Integer Quantum Hall Effect at Finite Temperature

Abstract

On the basis of the field theoretical model of the integer quantum Hall effect developed by Toyoda, Gudmundsson and Takahashi [Physica 132A, 164 (1985)] we examine the temperature dependence of the Hall conductivity. The fundamental hypothesis in the theory is the existence of the electron reservoir, which has been experimentally confirmed recently. We discuss the continuous transition from the quantum regime to the classical regime as the temperature increases from 1K to 14K in the MOSFET experiment. The quantized plateaus disappear around T=3K.Since the discovery of the integer quantum Hall effect in 1980 [1], more than two decades have passed. The discovery made an enormous impact not only on the solid state physics but on an extremely wide range of physics, from applied physics to theoretical high energy physics. That a physical constant can be measured with such an amazing accuracy in a rather simple semiconductor experiment was genuinely an astonishing discovery. Naturally a considerable number of physicists have tried to find the fundamental theory to explain the mechanism of the integer quantum Hall effects. And yet, the microscopic mechanism of the effect has not been clarified. No consensus has been reached on the fundamental mechanism. If we carefully examine the proposed theoretical attempts, there seems to be two fundamentally different approaches in the theoretical attempts so far proposed. The crucial criterion that divides the two different approaches is the relation between the gate voltage and the electron number density in MOSFET experiment. In the theory based on the electron localization hypothesis [2], a simple linear relation between these two quantities is assumed. On the other hand, in the theory based on the electron reservoir hypothesis [3][4][5], the electron number density is given as the quantum statistical expectation value of the grand canonical ensemble, where the gate voltage appears as the shift of the chemical potential. These two different interpretations of the relation between the gate voltage and the electron density in the experiments lead to very different theoretical formulations. Therefore, the search for the experimental evidence that would decide the right approach among those two different ones has been sought after for years. Now the latest experimental results seem to support solidly the electron reservoir hypothesis [6]. Encouraged by this recent experimental progress, in this paper we examine the effects of finite temperature on the Hall conductivity on the basis of the reservoir hypothesis. Following Ref.[5], we consider a two-dimensional interacting electron gas under magnetic field and electric field. We also include the interaction between the electrons and impurity atoms. The current density operator of the electron system is