Physics Reports vol.387

Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide r...

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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one speciﬁc subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also ﬁnd a suﬃcient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected] M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected]

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Instructions to authors

A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] H. ORLAND (Statistical physics and ﬁeld theory), Service de Physique Theorique, CE-Saclay, CEA, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientiﬁca, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insuﬃcient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be suﬃciently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of ﬁgure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classiﬁcation codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Instructions to authors

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[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

Ablowitz, M.J., D.J. Kaup, A.C. Newell and H. Segur, 1974. The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315. Abramowitz, M. and I. Stegun, 1965, Handbook of Mathematical Functions (Dover, New York). Ziegler, B., 1986, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York) p. 293. Ranking of references. The references in Physics Reports are ranked: crucial references are indicated by three asterisks, very important ones with two, and important references with one. Please indicate in your ﬁnal version the ranking of the references with the asterisk system. Please use the asterisks sparingly: certainly not more than 15% of all references should be placed in either of the three categories. Formulas. Formulas should be typed or unambiguously written. Special care should be taken of those symbols which might cause confusion. Unusual symbols should be identiﬁed in the margin the ﬁrst time they occur.

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from the version accepted by the Editor of the journal are permissible without the prior and explicit approval by the Editor. Such changes should be clearly indicated on an accompanying printout of the ﬁle.

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Available online at www.sciencedirect.com

Physics Reports 387 (2003) 1 – 149 www.elsevier.com/locate/physrep

Theory of superconductivity in strongly correlated electron systems Yoichi Yanasea , Takanobu Jujob , Takuji Nomurac , Hiroaki Ikedad , Takashi Hottae;∗ , Kosaku Yamadad a

Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan b Nara Institute of Science and Technology, Ikoma, Nara 630-0101, Japan c Synchrotron Radiation Research Center, Japan Atomic Energy Research Institute, Mikazuki, Hyogo 679-5148, Japan d Department of Physics, Kyoto University, Kyoto 606-8502, Japan e Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan Accepted 16 July 2003 editor: D.L. Mills

Abstract In this article we review essential natures of superconductivity in strongly correlated electron systems (SCES) from a universal point of view. First we summarize experimental results on SCES by focusing on typical materials such as cuprates, BEDT-TTF organic superconductors, and ruthenate Sr 2 RuO4 . Experimental results on other important SCES, heavy-fermion systems, will be reviewed separately. The formalism to discuss superconducting properties of SCES is shown based on the Dyson–Gor’kov equations. Here two typical methods to evaluate the vertex function are introduced: One is the perturbation calculation up to the third-order terms with respect to electron correlation. Another is the Auctuation-exchange (FLEX) method based on the Baym–KadanoC conserving approximation. The results obtained by the FLEX method are in good agreement with those obtained by the perturbation calculation. In fact, a reasonable value of Tc for spin-singlet d-wave superconductivity is successfully reproduced by using both methods for SCES such as cuprates and BEDT-TTF organic superconductors. As for Sr 2 RuO4 exhibiting spin-triplet superconductivity, it is quite diDcult to describe the superconducting transition by using the FLEX calculation. However, the superconductivity can be naturally explained by the perturbation calculation, since the third-order terms in the anomalous self-energy play the essential role to realize the triplet superconductivity. Another important purpose of this article is to review anomalous electronic properties of SCES near the Mott transition, since the nature of the normal state in SCES has been one of main issues to be discussed. Especially, we focus on pseudogap phenomena observed in under-doped cuprates and organic superconductors. A variety of scenarios to explain the pseudogap phenomena based on the superconducting and/or spin Auctuations are critically ∗

Corresponding author. Tel.: +81-29-284-3521; fax: +81-29-282-5939. E-mail address: [email protected] (T. Hotta).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.07.002

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

reviewed and examined in comparison with experimental results. According to the recent theory, superconducting Auctuations, inherent in the quasi-two-dimensional and strong-coupling superconductors, are the origin of the pseudogap formation. In these compounds, superconducting Auctuations induce a kind of resonance between the Fermi-liquid quasi-particle and the Cooper-pairing states. This resonance gives rise to a large damping eCect of quasi-particles and reduces the spectral weight near the Fermi energy. We discuss the magnetic and transport properties as well as the single-particle spectra in the pseudogap state by the microscopic theory of the superconducting Auctuations. As for heavy-fermion superconductors, experimental results are reviewed and several theoretical analyses on the mechanism are provided based on the same viewpoint as explained above. c 2003 Elsevier B.V. All rights reserved. PACS: 74.20.Mn; 74.25.Fy; 74.72−h; 74.70.Kn; 74.70.Pq; 74.70.Tx Keywords: Unconventional superconductivity; Strongly correlated electron systems; Fermi-liquid theory; Dyson–Gor’kov equations; Pseudogap phenomena; Spin Auctuations; Superconducting Auctuations

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experimental view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Symmetry of Cooper-pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6. Magnetic Meld penetration depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Organic superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Sr2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Microscopic mechanism of superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dyson–Gor’kov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. High-Tc cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Spin Auctuation-induced superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. FLEX approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Higher-order corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Organic superconductor -(ET)2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Electronic property and tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Application of the microscopic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Sr 2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Perturbation theory for the triplet superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3. IdentiMcation of the internal degree of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Pseudogap phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Basic mechanism of the pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 9 9 9 10 12 14 16 17 18 20 22 22 29 29 30 33 36 41 43 43 44 48 48 49 55 59 59 65 65

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 4.2.2. ECect of three-dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. ECect of the magnetic Meld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Another candidate: organic superconductor -(ET)2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5. Higher-order corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Microscopic theory: FLEX+T-matrix approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Single-particle properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Doping dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5. Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Heavy-Fermion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Experimental view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Ce-based compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. U-based compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Microscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Application to materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. CeCu2 X2 (X = Si and Ge) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. CeTIn5 (T = Co, Rh, and Ir) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. CeIn3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. UM2 Al3 (M = Pd and Ni) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Concluding remarks and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Fermi-liquid theory on the London constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. ECects of higher-order perturbation terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction In recent decades, elucidation of unconventional superconductivity in strongly correlated electron systems (SCES) has been one of the central issues both in experimental and theoretical research Melds of condensed matter physics. As is well known, SCES form a vast category, including varieties of materials such as transition metal oxides, molecular conductors, and f-electron compounds. Here we brieAy introduce these superconducting materials in this order. Among transition metal oxides, cuprate superconductors have certainly attracted the most attention of researchers in the condensed matter physics since the discovery in 1986 [1,2], due to the high superconducting transition temperature Tc as well as several kinds of anomalous behaviors in the electronic properties. In fact, the high-Tc cuprate is one of main targets of this review article. It is emphasized here that varieties of superconducting materials have been also discovered in other transition metal oxides. Especially, Sr 2 RuO4 [3] with isostructure of La2 CuO4 (the parent compound of high Tc superconductors) exhibits the superconductivity with triplet pairing, which has been conMrmed by nuclear magnetic resonance (NMR) measurements [4]. The ruthenate has attracted attention in spite of its low Tc as large as 1:5 K, since it is one of rare materials that show triplet superconductivity in the solid state. The ruthenate will be also discussed in detail in this review article.

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In addition to d-electron systems, materials composed of atoms with lighter mass are also the member of SCES. In general, those are called molecular conductors, including organics and fullerides. In 1980, (TMTSF)2 PF6 has been discovered as the Mrst organic superconductor [5], which has triggered the vigorous experimental researches on these materials. A characteristic issue of the molecular conductors is that it is possible to construct artiMcial molecule by controlling the synthesis in the atomic level. Then, much eCort has been made to elevate Tc by using several techniques to synthesize new molecular superconductors. Let us turn our attention to f-electron materials including rare-earth or actinide ions. The pioneering discovery of superconductivity in the heavy fermion material CeCu2 Si2 [6] has triggered the rapid increase of investigations on exotic properties of f-electron superconductivity, leading to a chain of further discoveries of superconductivity in uranium and cerium compounds. As we will review in Section 5, numbers of new superconducting materials in heavy fermion systems have been discovered in recent years. Here it is noted that new superconducting states such as coexisting one with magnetic orders are increasing. Even in the above brief survey, there are several kinds of superconductors categorized in SCES. It is quite natural for experimentalists to make the research Meld rich, as a result of high activities to synthesize new superconducting materials. However, the purpose in the theoretical research is not to pursue the variety in materials, although at some stage it is necessary to investigate a particular material as a typical example. There should exist a universal picture to explain the common essence in all SCES. A task imposed on theoreticians is to unveil this universal concept, and to clarify the interesting aspects in materials. We believe that we have arrived at a uniMed view on the superconductivity in SCES. Thus, the main purpose of this review article is to convey this viewpoint by showing explanations for typical materials. Before proceeding to the clariMcation of our uniMed viewpoint, let us consider Mrst the conditions on the theory of superconductivity in SCES. Without any restrictions, the present review article may be just the exhibition of previous theories for SCES. The most exotic possibility for the superconducting mechanism was superconductivity due to single-electron condensation, but the experimental results on the unit of magnetic Aux in high-Tc materials have conMrmed that the existence of Cooper-pair in high-Tc cuprates [7]. Also in heavy-fermion superconductors, the existence of Cooper-pair has been experimentally conMrmed. Those are quite important, since the basic point of the BCS theory for superconductivity are invariant even for SCES. Thus, in this review, issues on exotic superconductivity based on the non Cooper-pair formation are simply ignored. Furthermore, here we clarify our strategy to understand the electronic properties in the normal state before discussing the pairing mechanism. In the normal state of strongly correlated materials, “anomalous” metallic behaviors have been frequently observed. Namely, behaviors of physical quantities are sometimes deviated from those understood from the Fermi-liquid theory. Especially in high-Tc cuprates, such “anomalous” behaviors have provided many challenging issues and stimulated much interests. Indeed, the understanding of the “anomalous” metallic state is one of the central issues in this review. One way to explain such non Fermi-liquid behaviors is, of course, to pursue the non Fermi-liquid ground state, appearing due to the combined eCects of strong correlation and low dimensionality. In fact, as is well known in one dimension, there appears the Tomonaga– Luttinger liquid state, essentially diCerent from the Fermi-liquid state, due to the restriction in the phase space of one dimension. No doubt is cast on the results in one dimension and we know that there exist several approaches based on the non Fermi-liquid ground state in order to understand the

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anomalous normal-state properties of strongly correlated metals. However, it should be noted that non Fermi-liquid behaviors do not immediately indicate the breakdown of the Fermi-liquid theory in strongly correlated materials with two or three dimensionality. We believe that the Fermi-liquid state is a good starting point to understand the non Fermi-liquid behaviors in some compounds. If there is no discontinuity from the Fermi-liquid state, the deviation from the Fermi-liquid behaviors should be derived from such a starting point. In this review, we will actually explain the non Fermi-liquid behaviors from this point of view, but it is worth while to stress here that the Fermi-liquid state is a robust concept in two or three dimensions. The essential issue of the Fermi-liquid state is the continuity principle, which means that quasi-particles can be obtained by the adiabatic continuations from the non-interacting systems. Even if the overlap between bare electron state and dressed quasi-particle state is vanishingly small like in f-electron materials, the continuity principle can be still eCective. Here readers may have a naive question in their minds: Then, how to understand anomalous behaviors in the normal state? First note that the conventional Fermi-liquid behaviors may be restricted to the very low-temperature region, even if the ground state is a Fermi liquid. Such a situation is considered to be realized in some heavy-fermion compounds, owing to a strong renormalization of the Fermi energy. Even if the quasi-particle renormalization is not so strong like in high-Tc cuprates, this situation can be caused by some kinds of Auctuations, which give corrections to the Fermi-liquid behaviors. The latter eCect possibly appears more signiMcantly, when the long-range order exists in the ground state. Since the Fermi-liquid state is eventually destroyed by the long-range order, it is quite natural that the Fermi-liquid behaviors are altered by its precursor, namely, by the Auctuation. We will attribute the non Fermi-liquid behaviors in the high-Tc cuprates and some organic superconductors to the latter origin. In particular, the superconducting (SC) Auctuation as well as the anti-ferromagnetic (AF) spin Auctuation should be taken into account. In those compounds, the effects of such Auctuations are signiMcantly enhanced owing to quasi-two-dimensionality. The eCective inclusion of Auctuations in the Fermi-liquid state will lead to understandings of anomalous behaviors in the normal state. A trial to take into account the eCect of AF spin Auctuations in the Fermi-liquid normal state has been developed for a long time, since the AF insulating phase can be ubiquitously found in strongly correlated materials. The AF Auctuations are especially important near the phase boundary between metallic and AF insulating phases. For instance, the self-consistent renormalization (SCR) theory developed by Moriya and co-workers is one of the powerful methods to include eCects of spin Auctuations, but the details will be simply skipped, since readers can consult with the textbook [8]. It is commented here that the AF spin Auctuation theory should be interpreted as an extension of the Fermi-liquid theory [9], which is called “nearly anti-ferromagnetic Fermi-liquid theory” at present. The AF spin Auctuation is actually eCective in high-Tc cuprates, in which the NMR and neutron scattering measurements have clearly observed the spin Auctuations. For example, the anomalous temperature dependence of the electric resistivity is explained by including the AF spin Auctuation. Interestingly, the cross-over from the conventional T -square to the anomalous T -linear resistivity has been observed in Tl-based cuprate superconductors [10], which occurs together with the enhancement of the AF spin Auctuation. Then, a coherent understanding is obtained by extending the Fermi-liquid theory to take into account the AF spin Auctuation. Such a continuity clearly indicates that the theory based on the Fermi-liquid picture is quite useful and eCective for strongly correlated systems.

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Another eCort has been devoted to include the eCect of SC Auctuations. This is closely related to the most challenging issue in the “anomalous” metallic state, the pseudogap phenomenon, which is focussed in this review. Among many theoretical proposals suggested before, we will introduce an understanding based on the SC Auctuation. Since high-Tc cuprates and organic superconductors have the superconducting ground state, it is expected that the SC Auctuation should be active in these materials at least near the phase transition. It has been common knowledge that the SC Auctuation is usually negligible, but the recent theoretical eCorts, which has been stimulated by the recent experimental results, have revealed the importance of the SC Auctuation for the electronic properties. We will clarify the condition for the appearance of the SC Auctuation and show that the condition is actually satisMed in high-Tc cuprates and some organic superconductors. Then, the precursor of the phase transition destroys the Fermi-liquid state and induces the excitation gap. Many aspects of the anomalous properties, including the magnetic and transport properties, are successfully explained by starting from the Fermi-liquid state and taking into account appropriately the eCect of AF spin Auctuation and/or SC Auctuation [11], as will be explained in Section 4 in details. The scenario based on the Fermi-liquid theory is rather simple, but the simple and uniMed scenario makes it easy to understand anomalous behaviors in SCES. Now our footing of this review article becomes clear, and the next point is how to clarify the mechanism of the Cooper-pair formation. First it is emphasized that a couple of quasi-particles form the Cooper-pair. Typically, in heavy-fermion superconductivity, the Cooper-pair is composed of heavy quasi-particles themselves, as conMrmed by the large jump in the speciMc heat at Tc . The investigation of pairing mechanism is reduced to the determination of the residual interaction among quasi-particles. As easily understood, non s-wave pairing should appear for superconductivity in highly correlated systems due to the eCect of strong short-range Coulomb interaction. In fact, anisotropic Cooper-pair has been found in common in strongly correlated superconductors, experimentally suggested by the power-law behavior of physical quantities in the low-temperature region. The value of the power sensitively depends on the node structure of the gap function on the Fermi surface. Thus, by analysing carefully the temperature dependence of physical quantities in experiments, it is possible to deduce the symmetry of the Cooper-pair under the group-theoretical restriction. The phenomenological theory, which is not focused in this review, plays an important role for the determination of the pairing symmetry. Since the superconducting transition is the second-order phase transition even for SCES, the Ginzburg–Landau theory has been still applicable to those systems, by paying due attention to the symmetry of Cooper-pair as well as the group-theoretical restrictions on the crystal structure. In fact, there have been signiMcant advances in the phenomenological understandings on unconventional superconductivity without special knowledge on the mechanism of Cooper-pair formation [12]. In particular, when the degeneracy remains in the internal degree of freedom in the triplet superconductor, the phenomenological Ginzburg–Landau theory has been quite useful to identify the pairing symmetry. When ones tried to discuss the microscopic aspects of the mechanism of unconventional superconductivity, again AF spin Auctuations were considered to play crucial roles to induce singlet d-wave superconductivity, consistent with the node structure in high-Tc cuprates. Indeed, the theory based on the AF spin Auctuations has provided many important understandings in high-Tc superconductors, as will be explained later. However, in order to arrive at the uniMed understanding of superconductivity in SCES, more general viewpoint should be considered, since the pairing potential originating from the residual interaction among quasi-particles is not always dominated by the spin Auctuations. The

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most general conclusion is that the origin of the Cooper-pair formation is the momentum dependence of the residual interaction. We believe that this point of view is important for the comprehensive understanding of superconductivity in SCES including high-Tc cuprates. A clear example for our belief can be found in the triplet superconductivity conMrmed in some d- and f-electron systems such as Sr 2 RuO4 and UPt 3 [4,13]. In order to understand the triplet pair formation, one may naively consider the eCect of ferromagnetic spin Auctuations, which was proposed for the origin of spin-triplet superAuidity in 3 He. However, in the triplet superconductors, paramagnons are not always dominant in the spin Auctuation spectrum. The ruthenate Sr 2 RuO4 is a typical example. We have found that the incommensurate AF spin Auctuation is enhanced, as observed in the neutron scattering experiment [14]. Then, in contrast to the naive expectation, paramagnons do not seem to play a central role in the occurrence of triplet superconductivity. In such a case, we have to consider the momentum dependence of the residual interaction from the general point of view. As we will discuss in detail in this review, it is possible to explain triplet as well as singlet superconductivity, in addition to several anomalous behaviors in the normal state, based on the microscopic Hamiltonian, leading to the uniMed picture for unconventional superconductivity in SCES. For the purpose, it is indispensable to choose appropriately the method for the calculation. One is the application of numerical techniques such as exact diagonalization and quantum Monte Carlo simulations. There is a clear advantage that in principle we can include the eCect of electronic correlation correctly, but in the exact diagonalization, the size of the model is severely restricted due to the limitation in computer memory. Also in the quantum Monte Carlo simulations, it is quite diDcult to increase the strength of correlation because of the negative sign problem and the system size is still restricted. On the other hand, recently developed technique such as density matrix renormalization group method is very powerful to analyze quantum systems even with frustrations. However, the target material is essentially limited to the one-dimensional system, even though it may be possible to treat ladder-like compounds. Complementary to the numerical method, another traditional technique is the quantum Meld theory, or more speciMcally, the Green’s function method, based on the assumption that it is allowed to perform the perturbation expansion in terms of the interaction. It is an advantage that we can calculate, in principle, physical quantities in the thermodynamic limit, while it is inevitable to resort to approximations for actual calculations, since it is quite diDcult to carry out the exact calculation based on the Green’s function method. Another important advantage is that the physical picture is to be clariMed. In this article, we focus on the Green’s function technique and the results obtained by this method. It matches our principle based on the Fermi-liquid theory. We will introduce an understanding obtained by the combination of the numerical techniques and the Green’s function approaches [15], but the techniques of numerical methods will not be introduced in this review article. Readers consult with other review articles regarding those issues (see, for instance, Ref. [16]). Next, we need to deMne a microscopic model Hamiltonian. In this paper, the Hubbard Hamiltonian will be focussed, since it is widely recognized as a canonical model for SCES, although for f-electron systems, we need to pay due attention for its application. We will also consider the multi-orbital Hubbard model, in which orbital degree of freedom of d electrons is explicitly included, especially for the analysis of ruthenate. For the purpose to study electronic properties of SCES, ones sometimes considered the so-called t–J model, obtained by the strong-coupling expansion in the Hubbard or

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d–p model. The prohibition of double occupancy at each site is imposed on the model, which is an essential point to include the strong correlation eCect in the t–J model. Since the Hilbert space becomes smaller than that of the Hubbard model, the t–J model has been frequently used for the analysis based on numerical techniques. However, in the Green’s function method or even in simpler mean-Meld calculations, it is diDcult to include correctly the prohibition of double occupancy. Thus, results and analysis on the t–J model are out of the scope of this review article. Readers interested in the t–J model can also consult with other previous review papers [16]. In this review article, then we will show the uniMed picture to understand unconventional superconductivity in SCES based on the Fermi-liquid framework by using the Hubbard (or Hubbard-like) Hamiltonian and the Green’s function method. The calculations will be done based on the Dyson– Gor’kov equations [17], composed of normal and anomalous Green’s functions to characterize the normal and superconducting states. Formally those are the coupled equations, related by the irreducible four-point vertex functions representing the interaction processes among quasi-particles. Depending on the eCective strength of the interaction, we change our approaches to evaluate the four-point irreducible vertex. For the weak correlation systems, we adopt the third-order perturbation theory with respect to the on-site Coulomb interaction U [18,19]. By solving the linearized Dyson–Gor’kov equations, we can determine Tc , which will well reproduce the critical temperatures for cuprates from overdoped to optimally doped regions. Moreover, this perturbation theory gives reasonable values of Tc also for organic superconductors -type (BEDT-TTF)2 X [20]. The symmetry of these two superconductors is considered to be the d-wave one from various experimental results. Theoretical calculations also predict the d-wave symmetry. In addition to the above results, it should be stressed that the perturbation theory gives also a reasonable explanation of the mechanism of triplet superconductivity in Sr 2 RuO4 [21]. The third-order terms in the vertex function play main roles in realizing the triplet superconductivity. Note that these terms are not attributed to the contribution from the spin Auctuations. This fact is in sharp contrast with the spin Auctuation mechanism adopted in cuprates, while it is consistent with the absence of paramagnon peak in neutron diCraction experiments on ruthenate. For the intermediate coupling systems such as optimally doped cuprates, we adopt the Auctuationexchange (FLEX) approximation [22] to evaluate the irreducible four-point vertex parts. This theory based on the spin Auctuations has been developed by many groups and used to give reasonable value of Tc for the d-wave superconductivity. We conMrm the reliability of these calculations by comparing the FLEX and the third-order perturbation calculations with each other. From the above calculation we conclude that the momentum dependence in the eCective interaction between quasi-particles originating initially from the on-site Coulomb repulsion U induces the superconductivity. The FLEX approximation is again used for the analysis of the anomalous properties in the normal state. In order to take into account the eCect of the superconducting Auctuations, the FLEX approximation is modiMed to be combined with the T-matrix approximation. By using this FLEX + T-matrix approximation, we develop a microscopic theory on the SC Auctuation and analyze the under-doped cuprates. The organization of this review article is as follows. In Section 2, experimental review will be provided regarding high-Tc cuprates, organic superconductors, and ruthenate. The interests from the theoretical point of view will be also explained. As is easily understood, those materials should include millions of references, and it is almost impossible to cite all papers. Thus, we will refer several

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papers, which will be relevant to the later theoretical discussions. In Section 3, theoretical results are reviewed based on the Green’s function techniques. First we introduce some basic equations which will be needed for the later analysis. Then, the results for cuprates, organic superconductors, and ruthenate will be discussed in this order. In Section 4, as a typical result in the anomalous normal-state properties of high-Tc cuprates, pseudogap phenomena will be discussed somewhat in detail. In Section 5, after the brief review of experimental results on f-electron superconductors, we will discuss the mechanism of superconductivity in heavy-fermion materials. In Section 6, we summarize this review article. In Appendix A, the correct derivation of the generalized Fermi-liquid theory on the London penetration depth is presented and we provide our picture to understand the so-called Uemura plot by taking account of the quasi-particle interaction. In Appendix B, the convergence of the perturbation expansion with respect to U is examined by calculating the fourth-order terms. The convergence is satisfactorily good for d-wave pairing cases, while for p-wave pairing cases Tc shows oscillatory behavior depending on the calculated order. By summing up the ladder diagrams up to inMnite order, we can eliminate this oscillation and recover the convergence also for p-wave pairing case. 2. Experimental view 2.1. Cuprates The discovery of high-Tc cuprates [1,2] was a trigger of the intensive studies on superconductivity in strongly correlated electron systems (SCES). The theoretical eCorts stimulated by high-Tc superconductivity have clariMed many basic properties of strongly correlated materials. Especially the understanding of the quasi-two-dimensional system has been developed. High-Tc cuprates have the perovskite-type crystal structure and the parent compounds are the Mott insulators with anti-ferromagnetic (AF) order. Carrier doping into this Mott insulating state induces high-Tc superconductivity, which essentially occurs in the two-dimensional (2D) CuO2 plane. Note here that both hole and electron dopings induce superconductivity [23]. From the early stage of the research, high-Tc cuprates have been recognized to be one of SCES [24,25]. Now the cuprate is conMrmed to be the most established unconventional superconductor. Namely, it is predominantly believed that Cooper-pairs with dx2 −y2 -wave symmetry originates from the electronic mechanism. We can understand also several normal-state anomalous properties such as T -linear electric resistivity [26] and strongly enhanced AF spin correlation [27], considering the eCect of strong electron correlation. In the following, we survey these experimental results on unconventional superconductivity and anomalous normal-state properties in high-Tc cuprates. 2.1.1. Symmetry of Cooper-pair First let us see the results on pairing symmetry, since the symmetry of the Cooper-pair is an important issue to consider the superconducting mechanism. The non-s-wave pairing is a direct evidence for the unconventional superconductivity. According to the crystal symmetry of the square lattice composed of copper and oxygen ions, the pairing symmetry is classiMed as s-wave, dx2 −y2 -wave, dxy -wave, and so on. Among them, in the theoretical point of view, the appearance of the conventional s-wave superconductivity is suppressed, because the strong on-site repulsion easily destroys

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the s-wave pairing. In fact, the theoretical studies on the electron correlation have predicted the dx2 −y2 -wave symmetry: For instance, in the literatures, we Mnd several works based on the RPA [28,29], scaling theory [30–32], variational method [33,34], and quantum Monte Carlo simulation [35]. In 1980s, however, the many experimental results supported the s-wave symmetry. Only a few experiments including the NMR measurement have supported the d-wave superconductivity [27]. This discrepancy has been clearly resolved by the intensive investigations in 1990s, which have demonstrated the various evidence for the anisotropic gap structure with line node [36–39]. Some deviations from the clean d-wave superconductivity has been explained by taking account of the strong impurity scattering [40]. Finally, the phase-sensitive measurements have clearly conMrmed the dx2 −y2 -wave symmetry [41–43]. Nowadays the momentum dependence of the excitation gap is directly observed in the angle resolved photoemission spectroscopy (ARPES), where the line node appears in the diagonal direction of kx = ky [44–47]. On the other hand, the pairing symmetry in electron-doped systems has not been settled for a long time, because node-less behaviors were reported by several groups [48]. However, recent experiments have Mnally concluded that the electron-doped systems are also the dx2 −y2 -wave superconductors [49–53]. Also in this case, crucial roles have been played by the phase-sensitive measurement [51] and ARPES [52,53]. These results seem to be natural, since from the theoretical point of view, the same pairing mechanism is expected both for hole- and electron-doped materials. Note, however, that interesting particle–hole asymmetry has been found in some aspects. For instance, in electron-doped systems, we have observed that (i) the transition temperature is relatively low, (ii) the superconducting region in the phase diagram is narrow, and (iii) the AF order is robust in comparison with hole-doped compounds. Later in this review, this asymmetry will be discussed in our theoretical approach by considering the detailed electronic structure. Now we can conclude that the issues on the paring symmetry in high-Tc cuprates have been settled. In the next stage, the following two issues are especially important. One is, of course, the pairing mechanism for high-Tc cuprates. This theoretical subject will be discussed in Section 3. Particularly we focus on the intensive studies from the microscopic point of view, which have opened a new way to understand the superconductivity in SCES. Another issue is to explain the anomalous properties in the normal state, which is one of the main points of this review. This subject will be discussed in detail in Section 4 with our main interests on the pseudogap phenomena. As for the latter problem, intensive experimental investigations have been performed and lots of important results have been piled up. Thus, here we survey the experimental results on the normal-state properties in the following subsections [54]. 2.1.2. Phase diagram Before proceeding to the review of normal-state properties of high-Tc cuprates, it is instructive to explain Mrst the outline of the T – phase diagram. Both the theoretical and experimental investigations have been focussed on the hole-doped systems, but recently, the electron-doped systems also have been studied intensively. Then, the whole phase diagram as shown in Fig. 1 has been clariMed. Let us see some characteristic points of this phase diagram in the following. In general, the properties of high-Tc cuprates are controlled by the carrier doping concentration , which is deMned as = 1 − n and n is the carrier number per copper site. The system is the AF Mott insulator at half-Mlling ( = 0) and the AF order is easily destroyed by the slight amount of

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11

T

T0

T*

PG AF SC

electron-doping

0

hole-doping

δ

Fig. 1. The phase diagram of high-Tc superconductors. The horizontal and vertical axes indicate the doping concentration and the temperature, respectively. “AF”, “SC”, and “PG” denote anti-ferromagnetic, superconducting, and pseudogap state, respectively. The onset curve for the spin Auctuation (T = T0 ) and that for the pseudogap formation (T = T ∗ ) show the typical cross-over temperatures.

hole-doping ( ¿ 0). Then, there appears the metallic phase with the superconducting ground state. The transition temperature Tc takes its maximum value (∼ 100 K) at the optimally-doped region ( ∼ 0:15) and decreases in the over-doped region. Here it is stressed that anomalous behaviors in the normal-state properties are signiMcant from the optimally-doped to the under-doped region, although the meaning of “anomalous behavior” will be discussed in detail in the following subsections. As indicated by thin and thick dashed curves in Fig. 1, there exist two characteristic temperatures, determined from the several experimental results. One is T0 , shown by the thin dashed curve, a temperature at which the AF spin correlation begins to develop with decreasing temperature. For instance, the NMR 1=T1 T and Hall coeDcient RH increase from the temperature T0 . Another characteristic temperature is T ∗ , indicated by the thick dashed curve. This is a temperature for the onset of the opening of pseudogap, observed in ARPES, NMR 1=T1 T , tunnelling measurements, and so on. The enormous investigations have been dedicated to the clariMcation of the pseudogap because this issue has been regarded as a central problem in high-Tc superconductors. Note that these experiments have indicated some similarities between the pseudogap and superconducting gap. We consider that the SC Auctuation becomes apparent around T = T ∗ . As shown in Section 4, our interests on the pseudogap phenomena are concerned with this lower cross-over curve. Note also that somewhat uncertainty may be included in the deMnitions of T0 and T ∗ , since two curves just denote the cross-over temperatures, not the phase transition. Nevertheless, we believe that the above classiMcations are useful to understand the anomalous properties in a coherent way. As already mentioned, the electron-doped systems ( ¡ 0) are also superconducting. However, some diCerent features from hole-doped ones are observed, as explained in the previous subsection. For instance, the AF order is robust for the carrier doping in these systems and superconductivity occurs in the narrow doping region with relatively low transition temperature. Especially, in the

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Fig. 2. The experimental data of the NMR 1=T1 T [55].

normal metallic state, the pseudogap phenomena and the under-doped region have not been observed in electron-doped systems. In Sections 3 and 4, we provide our understanding of the phase diagram including this particle–hole asymmetry. 2.1.3. Magnetic properties As is well known, NMR and neutron scattering measurements are the powerful methods to investigate spin correlations. These experiments have played crucial roles to reveal that the magnetic properties in the high-Tc cuprates are “anomalous” in many aspects. Here let us summarize these anomalous behaviors in the normal state. First we emphasize that the pseudogap phenomenon has been Mrst discovered in the NMR measurements [55]. As shown in Fig. 2, the spin–lattice relaxation rate over temperature 1=T1 T exhibits the peak around T = T ∗ and it begins to decrease below T ∗ . This is a typical pseudogap behavior observed in 1=T1 T , indicating the suppression of magnetic excitations in the low-energy part. The detailed investigations have subsequently observed this phenomenon in most of the under-doped compounds [55–64] and the doping dependence of T ∗ is found to be consistent with Fig. 1. In the conventional metal well described by the Fermi-liquid theory, 1=T1 T should be a constant for the temperature higher than Tc . On the other hand, in high-Tc cuprates, 1=T1 T is not constant for T ¿ Tc . Rather, the Curie–Weiss law 1=T1 T ˙ (T + )−1 has been observed for T ¿ T ∗ [27,65], where is a Weiss temperature. This fact clearly indicates that strong spin Auctuations exist in high-Tc cuprates. Note that this spin Auctuation has anti-ferromagnetic nature [66], since the ratio (1=T1 T )=K 2 is larger by one order than the usual value of the Korringa law, where K is the NMR Knight shift. For Tc ¡ T ¡ T ∗ , as explained above, 1=T1 T begins to decrease even before the system becomes superconducting. Since those behaviors are signiMcantly diCerent from the results in the conventional Fermi-liquid theory, they are regarded to be anomalous normal-state properties. Note that the Curie–Weiss law itself can be explained by the self-consistent renormalization (SCR) theory [8], which is one of the spin Auctuation theory. However, it is diDcult to understand the anomalous temperature dependence including pseudogap behavior only from the spin Auctuation theory.

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13

Fig. 3. The experimental data of the Knight shift [64].

The NMR Knight shift K is proportional to the uniform spin susceptibility K ˙ (0; 0). Although the Knight shift satisMes the Korringa-relation K 2 ˙ 1=T1 T in usual metals, in high-Tc cuprates, it shows a diCerent temperature dependence from the Korringa-relation. Namely, the Knight shift gradually decreases below T0 [57,67,68]. This behavior has been observed also in the uniform susceptibility [69,70]. Concerning the pseudogap, the Knight shift also shows an anomaly around T ∗ ; the decrease becomes rapid below T ∗ , as observed in Fig. 3 [64]. The spin-echo decay rate 1=T2G measures the momentum summation of the square of the static spin susceptibility (see Eq. (93)). The experiments in early years have shown that the NMR 1=T2G keep increasing below T ∗ [58,59,61]. However, the recent experiments have revealed the diCerent behavior depending on the number of CuO2 layers [60,62,63,71,72]. This diCerence is probably attributed to the eCect of the interlayer coupling [63,71]. It is important to note that the 1=T2G decreases below T ∗ in the single layer compounds [60,62,63,71]. Then, the decrease of 1=T2G is weaker than that of 1=T1 T . Now we turn our attention to the neutron scattering experiments, in which the wave vector of the spin Auctuation can be directly observed by measuring the form factor in proportion to the imaginary part of dynamical spin susceptibility, Im (q; !). In high-Tc cuprates, the peak of the spin–spin correlation function is located around q = (; ) [73]. The peak position depends both on the temperature and doping concentration, but interestingly enough, it can be incommensurate as q = (; ) in some materials [74–80]. The interesting temperature dependence of this incommensurability has been observed and discussed in relation with a stripe order [77,78], which is found in La-based compounds around = 1=8 [81]. We will comment on this problem in Section 4.3.3. Note that in the spin Auctuation theory, the detailed structure in spin correlation function is not important, since the main part of the magnetic excitation always locates around q = (; ). The pseudogap is clearly observed also in the spectral weight of the spin Auctuation, which is directly measured by the neutron scattering experiments. The spectral weight in the low frequency part is suppressed and at the same time, it is shifted to the high frequency part, as shown in

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Fig. 4. The experimental data of the dynamical spin susceptibility at the anti-ferromagnetic wave vector Im (Q; !) [82].

Fig. 4 [82]. This frequency dependence is anomalous in comparison with the results of the conventional Fermi-liquid theory and also those of the spin Auctuation theory. As will be discussed in Section 4.3.3, this behavior means that a new energy scale appears in the pseudogap state. We will attribute it to the energy scale due to the superconductivity. 2.1.4. Transport properties In the previous subsection, anomalous behaviors in magnetic properties have been brieAy reviewed, but transport properties are also anomalous in comparison with the conventional Fermi-liquid theory. In this subsection, let us see anomalous features of in-plane electric resistivity ab , Hall coeDcient RH , and c-axis resistivity c . It is well known that the Fermi-liquid theory predicts the T 2 -law for the electric resistivity in the low-temperature region, but in optimally- and under-doped region of high-Tc cuprates, it has been observed that ab ˙ T , [26,83] as shown in Fig. 5. However, when the amount of doping concentration is further increased, the temperature dependence in the in-plane resistivity gradually changes and eventually the T 2 -law recovers in the over-doped region [10]. This crossover behavior in the temperature dependence of the in-plane resistivity is well described by the nearly anti-ferromagnetic Fermi-liquid theory, as will be explained in Section 4.3.4 [84–86]. In the pseudogap state, a characteristic behavior can be observed in the in-plane resistivity. Namely, it changes its slope at T ∗ and slightly deviates downward [87–89]. This rather weak deviation has been one of the puzzling issues among anomalous normal-state properties of high-Tc cuprates. The transport properties are generally insensitive to the pseudogap in comparison with the magnetic properties [26]. Therefore, the pseudogap was sometimes called “spin gap”, in the sense that the gap occurs only in the spin excitations. These properties were complicated and diDcult to understand, but it is now consistently explained by including simultaneously the spin and superconducting Auctuations, as will be described in Section 4.3.4 [90].

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15

Fig. 5. The experimental data of the in-plane resistivity [89].

Fig. 6. The experimental data of the Hall coeDcient [87].

In the conventional Fermi-liquid, the Hall coeDcient RH is independent of T , while in high-Tc cuprates, it strongly depends on the temperature, as show in Fig. 6 [26,87,91,92]. Moreover, in the under-doped region, the Hall coeDcient takes much larger value than that expected from the band-structure calculation results. As seen in the in-plane resistivity, these anomalous behaviors

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gradually change to those in the conventional Fermi-liquid in the over-doped region. The enhanced Hall coeDcient was sometimes interpreted as an evidence for the low-carrier density n, because the relation RH ˙ 1=n is derived in the isotropic electron system. However, this interpretation is just a superMcial expectation in the anisotropic system such as the Hubbard model on the square lattice (see Section 4.3.4). In fact, the enhanced Hall coeDcient has been explained within the nearly AF Fermi-liquid theory [93,94]. In the pseudogap state, the Hall coeDcient remarkably deviates downward and decreases with temperature [87,88,91,95,96]. The anomaly around T ∗ is much clearer than that in the resistivity. A theory based on the d-wave superconducting (SC) Auctuation alone expects the enhancement of the Hall coeDcient [97]. We will also show that this discrepancy can be resolved by the simultaneous consideration of AF and SC Auctuations (Section 4.3.4). Another interesting property of high-Tc cuprates is the strongly anisotropic transport, originating from the layered structure. Typically the ratio of the c-axis and in-plane resistivity, c =ab , increases in the under-doped and/or low-temperature region [26,98]. Moreover, the pseudogap remarkably enhances the c-axis resistivity [98]. This response to the pseudogap, qualitatively diCerent from that of the in-plane resistivity, can be explained by considering the d-wave SC Auctuation combined with the characteristic band structure [97,99] (see Section 4.3.4). The c-axis optical conductivity c (!) shows no Drude peak in the under-doped region [100–102], which is consistent with the incoherent nature of the c-axis resistivity. As will be explained in Section 4.3.4, owing to the momentum dependence of the c-axis hopping matrix, the coherent transport along the c-axis is disturbed by the pseudogap. The gap structure appears in c (!) in the pseudogap state. This pseudogap smoothly changes to the SC gap and this behavior indicates the close relation between the pseudogap and the SC gap. 2.1.5. Spectroscopy The ARPES directly measures the single-particle spectral weight at the selected momentum. Therefore, many pieces of clear information on the single-particle excitation can be obtained, while some cares are required for the resolution. Especially, the ARPES has provided us an important suggestion on the pseudogap phenomena [103–106], such as the leading edge gap observed above Tc , as shown in Fig. 7. This experimental result indicates the suppression of the single-particle spectral weight near the Fermi energy, which clariMes following two important natures of the pseudogap: (i) The momentum dependence of the pseudogap is similar to that of the SC gap, i.e., the pseudogap has the dx2 −y2 -wave form. (ii) The magnitude of the pseudogap does not change through the superconducting transition. Below Tc , the coherent quasi-particle peak appears at the gap edge, and the gap structure becomes sharp with keeping its magnitude. These results have clearly suggested the close relation between the pseudogap phenomena and superconductivity, leading to the pairing scenarios in which the pseudogap is a precursor of superconductivity (see Section 4.1). Note that the qualitatively same suggestion has been obtained from the tunneling spectroscopy which measures the electronic density of states (DOS). The suppression of the DOS near the Fermi level is observed above Tc [107–109]. The magnitude of the pseudogap in the DOS is nearly the same (slightly larger) as that of the SC gap. Again we observe that the gap structure becomes sharp below Tc .

Y (d)

c b a Γ

Midpoint (meV)

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M

17

(e)

20 10

a b c

0 -10 0

50

100

150

T (K)

180 150 120 T(K)

95

70

(a)

(b)

50

0

(c)

50

0

14 50

0

Binding energy (meV) Fig. 7. The results of the ARPES [105].

The measurements of the SC gap at T Tc have revealed that the SC gap increases in the under-doped region in spite of the decrease in Tc [89,110]. If we ignore quantum Auctuations, the SC gap at T = 0 is in proportion to the transition temperature in the mean Meld theory TcMF , which is increased with the decrease of the amount of doping even in the under-doped region. It is expected that the thermal SC Auctuation suppresses Tc in the under-doped region, since it should become strong with the decrease of . Concerning the cross-over around T = T0 , both the tunneling spectroscopy [107–109] and the ARPES [111] have observed the slight and broad suppression of the DOS from T = T0 . This suppression is called “large pseudogap” and its energy scale is 3– 4 times larger than the SC gap. The large pseudogap should not be attributed to the precursor of superconductivity, and may have somewhat magnetic origin. The pseudogap below T ∗ is sometimes called “small pseudogap” in contrast with the large pseudogap. 2.1.6. Magnetic :eld penetration depth Finally in this subsection, let us discuss the magnetic Meld penetration depth which is identical to the London penetration depth L in the type II limit. This is not the quantity in the normal state, but it indicates an interesting property in the SC state. The doping dependence of the magnetic Meld penetration depth at T = 0 is known as “Uemura plot” [112,113]. The London constant = 1=4L2 is roughly proportional to the hole-doping and Tc ( ˙ ˙ Tc ) in the under-doped region. This constant shows a peak around the optimally-doping and decreases with doping in the over-doped

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region [114]. The London constant is sometimes expressed by the “superAuid density” ns as = ns =m∗ , where m∗ is the eCective mass of carrier. We focus on the London constant instead of the superAuid density in order to avoid any confusion, since it will be shown that the superAuid density is not related to the electron density. The London constant corresponds to the stiCness of the superconducting phase variable, which is generally related to the SC Auctuation in the ordered state. The Uemura plot exhibits that the phase Auctuation is softened in the under-doped region. It has been proposed along this line that the phase disordered state is a possible pseudogap state [115]. The temperature dependence of the London constant has also attracted much interests. The London constant exhibits T -linear dependence as (T ) = (0) − aT in the low-temperature region [39,113,116], which is a characteristic behavior of the d-wave superconductor. This T -linear law reAects the nodal quasi-particles around k = (=2; =2). Interestingly, the coeDcient a is almost independent of the doping concentration in sharp contrast to the drastic change of (0) [113]. These anomalous behaviors have much stimulated theoretical insights [117,118]. In order to provide a systematic understanding for the anomalous behaviors, we will explain the microscopic derivation of the London constant based on the Fermi-liquid theory, as shown in Appendix A [119,120]. Then, it is emphasized that the Fermi-liquid correction as well as the reduced symmetry in the square lattice plays an essential role [118]. The c-axis London constant is much smaller than the in-plane one and the anisotropy c = is reduced with under-doping [113,116]. These behaviors are consistent with the transport properties in the normal state. The power of the temperature dependence of c is larger than unity in the low-temperature region [113,116]. The origin of this nature is in common with the incoherent c-axis conductance in the pseudogap state. Note that in the clean limit T 5 -law is expected, but sample dependence is frequently observed, probably owing to the eCect of disorder. The London constant just below Tc shows a rapid growth rather than that obtained in the BCS theory [116,121]. This is regarded as an appearance of the critical Auctuation. The critical behavior of the 3D-XY universality class has been conMrmed [121]. This clear appearance of the SC Auctuation supports the pairing scenario on the pseudogap phenomena. The growth of c below Tc is more rapid than that of [116]. This is also because of the momentum dependence of the inter-layer hopping matrix. 2.2. Organic superconductor As mentioned in the introductory section, the discovery of organic superconductor (TMTSF)2 PF6 [5] has made a great impact on the community of superconductivity. The intensive exploitation in this Meld has revealed a universality of superconducting phenomena and attached much interests on the physical aspects of the organic materials [122]. Among them, one of the most interesting superconductors is -type (BEDT-TTF)2 X. In the following, it is expressed as -(ET)2 X for an abbreviation. These compounds have quasi-two-dimensional electronic structures, which have been conMrmed by the Shubnikov–de Haas experiments [123] and by the strong anisotropy in the electronic transport [124]. The conduction band is mainly constructed from ET molecules. A simpliMed model for the electronic state can be constructed by noting the molecular -orbitals (see Section 3.3.1), although the ET molecule has a complicated structure. -(ET)2 X compounds are one of the central objects in this area, because of their typical features as SCES. The typical phase diagram is shown in Fig. 8. We have two ordered phases by tuning the

19

(NCS)2

2

[N(CN)2 ]Br,h8

[N(CN) ]Br,d8

2

[N(CN) ]Cl

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T PI

PM

AFI

SC

P Fig. 8. Schematic phase diagram of -(ET)2 X compounds.

pressure P. One is the AF insulating state in the lower pressure region and the other is superconducting phase in the higher pressure region [125,126]. It should be noticed that the superconductivity occurs when the AF order disappears. The transition temperature decreases with increasing P. It is true that this phase diagram includes similar aspects to that of high-Tc cuprates, but several diCerences are observed. First, the phase transition from the AF to the SC state is the Mrst order. Second, the carrier number in the conduction band is always half-Mlling per dimer, independent of the control parameter P. The pressure widens the band width W and thus, it controls the parameter U=W . The decrease in U=W by the pressure has been conMrmed by many experimental facts. For instance, the resistivity becomes smaller together with the superconducting Tc [125–127]. Therefore, the Mott transition in these compounds is regarded to be “band width controlled”, which is contrasted with the “Mlling controlled” Mott transition, as observed in cuprate superconductors [128]. A series of these compounds have diCerent properties at ambient pressure due to the kinds of anions X: -(ET)2 Cu[N(CN)2 ]Cl is the AF insulator [129]. The deuterated -(ET)2 Cu[N(CN)2 ]Br is located on the boundary between the two phases [130]. -(ET)2 Cu[N(CN)2 ]Br and -(ET)2 Cu(NCS)2 show the superconductivity at the ambient pressure. This systematic change due to X is regarded as an eCect of the chemical pressure, as described in the phase diagram of Fig. 8. It is predominantly believed with a few objections that the d-wave superconductivity occurs in these compounds like high-Tc cuprates. NMR experiments again have played an important role in the identiMcation of the pairing symmetry [131–133]. The measurements below Tc have shown (i) no coherence peak, (ii) the decrease of the Knight shift, and (iii) T 3 -law of the NMR 1=T1 in the low-temperature region. From these results, the singlet pairing with line node has been suggested. The same conclusion is obtained by the measurement of the electronic speciMc heat, which is proportional to T 2 at low temperature [134]. These results indicate that the pairing symmetry is not the s-wave, but probably the d-wave.

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The highest Tc among -(ET)2 X compounds is about 13 K. From the theoretical point of view, this value is the same order as that of cuprate superconductors, when Tc is scaled by the band-width. In general, organic materials have smaller band-width by an order, because they are constructed from the molecular orbitals. The scaling between Tc and W have suggested a similar pairing mechanism to the high-Tc cuprates. We will review in detail the theoretical results in Section 3.3. Finally, we note that the similarity between organic and high-Tc superconductors should be extended to the anomalous properties in the normal state. In particular, the pseudogap has been also observed in the NMR 1=T1 T [135,136] with T ∗ ∼ 50 K. However, it is an important diCerence that T ∗ is much higher than Tc and the electronic state is almost incoherent above T ∗ . Thus, the diCerent nature around T ∗ is expected. As will be discussed in Section 4.2.4, however, the similar properties in the electronic state indicate the manifestation of the pseudogap with the same origin. Recently, Kanoda’s group has given a clear understanding on this problem [137] by measuring the magnetic Meld dependence of the NMR 1=T1 T ; the SC Auctuation appears from the new cross-over temperature Tc∗ which is between Tc and T ∗ . The electronic state below Tc∗ is regarded as the pseudogap state induced by the SC Auctuation as in the under-doped cuprates. The details will be discussed in Section 4.2.4. 2.3. Sr2 RuO4 Superconductivity in Sr 2 RuO4 has been discovered by Maeno et al. in 1994 [3] and the intrinsic Tc is now considered to be as large as 1:5 K in the high purity sample. This compound possesses the same crystal structure as La2−x Sr x CuO4 , one of the high-Tc superconductors, and it similarly has the quasi two-dimensional nature. For example, the resistivity exhibits a large anisotropy; the ratio c =ab is in the order of several hundreds [3,138]. The quantum oscillation measurement has also clearly shown the quasi-two-dimensional Fermi surfaces [139]. Then, RuO2 layers are expected to be essential for the metallic behavior and superconductivity, as CuO2 layers in high-Tc cuprates. In contrast to cuprates, Sr 2 RuO4 is considered to be an ideal two-dimensional Fermi-liquid [138], since there is no anomalous behavior in the normal state. Since Tc is much lower than that of cuprates and -(ET)2 X, the superconductivity is easily destroyed by the small perturbation or disorder. For ruthenate it is diDcult to Mnd, at least at present, a well-deMned controlling parameter for the appearance of the superconductivity such as doping in cuprates. Thus, our interest is here focused only on the superconductivity in Sr 2 RuO4 . According to the Mrst-principle band-structure calculations [140,141], it has been clariMed that the electronic states near the Fermi level mainly consist of the Ru4d” orbitals, although the Ru4d” and O2p orbitals hybridize with each other. Since it is expected that electrons strongly correlate through Coulomb interactions at Ru sites, Sr 2 RuO4 belongs to a class of SCES. In fact, the Mott insulating state has been found in the related compound Ca2 RuO4 [142], suggesting the importance of strong correlation eCect. Compared to the other compounds focused in this review, Sr 2 RuO4 has somewhat diCerent nature both in the electronic structure and in the superconducting properties. One interesting issue is that this compound is a multi-band system. The quantum oscillation measurement has shown three quasi-two-dimensional Fermi surfaces, which are deMned as ', (, and ) sheets [139]. Note that the observed Fermi surfaces are in good agreement with the Mrst-principle band-structure calculation results [140,141] and the recent ARPES measurement [143]. Another remarkable diCerence from

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21

cuprates and -(ET)2 X is the electron Mlling. Since the valence of ruthenium ion is Ru4+ , four electrons occupy Ru-site on the average. Again according to the quantum oscillation measurement, about 4=3 electrons are included in the ) band and remained part is included in ' and ( bands. Thus, it is regarded that this system is far from the half-Mlling. The most outstanding and interesting diCerence is the pairing symmetry, which has recently been clariMed to be spin-triplet due to excellent experiments [144]. The most important experimental evidence suggesting the spin-triplet pairing has been obtained by NMR [4,145]. Ishida et al. have measured the 17 O-NMR and Ru Knight shift by applying the magnetic Meld parallel to the ab-plane, and observed no suppression in the spin susceptibility below Tc [4,145]. This result excludes the possibility of the spin-singlet pairing, and at the same time, indicates the d-vector along the z-axis. ˆ The d-vector is an usual expression for the internal degree of freedom, which is an interesting subject in the triplet superconductivity [12,146]. Recent inelastic polarized neutron scattering experiment has suggested the same results [147]. Now the spin-triplet pairing in ruthenate has been experimentally conMrmed. Slight portion of non-magnetic impurities drastically suppresses the superconductivity [148], in sharp contrast to the impurity eCects in conventional s-wave superconductors. The NMR and NQR relaxation rates exhibit no coherence peak just below Tc [149,150]. The ,SR measurement has shown that an internal magnetic Meld is spontaneously turned on below Tc [151], indicating that the time-reversal symmetry is broken in the SC state of ruthenate. From this result, the chiral state dˆ = (kx ± iky )zˆ has been suggested [152]. The temperature dependence of the critical current in Pb/Sr 2 RuO4 =Pb junction is also consistent with the p-wave pairing state [153]. Theoretically, Rice and Sigrist pointed out a possibility of spin-triplet superconductivity, immediately after the discovery of superconductivity in Sr 2 RuO4 [154]. Their insights have been based on the following facts. First some Fermi-liquid parameters are similar to those of 3 He, which is conMrmed to be a spin-triplet p-wave superAuid [146]. Second the three-dimensional analogous compound SrRuO3 exhibits the ferromagnetism with a Curie temperature TC = 160 K. They have considered that the Hund’s rule coupling among Ru4d” orbitals stabilizes the spin-triplet pairing rather than the spin-singlet one. Although any microscopic justiMcation for above two insights has not yet been obtained up to now, their excellent prediction itself has obtained a great success. At the present stage, the theoretical interests on Sr 2 RuO4 are focused on the two fundamental aspects of superconductivity, namely the pairing symmetry and the pairing mechanism. Concerning the pairing symmetry, the origin of the power-law behaviors is a challenging subject. As is mentioned above, the chiral state without time-reversal symmetry is expected for the internal degree of freedom. Then, assuming the simple momentum dependence of the SC gap, for instance, -(k) ˙ sin kx [155], the excitation gap opens on the whole Fermi surface. On the contrary, the gap-less power-law behaviors, suggesting the existence of the line node, have been observed in common among the several experimental results on the speciMc heat [156], NMR 1=T1 T [150], magnetic Meld penetration depth [157], thermal conductivity [158,159], and ultrasonic attenuation rate [160]. We should note that only the point node is derived from the symmetry argument, even if the three-dimensional degree of freedom is taken into account [161]. Thus, if we assume the chiral state, the line node should appear only accidentally. Namely, the theoretical proposal on this problem has to rely on somewhat an accidental reason. Among them, the three-dimensional f-wave symmetry [162,163] has been supported by the thermal conductivity measurement [159,164]. The more improved proposal based on the multi-band eCect has been proposed along this line [165]. The essential assumption of

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this proposal is that the zeros of the order parameter corresponding to the symmetry -(k) ˙ kx is parallel to the plane. When the zeros have a slope, the point node is expected. The pairing state assumed here is generally diDcult in view of the pairing mechanism, because the Fermi surface of Sr 2 RuO4 is clearly two-dimensional. Therefore, we consider that this problem should be resolved within the two-dimensional model. We will provide a diCerent proposal along this line on the basis of the microscopic theory (see Section 3.4.2). Then, the node-like structure appears in the ( band. This is also an “accident”, but derived from the microscopic model. The power-law behaviors can be explained, although a Mtting of the parameters is required. The advances in the theoretical studies on the pairing mechanism will be reviewed in Section 3.4.1. Since this issue is one of the main subjects of this review, we will discuss it in detail. Then, we show the results of the microscopic investigation based on the perturbation theory in Section 3.4.2. Subsequently, the microscopic mechanism of stabilizing the chiral state will be investigated in Section 3.4.3. The microscopic study on the internal degree of freedom becomes possible owing to the relatively simple electronic state. Finally we mention that such study was very diDcult previously because triplet superconductors were basically observed only in the heavy-fermion compound. We close this section by noting that the discovery of Sr 2 RuO4 has accelerated the theoretical understanding on the triplet superconductivity.

3. Microscopic mechanism of superconductivity In this section, we review the theoretical investigations on the mechanism of unconventional superconductivity based on the microscopic Hamiltonian such as the single- and multi-band Hubbard model. Before discussing the particular superconductors, in order to make this review article self-contained, in Section 3.1 we brieAy explain the theoretical tools which will be used in the following subsections. Readers who may not be interested in the theoretical formulation can simply skip this subsection. In Section 3.2, high-Tc superconductivity will be discussed in detail. The basic properties and typical results of microscopic theories will be reviewed. The application to the organic superconductor is discussed in Section 3.3. Then, the applicability of the microscopic theory for the molecular materials are clearly shown. In Section 3.4, the microscopic theory is extended to the triplet superconductivity in Sr 2 RuO4 , where the qualitatively diCerent results are derived from the characteristic electronic structure. 3.1. Dyson–Gor’kov equation V In the following subsections, the Dyson–Gor’kov equation and Eliashberg theory [166] are used to discuss superconductivity. This formulation is suitable for the diagrammatic techniques in the quantum Meld theory, although some approximations are usually needed for the actual calculations. However, this approach is eCective to clarify the physical picture and in principle, it is free from the Mnite size eCect, compared with the numerical methods. This subsection is devoted to the introduction V of the linearized Eliashberg theory, which is used to determine the superconducting transition. For simplicity, we show the explicit expressions only for the single-band case, but the extension to the multi-band system is straightforward.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

23

G

= F

=

+

Σ

Σ

+

+

∆

∆

Fig. 9. Diagrammatic representation of the Dyson–Gor’kov equation.

In the Dyson–Gor’kov equation, the superconducting state is described by introducing the normal and anomalous Green functions, symbolically expressed as G and F, respectively [17]. In the homogeneous system, they are deMned as ( † G(k; i!n ) = − d0ei!n 0 T0 ck (0)ck ; (1) 0

F(k; i!n ) = †

F (k; i!n ) =

(

0

0

d0ei!n 0 T0 ck↑ (0)c−k↓ ; (

† † d0ei!n 0 T0 c− k↓ (0)ck↑ ;

(2) (3)

where ck (0) = eH0 ck e−H0 with a Hamiltonian H , ck is an annihilation operator for electron with spin and momentum k, ( = 1=T , and !n = T (2n + 1) with an integer n is a fermion Matsubara frequency. The symbol · · · means the operation to take statistical average and T0 is an ordering operator with respect to 0. Note that the following formulation is common to the singlet and triplet pairing cases unless we explicitly mention. The Green functions are expressed by normal and anomalous self-energies through the Dyson– Gor’kov equation, diagrammatically expressed in Fig. 9. Readers can Mnd a clear derivation of the Dyson–Gor’kov equation in Ref. [17] and more general expression for the inhomogeneous state in Ref. [167]. The normal and anomalous self-energies are exactly obtained from the Gor’kov equation [167,168] in the homogeneous case through the Fourier transformation and the Dyson–Gor’kov equation is written in the matrix form as −1 (0) −1 G(k) F(k) -(k) G (k) − 2n (k) : (4) = F † (k) −G(−k) -∗ (k) −G (0) (−k)−1 + 2n (−k) Here, 2n (k) and -(k) are the normal and anomalous self-energies, respectively, G (0) (k) is the non-interacting Green function, given by G (0) (k) = [i!n − ”(k)]−1 , ”(k) is the one-electron dispersion energy, and k is a shorthand notation as k = (k; i!n ). Note that a chemical potential is included in ”(k) in our notation. The normal and anomalous self-energies can be expressed by the perturbation series, which is obtained in a usual manner [17]. Since an approximation is usually required for an explicit estimation, we will introduce three approximations such as third-order perturbation (TOP), random phase approximation (RPA), and Auctuation-exchange (FLEX) approximation, as will be explained later

24

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 Va

Va (b)

(a)

(c)

Fig. 10. (a) The one-loop approximation for -(k) in the electron–phonon system. The wavy line represents the phonon propagator. Diagrammatic representation of (b) the anomalous vertex and (c) the anomalous self-energy.

in detail. In the other derivation, 2n (k) and -(k) are derived from the functional derivatives using the Luttinger’s functional 3[G; F † ] [169,170] as 2n (k) =

1 3 ; 2 G(k)

-(k) = −

3 : F † (k)

(5)

Then, the variational conditions about the free energy 4 are satisMed as 4 4 = =0 : 2n (k) -(k)

(6)

If above conditions are satisMed, the calculation is called “conserving approximation” [171]. Note that the perturbation scheme does not necessarily satisfy the above conditions. For the conventional s-wave superconductivity, the anomalous self-energy is obtained by the electron–phonon coupling. Since the typical phonon frequency is smaller than the Fermi energy in conventional metals, the Migdal’s theorem holds and the vertex corrections can be ignored. Thus, the one-loop approximation shown in Fig. 10(a) is valid. In the case of unconventional superconductivity arising from electron correlations, the irreducible vertex Va (k; k ) in the particle–particle channel (Fig. 10(b)) is derived from the many-body eCects. In analogy with the electron–phonon mechanism, this vertex is regarded as the eCective interaction for the pairing. Thus, the anomalous self-energy, represented formally by the diagram in Fig. 10(c), is expressed by -(k) = − Va (k; k )F(k ) : (7) k

Here the summation is deMned as k =(T=N ) k; n , where N is the number of sites. It is considered that the unconventional superconductivity arises from the momentum dependence in the eCective interaction Va (k; k ). The theoretical search for the pairing mechanism is then reduced to the identiMcation of the eCective interaction. The expression for the anomalous self-energy Eq. (7) is a self-consistent equation, which is one of the mean-Meld equations. We can reproduce the result of the weak-coupling BCS theory [172], if we ignore the normal self-energy as well as the frequency dependence of Va (k; k ). The self-consistent equation is then transformed to the gap equation as tanh (E(k )=2T ) Va (k; k ) -(k) = − W(k ) ; ) 2E(k E(k) =

(8)

k

”(k)2 + W(k)2 :

(9)

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

Here we further assume the BCS approximation for the interaction as −V |”(k)|; |”(k )| ¡ !c ; Va (k; k ) = 0 otherwise ;

25

(10)

where !c is a cut-oC energy. Then, we obtain the well-known formula for Tc as Tc = 1:13!c exp (−1=V ) ;

(11)

where is the DOS at the Fermi level. V Also in the Eliashberg theory, the superconducting order is determined by the non-trivial solution of the self-consistent equation. In order to determine the critical temperature and corresponding pairing symmetry, the Dyson–Gor’kov equation is linearized with respect to W(k) as G(k)−1 = G (0) (k)−1 − 2n (k) ;

(12)

F(k) = |G(k)|2 -(k) :

(13)

where G(k) is the dressed Green function, explicitly written as G(k) = [i!n − ”(k) − , − 2n (k)]−1 . Here the chemical potential shift , is determined by the conservation for the particle number as (G(k) − G (0) (k))ei!n 8 = 0 ; (14) k

where 8 is a positive inMnitesimal. The self-consistent equation for -(k) is expressed as Va (k; k )|G(k )|2 -(k ) : -(k) = −

(15)

k

V This linearized equation is called the Eliashberg equation, which is valid just at T =Tc . The transition temperature is practically estimated by solving the eigenvalue equation Va (k; k )|G(k )|2 -(k ) : (16) e W(k) = − k

The maximum eigenvalue becomes unity, e =1, at T = Tc and the anomalous self-energy -(k) plays a role of the eigenfunction. The pairing symmetry is determined by the momentum dependence of -(k). Since the pairing state with maximum Tc is usually realized in the ground state, we can get V knowledge on the behaviors below Tc from the Eliashberg equation. The momentum dependence of the quasi-particle energy gap is approximately described by the absolute value |-(k)| at !n = Tc . Note that strictly the excitation gap -ex (k) is renormalized as -ex (k) = z(k)|-(k; -ex (k))|, where z(k) = (1 − 9 Re 2n (k; !)=9!|!=0 )−1 is the renormalization factor. V The Eliashberg theory includes the normal self-energy, which generally induces de-pairing eCects. In particular, the quasi-particle damping gives rise to the pair-breaking and suppresses Tc . Note that the de-pairing eCects are not expected to alter the pairing symmetry. The frequency dependence V of the eCective interaction represents the retardation eCect, which also reduces Tc . The Eliashberg theory is usually called “strong-coupling theory” in contrast to the weak-coupling theory in Eqs. (8) and (9). We should note that the “strong-coupling superconductivity” discussed in Section 4 has a diCerent meaning.

26

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

(a)

(b)

(c)

Fig. 11. Diagrammatic representation of the normal self-energy within the TOP.

In the speciMc calculations, the normal and anomalous self-energies are evaluated by using an approximation. For the convenience in the following sections, we summarize the expressions for 2n (k) and Va (k; k ) in several approximations. First we show the perturbation series within the third order in terms of U [18,19], which will be used in Sections 3.2.3, 3.3.2, and 3.4.2. The normal self-energy is separated into three terms as 2n (k) = 2n(2) (k) + 2n(3RPA) (k) + 2n(3VC) (k) : These terms are represented in Figs. 11(a–c), written as 2n(2) (k) = U 2 0 (q)G (0) (k − q) ; q

2n(3RPA) (k)

= U3

(17) (18)

0 (q)2 G (0) (k − q) ;

(19)

:0 (q)2 G (0) (q − k) ;

(20)

q

2n(3VC) (k) = U 3

q

where 0 (q) and :0 (q) are, respectively, given by 0 (q) = − G (0) (k + q)G (0) (k) ;

(21)

k

and :0 (q) =

G (0) (q − k)G (0) (k) :

(22)

k

Here q denotes a shorthand notation as q=(q; i4n ), where 4n =2Tn is a boson Matsubara frequency. Within the third-order perturbation, the eCective interaction in the singlet channel is expressed as Vas (k; k ) = U + Va(2) (k; k ) + Va(3RPA) (k; k ) + Va(3VC) (k; k ) ;

(23)

where the Mrst, second, third, and fourth terms are represented in the diagrams in Fig. 12(a), (b), (c)+(d), and (e)+(f)+(g)+(h), respectively. Explicitly, those terms are written as Va(2) (k; k ) = U 2 0 (k − k ) ;

(24)

Va(3RPA) (k; k ) = 2U 3 0 (k − k )2 ; Va(3VC) (k; k ) = 2U 3 Re G (0) (k + q)G (0) (k + q)[0 (q) + :0 (q)] :

(25)

q

(26)

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

(a)

(b)

(e)

(c)

(f )

(g)

27

(d)

(h)

Fig. 12. Diagrammatic representation of the anomalous self-energy within the TOP.

Note that the Mrst three terms in Vas (k; k ) are included in the RPA, while the last term is called “vertex correction”. The eCective interactions for the triplet channel are given by the same diagrams as Fig. 12. Note here that these diagrams are obtained in case of dˆ z, ˆ where dˆ is the d-vector [146], but the SU(2) symmetry ensures the same results for the other d-vector. The explicit expressions for the triplet channel are given by Vat (k; k ) = Va(2) (k; k ) + Va(3VC) (k; k ) ;

(27)

Va(2) (k; k ) = −U 2 0 (k − k ) ;

(28)

where

Va(3VC) (k; k ) = 2U 3 Re

G (0) (k + q)G (0) (k + q)[0 (q) − :0 (q)] :

(29)

q

Note that for the triplet channel, the third-order RPA terms cancel each other and only the vertex correction terms remain. Next let us show the expressions for the RPA, which corresponds to the partial summation, as shown in Figs. 13 and 14 [173,174]. The normal self-energy is given as 2n (k) = Vn (q)G (0) (k − q) ; (30) q

where Vn (q) is given in the RPA as 1 2 3 s (q) + c (q) − 0 (q) : Vn (q) = U 2 2

(31)

Here s (q) and c (q) are the spin and charge susceptibilities in the RPA, respectively, given by s (q) =

0 (q) ; 1 − U0 (q)

c (q) =

0 (q) : 1 + U0 (q)

(32)

The eCective interaction for the singlet and triplet channel is given by Vas (k; k ) = U +

3 2 1 U s (k − k ) − U 2 c (k − k ) ; 2 2

(33)

28

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

+

+

+.....

+

+...

Fig. 13. Diagrammatic representation of the normal self-energy within the RPA or FLEX approximation. The bare (dressed) Green function is used in the RPA (FLEX).

+ +

+

+

+...

+.....

Fig. 14. The eCective interaction within the RPA or FLEX approximation.

and 1 1 Vat (k; k ) = − U 2 s (k − k ) − U 2 c (k − k ) ; (34) 2 2 respectively. Because 0 (q) ¿ 0 at 4n = 0, the spin part generally gives larger contribution than the charge part. In particular, the contribution from the charge susceptibility is suppressed in the vicinity of the magnetic instability (0 (q) ∼ 1). The closeness to the magnetic instability is an implicit assumption for the RPA. Then, the RPA in the Hubbard model is a description for the spin Auctuation theory (Section 3.2.1). The FLEX approximation [22], which is one of conserving approximations, has been used very widely. In this paper, Section 3.2.4 will be devoted to the review of the FLEX approximation, but here we provide a short comment on the formulation. The FLEX approximation can be also considered as one of the modiMcations of the RPA, in the sense that the dressed Green function G is used in Eqs. (21), (22), (30)–(34), instead of the bare Green function G (0) . Then, the Green function, normal self-energy, spin and charge susceptibility are determined self-consistently. The numerical calculation is used to take a summation in these diagrammatic techniques. The Mgures in this review show the results with 128 × 128 points in the Mrst Brillouin zone and 2048

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

29

Matsubara frequency. In same cases, the calculations have been performed for smaller number of meshes, when we have conMrmed that there is no problem in accuracy of the calculations. Note that Tc should be zero in the strict two-dimensional system due to the Mermin–Wagner theorem, although the estimation of Tc has been frequently performed in two dimensions. This point will be brieAy discussed later in this article. 3.2. High-Tc cuprates 3.2.1. Overview A number of the pairing mechanisms have been proposed for high-Tc superconductivity. After intensive investigations over more than a decade, the magnetic mechanism based on the spin Auctuation theory has been accepted most predominantly. This mechanism has been Mrst discussed for the superAuidity in 3 He, where the ferromagnetic paramagnon mediates the p-wave pairing interaction [146,175]. Next it has been pointed out that the d-wave superconductivity is most favorable when the spin Auctuation is anti-ferromagnetic [173,174]. After the proposals on this mechanism for cuprates [28,176] and the detailed investigations on the Hubbard model [22,29,177], the signiMcant developments have been given by the phenomenological theory [178–185]. Subsequently, the microscopic theory using the FLEX approximation [22,177, 186–193] has clariMed the detailed and important properties. The calculated results have succeeded in the quantitative agreement on the transition temperature. Moreover, the various properties both in the normal and superconducting state have been explained [194,195], except for the pseudogap phenomena. Note that the dx2 −y2 -wave superconductivity is obtained also in the t–J Hamiltonian near the half-Mlling as a result of the variational Monte Carlo simulation [33,34], exact diagonalization [196], and Green function Monte Carlo simulation [197]. Then, the pairing mechanism should be classiMed into the magnetic one, since the t–J Hamiltonian directly includes the anti-ferromagnetic interaction. At present, the magnetic mechanism should be regarded as one of the limiting cases of the electronic mechanism, in which the momentum dependence of the residual interaction among the quasi-particles inevitably gives the superconducting ground state. This idea has been Mrst given by Kohn and Luttinger in 1965 [198]. They discussed the possibility of the non-s-wave pairing state in the three-dimensional fermion gas model within the second-order perturbation. This general concept is particularly important for the comprehensive understanding from the under-doped to the over-doped region. This is because the spin Auctuation theory loses its justiMcation in the over-doped region, where the spin Auctuation is not clearly observed but Tc remains substantially. The results of the perturbation theory [18,19] (see Section 3.2.3) have been quite instructive, in the sense that the roles of the RPA and non-RPA terms are clariMed. The spin Auctuation theory corresponds to the partial summation of the perturbation series. In other word, the RPA terms (including some renormalization) are included in the spin Auctuation theory. The perturbation theory has revealed that the RPA terms favor the dx2 −y2 -wave superconductivity in the Hubbard model near the half-Mlling, while it is suppressed by the non-RPA terms. Therefore, it is expected that the RPA terms play a major role in the pairing interaction, even far from the magnetic instability. Again, the perturbation theory has revealed that the corrections from the non-RPA terms are not important for the case of cuprates. This is a microscopic justiMcation why the concept of the magnetic mechanism survives in the over-doped region.

30

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

Roughly speaking, the perturbation theory is appropriate to the over-doped region, while the spin-Auctuation theory is appropriate to the optimally-doped region. Also in the under-doped region, the main pairing mechanism should be in common with that in the optimally-doped region, although the SC Auctuation remarkably reduces the transition temperature. In our understanding, the SC Auctuation is the origin of the pseudogap phenomena, which has been a challenging issue in the study of high-Tc cuprates. Therefore, we will describe in detail the theory including SC Auctuations in Section 4 in order to understand the under-doped region. In this section, we focus on the pairing mechanism in the optimally- and over-doped region. In Section 3.2.2, we provide a simple explanation on the dx2 −y2 -wave superconductivity induced by the AF spin Auctuation. The important scattering process for the pairing is explained. In Section 3.2.3, we review the results on the perturbation theory which is performed within the third order. The obtained results are qualitatively similar to those of the spin Auctuation theory. The corrections to the spin Auctuation theory are discussed. In Section 3.2.4, we show the results of the FLEX approximation, which is a microscopic description for the spin Auctuation theory. The reasonable Tc ∼ 100 K is obtained near the magnetic instability. We discuss the eCects of higher-order corrections in Section 3.2.5. Some justiMcations for the spin Auctuation theory will be shown. 3.2.2. Spin ?uctuation-induced superconductivity First let us explain the phenomenological theory on the spin Auctuation-induced superconductivity [178–181]. The eCective interaction Va (k; k ) is phenomenologically given in this theory. Here we use the weak-coupling theory for simplicity. This treatment is quantitatively insuDcient, but it is enough to grasp the basic idea of spin Auctuation-induced superconductivity. A simple form of the eCective Hamiltonian is described as † † HeC = ”(k)ck ck − g2 s (q)'( · ) ck+q' ck† −q) ck ck( ; (35) k

k;k ;q

where =(x ; y ; z ) are the Pauli matrices and g is the eCective coupling constant for the interaction exchanging the spin Auctuation. We denote the static spin susceptibility as s (q) = s (q; 0). The spin susceptibility s (q; 4) near the magnetic instability is phenomenologically expressed as [199,200] s (q; 4) =

'=2 ; 1 + =2 (q − Q)2 − i>q2−z 4

(36)

where = is the correlation length of the spin Auctuation, Q is the wave vector deMned as Q = (; ) (Q = (0; 0)) and z is the dynamical exponent z = 2 (z = 3) for the anti-ferromagnetic (ferromagnetic) case. A large value of = is a hypothesis of the spin Auctuation theory, which is expected around the magnetic instability. The strong enhancement of the static susceptibility around q ∼ Q essentially induces the unconventional superconductivity. The dissipation term > describes the time scale of the spin Auctuation. The diCusive dynamics of the spin Auctuation is a characteristic property in the normal state. This is not the case in the SC state or in the AF state. The 4-dependence induces the retardation eCect, but it is not included in the weak-coupling theory. In the weak-coupling theory, the gap equation is written as tanh (E(k )=2T ) -(k) = − -(k ) ; Vas; t (k − k ) (37) ) 2E(k k

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

31

g

χ(q)

g

Fig. 15. ECective interaction which corresponds to the single paramagnon exchange. The wavy line represents the propagator of the paramagnon.

where the pairing interaction V s; t is given as Vas (k − k ) =

3 2 g s (k − k ) ; 2

(38)

for the singlet pairing and 1 Vat (k − k ) = − g2 s (k − k ) ; 2

(39)

for the triplet pairing. This is a result in the level of the one-loop approximation for the eCective model Eq. (35). Here note that the eCective interaction Vas; t (k − k ) corresponds to the single paramagnon exchange, which is diagrammatically represented in Fig. 15. The irreducible vertex estimated from the quantum Monte Carlo simulation is in good agreement with the single paramagnon exchange at least in the high-temperature region [15,201]. This form of the eCective interaction can be derived from the RPA or FLEX approximation for the Hubbard model, when g is considered as the on-site Coulomb interaction U . In this sense, the RPA and FLEX approximations are regarded as a microscopic description of the spin Auctuation theory. An advantage of the phenomenological theory is its universality, which does not depend on the microscopic details. It should be considered that the renormalization from the high-energy excitation and the higher-order corrections are eCectively included in the phenomenological parameters. By integrating the momentum perpendicular to the Fermi surface, introducing the cut-oC energy !c and transposing the velocity v(k) = |9”(k)=9k| as the average on the Fermi surface, the transition temperature is obtained as Tc = 1:13!c exp(−1=|Vsc |) ; where the eCective coupling constant Vsc is given by d k d k -(k)Vas; t (k − k )-(k ) : Vsc = F

F

(40)

(41)

Here the gap function is normalized as F d k|-(k)|2 = 1. The integration F is performed along the Fermi surface. In actual systems, it is considered that the pairing symmetry with maximum Tc is stabilized. For example, the dx2 −y2 -wave superconductivity is stabilized when Vsc is attractive (Vsc ¡ 0) and

32

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

θ

Fig. 16. Typical Fermi surface of high-Tc cuprates. The order parameter of the dx2 −y2 -wave superconductivity has positive (negative) sign in the shaded (light) region. The scattering process exchanging the AF spin Auctuation is shown by the dashed line.

smallest for the gap function -(k) = -d (k) ˙ cos kx − cos ky . This is the case of high-Tc cuprates where anti-ferromagnetic spin Auctuations are active. We can easily understand this result from the typical Fermi surface, as shown in Fig. 16. The scattering process from k1 ∼ (; 0) to k2 ∼ (0; −) is strongly enhanced and attractive for the dx2 −y2 -wave symmetry (-d (k1 )Va (k1 − k2 )-d (k2 ) ¡ 0). It is essential that the order parameter changes its sign from k1 to k2 and the eCective interaction Va (k1 − k2 ) is enhanced around k1 − k2 ∼ Q. In fact, it has been shown that the dx2 −y2 -wave symmetry is most favorable in the weak-coupling theory [178–181]. After that, the quantitative estimation for the transition temperature was performed on the basis of the strong-coupling theory where the phenomenological parameters were determined from the results of the NMR and resistivity. Then, Tc was estimated to be a reasonable value as Tc ∼ 100 K [182–185]. The quantitative agreement between theoretical and experimental Tc ’s has strongly supported the validity of the spin Auctuation mechanism. Before closing this subsection, let us comment on the ferromagnetic spin Auctuation, where s (q) is enhanced around q = (0; 0). Here we consider the three-dimensional fermion gas model as a typical example. In this case, the triplet p-wave superconductivity is favored, since the eCective interaction in the triplet channel Eq. (39) is strongly attractive for the scattering process from k to k ∼ k. Note that the detailed band structure is not important in this case. It is widely believed that the ferromagnetic spin Auctuation stabilizes the Anderson–Brinkman–Morel state in the superAuid 3 He [146,175]. The estimation by using the phenomenological theory [202] and the FLEX approximation [203] shows that Tc in the ferromagnetic case is generally low compared with the anti-ferromagnetic case. The Mrst proposal for the pairing mechanism in Sr 2 RuO4 was superconductivity mediated by ferromagnetic spin Auctuations [204,205] (see Section 3.4.1). However, this naive expectation was denied by the experimental results which did not observe the ferromagnetic spin Auctuation [14]. This is not a mystery from the point of view on the electronic mechanism. In such a case, we should restart the discussion from more general framework, as will be shown in Section 3.4.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

33

3.2.3. Perturbation theory In the following subsections we explain the microscopic theories. First let us review the perturbation theory, which is a method for the systematic estimation of the eCective interaction, certainly justiMed in the weak-coupling region. The expansion parameter here is U=W , where U is the on-site Coulomb repulsion and W is the band width. The principle of the adiabatic continuity in the Fermi-liquid theory requires the regularity of the expansion. For instance, a good convergence has been conMrmed in the Kondo problem based on the Anderson Hamiltonian, while the expansion with respect to the Kondo exchange coupling J is singular in the s–d model [206]. Also in the periodic system, the perturbation method is expected to be useful to understand the qualitative natures in the weak coupling region U ¡ W , as long as any long-range order does not occur. Note that the low-dimensional system d ¡ 2 is beyond our scope in this review (d is the dimensionality). We can see the applicability of the perturbation theory in the case of d = 2 in Refs. [207,208]. As is mentioned before, the spin Auctuation theory can be described by the perturbation scheme. The phenomenological form of the eCective interaction in Eqs. (38) and (39) is derived from the partial summation of the RPA terms. We should note that the partial summation is sometimes dangerous as a microscopic estimation, since it is generally expected that RPA terms are considerably canceled by the neglected terms, e.g., vertex corrections. In contrast to the spin Auctuation theory, the perturbation theory has a well-deMned basis. Namely, all kinds of the terms are estimated on an equal footing. The contribution from the non-RPA terms Mrst appears in the third order. Therefore, it is expected that the third-order perturbation (TOP) theory clariMes the general tendency of the vertex corrections. The application of the perturbation theory to the estimation of Tc in high-Tc superconductors has been Mrst performed by Hotta. He has performed both the second-order [18] and third-order calculation [19] for the d–p model. In the following, we show the calculated results for the Hubbard model, because this is simpler than the d–p model and it provides the qualitatively same results. The Hubbard Hamiltonian is expressed as † H= ”(k)ck ck + U ni ↑ ni ↓ ; (42) k

i

†

where ni =ci ci at site i. The two-dimensional dispersion relation ”(k) is given by the tight-binding model for the square lattice as ”(k) = −2t(cos kx + cos ky ) + 4t cos kx cos ky − , ; t

(43)

where t and represent the nearest-neighbor and next-nearest-neighbor hopping amplitudes, respectively. In Sections 3.2 and 4, we take the energy unit as 2t = 1, indicating that the band width W is given by W = 4. If we choose W = 4 eV according to the band-structure calculation [209], the experimentally observed Tc = 100 K corresponds to Tc = 0:01 in our unit. The next nearest-neighbor hopping, which corresponds to the p–p hopping in the d–p model, is needed to reproduce the typical Fermi surface of cuprates. The reasonable value for t is considered to be t =t = 0:1– 0.4, leading to the Fermi surface consistent with the experimental observation [45,210–216] and the band-structure calculation [209]. V Following the Eliashberg theory (Section 3.1), the superconducting transition temperature is V determined by solving the Eliashberg equation Eq. (16). Within the TOP, the normal self-energy is estimated as Eqs. (17)–(20), and the eCective interaction for the singlet channel is estimated as

34

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.03 TOP TOP(P) TOPWOVC TOPWOVC(P) SOP(P) RPA(P)

Tc

0.02

0.01

0.00

1

2

3

4

U/t

Fig. 17. Tc in the various approximations for t =t = 0:15 and = 0:1.

1.5

10% 15% 19%

χ 0 (q)

1.0

0.5

0.0 (0,0)

(0,π)

(π,π)

(0,0)

q

Fig. 18. Irreducible susceptibility 0 (k) at the 10%, 15%, and 19% doping, respectively, for t =t = 0:15 and T = 0:01.

Eqs. (23)–(26). It should be again noted that the second-order term Va(2) (k; k ) and a part of the third-order terms Va(3RPA) (k; k ) contribute to the RPA terms. The non-RPA terms Va(3VC) (k; k ) exist in the third order, leading to the lowest-order corrections to the spin Auctuation theory. In the TOP, the most favorable pairing symmetry is the dx2 −y2 -wave around the half-Mlling. Thus, we show only the results for the dx2 −y2 -wave symmetry in the following. In Fig. 17, Tc is depicted as a function of U . For comparison, we also show the results for the second-order perturbation (SOP), third-order perturbation without vertex correction (TOPWOVC), and RPA. The thin curves with labels “P” in Fig. 17 denote the results in which the normal self-energy is simply ignored. We can see that the suDciently high Tc = 0:01–100 K is obtained in the moderate coupling region. Here the attractive interaction in the dx2 −y2 -wave channel is mainly from the RPA terms. In Fig. 18, we show the irreducible spin susceptibility, which exhibits the peak around (; ). Then, the momentum dependence of the RPA terms is moderate, but qualitatively similar to the

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

35

ImΣ(kF ,0)

0.000

-0.005

-0.010

-0.015 0.0

SOP TOPWOVC TOP

0.5

θ

1.0

1.5

Fig. 19. The imaginary part of the self-energy on the Fermi surface Im 2nR (kF ; 0) for t =t = 0:15; = 0:1; U=t = 3, and T = 0:01. The horizontal axis is shown in Fig. 16. The analytic continuation from the Matsubara frequency to the real frequency is carried out by the PadVe approximation [222] through this review.

phenomenological theory (see Eqs. (36) and (38)). Therefore, the same scattering process as shown in Fig. 16 mainly contributes to the dx2 −y2 -wave superconductivity. It should be stressed that so strong enhancement of the spin Auctuation is not necessary for the appearance of superconductivity. The perturbative renormalization group theory has consistently expected the dx2 −y2 -wave superconductivity arising from the same scattering process [30–32,217–221], where the scattering amplitude from (k1 ; −k1 ) to (k2 ; −k2 ) is enhanced in the low energy eCective action. The comparison between the TOP(P) and the TOPWOVC(P) shows that the vertex correction terms suppress the value of Tc . Thus, it is generally expected that the RPA terms are considerably cancelled by the vertex correction terms. However, it is understood from the comparison between the TOP(P) and SOP(P) that the RPA terms overcome the vertex corrections: The third-order terms totally enhance the dx2 −y2 -wave superconductivity. In other words, the RPA-terms are not completely suppressed by the vertex corrections even far from the magnetic instability. Thus, we conclude that the vertex correction is not so severe in this case. Qualitatively the same mechanism as the spin Auctuation-induced superconductivity is expected in the weak coupling region, which corresponds to the over-doped cuprates. Simultaneously, this continuity conMrms the applicability of the spin Auctuation theory in the optimally-doped region. Next we discuss the eCect of the normal self-energy. We can see from Fig. 17 that the value of Tc is reduced by the de-pairing eCect. Because the reduction factor is about a half in the TOP, the same order of the magnitude of Tc still remains, even if the de-pairing eCect is considered. The realistic value Tc = 0:01–100 K is found around U=t = 3:2, where U=W = 0:4. On the other hand, Tc is remarkably reduced in the TOPWOVC, since the RPA terms intensify each other and give rise to a large self-energy. Indeed, the RPA terms in the normal self-energy are considerably compensated by the vertex correction. This is the reason why the reduction of Tc is moderate in the TOP. The third-order terms in the normal self-energy completely cancels each other in the particle–hole symmetric case, namely 2n(3RPA) (k) + 2n(3VC) (k) = 0 for t = 0 and = 0. The remained contribution in the particle–hole asymmetric case rather reduces the quasi-particle damping (see Fig. 19). As a result, the value of Tc in the TOP can be higher than that in the TOPWOVC owing

36

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 t’/t=0.0 t’/t=0.15 t’/t=0.25

0.04

Tc

0.03

0.02

0.01

0.00 -0.2

-0.1

0.0

0.1

0.2

0.3

δ Fig. 20. The doping dependence of Tc calculated by the TOP for U=t = 4. The normal self-energy is ignored for simplicity.

to the de-pairing eCect. Thus, it is generally expected that the normal self-energy is overestimated in the spin Auctuation theory, and thus, the magnitude of Tc is underestimated. Finally, we show the doping dependence of Tc in Fig. 20. For simplicity, here we ignore the normal self-energy, which is not important to see the qualitative behavior within the TOP. If t =t = 0, Tc takes the maximum value just at half-Mlling, because both the staggard susceptibility 0 (Q) and the electronic DOS becomes largest at = 0. By introducing the next nearest-neighbor hopping, the peak position is shifted to the hole-doped region. This is mainly due to the fact that the DOS takes its maximum value in the hole-doping side, since the Fermi surface crosses the van Hove singularity. These features will again appear in the FLEX approximation. 3.2.4. FLEX approximation It is considered that the spin Auctuation plays an important role in the optimally- and under-doped region, where the strong enhancement of the spin Auctuation is observed [27,65,66,73]. Since the perturbation theory is not suDcient to describe the strong spin Auctuation, some approximation beyond the TOP is required. A simple microscopic theory on the spin Auctuation is the RPA, but the tendency of the magnetic order is seriously overestimated in the RPA. Thus, the strong spin Auctuation in the quasi-two-dimensional systems is not correctly described, unfortunately. Among the several modiMcations of the simple RPA, Auctuation-exchange (FLEX) approximation has been used most widely. This approximation includes the renormalization of the spin Auctuation within the one-loop order, but the theory is surprisingly improved at this level. The magnetic properties in the nearly AF Fermi-liquid [178–181,194,195] are appropriately reproduced, since the mode coupling eCect is partly included [194]. For instance, the Curie–Weiss law in the NMR 1=T1 T is obtained in the wide temperature region, as shown in Fig. 60 (see also [188]). Applying the FLEX approximation to the Hubbard model [22,177,186–189] or d–p model [190–193], the doping dependences are properly reproduced at least from the over- to optimally-doped region. Thus, the results of the FLEX approximation are reviewed in this subsection and the following discussion is complementary with the perturbation theory in Section 3.2.3. The higher-order corrections beyond the FLEX approximation will be discussed in the next subsection.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

37

It should be mentioned that the accuracy in the quantitative estimation from the microscopic level is still questionable, because the FLEX approximation is also a partial summation of the perturbation series. However, we consider that the FLEX theory has a robust meaning as a semi-phenomenological theory, since the eCect of spin Auctuations is qualitatively well grasped in this approximation. Moreover, it is noted that unphysical results inherent in the phenomenological theory are considerably excluded. The FLEX approximation is one of the conserving approximation, formulated in the scheme of Baym and KadanoC [169,171] (see Eqs. (5) and (6)). The well-deMned basis makes it possible to perform the systematic calculation for the single- and two-particle properties in a coherent way. In fact, various quantities in the normal state have been calculated by using the FLEX approximation [93,188,223]. In general, anomalous properties arising from the AF spin Auctuation have been well explained within the FLEX approximation. The applicable region of the FLEX approximation is roughly between T0 and T ∗ in Fig. 1, since the SC Auctuation plays an essential role in the pseudogap state. We review the results of the normal-state properties together with the pseudogap phenomena in Section 4.3. This subsection is devoted to the review on the superconducting instability within the accuracy of mean Meld theory. Let us move on to the formulation of the FLEX approximation. In the FLEX approximation, the normal self-energy is expressed as 1 2 3 2F (k) = s (q) + c (q) − 0 (q) G(k − q) ; U (44) 2 2 q where s (q) and c (q) are given by Eq. (32). Note that 0 (q) is the irreducible susceptibility, which should be deMned by using the dressed Green function as G(k)G(k + q) : (45) 0 (q) = − k

Note that G(k) is self-consistently determined with the self-energy and susceptibilities. Throughout this self-consistent iterations, the renormalization eCect on the spin Auctuation is included in the self-energy. The spin Auctuation gives a large contribution to the self-energy in the quasi-2D systems, and thus, the magnetic order is suppressed. As a result, the FLEX approximation provides a wide critical region, in which the spin Auctuation is strongly enhanced. For an expression of the spin susceptibility Eq. (32) does not include the vertex corrections, which should be required in the conservation scheme, but it is considered that this correction is not severe [188]. Note that even in the conservation scheme, this vertex correction is not needed in the single-particle properties and superconducting Tc . V The superconducting transition is determined by solving the Eliashberg equation, Eq. (16), where the eCective interaction in the singlet channel is given by Eq. (33) with the self-consistently determined susceptibility. Near the magnetic instability, the second term in the right hand side of Eq. (33) becomes dominant. Then, the expressions are similar to those of the phenomenological theory with g replaced by U . It should be stressed again that the expressions are determined from the microscopic model. The FLEX approximation was Mrst performed by Bickers et al. [22,177] for the 2D Hubbard model with only the nearest-neighbor hopping. They have shown that the AF order exists near the half-Mlling and the dx2 −y2 -wave superconductivity occurs in the vicinity to the AF state. After that,

38

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 40

χs(q)

30

10% edope 10% hdope 16% hdope

20

10

0 (0,0)

(0,π)

(π,π)

(0,0)

q

Fig. 21. The momentum dependence of the static spin susceptibility in (a) electron-doped (10%), (b) under-doped (10%), and (c) optimally-doped (16%) region. Here, t =t = 0:25, T = 0:01 and U=t = 3:2 for hole-doped case, U=t = 3 for electron-doped case.

1.0

∆(k)

0.5

cos(k x)-cos(k y ) 10% hdoping 10% edoping

0.0

-0.5

-1.0 0.0

0.5

1.0

1.5

θ

Fig. 22. The momentum dependence of the order parameter -(k) = -(k; iT ) on the Fermi surface, for U=t = 3, t =t = 0:25 and T = Tc . The conventional form -(k) ˙ cos kx − cos ky is also shown for the comparison. The horizontal axis is shown in Fig. 16.

intensive studies on the Hubbard model [186–189] and on the d–p model [190–193] have commonly obtained the dx2 −y2 -wave superconductivity. We show the momentum dependence of the static spin susceptibility in Fig. 21, where the results for the optimally-doped, under-doped, and electron-doped cases are shown. It is observed that the spin Auctuation is strongly enhanced around q = (; ). Thus, the dx2 −y2 -wave superconductivity is mainly induced by the scattering process shown in Fig. 16. The order parameter is deformed, but similar to the conventional form -(k) ˙ cos kx −cos ky , as shown in Fig. 22. Since the scattering process between the zone boundary region

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.010

39

0.012 t’/t=0.25 U/t=3.2 t’/t=0.25 U/t=3

0.008

t’/t=0.15 t’/t=0.25 TOP

0.010

t’/t=0.35 U/t=4 t’/t=0.35 U/t=3.8

0.008

Tc

Tc

0.006 0.006

0.004 0.004 0.002

0.000 -0.20

0.002

-0.10

0.00

0.10

0.000 1.5

0.20

δ

(a)

2

2.5

3

3.5

4

U/t

(b)

Fig. 23. Tc in the FLEX approximation for the Hubbard model [11]. (a) Doping dependence and (b) U -dependence for t =t = 0:25 and = 0:1.

0.0080

Tc

0.0060

0.0040

0.0020

0.0000 0.00

d x 2 -y 2

εd -εp =-1.8 tdp =1.0 tpp =0.3 U=4.0 0.05

0.10

0.15

0.20

0.25

0.30

δ Fig. 24. Tc in the FLEX approximation for the d–p model [193].

(the circle in Fig. 16) plays a dominant role, the order parameter takes large absolute value around = 0 or =2. This deformation is in agreement with the experimental results of ARPES [224]. We show the results of Tc for the Hubbard model and those for the d–p model in Figs. 23 and 24, respectively. Fig. 23(a) shows the results for the electron-doped case, together. In general, the transition temperature increases with the development of the spin Auctuation, namely with decreasing and/or increasing U . In contrast to the TOP, Tc tends to saturate near the magnetic instability, since the de-pairing eCect becomes stronger in that region. The maximum value of Tc is commonly obtained as Tc=t = 0:01– 0.02 [186–189,191–193], which is consistent with the experimental value Tc ∼ 100 K in order of magnitude. It is straightforward to extend the FLEX approximation to the superconducting state. Actually, the calculation has been performed for the Hubbard model [186–188] and for the d–p model [193]. The ratio 2-=Tc =10–12 is commonly obtained, which is much larger than the BCS value 2-=Tc =3:5. This is mainly due to the feedback eCect on the spin Auctuation: Since the de-pairing eCect is remarkably

40

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

π

ky

10% hdoping 10% edoping

0

π

0 kx

Fig. 25. The Fermi surface obtained in the FLEX approximation for the hole- (thick line) and electron-doped (thin line) cases. Dashed lines show the non-interacting Fermi surface. The nesting nature is enhanced by the AF spin Auctuation [86].

reduced by the Mnite excitation gap, the order parameter at T = 0 exceeds the BCS value. This feature is qualitatively consistent with the experimental value 2-=Tc = 7–8 in the optimally-doped region. This value is over-estimated in the FLEX approximation probably because the de-pairing eCect around T = Tc is over-estimated (see Section 3.2.3). The other interesting property in the superconducting state is the resonance peak. When the system is near the magnetic order, a sharp resonance peak appears in the magnetic excitation with a smaller energy than the maximum gap 2- [188,193,225]. It is considered that the resonance peak corresponds to the 41 meV peak observed in the neutron scattering experiments in YBa2 Cu3 O6+ [226–230]. The inAuence of the resonance peak appears in the dip–hump structure in the quasi-particle spectrum and “kink” in the quasi-particle dispersion around (; 0) [193,225,231], which is also consistent with the ARPES measurements [45,106,232]. Finally let us review the application of the FLEX approximation to the electron-doped cuprates which has been recently performed [11,93,233,234]. The FLEX approximation also indicates the dx2 −y2 -wave superconductivity in the electron-doped region, which has been conMrmed by the recent experiments [49–53]. Moreover, the drastic particle–hole asymmetry in the phase diagram [23] is explained simply by taking account of the next-nearest-neighbor hopping t and choosing the electron number. In general, the electron correlation is relatively weak in the electron-doped region, because the electronic DOS decreases with electron-doping. Therefore, more conventional behaviors are expected in the normal state [11]. The detailed discussion will be given in Section 4.3.2. On the other hand, the AF order is robust for the carrier doping (Fig. 23(a)) because of the nesting property of the Fermi surface, as shown in Fig. 25. An interesting nature of the AF state appears in the electron-doped region [235], which has been well explained by the numerical calculation on the t–t –J model [236]. More interestingly, the superconducting Tc is very low and the doping region possessing the superconductivity is narrow [11,234]. This is understood by the following two reasons: One is the localized nature of the spin Auctuation in the momentum space (see Fig. 21). Then, the total weight of the spin Auctuation becomes small. It should be noted that the eCective interaction |Vsc | is not

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

T

=

+

41

T

Fig. 26. The renormalization of the interaction corresponding to the Kanamori theory.

determined by the magnetic correlation length. The other is the small DOS due to the fact that the Fermi level is apart from the Aat dispersion around (; 0). Thus, the eCective coupling |Vsc | decreases in the electron-doped region, and thus, Tc becomes low. These features are consistent with the experimentally observed phase diagram [23], and lead to the no appearance of the pseudogap phenomena (see Section 4.3.2). Another interesting feature is the strong modulation in order parameter. Namely, the order parameter has its maximum magnitude around the magnetic Brillouin zone. This is an inevitable result of the spin Auctuation-induced superconductivity, because the commensurate spin Auctuation (Fig. 21) induces the strongest interaction between the magnetic Brillouin zone. Recent Raman scattering measurement has supported this modulation of the order parameter [237]. Such a detailed consistency supports the importance of the spin Auctuation in the electron-doped cuprates. Summarizing, some detailed and interesting properties including the particle–hole asymmetry are well reproduced by the microscopic theory starting from the Fermi-liquid state. We consider that this is an important suggestion for the wide applicability and possibility of the microscopic theory for other strongly correlated electron materials. In fact, Sections 3.3 and 3.4 are devoted to the application to the organic superconductor and Sr 2 RuO4 , respectively. 3.2.5. Higher-order corrections Thus far, we have not provided any explicit justiMcation of the spin Auctuation theory and/or the FLEX approximation. They correspond to a partial summation in the perturbation series. The role of the neglected terms is not clear. Needless to say, an absolute work on this problem is very diDcult. However, several types of the higher-order corrections have been estimated and the positive results have been obtained to some extent. On the basis of the perturbation theory, the corrections Mrst appear in the third-order terms, which have been discussed in Section 3.2.3. There are two types of the correction in the eCective interaction. One is represented by the Feynmann diagram in Figs. 12(e) and (f). The other includes Figs. 12(g) and (h). We have conMrmed that both types of the diagram reduce the pairing interaction, that is, both terms in Eq. (26) are repulsive in the d-wave channel. We furthermore Mnd that the reduction from the two terms is nearly the same magnitude. This eCect from the diagrams in Figs. 12(e) and (f) is natural, because they are the lowest-order vertex correction included in the Kanamori-type T-matrix diagram (Fig. 26), which represents the screening eCect [238]. For instance, the higher-order correction represented by the T-matrix was approximately estimated by Bulut et al. [201]. They have concluded that the bare interaction U=t = 4 is renormalized to UY =t ∼ 2, where UY denotes the renormalized interaction. The coupling constant g in the phenomenological theory, and further U in the FLEX approximation should be regarded as the renormalized coupling constant including the screening eCect. The combination of the diagrammatic techniques and the quantum Monte Carlo simulation has shown that the eCective pairing interaction and the spin susceptibility are consistently obtained by the

42

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

(a)

(b)

Fig. 27. (a) The lowest-order vertex correction in the phenomenological spin Auctuation theory. (b) The eCective interaction corresponding to the multi-paramagnon exchange. Here, the solid and wavy lines are the propagator of the fermion and spin Auctuation, respectively.

generalized RPA, where the renormalized particle–hole vertex UY (q) is used [15]. The renormalized vertex in the intermediate coupling region U=t = 4 is estimated as about 80% of U , namely UY (q) ∼ 0:8U . Then, the expression for the eCective pairing interaction, given by Vas (k; k ) = U +

3 Y2 U s (k − k ) 2

(46)

provides an appropriate estimation, when the renormalized coupling constant UY = 0:8U is used with the spin susceptibility, obtained by the quantum Monte Carlo simulation. These results have indicated that the other scattering process, such as the multi-paramagnon exchange (see Fig. 27(b)), is negligible at least in the high-temperature region T=t ∼ 0:25. We Mnd the other intensive investigations on the higher-order corrections, namely, the vertex correction arising from the spin Auctuation [195,201,239–246]. The lowest-order vertex correction from the spin Auctuation is shown in Fig. 27(a). The diagrams in Figs. 12(g) and (h) are included in these corrections, while only the higher-order terms than the fourth-order are included in the phenomenological spin Auctuation theory. The strong cancellation of the eCective vertex between the fermion and spin Auctuation was suggested by SchrieCer [247]. His suggestion raised a serious question on the spin Auctuation-induced superconductivity, but it was just based on the mean-Meld theory in the SDW state. On the contrary, the studies in the normal state with diCusive spin Auctuation have commonly concluded that the vertex correction represented in Fig. 27(a) furthermore enhances the eCective vertex at q = (; ) [195,239–241,243–246,248]. The numerical calculation has furthermore revealed that the momentum dependence is not so altered by the vertex correction [243]. That is, we can appropriately include the vertex correction by renormalizing the coupling constant as g → 'g. Here the enhancement factor ' increases with the development of the spin Auctuation, but is expected not to exceed 2 under the reasonable parameters [243,244] The eCect of the vertex correction on the superconducting Tc has also been estimated [239,245,246]. The explicit calculation is diDcult, but the enhancement of Tc has been suggested commonly. It should be noted that the diagrams in Figs. 12(g) and (h) provide an opposite contribution, i.e., reduce the eCective vertex. This is because only the longitudinal component of the spin Auctuation appears in the vertex correction within the TOP. The transverse component appears in the higher-order correction and changes the sign. Combined with the corrections included in the T-matrix

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

43

π b

b2 p q' q' p'

(a)

c

c'

c

p b2 q

b2

c'

ky

b1

p'

b1 q

b

-t

b1

-t’ -t

(b)

(c)

0

−π −π (d)

0 kx

π

Fig. 28. (a) The lattice structure of -(ET)2 X. b1, b2, etc. stand for the transfer integrals tb1 , tb2 , and so on. The unit cell consists of the hatched molecules [265]. (b) The structure of the dimerized model. The diCerence between tc and tc is usually neglected [265]. (c) The tight binding model corresponding to the anisotropic triangular lattice. (d) The typical Fermi surface.

(Fig. 26), it is expected that the vertex correction reduces the bare interaction U in the weak-coupling region. This expectation is consistent with the result in the high-temperature region [15,201]. On the other hand, the enhancement of the eCective vertex is expected when the spin Auctuation is strongly enhanced. The explicit calculation is an open problem up to now. Another possible correction is the contribution from the multi-paramagnon exchange, which is shown in Fig. 27(b). This process has been proposed as the “spin-bag” mechanism [249]. However, the numerical estimations have concluded that the pairing interaction arising from this process is almost negligible [243,246]. In short, from the above results, we expect that the theory based on the single paramagnon exchange is qualitatively justiMed, although the higher-order corrections are required in the quantitative estimations. 3.3. Organic superconductor -(ET)2 X In this section, we discuss -(ET)2 X compounds among many organic superconductors. The outline of the experimental results and interesting issues on these compounds have been reviewed in Section 2.2. The metallic conduction in this material is owing to the organic molecule, which is complicated at Mrst glance. It is, however, shown that the simpliMed tight-binding model gives a reasonable understanding for the superconductivity. The successful application of the Hubbard model for the molecular systems will extend the possibility of the microscopic theory. 3.3.1. Electronic property and tight-binding model First we review the electronic properties and introduce an eCective Hamiltonian. This series of the organic materials are systematically described well by the Hubbard model with a tight-binding Mtting [250,251]. Note that the site in the Hubbard model is not one atom, but corresponds to one dimer of the organic molecules. The procedure for this simpliMcation is given in the following way. The quasi-2D conduction band consists of the -orbitals in the ET-molecules. There are four ET-molecules and two holes in a unit cell, as shown in Fig. 28(a). Then, the system is quarter-Mlling (n = 1:5 per molecule). In the Mrst simpliMcation, a dimer is regarded as a structural unit. This procedure is justiMed because the transfer integral tb1 is twice larger than the other transfer integrals [251,252]. Then, the holes are contained in the anti-bonding orbitals in dimers. The bonding orbitals

44

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

are far below the Fermi level. Focusing on the anti-bonding orbital, each dimer is connected to the nearest neighbor sites by three kinds of transfer integrals (Fig. 28(b)). They are derived as tb = −tb2 =2, tc = (tq − tp )=2, and tc = (tq − tp )=2. In this dimer model, the system is half-Mlling (n = 1 per dimer). The AF state in the lower pressure region is a Mott insulator constructed from the dimers [251]. Because the diCerence between tc and tc is negligible (only about 2%), the eCective single-band model is derived by setting tc = tc . Note that the obtained model is mapped to the anisotropic triangular lattice shown in Fig. 28(c). Now superconductivity has been discussed on the basis of the Hubbard model on the anisotropic triangular lattice. Because the system is near the Mott transition, the on-site repulsion is expected to be most eCective. The Hubbard Hamiltonian has been given in Eq. (42), but the dispersion relation is now given as ”(k) = −2t(cos kx + cos ky ) − 2t cos (kx + ky ) − , :

(47)

Note that t =t deMnes the anisotropy of the system and typically, t =t = 0:6– 0.8 according to the extended HZuckel band calculation [123,251]. The Fermi surface in this model (Fig. 38(d)) is in good agreement with the result of the Shubnikov–de Haas eCect. This model has a strong frustration compared to the case of high-Tc cuprates. It should be noticed that the Coulomb repulsion U is very small, because a lattice site of this model is not an atom but a dimer of molecules. However, this system is regarded as a strongly correlated electron system since the transfer integral t is also small. The parameter U=W is expected to be in the intermediate coupling region, because the system is half-Mlling near the Mott transition. The transfer integral has been estimated from the quantum chemistry calculation [253] as t = 70–80 meV, which is just about 1=10 of the typical value in d-electron systems. Thus, Tc ∼ 10 K again corresponds to Tc =t ∼ 0:01, which is theoretically comparable to that for cuprate superconductors (Section 3.2). 3.3.2. Application of the microscopic theories Microscopic calculations based on the Hubbard model have been performed by using the RPA [254], FLEX approximation [255–257], TOP [20], and quantum Monte Carlo simulation [258]. Note that strictly speaking, some small corrections on the model (47) are included in several papers, but they do not aCect the results, qualitatively. The above studies have concluded that (i) the pairing symmetry is the dx2 −y2 -wave, (ii) qualitatively the same pairing mechanism to high-Tc cuprates is expected, and (iii) the comparable Tc =t ∼ 0:01 is obtained in the intermediate coupling region. Most of the above proposals are classiMed into the spin-Auctuation theory. Although the phase transition from the AF to SC state is in the Mrst order, the strong AF spin Auctuation is observed experimentally [125,126]. Thus, it is expected that the spin Auctuation theory can be applied in the metallic state. The typical results of the FLEX approximation are shown in Fig. 29 [255–257]. The phase diagram in the plane of t =t and U=t is shown. The SC state appears in the neighborhood of the AF state. By increasing t =t, the introduced frustration destroys the AF state, and then, the superconductivity appears. We see that the static spin susceptibility in the normal state is strongly enhanced around q = (; ) and q = (; −) (Fig. 30). We have already shown that the scattering process exchanging the AF spin Auctuation is attractive for the d-wave superconductivity. The dominant scattering process is similar to that shown in Fig. 16. Thus, qualitatively the same mechanism to the high-Tc cuprates is expected for the superconductivity in organic materials. Note here that the spin Auctuation-induced superconductivity in organic materials has been proposed in 1980s for

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

45

10 0.02

Tc/τ

8

U/τ

6 0.00

4

2

0 0.0

SC

τ'/τ=0.4 τ'/τ=0.6 τ'/τ=0.8

0.01

3

6

U/τ

9

12

AF

PM 0.2

0.4

0.6

0.8

1.0

τ'/τ Fig. 29. The phase diagram obtained by the FLEX approximation [256]. The inset shows Tc for the various t =t. Note that 0 and 0 in this Mgure denote t and t , respectively, in our notations.

10 t’/t=0.3 t’/t=0.5 t’/t=0.7 t’/t=1.0

8

χ (q)

6 4 2 0 (0,0) (π,0)

(π,π)

(0,0)

(-π,π)

(0,π)

(0,0)

q

Fig. 30. The spin susceptibility obtained by the FLEX approximation at T=t = 0:02. The value of U=t is changed as U=t = 2:5, U=t = 3:7, U=t = 5:5, and U=t = 6:5 with increasing t =t.

(TMTSF)2 X compounds [259,260]. The transition temperature for the superconductivity is gradually reduced by the frustration. This is because the spin susceptibility becomes structure-less, and thus, the spin Auctuation mechanism is ineCective. In particular, the superconductivity almost disappears in case of the isotropic triangular lattice (t =t = 1). As in high-Tc cuprates, qualitatively the same results are obtained by the TOP [20]. For example, the dx2 −y2 -wave superconductivity is stabilized for t =t 6 0:8. As shown in Fig. 31, Tc =t ∼ 0:01 is obtained in the intermediate coupling region U=t ∼ 6, corresponding to U=W ∼ 0:7. The roles of the respective terms are the same as those for high-Tc superconductors. The superconductivity is mainly induced by the RPA-terms. The irreducible susceptibility is moderate, but has a similar tendency to

46

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.03 t’/t=0.6 t’/t=0.7 FLEX(t’/t=0.7)

Tc /t

0.02

0.01

0.00

5

6

7

U/t

Fig. 31. Tc obtained by the TOP. The results of the FLEX approximation for t =t = 0:7 is shown for comparison.

0.8 t’/t=0.1 t’/t=0.3 t’/t=0.7 t’/t=1.0

χ 0 (q)

0.6

0.4

0.2

0.0 (0,0)

(π,0)

(π,π)

(0,0)

(−π ,π)

(0,π)

(0,0)

q

Fig. 32. The bare spin susceptibility for various values of t =t.

the FLEX approximation, as is shown in Fig. 32. The non-RPA terms in the eCective interaction reduce Tc . Fig. 33 shows that Tc is reduced by the frustration and almost disappears around t =t = 0:8. This feature is also found in the result of the FLEX approximation. Thus, the anisotropy in the triangular lattice plays an essential role for the appearance of superconductivity. The decrease in Tc for t =t ¡ 0:3 is caused by the normal self-energy, which is enhanced by the nesting nature of the Fermi surface. We have also shown the comparison with the result when we ignore the vertex correction term Va(3VC) (k; k ). It is a natural conclusion that the vertex correction is more important for the strongly frustrated case. This is simply because the RPA terms become ineCective and then, the non-RPA terms relatively become eCective. In general, the spin Auctuation theory such as SCR [8] is justiMed when the Fermi surface is strongly nested. Under the strong frustration, it is not

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

47

0.20 TOP (VC) Without Va

Tc /t

0.15

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

t’/t

Fig. 33. t =t-dependence of Tc in the TOP (circles). The squares are the results when the vertex correction in the anomalous self-energy is neglected.

allowed to ignore the vertex corrections simply. Thus, the spin Auctuation theory is less appropriate to the -(ET)2 X compounds for t =t = 0:6– 0.8 than to the high-Tc cuprates. Note that the TOP and FLEX approximation commonly show that Tc increases with U . This tendency is in good agreement with the phase diagram in Fig. 8. The pressure increases the band width (equivalently, the transfer integral). Then, the parameter U=W , and accordingly Tc decreases with the applied pressure. For the variation of anions, the frustration is enhanced from X=Cu[N(CN)2 ]Br to X=Cu(NCS)2 , namely in the right hand side of the phase diagram in Fig. 8 [261]. Therefore, the frustration furthermore reduces Tc in the metallic state, combined with the chemical pressure. Recent FLEX calculations have pointed out that the strong dimerization is essential for the above results. When the dimerization is not strong enough, the superconductivity is severely suppressed, even if the bonding band is below the Fermi level [265,266]. In some cases, the dxy -wave symmetry is more favorable, while Tc is very low [265]. The parameters obtained from the quantum chemistry calculation locate in the boundary region between the strong and weak dimerization. We mention that the reasonable Tc is obtained only for the strongly dimerized region. Some experimental efforts have been devoted to identify the symmetry. Note that the dx2 −y2 -wave symmetry has been supported by the angle-dependence of the high-frequency conductivity [267] while the dxy -wave symmetry has been supported by the thermal conductivity measurement [268] as well as the tunnelling spectroscopy [269]. Before closing this section, we comment on the other superconducting materials in the molecular conductors. Since the phonon excitation is generally strong in the organic conductors, a class of the materials should be s-wave superconductor due to the electron–phonon mechanism. For instance, superconductivity in alkali-metal-doped fullerides A3 C60 is considered to originate from the phonon-mediated attractive interaction. However, some class of the organic superconductor can be categorized into strongly correlated electron systems and the unconventional superconductivity is expected. For example, (TMTSF)PF6 is considered to be the case, where the possibility of not only the d-wave pairing, but also the triplet pairing has been proposed [270]. The present microscopic calculations [260,271–273] seem to indicate the predominance of the d-wave superconductivity.

48

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

We believe that the development of the microscopic theory for organic materials is one of interesting future issues. 3.4. Sr 2 RuO4 Now let us review the theoretical investigations on Sr 2 RuO4 with main interests on the pairing mechanism, since interestingly this material is conMrmed to be a triplet superconductor (see Section 2.3). The triplet superconductivity has been already discovered in heavy-fermion compounds such as UPt 3 [274]. Probably, recently discovered uranium compounds UGe2 [275] and URhGe [276] are also the triplet superconductors, since superconductivity coexists with ferromagnetism. However, the microscopic investigation on these materials is generally diDcult due to their complicated electronic structure, as will be shown in Section 5. On the other hand, the electronic structure of Sr 2 RuO4 is relatively simple, deduced from such points as two-dimensional Fermi surfaces, a few degenerate orbitals, weak spin–orbit coupling, and relatively weak electron correlation. Thus, both from experimental and theoretical points of view, Sr 2 RuO4 is considered to be a typical compound for the microscopic investigation on the triplet superconductivity, which will provide new understanding on the unconventional superconductivity. 3.4.1. Overview Referring to the successes for the pairing mechanism in cuprates and 3 He, the spin Auctuationinduced superconductivity was examined for Sr 2 RuO4 at the Mrst stage. Several authors discussed the ferromagnetic spin Auctuation, which can induce the triplet p-wave superconductivity [146] (see the last in Section 3.2.2). Along this line, Mazin and Singh have insisted that ferromagnetic spin Auctuation is suDciently strong to induce the triplet superconductivity in this compound [204,205]. Monthoux and Lonzarich discussed the p-wave superconductivity by phenomenologically introducing the ferromagnetic spin Auctuation in a square lattice model [202]. However, it has become apparent that the situation is not so simple. Neutron scattering measurement by Sidis et al. has revealed that the ferromagnetic Auctuation is not so enhanced. Instead of ferromagnetic Auctuation, a sizeable incommensurate AF Auctuation has been observed at the wave vector QIAF = (0:6; 0:6; 0) [14]. This incommensurate spin Auctuation is due to the Fermi surface nesting eCect, as predicted by the band calculation with the use of local-density approximation (LDA) [205] and also derived from simpler models [277–279]. Under those circumstances, it has been proposed that the triplet superconductivity is possibly derived by assuming the strong anisotropy of the AF spin Auctuation [280–282]. The experimental support for the strong anisotropy has been obtained from the NMR measurement [283], but according to the recent neutron scattering experiment by Servant et al. [284], the spin susceptibility is very isotropic at q = QIAF . This discrepancy should be resolved experimentally. Theoretically, such anisotropy arises from the spin–orbit interaction in the Ru ions [278,279]. It should be commented that the signiMcant exchange enhancement is needed to explain the anisotropy observed in NMR measurement. Note also that the eCective interaction in the Cooper channel does not coincide with the observable spin susceptibility, when the spin–orbit interaction is explicitly taken into account [285]. Thus, more explicit calculation is required for this scenario in the present stage. Takimoto has proposed that the orbital Auctuation is important for the triplet superconductivity in Sr 2 RuO4 [286]. This type of Auctuation is similarly due to the nesting eCect of the Fermi surface.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

49

He estimated the pairing interaction by using the RPA on the three-band model and obtained the following results: (i) The instability to the f-wave superconductivity is generally derived by the RPA. (ii) The ' and ( bands are mainly superconducting. (iii) The strong orbital Auctuation is necessary for the triplet pairing to overcome the singlet one. (iv) As a result, the triplet superconductivity is stabilized, when the inter-orbital repulsion is larger than the intra-orbital one. Among them, the condition for (iv) is diDcult to be satisMed and thus, somewhat convincing evidence may be needed for the scenario based on orbital Auctuations. The analysis based on the perturbation theory has been given on this issue. Nomura and Yamada have expanded the eCective interaction with respect to the Coulomb interaction [21,287], as has been performed for high-Tc cuprates (Section 3.2.3) and -(ET)2 X (Section 3.3.2). Precedently, several works have discussed the two-dimensional Fermi gas model within the third-order perturbation [288,289]. They have concluded that the p-wave pairing state is expected to provide the highest Tc . The important results obtained in Refs. [21] and [287] are summarized in the following: (i) The non-RPA terms neglected in the Auctuation theory are signiMcantly attractive in the p-wave channel. These terms are the same ones discussed in the two-dimensional Fermi gas model [289], while the singularity in the isotropic model disappears in the Hubbard model. (ii) The irreducible spin susceptibility derived from the ) band shows rather weak momentum dependence. This situation is in sharp contrast to high-Tc cuprates and -(ET)2 X, but it is similar to the two-dimensional Fermi gas model. The results of the perturbation theory and their diCerences from the d-wave superconductors are discussed in the next subsection. The failure of the spin-Auctuation theory will be clariMed there. Recently, Honerkamp and Salmhofer found the p-wave superconducting phase in the single-band Hubbard model on the basis of the one-loop renormalization group theory [290]. They have adopted a band structure similar to that of the ) band. According to their results, it seems that the momentum dependence of the eCective interaction is not dominated by the ferromagnetic spin Auctuation. Rather it is similar to the one obtained within the naive third-order perturbation theory, in spite of the adjacent ferromagnetic phase. Note that this renormalization group analysis has an advantage that the magnetic, pairing, and the other instabilities can be treated on an equal footing. 3.4.2. Perturbation theory for the triplet superconductivity In this subsection the results on the perturbation theory are reviewed. First let us show the results on the single-band Hubbard model corresponding to the ) band [287]. Referring to the fact that the ) band has the largest DOS, we assume that the superconductivity is mainly determined by the ) band. This assumption will be justiMed later from the results on the three-band model. The ) band is mainly constructed from the Ru4d xy and O2p orbitals, which hybridize each other. Since the correlation eCect is dominated by the Coulomb interactions at the Ru sites, the Hubbard model Eq. (42) can be a reasonable starting point. This simpliMcation of the model is similar to the case of high-Tc cuprates, where the d–p model is reduced to the Hubbard model. For ruthenate, the dispersion relation is given as @(k) = −2t1 (cos kx + cos ky ) − 4t2 cos kx cos ky − , :

(48)

Due to the wave function of the 4d xy -orbitals, the sign of the next-nearest neighbor hopping is opposite to the case of cuprates. Note that the unit such as t1 = 1 is used in this subsection.

50

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.1 Spin-Triplet

t 2 =0.375

Spin-Singlet

Tc

n=1.260 n=1.333

0.01

0.001 3.5

4

5

6

7

7.5

U

Fig. 34. The results for the single band model [287]. The calculated Tc for the triplet p-wave and singlet dx2 −y2 -wave states.

In Fig. 34, we show Tc obtained by the TOP for the p- and dx2 −y2 -wave superconductivity. We can see that the triplet pairing is more stable than the singlet one for the electron Mlling n = 1:33, corresponding to the case of Sr 2 RuO4 [139]. On the other hand, the singlet pairing gives higher Tc near the half-Mlling. This is because the AF component of the spin Auctuation is enhanced and induces the dx2 −y2 -wave superconductivity. The latter situation corresponds to the case of cuprate superconductors. Thus, the electron number away from the half-Mlling is essential for the appearance of the triplet superconductivity. The transition temperature in the triplet channel increases with the electron number. This trend is consistent with the recent experiment [291]. This n-dependence of Tc is dominated by the eCective interaction [292], not by the electronic DOS. Let us explain the detailed roles of the respective terms in the perturbation expansion [292]. It should be noted that the RPA terms cancel each other and do not appear in the third-order term of Vat (k; k ) (Eqs. (27)–(29)). That is, all of the third-order terms are non-RPA terms. In the present case, both the second- and third-order terms are attractive in the triplet channel. Namely, the RPA- and non-RPA terms cooperatively induce the triplet superconductivity. In contrast to high-Tc cuprates, however, the contribution from the RPA term is not large, since the momentum dependence of 0 (q) is insigniMcant. On the other hand, the momentum dependence of the third-order terms are V particularly attractive. Fig. 35 shows the development of the eigenvalue of the Eliashberg equation in various calculations. We can see that the contribution from the third-order terms is important even in the moderately weak-coupling region; it overcomes the contribution from the second-order term around U ∼ 3, where the experimental Tc = 1:5 K is obtained. Thus, an important role of the non-RPA terms is expected for the appearance of the triplet superconductivity. We have conMrmed that the averaged magnitude of the third-order term is still 1/3 of the second-order term in this region, which does not contradict with the perturbative treatment adopted here. The eCect of the higher-order corrections will be discussed in Appendix B. On the other hand, the RPA gives the observed value of Tc ∼ 1:5 K only in the vicinity of the mean-Meld magnetic instability (U = 2). This is mainly because the RPA term is not so suitable to the triplet pairing and the magnetic order is signiMcantly overestimated in the RPA. It is also important that the contribution from the magnetic Auctuation to the triplet pairing is 1/3 of that to

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

51

0.6 SOP TOP RPA FLEX

0.5

λe

0.4 0.3 0.2 0.1 0.0

0

1

2

3

4

U

Fig. 35. The eigenvalue e in the triplet channel [292]. The temperature is Mxed to T = 0:005 (∼ 15 K). Then, e ∼ 0:4 corresponds to the experimental value of Tc ∼ 1:5 K by the extrapolation. Here, the normal self-energy is ignored for simplicity.

the singlet pairing (see Eqs. (33) and (34)). Note that this situation is not improved in the FLEX approximation, as is shown in the Mgure. This is mainly because the de-pairing eCect is very strong, when the spin Auctuation has a local nature, namely, nearly q-independent. We have pointed out in Section 3.2 that the de-pairing eCect is usually overestimated in the FLEX approximation, which is quite serious for the ) band just due to the local nature of the spin Auctuation. Also from this reason, the spin Auctuation theory is not suitable for the ) band. In fact, the ' and ( bands provide larger value of e in the FLEX approximation at U ¿ 2:5, where the spin susceptibility has a sharp peak around q ∼ QIAF . The obtained e in this case is shown in Fig. 35, but too small to lead to the observed value such as Tc ∼ 1:5 K. These comparisons with the spin Auctuation theory indicate the importance of the non-RPA term. The results of the perturbation theory have supported this indication. Note that the TOP is the lowest-order theory for the non-RPA terms. Here we comment on the roles of the perturbative terms in the singlet channel. They are qualitatively the same as in the previous subsections for high-Tc cuprates and organic superconductors. However, there is one signiMcant diCerence; the third-order term is totally repulsive in the dx2 −y2 -wave channel. That is, the vertex corrections completely suppress the RPA terms in the third order. It is expected that the spin-Auctuation theory loses its justiMcation in this case. The electron Mlling far from the half-Mlling is also essential for this conclusion. We have compared in detail the stability of the p-wave, dx2 −y2 -wave, and dxy -wave superconductivity [292]. We found that the dx2 −y2 -wave symmetry is most stable within the SOP, but the p-wave symmetry becomes stable owing to the third-order terms even in the rather weak-coupling region (U ¿ 1:5). Note that if we increase the electron number furthermore, the p-wave symmetry is most stable even in the SOP. In other words, the parameter region stabilizing the triplet superconductivity is obtained in the lowest-order theory and it is much enlarged in the TOP. In the next stage, the discussion is extended to the three band model [21], given by H = H0 + H ;

(49)

52

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

where H0 is written as † † @l (k)ck; g(k)(ck; H0 = l; s ck; l; s + yz; s ck; xz; s + h:c:) + HLS ; k;l; s

(50)

k; s

with the spin–orbit coupling term Li · Si : HLS = 2

(51)

i

The Coulomb interaction term H is given by 1 2Si; l Si; l + ni; l ni; l ni; l; ↑ ni; l; ↓ + U ni; l ni; l + J H =U 2 i;l

+J

i;l=l

i;l¿l

i;l¿l

ci;† l; ↓ ci;† l; ↑ ci; l ; ↑ ci; l ; ↓ ;

(52)

where l denote the Wannier states (xy; yz; xz) corresponding to the Ru(4d xy ; 4d yz ; 4d xz ) orbitals. The band dispersions are chosen as @xy (k) = −2t1 (cos kx + cos ky ) − 4t2 cos kx cos ky − ,xy ;

(53)

@yz (k) = −2t3 cos ky − 2t4 cos kx − ,yz ;

(54)

@xz (k) = −2t3 cos kx − 2t4 cos ky − ,xz ;

(55)

g(k) = 4t5 sin kx sin ky :

(56)

For the present calculation, we take (t1 ; t2 ; t3 ; t4 ; t5 )=(1:0; 0:4; 1:25; 0:125; 0:2). Under these parameters, the kinetic energy term H0 well reproduces the three Fermi surfaces observed by the de Haas–van Alphen eCect [139]. The last term in Eq. (50) is the spin–orbit interaction in the Ru ions, which plays an essential role in stabilizing the chiral superconductivity. We will discuss this subject in Section 3.4.3. For the time being, we ignore this term for simplicity. The interaction term H represents the on-site Coulomb interactions including the intra-band repulsion U , inter-band repulsion U , Hund’s coupling term J , and pair hopping term J . The parameters satisfy the relation U ¿ U ¿ |J | ∼ |J | and J ¡ 0 in the ordinary situation. We similarly expand the normal and anomalous self-energies up to the third order with respect V to H , and determine the superconducting instability by solving the Eliashberg equation extended to the multi-band system. Then, the mixing term with coupling constant t5 is approximately treated for simplicity. As for details, readers can refer Ref. [21]. As will be mentioned below, this procedure does not aCect seriously the calculated results for Tc , which is shown in Fig. 36. We obtain the qualitatively same results as those for the single band model. The inter-band interactions enhance Tc further, which takes the value Tc ∼ 0:003–10 K in the moderate coupling region U = 3– 4. The realistic value Tc =1:5 K should be obtained in the weaker coupling region. Note here that we restrict the calculation above T ¿ 0:003t1 in order to avoid the Mnite size eCects. Fig. 37 shows the momentum dependence of the anomalous self-energy -a (k; iT ) (a = '; (; )). This Mgure clearly shows that the pairing symmetry is p-wave. An important point to be noted is that the ) band has the largest magnitude of the anomalous self-energy. It is suggested that the

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.1

Tc

Tc

0.1

53

0.01

0.01 |J|=|J’|=U’

|J|=|J’|=0.667U’ U’=0.100U

U’=0.100U

U’=0.300U

U’=0.300U U’=0.500U

U’=0.500U

0.001

0.001 3

3.5

4

4.5

5

5.5

3

U

(a)

3.5

4

4.5

5

5.5

U

(b)

Fig. 36. The results for the three-band model [21]. The calculated Tc for the spin-triplet superconductivity at nl = 1:4. (a) |J | = |J | = 0:667U . (b) |J | = |J | = U . 2π

π

4

4

2

2

0

ky

ky

2π

π

0 -2

-2

-4

-4 0

0

(a)

π kx

0

2π

0

π kx

(b)

2π

2π

10

ky 0.0008

π

0 -5

Gap magnitude

ky

5

(c)

0

π kx

γ

kx

Γ

ky

0.0004

φ

kx

0.0002 0

-10 0

γ

0.0006

ky

α

ky

β 0

π/2

Angle φ (rad)

2π

π

X

α

β kx

Γ

kx

(d)

Fig. 37. Contour-plots of -a (k; iT ) for (a) ', (b) (, and (c) ) band [21]. The thick lines represent the Fermi surfaces. The parameters are chosen as U = 3:385, U = |J | = |J | = 0:5U , and T = 0:003. In (d), the gap magnitude on each Fermi surface is shown as a function of the inplane azimuthal angle : with respect to the Ru–O bonding direction (The numerical data is from the work in Ref. [294]). Here we have assumed that the orbital symmetry of the pairs is represented by kx ± iky and the unit of energy is about 4500 K.

condensation energy is mainly gained in the ) band. We can show that this situation is very robust for Sr 2 RuO4 [292]. That is, the magnitude of the anomalous self-energy remarkably depends on the band. Agterberg et al. have proposed this situation, called “orbital dependent superconductivity”

54

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

2π 2

8

0

7

-2

π

ky

ky

2π

π

6

-4 5 -6

0 (a)

k’ 0

-8

π kx

0

2π (b)

4

k’ 0

π kx

2π

Fig. 38. Contour-plots of the eCective interaction on the ) band for (a) the triplet channel, V);t ) (k; k ) and (b) the singlet channel, V);s ) (k; k ) [21]. Here the Matsubara frequency is chosen as k0 = k0 = iT and k is Mxed as pointed by the arrow. The parameters are chosen as U = 3:385, U = |J | = |J | = 0:5U and T = 0:007.

(ODS) [293]. In the present case, the ) band is mainly superconducting, because the DOS is largest in ). If we adopt the small t3 , the ' and ( bands have larger DOS. Then, the situation is converse. In the realistic case, superconductivity is dominated by the ) band and the contribution from the other bands are small. This is the physical background by which the theoretical approach based on the single-band model is justiMed. The approximate treatment for the mixing term g(k) is simultaneously justiMed, because this term only couples the ' and ( bands. In order to clarify the leading scattering process in the Cooper channel, we show the contour-plots of the eCective interactions on the )-band (Fig. 38). It is shown that the eCective interaction in the triplet channel V);t ) (k; k ) takes the large value around k ≈ −k . This characteristic momentum dependence mainly originates from the third-order terms and is attractive for the triplet pairing. This nature is in common with that of the single-band model. Fig. 38(b) shows that the characteristic peak appears in the eCective interaction in the singlet channel V);s ) (k; k ) around k = k + QIAF . These peaks arise from the nesting nature of the quasi-one-dimensional Fermi surfaces, namely ' and ( bands. This momentum dependence appears through the inter-orbital interactions and favors the dx2 −y2 -wave state. Thus, the singlet pairing is enhanced by the multi-band eCect. Although the triplet pairing is relatively suppressed by the multi-band eCect, such an eCect is not signiMcant when the inter-orbital interactions are small compared with the intra-band one. Finally let us comment on the understanding of the power-law behaviors below Tc [150,156–160] on the basis ofthe above results. Below Tc , the Bogoliubov quasi-particle energy is obtained as Ea (k) = za (k) ”a (k)2 + -a (k)2 . Here za (k) is the renormalization factor for the a-band and -a (k) = |-Ra (k; Ea (k))|, where -Ra (k) is the analytic continuation of -a (k). If we consider the weak coupling case, i.e., (Tc ; -a (k))W , an approximation -a (k) |-a (k; iT )| becomes very accurate around the Fermi surface. Therefore, the momentum and orbital dependence of the excitation gap is obtained from |-a (k; iT )|, except for the factor arising from za (k). Since we found that the a and k-dependence of za (k) is not outstanding in Sr 2 RuO4 , the qualitative nature of the excitation spectrum below Tc is captured by Fig. 37. Strictly speaking, the momentum and orbital dependence of -a (k) is deformed below Tc . However, this deformation is usually small in the weak-coupling region Tc W [146]. Note that the anisotropy of the excitation gap tends to be smeared below Tc .

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

55

As shown in Fig. 37, the momentum dependence of the anomalous self-energy is highly anisotropic and cannot be Mtted by the simple form -(k) = sin kx and so on. Recently, we have successfully shown that the gap structure derived in the present formulation is consistent with the power-law behavior of the speciMc heat [294]. A node-like structure on the ( Fermi surface remains in the ˆ = (kx ± iky )zˆ and results in the power-law behavior at low temperature. Combination chiral state d(k) with the “orbital dependent superconductivity” gives a whole temperature dependence, in agreement with experiments, although a Mtting of the parameters is required. Thus, the power-law behaviors and the time reversal symmetry breaking can coexist within the two-dimensional model. One of the remaining problems is to clarify whether or not the other experimental results, such as the NMR 1=T1 T [150], magnetic Meld penetration depth [157], thermal conductivity [158,159], and ultrasonic attenuation rate [160] can be consistently explained. 3.4.3. Identi:cation of the internal degree of freedom In this subsection, we discuss the subject concerning the internal degree of freedom in the triplet superconductivity. Since the triplet superconductivity has the spin degree of freedom, an internal degeneracy remains under the crystal Meld. The spin part of the order parameter is assigned by the d-vector as -↑↑ (k) -↑↓ (k) dz (k) −dx (k) + idy (k) ˆ (57) = = id(k) ˆ y : dx (k) + idy (k) -↓↑ (k) -↓↓ (k) dz (k) According to the tetragonal crystal symmetry, six eigenstates are degenerate in the calculation in Section 3.4.2, because we have ignored the spin–orbit interaction described in the last term of Eq. (50). This degeneracy is lifted by the spin–orbit interaction and classiMed into four onedimensional representations and a two-dimensional representation [12,152]. The internal structure in the superconductivity is an attractive character which does not usually exist in the singlet pairing. Since the SC state is characterized by this structure, both theoretical and experimental interests are widely stimulated. Owing to the internal degree of freedom, the multiple phase diagram can appear, as has been found in 3 He and UPt 3 , to which many theoretical investigations have been devoted [12,146,295,296]. For 3 He, the weak dipole interaction works as the leading spin–orbit interaction which stabilizes the A-phase near Tc and explains the interesting properties in the NMR shift [146]. For the superconducting materials, the internal degree of freedom has been discussed in the phenomenological level [12,295,296], which classiMes the possible eigenstates and provides some experimental methods to identify the internal structure. It has been diDcult to develop the microscopic theory on this issue, since the previous triplet superconductors are basically heavy-fermion systems. We have expected that the relatively simple electronic structure makes Sr 2 RuO4 to be the Mrst example of such microscopic study. Here we show that such investigation provides some interesting conclusions [292]. The experimental investigation on the multiple phase diagram does not seem to be completed for Sr 2 RuO4 . It has been discovered that the second phase appears in the low-temperature and high magnetic-Meld region with H being precisely parallel to the plane [297]. The property of this second phase is not identiMed up to now. Recently, NMR measurement has been performed to resolve the nature under the perpendicular Meld [298]. Then, the d-vector perpendicular to the z-axis is indicated under the high Meld. The stabilized state under the zero magnetic Meld has been identiMed experimentally. It has been revealed from the phenomenological argument [152] that only

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ˆ the chiral state d(k) = (kx ± iky )zˆ is consistent with the present experiments [4,145,147,151]. Here, our microscopic analysis is focused on the internal structure under the zero magnetic Meld. The microscopic mechanism of the chiral superconductivity has been one of important issues, and the resolution of the internal structure will indicate the phase diagram under the magnetic Meld. First we brieAy explain the formulation. The matrix representation of the kinetic energy term H0 is described as    ig(k) − s −s ck; yz; s ”yz (k) †    † †     ”xz (k) ( ck; yz; s ck; H0 = ck; xz; s xy; −s )  −ig(k) − s   ck; xz; s  : (58) k; s ck; xy; −s −s ”xy (k) We denote the 3 × 3 matrix in Eq. (58) as Hˆ 0 (k; s). Note that a constant phase factor is multiplied to the 4dxz orbital in order to simplify the notation. This deMnition makes the pair hopping term J negative between the 4dxz and other orbitals. We ignore the hybridization term g(k) in the following, since this term is not important when the ) band is mainly superconducting. We can show that the conclusions are not aCected even if the ' and ( bands are mainly superconducting [292]. New quasi-particles are obtained by diagonalizing the matrix Hˆ 0 (k; s) through the unitary transformation. This procedure corresponds to the transformation of the basis as (a†k; 1; s ; a†k; 2; s ; a†k; 3; s )= † † † † ˆ (ck; yz; s ; ck; xz; s ; ck; xy; −s )U (k; s). New quasi-particles created by the operators ak; l; s are characterized by the pseudo-orbital l and pseudo-spin s. The Fermi surface of the new quasi-particle is consistent with the quantum oscillation measurement [139]. The rotational symmetry in the spin space is violated by the spin–orbit interaction. The anisotropy of the spin susceptibility is derived from this eCect [278,279,292]. We calculate the scattering vertex >(k; k ; a; a ; b ; b; s1 ; s2 ; s3 ; s4 ) in the Cooper channel by using the old basis tentatively, and then, apply the unitary transformation in order to obtain the eCective interactions in the new basis, ˜ k ; '; ' ; ( ; (; s1 ; s2 ; s3 ; s4 ) >(k; = ua;∗ ' (k; s1 )ua∗ ; ' (−k; s2 )>(k; k ; a; a ; b ; b; s1 ; s2 ; s3 ; s4 )ub ; ( (−k ; s3 )ub; ( (k ; s4 ) ;

(59)

a;a ;b;b

where we redeMne the up (down) spin in 4dxy -orbital as s=−1 (s=1). It should be noticed that many terms are added to the scattering vertex >(k; k ; a; a ; b ; b; s1 ; s2 ; s3 ; s4 ) through the oC-diagonal Green functions. Moreover, we have to calculate the oC-diagonal part, because it contributes through the unitary transformation. The diagonal part with respect to the pseudo-orbital V˜'; ( (k; k ; s1 ; s2 ; s3 ; s4 ) = ˜ k ; '; '; (; (; s1 ; s2 ; s3 ; s4 ) contributes to the superconducting instability. As a result, the Eliashberg V >(k; equation is extended in the following way: (60) V˜'; ( (k; k ; s2 ; s1 ; s3 ; s4 )|G˜ ( (k )|2 W(; s3 ;s4 (k ) : e W'; s1 ;s2 (k) = − (; k ; s3 ; s4

However, the full calculation for this equation is very tedious, since the eCective interaction V˜'; ( (k; k ; s1 ; s2 ; s3 ; s4 ) includes lots of terms. Then, we simplify the calculation by using two additional approximations: One is the perturbation with respect to the spin–orbit coupling and the other is based on the ODS argument. Since the spin–orbit interaction in Sr 2 RuO4 is much smaller than the band width,

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57

t 2 /t 1

0.45

0.40

(1) (3)

0.35

1.2

1.4

nγ

1.6

1.8

Fig. 39. The stabilized state in the perturbation theory. Circles and triangles represent the state (3) and (1), respectively. The solid line shows the realistic value n) = 1:33. The Fermi surface is electron(hole)-like in the left (right) side of the dashed line. The parameters are chosen as U = 5, U = 0:3U and |J | = |J | = 0:2U .

the perturbation with respect to is very accurate. We Mnd that the Mrst-order term vanishes and the second-order term is the lowest order. We restrict the estimation within this order. The practical procedure has been explained in Ref. [292]. We can show that the ODS is very robust in Sr 2 RuO4 , while the main band depends on the parameters [292]. Then, the eigenvalue e is almost determined by the interaction between the main orbital. Therefore, it is suDcient to take into account only the diagonal part of the interaction V˜' (k; k ; s1 ; s2 ; s3 ; s4 ) = V˜'; ' (k; k ; s1 ; s2 ; s3 ; s4 ), and investigate each case where the main pseudo-orbital is ' = 1; 2 or 3. Finally, the eigenstates are classiMed by using the d-vector representation. We Mnd the following ˆ ˆ ˆ eigenstates: (1) d(k) = kx xˆ ± ky y, ˆ (2) d(k) = kx yˆ ± ky x, ˆ and (3) d(k) = (kx ± iky )zˆ with remained two-fold degeneracy. Although other linear combinations are possible, we have chosen the symmetric states, which are expected to be stabilized in order to gain the condensation energy. The degeneracy in (1) and that in (2) is Mnally lifted by the weak mixing term g(k), but the main results are not aCected by this eCect [292]. We estimate the eigenvalue e for each eigenstate (1) – (3) and regard the state with maximum Tc to be stabilized. This procedure is surely correct around T ∼ Tc . In order to estimate >, we use the perturbation method with respect to the interaction term H and take into account all of the second-order terms as well as the third-order term with coupling constant U 3 . This approximation is justiMed when |J |, |J |, U ¡ U ¡ W . We include the third-order term in the estimation because it is important for the pairing mechanism, as is explained in Section 3.4.2. The results on the d-vector are almost not aCected by the third-order terms since they do not lift the internal degeneracy. Now let us show the results by the fully microscopic calculation. We Mnd that the “symmetry breaking interaction”, which violates the SU(2) symmetry in the d-vector space, requires the Hund’s coupling term J . Considering that the superconductivity is mainly induced by the intra-orbital repulsion U , we conclude that the triplet superconductivity and the chiral superconductivity have a quite diCerent origin. In the present case, the cross-term with coeDcient UJ is the leading contribution to the “symmetry breaking interaction” and stabilizes the d-vector parallel to the z-axis. Fig. 39 shows

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Table 1 Stabilized state for each symmetry and main band

p-wave p-wave (with node) fx2 −y2 -wave fxy -wave

)-band

'-, (-band

(kx ± iky )zˆ None (kx2 − ky2 )(ky xˆ ± kx y) ˆ kx ky (kx xˆ ± ky y) ˆ

ky xˆ ± kx yˆ ky xˆ ± kx yˆ (kx2 − ky2 )(kx xˆ ± ky y) ˆ None

the phase diagram with respect to the parameter t2 =t1 and the electron number in the ) band. We can see that the chiral state (3) is robustly stabilized in the experimentally relevant region, t2 =t1 ∼ 0:4 and n) ∼ 1:33. The other state (1) also appears, but only in the experimentally irrelevant case where the )-Fermi surface is hole-like. The diCerence of Tc for each eigenstate is estimated as 2– 4%, if we put the parameters as 2 = 0:1 eV and W = 2 eV. This value seems to be surprisingly small, but this is a natural result, since the “symmetry breaking interaction’ is in the second order with respect to . Note that the chiral state (3) is not stabilized, if the ' and ( bands are mainly superconducting. We Mnd that state (2) is more stable there. Here we comment on the phase diagram under the magnetic Meld. Since the d-vector along the z-axis is stabilized under zero magnetic Meld, the parallel magnetic Meld does not alter the d-vector. Thus, the second phase transition observed under the parallel magnetic Meld is not the rotation of the d-vector. This phenomenon may be a transition within the two-dimensional representation ˆ d(k) = (kx ± ikx )zˆ or that of the vortex conMguration. Our estimation indicates that the d-vector can be altered by the small magnetic Meld along the z-axis, since the splitting of the degeneracy is small. While the Hc2 is small in this direction as Hc2 ∼ 0:07 T, our estimation for splitting of the degeneracy corresponds to the magnetic Meld H ∼ 0:1 T. Since our estimation includes ambiguity in the order unity, the d-vector perpendicular to the z-direction is possible in the high-Meld region. In the next stage, we compare several pairing states from the viewpoint of the internal degree of freedom. Assuming the pairing symmetry and the main band, we investigate which pairing state is stabilized in the respective case. Then, the “symmetry breaking interaction” is microscopically estimated within the lowest order. We believe that this semi-phenomenological comparison is meaningful, for which the applicability of the perturbation theory may be wider. This is because the perturbation series of the “symmetry breaking interaction” has smaller coeDcient than the total eCective interaction. The obtained results will restrict the possible pairing symmetry, which is consistent with the time-reversal symmetry breaking [151]. We restrict the discussion to the two-dimensional model, because the three-dimensional pairing state is hardly promising. The obtained results are summarized in Table 1, where the stabilized state is shown for each pairing symmetry. We see that the chiral state is stabilized only when the symmetry is the p-wave and the main band is ). The other paring states including the f-wave symmetry [286,299,300] provide other d-vector, which is incompatible with the time-reversal symmetry breaking. After all, it is concluded that the most favorable pairing state is the p-wave on the )-band, as expected in the perturbation theory. In other words, the perturbation theory provides the probable pairing symmetry and furthermore the compatible d-vector along the z-axis. ˆ

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Here we point out that our systematic treatment for the spin–orbit interaction is necessary to discuss the internal degree of freedom. The spin–orbit interaction contributes to the symmetry breaking interaction through (i) the virtual process in the eCective interaction and (ii) the unitary transformation of the quasi-particles. When the chiral state is stabilized, the dominant contribution comes from the cross-term of (i) and (ii). Thus, we have to treat the two eCects on the same footing. The estimation of the eCect (i) only [285] or the eCect (ii) only [301] may be inadequate in the microscopic theory. We should further mention that the eCective interaction derived here is quite diCerent from the phenomenological assumption in Ref. [301]. Indeed, the chiral state is not stabilized, if we take account of only the eCect (ii). At the last of this section, we comment on the heavy-fermion superconductor UPt 3 , which is a spin-triplet superconductor exhibiting three diCerent superconducting phases under the magnetic Meld. The phase diagram has been explained by assuming the weak spin–orbit interaction [295] because this assumption is necessary to explain the NMR Knight shift [13]. However, this assumption has raised a serious question, since the spin–orbit coupling is generally strong in the heavy-fermion system [296]. We think that the result obtained here gives a clue to this question. It has been shown that the Hund’s coupling term is required for the violation of the SU(2) symmetry in the d-vector space. On the contrary, the SU(2) symmetry in the real spin space is violated even for J = 0; the spin susceptibility is anisotropic, for example. Therefore, the violation of the SU(2) symmetry can be much smaller in the d-vector space than in the real spin space. In other words, it is possible that the anisotropy is almost absorbed in the character of the quasi-particles, and only a weak anisotropy is remained in the residual interaction. The examination of this possibility for UPt 3 will be an interesting future issue. 4. Pseudogap phenomena 4.1. Overview In this section, we discuss the normal-state properties of high-Tc cuprates. As we have noted in the review of the experimental results in Section 2.1 [54], many anomalous aspects of the normal state have been central issues of high-Tc superconductors. We should clarify the nature of the normal state, because the comprehensive understanding of the whole phase diagram is highly desirable. Among them, the pseudogap phenomena, which are widely observed in the optimally- to under-doped region (see Fig. 1), have attracted much interests. Since the superconductivity arises through the pseudogap state in optimally- and under-doped systems, the resolution of the pseudogap state is an essential subject for the high-Tc superconductivity. Furthermore, the pseudogap phenomena have many interesting aspects, because they are in sharp contrast to the conventional Fermi-liquid theory [302]. The unusual nature has indicated an appearance of a new concept in the condensed matter physics. The introduction of our understanding on this issue is one of the purposes of this review. Among many theoretical proposals, we have adopted the “pairing scenario” in which the pseudogap is a precursor of the superconducting (SC) gap. This scenario has been indicated by several experimental results, as have been explained in Section 2.1. Since there are several kinds of “pairing scenario”, we will explain later them somewhat in detail to avoid confusion. Before introducing the pairing scenarios, we brieAy review other scenarios, which have been widely investigated.

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An interesting proposal is the appearance of the RVB state [24,25]. Historically, the RVB theory was proposed for the quantum spin system with frustration [303]. Motivated by the high-Tc superconductivity, the superconductivity arising from the RVB state has been investigated intensively [304–318]. Naively speaking, the spin-liquid state is the underlying state which results in the superconductivity if holes are lightly doped. Among the several descriptions, the slave-boson method for the t–J model has been widely used [308,311,312]. In this theory, two essential excitations, spinon and holon, appear in the mean-Meld level and couple through a gauge Meld. The pseudogap state is regarded as a singlet pairing state of the spinons and the superconductivity is described as Bose–Einstein condensation (BEC) of the holons. Some characteristics of the pseudogap state as well as the whole phase diagram are reproduced within the mean-Meld theory. The electric transport is explained by taking account of the U(1) gauge Meld [315,316]. The gauge-Meld theory has been extended to the SU(2) symmetry [318]. These approaches are essentially from the Mott insulating state. The metallic state is described as a “doped Mott insulator”. This starting point is contrastive to our approach from the Fermi-liquid state. Another class of the proposal is the “hidden order scenario”, where some long-range order exists in the pseudogap state. For example, the anti-ferromagnetism [249,319] and the “stripe” order [320] have been proposed as candidates. Recently, “the d-density wave” has been proposed as a new type of the long-range order [321]. If a Mnite value of some “hidden” order parameter is conMrmed experimentally, the corresponding scenario will be the best candidate, but it is not the case up to now. In other class of the scenarios, a Auctuation of the order parameter is considered as an origin of the pseudogap. The Auctuation-induced pseudogap has been investigated from the old days [322]. Then, the (quasi-) one-dimensional Peierls transition was mainly investigated [323–325]. Since any long-range order does not exist at Mnite temperature, an extraordinary wide critical region is expected in one dimension. The pseudogap in the spectrum is expected as a precursor of the long-range order. This mechanism may be generally expected in the low-dimensional system. The pairing scenario is also classiMed into the case. Another candidate is the AF spin-Auctuation [188,195,223,326–328]. The pseudogap in the single-particle spectrum can be derived from the spin-Auctuation theory. The pseudogap Mrst opens at the “hot spot”, which is qualitatively consistent with experiments. A naive and crucial problem is on the magnetic excitation which is observed in the NMR and neutron scattering (Section 2.1.3). It will be a subject how the decrease of the magnetic excitation is derived from the magnetic Auctuation itself. An alternative understanding for the comprehensive phase diagram is proposed by Emery et al. [329] on the basis of the Auctuating “stripe” state. In the Emery’s proposal, the proximity eCect from the insulating region induces the spin gap. Now let us introduce the pairing scenarios, which are classiMed into some kinds. Because this concept is adopted in this review, a detailed explanation is given, including the related theories. The superconducting phase transition is usually described by the BCS theory which is an established mean-Meld theory. As is well known, the superconducting correlation does not appear above Tc within the mean-Meld theory. Therefore, we have to consider the breakdown of the BCS theory in order to discuss the pairing scenario. There are following two origins for the breakdown. One is the strong-coupling superconductivity and the other is the low-dimensionality. The justiMcation of the BCS theory is based on the long coherence length, which is =0 = 102 –103 in the conventional superconductors and makes the Auctuation negligible. However, the strong-coupling nature of the superconductivity results in the short coherence length, which causes the softening of the Auctuation.

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61

Fig. 40. The correction to the thermodynamic potential in the NSR theory.

Here, the coupling of the superconductivity is indicated by the parameter TcMF =EF , which determines the coherence length in the clean limit. Note that the non-s-wave superconductivity is always clean. Here, TcMF is the transition temperature in the mean-Meld theory and EF is the Fermi energy renormalized by the electron correlation. The Auctuation is generally enhanced by the low-dimensionality. In the strictly two-dimensional case, Tc is always zero according to the Mermin–Wagner theorem, expect for the KT transition. Then, wide critical region is expected even in the weak-coupling case. While the three-dimensional long-range order occurs owing to the inter-layer coupling, the quasi-two-dimensionality induces the strong Auctuation. These two conditions are surely satisMed in the high-Tc cuprates, where =0 = 3–5. The Mrst proposal of the pairing scenario was based on the Nozi\eres and Schmitt-Rink (NSR) theory [330,331], where the central subject is the cross-over problem from the BCS superconductivity to the Bose–Einstein condensation (BEC). In the weak-coupling region, the Cooper-pairing and phase coherence occur at the same time according to the BCS theory. In contrast to that, in the strong-coupling limit, the fermions construct pre-formed bosons above Tc and the BEC occurs at Tc . The latter situation is called “real space pairing” in contrast to the “momentum space pairing” in the BCS theory. The cross-over of two regions was Mrst formulated at T = 0 by Leggett [330] and extended to the Mnite temperature by Nozi\eres and Schmitt-Rink [331]. After that, the application to the two-dimensional system has been investigated [332–335]. These two regions are continuously described by adjusting the chemical potential. We note that this argument does not prohibit the phase transition. The Mrst-order phase transition with phase separation has been reported in the dynamical mean-Meld theory [336,337]. The NSR theory at Mnite temperature takes the lowest-order correction to the thermodynamic potential (4B ), which is shown in Fig. 40. The particle number is obtained as n = nF + nB , where nF is the usual Fermion contribution and nB is the contribution from the Auctuation, namely nB = −94B =9,. The chemical potential is set below the conduction band in the strong-coupling region. Then, the fermionic excitation is fully gapped even in the anisotropic superconductivity [330]. In this limit, the system is regarded as a bosonic system with residual interactions. It should be noted that the NSR theory is basically justiMed in the low-density system, since the shift of the chemical potential is the leading eCect in the low-density limit. When the BCS-BEC cross-over was proposed for the pseudogap phenomenon [338–340], the pseudogap state was regarded as a cross-over region. The strong-coupling limit is evidently not relevant for cuprates, because the excitation is clearly gapless along the diagonal direction. Note here that this picture is adopted in the phenomenological theories which assume the coexistence of the fermions and bosons [341–343]. By taking advantage of this proposal, intensive studies have

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been devoted to the cross-over problem [338–340,344–354]. For example, the description beyond the original NSR theory has been given by the self-consistent T-matrix (SCT) approximation [344] and by the pairing approximation [353,354]. The electronic spectrum has been calculated for the essentially low-density model by the T-matrix approach [348,355–360]. In particular, Janko et al. have shown that the pseudogap in the single-particle spectrum appears near the cross-over region in the 3D jellium model. Then, the importance of the self-energy correction has been pointed out. The quantum Monte Carlo simulation has been performed as a non-perturbative method for the same problem [361–367]. Then, the results in the strong-coupling region commonly show the pseudogap in the single-particle and magnetic excitations both in the high- and low-density region. We, however, believe that the BCS-BEC cross-over is not an essential viewpoint for the high-Tc cuprates [368]. This point of view is especially important in order to clarify the origin of the pseudogap. In Section 4.2, we show that the excitation gap is induced by the SC Auctuation even far from the cross-over region, if we adopt an appropriate model [368]. Then, the anomalous behaviors of the self-energy correction, which is derived by the T-matrix approach, play an essential role. This scenario is rather conventional in the Auctuation theory [322–325] but several conditions are required in the SC Auctuation theory. The self-energy correction arising from the SC Auctuation has been investigated in 1970s [322,369], which shows the decrease of the electronic DOS due to the SC Auctuation. However, it seems that this idea has not attracted much interests, because this is a very weak eCect in the conventional superconductor. We would like to stress that this common knowledge should be altered for cuprates, where the SC Auctuation is remarkably enhanced by the short coherence length and the quasi-two-dimensionality. Then, the more pronounced phenomenon is derived; the pseudogap appears in the single-particle spectral function. The high-Tc cuprates are probably the Mrst example for the clear appearance of this phenomenon. It should be noticed that the T-matrix approach is commonly used for the NSR theory and for the estimation of the self-energy. Actually, if we regard the Fig. 40 as a Luttinger’s functional Eq. (5), the self-energy should be estimated by the SCT approximation. Two aspects of the T-matrix approach, the BCS-BEC cross-over and the pseudogap, appear depending on the selected microscopic model. Indeed, the essential diCerences arise from the electron density and the dimensionality. If we use the 3D jellium model, the pseudogap appears from the cross-over region [348,357]. Then, the leading scattering process is called “resonance scattering” since the SC Auctuation has a resonance nature in this region. On the other hand, if we use the quasi-2D square lattice model near the half-Mlling (the nearly half-Mlled system should be regarded as high density), the self-energy correction gives rise to the pseudogap under more moderate condition. This is mainly because the quasi-two-dimensionality leads to the strong Auctuation and induces the pseudogap for the relatively small value of TcMF =EF . Moreover, the high electron density makes the BCS-BEC cross-over diDcult. If we consider the repulsive Hubbard model as in Section 4.3, the BCS-BEC cross-over is practically impossible. These diCerences are schematically shown in Fig. 41. We see from this observation that the 3D jellium model is too simpliMed to discuss the high-Tc cuprates, while several aspects are common. In two dimensions, the resonance nature of the SC Auctuation is not necessary for the pseudogap, while the renormalization eCect induces more propagating character [368]. In the following part, however, we will commonly use the term “resonance scattering” keeping in mind that a kind of the resonance between quasi-particle state and Cooper-pairing state is expected.

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(a) 3D jellium model

Pseudogap BCS

cross-over

BEC

(b) 2D squere lattice model

Pseudogap BCS

BEC

High-Tc cuprates

Fig. 41. The schematic Mgure for the relation between the NSR theory and the pseudogap phenomena. (a) The three-dimensional jellium model and (b) two-dimensional square lattice model around the half-Mlling. The horizontal axis corresponds to the control parameter TcMF =EF for the Mxed electron number. It should be noticed that the Fermi energy EF is renormalized in the SCES.

Along this line, the T-matrix approach has been applied to the two-dimensional lattice model including the d-wave symmetry [368,370–374]. Engelbrecht et al. have applied the SCT approximation to the quarter-Mlled model [370] and the present authors have performed both non-self-consistent and self-consistent T-matrix approximations for the nearly half-Mlled case using the TDGL expansion [368,371]. Subsequently, the microscopic theory starting from the repulsive Hubbard model has been developed [11] (Section 4.3). Then, the pseudogap phenomena are systematically explained including their doping dependence. It should be stressed that this microscopic theory is a natural extension of the theories discussed in Section 3.2. Thus, we obtain a comprehensive understanding on the pairing mechanism and the normal state properties in Section 4.3. The T-matrix approximation is the lowest-order theory with respect to the SC Auctuation. Then, the pseudogap appears under the reasonable condition, namely intermediate coupling region with a suDcient Fermi degeneracy. Indeed, the chemical potential is almost not aCected by the Auctuation. Basically, the pairing symmetry is not important; the s-wave model shows the pseudogap in a similar way [374] except for the diCerence in the momentum dependence. The SCT approximation is one of the methods to include the higher-order correction, which is taken into account within the renormalization of the Green function. Then, the pseudogap in the single-particle spectrum is suppressed, while that of the DOS clearly remains. We will give a critical comment on the relevance of the partial summation performed in the SCT approximation (Section 4.2.5). The two-particle self-consistent (TPSC) approximation has been proposed for an improvement of the T-matrix approximation [375,376], where the coupling constants are adjusted to satisfy a sum-rule while the renormalization for the single-particle Green function is neglected. A similar, but another approach is the “phase Auctuation theory” [115,377–381]. In this approach, the amplitude of the SC order parameter is Mxed and only the phase variable is taken into account. This approach has been indicated [115] by the small London constant in the under-doped region [112,113], which is usually attributed to the small superAuid density ns . In the case of the Kosterlitz–Thouless (KT) transition [382], which is the sole phase transition in the two-dimensional system, the transition temperature TKT is proportional to the London constant. If we consider the

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actual three-dimensional critical temperature is close to TKT , the -dependence of Tc is qualitatively consistent according to the Uemura plot ˙ . The pseudogap state is described as the phase disordered state which is expected above TKT . The gap structure in the electronic spectrum is generally smeared by the phase Auctuation in the SC state [383]. It has been, however, shown that the gap structure remains in the phase disordered state [377–380]. We consider that the validity of the phase only theory on the electronic structure is not clear up to now. It is surely justiMed deeply below Tc where the phase Auctuation is dominant but aCects perturbatively, although the coupling to the density Auctuation should be considered there [384]. The description of the normal state may require further discussions; the amplitude Auctuation certainly aCects there. Here we stress that the phase Auctuation theory is not intrinsically incompatible with the T-matrix approach. We consider that the former is an approach from the low-temperature side and the latter is that from the high-temperature side, which will be complementary. Because the T-matrix approach is a perturbation theory with respect to the SC Auctuation, the precise description in the deeply critical region is diDcult [385]. The two dimension is clearly below the upper critical dimension. The phase only theory should be regarded as a phenomenological description for it, while the (renormalized) Gaussian Auctuation region cannot be described. An almost exact treatment is allowed for the one-dimensional model in which the fermions couple to the classical Meld (Lee–Rice–Anderson model) [323]. The classical Meld describes the Auctuating order parameter for which the static approximation is done. The statistical ensemble of the classical Meld is an essential assumption in this model; the Gaussian Auctuation model has been investigated, intensively. While the Peierls transition has been focused in early years, the AF spin Auctuation and SC Auctuation have been studied, motivated by the high-Tc cuprates. The pseudogap is commonly induced by the strong Auctuation which is characteristic in the one-dimensional system. In 1970s, Sadovskii has given an “exact” solution where the fermion self-energy is represented by the recursive continued fraction [325]. This method has been applied to the AF spin Auctuation with focus on the high-Tc cuprates [386]. The extension to the two-dimensional case is diDcult, but it has been performed by Schmalian et al. [387] by using the separable form of the spin susceptibility. However, Tchernyshyov has pointed out the technical error of the Sadovskii’s solution, which is actually an approximation [388]. A sophisticated numerical method has shown that the Sadovskii’s solution is qualitatively a good approximation for the complex order parameter [389,390]; the commensurate CDW is not the case, but the SC is the case. Then, the accuracy of the approximations usable in higher dimensions has been investigated [389]. It is shown that the SCT approximation properly reproduces the low-energy DOS, while the high-energy behavior ! ∼ - is not. Since the Gaussian Auctuation is inappropriate near the critical point, a more sophisticated treatment for the statistical ensemble is required for the critical behaviors. The cross-over from the Gaussian Auctuation region to the phase Auctuation region has been described on the basis of the Lee–Rice–Anderson model with a careful treatment of the statistical ensemble [391]. Then, the excitation gap appears more clearly in the phase Auctuation region, while the !2 -behavior appears in the Gaussian-Auctuation model with complex order parameter. Another solvable model is the BCS pairing model with suDciently long-range attractive interaction. This model is almost exactly solvable in any dimension with the use of the Sadovskii’s method [392]. Then, the calculation is technically reduced to the zero-dimensional problem and quantum Auctuation plays a dominant role for the low-energy spectrum. Note that the correction pointed out

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65

by Tchernyshyov [388] does not exist in this model. The result shows that the non-self-consistent T-matrix approximation appropriately reproduces the asymptotic behavior of the self-energy, while the SCT approximation does not. This model is a clear example in which the partial summation of the higher-order terms is dangerous. Although these simpliMed models cannot describe the phase space restriction which is characteristic in higher dimensions, the importance of the vertex correction will be common to the two-dimensional cases, at least quantitatively. It is expected that the exact solution will lie between the non-self-consistent and self-consistent T-matrix approximations which are used in the next section. The higher-order correction in the two-dimensional case will be discussed in the last subsection in Section 4.2. 4.2. General theory In this section, we clarify the basic mechanism of the pseudogap phenomena with the use of the attractive model. It is explained how the pseudogap appears from the SC Auctuation. The microscopic theory starting from the repulsive Hubbard model, as will be given in Section 4.3, basically justiMes this mechanism. In Section 4.2.1, the drastic eCect of the SC Auctuation on the single-particle properties is demonstrated. The basic idea of the “resonance scattering” is introduced. The importance of the quasi-two-dimensionality is discussed in Section 4.2.2. In Section 4.2.3, the pseudogap phenomena under the magnetic Meld are investigated. The coherent understanding on the doping dependence can be examined by the eCects of the magnetic Meld. The obtained results are qualitatively consistent with the high Meld NMR measurements including their doping dependence. From these results, -(ET)2 X compounds are suggested to be another candidate for an appearance of the pseudogap. Section 4.2.4 is devoted to the discussion on the -(ET)2 X compounds. Basically, the theoretical analysis is performed with the use of the T-matrix or self-consistent T-matrix approximation. The higher-order correction beyond the T-matrix approximation is discussed in Section 4.2.5. It is shown that the vertex correction enhances the pseudogap furthermore. 4.2.1. Basic mechanism of the pseudogap As is explained in Section 2.1, several experimental results have indicated the close relation between the SC gap and pseudogap. Among them, rich information obtained from ARPES has led us to the pairing scenario. ARPES clearly shows the pseudogap in the single-particle spectrum. In the theoretical point of view, the single-particle quantities are simple compared with the two-particle quantities such as the magnetic and transport properties. Therefore, the study on the single-particle properties will capture the basic mechanism of the pseudogap most clearly. We focus on the estimation of the single-particle Green function in this section. It is simply expected that the pseudogap in the two-particle spectrum is derived from the pseudogap in the single-particle spectrum. It is instructive to adopt the attractive model in order to investigate the fundamental roles of the SC Auctuation. † † † H= @(k)ck; Vk; k cq=2 (61) s ck; s + −k ; ↓ cq=2+k ; ↑ cq=2+k; ↑ cq=2−k; ↓ ; k; s

k;k ;q

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

T

= T +

(a)

T (b)

Fig. 42. (a) The scattering vertex represented by the ladder diagrams (T-matrix). (b) The diagrammatic representation of the self-energy in the T-matrix approach.

where Vk; k is the dx2 −y2 -wave pairing interaction, given in the separable form as Vk; k = g:k :k :

(62)

Here g is negative and :k is the dx2 −y2 -wave form factor, given by :k = cos kx − cos ky . We use the tight-binding dispersion @(k) deMned in Eq. (43) for t =t = 0:25 and = 0:10. The above Hamiltonian is an eCective model in which the paring interaction aCects the renormalized quasi-particles. Since the energy scale is renormalized, the magnitude of the gap and Tc are relatively larger than those in the original model. It should be noted again that the renormalized Fermi energy is used in the control parameter for the SC Auctuation, TcMF =EF . This attractive model is a very simpliMed one, but we expect that the fundamental features of the SC Auctuation and its eCects are included. The SC Auctuation is diagrammatically described by the T-matrix (Fig. 42(a)) which is a propagator of the SC Auctuation. The scattering vertex arising from the T-matrix is factorized into :k−q=2 t(q):k −q=2 , where t(q; i4n ) = [g−1 + p0 (q; i4n )]−1 ; and p0 (q; i4n ) = T

G(q=2 + k; i!m )G(q=2 − k; i4n − i!m ):2k :

(63)

(64)

k

The SC phase transition is determined by the divergence of the SC susceptibility t(0; 0), namely 1 + gp0 (0; 0) = 0. This is called “Thouless criterion”, which is equivalent to the BCS theory in the weak coupling limit. From early years, the eCects of the SC Auctuation on the two-particle correlation functions have been investigated intensively [369,393,394], which are observable in the conventional superconductors. On the other hand, eCects on the single-particle properties have not attracted interests except for the early investigation [322], probably because this eCect is very weak in the conventional (low-Tc and 3D) superconductors. However, the SC Auctuation seriously aCects the electronic state in the high-Tc superconductors in the following way. The anomalous contribution from the SC Auctuation generally originates from the enhanced T-matrix around q = 4 = 0. Therefore, we tentatively expand the reciprocal of the T-matrix as g : (65) t(q; 4) = t0 + bq2 − (a1 + ia2 )4

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

67

This procedure corresponds to the time-dependent Ginzburg–Landau (TDGL) expansion. The TDGL parameters are expressed for the Gaussian Auctuation as @ tanh T − Tc 2T (@) ∼ ; (66) t0 = 1 + g d@ = |g|d (0) d 2@ Tc 7C(3) 2 d (@)vF2 92 f(@) ∼ v ; (67) b = |g| d@ = |g|d (0) 16@ 9@2 32(T )2 F @ tanh 1 1 9p0 (0; 0) 2T (@) ∼ ; (68) a1 = |g| d@ = d (0)=d (0) ∼ = |g| d 2 (2@) 2 2 9, ; (69) a2 = |g|d (0) 8T where C(3) is the Riemann’s zeta function and vF is the averaged quasi-particle velocity on the Fermi surface. Here the SC Auctuation is mainly determined by the electronic state around (; 0), owing to the d-wave symmetry. Then, vF can be regarded as the velocity at the hot spot (see Fig. 65). The Aat dispersion observed around (; 0) means that the eCective Fermi energy EF = vF kF is small. We have deMned the eCective DOS as d (@) = A(k; @):2k ; (70) k

and d (0) is its derivative at @ = 0. We have denoted the spectral function as A(k; !) = −(1=) Im G R (k; !). The mass term t0 = 1 + g0 (0; 0) represents the closeness to the critical point. The Auctuation eCect gradually emerges as t0 is reduced. The parameter b is generally related to the coherence length =0 as b ˙ =20 ˙ vF2 =T 2 , and therefore suDciently large in the weak-coupling superconductor. This is the reason why the Auctuation is negligible in conventional superconductors. We have deMned the parameter TcMF =EF ∼ TcMF =vF kF as “superconducting coupling” which is an index of the Auctuation eCect. In case of high-Tc cuprates, TcMF is large and vF is renormalized, and therefore the coherence length is remarkably small =0 = 3–5. This is an essential background of the pseudogap phenomena. The parameter a2 represents the dissipation and expresses the time scale of the Auctuation. This parameter is also small in the strong coupling region owing to the high Tc and the renormalization eCect by the pseudogap [368,395]. The real part a1 is usually ignored because this term is in the higher order than the imaginary part a2 with respect to the small parameter TcMF =EF in the weak-coupling region. In the strong-coupling limit, the absolute value |a1 | can exceed a2 and then, the SC Auctuation has a “resonance nature” [348]. While this situation is somewhat extreme, there is no justiMcation to ignore a1 in the intermediate coupling region. The sign of the parameter a1 is related to the Hall conductivity close to Tc [396]. The anomalous sign of the Hall conductivity in high-Tc cuprates has stimulated much interests [397] and remains as an open problem. The anomalous properties in the single-particle spectrum [103–105] are described by the self-energy correction. Here, we estimate it within the one-loop order (Fig. 42(b)) as 2(k; i!n ) = T t(q; i4m )G(q − k; i4m − i!n ):2k−q=2 : (71) q;i4m

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

This procedure corresponds to the T-matrix approximation. In general, the T-matrix around q=0 gives rise to the anomalous properties and that far from q = 0 gives rise to the Fermi-liquid properties. The former process is very small in the weak-coupling case since b1. Then, the self-energy shows the Fermi-liquid behaviors except for just the vicinity to the critical point. On the other hand, the former overcomes the latter in the intermediate- or strong-coupling region, and then pseudogap appears. This criterion for the pseudogap is moderate in the quasi-two-dimensional system like high-Tc cuprates. We evaluate the anomalous contribution to the imaginary part by using the TDGL parameters [368], 1 =− T=GL GL (for 1D) ; 2 4bvk '2 =vk2 + =− GL T 1 (for 2D) ; = −|g|:2k 4bvk '2 =v2 + =−2

Im 2R (k; !) = −|g|:2k

= −|g|:2k

k

GL

Td q2 2 log 2 c 2 + =− GL 8bvk ' =vk

(72) (73)

(for 3D) ;

(74)

= b=t0 . These where we have deMned as '=!+@(k) and the GL correlation length is deMned as = GL 2 expressions are correct at |'| 6 2Mvk T ∼ T where 1=2M = b=max{a1 ; a2 }. We have estimated for the one- and three-dimensional cases for a comparison. Eqs. (73) and (74) clearly show that the anomalous part of the self-energy increases when the parameter b is small and/or the system is two-dimensional. The real part can be obtained by the Kramers–Kronig relation as Re 2R (k; !) = |g|:2k

T=GL '=vk 2 4bvk '2 =vk2 + =− GL

T 1 4bvk '2 =v2 + =−2 GL k  2'  2 T 1  log |g|:  k 1  2b ' =− GL vk =  T=2GL   '  |g|:2k 2bvk2

Re 2R (k; !) = |g|:2k

(for 1D) ;

(75)

 2 '2 =vk2 + =− GL  log  −2 2 2 '=vk − ' =vk + =GL 

'=vk +

(for 2D) ;

(76)

1 (|'|=− GL vk ) ;

(77) −1

(|'|=GL vk ) ;

These expressions correspond to the classical approximation, which is justiMed near the critical point. We note that the expression for the real part is not so accurate in the low-frequency region; the dynamics of the SC Auctuation aCects on it. However, the qualitative behaviors are correctly grasped by Eqs. (75) and (76). We show the typical behaviors in Fig. 43. It should be noticed that the real part of the self-energy shows the positive slope around ! = 0, and there, the imaginary part shows the sharp peak in its absolute value. Both features are very anomalous compared with the conventional Fermi-liquid theory. This is caused by the “resonance scattering” from the SC Auctuation, which is identiMed to be the origin of the pseudogap phenomena. Owing to the anomalous properties of the self-energy, the spectral function clearly shows the pseudogap (Fig. 43(c)).

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.8

0.0

69

1.0

0.0 -0.4 -0.8 -4

-2

(a)

0 ω

2

-0.4

(b)

0.5

-0.8

-1.2 -4

4

A(k,ω)

ImΣ(k,ω)

ReΣ(k,ω)

0.4

-2

0 ω

2

0.0 -4

4

(c)

-2

0 ω

2

4

Fig. 43. The characteristic behaviors of (a) the real part and (b) imaginary part of the self-energy induced by the “resonance scattering”. (c) The spectral function.

Let us provide a more simple explanation on these anomalous features. If we disregard the detailed structure around ! = 0, the self-energy is simply approximated as -2 :2k ; (78) 2R (k; !) = ! + @(k) + i -2 = − t(q; i4n ) : (79) q

This approximation is obtained by ignoring the q-dependence of the Green function and that of the form factor in Eq. (71). This form of the self-energy is assumed in the pairing approximation [353,354]. Also, the ARPES experiments have been analyzed by assuming the similar form of the self-energy [398]. If the self-energy is expressed as Eq. (78), the Green function has the same momentum and frequency dependence as the normal Green function in the SC state as ! + @(k) G R (k; !) = : (80) (! + @(k))(! − @(k)) − -2 :2k Then, the energy gap appears in the single-particle spectrum, where the quasi-particle energy is obtained as Ek = ± @(k)2 + -2 :2k : (81) The amplitude of the gap is determined by the total weight of the (thermal) SC Auctuation, while the gap is related to the order parameter in the mean-Meld theory. The above expression Eq. (78) is somewhat extreme and inaccurate around ! = 0 (see Eqs. (73) and (76)). We calculate Eqs. (63), (64), and (71) without using the TDGL expansion and show the self-energy, spectral function, and DOS in Fig. 44. Here :k−q=2 is replaced with (:q−k + :k )=2 in order to restore the periodicity, but the results are not aCected by this procedure. We see that the Fermi-liquid behaviors appear in the weak-coupling region (|g|=t = 0:5) and the anomalous behavior emerges in the intermediate-coupling region (|g|=t = 1–2). In the latter case, the pseudogap appears both in the spectral function (Fig. 44(c)) and in the DOS (Fig. 44(d)). The inset of Fig. 44(c) shows that the magnitude of the pseudogap is approximately scaled by the mean-Meld transition temperature TcMF . The pseudogap in the two-particle excitations including the NMR 1=T1 T is derived from the

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

0.0 0

-0.4

0.0

ImΣ(k,ω)

ReΣ(k,ω)

0.5

0.4

0.0

-0.5

0

-1.0

g/t=-0.5 g/t=-1 g/t=-1.5 g/t=-2

g/t=-0.5 g/t=-1 g/t=-1.5 g/t=-2

-0.5

-0.4

-2

-1

0

-2

0.0

-1

0.4

0

2∆

0.1

0.2

ρ(ω)

2

2

g/t=-0.5 g/t=-1 g/t=-1.5 g/t=-2

1.0

0 0.0

1

ω

(b)

1

g/t=-0.5 g/t=-1 g/t=-1.5 g/t=-2

3

-1.5

2

ω

(a)

A(k,ω)

1

MF

Tc

0.5 1

0

-1.5

(c)

-1.0

-0.5

0.0

ω

0.5

1.0

0.0

1.5

(d)

-2

-1

0

1

2

ω

Fig. 44. (a) The real and (b) imaginary part of the self-energy obtained by the T-matrix approximation. (c) The spectral function at k = (; 0:15). The inset in (a) and (b) shows the enlarged results for g=t = −0:5. The inset of (c) shows the relation between 2- and TcMF . (d) DOS. The temperature is chosen as T = 1:2Tc in all Mgures.

pseudogap in the single-particle excitations (see Sections 4.2.3 and 4.3.3). It should be noted that the shift of the chemical potential is negligible in this region. Thus, the scenario based on the NSR theory is irrelevant in the two-dimensional high-density system. As is mentioned above, the drastic change from the weak- to the strong-coupling region mainly arises through the parameter b. It is clearly understood from Fig. 45 that the total weight of the SC Auctuation is very small in the weak-coupling region, owing to the large value of b. In this case, the anomalous contribution from SC Auctuation is smeared by the Fermi-liquid contribution. This is why the pseudogap does not appear in the weak-coupling superconductor. The momentum dependence of the spectral function is shown in Fig. 46. The anomalous contribution to the self-energy is smeared around the gap node. Therefore, the “Fermi arc” appears around the diagonal direction and disappears as approaching to the critical point. These features are observed behaviors in the ARPES measurements [105]. Fig. 47 shows the temperature dependence. It is shown that the pseudogap is smeared as the temperature increases. The pseudogap disappears around T ∗ which is estimated as T ∗ ∼ 2TcMF at

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

0

6 5 4 3 2 1 0

−t(q,0)

−t(q,0)

7 6 5 4 3 2 1 0

0.2

0.4

0.6

0.8

qx

1

0

0.2

0.4

0.6

0.8

71

1

qy

0

0.2

0.4

0.6

qx

(a) g/t=-0.5

0.8

1

0

0.2

0.4

0.6

0.8

1

qy

(b) g/t=-1.5

Fig. 45. The static part of the T-matrix |t(q; 0)| at T = 1:2Tc . (a) g=t = −0:5. (b) g=t = −1:5.

4

3

A(k,ω)

π

A B C D

A B C

ky

0 2

D

0

kx

π

1

0 -1.0

-0.5

0.0

0.5

ω

1.0

Fig. 46. The momentum dependence of the spectral function for g=t = −1:5 and T = 1:2Tc . The momentum and the Fermi surface are shown in the inset.

2

0.8

ρ(ω)

A(k,ω)

T=1.1Tc T=1.2Tc T=1.5Tc T=2Tc T=3Tc

1.0

T=1.1Tc T=1.2Tc T=1.5Tc T=2.25Tc T=3Tc

1

0.6

0.4 0

-1.5

(a)

-1.0

-0.5

0.0

ω

0.5

1.0

-1.0

1.5

(b)

-0.5

0.0

0.5

1.0

ω

Fig. 47. The temperature dependence of (a) the spectral function and (b) DOS. Here, g=t = −1:5 and the momentum is A in the inset of Fig. 46.

72

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.3 Tc (mean field) Tc (SCT)

Tc

0.2

0.1

0.0

0.0

0.5

1.0

1.5

2.0

2.5

|g|/t Fig. 48. The phase diagram of the attractive model. The transition temperatures in the mean Meld theory and that in the SCT approximation are shown.

g=t = −1:5. It should be stressed that the pseudogap is closed by the broadening, but not by the disappearance of the gap amplitude. This feature is in contrast to Eq. (78) which is revealed to be a very rough estimate. This nature of the pseudogap closing is also consistent with the experiments [103–105,107,108]. The transition temperature suppressed by the Auctuation has been calculated by the SCT approximation [368], where the T-matrix, self-energy, and Green function are determined self-consistently. Here, Tc is determined by the condition t0 =0:02 by taking account of the weak three-dimensionality. If we choose a strictly two-dimensional model, Tc is always zero. We see from Fig. 48 that Tc is remarkably suppressed as the superconducting coupling increases, while the suppression rapidly disappears in the weak-coupling region. Thus, the wide critical region is expected in the strong-coupling superconductor. Since the under-doped cuprates correspond to the case, the pseudogap is widely observed. The pseudogap is basically obtained also in the SCT approximation. For example, the DOS is shown in Fig. 49. Exceptionally the single-particle spectral function shows no (or very weak) gap structure. On this point, the quantum Monte Carlo simulation for the attractive Hubbard model gives more similar results to those of the non-self-consistent T-matrix approximation, where the pseudogap is clearly observed in the spectral function. It seems that the estimation of the spectral function is not improved by the higher-order corrections within the SCT. We will discuss this point in Section 4.2.5. The extension of the SCT approximation to the superconducting state [383,399] has shown that the SC Auctuation is rapidly suppressed in the ordered state while the phase-mode remains gapless. Then, the order parameter grows more rapidly than the BCS theory. This property is a general consequence of the critical Auctuation. The rapid growth of the order parameter is commonly obtained in the FLEX approximation [186,187,193] in which the feedback eCect is its origin. These two eCects additively contribute in high-Tc cuprates. It should be noticed that we cannot directly measure the order parameter in high-Tc cuprates because the pseudogap already exists in the spectrum. Instead,

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

ρ(ω)

1.0

73

g/t=-1 g/t=-1.5 g/t=-2 g/t=-2.5 g/t=-3

0.5

0.0

-2

-1

0

1

ω

2

Fig. 49. DOS in the SCT approximation.

the London constant , which is proportional to the inverse square of the magnetic Meld penetration depth L ( = 1=4L2 ), is a better probe of this property [383]. This quantity reAects the long-range coherence of the order parameter and rapidly develops just below Tc with the increase of the order parameter. Experimental results have conMrmed the rapid growth, which precisely corresponds to the universality class of the 3D XY model [116,121]. 4.2.2. EEect of three-dimensionality So far, we have stressed that the strong SC Auctuation gives rise to pseudogap phenomena. In that case, it is necessary that the total weight of the SC Auctuation q |t(q; i4n )| is large compared with the conventional superconductor. The previous discussion in Section 4.2.1 has pointed out two conditions for the pseudogap phenomena. One is the strong-coupling superconductivity, namely the short coherence length. Another is the quasi-two-dimensionality. The former point has been clariMed in the previous subsection. In this subsection, we discuss the latter point in more detail. First, we show the expression for the imaginary part of the self-energy in case of the layered superconductor,   2 q2 + =−2 + '2 =v2 )q + ) c c Td GL k  ; Im 2R (k; !) = −|g|:2k log  (82) 8vk b) −2 2 2 2 2 −)q + ) q + = + ' =v c

c

GL

k

where ) is the anisotropy of the coherence length ) = =c0 ==0 and qc = =d is the cut-oC momentum along the c-axis. d is the inter-layer spacing. We can easily conMrm that Eq. (82) is reduced to Eq. (73) and Eq. (74) in the limit ) → 0 and ) → 1, respectively. It is clearly shown that the √ R quasi-particle damping −Im2 (k; 0) shows power-law anomaly in 2D case (˙ 1= t0 ), but only weak logarithmic anomaly in 3D case (˙ −log t0 ). This is evidently due to the restriction of the phase space by the dimensionality. Thus, the quasi-two-dimensionality generally enhances the anomaly. It is necessary for the two-dimensional behaviors that the c-axis coherence length is smaller than the inter-layer spacing =c0 6 d, which is well satisMed in high-Tc cuprates. Then, the critical

74

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0

2

-1

ImΣ (kF,ω)

ReΣ(kF,ω)

1

T=0.35 (tz/t=0.1) T=0.40 (tz/t=0.1) T=0.20 (tz/t=1.0) T=0.25 (tz/t=1.0)

0

-2 T=0.35 (tz/t=0.1) T=0.40 (tz/t=0.1) T=0.20 (tz/t=1.0) T=0.25 (tz/t=1.0)

-1 -3 -2 -10

(a)

0

ω

10

-10

(b)

0

ω

10

Fig. 50. The self-energy obtained by the T-matrix approximation for tz =t = 0:1 (TcMF =t = 0:337) and tz =t = 1 (TcMF =t = 0:195) [400]. (a) real part and (b) imaginary part at g=t = −2. The decrease of mean Meld TcMF by the three-dimensionality is not essential for the results.

behavior shows the dimensional cross-over at =cGL = )=GL ∼ d=2 ;

(83)

=cGL

is the GL correlation length along the c-axis. Note that the critical behaviors are always where three-dimensional near the critical point. Even in the three-dimensional region, the anisotropy enhances the self-energy through the factor )−1 . We note that the dimensional cross-over occurs for the variation of the frequency !; two-dimensional behavior is more robust at Mnite frequency. We have actually taken into account the weak three-dimensionality and numerically estimated the single-particle properties like Section 4.2.1 [400]. The anisotropy is represented by the parameter tz =t, where tz is the hopping matrix along the c-axis. We show the results of the T-matrix approximation for the diCerent value of tz =t. The self-energy is shown in Fig. 50. We see that the anomalous behaviors which are shown in Fig. 44 appear in the quasi-two-dimensional case, tz =t = 0:1. Thus, the pseudogap is robust for the weak three-dimensionality. This robustness also results from the strong coupling nature of the superconductivity, as is explained later. On the contrary, typical Fermi-liquid behaviors appear in the isotropic three-dimensional case, tz =t=1. In this case, the contribution from the SC Auctuation shows only a weak anomaly (Eq. (74)), which is almost smeared by the Fermi-liquid contribution. To make sure, the pseudogap phenomena are possible even in the three-dimensional system [401]. However, somewhat hard condition, such as very large coupling constant or just vicinity to the critical point, is needed. Recent calculation for the three-dimensional jellium model has concluded that the pseudogap appears in the vicinity of the BCS-BEC cross-over region [357]. It is concluded that the appearance of the pseudogap is quantitatively diDcult for the three-dimensional systems like heavy-fermion compounds. Furthermore we comment on the dimensional cross-over in the quasi-two-dimensional systems. The two-dimensional behaviors generally appear in the high-temperature region and three-dimensional ones are expected below the cross-over temperature. The condition for the dimensional cross-over has been given in Eq. (83). Then, the cross-over temperature (Tcr ) is estimated as (Tcr − Tc )=Tc ∼ 4(=c0 =d)2 ∼ 4=20 (tz =t)2 ;

(84)

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75

since )=d ∼ tz =t. If we choose tz =t = 0:05 and =0 = 4, the above expression results in (Tcr − Tc )=Tc ∼ 0:04. Thus, two-dimensional behaviors are widely expected in high-Tc cuprates. This is also owing to the short coherence length. The strong coupling nature of the superconductivity plays an essential role also in this stage. From the results in Sections 4.2.1 and 4.2.2, the doping dependence is understood in the following way: According to the tunnelling experiment [89,110], the gap amplitude at T Tc increases as decreases. This result indicates that TcMF increases with decreasing . Combined with the renormalization of the quasi-particle velocity, which is clearly shown by ARPES [45,106], the superconducting coupling increases with under-doping. Then, two-dimensionality simultaneously becomes enhanced [26]. As a result, the clear pseudogap is expected in the under-doped region, while the pseudogap gradually disappears with doping. Since the pseudogap onset temperature is scaled by TcMF in the intermediate-coupling region, T ∗ increases with decreasing . These features are quite consistent with the phase diagram in high-Tc cuprates. The microscopic knowledge is necessary for Tc , which is scaled by the renormalized Fermi energy. We will show the results of the microscopic calculation in Section 4.3, where the appropriate doping dependence is reproduced. 4.2.3. EEect of the magnetic :eld It is generally expected that superconducting phenomena show characteristic behaviors under the magnetic Meld. The quantum phase of the order parameter is modulated by the magnetic Meld, and then the characteristic value of the magnetic Meld is very small in the conventional superconductors. Such a remarkable response is contrastive to the magnetic properties where the magnetic Meld dependence is usually small, because the Zeeman term (∼10 K) is much smaller than the exchange coupling J (∼ 1000 K). Therefore, the behaviors under the magnetic Meld are one of the important tests for the theoretical scenarios based on the SC Auctuation. Actually, the magnetic Meld dependence of the pseudogap has been investigated by recent experiments. Among them, we discuss here the NMR 1=T1 T which has been measured for high-Tc superconductors [402– 405] and organic superconductors [137]. Important knowledge has been obtained for both systems. In this subsection, we focus on the high-Tc cuprates. The experimental results are summarized as follows. The magnetic Meld dependence is little observed in the under-doped region [402,403]. In particular, the onset temperature T ∗ does not depend on the magnetic Meld up to 20 T. The eCect of the magnetic Meld is observed only close to T = Tc . It is also observed that the transition temperature is suppressed by the magnetic Meld. On the contrary, the eCects of the magnetic Meld are clearly visible around the optimally-doped region in which only the weak pseudogap phenomenon is observed in the narrow temperature region [404,405]. The theoretical results for the magnetic Meld dependence are obtained in the following way [371]. Although the following explanation basically considers the Gaussian Auctuation, qualitatively the same tendency is expected for the critical Auctuation. We consider the magnetic Meld applied along the c-axis (B ˜c) in accordance with the experimental condition. The main eCect of the magnetic Meld is the Landau level quantization of the SC Auctuation, which is quasi-classically expressed by the replacement of the quadratic term in the T-matrix as q2 ⇒ 4eB(n + 1=2) [406,407]. The Landau quantization of the quasi-particle is ignored in the usual condition !c 01, where !c is the cyclotron frequency.

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1/(T1T)

0.60

0.50

4eB=0.01 4eB=0.02 4eB=0.05 4eB=0.10

0.40

0.30

0.00

0.05

0.10

0.15

0.20

T/Tc0 -1 Fig. 51. The calculated results for 1=T1 T under the magnetic Meld. The relatively strong coupling case g=t = −1:6. The horizontal axis is the temperature scaled by the Tc under the zero magnetic Meld.

The Landau quantization has the following two important eCects. One is the suppression of superconductivity. The mass term is increased by the zero-point oscillation energy as t0 ⇒ t0 +2beB, which corresponds to the lowest Landau level. The SC Auctuation is suppressed by this eCect. The other is the Landau degeneracy which generally enhances the Auctuation because the eCective dimension is reduced. The transition temperature is further reduced by the enhanced Auctuation. Considering at the Mxed temperature in accordance with the experiments, the dominant eCect is the former. Then, we 2 see that the characteristic magnetic Meld Bch for the SC Auctuation is expressed as Bch ∼ t0 =b = =− GL . That is, the eCects of the magnetic Meld are scaled by the magnetic Aux penetrating the correlated area =2GL . As we have stressed above, pseudogap appears in the short coherence length superconductors. Then, a large value of the characteristic magnetic Meld is expected. Moreover, the parameter t0 around the onset temperature T ∗ is not so small in the strong-coupling superconductors. Then, the characteristic magnetic Meld Bch around T =T ∗ is in the same order as Hc2 (0) which is over 100 T in the under-doped region. This is the simple but robust reason why the magnetic Meld dependence of the onset temperature T ∗ is not observed in the under-doped region. Because the characteristic magnetic Meld is generally small in the conventional superconductors, this magnetic Meld independence was sometimes interpreted as a negative evidence for the pairing scenario. However, this interpretation is inappropriate. The magnetic Meld eCects are apparent in the temperature region close to Tc , since the GL correlation length =GL diverges at the critical point. This eCect is actually observed even in the under-doped region [402]. These behaviors are conMrmed by the numerical calculation based on the T-matrix approximation [371], where the magnetic Meld with experimentally relevant order B ∼ 10 T is considered. Then, the Kubo formula is used to estimate the correlation function and the eCect of the SC Auctuation is included in the self-energy correction. The vertex correction corresponding to the Maki-term is usually not important for the d-wave superconductor [407,408]. Fig. 51 shows the results for g=t = −1:6 where the pseudogap is clearly observed (Section 4.2.1). We see the magnetic Meld independence below T ∗ and magnetic Meld dependence around Tc .

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1/(T1T)

0.58

0.48

4eB=0.01 4eB=0.02 4eB=0.05 4eB=0.10 0.38

0.00

0.05

0.10

0.15

0.20

T/Tc0-1

Fig. 52. The calculated results for 1=T1 T under the magnetic Meld. The weak coupling case g=t = −1:0.

It should be noted that the magnetic Meld dependence of Tc is explained by taking account of the critical Auctuation. The latter eCect “Landau degeneracy” signiMcantly enhances the Auctuation and suppresses the Tc . Therefore, the magnetic Meld dependence of Tc is more drastic than that expected from the mean-Meld theory. This is a qualitative explanation [371,409] of the diCerent behaviors between T ∗ and Tc [402,403]. On the contrary, larger magnetic-Meld dependence is expected in the relatively weak-coupling case. In this case, the onset temperature T ∗ may be reduced by the magnetic Meld. The result of the T-matrix approximation for g=t = −1:0 shows a clear magnetic Meld dependence below T ∗ (Fig. 52). This behavior corresponds to the optimally-doped region. These doping dependences are in good agreement with the experimental results. The detailed estimation on the magnetic Meld dependence in the optimally-doped region has been performed in Ref. [407]. The pseudogap phenomena should be comprehensively understood from the under- to optimallydoped region. Based on this belief, we consider that the magnetic Meld dependence of NMR 1=T1 T gives a strong support for the pairing scenario. The usual behaviors of the SC Auctuation in the optimally-doped region and the strong-coupling behaviors in the under-doped region are consistent with the understanding of the phase diagram (Fig. 1 and Section 4.2.2). Here note that we have ignored the Zeeman coupling term. This procedure is justiMed because the Zeeman term gives only a higher-order correction in the Auctuating regime [371]. The role of the Zeeman term can be clariMed by changing the direction of the magnetic Meld because the “orbital eCect” is suppressed when the magnetic Meld is applied along the plane. Recently, the magnetic Meld dependence of the c-axis conductivity is measured for the Bi-based compounds. Owing to the strong two-dimensionality, it is expected that the incoherent process dominates the c-axis conductivity in these compounds. Then, the c-axis DC conductivity basically measures the electronic DOS at ! = 0. Therefore, this probe additively detects the large pseudogap below T0 as well as the small pseudogap below T ∗ (Fig. 1). The c-axis resistivity increases below T0 especially in the under-doped region. This upturn of the c-axis resistivity coincides with the decrease of the uniform spin susceptibility [410]. Then, the magnetic resistance is negative [411] owing

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to the destruction of the pseudogap. Shibauchi et al. have reported the measurements for Bi-2212 compounds up to 60 T [412]. The result shows that the Zeeman term suppresses the large pseudogap in the over-doped region where the typical magnetic Meld is small. This result may be an evidence for the magnetic origin of the large pseudogap. Lavrov et al. have reported the measurements for Bi-2201 compounds from over-doped to heavily under-doped (insulating) region [413]. Then, two distinct responses to the magnetic Meld are observed in the under-doped (superconducting) region. This result implies two distinct origins of the suppression of the DOS which have been discussed in Section 2.1.5. In particular, the anisotropy of the magnet resistance will be a clear evidence for the appearance of the SC Auctuation. The sensitivity of the “orbital eCect” to the direction of the magnetic Meld is clearly observed in the vicinity of Tc , which is consistent with the pseudogap induced by the SC Auctuation. It should be noted again that the orbital eCect is not clear around T = T ∗ because the characteristic magnetic Meld is too large. Therefore, we have a question about the interpretation in Ref. [413], which concluded from the weak magnetic Meld dependence that T ∗ decreases with under-doping. Taking account of the short correlation length =GL at T = T ∗ , the T ∗ in the under-doped region will be higher than that determined in Ref. [413]. The analysis in the higher magnetic Meld is desirable to estimate T ∗ more precisely. Another interesting observation in Ref. [413] is that the magnetoresistance is positive in the non-superconducting region. This result implies the qualitative diCerence of the electronic state between the insulating and superconducting regions, which supports our theoretical approach to the high-Tc superconductivity from the metallic side. 4.2.4. Another candidate: organic superconductor -(ET)2 X In this subsection, we suggest another candidate for the pseudogap state induced by the SC Auctuation. The organic superconductor -(ET)2 X is a clear two-dimensional system with Tc ∼ 10 K. As is noted in Section 3.3, this value of Tc is comparable to that of the high-Tc cuprates when normalized by the Fermi energy. Thus, two important conditions explained in Section 4.2.2 can be satisMed in this compound. Therefore, we expect a similar appearance of the pseudogap. Concerning the pressure dependence, the pseudogap is expected in the lower pressure region because the Tc is large and EF is small. By applying the pressure, the superconductivity gradually becomes the weak-coupling one and therefore the pseudogap will disappear. This variation is similar to the doping dependence of high-Tc cuprates. The coherence length has been estimated from the Hc2 ^ and = = 53 ± 6(A) ^ for -(ET)2 Cu[N(CN)2 ]Br and -(ET)2 Cu(NCS)2 , respectively as = = 28 ± 5(A) [414]. This variation is consistent with the above expectation because -(ET)2 Cu[N(CN)2 ]Br corresponds to the lower pressure region (Fig. 8). These values of the coherence length are comparable to the length scale of the structural unit. This is just a condition for the pseudogap phenomena. The NMR measurements have actually observed the pseudogap in -(ET)2 X with T ∗ ∼ 50 K [135,136]. However, this T ∗ should not be identiMed to the onset of the SC Auctuation, because the pressure dependence is not consistent. The nature around T ∗ probably corresponds to the metal– insulator cross-over; the electronic state is almost incoherent above T ∗ . Then, the manifestation of the SC Auctuation is expected in the lower temperature region. This expectation is clearly supported by the recent NMR measurement by the Kanoda’s group [137]. They have measured the magnetic Meld dependence of the NMR 1=T1 T and shown that the SC Auctuation appears below the new cross-over temperature Tc∗ which is between Tc and T ∗ .

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

(a)

(b)

(d)

(e)

79

(c)

Fig. 53. The loop expansion of the self-energy within the 3-loop order.

It is an advantage of the organic materials that the superconductivity is destroyed by the magnetic Meld about 10 T which is experimentally practical. The experimental result shows that the NMR 1=T1 T increases above Tc and below Tc∗ ∼ 20 K by applying the magnetic Meld H ¿ Hc2 along the c-axis. On the contrary, the NMR 1=T1 T is not aCected by the same magnetic Meld along the plane. Hc2 is much larger than the applied Meld in this direction. The anisotropy is a clear evidence for the “orbital eCect” which is a characteristics of the SC Auctuation. Therefore, the further decrease of 1=T1 T below Tc∗ is attributed to the SC Auctuation. These results have revealed the hidden phenomena under the zero magnetic Meld. This eCect has been unclear since it is masked by the sizable decrease from 50 K. The magnetic Meld dependence has played a key role to identify the contribution from the SC Auctuation. Thus, the pseudogap induced by the SC Auctuation is a universal phenomenon for the quasi-two-dimensional short coherence length superconductors. 4.2.5. Higher-order corrections So far, we have discussed the eCect of the SC Auctuation within the 1-loop order (T-matrix approximation). At the last of Section 4.2, we brieAy discuss the eCect of the higher-order corrections. The SCT approximation, which is an extended version of the T-matrix approximation, shows the pseudogap in many aspects but only a weak one in the single-particle spectral function. This result implies that the qualitative role of SC Auctuation is robust but the higher-order corrections quantitatively suppress the pseudogap. However, because the SCT is a partial summation of the higher-order terms, particular eCects tend to be overestimated. Therefore, the eCect of the vertex correction is interesting to capture the qualitative tendency more precisely. In this subsection, we carry out the loop expansion within the 3-loop order [415]. The corresponding diagrams are shown in Fig. 53. The 1-loop term (Fig. 53(a)) is included in the T-matrix approximation. The 2-loop term (Fig. 53(b)) and a part of the 3-loop term (Fig. 53(c,d)) are included in the SCT approximation. Importantly, the lowest-order vertex correction term appears in the 3-loop order (Fig. 53(e)). Here, we ignore the higher-order corrections on the T-matrix. The corrections to the T-matrix are well expressed by the renormalization of the TDGL parameters (Eq. (65)) and qualitatively not important. The renormalization eCect further reduces the parameter b from Eq. (67), while Tc is reduced as is shown in Fig. 48. It is more important that the pseudogap in the spectral function is smeared in the SCT approximation through the terms like Fig. 53(b–d). This kind of the correction

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0.00 -0.02

(a)

10

5

-0.04 -0.06 -0.4

1loop 2loop 3loop (no VC) 3loop

15

A(k,ω)

0.02

ImΣ(kF,ω)

20

(a) (b) (d) (e)

-0.2

0.0

ω

0.2

0 -0.2

0.4

(b)

-0.1

0.0

ω

0.1

0.2

Fig. 54. (a) The imaginary part of the self-energy and (b) the spectral function at k = (; 0:13). Here, g=t = −1:4 and T = 0:049 = 1:04Tc . In (a), each contribution from the diagram in Fig. 63(a), (b), (d), and (e) is shown, respectively.

is not included in the two-particle self-consistent (TPSC) approximation [375,376], where the spectral function has a similar feature to the non-self-consistent T-matrix approximation. Here we use the static approximation where only the classical part of the T-matrix (t(q; i4n = 0)) is taken into account. This approximation is appropriate near the critical point because the singular contributions from the SC Auctuation (Eqs. (72)–(76)) are derived from the classical part. At least, this approximation is suDcient to capture the qualitative behaviors. In the following Mgure, we show the results in the s-wave case, namely :k = 1 for simplicity. Almost the same results are expected in the d-wave case. We show the imaginary part of the self-energy in Fig. 54(a). It is shown that the 1-loop term has similar features to the results of the T-matrix approximation, although the structure around ! = 0 is broadened. The 2-loop term gives an opposite contribution to the self-energy. Therefore, the spectral function within the 2-loop order becomes very sharp (see Fig. 54(b)), which indicates a well-deMned quasi-particle. To the contrary, the 3-loop terms enhance the 1-loop term, and therefore, pseudogap clearly appears in this order. This recurrence occurs in the SCT approximation and Mnally results in Fig. 49. It is important to note that the lowest-order vertex correction term enhances the pseudogap furthermore. Therefore, it is expected that the smearing of the spectral function and DOS in the SCT approximation is considerably canceled by the vertex correction. In other words, the eCects of the SC Auctuation are underestimated in the SCT approximation. Roughly speaking, the correct result will lie between the non-self-consistent and self-consistent T-matrix approximations. Then, Tc is reduced more remarkably than that in Fig. 48. This conjecture is consistent with the Quantum Monte Carlo simulation in the attractive Hubbard model [366,367]. Interestingly, the three kinds of the 3-loop term has qualitatively same feature. In particular, the comparison between the term in Fig. 53(d) and that in Fig. 53(e) may be interesting because they are equivalent in the Sadovskii’s method for the one-dimensional model [325] and for the BCS pairing model [392]. This equivalency is not satisMed in two dimensions, but Fig. 54(a) shows that it is qualitatively justiMed. We can see that the vertex correction term has larger contribution. This tendency is remarkable around the “hot spot” and insigniMcant around the “cold spot”. The diCerence increases with increasing the superconducting coupling TcMF =EF . It should be noticed that

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81

the asymptotic behavior of the spectral function in the Sadovskii’s method [392] is more similar to the non-self-consistent T-matrix approximation than to the SCT one. From the above observations, we expect that the correct result in the two-dimensional model will behave as in the T-matrix approximation. Then, the pseudogap clearly appears in the spectral function (see Section 4.2.1). 4.3. Microscopic theory: FLEX+T-matrix approximation In the previous section, we have discussed the general aspect of the pseudogap phenomena induced by the SC Auctuation. Then, a model with attractive interaction has been used as an eCective model. Of course, the relevant Hamiltonian for cuprates is the repulsive model, where the pairing interaction should be derived from the many-body eCects as performed in Section 3. Then, the microscopic theory starting from the repulsive model is highly desired for the pseudogap phenomena. It should be conMrmed that the eCective coupling of the superconductivity TcMF =EF is strong enough to lead to the pseudogap phenomena. The doping dependence will be an interesting consequence of such microscopic theory. We will see that the microscopic treatment is essential for the coherent understanding of the magnetic and transport properties, which have been stimulated much interests. The purpose of this section is a review of the microscopic theory based on the FLEX+T-matrix approximation, by which the SC Auctuation is microscopically taken into account. The Hubbard model (Eqs. (42) and (43)) is chosen as a microscopic model. We similarly Mx the parameter as 2t = 1 and t = 0:25t. The FLEX+T-matrix approximation was Mrst proposed in Ref. [416] where the NMR 1=T1 T is calculated by using the phenomenological form of the T-matrix. Fully microscopic treatment has been proposed in [417] and developed in Ref. [11]. We explain how the anomalous properties in the normal state are understood from this approach. 4.3.1. Single-particle properties Let us brieAy explain the basic formulation of the FLEX+T-matrix approximation. The characteristic behaviors of the single-particle properties are discussed in parallel. We show that the resonance scattering gives rise to the pseudogap in the under-doped region. The doping dependence including the electron-doped cuprates is consistently explained in Section 4.3.2. The magnetic and transport properties are discussed in Sections 4.3.3 and 4.3.4, respectively. First we describe the quasi-particles and AF spin Auctuation by using the FLEX approximation. The characteristic properties of the quasi-particles in the nearly AF Fermi-liquid [85,86] are summarized in the following way. Fig. 55 shows the typical results of the self-energy. The local minimum of the absolute value of the imaginary part at ! = 0 assures the Fermi-liquid behaviors. There clearly appear notable features of the nearly AF Fermi-liquid. The Mrst is the !-linear dependence of the imaginary part, which is in contrast to the !-square dependence in the conventional Fermi-liquid theory. The origin of the !-linear dependence is the low-energy spin excitation, which is also the origin of the T -linear law of the resistivity [85,86] (see Section 4.3.4). The second is the strong momentum dependence of the self-energy. For example, the damping rate of quasi-particles, 1=0(k) = −Im 2FR (k; 0) is large around (; 0) (“hot spot”) and is small around (=2; =2) (“cold spot”). This momentum dependence plays a crucial role in the transport properties, especially in the pseudogap state (Section 4.3.4). The renormalization factor zk−1 = 1 − 9 Re 2FR (k; !)=9!|!=0 has the qualitatively same momentum dependence. The quasi-particle velocity

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0.0

A B

-0.1

ImΣ(k,ω)

ReΣ(k,ω)

A B C 0.0

A B C

C

-0.2

-0.3

-0.4 -0.2

(a)

-2

-1

0

1

ω

2

-0.5

(b)

-2

-1

A(k,ω)

1

2

A B C

20

10

0

(c)

0

ω

-0.2

-0.1

0.0

ω

0.1

0.2

Fig. 55. (a) The real part and (b) imaginary part of the self-energy in the FLEX approximation. (c) The spectral function. The momentum A, B and C is shown in the inset of (b). Here, U=t = 3:2, = 0:10 and T = 0:01.

is considerably small at the hot spot because of the van Hove singularity and the large renormalization factor. This is just the Aat dispersion observed in ARPES [45,106]. We have commented in Fig. 25 that the AF spin Auctuation transforms the Fermi surface [86]. The Fermi surface is pinned to the Aat dispersion owing to the transformation of the Fermi surface. Because the eCective Fermi energy of the superconductivity is determined by the quasi-particle velocity around the hot spot as EF ∼ vF kF , the eCective coupling of the superconductivity TcMF =EF is enhanced by the Aat dispersion. Then, the coherence length = and the TDGL parameter b are much reduced. The spectral function A(k; !) shows the single peak structure (Fig. 55(c)), which justiMes the picture of the quasi-particles in the nearly AF Fermi-liquid. The spectral function is remarkably broad at the hot spot and sharp at the cold spot reAecting the momentum dependence of the quasi-particle damping. This is also observed in ARPES around the optimally-doping [106]. However, the pseudogap appears neither in the spectral function nor in the DOS. Thus, the FLEX approximation is insuDcient for the description of the pseudogap state. The SC Auctuation is derived from the eCective pairing interaction described in Eq. (33). We consider the T-matrix which is expressed by the ladder diagrams in the particle–particle channel (Fig. 56(a)). By restricting to the d-wave channel, the T-matrix is estimated by extending the

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

T

=

83

Va

T +

Va

T

(a)

(b)

Fig. 56. (a) The T-matrix. (b) The self-energy arising from the SC Auctuation, 2S (k).

V Eliashberg equation [11], g(q):(k1 ):∗ (k2 ) ; T (k1 ; k2 : q) = 1 − (q) :∗ (k)Va (k − p)G(p)G(q − p):(p) : (q) = −

(85) (86)

k;p

Here, the expression of the FLEX approximation is used for the eCective interaction Va (k) and the V Green function G(k). :(k) is the eigenfunction of the Eliashberg equation Eq. (16) with its maximum eigenvalue e . This function corresponds to the wave function of the Auctuating Cooper-pairs. This procedure is justiMed when the correlation length is long enough, namely in the vicinity V of the critical point T = Tc . Because this procedure is based on the Eliashberg equation with fully including the momentum and frequency dependence, the characteristics of the spin-Auctuationinduced superconductivity are included in the T-matrix. In Eqs. (85) and (86), the wave function is normalized as |:(k)|2 = 1 : (87) k

Then, the constant factor g is obtained as :∗ (k1 )Va (k1 − k2 ):(k2 ) : g=

(88)

k1 ; k 2

V It should be noted that the parameter (0) is equivalent to the maximum eigenvalue of the Eliashberg equation, namely (0) = e . Then, the divergence of the T-matrix is equivalent to the criterion in V the Eliashberg equation e = 1. The self-energy correction arising from the SC Auctuation is obtained within the 1-loop order (FLEX+T-matrix approximation), 2S (k) = T (k; k : q)G(q − k) : (89) q

The total self-energy is obtained by adding it to the contribution from the spin Auctuation 2(k) = 2F (k) + 2S (k). The self-energy in the under-doped region is shown in Figs. 57(a) and (b). We clearly see the anomalous behaviors which arises from the SC Auctuation. The similar structure to that in

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0.0

A B C

ImΣ(k,ω)

0.2

ReΣ(k,ω)

A B C

-0.1

0.0

-0.2 -0.3 -0.4 -0.5

-0.2

-1.0

-0.5

0.0

ω

(a)

-0.6

1.0

0.5

-1.0

-0.5

(b)

15

0.0

0.5

ω

1.0

0.9

A B C

0.8

ρ(ω)

A(k,ω)

10

0.7

1

0

1

0.6

5

FLEX FLEX+T

0.5 0

-0.2

(c)

-0.1

0.0

ω

0.1

0.2

(d)

0.4 -0.4

-0.2

0.0

ω

0.2

0.4

Fig. 57. (a) The real part and (b) the imaginary part of the self-energy in the FLEX+T-matrix approximation. (c) The spectral function and (d) DOS. The inset in (d) shows the same result in the large energy scale. The momentum A, B and C is shown in the inset in Fig. 55(b). Here, U=t = 3:2, = 0:10 and T = 1:2Tc .

the attractive model (Fig. 44) appears in the low-frequency region. In particular, the extremely large damping around the Fermi level gives rise to the clear pseudogap in the spectral function and in the DOS (Figs. 57(c) and (d)). This is a result of the “resonance scattering”. Thus, the superconductivity described by the FLEX approximation is strong coupling enough to induce the pseudogap in the under-doped region. It should be noted that the SC Auctuation is still over-damped, while an asymmetric structure appears owing to not so small a1 =a2 (see Eq. (65)). Because of the momentum dependence of the wave function :(k), the pseudogap has a d-wave form which is similar to the SC gap. The anomalous behaviors are smeared around the cold spot and the “Fermi arc” appears in the pseudogap region. This is an important character of the pseudogap observed by ARPES measurements [103–105]. It is an advantage of the microscopic calculation that the reasonable energy scale of the pseudogap is obtained, while much larger value is obtained in the attractive model. Here, the gap magnitude -pg and Tc are smaller by an order. This is because the superconductivity and the pseudogap take place in the renormalized quasi-particles near the Fermi surface. As a result, the energy scale takes a realistic value 2-pg ∼ 100 meV. We again stress that the eCective Fermi energy is renormalized by the electron correlation and therefore the eCective coupling TcMF =EF is enhanced.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 1.2

1.2 10%doping 13%doping 16%doping 19%doping

U/t=2.2 U/t=2.8 U/t=3.2 U/t=3.6 U/t=4.0

1.0

ρ(ω)

1.0

ρ(ω)

85

0.8

0.8

0.6 0.6 0.4 0.4 -0.4

(a)

-0.2

0.0

ω

0.2

0.4

-0.4

(b)

-0.2

0.0

ω

0.2

0.4

Fig. 58. (a) The doping dependence of the DOS at T = 1:2Tc and U=t = 3:2. (b)U -dependence of the DOS at T = 1:2Tc and = 0:10.

Before closing this subsection, we discuss the relation to the SC gap. The ratio 2-s =TcMF ∼ 12 has been obtained by the FLEX approximation in the optimally-doped region [186,187,193]. where -s is the maximum value of the SC gap. A larger value of this ratio is expected in the under-doped region. The experimental results have shown that the energy scale of the pseudogap is slightly larger than the SC gap [103–105,107]. Our result indicates the ratio 2-pg =Tc ∼ 20 in the under-doped region. Thus, the pseudogap obtained in the FLEX+T-matrix approximation has a relevant magnitude compared with the SC gap. 4.3.2. Doping dependence The FLEX+T-matrix approximation appropriately reproduces the doping dependence. In the previous subsection, a clear pseudogap is shown in the under-doped region. The anomalous properties gradually disappear as the doping concentration is increased. Consequently, the pseudogap in the DOS is Mlled up with hole-doping (Fig. 58(a)). This closing of the pseudogap is basically understood from the doping dependence of the TDGL parameter b (Fig. 59). The large value of b generally means the weak SC Auctuation, which is realized in the over-doped region. Owing to the relation b ˙ (vF∗ =T )2 , the parameter b rapidly develops in the weak-coupling region. Here vF∗ is the quasi-particle velocity around (; 0). In our notation, the parameter b includes the coupling constant g as b ˙ |g|d (0)(vF∗ =T )2 . Because the decrease of vF∗ =T is considerably canceled by the increase of g, the parameter b saturates in the under-doped region. This feature does not contradict with the development of the SC Auctuation. The similar closing of the pseudogap is caused by decreasing U (Figs. 58(b) and 59(b)). Thus, the pseudogap phenomena are one of the characteristics of the strongly correlated electron systems, where superconductivity tends to be strong coupling. The inset of Fig. 59(b) shows the TDGL parameter at the Mxed temperature. The result clariMes the role of the electron correlation, namely the renormalization of the quasi-particle velocity. We see that the TDGL parameter b and the coherence length is reduced by the electron correlation. At last, we discuss the electron-doped cuprates. Owing to the next-nearest neighbor hopping term t , the electronic DOS is relatively small in the electron-doped region. In particular, the Fermi level is not pinned to the Aat dispersion around (; 0). This feature means that the electron correlation is

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 40 25

10

30

b

8 6

15

b

b

20

20

2.0

2.4

2.8

3.2

U/t

10 10 5 0

0.05

(a)

0.10

0.15

δ

0.20

2.0

0.25

(b)

2.4

2.8

3.2

3.6

4.0

U/t

Fig. 59. (a) The doping dependence of the TDGL parameter b at T = Tc and U=t = 3:2. (b) The U -dependence at T = Tc and = 0:10. The inset shows the result at the Mxed temperature T = 0:0082.

eCectively weak in the electron-doped cuprates. Therefore, more conventional behaviors are expected in the normal state. The calculated results based on the FLEX approximation conMrm this naive expectation. The imaginary part of the self-energy shows the !2 -dependence, which is a conventional Fermi-liquid behavior. This feature indicates the T 2 -law of the resistivity which is conMrmed in Section 4.3.4 and experimentally observed in the wide temperature region [23]. Concerning the pseudogap, the SC Auctuation is very weak in the electron-doped region. This is because the quasi-particle velocity vF∗ is large around (; =4), and more importantly, because Tc is low. We have already explained the reason of low Tc in Section 3.2.4. Thus, the superconductivity is the weak-coupling one, which is indicated by the large TDGL parameter b ∼ 30. The comparable value to the over-doped region indicates that the pseudogap is very weak even if it is observed by some experiments. The large value of the coherence length = is consistent with the small Hc2 = 5–10 T [419]. These results are consistent with the recent experiments of the ARPES [418], neutron scattering [79], NMR 1=T1 T [419], and tunnelling spectroscopy [420]. These measurements do not show the “small pseudogap” in the electron-doped cuprates. Other authors have suggested the pseudogap in the electron-doped cuprates Nd 2−x Cex CuO4−y from the optical measurements [421]. They have shown that the characteristic structure of the frequency dependent scattering rate 1=0(!). The suggested pseudogap has much larger magnitude than the SC gap, and appears from the very high temperature T ¿ 300 K. It is therefore considered that this phenomenon is not attributed to the same origin as that of the small pseudogap in the hole-doped cuprates. Because a similar structure is observed even in the AF state, this phenomenon may be attributed to the “large pseudogap” with magnetic origin. The observation of the ARPES [418] indicates the broad suppression of the single particle spectrum around the magnetic Brillouin zone. Since this suppression has also large energy scale, the existence of the large pseudogap and importance of the AF spin Auctuation have been suggested. 4.3.3. Magnetic properties In the following subsections, we discuss the magnetic and transport properties which are calculated by using the Kubo formula. The estimation of the two-particle correlation functions are required for

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

87

20

2

T1T/T2G

0.3

1/T1T

15

0.0 0.00

10

0.00

0.04 10% doping 16% doping 23% doping

5

0

0.02

0.01

0.02

0.03

0.04

0.05

T Fig. 60. Temperature dependence of the NMR 1=T1 T and T1 T=(T2G )2 in the FLEX approximation for U=t = 3:2.

this purpose. First we focus on the magnetic properties, which have been the central issue among the several pseudogap phenomena. Many authors have used the FLEX approximation in order to investigate the magnetic properties in high-Tc cuprates [188]. The vertex correction is needed in the context of the conserving approximation [22,177], but usually it is ignored. The estimation in the early stage [188] has indicated that the vertex correction reduces the spin susceptibility but does not alter the qualitative behaviors. The characteristic behaviors of the nearly AF Fermi-liquid are well reproduced within the FLEX approximation. The NMR 1=T1 T shows the Curie–Weiss law (Fig. 60) and its relation to the NMR 1=T2G shows the magnetic scaling with the dynamical exponent z = 2 (see the inset in Fig. 60). It is microscopically shown that the AF spin correlation increases with decreasing . These features have been assumed in the phenomenological theory [178–181]. It should be, however, noted that the pseudogap is not derived within the FLEX approximation. The pseudogap in the NMR (see Section 2.1.3) is explained by taking account of the SC Auctuation, as is shown in the following way [11]. In the FLEX+T-matrix approximation, the dynamical spin susceptibility sR (q; 4) is obtained by extending the FLEX approximation 0 (q) ; 1 − U0 (q) 0 (q) = G(k)G(k + q) ;

(90)

s (q) =

(91)

k

where the eCects of the SC Auctuation are included in the self-energy 2S (k). The NMR spin–lattice relaxation rate 1=T1 and spin-echo decay rate 1=T2G are obtained by the following formula [27,422] 1 R Im s (q; !)|!→0 ; 1=T1 T = (92) F⊥ (q) ! q 2 (1=T2G )2 = [F (q) Re sR (q; 0)]2 − F (q) Re sR (q; 0) : (93) q

q

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 1.7

1/T1T

q=(π,π) q=(0,0)

6

0.75

1/T2G

1.6

0.76

0.0

1.2

(a)

0.02

0.01

χ(Q,0)

0.03

0.02

T

4 0.03

(b)

0.01

0.80

10 0 0.00

0.60 0.01

0.72

20

2

1.3

0.82

χ(0,0)

5

0.65

χ(0,0)

0.5

χ(Q,0)

1/T2G

1.4 T1T/T2G

1/T1T

0.74 0.70

1.5

0.02

0.02

0.04

0.70

0.78

0.03

0.68

0.04

T

Fig. 61. (a) The temperature dependence of the NMR 1=T1 T and 1=T2G in the FLEX+T-matrix approximation. (b) The static spin susceptibility sR (q; 0) at q =(0; 0) (open squares) and at q =(; ) (closed circles). Here, =0:10 and U=t =3:2. The inset of (b) shows the results of the FLEX approximation.

Here F⊥ (q)= 12 {A1 +2B(cos qx +cos qy )}2 + 12 {A2 +2B(cos qx +cos qy )}2 and F (q)={A2 +2B(cos qx + cos qy )}2 [423]. The hyperMne coupling constants A1 ; A2 , and B are evaluated as A1 = 0:84B and A2 = −4B [200]. We show the calculated results for the NMR 1=T1 T , 1=T2G , and static spin susceptibility in Fig. 61. We see that the pseudogap clearly appears in the NMR 1=T1 T (Fig. 61(a)). The decrease of the NMR 1=T1 T above Tc is the Mrst observation of the pseudogap phenomenon [55]. The 1=T1 T is reduced with approaching to the critical point because the dissipation of the spin Auctuation is suppressed by the pseudogap in the DOS. The NMR 1=T2G also shows the pseudogap with the onset temperature T ∗ close to that in the 1=T1 T (Fig. 61(a)). This is also an eCect of the SC Auctuation. So far, diCerent behaviors of 1=T2G have been reported for several high-Tc compounds [58–63,71,72]. At present, they are attributed to the eCects of the interlayer coupling [63,71]. The experimental data on the single layer compounds show the decrease of 1=T2G in the pseudogap state [62,63], which is consistent with our result in Fig. 61(a). It is interesting that the pseudogap in the NMR 1=T2G is moderate compared with the NMR 1=T1 T . This is because the NMR 1=T2G measures the static part which reAects the total weight of the spin Auctuation. It should be noticed that the pseudogap suppresses only the low-frequency component of the spin Auctuation, because the superconductivity has a smaller energy scale than that of the spin Auctuation. Thus, it is no wonder that the scaling relation of the spin Auctuation is violated in the pseudogap state (see the inset in Fig. 61(a) and experimental result [63]). These features are qualitatively consistent with the experimental results [62,63,71,72]. The obtained behavior of the 1=T2G is expected from the result in the superconducting state [424]. Then, the 1=T2G remains even at low temperature, although the 1=T1 T rapidly decreases. This is a characteristic behavior of the d-wave superconductivity and has played an important role for identifying the pairing symmetry [58]. Because the pseudogap is a precursor of the d-wave superconductivity, the above results in the pseudogap state are expected ones. Contrary to the other magnetic properties, experiments have shown that the uniform spin susceptibility decreases from much higher temperature than T ∗ [67,70]. The decrease becomes more rapid

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

T=0.05 T=0.016 T=0.0082

T=0.05 T=0.016 T=0.0082

4

Imχ(Q,Ω)

Reχ(Q,Ω)

6

89

4

2

2

0 0.0

(a)

0.1

0.2

Ω

0 0.00

0.3

(b)

0.10

0.20

0.30

Ω

Fig. 62. The dynamical spin susceptibility sR (Q; 4) in the FLEX+T-matrix approximation at = 0:10 and U=t = 3:2. (a) The real part. (b) The imaginary part.

below T ∗ [64]. We show the calculated results of the uniform susceptibility s (0; 0) and staggard susceptibility s (Q; 0) in Fig. 61(b). While the staggard susceptibility shows the pseudogap with the onset temperature T ∗ determined from the 1=T1 T , the decrease of the uniform susceptibility begins from much higher temperature and becomes rapid below T ∗ . Thus, the qualitatively diCerent behavior of the uniform susceptibility is consistently obtained by simultaneously taking account of the spin and SC Auctuations. The frequency dependence of the spin susceptibility well characterizes the magnetic properties in the pseudogap state. The results for the dynamical spin susceptibility at q = Q is shown in Fig. 62 [11]. We see the suppression of the real part in the low-frequency region, that is, the magnetic order is suppressed by the SC Auctuation. The imaginary part in the low-frequency region is more remarkably suppressed in the pseudogap state. At the same time, the spin excitation develops in the higher-frequency region. In other words, the pseudogap transfers the spectral weight of the spin Auctuation from the low- to high-frequency region. This is just the pseudogap phenomenon observed in the inelastic neutron scattering measurements [82]. This character in the frequency dependence indicates that the pseudogap has the smaller energy scale than that of the magnetic excitation. We stress that this is a natural consequence of the pairing scenario. These features are consistent with the above discussion on the NMR 1=T2G . It is notable that the spin Auctuation in the relatively wide frequency region contributes to the pairing interaction. The low-frequency component rather gives a de-pairing eCect through the self-energy. Therefore, the d-wave pairing is not suppressed by the pseudogap in the spin Auctuation. We have actually conMrmed that the feedback eCect rather enhances the d-wave superconductivity [11]. The detailed momentum dependence of the magnetic excitation is a current interest brought by the recent experiments. There are detailed analysis by using the FLEX approximation [188,425]. Then, the magnetic excitation is commensurate in the under-doped region and becomes incommensurate with hole-doping. It is robustly commensurate in the electron-doped case (Fig. 63). The long-range hopping term t plays an essential role in this structure. The qualitatively same results have been obtained in the RVB theory including the long-range hopping [312]. These features are qualitatively consistent with the inelastic neutron scattering for La2−x Sr x CuO4 [75,76] and Nd 2−x Cex CuO4−y [79].

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.50 0.40

δ

0.30 Γ

0.20 0.10 0.00 0.60

0.80

1.00

1.20

n Fig. 63. Doping dependence of the incommensurability [425], which is deMned by the peak of the magnetic excitation at (; (1 ± )) and ((1 ± ); ). The inset shows the Fermi surface in the electron-doped region.

The same result for the electron-doped cuprates has been reported by the numerical studies on the Hubbard model [16,426] and t–J model [427]. Recently, interesting temperature and frequency dependence has been reported for YBa2 Cu3 O6+ , Bi2 Sr 2 CaCu2 O8+ , and La2−x Sr x CuO4 [77,78,80]. The commensurate magnetic excitation is observed at high temperature and the incommensurability develops below the onset temperature above Tc . Here it is shown that these features can be explained by taking account of the pseudogap induced by the SC Auctuation [11]. First we point out that the SC Auctuation generally enhances the incommensurability. Although the commensurate peak is obtained by the FLEX approximation (Fig. 64(a)), it becomes incommensurate owing to the SC Auctuation (Fig. 64(b)). These features result from the d-wave momentum dependence of the pseudogap in the spectral function. The spectral gap around (; 0) reduces the spin excitation at q = Q more remarkably than the incommensurate component. The similar eCect is expected in the superconducting state. Thus, the incommensurate structure in the low-temperature region can be explained as an eCect of the pseudogap or SC gap. If so, the incommensurate structure disappears in the high-frequency region as is shown in Fig. 64(c), which has been observed in the experiments [77,78,80]. This is simply because the pseudogap appears only in the low-frequency region. The incommensurate structure disappears with increasing the temperature, owing to the closing of the pseudogap. Thus, the whole temperature and frequency dependences are consistent with the incommensurability induced by the pseudogap. It should be stressed that the incommensurate structure observed in inelastic neutron scattering does not necessarily mean the stripe order. 4.3.4. Transport properties In this subsection, we discuss the transport phenomena which have been one of the central issues of the cuprate superconductors. The understanding for the anomalous behaviors consistent with the magnetic properties has been a fundamental problem for a long time. A solution is provided by the FLEX+T-matrix approximation in the following way. Then, the anomalous properties above T ∗ are explained by taking the spin Auctuation into account [84–86,93,94], and those below T ∗ are explained by simultaneously taking account of the spin and SC Auctuations [90]. In the RVB theory, the electric conductivity around T ∗ is explained by taking account of the coupling between the spinons and holons through the gauge Meld [313,315,316]. The singlet pairing of the spinon

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

91

Fig. 64. The momentum dependence of the dynamical spin susceptibility Im sR (q; 4) at = 0:10, U=t = 3:2, and T = 0:01. (a) The FLEX approximation at 4 = 0:01. (b) The FLEX+T-matrix approximation at 4 = 0:01. (c) The FLEX+T-matrix approximation at 4 = 0:1.

weakly aCects the charge transport carried by the holon. On the contrary, our understanding is based on the characteristic momentum dependences of the quasi-particle properties. Before describing the microscopic theories, we introduce the general formula for the electric transport on the basis of the Fermi-liquid theory. According to the Kubo formula, the electric conductivity is expressed as the current–current correlation function R Im K,H (!) ; !=0 ! ( d0T0 J, (0)JH (0)ei!n 0 ; K,H (!n ) =

,H = e2 lim

0

(94) (95)

where !n = 2nT is the bosonic Matsubara frequency. Since the conductivity is inMnite in the non-interacting system, some procedure of the renormalization is required in the perturbation expansion. Then, the expression for K R (!) is generally V complicated in the process of the analytic continuation. However, Eliashberg has given a compact formula for the longitudinal conductivity xx by taking the most divergent terms with respect to the quasi-particle lifetime 0(k) = 1=|Im 2R (k; 0)| [428]. This procedure is correct in the coherent limit

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

lmax = v(k)0(k)|max 1, which is justiMed in the low-temperature region. The exceptional case is a system close to the phase transition. For example, the Aslamazov–Larkin (AL) term [393] in the SC Auctuation theory is higher order with respect to 1=lmax , but divergent at the critical point. We will estimate the AL-term afterward and conclude that its contribution is not important [90]. V Using the Green function, the Eliashberg formula is expressed as d@ (−f (@))|G R (k; @)|2 v˜, (k; @)J, (k; @) ; ,, = e 2

(96)

k

where v˜, (k; @) = v, (k) + 9 Re 2(k; @)=9k, is the velocity including the k-mass renormalization. Note that the k-mass renormalization corresponds to a part of the vertex correction. The total current vertex ˜J (k; @) is obtained by solving the Bethe–Salpeter equation d@ J, (k; @) = v˜, (k; @) + I22 (k; @ : k ; @ )|G R (k ; @ )|2 J, (k ; @ ) : 4i

(97)

k

The vertex function I22 (k; @ : k ; @ ) is obtained by the analytic continuation of the irreducible four-point particle–hole vertex [428]. The renormalization of the total current vertex ˜J (k; @) is usually the main contribution of the vertex correction. In the following, we use “the vertex correction” as the correction arising from the vertex function I22 (k; @ : k ; @ ). The expression for the Hall conductivity ,H corresponding to Eq. (96) has been given by Kohno and Yamada [429]: d@ (−f (@))|Im G R (k; ”)| |G R (k; ”)|2 ,H = −H e 3

k

×v˜, (k; ”)[J, (k; ”)9JH (k; ”)=9kH − JH (k; ”)9J, (k; ”)=9kH ] :

(98)

In case of the Hall conductivity, the most divergent term with respect to 0(k) is the quadratic term. In this section, the magnetic Meld is Mxed to be parallel to the c-axis H c, and the current J is Mxed to be perpendicular to the c-axis J ⊥ c. The vertex correction is generally required in order to satisfy the Ward identity which corresponds to the momentum conservation law [430–433]. When the current operator commutes with the Hamiltonian, the inMnite conductivity is generally expected. However, the Mnite conductivity is obtained when we consider only the self-energy correction. The inMnite conductivity is derived by taking into account the vertex correction according to the scheme of Baym and KadanoC. In the lattice system with Umklapp processes, however, the current operator does not commute with the Hamiltonian. Then, the vertex corrections are usually taken into account by only multiplying a constant factor, and have no important role [430]. This argument is based on the assumption that the temperature dependence of the four-point vertex is negligible, which is justiMed in the conventional Fermiliquid. However, the vertex correction sometimes gives a signiMcant eCect, when a collective mode induces a temperature dependence of the four-point vertex [93,94]. This case is realized in the underdoped region.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

93

Hot Spot J(k) v(k) ϕ

Cold Spot

θ

Fig. 65. The schematic Mgure of the band velocity ˜v and total current vertex ˜J . Here the hot and cold spots are shown. The dashed line indicates the magnetic Brillouin zone boundary, while the dotted line denotes the wave vector Q = (; ). Note that the angle is also used in other Mgures.

The above expressions (Eqs. (96) and (98)) are rewritten in the Fermi-liquid limit z(k))(k)T as xx = e2 z(k)(−f (”∗ (k)))v˜x (k; ”∗ (k))Jx (k; ”∗ (k))=)(k; ”∗ (k)) ; (99) k

∼ = e2

FS

d k v˜x (k) Jx (k)0(k) ; (2)2 v(k) ˜

H e3 z(k)(−f (”∗ (k)))v˜x (k; ”∗ (k)) 2 k 9Jy (k; ”∗ (k)) 9Jx (k; ”∗ (k)) ∗ ∗ − Jy (k; ” (k)) × Jx (k; ” (k)) 9ky 9ky H e3 9’(k) dk ˜ ∼ 0(k)2 : |J (k)|2 = 2 4 FS (2) 9k

(100)

xy = −

)(k; ”∗ (k))2 ;

(101) (102)

We have used the deMnition v˜, (k)=v˜, (k; 0), J, (k)=J, (k; 0), and so on. Above expressions are similar to the consequence of the Boltzmann equation; the velocity is replaced by the total current vertex J, . In Eq. (102), we have deMned the angle of the current vertex as ’(k) = Arctan(Jx (k)=Jy (k)) (see Fig. 65). It should be noticed that the Hall conductivity depends on the diCerential of the angle ’(k) with respect to the momentum along the Fermi surface k . In the isotropic systems, the current vertex is always perpendicular to the Fermi surface, and therefore the relation RH ˙ 1=n is proved. This expression is, however, very delicate. It is clearly understood from the above expressions (Eqs. (96)–(102)) that the Hall coeDcient is not directly related to the carrier number n. The electric transport is determined by the quasi-particles near the Fermi surface, not by the excitations deeply below the Fermi level.

94

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 k ε+ω

k' ε'+ω

k ε+ω

k' ε'+ω

k ε+ω

k' ε'+ω

kε

k' ε'

kε

k' ε'

kε

k' ε'

(a)

(b)

(c)

Fig. 66. The four-point vertex in the FLEX approximation. The wavy dashed line represents the spin Auctuation. (a) The “SPMT-term” which is dominant. The diagrams (b) and (c) are usually negligible.

In the following, we use Eqs. (96)–(98), although Eqs. (99)–(102) are expected to give qualitatively same results. The resulting resistivity and Hall coeDcient RH are obtained by the formula, 2 = 1=xx and RH = xy =xx H , respectively. Hereafter, the constant factor arising from the unit of charge e is omitted. It should be noted that the coherent transport is assumed in the above expressions which are based on the Fermi-liquid theory. This assumption seems to be incompatible with the pseudogap induced by the “resonance scattering”, where the extremely large damping is the origin of the pseudogap. However, this diDculty is removed by the characteristic momentum dependence; the pseudogap occurs at the hot spot, while the in-plane transport is determined by the cold spot as explained below. Because the coherent nature of the quasi-particles is suDciently maintained at the cold spot (Figs. 65 and 67), the above formula are justiMed even in the pseudogap state. Now let us explain the microscopic theories. First the results of the spin Auctuation theory are reviewed [90,93]. We show the calculated results of the resistivity and Hall coeDcient in the FLEX approximation. The four-point vertex in the FLEX approximation includes three terms (see Fig. 66). Because the spin-Auctuation Maki–Thompson (SPMT) term shown in Fig. 66(a) is dominant among them, only the SPMT-term is taken into account [90,93,94]. We provide a detailed explanation because a part of the following understanding is obtained in recent years. The T -linear law of the electric resistivity and the enhancement of the Hall coeDcient are discussed. These behaviors are the characteristics of the nearly AF Fermi liquid and observed in the experimental results above T ∗ . We point out three important properties from which the unconventional transport above T ∗ is explained. The Mrst is the momentum dependence of the quasi-particle lifetime 0(k). The typical results are shown in Fig. 67. We see that the lifetime is long at the cold spot ∼ =4 and short at the hot spot ∼ 0 or ∼ =2 [84–86]. When the exchange of the AF spin Auctuation is the dominant scattering process, the quasi-particle damping )(k) = −Im 2R (k; 0) is almost determined by the low-energy DOS around k = k + Q. Then, the lifetime is short around the magnetic Brillouin zone boundary and/or around the van Hove singularity. The electric transport is practically carried by the quasi-particles at the cold spot which is located around k = (=2; =2) (see Fig. 65). This momentum dependence of the lifetime aCects the spectral function, as is explained in Section 4.3.1, which is actually observed in the ARPES measurements [45,106,434,435]. Also the magnetic-transport measurement has conMrmed this momentum dependence [436]. The second is the T -linear dependence of the damping rate )c = )(kc ), where kc is the momentum at the cold spot. The T -linear resistivity originates from this T -linear dependence [84–86]. It have

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

95

200 δ=0.09 δ=0.15 δ=-0.10

τ(k)

150

100

50

0 0.0

0.5

1.0

θ

1.5

Fig. 67. The momentum dependence of the lifetime 0(k) on the Fermi surface. The circles and triangles correspond to T = 0:005 and T = 0:009, respectively and U=t = 3:2. The solid curve is a result in the electron-doped region ( = −0:10, U=t = 3 and T = 0:018).

0.8

4

RH

ρ

0.6

δ=0.09 δ=0.15 δ=0.09 (no VC) δ=-0.1

6

δ=0.09 δ=0.09 (no VC) δ=-0.1

0.4

2 0

0.2

-2 0.0 0.00

(a)

0.01

0.02

0.03

T

0.04

0.00

0.05

(b)

0.01

0.02

0.03

0.04

0.05

T

Fig. 68. The results of the FLEX approximation at U=t = 3:2. (a)The resistivity and (b) Hall coeDcient in under-doped ( = 0:09, circles), optimally-doped ( = 0:15, squares), and electron-doped ( = −0:1, stars) region.

been pointed out that the T 2 -resistivity is always obtained in the low-temperature limit even in the quantum critical point unless the Fermi surface is perfectly nested [86]. This is because the quasi-particles at the cold spot are not directly scattered by the AF spin Auctuation at q = Q. Note that the anomalous power-law can be induced by the slight impurity scattering near the quantum critical point [437]. However, the crossover temperature from T -square to T -linear resistivity is suDciently small in the under-doped region (Fig. 68(a)). The transformation of the Fermi surface plays a role to reduce the crossover temperature [86]. The third is the temperature dependence of the vertex correction [93,94]. We will see that the contribution from the SPMT-term to the total current vertex ˜J (k) signiMcantly enhances the Hall coeDcient. The schematic Mgure (Fig. 65) shows that the total current vertex ˜J (k) is signiMcantly altered from the band velocity ˜v(k). This is an eCect of the AF spin Auctuation. Then, the four-point

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vertex I22 (k; ” : k ; ” ) is signiMcantly enhanced around k − k = Q. In the vicinity of the magnetic instability, the diCerential 9’(k)=9k increases around the cold spot. Therefore, the Hall coeDcient increases with the development of the spin Auctuation (Fig. 68(b)). The vertex correction is not so important for the resistivity. The resistivity is enhanced by the SPMT-term, however, the qualitative temperature dependence is not altered (Fig. 68(a)). The nearly AF Fermi-liquid theory explains the transport phenomena also in the electron-doped region. Then, the cold spot is located around (; =4) and (=4; ), as is shown in Fig. 67, where the Fermi surface is far from the magnetic Brillouin zone (see Fig. 25). The large damping around the magnetic Brillouin zone is consistent with the observation in the ARPES measurements [418], which conMrms this picture. Then, the electric transport shows considerably diCerent behaviors. Combined with the sharp commensurate structure of the magnetic excitation (Fig. 21), the quasi-particle lifetime at the cold spot follows the usual T -square law. Note that the lifetime is much longer than the hole-doped case. Therefore, the resistivity shows the T -square law with a small magnitude [1]. More interestingly, the Hall coeDcient changes its sign because the diCerential of the angle 9’(k)=9k has negative sign around (; =4) [93,94]. These behaviors are consistent with the experimental results [23,26,91]. In the following part, we take account of the SC Auctuation and discuss the electric transport in the pseudogap state [90]. In order to make the discussion clear, we classify the eCects of the SC Auctuation in the following way. 1. The pseudogap in the single-particle properties. 2. The feedback eCects through the AF spin Auctuation. 3. The vertex corrections from the SC Auctuation. The AL-term is classiMed into them. The following calculation identiMes the eCect (2) as a main contribution. That is, the coupling between the spin and SC Auctuations plays an essential role for the electric transport. Therefore, we perform the self-consistent FLEX+T-matrix (SCFT) approximation [11] in which the dynamical spin susceptibility, T-matrix, and the Green function are determined self-consistently. We Mrst discuss the resistivity. The eCect (1) obviously reduces the longitudinal and transverse conductivities. However, the increase of the resistivity is not signiMcant because the pseudogap occurs at the hot spot which is not important for the transport phenomena. It should be noted that the self-energy at the cold spot is always dominated by the spin Auctuation. Then, the eCect (2) increases the conductivity and gives the larger contribution than (1). The quasi-particle damping arising from the spin Auctuation )F (k) = −Im 2FR (k; 0) is reduced by the pseudogap in the spin excitation. Thus, the SC Auctuation induces the downward deviation of the resistivity around T = T ∗ (Fig. 69(a)) through the feedback eCect. This downward deviation becomes more remarkable as decreasing the hole-doping, which is consistent with the experimental results [87–89,98]. It is another important property that the deviation of the resistivity is only a slight one, while the NMR 1=T1 T is remarkably reduced by the SC Auctuation. This is because the cold spot is not so sensitive to spin Auctuation at q=Q. As shown in Fig. 64, the suppression of the magnetic excitation is moderate for the incommensurate component q = Q. This is an origin of the incommensurate peak in the pseudogap state. Since the quasi-particle damping at the cold spot is determined by some average of the incommensurate component, the enhancement of the lifetime is not so signiMcant. Thus, the weak response of the resistivity to the pseudogap is explained by the detailed analysis

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 0.6 0.5

δ=0.09 δ=0.15

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δ=0.15 SCMT SCMT+AL 0.004

0.005

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T

Fig. 69. (a) The temperature dependence of the resistivity at U=t = 4:0 without the eCect (3). The closed and open symbols show the results with and without the SPMT-term, respectively. (b) The eCects of the vertex correction. The results with the SCMT-term and with the SCMT- and AL-terms are shown. k ε+ω

k' ε'

kε

k' ε'+ω

(a)

(b)

Fig. 70. (a) The four-point vertex in the lowest order with respect to the SC Auctuation (SCMT-term). (b) The Feynmann diagram representing the AL-term. The wavy dashed line represents the SC Auctuation.

of the momentum dependence. This explanation is in sharp contrast with that in the RVB theory [313,315,316]. We have clariMed the role of the vertex correction arising from the SC Auctuation [90]. Then, we V have estimated the lowest-order term within the Eliashberg theory (“SCMT-term”) and the AL-term V [393] beyond the Eliashberg theory (Fig. 70). Both contributions increase the conductivity. It should be noted that the SCMT term is not equivalent to the Maki–Thompson term [394] which has been investigated in the Auctuation theory [369,406,438]. The SCMT term includes the lowest order contribution with respect to 1=0(k), and take into account the vertex correction iteratively. Note that the diCusion propagator, which is important in the s-wave superconductor, is negligible in the d-wave superconductor. Fig. 69(b) shows that the SCMT-term does not aCect the temperature dependence, qualitatively. This is mainly because of the momentum dependence of the d-wave order parameter. Although the contribution from the SCMT-term increases as the temperature decreases, it is not visible in the temperature dependence of the resistivity because the other contribution to the conductivity also increases. The role of the AL-term may be interesting because this contribution has been intensively investigated in the Auctuation theory [369,406,438,439]. The AL-term is interpreted as the conductivity

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carried by the Auctuating Cooper-pairs [440], and should be written as the superconducting part s V in contrast to the normal part n included in the Eliashberg theory. We have concluded that the AL-term is almost negligible in the wide temperature range (see Fig. 69(b)). This is simply because the AL-term is higher order with respect to the parameter 1=lmax = 1=v(kc )0(kc ). The AL-term becomes dominant just above Tc because this term is more divergent with respect to 1=t0 =1=(1−(0)). However, this temperature range is narrow because of the long lifetime of the quasi-particle at the cold spot. The detailed analysis on the Auctuation conductivity in the vicinity of the critical point has been performed within the weak-coupling theory [369,406,438,439]. Here we stress that the downward deviation of the resistivity is not attributed to the AL-term, but to the feedback eCect through the suppression of the spin Auctuation. Here we discuss the role of the AL-term furthermore, since some characteristics of the high-Tc cuprates manifest. There are two important points in the above discussion on the AL-term. One is the strong-coupling superconductivity. While the resonance scattering is strong in the superconductor with short coherence length, the AL-term does not directly depend on the coherence length in two dimensions [393,430]. This is because the short coherence length means the small velocity of the Auctuating Cooper-pairs. Then, the pseudogap occurs under not so small value of t0 where the AL-term is still small. The other is the characteristic momentum dependences in high-Tc cuprates. For example, in case of the strong-coupling s-wave superconductors, the AL-term will be much more important. This is because the pseudogap opens on the whole Fermi surface and therefore the normal part n is remarkably suppressed. Let us comment on the c-axis transport, which shows qualitatively diCerent behaviors from the in-plane transport. That is, the c-axis transport is strongly incoherent in the pseudogap state, while the in-plane transport is suDciently coherent [98]. We can understand these qualitatively diCerent behaviors in a consistent way. Then, the momentum dependence of the inter-layer hopping matrix plays an essential role [86,97]. The result of the band calculation has shown the following behavior [99], t⊥ (k) = t⊥ (cos kx − cos ky )2 :

(103)

In short, the inter-layer hopping matrix vanishes at the cold spot and the c-axis transport is mainly determined by the hot spot. Therefore, the c-axis conductivity is signiMcantly reduced by the eCect (1). Actually, the incoherent nature of the c-axis optical conductivity has been obtained in the pseudogap state within the FLEX+T-matrix approximation, together with the coherent nature of the in-plane optical conductivity [441]. This is an experimentally observed behavior [100–102]. Naively, it may be expected that the AL-term is important in the c-axis transport instead of the quasi-particle transport. However, the AL-term is higher order with respect to the inter-layer hopping, 4 2 i.e., s ˙ t⊥ while n ˙ t⊥ [406,439]. Therefore, the AL-term is negligible in the quasi-2D system. We stress that the qualitative diCerence between the c-axis and in-plane transport is not attributed to the diCerence of the AL-term [406,439], but to the diCerence of the normal part n . In order to arrive at this conclusion, we have to consider the spin and SC Auctuations simultaneously and to take account of the momentum dependent inter-layer hopping matrix. Finally let us discuss the Hall coeDcient. The Hall coeDcient increases owing to the eCect (1), because the momentum dependence of the lifetime is enhanced by (1). Actually, the SC Auctuation was proposed to be the origin of the enhancement of the Hall coeDcient [97]. However, the situation is quite altered by taking account of the SPMT-term. The kernel of the Bethe–Salpeter equation

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Fig. 71. (a) The temperature dependence of the Hall coeDcient RH at U=t = 4:0 without the eCect (3). The circles and squares correspond to = 0:09 and = 0:15, respectively. (b) The eCects of the vertex correction. The meanings of the lines are the same as in Fig. 69(b).

(Eq. (97)) includes the four-point vertex arising from the spin Auctuation and the absolute value of the Green function. Therefore, the vertex correction is reduced by the feedback eCect (2) and furthermore by the eCect (1). As a result, the Hall coeDcient is reduced by the SC Auctuation through the SPMT-term. We have conMrmed that the feedback eCect (2) is dominant also for the Hall coeDcient. The calculated results in Fig. 71 clearly show the pseudogap behavior of the Hall coeDcient. The Hall coeDcient shows the peak around T = 0:006 and decreases with decreasing the temperature. This phenomenon becomes moderate with increasing the hole-doping. These results qualitatively explain the experimental results in the pseudogap state including the doping dependence [87,88,91,95,96]. The vertex correction arising from the SCMT-term enhances the momentum dependence of the angle ’(k) [90]. Therefore, the Hall coeDcient is enhanced by the vertex correction arising from the SC Auctuation. It is notable that this enhancement does not occur without the SPMT-term. In other words, the SCMT-term indirectly enhances the Hall coeDcient through the combination with the SPMT-term. Fig. 71(b) shows that this enhancement is not so signiMcant to alter the qualitative behaviors. This is also because of the wave function of the d-wave superconductivity. Including the AL-term in the longitudinal conductivity, the suppression of the Hall coeDcient becomes clearer. Thus, the transport coeDcients in the pseudogap state are explained by simultaneously taking into account the spin Auctuation and SC Auctuation. It is conMrmed that characteristic behaviors of the electric transport are mainly caused by the feedback eCect through the pseudogap in the spin Auctuation. This eCect is not outstanding in the resistivity, but rather apparent in the Hall coeDcient. This diCerence reAects the importance of the SPMT-term for these quantities. We stress that the conventional theory on the Auctuation conductivity is not satisfactory, but detailed knowledge on the electronic structure are required for the understanding of high-Tc cuprates. Recently, the theory including the spin and SC Auctuations has been extended to the thermal transport [442]. It is shown that the vertex correction from the SCMT-term combined with the SPMT-term signiMcantly enhances the Nernst coeDcient. This feature is also consistent with the experiments [443]. Although the vortex excitation has been considered as an origin of this behavior

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Tc

SCFT(U/t=3.2)

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δ Fig. 72. The phase diagram obtained by the FLEX and SCFT approximations.

[443], we consider that the contribution from quasi-particles is dominant in the wide temperature V region like for the electric conductivity. That is, the understanding within the Eliashberg theory will be relevant also for the thermal transport. The importance of the superconducting part in the vicinity of the critical point is common. The analysis on the superconducting part under the magnetic Meld has been recently given [444]. 4.3.5. Phase diagram At the last of this section, we show the phase diagram in Fig. 72. The SCFT approximation is used to estimate Tc suppressed by the Auctuation. The eCects of the SC Auctuation are similarly classiMed into the following two parts: (1) The pseudogap in the single-particle properties. (2) The feedback eCect through the spin Auctuation. We have concluded that the feedback eCect enhances the transition temperature [11]. The spin Auctuation has both the pairing and de-pairing eCects. Since the low-frequency component mainly aCects as a de-pairing source, the pseudogap in the spin Auctuation rather enhances the superconductivity (see also Section 4.3.3). However, the eCect (1) is dominant in the present case and signiMcantly suppresses Tc . Note here that strictly speaking, the transition temperature should be always zero (Tc = 0) in the two-dimension, which is known as the Mermin– Wagner theorem. However, the singularity of the two-dimensional system is always removed in the layered systems. Taking account of the weak inter-layer coupling, we determine the critical point as e = 0:98 instead of e = 1. This criterion corresponds to the 2D–3D crossover of the SC Auctuation with the anisotropy being =ab ==c = 10 and =ab = 2–3. It is shown that the suppression of Tc from the mean-Meld value becomes remarkable with under-doping. This is a natural consequence because the SC Auctuation becomes strong with underdoping (see Section 4.3.2). In other words, the pseudogap develops with under-doping and the reduced DOS gives the suppressed Tc . It is an interesting result that the transition temperature takes the maximum value at ∼ 0:11 and decreases with under-doping in the SCFT approximation for U=t = 4:4. This feature is in sharp contrast to the FLEX approximation where the Tc goes on increasing with decreasing (see also Section 3.2.4). Thus, the mean-Meld transition temperature develops

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with under-doping, however, the strong SC Auctuation decreases the Tc in the under-doped region. This picture is consistent with the experimental results from the tunnelling measurements [89,110], where the SC gap develops with decreasing in the under-doped region. We can see from Fig. 72 that the decrease of Tc in the under-doped region is only a slight one in case of U=t = 3:2. Thus, the strong electron correlation plays an essential role for describing the under-doped region. It is also important that we can treat the strong correlation by considering the SC Auctuation which suppresses the AF order. The stabilization of the metallic state by the SC Auctuation is actually expected in the hole-doped cuprates. It is worth while to write again that the strength of the SC coupling is indicated by the ratio MF Tc =EF , and not by Tc . Since the mean-Meld transition temperature TcMF increases and the eCective Fermi energy EF decreases with under-doping, the superconductivity becomes strong coupling in spite of the decreasing Tc . Thus, the decreasing Tc in the under-doped region is obtained by starting from the microscopic Hamiltonian and taking account of the SC Auctuation. This is an important development for the theory starting from the Fermi-liquid state. As has been discussed in Section 4.2.5, the SCFT approximation underestimates the eCects of the SC Auctuation. Therefore, the more signiMcant doping dependence is expected in the higher-order theory, which is beyond the scope of this review. It is quite clear that the SCFT approximation is not suDcient to show the disappearance of the superconductivity. If we describe the phase boundary between the SC phase and the spin glass phase, the loss of the metallic behavior will be an essential aspect. Then, the validity of the FLEX approximation will be a subject. The systematic treatment for the electron correlation and the disorder will be necessary for this issue. This is an interesting and open problem. 5. Heavy-Fermion systems In previous sections, we have discussed the superconductivity in high-Tc cuprates, organic superconductors, and Sr 2 RuO4 . The pairing mechanism and several physical properties of these materials have been explained. In the calculation of the transition temperature within the mean-Meld theory, we have adopted the third-order perturbation (TOP) theory with respect to the on-site Coulomb repulsion U for the weak-coupling case. When the anti-ferromagnetic (AF) spin Auctuation is strongly enhanced, the Auctuation-exchange (FLEX) approximation has been applied. In general, these two methods are useful for the weak and intermediate coupling regimes in U . Here note that the above approaches stand on a common basis of the Fermi-liquid state. As far as the system is in a metallic state, or more strictly, the existence of Fermi-liquid quasi-particles is insured, such methods have an established basis. This is related to the analytic property in U and the continuity principle stressed by P. W. Anderson [445]. In this case, the long-lived Fermi-liquid quasi-particles form the Fermi surface at low temperatures. The superconductivity is induced by the eCective interaction among quasi-particles and the paring symmetry is determined by the momentum dependence of the four-point vertex function, which is naturally caused by the short-range repulsion through the many-body eCect. Thus, the superconductivity in strongly correlated materials discussed above can be understood from such a uniMed view. On the other hand, the superconductivity in heavy-fermion compounds, which are the typical SCES materials, has not been explained from the microscopic point of view, mainly due to the complicated

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band structures and the strong correlation eCect. However, we believe that the key mechanism of the superconductivity in these compounds should be also explained on the same footing as described in previous sections. It is expected that such eCorts will extend the universality of the understanding on the superconductivity. Then, the identiMcation of the residual interaction between the quasi-particles is an essential task for this purpose. In this section, we introduce a strategy to treat heavy-fermion superconductors and discuss the possible applications to some Ce-based and U-based heavy-fermion superconductors. Then, we analyze the eCective single f-band model by choosing one dominant band. Further analysis based on more realistic models is a desirable future issue, but we believe that the essential physics will be unchanged and the discussion in this section will be a basis in the future progress. 5.1. Experimental view Most of intermetallic compounds with several localized f-electrons at each atomic site exhibit some kinds of magnetic transition by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. Some of Ce-based and U-based compounds, however, form coherent itinerant electron bands at low temperatures due to the mixing eCect with conduction electrons, while they possess magnetic characters at high temperatures. Such electron systems are frequently called “heavy-fermion systems” [446], since the eCective mass of the itinerant electron becomes several hundred times larger than free-electron mass m0 due to the strong electron correlation. In 1979, the superconductivity in CeCu2 Si2 , which is one of typical heavy-fermion systems, has been reported by Steglich et al. [6]. The discovery of the superconductivity in the material with remarkable magnetic characters has implied a scenario clearly diCerent from the electron–phonon mechanism in the conventional BCS theory. In fact, physical properties in the superconducting phase at low temperature have shown power-law temperature dependence, diCerent from exponential decay observed in the conventional s-wave superconductor. Thus, CeCu2 Si2 has become the pioneering discovery of unconventional superconductivity in SCES. It is now considered as an even-parity superconductor with line-nodes, probably d-wave pairing symmetry. Since then, many unconventional superconductors have been discovered in heavy-fermion systems. Multi-phase diagrams in UPt 3 [274,447,448] and U(Be1−x Thx )13 [12,449–451] indicates curious superconductivity with multi-components. In particular, UPt 3 is the odd-parity superconductor Mrst discovered in electronic systems [13]. UPd 2 Al3 and UNi2 Al3 [452,453] are unconventional superconductors coexisting with the AF phase, and are considered to have even- and odd-parity pairing states, respectively [454,455]. URu2 Si2 indicates coexistence of unconventional superconductivity and a hidden order [456]. Furthermore, recent progress in experiments under pressures have promoted discoveries of new superconductors: CeCu2 Ge2 [457,458], CePd 2 Si2 [459,460], CeRh2 Si2 [461], CeNi2 Ge2 [462], and CeIn3 [460,463]. These compounds are AF metals at ambient pressure, while under high pressures, the AF phases abruptly disappear accompanied by SC transitions. Except for cubic CeIn3 , all other materials have the same ThCr 2 Si2 -type crystal structure as in CeCu2 Si2 . In addition, quite recently, several kinds of new heavy-fermion superconductors have been discovered. One is a family of CeTIn5 (T=Co, Rh, and Ir) with HoCoGa5 -type crystal structure [464–466]. Since the discovery, a variety of experimental investigations have rapidly increased. These compounds possess relatively high transition temperature such as Tc = 2:3 K for CeCoIn5 , which is the highest among heavy fermion superconductors observed yet. The dominant AF spin Auctuations have been

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suggested by the existence of the neighboring AF phase in the pressure-temperature (P–T ) phase diagram, the power-law (T 1–1:3 ) behavior in the resistivity, and the magnetic behavior observed in the NQR/NMR 1=T1 . The situation is quite similar to other Ce-based superconductors mentioned above. Another is the coexistence of superconductivity and ferromagnetism in UGe2 at high pressure and URhGe at ambient pressure [275,276]. Much attention has been attracted to features of the superconductivity, since it is naively believed that a triplet pairing state coexists with the ferromagnetic phase. In fact, the unconventional T 3 -behavior has been observed in the NQR/NMR-1=T1 [467]. In addition, other new superconductors, including neither Ce nor U atom, have been also discovered and attracted attentions. In the Mlled skutterudite compound PrOs4 Sb12 (Tc = 1:85 K), the possibility of the double transition has been indicated [468]. In the Pu-based compound PuCoGa5 with the same crystal structure as a family of CeTIn5 , a very high transition temperature Tc = 18:5 K has been reported [469]. As introduced above, heavy-fermion superconductors show a great variety of ground states and oCer rich examples to investigate unconventional superconductivity in SCES. We cannot review here in detail the superconductivity in each heavy-fermion compound. Alternatively, in the following subsections, we overview characteristic features in Ce-based and U-based heavy-fermion superconductors, relevant to the key mechanism of the superconductivity. For more details of each material, readers can consult the review articles by Stewart [447], Grewe and Steglich [446], and Sigrist and Ueda [12]. 5.1.1. Ce-based compounds Ce-based heavy-fermion compounds possess the typical nature of what is known as “Kondo eCect” in the impurity case [206]. The impurity Kondo problem is a typical example of the Fermi-liquid formation in many-body systems. The local spin Auctuation at impurity (Ce) site, observed at high temperatures, is quenched at low temperature by the mixing eCect with the conduction electrons. The system forms the local Fermi-liquid state and has the singlet ground state as a whole. In the process, the resistivity shows the log T -behavior with decreasing T and eventually arrives at the value of the unitarity limit. The change of the behavior is marked by a characteristic temperature TK , which is called the Kondo temperature. In the actual case, there remains degeneracy in f orbitals. For instance, assuming that the crystal-Meld (CF) ground state is >7 doublet and the excited state is >8 quartet, we obtain the characteristic Kondo temperature as [470] 2 D TK = D exp(−N=|J |) ; (104) TK + where D, -, and |J |=N represent the conduction band-width, the CF splitting, and the Kondo coupling constant, respectively. For T ¿ -, the higher Kondo temperature TKH D exp(−N=3|J |) is eCective, while for T ¡ -, the lower Kondo temperature TKL (D=-)2 D exp(−N=|J |) becomes eCective. Then, the resistivity shows the broad peak around TKH and the typical log T dependence around TKL . This is the case also in the periodic lattice system. The diCerent point from the impurity case is that in the periodic lattice systems, the resistivity smoothly changes into a power-law decay (typically T 2 ) with a large coeDcient A at lower temperature. This corresponds to the formation of coherent quasi-particle states with a large eCective mass m∗ , typically m∗ 100m0 . Here note that there exists a heuristic relation between the eCective mass and the coeDcient of the T 2 -resistivity. Actually, the Kadowaki–Woods’ relation, A ˙ )2 , has been found in many heavy-fermion compounds

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[471], where ) denotes the Sommerfeld coeDcient in the electronic speciMc heat. For instance, the resistivity in CeCu2 Si2 exhibits two hump structures at T ∼ 100 K and ∼ 20 K. Then, it smoothly changes into T 2 dependence at low temperatures [6,446,451]. These two characteristic temperatures (T ∼ 100 K and ∼ 20 K) correspond to TKH and TKL , respectively. The metallic behavior with the T 2 -resistivity at low temperatures indicates the formation of the coherent itinerant band of heavy quasi-particles. The coeDcient A is also consistent with the large enhanced value of the electronic speciMc-heat coeDcient, ) = 0:7–1:1 J=K 2 mol. The compound CeCu2 Si2 undergoes the transition into the SC phase at Tc = 0:7 K [6], marked by the discontinuity in the speciMc heat. The ratio of the jump to the normal-state speciMc heat WC=)Tc is of the order of unity (∼ 1:4), which could be explained in terms of the conventional BCS theory. This fact is a strong evidence for the fact that the superconductivity in this compound is a cooperative phenomenon in the heavy quasi-particles. It is natural to consider that the heavy quasi-particles on the Fermi surface form the Cooper-pairs with use of the residual interaction. In this case, a couple of quasi-particles prefer to form some anisotropic pairing state, not an s-wave one, in order to avoid the strong on-site repulsion. Actually, CeCu2 Si2 in the SC phase exhibits the power-law dependence in physical quantities at low temperatures, such as T 3 -dependence in the NMR/NQR 1=T1 [472,473] and T 2 -dependence in the thermal conductivity [474]. All these facts imply zeros of the gap along lines on the Fermi surface. In addition, the Knight shift decreases irrespective of directions of the crystallographic axis [475]. Thus, the pairing symmetry is considered as an even-parity state with line nodes, and presumably d-wave symmetry from many similarities with the cuprates and the organic superconductors. One of the remarkable similarities is the existence of the neighboring AF phase in the P–T phase diagram and the phase diagram of chemical substitutions in a series of the isostructural compounds. The ground state of CeCu2 (Si1−x Gex )2 , with increasing x, changes from the SC phase to the AF phase through the coexistent phase, owing to negative chemical pressures by Ge substitution for Si [476,477]. This property is similar to that of the organic superconductor, as is explained in Section 2.2. An isostructural compound CeCu2 Ge2 is the incommensurate spin-density-wave (SDW) state with Q = (0:28; 0:28; 0:54) at ambient pressure [478]. Under high pressures around 7 GPa, the magnetic phase abruptly disappears, accompanied by the SC transition with Tc = 0:5 K [457,458]. CePd 2 Si2 [459,460], CeRh2 Si2 [461], and CeNi2 Ge2 [462] also exhibit the transition from the AF metals to the SC phases (Tc 0:5 K) under high pressures. This is also the case in CeIn3 [460,463] (Tc = 0:2 K) with the cubic AuCu3 -type structure. The remarkable AF spin Auctuation in these compounds is actually implied by some measurements. The power smaller than two of temperature (T 1–1:5 ) is observed in the resistivity [460], and the NMR/NQR 1=T1 T increases at low temperatures [476,479,480]. These properties are typical ones in the nearly AF Fermi-liquid in quasi-2D and 3D systems. The relationship of such AF spin Auctuation with the unconventional superconductivity has been intensively investigated, especially in the cuprates, from the microscopic point of view (see Section 3.2). A series of CeTIn5 (T = Co, Rh, and Ir) discovered recently [464–466] possess many similarities with the cuprates and the organic superconductors. Thus, these compounds oCer a great opportunity to bridge our understanding in Ce-based heavy-fermion superconductors, high-Tc cuprates, and organic superconductors. These compounds are quasi-2D materials with the layered structure, which has been conMrmed by the de Haas-van Alphen (dHvA) measurements [481–485]. The P–T phase diagram in CeRhIn5 [477] and the phase diagram in the alloy system CeRh1−x Ir x In5 [486] indicate that

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the superconductivity appears in the neighborhood of the AF phase. The power-law behavior of the resistivity in the normal phase with T 1:3 in CeIrIn5 (Tc = 0:4 K) [465] and T in CeCoIn5 (Tc = 2:3 K) [466] implies the strong quasi-2D AF spin Auctuation. In the SC phase, no coherence peak just below Tc , T 3 -behavior at low temperatures in the NMR/NQR 1=T1 , and the decrease in the Knight shift irrespective of directions indicate the anisotropic even-parity superconductivity, similar to other Ce-based heavy-fermion superconductors [487–492]. In addition, the four-fold symmetry in the thermal conductivity in CeCoIn5 [493] strongly suggests the dx2 −y2 -wave singlet pairing. Furthermore, the pseudogap phenomenon has been reported in the 115 In-NQR measurement in CeRhIn5 in the range of pressures (P = 1:53–1:75 GPa), under which the coexistent phase exists [479]. These similarities, especially the AF spin Auctuation and dx2 −y2 -wave pairing, indicate that the underlying physics in the superconductivity in these Ce-based heavy-fermion superconductors is in common with the cuprate superconductors. This is also inferred from the microscopic electronic state, where one f-electron exists per Ce site and the f-electron band itself is almost half-Mlled when the hybridization with the conduction electron band is not taken into account. In the latter section, we verify the validity of the argument by estimating Tc in the Ce-based heavy-fermion superconductors on the same framework as in the cuprates. We introduce neither the coexistent phase in the boundary between the SC and AF phases nor the A phase with anomalous magnetic features [494]. Although these are one of interesting issues of f-electron systems, here we focused our interests on the superconductivity itself. 5.1.2. U-based compounds Next we brieAy review U-based heavy-fermion superconductors. They exhibit various kinds of SC phases probably owing to multi f-band structures with two or three f electrons per U site. We will show a possibility of our argument even in such complicated systems. First we note that U-based compounds do not necessarily display clear log T dependence in the resistivity. This is attributed to the multi f-orbitals and relatively large hybridization terms with the conduction electrons. These properties also provide the complicated multi-band structures and many Fermi surfaces with dominant f character. The Fermi surfaces obtained by the band calculations well explain the dHvA measurements. One of the characteristics of U-based compounds diCerent from Ce-based compounds is the clear coexistence between the SC phase and some kind of magnetic ordered phase. UPt 3 is the odd-parity superconductor (Tc = 0:5 K) discovered for the Mrst time in the electronic systems [13,274,447,448]. This superconductor coexists with an unusual AF order below TN = 5 K [495], which is observed in the neutron scattering. Note that it has not been observed by the static and/or dynamically slow probes. This unusual AF order at Q = (0:5; 0; 1) becomes the long-range order below 20 mK within the resolution of the neutron scattering [496]. In URu2 Si2 , the unconventional superconductivity below 1:5 K also coexists with a hidden order with a clear jump at 17:5 K in the speciMc heat [497,498]. UPd 2 Al3 is an AF metal with Q =(0; 0; 0:5) below TN =14:5 K [499] and coexists with the anisotropic even-parity superconductivity below Tc = 2 K [452,454]. The isostructural compound UNi2 Al3 is in the SDW state with Q = (0:5 ± 0; 0; 0:5) and 0 = 0:11 ± 0:003 below TN = 4:6 K [500,501]. This SDW state coexists with the unconventional superconductivity below Tc =1:2 K [453], which may be the odd-parity state as indicated by the NMR/NQR measurement [455]. Recently, in UGe2 and URhGe, the superconductivity in the ferromagnetic phase has been reported at Tc = 0:75 K (for P = 11:4 kbar) and 0:25 K (ambient pressure), respectively [275,276].

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Although such a variety of interesting phenomena have been observed experimentally, there are little theoretical progress from the microscopic point of view. One of the causes is the complicated band structure, since U-based compounds possess several f electrons per U site and many Fermi-surface sheets originating from multi f-orbitals. On the one hand, the complicated nature is unfavorable for the theoretical eCort, but on the other hand, it is related to a great variety of the ground states. It seems to be natural to consider that part of the Fermi surfaces stabilizing the magnetic phase is diCerent from that leading to the SC state, even though we cannot completely decouple them. Then, we can discuss possibility of the superconductivity by investigating the remaining Fermi surfaces in the magnetic phase. We will introduce the simple application on the unconventional superconductivity in the AF phase in UPd 2 Al3 and UNi2 Al3 later. The former is considered as an even-parity superconductor, while the latter is an odd-parity one. We will discuss how this diCerence occurs. 5.2. Microscopic theory First we introduce a theoretical treatment of the heavy-fermion superconductors from the microscopic point of view. As indicated in a variety of experiments for heavy-fermion compounds, the superconducting transition occurs after the formation of the coherent quasi-particle state with heavy eCective mass. Here we formulate a way to calculate Tc on the quasi-particle description based on the Fermi-liquid theory. Let us start our discussion on the periodic Anderson model (PAM) as a typical model for heavy-fermion systems. This model well describes the dual nature of f-electron systems. The perturbation expansion in terms of the hybridization matrix element leads to the RKKY interaction between the localized f-electron spins. This is the origin of the magnetic transition in f-electron systems. On the other hand, if no magnetic transition occurs, the system goes to a singlet ground state as the whole. This is the Fermi-liquid state of quasi-particles with heavy eCective mass. In this case, we should treat the PAM by the perturbation theory with respect to the on-site Coulomb repulsion U between f electrons, expecting the analyticity about U as long as no phase transition occurs. The analytic property has been exactly proved in the impurity Anderson model [502–505]. Although such an exact proof does not exist for the PAM, the continuity principle is believed to justify the applicability of the perturbation theory with respect to U . From such a point of view, Yamada and Yosida have developed the Fermi-liquid theory for the heavy-fermion state based on the PAM [506]. The momentum independent part of the four-point loc vertex functions > between f electrons with spin and plays an important role. This dominant s-wave scattering part leads to the nearly momentum independent large mass enhancement factor )˜ = a−1 . In fact, when the momentum dependence of the mass enhancement factor can be ignored, the T -linear coeDcient of the speciMc heat in the PAM, is given by loc ) (2=3)2 kB2 f (0)2 >↑↓ = (2=3)2 kB2 f (0))˜ ;

(105)

where kB is the Boltzmann’s constant and f (0) is the f-electron density of states at the Fermi level. −1 ) is represented as f (0)>loc with use This indicates that the large mass enhancement factor )(=a ˜ ↑↓ of the s-wave scattering part of the four-point vertex. This result implies that the interaction between loc the quasi-particles a2 >↑↓ holds the order of magnitude of the eCective band-width of quasi-particles

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107

T0 a=f (0), where T0 is the characteristic energy scale of the heavy-fermion state. For T ¡ T0 , the low-energy excitation can be described by the Fermi-liquid theory. On the other hand, the imaginary part of the f-electron self-energy, which is proportional to the electric resistivity, is given by 2 2 ; -k = (4=3)(f (0))3 >↑↓ loc T

(106)

when the momentum dependence of the vertices can be ignored. Thus, the coeDcient A of the T 2 -term loc of the electrical resistivity is proportional to )2 through the vertex >↑↓ . This relation A ˙ )2 , which is well known as the Kadowaki–Woods’ relation [471], holds when the momentum dependence of the vertices is suDciently weak to be ignored. Note that the relation does not hold when the momentum dependence of the quasi-particle interaction is remarkable. The high-Tc cuprate is the typical case, as is explained in Section 4.3. In most of heavy-fermion systems, the Kadowaki–Woods’ relation has been conMrmed, indicating that the four-point vertex function, i.e., the interaction among quasi-particles, possesses rather weak momentum dependence. The heavy-fermion superconductivity appears under this situation for T ¡ T0 . The large s-wave loc repulsive part >↑↓ prevents an appearance of the s-wave singlet superconductivity. In this case, anisotropic pairing states such as p- or d-wave will be formed due to the remaining momentum dependence of the four-point vertices in the particle–particle channel. Here we discuss such anisotropic paring on the quasi-particles. First we divide the four-point vertex function > (p1 ; p2 ; p3 ; p4 ) into the large s-wave scattering loc part > and the non-s-wave part W> (p1 ; p2 ; p3 ; p4 ); loc > (p1 ; p2 ; p3 ; p4 ) = > + W> (p1 ; p2 ; p3 ; p4 ) ;

(107)

where p1 and p2 are the incident momenta, while p3 , p4 the outgoing. The second term W> (p1 ; p2 ; p3 ; p4 ) has remarkable momentum dependence leading to the anisotropic Cooper-pairing. Our purpose is to formulate how to calculate W> (p1 ; p2 ; p3 ; p4 ) for the heavy-fermion quasi-particles, loc which are renormalized by the large local part >↑↓ . Corresponding to these two terms of the vertex function, we can also separate the self-energy into the local and non-local parts as 2(k; !) = 2loc (!) + W2(k; !) :

(108)

In this case, the f-electron Green’s function below T0 is given by 1 ; G(k; !) = Vk2 loc ! − =k − 2 (!) − W2(k; !) − ! − @k a + Ginc (!) ; (109) = ˜k2 V ˜ !) − ! − E˜ k − 2(k; ! − @k loc ˜ where a is the mass √ renormalization factor, E k = a(=k + Re 2 (0)) is the renormalized f-electron ˜ !) = aW2(k; !) is the dispersion, V˜k = aVk is the renormalized hybridization term, 2(k; renormalized f-electron self-energy, and Ginc (!) denotes the momentum independent incoherent part of the Green’s function. The imaginary part of 2loc in proportion to !2 + (T )2 can be ˜ !), as indicated by Hewson for the imincluded in the remaining renormalized self-energy 2(k; purity Anderson model [507]. From the Ward–Takahashi identity, we can obtain the large mass

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enhancement factor as )˜ = a

−1

! d2loc (!) !! i loc 2 >↑↓ =1− = 1 − Ginc (!) d! : d! !!=0 2

(110)

loc In addition, we can set the large local vertex >↑↓ as a constant value at zero frequencies, as far as loc the frequency of external lines is less than T0 . In this case, the eCective on-site interaction a2 >↑↓ loc works on the quasi-particles. In the impurity Anderson model, a2 >↑↓ = a- = 4TK , where - and TK are the width of the virtual bound state and the Kondo temperature, respectively. Thus, we loc can assume that a2 >↑↓ is of the order of T0 in the periodic system. This is also consistent with the above-mentioned discussion of the speciMc-heat coeDcient. It should be noted again that the renormalized s-wave interaction among quasi-particles becomes comparable with the quasi-particle band-width, while the on-site repulsion among the bare f-electrons is suDciently larger than the bare f-electron band-width. ˜ !) and vertex funcNow we show that in the region ! 6 T0 , the renormalized self-energy 2(k; 2 ˜ tion > (p1 ; p2 ; p3 ; p4 ) = a W> (p1 ; p2 ; p3 ; p4 ) can be discussed with the perturbation scheme 2 loc with respect to the eCective s-wave interaction, which is expected to be in the order of >˜ loc ↑↓ = a >↑↓ . ˜ !) = aW2(k; !). Because we have divided First, let us consider the renormalized self-energy 2(k; ˜ the Green’s function into the two parts as aG(k; !) + Ginc (!) in Eq. (109), we can correspondingly divide all diagrams in the perturbative series into diagrams with and without the remarkable momentum dependence. The latter is contained in the local self-energy 2loc (!). On the other hand, the former can be rewritten as the expansion with respect to the eCective s-wave interaction >˜ ↑↓ by concentrating on diagrams with the same momentum dependence. For instance, let us consider the third-order self-energy 2(3) (k; !) = U 3 q (q)2 G(k + q), as shown in Fig. 73. The particle–hole line (q; H) = − p G(p + q)G(p) is divided into (q; ˜ H) = a2 G˜ G˜ and inc (H) including Ginc (! ). The former part contributes mainly for H 6 T0 , while the latter does not possess remarkable momentum dependence the p summation. Then, the third-order is also divided into three parts: after diagram ˜ and the remaining terms 20(3) =U 3 ˜ G, ˜2 aG˜ with only the coherent part, 2v(2) =2U 2 q H (Uinc )a with no remarkable momentum dependence. The last terms only contributes to the local self-energy. By ignoring the frequency dependence of inc (H), the second term 2v(2) has the same momentum dependence as that in the second-order diagram in U . Then, one of the bare vertex U is replaced by U 2 inc . This is one of vertex corrections for the second-order self-energy. Also in the higher-order diagrams, we can pick out diagrams with three lines of aG˜ producing the same momentum dependence. We get together such terms and reduce them to a second order diagram with respect to the ˜ !) and a in the deMnition eCective coupling constant >↑↓ . By counting a3 in three lines of aG(k; 2 ˜ ˜ ˜ G G G can be of the renormalized self-energy, the renormalized second-order diagram 2˜ (2) = a4 >↑↓ regarded as the second-order diagram with respect to the eCective s-wave interaction >˜ ↑↓ = a2 >↑↓ . Here we replace the eCective coupling constant for each reduced diagram to be >˜ ↑↓ and approximate it as >˜ loc ↑↓ . Then, we can estimate the renormalized self-energy in the same manner as explained in Section 3.1. Likewise, within the low-order diagrams, the vertex function >˜ (p1 ; p2 ; p3 ; p4 ) can ˜ !) and be reconstructed as the expansion with respect to the quasi-particle Green’s function G(k; loc ˜ the eCective s-wave interaction >↑↓ . Strictly speaking, in the higher order, there appear diagrams which cannot be reduced to the expansion in this manner. For instance, higher-order vertex generally

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

(a)

109

(b)

2 ˜ Fig. 73. (a) The third-order self-energy 2(3) (k; !) can be divided into three parts; 20(3) = U 3 ˜ aG, (2) 2 ˜ 2v = 2U ˜ G and the remaining terms with no remarkable momentum dependence. The second term q H (Uinc )a

2v(2) has the same momentum dependence as that in the second-order diagram in U . (b) The renormalized self-energy can ˜ be rewritten as the expansion with respect to the quasi-particle Green’s function G(p) and the eCective s-wave interaction loc 2 loc ˜ ˜ >loc = >↑↓ = a >↑↓ .

remains. Here we simply ignore such diagrams and discuss the renormalized eCective PAM. Then, we take a change of energy scale in the band-width and the interaction between quasi-particles. A better convergence is expected for this renormalized scheme rather than the original perturbation scheme. This procedure just corresponds to an approximation similar to the pseudo-potential method, such as the ladder summation of the divergent hard-core potential in 3 He shown by Galitskii [508]. The concept of our scheme corresponds to that the eCective Hamiltonian obtained by the renormalization group method can be written as the renormalized PAM. Then, the essential assumption is the locality of the interaction and the unimportance of the higher order vertex, when the renormalization is performed to the energy scale T0 . Of course, we must directly treat the expansion in the bare interaction U to describe the crossover from the localized feature at high temperatures to the itinerant feature at low temperatures, which is characteristic in heavy-fermion systems. This is one of the future problems. Next we formulate the Gor’kov equation for the SC transition on the quasi-particle description discussed above. As usual, the SC transition is marked by divergence of the full vertex in the Cooper channel. This is determined by evaluating the linearized Dyson–Gor’kov equation (see Section 3.1) as >(2) (p; p )|G(p ; i!n )|2 -(p ; i!n ) = -(p; i!n ) ; (111) p

where -(p; i!n ) is an anomalous self-energy and the Cooper-pairing eCective interaction >(2) (p; p ) is the particle–particle irreducible vertex. This equation includes the integral of |G(p ; i!n )|2 . The ˜ ; i!n )|2 . most important part of this integral comes from the part mediated by quasi-particles a2 |G(p

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Thus, the Gor’kov equation can be rewritten as ˜ ; i!n )|2 -(p ; i!n ) = -(p; i!n ) : a2 >(2) (p; p )|G(p

(112)

p

This a2 >(2) (p; p ) is the particle–particle irreducible vertex between quasi-particles with the external frequencies smaller than T0 . As discussed above, for such a vertex, we can also apply the perturbation loc expansion with respect to the renormalized s-wave interaction a2 >↑↓ between quasi-particles. Thus, in order to treat the heavy-fermion superconductivity, we start by introducing the quasiparticle state renormalized by the dominant s-wave scattering part, which itself does not yield the stable pairing interaction. Then, we calculate the momentum dependent interaction between the quasi-particles, using the perturbation expansion in terms of the renormalized on-site repulsion. This leads to the transition into unconventional SC phases as a cooperative phenomenon of the heavy-fermion quasi-particles. Note that the extension of the above procedure is straightforward, even if the AF spin Auctuation is relatively strong and the eCective interaction has rather strong momentum dependence as seen in a kind of Ce-based heavy-fermion superconductors and the cuprate superconductors. Now we can begin by the PAM and the Hubbard model describing the quasi-particle Fermi surface. Note also that such quasi-particle description implicitly includes the following two important arguments. The renormalized s-wave interaction among quasi-particles is expected to be moderate as compared with the quasi-particle band-width, even if the on-site repulsion among the bare f electrons is suDciently larger than the bare f-electron band-width. The electron–phonon interaction between quasi-particles is renormalized more severely, since it does not possess enhancement due loc to the vertex corrections as >↑↓ ˙ a−1 . Thus, the conventional superconductivity mediated by the electron–phonon interaction is excluded at this stage. 5.3. Application to materials We proceed to the results of the perturbation theory based on the renormalized formula in the typical heavy-fermion superconductors; CeCu2 X2 (X = Si and Ge), CeTIn5 (T = Co, Rh, and Ir), CeIn3 , and UM2 Al3 (M=Pd and Ni). It should be noted that the following calculations do not strictly obey the above renormalization scheme. For instance, we do not include explicitly the contribution from the imaginary part of the local self-energy and the cut-oC of the energy. Rather the perturbative scheme explained in Section 3.1 will be directly applied. However, these diCerences do not alter the qualitative results such as the pairing symmetry of the stable state. Note also that the multi-band system is frequently reduced to the eCective single-band model by taking into account the most important band triggering the superconductivity. 5.3.1. CeCu2 X2 (X = Si and Ge) As introduced in Section 5.1, the superconductivity in CeCu2 Si2 with Tc =0:7 K locates around the border with the AF phase at ambient pressure [6,473]. This is also conMrmed in terms of similarities with the P–T phase diagram in the isostructural compound CeCu2 Ge2 [509] (Section 5.1.1). The eCect of the dominant AF spin Auctuation has been observed as deviation from the conventional Fermi-liquid theory in the physical quantities, such as the resistivity, the speciMc heat [510], and the NMR/NQR 1=T1 [472,476]. Then, the AF spin Auctuation may be responsible for the anisotropic even-parity superconductivity, which is observed in these compounds. In the spin Auctuation theory,

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111

Fig. 74. The renormalized Fermi surfaces at U=t = 8, V=t = 2:0, tb =t = 0, E0f =t = −6 and T=t = 0:05, which is calculated by (, − =k − 2N (k; 0))(, − @k ) − V 2 = 0. (a) The electron Fermi surface around >-point. (b) The hole Fermi surface around Z-point. (c) The periodic zone at kz = 0. (a) and (b) correspond to light and dark hatches in (c), respectively.

the SC transition temperature usually takes maximum value near the AF phase (see Figs. 23, 24 and 29). CePd 2 Si2 and CeIn3 [460] are often cited as the typical examples. In CeCu2 X2 , however, the phase diagram is not so simple. With increasing pressures, Tc displays the curious enhancement in the range of P =2–3 GPa, not the monotonic decrease. The maximum value of Tc is about 2 K. Thus, there exists an optimum condition, such as the band structure, for the superconductivity. It indicates that the low-lying part of the quantum-critical AF Auctuation is not responsible for the formation of the superconductivity, as indicated in CePd 2 Si2 and CeIn3 [460]. On the other hand, recent systematic experiments on CeCu2 Ge2 have shown that the curious behavior seems to correspond to the peak structure of the residual resistivity and the drastic decrease of the T 2 -coeDcient A of resistivity [458]. As a model involving these properties, a new pairing mechanism mediating the critical f-valence Auctuations has been proposed [511]. However, here we investigate which kind of anisotropic superconductivity can be stabilized in the complicated band structure of CeCu2 X2 on the basis of TOP based on the quasi-particle description [512]. In addition, let us verify the validity of the simple uniMed view. The band calculation with the linearized augmented-plane-wave (LAPW) method indicates that CeCu2 X2 are compensated metals with large and small electron Fermi surfaces around the >-point and complex hole Fermi surfaces mainly around the Z-point [513]. The band structure near these Fermi surfaces are well represented by two bands, which are formed by the mixing between a quasi-2D f-band =k = −2t[cos(kx ) + cos(ky )] − 8tb cos(kx =2) cos(ky =2) cos(kz =2) + E0f ;

(113)

and a 3D conduction-band @k = 2tc [cos(kx ) + cos(ky ) − 0:8 cos(kx ) cos(ky ) − 2 cos(kx =2) cos(ky =2) cos(kz =2) − 0:5] : (114) Here t and tb are transfer integrals of f electrons, which are smaller than that of conduction electrons tc = 10t, and E0f = −6t is an f-electron site energy. When we choose tc = 10t = 0:025 Ryd: = 0:34 eV and the hybridization term without momentum dependence V = 0:068 eV, the band structure near the Fermi level can be almost explained except for some small Fermi surfaces as shown in Fig. 74.

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χ0(q,0)

0.6

tb/t=0.0 tb/t=0.3 tb/t=0.5

0.4

0.2

0 Z

Γ

Z

X

Γ N

P X

Fig. 75. The zeroth-order spin susceptibilities 0 (q; 0) at tb =t = 0:0, 0.3 and 0.5. The other parameters are the same as in Fig. 74. The peak structure around X-point is due to the nesting properties of the Fermi surfaces. This peak structure is suppressed with increasing tb .

We consider the PAM with this band structure as described by † f f † † H= @k ck ck + V (fk ck + h:c:) + =k fk fk + U ni ↑ ni ↓ ; k

k

†

k

(115)

i

†

where fk , ck (fk , ck ) are the annihilation (creation) operators for f and conduction electrons with the wave-vector k and a pseudo-spin index . The Green’s functions in the PAM are described in a 2 × 2 matrix form as −1 f G (k) G fc (k) i!n − =k + , − 2N (k) V = G(k) = : (116) V i!n − @k + , G cf (k) G c (k) Since the bare interaction works only between f electrons, the Dyson–Gor’kov equation for G f (k) is the same one as in single band case introduced in the previous chapters. G c (k) does not directly contribute to the equation like the p-electron Green function in the d–p model. Conduction electrons are incorporated into the SC state through the hybridization with f electrons. Within the simple TOP, we cannot obtain the large mass enhancement factor ), Y since U cannot be larger than the f band-width. In CeCu2 Si2 , most parts of the large mass enhancement factor comes from almost momentum independent self-energy, as indicated in the Kadowaki–Woods’ relation. Thus, the renormalization treatment introduced in Section 5.2 will be valid. In this case, the large mass enhancement is phenomenologically taken into account. Hereafter we consider loc the on-site Coulomb repulsion U as the renormalized quasi-particle interaction a2 >↑↓ as is discussed in Section 5.2. The most favorable pairing state within the TOP is dx2 −y2 -wave symmetry. The dominant attractive part originates from the RPA-type diagrams including the particle–hole bubble-type diagrams. As shown in Fig. 75, the bare f-electron susceptibility possesses the peak structure around (; ; qz ) for any qz . Thus, the AF spin Auctuation basically induces the unconventional superconductivity. Fig. 76 illustrates E0f dependence of Tc . If assuming )Y ∼ 10, we obtain the renormalized hopping integral t ∼ 39 K, and the maximum value of Tc ∼ 0:06 × 39–2:3 K. This is a reasonable value. Furthermore, since pressures relatively raise up the f-level E0f and reduce the f-electron number, we can consider that Fig. 76 shows pressure dependence of Tc . The region of E0f . −6t, where

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113

0.08

Tc /t

0.06 0.04 V/t=2.0 V/t=2.2 V/t=2.4 V/t=2.6 V/t=2.8 V/t=3.0

0.02 0

-8

-4

f

0

4

E 0 /t Fig. 76. Tc as a function of the f-level E0f for several values of V at U = 8t and tb = 0:0. Tc decreases with increasing V , although it is almost unchanged for E0f . −6t. This is because f-band character becomes stronger with decreasing E0f .

the hybridization dependence is weak, corresponds to the AF phase in CeCu2 X2 . In the region of E0f & −6t, on the other hand, Tc shows a hump structure around E0f ∼ 2t. Suppression of Tc for E0f . 2t originates from the mass renormalization of the normal self-energy due to the relatively strong correlation, while for E0f & 2t, from the weak pairing interaction due to the relatively weak correlation. This hump structure corresponds to the curious enhancement of Tc under high pressures in CeCu2 X2 . If we strictly follow the renormalization procedure in Section 5.2, the most of the mass enhancement is not included in the calculated normal self-energy. In fact, the renormalized quasi-particle band itself, which is phenomenologically introduced here, is sensitive to pressures. Pressures make the mass enhancement )˜ smaller and the eCective band-width larger. This will enhance the hump structure in T −E0f phase diagram illustrated in Fig. 76. Thus, pressure dependence of Tc in CeCu2 X2 can be naturally explained by taking the mass renormalization into account. 5.3.2. CeTIn5 (T = Co, Rh, and Ir) Recently, superconductivity in a series of CeTIn5 has been discovered [464–466]. CeRhIn5 is an AF state with TN = 3:8 K at ambient pressure, and coexists with a SC state under pressures P ¿ 1:5 GPa. For P ¿ 1:8 GPa, the AF phase vanishes and only the SC state with Tc = 2:1 K appears. CeIrIn5 and CeCoIn5 are superconducting at ambient pressure with Tc = 0:4 K and 2:3 K, respectively. CeCoIn5 has the highest Tc among heavy-fermion superconductors discovered up to now. These compounds are worthy of note, since the nature is reminiscent of the cuprates and the organic superconductors, and therefore, these compounds will bridge between our understanding of Ce-based heavy-fermion superconductors and that of the cuprates and organic superconductors. Such a viewpoint is suitable for the purpose of this review. The crystal structure of these compounds is HoCoGa5 -type tetragonal one with a layered structure, in which alternating layers of CeIn3 and TIn2 stack sequentially along the c-axis. In the dHvA measurements, the quasi-2D Fermi surfaces consistent with the band calculations have been observed [481–485]. In the SC phase, the results of the NMR/NQR measurements indicate the anisotropic even-parity pairing [487–492]. In addition,

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Tc /t

0.1

0.01

n=0.77 n=0.73 n=0.69 CeCoIn5 CeIrIn5

0.001 5

5.5

6

6.5

7

7.5

U/t Fig. 77. Tc as a function of U for several n. For comparison, we illustrate Tc corresponding to the transition temperatures of CeCoIn5 and CeIrIn5 .

the dx2 −y2 -wave pairing was suggested from the thermal conductivity in CeCoIn5 under the parallel magnetic Meld [493]. In the normal phase, both the transport [465,466] and magnetic properties imply the strong quasi-2D AF spin Auctuation. From the detail analysis of NMR/NQR T1 with the application of SCR theory [8], it has been shown that these compounds locate close to the quantum critical point (QCP). In particular, CeCoIn5 is considered to be just above the QCP. This is also inferred from the behavior C=T ˙ −ln T in the normal-state speciMc heat under the magnetic Meld above Hc2 [466,514,515]. The associated low-temperature entropy is consistent with the huge zero-Meld speciMc-heat jump at Tc , WC=)Tc = 4:5. Although this remarkable behavior in the speciMc heat has not been observed in the cuprates, the nature can be well understood as the Fermi-liquid state with the quasi-2D AF spin Auctuation. Thus, the superconductivity in these compounds should be explained on the same footing as that in the cuprates. We show below that x-dependence of Tc in CeIr x Co1−x In5 [516] is consistent with change of the carrier number in the main Fermi surface triggering the superconductivity, and that the large enhanced value of the speciMc-heat jump can be explained by the strong-coupling theory of superconductivity including the quasi-2D AF spin Auctuation. Let us discuss the x-dependence of Tc in CeIr x Co1−x In5 within TOP on the basis of the quasi-particle description [517]. The main part of the band structure can be approximated by the tight binding Mtting in a square lattice, =k = 2t(cos kx + cos ky ), which reproduces the large Fermi surface with the heaviest cyclotron mass [481–485]. The main Fermi surfaces in CeIrIn5 and CeCoIn5 are well reproduced for the electron number n = 0:69 and 0.77, respectively. Thus, we consider the Hubbard model with such quasi-particle band and the renormalized on-site repulsion. The resulting model is the same one as that was discussed in Section 3.2.3 with t =t = 0. We have already shown that TOP, in this case, gives the qualitatively same results with the spin Auctuation theory (Section 3.2). If using the PAM as discussed above, we may reproduce more parts of the Fermi surfaces, but here we simply consider the single-band model based upon a belief that quasi-particles on the main Fermi surface play an essential role for the pair formation. Multi-band Hubbard model for f-electron systems has been studied [518], but the multi-band eCect is not discussed here. The result of TOP is illustrated in Fig. 77. The third-order perturbation expansion with respect to U leads to the eCective pairing interaction for the dx2 −y2 -wave superconductivity as expected from

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the results in Section 3.2. Tc provides proper values for the moderate U . We see that Tc increases with approaching to the half-Mlling. This tendency is also consistent with experimental results in CeIr x Co1−x In5 [516]. Next we discuss the huge enhanced value of the speciMc-heat jump at Tc in CeCoIn5 , by analyzing the same model Hamiltonian within FLEX [519]. The thermodynamic potential 4(T; ,) in FLEX is given by the scheme of Baym and KadanoC, since FLEX is a kind of the conserving approximations. Then, the entropy S = −(94=9T ), is given by explicit derivative of 4(T; ,) with respect to T , since its implicit derivative through the self-energy vanishes owing to the stationary conditions (Eq. (6)) [169]. As a result, we obtain the entropy 9 T ˆ Tr ln G(k) : (117) S =− 9T N ! k

n

After the analytic continuation on the real axis, we obtain in the normal state 9f 1 1 [ln GRn − ln GnA ] ; d@@ − S =2 2iT N 9@

(118)

k

and in the superconducting state 9f 1 1 R A [ln Gsc d@@ − − ln Gsc ] ; S= 2iT N 9@

(119)

k

R; A 2 2 = [!± Zk − =Y2k − 2a2 ]−1 with =Yk = =k + and 2n (k; ±!) = ±!(1 − Zk ) + . The results are where Gsc illustrated in Fig. 78. Fig. 78(a) illustrates the maximum of the anomalous self-energy 2a (k; T ) as a function of T=Tc for U=t = 4:0, 4.5, and 5.0. The rapid increase of the anomalous self-energy below Tc is the characteristic behavior in the strong coupling theory, which has been already obtained in the FLEX approximation [186,187,193]. This is because the de-pairing eCect arising from the normal self-energy is suppressed below Tc . We can see that superconductivity becomes strong coupling, as U=t is larger and equivalently the largest s (Q) at Q ∼ (; ) is larger. In this case, the entropy S is evaluated as in Fig. 78(b). It shows the convex behavior from high temperatures, which originates from the AF spin Auctuation. Too small entropy below T=t = 0:005 may be due to the numerical errors. With use of the Mtting by the polynomial function of the entropy in the vicinity of Tc , we obtain the speciMc heat C and the enhanced jump of the speciMc heat WC=)Tc in Fig. 78(c). As U=t is larger, Tc and WC=)Tc are also larger. At U=t = 5:0, Tc =t = 0:017 and WC=)Tc = 4:6. It is not easy to obtain larger value than this, since the system becomes the AF phase. Increase of C=T in the normal state at lower temperatures can be considered as the precursor of −ln T dependence, which is predicted by the SCR theory for 2D AF spin Auctuations. Furthermore, if we set Tc =0:017t as 2:3 K, then t 135 K and C=T just above Tc corresponds to ∼ 200 mJ=mol K 2 (290 mJ=mol K 2 in the experimental data [466]). Since the bare t is expected to be 2–3 times larger than the present value 135 K, the renormalization inherent in the quasi-particle description is not large in this case. Thus, the anomalous behavior of the speciMc heat in CeCoIn5 can be almost explained by the quasi-2D AF spin Auctuation.

5.3.3. CeIn3 CeIn3 is only one heavy-fermion superconductor with the cubic symmetry. At ambient pressure, it is the AF state with an ordering vector Q = (0:5; 0:5; 0:5) and TN = 10 K, while at the critical

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∆

χ

116

(a)

(b)

(c)

Fig. 78. (a) 1=(Q) as a function of T=Tc and -=Tc = 2amax (k; T )=Tc as a function of T=Tc . (b) The entropy S and (c) the speciMc heat C in the normal and SC states. As U=t is larger, the jump of C=T at Tc is enhanced.

pressure Pc = 2:55 GPa, it becomes superconducting with Tc 0:2 K [460,463]. At P = 2:65 GPa, 1=T1 displays a signiMcant decrease below T ∗ =30 K, and T1 T =const. below TFL =5 K, which is the typical Fermi-liquid behavior. Recently, it has been conMrmed that the superconductivity possesses the unconventional nature from no coherence peak in the 115 In-NQR measurement [479,480]. We note again that the relative material CeRhIn5 , which is the quasi-2D material, has relatively high Tc = 2:1 K under pressure. This comparison indicates that the dimensionality is one of important factors for occurrence of the unconventional superconductivity. ECect of dimensionality on Tc has been already discussed for the spin-Auctuation mediated superconductivity. Both in the phenomenological models [185,520] and in the microscopic calculations on the basis of FLEX [521,522], it has been shown that the magnitude of Tc is higher in quasi-2D systems than in 3D systems. With increasing three-dimensionality on the energy dispersion, the AF phase is more stabilized owing to the suppression of the Auctuation, and then the SC phase shrinks. Moreover, the total weight of the spin Auctuation decreases with increasing the three-dimensionality. This is a general feature of the Auctuation theory as explained in Section 4.2.2. Then, Tc decreases because the pairing interaction

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117

t2=−0.2t1,n=0.9,U/W=3/4

Tc /t1

0.04

Tc /TcRPA 0.3 0.02 0.2 0.1 0 0 0

0

0.2

0.5 tz

1 0.4

tz

0.6

0.8

1

Fig. 79. Tc =t1 as a function of the anisotropy tz . In the inset, the ratio Tc =TcRPA is illustrated. Tc in 3D systems becomes one order smaller than that in 2D systems.

becomes eCectively weak. We consider that CeIn3 is a clear example for this suppression of Tc due to the three-dimensionality. We here estimate the superconducting Tc in cubic CeIn3 and its variation for the dimensionality within TOP [523]. We discuss relation between Tc and dimensionality. For simplicity, we consider the Hubbard Hamiltonian with the energy dispersion @k = −2t1 (cos kx + cos ky + tz cos kz ) + 4t2 (cos kx cos ky + tz cos ky cos kz + tz cos kz cos kx ) ;

(120)

where t1 and t2 denote the nearest-neighbor and the next-nearest-neighbor hopping integrals. Note that dimensionality is controlled by a parameter tz . The cases of tz = 0 and 1 correspond to the 2D square lattice and the 3D cubic lattice, respectively. In order to describe the main Fermi surface with a large volume in CeIn3 , we choose t2 = −0:2t1 and the electron density n = 0:9 for tz = 1 [524]. The Fermi surface possesses the nesting property, and the bare susceptibility exhibits a peak structure in the vicinity of Q =(0:5; 0:5; 0:5). The qualitative nature of the superconductivity is the same as in the quasi-2D systems. The main pairing interaction originates from the RPA-type diagrams. The most favorable pairing state has dx2 −y2 -wave symmetry. In 3D cubic systems, this state degenerates with d3z2 −r 2 -wave symmetry. According to the weak coupling theory, time-reversal-symmetry-breaking state, which is the linear combination of the two pairing states, is expected in the SC state. However, one or another may be preferred because of the feedback eCect. This subject will be clariMed in the NQR/NMR 1=T1 at very low temperatures. The obtained Tc = 0:003t1 for the moderate value of U=W = 3=4 with the band-width W 3D 12t1 can explain Tc 0:2 K in CeIn3 with t1 100 K. We show the role of dimensionality in Fig. 79. It can be seen that Tc in 3D systems becomes one order smaller than that in 2D systems, although Tc in quasi-2D systems is robust for tz . 0:5. These features are consistent with the results in the spin Auctuation theory and with the experimental fact that Tc = 0:2 K in CeIn3 and Tc = 2:1 K in CeRhIn5 . It is generally expected that the superconducting Tc is small in 3D systems, since the momentum dependence of the eCective interaction is relatively weak. This is the underlying physics in common with the spin Auctuation theory.

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Finally, let us comment on the quasi-1D system. In this case, Tc for the d-wave superconductivity is generally low, because the quasi-1D momentum dependence of the eCective interaction is not suitable for the d-wave superconductivity and the de-pairing eCect is enhanced by the nested Fermi surface. Thus, the quasi-2D systems like high-Tc cuprates, -(ET)2 X and CeTIn5 are the most favorable for the appearance of the d-wave superconductivity. 5.3.4. UM2 Al3 (M = Pd and Ni) UPd 2 Al3 and UNi2 Al3 [452,453] exhibit clear coexistence between an AF phase and an unconventional superconductor. UPd 2 Al3 is an AF metal [499] and coexists with the anisotropic even-parity superconductivity below Tc = 2 K [452,454]. UNi2 Al3 is the SDW state with Q = (0:5 ± 0; 0; 0:5) [500,501] and coexists with the odd-parity superconductivity at Tc = 1:2 K [453]. The AF state in UPd 2 Al3 possesses relatively large ordered moments 0:85,B =U as compared with 0:2,B =U in UNi2 Al3 , and the AF transition seems to be that of the localized f-electron system. The ordered Y direction on the c-plane, and alternate along the c-axis moments are aligned parallel to the [1120] with Q = (0; 0; 0:5). For Tc . T TN , both systems exhibit typical behaviors of heavy-fermion systems; the large enhanced coeDcient of the electronic speciMc heat () = 140 mJ=K 2 mol in UPd 2 Al3 and ) = 120 mJ=K 2 mol in UNi2 Al3 ) and T 2 behavior in the resistivity with the large coeDcient [452,453]. Thus, it has been indicated that, especially in UPd 2 Al3 , two separated subsystems, a localized part leading to the magnetic long-range order and an itinerant part forming the heavy-fermion state, seem to coexist in momentum space [525,526]. On the other hand, the inelastic peak observed in the neutron scattering measurement in UPd 2 Al3 implies a sizable interaction between two subsystems [528–530]. At T = 4:2 K, the spectrum at the AF zone center Q = (0; 0; 0:5) exhibits a quasi-elastic peak at ! = 0 and an inelastic peak (magnetic exciton) at ! = 1:5 meV. After the SC transition, the former peak shifts to the high-energy side, and develops into the inelastic (‘resonance’) peak centered at ! = 0:4 meV at T = 0:4 K. This is considered as a circumstantial evidence for the scenario that the dispersive magnetic exciton by localized 5f electrons is responsible for the unconventional superconductivity [531,532]. This mechanism indicates the horizontal line nodes on the AF zone boundary. However, it has been recently shown by Thalmeier that this mechanism rather favors an odd parity state [533]. In addition, it is suspicious whether such an indirect interaction as mediated by magnetic-excitons dominates the many-body eCect originating from the Coulomb repulsion between itinerant electrons. Here we consider that the Coulomb interaction between the itinerant electrons should be responsible for the pairing mechanism in this compound. In this case, the band structure of the itinerant electrons is important for the superconductivity, according to the renormalization procedure in Section 5.2. In UPd 2 Al3 , the dHvA eCect in the AF phase is in good agreement with the Fermi surfaces calculated by the band calculation [534]. The two dominant Fermi surfaces (‘party hat’ and ‘column’) with heavy cyclotron mass have quasi-2D nature to some extent. These quasi-2D Fermi surfaces will prefer the vertical line node rather than the horizontal one in the SC state, if their own Coulomb repulsion is a dominant interaction. This is another probable candidate for the unconventional superconductivity in UPd 2 Al3 . This mechanism is suitable for uniMed viewpoint on the unconventional superconductivity in SCES as frequently stressed in this review. Here we investigate a possible scenario for the appearance of the diCerent SC states in UPd 2 Al3 and UNi2 Al3 on the basis of the above standpoint.

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Tc /t

0.01 n=1.300 (B2u2 ) n=1.300 (B1g) n=1.200 (B1g) n=1.144 (B1g)

0.001 0.75

0.8

0.85

1.2 1 0.8 0.6 0.4 0.2 0.9

n=1.200 T=0.003t 2 B2u

λ

λ

(b)

0.9

0.95

1

tm/t

(a)

B1g 0.95 tm /t

1

0.8 0.7 B1g 0.6 0.5 2 B2u 0.4 0.3 0.9

(c)

n=1.144 T=0.003t

0.95 tm /t

1

Fig. 80. (a) Tc as a function of the anisotropy tm =t at U = 7:5t. (b) and (c) illustrate, respectively, the eigen values V of the Eliashberg equation for n = 1:200 and n = 1:144 at T = 0:003t. -B1g is stable in a wide range of tm =t, while -2B2u only in the vicinity of tm =t = 1. Although the region of -2B2u becomes wider as n decreases, the transition temperature is suppressed abruptly.

We investigate the 2D Hubbard model on an anisotropic triangular lattice, which corresponds to the c-plane in UPd 2 Al3 and UNi2 Al3 . Note that it is essentially the same model as investigated for organic superconductors (Section 3.3). For simplicity, here we introduce only an anisotropy of the Fermi surfaces as an eCect of the AF order. Then, the dispersion of the quasi-particles is represented by √ 1 3 @k = −4t cos kx cos ky − 2tm cos(ky ) ; (121) 2 2 reAecting that the Brillouin zone is reduced by the AF arrangement to a c-based-center orthorhombic, although the chemical unit cell is hexagonal [535,536]. Here t and tm are the nearest and next nearest hopping integrals, respectively. Note that the dispersion possesses the D2h symmetry for tm = t, while at tm = t, it is reduced to that of the triangular lattice with the D6h symmeV try. Now we evaluate Eliashberg equation within TOP to study stable superconducting states for the anisotropy tm =t = 0:75–1.0, the electron density n = 1:0–1.4, and the Coulomb repulsion U=t = 3:5–7.5. Among irreducible representations of D2h symmetry, we investigate the following probable √ four kinds of pairing symmetry, except A1g and ones including kz -dependence: -B1g = sin( 23 kx ) √

√

sin( 12 ky ), -1B2u = cos( 23 kx ) sin( 12 ky ), -2B2u = sin(ky ), and -B3u = sin( 23 kx ) cos( 12 ky ). We have found two kinds of stable state in a Mnite range of parameters, irrespective of n and U : One is the even-parity pairing state -B1g and another is the odd-parity pairing state -2B2u . As shown in Fig. 80, the former is stable in a wide range of tm , while the latter only in the vicinity of the symmetric point tm =t (D6h symmetry). Although the range of -2B2u becomes wider as n decreases, the transition

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temperature is suppressed abruptly. We propose these two stable states as the even-parity pairing in UPd 2 Al3 and the odd-parity pairing in UNi2 Al3 , respectively. This proposal is consistent with a feature in each compound, because we can expect that the distortion from the D6h symmetry under the magnetic order is larger in UPd 2 Al3 than in UNi2 Al3 , according to the magnitude of the ordered magnetic moment. Thus, the result obtained by TOP for this simpliMed model can well explain the emergence of the diCerent SC states in UPd 2 Al3 and UNi2 Al3 . Note that the simpliMcation of the model is obviously hypothetical and some future works will be required. For instance, the eCects of the AF order, three-dimensionality, multi-band eCect [527], and the relation with the ‘resonance’ peak at Q = (0; 0; 0:5) should be investigated in more details. 6. Concluding remarks and discussion In this review, we have discussed systems possessing the strong electron correlation. They are cuprate superconductors, organic superconductors, Sr 2 RuO4 , and heavy-fermion superconductors. We have analyzed the superconductivity realized in these substances on the basis of the singleand multi-orbital Hubbard Hamiltonian. In the single-band Hubbard Hamiltonian, the on-site electron correlation U is the only many-body interaction. Although the Coulomb repulsion suppresses the s-wave superconductivity, it induces various types of anisotropic superconductivity through the momentum dependence of quasi-particle interaction, which originates from the many-body eCect. While some inter-orbital interactions exist in the multi-orbital Hubbard Hamiltonian, anisotropic superconductivity is induced by essentially the same mechanism, namely the momentum dependence of quasi-particle interaction. This is an important understanding of the mechanism of superconductivity in strongly correlated electron systems (SCES). Only the above understanding is believed to be the uniMed one. In other words, some simpliMed ideas will be insuDcient for the purpose to provide a uniMed picture. For example, the pairing mechanism is frequently attributed to Auctuation of some order parameter. The spin Auctuation theory is a typical one. While this theory has obtained great success as is reviewed in Section 3.2, the uniMed understanding cannot be derived from this theory. We can easily understand this fact from the result on Sr 2 RuO4 , where the approach from the Auctuation theory clearly fails. As is explained in Section 3.4, the pairing mechanism of Sr 2 RuO4 is appropriately clariMed on the basis of the understanding proposed above. Now we emphasize that the quasi-particle in the Fermi-liquid theory is an essentially important concept to describe the SCES. The basis of this fact is the continuity principle existing in the Fermi liquid. This widely accepted idea not only ensures the applicability of the perturbative methods but also promises some universal understandings in the SCES. While the superconducting materials in SCES frequently show the non Fermi-liquid behaviors, the usefulness of the Fermi-liquid theory as a starting point is robust. In the general statement, the existence of Fermi-liquid quasi-particle is justiMed when the damping rate is smaller than its energy. If we adopt the nearly anti-ferromagnetic (AF) Fermi-liquid theory, the energy width of single-particle spectrum sometimes approaches to their energies, as seen in the vicinity of hot spots in the under-doped cuprates (Section 4.3.1). However, it is still possible to consider the quasi-particle states with strong damping eCects. The usefulness of the approach from the Fermi-liquid theory has been conMrmed by the arguments on the pseudogap phenomena (Section 4). In the argument of the pseudogap, we have calculated the normal self-energy due to superconducting (SC) Auctuations. The self-energy shows anomalous behaviors: The real part

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possesses positive slope around Fermi energy and the imaginary part shows a large peak around Fermi energy in its absolute value. These behaviors are in sharp contrast to the normal Fermi liquid. We have clariMed that these anomalous behaviors of the self-energy are the origin of the pseudogap. Because the superconducting long-range order violates the continuity principle, it is natural for its precursor to destroy the picture of quasi-particles by degrees. On the basis of the above standpoint, we have reviewed several topics. We have clariMed the mechanism of superconductivity for each system with the corresponding pairing symmetry. They are the d-wave pairing for cuprates and organic superconductors, p-wave pairing for Sr 2 RuO4 , and d- or p-wave pairing for heavy-fermion superconductors. These results have been obtained by solving the Dyson–Gor’kov equation derived by perturbation calculation or FLEX approximation. In particular, we should stress that the physical properties related to the cuprate high temperature superconductivity have been explained appropriately in the normal and superconducting states. The hole-doped and electron-doped systems have been explained in common on the basis of the same Hubbard Hamiltonian by only adjusting the carrier number to that in real systems. Moreover, pseudogap phenomena have been also explained by taking the SC Auctuations into account without any other assumptions. In this review we have shown the following uniMed view for strongly correlated electron systems: 1. The anisotropic superconductivity in SCES originates from the momentum dependence of the quasi-particle interaction, stemming from the on-site Coulomb repulsion. The triplet p-wave pairing in Sr 2 RuO4 also can be explained on the basis of this uniMed understanding. The obtained mechanism is a new one, which is diCerent from the paramagnon mechanism. In general, the d-wave superconductivity is stabilized near the half-Mlling, while the p-wave superconductivity is stabilized apart from the half-Mlling. 2. The pseudogap phenomena arise from the SC Auctuation in the quasi-two-dimensional system. In addition to the quasi-two-dimensionality, the strong-coupling superconductivity possessing short coherence length = is necessary for the appearance of the pseudogap phenomena. The SC Auctuation itself originates from attractive interactions induced by AF spin Auctuation. Starting with the repulsive Hubbard Hamiltonian, we have derived the pseudogap phenomena and succeeded in explaining not only the single particle properties but also the magnetic and transport ones consistently in the pseudogap region. It is important that the explanation of pseudogap is the natural extension of our theory applied to the other regions with diCerent hole- and electron-dopings. A kind of resonance between quasi-particle states in Fermi-liquid and Cooper-pairing states give rise to the pseudogap. This is an essentially new physical phenomenon discovered in quasi-two-dimensional SCES. The pseudogap above Tc gains the energy without superconducting long-range order. 3. Among the anomalous properties in high-Tc cuprates, we have made an important progress in the analysis of transport phenomena. It is shown that the vertex correction in the linear response theory (Kubo formula) sometimes plays an essential role in SCES. In other words, not only the properties of quasi-particles but also those of the residual interaction is necessary to understand the transport phenomena. For example, the Hall coeDcient, the magnetic penetration depth, and the Nernst coeDcient in high-Tc cuprates are the cases. This is one of the characteristic features in the SCES with signiMcant momentum dependence. The seemingly anomalous behaviors in cuprates are reasonably explained on the basis of the Fermi-liquid theory. 4. To describe the superconductivity in SCES, the following renormalization procedure oCers an important perspective. First we include the on-site Coulomb interaction and then, band energy

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and eCective interaction are renormalized. The momentum dependence of the eCective interaction between quasi-particles gives rise to the anisotropic superconductivity. This perspective gives the explanation on the enhanced jump of the speciMc heat at Tc in heavy fermion materials. Furthermore, important knowledge for the superconductivity in SCES is obtained. For instance, by considering the renormalization eCect due to electron correlation, we can show that the electron–phonon interaction is generally reduced by the renormalization factor z compared with the electron–electron interaction. Therefore, the s-wave pairing is almost impossible in heavy-fermion systems. The above argument is mainly related to a strategy to understand the superconductivity in heavy-fermion systems. By following this strategy, we have analyzed the superconductivity in the Ce-based and U-based heavy fermion superconductors. Concerning the systems possessing not so heavy eCective mass such as the d-electron and organic materials, we can treat simultaneously both the mass enhancement and the superconductivity, as has been done in this review. 5. Recently, the higher-order correction beyond the third-order perturbation theory, which is mainly used in this review, has been discussed. The fourth order perturbation with respect to U has been performed by Nomura and Yamada (Appendix B). It is shown that the eCective interaction for d-wave pairing converges very smoothly. The fourth-order terms considerably cancel each other and as a result, their contribution becomes small in total. The superconducting critical temperature also shows a good convergence. These results qualitatively justify the understanding reviewed in Section 3.2. On the other hand, the obtained critical temperatures for the p-wave pairing case show an oscillatory behavior with respect to the calculated order. This behavior originates from the oscillation in particle–particle scattering terms. For the p-wave case, the correction from the particle–particle ladder diagrams induces the important contribution to the quasi-particle interaction. As is well known, the particle–particle ladder terms can be collected up to the inMnite order to give a smooth function. It is shown that rather convergent results are obtained by applying this procedure. Thus, it is expected that the higher-order terms can be treated so as not to change the result obtained within the third-order perturbation theory. Of course, the theory to treat the electron correlation in a closed form is highly desirable. The results given in Appendix B imply a possibility to develop the theory of unconventional superconductivity in a closed form by starting with an appropriate renormalized form. New superconducting systems are continuously found in the various systems such as heavy-fermion systems. The superconductivity in heavy-fermion systems provides many interesting subjects which deserve theoretical eCorts. We close this review by noting that the heavy-fermion superconductivity will be an important and attractive future issue. Acknowledgements The authors would like to thank E. Dagotto, K. Deguchi, S. Fujimoto, H. Fukazawa, R. Ikeda, K. Ishida, K. Kanki, H. Kohno, S. Koikegami, H. Kondo, H. Kontani, K. Kuroki, Y. Maeno, T. Moriya, Y. Nisikawa, M. Ogata, T. Ohmi, M. Sigrist, Y. Takada, T. Takimoto, K. Ueda, H. Yasuoka, and K. Yosida for discussions on the present subject. Figs. 24, 28, and 29 were presented by courtesy of T. Takimoto, K. Kuroki, and H. Kondo, respectively. Y. Y. is supported by the Grant-in-Aid for ScientiMc Research from Japan Society for the Promotion of Science. T. N. is supported from the Japan Society for the Promotion of Science for Young Scientists. T. H. has been supported by the Grant-in-Aid for ScientiMc Research Priority Area from the Ministry of Education, Culture, Sports,

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Science, and Technology of Japan. He is also supported by the Grant-in-Aid for ScientiMc Research from Japan Society for the Promotion of Science. Appendix A. Fermi-liquid theory on the London constant As an interesting topic, in this appendix, we discuss the London constant in high-Tc cuprates. The London constant is described by the magnetic penetration depth as =1=4L2 and often described by the “superAuid density” as ns =m∗ . The experimental results for high-Tc cuprates have been reviewed in Section 2.1.6. The so-called “Uemura plot” [112] has particularly stimulated interests, because the relation Tc ˙ (0) indicates the Kosterlitz–Thouless (KT) transition which is established in the exactly two-dimensional system [382]. From this insight, the phase disordered state was proposed as a possible pseudogap state [11,377,378]. The situation is diCerent for the quasi-two-dimensional system where the long-range order occurs at Mnite temperature. Actually, the characteristic behavior for the KT transition, such as the Nelson–Kosterlitz jump, is not observed experimentally. Here, we do not enter into the validity of the phase-only model, but focus on the microscopic origin of the doping and temperature dependence of . Not only the Uemura plot but also the doping independence of a = −d =dT have attracted theoretical interests. Lee and Wen have shown that the doping and temperature dependences of are explained by the SU(2) formulation for the t–J model [117]. Another understanding was proposed on the basis of the phenomenological description for the Fermi-liquid theory [118,537]. The conventional Fermi-liquid theory for the isotropic system [538] is not available in this case. In the following, we derive the microscopic description for the anisotropic Fermi-liquid and propose an understanding based on the Fermi-liquid theory [119,120]. It is shown that the expected behaviors of the Fermi-liquid parameters are successfully reproduced by the FLEX approximation. The Fermi-liquid description for the London constant is obtained by the Kubo formula. The superconducting (SC) state is no longer the Fermi-liquid in a strict sense. However, the description based on the Bogoliubov quasi-particle is similarly possible. The London constant is obtained by subtracting the paramagnetic contribution from the diamagnetic contribution. Then, the London constant is derived from the current–current correlation function and therefore described by the quasi-particles as well as the quasi-particle interaction. The quasi-particle interaction is expressed by the vertex correction for the correlation function. At T =0, the interaction between the quasi-particles is taken into account in the same way as in the Drude weight [432,433] and in the cyclotron resonance frequency [539]. This is natural because the Drude weight at T =0 is equal to the London constant according to the f-sum rule. A careful treatment for the many-body eCect is needed at Mnite temperature. A systematic discussion is presented in Refs. [119,120]. Hereafter, we use the mean-Meld theory for the superconductivity. We ignore the SC Auctuation because it is not important at low temperature. In particular, it has been concluded that the thermal Auctuation does not contribute to the T -linear coeDcient a=−9 =9T [384]. While the importance of the quantum Auctuation has been pointed out [384], we do not take it into account. The doping dependence of the quantum Auctuation cooperatively aCects on the London constant together with the eCect discussed below. From the results in Ref. [119], the London coeDcient is written at Mnite temperature as dSk 2 v∗ (k)(1 − Y (kF ; T ))vY∗H (k; T ) ; (122) ,H (T ) = e 3 |˜ ∗ (k)| , 4 v FS

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

where FS dSk means the integral over the Fermi surface, , and H are the spatial indices, and Y (kF ; T ) is the Yosida function, given by

9f(E ∗ (k)) d” (k) − 9E ∗ (k) ∗

Y (kF ; T ) =

:

(123)

Here E ∗ (k) = ”∗2 (k) + -(k)2 and -(k) is the momentum dependent excitation gap renormalized by the many-body eCect. v,∗ (k) is the renormalized velocity of quasi-particles (˜v∗ (k) = 9”∗ (k)=9k) and vY∗H (k; T ) is determined by the following integral equation ∗

∗

vYH (k; T ) = jH (k) −

FS

dSk 3 4 |˜v∗ (k )|

f(k; k )Y (kF ; T )vY∗H (k ; T ) :

(124)

Here f(k; k ) = z(k)>! (k; k )z(k ) is the interaction between quasi-particles, where >! is obtained from the !-limit of reducible four-point vertex. The notation about !-limit and k-limit follows Ref. [17]. The current vertex j,∗ (k) in the collision-less region is usually written as

j,∗ (k) = v,∗ (k) +

f(k; k )(”∗ (k ))v,∗ (k ) :

(125)

k

It is easily understood that vY∗H (k; 0) = jH∗ (k) because Y (k; T = 0) = 0. Note that the current vertex j,∗ (k) is diCerent from J, (k) in Section 4.3.4. The latter is deMned in the hydrodynamic region. The above expression for the current vertex is rewritten by more convenient form as j,∗ (k) = v, (k) + z(k)w, (k)|@=0 ;

(126)

where w ˜ (k) is obtained from the following integral equations: w ˜ (k) = ˜u(k) + ˜u(k) =

k

k

d k I (k; k )[G(k )2 ]! w ˜ (k ) ;

2 !

d k I (k; k )[G(k ) ]

92(k ) 1− 9@

(127)

(˜v(k ) − ˜v(k)) ;

(128)

where I (k; k ) is the irreducible four-point vertex which satisMes the relation I (k; k ) := 2(k)=G(k ). We can show that u, (k) = 0 and therefore j,∗ (k) = v, (k) in the Galilei invariant system. Thus, the renormalization for the current vertex j,∗ (k) generally results from the Umklapp scattering. Finally we derive the coeDcient of the T -linear term from Eqs. (122) and (124) as a=−

d

!

,H (T ) !!

dT

!

T =0

=e

2

FS

dSk j ∗ (k) 2 2 |˜v∗ (k)| ,

9Y (kF ; T ) 9T

T =0

jH∗ (k) :

(129)

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

125

This coeDcient is determined by the current vertex around the node as is expected. This is contrasted from the behavior of (0) which is determined by the quasi-particles around the whole Fermi surface. We obtain the expression dSk 2 (130) v∗ (k)jH∗ (k) : ,H (0) = e 3 v∗ (k)| , FS 4 |˜ This is an underlying origin of the qualitatively diCerent doping dependence between (0) and a. Before discussing the realistic situation, we comment on the failure of the Fermi-liquid theory for the isotropic system [146,302]. If we apply the above expressions for the two-dimensional isotropic system where ”∗ (k) = k2 =2m∗ , the London constant is obtained in the low-temperature region as [538] n (131) (T ) = (1 + F1s =2) ∗ − (1 + F1s =2)2 'T ; m where ' is a factor in the order of unity. The simpliMed expression (0)=n=m∗ is obtained if F1s 1. If we apply Eq. (131) to the high-Tc cuprates, the Uemura plot requires the scaling behavior m∗ ˙ −1 or 1 + F1s =2 ˙ or n ˙ . The Mrst candidate clearly contradicts the ARPES measurement [540], where the mass-renormalization does not signiMcantly depend on the doping. The second candidate leads to the relation −d =dT ˙ 2 and clearly contradicts the experimental results (see Section 2.1.6). Thus, we cannot resolve the anomalous behaviors by taking account of the electron correlation on the basis of the isotropic Fermi-liquid theory. Contrary to that, the last candidate gives an appropriate result. This observation is sometimes suggested as an evidence for that the under-doped cuprate is a low-carrier system. This is one of the reasons why the NSR theory has been applied to the pseudogap phenomena (Section 4.1). However, this argument is too naive, because the appearance of the particle number n is an accidental result in the isotropic case. That is, the relation ˙ n is derived from the replacement n = kF2 =2 which is justiMed only in the isotropic case. From the expressions Eqs. (122)–(130), the London constant is generally determined by the quasi-particles near the Fermi surface. The excitation deeply below the Fermi level is not important. The similar observation has been obtained for the Hall coeDcient (Section 4.3.4). In the following, we show that the generalized Fermi-liquid theory resolves the anomalous behaviors. The Hubbard model is adopted as a microscopic Hamiltonian. We start from the usual case where the momentum dependence of the quasi-particle interaction is not essential. The over-doped cuprate is probably included in this case. Then, the current vertex is given as j,∗ (k) v,∗ (k) z(k)v, (k) :

(132)

Thus, the London constant is reduced by the renormalization factor z(k). If the momentum dependence of the velocity |˜v∗ (k)| is not so signiMcant, the conventional behavior is expected for the London constant. In order to study this case, namely, the over-doped region, we use the perturbation theory. This choice is in common with the discussion on the pairing mechanism (Section 3.2). In this case, the momentum dependence of the quasi-particle interaction exists, but is not strong. Fig. 81shows the results of the self-consistent second-order perturbation. It is shown that the London constant at T =0 decreases with doping. This is an observed behavior in the over-doped region [114]. We see that the vertex correction (quasi-particle interaction) is not important. Note that the above situation is irrelevant in the under-doped region, because the strong momentum dependence is an important property of the under-doped cuprates (see Section 4.3). Here, the

126

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 1.00

0.80 X=jx*(k) X=z(k)vx(k) X=vx(k)

0.60

0.40

0.20 0.10

0.15

0.20

0.25

0.30

0.35

δ

Fig. 81. Results of the self-consistent second order perturbation for FS (dSk =|˜v∗ (k)|)vx∗ (k)X . Here X = jx∗ (k) (circles), z(k)vx (k) (triangles), and vx (k) (squares). This quantity is proportional to xx (0) in the case of X = jx∗ (k).

π A

*

j (k)

hs1

*

v (k)

ky cs hs2

0 0

kx

B

π

Fig. 82. The schematic Mgure for the current vertex ˜j ∗ (k) in the under-doped region. The renormalized velocity ˜v∗ (k) is perpendicular to the deformed Fermi surface, while ˜j ∗ (k) is not.

most eCective momentum dependence should be derived from AF spin Auctuations. If we use the phenomenological description for the quasi-particle interaction, which is represented by the Feynmann diagram in Fig. 15, a rough estimation [120] shows that the current vertex behaves as in the schematic Mgure (see Fig. 82). The current vertex is signiMcantly reduced by the quasi-particle interaction f(k; k ) which is enhanced around k = k + Q. The reduction is especially signiMcant around the “hot spot”. Fig. 82 indicates that the London constant at T = 0 decreases with the development of the spin Auctuation. On the other hand, the coeDcient of the T -linear term a is not so doping-dependent because the “cold spot” is not directly aCected by the AF spin Auctuation at q = Q. These behaviors are expected in the phenomenological proposal on this subject [118,537], while the origin of the quasi-particle interaction has not been identiMed there. Note here that the Fermi-liquid parameter assumed in Ref. [537] is qualitatively diCerent from that in our microscopic treatment.

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

127

0.20 X=jx*(k) X=z(k)vx(k)

0.15

0.10

0.05

0.00 0.10

Fig. 83. Results of the FLEX approximation for

0.15

FS

0.20

0.25

δ

0.30

0.35

0.40

(dSk =|˜v∗ (k)|)vx∗ (k)X . Here X =jx∗ (k) (circles) and z(k)vx (k) (triangles).

0.40

j x*(k)

0.20

0.00 δ=0.10 δ=0.20

-0.20

-0.40

A

hs1

cs

hs2

B

Fig. 84. The doping dependence of the current vertex jx∗ (k) obtained by the FLEX approximation. The horizontal axis represents the momentum along the Fermi surface (see Fig. 82).

The FLEX approximation well reproduces the above estimation for the current vertex and the London constant [120]. Fig. 83 shows that the London constant at T = 0 decreases in the under-doped region like the Uemura plot. The development of the AF spin Auctuation is essential for this behavior. Note that the quasi-particle interaction f(k; k ) derived from I (k; k ) is related with the real part of the spin susceptibility, while the imaginary part is important in the hydrodynamic response (Section 4.3.4). Therefore, the vertex correction arising from the AF spin Auctuation is not so suppressed by the excitation gap. If we neglect the quasi-particle interaction, qualitatively diCerent doping dependence is obtained. We conclude that the Fermi-liquid correction in the nearly AF Fermi-liquid is essential for the Uemura plot. Fig. 84 shows that the current vertex around the gap node is almost independent of the doping concentration. Thus, the Fermi-liquid correction is not important for the doping dependence of a.

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

We stress that this result is in sharp contrast to Eq. (131). The detailed information of the SC gap is needed for the explicit estimation of a. The deformation of the gap function W(k), which is observed experimentally [224], may play a role to explain the doping independence of a. This tendency is consistent with the FLEX approximation (Fig. 22). We consider, however, that the essential origin of the doping independence of a is not this deformation but also the character of the Fermi-liquid correction. Thus, the anomalous behaviors of the London constant are explained on the basis of the general formulation for the Fermi-liquid theory. The vertex correction arising from the AF spin Auctuation plays an essential role, as in the hydrodynamic transport in the normal state (Section 4.3.4). These results excellently contribute to the coherent understanding of high-Tc cuprates. So far, we have used the two-dimensional model and discussed the in-plane transport. We brieAy comment on the qualitatively diCerent behavior of the c-axis London constant c [113,116]. Owing to the momentum dependence of the inter-layer hopping (see Eq. (103)), the coherent transport gives the T 5 -law of c in the clean limit [541]. However, the power is easily aCected by the randomness and then, the incoherent nature should be taken in the under-doped region. The lower power is observed in many cases [541]. The Fermi-liquid correction is probably not important for the c-axis London constant, because the kz -dependence of the quasi-particle interaction is very weak. Contrary to the above discussion on the low-temperature behaviors, the temperature dependence around Tc is dominated by the SC Auctuation. The critical Auctuation generates the rapid growth of the London constant, which is more remarkable along the c-axis [383]. These behaviors are also observed in the experimental result which has indicated the scaling behavior corresponding to the universality class of the 3D XY model [121]. Appendix B. E,ects of higher-order perturbation terms In Section 3.1, we have provided the perturbation expansion for the eCective pairing interaction or the anomalous self-energy up to third order of the on-site Coulomb repulsion U . However, the value of U leading to the realistic Tc is not always suDciently small compared with the kinetic energy of the system. Thus, the convergence of the perturbation expansion should be examined. Furthermore, it is interesting to investigate the momentum dependence of the higher-order perturbation terms, since the momentum dependence of the eCective interaction among quasi-particles is important for unconventional superconductivity in SCES. For these problems, recently two of the present authors, Nomura and Yamada, have performed the fourth-order expansion for the pairing interaction and investigated the momentum dependence of fourth-order contributions [542]. In this appendix, following their work, we brieAy discuss the eCects of the higher-order terms. Nomura and Yamada have considered two typical cases in the two-dimensional Hubbard model on the square lattice, given by Eqs. (42) and (43) in Section 3.2.3: One is the case (I) similar to the high-Tc cuprates, where the system is near the half-Mlling and AF spin Auctuations are strong due to the Fermi surface nesting. Another is the case (II) similar to the ) band in the spin-triplet superconductor Sr 2 RuO4 , where the system is away from the half-Mlling. In the both cases, the Fermi level is set close to the van Hove singularity. Note here that the higher-order expansions have been carried out by Efremov et al. for a fermion gas model with repulsive interaction in the isotropic three-dimensional space [543]. They have

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

129

evaluated the eCective interactions perturbatively up to the fourth order with respect to the s-wave scattering component of the bare interaction. Their discussions are applicable to the diluted liquid 3 He. The irreducible vertex function V1 2 ;3 4 (k; k ) is expanded in terms of U as (k; k )U + V(2) (k; k )U 2 + V(3) (k; k )U 3 V1 2 ;3 4 (k; k ) = V(1) 1 2 ;3 4 1 2 ;3 4 1 2 ;3 4 + V(4) (k; k )U 4 + · · · : 1 2 ;3 4

(133)

Then, we deMne the following quantities: V(1622);3 4 (k; k ) = V(1) (k; k )U + V(2) (k; k )U 2 ; 1 2 ;3 4 1 2 ;3 4

(134)

(k; k )U 3 ; V(1623);3 4 (k; k ) = V(1622);3 4 (k; k ) + V(3) 1 2 ;3 4

(135)

(k; k )U 4 : V(1624);3 4 (k; k ) = V(1623);3 4 (k; k ) + V(4) 1 2 ;3 4

(136)

These functions are used as perturbative approximate forms for the eCective pairing interaction V1 2 ;3 4 (k; k ). Note that in the Section 3, we have taken V(1623);3 4 (k; k ) as the approximate form. The diagrammatic expressions for the fourth-order perturbation terms are shown in Fig. 85. The analytic expressions for the diagrams are found in Ref. [542] (see also Ref. [543]). Since the weak coupling region is considered here, the normal self-energy is simply ignored and the bare Green’s function G (0) (k) is assigned to the internal lines in Fig. 85. When we include the normal self-energy correction in the internal lines, the transition temperature should be decreased due to the increase of quasi-particle damping. However, the momentum dependence of the eCective pairing interaction would not be aCected at least qualitatively and thus, the most probable pairing symmetry would not be changed. Now all fourth-order terms are at our hands, but unfortunately, it is not feasible task to perform faithfully the summation in momentum and frequency for the fourth-order terms. Up to the third order, we could carry out the summation numerically both in momentum and frequency by exploiting the fast Fourier transformation (FFT) algorithm, but the fourth-order terms include contributions which cannot be summed up in the same manner. In order to proceed to further calculations, we consider the momentum dependence only on the Fermi surface for V1 2 ;3 4 (k; k ) and the anomalous self-energy -(k). This simpliMcation is justiMed as far as we discuss low-energy phenomena including superconductivity, for which the quasi-particles in the vicinity of the Fermi level play the most important roles. We deMne NF points, kFi (1 6 i 6 NF ), on the Fermi circle. Then, the eCective interaction V (kF ; kF ) and the gap function -(kF ) are expressed as functions of the positions kF and kF on the Fermi surface. By Mxing momenta kF and kF on the Fermi surface, we can numerically perform the summation with respect to the internal momenta and frequencies up to the fourth order with the use of FFT algorithm. The above simpliMcation is summarized as follows: V1 2 ;3 4 (k; k ) → V1 2 ;3 4 (k; k ) ≡ (k; k ) → V(p) (k; k ) ≡ V(p) 1 2 ;3 4 1 2 ;3 4 -1 2 (k) → -1 2 (k) ;

lim

V1 2 ;3 4 (k = (k; i!); k = (k ; i! )) ;

(137)

lim

V(p) (k = (k; i!); k = (k ; i! )) ; 1 2 ;3 4

(138)

!=! →+0 !=! →+0

(139)

130

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

Fig. 85. The fourth-order diagrams for the eCective pairing interaction. The vertex points and the solid lines represent the bare interaction U and the bare Green’s function G (0) (k), respectively.

where p symbolically denotes p = 6 2; 6 3; 6 4; : : : ; etc. The quantity V1 2 ;3 4 (k; k ) is regarded as the scattering amplitude in the pair scattering process, where the two electrons (or Fermi-liquid quasi-particles) with the momenta and spins, (k; 1 ) and (−k; 2 ), are scattered to the states characterized by the momenta and spins, (k ; 4 ) and (−k ; 3 ), respectively. As a result of the simpliMcation for Eq. (16), we obtain the BCS-like gap equation as ! ) NF ! ! 9”(k) !−1 1 LF 2e W ! ! ln · -1 2 (kFi ) = − ! 9k ! (2)2 NF T k=kFj j=1 × V1 2 ;3 4 (kFi ; kFj )W4 3 (kFj ) ; 3 4

(140)

NF where the summation j=1 is performed only with respect to the Fermi surface points kFj , and NF is the total number of the Fermi surface points at which the vertex function V1 2 ;3 4 (kFi ; kFj ) and the gap function -(kFi ) are calculated. LF is the circumference of the Fermi circle, W is the cut-oC energy, and )(=0:5772 · · ·) is the Euler’s constant. The cut-oC energy is taken as

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

ky

S Q

0

(a)

P R

S Q

∆ ( k F)

2π

131

0 4th 3rd 2nd

2π

P R kx

Q

P

S

R kF

(b)

P

Fig. 86. (a) Fermi surface for the nearly half-Mlled case (I). (b) The gap function -singlet (kF ) as a function of the position kF on the Fermi circle. Note that kF traces the Fermi circle following the arrows in (a). The symmetry is dx2 −y2 -wave state for all cases. The parameters are T = 0:0100 and U = 2:066.

Tc

0.1

0.01 4th 3rd 2nd

0.001 1.5

2

2.5

3

3.5 U

4

4.5

5

5.5

Fig. 87. Transition temperature as a function of the repulsion U for the nearly half-Mlled case (I). The most probable pairing state is spin-singlet dx2 −y2 -wave.

W = 0:700 throughout the present calculation. We can determine the transition temperature at which the maximum eigenvalue max is equal to unity and the eigenfunction giving max is related to the most probable pairing symmetry. Details of the formulation are given in Ref. [542]. Now we show the numerical results of the gap function -1 2 (k) and the transition temperatures as functions of the repulsion U . For the case (I), we have taken the hopping parameters t = 1:00 and t = −0:100 and set the electron number as n = 0:98. The Fermi surface is depicted in Fig. 86(a). The obtained gap function -1 2 (k) exhibits the dx2 −y2 -wave state in all the cases of the second-, third-, and fourth-order perturbation theories. This may be natural if we recall that the momentum dependence of the pairing interaction is well approximated by using the susceptibility as g2 (k − k ) in Section 3.2.2 for the superconductivity induced by the strong anti-ferromagnetic spin Auctuations. We show the transition temperatures as functions of the repulsion U for the case (I) within the second-, third-, and fourth-order perturbation theories in Fig. 87. The values of Tc for the third- and fourth-order theories are quantitatively close to each other, although Tc for the second-order theory is very low. If we further take account of the perturbation terms beyond fourth order, the curve of Tc as a function of U is considered to converge to a single curve, leading to a better estimation for Tc than that of the present fourth-order calculation.

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Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149

+

+

+

+

+ ...

+

+ ...

Fig. 88. The strong contributions with particle–particle ladder in the vertex. The contributions enclosed by the dotted lines are added to V (64) (k; k ) in order to suppress ill-convergence.

Let us discuss the validity of the perturbation expansion by comparing the magnitude of the thirdand the fourth-order vertex functions, V (3) (k; k ) and V (4) (k; k ). This naive analysis justiMes the expansion in U up to U ∼ |V (3) =V (4) | ∼ 3:5 [542]. Although there are much more perturbation terms in the fourth order than in the second or third order, most of the fourth-order contributions cancel each other, and therefore the total magnitude of V (4) (k; k ) is smaller than that of V (3) (k; k ). Thus, the convergence of the perturbation expansion in U becomes good up to moderately strong U within the fourth-order perturbation theory. In addition, the momentum dependences of the functions, V (2) (k; k ), V (3) (k; k ), and V (4) (k; k ) are all similar to each other. Therefore the perturbation expansion would give a good estimation of the eCective pairing interaction and a reasonable analysis of the pairing symmetry up to moderately strong U in this case. Next we consider the other case (II). According to the recent work [542], the convergence of the perturbation expansion is not good for the spin-triplet channel, in contrast to the spin-singlet channel near the half-Mlled case (I). In one word, this is due to an oscillatory behavior in terms of U . As shown in Section 3.4.2, the third-order terms give the momentum dependence much favorable for the p-wave pairing. After the momentum dependences of all fourth-order terms are examined, it is found that dominant contributions have the momentum dependence similar to that of the third-order term, but their signs are opposite. Further analysis on higher-order terms has revealed that the contributions shown in Fig. 88 provide the dominant momentum dependence in each order. These contributions in each higher-order term have positive (negative) sign for odd (even) order of U and thus, these contributions can be summed up to the inMnite order. For the pairing interaction, then we use V (64 corr:) (k; k ), which is obtained by adding the higher-order corrections shown in Fig. 88 to V (64) (k; k ). Due to this improved procedure, the most probable pairing symmetry is triplet p-wave. In Fig. 89, we show the Fermi surface and the gap function for the case (II), where we have taken the hopping parameters t = 1:00 and t = −0:375 and set the electron number as n = 1:334. The gap functions obtained for the third-order and the above mentioned improved fourth-order perturbation theories show the highly anisotropic p-wave symmetry with the nodes on the line ky = 0, while the gap function obtained in the second-order theory has eight additional nodes on the Fermi surface. We show the transition temperature as functions of the repulsion U in Fig. 90. Note that the second-order perturbation theory gives only low Tc , because the momentum dependence of the eCective interaction V (62) (k; k ) is less favorable for the triplet p-wave pairing than V (63) (k; k ) and V (64 corr:) (k; k ). Finally we again emphasize that the momentum dependence induced by the electron correlations is important for the anisotropic pairing in SCES. In some cases, the picture that some

Y. Yanase et al. / Physics Reports 387 (2003) 1 – 149 4th corrected 3rd 2nd

P R

S Q

ky S Q

0

∆ ( k F)

2π

133

0

2π

P R kx

Q

P

(a)

S

R kF

(b)

P

Fig. 89. (a) Fermi surface for the case (II) away from the half-Mlled. (b) The gap function -triplet (kF ) as a function of the position kF on the Fermi circle. Note that kF traces the Fermi circle following the arrows in (a). The parameters are T = 0:0100, U = 3:428.

Tc

0.1

0.01

4th corrected 3rd 2nd 0.001

3

3.5

4

4.5 U

5

5.5

6

Fig. 90. Transition temperature as a function of the repulsion U for the case (II). The most probable pairing state is spin-triplet p-wave pairing state.

magnetic Auctuations induce the anisotropic superconductivity would be valid. However, we could not always consider that the essential momentum dependence of the pairing interaction originates from some strong Auctuations. The most essential point for anisotropic pairings is what momentum dependence the eCective pairing interaction could acquire as a result of many-body eCects or correlation eCects, as we have stressed throughout this review article. In general, we could expect that the perturbation theories developed in this review capture qualitatively well the essential momentum dependences of the pairing interaction. It is highly believed that the anisotropic pairing deduced within the weak-coupling perturbation theory evolves smoothly up to the realistically strong U . References [1] J.G. Bednorz, K.A. MZuller, Z. Phys. B 64 (1986) 189. [2] S. Uchida, H. Takagi, K. Kitazawa, S. Tanaka, Jpn. J. Appl. Phys. 26 (1987) L1. [3] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg, Nature 372 (1994) 532. [4] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori, Y. Maeno, Nature 396 (1998) 658.

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Available online at www.sciencedirect.com

Physics Reports 387 (2003) 151 – 213 www.elsevier.com/locate/physrep

Ab initio lattice dynamics of metal surfaces R. Heid∗ , K.-P. Bohnen Forschungszentrum Karlsruhe, Institut fur Festkorperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany Accepted 15 July 2003 editor: A.A. Maradudin

Abstract Dynamical properties of atoms on surfaces depend sensitively on their bonding environment and thus provide valuable insight into the local geometry and chemical binding at the boundary of a solid. Density-functional theory provides a uni.ed approach to the calculation of structural and dynamical properties from .rst principles. Its high accuracy and predictive power for lattice dynamical properties of semiconductor surfaces has been demonstrated in a previous article by Fritsch and Schr2oder (Phys. Rep. 309 (1999) 209). In this report, we review the state-of-the-art of these ab initio approaches to surface dynamical properties of metal surfaces. We give a brief introduction to the conceptual framework with focus on recent advances in computational procedures for the ab initio linear-response approach, which have been a prerequisite for an e7cient treatment of surface dynamics of noble and transition metals. The discussed applications to clean and adsorbate-covered surfaces demonstrate the high accuracy and reliability of this approach in predicting detailed microscopic properties of the phonon dynamics for a wide range of metallic surfaces. c 2003 Elsevier B.V. All rights reserved. PACS: 68.35.Ja; 68.47.De; 63.20.Dj; 71.15.Mb Keywords: Density-functional theory; Metal surfaces; Surface dynamics

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Microscopic theory of lattice dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Harmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Application to surface lattice dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. Tel.: +49-7247-82-3438; fax: +49-7247-82-4624. E-mail address: [email protected] (R. Heid).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.07.003

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2.2. Density-functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Pseudopotential methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Ab initio methods for calculating lattice dynamical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Frozen-phonon methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Linear-response approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Surface lattice dynamics of noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Surface phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Ag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Au . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Surface force constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Relaxation versus bond-breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Relation between surface phonons and surface stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The phenomenon of the longitudinal resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Surface lattice dynamics of fcc transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Phonons of Pd and Pt surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Phonons of Rh and Ir surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Surface lattice interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Surface lattice dynamics of non-fcc transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Surface phonons of W(110) and Mo(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Phonon dynamics of Ru(0001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Surface phonons of Zr(0001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Surface lattice dynamics of nearly-free electron metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Phonons of Al surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Surface phonons of Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H 6.3. Phonon dynamics of Be(0001) and Be(1010) ...................................................... H ...................................................... 6.4. Surface phonons of Mg(0001) and Mg(1010) 7. Lattice dynamics of adsorbate-covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Surface phonons of H/W(110) and H/Mo(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Surface phonons of (1 × 1)-O/Ru(0001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Substrate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Adsorbate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Concluding remarks and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

158 158 160 161 161 162 165 166 167 169 170 172 174 175 177 182 182 183 190 192 192 193 195 195 195 196 196 199 201 201 202 203 204 205 207 207

1. Introduction Vibrations of atoms and their interaction with electrons constitute the basis for a variety of physical properties of a crystal. Examples are elastic, optic, and thermodynamic properties (speci.c heat, thermal expansion). Coupling to electrons inJuences phenomena like transport and superconductivity. Since lattice vibrations depend sensitively on the binding properties and on the local atomic arrangements, they constitute a valuable probe for the investigation of structural modi.cations. Many physical and chemical processes of technical relevance take place at surfaces of crystals, like catalysis or corrosion. A deeper understanding of these processes on a microscopic level demands a proper knowledge of structural, electronic, and dynamical properties of the underlying surface. The presence of a surface often provokes large modi.cations in the interatomic binding, which have

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a signi.cant eKect on the vibrational properties of atoms in the vicinity of the surface. The study of the lattice dynamics at surfaces thus provides an important tool to characterize physical properties of a surface, which is complementary to investigations of structural and electronic properties. Furthermore, important surface phenomena like adsorption, diKusion or growth can be signi.cantly inJuenced by the dynamics of surface atoms. Since the beginning of the 1980s, the interest in surface dynamical properties has been stimulated by the development of two spectroscopical methods for the detection of surface phonons: the inelastic scattering of neutral atoms, mostly helium (HAS = helium atom scattering) [1], and the inelastic scattering of electrons (EELS = electron energy loss spectroscopy) [2,3]. Both techniques have been described extensively in a review article by Toennies [4]. HAS is an extremely surface sensitive tool, because the scattering properties of thermal He atoms (with kinetic energies of 20 –60 meV) are mainly determined by the vibrations of the outermost surface layer. The kinetic energy of electrons used for EELS is much higher (20 –300 eV) which therefore penetrate more deeply into the surface. Modern high-resolution realizations of this method (HREELS) [5,6] reach resolution below one meV and also allow investigation of low-frequency surface modes. Collections of experimental phonon spectra for clean metal surfaces can be found in two earlier review articles of Kress [7] and of Wallis and Tong [8], and for adsorbate-covered surfaces in a recent review by Rocca [9]. For both methods, the interpretation of measured spectra requires a complex theoretical analysis. In the case of HAS the di7culty lies in the theoretical description of the potential governing the He-surface interaction [10,11]. For EELS multi-scattering events are very important, which result in a sensitive dependence of the scattering intensity on the energy of the incoming electrons [12–14]. Therefore, a detailed understanding of the spectra relies not only on the phonon frequencies, but also on the phonon polarization, which demands an accurate theoretical description of the surface phonon dynamics. In recent years, signi.cant progress in the theoretical predictability of surface phonons has been achieved by the application of ab initio numerical methods to surface dynamical problems. A central role is played by methods based on density-functional theory, which provide an accurate calculation of electronic structure and binding properties of solids and surfaces without the need to resort to material-speci.c parameters. Due to numerical costs, early real-space methods (frozen-phonon approach) have supplied only limited informations, but the development of perturbational techniques and the availability of ever faster computer resources have now opened the door for a detailed analysis of surface dynamical properties. Many applications have been devoted to semiconductor and insulator surfaces, which have been the subject of a previous review article by Fritsch and Schr2oder [15]. The purpose of the present article is to give a review of the application of ab initio methods to the lattice dynamical properties of metal surfaces. It tries to achieve two main goals: (i) to demonstrate the usefulness and versatility of the current theoretical methods, and (ii) to provide an up-to-date reference to theoretical results obtained for phonons on metal surfaces. In the following, we will brieJy recapitulate the achievements and recent developments in the theoretical description of surface lattice dynamics. 1.1. Historical overview The .rst theoretical works concerning the surface phonon dynamics of rare gases [16,17] and ionic crystals [18] appeared already in the 1960s. A widely used systematic classi.cation of surface

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localized modes for low-indexed face-centered cubic (fcc) surfaces has been introduced by Allen et al. in 1971 [19,20]. A major impetus in the .eld was evoked by the improvements of the spectroscopical methods at the beginning of the 1980s, which forced the developments of theoretical approaches with the goal of providing quantitative and material-speci.c predictions. The majority of this work had been of phenomenological nature, based either on classical force-constant models (Born–von-KOarmOan) or on more re.ned approaches taking into account explicitly the electronic degrees of freedom (pseudo-charge model). As the model parameters had to be adjusted to experimental results for the bulk material, they did not allow predictions of the structural modi.cations and lattice dynamical couplings at the surface. The latter often were adjusted ad hoc using experimental surface phonon spectra. These changes often sensitively depend on the chosen parametrization, making their physical interpretation unreliable. A more consistent description of static and dynamical properties at surfaces is provided by approaches based on the total energy of a con.guration of atoms as the central quantity. Structure and dynamics are then obtained by minimizing the total energy and by analyzing the second-order changes at the minimum. First approaches based on pair potentials adequate for noble-gas solids [16], could not be transferred to metals, as the metallic bond possesses a signi.cant many-body character. The most successful methods of this class for metals have been semi-empirical approaches describing the total energy of a metal as the sum of a pair potential and an atomic energy. Borrowing ideas from density-functional theory, the latter quantity depends only on the electron density at the location of the atom and is supposed to account for the embedding energy of an atom in an environment determined by neighboring atoms. Examples are the embedded-atom method (EAM) [21–23], the eKective-medium theory (EMT) [24], and the glue model [25]. For the functional dependences of quantities entering the total energy, EMT uses quantities derived from the homogeneous electron gas, while EAM and the glue model adopt empirical forms adjusted to various bulk properties. The glue model was used for structural investigations of Au surfaces [26–28] and the Pb(110) surface [29], with the only application to dynamical properties in the case of the Au(110) surface. In contrast, EMT and EAM are still widely used tools for investigations of metallic surfaces due to their high numerical speed. Applications range from lattice dynamics of clean and adsorbate covered surfaces [30–33], dynamics of stepped (vicinal) surfaces [34–36] to diKusion of adatoms [37,38]. A basic weakness of these methods is that the charge density is not treated self-consistently. Therefore, rearrangements of the electron density and changes in the screening properties cannot properly be taken into account. This leads to quantitatively, in some cases even to qualitatively wrong predictions for structural modi.cations at a surface with resulting errors for the surface phonons [39–41]. In addition, these methods are restricted to the total energy only and do not allow investigation of further details of the electronic structure or binding properties. Methods based on density-functional theory avoid many of the approximations inherent to the semi-empirical approaches, and therefore possess a higher predictive power with respect to lattice dynamical properties. First work along this line has appeared in the mid-1980s applying the frozen-phonon method [42]. Use of proper supercells allowed the calculation of normal modes of thin .lms for high-symmetry points of the surface Brillouin zone and the determination of interplanar force constants. This method has been applied successfully to clean surfaces of various types of metals: for nearly-free electron metals (Al(110) [42,43], Al(100) [43], Al(111) [44], Na(110) [45]), for noble metals (Ag(100) [46,47], Ag(110) [41], Ag(111) [48]), Cu(100) [46–48], Cu(110) [48,49], Cu(111) [48], and the (1 × 2)-reconstructed Au(110) [39,40]), and for transition metals

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(Rh(111) [50], Ir(110) [51], Zr(0001) [52]). The work prior to 1993 in this .eld is covered by the review article of Bohnen and Ho [41]. These studies have proven that the lattice dynamics of metal surfaces can be described by density-functional theory with high accuracy. The frozen-phonon method has been also applied to the vibration of adsorbates on metal surfaces. The .rst calculations have been dealing with hydrogen adsorbates [53–58], while more recently different types of adsorbate atoms on various surfaces have been considered [59–61]. As these calculations are numerically expensive, with few exceptions [62] only the fundamental -point vibrations of the adsorbate atoms have been considered and often the coupling to the substrate modes has been neglected. Because the frozen-phonon method based on supercells is numerically expensive, most of its applications for clean surfaces have been restricted to a few high-symmetry points. To obtain a more complete picture of the surface dispersion curves, the ab initio interplanar force constants were .tted to a phenomenological model of the interatomic interaction. This additional step involved further simplifying assumptions with respect to the form of the interaction, and gives therefore only a rough approximation to the dispersion curves. It is inappropriate in cases where phonon anomalies away from high-symmetry points exist. There, an interpolation scheme is required which properly takes into account the inJuence of the electronic structure on the low-symmetry phonons. A numerical e7cient interpolation scheme of this kind is provided by tight-binding models. Static and dynamical instabilities of the clean and hydrogen-covered (001) surfaces of W [63] and Mo [64] have been analyzed in the framework of the non-orthogonal-tight-binding scheme [65–67]. The tight-binding parameters have been .tted to ab initio band structure calculations for bulk and surface materials. This method, however, is not a pure ab initio method since it still requires an empirical parametrization of the short-range lattice interaction. Signi.cant progress with respect to the frozen-phonon method has been achieved by applying linear-response techniques within density-functional theory. This approach aims at calculating the dynamical matrix for a given wavevector using perturbation theory. The big advantage is that it does not require supercells and allows the calculations of phonons at arbitrary wavevectors without increasing the numerical cost. The .rst attempt to use this kind of technique for surface phonon spectra was undertaken by Eguiluz and coworkers [68,69]. Their approach was based on the calculation of density-response functions using pseudopotentials for the electron-ion interaction, while treating the electron-electron interaction within local-density approximation. This allowed to access the dynamical matrix for arbitrary wavevectors. While in work related to phonons of Al surfaces the application of the pseudopotential was treated perturbatively [70,71], this approximation could be avoided in later work on Na(100) [72]. However, this formalism worked only in combination with local pseudopotentials and was therefore restricted to applications on nearly-free electron systems. A crucial impetus to the .eld of ab initio calculations of surface phonons was provided by a new formulation of the linear-response approach within density-functional theory, which is free of the restrictions mentioned above. This density functional perturbation theory was proposed independently by Baroni et al. [73,74] and by Zein [75–77], and opened the way for e7cient calculations of lattice dynamical properties [78]. Numerous studies have been devoted to the lattice dynamics of semiconductors, which have been reviewed recently by Fritsch and Schr2oder [15]. On the contrary, for metal surfaces applications were rare. A remarkable work has been dealing with the development of a Kohn anomaly of the Rayleigh branch of the W(110) surface after adsorption of hydrogen [79].

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Complete phonon spectra via density functional perturbation theory have been obtained for Ag(111) H [83] and Mg(0001) [84]. More recently, the present [80], Cu(100) [81], Be(0001) [82], Be(1010) authors have dealt with the surface lattice dynamics of clean and oxygen covered Ru(0001) surfaces [85,86]. Another area of application is given by calculations of thermodynamical properties, especially of the thermal expansion of surfaces in the framework of the quasiharmonic approximation. So far, studies have been reported for Be(0001) [87] and Ag(111) [80], and very recently for the more H [88] and Be(1010) H [89]. complex multi-layer thermal expansion of Mg(1010) A further class of an ab initio approach to surface dynamics constitutes the cluster method, where the surface is represented by a small number of atoms. These .nite clusters are accessible to quantum-chemical techniques, by which the electronic structure can be treated more accurately than by the standard approximations to density-functional theory. Because of its rather crude representation of a surface, this approach is mainly useful for the study of vibrational properties of isolated objects on a surface, such as adsorbed atoms or clusters. But it is inappropriate for investigations of collective dynamical properties of a surface or of an adsorbate layer. As the present article focuses on the latter, we will not discuss the cluster approach further, but refer to the review article of Whitten and Yang [90] for an overview of its application to surface dynamical properties. 1.2. Scope and outline The present report should be considered as a successor to the review article of Fritsch and Schr2oder [15] addressing ab initio calculations of surface phonons of semiconductors. As these authors have discussed extensively the general principles and techniques underlying the ab initio approach, we will present in Section 2 only a brief survey of the methods with focus on those aspects speci.c to metal systems and of relevance for the understanding of the following chapters. Sections 3–6 are devoted to surface lattice dynamics of clean metallic surfaces. We have attempted to collect all available ab initio results on surface phonon modes to provide an up-to-date summary for future reference. Not covered is work on vicinal surfaces. In Section 3, we discuss the surface dynamics of noble metals, which can be considered as model cases because they allow for a transparent description of surface induced changes in the phonon spectrum. In the light of the theoretical results, we will discuss the relationship between surface phonons and surface stress in Section 3.2.2. Also we will take a closer look at the mystery of additional modes seen in HAS experiments, which are not accounted for in simple lattice dynamics models and have been puzzling since their .rst observation almost 30 years ago. In Section 4, we present a comprehensive study of all low-indexed surfaces of transition metals with fcc structure. Here the focus is primarily on observable trends in the surface interactions and the resulting phonon spectra. This is followed by the discussion of the surface dynamics for selected transition metals of body-centered cubic (bcc) and hexagonal close-packed (hcp) structure in Section 5. Work on surface dynamics of nearly-free electron metals is reviewed in Section 6. The lattice dynamics of adsorbate-covered surfaces is the subject of Section 7. Here, we restrict the discussion to the density functional perturbation theory investigations reported until now. No attempt is made to address the numerous frozen-phonon studies, which often concentrate on the adsorbate vibrations only. Finally, a brief summary and outlook is given in Section 8.

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2. Theoretical background 2.1. Microscopic theory of lattice dynamics 2.1.1. Harmonic approximation The underlying assumption of all ab initio applications to lattice dynamical properties is the adiabatic approximation or Born–Oppenheimer approximation [91], which allows for a decoupling of the electronic and nuclear degrees of freedom due to their large mass ratio. It follows that the statics and dynamics of the nuclei is governed by an eKective potential ({R}) = Vnn ({R}) + Ee ({R}) ;

(1)

which depends only on the coordinates R of the nuclei. Vnn is the nucleus–nucleus coulomb potential, while Ee ({R}) denotes the electronic ground-state energy for a given con.guration {R} of the nuclei. The eKective potential builds the starting point of the microscopic theory of lattice dynamics, which has been outlined in a number of review articles [92–94]. Dynamical properties are derived by a systematic expansion of for atom displacements u around a chosen reference con.guration Ri = Ri0 + ui ; ({R}) = ({R0 }) +

(2) ia

a (i)uia +

1

ab (i; j)uia ujb + · · · : 2

(3)

iajb

Latin indices a and b denote Cartesian coordinates, while i and j are atom indices. The term of .rst order vanishes, if one chooses as reference the equilibrium con.guration, which minimizes . The harmonic approximation is based on truncating the sum after the second order. The second order coe7cients are the harmonic force constants

ab (i; j) =

92 : 9Ria 9Rjb

(4)

In periodic crystals, the atoms are characterized by two indices i = (L), which denote the unit cell (L) and the atoms inside a unit cell (), respectively. For periodic boundary conditions, the Fourier transform of the force constant matrix is related to the dynamical matrix 1 Da a (q) = √

ab (L; 0 )eiq(RL −R0 ) ; (5) M M L which determines the equation for the normal modes or phonons, Da a (q)a (q) = !2 (q)a (q) :

(6)

a

!(q) and a (q) denote the energy and polarization of the normal mode determined by the wavevector q and branch index . Here, as in the entire theory section, we adopt Rydberg atomic units, de.ned by ˝2 = 2m = e2 =2 = 1. Energies are then given in units of Rydberg (13:606 eV), R and masses in units of the electron mass. lengths in units of the Bohr radius (0:529 A),

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A complete characterization of the harmonic vibrational spectrum requires the knowledge of either the normal modes for the whole Brillouin zone, or the force constants for all atom bonds. For a metallic system, the latter representation is often economical since the lattice interaction in real space is rather short ranged due to electronic screening. 2.1.2. Application to surface lattice dynamics The presence of a surface destroys the three dimensional periodicity of a crystal, and leads to additional vibrational modes not present for the bulk. They are characterized by large displacements of atoms close to the surface and can be distinguished in two classes: (i) Surface localized modes, whose frequencies are outside the bulk spectrum of the same symmetry. These modes are well localized near the surface, and its polarization decays rapidly away from the surface. (ii) Resonances, whose frequencies fall into the bulk spectrum, and thus hybridize with bulk modes. They possess less vibrational weight near the surface and deeply penetrate into the bulk. For the theoretical treatment of surfaces lattice dynamics, essentially two diKerent methods have been used in the past. In the Greens-function method the surface is considered as a perturbation of the three dimensional crystal and the .nding of surface modes is formulated as a scattering problem [95]. This approach is especially useful to detect surface resonances, but is technically involved. A more straightforward approach is the slab method, which simulates the semi-in.nite crystal by a slab of .nite thickness [96]. The thickness of the slab has to be large enough that the two surfaces decouple dynamically. This ensures a proper representation of resonances deeply penetrating into the bulk. In the context of ab initio approaches, the slab method has proven to be very convenient, and is typically applied in two steps: (i) A numerically expensive ab initio calculation is performed for a periodic-slab geometry. It consists of a periodic arrangement of thin slabs with a few atomic layers separated by vacuum regions along the direction of the surface normal. From these calculations, information about the geometry and the dynamical couplings at the surface are extracted. (ii) The surface phonon spectrum is then simulated by a thick slab, where the force constants near the surface are taken from the thin-slab calculation, while the inner part of the slab is represented by bulk force constants. The latter often are extracted from ab initio bulk calculations. This process of slab enlargement is also called slab :lling. Since the periodicity of the slab is two-dimensional, vibrational modes are classi.ed by a wavevector from the two-dimensional surface Brillouin zone. For a more extensive outline of the slab method, see [97,15]. As seen from Eq. (1), lattice dynamical properties can be derived from the knowledge of the electronic ground-state energy Ee ({R}) for a given atom con.guration. This task is performed by density-functional theory, which will be described in the following section. 2.2. Density-functional theory 2.2.1. Basic principles The foundations to density-functional theory (DFT) have been worked out by Hohenberg et al. [98,99] in the mid-1960s, and are outlined in numerous reviews [100–102]. In DFT, the ground-state energy of a system of interacting electrons moving in an external potential vext is obtained by

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minimizing the functional E[n] = F[n] + d 3 r vext (r)n(r)

159

(7)

with respect to the electron density n(r). At the minimum, n(r) is the true electron density of the interacting system. The functional F[n] is universal, i.e. independent of the external potential. The minimum principle allows to map the complex many-body problem onto a .ctitious system of non-interacting electrons, which in its ground state possesses the same inhomogeneous density as the interacting system [99]. The wavefunctions of the .ctitious electrons obey a single-particle equation (Kohn–Sham equation) {−∇2 + veK (r)} i (r) = i i (r) :

(8)

The eKective potential veK is a sum of the external potential and a screening potential veK = vext + vscr = vext + vH + vXC : The Hartree potential n(r ) vH (r) = d 3 r |r − r |

(9)

(10)

describes an average electrostatic potential originating from the other electrons, while the so-called exchange-correlation potential vXC contains all remaining contributions. Both quantities depend on the electron density fi | i (r)|2 ; (11) n(r) = i

where fi is the occupation number of the single-particle state i . By this formulation, the original many-body problem has been cast into a set of single-particle equations (8)–(11), which has to be solved self-consistently. The complexity of the original manybody problem is transferred to the task of determining the exchange-correlation potential vXC . The big success of DFT partly rests on the empirical fact that already simple approximations to vXC give in many cases very accurate results. The most widely used ansatz is the local-density approximation (LDA) hom d(nXC (n)) LDA vXC (r) = ; (12) dn n=n(r) hom (n) represents the exchange-correlation energy density of the homogeneous interacting where XC hom electron gas. For XC various parametrizations derived from analytical and numerical studies exist [101]. Another popular ansatz is the generalized-gradient approximation (GGA), where in addition to LDA a dependence of vXC on the local gradient of the electron density is considered to better account for inhomogeneous density distributions [103–105].

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2.2.2. Pseudopotential methods Solving the single-particle equations (8) for all electrons of a crystalline solid on equal footing is still a formidable task, because one has to represent localized core states as well as delocalized valence states with su7cient accuracy. In the last 30 years various all-electron schemes have been developed to deal with this situation. Among them, linear methods like the linear-augmented-plane-wave method (LAPW) [106] or the linear-mu7n-tin-orbitals method (LMTO) [107] have been most successful. Modern variants (full potential LAPW [108,109] and full potential LMTO [110,111]) are very accurate, but the numerical eKort increases rapidly with increasing number of electrons. For this reason, applications to surface dynamics have been restricted until now to frozen-phonon calculations (see Section 2.3.1). Most results for surface phonons presented in this article have been obtained with pseudopotential methods. They take advantage of the fact that the core electrons have often a negligible inJuence on many physical or chemical properties of a crystal. In the pseudopotential methods these core states are frozen-in, and only the valence orbitals are taken into account explicitly. The interaction of the core electrons and the nucleus with the valence electrons is described by an eKective potential (pseudopotential). Simultaneously, one can eliminate the oscillatory behavior of valence states in the core region, which is forced by their orthogonality to the core states, by introducing node-less pseudo-valence functions, which facilitates their expansion into a crystal-adapted basis set. Modern ab initio pseudopotentials are constructed on the basis of an atomic density-functional theory calculation. DiKerent forms and construction schemes for pseudopotentials have been proposed in the past. The commonly used norm-conserving pseudopotentials [112–117] possess a semi-local representation of the form vPS (r) = vl (r)Pl ; (13) l

where Pl denotes the projection operator onto the angular momentum l. The norm-conservation property states that the pseudo-valence functions possess the same charge as the true valence functions, and guarantees a good transferability of the pseudopotential into diKerent chemical environments [118–120]. A second widely used type of pseudopotentials are so-called fully separable pseudopotentials, where also the radial dependence obtains a non-local form [121]. This form allows an easy application of its Fourier transform and has numerical advantages in crystal calculations in connection with plane-wave basis sets. Further details about pseudopotentials can be found in the article of Pickett [122]. Pseudopotential band structure methods commonly use a basis-set expansion of the Bloch wavefunctions of the crystal to obtain an explicit solution for the single-particle equation (8) c! (k )#!k (r) ; (14) k (r) = !

with a proper choice of the basis functions #!k (r) for wavevector k. Here denotes the band index of the electronic eigenstate. This ansatz reduces the search for eigenstates to an algebraic eigenvalue problem for the expansion coe7cients c! (k ). The conceptually simplest choice for a basis set consists of plane waves 1 k (r) = √ ei(k+G)r ; (15) #G V

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161

where G represents a reciprocal lattice vector and V is the crystal volume. Plane-wave methods have been applied successfully for many systems, especially for semiconductors. However, for elements with partially .lled 2p shells and for transition metals with partially .lled d shells, the valence states tend to be stronger localized near the atom core and therefore require a large number of plane waves. In these cases, it is more e7cient to augment the plane-wave basis set with Bloch states build from localized functions which are centered at the atomic sites R 1 ik(R+R ) k #lm (r) = √ e ’lm (r − R − R ) : (16) N R The sum is taken over the lattice vectors. Although this mixed-basis (MB) representation [123–125] is technically more involved than the plane-wave method, it allows to drastically reduce the size of the basis set. Due to the resulting computational speed-up it extends the applicability range of the pseudopotential method to larger systems, which is often a prerequisite for surface dynamical studies. A diKerent road is taken by the ultrasoft-pseudopotential (USPP) approach introduced by Vanderbilt [126,127]. Here, the norm-conserving condition for the pseudopotential is relaxed to facilitate the construction of much softer pseudopotentials, which require less plane waves in the expansion of the valence states. This method bears some formal similarities to the mixed-basis method, as a generalized eigenvalue problem has to be solved, and local augmentation charges centered at atomic sites have to be introduced to restore charge neutrality. Both methods are therefore comparable with respect to their numerical e7ciency and range of applicability. 2.3. Ab initio methods for calculating lattice dynamical properties As we have seen in Section 2.1, lattice dynamical properties are determined by the adiabatic lattice potential , which equals the ground state energy for a .xed atom con.guration. Hence, lattice dynamics is in principle accessible in the framework of density-functional theory without need to resort to material-speci.c parameters. An overview of the various methods to extract lattice dynamical properties from ab initio calculations have been given in [15]. In the following, we will only address those schemes used in phonon calculations for metal surfaces. 2.3.1. Frozen-phonon methods The frozen-phonon (FP) technique is the conceptually simplest and historically .rst applied method to obtain phonon properties within density-functional theory. It is based on ground state calculations for the ideal crystal and for geometries with atoms displaced from their equilibrium position. The use of supercells allows to access phonons with nonzero wavevector q. Several variants exist: (i) Phonon frequencies are directly extracted from the change of the total energy using a displacement pattern which corresponds to a normal mode [128]. Since this requires a priori knowledge of the phonon eigenvector, it can only be used in cases where symmetry completely determines its form. Because in general the symmetry of slabs employed in surface studies is rather low, this technique has been only applied in special cases where light atoms adsorbed on a substrate build by heavier atoms allow an approximate decoupling of the adsorbate motion.

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(ii) A more e7cient scheme is based on the calculation of forces acting on atoms in a unit cell after displacing an atom [129–131]. This can be achieved with little numerical expenses, as forces can be derived directly from quantities obtained in a ground state calculation with the help of the Hellman–Feynman theorem. A single calculation then gives information about a complete row of the dynamical matrix. The complete dynamical matrix can be constructed using a few appropriately chosen displacements. Hence, this approach does not require any a priori information about the normal modes. The second variant has been .rst applied to surface phonon calculations by Ho and Bohnen [42] in the case of the Al(110) surface, and has been widely used since then for various clean and adsorbate covered metallic surfaces. Since frozen-phonon calculations employ .nite displacements of atoms, their results principally contain all anharmonic eKects, which could be used to extract higher anharmonic coupling constants. For bulk materials they are of importance for a microscopic description of displacive phase transitions or vibrational properties near defects [132,133]. In surface calculations, applications have mainly focused on anharmonic couplings for light adsorbates like hydrogen [134]. A big disadvantage of the frozen-phonon method is the need to resort to supercells to extract properties for non-zero wavevector phonons. As the numerical eKort scales with the third power of the number of atoms in the supercell, slab calculations are often limited to the zone center and a few zone boundary modes. One has to resort to certain interpolation techniques to obtain the phonon spectrum at other points in the surface Brillouin zone. Most commonly, this is done by .tting a suitable force constant model to the ab initio dynamical matrices. Since this procedure rests on additional assumptions on the interaction form, it gives only a crude estimate of the phonon spectrum. A complete determination of the phonon spectrum would require supercells larger than the eKective range of the lattice interactions [135–138]. By combining the informations obtained from diKerent supercell calculations, this condition can be somewhat relaxed [139]. In all cases, however, these applications have been limited to bulk systems with high lattice symmetry. 2.3.2. Linear-response approach 2n + 1 theorem. This technique is complementary to the frozen-phonon ansatz as it aims to calculate directly the higher derivatives of the total energy within perturbative schemes. For practical purposes it is important that the second derivatives (force constants) require only the knowledge of the .rst-order variations of the electron density. In the case of a local external potential this property can be shown easily with the help of the Hellman–Feynman-Theorem [140]. It states that the .rst derivative of the total energy is determined by the ground state density (E (vext (r) = d 3 r n(r) ; (17) (i (i where the parameter set (i ; i = 1; : : : ; p) characterizes the adiabatic perturbation of the external potential, e.g. induced by atomic displacements. Then (2 E (n(r) (vext (r) (2 vext (r) = d3 r + d 3 r n(r) : (18) (i (j (j (i (i (j

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This result is a special case of the so-called 2n + 1 theorem, which states that all derivatives of the total energy up to (2n + 1)th order with respect to the adiabatic perturbation can be calculated from the knowledge of all derivatives of the Kohn–Sham eigenstates and density up to only nth order. In the framework of density-functional theory this theorem also holds for nonlocal external potentials and is thus applicable within pseudopotential methods. The proof given by Gonze et al. [141–143] essentially rests on the variational property of the energy functional. As a corollary of this theorem, harmonic as well as third-order anharmonic force constants only require a calculation of the linear variations of the Kohn–Sham eigenstates and the density, and are thus accessible by linear-response calculations. Dielectric matrix approach. First work along these lines made use of an inversion of the dielectric matrix [144,145]. The starting point is the expression for the linear density-response evoked by an external perturbation, which is given by (n(r) = d 3 r #(r; r )(vext (r ) : (19) The response function # is the polarizability of the electron system. In the Kohn–Sham formalism, one can immediately get the non-interacting susceptibility #0 , which connects the density response with the variation of the eKective potential (n(r) = d 3 r #0 (r; r )(veK (r ) : (20) #0 can be expressed solely by ground-state quantities of the Kohn–Sham system [146] fi (1 − fj ) #0 (r; r ) = [ i∗ (r) j (r) j∗ (r ) i (r ) + c:c:] : − i j ij

(21)

In the case of a periodic system, this is just the well-known Adler–Wiser form [147,148]. Although obtained by perturbation theory, Eq. (21) is exact because the Kohn–Sham equations describe noninteracting electrons. Using (veK = (vext + (vscr ; (vscr (r) =

d 3 r I (r; r )(n(r ) ;

(22)

where I (r; r ) =

((vH (r) + vXC (r)) (n(r )

(23)

one can derive a relationship between # and #0 # = #0 (1 − I#0 )−1 = #0 −1 :

(24)

The static dielectric matrix de.ned by = 1 − I#0 describes the screening of the external potential via (veK = −1 (vext .

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The original formulation based on local potentials has been extended more recently to the general case of nonlocal potentials [149]. Direct application of these equations, however, has several practical disadvantages. It requires an inversion of the Matrix (r; r ), which for periodic systems is most conveniently done in Fourier space. This inversion turns out to be the bottleneck of this scheme, as a proper convergence often requires a large number of Fourier components. Attempts to perform this inversion in direct space using a Wannier representation did not lead to signi.cant improvements [150]. In the calculation of #0 in Eq. (21) also unoccupied orbitals enter, which are not available for bandstructure methods employing minimal basis sets (e.g. LMTO). Alternatively, the static dielectric matrix can be extracted directly from calculations in direct space using supercells [151,152]. This approach shares the disadvantage with the frozen-phonon methods of being restricted to a few high-symmetry points in reciprocal space because of numerical costs. Density functional perturbation theory. An important progress has been achieved by a new formulation of the linear-response approach for the lattice dynamics, which avoids some of the aforementioned problems of the dielectric matrix approach. This so-called density functional perturbation theory (DFPT) has been proposed independently by Zein et al. [75–77] and Baroni et al. [73,74]. A concise description can be found in [78]. We will give a short outline for the case of a non-metallic system. It rests on the idea that the change of the self-consistent electron density induced by varying the eKective potential can be expressed as (n(q + G) = −

4 kv|e−i(k+G)r |k + qc k + qc|(veK |kv : V c (k + q) − v (k) vc

(25)

k

q is the wavevector of the periodic perturbation, G is a reciprocal lattice vector, and the indices v and c refer to valence and conduction states, respectively. This expression is a generalization of Eq. (20) to nonlocal potentials, and is exact for the system of non-interacting Kohn–Sham electrons. The variation of (veK depends linearly on (n I (q + G; q + G )(n(q + G ) : (26) (veK (q + G) = (vext (q + G) + G

Eqs. (25) and (26) represent a self-consistent set of equations for (n(r), which can be solved iteratively, thus avoiding to build and invert a potentially large matrix . In addition, one introduces an auxiliary wave function |k + qc k + qc|(veK |kv ; (27) |*kv = c (k + q) − v (k) c which can be obtained as the solution of an inhomogeneous linear equation (HKS − v (k))|*kv = (Pv − 1)(veK |kv :

(28)

Here HKS = −∇2 + veK is the Kohn–Sham operator, and Pv denotes the projector onto the valence states. Then the variation of the density is expressed as 4

kv|e−i(k+G)r |*kv : (29) (n(q + G) = − V v k

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Furthermore, the electronic contribution to the force constant matrix can by expressed by |*kv . By this reformulation, only occupied valence states enter these equations, and expensive summations over unoccupied conduction states are avoided. Extensions and implementations of DFPT. Originally, the scheme was formulated for non-metallic systems and .rst applied in the framework of the plane-wave pseudopotential method. An extension to metallic systems has been derived by de Gironcoli, which contains essentially technical modi.cations related to the appearance of fractional occupation numbers for electronic states with energies close to the Fermi energy [153]. An alternative way to the iterative procedure described above has been proposed by Gonze et al. They derived a representation of the force constant matrix which is stationary with respect to .rst-order variations of the single-particle states [142,154–156]. This variational principle allows the application of e7cient search algorithm, e.g. the conjugate-gradient method [157]. The DFPT scheme has been also transferred to other bandstructure methods. Implementations exist for LMTO [158,159] and LAPW [160,161], which can deal with stronger localized electrons. They share an enhanced complexity of the perturbation formulation, and applications have been restricted so far to rather small systems (up to 7 atoms per unit cell [162]). Thus, they have not yet been applied to surface phonon studies. In studies of surface vibrations the slab thickness cannot be made too small to avoid arti.cial surface-surface couplings. Hence, one has to handle larger unit cells, which currently are adequately addressed only by pseudopotential methods. The plane-wave pseudopotential implementation of DFPT has lead to a wide range of application. For two further pseudopotential methods, MB and USPP, implementations of DFPT have been worked out [163,164]. As outlined in Section 2.2, these schemes are better suited to treat localized valence states than a pure plane-wave expansion. The MB formulation [163] has enhanced complexity with respect to the plane-wave scheme. This complexity has its origin in the non-orthogonality of the basis set, and in the local functions of the basis set, which depend on the actual position of the atoms and thus on the perturbation. Both properties are the source for a variety of correction terms. The scheme described in [163] extents the variational form of the force constant matrix given by Gonze et al. [154,142] to the mixed-basis set, resulting in a faster convergence of the iterative procedure. In the case of the USPP-implementation, similar complications arise due to the local augmentation charges and the generalization of the eigenvalue problem [164,81]. Therefore, the correction terms are very similar to those of the MB scheme. The USPP-DFPT has been recently extended to GGA and spin-polarized systems [165,166,81]. Some of the aforementioned implementations of DFPT are currently available on the Internet. The package PWSCF contains a PW code written by Baroni and coworkers, and also incorporates the USPP-DFPT by Dal Corso [167]. The ABINIT package written by Gonze and coworkers contains a PW implementation which is based on the variational form of the second derivatives [168]. Finally, the LMTO code PHN by Savrasov is also made accessible [169]. 3. Surface lattice dynamics of noble metals Surfaces of noble metals have enjoyed an intensive investigation of their properties in the past. There are several reasons for this:

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On the experimental side, clean surfaces of noble metals can be prepared easily and are rather stable due to their chemical inertness. For Ag and Cu, all low-indexed surfaces do not reconstruct. Furthermore, the lattice interaction in the bulk is very short ranged and to a good degree dominated by the longitudinal coupling to the nearest neighbor. This property simpli.es signi.cantly the analysis of changes in the interatomic coupling at a surface. On the theoretical side, the treatment of the electronic structure of noble metals is complicated by the fact that the d-states, although fully occupied, inJuence various physical properties and have to be treated explicitly as valence states. Because of this property, ab initio surface dynamics calculations have been carried out for a long time exclusively within a frozen-phonon approach using the mixed-basis-method [39,40,47–49,170]. Recently, calculations of complete surface phonon spectra have been obtained for Ag(111) [80] and Cu(100) [81] using DFPT, and for Ag(110) with an elaborate frozen-phonon approach [171]. In this chapter, we present a comprehensive study of the lattice dynamics of all low-indexed surfaces of Ag, Cu and Au. The results have been obtained by the present authors using the MB-DFPT method [163,172,173]. We will .rst discuss the surface phonon spectra which will also serve as reference for the more complicated case of transition metal surfaces presented in the next chapter. This is followed by an analysis of the modi.cations of the interatomic couplings near the surface and its relation to structural modi.cations or surface stress. Finally, the long-standing problem of the longitudinal resonance observed on various fcc surfaces is analyzed in view of the present ab initio results. 3.1. Surface phonons The calculations of the following surface phonon dispersion curves were based on slabs of 11 atom layers and 5 vacuum layers for the (100) and (110) surfaces, while 9 atom layers and 6 vacuum layers have been used for the (111) surfaces. Symmetric relaxations of 3, 4, and 3 outer layers for (100), (110), and (111) surfaces have been performed. The dispersion curves are obtained by interpolation of (4 × 4) and (4 × 6) q-point meshes for the rectangular surface Brillouin zones (SBZ) of (100) and (110) surfaces, respectively, while a hexagonal (6 × 6) mesh was adopted for the (111) surfaces. For slab .lling, surface force constants are combined with ab initio bulk force constants extracted from a similar bulk calculation using fcc (4 × 4 × 4) q-point meshes. In all cases, the theoretically R Cu: 3:607 A, R Au: 4:076 A). R Relaxation optimized bulk lattice constant has been used (Ag: 4:106 A, results are summarized in Table 1. All surfaces exhibit an oscillatory relaxation starting with an inward relaxation of the surface layer. The only exception is Au(111) which shows a slight outward movement of the .rst layer. The size of relaxations increases among each element when going from (111) to (100) to (110) surface, i.e. according to a decreasing atom density per layer. This behavior and the oscillatory relaxation pattern are well known for many metal surfaces [41]. For more open surfaces, the oscillatory relaxation can be qualitatively explained by the tendency of the electrons to smoothen the charge distribution at the surface in order to lower the kinetic energy (so-called Smoluchowski smoothing, see, e.g., discussion in Ref. [174] and references therein). Since for all noble metals the bulk lattice interaction is of short-ranged nature, the spectra are qualitatively very similar to those obtained for a model fcc crystal studied in the famous work of

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Table 1 Calculated relaxation of the low-indexed surfaces of noble metals Ag

*12 *23 *34 *45

Cu

Au

(100)

(110)

(111)

(100)

(110)

(111)

(100)

(110)

(111)

−2.1 +0.1 −0.1

−7.3 +2.7 0.0 0.0

−0.6 +0.1

−3.0 +0.4 +0.1

−8.6 +3.0 −0.5 +0.6

−1.0 −0.2 +0.3

−1.6 +0.4 +0.3

−12.3 +6.9 −2.4 +1.2

+0.7 −0.1 −0.3

Changes in the interlayer distances are given in % of the bulk value.

Y

X Γ

(100)

M Γ

(110)

S X

M Γ

K

(111)

Fig. 1. Surface-Brillouin zones of the fcc (100), (110), and (111) surfaces.

Allen et al. [20]. We therefore closely follow their nomenclature for the various surface modes. Our convention for the surface Brillouin zones is shown in Fig. 1. All spectra shown in the following correspond to 50-layer slabs. We used asymmetric slabs, where one surface represents the fully relaxed surface, while the other surface corresponds to the ideal bulk-truncated case. The phonon spectrum of such a slab thus contains surface modes related to both types of surfaces. Modes emphasized by black dots in the spectra are surface modes belonging to the true (relaxed) surface. They are identi.ed by a weight larger than 20% in the .rst two layers. 3.1.1. Ag The phonon dispersion curves of the three low-indexed surfaces of Ag are shown in Fig. 2. All surfaces exhibit the so-called Rayleigh wave (RW), which can be usually found below the bulk H this mode corresponds to essentially vertical deformaspectrum. At long wavelengths (i.e. near V) tion vibrations of the surface. At short wavelengths, however, its vibrational character can change signi.cantly, because it is more sensitive to short ranged interatomic couplings. In most cases, the RW corresponds to the lowest surface mode S1 , except for the VX direction of (100), where it is denoted by S4 . Here, the lowest mode S1 is shear-horizontally polarized. The dispersion curves of the theoretical RW agree well with HAS and EELS measurements for all surfaces. For the (110) surface, theory also reproduces the measured dispersion of the second sagittal mode denoted MS0 (VX) and S3 (VY), respectively, as well as the resonance MS7 in the vicinity H All theoretical frequencies are lying slightly above the experimental ones, which have been of V. obtained at room temperature. The diKerences are consistent with the observed hardening of surface modes along VY with decreasing temperature [182], which is a normal anharmonic behavior. Also

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25

Γ

Γ

M

X

ω (meV)

20 S6 S2

15 S2

10

S4

5 S1

S1

Ag(100)

0

25

Γ

X

S

Γ

Y

20

ω (meV)

S6 S7

15 MS7

MS7

S2

10

S5 S3

MS0

S2

5 Ag(110)

S1

S1

0 25

Γ

M

Γ

K

20

S4

ω (meV)

S2

15

S3

MS3

10

5

S1

S1

Ag(111)

0

40 Fig. 2. Theoretical surface phonon spectra of Ag(100), Ag(110), and Ag(111). Black dots denote theoretical surface modes. Open symbols show experimental data (circles=HAS, squares=EELS) taken from [175–179] for (100), (110), and (111) surface, respectively.

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

169

a large temperature dependence is observed for the frequency of the MS7 resonance, which increases by about 2 meV when cooling to 160 K [180]. For both Ag(100) and Ag(111) there is a clear discrepancy between the present theoretical results and HAS measurements with respect to the higher frequency surface modes observed experimentally. For Ag(100)-VM and Ag(111)-VK, the experimental data are close to the calculated modes S2 and MS3 , respectively. While the MS3 mode is predominantly shear-horizontally polarized with only weak vertical and longitudinal components (¡ 15%), the S2 mode is a pure shear-horizontal phonon. Both modes could not have been observed under the applied experimental conditions. In the other two cases, Ag(100)-VX and Ag(111)-VM, no theoretical surface mode is predicted in the vicinity of the experimental data. This phenomenon of additional modes, which are commonly referred to as longitudinal resonances, has been observed also for other fcc metal surfaces, and has been discussed controversially since its .rst discovery for the Ag(111) surface [179]. Section 3.3 is devoted to a more detailed discussion of this still open question. The lattice dynamics of the Ag(111) surface has been previously studied within a plane-wave DFPT method by Xie et al. [80]. While the authors focused on the thermal expansion properties of the surface layers, they also presented a surface phonon spectrum for the surface geometry obtained for 300 K, which is qualitatively the same as the one shown in Fig. 2. In particular, they also do not .nd surfaces modes corresponding to the longitudinal resonances mentioned above. In Table 2, the frequencies of surface modes at high-symmetry points of the SBZ are compared with previous ab initio and experimental results. The present DFPT results are very similar to those obtained in earlier frozen-phonon calculations using the mixed-basis method [48,170,47,41], where the largest diKerence occurs for the Ag(111)-RW at K. The values for the Ag(111) modes from the DFPT study of Xie et al. refer to a room temperature geometry, which takes into account thermal expansion eKects on the surface relaxation. The .rst interlayer distance is increased with respect to the low-temperature surface, which may explain the slightly softened surface modes. The lattice dynamics of Ag(110) has been investigated recently by Narasimhan within an elaborate frozen-phonon approach using a plane-wave pseudopotential method [171]. From a variety of H X, and Y the author extracted interlayer force constants, which he frozen-phonon calculations at V, then .tted to a nearest-neighbor (NN) axial force constant model with two parameters (longitudinal and transverse force constant). In a similar way, ab initio bulk force constants were extracted from frozen-phonon calculations. The resulting Ag(110) surface modes are in good agreement with the ones from the present DFPT study shown in Fig. 2. Some smaller numerical diKerences may be due to diKerent slab thickness (7 layers in the FP study versus 11 in our DFPT study). Also the optimized interlayer distances (*12 = −6:9%, *23 = +2:3%, *34 ; = − 1:2%) diKer especially for *34 with the present ones shown in Table 1. 3.1.2. Cu The phonon spectra of the low-indexed surfaces of Cu, shown in Fig. 3, are qualitatively similar to those of Ag. This comes as no surprise in view of the similarities between the bulk lattice dynamics (dominated by the NN longitudinal coupling) and surface relaxations of Cu and Ag (Table 1). The main diKerence lies in the frequency scale of the spectrum, which is about 45% larger for Cu due to a smaller nuclear mass and a 23% larger longitudinal NN coupling constant in the bulk. Dal Corso has carried out a detailed DFPT analysis of the Cu(100) surface dynamics [81] within the ultrasoft-pseudopotential method [167]. To study the inJuence of the exchange-correlation

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Table 2 Surface-mode frequencies (in meV) of Ag(100), Ag(110), and Ag(111) at selected high-symmetry points Ag(100)

X S1 6.4 5.8

Ag(110)

VH MS7 14.1 13.8–14.6 12.9 14.9 Y S1 5.2 4.4 5.0

Ag(111)

M S1 9.1 8.9 9.0 8.7

S4

S2

9.0 8.7 8.7 9.1 X S1

9.8 9.1

MS0 9.3 9.0 8.9

19.0 18.9 18.8

M S1 11.6 10.3 9.8 11.6

8.4 8.6 7.9 K S1 10.2 9.6 8.6 ≈9:1

S2

Method

14.4 13.2–14.1

DFPT FP [47] EELS [175] HAS [176]

S7

Method

15.2 16.1

16.7 18.3

DFPT FP [171] HAS [177,178] EELS [180]

S5

S S1

14.3 13.6

S2 11.8 12.4

14.8

15.2

S3

7.6 7.8

S2

17.5 17.0 12.1

9.0 9.0 8.3

S2

S6

10.3 9.5

10.6 10.9

11.1

13.6 14.6

8.1

S2

S7

Method

9.0

17.4

DFPT FP [171] HAS [177,178]

S3

Method

15.6 15.2 15.4

DFPT DFPT [80] FP [48,170] HAS [179,181]

FP denotes the frozen-phonon method.

functional, he performed calculations with both LDA and GGA. The LDA calculation gave a ≈15% harder spectrum than the GGA calculation, consistent with a smaller equilibrium lattice constant R GGA: 3:68 A). R The GGA showed a signi.cantly better agreement with in the bulk (LDA: 3:55 A, experimental data. Otherwise, the same qualitative trends in the surface interaction and phonon spectrum was found. The values given in Table 3 correspond to the GGA calculation. 3.1.3. Au Contrary to Ag and Cu, all low-indexed surfaces of Au do reconstruct. Despite this fact, the theoretical spectra displayed in Fig. 4 show that the relaxed ideal surfaces are dynamically stable. However, the bulk truncated calculation predicts an unstable RW at long wavelengths. In Table 4, the phonon frequencies at high-symmetry points are summarized. Noticeably, a frozenphonon study has been performed for the (1 × 2) missing-row reconstructed Au(110) surface, with good agreement to EELS measurements [40].

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

35

Γ

X

171

Γ

M

30

ω (meV)

25

S6 S2

20 S2

15 S4

10

S1

S1

5

Cu(100)

0

35

Γ

S

X

Γ

Y

30 S6

ω (meV)

25

S7

MS7

20 15

MS7

S2

S5

MS0

S3 S2

10 5

Cu(110)

S1

S1

0 35

Γ

M

30

S4 S2

25 ω (meV)

Γ

K

S3

20 15

MS3

10 5

S1

S1

Cu(111)

0

45

Fig. 3. Theoretical surface phonon spectra of Cu(100), Cu(110), and Cu(111). Black dots denote theoretical surface modes. Open symbols show experimental data (circles=HAS, squares=EELS) taken from [183–185,47], [186], and [181,187,188] for (100), (110), and (111) surface, respectively.

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Table 3 Surface-mode frequencies (in meV) of Cu(100), Cu(110), and Cu(111) at selected high-symmetry points Cu(100)

X S1 9.9 9.5 ≈9:1

Cu(110)

S2

S6

14.0 12.8 13.0 13.4

15.0 14.1

26.1 25.6 23.8

VH MS7

X S1

21.0 –21.3

MS0

S2

19.4

13.8 13.1 13.0

14.3 13.2

18.2 18.2

Y S1

S2

S3

11.9 12.0

12.8 12.8 12.0

7.6 7.1 7.0 Cu(111)

S4

M S1

S2

13.5 13.8 13.2

27.7 27.3 26.1

19.4

M S1 17.9 17.8 16.7 16.7 25.2

22.0 21.3

15.7

16.4

K S1

S3

15.3 13.9

23.3 22.5

S2 21.1–21.7 20.3

23.3

Method

20.2

DFPT FP [47] DFPT [81] EELS [185,184] EELS [47]

S7

Method

25.0 24.4 24.0

DFPT FP [49] HAS [186]

20.7

S5

S S1

S2

S7

Method

20.4 20.2

12.5 11.6

13.8 12.9

25.8 23.5

DFPT FP [49] HAS [186] Method

24.1

DFPT FP [48] EELS [187,188]

The Au(111) surface is known to exhibit a complex (23 × 1)-reconstruction [189], but the disturbance with respect to the ideal (111) surface is small enough so that to a .rst approximation, the reconstruction can be ignored for the surface lattice dynamics. Therefore, we have included the HAS data in Fig. 4 for comparison. 3.2. Surface force constants The DFPT study gives direct access to the changes of the interatomic coupling for bonds near the surfaces. Results for the NN longitudinal force constant are shown in Table 5. As a general trend, the coupling within the .rst surface layer softens, but in all cases this softening is less than 30%. As expected, the degree of softening is closely related to the compactness of a surface, largest for the most open (110) surface and smallest for the most compact (111) surface. In contrast, the hardening of interlayer force constant is linked to a shortening of the bonds. The present results are consistent with the .ndings of Dal Corso in his DFPT study of Cu(100), where he obtains a softening of ≈11% for the .rst intralayer coupling and a stiKening of about 15 –20% of the interaction between the surface and subsurface layer [81].

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

25

Γ

X

173

Γ

M

ω (meV)

20

15

S6 S2

10 S2 S4

5

S1

Au(100)

S1

0

25

Γ

X

S

Γ

Y

20

ω (meV)

S7

S6

15 MS7

MS7

10

S5

S2

MS0

S2

5

S3

Au(110)

S1

S1

0 25

Γ

M

Γ

K

20

ω (meV)

S4

15

S2 S3

10

5

S1

Au(111)

MS3

S1

0

Fig. 4. Theoretical surface phonon spectra of Au(100), Au(110), and Au(111). Black dots denote theoretical surface modes. The HAS results for Au(111) (open circles) are taken from [181].

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R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

Table 4 Surface-mode frequencies (in meV) of Au(100), Au(110), and Au(111) at selected high-symmetry points Au(100)

X S1 5.6

Au(110) (1×1)

X S1

11.2

12.0

Y S1

S3

VH Z 6.6 6.2

Au(111)

7.0

VH MS7

3.5 Au(110) (1 × 2)

S4

M 7.6

7.4

S2 7.7

MS0 6.7

S6

M S1

15.1

10.3

S2 10.3

10.6

S2

10.8

S2

Method

10.9

DFPT

S7

Method

13.0

14.9

19.3

DFPT

S5

S S1

S2

Method

5.9

6.7

8.3

9.3

13.1

7.6

6.1

7.5

Y

X

Y

Y

X Z

Y

Y

X

6.6

7.4

7.3

11.6

15.0

S1 7.6 7.5

S2 16.3

14.5 15.5 –18.0

19.3

K S1 8.1 7.5

7.3 6.9 S3

9.1

12.4

DFPT Z

X

Method

16.4 17.4 –18.2

19.3

FP [40,39] EELS [40] Method

13.3

DFPT HAS [181]

In the case of the (1×2) reconstructed Au(110) surface, the modes are classi.ed according to their dominant polarization H with X = [110], Y = [001], and Z = [110] (surface normal).

In the frozen-phonon study of Ag(110), Narasimhan found very similar changes of the longitudinal force constants by a .t of his ab initio data to an axial-symmetric model [171]. From a similar analysis of the interatomic interactions for Cu(110) and Al(110), he found that these stiKenings become stronger for Cu in agreement with the present results, and even larger (up to 85%) for Al [190,191]. The author argues that the large increase of the coupling between the .rst and third layer is responsible for an unusually small vertical mean-square displacement of a surface atom. 3.2.1. Relaxation versus bond-breaking Changes in the interatomic force constants near the surface are evoked by two eKects. The .rst is the breaking of bonds due to the reduced number of neighbors for surface atoms. The second is the layer relaxation which leads to modi.cations of the bond lengths. Both induce charge redistributions and related changes in the electronic screening, which inJuence the lattice dynamical couplings. The surface phonon spectra, shown in the previous section, correspond to asymmetric slabs, which contain beside the dispersion curves of the full relaxed surfaces also those of the truncated bulk surfaces (light lines separated from the bulk background). Their diKerences represent the summary of these two eKects.

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

175

Table 5 Calculated changes WF in the nearest-neighbor longitudinal force constant for bonds near the surface Bond

L1–L1 L1–L2 L1–L3

Ag

WF WF WR WF WR

Cu

Au

(100)

(110)

(111)

(100)

(110)

(111)

(100)

(110)

(111)

−13 +14 −1.0

−15 +21 −1.8 +36 −2.3

−9 +3 −0.4

−10 +21 −1.5

−14 +27 −2.1 +43 −2.8

−10 +7 −0.6

−30 +7 −0.8

−30 +39 −2.9 +46 −2.7

−20 −11 +0.5

The values are changes in % with respect to the bulk values (Ag: 24:9 × 103 dyne=cm; Cu: 30:6 × 103 dyne=cm; Au: 38:9 × 103 dyne=cm). A bond is characterized by the layer indices of the bonded atoms. WR denotes relaxation induced changes of the bond lengths (in % of the bulk value). Table 6 Changes WF in the longitudinal force constant for .rst and second neighbor bonds in the vicinity of the Ag(110) surface Neighbor

1st

2nd

Bond

WF

WR

unrelaxed

relaxed

L1–L3 L1–L2 L2–L3 L2–L4

−5 −10 −1 −1

+36 +21 −11 −20

−2.3 −1.8 +0.7 +1.3

L1–L1 L1–L1

−17 −9

−15 −8

— —

Results for the unrelaxed and relaxed surface are shown. A bond is characterized by the layer indices of the bonded atoms. WR denotes relaxation induced changes of the bond lengths (in % of the bulk values).

A calculation for an unrelaxed surface allows to discriminate between these two mechanisms. The results obtained by the present authors for the Ag(110) surface are shown in Table 6. They clearly exhibit a bond selective behavior: the interaction within the .rst surface layer is essentially independent from the relaxation, but is almost completely determined by the bond-breaking eKect. Remarkably, also a signi.cant coupling to the second neighbor within the surface layer is induced. This coupling, which is negligible in the bulk, mainly inJuences mode frequencies at Y [192]. On the other hand, relaxation modi.es especially interlayer couplings whose corresponding bond lengths are strongly altered. The shorter the bonds, the stronger the coupling becomes, but the ab initio data do not follow a power-law behavior F ˙ r ! as proposed by Badger [193]. The same behavior has been found in the frozen-phonon study of Ag(110) of Narasimhan [190,171]. 3.2.2. Relation between surface phonons and surface stress Surface stress is an important macroscopic property of a surface which can have important impact on various processes as, e.g., reconstruction or epitaxial growth (see the review article of

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R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

Table 7 Interatomic force constants for nearest-neighbor (NN) and 2nd NN surface-layer bonds NN

Bond direction

Force constants (103 dyne=cm) LL

VV

TT

LV

Ag(100)

1 2

011 001

21.68 −1.17

0.17 0.24

−0.62 −0.01

−2.32 −0.60

Ag(110)

1 2

H 110 001

21.18 −1.74

0.62 0.42

−0.10 0.12

2.30 0.55

Ag(111)

1

H 110

22.53

0.34

−0.84

−1.93

Cu(100)

1 2

011 001

27.44 −1.94

1.28 0.71

−1.11 0.17

−2.27 −1.09

Cu(110)

1 2

H 110 001

26.37 −2.82

1.50 1.01

0.79 0.36

2.44 0.94

Cu(111)

1

H 110

27.50

1.07

0.17

−2.45

L denotes the direction of the bond, V the direction of the surface normal, and T the direction orthogonal to L and V.

Ibach [194]). Experimentally, however, it is rather di7cult to obtain an absolute value for this quantity. The commonly applied measurement techniques only allow to determine stress diKerences, as for example between clean and adsorbate-covered surfaces. This shortcoming initiated discussions about the possibility to extract absolute values for the surface stress from measurements of speci.c surface phonons [184,195]. This approach was based on the following considerations. The surface stress tensor is de.ned as the .rst derivative of the surface energy with respect to the surface strain tensor. If the lattice interaction between surface atoms can be described by axial-symmetric pair potentials, the transverse force constant of the nearest-neighbor coupling determines the surface stress as well as certain surface modes signi.cantly. Following this line of reasoning, Lehwald et al. attempted to determine the anisotropic stress tensor for the Ni(110) surface [195] from the Rayleigh H direction should be dispersion curves. They found that the stress along the densely packed [110] smaller than along [100]. However, this .nding contradicted simple heuristic arguments, which suggest a larger stress for the densely packed direction. Later measurements shed doubt on the validity of the extracted force-constant model, as it fails to describe the dispersion of the shear-horizontal mode [196]. Recent .rst principles calculations for Pt(110) [197] also do not support the unexpected anisotropy. The weakness of this approach rests in the fact that stress is given by the .rst derivative of the total energy, while phonons are related to second derivatives. Therefore, in general, no relationship between phonon frequencies and stress can be established. Only in the special case that both phenomena are dominated by pair potential interactions, a quantitative relationship may be established. The present DFPT calculations allow a check of this assumption, as they provide the full coupling tensor without any assumptions on its shape. In Table 7, the results for the diKerent components

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213 F

F

177

L1

u L2

[110]

L3 [110]

L4

Fig. 5. Schematic picture of the tensorial coupling (LV) within the surface layer of a fcc(110) surface. F and u denote the force and displacement of an atom, respectively.

of the coupling tensor for the shortest bonds in the surface layer are presented. In all cases, a signi.cant coupling between vertical and longitudinal (parallel to the bond) motion is found. Tensor components of comparable magnitude have been also reported by Dal Corso for the Cu(100) surface [81]. These .ndings are in qualitative agreement with model-pseudopotential studies on the Al(100) surface [70]. An illustration of this LV coupling for Ag(110) is shown in Fig. 5. Interestingly, also the coupling to the second neighbor in the surface layer for the (100) and (110) surfaces exhibits the same type of tensor coupling. The existence of a signi.cant tensor component which amounts to ≈10% of the large longitudinal component, invalidates the central pair potential ansatz. As a consequence, no quantitative relationship between surface phonons and surface stress should be expected for all examined surfaces of Ag and Cu. 3.3. The phenomenon of the longitudinal resonance The phenomenon of the longitudinal resonance (LR) has been already observed in the .rst measurement of a surface-phonon dispersion. For the Ag(111) surface, Doak et al. found, at frequencies above those of the Rayleigh wave, a second structure in the HAS spectrum [179]. Since then, similar observations have been made for several other metal fcc(111) surfaces: besides the noble metals Cu and Au [181] also for transition metals Pt [198,199] and Rh [200]. In addition, similar HAS-resonances exist for (100) surfaces of Cu [183,201] and Ag [176], where under certain scattering conditions, the resonance even dominates the HAS spectrum over the RW. Contrary to early speculations, the LR-phenomenon is not limited to d-metals, as similar weak resonant structures have been found in HAS-spectra of all low-indexed surfaces of Al [202]. These observations were surprising, since simple lattice dynamical models did not predict any surface mode in this energy range. Only after drastic modi.cations of surface couplings these models exhibit a longitudinally polarized surface resonance, which is split oK from the longitudinal bulk band. In all cases, this explanation required large softenings of the .rst-layer coupling by 30 –70% [203,204,199,205,4]. Such changes have been astonishing in view of the usually small relaxations of the compact fcc(111) surfaces (typically the .rst interlayer distance changes by less than 2% [41,206,207], with the exception of the complex reconstruction of Au(111) [174]). Several mechanisms have been proposed to explain this apparent softening. The phenomenological pseudo-charge model was built on the idea that the reorganization of charge density at a surface may lead to a larger change in the dynamical response of the electrons. Following ideas of bond–charge

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R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

and shell models, which have been rather successfully applied for semiconductors [208,209], this model treated explicitly the coupling of atomic movements with electronic degrees of freedom by introducing pseudo-charges and their Juctuating multipole moments in the vicinity of the surface [210,211]. While appropriate adjustments of the phenomenological parameters allowed a proper description of the LR dispersion, elimination of the electronic degrees of freedom still resulted in large softenings of the eKective nearest-neighbor coupling by more than 30% for the (111) surfaces [210], and about 30% and 23% for the (100) surfaces of Cu and Ag, respectively [183,176]. Doubts on this picture of an anomalous surface dynamics have been put forward both from experimental and theoretical side. EELS studies by Hall et al. on Cu(111) also found a second mode along VM in the same frequency range as the LR seen by HAS. They could show that this mode is not related to the longitudinal gap mode S2 , which was found at a much higher frequency. The EELS cross section could be reasonably described with a Born–von-KOarmOan model using moderately softened NN couplings by 15%. The idea of an anomalous softening of longitudinal lattice couplings on (100) and (111) surfaces of noble metals is also at variance with all presently available ab initio calculations. Previous frozen-phonon studies on Ag and Cu found moderate softenings of 13% and 8% in the case of the (111) surfaces [48,170], and even smaller ones (¡5%) for the (100) surfaces [47]. These estimates were obtained by .tting frozen-phonon calculations to axial Born–von-KOarmOan models. The DFPT studies from the present authors, which avoid the uncertainties introduced by this .tting procedure, gave in all cases softenings smaller than 13% (see Table 5). This is also in agreement with the .ndings of dal Corso for Cu(100) [81]. Consequently, the ab initio spectra do not show a well de.ned surface mode which could be attributed to the LR resonance. There is also the possibility that the features in the HAS spectra indicate a weak surface resonance. Weak resonances usually are identi.ed from an analysis of the surface projected phonon density-of-states (SPPDOS). As the He-scattering process happens far above the surface, the relevant quantity is the .rst-layer SPPDOS de.ned by .1; a (q; !) =

|=1; a (q)|2 ((! − !q ) ;

(30)

which is shown for Ag(100)-VX in Fig. 6. The spectra do not exhibit any structure which could be assigned to the HAS maximum indicated by the black bar. The same conclusion for Cu(100) has been reached by dal Corso from his DFPT analysis of the SPPDOS along VX [81]. In contrast, for VM he could identify a weak maximum in the longitudinal part of the SPPDOS in the vicinity of the experimental features. What is the cause of this apparent contradiction between ab initio theory and experiment? There are two possibilities: (1) There exist well de.ned LR modes on (100) and (111) surfaces which are not accounted for by the present ab initio calculations. This would imply that theory wrongly predicts important interatomic couplings. However, such a failure would be very surprising in view of the good description obtained for the (110) surface dynamics. (2) Some structures in the HAS spectra are caused by special properties of the He-surface interaction and are not related to any surface mode or resonance.

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213 40

V L SH

first-layer phonon DOS (1/meV)

20

x=0.5

179

X

0 40

x=0.4

20 0 40

x=0.3

20 0 40

x=0.2

20 0 40

x=0.1

20 0

0

5

10

15

20

25

E (meV)

Fig. 6. Surface projected phonon density-of-states of Ag(100) for several momenta along VX (x; 0). A gaussian broadening of 0:5 meV has been used. The black bars indicate the position of the second maximum (LR) in the HAS spectra [176] (for x = 0:1, it denotes the second weak structure in the HAS spectrum).

Usually, a model analysis of HAS measurements involves the simulation of the He scattering cross section. As the speci.c form of the He-surface interaction potential is not known very well, various simplifying assumptions have been adopted (see, e.g., the review by Celli [10]). Recent work showed that the information about the lattice dynamics extracted from such analysis may depend sensitively on these assumptions. Franchini et al. have demonstrated that with an anisotropic He-surface interaction, the HAS spectra could be described with moderate force constant changes of 15%, much closer to the ab initio values [201]. There are also strong indications that the commonly adopted approach that the He-surface interaction potential is proportional to the electronic density (Esbjerg-NHrskov [212]) may be violated in several cases. These are based on observations that for Rh(110) and Ni(110) surfaces the turning point of a He atom lies closer to the surface when scattered from an on-top site than from a bond site [213], and are supported by ab initio studies [214]. Santoro et al. have shown that this anticorrugation eKect [215] can be accounted for by an additional structure factor in the expression for the He scattering cross section. This typically leads to an enhancement of the LR intensity with respect to those of the RW. With this modi.cation, the authors obtained a reasonable description of the HAS spectra for Rh(111) [200] and the (111) and (100) surface of Cu [216] with only moderate force constant changes of less than 20%. These .ndings demonstrate the need for a proper simulation of the He scattering cross section when comparing theory and experiment. The present authors have performed such a simulation based on the distorted wave Born approximation [217] using the ab initio results as input. Following standard approximations [203–205,71], the relevant 1-phonon contribution to the reJexion coe7cient has the form I1−Phonon (!) ˙ W (ki ; kf ; !).HAS (q; !) :

(31)

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Fig. 7. Left panel: comparison of experimental peak positions for Ag(100)-VX (open circles, after [176]) with maxima of simulated HAS spectra. The black line indicates the scan curve (Ei = 24:4 meV, 1i = 37◦ , T = 150 K) corresponding to the spectrum shown as thick line in the upper right panel. There, the thin line takes into account the anticorrugation eKect (see text). The corresponding spectral density .HAS and its decomposition is shown in the lower right panel. The spectra are broadened by 0:5 meV.

Here, we consider the scattering of a He atom from an initial state with energy/momentum (Ei ; ki ) to a .nal state (Ef ; kf ). ! denotes the energy transfer from the phonon system, q the momentum of the created (! ¡ 0) or annihilated (! ¿ 0) phonon. The prefactor in (31) contains scattering matrix elements and the thermal occupation of the phonon states. The surface lattice dynamics enters the spectral density .HAS , which can be expressed as 2 q q q 2 |z (q) − i L (q)| ((! − !q ) = .zz + .LL + .zL .HAS (q; !) = 5 5 5

.zL (q; !) = 2

Im[z (q)∗ L (q)]((! − !q ) ;

(32)

ˆ are the vertical and longitudinal components of the .rst-layer polarization ˆ and L = q” where z = z” vector of the phonon, respectively, and .zz and .LL are the corresponding SPPDOS. 5 is the so-called softness parameter, which describes the exponential increase of the He-surface potential for short R −1 [218] and thus q=51, which leads to a strong suppression distances. For Ag surfaces, 5≈2:4 A of the contribution from .LL . In contrast, the interference term proportional to .zL can be responsible for structures in the HAS spectrum. This has been observed already by Franchini and coworkers in their analysis of Al surface spectra [71,201]. Results of the simulation for the Ag(100)-VX direction are summarized in Fig. 7. The right-hand side shows a characteristic HAS spectrum together with the corresponding spectral function .HAS and its decomposition according to (32). Both are shown as function of the transfered energy for a typical scan curve !(q), indicated by the black line in the .gure on the left-hand side. A scan curve is determined by energy and momentum conservation and by the adopted scattering geometry. The

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181

Fig. 8. Results for simulated HAS spectra for Ag(100)-VM. For the meaning of symbols see Fig. 7. The sample spectrum shown on the right-hand side corresponds to the scattering conditions Ei = 24:4 meV, 1i = 31◦ , and T = 150 K (broadened by 0:5 meV).

spectrum exhibits, besides the RW peak at 6:4 meV, a maximum at 7:6 meV, which has its origin in the interference term .zL , and a very broad maximum at 8:8 meV due to an antiresonance in .zz . The plot on the left-hand side of Fig. 7 shows the position of the maxima extracted from a series of similar spectra, with scattering conditions chosen to simulate the experiment by Bunjes et al. [176]. The position of the second peak is in very good agreement with the observed HAS feature. Also the third broad maximum in the simulated spectra closely follows the positions of the weak second peak seen in the experiment [176]. These results indicate that some features in the HAS spectra are produced by the peculiar interference term, and cannot be assigned to any surface resonance. In particular the inJuence of the longitudinal SPPDOS .LL is negligible. In the case of Ag(100)-VM, shown in Fig. 8, the interpretation is diKerent. Both .zz and .zL contribute to the higher frequency peak, which in accordance with experiment can obtain a larger weight than the RW. The peak position agrees very well with the observed LR feature. Here, one can talk about a resonance showing up mainly in the vertical SPPDOS with a small longitudinal admixture. It is diKerent from the shear-horizontal resonance S2 with appears in the same frequency range. The eKect of the anticorrugation factor proposed by Santoro et al. [200,216] (shown in Figs. 7 and 8 as dashed lines) typically enhances the weight of the higher frequency peaks with respect to those of the RW. This enhancement is especially strong in the case of Ag(100)-VM and improves the agreement between theoretical and experimental peak intensities. The presented results suggest that in some cases the LR peaks do not correspond to a surface resonance but are solely due to the interference term appearing in the He-surface scattering cross section. Thus the name “longitudinal resonance” is not justi.ed. In this interpretation the HAS measurements for Ag(100) are not in contradiction with small changes in the force constants and do not imply an anomalous softening. This would explain why the observation of LR features by HAS is a common phenomenon for fcc(100) and fcc(111) surfaces, and also why they are not found in EELS measurements, with the only exception of Cu(111)-VM [188].

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In view of the various uncontrolled approximations inherent in all current simulations of HAS spectra, .rst principles calculations of the He-surface interaction would be highly desirable. This would also lead to a better founded description of the anticorrugation eKect. First steps in this direction have been undertaken recently [219–221].

4. Surface lattice dynamics of fcc transition metals In this section we extend the discussion of the surface lattice dynamics of elemental fcc metals to the class of transition metals (TM). The fcc lattice structure is realized only for the late TM, i.e., for those which possess almost .lled d shells. We present a comparative study for all low-indexed surfaces of the 4d and 5d elements Pd, Pt, Rh, and Ir. Among the surfaces considered, the (100) and (110) surfaces of the 5d elements Pt and Ir are indeed metastable or unstable. Similar to Au, the (100) surfaces exhibit a stable (5×1) superstructure, while the (110) surfaces undergo a (1 × 2) missing-row reconstruction. In the study presented below, we have only considered the ideal surfaces to allow for a better comparison with the other cases. Also, for Ir(100) there exist experimental phonon data for the metastable (1 × 1) phase. The results are grouped into two classes, Pd/Pt and Rh/Ir, as the lattice dynamics turns out to be qualitatively similar for the elements with formally the same number of d electrons in the un.lled d shell. The following results have been obtained with the mixed-basis density functional perturbation theory using relaxed slabs of 11, 11, and 9 layers for the (100), (110), and (111) surfaces, respectively. The calculations for Pt have been performed by Hong et al. [222]. The same q meshes as for the noble metal calculations have been adopted. As the range of lattice interaction in the bulk is larger for the TM than for the noble metals, it was necessary to use a denser (8 × 8 × 8) q-point grid in the DFPT calculation of the bulk force constants. Like in the previous chapter, the following surface-phonon spectra correspond to asymmetric slabs of 50 layers. Surface modes are identi.ed by a weight larger than 20% in the .rst two layers. Where possible, the standard notation for the surface modes has been adopted [20]. 4.1. Phonons of Pd and Pt surfaces The bulk phonon spectra of both Pd and Pt are dominated by the NN coupling and thus resemble closely the simple fcc spectra of Ag or Cu. Further couplings amount to less than 5% of the NN coupling for Pd and to less than 10% for Pt, but they account for the observed weak anomalies in transverse branches along the [110] direction. Table 8 shows theoretical results for the surface relaxation. All surfaces exhibit the usual inward relaxation of the .rst layer, with the remarkable exception of both (111) surfaces. Similar to the trends observed for noble metals, the size of the relaxation increases with decreasing compactness of the surface type, i.e. from (111) to (100) to (110), and is larger for the 5d element. The inward relaxation of Pd(100) is consistently found with diKerent theoretical methods, but is at variance with experiments suggesting an outward relaxation. The origin of this discrepancy is not known yet (see discussion in [223,224]). A small outward relaxation of Pt(111) has been indeed observed in LEED experiments (see Ref. [206] and references therein).

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183

Table 8 Calculated surface relaxation for Pd and Pt Pd

*12 *23 *34 *45

Pt

(100)

(110)

(111)

(100)

(110)

(111)

−1.3 0.0 −0.3

−9.1 +3.8 −0.4 +0.3

+0.1 −0.3 0.0

−2.3 −0.4 +0.4 +0.5

−14.9 +7.0 −2.2 −0.7

+0.5 −0.5 −0.3

Changes in the interlayer distances are shown in % of the bulk value.

Surface phonon dispersion curves are shown in Figs. 9 and 10, and mode frequencies at highsymmetry points are collected in Tables 9 and 10. For the same surface type, the theoretical spectra for Pd and Pt are very similar. All surfaces are found to be dynamically stable. This is especially true for the ideal (100) and (110) surfaces of Pt, which do normally reconstruct and thus do not represent the thermodynamical ground state. In contrast, the truncated bulk calculations of all three Pt surfaces exhibit a RW mode which is unstable at long wavelengths. Various diKerences with respect to the spectra of noble metals can be observed. For the (100) surface, an additional surface mode appears above the S4 mode at short wavelengths (mainly along XM). For all Pd surfaces, pronounced dips in the RW branch (S1 ) occur along high-symmetry lines H These are .ngerprints of the bulk anomalies which cause a softening of the bulk which connect to V. background spectrum derived from the transverse acoustic branch. In the case of Pt, this anomalous softening is so pronounced that the RW branch immerses early into the bulk spectrum and exists only as a resonance for longer wavelengths. The same behavior has been found in an earlier theoretical study of Pt(111) based on a force-constant model [205]. Available experimental data are in satisfactory agreement with the calculation except for two cases. For Pt(111), the theoretical RW branch emerges from the bulk continuum only close to the SBZ boundary, whereas experimentally it was clearly observed for most part of the VM and VK lines. Also, pronounced wiggles in the RW branch have been identi.ed by HAS along the VK line [198]. Furthermore, no theoretical surface mode is found to explain the second experimental surface mode. It represents another example of the phenomenon of longitudinal resonance discussed in detail in Section 3.3. For Pd(111), the EELS data of Hsu et al. suggests that theory overestimates the frequency of the RW mode near M [226]. However, the measurements have been done for a slightly hydrogenated surface (¡ 10% H coverage), and no change in the RW dispersion was observed when going over to full coverage. As it is rather common that adsorbates induce a softening of the RW branch (see, e.g. discussion in [9]), a study of the truly clean (111) surface would be desirable to clarify this point. 4.2. Phonons of Rh and Ir surfaces The bulk lattice dynamics of Rh and Ir have already been the subject of an extended frozen-phonon study by the present authors based on a generalized supercell approach [139,229]. For both cases,

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35

Γ

X

Γ

M

30

ω (meV)

25 S6

20

S2

S2

15 S5

10 S4

5

S1

Pd(100)

S1

0

35

Γ

X

S

Γ

Y

30 S6

ω (meV)

25 S7

20 MS7

MS7 S5

S2

15 MS0 S2

10 5

Pd(110)

S1

S3

S1

0 35

Γ

M

Γ

K

30 S4

25 ω (meV)

S2

20

S3 S5

15

S5 MS3

10 S1

5

S1

Pd(111)

0

Fig. 9. Theoretical surface phonon spectra of Pd(100), Pd(110), and Pd(111). Black dots denote theoretical surface modes. Squares represent EELS data for Pd(100) and Pd(111) taken from Refs. [225,226], respectively. The latter measurement has been performed on a slightly hydrogenated surface (¡ 10% coverage). The HAS results for Pd(110) (open circles) are taken from [227].

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

30

Γ

X

185

Γ

M

25

ω (meV)

20

S6 S2

15

S2 S5

10

S1

S4

S1

5

Pt(100)

0 30

Γ

X

S

Γ

Y

25

ω (meV)

S6

S7

20

MS7

15 MS7 10

S5

S2

MS0

S3

S2

5

S1

Pt(110)

S1

0 30

Γ

M

Γ

K

25 S2

ω (meV)

20 15

S4

S3 S5 MS3

10 5

S1

Pt(111)

S1

0

Fig. 10. Theoretical surface phonon spectra of Pt(100), Pt(110), and Pt(111). Black dots denote theoretical surface modes. Open circles for the (111) surface denote HAS data from [198].

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Table 9 Surface-mode frequencies (in meV) of Pd(100), Pd(110), and Pd(111) at selected high-symmetry points Pd(100)

X S1

S4 9.2 9.5

Pd(110)

11.2 12.0 11.2

S2

S6

M S1

13.0

13.6 12.8

22.6 21.9

13.1 13.6

VH MS7

X MS0

S1

S2

18.1 ≈16:8

11.4 ≈11:4

11.5 10.6

15.7

Y S1

S2

S3

7.6 ≈6:7 Pd(111)

S5

M S1

10.9

10.1 9.3

S5

13.0 ≈11:0

14.1

20.7

18.0

S2

Method

19.3

DFPT FP [223] EELS [225]

S7

Method

22.0

DFPT HAS [227]

S5

S S1

S2

12.8

12.2

14.3

14.7

18.2

S2

K S1

S5

S3

24.4

13.8

15.1

18.9

13.6

14.1

S7

Method

23.4

DFPT HAS [227] Method

20.8

DFPT EELS [226]

Table 10 Surface-mode frequencies (in meV) of Pt(100), Pt(110), and Pt(111) at selected high-symmetry points Pt(100)

Pt(110)

X S1

S4

S5

8.4 7.9

10.7 9.9

10.3 9.1

VH MS7

X MS0

S2

13.6

7.8

Y S1 6.1 Pt(111)

M S1 11.1 10.8

S2 8.9

13.3

S2 12.2

14.3

S6

M S1

18.4 18.2

12.6 11.6

16.9

S5

S2

11.4

19.3

12.2 K S1 11.9 ≈11:1

Method

16.5

DFPT [222] FP [228]

S7

Method

17.4

DFPT [222]

S5

S S1

12.8

15.6

10.8

S5

S3

11.1

14.1

S3 7.5

13.7

S2

16.2

S2 9.8

10.0

10.5

S7

Method

19.0

DFPT [222]

S4

Method

19.6

DFPT [222] HAS [198]

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187

Table 11 Calculated surface relaxation for Rh and Ir Rh

*12 *23 *34 *45

Ir

(100)

(110)

(111)

(100)

(110)

(111)

−3.9 +0.1 +0.1

−11.0 +2.8 +0.9 −2.2

−1.8 −0.7 +0.2

−5.1 +0.8 +0.1

−13.5 +5.3 −0.4 −1.9

−2.1 −0.5 +0.2

Changes in the interlayer distances are shown in % of the bulk value.

pronounced anomalies in the transverse acoustic branches along [110] have been predicted, which were subsequently con.rmed experimentally [139,229]. The appearance of anomalies could be linked to an unusually large coupling to the 9th neighbor. While the overall shape of the spectra is still dominated by the NN longitudinal coupling, the second-neighbor force constant becomes more important and amounts to about 20% and 30% for Rh and Ir, respectively. These .ndings are con.rmed by the present DFPT study. The theoretical surface relaxation of Rh and Ir, shown in Table 11, exhibit the same trends as for Pd and Pt, but is generally larger. Furthermore, a signi.cant inward relaxation is found for both (111) surfaces. The results for Rh are in reasonable agreement with previous theoretical .ndings [207]. As expected, these large structural modi.cations result in more complicated surface phonon spectra (see Figs. 11 and 12, and Tables 12 and 13). This is most prominent for the (110) surfaces, where an increased number of surface modes or surface resonances appears. Here the RW branch is softened with respect to a truncated bulk calculation in contrast to the other surface types. This softening is strongest for Ir(110), but the ideal surface still remains dynamically stable despite its thermodynamical instability. The same holds for the metastable Ir(100), where the measured RW branch is in reasonable agreement with the present calculation. For the other case of a measured phonon spectrum, Rh(111), the experimental RW branch perfectly agrees with the theoretical one. But again, the observation of a longitudinal resonance above the RW lacks a corresponding theoretical surface mode (see Section 3.3). Contrary to the case of Pd and Pt, the bulk anomalies of Rh and Ir do not show up clearly in the surface spectra. They are located closer to the bulk Brillouin zone boundary and are thus more inJuential for short-wavelength vibrations. Surface modes with short wavelengths, however, mainly feel the modi.ed couplings at the surface and are less sensitive to the couplings deeper in the bulk. This may explain the absence of any sharp dip in the surface phonon dispersion curves. All low-indexed Rh surfaces were examined previously with the frozen-phonon technique. The frequencies for the (100) and (110) surfaces published by Xie and Sche]er are typically softer by 10% with respect to the present one [230]. Their calculation, however, was performed for a surface geometry, where thermal expansion of the .rst layer has been taken into account. The published values correspond to a theoretical geometry for a temperature of 300 K. For this geometry, the authors obtained a decrease in the corresponding interplanar force constants by about 10% with

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35

Γ

X

Γ

M

30

ω (meV)

25

S6

20

S2

S2 S5

15

S4

10

S1

S1

5

Rh(100)

0

35

Γ

X

S

Γ

Y

30 S6

ω (meV)

25

S5

S7

MS7

20

MS7

S2 S3

15 MS0

S2

10 S1

S1

5

Rh(110)

0 35

Γ

M

Γ

K

30 S2

ω (meV)

25 20

S5

S3

15 MS3

10

S1

S1

5

Rh(111)

0

Fig. 11. Theoretical surface phonon spectra of Rh(100), Rh(110), and Rh(111). Black dots denote theoretical surface modes. Open circles for the (111) surface denote HAS data taken from [231].

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

30

Γ

Γ

M

X

189

25 S6

ω (meV)

20

S2

S2

S5

15 S4

10 S1

S1

5

Ir(100)

0

30

Γ

X

S

Γ

Y

25 20 ω (meV)

S6

S7

S5

MS7 MS7

15

S3

S2

MS0

10

S2 S1

5

S1

Ir(110)

0 30

Γ

M

Γ

K

25 S4

S2

ω (meV)

20 S3 S5

15

MS3

10 S1

S1

5

Ir(111)

0

78 Fig. 12. Theoretical surface phonon spectra of Ir(100), Ir(110), and Ir(111). Black dots denote theoretical surface modes. Open squares represent EELS data from [232] obtained for the metastable (100) surface.

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R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

Table 12 Surface-mode frequencies (in meV) of Rh(100), Rh(110), and Rh(111) at selected high-symmetry points Rh(100)

VH 29.9

Rh(110)

VH MS7 20.1

X S1

S4

S5

S2

S6

15.5 12.3

16.6 13.9

15.5

17.9 14.1

26.4 22.3

X S1

MS0

S2

S7

17.3

18.2 14.6

21.2

14.4 14.2

Y

S5 18.8

Rh(111)

VH

19.2

22.0 18.7

32.1

S5

S3

16.4 16.3 16.2

16.6 16.0

21.5

S5

S3

18.4 18.2

18.8 18.7

M S1 17.5

K S1 17.1 17.8 ≈17:4

M S1 31.9

S2

21.6 18.4

21.5 18.6

Y S1

S2

S3

Method

30.4

9.8 10.3

13.4 10.8

15.5 15.1

DFPT FP [230]

S S1

S2

S3

S7

Method

14.3 14.6

15.2 14.9

16.1 15.6

S2 23.4

25.9 25.4

16.2

24.6 21.8

23.3

Method

27.2

24.2

DFPT FP [230]

DFPT FP [230] Method

31.8

DFPT FP [50] HAS [231] Method

21.7

24.9

27.5

28.2

DFPT FP [50] HAS [231]

The frozen-phonon (FP) results for the (100) and (110) surfaces were obtained for a thermally expanded surface geometry corresponding to T = 300 K.

respect to the static relaxation. This may partly explain the discrepancy. For Rh(111), the DFPT results agree well with those of a previous frozen-phonon study by Bohnen et al. except for the high-frequency part of the spectrum [50]. The latter may be due to the fact that in the DFPT study, larger modi.cations in the dynamical coupling also appear within the second layer and between the second and third layer, which have not been considered in the frozen-phonon work. 4.3. Surface lattice interaction As already suggested by the increased number of surface modes in the spectra shown above, the interatomic couplings at all considered TM surfaces exhibit stronger deviations from the bulk values than found for the noble metals. The theoretical changes for the NN coupling within the surface layer are collected in Table 14. Since here the lattice interaction can no longer be described by

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

191

Table 13 Surface-mode frequencies (in meV) of Ir(100), Ir(110), and Ir(111) at selected high-symmetry points Ir(100)

VH

X S1

24.5

13.7

M S1 18.1 16.4 Ir(110)

16.9

S5

S2

13.5 13.6

12.1

15.4

S2

S6

18.9

20.9

VH MS7 14.8

18.5 18.6

23.9 24.4

S2

S3

11.1 11.0

13.0 12.4

S S1

S2

S3

9.7 10.5

12.9 10.9

12.3 12.5

13.4 13.5

VH

M S1

S5

S3

14.3

13.5

13.6

17.3

Y S1 7.5 8.8

Ir(111)

S4

24.7

S6 17.6

21.9

Method 25.5

DFPT EELS [233,232] Method DFPT EELS [233,232]

22.3 X S1

MS0

S2

10.8 13.2

15.2 14.2

15.2 15.6

S7 17.0

23.2

Method 24.6

24.3

S5 13.2

15.2 13.4

14.2

18.7 19.1

Method 26.3 25.5

DFPT FP [51]

S7 20.1 19.2

Method 23.0 21.9

DFPT FP [51]

S2 18.4

DFPT FP [51]

20.7

25.3

K S1

S5

S3

14.0

15.3

15.5

Method 22.2

DFPT

a single longitudinal force constant, we have made use of an average coupling of a bond b de.ned by [229] I (b) =

1 2

!5 (b ) ; 3

(33)

!5

where !5 (b) denotes the force constant matrix. In all cases, a weakening of this coupling occurs, which is substantially stronger than for Ag and Cu. The softening tends to increase with decreasing surface compactness, and is larger for the 5d elements. As expected from the observation for Ag discussed in Section 3.2, no correlation with the size of the layer relaxation can be established.

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Table 14 Calculated changes of the average coupling for nearest-neighbor bonds within the outermost layer (in % of the bulk value)

Pd Pt Rh Ir

(100)

(110)

(111)

−32 −40 −24 −23

−24 −36 −42 −51

−21 −32 −29 −29

Larger changes are also found for the intralayer coupling of the subsurface layer. For the (110) surfaces, a softening of 10 –30% occurs with respect to the bulk. For (100) and (111), Rh and Ir exhibit an increase in the range of 5 –20%. This stiKening is responsible for the appearance of surface modes above the bulk spectrum, which mainly consist of in-plane motions of the second-layer atoms. In contrast, the on-top modes for the (110) surfaces of Rh and Ir possess stronger vertical character, and are caused by an increased interlayer coupling of the surface atoms to the second and third sublayer. Similar to the case of noble metals, these modi.cations closely follow the relaxation-induced changes of the bond lengths, but are much larger for the TM with changes up to 130%. A comparison of TM results with those for the noble metals demonstrate that the presence of an open d shell invokes an increased sensitivity of the bonding properties and of the dynamical couplings to disturbances of the bulk periodicity. For the late TM, this sensitivity increases with decreasing number of d electrons, and is larger for the 5d elements than for the 4d elements. 5. Surface lattice dynamics of non-fcc transition metals 5.1. Surface phonons of W(110) and Mo(110) Hydrogenated W(110) and Mo(110) surface exhibit very pronounced phonon anomalies, which have been the subject of several experimental studies [234–237], and also initiated investigations of the clean surfaces. Here, we brieJy discuss the theoretical results for the clean cases. We defer a discussion of the more interesting case of hydrogenated surfaces to Section 7.1. The surface dynamics of W(110) has been investigated by Bungaro et al. [79]. This work is rather remarkable as it constitutes the .rst application of the DFPT method to a metallic surface. It has been performed using a pure plane-wave basis in conjunction with a norm-conserving pseudopotential. The calculations provided a good description of the bulk phonon dispersion curves including certain anomalous features. Relaxation of the (110) surface was found to be mainly con.ned to the outermost layer which moves inward by 2.9%. Part of the calculated dispersion curves for W(110) are shown in Fig. 17, and surface mode frequencies at high-symmetry points are reported in Table 15. The low-frequency part exhibits three well de.ned surface modes. Good agreement with experimental data is found for the RW and for a second higher-frequency mode of mainly longitudinal character. The third mode with shear-horizontal polarization was not observed experimentally. In addition, some resonant structures in the energy range of the bulk modes agree well with features seen in EELS spectra [236]. The calculations gave strong indication that the lattice dynamics of the clean W(110) behaves rather normally.

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193

Table 15 Surface-mode frequencies (in meV) of W(110) at selected high-symmetry points VH

H

N

24.4

16.6

18.6

≈23.7

15.9

18.5

18.6

21.0 ≈21:3

14.7 15.4 14.4

S 15.6

19.9

21.6

19.9

21.0

13.2

15.3 18.3 16.2

21.8

DFPT [79] FP [62] EELS [236]

The clean surfaces of W(110) and Mo(110) have also been the subject of a frozen-phonon study by Kohler et al. [62]. As the authors were interested in the anomalous electron-phonon coupling induced by H adsorption, they concentrated on the RW mode at the SBZ boundary points S and N. Because these frozen-phonon calculations required laterally expanded supercells, the authors took into account only vertical motions of the two outermost layers to reduce the numerical cost. The RW frequencies obtained for W(110), shown in Table 15, are slightly harder than those found experimentally. For Mo(110), the theoretical RW frequency at the S point was 22:7 meV as compared to ≈19 meV found by EELS [237]. 5.2. Phonon dynamics of Ru(0001) Ru is currently the only transition metal with hexagonal close-packed (hcp) structure, whose surface lattice dynamics has been studied in some detail. An inelastic He-atom scattering (HAS) measurement of the clean and the hydrogenated Ru(0001) surface has been carried out by Braun et al. [238]. For the clean surface, they observed the RW and a second surface mode, which could be described by a simple force constant model without adjusting the bulk values for bonds near the surface. Doubt on the validity of this interpretation was shed by a combined DFPT and neutron scattering investigation [239]. It revealed that the lattice dynamics of bulk Ru exhibits various anomalies, requiring long-range force constants and signi.cant non-central couplings for an accurate modeling. The HAS measurements have been limited to modes with longer wavelengths due to kinematic restrictions. A more complete picture of the lattice dynamics of Ru(0001) was provided recently by a combination of MB-DFPT calculations by the present authors and of high-resolution electron energy loss spectroscopy (HREELS) measurements performed by Widdra and his coworkers [86]. Experimental and theoretical results for the surface phonon spectrum are collected in Fig. 13. The higher-frequency mode seen in HAS and assigned to a longitudinally polarized resonance (LR) is con.rmed by HREELS and DFPT. Theory predicts even a well de.ned surface mode with predominantly longitudinal character. The dispersion curves reveal a very anomalous surface lattice dynamics especially in the vicinity of the SBZ boundary, which were not accessible in the HAS experiment. A prominent example is given by the RW branch, which exhibits a signi.cant dispersion along the SBZ boundary from K to M, followed by an upturn along MV. This behavior is in contrast to the two cases of hcp(0001) surfaces studied until now, Be(0001) [82] and Mg(0001) [84] (see Section 6). The RW branch follows closely the low-frequency edge of the surface projected bulk spectrum, where the large frequency diKerence between K and M is reminiscent of the anomalous dispersion of bulk Ru in the vicinity of the K and M points [239]. However, the bulk-terminated

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213 Γ

K

Γ

M

250

30

ω (meV)

25

200

LR

20 15

S3 LR

150

S2 S2

-1

35

ω (cm )

194

100

10

S1 5 0

M Γ

S1

50

K M

0

Fig. 13. Phonon dispersion curves of Ru(0001) along the high-symmetry direction VKM and VM according to Ref. [86]. Squares denote experimental results from HREEL measurements obtained at 300 K (open squares) and at 16 K (.lled squares). Open circles are HAS results from Ref. [238]. The theoretical phonon dispersion curves correspond to a 50 layer slab and are shown as dashed lines. Theoretical surface modes with a weight larger than 20% in the .rst two layers are denoted by .lled circles.

RW (shown as thin line in Fig. 13) shows a much weaker dispersion along KM, which indicates a very anisotropic stiKening. Similarly, the shear-horizontal mode S2 also has an unusual dispersion with a local minimum at M and a steep rise towards K. Another interesting feature is the dip structure of the RW branch along VK, which is resolved in the experimental data although less pronounced than predicted by theory. Such a dip structure could be the signature of a Kohn-type anomaly, as has been identi.ed to be the origin of the anomalous Rayleigh mode of 2H-TaSe2 (001) [240]. For Ru(0001), however, spectra taken at T = 16 K, do not indicate any signi.cant temperature dependence of this feature, thus ruling out this interpretation. Theory reveals that in the region of the dip, the RW polarization drastically changes from a vertical to an in-plane motion with a displacement pattern similar to those of the LR mode, while the vertical weight is partly transfered to the surface mode S3 . This strongly suggests a mode crossing of the RW and LR branches as the origin of the dip. The phenomenon of mode mixing or mode crossing is not restricted to the dip region. Along the entire SBZ boundary, the RW gains a strong longitudinal character, while its vertical weight is strongly reduced. This is in clear contrast to an almost purely vertical motion of the .rst-layer atoms in the case of the truncated-bulk calculation. From an analysis of the theoretical interatomic coupling constants, this anomalous exchange of mode character could be traced back to a 60% softening of the .rst-intralayer longitudinal coupling. It indicates that the large softening of in-plane vibrations is a general phenomenon at short wavelengths. In agreement with most previous LDA studies [241–244], the present work obtains a large inward relaxation of the surface layer by *12 = −4:1%. This seems to be at variance with several LEED experiments indicating a much smaller inward relaxation of *12 ≈ − 2% [243,245–247]. DFPT calculations for a smaller relaxation *12 = −1:6% showed that the anomalous softening is rather robust. In contrast, the unusual high-frequency mode above the bulk is most likely an artefact of the overestimation of *12 by LDA.

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195

Ru(0001) constitutes a clear case of a simple surface that exhibits signi.cant softening of a longitudinal surface mode for short wavelengths. It indicates a delicate balance between repulsive and attractive components in the interatomic binding of Ru. This example shows that large modi.cations of the lattice dynamical couplings can be present also for unreconstructed and rather compact surfaces. 5.3. Surface phonons of Zr(0001) The VH point of the Zr(0001) surface has been subject to a mixed-basis frozen-phonon study by Yamamoto et al. [52]. Only vibrations parallel to the hexagonal axis have been considered, which decouple from the transverse ones at this high-symmetry point. The authors found a strong increase of the zz-coupling between the surface and the subsurface layer by 70%, and an accompanying increase in the restoring force constant of the second-layer atom. As a consequence, a doubly degenerate mode at 23:6 meV is split oK from the top of the bulk continuum by about 1 meV, whose motion is localized predominantly in the second layer. The stiKening of the .rst interlayer coupling seems to be connected to the large inward relaxation of the .rst layer by −4:7%. This split-oK mode resembles the localized mode found above the bulk continuum in the case of Ru(0001) [86] (see Section 5.2), where atomic motions are also localized in the second layer. As mentioned above, this theoretical result is most likely an artefact of the overestimation of the lattice relaxation. In the case of Zr(0001), no experimental data is available to compare with the theoretical predictions. 6. Surface lattice dynamics of nearly-free electron metals 6.1. Phonons of Al surfaces The Al surfaces have been the .rst test cases for an application of ab initio methods to surface lattice dynamics. The frozen phonon technique has been applied by Ho and Bohnen to the (110) [42,43] and the (100) surface [248]. They determined interlayer force constants and surface modes at highsymmetry points of the SBZ. Later, a similar calculation has been performed for Al(111) by Sch2ochlin et al. [44]. It has been observed that the changes in the interlayer force constants mirror the degree of the surface relaxations, being largest for the open (110) surface and rather small for the more compact (100) and (111) surfaces. All three low-indexed surfaces of Al have been investigated within the dielectric screening (DS) approach by Eguiluz and coworkers [69,70,202]. This perturbative method is not restricted to highsymmetry points, and allows the determination of interatomic force constants. Surface dispersion curves have been reported for the relaxed and unrelaxed Al(110) surface [69] and for Al(100) [70]. The theoretical spectra have then been used in a detailed analysis of HAS-reJection coe7cients in an attempt to understand the origin of weak resonances observed in HAS experiments above the RW branch (longitudinal resonances, see Section 3.3) [202,71]. In the calculations of the force constants, however, only second-order contributions from the pseudopotential were taken into account, which was modeled by a local pseudopotential of the Heine–Abarenkov form. These approximations limited the accuracy of the predicted surface modes, and the authors suggested that these lead to an overestimation of the RW frequency of Al(100) by 9% [70].

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Results for the surface mode frequencies are summarized in Table 16. 6.2. Surface phonons of Na The bcc metal Na is currently the only alkali metal whose surface dynamics has been investigated by ab initio methods. H Y, and S have been studied with the frozen-phonon approach The surface phonons of Na(110) at V, by Rodach et al. [45]. Interatomic surface force constants obtained from a .t of an axially-symmetric force constant model to their interplanar force constants showed an unusual 15% stiKening of the radial NN coupling in the surface layer. Surprisingly, the same coupling between .rst and second layer increases by 30% despite a small inward relaxation of 1.6%. Quong et al. applied the dielectric screening approach to Na(100) [72]. In their work the approximations made in the calculations for the Al surfaces discussed above could be avoided. Surface relaxation has been neglected, since it was found to be within numerical accuracy. Substantial changes are observed mainly for tangential force constants, as evidenced by a 93% increase of the zz-component of the NN coupling in the surface layer. The theoretical surface modes for both cases are summarized in Table 17. No experimental data are available for comparison. @ 6.3. Phonon dynamics of Be(0001) and Be(1010) Although being an sp-bonded metal, Be exhibits various properties which are anomalous with respect to other sp elements. Bulk Be possesses a hcp structure with a strongly contracted c=a ratio as compared to the ideal hcp structure, and a small density-of-states at the Fermi energy. These properties are related to a strong s–p hybridization which is the cause of signi.cant non-central force constants necessary to describe the bulk phonon dispersion (see, e.g. the discussion in [253]). This hybridization sensitively depends on the local atomic arrangement therefore, it is likely to cause anomalous surface properties. The surface dynamics of Be(0001) has been studied by Lazzeri and de Gironcoli using a planewave based DFPT [82]. The (0001) surface exhibits a large outward relaxation of the .rst layer. This property tends to increase the c=a ratio, and has been attributed to a more free-electron like behavior of the density-of-states at the surface [253]. The theoretical phonon-dispersion curves are shown in Fig. 14. The RW dispersion curve nicely agrees with the experimental data. It correctly reproduces a slight downward dispersion from K to M, a feature which comes out even qualitatively wrong in a truncated-bulk calculation. The surface dynamics is found to involve many surface layers, indicated by several surface modes which are strongly localized in subsurface layers. Among them are two modes predicted to exist below the bulk spectrum in the vicinity of K, which have not been observed experimentally. The authors suggested that this may be caused by the closeness of the RW, which possesses a strong intensity and may mask the other two modes with only a small vibrational amplitude in the surface layer. Experimentally, besides the RW, only a shear-horizontal mode has been observed near M in good agreement with theory (see also Table 18). The authors found that the theoretical force constants near the surface still contain strong non-central components comparable to those in the bulk, thus not supporting the idea of a more free-electron like behavior at the surface.

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197

Table 16 Surface-mode frequencies (in meV) of Al(100), Al(110), and Al(111) at selected high-symmetry points Al(100)

X S4

S1

14.9 16.5 15.1 16.1 Al(110)

12.4 14.9

33.5 33.7

M S1

Method

20.3 20.8

FP [248] DS [70] EELS [187] HAS [249]

X S1 17.2 19.0 14.6

Al(111)

S6

16.7

M S1

25.3

26.9

31.8

S3

10.4 13.0 8.9

13.9 13.8 13.5

S 14.9

25.8

32.9

K S1

S2

17.0 ≈16:5 ≈17:1 17:2 ± 0:4

22.8

Y S1

35.2

Method FP [42,43] DS [69] HAS [250] Method

18.6

FP [44] DS [71] HAS [251] HAS [249]

≈17:7

DS denotes the dielectric screening approach.

Table 17 Surface-mode frequencies (in meV) of Na(100) and Na(110) at selected high-symmetry points Na(100)

X 4.0

Na(110)

Method 4.2

11.2

DS [72]

Y 3.3

S 10.3

14.5

16.5

9.9

Method 11.6

16.1

FP [45]

H surface dynamThe same authors have also performed a DFPT investigation of the Be(1010) ics [83]. The atoms at this surface are less densely packed than at the (0001) surface. The surface layer consists of rows of atoms similar to the fcc(110) surface. There exist two interlayer spacings which diKer by a factor of two. The open surface structure exhibits an oscillatory relaxation involving several layers, with an almost 25% contraction of the terminating short interlayer spacing. These large structural modi.cations induce a rather rich surface phonon spectrum, shown in Fig. 15. The authors were able to assign all surface modes observed in an EELS experiment [254] H This in turn was identi.ed to (Table 18), with the exception of the loss feature at 63:3 meV near V. be caused by an enhanced vibrational surface density-of-states and not by a surface localized mode.

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80

15

60

10

40 K

Energy (meV)

Frequency (THz)

20

M

5

20 Γ

0

Γ

Γ

M

K

0

Fig. 14. Phonon dispersion curves of Be(0001). Thin lines represent theoretical results for a 30-layer slab. Full and open dots denote intense and weak features from an EELS experiment [252], respectively. Thick lines indicate the boundaries of the bulk spectrum. (From Ref. [82]. Copyright (1998), with permission from Elsevier.)

Table 18 H at selected high-symmetry points Surface-mode frequencies (in meV) of Be(0001) and Be(1010) Be(0001)

K

M

≈46 H Be(1010)

≈44

≈44

≈65

VH

39.5 39.5

50.0 50.5

≈68

DFPT [82] EELS [252,253]

A 45.9 43

52.9 55

73.9

84.8

26.4

Method 32.3 33

63.7

DFPT [83] EELS [254]

M 41.4 40

Method 49.8 47

54.3

56.0

61.1

68.7

69.4

76.5

DFPT [83] EELS [254]

A bulk-truncated model calculation resulted in very diKerent surface modes, which indicates that the electronic screening at this surface is markedly diKerent from that of the bulk. In an extension of these works, Lazzeri and de Gironcoli investigated the surface thermal expansion H [89]. They calculated the phonon contribution to the free energy of Be(0001) [87] and Be(1010) in the quasiharmonic approximation from vibrational spectra obtained via DFPT. In the case of the (0001) surface, their calculated temperature dependence of the .rst interlayer spacing diKered noticeably from results of a previous analysis by Pohl et al. [255], where they considered only a few high-symmetry modes in the calculation of the free energy. This stresses the importance of a proper sampling of the phonon spectrum in determining thermodynamical properties.

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

80

15

60

10

40

5

0

A

Γ

A

L

Γ

M

Energy (meV)

Frequency (THz)

20

199

20

0

M

H Fig. 15. Phonon dispersion curves of Be(1010). Lines represent dispersion curves for a 104-layer slab. Modes localized more than 25% in the three topmost layers are denoted by dots, with the dot size proportional to this percentage. Full dots correspond to modes localized more than 25% in the topmost layer with polarization perpendicular to the surface. (From Ref. [83]. Copyright (2000), with permission from Elsevier.)

H surface many layers are involved in the thermal relaxation. This increases the For the Be(1010) complexity of the ab initio approach substantially. The authors therefore extended the standard DFPT approach by an e7cient calculation of third-order derivatives of the total energy [89]. For the .rst four interlayer spacings, they found an oscillatory thermal relaxation, which tends to increase the static relaxation pattern with increasing temperature. A similar thermal relaxation has been found H (see the following section). theoretically and experimentally for Mg(1010) @ 6.4. Surface phonons of Mg(0001) and Mg(1010) Most properties of bulk Mg can be well understood in a nearly free-electron like picture. Contrary to Be, the hcp structure of bulk Mg has an almost ideal c/a ratio, and the lattice dynamics is well described by an axially-symmetric force constant model. The surface lattice dynamics of Mg(0001) was studied with DFPT and EELS by Ismail et al. [84]. The results are reproduced in Fig. 16. Good agreement was found for the RW dispersion along the whole VKM-line of the SBZ. In addition, all experimentally resolved surface phonons at K and M could be classi.ed on the basis of the calculation. These are summarized in Table 19. To gain more insight into the physics behind the changes in surface force constants, the authors compared the RW dispersion of Mg(0001) with those of two other sp-bonded surfaces with hexagonal structure, Al(111) and Be(0001). An outward relaxation of the .rst layer is common among these three surfaces. Also in all cases, the RW exhibits only a small frequency diKerence between K and M, with a lower frequency at the latter point. This indicates that the surface dynamics of the relaxed surfaces are qualitatively similar. However, when the changes in the surface force constants are ignored, i.e. when performing bulk-terminated calculations, qualitative diKerences are revealed. For Mg(0001) and Ag(111) the RW frequency still decreases from K to M, while it increases for Be(0001). In addition, the transition from the ideal bulk-terminated to the completely relaxed surface

200

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

8 6 20 4 10

2

Γ

K

energyy (meV)

frequency (THz)

30

Γ

M

Fig. 16. Phonon dispersion curves of Mg(0001). Theoretical surface modes are shown as solid lines (well localized, with more than 30% in the .rst two surface layers) and short-dashed lines (weakly localized, with more than 15% in the .rst three layers). Dark-gray shaded regions denote theoretical resonances, while the surface-projected bulk spectrum is given by the large area shaded in grey. Filled (open) diamonds represent intense (weak) experimental losses. The dashed-dotted line is the Rayleigh mode obtained from a bulk-truncated calculation. (Reprinted from Ref. [84] with permission. Copyright 2000 by the American Physical Society.) Table 19 Surface-mode frequencies (in meV) of Mg(0001) at selected high-symmetry points K

M

K1

K2

K3

M1

M2

M3

M4

M5

Method

14.9 14.5

16.8 16.5

24.3 23.5

14.2 13.5

15.8–16.4 17.0

18.9 18.5

27.8 27.5

29.7

DFPT [84] EELS [84]

Notation for the modes follows Ref. [84]. The Rayleigh mode corresponds to K1 and M1 , respectively.

results in a stiKening of the RW throughout the SBZ for Mg(0001) and Ag(111), while for Be(0001) a softening near M occurs. This behavior was interpreted as a consequence of the diKerent nature of the lattice interaction in bulk Be. In a subsequent article, Ismail et al. have discussed a combined DFPT and LEED study of the H surface [88]. The theoretical analysis was based on the quasithermal expansion of the Mg(1010) H harmonic approximation using the same extension of the DFPT method as was applied for Be(1010) H surface possesses alternating short and long interlayer (see the previous section). The Mg(1010) spacings, and is terminated by a short one. It was found both experimentally and theoretically that H surface exhibits an oscillatory thermal relaxation, which increases the static relaxthe Mg(1010) ation pattern, i.e. thermal contraction of the short-interlayer spacing and thermal expansion of the H long-interlayer spacing. The same qualitative behavior was obtained theoretically for Be(1010). The H indigood agreement between experiment and .rst principles calculations in the case of Mg(1010) cates a high predictive power of the ab initio approach for thermodynamic properties even in the case of a rather complex multilayer thermal relaxation.

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201

7. Lattice dynamics of adsorbate-covered surfaces 7.1. Surface phonons of H/W(110) and H/Mo(110) Hydrogenated surfaces of W(110) and Mo(110) had been the focus of much attention in recent years because they display very unusual vibrational properties of the substrate. At full coverage, a bifurcation of the RW branch has been observed in HAS experiments [256,234,235] in the vicinity of an incommensurate wavevector qc along the [001] direction (VH). Both branches show indentations with respect to the RW dispersion curves of the clean surfaces. One branch is very deep and sharp. EELS experiments could con.rm the anomaly in the upper branch, but did not see any indication of the lower one [236]. Similar but less pronounced softenings were observed also at the S point. Theoretical analysis of surface electronic states of Mo(110) suggest the existence of nesting properties of the surface Fermi-surface with nesting vectors close to the critical wavevector qc of the vibrational anomalies [257]. This suggested that the surface phonon anomalies are caused by a Kohn-like mechanism. To test this hypothesis, Bungaro et al. have performed a lattice dynamics study of the H/W(110) surface using a plane-wave density functional perturbation theory method [79,258,167]. The application of this method to the clean W(110) surface has been already discussed in Section 5.1. The nesting feature of the surface Fermi surface represents a delicate theoretical problem, because it requires a smearing width for the electronic states much smaller than applied usually. To reach convergent results for the phonon modes, this would imply a very dense k point mesh of more than 300 points in the SBZ which was not feasible at the time of this work. To obtain an accurate estimate of the RW phonon frequencies near the critical wavevector, the authors resort to a iterative variation-perturbation method. They .rst calculated the dynamical matrix for a large smearing width, and treated the diKerence to the correct one as a perturbation. Within .rst-order perturbation theory, corrections to frequency and eigenvectors could be obtained. This procedure was repeated iteratively until convergence was reached. Details of the method have been outlined in Ref. [258]. Results for the phonon spectra of the clean and hydrogenated W(110) surface are reproduced in Fig. 17. The iterative perturbation scheme gave a very good description of the sharp dip in the RW and a similar indentation of the longitudinal surface mode at higher frequencies. A softening was also found at the S point. All .ndings were in excellent agreement with the EELS results. The theoretical RW dispersion also coincides with the upper branch seen by HAS. This theoretical analysis could not provide, however, an explanation for the second very deep branch seen in HAS. The authors suggest that this feature is related to disorder in the adsorbate layer, and corresponds to a plasmon-like excitation of a diKusive H motion. Another possibility is the coupling of the RW mode to electron–hole excitations. This mechanism has been favored by Kohler et al. [257,62], who have performed frozen-phonon studies of the RW mode for the H-covered W(110) and Mo(110) surfaces. Similar to their approach for the clean surfaces described in Section 5.1, the vibrational patterns have been restricted to vertical motions of the adsorbate and the .rst two substrate layers due to computational costs. Results for H/W(110) shown in Table 20 reproduce correctly the observed softening at S and hardening at N with H adsorption. For the S point of H/Mo(110), a similar softening from 22:7 meV of the clean surface down to 17:2 meV was found theoretically (EELS [237]: 18.8 and 14:9 meV, respectively).

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R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213 clean

hydrogenated

frequency (cm-1)

200

[110]

100

N S

[001] H

Γ

Γ

[112]

H

N

S

Γ

H

N

S

Fig. 17. Phonon dispersion curves of W(110) (left) and H/W(110) (right). Experimental data are represented by open diamonds (HAS, from Ref. [234]) and full dots (EELS, from Ref. [236]). Theoretical surface modes are shown as lines. (Reprinted from Ref. [79] with permission. Copyright 1996 by the American Physical Society.)

Table 20 Surface-mode frequencies (in meV) of H/W(110) at selected high-symmetry points H 16.9 16.9

N 19.3

20.6

14.5

S 15.8 16.4

17.4 17.1

17.6

21.4

10.4 11.2 11.0 12.0

15.9

22.2 21.6

DFPT [79] EELS [236] HAS [256] FP [92]

7.2. Surface phonons of (1 × 1)-O/Ru(0001) Oxygen is an example of a strong-binding adsorbate on many transition metal surfaces. The O/Ru(0001) is an especially fruitful system as diKerent well ordered adsorbate layers form depending on the coverage. Their dynamical properties have been the focus of several experimental studies [259,6,5]. Among the diKerent structures, the (1 × 1)-O high-coverage phase plays an important role for several reasons: (i) The dense adlayer maximizes the strong inJuence of the adsorbates on the properties of the .rst substrate layer. The strong binding is reJected in a large outward relaxation by 3.7% of the .rst Ru layer [260], which has to be compared to an inward relaxation of −2:3% for the clean Ru(0001) surface [247]. (ii) The periodicity of the substrate is not changed by the adsorbate overlayer. This simpli.es investigations of the adlayer-induced changes in the substrate dynamics without complications arising from symmetry reduction (backfolding) or substrate reconstruction. (iii) There is a large gap between the frequency range of substrate modes (¡ 35 meV) and of adsorbate modes (¿ 60 meV) because of the small mass ratio MO =MRu and the large dynamical coupling between O and Ru. This allows a clear separation of the substrate and adsorbate dynamics, and simpli.es the identi.cation of its mutual inJuence.

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

Γ

K

Γ

M

250

30

200

20

S5

MS3

150

15

S3

-1

ω (meV)

25

ω (cm )

35

203

100 10

S1

S1 Γ

0

50

M

5

K

M

0

Fig. 18. Phonon dispersion curves of the substrate modes of (1 × 1)-O/Ru(0001). Open symbols indicate the experimental data; theoretical dispersion curves for an asymmetric 50 layer slab are indicated by dashed lines, while .lled circles represent surface modes identi.ed by a weight larger than 20% in the .rst three layers. (After Ref. [85].)

7.2.1. Substrate dynamics A combined HREELS and DFPT investigation of the substrate lattice dynamics of the (1 × 1)-O/Ru(0001) surface has been recently conducted by the group of Menzel and by the present authors [85]. Results for the phonon spectrum are shown in Fig. 18. The theoretical spectrum is obtained from combining ab initio bulk force constants taken from Ref. [239] and surface force constants extracted from ab initio calculations for a symmetric slab consisting of 6 Ru layers and one O layer added on each side. Very good agreement with the HREELS data is observed. The most striking feature in the spectrum is the appearance of a new substrate phonon appearing above the bulk spectrum. It is peeled oK from the bulk bands around the K point and is located 2:3 meV higher than any bulk phonon mode at K. As truncated-bulk calculations do not show any surface phonon located above the bulk bands, this split-oK phonon reJects signi.cantly changed dynamical properties induced by the adsorbate layer. A mode analysis reveals that, although predominantly a substrate mode, also O atoms are involved in the vibration. This indicates some hybridization with the Ru–O bending modes despite the large frequency gap between substrate and adsorbate modes. At K this mode is almost completely localized in the .rst substrate layer and corresponds to circular in-plane movements of the Ru and O atoms. The origin of the high-frequency substrate mode is a giant stiKening of the intralayer force constant in the outermost Ru layer. The averaged nearest-neighbor coupling increases by a factor of 2.5 compared to the bulk, or even by a factor of .ve with respect to the clean Ru(0001) surface (see Section 5.2). The huge increase in the dynamical coupling points to a delicate balance between attractive and repulsive contributions in the Ru–Ru bond which is disturbed by the binding to the O atoms. Signi.cant changes also appear for non-central components of the force constant tensor.

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90

Γ

K

M

Γ

700

650

75

600

-1

80

ω (cm )

ω (meV)

85

70 550 65 500 60

Fig. 19. Phonon dispersion curves of the adsorbate modes of (1 × 1)-O=Ru(0001). Open symbols indicate the experimental HREELS data from Moritz et al. [6,263]. Theoretical dispersion curves indicated by thin lines are shifted by −2:3 meV.

This points to more complex adsorbate-induced modi.cations of the binding properties which involve many-body interactions as well. As a second unusual feature of the phonon spectrum, the authors identi.ed a hardening of the Rayleigh phonon with respect to the truncated bulk result. This behavior contrasts the usual expectation that an outward relaxation of the .rst substrate layer leads to a softening of the RW, as observed for oxygen on Ni(100) [261] or for clean Pd(100) [225]. Earlier theoretical work has shown that the large expansion of the .rst Ru interlayer spacing is the result of a weakening of the attraction between the .rst and second substrate layer due to an oxygen-induced charge removal from the Ru–Ru d states [260,262]. Consistent with this picture, the dynamical coupling between the .rst and second Ru layer weakens by 15%. This by itself would imply a Rayleigh phonon softening. Thus, the unusual increase of the RW frequency is also caused by the intralayer stiKening discussed above, which outweighs the interlayer softening. 7.2.2. Adsorbate dynamics Early investigations of the (1×1)-O=Ru(0001) surface found a peculiar soft vibrational mode which was explained as a soft phonon mode induced by the large outward relaxation of the .rst layer [259]. However, this mode could not be con.rmed by a subsequent HREELS study [6] questioning this interpretation. For the high-symmetry (1 × 1)-O overlayer, the adsorbate spectrum consists of three branches: a vertical O-Ru stretching vibration, and two bending modes with motions parallel to the surface. The dispersion curves are shown in Fig. 19. No low-lying adsorbate mode is found in agreement with the HREELS results. The theoretical dispersion agrees well with experiment except for a small overestimation of the overall scale by ≈3% (accounted for by a small shift of −2:3 meV of the theoretical curve in Fig. 19). The large dispersion of the upper branch, which corresponds to the vertical stretching mode, points to a signi.cant coupling among the vertical motions of the adsorbate atoms. For well separated adsorbates, the dominant interaction is given by the dynamical dipole–dipole (DDD) interaction [264]. A .t of their data for the dense (1 × 1)-O phase to the DDD model has been performed by

R. Heid, K.-P. Bohnen / Physics Reports 387 (2003) 151 – 213

2

3 1

0

205

2 1

3

Fig. 20. Schematic picture of the nearest neighbor shells of O atoms at (1 × 1)-O/Ru(0001).

Moritz et al. and is shown as dotted line in Fig. 19. It provides a very good description for the dispersion along VK, but fails to account for the anisotropy between the K and the M point. From the DFPT calculation, two diKerent sources for this discrepancy could be identi.ed. (i) The high frequency at M is partly caused by a mode coupling of the stretching mode to the upper bending mode. This mode hybridization is essentially absent along VK, but increases towards M and causes an increased splitting between the two modes. From the theoretical eigenvector, one can deduce a hybridization-induced increase of about 2 meV of the upper mode frequency at M with respect to the uncoupled stretching mode. (ii) The remaining discrepancy between DFPT calculation and DDD model results from a significant anisotropy of the short range coupling (the geometry of the O layer is sketched in Fig. 20). While the coupling to the second-neighbor O atom is small in both calculations, the ab initio analysis reveals a large coupling to the third-neighbor shell. In the DDD model the interaction strength decreases monotonically with increasing distance. Thus the observed large dispersion along the KM line indicates the presence of additional short-ranged interactions beyond the DDD picture. 8. Concluding remarks and outlook In this report we have reviewed the state-of-the art of density functional theory (DFT) calculations of lattice dynamical properties of metal surfaces. We have mainly focused on its most advanced technique, density-functional perturbation theory (DFPT). DFPT allows the calculation of surface phonons for the entire surface Brillouin zone, and therefore constitutes a powerful tool to obtain microscopic information of the lattice interaction and of the bonding properties. In recent years, the development of new DFPT implementations has laid the foundation for a wider application range including surfaces of noble and transition metals. In metals, the bonding of atoms and thus the dynamical couplings depend sensitively on the screening properties of the delocalized electrons. The applications discussed in this report demonstrate

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that the DFT approach can predict this electronic screening occurring at a surface with high accuracy for various types of metals ranging from simple sp-bonded metals to noble metals and to transition metals. As shown in Section 7, this predictive power is not limited to the case of clean metal surfaces, but the inJuence of adsorbates on the substrate phonons as well as the dynamics of the adsorbate layer itself are also well accounted for by this theory. One advantage of the DFT approach is that it provides a common framework for both structural and lattice dynamical properties. This enables an unbiased study of the interplay between local structural changes and dynamical couplings, which is crucial for an understanding of the modi.cations occurring at a surface. A nice example is given by the noble metal surfaces in Section 3, where a correlation between the relaxation induced bond length and the force constant could be established. Within the limitations discussed in Section 2, the DFT-based schemes represent an ab initio approach, which does not depend on material-speci.c parameters. This is a signi.cant improvement over model approaches to lattice dynamics, which have to resort to various phenomenological parameters determined by .tting experimental data or by using ad hoc assumptions. Often the choice of a parameter set is not unique. An example is the long-standing problem of the apparent large softening of longitudinal couplings on surfaces of noble and transition metals. These were suggested on the basis of model .ts to the longitudinal resonance observed in Helium atom scattering (HAS) experiments. The DFPT results presented in Section 3.3 clearly contradict this conjecture and point to a diKerent interpretation of the experimental features as being partly caused by an interference eKect of the scattering process. The rather complete and accurate information on the phonon spectrum and on the interatomic couplings provided by DFPT is a prerequisite for the determination of thermodynamic properties. First results reported for the thermal expansion of compact surfaces like Ag(111) and Be(0001), as H surfaces well as for the more complex case of multilayer thermal expansion in case of the (1010) of Be and Mg, con.rm the usefulness of this approach. Further promising applications of this kind are to be expected in the future. The reliable calculation of surface mode eigenvectors further opens the way for a more accurate simulation of inelastic scattering intensities, which is pertinent for a proper interpretation of experimental HAS or EELS spectra. With respect to the scattering by helium atoms, however, a detailed knowledge of the interaction potential is still lacking. Additional insight from an initio calculations would be highly desirable. All applications of DFPT discussed in this report were concerned with simple surface structures, as treatment of larger unit cells is still a formidable task. With advancements in computer technology, calculations for more complex surfaces, such as reconstructed surfaces or adsorbate phases with submonolayer coverage, will become manageable in the future. A .rst step beyond low-indexed surfaces are vicinal surfaces, where the introduction of periodic arrangements of steps and terraces allow for the study of local vibrational properties of atoms with diKerent bonding environments. Currently, this problem has been tackled almost exclusively with methods based on semi-phenomenological potentials [34–36], with the only exception of an ab initio frozen-phonon calculation of the VH phonons for the vicinal Cu(211) surface [265]. Another fruitful research .eld is given by structural transitions at surfaces, either driven by temperature or induced by adsorbates. Here, detailed ab initio calculations could help to clarify the role played by the phonons for these structural instabilities. Finally, another direction for future research is related to the electron–phonon coupling at metallic surfaces. The DFPT can easily be extended to obtain microscopic information about the

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electron–phonon matrix elements [266,267], which has been successfully applied to various bulk materials [268–270,239,271]. Transferring this concept to surface problems may help to gain more insight into recent photoemission results, which indicted that surface electronic states can exhibit couplings to the phonon system which are signi.cantly enhanced with respect to the bulk [272–274]. Such an analysis could also shed more light on the nature of Fermi-surface driven (Kohn-type) anomalies of surface-phonon branches as observed for the layered metal 2H-TaSe2 (001) [240] or for adsorbate systems of W(110) and Mo(110) [234–237,275,276]. Acknowledgements We are thankful to B. Meyer, C. Els2asser and M. F2ahnle for providing us with the newly written mixed-basis ground-state code, which greatly facilitated the programming of the perturbation scheme, and for enlightening discussions on the technical aspects of the method. We would like to thank W. Widdra and his group for valuable discussions and for providing us their EELS data for the adsorbate modes of O/Ru(0001). We are also thankful to Y. Nourani for her careful reading of the manuscript. References [1] G. Brusdeylins, R.B. Doak, J.P. Toennies, Phys. Rev. Lett. 44 (1980) 1417. [2] H. Ibach, D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations, Academic Press, New York, 1982. [3] S. Lehwald, J.M. Szeftel, H. Ibach, T.S. Rahman, D.L. Mills, Phys. Rev. Lett. 50 (1983) 518. [4] J.P. Toennies, in: W. Kress, F.W. de Wette (Eds.), Surface Phonons, Vol. 21: Springer Series in Surface Sciences, Springer, Berlin, Heidelberg, 1991, p. 111 (Chapter 5). [5] K.L. Kostov, M. Gsell, P. Jakob, T. Moritz, W. Widdra, D. Menzel, Surf. Sci. 394 (1997) L138. [6] T. Moritz, D. Menzel, W. Widdra, Surf. Sci. 427– 428 (1999) 64. [7] W. Kress, in: W. Kress, F.W. de Wette (Eds.), Surface Phonons, Vol. 21: Springer Series in Surface Sciences, Springer, Berlin, Heidelberg, 1991, p. 209 (Chapter 8). ∗ [8] R.F. Wallis, S.Y. Tong, in: G. Chiarotti (Ed.), Landolt-B2ornstein: Physics of Solid Surfaces, Vol. 24b, New Series, Group III, Springer, Berlin, 1994, p. 433 (Chapter 4.1). ∗ [9] M.A. Rocca, in: H.P. Bonzel (Ed.), Landolt-B2ornstein: Physics of Covered Solid Surfaces, Vol. 42A2, New Series, Group III, Springer, Berlin, 2002, p. 352 (Chapter 4.5). ∗ [10] V. Celli, in: W. Kress, F.W. de Wette (Eds.), Surface Phonons, Vol. 21: Springer Series in Surface Sciences, Springer, Berlin, Heidelberg, 1991, p. 167 (Chapter 6). [11] B. Gumhalter, Phys. Rep. 351 (2001) 1. [12] C.H. Li, S.Y. Tong, D.L. Mills, Phys. Rev. 21 (1980) 3057. [13] S.Y. Tong, C.H. Li, D.L. Mills, Phys. Rev. Lett. 44 (1980) 407. [14] D.L. Mills, S.Y. Tong, J.E. Black, in: W. Kress, F.W. de Wette (Eds.), Surface Phonons, Vol. 21: Springer Series in Surface Sciences, Springer, Berlin, Heidelberg, 1991, p. 193 (Chapter 7). [15] J. Fritsch, U. Schr2oder, Phys. Rep. 309 (1999) 209. ∗ ∗ ∗ [16] R.E. Allen, F.W. de Wette, Phys. Rev. 179 (1969) 873. [17] R.E. Allen, F.W. de Wette, A. Rahman, Phys. Rev. 179 (1969) 887. [18] S.Y. Tong, A.A. Maradudin, Phys. Rev. 181 (1969) 1318. [19] R.E. Allen, G.P. Alldredge, F.W. de Wette, Phys. Rev. B 4 (1971) 1648. ∗ ∗ ∗ [20] R.E. Allen, G.P. Alldredge, F.W. de Wette, Phys. Rev. B 4 (1971) 1661. ∗ ∗ ∗ [21] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285.

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CONTENTS VOLUME 387 Y. Yanase, T. Jujo, T. Nomura, H. Ikeda, T. Hotta, K. Yamada. Theory of superconductivity in strongly correlated electron systems

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R. Heid, K.-P. Bohnen. Ab initio lattice dynamics of metal surfaces

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doi:10.1016/S0370-1573(03)00388-0