# HW-8 - Theory of Solids HW-8 Solution By Qifeng Shan...

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1 Theory of Solids HW-8 Solution By Qifeng Shan Ashcroft and Mermin’s book 8-2 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)) g ( ε F ) = mk F / ħ 2 π 2 . Show that the general form (8.63) reduces to this when ε n ( k ) = ħ 2 k 2 / 2 m and the (spherical) Fermi surface lies entirely within a primitive cell. (b) Consider a band in which, for sufficiently small k , ε n ( k ) = ε 0 + ( ħ 2 /2)( k x 2 / m x + k y 2 / m y + k z 2 / m z ) (as might be the case in a crystal of orthorhombic symmetry) where m x , m y , and m z are positive constants. Show that if ε is close enough to ε 0 that this form is valid, then g n ( ε F ) is proportional to ( ε ε 0 ) 1/2 , so its derivative becomes infinite (van Hove singularity) as ε approaches the band minimum. (Hint: use the form (8.57) for the density of levels.) Deduce from this that if the quadratic form for ε n ( k ) remains valid up to ε F, then g n ( ε F ) can be written in the obvious generalization of the free electron form (2.65): 0 2 3 F F n n g (8.81) where n is the contribution of the electrons in the band to the total electronic density. (c) Consider the density of levels in the neighborhood of a saddle point, where ε n ( k ) = ε 0 + ( ħ 2 /2)( k x 2 / m x + k y 2 / m y k z 2 / m z ) where m x , m y , and m z are positive constants. Show that when ε ε 0 , the derivative of the density of levels has the form . , ; constant, 0 2 / 1 0 0 ' F n g Solution: (a) 3D DOS per unit energy and unit volume for isotropic parabolic dispersion relation is:       2 2 2 2 3 3 4 4 1 4 1 k m m k dS g Surface n n k k k (1) Thus, at Fermi energy level,

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2 2 2 F F n k m g (2) ε n ( k )is periodic in the reciprocal lattice. The Fermi surface lies entirely in the primitive
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