MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 1 8 512 Theory of Solids Problem Set 10 Due May 6 2004 Consider the non interacting problem of an impurity d orbital at energy d hybridizing with matrix element Vk with a conduction band In class we showed the relation nd F 1 where E tan 1 d d E E k d E E 2 and Vk 2 E k i 3 You should have complained that Eq 1 cannot be correct because according to Friedel sum rule the phase shift at the Fermi energy is related to the total charge accumulation around the impurity site not just the charge nd which sits on the impurity site The conduction electron charge accumulated around the impurity is missing in Eq 1 The goal of this problem is to account for this missing charge 1 Show that E Im ln Gd E where Gd E d E H i 1 d 2 Show that F ln Gd E dE E Hint Start with the identity E 1 ln G d F d dEGd E 1 E 1 G d E 4 2 3 By taking the imaginary part of both sides of Eq 4 show that F nd Im F dEGd E 5 The second term on the right hand side is the correction of Eq 1 Note that in class we have assumed and to be constant which explains why this term was missing in Eq 1 4 Show that the conduction electron charge accumulated in the vicinity of the impurity is given by c Im k F dE Gk E G0k E 6 where Gk k E H i 1 k and G0k E k i 1 is the conduction band Green function in the absence of the impurity orbital 5 Show that the second term on the R H S of Eq 5 is just c so that the Friedel sum rule is satisfied Hint Write down an equation for Gk following the method we used to derive the Gd equation 6 Estimate the energy dependence of and argue that the correction to Eq 1 is small when F i e when the resonance is narrow
View Full Document
Unlocking...