Enhanced charge excitations in electron-doped cuprates by

resonant inelastic x-ray scattering

Takami TohyamaTokyo University of Science, JAPAN

K. Ishii (SPring-8,JAEA)M. Fujita, T. Sasaki, K. Tsutsumi, K. Sato (Tohoku Univ.)M. Yoshida, M. Kurooka, J. Mizuki (Kwansei Gakuin Univ.)R. Kajimoto (J-PARC Center)K. Ikeuchi (CROSS)K. Yamada (KEK)M. Minola, G. Dellea, C. Mazzoli, G. Ghiringhelli, L. Braicovich (Politecnico di Milano)K. Kummer (ESRF)

K. Tsutsui (SPring-8,JAEA)M. Mori (JAEA)S. Sota, S. Yunoki (AICS,RIKEN)

Theory

Experiment

2. Charge degree of freedom in electron-doped cuprates

Outline

3. Q-dependent “coherent” charge excitations

Density-matrix renormalization group (DMRG)

K. Ishii, M. Fujita, T. T. et al., Nat. Commun. 5, 3714 (2014)T. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, 014515 (2015)T.T., J. Electro. Spector. and Related Phenome., 200, 209 (2015)

Charge orderIncoherent charge excitation t-t’-J model

Theoretical prediction t-t’-U Hubbard model

Low-energy charge mode, detectable by RIXS

1. Electron-hole asymmetry in cuprates

Resonant inelastic x-ray scattering (RIXS)

CuO2 plane

La2CuO4 Nd2CuO4(hole-doping) (electron-doping)

Crystal structure of La2CuO4 and Nd2CuO4

● Cu2+

○ O2-

Nd

localized spin → antiferromagnetic exchange interaction

Cu2+ 3d9 1 hole on each x2-y2 orbital

J~1400K

Phase diagrams T.T., JJAP 51, 010004 (2012)

- 3d x2-y2 : single-band material- doped-Mott insulator; d-wave super.- asymmetric: electron v.s. hole

(J = 0.14 eV, t = 0.35 eV)

La2-xSrxCuO4:J= 0.4

t= 1, t’ = -0.12hole

Bi2Sr2Ca1-yRyCu2O8+z : t= 1, t’ = -0.34

electron Nd2-xCexCuO4 (NCCO): t= -1, t’ = 0.25

1 2

, , , ,, , , ,

'tJ i j i j i ii j i j i

H t c c t c c Jσ σ σ σ δσ σ δ

+ ++

< > < >

= − − + ⋅∑ ∑ ∑S S

t

J

t' hole-doping electron-doping

t > 0, t’ < 0 t < 0, t’ > 0

No double occupation

Lower Hubbard band

t-t’-J model

Upper Hubbard band

|G.S>~ c1 x + c2 x + ・・・

The larger |c1| is, the stronger the AF correlation is.

What is the effect of t’ on the magnitude of c1?

– t’

No change of spin configuration

Self-energy gain –t’

-t’ < 0hole-doping electron-doping

|c1| increase

AF correlation increase

> 0

decrease

decrease

Effect of next-nearest-neighbor hopping t’ on AF correlation

Effect of t’ on many-body electronic states

Schematic energy level diagram

t-J t-t’-Jelectron-doping

t-t’-Jhole-doping

Energy

t’ = 0 t’ > 0t’ < 0

AF state

uniform statestripe state

inhomo. state

T.T., Jpn. J. Appl. Phys. 51, 010004 (2012)

Phase diagrams T.T., JJAP 51, 010004 (2012)

Physics of doped-Mott insulator

2p

3d

absorption process(=XAS)

emission process(=XES)intermediate state

ground state

Resonant Inelastic X-ray Scattering (RIXS) for L-edge

L. J. P. Ament et al., Rev. Mod. Phys. 83, 705 (2011)

single magnon

Spin-orbit interaction at 2p

2p

3d

absorption process(=XAS)

emission process(=XES)intermediate state

ground state

RIXS for L-edge

( ) ( )2

, ,1, ~ K K h go o i i

g ih

i o

i o

I q h D D g E EH E i

q K K

+ε ε∆ω δ ∆ω− +

− −ω − Γ

∆ω = ω −ω= −

∑

g : ground state (no Cu2p hole)h : ‘final’ state with momentum q

: dipole transition operator,KD ε

Two contributions to RIXS spectrum

Non spin-flip excitation

Spin-flip excitation

Dynamical charge structure factor

Dynamical spin structure factor

Large Γ limit

Dynamical spin structure factor in 2D t-t’ Hubbard model[C. J. Jia et al., Nature Commun. 5, 3314 (2014)]

Quantum Monte Carlo, β=3/tU/t=8, t’/t= -0.3t=400meV

- Increase of spin-excitation energy for n>1 (electron doping)

3-site hopping terms (order of J)

Comparison between t-t’-U model and t-t’-J model

t-t’-J vs. t-t’-U Hubbard

[C. J. Jia et al., Nature Commun. 5, 3314 (2014)]

[W. S. Lee et al., Nat. Phys. 10, 883 (2014)]

[K. Ishii et al., Nat. Commun. 5, 3714 (2014)]

Spin-flip excitations observed by RIXS in electron-doped cuprates

Increase of spin-flip excitation energy with dopingConsistent with theory

Charge dynamics seen in RIXS for Nd2-xCexCuO4

spin

charge

[Ishii et al., Nat. Comm. 5, 3714 (2014)]

Same as K-edge RIXS[Ishii et al., PRL 91, 207003 (‘05)]

Similar structure in overdoped LSCO[Wakimoto et al., PRB 91, 184513 (2015)]

Dynamical charge structure factor in the 20-site t-t’-J model T.T., J. Electro. Spector. and Related Phenome., 200, 209 (2015)

Hole motion is affected by spin background: spin-charge coupling

Square lattice in 2D

Doped Mott Insulator

Incoherent charge excitation

Phase diagrams T.T., JJAP 51, 010004 (2012)

Physics of doped-Mott insulator

Charge order

Charge order of hole-doped cuprates seen by RIXS (Y,Nd)Ba2Cu3O6+x

[G. Ghiringhelli et al., Science 337, 821 (2012)]

Long-range incommensurate charge order

Charge order of electron-doped cupratesNd2-xCexCuO4

[E. H. da Silva Neto et al., Science 347, 282 (2015)]

Resonant x-ray scattering

Static charge structure factor in the 20-site t-t’-J model

Enhancement of N(q) at small q in underdopded electron system

( ) ( , )N N dω ω= ∫q q

c.f. Enhancement of charge fluctuations near the phase separation

t-J model (many groups), Hubbard model (e.g., T. Misawa, M. Imada, PRB 95,115137 (2014))

T.T., J. Electro. Spector. and Related Phenome., 200, 209 (2015)

2. Charge degree of freedom in electron-doped cuprates

Outline

3. Q-dependent “coherent” charge excitations

Density-matrix renormalization group (DMRG)

K. Ishii, M. Fujita, T. T. et al., Nat. Commun. 5, 3714 (2014)T. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, 014515 (2015)T.T., J. Electro. Spector. and Related Phenome., 200, 209 (2015)

Charge orderIncoherent charge excitation t-t’-J model

Theoretical prediction t-t’-U Hubbard model

Low-energy charge mode, detectable by RIXS

1. Electron-hole asymmetry in cuprates

Resonant inelastic x-ray scattering (RIXS)

Density-matrix renormalization group（DMRG)

[S. R. White, PRL 69, 2863 (1992)]

・・・

・・・

・・・

System |i> Environment |j>Renormalize the states of the Environment into those of the System for each step, by using the density-matrix given by the ground-state wave function.

∑=ji

ij ji,

ψψ

ii ij i jj

ρ ψ ψ′ ′= ∑

ground-state wave function

density matrix

1Tr( )

: eigenstate of

( 0): eigenvalue of

m

A ρA A Au u u u

ρuρ

α α α αα α

α α

α

α

ω ω

ω

=

= = ≈

≥

∑ ∑

discard unimportant states: 0αω ≈m: truncation number

Dynamical density-matrix renormalization group (DMRG)

system environmentijψi j

∑∑∑ ==α

αααα

α ψψρ 1,,,pp

ji’jijii’

Multi target: α

[E.Jeckelmann, PRB66, 045114 (2002)]

0

1 1( , ) Im 0 0q qS q S SE H i

ωπ ω γ−=

− + −

10

0( ) 0 ( )

( ) 0q

q

SE H i S

α ω ρ ωψω γ −

= ⇒ − + −

The last target depends on the energy ω.

Density matrix

DMRG in 2 Dimensions

Introducing long-range interactions to an 1D system.

periodic boundary conditions

system environmentx

added sites

: 1D system

: long-range interactions

DMRG

DMRG method can be applied to 2-D systems

0

2

4q=(π/7,π/3)

x=0 x=0.06 x=0.11 x=0.22

S(q

,ω) (

arb.

uni

ts)

0

2

4 q=(π/7,2π/3)

0.0 0.2 0.4 0.60

2

4 q=(π/7,π)

ω (eV)

Dynamical spin structure factor S(q,ω) by DMRG

Doping dependence of S(q,ω) in electron-doped 2D Hubbard model

6x6 sites

Parameters:t=0.3eVU/t=8, t’/t=-0.3

x: carrier concentration

[C. J. Jia et al., Nat. Commun. 5, 3314 (2014)]

☑ Consistent with quantum Monte Carlo calculations

☑ Shift of peak toward high energy with x

Consistent with experiment

[K. Ishii et al., Nat. Comm. 5, 3714 (2014)][W. S. Lee et al., Nat.

Phys. 10, 883 (2014)]

0.0

0.5

1.0

1.5 q=(π/7,π/3)

x=0.06 x=0.11 x=0.16 x=0.22

N(q

,ω) (

arb.

uni

ts)

0.0

0.5

1.0

1.5 q=(π/7,2π/3)

0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

q=(π/7,π)

ω (eV)

0.0

0.5

1.0

1.5

x=0.06 x=0.11 x=0.22

q=(π/7,π/3)

N(q

,ω) (

arb.

uni

ts)

0.0

0.5

1.0

1.5 q=(π/7,2π/3)

0.0 0.2 0.4 0.60.0

0.5

1.0

1.5 q=(π/7,π)

ω (eV)

Hole-doping

☑ Strong intensity at low-q, low-energy

Electron-doping

Prediction for experiments

Dynamical charge structure factor N(q,ω) by DMRG

T. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, 014515 (2015)

Doping dependence of N(q,ω) in electron-doped t-t’-U Hubbard model

☑ Lower in energy than spin excitations

Peak in S(q,ω)

Charge motion in doped Mott insulator

Incoherent motion energy scale:

hopping t

already observed by RIXS (Ishii et al.)

Coherent motion energy scale:

magnetic J

Prediction to RIXS (T. T. et al.)

[G. Khaliullin, P. Horsch, PRB 54, R9600 (1996)]

N(q,ω) of t-J model

Spin and charge velocities in the Hubbard-type model

0 10.5n

1D: spin-charge separation

Velo

city

spin velocity vs

charge velocity vc

0 10.5n

2D: approximate spin-charge separation

Velo

city

vs

vc[T. T. and S. Maekawa, JPSJ 65, 1902 (1996)]

2. Q-dependent “incoherent” charge excitations

Summary

3. Q-dependent “coherent” charge excitations

Density-matrix renormalization group (DMRG)

Resonant inelastic x-ray scattering (RIXS)

Theoretical prediction

Low-energy charge mode, detectable by RIXST. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, 014515 (2015)

K. Ishii, M. Fujita, T. T. et al., Nat. Commun. 5, 3714 (2014)T.T., J. Electro. Spector. and Related Phenome., 200, 209 (2015)

O K-edge RIXS for charge mode

Perspectives

1. Electron-hole asymmetry in cuprates