Two dimensions, Bloch electrons

So far we have derived Eq. (32) only for free electrons in two dimensions. The same result is also true for Bloch electrons (i.e. electrons in a periodic potential) as we shall now show. Already that the fact that it does not depend on the electron mass, which could be altered to an effective mass by band structure effects, gives one a suspicion that this might be so. In this section we still consider two dimensions.

The first stage in the proof is to derive Ashcroft and Mermin, Eq. (12.42), relating the period of an orbit of a Bloch electron in a magnetic field to the change in the area of the orbit with energy. We start with the semi-classical equations of motion:

$${\bf v}({\bf k}) = {1 \over \hbar} {\partial \epsilon({\bf k}) \over \partial {\bf k}}$$ (38)

$$\hbar \dot{\bf k} = (-e) {1\over c} {\bf v}({\bf k}) \times {\bf H},$$ (39)

from which it follows that

$$\left\vert \dot{\bf k} \right\vert = {e H \over \hbar^2 c} \... ...n({\bf k}) \over \partial {\bf k}} \right)_\perp \right\vert ,$$ (40)

where $(\partial \epsilon({\bf k}) / \partial {\bf k})_\perp$ is the component of $\partial \epsilon({\bf k}) / \partial {\bf k}$ perpendicular to the field, i.e. its projection in the plane of the orbit. (Note this projection is not necessary in two dimensions considered here, but we include it so that our derivation of Eq. (44) below will also apply in three dimensions.) Hence the period of an orbit is given by

$$T = \oint {dk \over \left\vert \dot{\bf k} \right\vert } = {... ...ial \epsilon({\bf k}) / \partial {\bf k})_\perp \right\vert} ,$$ (41)

where $d k$ is the magnitude of a small element of the orbit, and we integrate around the closed orbit. If we consider two orbits whose difference in energy $\Delta \epsilon$ is small then the region between them in the $k_x$-$k_y$ plane is a ribbon of width ${\bf\Delta}({\bf k})$, see AM Fig. 12.9, where

$$\Delta \epsilon = \left\vert\left( {\partial \epsilon({\bf k... ...k}} \right)_\perp \right\vert \vert{\bf\Delta}({\bf k})\vert .$$ (42)

Hence the period $T$ can be written as

$$T = {\hbar^2 c \over e H} {1 \over \Delta \epsilon} \oint \vert{\bf\Delta}({\bf k})\vert \, dk.$$ (43)

The integral in the last equation is just the area between the two neighboring orbits, $\Delta A$, and so

$$T = {\hbar^2 c \over e H} \left({\Delta A \over \Delta \epsilon} \right).$$ (44)

Note that with a free electron band structure, $\epsilon = \hbar^2 k^2 / 2m, \ A = \pi k^2$, this gives $T = 2\pi / \omega _c$, where $\omega _c$ is the cyclotron frequency, as expected.

The second stage of the proof is to use another relationship involving the period of the orbits, the Bohr-Sommerfeld correspondence principle, which states that if $\epsilon_p$ and $\epsilon_{p+1}$ are two adjacent energy levels with quantum numbers $p$ and $p+1$ where $p$ is assumed large, then

$$\epsilon_{p+1} - \epsilon_p = {h \over T} ,$$ (45)

where $T$ is the period of the motion of a semiclassical wavepacket on an orbit with energy centered on $\epsilon_p$. This can be derived by noting that the wavepacket is built up out of adjacent energy levels, and its motion comes from interference between different levels. This requires that the levels around $\epsilon_p$ be uniformly spaced with spacing $\hbar\omega = h/T$, where $\omega$ is the angular frequency of the orbit. (Note that the ``correspondence principle'' really only comes in with the further remark that $T$ is also the period of a purely classical particle of the same energy.)

Applying this relation to Eq. (44) in which we take $\Delta \epsilon$ to be the difference in energy between adjacent Landau levels, $\epsilon_{p+1} - \epsilon_p$, then the area between the semiclassical orbits of two adjacent Landau levels is given by

$$\Delta A = {2 \pi e H \over \hbar c} .$$ (46)

This elegant result, first obtained by Onsager, can be reexpressed by stating that at large $p$ the area $A_p$ inside the Landau level $p$ is given by

$$A_p = (p + \lambda) {2 \pi e H \over \hbar c} ,$$ (47)

where $\lambda$ is some number independent of $p$. This is the same as Eq. (35) for free electrons, except that $\lambda$ need not equal 1/2. The periodicity in $1/H$ in Eq. (37), obtained from Eq. (35), follows equally from Eq. (47).