Quasi-particle Lifetimes in a D X 2 −y 2 Superconductor Typeset Using Revt E X

We consider the lifetime of quasi-particles in a d-wave superconductor due to scattering from antiferromagnetic spin-fluctuations, and explicitly separate the contribution from Umklapp processes which determines the electrical conductivity. Results for the temperature dependence of the total scattering rate and the Umklapp scattering rate are compared with relaxation rates obtained from thermal and microwave conductivity measurements, respectively. Typeset using REVTEX [email protected] [email protected] [email protected] 1 It is now widely accepted that the superconducting state of the high Tc cuprates is characterized by a dx2−y2-order parameter [1,2]. Furthermore, various transport properties such as NMR relaxation times [3,4] and the microwave conductivity [5] have been calculated within a quasi-particle BCS-RPA framework which phenomenologically takes into account the interplay of both the dx2−y2 pairing and the antiferromagnetic spin correlations. In particular, this approach has previously provided reasonable fits of the temperature dependence of the NMR T1 and T2 relaxation times in optimally doped YBa2Cu3O7−δ [6,7]. With recent advances in the preparation of YBCO crystals [8,9], microwave measurements [10] at a variety of higher frequencies, and κxy thermal conductivity measurements [11], one has new information on the quasi-particle lifetime involved in the electrical conductivity and the mean-free path associated with the thermal conductivity. Several puzzles have presented themselves. First, microwave measurements appear to indicate that the relaxation rate, long known to collapse rapidly below the critical temperature [12–14], varies as T , or possibly exponentially, at temperatures T ≪ Tc. On the other hand, the theory of the quasi-particle relaxation rate in a d-wave superconductor due to electron-electron interactions predicts a T 3 variation above a scale set by impurity scattering [3,15,16]. A solution to this discrepancy has recently been suggested by Walker and Smith [17], who noted that in the absence of disorder, the transport lifetime entering the microwave conductivity arises solely from Umklapp processes, which are always gapped below Tc even in a d-wave superconductor, leading to an exponential decay 1/τU ∼ exp(−∆U/T ). Secondly, while early thermal conductivity experiments extracted quasi-particle mean-free paths in apparent agreement with microwave measurements, recent measurements indicate that on cleaner crystals, where the mean-free path due to inelastic scattering can be probed to lower temperatures, the extracted meanfree path is shorter than in the microwave case [11]. Furthermore, the inelastic contribution to the inverse thermal mean free path appears to follow a T 3 temperature dependence at low temperatures (see below). Motivated by the proposal of Walker and Smith [17] regarding the role of Umklapp pro-