Department of physics, NTNU TFY4340 Mesoscopic Physics Spring 2010

Exercise 1

Assistance: Monday January 25.

Question 1

Show that V_{3}Ṽ_{3} = (2π)^{3}. Here, V_{3} = | (a_{1} × a_{2}) · a_{3}| and Ṽ_{3} = | (b_{1} × b_{2}) · b_{3}| are the primitive cell volumes in direct and reciprocal space, respectively, and b_{1} = 2πa_{2} × a_{3}/V etc. Hint: Use the vector identity

(A × B) × C = (A · C)B − (B · C)A.

Question 2

Calculate the reciprocal lattice of graphene, i.e., a single layer of graphite, and construct its first Brillouin zone (1BZ). Check that AÃ = (2π)^{2}. Note that the basis consists of two carbon atoms:

Hint: b_{i} · a_{j} = 2πδ_{ij}, where δ_{ij} is the Kronecker delta: δ_{ij} = 1 if i = j, δ_{ij} = 0 if i ≠ j.

Question 3

We have calculated the density of states (DOS) in a 3D system of volume V:

D_{3}(E) = \frac{V}{2π^{2}} · (\frac{2m}{h̄^{2}})^{3/2} · E^{1/2}.

Show that the DOS in a 2D and in a 1D system are proportional with E^{0} (i.e., independent of energy E) and E^{−1/2}, respectively. Use the free–electron model (V = 0), assume systems of area A = L^{2} in 2D and length L in 1D, and determine the prefactors C_{2} and C_{1} in D_{2}(E) = C_{2}E^{0} = C_{2} and D_{1}(E) = C_{1}E^{−1/2}, respectively.