Quenching of the Hall effect in strongly modulated two-dimensional electronic systems

The longitudinal (${\mathrm{\rho }}_{\mathit{x}\mathit{x}}$) and Hall (${\mathrm{\rho }}_{\mathit{x}\mathit{y}}$) resistivities are calculated for a strongly modulated two-dimensional electron gas. This modulation is simulated by a square array of δ-potential scatterers. It is shown that when the strength of the potential is sufficiently strong ${\mathrm{\rho }}_{\mathit{x}\mathit{y}}$ can be negative and quenched and ${\mathrm{\rho }}_{\mathit{x}\mathit{x}}$ has commensurate oscillations. Similar features have been observed experimentally for a square array of antidots. The quenching of the Hall resistivity is shown to be due to the collimated states from resonant scattering of the electrons by a lattice potential in a magnetic field when the magnetic and lattice Brillouin zones become commensurate. Impurities do not qualitatively change ${\mathrm{\rho }}_{\mathit{x}\mathit{y}}$ but they quantitatively change ${\mathrm{\rho }}_{\mathit{x}\mathit{x}}$. Numerical results are presented to show how the lattice, impurity scattering, and the electron density affect the quenching of the Hall effect and the commensurate oscillations.