Set of supplementary notes.

5.3. SEMICLASSICAL DYNAMICS 81
Figure 5.8: The valence charge density for Ge, GaAs, and ZnSe from an early pseudopotential
calculation, plotted along a surface in a 110 plane that contains the two atoms of the unit
cell. Note the (pseudo-)charge density shifting from the centre of the bond in Ge to be almost
entirely ionic in ZnSe. [M.L.Cohen, Science 179, 1189 (1973)]
If a force F is applied to a particle, the rate of doing work on the particle is
which leads to the key relation
dɛ k
dt = dɛ dk
dk dt = F v g (5.15)
h¯ dk
dt = F = −e(E + v ∧ B) = −e(E + h¯ −1 ∇ k ɛ(k) ∧ B) (5.16)
where we have introduced electric E and magnetic B fields.
The effect of an electric field is to shift the crystal momentum in the direction of the field,
whereas the effect of a magnetic field is conservative - the motion in k-space is normal to the
gradient of the energy. Thus a magnetic field causes an electron to move on a line of constant
energy, in a plane perpendicular to the magnetic field. This property is the basis of magnetic
techniques to measure the fermi surface of metals.
Bloch oscillations
Suppose we have a one-dimensional electron band, such as shown in in Fig. 5.9. The group
velocity is also shown — note that it reaches maximum size about half way to the zone boundary,
and then decreases to zero at the zone boundary. If an electron in this band were subject to a
constant electric field, we get
k(t) = k(0) − eEt , (5.17)
h¯