Electronic structure

The electronic structure of the parent compound

To discuss the electronic structure, I begin with the CuO₆-octahedron: Each Cu-atom has four O-atoms as nearest neighbors in the plane and two apical O-atoms. For Bi2201, the Cu-atom is covalently bound to the O-atoms in the plane with a bond length of about 1.9 Å. In apical direction, the CuO-bond is only weakly covalent with a typical bond length of about 2.3 Å [84].

A schematic picture of the CuO binding is shown in Fig. 2.7. From the atomic O-p_x,y and Cu-d_x2−y²-orbits one accounts for the crystal field splitting and the Jahn-Teller distortion [85]

and results in CuO-hybrids of pdσ -symmetry. According to this simple model, it is evident that the electronic structure of the region around the Fermi-energy is composed mainly of these CuO pdσ orbits. Because of this, the investigations have been concentrated on the CuO₂-plane.

In principle, four steps are necessary to to illustrate the current level of knowledge about its electronic structure:

Figure 2.7: From [86]: Crystal-field splitting and hybridization produces CuO-hybrids of pdσ -symmetry. On the upper left a side view of the atomic place-ment due to the different pro-cesses is shown.

The first step is to calculate the electronic band structure within the local density approximation (LDA). In the case of an undoped CuO₂-plane, this will give an antibonding CuO₂ pdσ-hybrid band which is half filled. Thus, band structure calculations propose a metal shown by the density of states (DOS) in Fig. 2.8 (1). In contrast to this, it is well known that the undoped CuO₂ -plane is an antiferromagnetic insulator. For example, in polycrystalline Bi₂Sr₁La₁CuO_6+δ, the Néel temperature is approximately T_N=270K [87]. Although LDA model calculations obviously fail in this case, they propose for high temperatures and high hole-dopings quite realistic Fermi

2.2 The cuprate high-temperature superconductors

Figure 2.8: Qualitative overview of the density of states expected in different models for the electronic structure of aCuO2-plane:

(1) The LDA band structure calculation produces a half-filled conduction band according to a metal.

(2) The DOS of a Mott-Hubbard insulator. The upper Hubbard band (UHB), which is empty when half-full, and the filled lower Hubbard band (LHB) are visible.

(3)

The extension of the Mott-Hubbard model, taking into account also the O2p_x,y-orbits, is the three-band Hubbard model. An additional DOS peak between the UHB and LHB is produced.

This peak is often associated with the non-bonding O2p band and denoted as NBO.

(4) Increasing the number of holes and including the interaction between the 2p- and 3d- holes, an additional correlated state (CS) splits off from the NBO band.

surfaces⁴ in the normal-state (see, e.g., [89, 90]).

The reason why the LDA band structure calculations fail in describing the electronic character of the CuO₂-plane is because the correlations of the electronic system are not adequately taken into account. The strong correlations may be considered at first only on the Cu sites and not the Oxygen sites. This setup can be described within the Mott-Hubbard model [91, 92].

The electrons within the Cu-orbital at Cu-atom position i are constructed within the second quantization formalism by the operatord⁺_i,sand destructed by the operatord_i,s. The mean energy of each electron is represented byd. The strong correlations due to Coulomb interactions for two electrons on the same Cu-site are included by the Hubbard interaction energyU_d. The hopping of an electron between different sites with positioni,j is described byt_d. The Hamiltonian then

4Let me note here that in the case of the Bi-cuprates the LDA derived Fermi surfaces show BiO-pockets, which until now could not resolved by ARPES. A good starting point for the ’BiO-pocket problem’ is [88].

yields summation only over next neighbors. At half-filling (Cu3d⁹-configuration) this model produces two bands split by the energy gap U_d. The Fermi energy of this hypothetically correlated Cu 3d⁹ system lies within the gap. Fig. 2.8 (2) shows the DOS of thisMott-Hubbard insulator.

The extension of the Mott-Hubbard model also including the O2p_x,y -orbits is the three-band Hubbard model. In the same notation as above, the includedp electrons are constructed by p⁺_l,s and destructed by p_l,s. The energy of each electron is _p. The Coulomb correlation within one O2p -orbital is Up, while the correlation between next neighbor p- and d-orbits is given byUpd. The hopping between different O 2p-orbits is described by tp. The hopping between the O2p -and the Cu3d-orbits is represented byt_pd. The Hamiltonian can be written as

H = _d^X

Based on the approximation by Zaanen, Sawatzky and Allen [93], it became possible to determine the character of the DOS as a function of the ratioU_d/W. Common parameters for the CuO₂ -plane are [94]:

Ud≈6 eV, Upd≈0 - 2 eV, Up ≈3 eV, t_pd≈1.3 - 1.5 eV, t_p ≈0.5 eV, and W ≈1 - 3 eV.

In the case of W < Ud, the system is called a charge transfer insulator. The model produces an additional peak in the DOS between the lower occupied and the upper empty Hubbard band, which is associated with the valence band composed mainly of non-bonding O2p-orbits (NBO). This is shown in Fig. 2.8 (3). As a consequence of the charge transfer insulating regime, hole doping from the antiferromagnetic insulator energetically favors the Oxygen sites. Double occupancy of holes on the Cu site is impeded due to the strong intersite exchange interaction Ud. This behavior of the holes is also known from experiments [95, 96, 97].

For theoretical calculations, instead of the Hubbard model, the tJ-model is often used, which can be derived from the Hubbard modell by second order perturbation theory (see, e.g., [98]).

Let me now come to the last point of discussing the electronic structure, which is the change caused by doping the holes in the charge-transfer insulator:

2.2 The cuprate high-temperature superconductors

Change of the electronic structure due to hole-doping

Figure 2.9: A Zhang-Rice singlet located within a CuO2-plaquette (green square) is formed by the interaction of the doped holes on Oxygen- and Copper-site.

There were several models discussed for the changes of the electronic structure upon hole-doping. Of-ten discussed is the Zhang-Rice singlet. This con-struction can be seen as a mapping of the topmost excitations of the system to an effective one-band Hubbard model. Zhang and Rice[99] started from a two-band Hubbard Hamiltonian and added the hy-bridization introduced by the interaction of a hole on an O 2p site with the neighboring Cu²⁺ ion.

This strong coupling results in a singlet and a triplet state. Increasing the doping of holes splits off spec-tral weight from NBO into the charge transfer gap yielding a correlated state (CS) at the Fermi energy which has no analogue in the band structure. This can be seen in Fig. 2.8 (4). This feature can be interpreted as the singlet state (band) due to inter-actions between the Cu hole and the doped O hole [94]. This Zhang-Rice singlet is sketched in Fig.

2.9. In addition to the singlet states band, a triplet state also appears, which is located at higher binding energy near the bottom of NBO. Taking into consideration several models for the doping-dependent electronic structure, this model seems to be in closest correspondence with the experiment [100].