Physics 617 Problem Set 4
Due Weds, March 9
[1] Ashcroft-Mermin #17.4
[2] Consider a simple cubic lattice with an energy band similar to our result for an s-orbital tightbinding band:
ε(k) = –2γ (coskxa + coskya + coskza).
(a) Find a general expression for the real-space trajectory of an electron in this band, in the
semiclassical model, given a uniform electric field E (including an arbitrary general direction;
write answer in terms of the components). Assume for simplicity that the electron starts at k = 0
at t = 0.
(b) Find the specific trajectory in real space for the case of E which points along (3, 1, 0), and
sketch this trajectory.
(c) If the electric field has components (3.0 V/cm, 1.0 V/cm, 0), and the total bandwidth of the
band is 2 eV, what is the real-space spatial extent of the Bloch-oscillating electron in this case?
[3] For the tight-binding band described in the above problem,
(a) find the effective mass tensor, for a general vector k in the first Brillouin zone.
(b) For k very close to the zone center, show that the effective mass is isotropic and positive.
(c) show that for a Fermi surface just below the top of the band, the particles are holes, with
isotropic negative effective mass tensor, and the hole pockets are spherical.
(d) find a saddle point for this band, and find the effective mass tensor at that point.