- Email: [email protected]
Theory of resonant inelastic X-ray scattering in cuprates
Journal of Physics and Chemistry of Solids 67 (2006) 262–265 www.elsevier.com/locate/jpcs
Theory of resonant inelastic X-ray scattering in cuprates Takuji Nomura a,*, Jun-ichi Igarashi b a
Synchrotron Radiation Research Center/SPring-8, Japan Atomic Energy Research Institute, Hyogo 679-5148, Japan b Department of Mathematical Sciences, Ibaraki University, Ibaraki 310-8512, Japan
Abstract We analyze theoretically resonant inelastic X-ray scattering near the Cu absorption K edge in some cuprates (La2CuO4, Sr2CuO3, SrCuO2). We explain the drastic momentum dependence of the spectra observed recently in experiments. The antiferromagnetic ground state is described by using the Hartree–Fock approximation, and scattering by the Cu1s core hole is treated within the Born approximation, and the electron correlation effects are taken into account within the random phase approximation, using the time-dependent perturbation theory. The main spectral weight originates predominantly from the charge transfer process. The electron correlations between Cu3d electrons are essential for the drastic momentum dependence. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Oxides; C. Raman spectroscopy; D. Optical properties
1. Introduction Charge excitation properties have been one of the most important and intriguing issues in the physics of strongly correlated electron systems. There are a lot of spectroscopic approaches for clarifying charge excitation properties in solids. Among them, resonant inelastic X-ray scattering (RIXS) [1] in the hard X-ray regime are recently providing a promising and powerful tool to clarify the relatively high-energy charge excitation properties, since they have indeed succeeded in obtaining momentum-dependent charge excitation spectra in some materials [2–6]. It is naturally expected that we can obtain an insight into the electronic structure and the excitation properties of solids by inspecting the detailed momentum dependence of the spectra. An impressive example of strongly momentum-dependent spectrum is given by the RIXS near the Cu absorption K edge for some insulating cuprates [3–6], where the incident photon energy is set around the Cu1s–4p absorption energy. The incident photon (ui, qi) with the resonance energy (uiz34pK 31s) is absorbed to excite the Cu1s core electrons to the Cu4p band (Process 1, in Fig. 1). The created Cu1s hole plays a role similar to nonmagnetic impurity for the electron system. The Cu3d electrons are scattered to some excited states by the core * Corresponding author. Tel.: C81 791582701; fax: C81 791582740. E-mail address: [email protected] (T. Nomura).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.045
hole through the inter-orbital Coulomb interaction (Process 2, in Fig. 1). Finally the Cu4p electron is annihilated together with the Cu1s core hole, emitting a photon (uf, qf) (Process 3, in Fig. 1). Compared with the initially absorbed photon, the emitted photon loses an equal amount of energy to that required to excite the Cu3d electrons. Investigating the transferred momentum (qZqiKqf) and the energy loss (uZuiKuf) of the photon systematically, momentum-dependent excitations are clarified. Cu1s–4p RIXS experiments were performed for La2CuO4 by Kim et al. [4] and quasi-one-dimensional (Q1D) copper oxides, SrCuO2 and Sr2CuO3 by Hasan et al. [5] and Kim et al. [6] In La2CuO4 characteristic two-peak structure was observed around 2 and 4 eV energy loss. It is interesting that the 2 eV peak shows a relatively large dispersion (w1 eV) along the symmetry line qZ(0,0)–(p,0), and is drastically suppressed around qZ(p,p), where q is the in-plane transferred momentum of the photon. In the Q1D cuprates, characteristic two-peak structure was observed around 2 and 5.7 eV energy loss. The 2 eV peak shows a relatively large dispersion. The 2 eV peak around qZ0 shifts by about 1.1 eV at qZp. In the present short article, we analyze the RIXS in these cuprates theoretically. The obtained momentum-dependent spectral structure agrees with the recent experiments semi-quantitatively. 2. Theory We consider the total Hamiltonian of the form, H Z Hdp C H1sK3d C H1s C H4p C Hx . The dp Hamiltonian Hdp describing the electronic properties of the Cu3d–O2p bands is given in the form, HdpZH0CH 0 . H0 and H 0 are the
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
263
processes from the ground state within the random phase approximation (RPA) in Ud, where the antiferromagnetic ground state is described by the Hartree–Fock (HF) approximation for H 0 . Diagrammatic representation of the RIXS intensity is given in Refs. [7,8]. The Cu1s–4p RIXS intensity is given by the formula, X Wðui qi ; uf qf Þ Z ð2pÞ3 Njwj4 dðEj ðkÞ C u kjj 0
KEj0 ðk C qÞÞnj ðkÞð1Knj 0 ðk C qÞÞ X † U ðkÞLss0 ðu; qÞUds ! 0 ;j 0 ðk C qÞ 0 j;ds ss
Fig. 1. Schematic figure of the Cu1s–4p RIXS. The core hole scatters the Cu3d electrons through the inter-orbital Coulomb interaction V.
noninteracting and the interacting parts, respectively. X X † H0 Z 3d dks dks C 3p[ ðkÞp†k[s pk[s ks
C
† ðVdp[ ðkÞdks pk[s C h:c:Þ;
k1
2 V ; ðui C 31s C iG1s K34p ðk1 ÞÞðuf C 31s C iG1s K34p ðk1 ÞÞ (5)
k[s
X
X
(1)
k[s
where qi and qf (ui and uf) are the momenta (energies) of the initially absorbed and the finally emitted photons, respectively. qZqiKqf and uZuiKuf are the transferred momentum and the energy loss, respectively. Ej(k) and nj(k) are the energy dispersion and the electron occupation number of band j,
† and p†k[s ) are annihilation (creation) where dks and pk[s (dks operators for Cu3dx2Ky2 and O2p[ ð[Z x; yÞ orbitals, respectively. The parameters are determined from the first-principle band calculations. The interacting part is given by X † † dkK (2) H 0 Z Ud q[dk 0CqYdk 0Ydk[:
(a)
q=(0,0)
B
Cu O
A qy
kk 0 q
kk 0 qss 0
where sks ðs†ks Þ is the annihilation (creation) operator for the Cu1s electrons with momentum k and spin s, and V is the corehole potential. H4p and H1s describe the kinetics of the electrons on the Cu4p and Cu1s bands, respectively. Since the Cu1s electrons are well localized, we take completely flat dispersion for them. For the Cu4p electrons, we use simple two-dimensional cosine-shaped band for simplicity. This simplification does not affect the spectral shape drastically, because the factor containing the Cu4p dispersion function is integrated in momentum. Hx describes the transitions between the Cu1s and Cu4p states, involving the photon absorption and emission processes. Hx is of the form X 0† ðwðq; kÞpkCqs sks C h:c:Þ; (4) Hx Z kqs 0
where p † is the creation operator of the Cu4p electron. In the present study, we neglect the momentum dependence of the matrix elements w(q;k), i.e., w(q;k)Zw. We consider only the lowest order in the core hole potential V (i.e., Born scattering), and take account of the excitation
(b)
RIXS Intensity (arb. units)
The Coulomb energy is taken to be UdZ11 eV in the present study. The scattering of the Cu3d electrons by the Cu1s core hole is described by X dk†0Cqs s†kKqs0 sks0 dk 0 s ; (3) H1sK3d Z V
qx
q=(π,0) A
(c)
0
B
q=(π,π)
2
B
4 6 8 Energy loss (eV)
10
12
Fig. 2. The calculated RIXS spectra for La2CuO4 as a function of photon energy loss u. The arrows indicate the positions of the characteristic peaks A and B. Three cases of in-plane transferred momenta qZ(0,0), (p,0), (p,p) are shown. The inset shows the schematic figure of the CuO2 plane.
264
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
respectively, obtained by diagonalizing the HF Hamiltonian for Hdp. Uj,ds(k) is the (j,ds)-element of the unitary matrix for the diagonalization. 31s and 34p(k) are the kinetic energies of the Cu1s and Cu4p electrons, respectively. Lss 0 (u,q) is evaluated within RPA. In the present study the incident photon energy ui is tuned to the Cu1s–4p absorption energy uiz34p(0)K31s, for which the intensity of the spectrum is enhanced by the resonance. The decay rate G1s of the core hole is 0.8 eV in the present study.
3.1. La2CuO4 The electronic properties of the Cu3d–O2p system in La2CuO4 are well described by two-dimensional dp Hamiltonian. The parameters for the dp model are tdpZK1.3 eV, tppZK0.65 eV as determined from first-principle calculations [9]. The charge transfer energy is 3HF d K3p ZK0:7 eV, where the 3HF d is the renormalized Cu3d one-electron energy within HF theory. The characteristic two-peak structure observed in experiments [4] is reproduced in the numerical results of the present study (Peaks A and B in Fig. 2), although the position of the peak B deviates from the experimental position (w4 eV) maybe due to the crude tight-binding fitting. These two peaks are attributed to the charge transfer (CT) process from the Cu3d–O2p bonding and anti-bonding bands to the unoccupied upper Cu3d band, which is clarified by inspecting the Cu3d partial density of states in each band. The 2 eV peak A exhibits a relatively large shift to the high energy region by around 0.8 eV along the line qZ(0,0)–(p,0) (Fig. 2a and b), and becomes remarkably small around qZ(p,p) (Fig. 2c). On the other hand, the high energy peak B shows only a small dispersion. These characteristic behaviors are in semi-quantitative agreement with experimental results. Actually it is reasonably expected that the small weight around uz9 eV is not observable [7].
O(1)
B
q=0
Cu O(2)
q A
(b)
RIXS Intensity (arb. units)
3. Results
(a)
B
q=π/2
A
B
(c)
q=π A
0
2
4 6 8 Energy loss (eV)
10
12
Fig. 3. The RIXS intensity calculated for the corner-sharing one-dimensional cuprate as a function of photon energy loss u. Three cases of transferred momenta qZ0,p/2,p are shown.
behaviors are in qualitative agreement with experimental results, although the position of the peak B deviates from the experimental position (z5.7 eV) maybe due to the crude tightbinding fitting. 3.4
3.2. Q1D copper oxides
3 Energy Loss ω (eV)
The electronic properties of the Cu3d–O2p system in Sr2CuO3 and SrCuO2 are well described by the onedimensional corner-sharing dp Hamiltonian. The calculated RIXS intensity for three cases of transferred momenta qZ0, p/2, p is displayed in Fig. 3. The parameters for the dp model are td–O(1)pZK1.45 eV along the chain, td–O(2)pZK1.8 eV perpendicular to the chain, and tO(1)p–O(2)pZK0.70 eV, as determined from first-principles calculations [10]. The oneparticle energies are 3HF d K3Oð1Þp ZK0:5 eV and 3O(2)p K 3O(1)pZ0.5 eV. The characteristic two-peak structure observed in experiments [5,6] is reproduced in the numerical results of the present study (Peaks A and B in Fig. 3). The 2 eV peak A exhibits a relatively large shift to the high energy region by around 1.1 eV. On the other hand, the high energy peak B does not show such a dispersion (Fig. 3). These characteristic
3.2
2.8 2.6 2.4 2.2 2 1.8 1.6
0
π/2 Momentum Transfer q
π
Fig. 4. Calculated spectral intensity around uw2 eV as a function of transferred momentum q along the Cu–O chain. The light (dark) region corresponds to strong (weak) RIXS intensity. The empty circles and squares are the peak positions experimentally obtained by Hasan et al. [5].
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
The position of the peak A shows a sinusoidal dispersion (Fig. 4). Our recent work [8], where the results with and without the RPA corrections are compared, showed that manybody correction is essential for this sinusoidal dispersion.
4. Concluding remarks The main spectral feature characterized by two peaks in La2CuO4 and the Q1D copper oxides are attributed to the CT process from the Cu3d–O2p bonding and anti-bonding bands to the unoccupied upper Cu3d band. Thus discussions based on simple Hubbard model neglecting the O2p bands [11] are inappropriate. In conclusion, we have theoretically reproduced the twopeak structure and the strong momentum dependence of the Cu K edge RIXS spectra observed in a two-dimensional insulating cuprate La2CuO4 and one-dimensional ones Sr2CuO3 and SrCuO2.
265
References [1] For a review, A. Kotani, S. Shin, Rev. Mod. Phys. 73 (2001) 203. [2] P. Abbamonte, C.A. Burns, E.D. Isaacs, P.M. Platzman, L.L. Miller, S.W. Cheong, M.V. Klein, Phys. Rev. Lett. 83 (1999) 860. [3] M.Z. Hasan, E.D. Isaacs, Z.X. Shen, L.L. Miller, K. Tsutsui, T. Tohyama, S. Maekawa, Science 288 (2000) 1811. [4] Y.J. Kim, J.P. Hill, C.A. Burns, S. Wakimoto, R.J. Birgeneau, D. Casa, T. Gog, C.T. Venkataraman, Phys. Rev. Lett. 89 (2002) 177003. [5] M.Z. Hasan, P.A. Montano, E.D. Isaacs, Z.-X. Shen, H. Eisaki, S.K. Sinha, Z. Islam, N. Motoyama, S. Uchida, Phys. Rev. Lett. 88 (2002) 177403. [6] Y.J. Kim, J.P. Hill, H. Benthien, F.H.L. Essler, E. Jeckelman, H.S. Choi, T.W. Noh, N. Motoyama, K.M. Kojima, S. Uchida, D. Casa, T. Gog, Phys. Rev. Lett. 92 (2004) 137402. [7] T. Nomura, J. Igarashi, Phys. Rev. B 71 (2005) 035110. [8] T. Nomura, J. Igarashi, J. Phys. Soc. Jpn 73 (2004) 1677. [9] M.S. Hybertsen, M. Schlu¨ter, Phys. Rev. B 39 (1989) 9028. [10] R. Neudert, S.-L. Drechsler, J. Ma´lek, H. Rosner, M. Kielwein, Z. Hu, M. Knupfer, M.S. Golden, J. Fink, N. Nu¨cker, M. Merz, S. Schuppler, N. Motoyama, H. Eisaki, S. Uchida, M. Domke, G. Kaindl, Phys. Rev. B 62 (2000) 10752. [11] K. Tsutsui, T. Tohyama, S. Maekawa, Phys. Rev. Lett. 83 (1999) 3705.
Theory of resonant inelastic X-ray scattering in cuprates Takuji Nomura a,*, Jun-ichi Igarashi b a
Synchrotron Radiation Research Center/SPring-8, Japan Atomic Energy Research Institute, Hyogo 679-5148, Japan b Department of Mathematical Sciences, Ibaraki University, Ibaraki 310-8512, Japan
Abstract We analyze theoretically resonant inelastic X-ray scattering near the Cu absorption K edge in some cuprates (La2CuO4, Sr2CuO3, SrCuO2). We explain the drastic momentum dependence of the spectra observed recently in experiments. The antiferromagnetic ground state is described by using the Hartree–Fock approximation, and scattering by the Cu1s core hole is treated within the Born approximation, and the electron correlation effects are taken into account within the random phase approximation, using the time-dependent perturbation theory. The main spectral weight originates predominantly from the charge transfer process. The electron correlations between Cu3d electrons are essential for the drastic momentum dependence. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Oxides; C. Raman spectroscopy; D. Optical properties
1. Introduction Charge excitation properties have been one of the most important and intriguing issues in the physics of strongly correlated electron systems. There are a lot of spectroscopic approaches for clarifying charge excitation properties in solids. Among them, resonant inelastic X-ray scattering (RIXS) [1] in the hard X-ray regime are recently providing a promising and powerful tool to clarify the relatively high-energy charge excitation properties, since they have indeed succeeded in obtaining momentum-dependent charge excitation spectra in some materials [2–6]. It is naturally expected that we can obtain an insight into the electronic structure and the excitation properties of solids by inspecting the detailed momentum dependence of the spectra. An impressive example of strongly momentum-dependent spectrum is given by the RIXS near the Cu absorption K edge for some insulating cuprates [3–6], where the incident photon energy is set around the Cu1s–4p absorption energy. The incident photon (ui, qi) with the resonance energy (uiz34pK 31s) is absorbed to excite the Cu1s core electrons to the Cu4p band (Process 1, in Fig. 1). The created Cu1s hole plays a role similar to nonmagnetic impurity for the electron system. The Cu3d electrons are scattered to some excited states by the core * Corresponding author. Tel.: C81 791582701; fax: C81 791582740. E-mail address: [email protected] (T. Nomura).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.045
hole through the inter-orbital Coulomb interaction (Process 2, in Fig. 1). Finally the Cu4p electron is annihilated together with the Cu1s core hole, emitting a photon (uf, qf) (Process 3, in Fig. 1). Compared with the initially absorbed photon, the emitted photon loses an equal amount of energy to that required to excite the Cu3d electrons. Investigating the transferred momentum (qZqiKqf) and the energy loss (uZuiKuf) of the photon systematically, momentum-dependent excitations are clarified. Cu1s–4p RIXS experiments were performed for La2CuO4 by Kim et al. [4] and quasi-one-dimensional (Q1D) copper oxides, SrCuO2 and Sr2CuO3 by Hasan et al. [5] and Kim et al. [6] In La2CuO4 characteristic two-peak structure was observed around 2 and 4 eV energy loss. It is interesting that the 2 eV peak shows a relatively large dispersion (w1 eV) along the symmetry line qZ(0,0)–(p,0), and is drastically suppressed around qZ(p,p), where q is the in-plane transferred momentum of the photon. In the Q1D cuprates, characteristic two-peak structure was observed around 2 and 5.7 eV energy loss. The 2 eV peak shows a relatively large dispersion. The 2 eV peak around qZ0 shifts by about 1.1 eV at qZp. In the present short article, we analyze the RIXS in these cuprates theoretically. The obtained momentum-dependent spectral structure agrees with the recent experiments semi-quantitatively. 2. Theory We consider the total Hamiltonian of the form, H Z Hdp C H1sK3d C H1s C H4p C Hx . The dp Hamiltonian Hdp describing the electronic properties of the Cu3d–O2p bands is given in the form, HdpZH0CH 0 . H0 and H 0 are the
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
263
processes from the ground state within the random phase approximation (RPA) in Ud, where the antiferromagnetic ground state is described by the Hartree–Fock (HF) approximation for H 0 . Diagrammatic representation of the RIXS intensity is given in Refs. [7,8]. The Cu1s–4p RIXS intensity is given by the formula, X Wðui qi ; uf qf Þ Z ð2pÞ3 Njwj4 dðEj ðkÞ C u kjj 0
KEj0 ðk C qÞÞnj ðkÞð1Knj 0 ðk C qÞÞ X † U ðkÞLss0 ðu; qÞUds ! 0 ;j 0 ðk C qÞ 0 j;ds ss
Fig. 1. Schematic figure of the Cu1s–4p RIXS. The core hole scatters the Cu3d electrons through the inter-orbital Coulomb interaction V.
noninteracting and the interacting parts, respectively. X X † H0 Z 3d dks dks C 3p[ ðkÞp†k[s pk[s ks
C
† ðVdp[ ðkÞdks pk[s C h:c:Þ;
k1
2 V ; ðui C 31s C iG1s K34p ðk1 ÞÞðuf C 31s C iG1s K34p ðk1 ÞÞ (5)
k[s
X
X
(1)
k[s
where qi and qf (ui and uf) are the momenta (energies) of the initially absorbed and the finally emitted photons, respectively. qZqiKqf and uZuiKuf are the transferred momentum and the energy loss, respectively. Ej(k) and nj(k) are the energy dispersion and the electron occupation number of band j,
† and p†k[s ) are annihilation (creation) where dks and pk[s (dks operators for Cu3dx2Ky2 and O2p[ ð[Z x; yÞ orbitals, respectively. The parameters are determined from the first-principle band calculations. The interacting part is given by X † † dkK (2) H 0 Z Ud q[dk 0CqYdk 0Ydk[:
(a)
q=(0,0)
B
Cu O
A qy
kk 0 q
kk 0 qss 0
where sks ðs†ks Þ is the annihilation (creation) operator for the Cu1s electrons with momentum k and spin s, and V is the corehole potential. H4p and H1s describe the kinetics of the electrons on the Cu4p and Cu1s bands, respectively. Since the Cu1s electrons are well localized, we take completely flat dispersion for them. For the Cu4p electrons, we use simple two-dimensional cosine-shaped band for simplicity. This simplification does not affect the spectral shape drastically, because the factor containing the Cu4p dispersion function is integrated in momentum. Hx describes the transitions between the Cu1s and Cu4p states, involving the photon absorption and emission processes. Hx is of the form X 0† ðwðq; kÞpkCqs sks C h:c:Þ; (4) Hx Z kqs 0
where p † is the creation operator of the Cu4p electron. In the present study, we neglect the momentum dependence of the matrix elements w(q;k), i.e., w(q;k)Zw. We consider only the lowest order in the core hole potential V (i.e., Born scattering), and take account of the excitation
(b)
RIXS Intensity (arb. units)
The Coulomb energy is taken to be UdZ11 eV in the present study. The scattering of the Cu3d electrons by the Cu1s core hole is described by X dk†0Cqs s†kKqs0 sks0 dk 0 s ; (3) H1sK3d Z V
qx
q=(π,0) A
(c)
0
B
q=(π,π)
2
B
4 6 8 Energy loss (eV)
10
12
Fig. 2. The calculated RIXS spectra for La2CuO4 as a function of photon energy loss u. The arrows indicate the positions of the characteristic peaks A and B. Three cases of in-plane transferred momenta qZ(0,0), (p,0), (p,p) are shown. The inset shows the schematic figure of the CuO2 plane.
264
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
respectively, obtained by diagonalizing the HF Hamiltonian for Hdp. Uj,ds(k) is the (j,ds)-element of the unitary matrix for the diagonalization. 31s and 34p(k) are the kinetic energies of the Cu1s and Cu4p electrons, respectively. Lss 0 (u,q) is evaluated within RPA. In the present study the incident photon energy ui is tuned to the Cu1s–4p absorption energy uiz34p(0)K31s, for which the intensity of the spectrum is enhanced by the resonance. The decay rate G1s of the core hole is 0.8 eV in the present study.
3.1. La2CuO4 The electronic properties of the Cu3d–O2p system in La2CuO4 are well described by two-dimensional dp Hamiltonian. The parameters for the dp model are tdpZK1.3 eV, tppZK0.65 eV as determined from first-principle calculations [9]. The charge transfer energy is 3HF d K3p ZK0:7 eV, where the 3HF d is the renormalized Cu3d one-electron energy within HF theory. The characteristic two-peak structure observed in experiments [4] is reproduced in the numerical results of the present study (Peaks A and B in Fig. 2), although the position of the peak B deviates from the experimental position (w4 eV) maybe due to the crude tight-binding fitting. These two peaks are attributed to the charge transfer (CT) process from the Cu3d–O2p bonding and anti-bonding bands to the unoccupied upper Cu3d band, which is clarified by inspecting the Cu3d partial density of states in each band. The 2 eV peak A exhibits a relatively large shift to the high energy region by around 0.8 eV along the line qZ(0,0)–(p,0) (Fig. 2a and b), and becomes remarkably small around qZ(p,p) (Fig. 2c). On the other hand, the high energy peak B shows only a small dispersion. These characteristic behaviors are in semi-quantitative agreement with experimental results. Actually it is reasonably expected that the small weight around uz9 eV is not observable [7].
O(1)
B
q=0
Cu O(2)
q A
(b)
RIXS Intensity (arb. units)
3. Results
(a)
B
q=π/2
A
B
(c)
q=π A
0
2
4 6 8 Energy loss (eV)
10
12
Fig. 3. The RIXS intensity calculated for the corner-sharing one-dimensional cuprate as a function of photon energy loss u. Three cases of transferred momenta qZ0,p/2,p are shown.
behaviors are in qualitative agreement with experimental results, although the position of the peak B deviates from the experimental position (z5.7 eV) maybe due to the crude tightbinding fitting. 3.4
3.2. Q1D copper oxides
3 Energy Loss ω (eV)
The electronic properties of the Cu3d–O2p system in Sr2CuO3 and SrCuO2 are well described by the onedimensional corner-sharing dp Hamiltonian. The calculated RIXS intensity for three cases of transferred momenta qZ0, p/2, p is displayed in Fig. 3. The parameters for the dp model are td–O(1)pZK1.45 eV along the chain, td–O(2)pZK1.8 eV perpendicular to the chain, and tO(1)p–O(2)pZK0.70 eV, as determined from first-principles calculations [10]. The oneparticle energies are 3HF d K3Oð1Þp ZK0:5 eV and 3O(2)p K 3O(1)pZ0.5 eV. The characteristic two-peak structure observed in experiments [5,6] is reproduced in the numerical results of the present study (Peaks A and B in Fig. 3). The 2 eV peak A exhibits a relatively large shift to the high energy region by around 1.1 eV. On the other hand, the high energy peak B does not show such a dispersion (Fig. 3). These characteristic
3.2
2.8 2.6 2.4 2.2 2 1.8 1.6
0
π/2 Momentum Transfer q
π
Fig. 4. Calculated spectral intensity around uw2 eV as a function of transferred momentum q along the Cu–O chain. The light (dark) region corresponds to strong (weak) RIXS intensity. The empty circles and squares are the peak positions experimentally obtained by Hasan et al. [5].
T. Nomura, J.-i Igarashi / Journal of Physics and Chemistry of Solids 67 (2006) 262–265
The position of the peak A shows a sinusoidal dispersion (Fig. 4). Our recent work [8], where the results with and without the RPA corrections are compared, showed that manybody correction is essential for this sinusoidal dispersion.
4. Concluding remarks The main spectral feature characterized by two peaks in La2CuO4 and the Q1D copper oxides are attributed to the CT process from the Cu3d–O2p bonding and anti-bonding bands to the unoccupied upper Cu3d band. Thus discussions based on simple Hubbard model neglecting the O2p bands [11] are inappropriate. In conclusion, we have theoretically reproduced the twopeak structure and the strong momentum dependence of the Cu K edge RIXS spectra observed in a two-dimensional insulating cuprate La2CuO4 and one-dimensional ones Sr2CuO3 and SrCuO2.
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