When graphene is placed on hexagonal boron nitride (hBN) or transition metal dichalcogenides (TMDCs), the resulting heterostructure exhibits unique quantum Hall phenomena due to the interplay between the moiré superlattice potential and the relativistic charge carriers in graphene. The periodic potential introduced by the substrate modifies the electronic band structure, leading to the emergence of Hofstadter’s butterfly and fractional quantum Hall states, which are absent or less pronounced in monolayer graphene under moderate magnetic fields.

The Hofstadter’s butterfly spectrum arises when a two-dimensional electron system is subjected to both a periodic potential and a perpendicular magnetic field. In graphene/hBN or graphene/TMDC heterostructures, the moiré pattern creates a superlattice with a periodicity significantly larger than graphene’s lattice constant. When a magnetic field is applied, the magnetic flux through the superlattice unit cell becomes commensurate with the flux quantum, leading to a fractal energy spectrum as a function of the magnetic field. This spectrum is characterized by a self-similar pattern of energy gaps and Landau levels, forming the butterfly structure. The observation of Hofstadter’s butterfly in these systems requires high-quality interfaces and relatively low disorder, as the moiré potential must dominate over extrinsic scattering mechanisms.

In contrast, monolayer graphene in a magnetic field exhibits a conventional quantum Hall effect with Landau levels that follow a square-root dependence on the field strength and level index, a consequence of its linear Dirac dispersion. The Landau levels in monolayer graphene are fourfold degenerate due to spin and valley symmetry, leading to quantum Hall plateaus at filling factors ν = ±4(n + 1/2), where n is an integer. The moiré potential in heterostructures breaks some of these symmetries, lifting degeneracies and introducing additional features in the quantum Hall sequence.

Fractional quantum Hall states, typically observed in high-mobility two-dimensional electron gases under strong magnetic fields, also emerge in graphene/hBN and graphene/TMDC systems. These states result from electron-electron interactions, which become significant when the kinetic energy is quenched by the magnetic field. In monolayer graphene, fractional states are rare and usually require very high magnetic fields due to the large energy spacing between Landau levels. However, in heterostructures, the moiré potential enhances the effective interaction strength by flattening the bands and increasing the density of states, making fractional states observable at lower fields. The presence of a superlattice also introduces new fractional states tied to the moiré periodicity, which are absent in monolayer graphene.

The role of the substrate in modifying the quantum Hall physics is critical. Hexagonal boron nitride, with its atomically flat surface and negligible charge disorder, provides an ideal environment for preserving graphene’s electronic quality while introducing a weak periodic potential. TMDCs, on the other hand, can induce stronger spin-orbit coupling and additional symmetry breaking, further enriching the quantum Hall phenomena. For example, graphene on WSe2 exhibits spin-polarized edge states due to proximity-induced spin-orbit effects, which are absent in graphene/hBN systems.

Experimental observations of these effects rely on precise control of the twist angle between graphene and the substrate, as the moiré periodicity depends sensitively on the relative orientation. Small-angle twists produce larger superlattices, which require lower magnetic fields to reach the Hofstadter regime. Advances in van der Waals assembly techniques have enabled the fabrication of heterostructures with twist angles controlled to within fractions of a degree, allowing systematic studies of the quantum Hall phenomena as a function of the moiré periodicity.

Theoretical models of these systems combine the Dirac equation for graphene’s charge carriers with the periodic potential of the moiré superlattice. The resulting Hamiltonian predicts the formation of mini-bands and the evolution of Landau levels into the fractal Hofstadter spectrum. Numerical simulations have successfully reproduced the observed butterfly patterns and provided insights into the conditions required for fractional state formation. These models highlight the importance of lattice relaxation and strain in determining the effective moiré potential, as local deformations can significantly alter the electronic structure.

The quantum Hall phenomena in graphene heterostructures not only deepen the understanding of two-dimensional electron systems but also offer opportunities for exploring novel correlated states. The tunability of the moiré superlattice through twist angle and applied pressure provides a versatile platform for engineering quantum phases. For instance, recent experiments have demonstrated the emergence of Chern insulators in graphene/hBN at specific filling factors, where the system exhibits quantized Hall conductance without an external magnetic field.

Comparisons with monolayer graphene underscore the transformative effect of the moiré potential. While monolayer graphene serves as a paradigmatic system for studying Dirac fermions in a magnetic field, heterostructures introduce additional degrees of freedom and interaction mechanisms, leading to richer physics. The interplay between the superlattice and the magnetic field creates a complex energy landscape where single-particle and many-body effects coexist, offering a fertile ground for discovering new quantum states.

Practical implications of these findings extend to the development of quantum materials with tailored electronic properties. The ability to control the quantum Hall response through substrate engineering opens avenues for designing devices with unconventional transport characteristics. For example, heterostructures with robust fractional states could be exploited for topological quantum computation, although significant challenges remain in maintaining coherence at elevated temperatures.

In summary, graphene/hBN and graphene/TMDC heterostructures exhibit quantum Hall phenomena that are qualitatively distinct from those in monolayer graphene. The Hofstadter’s butterfly spectrum and fractional states arise from the interplay of the moiré potential and electron interactions, highlighting the profound influence of the substrate on graphene’s electronic behavior. These systems provide a unique platform for exploring the frontiers of condensed matter physics, where the synergy between periodicity, magnetism, and correlation leads to emergent quantum phenomena.