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 II Problem Set 7 Due April 6 2009 1 Optical conductivity of disordered superconductors Following our discussion of disordered metals the optical conductivity of a disordered superconductor is given by the Kubo formula which is easily derived by considering the rate of absorption of electromagnetic radiation 1 0 q 0 n drjxP r n 2 En E0 1 where is the volume The paramagnetic current operator is written in the exact eigenstate representation as drjxP r e v c c 2 and v 1 m dr x i 3 is the velocity matrix elements between exact eigenstates of the Hamiltonian H1 which describes free fermions with a disordered potential H1 4 In Eq 1 0 and n are the ground and excited states of the BCS mean field Hamiltonian in the presence of disorder a Using the Bogolinbov transformation show that q 0 where e2 u v v u 2 v 2 E E 5 2 1 1 2 E 1 2 v 1 2 E E 2 2 u2 6 7 8 By defining f 1 v 2 9 show that Eq 5 can be written as e2 q 0 d d uv vu 2 f E E 10 where the relation between u v E and is given by Eqs 6 8 b Show that 1 uv vu 2 2 2 1 EE EE 11 c Note that f depends only on the normal state properties Indeed it appeared in our treatment of disordered metals By factorizing the impurity average argue that f can be approximated by a constant for small and i e for energies near the Fermi level More accurately 1 Amuse yourself by trying to point out at what step in the argument was this condition imposed By taking the limit 0 show how the expression we derived for the normal state conductivity N can be recovered Note how the spin sum is magically included d Show that Eq 10 simplifies to 1 q 0 N d 0 d 0 2 1 EE E E 12 This is known as the Mattis Bardeen formula By changing the integration vari able from to E Eq 12 reduces to a one dimensional integral which can be 3 done numerically or by mathematics Sketch N and comment on the key fea tures We emphasize that the Mattis Bardeen formula is valid only for disordered superconductors and for 1
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