Fig. 1.3. Structure of the electron–doped high–Tc cuprate Nd2−x Cex CuO2 (NCCO). This structure is similar to that of LSCO, but no apical oxygen is present.

To briefly summarize this section, we have demonstrated that the main ingredient of the strong electronic correlations that yield high–Tc superconductivity is the CuO2 planes. In the following we shall assume that the main physics and most important properties of cuprates are intimately related to the electronic correlations within one CuO2 plane. The regions between the CuO2 planes are believed to act mainly as a charge reservoir. Bilayer effects are treated elsewhere [16]. Thus the general phase diagram, the pairing mechanism, the important transport and optical properties, etc. should be independent of the number of CuO2 layers per unit cell, in principle. These general questions, motivated by experiment, will be asked in the next section.

1.2 General Phase Diagram of Cuprates

and Main Questions

One fundamental problem which one has to solve is the theoretical description and understanding of the general phase diagrams of both hole–doped and electron–doped cuprates, which are shown in Figs. 1.4 and 1.5, respectively. Although details of the T (x) diagram may di er from material to material, for practical purposes Fig. 1.4 describes all of the main features of hole– doped cuprates. As already mentioned, high–Tc superconductivity in hole– doped cuprates always occurs in the vicinity of an antiferromagnetic (AF) phase transition, and has its highest Tc for an optimum doping concentration of around xopt 0.16. The regions in the phase diagram where x < xopt and x > xopt are called “underdoped” and “overdoped”, respectively. In Fig. 1.5, we compare the phase diagram of electron–doped NCCO with that

81 Introduction

Fig. 1.4. Schematic generic phase diagram of hole–doped cuprates. High-Tc superconductivity always occurs in the vicinity of an antiferromagnetic (AF) phase transition, and the superconducting transition temperature as a function of the hole concentration, Tc (x), has a characteristic (nearly parabola–like) shape [17]. Below Tc, the corresponding superconducting order parameter is of d–wave symmetry. The normal state can be separated into two parts. In the overdoped region, i.e. x > 0.15, the system behaves like a conventional Fermi liquid, whereas in the underdoped regime, below the pseudogap temperature T , one find strong antiferromagnetic correlations. As is discussed in the text, Cooper pairing can be mainly described by the exchange of AF spin fluctuations (often called paramagnons), which are present everywhere in the system. In the doping region between Tc and Tc (shaded region) local Cooper pair formation occurs. Below Tc these pairs become phase–coherent and the Meissner e ect is observed.

of hole–doped LSCO. The similarities between the two phase diagrams are remarkable. In particular, both cases reveal an antiferromagnetic phase with a similar N´eel temperature and a superconducting phase in its vicinity. In the following, we shall describe these phase diagrams in more detail.

1.2.1 Normal–State Properties

It is widely believed that understanding the normal–state properties of high- Tc cuprates will also shed some light on the mechanism of superconduc-

Fig. 1.5. Phase diagrams of the electron–doped superconductor NCCO and of hole–doped LSCO. Superconductivity in the electron–doped cuprates occurs only in a narrow doping range and has a smaller Tc .

tivity. One important fact which we shall analyze is the asymmetry of the cuprate phase diagram with respect to hole and electron doping. In the case of electron–doped cuprates, the antiferromagnetic phase persists up to higher doping values and superconductivity occurs only in a narrow doping region. Also, Tc in the electron–doped case is usually smaller than in hole–doped cuprates, namely approximately 25 K.

Let us start with the analysis of the elementary excitations. Important data are provided by angle–resolved photoemission (ARPES) studies, which provide detailed information about the spectral function A(k, ω) (i.e. the local density of states) of the quasiparticles. Owing to recent developments in ARPES, A(k, ω) can be studied with high accuracy versus frequency for a fixed momentum (energy distribution curve, EDC) and as a function of momentum at a fixed frequency (momentum distribution curve, MDC). One of the most important results that one obtains by analyzing MDCs and EDCs is the renormalized energy dispersion ωk, which is shown in Fig. 1.6. These experiments reveal a so–called “kink feature”, which reflects a change of the quasiparticle velocity below kF due to strong correlation e ects. The kink is seen in various hole–doped cuprates, but not in electron–doped ones [18, 19, 20]. It has been argued in [18] that the kink is seen along all directions in the Brillouin zone. However, in most of the studies the kink feature has

10 1 Introduction

Fig. 1.6. ARPES results for the renormalized energy dispersion ωk along di erent directions in the first Brillouin zone, as shown in the inset. Taken from [18].

been investigated only along the (0, 0) → (π, π) direction. This is connected to the fact that along the (0, 0) → (π, 0) direction there are additional e ects such as matrix elements and bilayer splitting which complicate the analysis of experimental data. Originally, the kink feature was attributed to a coupling of itinerant quasiparticles to phonons, in particular to a longitudinal phonon mode at 70 meV which behaves anomalously in several experiments [21]. However, this interpretation has several di culties. The first relates to the fact that the in–plane resistivity ρab in hole–doped cuprates (at the optimal doping) is linear with frequency or temperature (whichever dependence gives the larger value), which is hard to explain within conventional electron– phonon coupling, which predicts ρab T 2 or ρab ω2. At the same time, in electron–doped cuprates no kink is observed [18], and the resistivity is quadratic in temperature. Thus, it is not clear whether both sets of results can be explained assuming the same electron–phonon coupling. In this book we shall study the spectrum of the elementary excitations, and thus the kink feature due to coupling of holes or electrons to spin fluctuations. Spin excitations result in a frequency and momentum dependence of the quasiparticle self–energy which di ers from the phonon case. We shall demonstrate that the kink feature is one of the key facts that can be explained by coupling of holes to spin fluctuations. Furthermore, the anisotropy in k–space and the doping dependence of the kink might be seen as a fingerprint of the coupling to spin fluctuations, too. This will be discussed in detail later.

In general, the normal state can be separated into two parts. In the overdoped region the system behaves mainly like a conventional Fermi liquid, whereas in the underdoped case, in particular below the pseudogap temper-

ature T , the system reveals some unusual properties. For example, a gap is present in the elementary excitations, strong anisotropies are observed (caused mainly by the 2D nature of the system), and local magnetic phases exist. To be more precise, important examples are provided by the 63Cu spin–lattice relaxation rate and the inelastic neutron scattering intensity in hole–doped cuprates: while in the overdoped regime the spin–lattice relaxation rate 1/T1T increases monotonically as T decreases to Tc, one finds in the underdoped case that 1/T1T passes through a maximum at the spin gap temperature T with decreasing T (see [22] for a review). These results are confirmed by INS data, where in the underdoped regime, Im χ(Q, ω) at fixed small ω ( 10–15 meV) also passes through a maximum at T with decreasing T [23]. Thus, one of the main theoretical questions for hole–doped cuprates is to explain the origin of this spin gap temperature in the normal state and its relation to the underlying mechanism of Cooper pairing. In addition, ARPES experiments on underdoped Bi2Sr2CaCu2O8+δ show the presence of a gap with dx2−y2 –wave symmetry well above Tc in the charge excitation spectrum [24, 25]. This gap also opens below the temperature T and thus seems to coincide with the spin gap temperature. Furthermore, recently several experiments, including measurements of heat capacity [26], transport [27], and Raman scattering [28], and, in particular, scanning tunneling spectroscopy [29, 30] have confirmed the existence of a gap in the elementary excitations below T . Thus T is usually called the pseudogap temperature. Whether a pseudogap is present in electron-doped cuprates is still a subject of debate. While measurements of the optical conductivity report a pseudogap similarly to the hole–doped case [31], tunneling data reveal a pseudogap (i.e. a reduction of the spectral weight at the Fermi level) only below Tc and when a high magnetic field (> Hc2) is applied [32, 33].

The existence and origin of the pseudogap are another fundamental question which we shall address in this book. So far, a few phenomenological models, such as marginal–Fermi liquid (MFL) [34], nested–Fermi liquid (NFL) [35, 36], and nearly–antiferromagnetic–liquid (NAFL) [37] models, have been developed in order to understand the unusual Fermi liquid properties in the normal state. At the moment it is not clear whether these concepts can also be applied to electron-doped superconductors.

Another important energy scale is the temperature Tc , which is only present in the underdoped region and close to Tc. Below Tc , local Cooper pairs without long-range phase coherence are found [40, 41, 42] (“preformed pairs”), which become phase–coherent only for temperatures T < Tc where the Meissner e ect is observed. Tc and T seem to be crossover temperatures rather than true phase transitions (although this is a subject of debate [38, 39]). As we shall discuss later, Tc is connected with the fact that, in the doping behavior of the superconducting transition temperature Tc(x), a maximum around x = 0.15 is found. It has to be clarified whether Tc exists also in the case of electron–doped cuprates.