National Institute of Advanced Industrial Science and Technology (AIST)Nanoelectronics Research Institute

T Yanagisawa M Miyazaki

K Yamaji

Lattice distortions, stripes andsuperconductivity in high-Tc

cuprates

1. Introduction Structural transition in cuprates LTO,LTLO,LTT phases

Incommensurate spin correlation

2. Correlated d-wave State Vertical Stripes SC Condensation Energy

3. Stripes in the LTT and LTO phases Diagonal stripes

4. Summary

Contents

0

100

200

300

T(K)

Concentration x in Ln2 - x

MxCuO

4 - y

SC

AFAF

SC

Hole dopedElectron doped

0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0

100

200

300

0 0.1 0.2 0.3 0.4

T

Hole concentration

Non-Fermi Liquid

Fermi Liquid

Pseudogap

SC

AF

Phase diagram Theoretical suggestions

High-Tc Superconductor: Phase diagram

LTT,LTO,LTLO,HTT

M.Fujita et al. Phys. Rev.B65,064505(‘02)

Structural transitions: Lattice distortions

Stripes: suggested by Incommensurability

N.Ichikawa et al. PRL85, 1738(‘00)

1. Introduction: Stripes and structural transition

Quantum Variational Monte Carlo method

S.Wakimoto et al. PRB

1. Stable stripes in LTT phase Vertical stripes Coexistence of stripes and SC SC Condensation energy Stripes and tilt axis2. Stripes and Spin-Orbit in LTO phase Diagonal stripes for lightly doping Flux phase

CoexistenceSC+stripes

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

LTLO

Lattice distortions and stripes

3-band Hubbard model(d-p model)

pd

Vpotential = [ρdiiσ∑ − σ(−1)xi +yi mi ]diσ

+ diσ

Gutzwiller function

Hdp0 = diσ

+ pi+x / 2σ+ pi+y / 2σ

+( )ijσ∑ Hijσ

0( )djσ

pj +x / 2σ

pj +y / 2σ

(H 0 + V )ij ujλ = Eλ ui

λ

weightw = det(φ+PGPGφ) φjλ = uj

λ

Hubbard-Stratonovich variables

To include SC order parameter, weSolve the Bogoliubov-de Gennes eq.

PG = Gutzwiller operator

2. Correlated wave functions

SC state in the strongly correlated electron system

ψCdS = PG (ukk

∏ + vkck↑+ c−k↓

+ ) 0

Gutzwiller Projection PG

To control the on-site strongcorrelation

Weight g Weight 1

Coulomb +U Parameter 0<g<1

Essentially equivalent toRVB state (Anderson)Gossamer SC (Laughlin) t-J, t-U-J model

Superconducting state: Gossamer state

SC Condensation energy

∆ESC = Ωn − Ωs = (Sn

0

Tc

∫ − Ss )dT

= (Cs0

Tc

∫ − Cn )dT

Loram et al. PRL 71, 1740 (‘93)

T

C/T

SC Condensation energy ~ 0.2 meV

optimally doped YBCO

Entropybalance

Superconducting Condensation Energy

SC condensation energy

1-band Hubbard model

3-band: d-wave

3-band: stripes+d-wave

Variational Monte Carlo evaluations

~0.2meV

∆Esc = 0.00117t = 0.59 meV/site (ρ=0.86, t’=-0.2, U=8)

Experiments0.26 meV/site

critical field Hc

0.17~0.26 C/T

Agreement is good!

SC Condensation energy in the bulk limitSC Condensation energy in the bulk limit

Yamaji et al., Physica C304, 225(‘98)T.Yanagisawa, Phys. Rev.B67,132408

SC Condensation Energy in VMC

0

0.001

0.002

0.003

0.004

0 0.005 0.01

∆ E/N

1 / Na

Superconductivity and Antiferromagnetism:Competition

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2

SDW(U=6,L=10)

SDW(U=6,L=12)

SC(U=6,L=10)

SC(U=6,L=12)

Eco

nd

t'

Fig. 1

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0.002 0.004 0.006 0.008 0.01

Fig. 2SC E

cond (ρ ~.84, U=5, t'=−.05)

Eco

nd

1/Ns

Size dependence ofSC condensation energySize dependence ofSize dependence ofSC condensation energySC condensation energy

SC and AFSC and AF

Pure d-wave SC

0

0.0001

0.0002

0.0003

0 0.1 0.2 0.3

∆ E/N

Hole density

SC Condensation energy

3-band model

1-band Hubbard

Stripe effect

VMC

M.Ido et al. LT23

Loram et al.

Hokkaido

Decrease of ∆Esc due to Stripe order

Decrease of ∆Esc due to Stripe order

Carrier density dependenceof condensation energy

AFspin

SC order parameter

Inhomogeneous SC state

Charge

Vanishing SC order parameter in AF (hole poor) domain

SC

Relative π-phase shift across the AF domain

θ1 θ2

θ1−θ2= π

Coexistence of d-wave SC and stripes

LTO (Q1=0,Q2LTO (Q1=0,Q2 0)0) LTLO (Q1LTLO (Q1 Q2Q2 0)0) LTT (Q1= Q2 LTT (Q1= Q2 0)0)

1

CuOxygen

Lanthanide (La, Nd, Eu)Sr, Ba

Up

Down

3. Lattice distortions

Cf. A. P. Kampf et. al. PRB 64 (2001) 052509

LTT structural transitions stabilize stripes.

• = ty / tx

• X

• Y

One-band Hubbard model (Miyazaki)

Anisotropy of the transfer integrals in LTT phase

• X

tpdx=1+u

tpdy=1

tilt axisParallelstripes E

Commensurate SDW

Stripes: 8-lattice period

Perpendicular

T.Y. et al., J.Phys.C14,21(‘02)

-0.1

0

0.1

0.2

0 0.02 0.04 0.06 0.08 0.1

LTT-HTTParallel //stripesPerpendicular

∆ E/N

u

(E(u=

0)-E(u))/N

Vertical Stripes in LTT

tx

LTLO LTTLTTLTLO

E. S. Bozin et. al. PRB 59 (1999) 4445Lanzara et. al. J. Phys. Cond. Mat 11 (1999) 541

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

tx

ty

LTLO

Mixed phase ofMixed phase ofLTT and LTLOLTT and LTLO Stripes // tilt axisStripes // tilt axis

Hole richdomain

Hole poordomain

Stabilize stripes

Possible Stripe Structures 1

tx

LTT HTTHTTLTT

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

tx=1+ucos(Qyy)

ty

LTLO

LTT and HTTLTT and HTT

Stripes perpendicularStripes perpendicularto tilt axisto tilt axis

Oscillation of tilt angleB.Buchner et. al. PRL 73(1994)1841H.Oyanagi et al., T.Y.et al. LT23 Proc.

stable

StripesSmall tilt angles

Possible Stripe Structures 2

tx

LTT HTTHTTLTT

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

tx

ty

LTLO

Mixed phase of LTTMixed phase of LTTand HTTand HTT

Stripes perpendicularStripes perpendicularto tilt axisto tilt axis

StableH. OyanagiA. Bianconi

Charge poor

Charge rich

Possible Stripe Structures 3

-0.1

0

0.1

0.2

0 0.02 0.04 0.06 0.08 0.1

LTT-HTTParallel //stripesPerpendicular

∆ E/N

u

LaLa2-x2-xSrSrxxNiONiO44

LaLa2-x2-xSrSrxxCuOCuO44

LaLa2-x-y2-x-yNdNdyySrSrxxCuOCuO44

Diagonal stripes are observed for

in the lightly doped region.

Charge density

DV

D

U=8tt’=-0.2t

1-bandHubbard

Diagonal stripesare unstable.

Diagonal stripes in lightly doped region

15x15 ~ 3% doping

Anisotropy in tpp transferstabilizes diagonal stripes.

-3.75

-3.7

-3.65

-3.6

-3.55

-3.5

-0.02 0 0.02 0.04 0.06 0.08 0.1

verticaldiagonal

E/N

u

-3.6

-3.5

-3.4

-3.3

-3.2

0.06 0.08 0.1 0.12 0.14 0.16 0.18

verticaldiagonal

E/N

u

tpd tpd (1-u) tpp tpp (1-u)(one direction)

16x15 ~ 3% dopingLTO (Q1=0,Q2LTO (Q1=0,Q2 0)0)

Up

Down

Diagonal

Vertical

Diagonal

Vertical

Diagonal Stripes in LTO structure

Hkin = − (tij +ijσ∑ icσθij )diσ

+ djσ

Stripesφ

φ

φ

−φ

−φ

φ

φ

φ

Buckling in the Cu-O planeInduces spin-orbit coupling.

Flux state and Diagonal Stripes in underdoped (lightly) region

flux

(Bonesteal et al.,PRL68,2684(‘92))

String-density wave

-0.575

-0.570

-0.565

-0.560

-0.555

-0.550

0.00 0.02 0.04 0.06 0.08 0.10

CommensurateVerticalDiagonal

E/N

φ/π

4. Spin-Orbit Coupling

φ

φ

φ

−φ

−φ

φ

φ

φ

Excitation: Dirac fermionLinear dispersion

Fermi point (half-filling)

Insulating or Bad metal state (small Fermi surface)

EkE(kx ,ky ) = ±eiφ/ 4eikx + e−iφ/ 4eiky

+eiφ/ 4e−ikx + e−iφ/ 4e−iky

φ

Flux state

1. SC condensation energy agrees with experiments based on the Correlated d-wave SC function (Gossamer state).2. Vertical stripes are stable in the LTT phase. Mixed LTT-HTT is most stable.

δ=x

x~0.05

Vertical stripesSC (Coexistence)

LTT

Inco

mm

ensu

rabi

lity

δ

DiagonalLTO

Insulator

δ=x/2?

spin-orbitcoupling

Flux state

LTT

LTO

lightly under optimal over

Vertical

Diagonal

Vertical

Vertical

(V?)

(V?)

Comm.

Comm.

SC condensationenergy

∆Esc ~ 0.2 meV

(optimally doped)

5. Summary