Angle-Resolved Two-Photon Photoemission of Mott Insulator

Transcription

1 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/53267

2 OUTLINE Introduction Mott insulator: Copper oxides Why is the upper Hubbard band important? Two-photon photoemission (2PPES) The ω and 2ω processes The simultaneous and cascade processes Angle-resolved 2PPES for insulating cuprates in two dimensions Pump photon Zhang-Rice singlet band Non-bonding oxygen band Predictions for future experiments

3 Crystal structures of insulating cuprates La 2 CuO 4 (two dimension) Sr 2 CuO 3 (one dimension) Cu 2+ O 2- CuO 2 plane CuO 3 chain Cu 2+ 3d 9 1 hole on each x 2 -y 2 orbital localized spin antiferromagnetic exchange interaction J~1K-2K

4 E Photo excitation: Band insulator vs. Mott insulator Band Mott conduction E upper Hubbard ε F hole electron k ε F k valence lower Hubbard V Band (r) V Mott (r) exciton picture new concept!

5 Si Band structure of semiconductors Ge GaAs indirect gap direct gap [J. R. Chelikowsky and M. L. Cohen: Phys. Rev. B 14, 556 (1976)]

6 Band structure of Mott insulators: Theoretical prediction one dimension (14-site t-u) two dimension (4X4 t-t -t -U) 4. doublon spinon 2. spinon holon A(k,ω). -2. ω/t -4. π/4 π/2 3π/4 π momentum k direct gap (,) (π/2,π/2) (π,π) (π,π/2) (π,) (π/2,) (,) k indirect gap because of t and t [K. Tsutsui, T. T. and S. Maekawa: Phys. Rev. Lett. 83, 375 (1999); Phys. Rev. B 61, 718 (2)]

7 Why is the upper Hubbard band (UHB) important? UHB characterizes the particle-hole excitation. It is necessary to understand its nature for future application of the Mott insulators. 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) Indirect measurements Resonant inelastic X-ray scattering (RIXS) [2D cuprates: PRL83, 375(1999); Science288, 1811(2)]

8 Energy diagram for 2PPES Time-resolved Energy-resolved [P. M. Echenique et al., Surf. Sci. Rep.52, 219 (24)]

9 Two-photon photoemission Image-potential states at metal surfaces and their decay Time, energy- and angle-resolved modes Reviews: M. Weinelt, J. Phys. Condens. Matter 14, R199 (22) P. M. Echenique et al., Surf. Sci. Rep.52, 219 (24) High-Tc cuprate superconductor: Bi 2 Sr 2 CaCu 2 O 8+δ Time-resolved mode W. Nessler et al., Phys. Rev. Lett. 81, 448 (1998) Energy-resolved mode Y. Sonoda and T. Munakata, Phys. Rev. B 7, (24) No experimental works in Mott insulators!

10 ω 2 E kin Y. Sonoda and T. Munakata Phys. Rev. B 7, (24) ω 1 sample vacuum level ω 2 unoccupied state occupied state ω 1 A:excitations from lower-hubbard band B:excitations from upper Hubbard band C,D: excitations perpendicular to CuO 2

11 The 2ω and ω processes vacuum level unoccupied state Fermi level e UO occupied state e OC The 2ω process Information on occupied state E kin = e OC + 2ω The ω process Information on unoccupied state E kin =e UO + ω Possible to separate the two states from the ω dependence

12 Formulation for two-photon photoemission Three-level model final state probe photon Optical Bloch equation Intermediate state pump photon Initial state ρ: density operator, where the superscript (#) denotes the order of the electric field. Γ: retaliation rate

13 simultaneous process Virtual excitation to m ρ ii ρ im ρ if ρ mf ρ ff I (s) Γ im : phase-relaxation rate sequence or cascade process Real excitation to m ρ ii ρ im ρ mm ρ mf ρ ff I(c) Γ mm : energy-relaxation rat

14 Density of states for insulating cuprates Upper-Hubbard band Mixing O2p and Cu3d Lower-Hubbard band Emergence of a bound state: Zhang-Rice singlet band

15 Mapping from three-band model to single-band model d 1 state Zhang-Rice Singlet d 8 Cu O Upper Hubbard band Lower Hubbard band Upper Hubbard band Lower Hubbard band

16 t-t -t -U Hubbard model ij ij U/t = 1 ( + ) ( + ),, h.c. i σ j σ i, σ j, σ h.c. H = t c c + t c c +, σ ij ', σ ", σ ( + h.c. ) σ σ t c c + + U n n i, j, i, i, i t t'' t' Cuprates in 2D: Ca 2 CuO 2 Cl 2 t =.35 ev, t = -.12 ev, t =.8 ev

17 Angle-resolved 2PPES for cuprates simultaneous process Fc M M j I (,,, ) ( ) ( s ) k x I ω 1 ω 2 E kin k = δ ω 1 + ω 2 + E I E F E kin F M E M E I ω 1 i Γ IM 2 sequence or cascade process 2 2 Fc M M j I ( ) ( ) ( c ) k x I ( ω1, ω2, Ekin, k ) = δ EM EI ω1 δ ω1 + ω2 + EI EF Ekin F M Γ MM The spectra are calculated for finite-size clusters such as a 4x4 lattice. Initial state : the standard Lanczos method I ( s) ( c) I, I : the correction-vector method based on the conjugate gradient technique Γ MM = 2Γ IM =.4t, neglecting the pure dephasing

18 Calculation of spectrum in the second-order process Correction-vector method Fc M M j I (,,, ) ( ) ( s ) k x I ω1 ω2 Ekin k = δ ω1 + ω2 + EI EF Ekin F M EM EI ω1 iγim 2 Correction vector: φ = 1 H E ω iγ I 1 IM j x I ( ω ) H E iγ φ = j I I 1 IM x This is evaluated by the conjugate-gradient method. 2 ( s ) I ω1 ω2 Ekin k = F ck φ δ ω1 + ω2 + EI EF Ekin F (,,, ) ( ) This is easily calculated by the standard Lanczos algorithm (recursion method).

19 Calculation of spectrum in the second-order process Fc M 2 2 ( ) ( ) ( c) k x I ( ω1, ω2, Ekin, k) = δ EM EI ω1 δ ω1+ ω2+ EI EF Ekin F M ΓMM F M'1 = MM M j I 2 2 M ' max Fck M' M' jx I δ ω δ ω + ω + Γ ( E E ) ( E E E ) M' I I F kin M : approximate eigenstates with large value of M j x I 2, which are evaluated by the standard Lanczos algorithm M max ~ 2 For each M, we perform the Lanczos process.

20 Possible excitation due to the pump photon ω 1 Excitation from Zhang-Rice singlet band Single-band picture ω 1 ~ 2-3eV ω 1 Current operator in the Hubbard model

21 Excitation from Zhang-Rice band 4x4 Hubbard model U=1t t =-.343t t =.229t ω I (s) I (c) Absorption spectrum ε I (ω) ω 2 (arb.unit) 2ω ω/t ω 1

22 Single-particle excitation A(k,ω) at half filling Momentum dependence of I (s) at the two photon energies ω 1 =6t 2ω Global feature is similar to that of LHB. ω 1 =6t ω 1 =9t ω 1 =9t ω Global feature is similar to that of UHB, but it is difficult to identify the bottom of UHB due to diffusive features.

23 Possible excitation due to the pump photon ω 1 Excitation from non-bonding band We assume no dispersion of NB and no interactions with other bands. Single-band picture ω 1 ω 1 ~ 4-5eV Electron-addition operator in the Hubbard model Momentum dependence comes from the construction of Wannier orbital, following by Zhang and Rice.

24 Excitation from non-bonding band A(k,ω) in UHB ω 1 =2.4t+ε p AR-2PPES: I (s) 2-site t-t -t -J model U=1t, t =-.343t, t =.229t Emission from the (,π) state. additional structures spin-wave excitation Absence in the spectral function I : N (half-filling) M : N+1 F : N

25 (,) (2π /5,π /5) (-π /5,2π /5) A (+) (arb.unit) (π /5,3π /5) (-3π /5,π /5) (π,) (,π ) (4π /5,2π /5) (-2π /5,4π /5) (3π /5,4π /5) (-4π /5,3π /5) (π,π ) ω 1 =3.7t+ε p ω B /t (,) 2 ω 1 =4.t+ε p (,).8 I (s) 2PPE (arb.unit) (4π/5,2π/5) (-2π/5,4π/5) (3π/5,4π/5) (-4π/5,3π/5) (π,π) E' kin /t (2π/5,π/5) (-π/5,2π/5) (π/5,3π/5) (-3π/5,π/5) (π,) (,π) I (s) 2PPE (arb.unit) (2π/5,π/5) (-π/5,2π/5) (π/5,3π/5) (-3π/5,π/5) (π,) (,π) (4π/5,2π/5) (-2π/5,4π/5) (3π/5,4π/5) (-4π/5,3π/5) (π,π) E' kin /t With increasing ω 1, the highest-energy position follows UHB dispersion, but accompanied with the spin-related excitation

26 Experimental conditions to detect the bottom of UHB at (π,) Maximum kinetic energy of photoelectron from the bottom of UHB E = E 2+ ω Egap = 4.8 t ~ 2eV max kin gap 2 (i) From Zhang-Rice band Max. intensity appears at ω 1 ~9t~3 ev. E kin ~4 ev max If ω 1 =ω 2, then. (ii) From non-bonding band ω 1 =2.4t+ε p ~5 ev. E kin ~6 ev max If ω 1 =ω 2, then. Minimum kinetic energy necessary to reach (π,) 2 2 From // ( kin vac ) sin 2m k = E E θ, using d Cu-Cu ~.4 nm and E vac ~4 ev. min E kin ~6 ev k // Photoelectron k,e kin ) The case (ii) critically satisfies this condition. If the condition ω 1 < ω 2 is used, it is easy to observe the bottom of UHB.

27 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,), it may be necessary to use either the excitation from non-bonding band or two photons with different energies.