1 Introduction

One of the most exciting and fascinating fields in condensed matter physics is high-temperature and unconventional superconductivity, for example in holeand electron-doped cuprates, in Sr2RuO4, in organic superconductors, in MgB2, and in C60 compounds. In cuprates, the highest transition temperature (without application of pressure) Tc 134 K has been measured in HgBa2Ca2Cu3O8+δ , followed by – to name just a few – Bi2Sr2CaCu2O8+δ (δ = 0.15 ↔ Tc 95 K), YBa2Cu3O6+x (x = 0.93 ↔ Tc 93 K), Nd2−xCexCuO4 (x = 0.15 ↔ Tc 24 K), and La2−xSrxCuO4, where, for an optimum doping concentration x = 0.15, a maximum value of Tc 39 K occurs. Since 77 K is the boiling temperature of nitrogen, it is now possible that new technologies, based for example on SQUIDs (superconducting quantum interference devices) or Josephson integrated circuits [1], might be developed. However, at present, the critical current densities are still not high enough for most technology applications. A recent overview an account of the possible prospects can be found in [2] and references therein.

Throughout this book, we shall focus mainly on Cooper pairing in cuprates and in Sr2RuO4. All members of the cuprate family discovered so far contain one or more CuO2 planes and various metallic elements. As we shall discuss in the next section, their structure resembles that of the perovskites [3]. It is now fairly well established that the important physics related to superconductivity occurs in the CuO2 planes and that the other layers simply act as charge reservoirs. Thus, the coupling in the c direction provides a three–dimensional superconducting state, but the main pairing interaction acts between carriers within a CuO2 plane. The undoped parent compounds are antiferromagnetic insulators, but if one dopes the copper–oxygen plane with carriers (electrons or holes), the long-range order is destroyed. Note that even without strict long-range order, the spin correlation length can be large enough to produce a local arrangement of magnetic moments that di ers only little from that observed below the N´eel temperature in the insulating state. In the doped state the cuprates become metallic or, below Tc, superconducting.

As mentioned above, in hole–doped cuprates Tc is of the order of 100 K and in electron–doped cuprates one finds Tc 25K (as will be explained later), and thus much larger values of Tc are obtained than in conventional

D. Manske: Theory of Unconventional Superconductors, STMP 202, 1–32 (2004)c Springer-Verlag Berlin Heidelberg 2004

21 Introduction

strong–coupling superconductors such as lead (Tc = 7.2 K) or niobium (Tc = 9.25 K). Therefore, the phenomenon of high–Tc superconductivity in cuprates that occurs in the vicinity of an antiferromagnetic phase transition suggests a purely electronic or magnetic mechanism, in contrast to the conventional picture of electrons paired through the exchange of phonons. For example, the simplest idea to explain such high critical temperatures might be to introduce a higher cuto energy ωc due to electronic correlations in the system instead of integrating over an energy shell corresponding to ωD (the Debye frequency), i.e.

where λ denotes the usual coupling strength for a given symmetry of the gap function. In the BCS theory [4], λ is equal to N (0)V , where N (0) is the density of states (per spin) at the Fermi level and V = const is the attractive pairing potential acting between electrons, leading to the superconducting instability of the normal state. If the relevant energy cuto ωc of the problem is of the order of electronic degrees of freedom, e.g. ωc 0.3 eV ≈ 250 K [5], one can easily obtain a transition temperature of the order of 100 K. However, as we shall discuss below, in a more realistic treatment the relation between Tc and λ is not as simple as in (1.1).

Superconductivity in strontium ruthenate (Sr2RuO4) is also very exciting because its structure is similar to that of the high–Tc cuprate La2−xSrxCuO4 (RuO2 planes instead of CuO2 planes), but its superconducting properties resemble those of 3He. As will be discussed later in detail, Sr2RuO4 is in the vicinity of a ferromagnetic transition and thus is a triplet superconductor. It has a Tc 1.5 K. Furthermore, in contrast to cuprates, its normal–state behavior follows the standard Fermi liquid theory. All this makes the theoretical investigation of Sr2RuO4 very interesting.

In this book, we present a general theory of the elementary excitations and singlet Cooper pairing in hole– and electron–doped high–Tc cuprates and compare our results with experiment. Then, we apply our theory also to the novel superconductor Sr2RuO4, where triplet pairing is present. We shall present the structures and electronic properties of the most important compounds and their possible theoretical descriptions, and then use those descriptions in the rest of the book. We shall point out some general features of many unconventional superconductors and give the main ideas and concepts used to describe Cooper pairing in these materials. Although it is known that organic superconductors, heavy–fermion superconductors, and some other materials cannot be described by the BCS model [4], we consider the theory of BCS–like pairing (or its strong–coupling extension, i.e. the Eliashberg theory) as a broader and still valid concept in many–body theory. However, the source of the corresponding pairing interaction has to be calculated from a microscopic theory. This is one important goal of this book.