191. Integer quantum Hall effect – theory

I guess, my reader, you are either a) an undergrad, b) a graduate student, c) a postdoc (or even professor? Cheers, Joe) or d) you are just interested in science and theoretical physics in particular (maybe, you even had a couple of courses related to physics in college). If b) or c) holds, you will probably fail to find anything interesting in this post, but if I am lucky, then a) or d) holds

The reason, as I said before, is that studies of quantum Hall effect remain a very hot subject in condensed matter theory after almost 20 years. In a sense, the subject provides a kind of theoretical laboratory, where new non-trivial theoretical ideas constantly appear, get estimated and finally get dismissed to reappear in condensed matter theory in some other context

Since you already have a first impression what physics of the quantum Hall effect is about, it is time to discuss theories that might have something to do with this physics. I’ll start with integer quantum Hall effect which is fairly understood and discuss the fractional quantum Hall effect tomorrow.

But before proceeding to further discussion, let me ask you to solve the following

Exercise 1. Consider a classical Hall effect as discussed in my previous post on the subject – that is, two-dimensional metallic plate, the electric current flows along x-direction, magnetic field is applied along z-axis. As you remember, magnetic field curves the trajectories of carriers: trajectories of positive and negative charges are getting curved in opposite directions. As a result, positive and negative charges accumulate on the opposite faces of the material along y-axis, and effective electric field is created. Try to calculate the Hall resistivity defined as

.

This is a rather simple problem – all you need in order to solve it is to recall the basics of electromagnetism. This will also give you some intuition regarding the quantum Hall effect, too.

But let me actually turn to the integer quantum Hall effect. It is possible to explain it using free electron gas model, more accurately – 2 dimensional free electron gas model, since we are essentially dealing with quasi-two dimensional sample – carriers can move freely only in the plane of the sample.

Without magnetic field the spectrum of carriers is continuous. If a magnetic field is applied perpendicular to the plane of the sample, the electron spectrum becomes discrete – it consists of equidistant Landau levels:

, (1)

where is cyclotron frequency and is (effective) electron mass.

The magnetic flux is also quantized – in units

, (2)

and the density of states for every Landau level is given by the density of quanta of magnetic flux:

. (3)

The density of states has the dimension , and there is some area corresponding to every state on every Landau level.

Exercise 2. Calculate this area.

What we always have to recall is that we deal with electron gas at very low temperatures (that’s the regime where the quantum Hall effect is observed) – this means that only electron states with energies below the Fermi energy are occupied.

Imagine now that we change the magnetic field . The position of Landau levels changes (see the formula (1) above), so does its relative position w.r.t. the Fermi energy . If is between neighbouring Landau levels and , all low laying states with , are occupied.

Therefore, one can estimate the overall density of carriers in the sample as

. (4)

If you substitute (4) into the expression for in classical Hall effect, you'll find that the Hall resistivity is equal to

, (5)

where , the formula you saw in my previous post on the subject.

What do we learn from (5)? First of all, we only needed the basics of quantum mechanics – Landau quantization in magnetic field and Fermi-Dirac statistics to derive it. The first lesson from my point of view is that it is always good to look for simpler explanations – although it seems to be the whole particle zoo in the picture below,

the physics behind it is just one of a harmonic oscillator.

The second lesson is that it is relatively clear from these considerations how to shut the effect down. For example, we can increase temperature , and the energy levels will get also filled above . Another way to affect our considerations is to increase the interaction between carriers. As a result, Landau levels will acquire some width, and discreteness in will be killed, if this width becomes comparable to the distance between adjacent Landau levels.

I’ll consider the fractional Hall effect that turns out to be way more complicated tomorrow, but in the mean time please feel free to drop your thoughts on integer quantum Hall effect in comments