Spectroscopic signatures of
two energy scales in
superconducting underdoped
cuprates
B. Valenzuela
Instituto de Ciencia de Materiales de Madrid
(ICMM-CSIC)
In collaboration with:
Elena Bascones
(ICMM-CSIC)
Outline
• Conventional superconducting phase in high Tc
superconductors?
•Experimental evidence in ARPES and Raman of
deviation from d-wave BCS in superconducting
underdoped cuprates: two-scales
•Introduction to the phenomenological model proposed
to describe the pseudogap (Yang, Rice & Zhang ‘06)
•Results on how two energy scales appear in underdoped
superconducting cuprates in ARPES,
autocorrelated ARPES and Raman.
Pseudogap State: Fermi arcs and
Nodal-Antinodal Dichotomy
Antinodal
region
Nodal
region
Shen et al, Science 307, 901 (2005)
Superconducting phase
Conventional BCS
d-wave
superconductivity?
Recently nodalantinodal dichotomy
has also been
observed in
underdoped
Scenarios for the High-Tc
cuprates
QCP at xc
Standard BCS
xc
Pseudogap and
superconductivity
have a common origin
Pseudogap and superconductivity are
different instabilities which compete
d-wave BCS superconductor:
BCS:
E(k )  ( (k )   S (k ))1/ 2
2
2
d-wave:
 S (k )   S (cos k x  cos k y )   S cos( 2 )
d-wave BCS: A single energy
scale s
E
Gap depends linearly
on cos(2): V-shape
coskx-cosky
Nodal velocity v=1/2(ds()/d)|=p/4=s
Antinodal gap, max=s(=p/2)=s
ARPES deviations from d-wave BCS
in Underdoped SC Cuprates
U-shape
K. Tanaka et al,
Science 314, 1910
(2006)
Two scales in Energy spectrum
with underdoping v decreases
max increases
Two energy scales in Raman Spectrum in
the SC State of Underdoped Cuprates
Energy scale of peak in
antinodal (nodal) region
increases (decreases) with
decreasing doping in
underdoped cuprates.
Pair breaking peak
intensity decreases with
underdoping in antinodal
region (opposite behavior
expected from increasing
energy scale)
Le Tacon et al, Nat. Phys. 2, 537 (2006)
Evolution of Nodal and
Antinodal energy scales with x
Le Tacon et al,
Nat. Phys. 2, 537 (2006)
Energy
Evolution of Nodal and
Antinodal energy scales with x
2
2
Doping
BV and E. Bascones PRl 98, 227002 (2007)
Also able to reproduce the decrease in intensity of antinodal Raman peak with underdoping
Phenomenological model for doped spin
liquid+QCP to describe the pseudogap state
Coherent + Incoherent part
G
RVB
gt
(k ,  ) 
     (k )   R (k ,  )
Only diagonal
 0 (k )  2t ( x)(cos k x  cos k y )
 (k)   0 (/2
k )  2t ' ( x) cos k x cos k y  2t ' ' ( x)(cos 2k x  cos 2k y )
 R (k ,  )  2R ( x) /(   0 (k ))
 R (k )   R ( x)(cos k x  cos k y )
gt ( x)  2 x /(1  x)
Yang, Rice,Zhang PRB 73, 174501 (2006)
QC
“Gapless Fermi arcs”
X=0.05
X=0.14
ky
ky
ky
E
coskx-cosky
kx
kx
kx
E
X=0.20
E
coskx-cosky
coskx-cosky
Doped spin liquid in the SC State

G (k ,  )  g t /      k   R (k ,  )   S (k ) /(   R (k , ))
YRZ
SC
2
X=0.10
Pseudogap physics
(and scale R)
present, if x<xc, in
SC state
X=0.14
X=0.18
X=0.20
Four bands
with energies ±E±
for x<xc
X=0.25

Two scales in the Raman spectra
BV and E. Bascones PRl 98, 227002 (2007)
X=0.10
X=0.14
X=0.18
X=0.20
Antinodal (B1g) peak shifts
to higher energy and
its intensity decreases
with underdoping.
X=0.10
X=0.14
X=0.18
X=0.20
Nodal (B2g) peak shifts
to lower energy with
weaker effect on intensity
with underdoping.
Two pair-breaking transitions below xc
BV and E. Bascones PRl 98, 227002 (2007)
X=0.10
X=0.14
Pair breaking transitions
with energy 2E+ and 2Eappear for x<xc when entering
the superconducting state
X=0.18
X=0.20
X=0.25
Only one pair breaking
transition can be
distinguished for x≥xc
Two pair-breaking transitions below xc
BV and E. Bascones PRl 98, 227002 (2007)
X=0.14, E-
X=0.14, E+
X=0.20, E-& E+
Gap at the maximum intensity surface: Ushape in ARPES
X=0.14 (x<xc)
BV and E. Bascones PRl 98, 227002 (2007)
X=0.20 (x ≥ xc)
E
ky
E
coskx-cosky
kx
U-shape
v decreases
max increases
with underdoping
coskx-cosky
ky
kx
V-shape
Single scale
v=max
The convergence of the two energy scales
and the possible phase-diagram scenarios
T=0
QCP scenario
BV and E. Bascones PRl 98, 227002 (2007)
Do not confuse convergence
of scales below Tc with
convergence of T* and Tc
xc
Autocorrelation of ARPES data (AC-ARPES)
Dispersive peaks in
Superconducting State
Non-Dispersive peaks in
Pseudogap
from momenta joining the
along
tips of the Fermi arcs
bond
Chatterjee et al,
PRL 96, 067005
(2006)
Suggest similar origin for
dispersive and non-dispersive peaks
AC-ARPES in the absence of Pseudogap
Correlations (beyond xc)
Dispersive
peaks in
SC State
Calculated
AC-ARPES
spectra
EXPERIMENT
AC-ARPES in the Pseudogap State (below x )
c
along
bond
EXPERIMENT
Chatterjee et al,
PRL 96, 067005
(2006)
Calculated
AC-ARPES
spectra
2p
3(2p
as origin of ¾ substructure
0
Suggests q*5
E. Bascones and B. V.
cond-mat/0702111
q
Dispersive
and/or nondispersive peaks
can appear in
the SC state
below xc ->
confirmed in
Chatterjee
arXiv:0705.4136
E. Bascones and B. V. cond-mat/0702111
Summary
Two energy scales (nodal and antinodal) in the Raman and
ARPES spectra appear naturally in some QCP models below
xc
With the YRZ Green’s function scenario v is a good measure
of the superconducting order parameter
In this picture the suppression of intensity in B1g channel with
underdoping is a consequence of the competition between
pseudogap and superconductivity
These results suggest that there is a QCP under the
superconducting dome in the high-Tc phase diagram
Other experiments? Autocorrelated ARPES, Prediction:
Dispersive and non dispersive peaks in underdoped SC
cuprates ->confirmed in experiments
Doping independent slope in B2g at low
frequencies
X=0.14
X=0.18
X=0.20
BV and E. Bascones PRl 98, 227002 (2007)
Le Tacon et al, Nat. Phys. 2, 537 (2006)
Hole-doped High-Tc Superconductors
Cu
O
Norman et al, Nature 392, 157 (1998)
X=0 (undoped)
Mott insulator
Optimally doped
(Highest Tc)
(added holes per Cu ion)
Overdoped
How to fulfill Luttinger sum
rule?
X=0.05
X=0.18
X=0.10
Luttinger
surface
X=0.14
X=0.20
Yang, Rice,Zhang PRB 73, 174501 (2006)
Hole pockets
Topological QCP
A third “crossing” transition is
expected below xc
Superconducting state
X=0.14
BV and E. Bascones PRl 98, 227002 (2007)
A transition with
energy E-+E+
is expected in both
superconducting and
pseudogap states
Pseudogap state
X=0.14
Small effect of this transition
in the subtracted response
Total Raman response in the SC state
BV and E. Bascones PRl 98, 227002 (2007)
The crossing transition is hardly distinguished in the superconducting state
And what else?
BV and E. Bascones PRl 98, 227002 (2007)
B1g: antinodal region participates in
superconductivity.
What about QCP models with symmetry-breaking?
Not absolutely ruled out by these experiments
but work worse and no clear evidence of phase
transition from other measurements.
Pseudogap without long-range, hole pockets,
Luttinger surface, QCP, two gaps in the SC
state and v ~S also in cellular DMFT