Raman Scattering As a Probe of Raman Scattering As a Probe of Unconventional Electron Dynamics in Unconventional Electron Dynamics in

the Cupratesthe CupratesT. 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.

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), Cv(T), superfluid(T) dominated by transport along zone diagonals. Hot qps?

Raman:

Light scattering amplitude

B1g: (k) ~ cos(kxa)-cos(kya)

B2g: (k) ~ sin(kxa) sin(kya)

Clear, simultaneous view of hot (B1g ) and cold (B2g ) qps evolution with temperature and doping.

Review of Raman Data on the Cuprates:

Normal State:

Low frequencies:

• B2g - intensity largely independent of doping.

• B1g - loss of low frequency spectral weight with underdoping.

J. G. Naeini et al., PRB 1999

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) –

kkk

kqp scattering rate.

k qp residue.

<…> average over Fermi surface.

Inverse of the Raman slope determines the T-dependence of the qp scattering rate.

Exp. Review (cont.):M. Opel et al., PRB 2000

B2g:

• (T) as T , same magnitude for all doping.

• follows DC transport behavior.

B1g:

• (T) as T , except for overdoped.

• qps increasingly gapped with underdoping.

• distinctly non Fermi liquid-like -> likely due to T-dependence of qp residue.

FeSi L. Cooper, 1995

Cold spot analysis of QP scattering rate:Amended cold spot model:

k(T) = c(T) + h [cos(kxa)-cos(kya)]2

c(T)= T2/T0, T0 ~ 40 meV

Zk(T) = Z0 exp{-Eg [cos(kxa)-cos(kya)]2/T}

Slightly Underdoped:

h = 470 cm-1, Eg = 140 cm-1

Slightly Overdoped:

h = 410 cm-1, Eg= 20 cm-1

Appreciably Overdoped:

h = 8 cm-1, Eg = 0 cm-1

Superconducting State:

Near optimal doping, relative peak positions, low frequency power-laws consistent with dx

2-y2

pairing.

TPD and A. Kampf, IJMPB 97

S. Sugai and T. Hosokawa, cond-mat/9912232

B2g: always

shows reorganization at Tc.

B1g: reorganization disappears with underdoping.

Bi 2212

Tc = 86K

Superconducting State (cont.):

B2g:

• shows reorganization for all doping.

• clear superconducting feature at all dopings at Tc.

• peak/Tc ~ 6 for all doping.

B1g:

• only shows reorganization for optimal and overdoped.

• no clear superconducting feature for underdoped.

• peak/Tc ~ 8 for optimal

doping, less for overdoped.

Should be viewed as having large error bars.

Summary (exp. data):

Distinctly different behavior of dynamics of BB1g1g and BB2g2g Raman response -> “hot”

and “cold” qps.

BB2g2g::

• relatively independent of doping.

• follows transport in normal state.

• shows superconducting gap proportional to Tc for all doping.

BB1g1g::

• 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.

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.

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.

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 ~ t2/U.

Fixed Temperature

Charge transfer peaks.

small band of qps

Ram

an r

espo

nse

[arb

. uni

ts]

Raman results (cont):

• low frequency spectral weight begins to deplete at T ~ t2/U.

• weight piles up at U, charge transfer energy.

• reorganization of spectral weight in pseudo-gap and insulating cases.

• qualitatively similar to B1g Raman response in the cuprates.

Ram

an r

espo

nse

[arb

. uni

ts] U=2t

I low fr

eq. /

I hi

gh f

req.

Integrated Intensity

U=t

U=1.5t

U=2t

U=4t

Raman inverse slope:

U=2t

U=1.5t

U=t

U=0.5t

U=0.25t

0.1 1.0

Temperature [t]

Ram

an in

vers

e sl

ope

[arb

. uni

ts]

weakly interacting

insulating

pseudogap phase

Qualitatively similar to B1g in cuprates.

Conclusions and Summary:

Raman gives detailed evidence for 2 distinct types of dynamics related to hot and cold qps.

• 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.