Activated resistivities in the integer quantum Hall effect

Abstract

We have determined the off-diagonal and diagonal conductivities for a quantum Hall effect system at exactly integer filling at finite temperatures and in the presence of weak short ranged disorder potential within the self consistent Born approximation. We find that there is a finite temperature contribution to off-diagonal conductivity $\sigma_{xy}$ which is `anomalous' in nature as it survives even in the zero impurity limit. The diagonal conductivity $\sigma_{xx}$ survives only when both temperature and disorder is non zero. At low temperatures, $\sigma_{xx}$ activates with a temperature dependent prefactor. Inverting the conductivity matrix, we determine the resistivities. The deviation of the off-diagonal resistivity $\rho_{xy}$ from its zero temperature value and the diagonal resistivity $\rho_{xx}$ activate with a temperature dependent prefactor at low temperatures, in agreement with experiments. Further, we find two physical regimes both of which are at low temperatures and low broadening, which provide the experimentally observed linear relationship between the deviation of $\rho_{xy}$ and the $\rho_{xx}$ with different signs. We have also estimated the effective masses from the experimental data of $\rho_{xy}$ and find them to be reasonable. Finally, our result on compressibility as a function of temperature shows that there is no phase transition involved in the system as far as the temperature is concerned.