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 5Due March 18, 20041. Consider a two dimensional superconductor with a d-wave energy gap given by∆(φ) = ∆0cos 2φ .Assume an isotropic energy band with Fermi velocity νFin the normal state. Thequasiparticle spectrum is given byE(k) =!ν2F(|k| − kF)2+ ∆2(φ) .(a) Show that the energy gap vanishes at 4 points on the Fermi surface. In thevicinity of these nodal points, show that the quasiparticle dispersion is given byE(k) =!ν2Fk21+ ν22k22,where k1and k2are momentum components perpendicular and parallel to theFermi surface me asured from the nodal points. What is ν2in terms of ∆0andkF? Show that the density of states at energy E per node per spin is12πνFνE.2(b) Show that at low T , thermal excitation of the quasiparticles leads to a linear Treduction of the superfluid densityρsρs(T ) =m2 ln 2(T = 0)m−νFπT .ν2The integral you encounter can be done by a change of variable y = e−x.(c) In the presence of A and ∇θ where θ is the phase of the order parameter, thequasiparticle spectrum is changed by1E(k, A) = E(k) + νF·2θ2"e∇ +Ac#The last term is the gauge invariant generalization of the term we discussed inclass. Consider a single vortex and assume the superconductor is extreme type II.2At a distance R away from the vortex core in the ˆx direction, calculate the densityof states which is generated at the Fermi level. (Assume ξ0# R # λL.) How isyour answer different if you approach the vortex core in the (1,1) direction?(d) In an external field H, a triangular vortex lattice is formed. Show that thedensity of states found in (c) gives rise to the following unusual contribution tothe specific heatcν= α√HT ,Make a crude estimate of the coefficient α.For an experimental confirmation of the prediction first made by G. Volovik,JETP Lett. 58, 469 (1993), see K. Moler et al., Phys. Rev. Lett. 73, 2744(1994).2. Make a table for the real part of the transverse and longitudinal response functionsK⊥and K||. Give the limits ω = 0, q → 0, and q = 0, ω → 0 for a perfect metal,a disordered metal, and a superconductor with or without disorder (16 quantities inall!). Write the leading nonvanishing contributions in terms of physical quantities suchas Landau diamagnetism, conductivity and scattering
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