- •Preface
- •Contents
- •1 Introduction
- •Layered Materials and Their Electronic Structure
- •General Phase Diagram of Cuprates and Main Questions
- •Superconducting State: Symmetry of the Order Parameter
- •Triplet Pairing in Strontium Ruthenate (Sr2RuO4): Main Facts and Main Questions
- •From the Crystal Structure to Electronic Properties
- •Spin Fluctuation Mechanism for Superconductivity
- •References
- •Generalized Eliashberg Equations for Cuprates and Strontium Ruthenate
- •Theory for Underdoped Cuprates
- •Extensions for the Inclusion of a d-Wave Pseudogap
- •Derivation of Important Formulae and Quantities
- •Elementary Excitations
- •Raman Scattering Intensity Including Vertex Corrections
- •Optical Conductivity
- •Comparison with Similar Approaches for Cuprates
- •The Spin Bag Mechanism
- •Other Scenarios for Cuprates: Doping a Mott Insulator
- •Local vs. Nonlocal Correlations
- •The Large-U Limit
- •Projected Trial Wave Functions and the RVB Picture
- •Current Research and Discussion
- •References
- •The Spectral Density Observed by ARPES: Explanation of the Kink Feature
- •Raman Response and its Relation to the Anisotropy and Temperature Dependence of the Scattering Rate
- •A Reinvestigation of Inelastic Neutron Scattering
- •Collective Modes in Electronic Raman Scattering?
- •Elementary Excitations and the Phase Diagram
- •Optical Conductivity and Electronic Raman Response
- •Brief Summary of the Consequences of the Pseudogap
- •References
- •4 Results for Sr2RuO4
- •Elementary Spin Excitations in the Normal State of Sr2RuO4
- •The Role of Hybridization
- •Comparison with Experiment
- •Symmetry Analysis of the Superconducting Order Parameter
- •Triplet Pairing Arising from Spin Excitations
- •Summary, Comparison with Cuprates, and Outlook
- •References
- •5 Summary, Conclusions, and Critical remarks
- •References
- •References
- •Index
4 Results for Sr2RuO4
In this chapter, we focus on our results for the elementary excitations and Cooper pairing in strontium ruthenate (Sr2RuO4). The novel spin-triplet superconductivity with Tc = 1.5 K observed recently in layered Sr2RuO4 is a new example of unconventional superconductivity [1]. Clearly, it is important and of general interest to analyze in more detail the origin of the superconductivity, and to calculate the transition temperature Tc and also the symmetry of the order parameter on the basis of an electronic theory. This is di cult, since there are three Ru4+ t2g bands that cross the Fermi level, with approximately two–thirds filling of every band in Sr2RuO4. The coupling between all three bands seems to cause a single Tc. Furthermore, the presence of incommensurate antiferromagnetic (IAF) and ferromagnetic spin fluctuations, confirmed recently by inelastic neutron scattering [2] and the NMR 17O Knight shift [3], respectively, suggests a pairing mechanism for Cooper pairs due to spin fluctuations. Also, a non–s–wave symmetry of the order parameter has been observed. This makes the theoretical investigation of ruthenates very interesting.
To be more precise, recent studies by means of INS [2] and NMR [4] of the spin dynamics in Sr2RuO4 reveal the presence of strong incommensurate fluctuations in the RuO2 planes at the antiferromagnetic wave vector Qi = (2π/3, 2π/3). As was found in band structure calculations [5], these fluctuations result from the nesting properties of the quasi–one–dimensional dxz and dyz bands. The two–dimensional dxy band contains only weak ferromagnetic fluctuations. The very recent observation of the possibility of line nodes between the RuO2 planes [6, 7] suggests strong spin fluctuations between the RuO2 planes in the z direction also [8, 9]. However, INS [10] shows that the magnetic fluctuations are purely two–dimensional and originate from the RuO2 planes. Both behaviors could be a consequence of the magnetic anisotropy within the RuO2 planes, as indeed was observed in recent NMR experiments by Ishida et al. [11]. In particular, by analyzing the temperature dependence of the nuclear spin–lattice relaxation rate for 17O in the RuO2 planes at low temperatures (but still in the normal state), Ishida et al. have demonstrated that the out-of-plane component of the spin susceptibility can become almost up to three times larger than the in–plane component. This
D. Manske: Theory of Unconventional Superconductors, STMP 202, 177–199 (2004)c Springer-Verlag Berlin Heidelberg 2004
178 4 Results for Sr2RuO4
strong and unexpected anisotropy disappears with increasing temperature [11].
Below Tc, NMR [12, 13] and polarized neutron scattering [14] measurements indicate spin-triplet state Cooper pairing. From the analogy to 3He, this led Rice, Sigrist, and coworkers [15, 16], as well as Tewordt [17, 18] and others [19, 20, 21], to conclude that p–wave superconductivity is present. However, by fitting the specific heat and the ultrasound attenuation, Maki and coworkers [22], as well as others [23, 24, 25], found reason to doubt the presence of p–wave superconductivity and have proposed an f –wave symmetry of the superconducting order parameter. A similar conclusion was drawn in [26]. Recently it has been reported that thermal–conductivity measurements are also most consistent with f –wave symmetry [7]. In view of these facts, we shall reexamine the previous theoretical analysis of the gap symmetries and the competition between p– and d–wave superconductivity [5, 25, 27]. We shall also investigate superconductivity within an electronic theory and derive the symmetry of the order parameter from general arguments.
This chapter is organized as follows: In the next section, we shall analyze the normal–state spin dynamics of Sr2RuO4 using the two–dimensional threeband Hubbard Hamiltonian for the three bands crossing the Fermi level. In the first subsection, we calculate the dynamical spin susceptibility χ(q, ω) within the random–phase approximation and show that the observed magnetic anisotropy in the RuO2 planes arises mainly from the spin–orbit coupling. Its further enhancement with decreasing temperature is due to the vicinity of a magnetic instability. Thus we demonstrate that, as in the superconducting state [28], the spin–orbit coupling also plays an important role in the normal–state spin dynamics of Sr2RuO4. Then, in Sect. 4.1.2, we present an electronic theory which takes into account only hybridization between the three bands. This is enough to demonstrate how triplet pairing due mainly to AF spin excitations is possible. For this purpose, we calculate the Fermi surface (FS), the energy dispersion, and the spin susceptibility χ including all cross-susceptibilities. In the last subsection we compare our results with experiment. By analyzing experimental results for the 17O Knight shift and INS data as well as the FS observed by ARPES [29], we can obtain values for the hopping integrals and the e ective Coulomb repulsion U . Taking this as an input to the pairing interaction, we analyze the p–, d– and f –wave superconducting gap symmetries in Sect. 4.2 and demonstrate how triplet pairing is possible. This will be compared with the result obtained from the inclusion of spin–orbit coupling alone (i.e. without hybridization). In this case the parameters of the Hamiltonian are taken from band structure calculations. The delicate competition between weak ferromagnetic spin fluctuations and relatively strong incommensurate antiferromagnetic spin fluctuations due to nesting of the FS causes triplet Cooper pairing if spin–orbit coupling is taken into account. We shall also demonstrate that singlet dx2−y2 –wave symmetry (which is present in cuprates) is energetically less favorable. We summarize