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 SolidsProblem Set 6Due April 6, 20041. (a) We can include the effects of Coulomb repulsion by the following effective poten-tial:V (ω) = Vp(ω) + Vc(ω)where Vp= −V0for |ω| < ωDis the phonon mediated attraction and N(0)Vc=µ > 0 for |ω| < EFrepresents the Coulomb repulsion. Write down the self-consistent gap equation at finite temperature. Show that ∆(ξ) is frequency de-pendent even near Tcso that the Tcequation b e comes1 2f(ξ!)∆(ξ) = −N(0)!dξ!V (ξ − ξ!)∆(ξ!)−(1)2ξ!This integral equation is difficult to solve analytically, but we may try the fol-lowing approximate solution:∆(ω) = ∆1, |ω| < ωD= ∆2, |ω| > ωDNow rewrite Eq.(1) as1 2f(ξ!)∆(ξ) = −N(0)!dξ!Vp(ξ!− ξ)∆(ξ!)−+ A (2)2ξ!where!1 − 2f(ξ!)A(ξ) = −N(0) dξ!Vc(ξ!− ξ)∆(ξ!) (3)2ξ!Convince yourself that A(ξ) is a slowly varying function of ξ for ξ << EF, sothat we may approximate A(ξ) by A(0) in Eq.(2). Pro duce an argument to showthat in the region ξ > ωDthe first term in the R.H.S. of Eq.(2) is small comparedwith A so that in fact ∆2≈ A(0). In the same spirit show that2∆1∼ N(0)V0∆1lnωDkTc+ ∆2Combining this with an equation for ∆2using Eq.(3), show that the Tcequationbecomes1 = ln"ωDkTc#(N(0)V0− µ∗) (4)where µ∗=µ1+µ ln(EF/ωD). µ∗< µ is called the renormalized Coulomb repulsion.It can be thought of as an effective repulsion with a cutoff at ωDinstead of EF.Equation (4) shows that the condition for superconductivity is N(0)V0> µ∗andnot N(0)V0> µ. For screened Coulomb repulsion, estimate µ and µ∗for a typicalmetal.(b) Upon isotope substituting M → M + δM, how is the Debye frequency affectedto leading order? Assuming that this is the only effect, how is δTc/Tcrelatedto δM/M, (i) in the absence of Coulomb repulsion, and (ii) including
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