Physics ± Uspekhi 44 (5) 515 ± 539 (2001) # 2001 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences REVIEWS OF TOPICAL PROBLEMS PACS numbers: 74.20.Mn, 74.72. ± h, 74.25. ± q, 74.25.Jb Pseudogap in high-temperature superconductors M V Sadovski|¯ DOI: 10.1070/PU2001v044n05ABEH000902 Contents 1. Introduction 515 2. Basic experimental facts 516 2.1 Specific heat and tunneling; 2.2 NMR and kinetic properties; 2.3 Optical conductivity; 2.4 Fermi surface and ARPES; 2.5 Other experiments 3. Theoretical models of the pseudogap state 522 3.1 Scattering on short-range-order fluctuations: Qualitative ideas; 3.2 Recurrent procedure for the Green functions; 3.3 Spectral density and the density of states; 3.4 Two-particle Green function and optical conductivity 4. Superconductivity in the pseudogap state 529 4.1 Gor'kov equations; 4.2 Transition temperature and the temperature dependence of the gap; 4.3 Cooper instability; 4.4 Ginzburg ± Landau equations and the basic properties of superconductors with a pseudogap near Tc; 4.5 Effects of the non-self-averaging nature of the order parameter 5. Conclusion. Problems and prospects 536 References 538 Abstract. The paper reviews the basic experimental facts and a the very unusual properties of these systems in the normal number of theoretical models relevant to the understanding of (nonsuperconducting) state, and without a clear understand- the pseudogap state in high-temperature superconductors. The ing of the nature of these properties there is little hope for state is observed in the region of less-than-optimal current- completely elucidating the microscopic mechanism of high-Tc carrier concentrations in the HTSC cuprate phase diagram superconductivity. In recent years one of the main aspects of and manifests itself as various anomalies in the electronic high-Tc superconductor physics has been the study of the properties, presumably due to the antiferromagnetic short- anomalies of what is known as the pseudogap state [1], which range-order fluctuations that occur as the antiferromagnetic is observed in the region of the phase diagram corresponding region of the phase diagram is approached. The interaction of to charge-carrier concentrations lower than optimal one (i.e. current carriers with these fluctuations leads to an anisotropic the concentration corresponding to the maximum tempera- transformation of the electron spectrum and causes the system ture of the superconducting transition, Tc); it is usually called to behave as a non-Fermi liquid in certain regions of the Fermi the `underdoped' region. In this region there are many surface. Simple theoretical models for describing the basic anomalies of the electronic properties in both the normal properties of the pseudogap state, in particular renormaliza- and the superconducting state, which are related to the drop tion-induced anomalies in the superconducting state, are dis- in the single-particle density of states and to the anisotropic cussed. transformation of the charge-carrier spectral density. The understanding of the nature and properties of the pseudogap 1. Introduction state is the central problem of any approach to the description of the complex phase diagram of HTSC systems. Hundreds of The study of high-temperature (or high-Tc) superconductiv- experimental and theoretical papers have been devoted to this ity (HTSC) in copper oxides remains one of the central problem 1. avenues of research in the physics of the condensed state. The goal of the present review is to present the main Despite enormous efforts of both experimenters and theore- experimental facts pertaining to the observations of the ticians, the nature of this phenomenon remains unclear. It is a pseudogap state in underdoped high-Tc cuprates and to well-known fact that the main difficulties here are related to discuss a number of simple theoretical models of this state. The review is not exhaustive in either describing the experi- mental data or giving the full theoretical picture. The M V Sadovski|¯ Institute of Electrophysics, Ural Branch of the Russian Academy of Sciences experimental data are presented fairly briefly, since the ul. Amundsena 106, 620016 Ekaterinburg, Russian Federation literature already has a number of good reviews [2 ± 6] Tel. (7-3432) 67 87 86. Fax (7-3432) 67 87 94 E-mail: [email protected] 1 Suffice it to say that the well-known E-archive of preprints, cond-mat, Received 27 November 2000, revised 8 February 2001 contains more than 600 papers devoted, in one way or another, to the Uspekhi Fizicheskikh Nauk 171 (5) 539 ±564 (2001) physics of the pseudogap sate. At the major conference on superconduc- Translated by E Yankovsky; edited by A V Getling tivity held in Houston, M 2S ± HTSC-VI ( February 2000), four section meetings were devoted to this problem Ð more than to any other problem of HTSC physics. 516 M V Sadovski|¯ Physics ± Uspekhi 44 (5) devoted to these problems in one way or another. The hundred kelvins and vanishes at hole concentrations p lower theoretical part is also fairly subjective and, to a large or of order 0.05, with the system becoming a (poor) metal. A extent, reflects the author's viewpoint. There are two basic further increase in the hole concentration transforms the theoretical scenarios for explaining the pseudogap anomalies system into a superconductor, with the transition tempera- of HTSC systems. The first one is based on the model in which ture increasing with the carrier concentration and going Cooper pairs are formed above the transition temperature through a characteristic maximum at p0 0:15 ± 0.17 (opti- [7 ± 10], with the subsequent establishment of their phase mum doping) and then decreasing and vanishing at coherence at T < Tc. The second scenario assumes that the p 0:250:30, although in this (overdoped) region the pseudogap state emerges due to `dielectric' type, short-range- metallic behavior is still quite evident. Here, at p > p0 the order fluctuations, which are present in the region of under- metallic properties are fairly traditional (Fermi-liquid beha- doped compositions in the phase diagram. The most popular vior), while at p < p0 the system becomes an anomalous picture here is that of antiferromagnetic (AFM) fluctuations metal, whose behavior (as most researchers believe) cannot [11 ± 15], although one cannot exclude that fluctuations of be described by the Fermi-liquid theory 2. charge-density waves (CDW), structural distortions, or phase The anomalies of physical properties currently attributed separation on a microscopic scale can play a similar role. I to the formation of a pseudogap state are observed in the à believe that many experiments conducted in recent years metallic phase at p < p0 and at temperatures T < T , where à speak in favor (fairly persuasively) of the second scenario. T decreases from values of order TN at p 0:05 and vanish Hence in our theoretical discussion we will limit ourselves to at a certain `critical' carrier concentration pc somewhat higher relevant models, giving a quite full description of the work of than p0 (Fig. 1a). For instance, Tallon and Loram [6] stated the present author and his collaborators, and also of the work that this happens at p pc 0:19. According to some of other researchers whose main ideas and approaches are in researchers (mostly adherents of the idea of the superconduct- the same vein. Accordingly, in the most part of the review we ing nature of the pseudogap), T à merges with the line of the will deal with the model of antiferromagnetic fluctuations, superconducting-transition temperature Tc near the optimum although its validity cannot be thought of as fully corrobo- concentration p0 (Fig. 1b). Below we will see that the majority rated. of new experimental data most likely support the variant of Neither can we claim that the list of references is complete the phase diagram depicted in Fig. 1a (for more details see and, the more so, that it reflects priorities in this field. It is Ref. [6]). Note that most researchers agree that T à does not assumed that the reader will find the necessary literature from have the meaning of the temperature of a phase transition; the papers in the list of references. The author would like to rather, it simply defines a characteristic temperature scale apologize to the numerous authors of papers not cited here Ð below which pseudogap anomalies appear in the system. In the aim was to make the list not too long. this region of the phase diagram a system exhibits no features characteristic of phase transitions in the behavior of thermo- 2. Basic experimental facts dynamic quantities 3. The general idea is that all these anomalies, to put it simply, are related to the suppression (in Typical variants of the phase diagram of HTSC cuprates are the given region) of the density of states of single-particle depicted in Fig. 1. Depending on the concentration of charge excitations near the Fermi level, which is what the general carriers (mainly holes), several phases and regions with concept of the pseudogap means 4. Here T à is simply anomalous physical properties can be observed in the high- proportional to the size of the pseudogap on the energy conductivity CuO2 plane. In the region of low hole concentra- scale. Sometimes another characteristic temperature scale is à tions all known HTSC cuprates are antiferromagnetic specified, T2 (see Fig. 1b), which is related to the transition insulators. As the current-carrier concentration increases, from a `weak'- to a `strong'-pseudogap regime [14]. The the Ne el temperature TN rapidly drops from values of several reasoning here is based on a change in the spin response of the system in the vicinity of this temperature. In the present 500 review we almost never touch on the corresponding details.