9780030839931, Chapter 8, Problem 2P

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Chapter 8, Problem 2P Problem Density of Levels (a) In the free electron case the density of levels at the Fermi energy can be written in the form (Eq. (2.64)) = Show that the general form (8.63) reduces to this when = h2k2/2m and the (spherical) Fermi surface lies entirely within a primitive cell. (b) Consider a band in which, for sufficiently small = + (h2/2)(kx2/mx + k2/my + k2/mz) (as might be the case in a crystal of orthorhombic symmetry) where mx, my, and mz are positive constants Show that if is close enough to that this form is valid, then gn(ε) is proportional to - so its derivative becomes infinite (van Hove singularity) as E approaches the band minimum. (Hint: Use the form (8.57) for the density of levels.) Deduce from this that if the quadratic form for en(k) remains valid up to EF then can be written in the obvious generalization of the free electron form (2.65). = 2 3 n (8.81) where n is the contribution of the electrons in the band to the total electronic density. (c) Consider the derivative of the density of levels in the neighborhood of a saddle point, where = + (h2 /2)(kx2/mx + ky2 kz2/mz) where mx, my, mz are positive constants. Show that when the derivative of the density of levels has the form ≈ constant, ≈ (ε₀ , (8.82) Step-by-step solution There is no solution to this problem yet. Get help from a Chegg subject expert. Ask an expert