Square lattice electron gas

[noospace]

Hi all,

I'm struggling to understand the relationship between electrical conductivity and Bragg reflection in a 2D square lattice free electron gas with lattice spacing [itex]a[/itex].

Is it the case that Bragg reflection in the electron gas results in electrical resistivity?

My understanding of Bragg reflection is that it will occur when the change in wave-vector is equal to some reciprocal lattice vector.

But what happens if the radius of the Fermi circle is less than half the spacing of the reciprocal lattice points [itex]2\pi/a[/itex]? In this case it should be impossible for any wavevector in the electron gas to undergo Bragg reflection. Does that mean that the electron gas has zero resistance in any direction?

If we compress the lattice into a rectangular shape, then it may be the case that Bragg reflection can occur in a specific direction. Would this then increase the resistance in the same direction?

If anyone would be able to comment on this logic it would be much appreciated.

 

[Jhero]

Hi there,

Your understanding of Bragg reflection is correct - it occurs when the change in wave-vector is equal to a reciprocal lattice vector. In a 2D square lattice, the reciprocal lattice vectors are given by 2\pi/a in both the x and y directions. So, if the radius of the Fermi circle is less than half this value, then no wavevector in the electron gas will undergo Bragg reflection. This means that the electron gas will have zero resistance in any direction, as there will be no scattering of electrons due to the lattice.

However, if we compress the lattice into a rectangular shape, the reciprocal lattice vectors will change and Bragg reflection may occur in a specific direction. This would result in an increase in resistance in that specific direction, as the electrons will experience scattering due to the lattice.

In general, the relationship between electrical conductivity and Bragg reflection in a 2D square lattice free electron gas is complex and depends on various factors such as the Fermi energy, lattice spacing, and lattice shape. It is important to consider all of these factors when trying to understand the behavior of electrons in such a system.

I hope this helps to clarify things for you. Let me know if you have any further questions.