Superconducting quasiparticle lifetimes due to spin-fluctuation scattering

PHYSICAL REVIEW B VOLUME 49, NUMBER 2 JANUARY 1994-II 1 lifetimes due to spin-fluctuation scattering S. M. Quinlan and D. J. Scalapino Department of Physics, University of California, Santa Barbara, California 93106-9530 N. Bulut Department of Physics, University of Illinois at Urbana C— hampaign, Urbana, Illinois 61801-3080 Superconducting quasiyarticle (Received 27 August 1993) Superconducting quasiparticle lifetimes associated with spin-fluctuation scattering are calculated. A Berk-Schrieffer interaction with an irreducible susceptibility given by a BCS form is used to model the quasiparticle damping due to spin fluctuations. Results are presented for both s-wave and d-wave gaps. Also, quasiparticle lifetimes due to impurity scattering are calculated for a d-wave superconductor. In the traditional low-temperature superconductors, the dominant dynamic quasiparticle relaxation processes involve the electron-phonon interaction. In these materials there are inelastic-scattering events in which a quasiparticle emits or absorbs a phonon, as well as events in which two quasiparticles recombine to form a pair or in which a pair is broken into two quasiparticles with the emission or absorption of a phonon, respectively. In some materials with large Debye energies, dynamic electron-electron scattering processes can play a role at low temperatures. In the cuprates, however, the dynamic electron-electron scattering processes are enhanced by the existence of strong short-range antiferromagnetic spin fluctuations. We have previously found' that the existence of such spin fluctuations can account for the terntime perature dependence of the nuclear-spin-relaxation T, . Below ?; the temperature dependence of T, could form for be fit by using a random-phase-approximation y(q, to) with an irreducible susceptibility given by the BCS susceptibility with a d-wave gap. Here we extend this approach to determine the scattering and recombina- tion quasiparticle lifetimes for both s- and d-wave gaps. We compare our results with recent transport lifetimes reported by Bonn and co-workers. We also examine the influence of impurity scattering on the low-temperature quasiparticle lifetime for the d-wave case. Our approach is similar to the quasiparticle lifetime calculations of Kaplan et al. for the low-temperature superconductors except that we assume that antiferromagnetic spin fluctuations rather than phonons provide the dominant relaxation mechanism. Specifically, we consider a Hubbard model on a two-dimensional lattice with a near-neighbor hopping t and an on-site Coulomb interaction U. We approximate the spin-fluctuation interaction V(q, co) which enters the self-energy by =— ', U/[1 V(q, co) — —Uy (q, co)] . Here U is a reduced effective interaction chosen along with the filling ( n ) = ( n; t+ n; i ) to adjust the strength of the antiferromagnetic is the spin fluctuations, and go BCS susceptibility I (q~)=, Bcs Xo d 2p l — 1+ p+q 1— +— 4 E'p + p +q E.+.Ep +q Er ' p ilmk —f (Ep) z)+iO+— (Ev+q E— 1 f (Ep+q ) —f—(Ev ) f (Eviq p Ep+q Ep 2 (2m) p+q p to La! p ) to+(Ez+q+Ez)+i0+ —1 iO+ to (Ez+q+Ez)+ — E.+.Ep with E~ = Qe&+b, and ez= — 2t(cosp„+cosp~ ) —JLt. 1— p+. +— 4 +~p+q~p f (Ev+q )+f (Ep ) (2) Results will be presented This has the usual coherence factors z for both an s-wave gap 5, =bz(T) and a d-wave gap bd(p) =(bo(T)/2)(cosp„—cosp ). In the following, we will use the parameters U/t =2 and (n ) =0. 85, which are similar to those used previously in our analysis of the NMR data. ' We also assume that b, z( T) has the usual mean-field temperature dependence. With the interaction given by Eq. (1) and following the approach described in Ref. 3, we find that the inverse lifetime for a quasiparticle of energy ~ and momentum p is given by co — iA, 6 Ap p', Q)5(co —Q — '(p, co)= E ) 1+ dQImV(p — — [n(Q)+1][1—f(co —Q)] T ' f (2~)' f i ~ to(to Q) co) [n (Q)+ 1] (Q — f + f 0 dQImV(p 0163-1829/94/49(2)/1470(4)/$06. 00 —p', Q)5(Q+to —E, ) 1+ 49 1470 co(Q+co) n( )[Q1 f(Q+to)] 1994 ' . —(3) The American Physical Society BRIEF REPORTS Here a quasiparticle renormalization factor has been absorbed into V, and n (0) and (co) are the usual Bose and Fermi factors. The first and third terms give contributions to the quasiparticle scattering rate ~, arising from the emission and absorption of a spin-fluctuation, respec' tively. The second term is the recombination rate T, to a process in which a quasiparticle corresponding recombines with another quasiparticle to form a pair with the excess energy emitted as a spin fluctuation. Note that the signs in the coherence factors for the antiferromagnetic spin-fluctuation interaction are opposite to those for the phonon case. As seen in neutron-scattering experiments, ' when the the low-frequency co(26 system goes superconducting, spin-fluctuation spectral weight is reduced over most of the Brillouin zone. It follows from Eq. (3) that this will lead to a decrease in the quasiparticle decay rate. Nuss et al. and Littlewood and Varma suggested that this was responsible for the peak observed in the far-infrared conductivity rather than a coherence factor. This may be contrasted with the phonon case in which the lowfrequency phonon spectral weight is essentially unchanged in the superconducting state. The momentum integrations in Eqs. (2) and (3) cover the Brillouin zone. They are carried out by dividing up the momentum space into a grid of small squares and evaluating the integrands at the center of each square. The square size is then reduced until the finite-size effects become negligible. The exception to this procedure is the evaluation of Imago(q, co). Since this involves an integral over a 5 function, a slightly different procedure is required. For this integration, the small squares covering the momentum space are further divided onto four triangular regions each. The argument of the 5 function involved in the integral is then evaluated at the three vertices of each triangle. This allows the argument to be replaced by a linear approximation of itself within a given triangle. Once this replacement is done, the integral of f 1471 the 5 function over the triangular region may be done analytically. Repeating this procedure over the whole zone completes the momentum integration. Results for the quasiparticle decay rate obtained from Eq. (3) for s- and d-wave gaps with 2b, o(0)/kT, =6 are shown in Figs. 1 and 2. Here we have taken T, =0.2t in order to examine the behavior of the decay rate at small values of T/T, . We have normalized r( T) by its value at T, . For both cases we have set p on the Fermi surface with p =p . For the d-wave gap this corresponds to looking at a quasiparticle in the region of the node where the gap vanishes. In this case we have set co equal to the thermal energy kT. This is representative of a typical quasiparticle energy and momentum at lower temperatures where the nodal regions of the Fermi surface are occupied. For the s-wave gap below T, predominantly we have set co=6,o(T). 0— 2 0 I I I I I I I I I I I I I I I I j I I I I I I I I I I ) ~09io(T/T I I -0.5 —1 I I I I I I I I I I I l I I 1 I I I I I 0 — ' 2 I- Ql 0 —" —10 l IC) 0 6 —1.2 I -1 I I —0.8 I I -0.6 ~0910(T/T —20 FIG. 2. I 0 i i i I i i i I s I 8 i i s I 10 s s 12 FIG. 1. Plot of log&0(w(T, )/~„(T)) versus T, /T for an swave gap with 260(0)/kT, =6 and T, =0.2t. The solid line shows the quasiparticle scattering rate v; '. The dashed line shows the recombination rate ~, '. At low temperatures both curves fall as exp( 5/T). — l -0.4 I -0.2 ) (a) Plot of log&0(~(T, )/s(T)) versus log, o(T/T, ) for a d-wave gap with 260(0)/kT, =6 and T, =0.2t. At low temperatures v '(~) varies as ( T/T, ) . The dashed line displays a slope of three on the log-log plot as a guide to the eye. The symbols represent ~ ' values calculated using different momentum lattice sizes: 512 X 512 (circles), 256 X 256 (triangles), and 128X128 (crosses). (b) Plot showing separately the scattering rate v; ' (solid line) and the recombination rate v, ' (dashed line). BRIEF REPORTS 1472 In Fig. 1 we show the temperature dependence of the quasiparticle scattering and recombination rates for an swave gap on a semilog plot. This clearly shows that both rates decrease exponentially as e due to the opening of a gap in the spin-fluctuation spectrum. In Figs. 2(a) and 2(b} we show similar results for the d-wave case on a log-log scale. This shows that at very low temperatures the relaxation rates for a d-wave gap have a T dependence associated with the available phase space for scattering in the nodal regions. The scattering and recombination rates in the nodal region are comparable over most of the temperature range shown, due to the small quasiparticle frequency. In order to show the magnitude of the finite lattice size error in this calculation, Fig. 2(a) includes points showing the r values calculated using a few different momentum lattices sizes. Using microwave surface-resistance and penetrationdepth data, Bonn and co-workers find that the real part of the conductivity of YBa2Cu06 95 exhibits a broad peak around 40 K, which has a height 10—20 times the value of the conductivity at T, . Within the framework of a generalized two-Quid model in which the conductivity of the normal fluid is modeled by a Drude form, they extract a transport lifetime and find that the inverse of this lifetime decreases rapidly with decreasing temperature as the temperature drops below T, . At temperatures below T, /2 they find that this lifetime reaches a temperatureindependent limiting value. Semilog plots of the quasiparticle decay rates for s- and d-wave gaps are shown in Figs. 3(a) and 3(b) along with data from Ref. 2. The dwave gap results with 260(0)/kT, —6 to 8 appear to roughly follow the microwave results above T/T, =0. 5. Thus we find that a spin-fluctuation interaction with calculated within a BCS framework using a d-wave a quasiparticle decay rate which decreases rapyields gap idly with decreasing temperature below T, . %ell below T, this decay rate exhibits a T temperature dependence. Above T, /2 this decay rate is in rough agreement with dependence of the inverse the observed temperature transport lifetime r«( T) obtained by Bonn and coworkers. A similar calculation using an s-wave gap does not appear to provide as satisfactory a fit. This suggests that, assuming the transport lifetime is indeed driven by are fluctuations these scattering, spin-fluctuation suppressed, but apparently not gapped, below T, . Bonn and co-workers suggest that the limiting value that the transport lifetime reaches at low temperatures may be due to impurity scattering. To investigate this possibility, we have calculated the impurity scattering rate for a d-wave superconductor with model parameters as given above. In the dilute impurity concentration limit the quasiparticle scattering rate is given by' N(0) (2m ) 2 7 (6) where n; is the impurity concentration, N (0) is the normal phase density of states, and 60 is the scattering phase " shift. We have solved these equations for the case of a dilute concentration of impurities such that I /b, o(0)=10 The results are plotted in Fig. 4 for several values of the phase shift parameter c. Since the impurity scattering rate is strongly dependent on the quasiparticle frequency, a weighted average of this scattering rate will enter the transport lifetime. For illustrative purposes we have set co= T in the impurity scattering rate and added this to the spin-fluctuation driven quasiparticle decay rate shown previously. In Fig. 5 this combined relaxation rate is compared to the results from Ref. 2. It appears that for any choice of the phase shift parameter we find more structure at low temperatures in the impurity scattering rate than obtained in the phenomenological analysis of Ref. 2. I x I I 3.64 GHz 34.8 GHz I ) I I I ~ —0.5 CI 0 II —1,5 «X X XX ««««««« XXX : «k ~„ «, g I ) I 0.5 x 3.64 GHz 34.8 GHz ~ —0.5 I- CI —1 0 «X —2 —Go(co) I =n, /(mN(0)), c =cot5o, m —Xo(& } 2 2 E— [co — Xo(co) ) p m d 2p XXX with the self-energy Xo(~) Here Go(co) = ""/ g««««««««XX r; '(co)= —2lmXO(co) Xo(co)=l Go(co)/[c 49 ) and . 0 X / 0.5 FIG. 3. Plot of log lo{~( T )/~( T) ) versus T /T„where T, =0. 1t, for (a) an s-wave gap with 260(0)/kT, =4 (dotted), 6 (dashed), and 8 (solid) and (b) a d-wave gap with 25 0(0)/kT, =6 (dashed) and 8 (solid). The symbols are data from Ref. 2. BRIEF REPORTS 49 I I I I I ) I I I [ I I I 1473 I I f I I [ I I I I f l I 0.03 0.02 a E 0.01 0 I 0 0.05 «» xx" —2 I 0. 1 0. 15 0.5 0 T/T, FIG. 4. Plot of impurity scattering rate 1/(~; ~(co)50) versus frequency ~/60 at T=O for I /6&(0)=10 '. This scattering rate is shown various phase shift parameter values: c =0 (the unitary limit, solid line), c =0. 1 (dashed line), and c =0.2 (dotted line). The authors would like to thank the authors of Ref. 2 for allowing their data to be shown in Figs. 4 and 5. We would also like to acknowledge useful discussions with D. Bonn, W. N. Hardy, P. J. Hardy, P. J. Hirschfeld, and %. O. Putikka. We thank H. B. Schiittler for the suggestion of the algorithm used to calculate Imago(q, co). One 'N. Bulut and D. J. Scalapino, Phys. Rev. Lett. 68, 706 (1992). D. A. Bonn, P. Dosanjh, R. Liang, and W. N. Hardy, Phys. Rev. Lett. 68, 2390 (1992); D. A. Bonn, R. Liang, T. M. Riseman, D. J. Baar, D. C. Morgan, K. Zhang, P. Dosanjh, T. L. Duty, A. MacFarlane, G. D. Morris, J. H. Brewer, W. N. Hardy, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 47, 11 314 {1993). 3S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey, and D. J. Scalapino, Phys. Rev. B 14, 4854 (1976). 4T. E. Mason, G. Aeppli, and H. A. Mook, Phys. Rev. Lett. 68, 1414 (1992). 5J. M. Tranquada, P. M. Gehring, G. Shirane, S. Shamoto, and M. Sato, Phys. Rev. B 46, 5561 (1992). 6M. C. Nuss, P. M. Mankiewich, M. L. O' Malley, E. H. Westerwick, and P. B. Littlewood, Phys. Rev. Lett. 66, 3305 (1991). B. Littlewood and C. M. Varma, J. Appl. Phys. 69, 4979 (1991). sB. W. Statt and A. Griffin [Phys. Rev. B 46, 3199 (1992)], on 7P. the other hand, suggested that a uniform suppression below T, of the spin-fluctuation spectral weight at all frequencies can account for this peak. FIG. 5. Plot of combined quasiparticle decay rates due to impurity and spin-fluctuation scattering versus temperature compared with results from Ref. 2. Here we have used T, =0. 1t, 250(0)/kT, =8, and I /60(0)=10 . The three curves shown are for various phase shift parameter values: c =0 (the unitary limit, solid line), c =0. 1 (dashed line), and c =0.2 {dotted line). of the authors (N. B.) acknowledges financial support from IBM. This work was partially supported by the National Science Foundation under Grants No. DMR9002492 and No. PHY89-04035. The numerical calculations reported in this paper were performed at the San Diego Supercomputer Center. Previous calculations using the interaction given by Eq. {1) have shown that spin fluctuations lead to the right order of magnitude for the quasiparticle relaxation rate in the normal state with r '(T, [S. Wermbter and L. Tewordt, Phys. Rev. B 43, 10530 (1991), and N. Bulut, H. Morawitz, and D. J. Scalapino (unpubhshed)]. Furthermore, this same strength of interaction accounts for the correct size of the spin-lattice relaxation rate' T& '{T,). 0P. J. Hirschfeld, P. Wolfle, and D. Einzel, Phys. Rev. B 37, 83 )-T, (1988). ' Note that to obtain the above equations for the impurity scattering rate we must assume both that the average of the superconducting gap hz over the Fermi surface vanishes and that the system has particle-hole symmetry (Ref. 10). This second assumption does not hold for the tight-binding band used in our model. However, we are only interested in the impurity scattering at temperatures and frequencies small compared to the superconducting gap ho. At such a small energy scale particle-hole symmetry is approximately obeyed, and corrections due to the lack of such symmetry will be small.