Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system

Details Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system

2009-07-30

arxiv.org/abs/0907.5253v1

J. Phys.: Condens. Matter 21 (2009) 345803

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

21 pages, 8 figures, accepted for publication in J. Phys.: Condens. Matter

Scientific paper

10.1088/0953-8984/21/34/345803

We investigate the relation between the diagonal ($\sigma_{xx}$) and off-diagonal ($\sigma_{xy}$) components of the conductivity tensor in the quantum Hall system. We calculate the conductivity components for a short-range impurity potential using the linear response theory, employing an approximation that simply replaces the self-energy by a constant value $-i \hbar /(2 \tau)$ with $\tau$ the scattering time. The approximation is equivalent to assuming that the broadening of a Landau level due to disorder is represented by a Lorentzian with the width $\Gamma = \hbar /(2 \tau)$. Analytic formulas are obtained for both $\sigma_{xx}$ and $\sigma_{xy}$ within the framework of this simple approximation at low temperatures. By examining the leading terms in $\sigma_{xx}$ and $\sigma_{xy}$, we find a proportional relation between $\mathrm{d}\sigma_{xy}/\mathrm{d}B$ and $B \sigma_{xx}^2$. The relation, after slight modification to account for the long-range nature of the impurity potential, is shown to be in quantitative agreement with experimental results obtained in the GaAs/AlGaAs two-dimensional electron system at the low magnetic-field regime where spin splitting is negligibly small.

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