Physics 223b: Problem Set 5 - Excitons and Holes in Semiconductors, Assignments of Chemistry

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Physics 223b: Problem Set 5

due Feb. 25, 2009

1. Excitons: Consider a single electron and a hole in a semiconductor, within the effective mass approximation.

(a) Write down the Schr¨odinger equation for the two-particle wavefunction. Solve it for the spectrum of electron-hole bound states. These bound states are called excitons. (b) Estimate the binding energy of an exciton (i.e. the lowest-energy bound state) for silicon and gallium arsenide. Estimate the radius of the exciton. (c) Unlike atoms, excitons can decay, because the electron and hole can annihilate one another. Why might the decay rate for such annihilation processes be slower in an indirect gap semiconductor (e.g. silicon) than in a direct-gap material (e.g. gallium arsenide)?

2. Holes in III-V and similar materials: In many popular semiconductor materials, the states at the top of the valence band arise primarily from the p-orbitals (orbital angular momentum L = 1) of the constituent atoms. In these materials, spin-orbit coupling is not negligible, and already in a single atom, the eigenstates are better thought of as states of total angular momentum J = L ± 1 /2 = 3/ 2 , 1 /2. The states with J = 3/2 turn out to be of higher energy, i.e. the highest occupied states making up the true “valence band”. According to band theory, therefore, the Bloch states can be described by their quasimomentum ~k and some quantum state of total angular momentum J = 3/2. Near the top of valence band, the hamiltonian can be expanded around the maximum energy. Assuming this occurs at ~k = 0 (the “Γ” point – this is indeed the band maximum in many materials), and expanding to second order only, one finds the form

H ≈ Ev −

2 m

(

(γ 1 +

γ 2 )k^2 − 2 γ 2 (~k · J~)^2

)

, (1)

where γ 1 , γ 2 are material-dependent parameters satisfying 0 < γ 2 < γ 1 /2. (This is the so-called “spherical approximation”. Actually another term is allowed by cubic symmetry, which I have dropped to keep it simple.) Here H should be regarded as a 4 × 4 matrix in the J = 3/2 space, and J~ is the 3-component vector of angular momentum operators. They satisfy as usual J~ · J~ = J(J + 1), [Jμ, Jν^ ] = iμνλJλ, with μ, ν, λ = 1, 2 , 3 = x, y, z. You can look up the explicit forms if need be.

(a) By diagonalizing this matrix, find the energies of 4 bands, n(~k), with n = 1, 2 , 3 , 4. Show that they have the usual effective mass form, but appear in two pairs with two different masses. Find these m∗ hh, m∗ lh, for the so-called “heavy holes” and “light holes”, respectively. (b) What is the direction of the average value of the angular momentum (colloquially called “spin”) in each Bloch state with a given ~k and band n, i.e. what is the direction of 〈n; ~k| J~|n~k〉? Sketch this “spin” vector field on a sphere of fixed |k| for one of the heavy-hole bands.