angle-resolved two-photon photoemission of mott insulatorcorpes05/presentations/tohyama... · 2005....
TRANSCRIPT
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Angle-Resolved Two-Photon Photoemission of Mott Insulator
Takami Tohyama
Institute for Materials Research (IMR)Tohoku University, Sendai
Collaborators
IMR: H. Onodera, K. Tsutsui, S. Maekawa
H. Onodera et al., cond-mat/0503267
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OUTLINE
Introduction
Why is the upper Hubbard band important?Mott insulator: Copper oxides
Two-photon photoemission (2PPES)
The “ω” and “2ω” processesThe “simultaneous” and “cascade” processes
Angle-resolved 2PPES for insulating cuprates in two dimensions
Zhang-Rice singlet bandPump photon
Non-bonding oxygen band
Predictions for future experiments
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Crystal structures of insulating cuprates
CuO3 chain
Sr2CuO3 (one dimension)La2CuO4 (two dimension)
CuO2 plane
● Cu2+
○ O2-
localized spin → antiferromagnetic exchange interactionCu2+ 3d9 1 hole on each x2-y2 orbital
J~1000K-2000K
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new concept !
Band Mott
k
conduction
valence
k
upper Hubbard
lower Hubbard
EE
εF
EE
εF
Photo excitation: Band insulator vs. Mott insulator
electronhole
VBand(r)
exciton picture
VMott(r)
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Band structure of semiconductors
Si GaAsGe
indirect gap direct gap
[J. R. Chelikowsky and M. L. Cohen: Phys. Rev. B 14, 556 (1976)]
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-4.0
-2.0
0.0
2.0
4.0
k
A(k
,ω)
(0,0) (π/2,π/2) (π,π) (π,π/2) (π,0) (π/2,0) (0,0)
ω/t
two dimension (4X4 t-t’-t’’-U)
indirect gapbecause of t’ and t’’
Band structure of Mott insulators: Theoretical prediction
0 π/4 ππ/2 3π/4momentum k
holon
doublonspinon
spinon
one dimension (14-site t-U)
direct gap
[K. Tsutsui, T. T. and S. Maekawa: Phys. Rev. Lett. 83, 3705 (1999); Phys. Rev. B 61, 7180 (2000)]
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Why is the upper Hubbard band (UHB) important?
It is necessary to understand its nature for future application of the Mott insulators.
UHB characterizes the “particle-hole” excitation.
We propose a method to see the momentum-dependent UHB.
Angle-resolved two-photon photoemission(AR-2PPES)
cf. Inverse photoemission low-energy resolution
Electron-energy loss spectroscopy (EELS)Resonant inelastic X-ray scattering (RIXS)
Indirect measurements
[2D cuprates: PRL83, 3705(1999); Science288, 1811(2000)]
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Energy diagram for 2PPESTime-resolved
Ene
rgy-
reso
lved
[P. M. Echenique et al., Surf. Sci. Rep.52, 219 (2004)]
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Two-photon photoemission
Image-potential states at metal surfaces and their decay
Time, energy- and angle-resolved modes
M. Weinelt, J. Phys. Condens. Matter 14, R1099 (2002)Reviews:P. M. Echenique et al., Surf. Sci. Rep.52, 219 (2004)
High-Tc cuprate superconductor: Bi2Sr2CaCu2O8+δ
Time-resolved modeW. Nessler et al., Phys. Rev. Lett. 81, 4480 (1998)
Energy-resolved modeY. Sonoda and T. Munakata, Phys. Rev. B 70, 134517 (2004)
No experimental works in Mott insulators!
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Y. Sonoda and T. MunakataPhys. Rev. B 70, 134517 (2004)
A:excitations from lower-Hubbard bandB:excitations from upper Hubbard bandC,D: excitations perpendicular to CuO2
ω1
ω2
sample
Ekin
occupied state
unoccupied state
vacuum levelω2
ω1
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The “2ω” and “ω” processes
The “2ω” processInformation on unoccupied stateEkin=eUO+ ω
Information on occupied stateEkin=−eOC+ 2ω
unoccupiedstate
Fermi level
vacuum level
eUO
eOCoccupiedstate
The “ω” process
Possible to separate the two states from the ω dependence
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Formulation for two-photon photoemission
Initial state
Intermediate state
final state
pump photon
probe photon
Three-level model
Optical Bloch equation
ρ: density operator, where the superscript (#) denotesthe order of the electric field.
Γ: retaliation rate
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“simultaneous” process Virtual excitation to m
ρii → ρim → ρif → ρmf → ρff ⇒ I(s)
imΓ : phase-relaxation rate
“sequence” or “cascade” process Real excitation to m
ρii → ρim → ρmm → ρmf → ρff ⇒ I(c)
mmΓ : energy-relaxation rat
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Density of states for insulating cuprates
Lower-Hubbard band
Upper-Hubbard band
Mixing O2p and Cu3d
Emergence of a bound state:Zhang-Rice singlet band
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Cu O
② Zhang-Rice Singlet
E BU
E F Upper Hubbardband
Lower Hubbardband
d8
① d10 state
d
E B
nonbonding
UΔ
①
E Foxygen
②
Mapping from three-band model to single-band model
Upper Hubbard band
Lower Hubbard band
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t-t’-t’’-U Hubbard model
t
t't''
U/t = 10
( ) ( )
( )
, , , ,, ',
, , , ,",
h.c. h.c.
h.c.
i j i jij ij
i j i iij i
H t c c t c c
t c c U n n
σ σ σ σσ σ
σ σσ
+ +
+↑ ↓
′= − + − +
′′− + +
∑ ∑
∑ ∑
t = 0.35 eV, t’ = - 0.12 eV, t’’ = 0.08 eVCuprates in 2D: Ca2CuO2Cl2
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Angle-resolved 2PPES for cuprates
“simultaneous” process
2
( )1 2 kin 1 2 kin
1
( , , , ) ( )xsI F
F M M I IM
F c M M j II E E E E
E E iω ω δ ω ω
ω= + + − −
− − − Γ∑ ∑ k
2
( )1 2 kin 1 2 kin
1
( , , , ) ( )xsI F
F M M I IM
F c M M j II E E E E
E E iω ω δ ω ω
ω= + + − −
− − − Γ∑ ∑ kk
“sequence” or “cascade” process
( ) ( )2 2
( )1 2 kin 1 1 2 kin( , , , ) xc
M I I FF M MM
F c M M j II E E E E E Eω ω δ ω δ ω ω= − − × + + − −
Γ∑ ∑ kk
Initial state : the standard Lanczos method I
The spectra are calculated for finite-size clusters such as a 4x4 lattice.
( ) ( ),s cI I : the correction-vector method based on the conjugate gradient technique
2 0.4MM IM tΓ = Γ = , neglecting the pure dephasing
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Calculation of spectrum in the second-order processCorrection-vector method
2
( )1 2 kin 1 2 kin
1 IM
( , , , ) ( )xsI F
F M M I
F c M M j II E E E E
E E iω ω δ ω ω
ω= + + − −
− − − Γ∑ ∑ kk
Correction vector:
1 IM
1x
I
j IH E i
φω
=− − − Γ
( )1 IMI xH E i j Iω φ− − − Γ =
This is evaluated by the conjugate-gradient method.
2( )1 2 kin 1 2 kin( , , , ) ( )s
I FF
I E F c E E Eω ω φ δ ω ω= + + − −∑ kk
This is easily calculated by the standard Lanczos algorithm(recursion method).
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Calculation of spectrum in the second-order process
( ) ( )
( ) ( )max
2 2
( )1 2 kin 1 1 2 kin
2 2'
' 1 1 2 kin' 1
( , , , )
' '
xcM I I F
F M MM
Mx
M I I FF M MM
F c M M j II E E E E E E
F c M M j IE E E E E
ω ω δ ω δ ω ω
δ ω δ ω ω=
= − − × + + − −Γ
≈ − − × + + − −Γ
∑∑
∑ ∑
k
k
k
|M’ ⟩: approximate eigenstates with large value of |⟨M’ |jx | I⟩|2, whichare evaluated by the standard Lanczos algorithm M’ max~ 20
For each |M’ ⟩ , we perform the Lanczos process.
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Possible excitation due to the pump photon ω1
Excitation from Zhang-Rice singlet band
Single-band picture
ω1 ~ 2-3eV ω1
Current operator in the Hubbard model
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4x4 Hubbard modelU=10tt’=-0.343tt’’=0.229t
“ω”
0 5 10 15 20
ε I(ω) ω
2 (arb
.uni
t)
ω/t
Absorption spectrum
I(s) I(c)
Excitation from Zhang-Rice band
“2ω”
ω1
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ω1=6t ω1=9tMomentum dependence of I(s)
at the two photon energies
Single-particle excitation A(k,ω) at half filling
ω1=6t Global feature is similar to that of LHB.
ω1=9t Global feature is similar to that of UHB, but it is difficult to identify the bottom of UHB due to diffusive features.
“ω”
“2ω”
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Possible excitation due to the pump photon ω1
Excitation from non-bonding band
Electron-addition operator in the Hubbard modelω1 ~ 4-5eV
Single-band picture
ω1
Momentum dependence comes from the construction of Wannier orbital, following by Zhang and Rice.
We assume no dispersion of NB and no interactions with other bands.
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Excitation from non-bonding band 20-site t-t’-t’’-J modelU=10t, t’=-0.343t, t’’=0.229t
ω1=2.4t+εp
A(k,ω) in UHBAR-2PPES: I(s)
Emission from the (0,π) state.
“spin-wave”excitation
additional structures
Absence in the spectral function
|I⟩: N (half-filling)|M⟩: N+1|F⟩: N
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5 6 7 8 9 10 110
0.80
0.8
(π,π)
(3π/5,4π/5)(-4π/5,3π/5)
0
0.8(4π/5,2π/5)(-2π/5,4π/5)
0
0.8
I(s) 2P
PE (a
rb.u
nit)
0
0.8
(π,0)(0,π)
(π/5,3π/5)(-3π/5,π/5)
0
0.8(2π/5,π/5)(-π/5,2π/5)
0
0.8(0,0)
E'kin/t
4 5 6 7 8 9 100
20
2
(π,π)
(3π/5,4π/5)(-4π/5,3π/5)
0
2(4π/5,2π/5)(-2π/5,4π/5)
0
2
I(s) 2P
PE (a
rb.u
nit)
0
2
(π,0)(0,π)
(π/5,3π/5)(-3π/5,π/5)
0
2(2π/5,π/5)(-π/5,2π/5)
0
2
(0,0)
E'kin/t
1 2 3 4 5
(π ,π )
(3 π /5 ,4 π /5 )( -4 π /5 ,3 π /5 )
A(+
) (arb
.uni
t)
(4 π /5 ,2 π /5 )( -2 π /5 ,4 π /5 )
(π ,0 )(0 ,π )
(π /5 ,3 π /5 )( -3 π /5 ,π /5 )
(2 π /5 ,π /5 )( -π /5 ,2 π /5 )
(0 ,0 )
ω B / tω1=3.7t+εp
With increasing ω1, the highest-energy position follows UHB dispersion, but accompanied with the spin-related excitation.
ω1=4.0t+εp
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Experimental conditions to detect the bottom of UHB at (π,0)
Maximum kinetic energy of photoelectron from the bottom of UHBmaxkin gap 22+E E ω= gap 4.8 ~ 2eVE t=
If ω1=ω2 , then .
(i) From Zhang-Rice band → Max. intensity appears at ω1~9t~3 eV.maxkin ~ 4 eVE
If ω1=ω2 , then .
(ii) From non-bonding band → ω1=2.4t+εp~5 eV.maxkin ~ 6 eVE
θ
k//
Photoelectron(k,Ekin)Minimum kinetic energy necessary to reach (π,0)
( )2
2// kin vac sin
2k E E
mθ= −From , min
kin ~ 6 eVE
using dCu-Cu~0.4 nm and Evac~4 eV.
The case (ii) critically satisfies this condition.
If the condition ω1 < ω2 is used, it is easy to observe the bottom of UHB.
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Summary
We proposed angle-resolved two-photon photoemission spectroscopy (AR-2PPES) as a new technique to detect the location of the bottom of the upper Hubbard band (UHB) in two-dimensional insulating cuprates.
When the pump photon excites an electron from the Zhang-Rice singlet band, the bottom of UHB is less clear, because of diffusive features in the spectra.
When the pump photon excites an electron from the non-bonding band, the bottom of UHB can be clearly identified.
In addition to information of UHB, additional excitations related to spin degree of freedom emerge in the spectrum, which are characteristic of strongly correlated system,
To detect the bottom of UHB at (π,0), it may be necessary to use either the excitation from non-bonding band or two photons with different energies.