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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) × (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?