Impurities

We have a wave function, that is fine, but where is the quantum Hall effect? The clue to understand the quantum Hall effect is the finite number of states in a Landau level and the fact that a quantum Hall sample is never perfect; it has impurities.

In the previous section we argued that the many-body wave function is a total anti-symmetrized function of the one-particle states. This obviously means that we need as many one-particle states as we have particles in our system. So we can only accommodate a finite number of particles in a Landau Level. It is now useful to define a quantity that measures the number of filled landau levels, the filling fraction. Hence it is defined as the number of electrons in the sample divided by the number of states in a Landau level,

$$\nu = \frac{N_{e}}{N_{LL}}=\frac{2\pi n_{e}}{B}. <tex2html_comment_mark>43$$ (2.18)

If we start filling the integer quantum Hall system with electrons, we can accommodate all electrons for 'free', since all states in a Landau level have the same energy. When a Landau level is completely filled, in other words the filling fraction becomes an integer, we have a new situation. The first state available has considerable higher energy, this means that the Fermi energy lays in a big gap. In this situation scattering becomes impossible and we are allowed to send $\tau$ to infinity in (2.5).

It seems that we now have captured some of the features of the quantum Hall effect, but it is not yet complete. First of all the arguments in last paragraph hinge on a single electron. If we add one more electron the Fermi level does not lay in a gap anymore and our arguments are no longer valid. We should thus only see the off-diagonal conductance when we have the system tuned up to one electron. In a system of approximately $10^{20}$ electrons that is of course an absurd assumption. A second thing is that we have no explanation for the plateaus yet. These both things are very close related. To understand these matters we have to include impurities into our model. A real condensed matter system always has dislocations, foreign atoms and such. These imperfections can be modeled by adding a disorder potential to the Hamiltonian.

The effect of adding these impurities to the landau level system is that the nature of the states gets changed, but their number stays the same. We will not show this here, because these are actually complicated matters. We will only use the well known results. After adding the impurities, we can divide the states in two groups; localized and extended states. Localized states are basically electrons that are bound to an impurity. Their energy is shifted from the value found from the 'clean' Hamiltonian. These states are localized in space and do not contribute to a current flowing through the system. The second class of states are the extended states. These states are like the states we found in the previous system. They extend all over the sample and support a net current. Because they stay clear of impurities, their energy is practically unshifted from the value found for the clean system.

We now have a different density of states. It is changed from peaks with infinitesimal width at the Landau energies to a smooth peak (see figure 2.3.1). In the middle of the peak we find the extended states, their energy was least effected by the impurities, around them we find the the localized states.

Figure 2.3: Density of states for a sample with and without impurities.

The arguments above are changed by these new states. Let us reexamine these arguments, starting with a system with integer filling fraction. The Fermi level is again half way two landau energies. If we now change the occupation number we change only the occupation number of localized states. Since these states do not contribute to the current, the conductivity stays the same. The Fermi level now does not lay in a energy gap, as was the case in the arguments above, but we have a similar effect. For this reason the Fermi level is said to lay in a mobility gap. Only when we the number of occupied extended states changes, the Hall conductance can increase. This happens when the Fermi energy crosses a Landau energy. Hence we see at half-integer filling fraction a sudden jump in the Hall conductance; we take a step to the next plateau.

With the results obtained from our quantum mechanical model and these considerations involving impurities we can now sketch figure 2.2. It seems that we have a pretty good idea of what is going on in the integer quantum Hall effect.