Solid State Communications, Vol. 14, pp. 1251—1252, 1974. Pergamon Press Printed in Great Britain DENSITY OF STATES IN DIRTY TYPE II SUPERCONDUCTORS NEAR T~ Lorenz Kramer, WerneT Pesch* and Richard Watts..Tobin’~ Institut für Festkörperforschung der Kemforschungsanlage Jiilich, 517 Jtilich, West-Germany (Received 19 February 1974 by G. Lie! ned) We have determined the behavior of the density of states in the mixed state of superconducting alloys for T -. T~. The local density of states tends towards the BCS expression with the order parameter playing the role of the energy gap. The singularities are smeared Out by the spatial average The effective normal core radius of a vortex diverges like (1 — T/T~)’~ for T -+ T~ unlike the coherence length which diverges like (1 — Tf THE DENSITY of states N(~) in dirty type II super. N(w) = N 0 Re n/(t~ 2 — ~2)l/2 ; i~ = ~ + ie~/2 (1) conductors near the upper critical field H~ 2 has been calculated for the first time by De Gennes.’ his ex- where N0 is the normal density of states at the Fermi pression corresponds to the leading terms of a series surface. expansion in powers of ‘y= (A/co )2, With A = ((A 2(r)))”2 ((...> denotes the spatial average) Taking the limit t -+ 1 and keeping B/HC 2 fixed and co = 4irBf~0. Here B is the spatial average of the equation (1) goes over into the BCS density of states magnetic field and $~ is the flux quantum (the with the characteristic square-root singularity at reduced units introduced previously 2 are used through- (A) A. At ~o = 0 the density of states vanishes like out this paper). Note that 4ir/e 0 corresponds to the (1 — r)” 2 for t -~ 1. area of the unit cell of the vortex lattice. I~ the following we use an alternative approach Near T~ the expansion breaks down because based on the Eilenberger equations,7 as simplified by diverges like (1 — t)’ (t = T/T~) if one keeps the Usadel8 for the dirty limit. Equation (I) can then be ratio BfH~ fixed. One is indeed faced with a similar rederived using some reasonable looking approxi- situation in the clean case where the corresponding mations.9 We show, however, that the density of expansion parameter near H~, 2 is A/w. In order to states given by equation (1) has the wrong behavior circumvent the obvious breakdown for ~ near T~ , so that the method of reference 6 has not in frequencies 3 a kind of partial summation has been fact resolved the difficulties near T~. For simplicity applied more heuristically by Maki4 and systemati- we argue within the circular cell approximation of the cally by Brandt et aL5 Extending the method of vortex lattice.2 The results, based mainly on scaling reference 5 to the dirty limit Takayama and Maki6 arguments, are not affected by this approximation. have recently proposed the following expression for In general the density of states N(~) is given by the density of states per unit area Im<GQ.s.,, r, r)) where G is the analytic continuation of the thermal Green’s function onto the real axis. *On Leave from Abteilung fir Theoretische Fest- In our case this reduces to2 körperphysik am Institut für Angewandte Physik der Universit~t Hamburg, Germany ton Leave from Department of Physics, University of N(~) = Re (cos $ (r, )) (2) Lancaster, England. where 1’ satisfies the equation of motion 1251