Imaging Doped Holes in a Cuprate Superconductor with High-Resolution Compton Scattering

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Science  06 May 2011:
Vol. 332, Issue 6030, pp. 698-702
DOI: 10.1126/science.1199391


The high-temperature superconducting cuprate La2−xSrxCuO4 (LSCO) shows several phases ranging from antiferromagnetic insulator to metal with increasing hole doping. To understand how the nature of the hole state evolves with doping, we have carried out high-resolution Compton scattering measurements at room temperature together with first-principles electronic structure computations on a series of LSCO single crystals in which the hole doping level varies from the underdoped (UD) to the overdoped (OD) regime. Holes in the UD system are found to primarily populate the O 2px/py orbitals. In contrast, the character of holes in the OD system is very different in that these holes mostly enter Cu d orbitals. High-resolution Compton scattering provides a bulk-sensitive method for imaging the orbital character of dopants in complex materials.

The evolution of the electronic structure as well as the orbital character of the doped carriers is a key ingredient for understanding the physics of the cuprates and the mechanism of high-temperature superconductivity. Photoemission has succeeded in obtaining detailed information on the electronic dispersion and Fermi surface topology (13). Concerning the orbital character, comprehensive studies have concluded that the doped holes predominantly enter the oxygen 2p orbital in the cuprates, at least up to optimal doping (47). As a result, the unusual physical properties of underdoped (UD) cuprates have been analyzed mainly by ascribing a single orbital character to the doped holes. However, in the overdoped (OD) cuprates the orbital character is not fully understood, even though distinct doping dependencies of x-ray absorption (8, 6) and optical reflectivity spectra (9) suggest a change in the oxygen 2p orbital character with overdoping. In order to detect changes in orbital character with doping, we need to consider spectral differences, which requires high-quality data for extracting weak wave function effects. Moreover, it is essential to measure a physical quantity that is connected to wave functions, such as the electron momentum density (EMD). Compton scattering is one of the most promising techniques for probing the orbital character of doped holes because it allows direct access to the EMD (10). Advantages of Compton scattering over other spectroscopies are that we do not need a nearly defect-free single crystal or a clean surface or ultrahigh vacuum and that the matrix element involved is much simpler than in other highly resolved spectroscopies such as photoemission (11), resonant inelastic x-ray scattering (12), scanning tunneling (13), and positron annihilation (14). X-ray Compton scattering has established itself as a viable technique for investigating orbital character (15, 16) and Fermi surface geometry of the bulk system (1720) in wide classes of materials.

We have obtained two-dimensional electron momentum densities (2D-EMDs), which represent one-dimensional integrals along the c-axis of three-dimensional EMDs for single crystalline samples of La2−xSrxCuO4 (LSCO) with four different hole dopings, x = 0.0, 0.08, 0.15, and 0.30 at room temperature. The samples and measurements are described in (21), and the experimental 2D-EMDs are presented in fig. S1. To clarify the evolving character of doped holes, we took the differences in 2D-EMDs between two samples with different doping levels. This subtraction technique (Fig. 1) provides information on changes in orbital occupation numbers associated with doped holes (22). Subtraction acts as a projector on an energy slice near the Fermi level with the advantage of eliminating the large isotropic contribution of the core as well as irrelevant valence electrons. Each electronic state has its own angular dependence in momentum space, which facilitates the detection of the state. For an atomic orbital, the EMD, which is the squared modulus of the momentum-space wave function, has the same point symmetry as the corresponding charge density. This result carries over to molecular states (23) and is equally applicable to solid-state wave functions (24). The radial behavior of the momentum-space wave function for each atomic orbital is determined by the spherical Bessel function, which behaves as pl at small momenta p, where l is the orbital quantum number (25). Therefore, oxygen 2p bands contribute to the EMD at low momenta, whereas the contribution of the Cu 3d bands increases as p4. This means that O 2p states are more visible in the first Brillouin zone (BZ), whereas Cu 3d states are better seen in higher BZs.

Fig. 1

Experimental difference 2D-EMDs in LSCO between two hole doping concentrations: (A) Nondoped (x = 0.0) minus UD (x = 0.08); (B) nondoped (x = 0.0) minus optimal-doped (x = 0.15); (C) optimal-doped (x = 0.15) minus heavily OD (x = 0.30); and (D) nondoped (x = 0.0) minus heavily OD (x = 0.30). (E) Error map of the difference 2D-EMD (the errors are typically 10 times smaller than the amplitudes in the 2D-EMD differences except very close to the origin). a.u., atomic units.

An important question regarding the nature of the electronic ground state of LSCO is the character of the extra holes introduced when we dope the half-filled insulator. Zhang and Rice (7) suggested that the holes reside in a molecular orbital state, PZR = P1xP2yP3x + P4y (Fig. 2A), where numbers 1 to 4 label the four O atoms and subscripts x and y denote the direction of the O 2p orbital involved in the plaquette of four O atoms surrounding a Cu atom in the CuO2 plane. The molecular orbital PZR couples with the Cu 3dx2y2 state and forms the so-called Zhang-Rice singlet. By enhancing the O character of the doped hole near half filling, the Zhang-Rice singlet has a strong impact on Compton scattering. In cuprates the angular dependence of the wave functions is primarily set by the d orbitals of Cu, which hybridize with properly symmetrized combinations of p orbitals on nearest-neighbor oxygens. Hence we separate the strong peaks in Fig. 1 into peaks along the [100] axes associated with predominantly Cu 3dx2y2 states and other peaks along the diagonal directions assigned to the Cu 3dz2 orbital hybridized with the molecular orbital P0 = P1x + P2y + P3x + P4y (Fig. 2D). The Cu 3dz2 state can also mix with the pz atomic orbital from the apical oxygen. A two-orbital model that incorporates both eg bands, namely the z2 and the x2y2 hybrid bands, captures several crucial properties of LSCO (26).

Fig. 2

Directional symmetry of Cu-O octahedral molecular orbitals. 2D-EMD projected onto the xy plane of the molecular orbitals: (A) Diagram of the PZR = P1xP2yP3x + P4y state; (B) PZR state (only O 2px/y orbitals); (C) PZR state hybridized with Cu 3dx2y2; (D) diagram of the P0 = P1x + P2y + P3x + P4y state; (E) P0 state (only O 2px/y orbitals); (F) P0 state hybridized with Cu 3dz2. a.u., atomic units.

Figure 2 shows that a number of features of the momentum density of Fig. 1 can be understood in a simple molecular orbital picture (22) in which band structure details are neglected. By varying the relative Cu character of the orbital, the model properly describes the doping evolution of the Zhang-Rice characteristics (Fig. 2, B and C) as well as features of the z2 states (Fig. 2, E and F). However, because of the finite molecular size, the calculations do not quantitatively reproduce the radial positions of O and Cu features in momentum space. These features are more correctly described by band structure calculations, despite the fact that these calculations do not properly reproduce the doping evolution of Cu-O hybridization. Details of the EMD patterns are determined by competition between two opposing tendencies. Coulomb repulsion tends to localize d electrons on the Cu atoms, whereas mixing with the oxygen p electron states tends to delocalize these same electrons. Band structure and molecular orbital calculations are thus both useful for identifying various features in the Compton spectra. The band diagram in Fig. 3A shows the two main bands near the Fermi level. The band with x2y2 symmetry is higher in energy than the band of z2 symmetry. By calculating difference spectra over appropriate energy intervals, we can determine the patterns associated with particular orbitals. As shown in Fig. 3B, experiment and theory show a reasonable agreement regarding the difference of the directional Compton profiles [100]-[110] for x = 0.15. The present theoretical Compton profiles are computed from the Green functions obtained by Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA) calculations within the local density approximation (LDA) (27, 28, 21), which yields Fermi surface properties in agreement with angle-resolved photoemission spectroscopy (ARPES) experiments (1). Figure 3 also shows the contribution of the 2D-EMDs between –0.4 eV and +0.1 eV from the x2y2 band (Fig. 3C), between –0.8 eV and –0.4 eV from the z2 band (Fig. 3D), and between –0.8 eV and +1.3 eV from both bands (Fig. 3E). These calculations reproduce trends related to the Cu-O states. Notably, in Fig. 3E the two diagonal features approximately agree with the experimental Fig. 1C despite some discrepancies regarding the position of the Cu d features along px and py. Interestingly, other small experimental features along these directions, which are related to dxz and dyz orbitals, could be found at lower energy in the theory. In any event, the present analysis in the energy range between –0.8 eV and +1.3 eV captures the dominant features of the two-orbital band model (26). Part of the disagreement between theory and experiment is related to the fact that our LDA computations tend to underestimate the dz2 character of states at the Fermi energy (29). Our analysis shows that for small Sr concentrations the doped holes have a substantially greater O 2p character than predicted by band theory. These effects are difficult to describe within the conventional Fermi liquid picture and would likely require the treatment of electron correlations involving the formation of Zhang-Rice singlets (7). We note as well that lattice perturbations affecting the bonding properties, such as Jahn-Teller distortions and polaronic displacements, are not included in the present calculations.

Fig. 3

Theoretical electronic structure: (A) Band structure; (B) anisotropy of Compton profiles; (C) 2D-EMD contribution between –0.4 eV and +0.1 eV; (D) 2D-EMD contribution between –0.8 eV and –0.4 eV; and (E) 2D-EMD contribution between –0.8 eV and +1.3 eV. Error bars in (B) indicate SEM. a.u., atomic units.

Our study shows the use of high-resolution Compton scattering for direct, bulk-sensitive imaging of the orbital character of dopants in the ground state of complex materials. This information cannot be obtained from other highly resolved spectroscopies. Features in Compton spectra are extremely robust because they are not significantly affected by defects, surfaces, or impurities.

Supporting Online Material

Materials and Methods

Fig. S1

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. Acknowledgments: The work at JASRI was supported by a Grant-in-Aid for Scientific Research (no. 18340111) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, and that at Tohoku University was supported by a Grant-in-Aid for Scientific Research (nos. 16104005, 19340090, and 22244039) from the MEXT, Japan. The work at Northeastern University (NU) was supported by the Division of Materials Science and Engineering, Office of Science, U.S. Department of Energy, and it benefited from the allocation of supercomputer time at the National Energy Research Scientific Computing Center, the NU's Advanced Scientific Computation Center (ASCC), and the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF). The Compton scattering experiments were performed with the approval of JASRI (proposals 2003B0762, 2004A0152, 2007B1413, 2008A1191, and 2010A1907).
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