2. Raman Scattering As a Probe of Unconventional Electron Dynamics in the Cuprates Raman Scattering As a Probe of Unconventional Electron Dynamics in the Cuprates T. P. Devereaux University of Waterloo Generic Phase Diagram Fermi liquid or non Fermi liquid? Doped AF or AF correlated metal? Underlying quantum critical point(s) and crossovers? Issues/Questions: Key problem: to understand dynamics as a function of temperature and doping.

3. A Discussion of Probes: ARPES: Well documented evidence for strongly anisotropic spectral functions -> “hot” and “cold” qps. Z.-X. Shen and J. R. Schrieffer, PRL 97. Reveals 1-particle properties, but limited view on dynamics. Transport:  (optical, thermal), C v (T),  superfluid (T) dominated by transport along zone diagonals. Hot qps? Raman: Light scattering amplitude  B 1g :  (k) ~ cos(k x a)-cos(k y a) B 2g :  (k) ~ sin(k x a) sin(k y a) Clear, simultaneous view of hot (B 1g ) and cold (B 2g ) qps evolution with temperature and doping.  

4. Review of Raman Data on the Cuprates: Normal State: Low frequencies: B 2g - intensity largely independent of doping. B 1g - loss of low frequency spectral weight with underdoping. J. G. Naeini et al., PRB 1999

5. Exp. Review (cont.): B 1g : spectral weight shifts to 2-magnon energies ~ 3J. T- dependence: General form for low frequency Raman response (independent of microscopic theory) –     k    k  k    k  qp scattering rate.   k qp residue.  average over Fermi surface. Inverse of the Raman slope determines the T- dependence of the qp scattering rate.

6. Exp. Review (cont.): M. Opel et al., PRB 2000 B 2g :  (T) as T, same magnitude for all doping. follows DC transport behavior. B 1g :  (T) as T, except for overdoped. qps increasingly gapped with underdoping. distinctly non Fermi liquid-like -> likely due to T-dependence of qp residue.

7. FeSi L. Cooper, 1995

8. Cold spot analysis of QP scattering rate: Amended cold spot model:  k (T) =  c (T) +  h [cos(k x a)-cos(k y a)] 2  c (T)= T 2 /T 0, T 0 ~ 40 meV Z k (T) = Z 0 exp{-E g [cos(k x a)-cos(k y a)] 2 /T} Slightly Underdoped:  h = 470 cm -1, E g = 140 cm -1 Slightly Overdoped:  h = 410 cm -1, E g = 20 cm -1 Appreciably Overdoped:  h = 8 cm -1, E g = 0 cm -1

9. Superconducting State: Near optimal doping, relative peak positions, low frequency power-laws consistent with d x 2 - y 2 pairing. TPD and A. Kampf, IJMPB 97 S. Sugai and T. Hosokawa, cond- mat/9912232 B 2g : always shows reorganization at T c. B 1g : reorganization disappears with underdoping. Bi 2212 T c = 86K

10. Superconducting State (cont.): B 2g : shows reorganization for all doping. clear superconducting feature at all dopings at T c.  peak /T c ~ 6 for all doping. B 1g : only shows reorganization for optimal and overdoped. no clear superconducting feature for underdoped.  peak /T c ~ 8 for optimal doping, less for overdoped. Should be viewed as having large error bars.

11. Summary (exp. data): B 1g B 2g Distinctly different behavior of dynamics of B 1g and B 2g Raman response -> “hot” and “cold” qps. B 2g : relatively independent of doping. follows transport in normal state. shows superconducting gap proportional to T c for all doping. B 1g : strongly doping dependent. spectral weight transfer to higher energies for low dopings indicative of gapped response at different energy and temperature scale than B 2g. Merges with B 2g behavior for overdoped samples.

12. Raman Theory – what drives gapping of “hot” qps? Theory for inelastic light scattering exists for Antiferromagnetic insulators. (e.g., A. V. Chubukov and D. Frenkel, PRL 95) Antiferromagnetically correlated metals. (e.g., TPD and A. P. Kampf, PRB 99) Theory lacking which takes one across MIT. Model Calculation: Spinless Falicov-Kimball model (with J. K. Freericks) - exactly solvable model on a hypercubic lattice in infinite dimensions using dynamical mean field theory. - possesses homogeneous, commensurate/incommensurate CDW phases, phase segregation, and MIT transitions. - Raman response can be constructed formally exactly.

13. Light Scattering Processes: Incoming photon  i Costs energy U (charge transfer energy). Electron hops, gains t. Outgoing photon  f For finite T, double occupancies lead to small band of low energy electrons.

14. Raman results at ½ Filling: through MIT Weakly interacting Pseudogap phase insulating MIT Homogeneous phase 1- particle DOS is T- independent, shows MIT. Spectral weight shifts into charge transfer peak for increasing U. Low frequency spectral weight ~ t 2 /U. Fixed Temperature Charge transfer peaks. small band of qps Raman response [arb. units]

15. Raman results (cont): low frequency spectral weight begins to deplete at T ~ t 2 /U. weight piles up at U, charge transfer energy. reorganization of spectral weight in pseudo-gap and insulating cases. qualitatively similar to B 1g Raman response in the cuprates. Raman response [arb. units] U=2t I low freq. /I high freq. Integrated Intensity U=t U=1.5t U=2t U=4t

16. Raman inverse slope: U=2t U=1.5t U=t U=0.5t U=0.25t 0.1 1.0 Temperature [t] Raman inverse slope [arb. units] weakly interacting insulating pseudogap phase Qualitatively similar to B 1g in cuprates.

17. Conclusions and Summary: Raman gives detailed evidence for 2 distinct types of dynamics related to hot and cold qps. B 2g (cold) minor dependence on doping, follows transport in normal state. reveals superconducting gap which tracks Tc for all doping. B 1g (hot) strongly dependent on doping, shift of spectral weight -> qps gapped in normal state (presumably precursor SDW). superconducting gap only appears for optimal and overdoped samples -> suggests a competition for qps.