The integer and fractional quantum Hall effects (QHE) in graphene and transition metal dichalcogenide (TMDC) heterostructures represent a cornerstone of modern condensed matter physics, revealing profound quantum phenomena under high magnetic fields. These systems exhibit quantized conductance plateaus, topological edge states, and exotic quasiparticle statistics, making them indispensable for quantum resistance standards and future quantum technologies.

In graphene, the quantum Hall effect arises due to the formation of Landau levels, which are quantized energy states of electrons in a perpendicular magnetic field. The unique linear dispersion of graphene near the Dirac points leads to a Landau level spectrum given by En = ±vF√(2eħB|n|), where vF is the Fermi velocity, B is the magnetic field, and n is the Landau level index. Unlike conventional semiconductors, graphene exhibits a zero-energy Landau level (n=0), contributing to its anomalous quantum Hall sequence with conductance plateaus at σxy = ±4e²/h (n + 1/2), where the factor of 4 accounts for spin and valley degeneracy. At high magnetic fields (typically above 10 T), these plateaus become exceptionally sharp, enabling resistance quantization with metrological precision.

TMDC heterostructures, such as those based on MoS2 or WSe2, display distinct Landau level structures due to their massive Dirac fermions and strong spin-orbit coupling. The Landau levels in monolayer TMDCs follow En = ±√(Δ² + 2nħeBvF²), where Δ is the bandgap. The large spin-orbit splitting in TMDCs lifts the spin degeneracy, leading to spin-polarized Landau levels and enabling the observation of the quantum Hall effect at lower magnetic fields compared to graphene. Heterostructures combining TMDCs with hexagonal boron nitride (hBN) further enhance mobility, reducing disorder and sharpening the quantization.

Edge states play a pivotal role in the quantum Hall effect. In graphene and TMDCs, the bulk Landau levels are accompanied by chiral edge states that carry current without dissipation. These states are topologically protected, meaning backscattering is suppressed, ensuring precise quantization of the Hall conductance. In graphene, the edge states exhibit valley polarization, while in TMDCs, spin-polarized edge states emerge due to strong spin-orbit coupling. Device architectures exploiting these edge states include Hall bar and Corbino geometries, where contacts are carefully placed to isolate edge transport from bulk contributions.

The fractional quantum Hall effect (FQHE) occurs at higher magnetic fields and lower temperatures, where electron-electron interactions dominate. In graphene, FQHE states appear at filling factors ν = p/q (p, q integers), such as ν = 1/3 or 2/5, corresponding to the formation of correlated electron states like Laughlin states. TMDC heterostructures also host FQHE states, with additional complexity due to their strong spin-valley coupling. The presence of anyonic statistics in these systems is particularly notable: quasiparticles in FQHE states obey fractional exchange statistics, a key feature for topological quantum computation.

Device architectures for quantum resistance standards leverage the precision of the integer quantum Hall effect. Graphene-based standards operate at lower magnetic fields than traditional GaAs devices due to graphene’s larger energy spacing between Landau levels. Dual-gated structures are employed to tune carrier density and maintain quantization at ν = 2 or ν = 6, ensuring robustness against disorder. TMDC heterostructures, with their spin-polarized edge states, offer potential for spin-resolved quantum Hall standards, though challenges remain in achieving sufficiently high mobility.

Experimental observations in graphene reveal quantum Hall plateaus with accuracy exceeding one part per billion, meeting the criteria for primary resistance standards. The quantum Hall resistance RH = h/νe² is reproduced with extreme fidelity, where ν is the filling factor. In TMDCs, the interplay between spin, valley, and Landau level quantization introduces additional plateaus, though their metrological utility is still under investigation.

The anyonic statistics of fractional quantum Hall states are probed through interferometry and noise measurements. In graphene, Fabry-Pérot interferometers have been used to detect phase shifts consistent with fractional statistics. TMDC heterostructures offer a platform for studying non-Abelian anyons, which could enable fault-tolerant quantum computing, though conclusive evidence remains elusive.

Key challenges in these systems include disorder mitigation and temperature stability. Graphene devices encapsulated in hBN achieve mobilities exceeding 10⁶ cm²/Vs, essential for observing fragile FQHE states. TMDC heterostructures require further optimization to reduce charge inhomogeneity. Cryogenic temperatures below 1 K are typically necessary to resolve fractional states, though graphene’s larger energy scales allow observations at slightly higher temperatures compared to GaAs.

Future directions include integrating quantum Hall devices with superconducting circuits for hybrid quantum systems and exploring twist-engineered heterostructures for novel correlated states. The precision of graphene-based resistance standards continues to drive advancements in metrology, while TMDCs offer a rich playground for spin-valley quantum Hall physics.

In summary, the integer and fractional quantum Hall effects in graphene and TMDC heterostructures under high magnetic fields reveal a wealth of quantum phenomena, from topological edge states to anyonic quasiparticles. These systems not only provide the foundation for quantum resistance standards but also open avenues for exploring exotic quantum phases and future quantum technologies.