Integer quantum hall effect - edge states/bulk effects

[matt_s]

Hi there, I am currently learning about the quantum hall effect and am a bit confused about the edge states picture and how this fits in with the rest of the theory.

In most books/review texts the theory is dicussed from the point of view of an infinite 2D system the magneteic field collapses the density of states in landau levels and then disorder in the system breaks the degeneracy of these landau levels. This broadens the landau levels so that extended states exist at the centtre of the landau levels and localized states in the tails. These localised states mean that there is a mobility gap between the different landau levels and this can then explain some of the features of the experimental data. After books have discussed this idea they present a real system for example hall bar where the energy of the lada level rises at the edges of the smaple becasue of the confinement. Where these landau levels meet the fermi level then 1D channels are formed and these are presented as the means by which curren traveles through the sample. What I am struggling to understand is how this edge state model fits with the infinite sample idea where you get extended staes percolating through the bulk and mobility gaps form localised states. Do you just ignore all these ideas in a real system with confinement at the edges of the sample? How do locallised states fit with edge states? Are the landau states at the edges broadened by disorder?

Thanks

Matt

 

[tayyaba aftab]

hi
m working on quantum hall effect.i have to present it in my department.i don't know where to start from
if some body can help me please
even don't have basics physicsl concepts about the quantum hall effect.
pleasezzzzzzzzzzzzzzzzzzzzzzzzzzzz help me
thanking
tayyaba

 

[xepma]

Hi there, I am currently learning about the quantum hall effect and am a bit confused about the edge states picture and how this fits in with the rest of the theory.

In most books/review texts the theory is dicussed from the point of view of an infinite 2D system the magneteic field collapses the density of states in landau levels and then disorder in the system breaks the degeneracy of these landau levels. This broadens the landau levels so that extended states exist at the centtre of the landau levels and localized states in the tails. These localised states mean that there is a mobility gap between the different landau levels and this can then explain some of the features of the experimental data. After books have discussed this idea they present a real system for example hall bar where the energy of the lada level rises at the edges of the smaple becasue of the confinement. Where these landau levels meet the fermi level then 1D channels are formed and these are presented as the means by which curren traveles through the sample. What I am struggling to understand is how this edge state model fits with the infinite sample idea where you get extended staes percolating through the bulk and mobility gaps form localised states. Do you just ignore all these ideas in a real system with confinement at the edges of the sample? How do locallised states fit with edge states? Are the landau states at the edges broadened by disorder?

Thanks

Matt

What happens at the edge is actually one of the main reasons what makes the quantum Hall effect so special. It is an example of a topological insulator. The quantum Hall effect has a bulk state, which is in a topological phase. One of the characteristic features of such a phase is that this bulk phase has a gap, in this case a mobility gap. Furthermore, as a result of this 'topological behavior', if you put the system next to another phase (such as the vacuum) the system always develops massless degrees of freedom on the edge. These edge states cannot be destroyed or localized by any local perturbation. They are protected.

So the existence of the delocalized states in the bulk and the massless degrees of freedom on the edge are, in fact, both reflections of the special 'topological phase' the system is in.

Inside the bulk there are extended states present. Note however, these states do not contribute to the conductivity of the sample. The edge states are responsible for that. But the edge states only occur because of the extended states in the bulk. The localised states do play some role in this. Namely, if it wasn't for these localised state the special phase could only be reached through fine-tuning of the chemical potential (which needs to sit right in the gap). With the presence of the localised states the chemical potential does not need to be fine-tuned.