $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$0–$$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$1, whereas comparable W1 line-defect photonic-crystal waveguides displayed bend-induced dips, slow-light degradation, and multimode penalties (Shalaev et al., 2017, Yamaguchi et al., 2023).

The protection mechanism has clear geometric limits. Armchair interfaces project $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$2 and $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$3 onto the same one-dimensional edge momentum and therefore allow intervalley mixing. In femtosecond-written honeycomb waveguide arrays with sublattice detuning, armchair domain walls were found to be gapped for any detuning, while zigzag walls remained ungapped until a finite cutoff was reached. The same distinction recurs across VPhC platforms: zigzag-like interfaces are preferred because their Fourier content does not efficiently supply the $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$4 momentum transfer needed for backscattering (Noh et al., 2017).

A common misconception is therefore that VPhC transport is absolutely immune to reflection. The literature instead supports a more specific statement: VPhC edge transport is robust against smooth disorder, sharp bends, and moderate imperfections so long as the interface geometry, disorder spectrum, and operating band suppress intervalley scattering. Armchair segments, sharp lattice-scale defects, or sufficiently strong disorder can degrade that protection (Chen et al., 2016, Zhang et al., 2024).

4. Material platforms and spectral realizations

VPhCs have been realized in a wide range of photonic media. All-dielectric rod-in-air systems established the original valley OAM physics at normalized frequencies such as $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$5 and $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$6 using silicon rods with $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$7 (Chen et al., 2016). Microwave implementations then used high-permittivity ceramic rods in parallel-plate waveguides, with a valley extremum at $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$8, to demonstrate dynamic light-flow control and boundary-engineered edge dispersion (Chen et al., 2018).

Silicon slab implementations brought VPhCs into the telecommunication band. One route used unequal triangular air holes in suspended membranes or SOI, with experimentally observed valley-guided transport in the $$H_\tau(\mathbf{k}) = v_D (\tau k_x \sigma_x + k_y \sigma_y) + \Delta \sigma_z,$$9 range (Shalaev et al., 2017, Yamaguchi et al., 2023). Another used a planar InGaAsP breathing Kagome lattice with lattice constant $m$0, hole diameter $m$1, and perturbations $m$2 or $m$3, producing a projected edge-state window from $m$4 to $m$5, of which $m$6–$m$7 lies below the light line (Gong et al., 2020).

Beyond conventional dielectric slabs, several nonstandard VPhC platforms have been developed. A plasma photonic crystal formed from gas-discharge tubes embedded in ring-shaped dielectric columns exhibited a valley-dependent bandgap at $m$8–$m$9 for electron density $\Delta$0, with the edge-mode frequency tunable by $\Delta$1; at $\Delta$2, the Dirac cone in the symmetric honeycomb case was broken by plasma dispersion itself, opening a topologically trivial gap (Li et al., 2023). A vertically coupled designer-surface-plasmon structure introduced a layer pseudospin degree of freedom and realized conventional and layer-polarized valley-Hall phases in the $\Delta$3–$\Delta$4 range (Wu et al., 2018).

Gyromagnetic and hybrid topological systems have also incorporated valley physics. In a YIG-based hexagonal microwave crystal under static magnetic bias, an upper bandgap between $\Delta$5 and $\Delta$6 became a valley-Hall gap with integer valley Chern numbers associated with an unpaired quadratic valley point, while a lower gap remained quantum anomalous Hall; this enabled frequency-multiplexed edge transport of distinct topological origin on the same network (Wang et al., 2022).

These realizations show that the VPhC concept is not tied to a single material class. What is preserved across platforms is the valley-resolved gap opening, the existence of domain-wall modes, and the need to control symmetry, band alignment, and intervalley coupling. The specific invariant, field texture, and transport bandwidth, however, can change significantly with the lattice type and with whether the system is dielectric, gyromagnetic, dispersive, or multilayer (Wang et al., 2022, Li et al., 2023, Wu et al., 2018).

5. Devices, functional extensions, and application-specific architectures

VPhCs have evolved from edge-transport demonstrators into device platforms. Source-controlled beam steering was shown in a microwave valley crystal, where a three-element antenna array continuously tuned the power split between left and right exits with measured ratios from $\Delta$7 to $\Delta$8 (Chen et al., 2018). In silicon slabs, efficient chip-scale coupling to topological slow-light edge modes has been realized by filling the air holes adjacent to a bearded interface over a short length $\Delta$9, giving an average simulated coupling efficiency of $\tau=\pm1$0 and an experimental value of approximately $\tau=\pm1$1 over the slow-light range $\tau=\pm1$2–$\tau=\pm1$3 (Yoshimi et al., 2023).

Slow-light engineering has become a substantial subfield in VPhCs. A heterostructure $\tau=\pm1$4, with a gapless photonic-graphene region inserted between two topologically distinct VPhCs and reduced unit-cell spacing near the domain walls, was used to realize wide-mode-area slow-light modes. Measurements on silicon slabs showed group indices exceeding $\tau=\pm1$5, with simulations reaching $\tau=\pm1$6, while the mode width increased approximately linearly with the number $\tau=\pm1$7 of graphene-like layers for both fast-light and slow-light operation (Zhang et al., 2024, Zhang et al., 21 May 2025).

Active and nonlinear extensions are equally prominent. In an all-dielectric InGaAsP Kagome lattice, topological edge states below the light line formed equilateral triangular laser cavities whose ring-resonator modes reached a maximum $\tau=\pm1$8 of $\tau=\pm1$9 at $K'$00, with lasing near $K'$01 under a four-level gain model (Gong et al., 2020). In another all-dielectric honeycomb VPhC with two valley gaps around $K'$02 and $K'$03, nonlinear interaction between double valley-Hall kink modes produced phase-matched second-harmonic generation, and a directional-dichroism metric reached approximately $K'$04 at normalized frequency $K'$05 (Lan et al., 2020).

Higher-functionality routing has also been achieved by combining multiple topological channels. A $K'$06 tri-sublattice photonic crystal supported one pseudospin edge band and two valley edge bands in three separate gaps, and a four-channel system used source chirality or phase matching to route waves through sharp $K'$07 bends at $K'$08, $K'$09, and $K'$10 (Chen et al., 2020). Hybrid topological crystals combining quantum anomalous Hall and valley-Hall gaps further enabled frequency-selective routing, with low-frequency signals following one edge network and high-frequency signals another (Wang et al., 2022).

Application-driven designs now include sensing. A THz VPhC with silicon rods in air and alternating diameters $K'$11, $K'$12 supported robust transport in both linear and $K'$13-shaped waveguides and formed a hexagonal topological cavity for biosensing. In that design, carcinoma-cell detection was associated with a maximum quality factor of $K'$14 and a maximum sensitivity of $K'$15 (Hossain et al., 6 Sep 2025). A plausible implication is that the same interface confinement and bend robustness that originally motivated VPhCs for routing can also be exploited to stabilize resonant sensing architectures.

6. Conceptual extensions, “valley-Hall-like” regimes, and unresolved distinctions

The most persistent conceptual debate concerns how strictly the term “valley-Hall” should be applied in photonics. In conventional honeycomb VPhCs, the language is straightforward: inversion-symmetry breaking creates local Berry curvature near $K'$16 and $K'$17, valley Chern numbers are opposite, and a domain wall between opposite masses supports kink states (Chen et al., 2016, Noh et al., 2017). However, several later works showed that closely related edge phenomena can persist when this simple characterization becomes incomplete.

One important qualification is that the valley Chern number in photonics is not always a strictly quantized bulk invariant in the same sense as in electronic topological phases. In slab structures continuously deformed from perturbed honeycomb lattices to triangular lattices, edge modes were shown to persist even as the Berry curvature approached zero in the triangular limit; these “zero-Berry-curvature edge modes” still propagated with extremely low bending loss in a wide photonic bandgap (Yang et al., 2020). Relatedly, a $K'$18-symmetric triangular photonic crystal supported valley-polarized edge states via a local valley Hall effect despite globally vanishing Berry curvature near the valleys, because defect-induced sublattice asymmetry activated location-dependent phase-vortex selection rules (Bisharat et al., 2023).

A second extension replaces valley Chern numbers with other invariants. In a $K'$19-symmetric Kagome lattice of circular air holes, the relevant bulk invariant was a quantized electric polarization rather than a valley Chern number, yet the system still exhibited valley-Hall-like edge transport and, additionally, second-order corner states at oblique corners (Zhang, 2019). This has made “valley-Hall-like” a useful term for systems that preserve valley-momentum locking and reflection-suppressed routing while being topologically classified by polarization, band inversion, or symmetry indicators rather than by a conventional valley Chern number.

A third line of work expands the valley degree of freedom itself. Layer pseudospins in vertically coupled designer surface plasmon crystals produced conventional valley-Hall and layer-polarized valley-Hall phases, and enabled broadband layer convertors and layer-selected delay lines with measured group velocities $K'$20 and $K'$21 (Wu et al., 2018). Hybrid $K'$22 lattices combined pseudospin and valley Hall mechanisms in one crystal (Chen et al., 2020), while gyromagnetic platforms produced valley phases with integer rather than half-integer valley Chern numbers because the relevant valley point was quadratic rather than linear (Wang et al., 2022).

Finally, recent inverse-design work has turned these distinctions into design variables. By optimizing a frequency-domain objective based on TSAM at the $K'$23 point, dual-band VPhCs with arbitrary pseudospin states were designed so that one interface supported routing near $K'$24 and another near $K'$25, with reported output-to-input power ratios of $K'$26 and $K'$27, respectively (Sato et al., 28 Mar 2025). This suggests that the future taxonomy of VPhCs may be organized less by a single canonical lattice and more by which symmetry, local field texture, or effective topological invariant is being engineered.

In that sense, VPhCs now denote both a specific topological mechanism—valley-contrasting Dirac-gap photonics—and a broader design family centered on valley-resolved wave control. The literature supports both usages, provided the underlying invariant and the scope of the protection claim are stated precisely (Yang et al., 2020, Zhang, 2019, Bisharat et al., 2023).