Magnetoresistance for Open Orbits

A good feel for what saturation/non-saturation of magneto-resistance represents can be found around p235-239 in Ashcroft and Mermin.

For the open orbit case we discussed in the lecture the form of the current:

j = sigma(0)n(n.E) + sigma(1).E

Where n unit vector in the direction of the open orbit. It was argued that the component sigma(1) vanishes in the high-field limit, whereas the sigma(0) contribution does not. This means that unlike the closed-orbit case the magneto-resistance will not approach a constant.

To see this last assertion it is useful to examine a simple case where in the high-field limit the current flow is directed in a direction =/ n, say n'. From our statement above this means that the projection of the electric field, E.n = 0.

Following the working in Ashcroft and Mermin, this gives a particular form for the electric field which I will not write here. Suffice to say, defining the magneto-resistance rho as;

rho = E.j/|j|

One can arrive at the expression:

rho = (n'.j)^2/(n'.sigma(1).n')

Now as we have previously argued that sigma(1) vanishes in the high-field limit we clearly see that the magneto-resistance must diverge, i.e. it grows with increasing magnetic field.