IMSS Symposium – Materials Structure Science using Synchrotron Radiation,
Neutrons and Muons - October 16-17, 2008

Angle-resolved photoemission spectroscopyof high-temperature superconductors:
Present status and outlook
Atsushi Fujimori
University of Tokyo

Collaborators
T. Yoshida, S. Ideta, M. Hashimoto, M. Ikeda, W. Malaeb (Univ. of Tokyo)
K. Tanaka, X.-J. Zhou, D.H. Lu，Z.-X. Shen (Stanford Univ.)Z. Hussain, W.L. Yang (Advanced Light Source)
K. Ono, M. Kubota (KEK-PF)
A. Ino, M. Arita, H. Namatame, M. Taniguchi (HiSOR, Hiroshima Univ.)
S. Uchida, K. Kojima, K. Takashima, T. Kakeshita (Univ. of Tokyo)
H. Eisaki (AIST)
Y. Ando, K. Segawa (Osaka Univ.), S. Komiya, S. Ono (CRIEPI)High-resolution ARPES station BL-28A
at Photon Factory
ARPES endstation

Sample manipulator with two-axis rotation

Y. Aiura(AIST)
Staff: K. Ono, M. Kubota, N. Kamakura (KEK-PF)
User group: A. Fujimori, T. Takahashi, Y. Aiura,
T. Saitoh, K. Ozawa, ….

hn = 35-200 eV
K. Amemiya (KEK-PF)

Angle-Resolved Photoemission
Spectroscopy （ARPES）
EF

below EF

Energy distribution curves
Superconducting gap(EDC)
Energy distributioncurves
D(EDC)

Band dispersion

kky

Fermi surface

Momentum distribution curves
(MDC)

kx

Band structure and Fermi surface inhigh-TC cuprates
Band dispersion

Fermi surface

EtmF

ermilev
el

Holet’

Hole
ElectronHole

Holet’’
Band structure: E(k) = -2t(cos kxa+cos kya)
- 4t’cos kxa cos kya - 2t”(cos2kxa+cos2kya)d-wave superconducting gap inhigh-TC cuprates
d-wave order parameterd-wave superconducting gap
00

Anti-nodeAnti-nodeeodN

Nodedx2-y2 symmetry

Order parameter
D(k) = D0(coskxa - coskya)

BCS theory of d-wave superconductor
Density of statese! 0 (0) =Superconducting gap D0(T)

4.3k BTc
2m

2D0D0TTc
Electronic specific heatDOS

D(k) = D0(coskxa - coskya)

C/TSuperfluid density= Fermi surface volume/m*Tc

TOutline• Pseudogap, Fermi arc and superconducting gap• Coupling of electron to Boson excitations

Pseudogap, Fermi arc andsuperconducting gapTemperature-dependent pseudogap opening
ARPES of underdoped Bi2212
300

T*T*AFI

D*T (K)

T200
c100Fermi liquid
pseudogap

DLEM0Tc
d-SC

0.1

Hole/Cu
Temperature dependence of gaps

M. R. Norman et al. Nature ’98

0.2

0.3

Temperature-dependent pseudogap opening

STM of underdoped Bi2212

300

2D0Tc

TN

T*

AFIT (K)

2001000

Fermi liquid
pseudogap

Tc
d-SC

0.1

0.2

Hole/Cu
2D*

Ch. Renner et al., PRL, ‘98

T*0.3

Temperature-dependent pseudogap opening
Electronic specific heatLa2-xSrxCuO4300TNT*

AFIT (K)

2001000Fermi liquid
pseudogap

Tc
d-SC

0.1

0.2Hole/CuTCT*

N. Momono et al., JPSJ ’02
J. Loram et al., Physica C

0.3

Pseudogap phenomena in high-Tc cuprates
D*

Density of states

Superconducting gap D0(T)e! 0 (0) =m

2D0

4.3k BTc
2D02D*
TTc

T*

Electronic specific heatDOS

D(k) = D0(coskxa - coskya)

C/TSuperfluid density=<<Fermi
Fermisurface
surfacevolume/m*
volume/m*∝ Tc : Uemura relationTc

TT*

Pseudogap opening and Fermi “arc” formation

300T*NAFIT (K)200 T

100Fermi liquidpseudogap

Tcd-SC

00.1

0.2

0.3

Hole/Cu(0,p.)(0,0)

M. R. Norman et al. Nature ’98

x=0.03(p.p)(0,p.)(p.0) (0,0)

x=0.07(p.p) (0,p.)(p.0) (0,0)

x=0.15

(p.p)

(p.0)

T. Yoshida et al., PRB‘06

Distinct superconducting gap andpseudogap?
Andreev reflection

m-on measurements of penetration depthtunneling

Andreev reflection

D(k) = D0(coskxa - coskya)
G. Deutcher, Nature ‘99

C. Panagopoulos, PRL ‘98

Two gap energy scales D* and D0 in
underdoped Bi2212
D*: larger gap near (p,0)

D*D(k) = D0(coskxa - coskya)
near nodeD = 4.3 kBTD0Tc

K. Tanaka et al., Science ‘06

Two gap energy scales D* and D0 in
underdoped La2-xSrxCuO4
ARPES spectra on Fermi surface
D=2.2DLEM

T. Yoshida et al.

Two gap energy scales D* and D0 in
underdoped cuprates

D*D0

D(k) = D0(coskxa - coskya) near node
K. Tanaka et al., Science ‘06

Temperature dependence of superconducting
gap/pseudogap in underdoped Bi2212

T<<Tc

D*~ D0T > Tc

DscFermi arc

D*Universal Fermi arc length(Yoshida et al.)(Ideta et al.)

T<<Tc

D0

Dsc T > Tc
Fermi arc
W.S. Lee et al., Nature ‘07

S. Ideta et al

Superconducting gap Dsc vs Tc
Dsc vs Tc

T(p,p)(p,0)
Fermi surface

T*

D*E
(p,p)

n=3
n=2
(p,0)

Tc

n=1Fermi arc
arc length: LaE

D*(p,p)Dsc

D0d-wave supercond. gap
M. Oda et al, JPSJ ’00
P.A. Lee and X.G. Wen, PRL ’97

Dsc ~ D0(a/2√2)La

2Dsc/kBTc ~ 4.3
S. Ideta et al

Superconducting gap/pseudogap insingle, double and triple layer cuprates
n=2

n=1

n=3

T*, D*

T*, D*

T*, D*

D0

D0

Dsc

Dsc
D0

Dsc

Tc

Tc

Tc

2D*/kBT* ~ 4.3

D0: pairing strength
La ~ superfluid density

Dsc ~ D0(a/2√2)La << D0

2Dsc/kBTc ~ 4.3

T. Yoshida et al.
S. Ideta et al.

Dependence of D0 and Tc,max on the CuO2-layer
numberD*, T*: independent of CuO2-layer #Property of a single CuO2 layer (U, J, t, ….)t

D0, Dsc, Tc: dependent on CuO2-layer #Apical oxygen (t’, t”,…)
Out-of-plane disorder (Eisaki, Uchida)
Interlayer coupling – Copper pair tunneling

t’t’’

small

|t’/t|
E. Pavarini et al., PRL ’01

large
T.Tohyama and S. Maekawa,Supercond. Sci. Technol. ‘00

Dependence of Tc,max on material parameters
Relationship for optimally-doped LSCO, Bi2212, Bi2223

n=3n=2n=1D*D0-t’/t

D0: paring strength

S. Ideta

Possible origin of pseudogap?
(1) Antiferromagnetic fluctuations
Generalized t-J model calcT

P. Prelovsek and A. Ramsak, PRB ’02
Pseudogap

Possible origin of pseudogap?
(2) d-wave pairing fluctuations
Nernst effect

T*, D*/kB

D0/kB

Signals too weakDrops toward x 0 unlike T*
Y. Wang et al. PRB ‘01

Possible origin of pseudogap?
(3) RVB - Resonating Valence Bonds
Slave-boson mean-field theories~T*
H. Fukuyama
N. Nagaosa and P.A. Lee
G. BaskaranG. Kotliar

SU(2)

T*

X.G. Wen and P.A. Lee, PRL ‘98

Possible origin of pseudogap?
(4) Time-reversal symmetry breaking
d-DW

S. Chakraverty et al., PRB ‘01

Circulating current in p-d model Fermi arc

C.M. Varma PRB ’06
C.M. Varma and L. Zhu, RL ‘07

Possible origin of pseudogap?
(5) k-dependent Mott transition
CDMFT calc

M. Civelli et al., PRL ’05
Y.Z. Zhang and M. Imada, PRB ‘07

Possible origin of pseudogap?
(6) CDW, polaron effects
Fermi arc in NaxCa2-xCuCl2O2

Fermi surface nesting?

Checkerboard pattern?

K.M. Shen et al., Science ’05

Cluster glass in STM?

Y. Kohsaka et al., Science ‘07

Short summary 1• Superconductivity in high-Tc cuprates occurs on theFermi arc of length La.
Tc is determined by the paring strength D0 and theavailable electron density ∝ La: Tc ∝ La D0., which
explains most of peculiar features of the high-Tc
cuprates.
• Pseudogap magnitude D* and La are independent ofthe CuO2 layer number while D0 and hence Tc aredependent on the CuO2 layer number.
• Origin of the pseudogap/Fermi arc???

Coupling of electron to Bosonexcitations

Boson-mediated d-wave superconductivity
Superconducting gap + boson structure

Cooper pair formation-k↓ +--k’↑=-k+q↑

Anti-nodeeNode

Boson Wq

Anti-node-qqk’↓=k-q↓

Cooper p
air--Fermi
level2DD2D-D+ -k↑

Fermi surface

Fermi
kmomentum FNodem#BCS = $ (u! + v! a!+," a"+! ,! ) 0!

Gap equation

kFk" = ± " (! ) 2 + !(! ) 2
D(k) = D0(coskxa - coskya)

DOSScattering of quasi-particle by Boson excitation
Electron-Boson couplingk↓ +-odeN

Anti-nodee-Wq

Fermilevel-Nodekink

Boson Wq

Anti-node-qk’↓=k-q↓

Superconducting gap + boson structure-Fermi surface

m2D02DWq

kink+-qFermi
momentumkWqk’

Wq: Boson energy

D0+Wq
D0-D0-D0-Wq

kDOS

Evidence for electron-phonon coupling fromtunneling spectroscopy
S/I/S tunneling barrier

Tunneling spectrum of Pb/MgO/Pb junction
d2I/dV2D+WmD+We|V|De

2D

D-D-D-Wm-D-D-W

2D+W
Phonon DOS

Superconductor

Superconductor
Insulator

0I. Giaever et al., PR ‘62

510Energy (meV)

W.L. McMillan and J.M. Rowell, PRL ‘65

~70 meV kink in the nodal direction
New interpretation
W2~70meVPrevious interpretation“sub kink”

Normalized Re S

70 meVT > Tc
W1~ 35 meV

T< Tc
Bi2212

D0~ 35 meV-0.10

E – EF (eV)
Boson energies

Breathing mode
W2~ 70 meV
A. Lanzara et al., Nature ‘01

0.00

W1 ~ 35 meV
W2 ~ 70 meV
W. S. Lee et al., PRB ‘08
Theory: A. W. Sandvik et al., PRB ‘04

~95(!) meV kink in the trilayer cuprate Bi2223~70 meV

W2OP

T>Tc

IP

“sub kink”
OPT<Tc

Bi2223
Energy gap (eV)W1 ~35 meV

D0OPIP

Bi2223

D0IP ~ 60meV

T>Tc
D0IP

D0OP ~ 40meV

W1

T<Tc
-0.10

-0.05

E – EF (eV)S. Ideta et al.

Phonon scenario of ARPES kinks
Nodal quasi-particle

Antinodal quasi-particle

Buckling mode
W1 ~ 35 meVBreathing mode
W2~ 70 meVKink at ~ W1+D0W

W1+D0Boson WBosonT. P. Deveraux et al. PRL ‘04

WBoson WonsBo

W2W2+D0 but weakKink in ARPES spectra
Nodal region

Nodek↓ +---ode

Anti-nodal regionk↓ +Boson Wq

Anti-node -qk’↓=k-q↓

N-

Boson Wq

W1 ~ 35 meV
W2 ~ 70 meV

Anti-node-q-+Bi2212k’↓=k-q↓-Bi221270 meV

Boson energies+

D0Wi+Dk

W1+D0

W1 ~ 35 meV
D0 ~ 40 meV

W1 ~ 35 meV
D0 ~ 40 meV

W2 ~ 70 meV
Dnode~ 0 meV

T. Yamasaki et al., PRB ‘07

T. Cuk et al., PRL ’04

Coupling to magnetic resonance at W1~ 40 meVInelastic neutron scattering spectracf) Phonon scenario~ W1

Broadening of Boson
spectrum above Tc

Smearing of ARPES kinkaround the anti-node

P. Bourges et al., in High Temperature
Superconductivity,
ed. S.E. Barnes et al. (AIP, ’99)

Broadening of electronic
spectrum above Tc

T. P. Deveraux et al. PRL ‘04

Phonon anomalies in conventional BCS
superconductors
Phonon dispersions in NbC by neutron scattering

softening

Breathing LA phonons:q ~ (+p/2a,0,0)
W~ 15 meVqW. Hanke et al., PRL ‘76hk’=k-qekq

Corresponding W2~ 70 meV phonon
anomalies in high-Tc cuprates
Neutron scattering of La2-xSrxCuO4

X-ray scattering of La2-xSrxCuO4

softeningsofteningW2~ 70 meVBreathing LO phononsq ~ (+p/2a,0)

broadening
L. Pinchovius and M. Braden, PRB ‘99

T. Fukuda et al., PRB ’05

Corresponding W1~ 35 meV phonon
anomalies in high-Tc cuprates
Raman shifts in YBa2Cu3O7-d2DTc
hqetc
W1 ~ 35 meVek’=k-qqk
C. Thomsen et al., SSC ‘90

Short summary 2• Electrons are coupled to Boson excitations, whichyield kinks in ARPES spectra and may provide glue
for Cooper pairing.
• Kinks at W2 ~ 70 meV are due to breathing-type
optical phonons.
• Kinks at W1 ~ 35-40 meV are due to bond-bucklingphonons or magnetic resonance.

Outlook• Understanding the origin of the pseudogap/Fermiarc
and the CuO2 layer number dependences of thepairing strength D0 and hence of Tc.max will provide a
key to elucidate the mechanism of high-Tc
superconductivity.
• Understanding the momentum, temperature, doping,and CuO2 layer number dependences of kinks in
ARPES spectra will also provide a key tounderstand the mechanism of high-Tc
superconductivity. Close collaboration between
ARPES, neutron and x-ray studies areindispensable.