Stability of the vortex lattice in D-wave superconductors

Use is made of Onsager’s hydrodynamic equation to derive the vibration spectrum of the vortex lattice in d-wave superconductor. In particular the rhombic lattice (i.e. the 45 tilted square lattice) is found to be stable for B > Hcr(t). Here Hcr(t) denotes the critical field at which the vortex lattice transition takes place. Typeset using REVTEX 1 Since the discovery of the triangular vortex lattice in type II superconductors by Abrikosov [1] and others [2,3], the vibrational modes of the vortex lattice have been studied by a number of people [4]. In these works the stability of the triangular vortex lattice in an s-wave superconductor is established. The discovery of hole-doped high Tc cuprate superconductivity by Bednorz and Müller [5] in 1986 and the recent realization [6] that d-wave superconductivity is involved may give a new twist on the whole subject. We have shown earlier that the rhombic vortex lattice (or the 45 tilted square lattice) is stable in the vicinity of the upper critical field in a d-wave superconductor in a magnetic field parallel to the c axis [7]. More recently we have shown [8] the vortex lattice transition from the triangular lattice to the square lattice takes place at a small magnetic field Hcr(t) ∼ κHc2(t), which implies that the vortex lattice should be rhombic in the overwhelming region in the B-T phase diagram (see Fig. 1). Here κ is the Ginzburg-Landau parameter and we have Hcr about a few Tesla in YBCO and Bi2212 at low temperatures. Indeed this critical field Hcr(t) is consistent with the observation of rhombic vortex lattice by SANS [9] and by STM imaging [10] in monocrystals of YBCO at 3 Tesla. The object of this paper is to study the vibrational spectrum of the 45 tilted square vortex lattice in a magnetic field. First following Fetter et al. [4], we study Onsager’s and Landau’s hydrodynamic equation [11,12] for a vortex lattice. Basically we assume that the vortex moves with the local velocity generated by other vortices. We find that the square vortex lattice is unstable when B < Hcr(t) but becomes stable for B ≥ Hcr(t). The vibrational spectrum are determined in the whole Brillouin zone. Second we analyze the vibration spectrum within the time dependent Ginzburg-Landau equation [14] with the Aranov-Hikami-Larkin term [15]. In this limit we have now a set of damped oscillation modes rather than oscillation modes. However the stability condition of the square vortex lattice is the same as in the analysis using Landau’s hydrodynamic equation as expected. For example this new vibration spectrum will be crucial in determining the melting transition line where the vortex lattice melts into a vortex liquid. I. Vibration modes in the square lattice (hydrodynamic limit) As is well known the Landau’s and Onsager’s hydrodynamic equation applies when motion of vortices involves no energy dissipation. Unfortunately this condition is never realized for vortices in a d-wave superconductor since there are the low energy extended states attached to every vortex and they certainly dissipate energy whenever the vortex is in motion [16]. Nevertheless it is of great interest to study this idealized limit. The extended Ginzburg-Landau free energy for the vortex state with ξ ≪ d ≪ λ in d-wave superconductors reduces to [8] Ω = 2πnφξ 2 κ ∑ L φ(rL − r0), (1) φ(r) = −ǫa2ξκ cos 4θr |r|4 +K0 ( r λ )