Precision of quantization of the hall conductivity in a finite-size sample: Power law
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IMENSIONAL SYSTEMS
Precision of Quantization of the Hall Conductivity in a Finite-Size Sample: Power Law A. A. Greshnov, É. N. Kolesnikova, and G. G. Zegrya^ Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia ^e-mail: [email protected] Submitted May 25, 2005; accepted for publication May 27, 2005
Abstract—A microscopic calculation of the conductivity in the integer quantum Hall effect (IQHE) mode is carried out. The precision of quantization is analyzed for finite-size samples. The precision of quantization shows a power-law dependence on the sample size. A new scaling parameter describing this dependence is introduced. It is also demonstrated that the precision of quantization linearly depends on the ratio between the amplitude of the disorder potential and the cyclotron energy. The data obtained are compared with the results of magnetotransport measurements in mesoscopic samples. PACS numbers: 73.43.Cd, 75.47.-m DOI: 10.1134/S1063782606010167
efficient methods are available, which can be used to calculate the localization length for rather large samples. Calculations of this kind have been employed to confirm the theory of finite-dimension scaling, with the scaling indices obtained being in good agreement with the experiment [3, 4]. Unfortunately, these methods are inapplicable to the calculations of the Hall conductivity, because complete information about the carrier spectrum and wave functions is necessary in this case. In this study, we calculate ab initio the Hall conductivity of a 2D electron gas in a strong magnetic field. The results obtained suggest that the precision of quantization of the Hall conductivity on the plateau shows a power-law dependence on the sample size and is directly proportional to the ratio between the amplitude of the disorder potential and the cyclotron energy. Below, we describe our model of the disorder potential and the procedure used to calculate the conductivity. Then we present the results of our numerical calculations and their analysis. Finally, we compare the theoretical results with data obtained in magnetotransport measurements on mesoscopic samples.
1. INTRODUCTION Despite the considerable progress in the understanding of the quantum Hall effect (QHE), no consistent microscopic theory of this phenomenon has been developed so far. It will be recalled that the Hall resistance RH = h/νe2 is quantized in a strong magnetic field directed perpendicularly to the plane of a two-dimensional (2D) semiconductor sample [1]. Here ν is an integer, with the precision of quantization at sufficiently low temperatures commonly limited only by the measurement error, being as good as a millionth of a percent [2]. It is important that the quantization occurs in a certain range of magnetic field strengths or carrier concentrations (on the Hall plateaus). Such behavior of the conductivity of a 2D electron gas in a strong magnetic field contradicts the conclusions of the classical kinetic theory and the diagram technique of averaging over disorder
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