Homework #6

Electronic band structure

1. Compute and plot E(k) for the free electron model along the following directions in the first Brillouin zone:

(a) (2 pt.) Simple cubic, from Γ to R,

(b) (2 pt.) Body-centered cubic, from Γ to N,

(c) (2 pt.) Face-centered cubic, from Γ to X,

(d) (2 pt.) Hexagonal close-packed, from Γ to A.

Information about high-symmetry points in the different lattices is listed HERE. Consider also contributions from k points in the higher Brillouin zones of the reciprocal lattice [using E(k)=E(k+K)] and determine few lowest lying energy eigenvalues at each of the above high-symmetry points.

2. Kronig-Penney Model.
Consider a l-dimensional crystal, where the potential is given by a chain of δ-functions:

V(x)=aV ∞ Σ n=-∞  δ (x-na) .

(a) (4 pt.) Show that the solutions of the Schrödinger equation satisfy the dimensionless Kronig-Penney equation

cosK = cosα+ P sinα α   ,

where P=Vma²/(^h/_2π)², α² =(2ma²/(^h/_2π)²)E, and K=ka. (Hint: Solve the Schrödinger equation in regions V(x)=0 to obtain a continuous wave function ψ(x), noting that ψ'(x) is discontinuous at x=na. Use the Bloch theorem ψ(x+a)=e^ika ψ(x) to determine the remaining free coefficients).

(b) (2 pt.) Plot f(α)= cosα+P (sinα)/ α as a function of α. Show that for V≠0, the Kronig-Penney equation cos K = f(α) leads to forbidden α regions (energy gaps).

(c) (2 pt.) Show E(k) (in the reduced zone scheme) in the first Brillouin zone and compare it to the free electron model.

(d) (2 pt.) For the lowest band, expand E(k) in a Taylor series around k=0. Show that E(k)≈E₀(k)+((^h/_2π)²/2m^*)k², where E₀(k) is the free electron value. Determine the effective electron mass m^* and show that m^*>m.

(e) (2 pt.) Show that the magnitude of the gaps at high energy is twice the potential.