195. Integer quantum Hall effect – plateaus in Hall resistivity

This post is my attempt to give an answer to tg’s question: Ok, we a kind of proved that the Hall resistivity is quantized in the integer quantum Hall effect, but how to explain plateaus in the Hall resistivity and deep minima in diagonal resistivity as shown on the Fig.

We have to introduce quite a non-trivial idea to explain both effects – we will say that electronic bound states exist in the material that do not contribute to the current and conductivity. There are several mechanisms for these bound states to appear – for example, Wigner cristallization or Anderson localization of electrons on the fluctuations of effective potential (generated, for example, by impurities in the material). The latter actually allows to explain almost all experimental data.

How does the Anderson localization work? Imagine that there are impurities built into the material of the 2-dimensional sample. In the presence of the random potential generated by them the energy of every Landau level becomes a function of coordinates . For every electron, its wave function is localized near equipotential surface

In places where random potential fluctuates strongly (say, near some point – I’ll choose it by hands), equipotential trajectories are closed, and wave function of the corresponding electron is also localized near . As a result, this electron cannot contribute to the overall current (at ) simply because it cannot pass from one edge of the sample to the other.

Now, let us average over the random potential generated by impurities. What would be the result of such averaging? The Landau levels will acquire some width (related to the mean free path and, therefore, the strength of the random potnetial – and we will consider the case of the weak potential), that is, the density of states will be peaked near the Landau levels , but there also will be (localized) states with energy slightly larger or smaller than .

Let us consider the behaviour of the Hall resistivity changing the Fermi energy . First, if is located in between Landau levels and , the contribution into the overall current is given only by delocalized states with energies ,, and is given by the formula (5) that we derived in the previous post.

Let us slowly change the value of the Fermi energy . It is clear that as long as is in the interval of localized states near the Landau level , the Hall resistivity remains the same, since localized states do not contribute into the current. The situation finally changes when enters the interval of delocalized states near the Landau level . Concentration of carriers will rapidly change (new delocalized states are getting occupied!), and this will correspond to a sudden jump in the value of . If will grow even further, it will again enter the interval of localized states, near the level this time, and the Hall resistivity will remain the same.

How to explain the behaviour of diagonal resistivity ? Actually, I said enough for you to be able to find the answer yourselves – can you do that?

Update. My explanation of the fact that vanishes is the following (see also in comments). Non-vanishing resistivity comes from dissipation effects. In order for the energy to be dissipated, the electron has to jump from a level of higher energy to a level of lower energy. Then, the difference can be transferred to the fluctuations of the lattice (to heat, in other words).

Now, if the temperature is not completely zero, there are always some empty states below the energy of a given electron. However, all these states (as well as the current state of electron) are localized – that is, separates in space, and the distance between them is way larger than the mean free path for electrons in the sample. So, basically, electrons near (and recall that all the states below are filled) cannot jump to lower levels, and the current flows unaffected by longitudinal resistivity.

What happens if approaches a Landau level ? The states in the closest vicinity of are delocalized, and electrons can jump from one state to another in the narrow region near . This leads to a slight jump in dissipation and diagonal resistivity. As you can imagine, the jump should be related to the width of the band of extended states near .