Vortex contribution to specific heat of dirty d-wave superconductors: Breakdown of scaling

Abstract

We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution δC(H) ∼ H 1/2 . We show that the extended states continue to dominate the dirty limit, but lead to a H log H behavior at the lowest fields, Hc1 < ∼ H ≪ Hc2. This crossover may explain recent discrepancies in specific heat measurements at low temperatures and fields in the cuprate superconductors. We discuss the range of validity of recent predictions of scaling with H 1/2 /T in real samples.PACS Numbers: 74.25.Fy,74.25.Jb Introduction. With the growing consensus that the symmetry of the cuprate superconductors is d-wave [1] has come a renewed interest in the properties of the vortex state in "unconventional" superconductors with order parameter nodes. Many of the basic ideas about how this state differs from the conventional Abrikosov state in classic superconductors were worked out already in the context of rotating 3 He and heavy fermion superconductors. Recently, however, a number of novel features of the problem peculiar to those systems with Dirac spectrum (line nodes in 3D or point nodes in 2D with order parameter vanishing linearly with angle on the Fermi surface) have been pointed out. Volovik [2] showed that, in contrast to conventional superconductors, extended quasiparticle states with momentum k near order parameter nodal directions k n dominate the density of states at zero energy. This leads to a specific heat which varies as δC(H) ∼ H 1/2 , in contrast to classic superconductors, where localized quasiparticle states in vortex cores lead to a scaling of δC(H) ∼ H since the number of vortices scales proportionally to the field. Simon and Lee [3] then showed that thermal and transport properties exhibit a scaling with H 1/2 /T , again arising simply from the low-energy Dirac form of the electronic spectrum.The predicted proportionality of the electronic specific heat to √ H was in fact identified in measurements on high quality single crystals by Moler et al.,[4] one of the crucial early experiments lending credence to the d-wave hypothesis. However, the interpretation of the observed √ H dependence has been questioned by Ramirez [5] who points out that there are well-known cases where classic superconductors show a similar "nonanalytic" behavior sufficiently close to the lower critical field H c1 . Furthermore, experimental results of Fisher et al [6] and Revaz et al. [7] cannot be well fit by a √ H form. The above scaling predictions hold, strictly speaking, for clean d-wave superconductors and for energy scales small compared to the maximum gap scale ∆ 0 . To make realistic predictions for experiments, deviations from scaling due to disorder and other real-materials effects must be accounted for. In this work, we study the

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“…(4),(5) still hold in the superclean limit where (∆Γ) 1 2 v √ eH with Γ denoting the quasiparticle scattering rate in the normal state [3,4]. There are only two energy scales in this limit, namely v √ eH and T. Therefore the thermodynamic functions obey the scaling law as proposed by Simon and Lee [26] and shown for a particular case by Kübert and Hirschfeld [19].…”

“…Here ... means average over the Fermi surface and over the unit cell of the vortex lattice. In the present configuration this is readily done [2,19] …”

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“…Overall, γ v (H) is better represented by the solid straight line, which has a slope 0.75 mJ K -2 mol -1 T -1 , than the dashed curve for H 1/2 . (The HlnH dependence suggested for a dirty d-wave superconductor 27 would not give a better fit.) For this reason, and particularly because the low-field data suggest a finite limiting slope, these results are more consistent with an isotropic gap than with the low-energy excitations associated with nodes.…”

“…It has been suggested that this H 1/2 dependence is modified to HlnH at low fields in a dirty superconductor. 27 Modifications of the H-proportional dependence in the case of an isotropic gap, negative curvature in high fields, have also been suggested 28 . The H dependence of γ v (H) is compared with H and H 1/2 dependences in Fig.…”

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“…This work has been extended into several directions: a) thermodynamic functions; b) thermal conductivity; c) scaling relations; d) for arbitrary field orientation; and e) for a variety of gap functions ∆(k) ′ s [14,15,16,17,18,19]. As is well known the gap symmetry and the gap function has been the central issue since the discovery of the heavy-fermion superconductors [20].…”