High Temperature Superconductivity:High Temperature Superconductivity:

D. OrgadD. Orgad Racah Institute, Hebrew University, JerusalemRacah Institute, Hebrew University, Jerusalem

Stripes: What are they and why do they occurStripes: What are they and why do they occur

Basic facts concerning the cupratesBasic facts concerning the cuprates

Experimental signatures of stripesExperimental signatures of stripes

Consequences of stripe formationConsequences of stripe formation::

• FractionalizationFractionalization

• ConfinementConfinement

OutlineOutline::

Are stripes good or bad for superconductivity ? Are stripes good or bad for superconductivity ?

S. KivelsonS. Kivelson V. EmeryV. Emery

E. CarlsonE. Carlson M. GranathM. Granath

V. OganesyanV. OganesyanX-J. ZhouX-J. Zhou

Z-X. ShenZ-X. Shen

The Cuprates: Basic StructureThe Cuprates: Basic Structure

• Universal element – CuO planes • Parent (undoped) compounds – Heisenberg antiferromagnets• Hole doping by chemical substitution / Oxygen doping

4xx)-(2 CuOSrLa

The Cuprates: Typical Phase DiagramThe Cuprates: Typical Phase Diagram

x

T

SC

AF

under optimal overdoping

The Central Question: What happens to an AF upon doping with holes?

UD Bi2212

tunneling ARPES NMR DC resistivity

Optical conductivity

Pseudogap

Neutron scattering, Specific heat…

Renner et al. Harris et al. Warren et al.

Puchkov et al.

Takagi et al.

Holes in an AF : Why Do Stripes Occur?Holes in an AF : Why Do Stripes Occur?

PHASE SEPARATIONPHASE SEPARATION

Coulomb Interactions

Kinetic EnergyFrustration

STRIPESSTRIPES

)4

(.).( jij

ijijs

sijis

nnSSJchcctH

txJt

Stripes in Other Systems: Stripes in Other Systems: Competing Interactions Competing Interactions

Ferrofluid between glass plates Ferromagnetic garnet film

Ferromagnetic garnet film Block copolymers film

cm m

m

Stripe Signatures in S(k,Stripe Signatures in S(k,Real Space Momentum Space

ky

kx

css

c

c

s

ky

kx

s 2

Experimental Evidence for Stripes: Experimental Evidence for Stripes: Neutron ScatteringNeutron Scattering

Static stripeStatic stripe order (LNSCO)order (LNSCO)

ky

kx

Dynamic stripesDynamic stripes))YBCOYBCO((

Mook et al.

E=24.5meV

Tranquada et al.

0.25

0.12

Experimental Evidence for Stripes: Experimental Evidence for Stripes: ARPESARPES

Angle Resolved PhotoEmission Spectroscopy measures the single hole spectral function

)0,0(),(),( )( txedtdxkA tkxi

LNSCOLNSCO

),()( kAdkn

Experimental Evidence for Stripes: Experimental Evidence for Stripes: Tunneling MicroscopyTunneling Microscopy

Hoffman et al.

Howald et al.

B=5TB=0

Consequences of Stripe Formation: Consequences of Stripe Formation: Spin-gap and Enhanced SC CorrelationsSpin-gap and Enhanced SC Correlations

Stripes Doped Spin Ladders: known to be spin-gapped

SCx

PG

AF

T

• The spin-gap creates an amplitude of the SC order parameter

• Provides high pairing scale (avoid Coulomb repulsion)

ws Je

cisLRRL e

2)2cos(

cis e

2

A ProblemA Problem… …

In 1D a spin-gap enhances pairing:)2( 1 cK

ssc TGood News:

divergent for Kc>1/2)Kc<1 for repulsive interactions(

)2/(1 1

)/(~ K

cCEgET FSCF

Bad News:It also enhances CDW correlations:

)2( cKsTCDW

more divergent !

Old problem from search for organic superconductors

)2/(1)/(~ Kc

CEgET FCDWF

L1 L2

y

x

y2y1

Stripe fluctuations dephase CDW coupling:

22)(221

LkLLikFF ee

Stripe fluctuations enhance phase coupling:

22212|| yyy

ee

SC

PG

x

T

Phase Stiffness

static fluctuating dissolved

Nematic?

Stripe fluctuations (quantum, thermal or quenched)are necessary for high Tc!

Phase Stiffnes

s

AF

Yamada et al.

… … And Its ResolutionAnd Its Resolution

Consequences of Stripe Formation:Consequences of Stripe Formation:Electron Fractionalization Above TElectron Fractionalization Above Tcc

In a Fermi liquid the elementary excitations have the quantum numbers of an electron

Valla et al.

Mo surfacestate

qp peak

kk Fv

multi-qpbackground

In a Luttinger liquid the excitations come in 4 flavors

sc

RL

||vs k

k

||vc k

k

ksv kcv ksv|| kcv|| ),( sL),( cL

MDC

EDC

MDC )0( EDC )0( k

3.0c

0c

5.0c

Evidence for FractionalizationEvidence for Fractionalization

Sharp in Momentum Broad in Energy

1DEG

A5.3v

A7.0v

5.0,0

c

s

eV

eV

cs

Orgad et al.

ARPES in LaARPES in La1.251.25NdNd0.60.6SrSr0.150.15CuOCuO44 Breakdown of W-F LawBreakdown of W-F Law

0

22

3L

ek

TB

in Prin Pr1.851.85CeCe0.150.15CuOCuO44

Hill et al.

Below TBelow Tcc: A Coherent Peak: A Coherent PeakOptimally Doped BSCCO (Tc=91K)

Not a Conventional QPNot a Conventional QP

• Not present above Tc

• Intensity grows below Tc

• Energy and lifetime not temperature dependent

Fedorov et al.

Josephson Coupling Confines 1D SolitonsJosephson Coupling Confines 1D Solitons

The electronic operator sscci

L e

2 creates kinks in

s and c

x

cs ,

2

Frustrated Josephson Coupling .].[ chJH jiJosephsonij SC

between solitonsBound spin-charge soliton pair

Charge and spin solitons create phase shift in pair fieldci

s e 2)2cos(

s c

AA<<(k(k in the Superconducting Phase in the Superconducting Phase

)(),( EZkA

incoherent

)2/12(),(),( cxTxTZ

• Quasiparticle weight depends on superfluid density:

Feng et al.

ConclusionsConclusions

• Stripes are ubiquitous in the cuprate high temperature Stripes are ubiquitous in the cuprate high temperature superconductorssuperconductors

• They are important for high temperature They are important for high temperature superconductivitysuperconductivity

• There is evidence that the normal state of the cuprates There is evidence that the normal state of the cuprates is fractionalizedis fractionalized

• In a quasi-one-dimensional superconductor TIn a quasi-one-dimensional superconductor Tcc also also marks a confinement transitionmarks a confinement transition

Landau Theory of Stripe PhasesLandau Theory of Stripe PhasesCoupled charge (CDW) order and spin (SDW) order k

qQS

aa

,

2*x

4242 ||||||21

||||21

qQqQqQSqQSkk SSUSUSrUrF

2221 ||||.].)[( kqQkqQqQ SchSS

qk

2

Stripes are “charge driven” : 0 0S

Spin order is secondary and may be absent

Zachar et al.

Spin-gap Proximity EffectSpin-gap Proximity EffectSingle particle tunneling irrelevantFF kk

~

Pair tunneling FFFF

KKKK ~~ possible

“system” “environment”

tunneling

Fk Fk~

)]~

(2cos[)~

2cos()2cos( ccsspair tH is relevant.

When 1~~~

41

ss

ccpair KK

KK

The spin modes and the relative charge phase mode are gapped. The only gapless mode involves the total SC phase cc ~

• Kinetic energy driven pairing• Repulsive interactions within system and environment increase • Repulsive interactions between system and environment decrease • Pre-existing spin-gap in environment decreases

ARPES and StripesARPES and Stripes

Angle Resolved PhotoEmission Spectroscopy measures the single hole spectral function

)0,0(),(),( )( txedtdxkA tkxi

LNSCOLNSCO

Zhou et al.

LNSCOLNSCO LSCOLSCO

),(30

kAd

meV

)(kn),( kAd

Disordered Stripe Array: Spectral WeightDisordered Stripe Array: Spectral Weight

Granath et al.

Low Energy Spectral Weight)()'()(

1

',

)'(

2.0nn

nn

rr

rrki ErreS

d

)(

)(

)(

Disordered Stripe Array: Disordered Stripe Array: Interacting Spectral FunctionInteracting Spectral Function

Granath et al.

A Model:A Model:Quasi-one-dimensional SuperconductorQuasi-one-dimensional Superconductor

Charge: Gapless

Spin: Gapped

Weak Pair Tunneling)Couples charge and spin(

Prediction: New Magnetic ResonancePrediction: New Magnetic ResonanceNeutron scattering measures the spin-spin correlation function:

)0,0(),( 22)(

FFkk

txki StxSedtdx

''',

2 21

LRkF

S creates 2 spin solitons and 2 charge solitons

Treat more massive spin solitons as static and solve for the charges:

)2cos()()(

)(2v ][

22c

ccc

cxcxcc x

KKH

)(xcs s

Get effective Schrodinger equation for spins:

)(2

v2 21

2

12

22s xxV

xH

j jss

eff

•Spin 1 mode that exists below 0.4 Tc

•2kF mode: should appear around

•Threshold at 2s

0,

2