Pii: S0038-1098(99)00292-6

We study the localization properties of a two-dimensional noninteracting electron gas in the presence of randomly distributed short-range scatterers in very high magnetic fields. We evaluate the participation number of the eigenstates obtained by exact diagonalization technique. At low impurity concentrations we obtain self-averaged values showing that all states, except those exactly at the Landau level, are localized with finite localization length. We conclude that in this dilute regime the localization length does not diverge. We also find that the maximum localization length increases exponentially with impurity concentration. Our calculations suggest that scaling behavior may be absent even for higher concentrations of scatterers. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: D. Quantum Hall effect; D. Quantum localization There has been a long lasting interest in understanding the localization problem in two-dimensional (2D) systems. According to scaling theory of localization [1,2], all states in a 2D system are localized if a disordered potential is present. However, in the presence of a strong perpendicular magnetic field, where the time reversal symmetry is broken, extended states appear in the center of impurity-broadened Landau bands [3]. If the scattering between Landau levels can be neglected, these extended states exist only at a single energy [4,5]. The width of the quantized plateaus of the integer quantum Hall effect (QHE) depends on the ratio of number of localized to extended states [6]. In analogy with the quantum critical phenomena and other localization transitions, it has been proposed that localization length j…E† diverges as E approaches the critical energy Ec, which is equal to Landau level energy, so that j…E† / uE 2 Ecu …1† where n is the localization critical exponent [7,8]. After the initial calculations of Aoki and Ando [9–11], several groups attempted to determine this critical exponent [12–23]. Some of the experimental results [24–28] are generally in good agreement with the calculated values. Various techniques have allowed the computation of the exponent n, and they strongly suggest a universal value n ˆ 2:3 ^ 0:1 for the lowest Landau level (LLL). However, in spite of a great deal of experimental evidence and numerical simulations in its favor, there is no rigorous derivation of power law divergence in the localization length. Furthermore, even if the power law divergence is true, it is not clear whether the localization critical exponent is universal, independent of impurity concentration or parameters of the disordered potential [21,29–32]. Recently, we developed a method for a particle in the LLL moving in an arbitrary potential [33], where we investigate the energy spectrum and eigenvalues for periodic distribution of point scatterers. In this study we apply the method, which is basically an exact diagonalization technique, to a potential formed by randomly distributed shortrange scatterers. We concentrate on low impurity concentrations, i.e. large average impurity–impurity separations in comparison to the magnetic length, where it is difficult to perform calculations by other methods due to the presence of zero eigenvalues associated with the extended states at the band center. At low concentrations, we obtain self-averaged values where energy spectrum or localization property of eigenstates do not change with increasing system size. Contrary to the widely accepted view, localization length Solid State Communications 112 (1999) 157–160 0038-1098/99/$ see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(99)00292-6 PERGAMON www.elsevier.com/locate/ssc * Corresponding author. does not diverge but instead the maximum localization length grows exponentially with impurity density. Extrapolation to less pure systems suggests that localization length can become as large as the sizes of the samples used in QHE experiments which explains the observed divergence in measurements. The Hamiltonian for a particle of mass m and charge q, moving in 2D in the presence of magnetic field B ˆ 7 × A perpendicular to the plane and potential V , is given by H ˆ H0 1 V where