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 5 Due March 18 2004 1 Consider a two dimensional superconductor with a d wave energy gap given by 0 cos 2 Assume an isotropic energy band with Fermi velocity F in the normal state The quasiparticle spectrum is given by E k F2 k kF 2 2 a Show that the energy gap vanishes at 4 points on the Fermi surface In the vicinity of these nodal points show that the quasiparticle dispersion is given by E k F2 k12 22 k22 where k1 and k2 are momentum components perpendicular and parallel to the Fermi surface measured from the nodal points What is 2 in terms of 0 and kF Show that the density of states at energy E per node per spin is 1 E 2 F 2 b Show that at low T thermal excitation of the quasiparticles leads to a linear T reduction of the superfluid density s 2 ln 2 F s T T 0 T m 2 m The integral you encounter can be done by a change of variable y e x c In the presence of A and where is the phase of the order parameter the quasiparticle spectrum is changed by 1 E k A E k F 2 2e A c The last term is the gauge invariant generalization of the term we discussed in class Consider a single vortex and assume the superconductor is extreme type II 2 At a distance R away from the vortex core in the x direction calculate the density of states which is generated at the Fermi level Assume 0 R L How is your 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 the density of states found in c gives rise to the following unusual contribution to the specific heat c HT Make a crude estimate of the coefficient 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 functions K and K Give the limits 0 q 0 and q 0 0 for a perfect metal a disordered metal and a superconductor with or without disorder 16 quantities in all Write the leading nonvanishing contributions in terms of physical quantities such as Landau diamagnetism conductivity and scattering lifetime
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