Hall Effect

The Integer Quantum Hall Effect

In our previous discussion, we deal with resistivities and conductivities, but note that the conductance and resistance are the fundamental quantities of interest both experimentally and theoretically. If it were the conductivity rather than the conductance which were quantized then precision measurements would be impossible since one would have to invoke assumptions of a homogeneous medium with a well-defined geometry in order to infer the microscopic conductivity from the macroscopic conductance. It is a remarkable feature of the QHE that this is not necessary.

According to Streda, whenever the Fermi level lies in a gap the Hall conductance will be given by
 
For a two dimensional system, the density of states at the abscence of magnetic field is g(E)=m/2. After applying a magnetic field, the energy states contract into seperate Landau levels. Each Landau level is degenerate, including  states. If the electrons completely occupy all the i levels-leaving all other levels empty, then the charge density
 
Eq. (20) and Eq. (21) may be combined to yield
 
Note that the Hall resistance is actually inversely proportional to the charge density. Eq. (22) is only correct in certain speicific n values. For the inversion layer of Si-MOSFET, n is proportional to the gate voltage V. So Hall resistance should be inverse proportional to the gate voltage. But in 1980, K.von Klitzing etc. discovered quantized Hall plateaus. Using a Hall voltage method suitable for precision measurements they obtained good steps in high-mobility Si-MOSFET devices and found that R(i)=h/i to an accruacy of at least 5 parts-per-million(ppm). Figure 6 is the experimental setup.

Fig.5 IQHE observed in Si-MOSFET

Fig.6 Experiment setup

The experiments show that between two adjacent Landau levels, the Hall resistance has fixed values and the longitudinal resistance R vanishes, which means that the electrons are localized in this region. Localization is a key point to interpret IQHE.

Due to impurity, the density of states will evolve from sharp Landau levels to a broader spectrum of levels(Figure 7). There are two kinds of levels , localized and extended , in the new spectrum , and it is expected that the extended states occupy a core near the orighinal Landau level energy while the localized states are more spread out in energy. Only the extended states can carry current at zero temperautre. Therefore, if the occupation of the extended states does not change, neither will the current change. An argument due to Laughlin(1981) and Halperin(1982) shows that extended states indeed exist at the cores of the Landau levels and if these states are full, (i.e., the Fermi level is not in the core of extended states) then they carry exactly the right current to give Eq. (22).

Fig.7 Diagram of Landau levels

The existence of the localized states can explain the appearance of plateaus. As the density is increased (or the magnetic field is decreased) the localized states gradually fill up without any change in occupation of the extended states, thus without any change in the Hall resistance. For these densities the Hall resistance is on a step in the Figure and the longitudinal resistance vanishes(at zero temperatrue). It is only as the Fermi level passes through the core of extended states that the longitudinal resistance becomes appreciable and the Hall resistance makes its transition from one plateau step to the next.

Finally, at finite temperature there is a small longitudinal resistance due to hopping processes between localized states at the Fermi level.

Ider Guerrero

EES

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