Quantum hall effect and disorder

[wasia]

Hello fellow PFers!

I have been trying to understand the Integer Quantum Hall Effect for quite a while. Many things seem to be understandable, however, I cannot create a satisfactory explanation of why IQHE (i.e. plateaux in the the Hall resistance R as a function of the magnetic field B) does not happen in the absence of impurities in the sample.

I understand that the problem is purely theoretical, as perfect samples do not happen in the real world. Moreover, it could be the case that I misunderstand many things at the same time, therefore, I would be very thankful if you pointed out every error.

Consider a 6-terminal configuration (in T=0K), which has a current source attached to 1 and 4 and there is magnetic field B in the direction perpendicular to the plane of the figure. We measure the longitudinal resistance R_xx using e.g. 2&3 and the Hall resistance R using 2&6.

This creates a situation where electron number in the sample is constant (after a sufficiently long time).

The (bulk) Landau levels are degenerate with (Girvin,arXiv:cond-mat/9907002):
$$N = \frac{L_x L_y}{2\pi l_B^2}$$
states in every Landau level, where L_x and L_y describe the size of the sample and l_B is the magnetic length
$$l_B = \sqrt{\frac{\hbar c}{eB}} .$$

The distance between the Landau levels is equal to the cyclotron energy
$$\Delta E_L = \hbar \omega_C = \frac{\hbar e B}{m c}.$$

Let's consider the trajectory of an electron in the bulk. We know it will follow an equipotential line (red line in this case) performing (classically) a cyclotronic motion around the red line, which I tried to draw as a black line with an arrow. Quantum mechanically, the wavefunction is a plane wave in y direction and a harmonic oscillator in x direction. It seems that these electrons contribute to both the Hall conductivity (and therefore R) and to the scattering of edge states, and therefore R_xx.

There also are equipotential lines along the edges of the sample (blue) due to the confining potential. However, every such line contributes two 'straight-line' states instead of one 'closed' state due to the effect of the electrodes 1 and 4. Those two states are the states of opposite chirality, i.e. each of them transports an electron only in one direction. Moreover, they are topologically protected and will not be destroyed upon addition of the impurity potential. The degeneracy of the edge states scales as l_B/L_y (Yoshioka, 1998). It therefore seems that in the thermodynamic limit there are much more bulk states than edge states. Moreover, the edge states are said to be gapless. My attempt to draw an energy spectrum from this data is attached (L denotes Landau levels and the grey continuum represents the edge states; the density of the edge states is greatly exaggerated).

Imagine a situation, when B is such that the Fermi level coincides with L1. Then L1 is exactly filled and so is every state below it. The longitudinal resistivity R_xx is very high due to the backscattering of edge states (the bulk states percolate the sample), while R is equal to h/(2e^2), where 2 comes from the fact that the L0 & L1 are filled.

Upon lowering B a little bit, we get decrease in the distance between Landau levels and decrease in their degeneracy. This is the place where I am utterly confused. Fermi level remains at the same place and therefore a few electrons from L1 to go to the edge states and therefore R_xx decreases (more edge channels!) while R increases (less electrons are going along the red lines). I had expected R ~ B and R_xx independent of B.

Electrons in the LLs contribute to 1/R. Do they contribute individually (1/R ~ sum of electrons in all LLs) or does one LL contribute one conductance quantum only when the LL is filled?

 

[BigJDubs]

How is it that the Hall resistance changes in spite of Fermi level being in the same LL? Is it because of the decrease in the degeneracy or something else?I apologize for the long post, but I cannot seem to find a satisfying answer anywhere.