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