# Physics 489 Problem Set 7 Due Thursday, Nov. 12

```Physics 489 Problem Set 7
Due Thursday, Nov. 12
(1) Coulomb blockade energies: Kittel 18.5.
(2) MOSFET sub-bands: Kittel 17.2.
(3) Cyclotron orbits: Kittel 8.4.
(4) Consider a simple cubic lattice with an energy band similar to our result for an s-orbital
tight-binding band:
ε (k) = −2γ coskx a + cosky a + coskz a .
(
)
(a) Find a general expression for the real-space
trajectory of an electron in the semiclassical

model, given a uniform electric field E (direction arbitrary). Assume for simplicity that the
electron starts at k = 0 at t = 0. For this, note that you first need to find k vs. time, leading to a
relation for the velocity vs. time, and this needs to be integrated
 to obtain position vs. time.
(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 is E = (1.0 V/cm) &times; (3, 1, 0 ) , and the total bandwidth of this band is 2
eV, what is the real-space dimension of the Bloch oscillation in this case?
(5) For the tight-binding band described in the above problem,

(a) find the effective mass tensor (for a general k ).

(b) for k very close to the zone center, show that the effective mass is isotropic and
positive, and find the value.
(c) Similar to (b), for a Fermi surface just below the top of the band, find the value of the
isotropic effective mass for holes. Also find a saddle point for the energy vs. k for this band,
and find the effective mass tensor at that point.
(d) What range of Fermi energies will exhibit open orbits for this band?
```