2. Department of Physics University of Toronto Low Temperature Thermal Transport Across the Cuprate Phase Diagram Mike Sutherland Louis Taillefer Rob Hill Cyril Proust Filip Ronning Makariy Tanatar R.Gagnon, H.Zhang D.Bonn, R.Liang, W.Hardy P.Fournier, R.Greene A.P.Mackenzie, D. Peets, S. Wakimoto Christian Lupien Etienne Boaknin Dave Hawthorn J. Paglione M. Chiao

3. Carrier concentration superconductor metal magnetismmagnetism Temperature pseudogap What questions can we address by studying low temperature thermal conductivity as a function of doping in the cuprates ? How well does d -wave BCS theory describe the superconducting state ? Is the superconducting order parameter pure d-wave throughout the phase diagram? How does the pseudogap influence the behaviour of low-energy quasiparticles?

4. The density of states in a d-wave superconductor impurity effects Finite density of delocalised states at zero energy density of states  impurity bandwidth presence of nodes  quasiparticles at low T Linear density of states at low energy - governs all low temperature properties clean limit

5. Fermi Liquid Theory of d-wave Nodal Quasiparticles With: The quasiparticle excitation spectrum near the nodes takes the form of a ‘Dirac cone’ : d-wave gap:  =  0 cos(2  ) E

6. Thermal Conductivity Primer l A  Q Kinetic theory formulation:   =  electrons +  phonons electrons ~ T phonons ~ T 3

7. d-wave BCS theory of thermal conductivity Electronic heat transport provided solely by quasiparticles ( T  0, T<<  ) A. Durst and P. A. Lee, Phys. Rev. B 62, 1270 (2000). M. J. Graf et al., Phys. Rev. B 53,15147 (1996). universal This result is universal with respect to impurity concentration Cooper pairs carry no heat Δ o from κ 0 /T

8. Optimally Doped Bi-2212  Ding et al. PRB 54 (1996) R9678 Mesot et al. PRL 83 (1999) 840 ARPES: Nodal quasiparticles in optimally doped Cuprates Weak Coupling BCS:  0 = 2.14k B T c = 17 meV  0 = 30 meV Increase Coupling:  0  4k B T c M.Chiao et al. PRB 62 3554 (2000)

9. doping dependence: v F X.J. Zhou et. al. Nature 423 398 ( 2003 ) LSCO(x)    essentially doping independent

10. Nodal quasiparticles in overdoped Cuprates How do we estimate hole concentration [p]? overdoped Tl 2201 Doping T c =15 K sample: Proust et al., PRL 89 147003 (2002). T c = 15K T c = 27K T c = 89K T c = 85K  0 = 4k B T c Other samples: Hawthorn et. al. to be published

11. underdoped YBCO Nodal quasiparticles in underdoped Cuprates v F /v 2 as doping  0 /T as doping decrease simple BCS theory violated: Δ o does not follow Δ BCS ! Sutherland et al. PRB (2003)

12. The pseudogap in underdoped Cuprates    pseudogap is : (i) quasiparticle gap (ii) must have nodes (iii) must have linear dispersion Campuzano et. al. PRL 83 (1999) 3709 Norman et. al. Nature 392, 157 (1998) White et. al. Phys. Rev. B. 54, R15669 (1996) Loeser et. al. Phys. Rev. B. 56, 14185 (1996) T = 15 K

13. Underdoped La 2-x Sr x CuO 4 Presence of static SDW order? Large intrinsic crystalline disorder?

14. Summary and Outlook doping dependence of superconducting gap maximum : overdoped – optimal doped:  0 scales with T c (BCS theory) optimal doped – underdoped:  0 increases while T c decreases (Failure BCS theory) Question: What happens near the AF – SC boundary? existence of nodes throughout the phase diagram: no evidence for quantum phase transition to d+ix in the bulk

15. doping dependence: v F essentially doping independent ARPES data Z.X.Shen LSCO(x)   

16. Specular Reflection of Phonons Specular reflection l ph = f(T)  ph /T~ T ,  <2 R. O. Pohl* and B. Stritzker, PRB 25, 3608 (1982). Sapphire V 3 Si s-wave SC (thermal insulator,  el =0) Fit data to  /T =  o /T + BT   o /T = 0  = 1.7