Transcript Raman scattering in correlated metals and insulators

Inelastic light scattering in
strongly correlated metals and
insulators
T. P. Devereaux
With J. Freericks (Georgetown), R. Bulla (Augsburg) and R. Hackl (Garching)
Work supported by NSERC, PREA, US-CDRF
Brock, 7/16/2015
T. P. Devereaux
1
Strong Electron Correlations
Transition metals,
oxides, mixed
valent compounds –
SmB6 , MnO, CuO,
La2-xSrxCuO4 , 3He.
Simple metals –
Na, Al -> realm
of Fermi Liquid
Theory.
0
Heavy fermions,
Kondo insulators, –
UPt3, FeSi, UPd2Al3,
CeCu2Si2.
Coulomb interaction/Bandwidth (U/W)
Brock, 7/16/2015
T. P. Devereaux
∞
2
Quantum Phase Transitions
A continuous
transition at T=0 as a
parameter of the
Hamiltonian is varied
(magnetic exchange,
disorder, electron
density, etc…).
-> Anderson – Mott transition
Si:P, T. F. Rosenbaum et al, 1983.
Brock, 7/16/2015
T. P. Devereaux
3
Quantum Critical Points
Cuprates phase
diagram
-one particle
properties may be
uncritical.
- two particle
properties may not.
EXAMPLE:
(Anderson) metalinsulator transition
1/t , DOS – non-
critical
s - falls to zero at
MIT.
Brock, 7/16/2015
T. P. Devereaux
4
Discussion of Probes
ARPES:
• Well documented evidence for strongly anisotropic
spectral functions -> “hot” and “cold” qps.
(0,p)
(p,0)
• Reveals 1-particle properties,
Z.-X. Shen and J. R. Schrieffer, PRL 97.
but limited view on dynamics.
Transport:
r(T), s(w,T), ĸ(T), Cv(T), rs(T) dominated by transport along zone diagonals. Hot qps?
Raman, X-ray:
Light scattering amplitude
B1g:
g
g(k) ~ cos(kxa)-cos(kya)
B2g: g(k) ~ sin(kxa) sin(kya)
Brock, 7/16/2015
Clear, simultaneous view of
hot (B1g ) and cold (B2g ) qps
evolution with temperature
and doping.
T. P. Devereaux
5
Experimental Review
J. C. Irwin et al., 1997.
• B2g - intensity largely independent of doping.
• B1g - loss of low frequency spectral weight with underdoping.
Brock, 7/16/2015
T. P. Devereaux
6
Experimental Review (cont.)
B1g: spectral weight
shifts to 2-magnon
energies ~ 3J.
- Depletion of low
energy spectral
weight.
- Development of an
isosbestic point (no
T dependence).
J. C. Irwin et al., 1998.
Brock, 7/16/2015
T. P. Devereaux
7
Common to other systems?
FeSi – Kondo Insulator
SmB6 – mixed valent insulator
• transfer of spectral weight from low frequencies to high as T reduced.
• occurrence of “isosbestic point” (spectrum independent of T).
• qualitatively similar to B1g in underdoped cuprates.
Brock, 7/16/2015
T. P. Devereaux
8
Low energy features.
F. Venturini et al, 2002.
Brock, 7/16/2015
T. P. Devereaux
9
T-dependence of low frequency
response
Χ//static(T)~
Ω <Z2k(T) γ2(k)/ Γk(T)>
Γk(T) - qp scattering rate.
Zk(T) - qp residue.
γ– Raman form factor.
<…> - average over Fermi surface.
Inverse of the Raman slope
determines the T-dependence
• B1g very doping dependent.
of the qp scattering rate and
• B2g shows no doping dependence. residue on selected regions of
the Fermi surface.
Brock, 7/16/2015
T. P. Devereaux
10
Raman Inverse Slope
M. Opel et al., PRB 2000
B2g: Γ(T)
B1g: Γ(T)
Brock, 7/16/2015
as T
, same magnitude for all doping.
(follows DC transport).
as T
, except for overdoped.
- qps increasingly gapped with underdoping.
- distinctly non Fermi liquid-like.
T. P. Devereaux
11
Shows a clear break
in behavior at a
doping pc ~ 0.22.
Indicates that the “hot”
qps become incapable of
carrying current.
-> unconventional quantum
critical metal – insulator
transition for p=pc.
Venturini et al, 2002.
Brock, 7/16/2015
T. P. Devereaux
12
Theory for light scattering
• The insulating limit has been analyzed by
Chubukov and Frenkel (PRL, 1995).
• The antiferromagnetically correlated metal has
been described by Devereaux and Kampf (PRB,
1999).
• But no theory exists that can connect these two
regimes and carry one through the quantum critical
point of a metal-insulator transition.
• Here we show how one can solve for Raman
scattering through a metal-insulator transition in
both the Falicov-Kimball model and the Hubbard
model.
Brock, 7/16/2015
T. P. Devereaux
13
Light scattering processes
Incoming photon wi
Costs energy U
(charge transfer
energy).
Outgoing photon wf
Electron hops,
gains t.
Brock, 7/16/2015
For finite T, double
occupancies lead to
small band of low
energy electrons.
T. P. Devereaux
14
Spinless Falicov-Kimball Model
wi
U
•exactly solvable model on a hypercubic lattice in infinite
dimensions using dynamical mean field theory.
•possesses homogeneous, commensurate/incommensurate
CDW phases, phase segregation, and MI transitions.
•Raman response can be constructed formally exactly.
Brock, 7/16/2015
T. P. Devereaux
15
Metal-Insulator transition
• Correlationinduced gap drives
the single-particle
DOS to zero at
U=1.5
• Interacting DOS is
independent of T
in DMFT (Van
Dongen, PRB, 1992)
• Examine Raman
response through
the (T=0) quantum
phase transition.
Brock, 7/16/2015
T. P. Devereaux
16
Results: Falicov-Kimball
Fixed Temperature
• Spectral weight
Charge
shifts into charge
transfer
transfer peakpeaks.
for
increasing U.
• Low frequency
spectral weight ~
2/U.
t
small
band of
qps
Brock, 7/16/2015
Fixed U=2t
Spectral
weight
shifts into
charge
transfer
peak for
increasing
U or
decreasing
T.
T. P. Devereaux
Charge
transfer
peaks.
17
Integrated spectral weight and inverse
Raman slope
• The Raman response is
sharply depleted at
low-T.
• The inverse Raman
slope changes from
nearly constant
uncorrelated metallic
behavior to a rising
pseudogap or insulating
behavior as the
correlations increase.
Brock, 7/16/2015
T. P. Devereaux
18
Hubbard Model
t
*
H 
c

is c js  U  ni  ni 
2 d
U
Both electrons are now mobile
•Exactly solvable model on a hypercubic lattice in infinite dimensions
using dynamical mean field theory (but requires NRG calculations to
extract real frequency information).
•The irreducible charge vertex is problematic to calculate because it
possesses too large a dynamic range for the max-ent techniques.
•Hence, the Raman response can be constructed formally exactly
for the nonresonant B1g channel only.
Brock, 7/16/2015
T. P. Devereaux
19
Nonresonant B1g Raman scattering (n=1,U=2.1)
• Note the charge
transfer peak as
well as the Fermi
liquid peak at low
energy. As T goes
to zero, the Fermi
peak sharpens and
moves to lower
energy.
• There is no low
energy and low-T
isosbestic point,
rather a high
frequency
isosbestic point
seems to develop.
Brock, 7/16/2015
T. P. Devereaux
20
Nonresonant B1g Raman scattering (n=1,U=3.5)
• A MIT occurs as a
function of T.
Note the
appearance of the
low-T isosbestic
point.
• The low energy
Raman response
has rich behavior,
with a number of
low energy peaks
developing at lowT, but the low
energy weight
increases as T
decreases.
Brock, 7/16/2015
T. P. Devereaux
21
Nonresonant B1g Raman scattering (n=1,U=4.2)
• Universal behavior
for the insulator--the low-energy
spectral weight is
depleted as T goes
to zero and an
isosbestic point
appears.
• The temperature
dependence here is
over a wider range
than for the FK
model due to the
T-dependence of
the interacting
DOS.
Brock, 7/16/2015
T. P. Devereaux
22
Summary and Conclusions
• Shown some exact solutions for Raman
scattering across a MIT.
• Insulating state, depletion of low energy
spectral weight into charge transfer
peak – universal behavior.
• Metallic state, development of low
energy peak reflecting qp coherence.
• Elucidates dynamics near and through a
quantum critical point.
Brock, 7/16/2015
T. P. Devereaux
23
Inelastic X-ray scattering
M. Hasan et al, 2001 – Ca2 Cu O2 Cl2
Brock, 7/16/2015
T. P. Devereaux
24
Inelastic X-ray scattering: Falicov-Kimball model
Brock, 7/16/2015
T. P. Devereaux
25
Peak position and width
Low energy feature High energy feature
Filled
symbols –
peak
positions.
Open
symbols –
peak
widths.
Brock, 7/16/2015
T. P. Devereaux
26