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MIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.€€€8.512 Theory of Solids Problem Set #2 Due: February 24, 2004 1.) Estimate the mean free path for plasmon production by a fast electron through a metal by the following steps: (a) for small q it is a good approximation to assume that ε−1 (q,ω) is dominated by the plasmon hole, i.e. 1Im− ≈ A(q)δ( ω−ωε(q,ω) pl) .Determine the constant A(q), using the f-sum rule. (b) Using (a), write down an expression for the probability of scattering into a solid angle Qˆ by emitting a plasmon. (c) Estimate the mean free path for plasmon emission. Put in some typical numbers (electron energy = 100 KeV, etc.) 2.) Dielectric constant of a semiconductor. (a) In a periodic solid, show that the dielectric response function is given within the random phase approximation by r r r r r r 2k q G eiq ⋅ r k f ( )f 4+ e2 + ( εk −π(εr r r k + q +G)ε)(q,ω)= 1+ 2 ∑(1) qk,G εr r r r + kiηk + −ε−ω−q Gr where G are the reciprocal lattice vectors. (b) We will evaluate Eq. (1) in an approximate way for a semiconductor in the limit ω = 0 and q → 0. We will work in the reduced zone scheme. Argue that the energy denominator in Eq. (1) may be replaced by the energy gap Δ. To estimate the numerator, derive the following theorem: €€€€€€r r r q⋅ a 22 h2q= (2)b ei∑ (εb −εa ) 2mb where ab and are the eigenstate of a Hamiltonian H with a kinetic energy term −h∇2/2m. This is a generalization of the f-sum rule in atomic physics. It is proven by evaluating the expectation value of [[ H,eir q ⋅ r r ],e−ir q ⋅ r r ] in the state a . (c) By making the further approximation that the energy difference in Eq. (2) may be replaced by the energy gap Δ, show that for a semiconductor ε(q → 0, ω = 0)= 1 +  hωpl  2 Δ where ωpl is the plasma frequency. Estimate ε for Si and Ge and compare with


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