1.4 From the Crystal Structure to Electronic Properties

1.4 From the Crystal Structure to Electronic Properties

After we have discussed the structure and the phase diagram of high-Tc cuprates and the main facts about triplet pairing in Sr2RuO4, the next step is to write down a Hamiltonian and to describe the elementary excitations with an electronic theory. Owing to the complexity of their structure, this is di cult. Instead, we need some reasonable simplifying approximations, for example we may construct a Hamiltonian for only a CuO2 plane or RuO2 plane. The very strong and important Cu-O–bonds in the conducting planes of cuprates and Ru-O–bonds in ruthenates justify this approximation.

1.4.1 Comparison of Cuprates and Sr2RuO4: Three–Band

Approach

Many researchers believe that the general phase diagram for hole–doped cuprates presented in Fig. 1.4 (and also the phase diagram for electron– doped cuprates) can be explained within an approximation that focuses only on one CuO2 plane. Why this is the case? As already mentioned, in the absence of doping the cuprates can be described well by mainly localized spin-1/2 states, which give these materials their antiferromagnetic character. The corresponding Cu and O orbitals are schematically shown in Fig. 1.10. The simplest microscopic model which can account for the calculated LDA band structure consists of two filled oxygen px,y orbitals and one half–filled dx2−y2 copper orbital. The bond lengths in the x and y directions are assumed to be identical. The corresponding tight–binding Hamiltonian reads

Here, tijpd and tijpp are the corresponding hopping integrals between the orbitals shown in Fig. 1.10, and the sums are performed over copper and oxygen lattice positions labeled by i, where i, j denotes nearest–neighbor pairs, and σ is the spin index. Note that in the hole representation, the so–called charge– transfer gap ∆ = p − d is positive.

However, the key ingredient missing in (1.2) is the strong Coulomb interaction in the Cu 3d wave functions. Thus, double occupancy is less energetically favored. A resultant Mott–Hubbard insulator was suggested early by Emery and coworkers [91, 92], which may be treated by a three-band version of the Hubbard Hamiltonian

H3−band = H3−band +Ud nd↓nd↑ +Up np↓np↑ +Upd npnd , (1.3) 0 i i i i i j

Fig. 1.11. Electronic structure of Sr2RuO4. The Ru ion is in the oxidation state Ru4+, which corresponds to a 4d4 level. In addition, the 4d level is split in the RuO6 crystal field into the eg and t2g subshells. The latter subshell, which consists of dxy , dxz , and dyz , crosses the Fermi level. In addition, the spin–orbit coupling seems to play an important role and provides the mixing of the spin and orbital degrees of freedom.

of the order of 150 meV and can be determined by two–magnon Raman scattering, for example [13].

In the case of Sr2RuO4, it is also necessary to employ a three–band Hubbard Hamiltonian because three bands cross the Fermi level; see the electronic structure of Sr2RuO4 in Fig. 1.11. Thus we start from

where ak,lσ is the Fourier-transformed annihilation operator for the dl orbital electrons (l = xy, yz, zx) and Ul is the corresponding on–site Coulomb repulsion. tkl denotes the energy dispersions of the tight–binding bands, calculated as follows: tkl = − 0 − 2tx cos kx − 2ty cos ky + 4t cos kx cos ky . For our calculations, we chose the values for the parameter set ( 0, tx, ty , t ) as (0.5, 0.42, 0.44, 0.14), (0.23, 0.31, 0.055, 0.01), and (0.24, 0.045, 0.31, 0.01) eV for the dxy , dzx, and dyz orbitals, respectively, in accordance with band structure calculations [78]. The electronic properties of this model applied to Sr2RuO4 were studied recently and were we found to be able to explain some features of the spin excitation spectrum of Sr2RuO4 [81, 96, 97]. However, this model fails to explain the observed magnetic anisotropy at low temperatures [85] and the possible line nodes in the superconducting order parameter below Tc. In contrast to cuprates, it is known that the spin–orbit coupling plays an important role in the superconducting state of Sr2RuO4 [96]. This is further confirmed by the recent observation of a large spin–orbit coupling in the insulator Ca2RuO4 [98]. Therefore, we shall include in our theory the spin–orbit coupling

Hso = λ LiSi , (1.6)

i

22 1 Introduction

where the angular momentum Li operates on the three t2g orbitals on the site i. Similarly to an earlier approach [96], we shall restrict ourselves to these three orbitals, ignoring e2g orbitals, and choose the coupling constant λ such that the t2g states behave like an l = 1 angular–momentum representation.

To summarize, a three–band Hubbard Hamiltonian provides a reasonable description of a CuO2 plane in cuprates and of an RuO2 plane in Sr2RuO4. In both cases the local Coulomb correlations play an important role in describing the electronic properties. This is because both classes of materials are in the vicinity of a magnetic transition (as described earlier). It turns out that for Sr2RuO4 no e ective one–band approach can be applied, because all three bands cross the Fermi level and show signs of hybridization and strong spin– orbit coupling. On the other hand, for cuprate superconductors, an e ective one-band theory is possible.

1.4.2 E ective Theory for Cuprates: One–Band Approach

Because a three-band Hamiltonian is di cult to solve, it is desirable to reduce it to a simpler model. It is generally believed that this is indeed possible for cuprate superconductors. Zhang and Rice [99] analyzed hole-doped cuprates and made progress in this direction by combining a Cu hole with an added hole (nearly on oxygen sites) to form a new spin singlet state and have shown that it is possible to work within this singlet subspace without changing the physics of the problem. In their description, the hole originally located at the oxygen has been replaced by a new (spin singlet) state at the copper. Thus, in this analysis, oxygen atoms are no longer present in an e ective theory. Note this is not equivalent to simply removing one Cu spin–1/2 state, because frustration of spins is also induced owing to doping of holes on oxygen sites. A removal of one Cu spin–1/2 state takes place only in the case of electron– doped cuprates, where the additional electron goes directly on the copper site, yielding a dilute antiferromagnet.

The analysis of Zhang and Rice leads to the so–called t–J model (originally introduced by Anderson [100]),

which is the main model that we shall use for cuprates in this book. Here, as usual, c†iσ is a fermionic operator that creates an electron or hole at site i with