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.18.512 Theory of Solids Problem Set 10 Due May 6, 2004Consider the non-interacting problem of an impurity d orbital at energy !dhybridizing withmatrix element Vkwith a conduction band. In class we showed the relation!nd" = δ(!F)/π (1)where∆δ(E) = tan−1, (2)!"d− E!"d= !d+ Σ"(E) , ∆ = −Σ""(E) and2Σ(E) =!|Vk|k. (3)E − !k+ iηYou should have complained that Eq.(1) cannot be correct, because according to Friedelsum rule, the phase shift at the Fermi energy is related to the total charge accumulationaround the impurity site, not just the charge !nd" which sits on the impurity site. Theconduction electron charge accumulated around the impurity is missing in Eq.(1). The goalof this problem is to account for this missing charge.1. Show thatδ(E) = Im{ln Gd(E)}where Gd(E) = !d|(E − H + iη)−1|d" .2. Show that"!F∂dE−∞!ln Gd(E) =∂E−"FdEGd(E)−∞#dΣ1 −(4∂E$)Hint: Start with the identity∂11∂Eln G−∂G−dd∂E23. By taking the imaginary part of both sides of Eq.(4), show thatδ(!F) = π!nd" − Im"!F−∞dEGd∂Σ∂E(5)The second term on the right hand side is the correction of Eq.(1). Note that in classwe have assumed ∆ and Σ"to be constant, which explains why this term was missingin Eq.(1).4. Show that the conduction electron charge accumulated in the vicinity of the impurityis given byδρc= Im!k"!F−∞dE%Gk(E) − G0k(E)&(6)where Gk= !k|(E − H + iη)−1|k" and G0k= (E − !k+ iη)−1is the conduction bandGreen function in the absence of the impurity orbital.5. Show that the second term on the R.H.S. of Eq.(5) is just π(δρc) so that the Friedelsum rule is satisfied. Hint: Write down an equation for Gkfollowing the method weused to derive the Gdequation.6. Estimate the energy dependence of Σ and argue that the correction to Eq.(1) is smallwhen ∆ $ !F, i.e., when the resonance is
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