Generic Phase Diagram

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

2. A Discussion of Probes: • Well documented evidence for strongly anisotropic spectral functions -> “hot” and “cold” qps. ARPES: (0,p) (p,0) • Reveals 1-particle properties, but limited view on dynamics. Z.-X. Shen and J. R. Schrieffer, PRL 97. Transport: • r(T), s(w,T) (optical, thermal), Cv(T), rsuperfluid(T) dominated by transport along zone diagonals. Hot qps? Raman: Clear, simultaneous view of hot(B1g ) and cold (B2g ) qps evolution with temperature and doping. Light scattering amplitude g B1g: g(k) ~ cos(kxa)-cos(kya) B2g: g(k) ~ sin(kxa) sin(kya)

3. Review of Raman Data on the Cuprates: Normal State: Low frequencies: J. G. Naeini et al., PRB 1999 • B2g - intensity largely independent of doping. • B1g - loss of low frequency spectral weight with underdoping.

4. Exp. Review (cont.): B1g: spectral weight shifts to 2-magnon energies ~ 3J. T- dependence: • General form for low frequency Raman response (independent of microscopic theory) – • c//( T >> W ~ 0) ~ W <Z2k g2(k)/ Gk(T)> • Gk(T) qp scattering rate. • Zk qp residue. • <…> average over Fermi surface. Inverse of the Raman slope determines the T-dependence of the qp scattering rate.

5. M. Opel et al., PRB 2000 Exp. Review (cont.): • B2g: • G(T) as T , same magnitude for all doping. • follows DC transport behavior. • B1g: • G(T) as T , except for overdoped. • qps increasingly gapped with underdoping. • distinctly non Fermi liquid-like -> likely due to T-dependence of qp residue.

6. FeSi L. Cooper, 1995

7. Amended cold spot model: Gk(T) = Gc(T) + Gh [cos(kxa)-cos(kya)]2 Gc(T)= T2/T0, T0 ~ 40 meV Zk(T) = Z0 exp{-Eg [cos(kxa)-cos(kya)]2/T} Cold spot analysis of QP scattering rate: Slightly Underdoped: Gh = 470 cm-1, Eg = 140 cm-1 Slightly Overdoped: Gh = 410 cm-1, Eg= 20 cm-1 Appreciably Overdoped: Gh = 8 cm-1, Eg = 0 cm-1

8. Superconducting State: TPD and A. Kampf, IJMPB 97 Bi 2212 Tc = 86K Near optimal doping, relative peak positions, low frequency power-laws consistent with dx2-y2 pairing. S. Sugai and T. Hosokawa, cond-mat/9912232 B2g: always shows reorganization at Tc. B1g: reorganization disappears with underdoping.

9. Superconducting State (cont.): Should be viewed as having large error bars. • B2g: • shows reorganization for all doping. • clear superconducting feature at all dopings at Tc. • wpeak/Tc ~ 6 for all doping. • B1g: • only shows reorganization for optimal and overdoped. • no clear superconducting feature for underdoped. • wpeak/Tc ~ 8 for optimal doping, less for overdoped.

10. Summary (exp. data): • Distinctly different behavior of dynamics of B1g and B2g Raman response -> “hot” and “cold” qps. • B2g: • relatively independent of doping. • follows transport in normal state. • shows superconducting gap proportional to Tc for all doping. • B1g: • strongly doping dependent. • spectral weight transfer to higher energies for low dopings indicative of gapped response at different energy and temperature scale than B2g. • Merges with B2g behavior for overdoped samples.

11. 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. Raman Theory – what drives gapping of “hot” qps? 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.

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

13. Raman results at ½ Filling: through MIT Homogeneous phase Weakly interacting 1- particle DOS is T-independent, shows MIT. Pseudogap phase insulating MIT Fixed Temperature • Spectral weight shifts into charge transfer peak for increasing U. • Low frequency spectral weight ~ t2/U. Charge transfer peaks. Raman response [arb. units] small band of qps

14. U=2t Raman results (cont): • low frequency spectral weight begins to deplete at T ~ t2/U. • weight piles up at U, charge transfer energy. Raman response [arb. units] Integrated Intensity • reorganization of spectral weight in pseudo-gap and insulating cases. • qualitatively similar to B1gRaman response in the cuprates. U=t U=1.5t Ilowfreq. /I high freq. U=2t U=4t

15. Raman inverse slope: insulating U=2t U=1.5t Raman inverse slope [arb. units] pseudogap phase U=t U=0.5t U=0.25t weakly interacting 0.1 1.0 Temperature [t] Qualitatively similar to B1g in cuprates.

16. Conclusions and Summary: • Raman gives detailed evidence for 2 distinct types of dynamics related tohotandcoldqps. • B2g (cold) • minor dependence on doping, follows transport in normal state. • reveals superconducting gap which tracks Tc for all doping. • B1g (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.