- Email: [email protected]
Theoretical aspects of photoemission spectroscopy on strongly correlated electron systems
Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
Theoretical aspects of photoemission spectroscopy on strongly correlated electron systems Norikazu Tomita ∗ , Masahiro Yamazaki, Keiichiro Nasu Institute of Materials Structure Science, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Available online 16 February 2005
Abstract We will clarify, from the theoretical viewpoint, how electron correlations affect the density of states (DOS) and the angle-resolved photoemission spectra (ARPES). The two-dimensional half-filled Hubbard model is employed as a prototype of the interacting electron system. In the weak interaction regime, the DOS has a multi-peak structure, which consists of a sharp one-body coherent component and a broad many-body incoherent component. Accordingly, the ARPES has a sharp coherent peak near the Fermi level. We will show that a short-range spin correlation causes the multi-peak structure in the DOS. On the other hand, as the electronic interaction increases, the DOS and APRES are dominated by the many-body incoherent component even near the Fermi level. It will be shown that the strong coupling between the collective spin excitations and photo-generated hole causes the incoherent component. © 2005 Elsevier B.V. All rights reserved. Keywords: Photoemission spectroscopy; Mott transition; Quantum Monte-Carlo simulation
1. Introduction The photoemission spectroscopy is a powerful tool to clarify one-body occupied states in solids. The recent progress on the experimental techniques, such as the fine energyresolution and bulk-sensitivity, allows us to see the detailed structures in the density of states (DOS) and angleresolved photoemission spectra (ARPES) of the correlated electron systems. As a result, we are now recognizing that the electron correlations cause some drastic changes in the DOS and ARPES from the conventional one-body picture. CaVO3 is one good example [1–6]. The recent bulksensitive soft X-ray photoemission experiment [2] has revealed that the DOS has two peaks, a sharp peak slightly below the Fermi level and a broad peak on the high bindingenergy side. The sharp peak is regarded as a coherent peak, reflecting the one-body band component, while the broad peak reflects a many-body incoherent component. The dynamical mean-field theory predicted a multi-peak structure [7], but in this theory, a sharp coherent peak should lie just ∗
Corresponding author: Tel.: +81 298 64 5593; fax: +81 298 64 2801. E-mail address: [email protected] (N. Tomita).
0368-2048/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2005.01.001
on the Fermi level. Therefore, the observed coherent peak below the Fermi level is a new finding by the photoemission experiment. On the other hand, a copper-oxide is another example [8]. In spite of the fine energy-resolution, the ARPES has only a broad peak and we cannot see a sharp coherent peak for the insulating or metallic copper-oxide. This contradicts the conventional one-body-based picture of the photoemission spectroscopy. In both cases mentioned above, the electron correlations play quite an important role on the photoemission spectroscopy. In this research, we will show how the electron correlations affect the DOS and ARPES by using the quantum Monte-Carlo simulation of path-integral form [9–11]. We will see that there exists a significant coherent component in the DOS and ARPES of the weakly interacting system while an incoherent component dominates the spectrum of the strongly interacting system.
2. Model and method In the following, we study the electron correlation effects on the DOS and ARPES of the two-dimensional (2D)
1238
N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
half-filled Hubbard model. Its Hamiltonian is given by + + H = −t (aiσ ajσ + ajσ aiσ ) + U ni↑ ni↓ ,σ
(1)
i
where i,j represents the set of the nearest-neighbor sites, and U represents the on-site Coulomb repulsion. The system size is 12 × 12 and a periodic boundary condition is imposed. As typical examples, we show two cases, U/t = 2 and 5. We will see a sharp coherent peak at U/t = 2, while no sharp peak appears in the ARPES at U/t = 5 and the spectrum seems dominated by the incoherent component. The many-body incoherent component would be caused by the nonlinear coupling between a photo-generated hole and gapless collective spin excitations. To make this point clear, we add a staggered onebody magnetic field to the Hubbard model at U/t = 5, such as + H = −t (alσ al σ + al+ σ alσ ) + U nl↑ nl↓ l,l
+∆
l
l
(−1) (nl↑ − nl↓ ).
nent near the Fermi level. On the other hand, the peak becomes quite broad as its wave-number comes close to the bottom of the band. This strongly suggests that the ARPES is dominated by the many-body incoherent component near the bottom of the band, even if the electron interaction is weak. Now, let us discuss why the DOS has a multi-peak structure in the weak interaction regime. The ground state of the half-filled Hubbard model is qualitatively described as a spin density wave (SDW) plus quantum fluctuations. However, in the case of the weakly interacting system, its spin correlation is quite short. Therefore, in such a short-range limit of the magnetic field, we can approximate the system by a single-impurity model, where the magnetic field works only on a certain single site. Such a single-impurity model is given by + + z (alσ al σ + al+ σ alσ ) + Ue H = −t a0σ σ a0σ , (3) ll σ
(2)
l
Due to this ∆, the energy gap artificially opens in the collective spin excitation. In fact, we will see that the ARPES has a sharp coherent peak in the case of finite ∆. Thus, the coherent component is well separated from the incoherent one coupled with the magnetic excitations. We determine the weight of the coherent component in the Hubbard model from its ∆-dependence by taking the limit of small ∆ [12].
3. Results and discussion First, we show, in Fig. 1a, the DOS at U/t = 2, where we can see a sharp peak slightly below (or above) the Fermi level, as well as a broad incoherent peak. Fig. 1b shows the momentum-specified Lehmann’s spectra, corresponding to the ARPES, from the bottom (k = (0,0)) to the top (k = (π/2,π/2)) of the band. We can see that the peak is very sharp near the Fermi level. This indicates that the DOS and ARPES are dominated by the one-body coherent compo-
σ
where σ z represents the z-component of the Pauli matrix, and Ue is the effective magnetic potential working only on the site m = 0. The lattice Green function at m = 0 for this singleimpurity model is given by G=
1 , G0 ± U e
(4)
where G0 is the lattice Green function at m = 0 for the noninteracting system, given by the first term of Eq. (3). It is explicitly written by ρ(E ) G0 = dE
. (5) E + iε − E
In Fig. 2(a), we show the real part of G0 . G has poles at the energies which satisfy ReG0 = ±
1 . Ue
(6)
The inner solutions, denoted by black circles, are called resonance states, which correspond to the coherent plainwave-like state. On the other hand, the outer solutions,
Fig. 1. DOS (a) and momentum-specified Lehmann’s spectrum (b) at U = 2.
N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
Fig. 2. The single-impurity picture for the multi-peak structure in the DOS. The real part of the lattice Green function G0 (a) and resultant DOS (b). In (b), finite temperature effects are included by the Lorentzian broadening.
denoted by white circles, are called isolated states, which correspond to the incoherent state. As a result, the DOS has four components, that is, the two coherent components near the Fermi level and two incoherent components on their highenergy sides, as shown in Fig. 2(b). Although this singleimpurity picture is too rough to describe the electron correlation effects sufficiently, it helps us understand the origin of the multi-peak structure of the DOS in the weak interaction regime. Then, in Fig. 3, we show the ∆-dependence of the Lehmann spectrum for U/t = 5 at k = (π/2,π/2). This state is closest to the Fermi level, and hence, it will have the largest coherent component in all the states. The Lehmann spectrum has a single sharp peak in the case of ∆ = 3.0, and its binding-energy coincides well with the corresponding Hartree–Fock (HF) energy level denoted by an upward arrow. This means that the sharp peak is a one-body coherent peak. Electron correlation effects are expected to be eminent as ∆ is decreased, since ∆ is a one-body potential. Actually, in the case of ∆ = 0.75, the incoherent component appears on the high binding-energy side of the coherent peak. We can see that the intensity of the coherent peak decreases with the decrease of ∆, and the incoherent component comes to dominate the spectrum. In the case of ∆ = 0.25, the coherent peak becomes so small that it looks just
1239
Fig. 3. ∆-Dependence of the Lehmann spectrum at k = (π/2,π/2). Black and white triangles denote the coherent and incoherent components, respectively. For each spectrum, the x-axis is shifted so that the position of the coherent peak comes to zero. The shifted energy is represented by Ecp .
like a shoulder on the low binding-energy side of the incoherent component. From such ∆-dependence of the coherent and incoherent components, we can infer that the incoherent component predominates the Lehmann spectrum of the usual half-filled Hubbard system. In order to clarify this point, we show how the coherent component, Z, depends on ∆ in Fig. 4. Here (2∆ + U) roughly corresponds to the total charge excitation gap. In this figure,
Fig. 4. ∆-Dependence of Z. The vertical axis is logarithmically scaled.
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N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
Z at ∆ = 0 is determined by the extrapolation from the nearest two points, and we have Z = 0.01. Thus, we can conclude that the incoherent component predominates the Lehmann’s spectrum of the 2D half-filled Hubbard model. Our results indicate that the peak observed in the ARPES of the Mott–Hubbard insulator originates from the incoherent component and it does not show the one-body state. 4. Summary We have clarified how the electron correlations affect the DOS and ARPES by using the 2D half-filled Hubbard model. The DOS has the sharp coherent peak, as well as the broad incoherent component, in the weak interaction regime. The coherent peak lies slightly below EF . In the strong interaction regime, the incoherent component predominates the Lehmann spectrum, and the coherent component makes only a minute contribution. Acknowledgements We thank Professor S. Suga, Professor A. Fujimori and Professor S. Shin for valuable discussion. This work was
supported by the NAREGI Nanoscience Project, Ministry of Education, Culture, Sports, Science and Technology, Japan.
References [1] I.H. Inoue, I. Hase, Y. Aiura, A. Fujimori, Y. Haruyama, T. Maruyama, Y. Nishihara, Phys. Rev. Lett. 74 (1995) 2539. [2] A. Sekiyama. H. Fujiwara, S. Imada, H. Eisaki, S. I. Uchida, K. Takegahara, H. Harima, Y. Saitoh, S. Suga, cond-mat/0206471. [3] R. Eguchi, et al., in preparation. [4] A. Liebsch, Phys. Rev. Lett. 90 (2003) 096401. [5] I.A. Nekrasov, G. Keller, D.E. Kondakov, A.V. Kozhevnikov, Th. Prushke, K. Held, D. Vollhardt, V.I. Anisimov, cond-mat/0211508. [6] N. Tomita, M. Yamazaki, K. Nasu, Phys. Rev. B68 (2003) 035117. [7] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [8] B.O. Wells, Z.X. Shen, A. Matsuura, D.M. King, M.A. Kastner, M. Greven, R.J. Birgeneau, Phys. Rev. Lett. 74 (1995) 964. [9] N. Tomita, K. Nasu, Phys. Rev. B56 (1997) 3779. [10] R. Blankenbecler, D.J. Scalapino, R.L. Suger, Phys. Rev. E24 (1981) 2278. [11] R.N. Silver, D.S. Sivia, J.E. Gubernatis, Phys. Rev. B41 (1990) 2380. [12] N. Tomita, K. Nasu, Phys. Rev. B60 (1999) 8602.
Theoretical aspects of photoemission spectroscopy on strongly correlated electron systems Norikazu Tomita ∗ , Masahiro Yamazaki, Keiichiro Nasu Institute of Materials Structure Science, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Available online 16 February 2005
Abstract We will clarify, from the theoretical viewpoint, how electron correlations affect the density of states (DOS) and the angle-resolved photoemission spectra (ARPES). The two-dimensional half-filled Hubbard model is employed as a prototype of the interacting electron system. In the weak interaction regime, the DOS has a multi-peak structure, which consists of a sharp one-body coherent component and a broad many-body incoherent component. Accordingly, the ARPES has a sharp coherent peak near the Fermi level. We will show that a short-range spin correlation causes the multi-peak structure in the DOS. On the other hand, as the electronic interaction increases, the DOS and APRES are dominated by the many-body incoherent component even near the Fermi level. It will be shown that the strong coupling between the collective spin excitations and photo-generated hole causes the incoherent component. © 2005 Elsevier B.V. All rights reserved. Keywords: Photoemission spectroscopy; Mott transition; Quantum Monte-Carlo simulation
1. Introduction The photoemission spectroscopy is a powerful tool to clarify one-body occupied states in solids. The recent progress on the experimental techniques, such as the fine energyresolution and bulk-sensitivity, allows us to see the detailed structures in the density of states (DOS) and angleresolved photoemission spectra (ARPES) of the correlated electron systems. As a result, we are now recognizing that the electron correlations cause some drastic changes in the DOS and ARPES from the conventional one-body picture. CaVO3 is one good example [1–6]. The recent bulksensitive soft X-ray photoemission experiment [2] has revealed that the DOS has two peaks, a sharp peak slightly below the Fermi level and a broad peak on the high bindingenergy side. The sharp peak is regarded as a coherent peak, reflecting the one-body band component, while the broad peak reflects a many-body incoherent component. The dynamical mean-field theory predicted a multi-peak structure [7], but in this theory, a sharp coherent peak should lie just ∗
Corresponding author: Tel.: +81 298 64 5593; fax: +81 298 64 2801. E-mail address: [email protected] (N. Tomita).
0368-2048/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2005.01.001
on the Fermi level. Therefore, the observed coherent peak below the Fermi level is a new finding by the photoemission experiment. On the other hand, a copper-oxide is another example [8]. In spite of the fine energy-resolution, the ARPES has only a broad peak and we cannot see a sharp coherent peak for the insulating or metallic copper-oxide. This contradicts the conventional one-body-based picture of the photoemission spectroscopy. In both cases mentioned above, the electron correlations play quite an important role on the photoemission spectroscopy. In this research, we will show how the electron correlations affect the DOS and ARPES by using the quantum Monte-Carlo simulation of path-integral form [9–11]. We will see that there exists a significant coherent component in the DOS and ARPES of the weakly interacting system while an incoherent component dominates the spectrum of the strongly interacting system.
2. Model and method In the following, we study the electron correlation effects on the DOS and ARPES of the two-dimensional (2D)
1238
N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
half-filled Hubbard model. Its Hamiltonian is given by + + H = −t (aiσ ajσ + ajσ aiσ ) + U ni↑ ni↓ ,
(1)
i
where i,j represents the set of the nearest-neighbor sites, and U represents the on-site Coulomb repulsion. The system size is 12 × 12 and a periodic boundary condition is imposed. As typical examples, we show two cases, U/t = 2 and 5. We will see a sharp coherent peak at U/t = 2, while no sharp peak appears in the ARPES at U/t = 5 and the spectrum seems dominated by the incoherent component. The many-body incoherent component would be caused by the nonlinear coupling between a photo-generated hole and gapless collective spin excitations. To make this point clear, we add a staggered onebody magnetic field to the Hubbard model at U/t = 5, such as + H = −t (alσ al σ + al+ σ alσ ) + U nl↑ nl↓ l,l
+∆
l
l
(−1) (nl↑ − nl↓ ).
nent near the Fermi level. On the other hand, the peak becomes quite broad as its wave-number comes close to the bottom of the band. This strongly suggests that the ARPES is dominated by the many-body incoherent component near the bottom of the band, even if the electron interaction is weak. Now, let us discuss why the DOS has a multi-peak structure in the weak interaction regime. The ground state of the half-filled Hubbard model is qualitatively described as a spin density wave (SDW) plus quantum fluctuations. However, in the case of the weakly interacting system, its spin correlation is quite short. Therefore, in such a short-range limit of the magnetic field, we can approximate the system by a single-impurity model, where the magnetic field works only on a certain single site. Such a single-impurity model is given by + + z (alσ al σ + al+ σ alσ ) + Ue H = −t a0σ σ a0σ , (3) ll σ
(2)
l
Due to this ∆, the energy gap artificially opens in the collective spin excitation. In fact, we will see that the ARPES has a sharp coherent peak in the case of finite ∆. Thus, the coherent component is well separated from the incoherent one coupled with the magnetic excitations. We determine the weight of the coherent component in the Hubbard model from its ∆-dependence by taking the limit of small ∆ [12].
3. Results and discussion First, we show, in Fig. 1a, the DOS at U/t = 2, where we can see a sharp peak slightly below (or above) the Fermi level, as well as a broad incoherent peak. Fig. 1b shows the momentum-specified Lehmann’s spectra, corresponding to the ARPES, from the bottom (k = (0,0)) to the top (k = (π/2,π/2)) of the band. We can see that the peak is very sharp near the Fermi level. This indicates that the DOS and ARPES are dominated by the one-body coherent compo-
σ
where σ z represents the z-component of the Pauli matrix, and Ue is the effective magnetic potential working only on the site m = 0. The lattice Green function at m = 0 for this singleimpurity model is given by G=
1 , G0 ± U e
(4)
where G0 is the lattice Green function at m = 0 for the noninteracting system, given by the first term of Eq. (3). It is explicitly written by ρ(E ) G0 = dE
. (5) E + iε − E
In Fig. 2(a), we show the real part of G0 . G has poles at the energies which satisfy ReG0 = ±
1 . Ue
(6)
The inner solutions, denoted by black circles, are called resonance states, which correspond to the coherent plainwave-like state. On the other hand, the outer solutions,
Fig. 1. DOS (a) and momentum-specified Lehmann’s spectrum (b) at U = 2.
N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
Fig. 2. The single-impurity picture for the multi-peak structure in the DOS. The real part of the lattice Green function G0 (a) and resultant DOS (b). In (b), finite temperature effects are included by the Lorentzian broadening.
denoted by white circles, are called isolated states, which correspond to the incoherent state. As a result, the DOS has four components, that is, the two coherent components near the Fermi level and two incoherent components on their highenergy sides, as shown in Fig. 2(b). Although this singleimpurity picture is too rough to describe the electron correlation effects sufficiently, it helps us understand the origin of the multi-peak structure of the DOS in the weak interaction regime. Then, in Fig. 3, we show the ∆-dependence of the Lehmann spectrum for U/t = 5 at k = (π/2,π/2). This state is closest to the Fermi level, and hence, it will have the largest coherent component in all the states. The Lehmann spectrum has a single sharp peak in the case of ∆ = 3.0, and its binding-energy coincides well with the corresponding Hartree–Fock (HF) energy level denoted by an upward arrow. This means that the sharp peak is a one-body coherent peak. Electron correlation effects are expected to be eminent as ∆ is decreased, since ∆ is a one-body potential. Actually, in the case of ∆ = 0.75, the incoherent component appears on the high binding-energy side of the coherent peak. We can see that the intensity of the coherent peak decreases with the decrease of ∆, and the incoherent component comes to dominate the spectrum. In the case of ∆ = 0.25, the coherent peak becomes so small that it looks just
1239
Fig. 3. ∆-Dependence of the Lehmann spectrum at k = (π/2,π/2). Black and white triangles denote the coherent and incoherent components, respectively. For each spectrum, the x-axis is shifted so that the position of the coherent peak comes to zero. The shifted energy is represented by Ecp .
like a shoulder on the low binding-energy side of the incoherent component. From such ∆-dependence of the coherent and incoherent components, we can infer that the incoherent component predominates the Lehmann spectrum of the usual half-filled Hubbard system. In order to clarify this point, we show how the coherent component, Z, depends on ∆ in Fig. 4. Here (2∆ + U) roughly corresponds to the total charge excitation gap. In this figure,
Fig. 4. ∆-Dependence of Z. The vertical axis is logarithmically scaled.
1240
N. Tomita et al. / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1237–1240
Z at ∆ = 0 is determined by the extrapolation from the nearest two points, and we have Z = 0.01. Thus, we can conclude that the incoherent component predominates the Lehmann’s spectrum of the 2D half-filled Hubbard model. Our results indicate that the peak observed in the ARPES of the Mott–Hubbard insulator originates from the incoherent component and it does not show the one-body state. 4. Summary We have clarified how the electron correlations affect the DOS and ARPES by using the 2D half-filled Hubbard model. The DOS has the sharp coherent peak, as well as the broad incoherent component, in the weak interaction regime. The coherent peak lies slightly below EF . In the strong interaction regime, the incoherent component predominates the Lehmann spectrum, and the coherent component makes only a minute contribution. Acknowledgements We thank Professor S. Suga, Professor A. Fujimori and Professor S. Shin for valuable discussion. This work was
supported by the NAREGI Nanoscience Project, Ministry of Education, Culture, Sports, Science and Technology, Japan.
References [1] I.H. Inoue, I. Hase, Y. Aiura, A. Fujimori, Y. Haruyama, T. Maruyama, Y. Nishihara, Phys. Rev. Lett. 74 (1995) 2539. [2] A. Sekiyama. H. Fujiwara, S. Imada, H. Eisaki, S. I. Uchida, K. Takegahara, H. Harima, Y. Saitoh, S. Suga, cond-mat/0206471. [3] R. Eguchi, et al., in preparation. [4] A. Liebsch, Phys. Rev. Lett. 90 (2003) 096401. [5] I.A. Nekrasov, G. Keller, D.E. Kondakov, A.V. Kozhevnikov, Th. Prushke, K. Held, D. Vollhardt, V.I. Anisimov, cond-mat/0211508. [6] N. Tomita, M. Yamazaki, K. Nasu, Phys. Rev. B68 (2003) 035117. [7] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [8] B.O. Wells, Z.X. Shen, A. Matsuura, D.M. King, M.A. Kastner, M. Greven, R.J. Birgeneau, Phys. Rev. Lett. 74 (1995) 964. [9] N. Tomita, K. Nasu, Phys. Rev. B56 (1997) 3779. [10] R. Blankenbecler, D.J. Scalapino, R.L. Suger, Phys. Rev. E24 (1981) 2278. [11] R.N. Silver, D.S. Sivia, J.E. Gubernatis, Phys. Rev. B41 (1990) 2380. [12] N. Tomita, K. Nasu, Phys. Rev. B60 (1999) 8602.