Unconventional Superconductors: Anisotropy and Multiband Effects

Springer Series in

materials science

153

Springer Series in

materials science Editors: R. Hull C. Jagadish R.M. Osgood, Jr. J. Parisi Z. Wang The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series ref lect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials.

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Iman Askerzade

Unconventional Superconductors Anisotropy and Multiband Effects

With 84 Figures

123

Iman Askerzade Center of Excellence of Superconductivity Research of Turkey Ankara University, 06100, Tandogan, Ankara ˘ and Institute of Physics, Azerbaijan National Academy of Sciences Baku-Az 1143, Azerbaijan [email protected]

Series Editors:

Professor Robert Hull

Professor J¨urgen Parisi

University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Universit¨at Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Straße 9–11 26129 Oldenburg, Germany

Professor Chennupati Jagadish

Dr. Zhiming Wang

Australian National University Research School of Physics and Engineering J4-22, Carver Building Canberra ACT 0200, Australia

University of Arkansas Department of Physics 835 W. Dicknson St. Fayetteville, AR 72701, USA

Professor R. M. Osgood, Jr. Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

Springer Series in Materials Science ISSN 0933-033X ISBN 978-3-642-22651-9 e-ISBN 978-3-642-22652-6 DOI 10.1007/978-3-642-22652-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941193 © Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To fatherland Garabag and my family

•

Preface

The epoch of high-Tc superconductivity began in 1986 when Bednorz and Muller found evidence for superconductivity at Tc  30 K in La–Ba–Cu–O ceramics. This remarkable discovery has renewed the interest in superconductive research. The late discovery of superconductivity in rare earth transition metal borocarbides in 1994 is still an intensive debate with respect to unusual features not observed for conventional superconductors. In the 2001 year, the discovery of superconductivity in MgB2 initiated an immediate broad research activity due to the high transition temperature Tc  40 K in a seemingly ordinary s  p metal. Apart from the high transition temperature of 40 K, two-band superconductivity was the other unexpected phenomenon in MgB2 which attracts increasing attention. In fact, at present it appears that MgB2 is the only superconductor with substantiated theoretical and experimental evidence for two-band superconductivity. In 2008, the discovery of a new family of high critical temperature iron and arsenic superconductors (AsFe) marked a new major revolution in the world of superconductivity. The new compounds, which do not contain copper (Cu) but which have oxygen (O), fluor (F) or arsenic (As), and iron (Fe) will help scientists to solve some of the mysteries in the area of solid-state physics. These compounds reveal many properties similar to high-Tc cuprates, and at the same time superconducting state has multiband character, likewise, to nonmagnetic borocarbides and MgB2 : Fortunately, the experimental investigations revealed a great variety of “exotic” physical properties in the above-presented compounds such as multiband and anistropic effects in superconducting state. Detailed comparison of the available data for new class of superconductors, especially with the high-Tc cuprates, might be helpful to improve our present incompetent understanding of challenging novel members of the rich and rapidly growing family of superconductors. This book deals with the new class of materials unconventional superconductors – cuprate compounds, borocarbides, magnesium diboride, and oxypnictides. It gives a systematic review of physical properties of novel superconductors. There is an increasing number of fundamental properties of these compounds which are relevant to future applications, opening new possibilities. The layout of this book consists of four chapters. Chapter 1 is devoted to the description of physical vii

viii

Preface

properties of newly discovered superconductors: cuprate superconductors, borocarbides, magnesium diboride, oxypnictides. We present briefly crystal structure, electronic properties, and related theoretical models for each group of superconductors. Anistropy and multiband effects are specially emphasized. Well-known and generally accepted results of computed Fermi surface of these compounds are presented. Results of order parameter symmetry investigations in this compounds are discussed. Chapter 2 gives a generalization of Ginzburg–Landau (GL) theory to the case of multiband and anisotropic superconductors. It is noted that single-band GL calculations were found to be inadequate for describing the temperature dependence of fundamental physical properties of these compounds, while the two-gap model was found to be successfully applied to determine the temperature dependence of superconducting state parameters in bulk nonmagnetic borocarbides MgB2 ; LuNi2 B2 C, and YNi2 B2 C. Presence of two-order parameters and their coupling play a significant role in determining its temperature dependence. The results of the calculations are in good agreements with experimental data for bulk nonmagnetic borocarbides and magnesium diboride. We also conclude that the two-band GL theory explains the reduced magnitude of the specific heat jump and the small slope of the thermodynamic magnetic field at critical temperature in MgB2 . It is shown that the relation between upper critical field and so-called surface critical field is similar to the case of single-band superconductors. Temperature dependence of surface critical field of two-band superconductors must give positive curvature. Quantization of magnetic flux in the case of two-band superconductors remains the same as in single-band superconductors. However, Little-Parks oscillations of Tc in two-band superconductors is different from one band case. The generalization of two-band GL theory to the case of layered anisotropy is presented. We have calculated anisotropy parameter of upper critical field Hc2 and London penetration depth  for magnesium diboride single crystals. The temperature-dependent anisotropy of upper critical field is shown, which in agreement with the experimental data for MgB2 and reveal opposite temperature tendency to anisotropy parameter of . Angular effects on the base two-band GL theory are also studied. Structure of single vortex in layered two-band superconductors is presented. Influence of dirty effects in two-band superconductors using GL approach is also included in this chapter. Results of investigation of upper critical fields, single vortex of d-wave superconductors, and nonlinear magnetization in d C s wave superconductors using d -wave version of GL theory are the scope of the Chap. 2. Very recent application of GL-like theory to cuprate superconductors for the calculation fundamental parameters is presented. Time-dependent two-band isotropic GL equations and corresponding (d Cs)-wave equations were applied for the study of vortex nucleation and dynamics effects. Briefly discussed is the very recent exciting discussion of 1.5 type superconductivity in literature. How we can introduce coexistence of superconductivity and antiferromagnetism in the framework of GL theory is also presented. Finally, application of GL equations to nanosize superconductors and possible new effects of vortex nucleation in mesoscopic superconductors is briefly discussed at the end of this chapter.

Preface

ix

In Chap. 3, we have summarized a number of recent investigations of layered superconductors using the microscopic electron–phonon Eliashberg theory. The critical temperature of layered superconductors was calculated using this theory, and the influence of nonadiabaticity effects on the critical temperature was considered. In the calculation of the effect of Coulomb repulsion on the critical temperature, arbitrary thicknesses of conducting layers were also taken into account. In the same approach, expression for the plasmon spectrum of layered superconductors with arbitrary thicknesses of the conducting layers was obtained. In addition, Bardeen– Cooper–Schrieffer (BCS) equations for layered superconductors were used for calculating the specific heat jump, which is smaller than in the isotropic case. The results are shown to be in qualitative agreement with experimental data for cuprate superconductors and the recently discovered MgB2 compound. Properties of two-band superconductors in isotropic (BCS) theory are investigated. The critical temperature, specific heat and upper critical field, influence of impurity, and doping effects on these parameters in two-band isotrop superconductors are considered. For the general outstanding nature of superconductivity in cuprates briefly presented results of d -wave BCS theory. The influence of nonadiabatic effects in two-band strong coupling superconductors is taken into account. Calculation of the spectrum of collective Legget mode in two-band superconductors and related experimental data is included in Chap. 3. Finally, nanosize two-band superconductors in framework of BCS theory and related results are considered. The last chapter of the book, Chap. 4, is devoted to fluctuation effects in new superconductors. There is an excellent book of Larkin and Varlamov about manifestation of fluctuation in isotropic and strong anisotropic superconductors. Here we study the fluctuation effects on specific heat in two-band superconductors taking the influence of external magnetic field into account. Diamagnetic susceptibility and fluctuation of conductivity neat Tc are calculated using two-band GL theory in application to new superconductors. Fluctuation of phase effects in layered superconductors on Tc studied using Lawrence–Doniach functional. Influence of post Gaussion fluctuation in superconductors is also considered. Finally, we present generalized GL theory for layered superconductors with small coherence lenght. I thank Professors of Physics F.M. Hashimzade, B.M. Askerov, A. Gencer, B. Tanatar, and R.R. Guseinov for the useful discussions, and Prof. Dr. SL Drechsler for initiation of study of many band effects in superconductors. I would like to acknowledge the support of Prof. Dr. C. Ascheron, Prof. Dr. F. Mikailzade and Prof. Dr. G.G. Huseinov in the stage of printing of book. Thanks go to Dr. Mehmet Canturk for helping in preparation of manuscript. I want to remember our colleague and friend, late Dr. Niyazi B. Gahramanov. I am grateful to the Abdus Salam ICTP and IFW Dresden for the hospitality during my stay as an associate member and financial support of NATO Post-Doc grant in Bilkent and Ankara University, NATO reintegration grant 980766, CNRS-ANAS grant No UNR5798, TUB˙ITAK No 104T522, 110T748 during the last years. Ankara, Baku, September 2011

Iman N. Askerzade

•

Contents

1

Physical Properties of Unconventional Superconductors .. . . . . . . . . . . . . . . 1.1 Cuprate Superconductors .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Atomic Structure and Classification . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Theoretical Models .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Borocarbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Crystal Structure and Tc . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 The Electronic Structure . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Magnesium Diboride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Crystal Structure.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Electronic and Band Structure . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Oxypnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Atomic Structure and Classification . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Electronic and Band Structure . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 GL Equations for Two-Band Isotropic Superconductors . . . . . . . . . . . . 2.1.1 Upper Critical Field Hc2 .T / . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Surface Magnetic Field Hc3 (T) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 Lower Critical Field Hc1 .T / . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.4 Upper Critical Field Hc2 of Thin Films . .. . . . . . . . . . . . . . . . . . . . 2.1.5 Magnetization of Two-Band Superconductors Near Hc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.6 Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.7 Little-Parks Effect in Two-Band Superconductors.. . . . . . . . . . 2.1.8 Thermodynamic Magnetic Field Hcm .T / and Specific Heat Jump C . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . CN 2.1.9 Critical Current Density jc .T / . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Anisotropy Effects in Two-Band GL Theory .. . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Layered Two-Band GL Equations .. . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Anisotropy Effect on Upper Critical Field in Layered Two-Band Superconductors ... . . . . . . . . . . . . . . . . . . .

1 1 2 6 10 13 13 15 16 17 19 21 24 27 28 30 34 37 39 41 42 42 43 45 47 47 49 xi

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Contents

2.3

2.4

2.5 2.6

2.7 2.8

2.2.3 Effects of Anisotropy on London Penetration Depth . . . . . . . . 2.2.4 Single Vortex in Two-Band Layered Superconductor . . . . . . . 2.2.5 Surface Magnetic Field in Anisotropic Two-Band GL Theory .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.6 Angular Effects in Two-Band GL Theory .. . . . . . . . . . . . . . . . . . . d-Wave GL Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Upper Critical Field of d-Wave Superconductors .. . . . . . . . . . . 2.3.2 Single Vortex in d-Wave Superconductor . . . . . . . . . . . . . . . . . . . . 2.3.3 Nonlinear Magnetization in d C s Wave Superconductors .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Vortex Lattice in d-Wave Superconductors . . . . . . . . . . . . . . . . . . GL-Like Theory in Application to Cuprate Superconductors . . . . . . . 2.4.1 Transition Temperature Tc .x/ . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 Superfluid Density. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Specific Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . GL Theory of Dirty Two-Band Superconductors . . . . . . . . . . . . . . . . . . . . Time-Dependent Two-Band GL Equations .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 Two-Band s-Wave Superconductors .. . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Time-Dependent d-Wave GL Equations .. . . . . . . . . . . . . . . . . . . . 2.6.3 1.5 Type Superconductivity .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Coexistence Antiferromagnetism and Superconductivity: GL Description .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Application of GL Equations to Nanosize Superconductors . . . . . . . .

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Single-Band Isotropic Eliashberg Equations and BCS Limit .. . . . . . . 3.2 Eliashberg Equations for Single Band Layered Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Eliashberg Equations for Two-Band Isotropic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Effects of Nonadiabacity in Layered Systems. . . .. . . . . . . . . . . . . . . . . . . . 3.5 Effect of Coulomb Repulsion in Layered Single-Band Superconductors with Arbitrary Thickness of Layers . . . . . . . . . . . . . . . 3.6 Plasmon Spectrum of Layered Superconductors .. . . . . . . . . . . . . . . . . . . . 3.7 Specific Heat Jump of Layered Superconductors . . . . . . . . . . . . . . . . . . . . 3.8 d -Wave Single Band BCS Theory .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.1 Critical Temperature . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.2 Specific Heat of Two-Band Superconductors . . . . . . . . . . . . . . . . 3.9.3 Upper Critical Field Hc2 in Two-Band Superconductors . . . . 3.10 Nanosize Two-Gap Superconductivity . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.11 Effect of Nonadiabacity in Two-Band Superconductors .. . . . . . . . . . . . 3.12 Leggett’s Mode in Two-Band Superconductors ... . . . . . . . . . . . . . . . . . . .

51 53 56 57 62 64 66 68 69 71 73 73 74 75 77 77 82 84 86 88 95 96 98 103 106 111 115 119 122 124 125 128 130 134 136 138

Contents

4 Fluctuation Effects in Anisotropic and Multiband Superconductors .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Fluctuations of Specific Heat in GL Theory.. . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Single-Band Isotropic GL Theory . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Fluctuation of Specific Heat in Two-Band Isotropic Superconductors . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 Influence of External Magnetic Field on Specific Heat Jump.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Diamagnetic Susceptibility T >Tc . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Fluctuation of Conductivity Near Tc . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Fluctuation Effects in Layered Superconductors .. . . . . . . . . . . . . . . . . . . . 4.4.1 Influence Phase Fluctuations on Critical Temperature in Layered Superconductors.. . . . . . . . . . . . . . . . . . . 4.5 Post-Gaussian Fluctuations in Superconductors .. . . . . . . . . . . . . . . . . . . . 4.6 GL Theory for Layered Superconductors with Small Coherence Length.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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141 141 141 143 145 147 149 151 153 155 157

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 159 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173

•

Chapter 1

Physical Properties of Unconventional Superconductors

The introduction part of the book is devoted to the description of physical properties of advanced classes of superconductors: cuprate superconductors, borocarbides, magnesium diboride, and oxypnictides. Description of crystal structure, electronic properties, and related theoretical models for each group of superconductors is presented. Anisotropy and multiband effects are specially emphasized. Well-known and generally accepted results of computed Fermi surface of these compounds and Fermi liquid properties are included to first part. Results of order parameter symmetry, its relation with possible pairing mechanism, and general consequences for experimentally accessible properties investigations in this class of superconductors are presented. Different vortex states in advanced classes of superconductors and experimental results of related phenomena are also considered. The rich variety of superconducting compounds in the advanced class of superconductors and the growing knowledge of their electronic structure can be helpful to clarify differences and similarities with other exotic superconductors.

1.1 Cuprate Superconductors In more than 24 years elapsed from the discovery of high Tc superconductivity in cooper oxides [1], a huge number of experimental and theoretical investigations of the physical properties of these materials have been done. The high-temperature superconductivity of cuprates was discovered in 1986 [1], when the highest superconducting transition temperature (i.e., critical temperature) characteristic of conventional superconductors (Tc D 23:2 K in Nb3 Ge) was substantially exceeded and a superconducting transition temperature Tc D 30 K was achieved in the ceramic La2x Bax CuO4ı . Within a year after this discovery, the record value of Tc exceeded 90 K (YBa2 Cu3 O7ı ceramic). The further search for and creation of new superconducting materials led to Tc D 138 K (Tl-doped HgBa2 Ca2 Cu3 O8ı compound) in 1994 and raised the question of room-temperature superconductivity. In spite of all these efforts, the mechanism of this new kind of superconductivity I. Askerzade, Unconventional Superconductors, Springer Series in Materials Science 153, DOI 10.1007/978-3-642-22652-6 1, © Springer-Verlag Berlin Heidelberg 2012

1

2

1 Physical Properties of Unconventional Superconductors

has not been clarified yet; it still remains one of the most enigmatic problems of the solid-state physics [2–4]. The description of their normal state properties [5, 6] has turned out to represent an even bigger challenge to solid-state physics theory. The difficulty of this problem is due to the complicated properties, including complicated crystalline structures of materials displaying high Tc , to the presence of a strong anisotropy, to the existence of non-adiabatic effects, to strong electronic correlations, and to a strong electron–phonon interaction. In these complicated materials, several phase transitions (structural, magnetic, superconductor, etc.) occur, and mixed states are allowed, for instance coexistence of superconductivity and ferromagnetism or vitreous spin state. The key structural element of layered quasi-two-dimensional cuprates is a (CuO2 ) plane (one or several in a unit cell); they differ from conventional superconductors not only in high values of Tc but also in a set of physical properties that cannot be described by the classical Bardeen–Cooper–Schriefffer (BCS) [7] scheme. In cuprates, charge carriers appear due to the doping of the CuO2 planes of a parent antiferromagnetic insulator upon nonisovalent atomic substitution or the creation of oxygen vacancies in charge reservoirs outside the conducting planes. The distance between equivalent CuO2 planes in neighboring unit cells is large compared to the in-plane distance between neighboring copper atoms, which results in a strong conductivity anisotropy at temperatures above Tc and the two-dimensional coherence of the superconducting state at temperatures below Tc .

1.1.1 Atomic Structure and Classification CuO2 planes plays a crucial role in superconducting in cuprate compounds. As shown in Table 1.1 crystal structure of cuprate compounds basically is tetragonal. Highest critical temperature reached in cuprates having flat and square CuO2 planes. The CuO2 planes in cuprates separated by the atoms as Bi, O, Y, Ba, La, etc. which plays role of charge reservoirs. In contrast to low temperature conventional

Table 1.1 Crystal structure and elementary cell of some cuprates [11] Compounds La2x Srx C uO4 .La=Sr  214/ YBa2 C u3 O7 .123/ BiSr2 CaC u2 O8 .Bi  2212/ T lBa2 Ca2 C u3 O9 .T l  1223/ T l2 Ba2 C uO6 .T l  2201/ T l2 Ba2 CaC u2 O8 .T l  2212/ T l2 Ba2 Ca2 C u3 O10 .T l  2223/ HgBa2 Ca2 C u3 O8 .Hg  1223/ HgBa2 Ca3 C u4 O10 .Hg  1234/

Crystal structure Tetragonal Orthorhombic Tetragonal Tetragonal Orthorhombic Tetragonal Tetragonal Tetragonal Tetragonal

Size of elementary cell,A

Tc ; K

a D b D 3:78; c D 13:2 a D 3:82; b D 3:88; c D 11:7 a D b D 5:4; c D 30:89 a D b D 3:85; c D 15:9 a D 5:468; b D 5:473; c D 23:24 a D b D 3:86; c D 29:3 a D b D 3:85; c D 35:9 a D b D 3:85; c D 15:9 a D b D 3:85; c D 19

37:5 90 95 120 90 112 125 133 127

1.1 Cuprate Superconductors

3

Fig. 1.1 Phase diagram of LaSrCuO (LSCO). As one can see in Fig. 1.1, Sr substitution for La in LSCO induces a structural phase transition from the high-temperature tetragonal (HTT) to low-temperature orthorhombic (LTO) and, at low temperatures, from the LTO phase to the low-temperature tetragonal (LTT) phase

superconductors, influence of carrier density on critical temperature Tc .n/ reveal nonmonotonic behavior [8]. There is an empirical expression of Tc .n/ dependence ˚  Tc .n/ Š .Tc /max 1  82:6.n  0:16/2 ;

(1.1)

where .Tc /max is the maximum critical temperature for certain compound. Superconductivity occurs within the interval 0:05  n  0:27, which vary slightly in different cuprates. In LaSrCuO compound at doping x D 18 , the curve Tc .n/ has a dip (Fig. 1.1). This dip is the so-called 18 anomaly and inherent to La2x Srx CuO4 . The insulating phase occurs at n  0:05 and usually called as an undoped region. The maximum critical temperature is observed at optimal doping x  0:16 [9]. The most prominent compound YBa2 Cu3 O7ı , the first high temperature superconductor discovered with a critical temperature Tc for the onset of superconductivity above the boiling point of liquid nitrogen, is traditionally abbreviated as “YBCO” or “Y  123”(YBa2Cu3 O7ı /. The orthorhombic unit cell of YBCO is presented in Fig. 1.2. The two CuO2 layers are separated by a single Y atom. The replacement of yttrium by many of the lanthanide series of rare-earth elements causes no appreciable change in the superconducting properties. Each copper ion is surrounded by a pyramid of five oxygen ions. YBCO is the only high temperature superconductor having the one-dimensional CuO chains. In YBa2 Cu3 O6 CuO2 chains are absent and this compound is the antiferromagnetic insulator [10]. So, oxygen content can be changed reversibly from 6.0 to 7.0 simply by pumping oxygen in and out of the parallel CuO chains running along the b axis. At low oxygen content, the lattice parameters a ¤ b (orthorhombic), while with increasing the oxygen content causes the unit cell to have square symmetry, a D b. At oxygen content of 6.4, the antiferromagnetic long-range order disappears and superconducting order parameter starts to grow. The maximum value of Tc is achieved at a doping level of about 6.95. From Fig. 1.3, we can see that at x  6:7

4

1 Physical Properties of Unconventional Superconductors

Fig. 1.2 Crystal structure of YBaCuO

Fig. 1.3 Phase diagram of YBaCuO

there is plateau at Tc  60K: One of the explanations of origin of this plateau is direct relation with 18 anomaly observed in LaSrCuO. In Bi 2212 in unit cell the CuO2 layer intercalated by C a. The unit cell also contains two semiconducting BiO and two insulating S rO layers. The family of the bismuth cuprate consists of three members: Bi 2202; Bi 2212, and Bi 2223 with the unit having 1, 2, and 3 CuO2 layers, respectively. Critical temperature increases with increasing number of CuO2 layers. The structure of the bismuth cuprates is very similar to the structure of thallium cuprates such as T l2202, T l2212, and T l2223. In these compounds, bismuth replaced by the thallium, and strontium replaced by barium. Bismuth, thallium, and mercury cuprates have the lattice constants a D b, there is no twinning within a crystal. In the Table 1.2 presented superconducting parameters of optimally doped cuprates: the coherence length ab;c and London penetration depth ab;c : As followed from the Table 1.2, cuprate superconductors reveals strong layered anisotropy of physical properties. In future investigations also important of introducing of

1.1 Cuprate Superconductors Table 1.2 Physical properties of cuprates [11]

5

Physical quantity ab c ab c

LSCO,A 33 2.5 2,000 20,000

YBCO,A 15 2 1,450 6,000

B2212,A 20 1 1,800 7,000

Hg1223,A 13 2 1,770 30,000

T QSS

QSS + BS

BS

T* TN

wPG

FL T *s sPG

Tc SC

AF x*

x opt

x*

x

Fig. 1.4 Generalized phase diagram of the hole doped cuprate superconductors

H

ab

anisotropy parameter of upper critical field Hc2 D Hc2; : As shown by measurec2I c ments for cuprate superconductors Hc2 changes in the interval 3–30 [11]. In the absence of an external magnetic field, the thermodynamic state of a doped cuprate compound can be described by the temperature Tc and the carrier concentration in the CuO2 plane (doping level) x. In the phase diagram (Fig. 1.4), the superconducting state field corresponds to a certain doping range x < x < x  inside which the superconducting transition temperature reaches its maximum value at the optimum doping xopt . Concentrations x  xopt correspond to underdoped cuprates, and concentrations x  xopt to overdoped cuprates. At x  xopt and T > Tc , cuprates are “bad” Fermi liquids, and at x  xopt ; over a wide temperature range Tc < T < Ts , they exhibit the pseudogap state, whose nature is unclear up to now [13]. The gap spectrum of quasiparticles at T > Tc demonstrates that the superconducting phase appears from a certain insulating state rather than from a Fermi liquid, such that the ground states of an insulator and a superconductor, with similar structures and energies, converge near the superconducting transition line. This is indication of that strong correlations in cuprates is very important [14]. The pseudogap state is divided into a strong pseudogap that is adjacent to Tc and exists over a wide temperature range Tc < T < Ts , and a weak pseudogap between Ts and T  : Ts corresponds to the breaking of a pair, and Tc corresponds to the appearance of phase coherence in the system of pairs. A consistent theory

6

1 Physical Properties of Unconventional Superconductors

of cuprate superconductivity should be able to explain both the high values of Tc and the physical properties of these compounds in a large neighborhood of the superconducting state that includes the strong and weak pseudogaps in the phase diagram.

1.1.2 Theoretical Models Theoretical models for the explanation origin of superconductivity in cuprate superconductors can be divided into two groups. First group of models supports the origin of superconducting state from dielectric state [3, 4, 15–18]. Second group of theoretical approaches related with modification of classical electron– phonon coupling mechanism taking into account peculiarities of superconducting state in cuprate compounds [19, 20]. Strong electron correlations and the unusual symmetry of the pseudogap and the superconducting order parameter in cuprates are arguments for a purely electron superconductivity mechanism (rather than a phonon mechanism, as in the BCS or Eliashberg theory). The studies of this mechanism Hubbard model and the related t J model are described in a number of reviews (for example, see [18] and references therein). The two-dimensional Hubbard problem has not been exactly solved and approximate solutions obtained by numerical methods are often in conflict, which leads to reasonable doubts about the usefulness of this approach, especially because the unusual isotopic effect in cuprates [21–23] indicates a nontrivial role of phonons in pairing-interaction formation. Another important moment in cuprate compounds is the growing of superconducting fluctuations [24]. Strong fluctuations can be described in framework of approach of the resonating valence bond [25, 26]. In accordance with resonating valence bond theory, the electrons which possess the opposite spins and located in the neighboring sites of the crystal lattice form pairs. At fixed temperature with decrease of the doping degree pairs are ordered in lattice and the state becomes antiferromagnetic. Vice versa, when the doping is fixed and temperature decreases, the pairs, being Bose particles, are condensed and the state becomes superconducting. The pseudogap manifests itself in the processes where the pairbreaking takes places. The Hamiltonian of Hubbard model has a form [15, 16]: HHubbard D

X

tij .ciC" ci " C ciC# ci # / C U

ij

X .ciC" ciC# ci " ci # /;

(1.2)

i

where U Coulomb interaction in one site, tij tunneling integral between neighbor atoms, c C and c creation and annihilation operators. Corresponding Hamiltonian for t  J model can be written as [25] Ht J D

X i

"0 ni  

1X J.Siz SizCı  ni ni C /; 2 i;i C

(1.3)

1.1 Cuprate Superconductors

7

where Siz projection operator of spin, J effective exchange integral. Despite of the simple form of the Hamiltonians (1.2) and (1.3), the quantitative results allowing the experimental verification still have not been obtained in their framework. Probably, the t  J model or of its modifications can describe the complete phase diagram presented in Fig. 1.4. No analytical solution of t  J model still does not find and the characteristic value of J and x, separating different domains of the phase diagram, can be found only numerically. For the value J D 0:3t numerical calculations was conducted in [27], and it was shown that the antiferromagnetic region restricted as x < 0:1: Some properties of the t  J model or Hubbard Hamiltonian ground state can be found by means of the variational prosedure. In this way, the d-wave symmetry of the superconducting pairing was successfully obtained in [27,28]. Calculated in this way order parameter remains finite even at zero doping while the superfluid density ns depends of the doping degree linearly [29]. In the typical phase diagram of hole-doped cuprates, the Neel (TN ) and superconducting transition (Tc ) temperatures, respectively, bound the long-range antiferromagnetic and superconducting order regions. Strong pseudogap (sPG) and weak pseudogap (wPG) regions are separated by a crossover temperature Ts . The temperature T separates the weak pseudogap from the normal Fermi liquid (FL). The regions in which the bound states (BSs) and quasi-stationary states (QSSs) of pairs appear are shown, and the region of coexisting BS and QSS is also depicted. In the case of overdoped and optimally doped high temperature superconductors, BCS type model can be useful for study cuprate superconductors. The paramagnon exchange between electrons results in their pairing. In contrast to the electron– phonon interaction in original BCS scheme, the electron (hole)–paramagnon interaction may be not weak. This leads to modification of BCS theory. The effect of soft paramagnons, with the characteristic energies small with respect to the pseudogap, on superconducting properties of high temperature superconductors became of the subject of work [30]. As shown in this study, influence of paramagnons is analogous to the effect of elastic impurities, which leads to the superconductivity with strongelectron–phonon coupling. Comparison paramagnon approach with the Migdal–Eliashberg [31, 32] theory was conducted in [30] and showed that despite m the small ratios of electron to ion mass M ratio justifying Eliashberg theory for phonons, an Eliashberg-type approach to the spin-fermion model is still allowed, but only at strong coupling [31]. The BCS superconducting instability of a Fermi system and Bose–Einstein Condensation (BEC) of bosons below a critical temperature can be unified by following the continuous evolution between these two limits as a strength of the fermion attraction increases. Within this approach, the phase diagram of cuprate superconductors can be interpreted in terms of a crossover from Bose–Einstein condensation of performed pairs to BCS superconductivity, as doping is varied. In the BEC limit, pairs form at high temperatures as result of strong coupling, and condense at some lower temperature, while in the BCS limit pairs form and condense at the same temperature. A large amount paper is devoted to the development of this idea [33–36]. The existence of pairs above Tc is the characteristic feature of a BEC

8

1 Physical Properties of Unconventional Superconductors

condensation, but their direct detection is so far an unsolved experimental problem. Other, fewer direct signatures of this mode of condensation that have been quoted in the literature include the power law dependence of Tc on the penetration depth ; Tc  2 ; interpreted as being due to the way the superfluid density ns varies with doping, ns (1.4) Tc   ; m assuming the effective mass m to be constant [37]. Another important moment is the existence of isotope effect on the pseudogap temperature T  . One of the popular scenario of theory of superconductivity in cuprate compounds is the bipolaron mechanism. In this model, electrons are coupled strongly in bipolarons [38] due to the electron–phonon interaction which is supposed to be with the characteristic energy considerably exceeding the Fermi energy. As a result, the mobile bipolarons are formed in normal state with their further Bose–Einstein condensation [38]. There are also purely phenomenological models in addition to above-mentioned semi-phenomenological models. These models related with a phenomenological description of the fermionic subsystem are possible after integration over all Bose fields. The first of such models is the marginal Fermi liquid model [39]. In this approach, the one-electron Green function has a usual form G.k; !/ D

1 !  ".k/ 

X ; .k; !/

(1.5)

X but self-energy part .k; !/ has an unusual structure [39]. Marginal Fermi liquid theory allows explaining the series of cuprate normal state anomalies, including the linear resistivity, specific heat peculiarities, a.c. conductivity. Another purely phenomenological model related with generalization GL theory, including both superconducting and antiferromagnetic fluctuations were taken into account [40]. This model also is called SO(5) model [40]. In this approach, five-dimensional order parameters were introduced for describing superconductivity and antiferromagnetism together. Speaking about phenomenogical model is necessary to underline that GL remains a very effective method for study superconductivity. In future chapters, we will discuss different versions of GL equations and their application for unconventional superconductors. One of the main and the most evident contradictions between theory and experiment was the linewidth of the Angle Resolved Photoemission (ARPES) peak in undoped cuprates. Although the dispersion of the peak is well reproduced by t t 0 t 00 J model, its width is very broad in experiment [41] and very narrow in theory [42]. Naively, contribution of electron–phonon interaction cannot explain the large width since the coupling to phonons, in addition to broadening, must also change the dispersion of the particle which, in turn, is already well described by the pure t t 0 t 00 J model. However, as was shown in [43,44], in the strong coupling regime of electron–phonon interaction the situation is exactly the same as in experiment. The

1.1 Cuprate Superconductors

9

polaron quasiparticle has very small weight and cannot be seen in experiment while shake-off Frank–Condon peak completely reproduces the dispersion of the pure magnetic model without electron–phonon interaction. Naturally, in such case the chemical potential must be pinned not to the observed broad shake-off peak but to the real invisible quasiparticle. Such decoupling of the chemical potential from the broad peak was observed in experiment [45] a few months after prediction had been made in [46]. Understanding the nature of the ground state and its low-lying excitations in the cuprate superconductors is a prerequisite for determining the origin of high temperature superconductivity. A superconducting order parameter with a predominantly dx 2 y2 symmetry is well established [47, 48]. However, there are several important issues that remain highly controversial. For example (in holedoped compound such as YBCO), various deviations from a pure d-wave pair state, such as the possibility of Cooper pairing with broken time-reversal symmetry or an admixed dx 2 y2 C s symmetry have been theoretically predicted [49, 50] and actively sought in numerous experimental studies [51, 52]. Furthermore, a transition of the pairing symmetry from d -wave behavior to s-wave-like behavior was also suggested as function of doping and temperature in various electron doped compounds [53, 54]. Further theoretical and experimental studies brought more evidences of the importance of electron–phonon interaction in cuprates. One of the evidences is the two-peak structure of the mid infra-red part of optical conductivity in the underdoped compounds, which is easily reproduced by taking the electron–phonon interaction into account [46]. Another confirmation is the anomalous temperature dependence of the width of the ARPES peak which can be explained only by the interplay of magnetic and lattice system [55, 56]. Various estimates for the electron–phonon interaction strength give the value   1 for undoped compounds [55, 57, 58]. The strength of electron–phonon interaction decreases with an increase of the concentration of holes reaching the intermediate coupling regime at optimal doping [46, 58]. In [59], the isotope effect in the Y 1x P rx BaC u3 O7d compound in all phases (superconducting, spin glass, and anti-ferromagnetic) was reported. Although somewhat debatable, there is experimental evidence [60] that electron– phonon interactions can be used to explain the fundamental mechanism operating in high-temperature cuprate superconductors. The dependence of the isotope shift ln Tc parameter ˛0 D  dd ln on the carrier density in cuprate superconductors is preM sented in Fig. 1.5. As shown in [11, 12, 60] in underdoped compounds isotope shift parameter is at the level 0.3. The existence of a strong electron–phonon interaction in cuprate superconductors was confirmed by the observation of a subgap structure in tunnel Josephson junction experiments [61]. As discussed in [62], similar phenomena occur due to the interaction of a Josephson current with phonons. As shown very recently in [63, 64], the electron–phonon mechanism explains many features of the low-energy relaxation process in cuprate superconductors, including the high values of the critical temperature; however, there are problems of coexistence of electron– phonon mechanism and unconventional symmetry of an order parameter.

10

1 Physical Properties of Unconventional Superconductors

Fig. 1.5 Isotope shift ln Tc parameter ˛0 D  dd ln M versus carrier density in cuprate superconductors [12]

1.0

- YBCO - LSCO - Bi2212

0.9 0.8 0.7

α0

0.6 ECS

0.5 0.4 0.3 0.2 0.1 0 0.05

0.1

0.15

0.2

0.25

hole concentration, p

1.2 Borocarbides The other class of new superconductors is rare-earth transition-metal borocarbides with the formula RN i2 B2 C (more general formula RTBC.N /I T  transition metal), which attracted the interest of many researchers, because of their wide variety of physical properties: compounds with R D Lu; Y exhibit fairly high superconducting transition temperatures, Tc , of about 15–16 K [65]; magnetism coexists with superconductivity for R D Dy; Ho; Er and T m [66]; whereas the only antiferromagnetic order occurs for R D P r; Nd; S m; Gd and T b [67]. These compounds show a layered structure, and therefore they are considered as possibly close to quasi-two-dimensional cuprates. However, various local density approximation band structure calculations [68–71] clearly indicated the three-dimensional electronic structure. Quantum oscillation measurements of nonmagnetic borocarbides LuN i2 B2 C and Y N i2 B2 C give clear evidence for a multiband character in the normal state [72]. The value of the gap ratio 2 varies from 0.45 to 3.2 [73–75]. Tc d-wave model of superconductivity was proposed for nonmagnetic borocarbides Y N i2 B2 C and LuN i2 B2 C for explanation of anisotropy effects [76, 77]. As mentioned by many authors, superconductivity in these materials is caused by phonons, as evidenced by specific heat [78, 79] and isotope effect [80, 81] measurements. It would then appear natural to relate the suppression of superconductivity, as R and the transition metal (T ) are varied, in terms of the BCS parameters, !D ; N.EF / and V (respectively, the Debye temperature, the density of states at the Fermi level, and some measure of the electron–phonon coupling strength): Tc D !D e

 N.E1

F /V

:

(1.6)

1.2 Borocarbides Table 1.3 Some superconducting characteristics of nonmagnetic borocarbides LuN i2 B2 C and Y N i2 B2 C [90]

11

Physical property Tc .K/ Hc2 .0/.T / Hc1 .0/.mT / .0/.A/ .0/.A/ .0/ D .0/=.0/ C.mJ =mol=K/ C=Tc eph

YNi2 B2 C 15.5 11 20–30 55 1200 22 460 1.83 1

LuNi2 B2 C 16.5 12 60 65 800 12 495 2.1 1.2

Since resistivity measurements indicate that V does not vary much with R in RN i2 B2 C [82], one is left primarily with !D and N.EF /. While !D generally increases as R goes from Lu to La, measurements of the Sommerfeld coefficient,  / N.EF /, lead to Lu  2La in the N i series. However, as the transition metal is varied, this simple parametrization no longer accounts for the trend of experimental data in an unambiguous way. Indeed, LaP t2 B2 C is a superconductor [and, for this P t series, so are the compounds with R D P r; Y [83], and, possibly [84], Nd ], even though it has a smaller N.EF / than nonsuperconducting LaN i2 B2 C :  D 5  8 mJ/molK2 , respectively [85]. There are any indications for unconventional pairing in nonmagnetic compounds LuN i2 B2 C and Y N i2 B2 C . Below presented list of several properties: a) a nonlinear H 1ˇ -dependence of the electronic specific heat in the superconducting state instead of the standard linear dependence, b) weak damping of the de Haas– van Alphen oscillations in the superconducting state, which can be related with vanishing gap at the parts of the Fermi surface [86]. c) a nonexponential and nonuniversal character of the temperature dependence of the electronic specific heat Cel  T ˇ ;

(1.7)

with ˇ  3 at low temperatures (in Y N i2 B2 C we have ˇ  3 [87], ˇ > 3 for LuN i2 B2 C and LaP t2 B2 C / d) the anisotropy of upper critical field within basal plane of LuN i2 B2 C [88] e) a quadratic flux line lattice at high fields has been observed not only for magnetic borocarbides but also for nonmagnetic compounds [89] f) deviations from the Korringa behavior of the nuclear spin lattice relaxation rate T11T D const have been ascribed to the presence of antiferromagnetic spin fluctuations on the N i site. Fundamental superconducting state parameters of nonmagnetic borocarbides Y .Lu/N i2 B2 C presented in Table 1.3. The coexistence or competition of magnetism and superconductivity in borocarbides with magnetic rare earth elements is one of the most challenging problems in the field. Most dramatic effects have been observed for HoN i2 B2 C , where below the onset of superconductivity at 8.8 K a suppression of superconductivity for magnetic field perpendicular top the c-axis and 4:5 < T < 5:5K has been observed in [91]. There are three magnetic structures shown in this region: caxis modulated commensurate cc, the spiral c-axis modulated incommensurate icc

12

1 Physical Properties of Unconventional Superconductors

Fig. 1.6 Magnetic structure observed for the compound HoNi2 B2 C

and a-axis modulated incommensurate ica ones. From the fact the icc structure and the superconductivity both occur within a narrow temperature range and that these effects have only been observed for HoNi2 B2 C , it has been supposed that the icc structure is the origin for the superconductivity (Fig. 1.6). However, replacing Ho partially by the nonmagnetic Y , the magnetic structures for Y1x Hox N i2 B2 C are shifted differently to lower temperatures. This has enabled researchers [92] to identify the i ca phase as the one responsible for the superconductivity phenomenon. The observed vector in the a-modulated incommensurate structure Q D 0:585a is close to the above-mentioned calculated nesting vector of 0:6a . A phenomenon, sometimes observed, is the so-called reentrant superconductivity, i.e., for decreasing temperature first material becomes superconducting and the a transition into the normal state observed before, at lower temperature, the superconducting state is reached again [92]. Influence of the interplay between helicoidal magnetic ordering and superconductivity on the differential conductance in Ho.N iB/2 C =Ag junctions was investigated in [93, 94]. Magnetic ordering in RN i2 B2 C compounds may result in a structural distortion caused by magnetostatic effects. Using high-resolution neutron scattering on a powder sample of HoNi2 B2 C , a tetragonal-to-orthorhombic distortion has been observed at low temperatures, where the Ho magnetic moment order in a c-axis modulated antiferromagnetic structure. Distortion is a shortening of the tetragonal unit cell in [110] direction. At 1.5 K, this shortening is 0.19% [95]. a similar tetragonal- to-orthorhombic phase transition driven by magnetostatic interaction has been also reported for ErNi2 B2 C [96]. The different types of antiferromagnetic order in RNi2 B2 C compounds have been determined by neutron diffraction [97]. The large variety of antiferromagnetic structures and the fact that in most cases they are not simple commensurate structures is related to the competition of crystalline electric field with Ruderman–Kittel–Kasuya–Yoshida exchange interaction, the modulation of which is not commensurable with the lattice structure. Now, it is

1.2 Borocarbides

13

Fig. 1.7 Crystal structure of YNi2 B2 C

generally accepted that the RN i2 B2 C compounds are three dimensional in their behavior, and thus are, in fact, quite different than the layered cuprates.

1.2.1 Crystal Structure and Tc The tetragonal layered crystal structure of the I 4=mmm or P 4=nmm types resolved so far for all well-characterized RTBC.N / compounds can be written schematically as .RC.N //n .TB/2 with n D 1; 2; 3: (Fig. 1.7). There are systematic dependences of critical temperature Tc with increasing T T distance, the transition metal component T : N i; P d; P t and the dopants replacing the T : C u; C o; V , etc., and the B  T  B bond angle. Finally, the number of metallic layers separating and doping the .N iB/2 networks also has a profound effect on the actual Tc value. Thus, for the single RC-layer .T D N i / compounds the highest Tc  14 to 16:6K values are obtained for R D Sc; Y; Lu, whereas for R D T h it is reduced to 8K and it vanishes for R D La. The double-layer Lu; Y -compounds exhibit very small transition temperatures of 2.9 K and 0.7 K [98], respectively, which however can be increased considerably replacing Ni by C u [99]. In the case of the twolayer boronitride .LaN /2 .N iB/2 ; triplelayer and quadro-layer .Y C /2 .N iB/2 so far no superconductivity has been detected, whereas the corresponding triple-layer compound exhibits a relatively high Tc  12K:

1.2.2 The Electronic Structure A typical band-structure calculation reveals sizeable dispersion in c-direction of the bands crossing the Fermi level and fluctuation magnetoconductance measurements clearly demonstrate the three-dimensional nature of the superconductivity under

14

1 Physical Properties of Unconventional Superconductors

consideration. Electronically, the coupling of the two-dimensional -(TB)2 networks is mediated mainly by the carbon and boron 2pz states. Further, important issues are the peak of the density of states N.0/ near the Fermi level EF D 0 and the intermediate strength of correlation effects [100]. The electronic structure near EF D 0 of all RTBC.N / compounds is characterized by a special band complex containing three or four bands total width about 1 eV; for the case of LaP tBC . Compared with Y N i2 B2 C and Lu N i2 B2 C for most of the other RTBC superconductors density of state near Fermi level are reduced. The comparison with the available specific heat data predicts that most RTBC.N / compounds exhibit intermediately strong averaged electron–phonon interaction elph  0:5 to 1.2, except LaT2 B2 C; T D N i; P d , which are weakly coupled and take place effect of pairbreaking. Physical properties such as the Hall conductivity, de Haas–van Alphen frequencies and related data, as well as the upper critical field Hc2 .T / are strongly determined by the shape of the Fermi surface. Band-structure of nonmagnetic borocarbides Y N i2 B2 C and Lu N i2 B2 C was calculated in [68–70]. Corresponding Fermi surface of LuN i2 B2 C is shown in Fig. 1.8 [67]. Fermi surfaces of both nonmagnetic compounds exhibit a similar geometry, characterized by a strong anisotropic behavior and special nestedregions along the a direction with vector qn  0:5 to 1.2 a . Nested and anisotropic properties for the Fermi surface of Lu N i2 B2 C close to the prediction [67]. Such topology has been observed by electron–positron annihilation radiation [101]. Very useful information about normal state of nonmagnetic borocarbides can be obtained from de Haas–van Alphen experiments [102]. In high quality Y N i2 B2 C crystals, six cross sections are found. The related Fermi velocities on extremal orbits can be grouped into two sets differing by a factor 4. These observations and sizeable anisotropy of the Hc2 for such crystals clearly indicate they are nearly in clean-limit regime.

Fig. 1.8 Fermi surface of nonmagnetic borocarbide LuNi2 B2 C. The magnitude of Fermi velocity F given at bottom

1.3 Magnesium Diboride

15

1.3 Magnesium Diboride MgB2 was discovered to be superconducting only in 2001 year [103], and despite that many of its characteristics have now been investigated and a consensus exists about its outstanding properties. First of all, this refers to its high Tc  40K/, which is a recordbreaking value among the s  p metals and alloys. It appears that this material is a rare example of the multi-band (at least two) electronic structures, which are weakly connected with each other. These bands lead to very uncommon properties. For example, Tc is almost independent of elastic scattering, unlike for other two-band superconductors [104]. Additionally, MgB2 has high potential to replace conventional superconducting materials in the electronic applications. Large critical densities are already reported for bulk samples [105] and bulk superconductivity is established immediately to support supercurrent transport between grains [106]. The material shows a pronounced isotope effect [107]. Measurements of the nuclear spin-lattice relaxation rate [108] indicate that MgB2 is a BCS type phonon-mediated superconductor. Calculations of the band structure and the phonon spectrum predict double energy gap [109, 110], a larger gap attributed to two-dimensional pxy orbitals and smaller gap attributed to three-dimensional pz bonding and anti-bonding orbitals. The maximal upper critical magnetic field can be made much higher than that for a single-band dirty superconductor [111]. The properties of M gB2 have been comprehensively calculated by the modern theoretical methods, which lead to a basic understanding of their behavior in various experiments. Any kind of disorder potentially changes the properties of MgB2 . Disorder can be introduced in a controlled way by doping or irradiation, but often arises from the preparation conditions [112]. Disorder generally decreases the transition temperature [112]. It has been suggested that intrinsic properties are affected by: (i) macroscopic particles that contribute to lattice distortion enhance both and scattering, (ii) disorder in the Mg sublattice (e.g., by Al addition) can increase the scattering, (iii) oxygen or carbon when substituting for B is expected to provide strong scattering [113, 114]. Superconductivity in the -band is suppressed at high magnetic fields, where the -band determines the magnetic properties and MgB2 behaves as a single-band superconductor [112]. The -band contributes significantly to the condensation energy and to the superfluid density only at low magnetic fields (below about 1 T in clean materials) [112]. Clean grain boundaries are no obstacles for supercurrents in MgB2 [112, 115–117]. This advantage compared to high temperature superconductors allows simple preparation techniques, but, the connections between the grains remain delicate, since dirty grain boundaries potentially reduce the critical currents [116, 118]. The main properties of MgB2 are presented in Table 1.4. Study of core structure of a single vortex in two-band superconductors was conducted in [120]. As shown in this work, at low temperatures a Kramer–Pesch effect occurs, i.e., a shrinkage of the size of the vortex core. Interestingly, this core shrinkage even exist, if only the  band is in the clean limit. This situation is believed to be realistic for high-quality M gB2 samples and opens the possibility to observe

16

1 Physical Properties of Unconventional Superconductors

Table 1.4 Some superconducting characteristics of MgB2 [119]

Parameters Carrier density (holes/sm3 ) Isotop shift ab .0/.T / Hc2 c .0/.T / Hc2 Hc1 .0/.mT / ab .0/.nm/ c .0/.nm/ .0/.nm/

Values .1:7–2:8/  1023 0.32 14–39 2–24 17–48 3.7–12 1.6–3.6 85–180

the Kramer-Pesch effect in this compound [121]. The temperature dependence of the microwave conductivity in M gB2 thin films at a fixed microwave frequency shows a peak at temperatures around 0:5Tc [122]. This is in contrast to conventional superconductors where a peak close to Tc appears and also in contrast to the cuprates where a huge conductivity peak at very low has been observed [123]. Another peculiarity of two-band superconductors is the occurrence of Leggett mode, which related with collective mode of small fluctuations of relative phase of the two superconducting order parameter [124]. Expression for Leggett modes in M gB2 given by the expression [125] 

  .12 C 21 / !0  2 11 22  12 21

1=2 ;

(1.8)

where ij is the interband and intraband electron–phonon interaction parameters. As followed from Exp.(1.8), Legget modes is determined by the interband interaction parameters and is absent in the case noninteracting bands. This mode is observable if !0 < 2 [125]. Experimental investigation of Legget modes conducted in Alx M g1x B2 [126] using Andreev spectroscopy method.

1.3.1 Crystal Structure Magnesium diboride, like other diborides M eB2 .M e D Al; Zr; T a; N b; T i; V etc.), crystallizes in a hexagonal structure, where honeycomb layers of boron are intercalated with hexagonal layers of magnesium locate above and below the centers of boron hexagons (Fig. 1.9). The bonding between boron atoms is much stronger than that between magnesium, and therefore the disordering in the magnesium layers appears to be much easier than in the boron layers. This difference in bonding between boron and magnesium atoms hinders the fabrication of M gB2 single crystals of appreciable size. Despite crystal structure of M gB2 being similar to that of a graphite intercalated compounds, M gB2 has a qualitatively different and uncommon structure of the conducting states.

1.3 Magnesium Diboride

17

Fig. 1.9 Crystal structure of MgB2

1.3.2 Electronic and Band Structure The electron band structure of M gB2 has been calculated using different ab initio methods yielding basically the same result [109, 111, 127–129]. The dispersion relations for boron pz character orbitals, which play a major role in transport and thermodynamic properties. The radii of the hollow circles are proportional to the -band character, which is made from pz boron orbitals, while those of the filled circles are proportional to the -band character, made from pxy orbitals. The most important is a quasi two-dimensional dispersion relation along the A (/ direction with a small Fermi energy  0:6eV , and accordingly, with a moderate Fermi velocity. The corresponding sheets of the Fermi energy form the cylindrical surfaces along the A direction seen in Fermi surface for MgB2 (Fig. 1.10). The corresponding electron transport is very anisotropic (c /ab 3.5 [130]) with the plasma frequency for the - band along the c (or z) axis being much smaller than that in the ab .xy/ direction [131]. The hole branch along A (/ experiences a huge interaction with the phonon E2g mode for carriers moving along the ab plane, although its manifestation is screened effectively by the much faster hole mobility in the -band [104]. An investigation of the charge density distribution would give a better understanding of how the superconductivity is related to the electronic and crystal structure M gB2 . Precise X-ray structure analysis [132–134] yielded accurate charge densities in M gB2 : The vortex density obtained at room temperature revealed a strong B  B covalent bonding feature. On the other hand, there was no bond electron between M g and B atoms, and M g atoms were found to be fully ionized and in the divalent state. As shown in [132], the value for M g atoms is very close to the number of electrons around M g 2C ions, so M g atoms are fully ionized in the M gB2 crystal at whole temperatures. On the other hand, the total numbers of electrons around boron two-dimensional sheets show significant difference, which can be attributed to the valence of the whole boron two-dimensional sheet changing from neutral to monovalent at 15K: This result suggests that the electrons transfer from  band (pz orbitals) to interplane  band (pxy orbitals) at 15K:

18

1 Physical Properties of Unconventional Superconductors

Fig. 1.10 Fermi surface of MgB2

Fig. 1.11 Experimentally measured two-gaps in MgB2

8

Energy gap (meV)

7

MgB2

Δσ

6 5 4 3

Δπ

2 1 0

0

5

10 15 20 25 Temperature (K)

30

35

40

Various theoretical and experimental problems of the family of doped M gB2 relative compounds are also widely investigated. The behavior of the superconducting gaps with doping seems interesting. The experimental situation can be followed [135, 136]. Theoretical study of superconductivity in doped M gB2 compounds was conducted in [137–140]. Last model applied for calculation in Alx M g1x B2 : The experimentally observed suppression of the  band gap is reproduced, whereas the three-dimensional  band gap shows no essential change until x D 0:4: The    bands crossing appears at xc D 0:485 where ; .0/ D 1:09 meV . Further, the gap decreases rapidly and vanish at the same Tc in accordance with the presence of interband pairing channel. The measured value for the merging point of the gaps is xc D 0:34 [141]. The leading  gap depression reflects the deviation from the optimal “self-doping” of pure M gB2 , which weakens the  intrabandpairing. Near xc the dimensionality of the  band changes as also the chemical potential relation to  the bands. The ratio  calculated by [140] seems to fall off to slowly as compared  with the measurement (Fig. 1.11) [136].

1.4 Oxypnictides

19

1.4 Oxypnictides In 2008, superconductivity at 26 K in the oxypnictide compound LaF eAs.O; F / was discovered [142]. The first communication on the superconductivity of LaOF eAs appeared as early as 2006, but the temperature Tc of the superconducting transition proved to be low, Tc D 3:5K. Similarly, for LaONiP, Tc D 4:5K was obtained [143]. Later, by substituting other rare-earth elements for La, several groups obtained considerably higher values (Tc D 41K in C eO1x Fx F eAs [144], Tc D 52K in P rO1x Fx F eAs [145]) and reached Tc D 55K in S mO1x Fx F eAs [146]. The parent compounds exhibit antiferromagnetic ordering of the iron moments, which are suppressed by doping in favor of superconductivity. The early awareness that magnetic order, even if in competition with superconductivity, is a key factor for determining superconductivity, drove the discovery within a short period of new iron-based superconductor families with different crystal structures such as (Ba; K/F e2 As2 [147], LiFeAs [148] and FeSe [149]. A large number of different compounds have now shown that superconductivity can be induced by carrier doping, both in the F e  As layer and in the spacing layer, and by external as well as by internal pressure. For simplicity in the following we will refer to the different families as: 1111 .REFeAs.O; F //; 122 .Ba; K/Fe2 As2 /; 11 Fe.Se; Te//; 111 .LiF eAs/ [150, 151]. A phase diagram on the plane T  x typical of the compounds of this type is shown in Fig. 1.12 [152]. This diagram resembles the diagrams characteristic of the cuprate superconductors, e.g., La2x Srx CuO4 . In cuprates, the superconductivity appears in compounds of the La2 CuO4 type as lanthanum is replaced by strontium. In both systems, the doping introduces charge carriers (electrons or holes), which suppresses the antiferromagnetic ordering and creates conditions for Cooper pairing. This analogy has led to the assumption that the high-temperature superconductivity of the new F eAs-type superconductors is caused by the proximity of the system

Fig. 1.12 Typical phase diagram of REO1x Fx FeAs compounds (RE is a rare-earth element) on the (T; x) plane. Ts is the structural transition temperature

20

1 Physical Properties of Unconventional Superconductors

to the magnetic phase transition; in this case, high values of Tc are caused by the pairing of charge carriers through spin fluctuations. In doped AF e2 As2 compounds, superconductivity at Tc D 38K was revealed immediately [147]. This assumption is confirmed by many detailed physical studies. The results of calculations of the electron–phonon coupling in these compounds showed that the high values of Tc in these new compounds cannot be explained in terms of the standard electron–phonon pairing mechanism. Using ab initio calculations of electron and phonon spectra for LaOFeAs was performed in [153]. Taking into account the value of the average logarithmic frequency of phonons [153] and neglecting Coulomb pseudopotential, Allen–Dynes formula [154, 155] gives the value of Tc D 0:5 K. In this study, numerical solution of Eliashberg equations with the calculated electron–phonon interaction function ˛2 F .!/ was used [153, 154]. Actually, to reproduce the experimental value of Tc D 26 K, coupling constant  should be approximately five times larger, even Coulomb repulsion is zero. It is well known that electron–phonon mechanism in Cooper pairing was always considered to be the observation of the isotope effect. Appropriate measurements were performed recently in [156] on S mF eAsO1x Fx , where 16 O was replaced by 18 O, and on Ba1x Kx F e2 As2 , where 54 F e was substituted for 56 F e. A finite shift in superconducting transition temperature was observed, which can be characterized in ln Tc a standard way by the isotope effect exponent ˛0 D  ln . For S mF eAsO1x Fx , M the isotope effect turned out to be small enough, with ˛0  0:00, which is quite natural as O ions reside exterior to the conducting FeAs layer. At the same time, the replacement of F e ions in FeAs layers in Ba1x Kx Fe2 As2 has led to a large isotope effect with a 0.4, which is close to the “ideal” value of 0.5. A superconductivity mechanism relying on the occurrence of nonmagnetic bipolaron in doped oxypnictides was proposed in [157] based on the Bose– Einstein Condensation of the bipolarons (see also [158, 159]). The applicability of another, so-called hole mechanism of superconductivity (see [160]) to the description of oxypnictides is analyzed in [161]. A “universal” superconductivity model proposed in [162] implies pairing of electrons with parallel spins and the existence of distinguished charge and magnetic bands (vortex lines) in oxyarsenides; this model was used to describe the effect of external pressure on the transition temperature. In [163], this model was used to calculate the optimum doping level of superconducting oxyarsenide. A possible superconductivity mechanism, including the electron–electron and electron–hole pairings as the respective consequence of electron–phonon and Coulomb interactions is reported in [164]. After a comparison among the families 122 comes out the most suitable for application with rather high Tc , upper critical field, low anisotropy, reduced thermal fluctuations and intrinsic pinning mechanisms. In particular, the C o-doped 122 compound with Tc of 22 K Hc2 (0) of greater than 50 T, has almost twice that of Nb3 Sn (30 T) with a Tc of 18 K. Although the N b-base materials are isotropic, C o-122 is almost isotropic .Hc2 < 2) too, making it potentially competitive as a low temperature superconductor. Even the highest Tc pnictides, Sm- and Nd1111 have anisotropies much smaller than typical cuprates (Hc2  30). However, a typical YBCO has Hc2  5, similar to the 1111. A clear drawback to present applications of the pnictides is their extrinsic and perhaps intrinsic granularity that

1.4 Oxypnictides Table 1.5 Some superconducting characteristics of oxypnictides [165]

21

Physical property Tc .K/ c 0 dHc2 =dT.T=K/ ab =dT.T=K/ 0 dHc2 Hc2 ab .nm/ c .nm/ Gi, Ginzburg number

Nd-1111 47.4 2.1 10.1 5 1.8 0.45 0.008

Ba122 22.0 2.5 4.9 1.91.5 2.4 1.2 0.00017

Fe11 14.5 14 26 1.91.1 1.2 0.35 0.0013

significantly restrict the critical current density of polycrystalline forms. However, only few years have passed since the first reports of Tc above 20 K in the pnictides, we should not expect that discoveries are yet over or that the final word on applications can yet be given. The main properties of oxypnictides are presented in Table 1.5. As mentioned above, phase diagrams on the T  x plane, where x is the concentration of the dopant (Fig. 1.12), are very similar to cuprate superconductors. Nevertheless, there is an essential difference between the cuprates and F eAs systems caused by electron correlations. Doped cuprate superconductors, in contrast to F eAs systems, are near the Mott–Hubbard metal-insulator transition, although in both cases superconductivity appears near the transition to the antiferromagnetic state. This means that the cuprates belong to materials with strong electron correlations, whereas F eAs systems are weakly (or moderately) correlated materials, although this question is still under discussion. But there is an essential difference between F eAs systems and the cuprates, which lies in the fact that the initial (stoichiometric) compounds are insulators in the case of cuprates and metals in F eAs systems. In both cases, however, the initial compounds are antiferromagnetic, although with localized magnetic moments in the first case and with itinerant-electron magnetism in the second. Upon doping the initial compounds, the temperatures of magnetic ordering TN decrease sharply and the compounds become superconducting at concentrations of a dopant of the order of 10%. The doping creates charge carriers in the main layers, which determine the interesting physics in cuprate materials. In the new class of oxypnictides, as in the cuprates, the doping can create either electrons or holes in the main layer. Depending on the relation between the valences of the ion to be substituted and the dopant, either electrons or holes appear in the FeAs layers. It is remarkable that there can occur a partial substitution for the atoms of the main layer, e.g., for F e atoms with C o atoms; in this case, superconductivity is induced in the doped compound, in contrast to cuprates, where the substitution for C u atoms located in the main layer destroys superconductivity.

1.4.1 Atomic Structure and Classification Under ordinary conditions, zero-defect stoichiometric LnOMP n (M D Mn, Fe, Co, Ni; Pn D P,As) phases have a layered tetragonal structure (ZrCuSiAs type, space

22

1 Physical Properties of Unconventional Superconductors

Fig. 1.13 Crystal structure of oxypnictide LaOFFeAs. The middle layer consists of Fe atoms (red spheres), which form a two-dimensional square lattice and As atoms (yellow). The second plane consists of O (green) and La (gray) atoms

group P4/nmm, Z D 2) [166] formed by stacking oppositely charged molecular layers .LnO/=.MP n/ along the caxis (Fig. 1.13). It is immediately noted that for superconductors (Tc > 26 K) obtained on the basis of matrix 111-phase LnOFeAs; the electron concentration in F eAs layers changes considerably pursuant as a result of the so-called modulation doping. For instance, the donor electrons in F eAs  layers can be due to a partial replacement of bivalent O 2 anions by univalent F1 in neighboring LnO layers, which results in an increase in the difference between the charge states of LnO and F eAs layers. Each MP n layer consists of a square net of M atoms, above and below which are the pnictogen P n atoms. The transition metal atoms have coordination number (CNs) equal to four, and their coordination polyhedrons are MP n4 tetrahedrons compressed along the c-axis. In other words, the MP n molecular layer can be represented as consisting of conjugate MP n4 tetrahedrons. In turn, for the atoms of rare-earth elements Ln with CN D 8, the coordination polyhedrons CPs are distorted square LnO4 Pn4 antiprisms; for oxygen atoms, CN D 4 and the CPs are OLn4 tetrahedrons. The crystal structure of tetragonal LnOMP n is characterized by the lattice parameters a.b D a/ and c and by two so-called internal parameters zLn and zP n , respectively, determining the Ln  O and M  P n interlayer spacings [166, 167]. The authors of [168] noticed a correlation between critical temperature and the LnOF eAs lattice parameters: the Tc value decreases considerably with increasing the parameter a. This dependence is sometimes interpreted in terms of the internal or chemical pressure created by rare-earth metal atoms in the lattice, but the origin of such a correlation actually remains unclear. Another interesting correlation was found between Tc and the angles ˛0 of As  F e  As bonds in F eAs4 tetrahedrons that form F eAs layers in the oxyarsenide structure. Critical temperature increases greatly as ˛0 approaches 109; 470, i.e., when F eAs4 groups assume a shape close to regular tetrahedrons. It was noted that such a structural transformation of FeAs4 groups, for instance, in the structure of NdOFeAs was favored by oxygen vacancies in the composition of NdO layers [169].

1.4 Oxypnictides

23

The analogy between F eAs systems and cuprates becomes even closer when we compare their crystal structures. FeAs systems are built of alternating FeAs planes separated by LaO planes, similar to how the alternating CuO2 planes are separated by LaBa or YBa planes in cuprates. Systems of both types are strongly anisotropic because of their layered structure, and the electron states in them are quasitwo-dimensional. Soon after the REOFeAs compounds, compounds of the AFe2 As2 type (A : Ba; Sr; Ca) were discovered, in which pairs of FeAs planes (bilayers), similar to the bilayers in cuprates of the YBa2 Cu3 O6 type, alternate. Thus, we now have three classes of compounds built from FeAs planes (LaOFeAs, AFe2 As2 , and LiFeAs) and analogous compounds of the FeSe type, in which superconductivity with relatively high Tc is revealed. Their physical properties are similar in many respects, which can be related to the similarity of their crystal structure, because in all cases it involves a common element, the FeAs layers. The highest values of Tc have been obtained in a number of doped REOFeAs compounds. At room temperature, all these compounds have a tetragonal structure belonging to the space group P4/nmm. Their crystal structure is formed by alternating F eAs and LaO layers. An FeAs layer actually consists of three closely spaced atomic planes, which are arranged in a square lattice of Fe atoms, above and below, which there are located square lattices of As atoms arranged relative to the Fe plane such that each Fe atom is surrounded by a tetrahedron of As atoms. In other words, an FeAs layer is formed by FeAs4 complexes. The spacing between F eAs and LaO layers is 1:8A. The crystal structure of LaOFeAs is shown in Fig. 1.13. It can be seen that the tetragonal unit cell has a rather elongated shape, which results in a strong anisotropy of all its properties and the quasi-two-dimensional nature of related electron states. The nearest neighbors of the Fe atoms are As atoms that separate the nearest adjacent Fe atoms, and hence the electron transfer over the iron sublattice is caused by FeAs hybridization; in this case, the exchange interaction between Fe atoms occurs not directly but through As atoms. Typical results of measurements of anisotropy parameter of upper critical field Hc2 on single crystals of 1111 [170] and 122 [171] systems are given in Fig. 1.14.

28 12.3 γ

24

T = 36.3 K

12.1 11.9

20

11.7

γ

0.6

Fig. 1.14 Anisotropy parameter of upper critical field in Sm oxypnictide. Shown in the inset is the field dependence of the anisotropy parameter

16

0.8

1.0 1.2 B (T)

1.4

SmFeAsO0.8F0.2 single crystal

12

B = 1.4 T 8 0

10

20

30 T (K)

40

50

24

1 Physical Properties of Unconventional Superconductors

Fig. 1.15 Temperature dependences of resistivity rab in the ab plane and of transverse resistivity rc in the orthogonal direction in a single crystal of BaFe2 As2 . Shown in the inset is the temperature dependence of the resistivity anisotropy

In particular, the anisotropy of the upper critical field Hc2 testifies to the quasi-twodimensional nature of the electronic subsystem in these superconductors, which is already evident from their crystal structure. At the same time, we can see that this anisotropy of critical fields is not too large. In Fig. 1.15, taken from [172] we depicted the temperature dependences of resistivity rab in the ab plane and of transverse resistivity rc in the orthogonal direction for a single crystal of a prototype (undoped) BaFe2 As2 system. It can be seen that the resistivity anisotropy exceeds 102 , which confirms the quasi-two-dimensional nature of the electronic properties of this system. This anisotropy is significantly larger than the value, which can be expected from simple estimates based on the above-mentioned anisotropy of Hc2 . In the data given in Fig. 1.15, we can also clearly see an anomaly in the temperature dependence of resistivity at Ts D 138 K, which is connected with the antiferromagnetic transition. The question concerning the anisotropy of electronic properties has become more acute after measurements already evident from their crystal structure. In the study of anisotropy parameter of upper critical field Hc2 in single crystals of Ba1x Kx Fe2 As2 [173] were performed in a much wider temperature interval than in [171], up to the field values on the order of 60T. According to [173], the anisotropy of Hc2 is observed only in the relatively narrow temperature interval close to Tc , changing to almost isotropic behavior as the temperature lowers.

1.4.2 Electronic and Band Structure First-principle calculations of the electronic structure of the LaOFeP compound, in which superconductivity (Tc D 4K) was revealed for the first time, were performed before the experimental detection of high Tc in this class of compounds [174]. The calculations for LaOFeAs were carried out almost simultaneously in a number of works [175–177]. The electronic structure of many REOFeAs compounds RE D LaI C eI S mI Nd I P rI Y was also calculated in [178]. The energy band structure is shown in Fig. 1.14, which is borrowed from the earlier work [174] for the sole reason that the corresponding figure in [175] is complicated by additional data due to variations in the coordinates of As atoms in the lattice; at the same time, the calculations for LaOFeP in [174] are identical to those for LaOFeAs. As can be seen

1.4 Oxypnictides

25

Fig. 1.16 Energy band structure of the LaOFeP compound. The unshifted band structure is indicated by the solid black line, while the shift away (towards) the Fe is indicated by the blue dotted lines (green dashed lines).The effect of As breathing along z taken into account. The symmetry points are D .0; 0I 0/; Z D .0; 0; 1=2/, X D .1=2; 0; 0/, R D .1=2; 0; 1=2/, M D .1=2; 1=2; 0/, A D .1=2; 1=2; 1=2/

from Fig. 1.16, there are 12 dispersion curves. The Fermi level intersects two-hole bands, which proceed from the point, and two electron bands, which proceed from the M point. We note the flatness of the curves in the direction Z, which indicates a weak dependence of the hole quasiparticles on the momentum kz ; therefore, the Fermi surface in the vicinity of has a cylindrical shape. The same relates to the Fermi surface sheets close to the M point (this follows from the flat segment of the curves on the line MA). Thus, in the case of the LaOF eP compound, two hole cylindrical sheets with their axes in the direction Z and two electron sheets with axes along MA are formed on the Fermi surface. This indicates the quasi-two-dimensional nature of the electron states formed by the dxz and dxy orbitals. In addition to these four cylindrical sheets, there is a three-dimensional hole pocket centered at the Z point of the Brillouin zone (Fig. 1.17). This three-dimensional pocket is formed by dz orbitals of F e hybridized with p states of As and La. In oxypnictides in a narrow (0,2 eV) energy interval around the Fermi level, where the superconducting state is formed [179,180]. It can be seen that electronic spectra of all systems in this energy interval are very close to each other. In the general case, the Fermi level is crossed by five bands formed by d-states of Fe. Of these, three form hole-like Fermi surface pockets close to the point, and the others form two-electron-like pockets at the corners of the Brillouin zone (note that the Brillouin zones of 1111, 111, and 122 systems are slightly different due to existing differences in lattice symmetry). It is not difficult to understand that this kind of a band structure leads to similar Fermi surfaces of these compounds presented in Fig. 1.17: there are three holelike cylinders at the center of the Brillouin zone and two electron-like ones at the corners. The almost cylindrical form of the Fermi surfaces reflects the quasitwo-dimensional nature of the electronic spectra in oxypnictide superconductors.

26

1 Physical Properties of Unconventional Superconductors

Fig. 1.17 Fermi surface of oxypnictide LaOFeAs: darker (blue) regions correspond to low velocity. The symmetry points are the same as in Fig. 1.16

The smallest of the hole-like cylinders is usually neglected in the analysis of superconducting pairings, as its contribution to electronic properties is rather small (the smallness of its phase space volume). At the same time, from the general picture of the electronic spectrum it is clear that superconductivity is formed in a multiple band system with several Fermi surfaces of different (electron or holelike) natures, which is drastically different from the simple single-band situation in cuprate superconductors. The study of the Fermi surface based on measurements of the de Haas-van Alphen effect, with quantum oscillations of magnetization in a magnetic field being was measured [181]. As the object of the study, the authors chose the compound LaOF eP (of the same crystal structure as LaOFeAs), with the superconducting transition temperature Tc D 7 K. This choice was made because a single-crystal sample, first, of high purity and, second, with a low value of the upper critical field (Bc2 D 0; 68T for the external magnetic field B k c and 7.2 T for B ? c) was needed in order to suppress the superconducting state in experimentally accessible magnetic fields. The results obtained showed that this compound has two cylindrical hole surfaces centered at the point and two electron surfaces centered at the M point, in complete agreement with the results of the LDA calculation in [174] but with higher effective masses (in the range 1,7–2.1me , where me is the mass of a free electron). The electron masses calculated for this compound are of the order of 0:8me: Therefore, we have an experimental confirmation (obtained with the aid of different techniques) of the basic conclusions of LDA calculations for FeAs systems. Thus, the main distinction of oxypnictide superconductors is their multiple band nature with anisotropy. An electronic structure in a narrow enough energy interval around the Fermi level is formed practically only from the d -states of Fe. The Fermi surface consists of several hole-like and electron-like cylinders and on each its “own” energy gap can be formed. Broadly speaking, this multiband character of superconductivity is not new and has already been analyzed in the scientific literature (see for details Chaps. 2 and 3). The specific band structure typical of FeAs layers, however, needs an additional analysis.

Chapter 2

Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Regardless of the origin of superconductivity, GL theory [182] has been found adequate for explaining the measurable macroscopic quantities. The temperature dependence of fundamental measurable quantities such as the lower critical field Hc1 and the upper critical field Hc2 are expected to help understanding the mechanism of superconductivity. The temperature dependence of Hc2 (T) and Hc1 (T) in cuprate superconductors, borocarbides, MgB2 and oxypnictides is different from that of the single band s-wave BCS theory and GL theory. The different temperature dependence may indicate a slight difference in pairing state of superconductor. Magnetic phase diagram of recently discovered superconductors, including MgB2 , LuNi2 B2 C,YNi2 B2 C and oxypnictides has also been of interest to researchers. In contrast to conventional superconductors, the upper critical field for a bulk MgB2 , LuNi2 B2 C,YNi2 B2 C and oxypnictides has a positive curvature near Tc [76,183–189]. To understand of the nature of the unusual behaviour at a microscopic level, a two-band Eliashberg model of superconductivity was first proposed by Shulga et al. [190] for LuNi2 B2 C and YNi2 B2 C and recently, for the MgB2 [191]. Here it is necessary to remark that the generalization of the BCS theory to the multiband case was first suggested in [192, 193] many years ago. Different aspects of two-band BCS theory described in monography [194]. Recent development of two-band BCS, theory, taking into account van Hove singularity of density of states presented in [195]. Temperature dependence of the thermodynamic critical magnetic field Hcm (T) remains to be determined theoretically. The temperature dependence of Hcm (T) is essential for the assessment of the behavior of specific heat at temperatures close to Tc . It is generally known that the BCS calculations, which implicitly incorporate an isotropic single-band Fermi surface, reveal that the jump in specific heat at Tc is constant at a magnitude of 1.43. The Eliashberg theory, assuming a strong electron– phonon coupling, would be expected to give a value greater than 1.43. Several groups have carried out measurements of specific heat on magnesium diboride, MgB2 ; [196, 197]. The measured specific heat shows a small jump at Tc , which is not explained within the standard BCS and Eliashberg theory. The calculation of

I. Askerzade, Unconventional Superconductors, Springer Series in Materials Science 153, DOI 10.1007/978-3-642-22652-6 2, © Springer-Verlag Berlin Heidelberg 2012

27

28

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

specific heat using the first principles of two-band Eliashberg theory were given by Golubov et al. [198]. In this Chapter, we use two-band GL theory and apply it to determine the temperature dependence of Hc2 (T), Hc1 (T), and Hcm (T) for nonmagnetic borocarbides and magnesium diboride. We show that the presence of two-order parameters in the theory gives a nonlinear temperature dependence, which is shown to be in a good agreement with experimental data for two-band superconductors MgB2 , LuNi2 B2 C, and YNi2 B2 C. Quantization of the magnetic flux, Little-Parks oscillations of critical temperature, relation between surface critical field Hc3 (T) and upper critical field Hc2 (T) in the framework of two-band GL theory also will be considered. First, we will derive the two-band GL equations for isotropic superconductors and these equations will be applied to the calculations of several physical quantities. In the end of this section, generalization of two-band GL theory to the case of layered anisotropy is considered. Anisotropy parameter of upper critical field Hc2 and London penetration depth  was calculated in this approach. d-wave GL equations also considered for the study any properties of cuprate superconductors. Coexistence of superconductivity and magnetism in framework GL theory is discussed. Angular effects on upper critical field on the base two-band GL theory also are studied. Structure of single vortex in layered two-band superconductors is presented. Influence of dirty effects in two-band superconductors using GL approach also included. Results of investigation of upper critical fields, single vortex in d-wave superconductors and nonlinear magnetization in d C s wave superconductors using d-wave version of GL theory in the content of second part. Modified version of GL-like theory very recent applied to cuprate superconductors for the calculation fundamental parameters and compared with available experimental data. Time-dependent twoband isotropic GL equations and corresponding (d C s)-wave equations were used for numerical modeling of single vortex nucleation and vortex dynamics discussion about 1.5 type superconductivity in two-band superconductors. Finally, application of GL equations to mesoscopic superconductors and possible new effects considered and peculiarities of vortex nucleation in such superconductors is briefly discussed.

2.1 GL Equations for Two-Band Isotropic Superconductors In the presence of two-order parameters in an isotropic s-wave superconductor, the GL functional free energy can be written as [199–201]: Z F Œ‰1; ‰2  D with Fi D

d 3 r.F1 C F12 C F2 C H 2 =8/;

  „2 2 i A ˇi j r ‰i j2 C ˛i .T /‰i2 C ‰i4 ; 4mi ˆ0 2

(2.1)

(2.2)

2.1 GL Equations for Two-Band Isotropic Superconductors

F12

  D " ‰1 ‰2 C c:c: C "1

29

     2 i A 2 i A  rC ‰1 r  ‰2 C c:c: : ˆ0 ˆ0 (2.3)

Here, mi denotes the effective mass of the carriers belonging to band i.i D 1I 2/. Fi is the free energy of separate bands. The coefficient ˛ is given as ˛i D i .T  Tci /, which depends on temperature linearly,  is the proportionality constant, while the coefficient ˇ is independent of temperature. H is the external magnetic field and H D curlA: The quantities " and "1 describe interband mixing of two-order parameters and their gradients, respectively. Minimization of the free energy functional with respect to the order parameters yields GL equations for two-band superconductors in one dimension for A D .0; Hx; 0/ „2  4m1



d2 x2  dx 2 ls4



 ‰1 C ˛1 .T /‰1 C "‰2 C "1

d2 x2  dx 2 ls4

 ‰2 C ˇ1 ‰13 D 0; (2.4a)



„ 4m2 2



2

2

d x  4 2 dx ls

 ‰2 C ˛2 .T /‰2 C "‰1 C "1

2

2

d x  4 2 dx ls

‰1 C ˇ2 ‰23 D 0; (2.4b)

„c is the so-called magnetic length. In deriving the last equations where ls2 D 2eH without losing generality, for simplicity we consider the case in which ‰ and A depend only on a single coordinate x: Boundary conditions to two-band GL equations have a form:

    2 i A 2 i A n r ‰1 C "1 n r  ‰2 D a‰1 C b‰2 ; ˆ0 ˆ0     2 i A 2 i A n r ‰2 C " 1 n r  ‰1 D c‰1 C d ‰2 ; ˆ0 ˆ0

(2.5a) (2.5b)

where a; b; c; d are constants. Here, it is necessary to remark that two-band GL equations was first discussed by Tilley and Moskalenko [202, 203]. However, in equations presented in [202, 203] terms similar to intergradient interaction in equations (2.4a) and (2.4b) are absent. In paper [202] was discussed only upper critical field Hc2 problem in linear approximation. Shown below, inclusion of term with intergradient interaction lead to interesting results. Considering ‰i .r/ D j‰i .r/j exp.ji .r// in (2.1)–(2.3), with i .r/ being the phase of order parameters, and j‰i .r/j the modulus of order parameter, we can obtain the equilibrium values for j‰i .r/j in the absence of any external magnetic fields as j‰10 j D  2

  ˛22 .T / ˛1 .T /˛2 .T /  "2 "2 ˇ2 ˛1 .T / C ˇ1 ˛23 .T /

;

(2.6a)

30

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

j‰20 j D  2

  ˛12 .T / ˛1 .T /˛2 .T /  "2 "2 ˇ1 ˛2 .T / C ˇ2 ˛13 .T /

:

(2.6b)

The phase difference of order parameter at equilibrium can be given as cos.1  2 / D 1I

if "<0,

(2.7a)

cos.1  2 / D 1I

if ">0.

(2.7b)

2.1.1 Upper Critical Field Hc2 .T / As it is well known, Hc2 .T / can be calculated as lower eigenvalue problem of linearized GL equation for the single-band superconductors [204]. In the vicinity Tc we can neglect cubic terms in (2.4a) and (2.4b). Then (2.4a) and (2.4b) can be rewritten as:  2   2  x2 x2 d d „2  4 ‰1 C ˛1 .T /‰1 C "‰2 C "1  4 ‰2 D 0; (2.8a)  4m1 dx 2 ls dx 2 ls 

„2 4m2



x2 d2  4 2 dx ls



 ‰2 C ˛2 .T /‰2 C "‰1 C "1

x2 d2  4 2 dx ls

 ‰1 D 0: (2.8b)

One can get identical equations for ‰1 .x/ and ‰2 .x/ from (2.8) by elimination. Therefore, we can suppose that ‰1 .x/ D C0 ‰2 .x/. The coefficient C0 is obtained by solution of (2.8a) and (2.8b). Substituting eigenfunction corresponding to lower 2 energy state ‰1 .x/ _ exp. ıx2 /; where ı D ls2 and relation ‰1 .x/ D C0 ‰2 .x/, we then obtain an expression for the coefficient C0 C0 D 

"  "1 ı „2 ı 4m1

C ˛1

; C0 D 

„2 ı 4m2

C ˛2

"  "1 ı

;

(2.9)

which is equivalent to the following equation: 

„2 ı C ˛1 4m1



„2 ı C ˛2 4m2

 D ."  "1 ı/2 :

(2.9’)

Using (2.9’), one obtains for the normalized upper critical field of two-band / Q superconductor in isotropic case hc2 D HHc2c2.T , Hc2 .0/ D cTc .1 m„e1 C2 m2 / : .0/   hc2 ./ D a01 c0 C .A 2 C B C c02 /0:5 I  D T =Tc  1; AD

a1 a2 rm2 1 m 1 Tc m2 "1 2 .rm 1/2 2 ; C A  I A D 64 I rm D I D 1 1 2 2 .rm C1/ .rm C1/ 2 m2 „2 "

(2.10) (2.10a)

2.1 GL Equations for Two-Band Isotropic Superconductors

BD

2.rm  1/ .a1 rm  a2 / Tci C .a1 C a2 /A1 2 C 2B1 I ai D 1  ; .rm C 1/2 Tc c0 D

.a1 rm C a2 / 16rm 2 "2 : C B1 I a0 D 1  .rm C 1/ 1 2 Tc2

31

(2.10b) (2.10c)

Critical temperature of two-band superconductor is determined from (2.9’) in the absence of external magnetic field: .Tc  Tc1 / .Tc  Tc2 / D "2 =1 2 :

(2.11)

Near Tc , we have an asymptotical behavior of normalized upper critical field hc2 of the form ˚  hc2 ./ D a01 .1  B=2c0 / C .A=2c0 / 2 : (2.12) The experimental data for the normalized upper critical field hc2 of MgB2 ; LuNi2 B2 C and YNi2 B2 C can be described with high accuracy by the simple expression hc2 ./ D

Hc2 ./1C˛ D ;  Hc2 .0/ f1  .1 C ˛/w C lw2 C mw3 g

(2.12a)

where w D .1 C /./1C˛ : This formula of (2.12) was applied for fitting the experimental data of hc2 in nonmagnetic borocarbides by Drechsler et al. [91]. The critical exponent ˛ determines the temperature dependence more effectively, which is very sensitive to disorder, i.e., to the quality of the samples. With increasing disorder, the positive curvature and correspondingly ˛ does decrease [183]. The saturation (negative curvature) at low temperatures is well described by the ratio of l=m, which is assumed in analogy as sensitive to the electronic structure. For the l > m, temperature region with negative curvature becomes wide. For l < m, we have an upper critical field changing almost linearly with temperature. In Fig. 2.1, we plot the experimental data of Freudenberger et al. [183] with full circle symbol for upper critical field Hc2 (T) in LuNi2 B2 C versus the reduced temperature T =Tc and also show the theoretical fits to this data by using (2.10) and (2.12a). The best fit to the experimental data was obtained by using (2.12a) (shown  by dotted lines) with fitting parameters 0 Hc2 .0/ D 7.75T, ˛ D 0:24; l D 3; m D 1. In the figure, the solid line display the two band GL fitting by using (2.10). For this fitting, GL parameters together with other fitting parameters are A D 0.66, B D 0:03; c0 D 0:19; rm D 5; 0 HQ c2 .0/ D 7.02 T, Tc1 D 9:8 K, Tc2 D 2:3 K. In Fig. 2.2, we plot the first and second order derivatives of hc2 of LuNi2 B2 C using (2.10) and (2.12) with respect to reduced temperature T =Tc . Two-band GL theory (open circles) gives a negative first derivative below Tc . On the other hand, the first derivative of the fitting formula (2.12a) (full circles) starts from zero at absolute zero, goes through a negative region and returns to zero at Tc . There is a good agreement in the temperature range from 0.4Tc to 0.95 Tc . The plots of the second derivatives are also given in the same figure. A good agreement in the

32

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Fig. 2.1 Temperature dependence of the upper critical field Hc2 (T) for LuNi2 B2 C versus reduced temperature, T =Tc . Full symbols of circles show the experimental data, while the lines is for GL theoretical expectations of Hc2 (T). Dotted line is for fitting formula (2.12a) calculations

same temperature range is observed for the second derivatives, too. The fit for the YNi2 B2 C is presented in Figs. 2.3 and 2.4. We found the following parameters [76]:  0 Hc2 .0/ D 8.6 T, ˛ D 0:28; l D 7; m D 13. and for the GL set of parameters: A D 0.71, B D 0:044; c0 D 0:157; rm D 5; 0 HQ c2 .0/ D 8T, Tc1 D 10 K, Tc2 D 1:825 K. Similar calculations of the bulk MgB2 using (2.10) and (2.12a) presented in [200] and a good agreement with experimental data [184] has been achieved. In the case of no intergradients of order parameter ( D 0), the GL curvature reaches its maximum at the point  D B=2A D 0.5. The inclusion of a negative intergradient interaction shifts this maximum to the region close to critical temperature. Physically, it means that in the vicinity of Tc , when both order parameters are small, their interaction becomes important for the behavior of the upper critical field. As shown by (2.11), the critical temperature of around Tc D 16 K can be obtained from Tc1 D 9:8 K and Tc2 D 2:3 K approximately with the interaction parameter 0.33 in the case LuNi2 B2 C compound. In the case of YNi2 B2 C, following parameters have been used: Tc1 D 10 K, Tc2 D 1:825 K and 0.33. We note that we must take two rather different critical temperatures Tc1>>Tc2 , in contrast to similar calculations for MgB2 [200]. Upper critical field hc2 is governed by the parameter " and "1 . As followed from the expression (2.10), the upper critical field is determined mainly with the larger mass m1 , while contribution from smaller mass is ignorable

2.1 GL Equations for Two-Band Isotropic Superconductors

33

6

Lu(NiB)2C

Derivatives of hc2

4

2

h''c2

0 h'c2

–2 0.0

0.2

0.4 0.6 Reduced temprature, T / Tc

0.8

1.0

Fig. 2.2 The first and second derivatives of (2.10) and (2.12a) versus reduced temperature for LuNi2 B2 C. The open circles show the derivatives using GL equations (2.10), while the full circles show the fitting formula (2.12a) derivatives

in the case of two different effective masses. In calculation of mass ratio parameter rm of LuNi2 B2 C and YNi2 B2 C borocarbides, we take the Fermi velocity in different bands as vF 1 D 0:85 107 cm/s, vF 2 D 3:8107 cm/s [190]. This yields that the parameter of mass ratio rm can be taken as 5, meanwhile for the MgB2 and rm is equal to 3 [200, 201]. Note that the fitting between theory and experiment shows a deviation at temperatures very close to Tc (see Figs. 2.3 and 2.5). It seems that we may have two main reasons for the mismatch between theory and experiment at temperatures very close to Tc . First, the thermal fluctuations may enhance the positive curvature near Tc , which is ignored in the GL theory as mean field theory. Fluctuations might be strongly enhanced due to the nested region on the Fermi surface, which contribute in an important way to the group of slow electrons described within our two-band model [190]. Second, in the solutions, we assumed a linear two-band GL theory for which there exists analytical solutions. For a better fitting, the cubic term must have been included in the two-band GL theory. Then the equations to be solves would necessitate the solutions for the vortex state and their symmetry in two-band GL theory. Application of two-band GL equations to oxypnictides up to now was not considered yet. As we will see in consideration of d-wave GL theory, positive curvature will be the result of d C s-wave interaction in cuprate superconductors near Tc .

34

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Y (NiB)2C

8

μ0 Hc2[T]

6

4

2

0 0.0

0.2

0.4 0.6 Reduced temperature, T / Tc

0.8

1.0

Fig. 2.3 Temperature dependence of the upper critical field Hc2 (T) for YNi2 B2 C versus reduced temperature, T =Tc . Full symbols of circles show the experimental data, while the lines are for GL theoretical expectations of Hc2 (T). Dotted line is for fitting formula (2.12a) calculations

2.1.2 Surface Magnetic Field Hc3 (T) In the case of single-band superconductivity, the surface critical field is determined from the GL equation with vector potential as A D H.x  x0 / in [205]. This procedure allows one to obtain an exact value of Hc3 .T /; but requires some sophisticated numerical calculations. However, a simple variational analysis provides us with an almost exact solution [206]. The accuracy of simple variational procedure is about 2 % and can be improved by choosing trial function [207]. As a result, the problem of solving GL equation may be presented as a variational problem of finding the minimum of free energy functional for the single band superconductor [206, 207]. 2 For trial solutions, we substitute functions of ‰1 .x/ _ exp. ıx2 / and ‰1 .x/ D C0 ‰2 .x/ into (2.8a) and (2.8b) with appropriate boundary conditions of ‰1 .x/ D 0;

d ‰1 .x/ .0/ D 0: dx

(2.13)

The coefficient ı must be determined from the minimum condition for the free energy functional (2.1). Taking into account expression for the vector potential A D H.x  x0 / and using the trial functions, we can rewrite expression (2.1) as

2.1 GL Equations for Two-Band Isotropic Superconductors

35

8 7 Y(NiB)2C 6

Derivatives of hc2

5 4 3 h''c2

2 1 0

h'c1

–1 –2 0.0

0.2

0.6 0.4 Reduced temperature, T / Tc

0.8

1.0

Fig. 2.4 The first and second derivatives of (2.10) and (2.12a) versus reduced temperature for YNi2 B2 C. The open circles show the derivatives using GL equations (2.10), while the full circles show the fitting formula (2.12a) derivatives

(here we neglected quadratic terms) # "   2H.x  x0 / 2 „2 d2 dx C C ˛1 .T /

F D 4m1 dx 2 ˆ0 0     1 „2 d 2 2H.x  x0 / 2 C 2 C C ˛ .T / 2 ˆ0 C0 4m2 dx 2  ) 2"1 2H.x  x0 / 2 ıx 2 2" e C 2 : C C0 ˆ0 C0 Z

1

(

(2.14)

First, we will find the minimum of the free energy (2.14) with respect to x0 after integration. Differentiation with respect to x0 gives: x0 D .ı/1=2 :

(2.15)

Next, we will then find the minimum of energy functional (2.14) with respect to ı. This gives a relation between ı and C0 as: C02

„2 „2 .˛1 .T / C ı/ C 2C0 ."  "1 ı/ C .˛2 .T / C ı/ D 0; 4m1 4m2

(2.16)

36

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

which is equivalent to the (2.9’). By substituting (2.15) and (2.16) into (2.14) with

F D 0, we can then obtain a formula for the surface critical field Hc3 : 

2H ˆ0

 2  2 D ı2; 1 

(2.17)

where ı is determined from (2.9’), which is consistent with the result found in [200]. Finally, the surface critical field can be given as Hc3 .T / D

 1=2 ˆ ı 0 D 1:66Hc2 .T /;  2 2

(2.18)

where Hc2 (T) is determined by the (2.10). From (2.18), one can see that the surface critical field Hc3 .T) has the same temperature dependence with Hc2 (T) for both single band and two-band GL calculations. The calculations favor the existence of the surface critical field. The first ARPES measurements on a MgB2 single crystal were reported in [208]. They observed several surface states. The effect of surface states for the superconductivity has been discussed in [209]. They claimed that the surface critical magnetic field would not exist. The surface electronic states can be responsible for the non-existence of the surface critical field [209]. As mention in this paper, for nonmagnetic borocarbides no Hc3 has been observed. Welp et al. [210] presented data of Hc3 (T) for a single crystal of MgB2 late. In their c-axis measurements, the surface critical field has a linear dependence, while it has a positive curvature for abplane measurements. Note that there are anisotropy in the measurements. However, the calculations presented here assume isotropic superconductivity for two-band GL theory. Although surface state influence on superconductivity is still controversial, it has been suggested that superconductivity at ab-surfaces is suppressed, while superconductivity along c-axis is unaffected. According to our calculations in the framework of isotropic two-band GL theory the upper critical field reveal positive curvature at temperatures close to Tc . With the parameters of two-band GL theory presented in [200], one can observe a surface critical field Hc3 in experiments with positive curvature similar to Hc2 for a bulk MgB2 superconductor. In our opinion preparation of bulk samples with smooth and clean surfaces would be enough for the measurements in MgB2 . Paramagnetic Meissner effect, associated with the existence of the surface critical field, was observed for the first time in a finite size MgB2 pellets and superconducting cores of the iron sheathed MgB2 wires in [211]. The paramagnetic Meissner effect can be obtained in the framework of self-consistent solution of single band GL equations for a cylinder of finite size [212]. Surface magnetic field in dirty twoband superconductors was calculated in [213] using BCS theory. It was shown that in the case of interaction of the two bands leads to a novel scenario with the ratio Hc3 =Hc2 varying with temperature. The results are applied to MgB2 and smaller than 1.6946 (see Sect. 2.2.6).

2.1 GL Equations for Two-Band Isotropic Superconductors

37

2.1.3 Lower Critical Field Hc1 .T / For temperatures near Tc and magnetic fields slightly larger than Hc1 , the influence of the field on the modulus of the order parameters ‰1 and ‰2 can be neglected and we assume j‰1 j D const, j‰2 j D const. Then the wave function can be written as ‰i .r/ D j‰i .r/j exp .ji .r//. Here, i .r/ are the phases of the order parameters and the GL free energy functional in (2.1), can be rewritten as Z F Œ1; 2  D

d 3r

    „2 d1 2A 2 d2 2A 2 „2 n1 .T / C n2 .T /   8m1 dr ˆ0 8m2 dr ˆ0

C " .n1 .T /n2 .T //0:5 cos .1  2 / C "1 .n1 .T /n2 .T //0:5     d1 2A d2 2A 2  cos.1  2 / C H =8 ;   dr ˆ0 dr ˆ0 (2.19) where n1 .T / D 2 j‰1 j2 and n2 .T / D 2 j‰2 j2 are the densities of superconducting electrons for the corresponding bands, respectively. The temperature dependencies of n1 .T /, n2 .T / and (1  2 ) are defined by the equilibrium value of order parameters j‰1 j and j‰2 j, which satisfy the two-band GL equations without linearization (see (2.6a), (2.6b), (2.7a), and (2.7b). The equations determining the equilibrium values of magnetic field and phases of the order parameters would be obtained by the minimizing the free energy functional in (2.19) with respect to the vector potential A and the phases 1 ; 2 . The equation for the vector potential takes the form     d1 2A d2 2A „2 „2 n1 .T / n2 .T / C   4m1 dr ˆ0 4m2 dr ˆ0     d1 2A d2 2A C :   C"1 .n1 .T /n2 .T //0:5 cos .1 2 / dr ˆ0 dr ˆ0

2 curlcurl A D 4 ˆ0



(2.20) By using the relevant Maxwell equation, curlH D 4 J (for the magnetostatic case), c (2.20) yields to the London equation (by taking into account the equilibrium value of the differences of phases 2.7a,b) of the form 2

d 2H  H D 0; dr 2

(2.21)

where  is the London penetration depth of the following form 4e 2  .T / D 2 c 2



n2 .T / n1 .T / 0:5 : C 2"1 .n1 .T /n2 .T // C m1 m2

(2.22)

38

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

It is well known that the lower critical field Hc1 can be obtained as in [204] Hc1 D

ˆ0 ln .T /: 42 .T /

(2.23)

We then introduce a dimensionless lower critical field in isotropic two-band case 2 / with Hc1 .0/ D ˆ0ce2 Tc . ˇ1m1 1 C ˇ2m2 2 /: We rewrite the normalized lower hc1 D HHc1c1.T .0/ critical field as: hc1 D B.T / ln .T /; (2.24) where B.T / D   DD

˚ 2  2 " C rm .  c1 /2 C 2rm "2 .  c1 / 1 rm C D  2 C .2  c1  c2 /  "2 D.  c2 / C .  c1 /3

;

(2.25)

ˇ1 22 Tc1;c2 T I c1 ;c2 D ; D : Tc Tc ˇ2 12

(2.26)

Temperature dependence of the normalized GL parameter .T / in two-band superconductors is given as .T / D .0/



hc2 .T / B.T /

1=2 :

(2.27)

Here, the upper critical field hc2 of two-band superconductors is given by the expression (2.10). In Fig. 2.5, we plot hc1 (T) versus reduced temperature T =Tc . Ghosh et al. [214] found that the lower critical field YNi2 B2 C of borocarbides follows a power law dependence of the form with hc1 D HHc1.0/ D 1  .T =Tc /2 , with c1  .0/ D 22 mT. Full circles in the graph of Fig. 2.5 show their results. Hc1 To our best knowledge, no measurements of lower critical field for LuNi2 B2 C. The full line denotes the results of calculations from two-band GL theory. Here, we used the (2.10), (2.24), and (2.27) with the parameters D D 1:5. The other GL set of parameters were also used to determine the temperature dependence of upper critical field hc2 in YNi2 B2 C (see above 2.1.1). The temperature dependence of the GL parameter .T /, obtained using (2.27) with the same fitting parameters, is presented in Fig. 2.6. As shown in Fig. 2.6 .T / varies little with temperature and as a result temperature dependence of hc1 is mainly determined by the London penetration depth .T / in (2.22). Calculations of hc1 for magnesium diboride MgB2 using two-band GL theory were carried out in [201] and also good agreement has been achieved. Note that the temperature dependence hc1 of the two-band GL theory is dominantly determined by the interaction parameters " and "1 . When the carriers have different effective masses in different bands (m1 >> m2 ), the lower critical field

2.1 GL Equations for Two-Band Isotropic Superconductors

39

1.0

Y(NiB)2C

0.8

hc1

0.6

0.4

0.2

0.0 0.6

0.7 0.8 0.9 Reduced temperature, T / Tc

1.0

Fig. 2.5 Temperature dependence of the lower critical field Hc1 (T) for YNi2 B2 C versus reduced temperature, T =Tc . Full symbols of circles show the experimental data [214], while the lines are for GL theoretical expectations of Hc1 (T)

hc1 can more effectively be determined by the small mass, in contrast upper critical field. The contribution from the larger mass is ignorable in such a case. As shown in Fig. 2.5, the theoretical data of two-band GL theory are in a good agreement with experimental data for nonmagnetic borocarbide [214].

2.1.4 Upper Critical Field Hc2 of Thin Films Assume now that the thickness of two-band superconductor film is d < eff ; , where London penetration depth  for two-band superconductors given by the expression (2.22), where j ‰10;20 .T / j2 is given by the (2.6a) and (2.6b). Then we may neglect variations of order parameter in the film and assume that magnetic field penetrates the film almost completely. The surfaces of the film are assumed to coincide with the planes x D ˙d=2: If neglected by changing order parameter in thin film, (2.4a) can be rewritten as 

"1 „2  4m1 C



h x2 "i ‰1 C ˇ1 ‰13 D 0: ‰ C ˛ .T / C 1 1 ls2 C

(2.28)

40

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

1.75

κ(T ) / κ(0)

1.50

1.25

1.00

0.6

0.7

0.8 T / Tc

0.9

1.0

Fig. 2.6 Temperature dependence of the GL parameter  for YNi2 B2 C versus reduced temperature, T =Tc based on (2.27)

Averaging last expression over the thickness of the film d , we can obtain the dependence of the order parameter ‰1 on the applied magnetic field j ‰1 j D  2

 ˛1 .T / C

" C



C



ˇ1

„2 4m1



"1 C



1 d2 ls4 12

:

(2.29)

Finally, using (2.9) and (2.29), we can get equation for upper critical field of the thin films  2  2     2  d2 "1 „ „ 1 „ 1 C ˛2 C "1 "  2 C˛2 ˛1 C 12.ls2 /2 4m1 4m2 ls2 ls 4m2 ls2   "1 (2.30) C " "  2 D 0: ls At small magnetic fields, we have  2   2  d2 „ „ ˛1 1 ˛2 C ""1 C C ""1 2 C .˛1 ˛2  "2 / D 0: 12.ls2 /2 4m1 4m2 ls

(2.31)

2.1 GL Equations for Two-Band Isotropic Superconductors

41

film

Final expression for Hc2 .T / has a form [215]: film

Hc2 .T /

D

„c 2e





„ 2 ˛1 4m2



C ""1 C

r

„ 2 ˛1 4m2

C ""1 h

d2 6

2

2

 4.˛1 ˛2  "2 / d12 i „2 ˛ C ""1 4m1 2

h

„2 ˛ 4m1 2

C ""1

i :

(2.32) For the films with d << eff ;  film

Hc2 .T / D 

„c .˛1 .T /˛2 .T /  "2 / h i : 2e „2 ˛1 .T / C 4""1 4

m2

(2.33)

„2

As followed from (2.32), upper critical field of the thin film of two band superconfilm ductors increases with decreasing thickness d 2 : It is well known that Hc2 .T / for 1 single-band superconductor films increases as d [204]. The common feature of (2.12) and (2.33) is the presence of positive curvature at the critical temperature Tc for bulk samples and films. Such conclusion in agreement with experimental data for bulk samplers of nonmagnetic borocarbides and magnesium diboride [200, 201] and for the MgB2 thin films [114]. Using (2.30), we can calculate slope of the upper critical field of bulk and thin film samples at Tc . For the estimation of the ratio film

of slopes at Tc

Hc2 .T /=d T m .T /=d T Hc2

, we will use (2.30)–(2.33) with the parameters which

were also used in [200, 201] to determine the temperature dependences of different physical quantities for a bulk samples MgB2 . With this choice of parameters, the H

film

.T /=d T

ratio of slopes at Tc Hc2m .T /=d T can be approximately 1.65. As followed from c2 [114, 216–218], ratio of slopes of bulk and thin films varies in the interval 1.5–2.2.

2.1.5 Magnetization of Two-Band Superconductors Near Hc2 It is well known that the magnetization per unit volume in the case of single-band superconductors may be written as [204] M.H; T / D 

1 Hc2 .T /  H ; 4 .2 2  1/ ˇA

(2.34)

where  is the GL parameter and ˇA D 1:16 for a triangular vortex lattice [204]. As followed from (2.34), the linearity of the experimentally observed M.T / curves in single-band superconductors is the result of linearity of upper critical Hc2 .T / for a fixed value of the external magnetic field H and temperature independent character

42

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

of GL parameter . From (2.27) for  for two-band superconductors followed that for the magnetization reveal nonlinear character. The temperature dependence of Hc2 .T / for two-band superconductors near Tc exhibits a positive curvature. It implies that the temperature dependence of the magnetization of two-band superconductors near Hc2 .T / shows nonlinear behavior at fixed external magnetic field. Nonlinear magnetization of M gB2 in vicinity Hc2 .T / was reported in [197] and late in [219]. As followed from (2.27), temperature changing of GL parameter  will lead to changing type of superconductor from II type to I type at the vicinity of critical temperature. Similar effects in cuprate superconductors were discussed in [220].

2.1.6 Flux Quantization This is a general feature of superconductors. It may be inferred by using (2.20) and assuming (2.7). Proceeding as usual (see [204, 207]), we find     d1 2A d2 2A „2 „2 n1 .T / n2 .T / C   4m1 dr ˆ0 4m2 dr ˆ0    d2 2A d1 2A C. :   C "1 .n1 .T /n2 .T //0:5 cos .1  2 / dr ˆ0 dr ˆ0 (2.35)

2 J D ˆ0



Let us now consider a hollow cylinder or, to put it differently, a tube with wall thickness larger than : We integrate this equation along a closed path, lying entirely within the superconductor cavity in the cylinder. In the loop passes inside of the wall, then J D 0 and integral on the right-hand is equal to zero. Taking into account 1 2 (2.7), we find that d D d : As a result, magnetic field passing through the loop dr dr may take a discrete series of values ˆ0 W ˆ D nˆ0 :

(2.36)

Experiments about quantum interferences devices on MgB2 were reported in [221, 222]. As followed from (2.36), quantization of the magnetic flux in twoband superconductors remains as the same in single band case. Results of these experiments [221, 222] confirm flux quantization (2.36). However, situation in mesoscopic samples will be different from this result (details see in 2.8).

2.1.7 Little-Parks Effect in Two-Band Superconductors Let us take a thin cylindrical film where the thickness is much less than the penetration depth  and apply the magnetic field along the axis of the cylinder.

2.1 GL Equations for Two-Band Isotropic Superconductors

43

In this case, superconductor cannot trap the magnetic flux because of its small thickness. However, the nonuniqueness of the phase leads to the oscillation of the critical temperature Tc with periodicity ˆ0 in single-band approximation [204]. In a similar manner with single-band superconductor case [204], using (2.19), (2.7a), 1 2 (2.7b) and d D d , we can show that the critical temperatures in different bands dr dr vary as:   „2 ˆ 2 0 ; (2.37a) n Tc1 D Tc1  4m1 R2 1 ˆ0   „2 ˆ 2 0 Tc2 D Tc2  ; (2.37b) n 4m2 R2 2 ˆ0 where R is the radius of cylinder. Then the critical temperature of the two-band superconductor cylinder is determined by following expression [see (2.11)]



"02 0 0 ; Tc  Tc1 Tc  Tc2 D 1 2

(2.38)

0

where " D " C R"12 .n  ˆˆ0 /2 [223]. As followed from (2.38), due to different period of oscillation of critical temperature in different bands (2.37a,b), strong periodicity in changing Tc of twoband superconductor is absent in contrast single-band superconductors. However, more accurate calculations [224] give general formula describing the dependence cˆ of relative shift of the superconducting transition temperature, tc .ˆ/ D Tc T , Tc on external magnetic flux, ˆ. As a particular case, this formula contains the classical Little–Parks effect for single-band superconductors. Contrary to the above statements, the dependence tc D tc (ˆ ) is strictly periodic, as in the classical effect. The main distinction from the classical effect, which allows for experimental observation, lies in the nonparabolic character of the dependence tc (ˆ). In the case where the physical parameters of both the bands coincide, the graph tc (ˆ) exhibits  additional features (Fig. 2.7) [224]. The case of  D 0 correspond to single-band  superconductors. With increasing of interband interaction ; peak of Little–Parks oscillations symmetrically suppressed. Unfortunately, no experimental data on Little–Parks effect are available in two-band superconductors MgB2 , LuNi2 B2 C, YNi2 B2 C, and oxypnictides.

2.1.8 Thermodynamic Magnetic Field Hcm .T / and Specific Heat Jump C CN Free-energy difference between normal and superconducting states can be written as

F D 

ˇ1 ˇ2 j‰1 j4  j‰2 j4  2"j‰1jj‰2 j: 2 2

(2.39)

44

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors 0.07 0.06

η=0

Δtc

0.05 η = 0.5

0.04

η = 0.8

0.03 0.02 0.01

0

0.5

1.0 Φ / Φ0

1.5

2.0

Fig. 2.7 Little-Parks effect in two-band superconductors

Last term in (2.39) is related with interband mixing and leads to increasing of freeenergy differences and consequently of critical temperature. On the other hand, the thermodynamic magnetic field is related to the free energy difference by 

2 Hcm D F: 8

(2.40)

In this paragraph, we use the notation Hcm for the thermodynamic critical field of a bulk two-band superconductor, which is different from that for thin films. The calculations for thin films are also important but are not considered here. Using (2.6a), (2.6b), (2.39), and (2.40), one can obtain the following formula with an appropriate manipulation of the thermodynamic magnetic field. p .˛ .T /˛2 .T /"2 / Hcm .T / D  4 "2 ˇ 1˛ .T /Cˇ 3 . 1 2 2 ˛1 .T //  !1=2  2 " ˇ1 ˛2 .T / C ˇ2 ˛13 .T / 4 4 2  ˇ1 " C ˇ2 ˛1 .T /  2" ˛1 .T / : .˛1 .T /˛2 .T /"2 /

(2.41)

.T / ; where Hcm .0/D If it introduce a dimensionless parameter of the form hcm D HHcm cm .0/ p 1 2 4Tc . 1=2 C 1=2 /; and we then have a normalized form of the thermodynamic ˇ1

magnetic field,

ˇ2

2.1 GL Equations for Two-Band Isotropic Superconductors

 4

45

 2



!1=2 ." /2 D.  c2 /C.  c1 /3

D ." / C .  c1 /  2." / p . 2 C.2 c1  c2 / / D

hcm ./ D  p 1C D ." /2 D .  c2 / C .  c1 /3     2 C .2  c1  c2 / ; 4

(2.42)

2

where ." /2 D  " T 2 : Here, note that all the parameters are dimensionless, which 1 2 c can be expressed as in (2.10) and (2.26). With this form, we apply the Ruthgers formula for the two-band case to account for the specific heat jump at Tc : 1

C D Tc 4



@Hcm @T

2 :

(2.43)

Tc

For the normalized specific heat jump at Tc , we obtain [225]: 

c D

@hcm @

2 ;

(2.44)

 D0

C C0 1 Hcm .0/ 2 where c D C I Tc D 4 . Tc / : 0 Note that the thermodynamic magnetic field Hcm (T) is not a directly measurable quantity. Fortunately, it can be calculated from the specific heat measurements. In Fig. 2.8, we plot the temperature dependence of Hcm (T) in MgB2 using (2.42) (open squares). To produce the data in Fig. 2.8, we have used the same parameters as in Refs.[200] and [201]. Empirical data for hcm (full squares) were extracted from the  results of Bouquet et al. [196] with Hcm .0/ D 0:36 T. Similar results were also observed experimentally in the work of Wang et al. [197]. Substituting the calculated value of @hcm =@ at the critical temperature into the (2.44), we can estimate the specific heat jump at Tc . The estimated value for the jump is 0.64, which is small compared to a value unity calculated from the single-band GL theory. However, the value is consistent with the experimental data in [196], accordingly to which C D 0:8 is smaller than the single-band BCS value of 1.43 CN [204]. Similar two-band BCS calculations for the reducing specific heat jump were conducted by Moskalenko et al. [226] and late [227, 228].

2.1.9 Critical Current Density jc .T / It is well known that critical current density of superconductors is determined by the expression [204, 207] (2.45) jc D 2ens .T /vc .T /;

46

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors 0.6

0.5

0.4

hcm

0.3

0.2

0.1

0

0.6

0.7

0.8

0.9

1.0

1.1

T / Tc

Fig. 2.8 Temperature dependence of thermodynamic magnetic field for MgB2

where ns .T /, is the superfluid density and vc .T / is the critical velocity of Cooper pairs. On the other hand, superfluid density ns .T / is related with London penetration depth by following formula [204] ns .T / 2 .0/ D 2 : ns .0/  .T /

(2.46)

London penetration depth in framework two-band GL given by the expression (2.22). Critical velocity of Cooper pairs vc is determined by the coherence length [204] 2„ vc D ; (2.47) m0 .T / where m0 is the effective mass of Cooper pairs, and coherence length given as:

2 .T / D

ˆ0 : 2Hc2

(2.48)

As a result of calculations, we can get final expression for normalized critical current density [229]: 2 .0/ jc .T / D 2 .hc2 .T //1=2 : (2.49) jc .0/  .T /

2.2 Anisotropy Effects in Two-Band GL Theory

47

0.5

jc (T ) / jc (0)

0.4

0.3

0.2

TB

0.1

0 0.6

SB

0.7

0.8

0.9

1.0

T / Tc

Fig. 2.9 Temperature dependence of critical current density for MgB2

Figure 2.9 shows the critical current density in (2.49) to compare with that given / by the single-band GL model [204] jjcc.T D .1  TTc /3=2 : It is clear that both .0/ curves exhibit positive curvature of critical current density at Tc . Triangle symbols in Fig. 2.9 show experimental data of recent measurements for MgB2 in [230]. It is clear that two-band GL gives a good approximation to experimental data [230] (see also review [231]). Similar measurement of critical current density was conducted in [232] for thin films of YNi2 B2 C. As followed from this study critical currernt density jc is the characteristic of superconductor and independent on the geometry of example. Theoretical derivation of such conclusion in general case can be found in [233].

2.2 Anisotropy Effects in Two-Band GL Theory 2.2.1 Layered Two-Band GL Equations Results presented above section are suitable for the explaining experimental data of a bulk polycrystalline samples. Studies with growths of single crystals of magnesium diboride, MgB2 and nonmagnetic borocarbides [90, 234], [235] show

48

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

anisotropy of physical properties. The mass anisotropy parameter  D . mmabc /1=2 of MgB2 in literature ranges from 1.2 to 9 in polycrystalline samples, 4.31–4.36 in single crystal [234]. From this point of view, derivation and calculations in framework of anisotropic two-band GL model seems attractive. Single-band s-wave GL theory for layered superconductors was developed in works [236–238]. One can write the two-band GL functional for layered superconductors in the form: F Œ‰1n; ‰2n  D

XZ

  d 2 r F1n C F1n;2n CF2n CF1n;1.nC1/ C F2n;2.nC1/ CH 2 =8 ;

n

(2.50) with Fi n

ˇ ˇ2  ˇ „2 ˇˇ 2 i A ˇi;n 4 2 ‰i n ˇˇ C ˛i;n .T /‰i;n ‰ ; D C r2d  ab ˇ ˆ0 2 i;n 4mi

(2.51)

   C c:c: F1n;2n D " ‰1;n ‰2;n      2 i A 2 i A  C "1 r2d C ‰1;n r2d  ‰2;n C c:c: ; (2.52) ˆ0 ˆ0 ˇ  ˇ „2 ˇˇ 2dAz ˇˇ2 Fin;i.nC1/ D (2.53) ‰i n  ‰i;.n˙1/ exp i ˇ ; 4mci d 2 ˇ ˆ0 where we choose x; y; z lying along the a; b and c crystallographic axes, respectively. Due to identical character of planes, we can write: ˛i n D ˛i ; ˇi n D ˇi . d is the distance between planes. The choice of the vector potential A as A D .0; H x; 0/ corresponds to the perpendicular component of the magnetic field H D .0; 0; H /: In this case, GL equations for two-band layered superconductors can be reduced to ? (2.8a, 2.8b). Calculation of Hc2 leads to ? .T / D Hc2

ˆ0 ; 2 2 ?

(2.54)

where effective coherent length eff of two-band superconductors is given by the expression 2 D

?

„2 " 4

1

#:   m1 ˛1 .T / C m2 ˛2 .T / C 8""1„m21 m2 q 2 C m1 ˛1 .T / C m2 ˛2 .T / C 8""1„m21 m2 4m1 m2 .˛1 .T /˛2 .T /"2 / (2.55) 

? At small values for the upper critical field Hc2 .T / is true:

2.2 Anisotropy Effects in Two-Band GL Theory

? Hc2 .T /

„c D 2e

„2 4

49

  ˛1 .T /˛2 .T /  "2 h i: ˛1 .T / ˛2 .T / 8""1 C C 2 m2 m1 „

(2.56)

k For the calculation Hc2 , we choose H D .0; H; 0/ and A D .0; 0; H x/: Then GL equations for two-band superconductors are reduced to the following form:

  2dH x 1  cos ‰1 D 0; ˆ0   (2.57a) d 2 ‰1 „2 d 2 ‰2 „2 2dH x  C ˛2 ‰2 C "‰1 C "1 C 2 c 2 1  cos ‰2 D 0: 4m2 dx 2 dx 2 4m2 d ˆ0 (2.57b) By elimination, we can get equations for ‰1 and ‰2 from (2.57a) and (2.57b), which turn out to be identical  2   2  „2 „2 d 4 ‰1 d ‰1 „2 2dH x „  ˛ C ˛ C ˛ ˛ ‰ C 1  cos 1 2 1 2 1 4m1 4m2 dx 4 4m2 4m1 dx 2 ˆ0      „2 „2 d 2 „2 „2 d 2 C ˛2 C 2 c 2  C ˛1 ‰1 2 c 2  2 4m1 d 4m2 dx 4m2 d 4m1 dx 2   d2 d4 D "2 C 2""1 2 C "21 4 ‰1 : (2.58) dx dx d 2 ‰2 „ 2 d 2 ‰1 „2  C ˛ ‰ C "‰ C " C 2 1 1 2 1 4m1 dx 2 dx 2 4mc1 d 2

4

By neglecting by high derivatives of order parameter ( ddx‰41 / and small terms, we k

can obtain the Mathieu equation for the calculation of upper critical field Hc2 :   2  „2 „2 „2 „2 d ‰1 ˛1 C ˛2 C 2""1 C 2 ˛ C ˛ 2 1 4m2 4m1 dx 2 4mc1 d 2 4mc2 d 2     2dH x (2.59) ‰1 D "2  ˛1 ˛2 ‰1 : 1  cos ˆ0 



2.2.2 Anisotropy Effect on Upper Critical Field in Layered Two-Band Superconductors ˆ0 At small magnetic field H << 2d 2 and after expansion of cosines in (2.59), we can get the final expression for anisotropy parameter of upper critical field

Hc2

Hk D c2 D ? Hc2

"

rm .T  Tc1 / C .T  Tc2 / C 8"2 rm Tc m2 1 r .T  Tc1 / C m .T  Tc2 / mc m mc 2

1

#1=2 :

(2.60)

50

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors k

ˆ0 At high magnetic field H > 2d 2 , upper critical field Hc2 can be defined from the lowest eigenvalue of Mathieu equation [239] and is given by the following expression [240]

˛2 4m„c d 2 C ˛1 4m„c d 2 1 2

2 „2 „2 „2 ˛ C 4m1 ˛2 C 2""1 ˛2 4mc d 2 C ˛1 4m„c d 2  4m2 1 2

k Hc2

ˆ0 D 2d

2

2

1

"2 ˛1 ˛2 2

1=2 : (2.61)

It means that k Hc2 

1 .T  T  /1=2

;

(2.62)

where T  is given by the following expression: T  D Tc 

„2 „2  : 4mc1 d 2 1 4mc2 d 2 2

(2.63)

Experimental study of anisotropy of the superconducting state properties in MgB2 was conducted in studies [241–243]. In Fig. 2.10, we plot anisotropy parameter Hc2 versus reduced temperature T =Tc : Experimental results from of Lyard [242] is given by the full symbols. The open points denote the results of calculations from above presented two-band layered GL theory. The same parameters were also used in [200, 201] to determine temperature dependence of superconducting state parameters in the framework of isotropic two-band GL theory. m1 2 Anisotropy mass parameters for single crystals m mc2 D 1:3 and mc1 D 0:03 are the same as in [244]. As followed from (2.60) influence of  (weak) band is effectively

16 14 12

gHc 2 (T )

10 8 6 4 2 0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

T / Tc

Fig. 2.10 Temperature dependence of anisotropy parameter for single crystals of MgB2 (full symbols exp.data from [242], open symbols anisotropic two-band GL theory)

2.2 Anisotropy Effects in Two-Band GL Theory

51

“switched of” and anisotropy parameter mainly defined by the  (strong) band. As a consequence, at small magnetic field there is good agreement with experimental data on investigation of anisotropy of upper critical field. Increasing of  with decreasing of temperature was observed experimentally by many groups [245,246]. Thus, there is consensus with understanding of temperature behavior of Hc2 : k

At high magnetic field, Hc2 goes to infinity as .T  T  /1=2 : It means that the orbital depairing effect of a magnetic field parallel to the layers does not destroy the superconductivity. This correspond to the case where the cores of the vortices fit between the superconducting layers and external magnetic field has no effects on the superconductivity. In fact, other magnetic mechanisms will limit the k divergence. The divergence of Hc2 at T  will be removed by taking into account spin-orbit scattering [247] and paramagnetic effect [248,249]. Similar anisotropy of upper critical field was observed for the other class of two-band superconductors– nonmagnetic borocarbides Y N i2 B2 C and LuN i2 B2 C [250]. Here, it is necessary to remark that similar two-band GL equations also was discussed in [251,252]. However, in equations presented in [251,252], terms similar to intergradient interaction in (2.4a), (2.4b), (2.8a), and (2.8b) are absent. As shown in [200, 201] maximal positive curvature of upper critical field of bulk samples can be achieved by inclusion of a intergradient interaction. In the case of no intergradients of order parameters  D 0, the curvature reaches maximum at the point of 0:5Tc. Intergradient interaction shifts this maximum to the region close to critical temperature. Such behavior is in good agreement with experimental data for a bulk samples (see Sect. 2.1). As we can see from (2.60), in the case of anisotropic GL equations, intergradient term also plays a crucial role in temperature dependence of anisotropy parameter Hc2 . Another version of GL approximation was presented in [253]. This approach corresponds to an effective single-band GL theory. In the framework of theory [253], the ratio of order parameters is temperature and field independent, i.e., is constant. It means that two-band GL theory is equivalent to effective singleband approximation. In contrast to [253], in our consideration the ratio of order parameters is temperature and field dependent [see also (2.4b), (2.4b), (2.8a) and (2.8b)]. Similar calculation using two-band GL theory with anisotropic mass tensor was carried out in paper [254].

2.2.3 Effects of Anisotropy on London Penetration Depth Using (2.20) and (2.21) in the case of H D .0; 0; H /, we can show that the London penetration depth along the superconducting layer ? is determined by the expression (2.22). For the calculation k ; we choose H D .0; H; 0/ and A D .0; 0; H x/: Taking into account equilibrium value of difference of phases (in  in˙g D 2 n/, we can get equation for vector potential

52

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

curlcurl A 2 D 4 ˆ0



  „2 2 „2 Az d : n1 .T / C n2 .T / sin 4mc1 d 4mc2 d ˆ0

(2.64)

ˆ0 At small magnetic field H  2d 2 and after expansion of sinuses in (2.64) and using Maxwell equation, we get final expression for the magnetic field, which is the same as (2.21) with replacement

2 k .T / D

4e 2 c2



n1 .T / n2 .T / : C mc1 mc2

(2.65)

Correspondingly, for the anisotropy parameter of London penetration depth  D k , we obtain the expression: ?  D

n1 .T / m1

C 2"1 .n1 .T /n2 .T //0:5 C n1 .T / mc1

C

n2 .T / mc2

n2 .T / m2

:

(2.66)

In general case, without expansion of sinuses, and high magnetic field H > ˆ0 ; (2.64) gives solution corresponding a single vortex, directed parallel to the 2d 2 superconducting layer. In this case, boundary condition requiring that the total magnetic field flux through the yz plane to be equal to the flux quantum ˆ0 . Furthermore, it seems interesting angular dependence of magnetic field in a vortex. Existence of the second-order parameter can lead to additional angular dependence of the magnetic field in a vortex. Using the solution of Ferrell–Prange equation [204] for the vector potential A, we can show that expression for the London penetration depth remains as in (2.65). It means that temperature dependent anisotropy of the London penetration depth  given by the (2.66) is true in all magnetic fields. In Fig. 2.11, we plot anisotropy parameter  versus reduced temperature T =Tc . Experimental data from Lyard [255] given by the full symbols. The open symbols denotes results of calculations using (2.6a), (2.6b), and (2.66). Due to negative sign of intergradient interaction  be decreasing of temperature, anisotropy factor of the London penetration depth  also decreased. Similar experimental results were obtained also in study Cubitt [256], Zehetmayer [235]. In papers [244, 257], within weak-coupling two-band anisotropic BCS model by introducing average parameters was calculated anisotropy parameters of Hc2 and : Results of this calculations also in agreement with above presented two-band GL theory calculations. The anisotropy parameter of London penetration depth  evaluated for two-band superconductors with arbitrary interband and intraband scattering times using Eilenberger theory in [258]. As shown by Bulaevskii [237] in the case of single-band layered superconductors k ˆ0 ? D 2 ˆ?0 k ; Hc2 D 2 upper critical field is defined by expressions: Hc2 2 : Note k

that in this case anisotropy parameter Hc2 is temperature independent. As stated in beginning, all coefficients ˛ and ˇ in GL model is field independent. Other generalization of considered model is related with introducing the field dependent

2.2 Anisotropy Effects in Two-Band GL Theory

53

Fig. 2.11 Temperature dependence of anisotropy parameter of London penetration depth for single crystals of MgB2 (full symbols exp.data from [255], open symbols anisotropic two-band GL theory)

parameters ˛ and ˇ. It is necessary to remark more recent study taking into account field-dependent two-band GL theory without intergradient interaction term [259].

2.2.4 Single Vortex in Two-Band Layered Superconductor For a single vortex centered at the origin the solution of equation for distances r  k is known to be given (see [204]) as ? D Hc1

  ˆ0 ? C  ln 0 :

k 42?

(2.67)

The quantity 0 corresponds to the “core” energy of the vortex filament, and 0  1 [204]. Structure of single vortex in perpendicular magnetic field is similar to twodimensional superconductors. For the magnetic field H D .H; 0; 0/, minimization of the free energy functional gives following equations   2ed P „2 1 @H 2 1 D n .T / sin    C A d C r.n1 .T /n2 .T / 2 i in inCg z 4 @y „c 4mci d ˆ   0  2dAz 2dAz  sin 1n  2nC1 C C sin 2n  1nC1 C ; ˆ0 ˆ0 (2.68)

54

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

   2Ay 1 @H 2e X ni .T / d1n  ; (2.69) D  4 @z „c mi dr ˆ0   1 „2 @2 1;n „2 2  C sin 1n  1nCg C g Az d  r.n1 .T /n2 .T / 2 4m1 @r 2 4mc1 d ˆ0      2dAz 2dAz C sin 2n  1nC1 C D 0;  sin 1n  2nC1 C ˆ0 ˆ0 (2.70)   „2 @2 2;n 1 „2 2  sin  C   C g A d  r.n1 .T /n2 .T / 2 2n 2nCg z 4m2 @r 2 4mc2 d ˆ0      2dAz 2dAz C sin 2n  1nC1 C D 0:  sin 1n  2nC1 C ˆ0 ˆ0 (2.71) The system of last equations is nonlinear. The elimination of 1n and 2n is carried out after expansion of sine function in (2.70)–(2.71). Taking into account discrete character of z variable and procedure of replacing the finite differences by differentiations in (2.70) and (2.71), we can get following system of equations: @2 H 2k .T / 2 @y

2k ˆ0 n1 .T / H D  1 2 @z 2 n1 .T / C n2 .T / ?

@2 H C2? .T / 2

@2 1n m1 @2 1n 4rm1 C  @y 2 mc1 @z2 „2 m2 @2 2n 4rm2 @2 2n C  @y 2 mc2 @z2 „2

 

n2 .T / n1 .T / n1 .T / n2 .T /

 12  12

!

@2 . 2n /; @z@y 1n (2.72)

.1n  2n / D 0;

(2.73)

.2n  1n / D 0;

(2.74)

where 2k is the penetration depth in the direction perpendicular to planes, determined as  4e 2 n1 .T / n2 .T / 4d 2 r 1=2 .T / D C C .n .T /n .T // : (2.75) 2 1 2 k c2 mc1 mc2 „2 Equation (2.72) gives solution corresponding a single vortex, directed parallel to the superconducting layer. In this case, boundary condition requiring that the total magnetic field flux through the yz plane to be equal to the flux quantum ˆ0 . As one you can see from (2.72) in contrast to single-band superconductors, equation for magnetic field in two-band superconductors is a nonhomogeneous. For the calculation of magnetic distribution in two-band superconductors, it is necessary to solve differential equations for 1n and 2n : The solutions of (2.72) and (2.73) in the case of small coupling between superconducting planes are a form:

2.2 Anisotropy Effects in Two-Band GL Theory

55

(

1

i n .y; z/ D tan

mi mci

1=2

) y : x

(2.76)

Using transformationy D k  sin  and x D ?  cos , we can rewrite (2.72) as 

  @ @H @2 H  2 H D g./;  C @ @ @ 2

(2.77)

where 2 sin2   cos2  22 sin2   cos2  C  g./ D 1  1 2  2 :  2 cos2  C 21 sin2  cos2  C 22 sin2 

(2.78)

In last equation were introduced notations:  i D

mi mci

1=2

k I i D 1; 2: ?

(2.79)

Using formula for the Fourier harmonics of right side of (2.77) g./ D where coefficients gn D

2 

Z

X

gn cos.n/;

(2.78)

g./ cos.n/d;

(2.79)



0

solution of the equation for the magnetic field (2.77) can be written as: H.; / D

X

hn ./ cos.n/:

(2.80)

The equation for the hn ./ has a form of nonhomogeneous Bessel equations. After some transformations under  << 1, we have following expression for magnetic field H.; / [260]   1 2 ˆ0  ln  C  cos.2/ ; H.; / D 2? k 2 where D

X

i

.1 C i /2 i D1;2

:

(2.81)

(2.82)

As followed from last equation (2.81), existence of two order of parameters and their anisotropy leads to an additional angular dependence of magnetic field in a vortex. Nonsymmetric behavior of magnetic field in a vortex in a single-band

56

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

layered superconductors was investigated by [237,247,261]. Transformation to .zy/ coordinates gives for magnetic field under conditions y<<k and z<<? "  2  2 !  2  2 !# y z ˆ0 z y  ln C : H.y; z/ D C  2? k k ? ? k (2.83) Using last expression for magnetic field in a single vortex H.y; z/, we can calculate the energy of the vortex. Due to that vortex lies between the superconducting layers, the lower limit of integration with respect to z must be equal to d . Result can be presented as k Hc1

      ˆ0 1 ? ' C2 2  :  ln 4? k d 

(2.84)

Structure of single vortex in two-band layered superconductors in arbitrary angle of applying external magnetic field seems very interesting problem and will be subject of future investigations.

2.2.5 Surface Magnetic Field in Anisotropic Two-Band GL Theory .l/

In study [262] in free energy functional was introduced effective mass tensor mj k , c3 where l D 1; 2 and j; k D x; y; z: Calculated ratio H using effective mass tensor Hc2 in two-band GL theory presented in Fig. 2.12. Case i / correspond to the relations .l/

.l/

.l/

.l/

mxx D ma D myy D mb and

.1/

mxx .1/ myy

D

.2/

mxx .2/ myy

D 1 and as a result

Hc3 Hc2

is constant.

Obtained result is similar to that for single-band superconductors [263]. Case (ii) – .l/

.l/

.l/

.l/

In this case, mxx D mc D myy D ma and

.1/

mxx .1/ myy

¤

.2/

mxx .2/ :The myy

ratio

Hc3 Hc2

strongly

depends on temperature with a minimum value at T  0:84Tc and a maximum value c3 of 1.6589 at T D Tc . From Fig. 2.12, it is evident that the ratio H is nearly constant Hc2 at low temperature. This is because, at low temperature, the fields Hc2 and Hc3 are dominated by the -band. With increasing temperature, the effect of the band on the fields Hc2 and Hc3 becomes stronger. At high temperature, the  and -bands together determine the fields Hc2 and Hc3 , which makes the dependence of Hc2 and c3 Hc3 on temperature complicated. Thus, it is reasonable that the ratio H depends on Hc2 Hc3 temperature. Figure 2.12 indicates that Hc2 for this case is less than that for case (i). Case (iii) – In this case, in equations we take into account angular effects on surface magnetic field. For the same parameters as those in case (ii), the results are plotted c3 in Fig. 2.12 as a dashed line. From Fig. 2.12 it is evident that the ratio H Hc2 is different when the superconductor surface coincides with the different crystallographic plane. Details of angular effects on Hc3 see following paragraph. This result agrees well

2.2 Anisotropy Effects in Two-Band GL Theory

57

1.66

Hc3 / Hc2

1.64 1.62 1.60 case(i) case(ii) case(iii)

1.58 1.56 0.0

0.2

0.4

0.6

0.8

1.0

T / Tc

Fig. 2.12 Surface critical magnetic field Hc3 (T) of two-band superconductors

with the experimental data obtained for MgB2 samples [264,265]. Similar result for dirty two-band superconductors was obtained in study [213].

2.2.6 Angular Effects in Two-Band GL Theory 2.2.6.1 Upper Critical Field Hc2 Let us consider discuss the out-off-plane behavior of the upper critical field Hc2: In the single-band GL theory [207], when magnetic field H tilted from c axis by an angle ; the upper critical field has an elliptic angulardependence: HSAGL .; T / D q

with Hc2 D

? Hc2 2 cos2  C Hc2 sin2 

;

(2.85)

k

Hc2

? Hc2

is the temperature independent constant and is determined by the

anisotropy of effective mass:

r Hc2 D

m? : mk

(2.86)

Experimental works on the study upper critical field in M gB2 have shown that not only Hc2 changes with temperature [235,242,246,266,267], but deviation from the elliptic angular dependence (2.85) grows with increasing temperature. Theoretical calculations of angular dependence using quasi-classical Uzadel equations was conducted in [268]. In this paragraph, we consider two-band GL theory with different anisotropy of masses in different bands. The method presented above for

58

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

the calculation of the upper critical field Hc2 is true in this case also. One needs only to replace effective masses m1 and m2 by an angular dependence 1 1 ) mi mi

1=2  mi 2 2 : cos  C c sin  mi

(2.87)

At the vicinity of critical temperature Tc expression for the upper critical field Hc2 can be expressed as: Hc2 .; T / D

„c 2e

„2 4



.˛1 .T /˛2 .T /  "2 /

1=2 ˛1 .T / ˛2 .T / 2 C m2 sin2  C cos cos2  C c m2 m m1 2

m1 mc1

sin2 

1=2

: 1 C 8"" „2

(2.88) At small angles  << 1, we obtain following expression for upper critical field Hc2 .; T /;



m1 m2 C rm .T  Tc1 / 1  2m .T  Tc2/ 1  2m c c Hc2 .; T / 1 2 1 '  2; ? .T  Tc2 / C rm .T  Tc1 / C 8"2 rm Tc Hc2 .T /

(2.89)

where was introduced dimensionless parameters as in [215]. At large tilt angles, cos  << 1; upper critical field is given by a formula Hc2 .; T / Hc2 .;T / k

Hc2 .T /

1

"



rm .T Tc1 /

mc2 m2



1=2 C.T Tc2 /

mc1 m1

1=2 #

 2 1 8 9   :



1=2 1=2 2 c c > m2 m1 2ˆ ˆ > 2 ˆ > ˆ > r .T  T / C .T  T / C 8" r T C c1 c2 m c < m = m2 m1   c 1=2 c 1=2   1 ˆ > 2 m m ˆ >  ˆ > ˆ > C .T  Tc2 / m11  rm .T  Tc1 / m22 :C ; 2 2 (2.90) For the intermediate angles, we obtain the following expression '

2

Hc2 .; T / k Hc2 .T /

1 ' 8 ˆ ˆ ˆ <0



c 1=2  3 1=2 m2 1 m2 r C7 .T  T /  c m c1 6 m 2 m2 2 6  2

c 1=2  7 1=2 4 5 cos  m1 1 m1 .T  Tc2 /  c m 2 m1 .rm .T  Tc1 /



1

1=2

m2 mc2 1=2

C .T  Tc2 /



mc1 m1

1=2

/

9: > > > =

1  c  c 1=2 m m rm .T Tc1 / m2 C.T Tc2 / m1 1 2 1 ˆ > 2 2 @ 2 ˆ  c 1=2  c 1=2 cos  A C 8" rm Tc > ˆ > : sin  C 2 .T ; m2 m1 rm Tc1 / C.T Tc2 / m2

m1

(2.91)

2.2 Anisotropy Effects in Two-Band GL Theory

59

7 6

SB GL,T=Tc SB GL,T=0.6 Tc TB GL,T=Tc TB GL,T=0.6 Tc

5 hc2 4 3 2 1 0

1 θ(rad)

Fig. 2.13 Angular dependence of the upper critical field in two-band superconductors

m1 As followed from expression (2.90) if each band has the same mass anisotropy( m c D 1 m2 /; due to intergradient interaction Hc2 .; T / deviate from elliptic dependence. mc2 Expression for upper critical field Hc2 .; T / shows that the deviation grows with the 1 2 disparity between m and m : On the other hand, disparity in two-band GL theory mc1 mc2 depends on the temperature. So it increases when departing from Tc : Examples of angular dependence of upper critical field at low and high temperatures presented in Fig. 2.13. Due to contribution from  band, one can see significant deviation from single-band anisotropic model at high temperatures. As input parameters we used the values Tc1 D 20 K, Tc2 D 10 K, "2 D 3=8; "1 D 0:0976 corresponds to MgB2 : According to the microscopical calculations ratio of masses in different bands is equal rm D 3. We also use following data for mass anisotropies in m2 1 different bands ( m mc1 D 1:3; mc2 D 0:03/: Similar value was used by Miranovich et al. [244]. As followed from Fig. 2.13 and (2.90) at high temperatures maximum deviation from single band theory is achieved around   75ı : At low temperatures, deviation from single band effective mass behavior is small (Fig. 2.13). In Fig. 2.14 presented temperature dependence of the anisotropy parameter of upper critical field, calculated from (2.89)–(2.91) using values at  D 0 and  D =2

H ab Hc2 .T / D c2c D Hc2

s

mc2 .T  Tc2 / C rm .T  Tc1 / C 8"2 rm Tc q c : m2 m2 m1 .T  T / C r .T  T / C 8"2 r T c2 m c1 m c m2 mc 1

(2.91a) It is necessary to note that even though the experiments have systematic differences among themselves, there is consensus on understanding of temperature behavior of anisotropy parameter Hc2 [242, 246], [267] there seems to be a general

60

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors 7 6

γHc 2

5 4 3

exp data Angst et al anisotropic TB GL exp data Shi et al exp data Lyard et al

2 1 0.5

0.6

0.7

0.8

0.9

1.0

T / Tc

Fig. 2.14 The temperature dependence of anisotropy parameter of upper critical field Hc2 (T)

decreasing trend in anisotropy parameter Hc2 at Tc approached. In contrast to Sect. 2.2.2, in this case, Hc2 .T / is limited, i.e., divergence at low temperatures is absent. The enhanced deviation from single-band GL theory can be characterized .;T / by the parameter max .1  A/ , where A D HHc2 ab .;T / is the ratio of the upper critical c2

field in the two-band and single-band models. The temperature dependence of this parameter is presented in Fig. 2.15. The result of two-band GL theory is given by the solid line, while experimental data of Lyard et al. [242] are presented by full circles. As seen in Fig. 2.15 at low temperatures deviation is small and with increasing temperature it first increases and as Tc approached it starts to decrease. At T =Tc  0:9, the deviation parameter reaches maximum .0:2).

2.2.6.2 Surface Magnetic Field Hc3 As mentioned above, angular effect on Hc3 (T) was calculated in [262] introducing relations 1.l/ ) . 1.l/ cos2  C 1.l/ sin2 / and 1.l/ D 1.l/ : Corresponding equations myy

mb

mc

mxx

ma

are solved numerically. In Fig. 2.16, the solid and dashed lines correspond to at T D 10 and 30 K, respectively. At T D 10 K, the minimum for ı

at   78 ; at T D 30 K, the ı

Hc3 . / Hc2 . /

Hc3 Hc2

Hc3 . / Hc2 . /

is obtained

curve monotonously decreases with  and

c3 ratios at T D 10 reaches a minimum at  D 90 : In Fig. 2.16, at  D 50ı , the H Hc2 and 30K are nearly constant. This can be explained by the fact that at small angles the anisotropy of the -band is almost the same as that of the -band. When  is large, the anisotropy of the -band ( ) is sharply different from that of the -band c3 ( ). Because the ratio  = changes with , the ratio H Hc2 depends on the angle .

2.2 Anisotropy Effects in Two-Band GL Theory

61

0.25

anisotropic TB GL exp data Lyard et al

max(1-A)

0.20

0.15

0.10

0.05

0.00 0.5

0.6

0.7

0.8

0.9

1.0

T / Tc

Fig. 2.15 The temperature dependence of deviation parameter of upper critical field A from single band superconductors

Fig. 2.16 Angular dependence surface magnetic field of two-band superconductors

Figure 2.16 also shows that the ratio Hc3 . / Hc2 . /

Hc3 Hc2

depends on temperature, because the curves

are different at T D 10 and 30 K, which agrees with the result for case (ii).

It is instructive to estimate region of validity of the GL theory applied to MgB2 within above presented approach. Due to large difference in anisotropy, c-axis 2 coherence length of second (weak) band 2c D  4mc„˛2 .T / is much larger than 2

62

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

coherence length of first (strong) band 1c D  4mc„˛1 .T / . In the case common 1 superconductivity in different bands, effective coherence length will be c << 2c at low temperatures. Applicability of GL approach is determined by the condition

c >> 2c , which is equivalent to 2

„2 =4



˛1 .T / m2

C

˛2 .T / m1



C 2""1

˛1 .T /˛2 .T /  "2

>

„2

: 4mc2 ˛2 .T /

(2.91b)

Using the numerical values of parameters, we can show that violation of above condition occurs for T  27K: This means that the temperature region is much wider than the narrow region suggested by Golubov–Koshelev [253]. As stated in the beginning, all coefficients ˛ and ˇ in GL model is field dependent. Another generalization of considered model is related with introducing the field-dependent parameters ˛ and ˇ. Possible inclusion field-dependent coefficients in framework two-band GL are subject of possible investigations. Another interesting problem is the study of Hc1 .; T / and .; T / dependences in the framework of anisotropic two-band GL theory. Thus, we conclude that the anisotropic two-band GL theory explains the deviation from elliptic law for the angular dependence upper critical field Hc2 .; T /:

2.3 d-Wave GL Equations Perhaps, the most gratifying event after the discovery of superconductivity in cuprate compounds [1] is that d-wave symmetry is established in hole-doped compounds [48, 49]. The order parameter characterizing the d-wave symmetry is given by

.k/ D cos.2/; (2.92) where  is the angle the vector k (within the a  b plane) makes from the a-axis. The angular dependence of .k/ shown in Fig. 2.17. In contrast to a variety of s-wave order parameter, there are two remarkable peculiarities. First, when  crosses =4; 3=4; 5=4 and 7=4; .k/ changes sign. So moving around  D 0 to 2, .k/ changes sign, 4 times. Second, associated with this crossing .k/ vanishes, .k/ D 0 in this four points. It means that there are four nodal lines running parallel to kz in .k/ for the cylindrical Fermi surface. In derivation of GL equations for d-wave superconductors, we assume that the d-wave superconductor is rather similar to the s-wave superconductor, except that the interaction potential favor the d-wave symmetry due to its Coulombian origin and is given by the expression [269] N0 V .k; k0 / D 2 cos.2/ cos.2 0 /  ;

(2.93)

2.3 d-Wave GL Equations

63

Fig. 2.17 The angular dependence of d-wave order parameter: dark region correspond to positive, while clear to negative sign of order parameter

where first term gives an attractive interaction, while the second term the on-site Coulomb repulsion. Here, N0 is the quasi-particle density of states in the normal state on the Fermi surface. Also for a homogeneous states ( .r/ D const), the Coulomb term plays no role. For derivation of d-wave GL equations in [270–272] general gap equation for order parameter was used

.x; x0 / D V .x  x0 /T

X

F C .x; x0 ; !n /;

(2.94)

where V .x  x0 / is the effective two-body interaction of weak coupling [273, 274]. Using Gorkov’s description of weakly coupled superconductors, taking effective interaction for spin-singlet pairing as

0

  0 V k  k0 D Vs C Vd kx2  ky2 kx2  ky2

(2.95)

and corresponding order parameter in similar way ‰  .R; k/ D ‰s .R/ C ‰d .kx2  ky2 /;

(2.96)

we can get GL equations for d-wave superconductors   

2Vs 1 2 2  1 2 2  2 1C ‰s C ˛s vF … ‰s C vF …x  …2y ‰d Vd 2 4   3 C 2 ‰s C 2.‰d /2 ‰s C .‰d /2 ‰s ;

(2.97)

64

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors







1 2 2  1 2 v … ‰d C vF .…2x  …2y /‰s 2 F 4  3 C 2.‰s /2 ‰d C .‰s /2 ‰d C .‰d /2 ‰d ; 4



d ‰d ln

Tc T

C ˛d

(2.98)

7.3/ where … D i rR  2eAR ; ˛ D 8 2 T 2 ; d D N.0/Vd =2: The equation for critical c temperature is determined from BCS relation

  d ln

2e  !D Tc

 D 1:

(2.99)

2.3.1 Upper Critical Field of d-Wave Superconductors In (2.95)–(2.98) Vs and Vd correspond, respectively, to the s-wave and d-wave interactions and are positive; N.0/ is the density of states at the Fermi surface, and vF is the Fermi velocity. Equations (2.97) and (2.98) are valid for two-dimensional d-wave superconductivity. In order to meet the reality, we use the effective mass to include z-direction effect such as 2 …2z ‰s in (2.97) and 2 …2z ‰d in (2.98). Such an approximation is valid for the coherent length larger than the layer spacing. Near the upper critical field Hc2 , the amplitudes of the order parameters are small. We may ignore the nonlinear terms and obtain the linearized GL equations [275] ˛s ‰s C 21 .…2x C …2y /‰s C 2 …2z ‰s C 1 .…2x  …2y /‰d D 0;

(2.100a)

˛d ‰d C 21 .…2x C …2y /‰d C 2 …2z ‰d C 1 .…2x  …2y /‰s D 0:

(2.100b)

By assuming the applied field H=H(i sin +jcos/, in which  is the angle from the z axis (c axis), the upper critical field Hc2 for all directions of the external field in the a-b plane as Hc2 .; T / D

2˛s ˛d ; q  q p 2 2 2 2 2e 1 2.2 cos . /C12 sin . //˛d C .cos . /C12 sin . //˛s  DD 

(2.101)

where 12 D DD

D 12

2 1

and

q

Cq

2.2 cos2 ./

C 12 sin .//˛d C 2

2

q .cos2 ./

C 12 sin .//˛s

2 2 ˛s ˛d c00 2 sin4  q ; 2.cos2 ./ C 12 sin2 .// .2 cos2 ./ C 12 sin2 .//

2

(2.102)

2.3 d-Wave GL Equations

a

65

3

θ=0

Hc2(t)

2

1

0

b

0.6

0.8

1.0

t 2.0

θ=π / 2

1.6

Hc2(t)

1.2

0.8

0.4

0.0

0.6

0.8

1.0

t

Fig. 2.18 Upper critical fields as a function of reduced temperature t: (a)  D 0 and (b)  D =2, where  is the angle between the applied field and the c axes

where c00 D

p u u 2. vss /1=4 . vdd /1=4 . uvss C uvss /1=2

: Result of calculations for upper critical field of d-wave

superconductors is presented in Fig. 2.18. In this figure, the solid line represents results [275] and the dashed line corresponds to the asymptotic solution. Parameters chosen as 1 D 1; 2 D 0:588; d D 1; ˛s D 2: It is easy to see that the upper critical field is positive curvature from the slope of Hc2 . Thus, there is similarity with two-band isotropic GL theory (Sect. 2.1), presence of two-order parameter in superconductor leads to positive curvature of upper critical field near Tc . As T!Tc ,

66

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors 12

10

-dHc 2 /dT

8

6

4

2

0

10

30

50

70

90

θ

Fig. 2.19 Dependence of dHc2 /dT evaluated at Tc D 90 K on angle from the z-axis corresponding to the ratio of 12 D 0:03 [275]

Hc2 .; T / D

d 1 

T Tc



q  : 2e 1 1 cos2 ./ C 2 sin2 ./

(2.103)

It means that the curves show linear temperature dependence. There is no admixture of s wave in this limit. Calculated in study [275], the slope of upper critical field dHc2 /dT near Tc at various angles is plotted in Fig. 2.19.

2.3.2 Single Vortex in d-Wave Superconductor In this paragraph, we determine the single-vortex structure for d-wave superconductors using above presented GL equations. Structure of vortex in d-wave superconductors is very different from s-wave [276] or p-wave [277]. It is useful to note that vortex structure of d-wave superconductors has been investigated by [278] using Bogoliubov–de Gennes theory. In contrast to [278], using approach presented in [270] we can get temperature dependence of the order parameter. The solution of rewritten in polar coordinates GL equations can be presented as [270]

s D

 1  ae i C be 3i ; r2

(2.104)

2.3 d-Wave GL Equations

67

Fig. 2.20 The equal-interval contours of magnitude of the induced s-wave parameter

y

x

d

De

 g0 

 1 i.a C 3b/ C sin 4 ; 2g0 r 2 2r 2   Tc ; g02 D ln T

i

(2.105) (2.106)

aD

˛s C 10g02 =3 g0 ; .˛s C 4g02 =3/2  .2g02 =3/2 4

(2.107)

bD

3˛s C 14g02 =3 g0 : 2 2 2 2 .˛s C 4g0 =3/  .2g0 =3/ 4

(2.108)

As followed from (2.104) induced s-wave component decay as r12 , which profile presented in Fig. 2.20. The d-wave component is also modified by the anisotropic terms proportional to r12 and shows fourfold symmetry (Fig. 2.21). Corresponding to local magnetic field around a d-wave vortex can be presented as [270]

BD

8 <

B0 

h : ˆ0 g02 ln   2  r 22

1 g02 r 2

 2 c02 C 2c12 r 2 z; r ! 0 i : (2.109)  ab  aC3b cos 4 z; 0 << r <<  r2 r2

1 4



The distribution of the local magnetic field around the vortex is plotted in Fig. 2.22, which exhibits fourfold symmetry. Nonuniform distribution of magnetic field also obtained in framework of anisotropy two-band GL theory [260]. As shown by further investigations, there are two types of vortices, which can be realized in .d C s/-wave superconductors: (1) singular vortices which have at least one point where the superconducting gap is zero [270, 281] and (2) nonsingular vortices [279,280], where the gap is nonzero anywhere in the flux core. When the parameters of the GL theory are chosen so that there is a pure d wave homogeneous state without a magnetic field, the structure of vortex a singular flux line in .d C s/-wave superconductors exhibits a fourfold symmetry of the arrangement of the swave

68

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Fig. 2.21 The equal-interval contours of magnitude of the induced d-wave parameter

y

x

Fig. 2.22 The equal-interval contours of local magnetic field around of d-wave vortex

unit vortices around the dominant d -wave unit vortex for the magnetic field applied along the caxis [270, 281–283]. Four swave unit vortices which are situated on the crystal axes at equal distances from the d wave unit vortex have the same winding number as the d wave vortex. One swave vortex with opposite winding number is situated in the center of the d wave vortex. It is the point where the superconducting gap is zero. As the magnetic field tilts from the c-axis, the fourfold symmetry of the flux line turns to the twofold one due to anisotropy of mass tensor [284, 285]. As the angle  between the magnetic field direction and the c-axis increases, two of the swave vortices move away from the center of the flux line. At the same time, the other swave vortices move toward the swave antivortex. When  approaches a certain critical angle, the latter two vortices merge with swave antivortex into central vortex. Finally, when magnetic field lies in the ab plane ( D =2/; there is only one swave unit vortex in the d wave vortex core.

2.3.3 Nonlinear Magnetization in d C s Wave Superconductors For the calculations beyond the London approximation, nonlinear version of GL equations (2.100) taken into account [286]. The solution of system of nonlinear GL

2.3 d-Wave GL Equations

69

equations seeks as  j‰d j D j‰d 0 j C

H Hc

 j‰s j D j‰s0 j C

H Hc

2

2x  px  px A1d e 2 D C A2d e   C A3d e 2 s ;

(2.110a)

2

2x  px  px B1s e 2 s C B2s e   C B3s e 2 d :

(2.110b)

The coefficients have been calculated in study [286] and they depend on the GL coefficients terms beyond the first order in intergradient interaction coefficient have been neglected. The thermodynamic field is Hc D p„c , H applied external 2 d  magnetic field. Final expression for London penetration depth has a form: "



eff D  1 C

H Hc

2 

2C3 C2 C p C p 4 = d C 2 2 = s C 2 2 2C1

# :

(2.111)

The expressions for coefficients Ci , i D 1; 2; 3 above are given in paper [286]. A microscopic theory of the nonlinear Meissner effect within a purely one-component d-wave scenario has been formulated in study [287]. Their emphasis was on low temperature behavior where they predict H linear field dependence. However, even at very low temperatures this model predicts a transition to a regime where the corrections are of the order H 2 . In addition to that, a very small admixture of another component is able to change the behavior of the penetration depth at very low temperatures as a function of temperature. Presented in [286] two-band picture is complementary to this in that it is only appropriate for relatively high temperatures.

2.3.4 Vortex Lattice in d-Wave Superconductors For the study of structure of the vortex lattice near Hc2 [281], the variational solution will be sought for the general case in the form of normalized ground state wavefunctions of the harmonic oscillator [204] 'k˙ .x/ D

r

˙ 2 .x  xk /2 =2l ; p exp ˙ l 

(2.111)

where 'kC D ‰d C ‰s ; 'k D ‰d  ‰s : The variational parameters  and C will be determined by minimization of the eigenvalue of system of linearized equations (C D  cos #;  D  sin #/: The minimum occurs for  2 D tan # C tan1 # , and is  1 T T 1 „!c ˚ hEi .1 C "v / tan # C .1  "v / tan1 #  D C

T

T 4 T 2

s 2 tan # ; 1 C tan2 # (2.112)

70

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

where !c D 2eH ; T D Td  Ts I T  D .Td C Ts /=2; "v D v =d : Minimization mc of Gibbs free energy can be obtained [281] hgi  hgin D 

1 .Hc2  H /2 ; 8 .2 2  1/ˇA

(2.113)

where Abrikosov ratio ˇA and GL parameter  are determined as ˇA D

hh2s i hf4 i I  2 D 4 2 : 2 hhs i hhs i

(2.114)

f4 stand for quadratic invariants in complete average free energy density. The Abrikosov ratio ˇA defined by (2.114) is independent of the coefficients ˇi in the quadratic part of the free energy and depends only on the shape of the unit cell in the vortex lattice. To the extent that  is independent of the specific lattice shape, the minimum Gibbs free energy corresponds to the minimum of ˇA , which generalizes the familiar Abrikosov result, apart from writing it in terms of magnetic field instead of the absolute squared order parameter. As mentioned above, in order to determine the shape of the vortex lattice, one needs to evaluate the averages of the fourth order terms hf4 i and hh2s i: The shape of the vortex lattice unit cell is determined by the ratio R D Lx =Ly . The value of R that corresponds to the thermodynamically stable configuration, Rmin , is obtained by requiring that the Gibbs free energy is minimum. Equation (2.113) shows that, at given external magnetic field H , the Gibbs free energy hgi is entirely determined by the two parameters given above, ˇA and . Numerical evaluation of these parameters confirms that  is only very weakly dependent on the particular lattice shape, as it is illustrated by Fig. 2.23. The dependence of the Gibbs free energy (2.113) on R is almost entirely contained in the Abrikosov ratio ˇA and thus, in most of the parameter space, the minimum of hgi coincides to a good accuracy with the minimum of ˇA . For example in the particular case displayed in Fig. 2.23, the minimum of ˇA differs by less than 2% from the minimum the full free energy. Figure 2.23 also shows a typical dependence of ˇA on R for different values of the mixed gradient coupling "V , as obtained by numerical evaluation [281]. When "v D 0, the superconductor is in a pure d -wave state with no s-wave component present. Within the phenomenological GL theory, this situation is identical to the case of a conventional superconductor p studied by Abrikosov. Thus, the state with minimum free energy has Rmin D 3, which corresponds to the usual triangular vortex lattice. In this limit obtained [281], the correct value of ˇA D 1:1596 [204] However, as soon as a nonzero coupling "v is introduced, the situation changes and p the minimum of ˇA shifts to the values Rmin < 3, signaling that an oblique vortex lattice is favored. The minimum Rmin varies continuously with "v and at certain value of "v , which depends on the other parameters in the GL free energy, Rmin reaches the value of 1, corresponding to the square lattice. Further increase of "v

2.4 GL-Like Theory in Application to Cuprate Superconductors 1.40

71

2

κ

1.3

1.35 1.1

βA

1.30 0.9 1.0

2.0 R

1.25

3.0

1.20

εv 0.0 0.4

1.15 1.0

0.6

1.5

2.0 R

2.5

3.0

Fig. 2.23 Dependence of ˇA and R parameter on intergradient interaction

then has no effect on the shape of the lattice, which remains square. The examples of the oblique vortex lattice are also investigated in study [281]. Contour plot of the amplitudes of d component and s component of the order parameter in the vortex lattice presented in Fig. 2.23. The same parameters are used as in Fig. 2.24 with V D 0:45 resulting in an oblique vortex lattice with Rmi n D 1:29 and the angle between primitive vectors  D 76ı . The oblique unit cell containing one flux quantum is marked by a solid line (see Fig. 2.24) [281].

2.4 GL-Like Theory in Application to Cuprate Superconductors Very recently in study [288], a phenomenological GL-like theory of cuprate high temperature superconductivity was developed. The cuprate free energy is expressed as a functional F of the complex spin-singlet pair amplitude ij D m D m exp(i m ) where i and j are nearest-neighbor sites of the square planar C u lattice in which the superconductivity is believed to primarily reside and m labels the site located at the center of the bond between sites i and j . The system is modeled as a weakly coupled stack of such planes and ( m , m ) are the real magnitude and phase of the pair amplitude at site m. The corresponding functional has the form

72

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Fig. 2.24 d and s order parameters in vortex structure of two-band d C s superconductors

F . m ; m / D F0 . m / C F1 . m ; m /;  X B A 2m C 4m ; 2 m X

m n cos .n  m / : F1 . m ; m / D C F0 . m / D

<mn>

(2.115a) (2.115b) (2.115c)

2.4 GL-Like Theory in Application to Cuprate Superconductors

73

The coefficients A; B, and C are determined from comparison with experimental result, i.e., as function of hole density x and temperature T: Specifically, the coefficients are as follows:      x T exp ; (2.116a) A.x; T / D A0 T  T0 1  xc Tp B D B0 T0 ; C.x/ D xC0 T0 ;

(2.116b)

where A0 ; B0 ; C0 > 0 and T0 D Tc .x D 0/I Tp is order of T0 :

2.4.1 Transition Temperature Tc .x/ Critical temperature determined as nonzero value for the superfluid density s .x; T / given by formula s D

  C X

m mC cos m  mC 2Nb m;   C2 X X 

m mC sin m  mC 2Nb T m

!2 :

(2.117)

The calculated curve of Tc is approximately of the same parabolic shape as that found experimentally (Fig. 2.25). Qualitative agreement for Tc of La2x S rx C uO4 (for this compound, hole concentration x can be inferred directly and unambiguously from chemical composition). The causes for the qualitative disagreement at both ends, namely quantum-phase fluctuationsand low-energy electronic degree of freedom.

2.4.2 Superfluid Density Superfluid density can be calculated analytically in mean-field approximation: + X   C X s D .

m mC cos m  mC D C hS˛ i20 : 2Nb m; ˛Dx;y *

(2.118)

At low temperatures, the calculated s .x; T / decreases linearly with T from its 0 zero temperature value, i.e., s .x; T / D s .x; 0/  s .x; T /; the coefficient of the 0 linear term, namely s .x/ remains more or less independent of x for small x and approaches a constant value as x ! 0 on the underdoped side. The same trend can

74

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Q

Fig. 2.25 Critical temperature Tc of cuprate superconductors from GL-like theory, Tc critical temperature in the case of quantum-phase fluctuations

be observed in the experimental data [289, 290] for in-plane magnetic penetration depth ab , where 2 ab / s :

2.4.3 Specific Heat Specific heat can be written as Cv D

1 1 1 @hF i D . .hF 2 i  hF i2 / Nb @T Nb T 2    @A X 1  2 h m F i  h 2m ihF i ; h m i2  C @T m T

(2.119)

@A D .f 2 exp.T =Tp / C A=Tp /: Second term in last equation related with where @T temperature-dependent coefficient A: Theoretical results presented in Fig. 2.26. It is clear that there is a sharp peak in Cv around critical temperature [291]. The peak amplitude increases as x increases, leading to BCS- like shape in the overdoped regime. In addition, there is a hump [292], relatively broad in temperature, centered around T  : The two features, namely the peak and the hump, and their evolution with x can be rationalized physically. The peak is due to the low-energy pairing degrees of freedom, which causes long-range phase coherence leading to superconductivity; these are phase fluctuations in the underdoped regime. The hump is mainly associated with the regime where the energy associated with order parameter

2.5 GL Theory of Dirty Two-Band Superconductors

75

Fig. 2.26 Specific heat of cuprate superconductors using GL-like theory

magnitude fluctuations changes rapidly with temperature. Since this change is a crossover centered around T  rather than a phase transition, there is only a specific heat hump, not a sharp peak or discontinuity. For small x; T >> Tc and so we see that the hump is well separated from the peak. As x increases, T  approaches Tc , and in the overdoped regime, these are not separated, and there is no hump, only a peak corresponding to the superconducting transition. We conclude that the GL theory proposed and developed in [288] not only ties together a range of cuprate superconductivity phenomena qualitatively and confronts them quantitatively with experiment, but also has the potential to explore meaningfully many other phenomena observed in them.

2.5 GL Theory of Dirty Two-Band Superconductors The effect of nonmagnetic impurities on two-band superconductors on the GL theory was considered in [293]. GL equations taking into account impurity effects can be obtained by replacing of coefficients in a gradient expansion of order parameters [274] by their averages over disorder potential. An impurity-averaged GL equations can be written as

i .r/ D

X  .0/  X  .0/ 1  .1/ K ij .T / j .r/ C K ij r 2 j .r/ C Lij kl j .r/ k .r/ l .r/; 2 j j kl

(2.120)

76

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors  .0/

 .1/

 .0/

where K ij , K ij and Lij kl are the impurity averaged coefficients in gradient expansion, for which there are expressions:  .0/

Kij 

X  X !D Deff .1/ I Kij  Vi k ij ln Vi k kj .r/ ; T Tc

(2.121)

1 X Vim Pm .0/Pj .0/Pk .0/Pl .0/Nt .0/; Tc2 m

(2.122)

k

 .0/

Lij kl  

k

ij  Pi .0/Pj .0/Ni .0/I P1.2/ .0/ D

N1.2/ .0/ I Deff D P1 .0/D1 C P2 .0/D2 : Ni .0/ (2.123)

v2

Di D Fd i is the diffusion constant for band i electrons. Two-band GL equations converted to effective single-band GL equations, which can be written

0 .r/ D a.T / 0 .r/ 

b 2 r 0 .r/  c j 0 .r/j2 0 .r/; 2

(2.124)

X where 0 .r/ D Vm Pm .0/ m.r/; a.T /  Vav Nt .0/=Tc2 with Vav D m X Pi .0/Vij Pj .0/ and Nt .0/ D N1 .0/ C N2 .0/: As followed from effective singleij

band GL theory, critical temperature is always lowered by disorder. However, the precise value of Tc is insensitive to the strength of disorder and depends only on the density of states of the Fermi surface. It means a reflection of Anderson theorem [294] applied to the effective single-band superconductor. Similar results obtained in Chap. 3 using microscopical approach. Thus, disorder has a stronger effect in multiband superconductors compared with single-band superconductors. Critical temperature reduces by disorder when Tc 1 in multiband superconductors, whereas Tc reduces by disorder only when Ef 1 in single-band superconductors ( is the electron scattering life time). Quasiclassic Uzadel equations for two-band superconductors in the dirty limit with the account of both intraband and interband scattering by nonmagnetic impurities are derived for any anisotropic Fermi surface in study [268]. A powerful tool for investigating inhomogeneous states of superconductors is the quasiclassic Eilenberger equations [295, 296] for the Green functions f .k; r; !/, f C .k; r; !/; g.k; r; !/ and the order parameters (k; r), which depend on the coordinates r, the Matsubara frequency !n = T .2n C 1/, and the wave vector k on the Fermi surface. The essential dependence of f .k; r; !/, f C .k; r; !/; g.k; r; !/ and (k; r) on the direction of k, makes them sensitive to the shape of the Fermi surface, which greatly complicates solving the nonlinear Eilenberger equations. However, in the dirty limit the Eilenberger equations reduce to much simpler Uzadel equations [296]. In Gurevich approach [268], all microscopic details are hidden in the electronic

2.6 Time-Dependent Two-Band GL Equations

77

˛ˇ

diffusivity tensors Dm for each m-th Fermi surface sheet and the interband scattering rate. The corresponding Uzadel equations derived have the form ˛ˇ

 g1 …˛ …ˇ f1 f1 r˛ rˇ g1 D 2 1 g1 C 12 .g1 f2  g2 f1 /; (2.124a)

˛ˇ

 g2 …˛ …ˇ f2 f2 r˛ rˇ g2 D 2 2 g2 C 21 .g2 f1  g1 f2 /: (2.124b)

2!f1  D1 2!f2  D2

From these equations, the GL equations are obtained [268]. Corresponding equation for upper critical field takes the form 2a0 .ln tU.h//.ln tCU.h//Ca2 .ln t C U.h// C a1 .ln t C U.h// D 0; (2.124c) 12 21 22 22 ; a1 D 1 C 11 ; a2 D 1  11 ; 20 D 2 C 412 21 ; where a0 D 11 22 0 0 0 2 h D H2ˆc20DT1 ;  D D : The results of work [268] show that the magnetic behavior D1 of two-gap superconductors can essentially depend on the intraband diffusivity 2 D D : This fact can have important consequences for MgB2 for which Hc2 can be D1 increased (Fig. 2.27) to a much greater extent than in single-band superconductors not just by the usual increase of normal state residual resistivity n; but also by tuning the ratio of intraband scattering rates via selective atomic substitutions on both Mg and B sites. The value Hc2 (0) can considerably exceed 0:7Tc dHc2 /dTc , which can have important consequences for applications of MgB2 [297]. Another 2 property for which the diffusivity ratio D is essential is the temperature dependence D1 of the anisotropy of upper critical field Hc2 . As shown by Gurevich [268], the anisotropy parameter Hc2 increases as T decreases for D  D , but decreases as T decreases for D >> D (see Fig. 2.28).

2.6 Time-Dependent Two-Band GL Equations 2.6.1 Two-Band s-Wave Superconductors Time-dependent equations in two-band GL theory can be obtained from (2.1)–(2.3) in the analogical way to [298]:  1

 ıF 2e @ C ' ‰1 D   ; @t „ ı‰1

(2.125a)



 ıF 2e @ 2 C ' ‰2 D   ; @t „ ı‰2   ıF @A n C r' D  : @t ıA

(2.125b) (2.125c)

78

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

a

1 0.9 0.8 0.7

D2 = 0.05D1

Hc2 / Hc2(0)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

b

0.2

0.4 T / Tc

0.6

0.8

1

0.4 T / Tc

0.6

0.8

1

0.4 T / Tc

0.6

0.8

1

1 0.9 0.8 0.7

Hc2 / Hc2(0)

0.6 0.5 0.4 D1 = D2

0.3 0.2 0.1 0 0

c

0.2

1 0.9 0.8 0.7

Hc2 / Hc2(0)

0.6 0.5 0.4 D2 = 200D1

0.3 0.2 0.1 0 0

0.2

Fig. 2.27 Temperature dependences Hc2 (T) for different ratios of D2 =D1 : Coupling parameters taken from Golubov et al. calculations (J. of Phys. Cond.Mat., 14,1353(2002) (for figure labels see paragraph 2.5)

2.6 Time-Dependent Two-Band GL Equations

79

Fig. 2.28 Temperature dependences of the anisotropy parameter of upper critical field of dirty superconductors (for figure labels see paragraph 2.5)

80

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Here, we use notations similar to [298]. In (2.125), ' means electrical scalar potential, 1;2 -relaxation time of order parameters, n -conductivity of sample in two-band case. Choosing corresponding gauge invariance we can eliminate scalar potential ' from system of equations (2.115) [298]. Under such a calibration and magnetic field in form H D .0; 0; H /, without any restriction of generality, timedependent equations can be written as 1

„2 @‰1 D @t 4m1



x2 d2  4 2 dx ls



 ‰1 C ˛1 .T /‰1 C "‰2 C "1

x2 d2  4 2 dx ls

 ‰2

C ˇ1 ‰13 D 0; (2.126a)     x2 x2 „2 d2 d2 @‰2  C ˛ .T /‰ C "‰ C "  ‰ ‰1 D 2 2 2 2 1 1 @t 4m2 dx 2 ls4 dx 2 ls4 C ˇ2 ‰23 D 0;  n

@A  r' @t



(2.126b)

  2  d1 2A „2 curlcurlA 2 „ n1 .T / n2 .T / C D C  4 ˆ0 4m1 dr ˆ0 4m2 ;   d2 2A C "1 .n1 .T /n2 .T //0:5 cos.1  2 /  dr ˆ0 (2.126c)

where 1;2 phase of order parameters, n1;2 .T /-density of superconducting electrons in different bands, expressions for which are presented in with so-called natural boundary conditions n

     2 i A 2 i A r ‰ 1 C "1 r  ‰2 D 0; ˆ0 ˆ0

(2.127a)

n

     2 i A 2 i A ‰2 C "1 r  ‰1 D 0: r ˆ0 ˆ0

(2.127b)

.n  A/  n D H0  n

(2.127c)

First, two conditions correspond to absence of supercurrent through boundary of two-band superconductor: third conditions correspond to the continuity of normal component of magnetic field to the boundary superconductor-vacuum. In the case of " D "1 D 0, we have time-dependent single-band GL equations. A huge numbers of works related with numerical solution of time-dependent single-band GL equations [299, 300]. Numerical modeling of two-band GL equations was conducted in recently [301, 302]. It is well known that, GL parameter  D  for single-band superconductors is temperature independent, while in two-band GL theory grows with decreasing of temperature. This implies about the possibility of changing of type of superconductivity with lowering of temperature. It means that dynamics of order parameters in two-band superconductors differs from those of

2.6 Time-Dependent Two-Band GL Equations

81

Fig. 2.29 Vortex nucleation in two-band superconductors

in single-band superconductors. In works [301, 302], numerical experiments was performed with two-band time-dependent GL system, and it was claimed that above model yields to realistic results for multiband superconductors. In [302], it was performed numerical modeling of vortex nucleation in the external magnetic field in two-band superconducting films MgB2 using two-band GL theory. The behavior of a homogeneous two-band superconducting film of a constant thickness, placed in a constant perpendicular magnetic field. At such a configuration, the model becomes two dimensional, and it is useful for further numerical analysis to introduce new variables related with gauge-invariance. In numerical experiments, the dimensions of a superconducting film were set at 40  40, where  is the magnetic field penetration depth for a two band superconductor, was used. To solve the corresponding discrete equations of the two-band GL theory, adaptive grid method was used [303]. The results of numerical calculations are given in the figure in the form of the space–time evolution of the superconducting current density in the film. It can be seen that with time (Fig. 2.29), the external

82

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

Fig. 2.30 Space–time evaluation of the vortices in two-band superconductors

magnetic field penetrates the superconducting film in the form of vortices. At the vortex center, the superconducting current density approaches zero. The numerical calculations show that the external magnetic field penetrates the superconducting film symmetrically from the sides. Calculations show that the final form of the equilibrium state depends on the value of external magnetic field H and GL parameter . Calculations also confirmed the existence of the Meissner state; i.e., at a fixed value of  and weak magnetic fields H < Hc1 , the vortices are not generated. The corresponding space–time evolution of the vortices in two-band superconductors presented in Fig. 2.30. From this figure, it is clear symmetrical character of vortex penetration to homogeneous sample in constant magnetic field. Steady-state vortex pattern study using two-band GL equations also was conducted. Results of calculations presented in Fig. 2.31. It was shown that the vortex configuration in the mixed state depends upon the initial state of the sample and that the system does not seem to yield hexagonal pattern for finite size homogeneous samples of uniform thickness with the natural boundary conditions. On the other hand, the time-dependent two-band GL equations leads to the expected hexagonal pattern, i.e., global minimizer of the energy functional.

2.6.2 Time-Dependent d-Wave GL Equations Time-dependent d-wave GL equations was derived in [304] for the study transport properties of singular and nonsingular vortex structures in cuprate superconductors. Corresponding equations are

2.6 Time-Dependent Two-Band GL Equations

83

Fig. 2.31 Steady-state vortex pattern using two-band GL equations

d

s



@‰d D ˛d ‰d C 21 …2x C …2y ‰s C 2 …2z ‰s @t

C 1 …2x  …2y ‰d C ˇd ‰d3 D 0;

(2.128a)



@‰s D ˛s ‰s C 21 …2x C …2y ‰d C 2 …2z ‰d C 1 .…2x  …2y /‰s @t

C ˇs ‰s3 D 0; (2.128b)      @ @ 2   d ‰d „ C 2ei' ‰d C s ‰s „ C 2ei' ‰s  cc : n ' D ˆ0 @t @t (2.128c) In study [304], the angular dependent correction to the viscosity tensor of the tilted singular flux line using the equations (2.128) is calculated. It is obvious that the internal structure of the flux line affects the viscosity tensor of flux line. An additional term of the viscosity, which depends on the magnetic field direction is proportional to the additional term of the loss function, which is due to the presence of the s-wave component in the vicinity of the flux line core. Authors consider that the temperature close to Tcd and perturbation theory with a small parameter ˛s = j˛d j [305]. As shown by calculations [304], the temperature dependence of correction has a positive curvature with a decreasing temperature, which reflects the increase of admixture of the s-wave order parameter component. Formation of nucleus of swave order parameter component in the center of a flux line leads to an increase of dissipation and, hence, viscosity of the nonsingular vortex in comparison with the singular one. Furthermore, this nucleus causes suppression of the pinning effect [304].

84

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

2.6.3 1.5 Type Superconductivity Terminology “1.5 type” superconductivity state was introduced in [306–308] for describing semi-Meissner state in two-band superconductors. In the opinion of these works [308], two-band superconductivity characterized by the three fundamental length scale, while single-band superconductivity described by a single parameter GL : As follows from [308] that in general there are three fundamental length scales in the problem (in contrast to the two length scales of one-component GL theory), which manifest themselves in the vortex asymptotics, namely 1= A , 1= 1 , and 1= 2 : 1= A can be interpreted as the London penetration length of the magnetic field. As a result interaction, potential can be presented as

 V .r/ D 2 q02 K0 . A r/  q12 K0 . 1 r/  q22 K0 . 2 r/ ;

(2.129)

where K0 .::/ denotes the zeroth modified Bessel function of the second kind, q0 ; q1 ; q2 are an unknown real constants. This formula reproduces the prediction explained in [306–308] the long-range interaction will be attractive if (at least) one of 1 , 2 is less than A : It is well known that in a II type superconductor the energy cost of a boundary between the normal and the superconducting state is negative, while the interaction between vortices is repulsive [204]. This leads to a formation of stable vortex lattices and liquids. In I type superconductors, the situation is the opposite; vortex interaction is attractive (thus making them unstable against collapse into one large vortex), while the boundary energy between normal and superconducting states is positive. Result of calculation of intervortex interaction potential presented in Figs. 2.32 and 2.33. Choosing of calculation parameters is given in study [306, 307]. In the limit of two condensates coupled only electromagnetically the length scales 1= 1 and 1= 2 are associated with standard GL coherence lengths. In [308] it was that introducing a nonzero Josephson and quadratic density–density couplings makes both density fields decay according to the same exponential law at very large distances from the core while, at the same time the system still possesses two fundamental length scales, which are associated now with variation of linear combinations of density fields rotated by a mixing angle. Rather, in a wide range of parameters, as a consequence of the existence of three fundamental length scales, there is a separate superconducting regime where vortices have the long-range attractive short-range repulsive interaction and form vortex clusters immersed in domains of two-component Meissner state. Recent experimental works [309, 310] have put forward the suggestion that this state is realized in the two-band material MgB2 : However, there are opposite opinions in respect existence of 1.5 type superconductivity in two-band superconductors. As discussed in review [311], several origins of vortex are conceivable, depending on the material, its purity, magnetic history, and temperature. Computer simulations using such a vortex interaction (short-range repulsive, long-range attractive) yield vortex arrangements that are

2.6 Time-Dependent Two-Band GL Equations

85

Fig. 2.32 Intervortex repulsive potential (for figure labels see [308])

Fig. 2.33 Intervortex interaction potential (for figure labels see [308])

similar to the observed ones [309, 310]. The details of the resulting clusters or chains also depend on the type and strength of vortex pinning that is always present and should be included in such simulations, in particular at very low induction B: As noted in [311], microscopical calculation based on quasiclassical Eilenberger method [312, 313] may have an attractive tail due to the complexity of the problem. More recently in the paper [314] using the GL theory for two-band superconductors, the surface energy s between coexisting normal and superconducting

86

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

states at the thermodynamic critical magnetic field was calculated. Close to the transition temperature, where the GL theory is applicable, it was demonstrated that the two-band problem maps onto an effective single-band problem. Similar effective single-band GL approach also was applied in papers [200,302]. According to [314], while the order parameters of the two bands may have different amplitudes in the homogeneous bulk, near Tc the coupling between the order parameters leads to the same spatial dependence of both order parameters near the interface. Results obtained in [314] puts into question the possibility of intermediate, so-called type1.5 superconductivity, in the regime where the GL theory applies.

2.7 Coexistence Antiferromagnetism and Superconductivity: GL Description Now we will consider cuprate superconductors as a system consisting of a superconducting subsystem in coexistence with an antiferromagnetic sublattice. Generalized GL functional for such system is F D FS C FA C FSA :

(2.130)

Fs is functional for superconducting system (see expression 2.2). FA functional accounts for the free energy of the antiferromagnetic phase, which given as Fi D a.T /jM.r/j2 C

b.T / jM.r/j4 C C jrM.r/j2 C 2jM.r/j2 ; 2

(2.131)

where M.r/ is the antiferromagnetic order parameter, which vanishes at the Neel temperature TN , and M is the internal magnetization field. The coefficients a.T /; b.T / are temperature dependent; for temperature below TN , a.T / is negative and b.T / is positive. In the vicinity of TN , a.T / D a0 .T  TN /=TN : The FSA functional accounts for the free energy of exchange is in interaction between superconducting and antiferromagnetic phases. FSA express as [315–318] FSA

   2e D  j‰.r/j jM.r/j C j i „r C A ‰.r/j2 jM.r/j2 2m c 2

2

C jrM.r/j2 j‰.r/j2 C HM:

(2.132)

The coefficients ; ;  are measures of the strength of the exchange interaction. Depending on the sign of the interaction coefficients, the exchange interaction mechanism can either lead to suppression or enhancement of superconductivity. The level of enhancement depends on the density of the order parameter jM.r/j2 and strength of the coefficient  [317]. Similar theory for calculation specific heat in heavy-fermion materials CeCoIn5 and UBe13 used in [319]. The recent analytically

2.7 Coexistence Antiferromagnetism and Superconductivity: GL Description

87

calculated [320] the upper critical field, the lower critical field and the critical magnetic field ratio of the anisotropic magnetic superconductor by GL theory that the Gibbs free energy of magnetic superconductors of Hampshire form [321] [322] are used. The effect of  (is the isotropic nonmagnetic GL parameter);  (magnetic susceptibility), and  (is the magnetic-to-anisotropic parameter ratio) on the critical c2 magnetic field ratio (H D H ) are presented in paper [320]. The diamagnetic Hc1 superconductor with highly anisotropic case is shown the highest value and the ferromagnetic superconductor with highly magnetic is shown the lowest value of the c2 critical magnetic field ratio H . The critical magnetic ratio can be used to predict Hc1 the magnetic parameters of superconductors. Finally, we will discuss the effect of a helicoidal structure on superconductivity. This question has been originally considered by Morosov [323] and also more recently in application to Ho borocarbides [324]. As it was shown [323, 324], using Bogoliubov transformations the gap parameter in the spectrum of electron quasiparticles becomes strongly anisotropic and vanishes at the boundaries of the breaks in the Fermi surface due to the Bragg planes generated by the magnetic ordering (i.e., when the Bragg planes intersect the Fermi surface). As pointed out by Morosov [324], the gap parameter of a superconductor in the presence of helical structure may be written as  

.k; T / D u2k  v2k .T /; where 

u2k



v2k



D

"k  "kCQ  2 "k  "kCQ C I 2 S 2

(2.133) !1=2 ;

(2.134)

in which I is the exchange interaction integral, S the average ion spin, "k the dispersion relation in the paramagnetic phase and Z!

.T / D 0 

0 1 Z / dS 0 .u2k0  v20

.T /.1  2nk / 1=2 @ 0 kˇ A ˇ ; d" 2 ˇ " C 2 .T / .2/3 ˇˇr 0  " 0ˇ MF S

k

(2.135)

k

where " k0 is the new dispersion relation [324] and nk takes into account the occupation of the electronic state. The last equation corresponds to the usual BCS self-consistent gap equation with an effective parameter eff .T / defined as the term in brackets. eff .T / depends on the underlying magnetic state through the Bogoliubov coefficients and the slope of the magnetic Fermi surface. Since all the anomalous magnetic wave vector dependencies come from the region where Fermi surface intersects the Bragg planes, the difference .T / D   eff .T / between the actual electron–phonon interaction  and its effective value, we can expand quantities in terms of IS="F . Using the results of band structure calculations [91] for borocarbide compounds the difference .T / is estimated by Amici, Thalmeier, and Fulde [325] as .T /= D 0:12. This result has been employed

88

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

in the explanation of the main anomaly (reentrant behavior) of the upper critical field Hc2 (T). Magnetic properties of helicoidal superconductors were investigated in [326] many years ago. Taking into account contribution of spin waves into bound energy of the helicoidal superconductor, lower critical field Hc1 was calculated: Hc1 D

  ˆ0 L 8m ln ; .1 C

e C D0 S 2 =4 42L

(2.136)

where L London penetration depth of clean superconductor, D0 is the parameter of anisotropy, e is the temperature of magnetic ordering due to electrons and moments, m D  20 n, 0 Bor magneton, n is concentration of electrons [326]. According to (2.136), suppression of London penetration depth of helicoidal superconductor L related with polarization of localized spins by external magnetic field. Specific heat of helicoidal superconductors at low temperature also is determined by the contribution of spin waves [326].

2.8 Application of GL Equations to Nanosize Superconductors Mesoscopic superconductivity bears a number of fascinating phenomena, ranging from specific vortex states, to enhancement of critical parameters by quantum tailoring. Interestingly, the form of the sample itself defines the confinement geometry for the superconducting state. The superconducting order parameter ‰, obeys the coupled GL equations. The shift of the transition temperature with magnetic field, Tc .H D 0/  Tc .H /; correspond to the lowest Landau level ELLL .H /; found as solution of the GL equation with the proper boundary condition. Using nanostructuring, the confinement geometry for superconducting samples can be tuned, thus leading to a substantial modification of ELLL .H / and to enhancement of Tc .H / [327]. Moreover, nanoengineering regular pinning arrays also provide the condition necessary for the dramatic increase of the critical current jc up to its theoretical limit, the depairing current jd : It is interesting to find Tc .ˆ/ dependence of mesoscopic single-band superconducting samples. The Tc .ˆ/ curve measured from the full square is similar to the result obtained from calculation for the mesoscopic disk in the presence of a magnetic field [328]. In that study, the linearized first GL equation is solved with the boundary condition for the ideal Superconductor/Insulator (SI) interface: .i „  2eA/‰?;b D 0:

(2.133)

The series of peaks in the Tc .ˆ/ curve correspond to transition between states with different angular momenta L ! L C 1: In paper [329], the Tc .ˆ/ phase boundary is studied theoretically and is compared the experimental data in [330]. Different types of microstructures (full, 4-antidot, 2 antidot) are investigated in paper [330].

2.8 Application of GL Equations to Nanosize Superconductors

peak position, Φ / Φ0

14

4-hole

7

2-hole

10

1-Tc(H) / Tc(0)

8

Theory full sq-

12

0.10

89

8

6

6 4

5

2 1

2

3

4 5 6 7 peak number

8

9

10

4

3

0.05 2 1

L=0

0.00

0

1

2

3

4

5

6 Φ / Φ0

7

8

9

10

11

12

Fig. 2.34 Tc .ˆ/ phase boundaries in reduced units of critical temperature and flux. The inset shows a comparison peak position in Tc .ˆ/ with the prediction for a mesoscopic disk

All the structures, in the high flux regime, have peaks in Tc .ˆ/ at the same values for ˆ > 5ˆ0 (Fig. 2.34). For the understanding coincidence of the peak positions at high fields, the Tc .ˆ/ curves were calculated for single circular dot and for an antidot in an infinite film, both of the same radius (Fig. 2.35). It is necessary to note that for a semi-infinite plane Hc3 D 1:695Hc2 (dotted line in Fig. 2.35). Since the dot has a larger Hc3 than the antidot, order parameter is expected to grow initially at the outer sample boundary, as the temperature drops below Tc . At slightly low temperatures, surface superconductivity should as well nucleate around the antidots. In the meantime, however, order parameter reached already a finite value over the whole width of the strips. The resistively measured Tc .ˆ/ curves, probably because of this substantially different Hc3 for a dot and an antidot, only show peaks related to the switching of the angular momentum L, associated with a closed contour along the outer sample boundary. At the Tc .ˆ/ boundary, in the high magnetic flux regime, there is no such closed superconducting path around each single antidot, and therefore the flux quantization condition does not need to be fulfilled for a closed contour encircling each single antidot. Two-band GL equations first was applied for the study of mesoscopic superconductivity in disks [331, 332]. They found by exact calculations within GL theory that fractional vortices, carrying arbitrary fraction of flux quantum (Fig. 2.36), exist in mesoscopic two-component superconductors as stable thermodynamic phases in

90

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors

a

6 5

R2 / ξ2(Tc)

4 3 2 1 0

b

0

1

2

0

2

3 Φ / Φ0

4

5

6

2.6 2.4 2.2

η

2.0 1.8 1.6 1.4 1.2 1.0

4

6 Φ / Φ0

8

10

Fig. 2.35 (a) Calculated phase boundaries for a circular antidot and a dot in normalized units temperature and magnetic flux (b) The enhancement factor  D Hc3 =Hc2 for the same structures

a broad range of superconducting parameters, temperature, and applied magnetic field. The origin of stabilization of these vortex phases is the different effect of the confinement on the two condensates exerted by the boundary of the sample, i.e., is a pure mesoscopic effect. This effect is counteracted by the coupling between the two condensates [term  " in free energy functional (2.1)] and the magnetic field induced by the screening currents, both favoring similar distributions of the order parameters. Fundamental properties of vortex matter in two-band superconductors mesoscopic disk was analyzed using GL theory very recently in [333]. In applied magnetic field, they derived the dependency of the size of the vortex on the sample size and the strength of the order parameter coupling: 

1 RV 0

2

 C

c RD

2

 D

1 RV  hRD

2 :

(2.134)

2.8 Application of GL Equations to Nanosize Superconductors

91

4 3 2

ΦΓ Φ0

1 40 0

30 (R / ξ1)2

20 2

4

6 Φ / Φ0

Fig. 2.36 The flux associated to vortices ˆ (where contour  lies inside of superconductor). Domains corresponding to the fractional vortex fluxes are clearly seen as nonhorizontal intermediate (noninteger) steps

Fig. 2.37 Vortex size as a function of the radius of the superconducting disk RD (RV CPD determined from decay of the Cooper-pair density, RV j determined from maximum of encircling current)

where RV 0 is the vortex size independent of the sample size, c is a length coefficient and h is the slope of RV vs RD for large RD . From fitting of numerical data [333], collected at different T ,  , ˛ and m, obtained fc D 1:90; h D 0g for RV;CPD and fc D 3:40; h D 0:006g for RV;j: These coefficients are valid for single-band superconductors as well as for two-band superconductors, even when the coherence lengths of the two condensates are very different. The result of calculations [333] presented in Fig. 2.37. When disks are too small, the vortex size is not always fixed by the disk size only, e.g., when T > Tcr and " < 0:1, when the coherence lengths differ much and the interaction of the vortex with the Meissner currents becomes too different in the two bands. However, we found that when RD > 10 these

92

2 Ginzburg–Landau Analysis of Multiband and Anisotropic Superconductors 4.0

T=1.0 T=0.8 T=0.6 T=0.4 T=0.2 T=0.0

3.5

Rv0

3.0

Tcr=0.4 Tcr=0.7 Tcr=1.0

2.5 2.0

Rv-0(from Hcr): Tcr=0.4 Tcr=0.7 Tcr=1.0

1.5 0.0 0.1 0.2 0.3 0.4 0.5 γ

0.6 0.7 0.8 0.9 1.0

Fig. 2.38 Vortex size as function of temperature and  < 1 parameter. ( is proportional to interaction between order parameters ") Choosing of calculation parameters see [333]

mesoscopic effects have only a minor influence and the correspondence between RV and RD becomes predictable again. In Fig. 2.38 the size-independent vortex size RV 0 is plotted versus interaction parameter ", for different temperatures T and Tcr . The dots represent the result of simulations [333]. The general behavior reveal the following trends. (1) Increasing T induces an increase in the vortex size whereas increasing has the opposite effect. (2) Deviation from the latter monotonic behavior occurs when T  Tcr and the coupling is weak, see, e.g., the curve at T D Tcr D 0:4: It is shown; that order parameter interaction " generally has an influence opposite to the one of temperature. In limiting cases, numerical findings of study [333] agree well with analytic expressions available in literature. They also found a fitting function, which gives an excellent estimate of the size of the vortex core as a function of the size of the mesoscopic disk. The existence of fractional states, i.e., states with different vorticity in the bands, depends strongly on the order parameter coupling ". They survive only at weak coupling " between the bands, while only integer flux states are possible at large " values. This is illustrated in Fig. 2.39, where is shown that the region of stability of fractional states shrinks with increasing coupling, but also that lower vorticity fractional states are more resilient to ". Parameters of the sample are RD / 10 = 4, ˛ D ı D m D 1 and T D 0 [333]. In insets at the top of the figure, we superimposed the logarithmic plots of the Cooper pair density in the two bands on each other (red/blue shades for condensates 1/2, respectively) for states indicated in the phase diagram by the red numbers. Another interesting aspect of fractional states is their strong affinity to asymmetry. In both condensates, vortices attempt to form a symmetric shell, but due to coupling and different respective number of vortices, the final state becomes asymmetric in most cases. For that reason, the

2.8 Application of GL Equations to Nanosize Superconductors

93

Fig. 2.39 The stability region in parameter space  of fractional states with different vorticities in the two bands

asymmetry is more apparent at larger coupling ": In [333], the existence and stability of fractional states, for which the two condensates comprise different vorticity, were demonstrated. Furthermore, in [333] pronounced asymmetric fractional states were found and shown their experimentally observable magnetic response. Finally, the magnetic coupling between condensates, and study in particular the case where one band is II type and the other is I type, i.e., the sample is effectively of 1.5 type [334], was introduced. The calculated M.H / loops show a clear signature of the mixed type of superconductivity, which strongly affected by the ratio of the coherence lengths in the two condensates [333].

•

Chapter 3

Anisotropic Eliashberg Equations and Influence of Multiband Effects

The two-band model of superconductivity using BCS approach has been developed by Moskalenko [192] and Suhl [193], obtaining a detailed description of thermodynamic and electromagnetic properties of multiband superconductors. The results have been communicated in a large number of articles and in several monographs (for example, [194]), indicated among the references of this book. Generalization of two-band BCS theory, including van Hove singularity of density of states presented in [195]. Eliashberg equations for isotropic two-band and anisotropic superconductors have been given by numerous authors for the study of MgB2 (see [198, 335, 336] and references therein). In Chap. 3 we, first, give the description of single-band isotrop Eliashberg equations [337]. Further, we present a number of recent investigations of layered superconductors using the microscopic Eliashberg theory. The critical temperature of layered superconductors was calculated using this theory, and the influence of nonadiabacity effects on the critical temperature was considered. In the calculation of the effect of Coulomb repulsion on the critical temperature, arbitrary thickness of conducting layers was also taken into account. In the same approach, expression for the plasmon spectrum of layered superconductors with arbitrary thickness of the conducting layers was obtained. In addition, BCS equations for layered superconductors were used for calculating the specific heat jump, which is smaller than in the isotropic case. The critical temperature, specific heat and upper critical field, influence of impurity and doping effects on these parameters in two-band isotrop superconductors are considered. For the general outstanding, we briefly presented results of d -wave BCS theory. Nanosize two-band superconductors in BCS theory and related results are considered. The influence of nonadiabatic effects in two-band strong coupling superconductors is taken into account. Spectrum of collective Legget mode in two-band superconductors and related experimental data is shown.

I. Askerzade, Unconventional Superconductors, Springer Series in Materials Science 153, DOI 10.1007/978-3-642-22652-6 3, © Springer-Verlag Berlin Heidelberg 2012

95

96

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

3.1 Single-Band Isotropic Eliashberg Equations and BCS Limit In general case interaction, Hamiltonian of electrons in metal can be presented as [337] Hint D Heph C HCoulomb ; (3.1) here Coulomb repulsion can be written as HCoulomb D

1 2

X

0

0

.p1 ; p2 jVCoulomb j p1 ; p2 /.‰pC1 3 ‰p0 /.‰pC2 3 ‰p0 /: (3.2)

0

1

0

2

p1 Cp2 Dp1 Cp2

In (3.2), functions: ‰p D ! the following notations was introduced for wave   " 1 0 C# " p C : Electron– C# ; ‰p D . p ; p /; 3 is Pauli matrix 3 D 0 1 p phonon interaction Hamiltonian has following form Heph D

1 X X gj .p; p0 /.‰pC 3 ‰p0 /.bqCj C bq0 /; j 2 0 j

(3.3)

pp Dq

bqCj ; bq0 is the phonon creation and annihilation operators, j polarization of j

phonons. Matrix element of electron–phonon interaction g.p; p0 / related with electron–ion interaction matrix element as X gj .p; p0 / D  p ˇ

1 2NM !j .q/

ˇ ˛ ˝ ˇ p ˇej .q; /rU .r/ˇ p0 eiq ;

(3.4)

!j .q/ and ej .q; / is the frequency and corresponding polarization vector of phonon branch of the number j ; U ;  ; M is the sliding potential, position and mass of ion number I N number of units structure in crystal [204]. For derivation of Eliashberg equations-microscopical equations of electron–phonon based superconductors [338], the generalized Green function method [274] was used. Accuracy of this equations in the order m D EF =!D (m is so-called Migdal parameter) [31, 32]. As shown by Migdal [31], in this approach in calculation of electron–phonon vertex function, it is enough to take into account only first term (Fig. 3.1). Finally, for the single-band isotropic superconductors, Eliashberg equations can be written as [337]

q,ω

Fig. 3.1 Bare vertex and first-order correction for e–ph scattering

+

3.1 Single-Band Isotropic Eliashberg Equations and BCS Limit

1 1  Z.!/ D  ! Z

97

Z1 0 ! 0 0 d! Kph .!; ! / Re p 0 sign! 0 ; ! 2  2 .! 0 / 0

1

0

0

d! .! / 0 sign! 0 0 Kph .!; ! / Re p 0 2 ! 0 !  2 .! 0 / Z !c 0 0 0 .! / d! ! Re p 0 ;  0 tanh ! 2T 0 ! 2  2 .! 0 /

Z.!/.!/ D

(3.5)

(3.6)

In (3.5) and (3.6), Z.!/ is the electron mass renormalization parameter due to electron–phonon interaction, .!/ is the energy gap. Electron–phonon interaction 0 kernel of these equations Kph .!; ! / is determined as (!c is the cut-off frequency of Coulomb interaction, which is equal to Fermi energyEF ) 1 Kph .!; ! / D 2 0

Z

(

1

2

dz˛ .z/F .z/ 0

) !0 !0 tanh 2T C cot anh 2Tz  cot anh 2Tz tanh 2T ;  !0 C z  !  i ı ! 0 z!  i ı (3.7)

where ˛ 2 .z/F .z/ is the averaged on Fermi surface Eliashberg e–ph interaction function Z Z Z ˇ2 d2 p d2 p0 X ˇˇ d2 p : (3.8) gj .p; p0 /ˇ ı.z  !j .q//= ˛ 2 .z/F .z/ D SF vp SF vp0 j SF v p Thus, Eliashberg equations (3.5)–(3.6) is the system of nonlinear coupled integral equations for the determination of parameters Z.!/ and .!/: Equation (3.5) related with renormalization of mass of electrons due to e-ph interaction. Second Eliashberg equation deals with anormal Green function, i.e., Cooper coupling of electrons. At temperature close to critical temperature Tc ; .!/ tends to zero and (3.5)–(3.6) can be rewritten as 1  Z.!/ D 

1 !

Z1 0 0 d! Kph .!; ! /;

(3.9)

0

Z

1

Z.!/.!/ D 0

0

0

.! / d! 0  0 Kph .!; ! / Re ! !0

Z 0

!c

0

0

0

! .! / d! : Re 0 tanh ! 2T !0 (3.10)

For further applications, it is useful to introduce electron–phonon interaction parameter  such as Z1 00 00 00 00 0 00  D 2 ˛ 2 .! /F .! /! d! K.!; ! ; ! /; 0

(3.11)

98

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

n o 0 00 1 1 where K.!; ! ; ! / D !C! 0 C! C . 00 0 00 Ci ı !C! C! i ı Analytical solving of Eliashberg equations in general case is impossible. As shown in [155, 337, 338] in weak e–ph coupling approach  < 0:3 Eliashberg equations transferred to BCS equations [7]. For the strong e-ph coupling ( > 1/ generally needs computer simulation [338] or McMillan approximation [339]. For the intermediate values of e–ph interaction parameter 0.3 < < 1; the system of integral equations can be solved by iteration procedure [155, 337].

3.2 Eliashberg Equations for Single Band Layered Superconductors In the case of layered systems, strong anisotropy of order parameter in directions perpendicular to layers leads to the additional dependence .pz , !). Taking into account this argument the Eliahberg equations can be written as [338]: 1 1  Z.pz ; !/ D  !  

h





0

Z Z.pz ; !/.pz ; !/ D

Z1 Z 0 d! d.pz0 / 2=d

Z

d.pz0 d / 0 pz; pz0 .!; ! / 2

1

0

0

(3.12)

0

! d! th ! 0 2Tc

i 0 0 0  pz; pz0 .!; ! /  pz; pz0 .EF  ! 0 / Re .pz ; ! //; (3.13) where the e–ph coupling parameter pz; pz0 for layered systems is given by the Z1 00 00 00 00 0 00 pz; pz0 .!; ! / D 2 ˛ 2 .! /Fpz pz .! /! d! K.!; ! ; ! /; 0

(3.14)

0 0

00

where expression for K.!; ! ; ! / presented above. In layered systems, anisotropic character of Coulomb repulsion also must be taken into account. Since the cuprates, magnesium diboride, borocarbides, and oxypnictides of recent interest consist of layered structures, we assume a dispersion relation appropriate for a layered system of the form [340–342]: E.k/ D

„2 .kx2 C ky2 / 2m

C 2tŒ1  cos .kz d / :

(3.15)

Here, m is the in-plane effective mass, t is the transverse transfer matrix element from one layer to another (or tunneling integral). t characterizes the intensity of

3.2 Eliashberg Equations for Single Band Layered Superconductors

99

Py

q P′

ϕ P π/d

–π/d

Pz

Px

Fig. 3.2 The Fermi surface of the layered superconductors

electron tunneling between the layers and must depend on the ratio a=d as t D F .a=d /, where a is the thickness of conducting layer, d is the characteristic distance of the order of unit cell size in the superconducting layers. The F .a=d / function rapidly decreases when the distance a increases. It is possible, in principle, to obtain an explicit expression for this function provided that the electron density distribution inside the superconducting layers is known. Such an energy spectrum of carriers was used by Jiang and Carbotte [343] (Ref. therein) for the calculation of various properties in a layered superconductors. For E > 4t, the Fermi surface is open and the density of states N.E/ is constant (Fig. 3.2). The phonon spectrum of the layered crystals is, generally speaking, anisotropic. The dispersion relation for longitudinal !L .q; qz ) and transverse phonons !T .q; qz ) is given by the following expressions: !L2 .q; qz / D u2jj .qx2 C qy2 / C 2

u2z .1  cos .qz d // ; d

(3.16a)

and

u2T (3.16b) Œ1  cos .qz d / ; d (the sound velocities satisfy the condition ujj >> uT ; uz ). As mentioned in [155, 337, 338], the functions appearing in generalized Eliashberg equations are defined by an averaging procedure over the Fermi surface. In the case of energy spectrum (3.15), this procedure is equivalent to integration !T2 .q; qz / D u2z .qx2 C qy2 / C 2

Z

Z

2

2p0

d'::: D 4 0

0

dq ::::;  q 2 g1=2

f.2p0 /2

(3.17)

100

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

where (p0 /2 D p0 2  4mt.1  cos pz d / and ' denotes angle between p and p 0 ; which is equal to p0 : It is clear that the region of phonon transfer momentums q D 2p0 makes a major contribution to the integrals. In power of last argument, generalized Eliashberg equations for layered systems presented in [340] with Einstein spectrum of effective frequency !0 : The corresponding effective frequency is determined by following expression (discussion of phonon modes in layered systems see below) 8 R =d 91=2 R 2p0 dq 2 d q < 2 ! 2 .q; qz / = =d dqz  N2d 0 f.2p0 /2 q 2 g1=2 L 2 D !0 D !av R =d R 2p0 dq : ; d dq 2 N =d

2

2d

z

 D 2u2jj .p02  4mt/ C

f.2p0 /2 q 2 g1=2

0

(3.18)

2  1=2

2uz d2

:

In the case of quasi-two-dimensional energy spectrum (3.15), Fermi-surface harmonics can be represented by cos .npz d / [338]. Anisotropic e-ph coupling parameter .pz ; pz0 / are expanded as 0

0

0

0

pz ;pz0 .!; ! / D 00 .!; ! / C 10 .!; ! / cos .pz d / C 01 .!; ! / cos .pz0 d / 0

C 11 .!; ! / cos .pz d  pz0 d / ;

(3.19)

R 0 0 00 0 00 where ij .!; ! / D ij I.!; ! / D d! K.!; ! ; ! /: As pointed out in [344] and Nakhmedov [345], the nondiagonal elements of the e–ph interaction in layered systems with electron spectrum of (3.15) are proportional to t=EF . Within the model of Fermi-surface harmonics, the order parameter takes the form .pz ; !/ D 0 .!/ C 1 .!/ cos .pz d /:

(3.20)

Taking into account (3.19), expression for Z.pz ; !/ has a form: Z.pz ; !/ D 1C00 .!/C10 .!/ cos .pz d / DZ00 .!/C10 .!/ cos .pz d /:

(3.21)

Substituting (3.20) and (3.21) into (3.12) and (3.13), we have a system of coupled integral equations: 10 .!/0 .!/ D Z00 .!/0 .!/ C 2

Z

1 0

Z

0

1

C 0

0

! d! th I.!; ! 0 /00 0 .!/  ~00 ! 0 2Tc 0

0

! d! th I.!; ! 0 /10 1 .!/  ~10 ; ! 0 2Tc (3.22)

3.2 Eliashberg Equations for Single Band Layered Superconductors

101

and Z00 .!/1 .!/ C

01 .!/0 .!/ D 2

Z

1 0

Z

0

1

C

0

d! ! I.!; ! 0 /01 0 .!/  ~01 0 th ! 2Tc

0

0

0

! 11 d! I.!; ! 0 / 1 .!/  ~11 : 0 th ! 2Tc 2 (3.23)

In (3.22) and (3.23) was introduced the following definitions: Z

EF

~ij D ij 0

0

0

d! ! th j .!/: ! 0 2Tc

(3.24)

Consequently, calculation of critical temperature lead to solving a system of singular integral equations [155, 338]. As mentioned above, for the intermediate values of e–ph interaction 0:3 <  < 1; the system of integral equations can be solved by iteration procedure [155, 337]. However, due to logarithmical singularity of the 0 00 kernel I.!; ! 0 / at ! D 0 and ! D ! I iteration prosedure will be diverged. Here Zubarev prosedure for the elimination of singularity was used [346]. Then system of (3.22) and (3.23) can be rewritten as Z00 .!/0 .!/ C Z

D

1

10 .!/0 .!/ 2

0

0

 d! ! ˚ I.!; ! 0 /  I.!; 0/I.0; ! 0/ 00 0 .!/  ~00 C I.!; 0/ 0 th ! 2Tc 0 Z 1 0 0 ! d! th I.!; ! 0 /10 1 .!/  ~10  f0 .0/ C ~00 g C 0 ! 2T c 0 C I.!; 0/ f1 .0/ C ~10 g ;

(3.25)

and Z00 .!/1 .!/ C Z

0

01 .!/0 .!/ 2 0

 d! ! ˚ I.!; ! 0 /  I.!; 0/I.0; ! 0 / 01 0 .!/  ~01 D 0 th ! 2Tc 0 Z 1 0 0 ! 11 d! C I.!; 0/ f1 .0/ C ~01 g C I.!; ! 0 / 1 .!/  ~11 0 th ! 2Tc 2 0 1

C I.!; 0/ f1 .0/ C ~11 g ; where j .0/ D I.!; 0/ C ~ij .I.!; 0/  1/ :

(3.26)

102

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Finally, system of integral (3.25) and (3.26) has no singularities and as result iteration procedure will be converged. Then the solution of the system of integral (3.25) and (3.26) gives expression for the critical temperature of superconductor. Calculation of integrals leads to a system of algebraic equations .Z00  .00  00 /x/0 C



10  2



10 10  2 2

  x 1 D 0 ;

(3.27)

and     11 11 x 1 D 0 ;  .01  .01  01 /x/0 C Z00  2 2 where x D ln

1:13!ln ; Tc

(3.28)

(3.29)

where logarithmically averaged phonon frequency !ln is defined as * R !ln D exphln !i D exp ij D

d! S.!/ ln ! R !S.!/ d! !

ij 1 C ij ln !ElnF

:

!+ ;

(3.30)

(3.31)

From the condition of vanishing of the determinant of system, (3.27) and (3.28), can be obtained by the following explicit expression for the critical temperature Tc D 1:134!lnexmin ;

(3.32)

where xmin determined as: xmin

˚  Z .  / Z00 .00  00 /C 00 112 11  12 .01 01 /10 C .10  10 /01 F 1=2 D ; .00  00 /.11  11 /  .10  10 /.01  01 / (3.33) Z00 .11  11 / 1  f.01 01 /10 C.10 10 /01 g/2 2 2   10 01 2  2..00  00 /.11  11 /  .10  10 /.01  01 // Z00  : 2 (3.34)

F D .Z00 .00  00 / C

Using the (3.32)–(3.34), we have calculated the critical temperature of MgB2 [347]. The matix elements taken from [198]: 00 D 1:017; 11 D 0:448; 01 D 0:212; 10 D 0:115: For the Coulomb pseudopotentials ij were used [198]:

3.3 Eliashberg Equations for Two-Band Isotropic Superconductors

103

00 D 0:21; 11 D 0:172; 01 D 0:095; 10 D 0:069: Logarithmically averaged value of the phonon frequency !ln taken from [197]: !ln D 480 K. For these parameters, the following results were obtained: Tc D 42:92 K. Late Mitrovich [335] used the following Coulomb pseudopotentials for calculations: 00 D 11 D 0:139; 01 D 10 D 0:027: The e–ph interaction parameters the same as in [198]. With this choice set of parameters and for the logarithmically averaged value of the phonon frequency !ln [348] 767–806 K, critical temperature using above presented expressions ranges in the interval 45.83–50.5 K. It is clear that these results overestimate the critical temperature of magnesium diboride. In our opinion, this is because we neglected the effects of nonadiabaticity in MgB2 . High phonon frequency of the boron atoms (!ph D 0:1 eV) indicates that MgB2 could be in the nonadiabatic regime of the e-ph interaction (Fermi level EF D 0:5 eV) [349]. In our opinion, inclusion of nonadiabatic effects on the consideration, would improve our results. The expression for the critical temperature of layered nonadiabatic superconductors was obtained in [340]. However, obtained results are true for the case 11 << 01 < 00 : Eliashberg equations for isotropic two-band and anisotropic superconductors have been given by numerous authors for the study of MgB2 [198, 335, 336]. In all cases was conducted numerical simulation of system of integral equations. In contrast to these works presented in this paragraph analytical approach applied and obtained formula for the calculation critical temperature Tc in the case of intermediate e-ph coupling.

3.3 Eliashberg Equations for Two-Band Isotropic Superconductors System of Eliashberg equations for two-band superconductors can be written as [338] Z1 1X 1  Zi .!/ D  ij .!; ! 0 /d! 0 ! j Zi .!/i .!/ D

XZ j

(3.35)

0

0

1

0 0 i d! ! h 0 0 ij .!; ! /  ij .EF  ! 0 / Re j .! //: 0 tan h ! 2Tc

(3.36) Analytical procedure in the solution of Eliashberg equations (3.35) and (3.36) for intermediate electron–phonon coupling is similar to previous paragraph calculations, however for xmin and F would following expressions: xmin D

Z1 .00  00 / C Z0 .11  11 /  F 1=2  ; 2 .00  00 /.11  11 /  .10  10 /.01  01 /

(3.37)

104

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Fig. 3.3 Critical temperature of Mg1x B2 Alx as a function of x

F D .Z1 .00  00 / C Z0 .11  11 //2  4Z0 Z1 ..00  00 /.11  11 /  .10  10 /.01  01 //:

(3.38)

In (3.37) and (3.38), notations Z0 and Z1 Z0 D 1 C 00 C 01 I Z1 D 1 C 11 C 10 :

(3.39)

We now present numerical results for the critical temperature Tc calculated for the material, parameters of Mg1x Alx B2 and MgB2x Cx . For the theoretical calculation of critical temperature in Mg1x Alx B2 , the values of ij and ij parameters are taken from papers [350, 351]. Final results of calculations are presented in Fig. 3.3. Full circles correspond to two-band Eliashberg calculations, while open circles to experimental data. Results for C-doped compound are presented in Fig. 3.4. Data for carbon-doped systems taken from [352,353]. The value of logarithmically averaged value of the phonon frequency !ln taken as !ln D 600 K to Mg1x Alx B2 and MgB2x Cx . As shown in [354, 355], !ln remains nearly constant for pure and doped MgB2: As you can see, there is satisfactory agreement between analytical calculations and experimental data. Parameters used for the calculation pressure effects on Tc are listed below. We have taken  D 0:1.00 D 01 D 10 D 11 /: Logarithmically averaged phonon

3.3 Eliashberg Equations for Two-Band Isotropic Superconductors

105

50 Theor. TB Eliasberg Exp. MgB2–xCx

Tc / K

40

30

20 0.00

0.05

0.10

0.15

0.20

0.25

0.30

x

Fig. 3.4 Critical temperature of MgB2x Cx as a function of x

Fig. 3.5 The pressure dependence of Tc in MgB2

frequency for MgB2 is !ln D 600 K and for YNi2 B2 C is !ln D 380 K. Here, we take for the interband coupling constants as 01 D 10 D 11 : Changing of ij by the external pressure presented in [355]. The pressure dependence of Tc in MgB2 is shown in Fig. 3.5. Similar calculations for YNi2 B2 C are presented in Fig. 3.6. As followed from evaluation of Tc under pressure in two band superconductors in satisfactory agreement with experimental data [356]. In the calculation of pressure effects, influence on averaged phonon characteristic frequences was neglected. As

106

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Fig. 3.6 The pressure dependence of Tc in YNi2 B2 C

shown in [190], superconducting properties of another nonmagnetic borocarbide LuNi2 B2 C is very similar to those for YNi2 B2 C. Unfortunately, we have no available experimental data on investigation of pressure effects on critical temperature in LuNi2 B2 C. In the above presented version of Eliashberg equations, we neglected anisotropy effects in two-band superconductors. Another important moment is the different character of anisotropy in different bands. Computer simulation of two-band anisotropic Eliashberg equations presented in [336] and similar generalization can be conducted in the case of our analytical approach.

3.4 Effects of Nonadiabacity in Layered Systems In the case of nonadiabatic anisotropic superconductors (m  1/, the generalized Eliashberg equations describing pairing in systems with cylindrical symmetry has the form [340]: Z

d.pz0 d / X  .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/ 2 .!n  !m /2 C !02  m

0 E 2 .pz ; !m / 2 ; (3.40) arctan  !0 j!m j  2Z.pz0 ; !m ; /j!m j

Z.pz ; !n /.pz ; !n / D Tc



3.4 Effects of Nonadiabacity in Layered Systems

107

and Z

d.pz0 d / X z .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/ 2 !m !0 2 j!m j .!n  !m /2 C !02  m

2 E  arctan ; (3.41)  2Z.pz0 ; !m /j!m j

Tc Z.pz ; !n / D 1 C !n



in which Z.pz ; !n / is the renormalization parameter, .pz ; !n / is the energy gap, E is the total bandwidth, so that the energy is defined in the interval E=2 < E < E=2, and Qc is the cut-off parameter for the phonon momentum transfer Qc D qc =2kF . !m D .2m  1/kB Tc with m D 0; ˙1; ˙2; : : : are the Matsubara frequencies. Here, the following usual notation for the effective couplings was used   .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/ D .pz ; pz0 / 1C2.pz ; pz0 /Pv

.pz ; pz0 ; !n ; !m ; Qc I !0 ; E/C .pz ; pz0 /Pc .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/ ;

(3.42)

z .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/  D .pz ; pz0 / 1 C .pz ; pz0 /Pv .pz ; pz0 ; !n ; !m ; Qc I !0 ; E/ :

(3.43)

and

The expressions for the so-called vertex and cross functions, Pv and Pc , respectively, in general case were given by Paci, Grimaldi, and Pietronero [357–359]. The vertex and cross functions are expanded in terms of Fermi-surface harmonics [338] which form a complete, orthonormal set of functions at the Fermi surface. In the case of model energy spectrum (3.15), Fermi-surface harmonics can be represented by cos .npz d /. Anisotropic e–ph coupling parameter .pz ; pz0 / without the corrections in (3.19) and (3.20) are expanded as by the expression (3.19) with 01 D 10 . As pointed out in [344] and [345], the nondiagonal elements of the e–ph interaction in layered systems with electron spectrum of (3.15) and quasitwo-dimensional phonon spectra (3.16a) and (3.16b) are proportional to t=EF . As shown in the previous works, [360,361] layered systems is characterized by the low frequency optical phonons, which corresponds to the oscillations of planes as rigid “molecules” in respect to each another. As pointed by Bergman and Rainer [362], Dubovskii and Kozlov [363], Allen and Dynes [364], low-frequency phonons plays significant role in superconductors with weak e–ph coupling limit. In the opposite case, i.e., in strong coupling limit critical temperature Tc is determined by the high frequency peculiarities in phonon spectrum. With these arguments in mind, we take into account the interaction of electrons with acoustical in plane phonons (3.16a) and (3.16b). For layered systems, the above condition implies that 11 << 01 < 00 , which suggests that in subsequent calculations, we can neglect terms of order 11 =01 and

108

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

11 =00 . For the calculation of 00 and 01 , the expression for e–ph interaction without the vertex correction in (3.42) can be used. In a more general situation, we have the following expressions for the vertex corrected interaction (for the convenience in (3.42) and (3.43), other argument is omitted) 2  .pz ; pz0 / D .pz ; pz0 / 41 C 2

X

3 .kz  pz /G.kz /G.pz0  pz C kz /5

kz

C .pz ; pz0 /

X

.kz  pz /G.kz /G.kz  pz  pz0 /

(3.44)

kz

and 2 0

z .pz ; pz0 / D .pz ; pz / 41 C

X

3 .kz  pz /G.kz /G.kz  pz C pz0 /5 :

(3.45)

kz

For the small parameter t=Tc << 1, and at temperatures close to Tc , the Green’s functions of electrons can be expressed as G.i !n ; p; pz / D



1 t cos .pz d / 1 1C ;  i !n  .p; pz / i !n  .p/ i !n  .p/

(3.46)

where .p; pz / D E.p; pz /  , and  being the chemical potential. Taking into account the expression given in (3.44)–(3.46), we obtain the final expression for the vertex corrected e–ph interaction  D 00 C 200 .2Pv C Pc / C 01 .1 C 200 .2Pv C Pc // cos .pz d / C 10  .1 C 00 .2Pv C Pc // cos .pz0 d / C 00 10 .2Pv C Pc / cos .pz d  pz0 d / (3.47) and z D 00 C 200 Pv :

(3.48)

Within the model of Fermi-surface harmonics, the order parameter takes the form of (3.20). As shown by Grimaldi, Pietronero, and Str¨assler [357], the critical temperature Tc can be obtained from the generalized Eliashberg equations by an analytical approach. The final expression for Tc beyond the adiabatic limit in s-wave isotropic superconductors for arbitrary momentum transfer is given by [357–359]



1:13 !0 m 1 C z =.1 C m/ Tc D exp exp  : .1 C m/e 1=2 .2 C 2m/ 

(3.49)

3.4 Effects of Nonadiabacity in Layered Systems

109

Substituting (3.47), (3.48), and (3.20) into (3.40) and (3.41), and making use of the McMillan approximation [339], we have a system of algebraic equations  and

 11 10 1 C 00 z =.1 C m/   x 0 C  x1 D 0 ;

(3.50)

  00 11 10  x0 C 1 C z =.1 C m/   x 1 D 0 ;

(3.51)

where xD

m .1  1Cm / ln .1:13/!0  ln .1 C m/  ; Tc 2

(3.52)

and 2 00 z D 00 C 00 Pv ;

(3.53)

2 00  D 00 C 00 .2Pv C Pc / ;

(3.54)

01 

D 01 C 200 01 .2Pv C Pc /;

(3.55)

11  D 00 11 .2Pv C Pc / :

(3.56)

From the condition of vanishing of the determinant of system of (3.50) and (3.51) and the condition t=EF << 1, we obtain the following explicit formula for the critical temperature   Tc (3.57) D exp ~.01 =00 /2 ; Tc0 where Tc0 is the critical temperature without the vertex corrections and ~D

1 1 C 00 .1 C 00 Pv /=.1 C m/ .1 C 00 Pv /=.1 C m/ : 2 00 .1 C 00 .2Pv C Pc // 00 .1 C 00 .2Pv C Pc //

(3.58)

The coefficient ~ embodies the effects of vertex corrections and anisotropy in determining Tc . The explicit forms of the vertex correctionPv and cross correction Pc for the two-dimensional case are presented by Paci et al. [365]. The main result for the effects of anisotropy on the critical temperature in layered nonadiabatic superconductors is given by (3.57). To assess these effects more quantitatively, we show in Fig. 3.7 Tc =Tc0 as a function of 01 =00 , for different values of Qc (where Qc is the cut-off parameter of phonon momentum transfer). The explicit expressions for 00 and 01 with energy spectrum of (3.15) were presented in [344]. These expressions embody microscopic parameters, which may be obtained from the experimental data (for example, ujj , uz , uT , EF ). However, final expression for the critical temperature Tc given in (3.57) contains only the ratio of the parameters 01 =00 . For the case of two-dimensional superconductors, we take 00 D 0:5. In this figure, the dashed curve denotes the behavior of Tc without the vertex corrections. Solid curves are for different Qc values in the range 0.1–0.9, from top to bottom, respectively. We observe that the nonadiabatic corrections become more prominent for small values of Qc . Note that ~ increases as Qc decreases. For values Qc D 0:9, the coefficient ~ becomes lower than that in the

110

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Tc / Tc0

3

2

1

0

0.2

0.4 l01 / l00

0.6

0.8

Fig. 3.7 Critical temperature versus 01 /00 . The dashed curve denotes the behavior of Tc without the vertex corrections. Solid curves correspond to different Qc values in the range 0.1–0.9, from top to down as Qc increases 2.0

1.8 Qc = 0.1 1.6 k 1.4 Qc = 0.9 1.2

1.0

0

0.2

0.4

0.6

0.8

1.0

m

Fig. 3.8 Dependence of ~ on the Migdal parameter m

adiabatic case. Thus, the vertex corrections have similar behavior in the anisotropic and isotropic superconductors when Qc is small. The critical temperature in the nonadiabatic case is enhanced compared to the solution without the vertex and cross corrections. Dependence of ~ on the nonadiabacity Migdal parameter m is displayed in Fig. 3.8 for two different values of Qc D 0:1 and Qc D 0:9. As can be seen in the figure, in both cases ~ decreases with increasing m. At small m, corrections become more significant and they are reduced by increasing m. The behavior of

3.5 Effect of Coulomb Repulsion in Layered Single-Band Superconductors

111

~ for other values of Qc is similar to that shown in Fig. 3.8. These results we obtain seem interesting and relevant in connection with cuprate compounds as layered nonadiabatic superconductors. Another popular layered superconductor is Sr2 RuO4 with Tc  1 K, which is rather low [366]. The layered structure of the system leads to the nearly cylindrical Fermi surface, which is open along the c-axis. However, there are various indications for strong correlation effects and nonadiabacity effects is absent in Sr2 RuO4 compounds. Thus, in isotropic single-band s-wave nonadiabatic superconductors vertex corrections are strongly dependent on the momentum transfer and small values of Qc lead to an enhancement of the critical temperature Tc [357–359].

3.5 Effect of Coulomb Repulsion in Layered Single-Band Superconductors with Arbitrary Thickness of Layers As followed from (3.33) and (3.37), in the calculation of critical temperature of e–ph-based superconductors, we need the average value of Coulomb potential . To average value  of the screened Coulomb potential VQ .k; kz / over the quasi-twodimensional Fermi surface (3.15), we will use the averaging procedure [341] D

D 2

Z

 D  D

dkz

2N2d .0/ 

Z

2p  F

0

dk VQ .k; kz /;  k 2 /1=2

..2pF /2

(3.59)

where N2d .0/ is the two-dimensional density of states on the Fermi surface, and pF is given by the expression (2pF /2 D .2pF /2  4mt.1  cos pz d /: D D a C d; where d is the thickness of the conducting layer and a is the distance between them. To calculate ; will be used the rewritten expression for the bare Coulomb potential V .k; kz / in a superlattice with different dielectric constants. Such a potential was obtained in [367] by Guseinov and has the form V .n.a C d /; k/ D where

1 D

2e .1 C 1 /.1  2 / exp.nk0 / ; 1 k . 1  2 /

exp.ka/  exp.k0 /.˛ exp.kd / C ˇ exp.kd // : exp.ka/  exp.k0 /.ˇ exp.kd / C ˛ exp.kd //

(3.60)

(3.61)

2 can be obtained from the expression for 1 by changing the sign of k0 ; and k0 is given as k0 D arccos h.cos h.k.a  d // C

2˛ 2 sin h.ka/ sin h.kd //: .2˛  1/

(3.62)

112

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

.1/ In (3.60)–(3.62), other dimensionless parameters read as ˛ D .1C/ 2 , ˇ D 2 , 1 and  D , where and 1 are the static dielectric constants of the metal and the dielectric, respectively. Using the expression for the Fourer transformation [368]

X

exp.nk0 / exp.i nkz .a C d // D

n

sin h.k0 / ; cos h.k0 /  cos kz .a C d /

(3.63)

finally we can get V .k; kz / D

˛ sin h.k.a C d // C ˇ sin h.k.a  d // 2e : 2 k ˛ cos h.k.a C d //  ˇ 2 cos h.k.a  d //   cos kz .a C d / (3.64)

In the case when a >> d , we get the expression for the Coulomb potential for a layered system filled by the medium with static dielectric constant 1 W V .k; kz / D

2e sin hka : 1 k cos hka  cos kz a

(3.65)

In the case a << d , the substitutions 1 ! and a ! d must be made in (3.65). The screened Coulomb potential VQ .q; qz I !/ can be expressed by using an electron polarizator ….q; qz I !/ as VQ .q; qz I !/ D

V .q; qz / : .1 C V .q; qz /….q; qz I !//

(3.66)

The polarization operator ….q; qz I !/ is given by the formula [274] ….q; qz I !/ D 2

X .n..p C q; pz C qz //  n..p; pz /// p;pz

.p; pz /  .p C q; pz C qz / C i !

;

(3.67)

where n.:/ is the Fermi distribution. In the case of the energy spectrum in (3.15) and at zero frequency, we receive 1 ….q; qz I !/ D1 2 ….0/ q where ….0/ D

Z

 aCd   aCd

dpz 2 1 .A  .2qq  /2 / 2 v.A2  .2qq  /2 /sgnA; (3.68) 2

m ; A D q 2 C 4mt „2 1 2mt.1  cos.pz  q2z /.a C d // 2 . For

/ sin. qz .aCd / sin.pz .a C d //; and q  D .qF 2  2

the functions v.x/ and sgn.x/; v.x/ D 1 when x > 0, v.x/ D 0 when x < 0, sgn.x/ D 1 when x > 0 and sgn.x/ D 1 when x < 0. From last expression, the polarization operator ….q; qz I 0/ remains constant over a wide range of q; and there are corrections in the vicinity of 2pF .

3.5 Effect of Coulomb Repulsion in Layered Single-Band Superconductors

113

For  (see (3.59)), we can get analytical expressions for different asymptotic cases. For the case of a > d; the value of  averaged over the Fermi surface Coulomb potential is given as [342] 0

1 1 1 ˛0 C 1 C ln 1=2 1 .˛0 C 1 =2pF a/ C 2˛0 B B .2pF a˛0 .2 1 C 2pF a˛0 // C .a;t/ D B C: A  @ 4˛0 .t=EF /1=2 1 C C 1C ˛0  1 1C ˛0 (3.69) 2

e is the ratio of average Coulomb potential to the Here, the parameter ˛0 D „v F kinetic energy of an electron on the Fermi surface, and usually ˛0 << 1: In the opposite asymptotic case of a < d; the replacements 1 ! and a ! d must be employed. Equation (3.69) shows that  decreases as the thickness d of the superconducting layers is increased. Such a result seems attractive for explaining the empirical Chu’s rule (see [369] and references therein). According to this rule, the critical temperature of cuprate superconductors can be calculated by using the formula

Tc .n/ Ð 40n .K/; where n is the number of CuO2 planes. However, at n > 5 saturation of Tc .n/ arises. In our model, the thickness of the conducting layer is increased by increasing the number of CuO2 planes. Our results are in good agreement with those obtained from Leggtett’s calculations [370, 371]. As shown in [370, 371], the difference between the transition temperature, Tc , for a homologous series of n layers and the singlelayer value is given by Tc D const.1  1=n/: Such a conclusion is related with the fact that the Coulomb energy in an n-layer structure is proportional to the number of acoustic modes (n  1 acoustic modes for n layers). Consequently, the saved energy per layer can be calculated as .n  1/=n D .1  1=n/ [370]. It is useful to remark that Leggett’s calculations are completely independent of any “model” or of the fundamental mechanism of superconductivity in the cuprates. Above presented approximation is related to straightforward calculations in the framework of the McMillan approach and takes into account the “hard” phonon spectrum of a cuprate superconductors. In both calculations, Tc saturates with increasing number of CuO2 planes (or ratio of the thickness of the conducting layer to the thickness of the dielectric, d=a). The values of a and d for different homologous cuprate series are presented in [372]. Using these data, we plot the ratio d=a as a function of the number n of C uO2 planes (Fig. 3.9). The ratio d=a for cuprate superconductors increases with increasing n; which corresponds to the region in Fig. 3.10, where the Coulomb repulsion changes crucially and means a considerable change in the critical temperature of layered superconductors is induced by changing the number of CuO2 planes. The value of d=a D 2:3 for n D 2 in Fig. 3.10 corresponds to the

114

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Fig. 3.9 The ratio d=a as a function of the number n of CuO2

Bi2 Sr2 CaCuO8 compound. For another two-layer superconductor YBa2 Cu3 O7ı ; the ratio d=a D 1:73 [372]. As shown in [19], for all cuprates, the lattice static dielectric constant varies in the range 6–10, and for YBa2 Cu3 O7ı we have a value of about 4 [372]. To estimate  D 1 ; we will use a value of in the range 4– 10, while 1 can be taken to be about 1. Consequently,  varies as 0.1–0.25. As Fig. 3.10 shows, despite the different values of d=a and  for YBa2 Cu3 O7ı and Bi2 Sr2 CaCuO8 [372], the Coulomb repulsion is the same for both compounds, and for this reason, the critical temperatures of these compounds are nearly the same. As a concluding remark, it is interesting to add some considerations of the magnesium diboride [103]. This material also has a layered structure with boron atoms forming layers of two-dimensional honeycomb lattices (single layer). Our results can also be applied to MgB2 in the limit when d=a tends to zero. It is interesting to discuss question related with conditions in which several atomic layers can be approximated by the continuum dielectric medium. It is well known that at the contact region of different layers in superlattice the crystal structure is deformed and as a result dielectric constant in this region is different from those for the bulk material. Due to that, for our purpose of finding the dependence of the critical temperature on the thickness of conducting and dielectric layers, dielectric constants and 1 presented here can be considered as effective dielectric constants of layers. In our opinion, introducing more realistic function for

3.6 Plasmon Spectrum of Layered Superconductors

115

Fig. 3.10 Coulomb repulsion in layered systems as a function of d=a for different values of 1 D

the change of dielectric constant (different from square-well-like changing, which was used in [367]) will change our results unconsiderably. Similar questions was discussed by [373], [374] many years ago in relation with exitonic superconductivity in “sandwich” structures.

3.6 Plasmon Spectrum of Layered Superconductors The plasmon modes of layered superconductors can be determined as poles of the Dyson’s equation for Coulomb potential, which has a form of expression (3.66). To calculate the plasmon spectrum, we use the expression for the bare Coulomb interaction V .q; qz / [367] of charged particles in a periodic layered system, consisting of alternating layers with different values of dielectric constant in the large wavelength approximation. It is clear that as the thickness of conducting layer increases the Coulomb repulsion is decreased. Using (3.64), (3.66), and (3.68), we can obtain the final expression for the plasmon spectrum in layered superconductors as

116

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects



 .qz D/ ! .q; qz / D vF q C 8t sin ….0/V .q; qz /; 2 2

2 2

2

2

(3.70)

where vF – velocity of electrons on the Fermi surface. At vanishing interplane tunneling integral between planes, t D 0 qz ! 0 and qD >> 1 we obtain the spectrum of two-dimensional plasmons in long wavelength approximation:  !.q/ D vF

2q ab

 12 ;

(3.71)

where ab Bor radius for free electron, ab D me1 2 . In the case qD << 1 and qz D << 1; the plasma frequency depends on the direction of wave vector:  !.q; qz / D

2 ab D

 12 h

2

2

v2F C .t D 2 v2F / cos

i1=2 ;

(3.72)

where is the angle between wave vector and normal vector to the layer. As follows from (3.72), the spectrum of plasmons is strongly anisotropic. Frequency of plasma oscillations with wave vector perpendicular to layer is tD << 1 times smaller than vF plasma frequency in the layer [375]. The plasmon frequency for a superlattice is given by the expression in (3.70). At long wavelength .q; qz ! 0/, we have an optical plasmon mode (bulk plasmon) ! 2 .0; 0/ D

8E F e 2 a C d : ˛ 2 .a C d /2  ˇ 2 .a  d /2

(3.73)

 In the other limit qz D D , we obtain in the lower branch an acoustic plasmon mode.  For the qz D D , q.a C D/ << 1; and EtF << 1, we obtain

!.q/ D !.0; 0/

˛ 2 .a C d /2  ˇ 2 .a  d /2 q: 2.a C d /

(3.74)

The plasmon spectrum of a layered superconductors has a rather complicated  form a band as shown in Fig. 3.11. structure. The plasmon modes for 0 < qz < D The size of the band is defined by the parameters  and ratio da . It is also important  to note that in the limit qz D !  the slope of acoustic plasmons d! dq .qz D D / is greater than in the case qz D D 0: In Fig. 3.12, we plot the dependence of the  normalized slope of acoustic plasmon modes d! dq .qz D D / versus ratio parameter d=a: It is clear that by increasing the thickness of conducting layer, the slope d! dq  .qz D D / is increased. Such conclusion is in good agreement with numerical calculations performed by Pindor and Griffin [376], where a periodic stacks of planes were considered.

3.6 Plasmon Spectrum of Layered Superconductors

117

4

ω(q,qz) 3

ω

qz=0 2

1

qzD=π

0 0

1 q

Fig. 3.11 The plasmon modes of layered superconductors for 0 < qz < =D; D D a C d is the superlattice period 4 η=0.1 η=0.3 η=0.6

dω/ dq(qzD =π)

3

2

1

0 0

5

10 d/a

15

20

25

Fig. 3.12 The dependence of the normalized slope of acoustic plasmon modes d!0 =dq .qz D =D) versus the ratio d=a for different values of  D 1

We can see that increasing the thickness of metallic layer leads to a decrease in the plasmon frequency !.0; 0/. These results can be useful for the explanation of the experimental data for YBa2 C u3 O7ı (!.0; 0/ D 2:3 eV) [377] and

118

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Bi2 S r2 C aC u2 O8 (!.0; 0/ D 1 eV) [378, 379] compounds. It is well known that in YBa2 Cu3 O7ı there are two CuO2 planes, while in Bi2 Sr2 CaCu2 O8 three CuO2 planes and as a result plasmon frequency is decreased. In our model, thickness of conducting layer increases by increasing the number of CuO2 planes. Ratio d=a for cuprate superconductors correspond to region at Fig. 3.10, where Coulomb repulsion changes crucially. Another interesting problem is related with influence of low-energy plasmon modes on superconductivity in layered systems. Consequence of the existence of plasmons on the superconductivity was discussed by [380]. As shown in this work, low-energy plasmons can contribute constructively to superconductivity. Bill et al. [380] (see also [381, 382]) considered the simplest form (3.65) for the Coulomb interaction in layered systems with conducting planes with zero thickness. The conducting sheets are stacked along the c axis and separated by spaces with dielectric constant M : The electrons moving within the superconducting sheets .t D 0/. The purpose of the work [380] was to study an increasing influence of the phonon–plasmon interaction on the electron pairing mechanism in framework Eliashberg theory. The plasmon contribution for superconductivity is shown to be dominant in newly discovered layered superconductor metal-intercalated halide nitrides [380]. In more early work [383] was reported study of plasmon modes in layered superconductors with conducting planes with zero thickness using kinetic equations for Green functions. It is shown that at the vicinity Tc plasma oscillations transformed to Carlson-Goldman mode [384] observed. Unlike other works, influence of the order parameter on the plasmon spectrum was also discussed. In [376], it was shown that plasmon modes which are expected in cuprate superconductors should be characteristic of a superlattice with a basis of several metallic sheets. Numerical results are given for the superlattice plasmon dispersion relations for two and three sheets/unit cell. Electron gas in metallic sheets is considered as two dimensional. If the spacing of the sheets is small compared to the superlattice period, it is shown that the low-frequency plasmon branch is essentially identical to those of an isolated bilayer or trilayer. Unlike the approaches presented above [376, 380, 383], in this paragraph was developed simplest model taking into account thickness d of the conducting sheets. The values of d and the thickness of dielectric layer a for different homologous cuprate series are presented in [372]. The ratio d=a for cuprate superconductors increases with increasing number of CuO2 planes in unit cell; which corresponds to the region in Fig. 3.10, where the Coulomb repulsion changes crucially and means a considerable change in the plasmon frequency of layered superconductors induced by changing the number of CuO planes. Studies with the growth of single crystals [235] show anisotropy of physical properties in MgB2 : Our results can be applied also to MgB2 in the limit, when d=a tends to zero. Calculation of plasma frequency in MgB2 ; using de Haas van Alphen data, was conducted by [385]. Another pecularity of plasmon modes in MgB2 ; related with two-band nature of superconductivity in this compound. In this case, the appearance of low-energy

3.7 Specific Heat Jump of Layered Superconductors

119

plasmon branches, so-called “demons” [386] appears as a result of two overlapping bands.

3.7 Specific Heat Jump of Layered Superconductors It is useful to note that manipulation by Eliashberg equations, presented in this part is not so easy, and sometimes it is hard to get physical results. From this point, BCS limits of Elashberg equations seem more convenient for obtaining qualitative results. It is well known that in the case of weak e-ph coupling Eliashberg equations (3.9) and (3.10) transferred to BCS equation [337]: Z.!/ D 1; Z  Z d.pz0 / 1 d2 p .pz / D .2/2  2=d 0 0 0 2 tanh @

. 2 .p

(3.75)

0

0

1

0

. .pp ;pz pz /C2 .pp ;pz pz //1=2 2T  p0 ; pz  pz0 / C 2 .p  p0 ; pz 

0

pz

//1=2

AV

0

pz; pz .pz /; 0

(3.76) where  is the energy spectrum (3.15) shifted from chemical potential  and pairing 0 potential V .pz ; pz / can be taken in similar way as in (3.19). In (3.76), we neglected by the Coulomb potential. Solution of linearized BCS equation at Tc is determined by the expression (3.20). At the vicinity of critical temperature Tc , the value of i can be expanded into series with small parameter t1=2 D .1  T =Tc /1=2 : i D 0 ci t 1=2 C ci t 3=2 C ::: Then from (3.76) we derive: .x00  1/c0 C

01 xc1 D 0; 2

.x11 =2  1/c1 C 01 xc0 D 0;

(3.77a) (3.77b)

where ij D N .0/Vij ; and N .0/ represent the two-dimensional density of states. This equation determined critical temperature Tc and ratio of order parameters c0 =c1 . Below we introduce ratio of order parameters at the temperature close to Tc   c0 .01 =2/x 0 1  .11 =2/x D D D D ; (3.78) 1 T DTc c1 1  00 x 01 x 2d

2d

where x D ln. !Tc0 / , !0 represent the Debye frequency. The behavior of order parameters 0 and 1 at temperatures close to critical can be calculated from (3.76) and then rewriting in Matsubara technique. After expansion on the right side on the powers

20 , Tc2

we have [274]:

120

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

(

) 0 2 .pz / 1 .pz / D T d.pz ; pz /.pz /  ; !n2 C  2 .!n2 C  2 /2  !0 n (3.79) where !n D .2n  1/kT with n D 0; ˙1; ˙2; ˙3; ˙4; ::: are the Matsubara frequences. Substituting the expression (3.20) for .pz / into last equation, and then calculating the integrals and equating the coefficients at the same harmonics, we have:   !0 01 7.3/ 0 D ln 00 0 C 1  T 2 8 2 Tc2    

3 4 10  00 30 C 0 21 C 320 1 C 31 ; (3.80) 2 2 3 XZ



d.pz0 / 2=d

Z

10

0

0

  7.3/ !0 11 01 0 C 1  1 D ln T 2 8 2 Tc2    

11 3 4 3 3 2 2 30 1 C 1 ;  01 0 C 0 1 C 2 2 3

(3.81)

where .x/ represent the Rietmann function. Taking into account the expression ln !T0 D x C ln.1 C t/ ' x C t and after same transformations, we can get following expression for c0 W 8 2 Tc2 84 C 42 : (3.82) c02 D 7.3/ 84 C 242 C 3 For the calculation of specific heat jump at critical temperature, we will use the expression from [387] CS  CN D

  1 X @2 .pz / exp.".p; pz /=T / ; 3 Tc p;p @.1=T / T DTc .1 C exp.".p; pz=T //2

(3.83)

z

where ˇ D 1=T: From summation to the integration over momenta with quasitwo-dimensional spectrum (3.15) under t= << 1; we arrive at: CS  CN D N2d .0/

8 2 Tc2 84 C 82 C 2 : 7.3/ 84 C 242 C 3

(3.84)

Taking into account specific heat in normal case [204], finally for the normalized specific heat jump of layered superconductors we can get [228]: 84 C 82 C 2 C S  CN D 1:43A./I A./ D 4 : CN 8 C 242 C 3

(3.85)

3.7 Specific Heat Jump of Layered Superconductors

121

(CS – CN) / CN

2

1

0

0

1

2 3 a = (c + 1) / (c –1)

4

5

Fig. 3.13 The behavior of the specific heat jump as a function of the anisotropy parameter a

As followed from the expression (3.85), A./ smaller than one and as a result, the normalized specific heat jump in layered superconductors is smaller than in the isotropic case. This result in qualitative agreement with the paper [388]. In last paper, it was shown that in general case A./ < 1 for anisotropic superconductors; however, the explicit expression for specific heat jump function was not obtained. Detailed behavior of the function A./ is determined by the anisotropy of order parameter. In the case of layered superconductors with pairing in neighbors planes, the order parameter is .pz / D 0 +1 cos pz d and it is convenient to introduce an anisotropy parameter aD

max 0 C 1 C1 : D D min  0  1 1

(3.86)

The behavior of the specific heat jump as a function of the anisotropy parameter is presented in Fig. 3.13. Presented the result can be used for the calculation anisotropy parameter in MgB2 : The experimental value of the specific heat jump in MgB2 N is CSCC D 1:18 [197]. Using (3.86), we find that a D 0:5 . D 3/ and N a D 2 . D 3/. Physical solution corresponds to the case of the positive ratio of order parameters  D 3: The similar result was obtained by computer simulation in the case of strong e-ph coupling limit in the framework of Eliashberg theory [389] for the cylindrical Fermi Surface with energy spectrum (3.15). As shown by calculations the specific heat jump decreased through the increase of the (see (3.78) and (3.85)). It is also necessary to state that the anisotropy ratio, 01 00 parameter obtained from our analytical calculations  D 3 is close to the computer

122

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

calculations performed by Golubov et al. [198] in the framework of isotropic two band microscopic Eliashberg theory (  =2.63). 

3.8 d-Wave Single Band BCS Theory In this paragraph, we will briefly discuss the properties of d-wave superconductors in framework of BCS theory. The order parameter characterizing the d-wave symmetry is given by expression (2.92). Interaction potential between electrons can be taken as (2.93). The gap equation is given by [390] !+   ZEc * E cos2 .2/ tanh 2 dE Re p D 1; 2T E 2  2 cos2 .2/

(3.87)

0

where Ec is cut-off energy and h....i means average over : From last expression, we can obtain   1 (3.88) ; 0 D 2:14Tc : Tc D 1:136Ec exp   Asymptotic expressions for temperature dependence of order parameter .T / has a form [390] 8  5  3 135 T T ˆ ˆ ˆ C 1:136Ec ; for T << Tc .5/ < 1  3.3/  8  0 0 .T /=0 D 1=2  ˆ 8 2Tc ˆ ˆ : .1  T =Tc /1=2 ; for T ' Tc : 0 21.3/ (3.89) Low temperature asymptotic form of specific heat is given by expression Cs .T / '

27.3/ T 2

N ; for T << Tc : 2 0

(3.90)

In contrast to exponential decreasing of specific heat at low temperatures in swave superconductors, d-wave superconductors reveal T 2 dependence. Specific heat jump of d-wave superconductors is smaller than in s-wave case (Fig. 3.14) (see also Fig. 3.13). Finally, the quasiparticle density of states is given by



N.E/=N0 D Re p

 E E 2 2 cos2 .2/

8 2 ˆ ˆ < xK.x/; for x < 1;  D ˆ ˆ : 2 K.1=x/; for x > 1; 

where x D E=; which is shown in Fig. 3.15.

(3.91)

3.8 d -Wave Single Band BCS Theory

123

2.5 d-wave s-wave

Cs(T)/ γNTc

2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1

T/Tc

Fig. 3.14 Behavior of specific heat in d -wave superconductors below Tc (s-wave behavior denoted as dotted line) 4 d-wave s-wave

N(E)/N0

3

2

1

0

0

0.5

1

1.5

2

E/Δ

Fig. 3.15 Behavior of density of states in d -wave superconductors below Tc (s-wave behavior denoted as dotted line)

It clears the fundamental differences from the case of s-wave superconductors for intergap structure. For cuprate superconductors, it is well established that Znsubstitution of Cu in the CuO2 planes produces dramatic effects. Inclusion of impurity effects can be taken into account using quasi-particle Green-function 

G 1 .k; !/ D !I  3  .k/1 ;

(3.92)

124

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

where 1 ; 3 are the Pauli spin matrices operating in the Nambu space, I is the unit  matrix, ! is the normalized frequency determined from [391] * q

+



!

2

! 2 cos2 .2/



ˆ ! D ! C iY

*

cot2

ıC

q



+!2 ;

(3.93)

!

2

! 2 cos2 .2/

ni ˆ ; ni is the impurity concentration and N0 is the electron density where YD N0 of states in the normal state on the Fermi surface per spin, and ı is the scattering phase shift at E D 0: As a result of influence of impurities, the superconducting transition temperature is rapidly suppressed [391]. Second effect related with rapid increasing of residual density of states (i.e., the quasi-particle density states at E D 0/ [391]. Perfect agreement of above presented BCS theory with experiment for cuprate superconductors is not expected; however, such models are useful for clarify many aspects of hole-doped cuprates.

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory In this paragraph, we will assume that, due to the crystal field, the overlapping bands are not degenerate, and each of them is characterized by an effective density of states near the Fermi level. We will also introduce the matrix elements of the electron–electron (phonon-mediated) interaction. These effective parameters are not evaluated inside this theory, and their values will be taken from experimental data. We also suppose that the bound Cooper pairs are formed inside the same energy band and that the paired electrons have opposite momenta and spins. Corresponding Hamiltonian of the multiband superconductivity model written as [194] H D

X n;k;

C "n .k/ank; ank; 

1 XX C C Vnm .k; k0 /ank;" ank;# amk0 ;# amk0 ;" (3.94) V n;m 0 kk

with n; m the band index; k is the quasi-momentum; n .k/ is the energy of an C C electron in band n;  is the electron spin index, taking the values " or #; ank;" ,ank;" are the creation and annihilation operators of electrons in a band n; V is the volume of the system. The quantities Vnm .k; k0 / are the matrix elements of the direct effective interaction of the electrons. Minimization of free energy [194] leads to the energy gaps that are given by the following equations (V12 D V21 ):

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory

125

    X 1 X 1 V11 V12 E1 .k/ E2 .k/ 1 D C ; 1 tanh 2 tanh 2V E1 .k/ T 2V E2 .k/ T k k (3.95a) 2 D

    X 1 X 1 V12 V22 E1 .k/ E2 .k/ C ; 1 tanh 2 tanh 2V E1 .k/ T 2V E2 .k/ T k

k

(3.95b) where Ei2 .k/ D 2i .k/ C "2i .k/: The study of the overlapping of energy bands yields not only the quantitative difference of the results from the case of a single-band superconductor, but, in some cases, qualitatively new results. Below we present any fundamental superconducting state parameters of two-band compounds.

3.9.1 Critical Temperature In two-band superconductors, high temperatures of the superconducting transition can take place not only in the case of the attractive interaction between electrons, but even in the case of repulsive one (Vnm < 0, nI m D 1  2), yet the relationship V11 V22 < V12 V21 is fulfilled [392]. This result can be obtained from the system of (3.95a) and (3.95b), using the expression of i can be written as a series expansion .1/ in the powers of the small parameter t D 1  .T =Tc /: i D Ci t 1=2 C Ci t 3=2 C    . In this case, the linearized equations take the form: E1 .k/ E2 .k/ V11 X tanh. Tc / V12 X tanh. Tc / C1 D C1 C C2 2V E1 .k/ 2V E2 .k/ k

C2 D

E1 .k/ E2 .k/ V22 X tanh. Tc / V12 X tanh. Tc / C1 C C2 : 2V E1 .k/ 2V E2 .k/ k

(3.96a)

k

(3.96b)

k

In the BCS approximation, we have the following expression for the critical temperature Tc Tc D 1:13!D ey ; (3.97a) where p N1 V11 C N2 V22 ˙ .N1 V11  N2 V22 /2 C 4N1 N2 V12 V21 : yD 2N1 N2 .V11 V22  V12 V21 /

(3.97b)

Above mentioned restrictiction V11 V22 < V12 V21 followed from (3.97b) for critical temperature.

126

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

3.9.1.1 Impurity Effects on Critical Temperature Let us consider the simplest situation, when the impurity has no spin and consequently, the diffusion on impurities do not modify the spin of electrons. In this case, in the isotropic single-band model of the superconductor, according to Anderson’s theorem [294], and to the results of Abrikosov and Gorkov [393], the nonmagnetic impurities do not influence essentially the thermodynamic properties of the superconductor and, in particular, do not influence the critical temperature Tc . However, due to the interband channel of the electronic diffusion on impurities, the nonmagnetic impurities essentially influence the critical temperature of two-band superconductors and in general, the thermodynamic properties of the multiband alloys. Taking into account impurıty effects leads to following expression for critical temperature [194] ln

 

Tc0 „ 1 „ D ~ ˙I C ; Tc kB Tc 212 221 ~˙ D  

(3.98a)

1 q 1 C 12 b02  4a 21

   

12 12 N2 12 .0/  N1 V12 1 C  ˙ a 1 C ;  V11 N1 C V22 N2 21 N1 21 21

(3.98b) b0 D V11 N1 C V22 N2 I a D N1 N2 .V11 V22  V12 V21 /:

(3.99)

In last expressions, it introduced notations: Tc0 is the transition temperature of the pure two-band superconductor for BCS, Tc is the transition temperature of the dirty two-band superconductor, nm is the relaxation time. The integral I.x/ is given by the formula Z1 dt tanh xt2 I.x/ D D t 1 C t2



1 C x= 2

 

  1 ; 2

(3.100)

0

where is the logarithmic derivative of  function. As followed from analysis [194,394], in impurity two-band superconductors, the Anderson theorem is violated at 1 ¤ 2 , and the dependence of the thermodynamic quantities on the concentration of a nonmagnetic impurity due to the interband scatteringof electrons on impurity atoms is observed. The dependence of Tc =Tc0 on 1=12 Tc0 is shown in Fig. 3.16 for different values of the parameters of the theory. Calculation parameters chosen as: 11 D 0:4; 12 D 0:1 for all curves. For curve 1 ˛ D 0:51 .n D 2; j D 0:5/; curve 2 ˛ D 0:375 .n D 1; j D 2/; curve 1 ˛ D 0:67 .n D 1; j D 0:5/ was taken. Parameter ˛ was calculated as

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory

127

1.0

0.8

0.6

Tc Tc 0

0.4

2

0.2

1

3 0

1

2

3 4 1/2t12Tc 0

5

6

Fig. 3.16 Critical temperature of two-band superconductors with nonmagnetic impurity using BCS theory

˛D

8 ˆ 1< 2ˆ :

p

1

11  2.11  12 n.1 C j /.1 C j n/ q 211 C 4212

9 > : > ;

1 =

(3.100a)

3.9.1.2 Tc of Two-Band Superconductors with Variable Charge Carrier Density The position of the Fermi level, which can be changed by doping, plays an important role in the determination of the thermodynamic and magnetic properties of a two-band superconductor. Having assumed the nonphonon pairing mechanism of superconductivity, as well as the phonon mechanism in the many-band systems with lowered densities of charge carriers, the regard for the singularities of density of states, is extremely crucial. Since we examine a system with the variable (including low) charge carrier density, let us supplement the system of (3.95) with a relation,  determining the chemical potential  (charge carrier density n): 

nD

X m



Zdm Nm dm

d"m

2 j"m .k/  j 1 Em .k/ j"m .k/  j ; C Em .k/ Em .k/ 1C exp.Em .k/=T / (3.101)

128

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

40 38 Tc , (K )

36 1

34

Mg1–xLixB2

32

Mg1–xCuxB2

30

Mg0,8Li0,2B2–xCx

28

Mg0,95Cu0,05B2–xCx

2

MgB2–xCx

26 –0,04

– 0,02

0

0,02

0,04

0,06

d

Fig. 3.17 Critical temperature versus doping level ı in two-band superconductors in BCS theory

where



!0n; dm D n ;

if n  !0n; ; if   !0n;

(3.101a)

!0n phonon cut-off parameter for each conducting band. In Fig. 3.17, the dependency of Tc from the doping parameter ı is given. This dependency was obtained on the basis of developed above expressions (curve 1) [395]. The same figure shows the points, which correspond to the experimental values of Tc for diverse substitutions of atoms of M g and B, taken from of the experimental work, and also the experimental dependency (curve 2) [396].

3.9.2 Specific Heat of Two-Band Superconductors Calculation of specific heat jump of two-band isotropic superconductors using BCS theory leads to result [194, 225]: C 12 A1 .z2c / D 1:43A1 .z2c /; D CN 7.3/ A1 .z2c / D

.N1 C N1 =z2c /2  1; .N1 C N1 =z4c /

(3.102a)

(3.102b)

1 / is the ratio of i in the critical temperature [194, 225]. Because where zc D .  2 c 2 A1 .zc / < 1, in a two-band superconductor, the relative jump of the specific heat at the transition point is smaller than in an isotrop, single-band superconductor, where this quantity has the universal value of 1.43. Heat capacity at low temperatures T ! 0 also calculated in two-band BCS approach:

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory

C ' 2kB

X

s Nn

n

  23n .0/ n .0/ exp  : .kB T /3 kB T

129

(3.103)

Each one-electronic energy band contributes additively to the specific heat. These /. Consequently, contributions are mainly given by the activation factors exp. knB.0/ T for the contribution of a given band, the density of electronic states Nn and the energy gaps n .0/ are very important. This second quantity determines how quickly the specific heat decreases with temperature. Following expressions are true for the value of gap parameters at zero temperature also [194, 225]: N2

1 .0/ ' 2!D .zc / N2 CN1 z2c exp.c /;

(3.104a)

N1 z2c

2 .0/ ' 2!D .zc / N2 CN1 z2c exp.c /; where c D

(3.104b)

N2 V22 V12 N2 =zc : a

3.9.2.1 Specific Heat in the Case Variable Carrier Density For the specific heat jump of two-band superconductors with variable carrier density is true the following expression [395] .N1 1 ./ C N2 2 ./=z2c /2 1 C D ; 4 CN 4.N1 F1 ./ C N2 F2 ./=zc / .N1 '1 ./ C N2 '2 .// where

Z1

(3.105)

x 2 dx ; .1 C ex /.1 C ex /

(3.106)



dcn 1 dn tanh ; n ./ D C tanh 2 2Tc 2Tc

(3.107)

'n ./ D .n /

1 Fn D 8

dZcn =Tc

dx dn =Tc

sinh x  x x cosh2 x=2

:

(3.108)

In Eq. (3.107) expression for the dn is given by (3.101a). For the parameter dcn is true following relation: dcn D

!0n; n ;

if &cn    !0n; if &cn    !0n;

(3.108a)

130

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

2.0

(CS – CN) / CN

1.5

1.0

0.5

0 –0.04

–0.02

0

0.02

0.04

0.06

d

Fig. 3.18 Specific heat jump of two-band superconductors versus doping level ı

Results of calculation based on expressions (3.105)–(3.108) presented in Fig. 3.18 for the doped MgB2 . As followed from this figure in all cases (ı < 0 and ı > 0/ specific heat jump grows with increasing of dopant concentration [395]. ı D 0 corresponds to the pure MgB2 ; for which C D 0:8: At high values of CN density of carriers, we have standard BCS value for single-band superconductors C D 1:43: CN

3.9.3 Upper Critical Field Hc2 in Two-Band Superconductors Calculation of upper critical field in single-band superconductors using BCS theory was conducted by [397–399] many years ago. Similar two-band microscopical theory was developed by [400] (see also [194] ). The main purpose of this paragraph is researching of pure two-band isotrop superconductor of the secondary type for arbitrary temperatures and external magnetic field close to the upper critical field and the definition of temperature dependence of the Hc2 value. If the exterior magnetic field is great enough, the order parameters m .m D 1; 2/ of two-band superconductor is small enough, and we can use equations [401] for pure two-band superconductor: m .x/ D

1 XX Vnm T ! 0 nn

Z

dygnn0 .y; x=!/ n0 .y/gn0n .y; x=  !/ :

(3.109)

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory

131

For infinitely small values m ; Green function defines by equation at presence of the magnetic field H [402] 0 0 gnn0 .r; r0 =!/ D exp.i'.r; r0 // gnn 0 .r; r =!/;

(3.110)

0 0 where gnn 0 .r; r =!/- Green function of an electron in normal metal without magnetic field. The presence of the magnetic field is taken into account by the phase multiplier Zr 0 A.l/dl : (3.111) '.r; r =!/ D e r0

Final result can be presented as [400] Hc2 .T / D

1 6 4 2 Tc2 2 2 e v1 1C v2 2 7.3/ 8 2 2 v1 ˆ v22 ˆ ˆ 2  6 < v2 1C v21 2 31 6 6 2  61 C v1  .5/ v2 ˆ 4 10 7.3/ ˆ ˆ 1C v1 2 : v2

where 1 D

93 > > > 3 =7 7 7; 5 2> > > ;

.1 C / .1  / 11 ; 2 D ; D ; 2 2 12 21 .1/

 2 .1/ D

.N1 V11  N2 V22 /2 C 4N1 N2 V12 V21 : .N1 N2 .V11 V22  V12 V21 //2

(3.112)

(3.113) (3.113a)

On the basis of (3.112), the value of the upper critical field in the two-band system can be calculated on the whole temperature interval 0  T < Tc : The analytic solutions of this equation were obtained for T ! Tc and T ! 0 [194, 400]. It is easy to notice that Hc2 depends on the correlations of the speeds v1 and v2 of the electrons on the Fermi surface, and on the constants of the electron–phonon coupling parameter nm (Fig. 3.19). It is easy to see that with growth of v1 =v2 , the curvature in this dependence changes. Curves 3 and 4 give the curvature, which was observed during the experiment in a row of cuprates and in other classes of new superconductors (see Chap. 2). So if there are heavy carriers in the second band (low speeds on the Fermi surface), the two-band model qualitatively describes the behavior of Hc2 as a function of temperature in these materials. At the same time, the regard for the overlapping of energy bands in the determination of the value of Hc2 leads to a qualitatively new result in the dependence Hc2 (T): this dependence exhibits a positive curvature in the vicinity of the superconducting transition temperature (curves 2, 3) unlike a single-band superconductor (curve 1). As mentioned in Chap. 2, positive curvature near Tc in two-band superconductors was obtained in framework of GL theory also.

132

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

Fig. 3.19 The temperature dependence Hc2 (T)/Hc2 (0) at 11 D 0:2, 22 D 0:0, 12 D 0:12, N1 /N2 D 0:8 and values v1 =v2 D 1; 2; 3 (curves 1–3 correspondingly). The dashed curve corresponds to the experimental dependence Eltsev, Yu., Nakbo, K., Lee, S., et al.: Physica C 378, 61 (2002)

3.9.3.1 Anisotropy Effects on Upper Critical Field of Two-Band Superconductors Consider the magnetic field H (which is parallel to the .ab/ plane) directed along the y-axis. Herein, it is possible to choose Az D  H20 .x C x 0 /I Ay D Az D 0 in the symmetric view, and on the basis of (3.109), we obtain 2'.r; r0 / D eH0 .x C x 0 /.z  z0 /:

(3.114)

In the case, when external magnetic field H is parallel to the c axis, it is possible to choose Az D H20 .x C x 0 /I Ax D Az D 0 and we have for the phase multiplier 2 '.r; r0 / D eH0 .x C x 0 /.y 0  y/:

(3.115)

Using (3.109), (3.114), and (3.115), calculation of anisotropy parameter Hc2 of upper critical field was conducted in [395] in framework of microscopical approach. The temperature dependence of anisotropy parameter Hc2 is presented in Fig. 3.20. These calculations also in agreement with GL calculations of Hc2 anisotropy parameter (see Sect. 2.2.2). In both cases with lowering of temperature, Hc2 is increased.

3.9.3.2 Two-Band System with a Variable Density of Charge Carriers For investigating the superconducting properties of the system with a variable density density of charge carriers, it is necessary to supplement equation (3.101)

3.9 Properties of Two-Band Isotropic Superconductors in BCS Theory

133

g Hc2

4,5

4,0

3,5

3,0

0,2

0,3

0,4

0,5 0,6 T / Tc

0,7

0,8

Fig. 3.20 Temperature dependence of anisotropy parameter of upper critical field using two-band BCS theory (choosing of parameters see in [395])

1,5

rc0, Tc /T 0c

2

1,0

1

0,5 3 0 0,70

0,72

0,74

0,76

0,78

0,80

m,eV

Fig. 3.21 The dependence of the ratio of the superconducting transition temperatures (curve 1) of doped and pure MgB2 and the critical fields Hc2 (ab) (curve 2) and Hc2 (c) (curve 3) as function of chemical potential 

with the correlation that defines the chemical potential. The paper [395] contains the basic equations that allow determining the behavior of the quantities Tc , Hc2 (ab), Hc2 (c), and Hc2 as a function of temperature and in a system with a variable density of charge carriers upon doping of the system with electrons or holes. In addition, we take into account the mechanism of filling of the energy bands upon replacement of atoms of Mg or B with other elements of the periodic table. Let us consider the results of numerical calculations obtained for the upper critical field using (3.109) for upper critical field and relation (3.101) defining the chemical potential. Figure 3.21 shows the dependence of the ratio of the superconducting transition temperatures of doped and pure MgB2 (curve 1) and the critical fields Hc2 (ab) (curve 2) and Hc2 (c) (curve 3) at the temperature T D 0 on the electron

134

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects 5,5

g Hc 2

5,0

4,5 1 4,0

2

3,5 0

10

20

30

40

T, K

Fig. 3.22 Temperature dependence of anisotropy coefficient for pure MgB2 (curve 1,  D 0:74 eV) and doped MgB2 (curve 2,  D 0:76 eV)

density (chemical potential) [395]. We can see that for  > 0:74 eV all quantities decrease with increasing electron density of charge carriers, remaining constant for  < 0:74 eV. Consequently, hole doping does not affect the superconducting transition temperature and the upper critical field. Figure 3.22 shows the temperature dependence of anisotropy coefficient in pure MgB2 ( D 0:74 eV) and in doped MgB2 ( D 0:76 eV). The circles correspond to the experimental data borrowed from [403]. These results are in satisfactory agreement with experimental data on the magnetic properties of intermetallic compound MgB2 (both pure and doped with electrons and holes), which indicates the effectiveness of the two-band model in describing the properties of real materials and in calculating the anomalies in physical properties associated with anisotropy of upper critical field.

3.10 Nanosize Two-Gap Superconductivity Recent experiments by Black et al. [404, 405] generated much interest in the size dependence of the superconductivity. Properties of ultrasmall superconducting grains have been theoretically investigated by many groups [406–408]. As mentioned by [294], the fundamental theoretical question in such systems related with the size dependency of superconductivity. The standard BCS theory gives good description of the phenomenon of superconductivity in large samples. With decreasing size, the BCS theory fails. In ultrasmall Al grains, the bulk gap has been discussed in relation to physical properties in ultrasmall grain such as the parity gap [407], condensation energy [408], electron correlation [409] with dependence

3.10 Nanosize Two-Gap Superconductivity

135

of the level spacing [406] of samples. Nanosize two-band superconductivity was considered by [410]. In nanosize grain of a superconductor, the quantum level spacing approaches the superconducting gap of a bulk sample. In the case of twoband superconductor, a model with two sublevels corresponding to two independent bands can be considered [410]. According to results of [410] in the case of two-band nanosize superconductors, the condensation energy can be written as ENC1 ;b1 IN2 ;b2 .1 ; 2 ; 12 / D ENG1 ;b1 IN2 ;b2 .0; 0; 0/  ENG1 ;b1 IN2 ;b2 .1 ; 2 ; 12 /  n1 1 d1  n2 2 d2 ;

(3.116)

where ENC1 ;b1 IN2 ;b2 .1 ; 2 ; 12 / denotes the ground state of (N1 C N2 / electron system, b is number of electrons on single occupied levels, n and  number of pair occupied level and dimensionless coupling parameter, respectively. d1 D 2!D D I d2 D 2! are the energy level spacing; N1I I N2J number of half-filled bands i: N1I N2J Calculations lead to final result [410]: ENC1 ;b1 IN2 ;b2 .1 ; 2 ; 12 / D ENC1 ;b1 .1 / C ENG2 ;b2 .2 /   

212 1 2  212

 22 2.1 2 C 1 2 / 21 C C ; p 1 d 1  2 d2 d1 d2 12

(3.117)

where ENC1 ;b1 .1 / and ENG2 ;b2 .2 / correspond to the condensation energy for the single-band case. Due to coupling constant 12 ; in the same phases of order parameters 1 and 2 condensation energy decreases. On the other hand, in the opposite phases 1 2 C 1 2 < 0, the condensation energy becomes large. To discuss the critical level spacing for two-band superconductors, which means that both gap functions vanish at a level spacing, 1 D 2 D 0: For the odd or even electron number parity in the grain critical level spacing becomes as: for odd numbers   1 d2 o o o D !D exp. / exp  D d1c ; (3.118a) I d2c d1c  d1 for even numbers e d1c

  1 d2 e e D 4!D exp. / exp  D d1c : I d2c  d1

(3.118b)

Analysis shows that the critical level spacing strongly depends upon 12 and the difference between the effective interaction constants for sublevels. The parity gap in a single band system is the difference between the ground state energy of a grain containing 2n C 1 electrons (odd parity) and the average ground state energy of grains containing 2n and 2n C 2 electrons. It is measure for the cost in energy of having one unpaired electron. The parity gap of nanosize two-gap superconductors is written as

136

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

 1 G G G 1p D E2n .1 ; 2 ; 12 /  E2n1 C2n2 ;0 C E2.n 1 C1C2n2 ;1 1 C1/C2n2 ;0 2   d1 1 1 ; (3.119a) D 1  4 2  1 G G G 2p D E2.n . ;  ;  /  C E E 1 2 12 2.n 2.n C1/C2n C1;1 C1/C2n ;0 C1/C2.n C1/;0 1 2 1 2 1 2 2   d2 32 1 (3.119b) D 2  4 1 where 12 the ratio of the density of states. Last expressions suggest two kinds of the dependence of the parity gap on the level spacing. The parity gap does not depend upon the effective interaction 12 . The structure around Fermi level plays an important role of the contribution to the size dependence on the parity gap [411].

3.11 Effect of Nonadiabacity in Two-Band Superconductors Theory of superconductivity in two-band nonadiabatic systems with strong electron correlations in the linear approximation over nonadiabaticity is considered in the article [412]. Nonadiabaticity here means that having determined the mass operators, diagonal Mn and nondiagonal †n in electron Green functionof twoband system (n D 1; 2) the additional in comparison with Eliashberg–Migdal theory diagrams with intersection of two lines of electron–phonon interaction, which corresponds to vertex functions PV n as well as to “intersecting” ones PC n (n D 1; 2), are taken into consideration. Applied weak coupling approximation (Tc =!D ; n =!D <<1) allows to obtain analytical formulas for these functions and shows that their behavior is determined by the value of transferred momentum q of electron–phonon interaction. At low values of cut-off momentum of electron– phonon interaction qc (qc 2pF ) functions PV n and PC n give positive contribution and at q 2pF they give negative contribution. Temperature of superconducting transition Tc is determined from the equation [412] (see also paragraph 3.4) 











.11 22  12 21 2 ; 12 /ˆ1 .Tc /ˆ2 .Tc /  11 ˆ1 .Tc /  22 ˆ2 .Tc / C 1 D 0; (3.120) 





where 11 D Z111 ; 22 D Z222 ; 12 D easily represented in the form:

12 ; Z1



21 D

21 Z2:

ˆn .Tc / D c C fn I c D ln

. The function ˆn .Tc / can be 2!D ; Tc

(3.121)

3.11 Effect of Nonadiabacity in Two-Band Superconductors

137

Tc / w 0

0.3

0.2 1

0.1 2 1′ 0.0

0.2

0.4

0.8

0.6

1.0

m

Fig. 3.23 Dependence of the superconducting transition temperature Tc on the Migdal parameter m. Curves 1 and 10 correspond to the respective cases of a nonadiabatic two-band system at Qc D 0:1 and Qc D 0:9; curve 2 corresponds to the case of an adiabatic system

fn D



1 1 m0n 1 mn C ln.1 C mn / C ln.1 C m0n /  C ; 2 2 4 1 C mn 1 C m0n

where mn D



!D



W n  n

; m0n D

!D  ; n



Wn D

Wn  ; n Zn

D

n ; Zn

(3.122)

Wn is width of band

number n: In calculations was considered the simple dispersion law of electron energy for nth band p2 "n .p/ D  : (3.123) 2mn The ratio Tc =!D as a function of m is presented in Fig. 3.23. The curve 2 on this figure corresponds to the case of two-band adiabatic system, curves 1 and 10 correspond to the case of nonadiabatic two-band system at Qc D 0; 1 and Qc D 0; 9, respectively. The comparison of these curves shows that the nonadiabatic effects and strong electron-correlation considerably increase the temperature of superconducting transition Tc at Qc 1. Moreover, the biggest value of Tc is reached in the area of small m .m 0:1/. At Qc 1 the Tc decreases and weakly depends on parameter m. The dependency of ratio Tc =!D on coefficient ı, which determines the proportional increasing of all electron–phonon interaction 0 constants in two-band model (nm D ımn ) is presented in Fig. 3.24. The cases m D 0; 2I Qc D 0; 1 (curve 1) and m D 0; 2I Qc D 0; 9 (curve 2) are considered. Increase of Tc follows from this figure with increasing the coefficient ı in both cases. These results point on the fact that in the systems with small values of the ratio Debye energy on Fermi energy the contribution of top approximations to the superconductivity can be essential if the transmitted momentum is small. Similar results was obtained for layered nonadiabatic systems (see paragraph 3.4).

138

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

1

Tc / w 0

0.4

0.2 2 1′ 0.0 0.5

1.0

1.5

2.0

d

Fig. 3.24 The dependence of Tc on the coefficient proportional to the increase in the values of the intraband (nn ) and interband .nn1 , n; n1 D 1; 2/ electron–phonon interaction constants for the Migdal parameter value m = 0.2. Labeling of the curves is the same as in Fig. 3.23

3.12 Leggett’s Mode in Two-Band Superconductors Density oscillations can couple to oscillations of the phase of the superconducting order parameter via the pairing interaction. In a neutral system, these collective sound like oscillations are called Bogolyubov–Anderson–Goldstone mode [413–415]. In a charged system, the frequency of this mode is pushed up to the plasma frequency due to the long-range Coulomb interaction. Leggtett’s mode in two-band superconductors is obtained using the modulus-phase variables in the path integral formalism in study [125]. The modulus-phase variables i and j in two bands are introduced exactly as: ˆi .; r/ D i .; r/ exp.i i .; r//;  ‰i .; r/ D

 exp.i i .; r/=2/ 0 Fi .; r//; 0 exp.i i .; r/=2/

(3.124a) (3.124b)

making the moduli i of the pairing field real. The effective thermodynamic potential is the sum of the energies of the phase fluctuations in each band [125] and can be presented as 0

1 21 22 2g12 1 1 B g C g  g g 1 2 cos. 1  2 /TrLnL1 TrLnL2  C 22 22 11 B 11 C B C pot .i ; i ; '/ D B Zˇ C; B 1 C @ ddr1 dr2 e'.; r1 /Vc1 e'.; r2 /Ci e'.; r1 /nı.r1 r2 / A 2 0

(3.125)

3.12 Leggett’s Mode in Two-Band Superconductors V2

139

V2

V2

where g11 D V11 .1 V1112V22 /; g22 D V22 .1 V1112V22 /; g12 D V12 .1 V1112V22 /: In (3.125), the Green’s function ( i ( i D 1, 2, 3) are Pauli matrices) L1 1

 D I @ C 3

 r2 C  C 1 i .; r//  †i .@ i /: 2mi

(3.125a)

Expressions for mass operators †i and repulsive part ' are presented in [125]. In pot , the most important for the appearance of Leggett’s mode term is the Josephson coupling energy of the condensates in two bands. This term explicitly depends on the relative . 1  2 / phase of two condensates. Finally, we arrive at s " # 1 N  N 1 2 2 2 2 2 4 2 2 2 2 2 .c  c2 /K 2 ; !0 C.c1 Cc2 /K ˙ !0 C .c1  c2 /K 4  2!0 ! D 2 N 1 C N2 1 2

(3.126) with !02 D

1 N1 C N2 8V12 1 2 : 2 N1 N2 V11 V22  V122

(3.127)

Legget modes correspond to solution ! 2 D !02 C v2 K 2 ; v2 D

N1 c22 C N2 c12 : N1 C N2

(3.128)

Introducing the dimensionless coupling constants, ij D Ni Vij that are often used for the description of the two-band model, (3.128) may rewrite in the following form

25

Mg1–x Alx B2, T = 4.2 K

ε20 (meV2)

20

15

10

5

Fig. 3.25 The dependence of excitation energy upon the product of energy gaps   : Experimental points taken from [126]

0

5

10

15

Δσ Δπ (meV2)

20

25

140

3 Anisotropic Eliashberg Equations and Influence of Multiband Effects

!02 D

11 C 22 41 2 : 11 22  12 21

(3.128a)

In two-bandsuperconductors, collective oscillations of the exciton-like Leggett mode [124] caused by phase fluctuations of order parameters from different bands are observed in [126] (see Fig. 3.25). As followed from Fig. 3.25, excitation energy at T D 4:2 K for Mg1x Alx B2 (0  x  0.45) samples in qualitative agreement with the prediction of [125].

Chapter 4

Fluctuation Effects in Anisotropic and Multiband Superconductors

The last Chapter of the book devoted to fluctuation effects in new classes of superconductors. There is an excellent book of Larkin and Varlamov [416] about manifestation of fluctuation in isotropic and strong anisotropic superconductors. Here, we study the fluctuation effects on specific heat in two-band superconductors without and taking into account of external magnetic field. Diamagnetic susceptibility and fluctuation of conductivity neat Tc are calculated using twoband GL theory in application to new class of superconductors. Fluctuation of phase of order parameter effects in layered superconductors on Tc studied using Lawrence–Doniach functional. Influence of post-Gaussian fluctuation in superconductors is also considered. Finally, we present generalized GL theory for layered superconductors with small coherence length for possible application to cuprate superconductors.

4.1 Fluctuations of Specific Heat in GL Theory 4.1.1 Single-Band Isotropic GL Theory In previous chapters, we neglect the effect of thermodynamic fluctuations on superconducting properties. Fluctuational effects expected to be important for some interval of temperature near critical temperature Tc : Order parameter ‰.r/ can be expanded as Fourier series X i kr ‰.r/ D (4.1) ke : k

For small deviation of order parameter ı‰ D ‰.r/  ‰0 from equilibrium state j‰0 j2 D  ˇ˛ [204] and using (4.1) for single-band GL theory, we can get expression for fluctuational part of free energy as [204]

I. Askerzade, Unconventional Superconductors, Springer Series in Materials Science 153, DOI 10.1007/978-3-642-22652-6 4, © Springer-Verlag Berlin Heidelberg 2012

141

142

Fluctuation Effects in Anisotropic and Multiband Superconductors

( ) X „2 k 2 2 Hf D V C ˛ C 3ˇ j‰0 j j‰k j2 : 4m

(4.2)

k¤0

The mean value square amplitude of a fluctuation of the order parameter with wave vector k can be calculated as Z1 Hf d2 j‰k j2 j‰k j2 e T hj‰k j2 i D

0

D

Z1 Hf d2 j‰k j2 e T

4m T ; „2 k 2 C  2 .T /

(4.3)

0 „ where  2 .T / D 4m˛.T coherence length of single-band superconductors. As / followed from (4.3), long wavelength fluctuations are important near Tc . The fluctuation contribution to the partition function may be written [207] 2

Z e



Ffl T

Z1 D

dVH



D‰.r/e

T

Z1 H.‰k ;‰k / D …k 2d2 j‰k j2 e T ;

0

(4.4)

0

where D‰.r/ means of path integration, which is reduced to integration amplitude of the order parameter. Then for fluctuational contribution to the free energy from (4.2) and (4.4), we derived 8 9 !( ) 1 = X < Z VX „2 k2 Ffl D T ln 2 d j‰k j exp  C ˛ C 3ˇ j‰0 j2 j‰k j2 : ; T 4m k

D T

X k

 ln

k

0

1 16 mT „2 V k2 C  2 .T /

 :

(4.5)

As a measure of the importance of fluctuations we will evaluate the fluctuation contribution to the heat capacity and compare with discontinuity at Tc [417–419]. This discontinuity at Tc given by C D T

˛2 @2 F @S D D T V: 2 @T @T ˇ

(4.6)

In evaluation of sum in (4.5), the introduction of a cut-off momentum km is requested (neglecting long wave-length fluctuations). In calculations, we ignore all temperature dependences except that arising from .T /

4.1 Fluctuations of Specific Heat in GL Theory

Sfl D 

143

1 @Ffl 4mT a X : D 2 C  2 @T „2 k k

(4.7)

Normalized fluctuational contribution to specific heat jump of three-dimensional superconductors is given by TF ' a2 Tc .0 kF /3 Tc V . ˇ / Cfl



Tc jT  Tc j

1=2 ;

(4.8)

where 0 D .T D 0/I n and TF are the electron density and Fermi temperature. For low temperature conventional superconductors with Tc < 23 K [416], influence of fluctuation effects on physical properties negligible small. For two-dimensional superconductors, (4.8) becomes  Cfl D

4mTc a „2

2

S  2 ; .2/2

(4.9)

As followed from (4.9), the heat capacity in two-dimensional superconductors 1 ; which means enhancement of fluctuations for low-dimensional diverges as jT T cj superconductors.

4.1.2 Fluctuation of Specific Heat in Two-Band Isotropic Superconductors For the calculation fluctuational specific heat in two-band isotropic superconductors, we will use (2.9). In the absence of any external magnetic field, H D 0, near Tc it is true the relation ‰1 .x/ D C0 ‰2 .x/ [200, 201], where C0 is determined as C0 D 

" ˛2 .Tc / D : ˛1 .Tc / "

(4.10)

Due to last relation, our two-band GL model in vicinity Tc is equivalent effective single-band GL theory equation 

„2 4m



x2 d2  dx 2 ls4



‰ C ˛ .T /‰ C ˇ  ‰ 3 D 0;

(4.11)

" ; ˇ  D ˇ1 : C0

(4.12)

with effective parameters [420]: m D

m1 1

4m1 "1 C 0 „2

; ˛ .T / D ˛1 .T / C

144

Fluctuation Effects in Anisotropic and Multiband Superconductors

Cfl(mJ/mol K)

Two bands Single band Experimental

101

10–3

10–2 T/Tc–1

Fig. 4.1 Enhancement of fluctuations of specific heat in two-band superconductors

Calculations similar to above presented method (paragraph 4.1.1) with effective parameters (4.12), we receive final expression for two-band superconductors [420] C . C /TB C . C /SB

Tc D Tc1 .1 C

1 4m1 ""1 3=2 / „2 ˛2 .T /

 1C

"2 ˛1 .T /˛1 .T /  "2

1=2 :

(4.13)

Results of the calculation using expression (4.13) are shown in Fig. 4.1 (circles). Experimental data for the fluctuations of specific heat in the two-band superconductor MgB2 taken from [421] (square symbols). The result of the single-band GL calculations [204, 207] in Fig. 4.1 is presented by straight solid line. As followed from last expression, in general case fluctuations in two-band superconductors 2 are enhanced by the factor .1 C ˛1 .T /˛"1 .T /"2 /1=2 > 1: On the other hand, another

enhancement factor related with the ratio TTc1c > 1; if take into account fitting parameters in (2.11). It means that as a result of interaction between order parameters (see (2.11)), the critical temperature of the two-band system increases, and as a result, the fluctuation part of specific heat also increases. Additional number

4.1 Fluctuations of Specific Heat in GL Theory

145

of enhancement factor related with sign of product ""1 : If ""1 < 0; then fluctuations additionally grows in respect single-band case. As shown in Chap. 2, in the case of MgB2 for the fitting of experimental data we take ""1 < 0: These arguments seem reasonable to explain [420] experimental data in [421]. It is presented calculations in framework in two-band GL theory also in agreement with calculations in [422]. Another useful moment in obtained expression (4.13) is the nonlinear character of temperature dependence specific heat near Tc [420, 421]. In analogy with results of Koshelev et al. [422], here anisotropy parameter of fluctuations 1zz >> 1 is replaced by TTc1c > 1: Enhancement of the fluctuations part of specific heat in two-band superconductors in comparison with single-band superconductors was also shown in [423]. Present calculations seems more in detail and in contrast to [422, 423] we use a free energy functional with an intergradient interaction parameter.

4.1.3 Influence of External Magnetic Field on Specific Heat Jump 4.1.3.1 Single-Band Approach The effect of a uniform magnetic field on the specific heat was first investigated theoretically in [424]. They neglected four-order term in (2.2) to calculate the fluctuation specific heat above Tc . In a uniform magnetic field, the fluctuating electron pairs move in Landau orbitals characterized by kz and n: Fluctuational contribution to the magnetic field dependent free energy F .H / can be described by linearized single-band GL functional (using above presented manner, see Sect. 4.1.1) [207] (

Z Ffl D

dV

) ˇ ˇ2  ˇ „2 ˇˇ 2 i eH 2 ‰.r/ˇˇ C ˛ j‰.r/j r 4m ˇ „c

(4.14)

Now the order parameter ‰.r/ will be represented as a Fourier series on eigenfunctions of the Schrodinger equation corresponding to previous (4.14), i.e., ‰.r/ D

X

cq

q .r/;

(4.15)

q

where q denotes three quantum numbers n; ky ; kz in spectrum of electron in external magnetic field   „2 kz2 1 C ; (4.16) "n;kz D „!0 n C 2 4m where !0 D eH m is the Larmor frequency of electrons in external magnetic field. Using (4.15), fluctuational free energy (4.14) may be written as

146

Fluctuation Effects in Anisotropic and Multiband Superconductors

Ffl D

X

ˇ ˇ2 ."n;kz C ˛/ ˇcn;kz ˇ :

(4.17)

n;kz

ˇ ˇ2 For expansion coefficients cn;kz is true ˇcn;kz ˇ D ."n;kTC˛/ : As followed from z above presented equations, for specific heat jump for single-band isotropic threedimensional superconductors in external magnetic field has a form [424] eH ˇ C ; D C  4˛H .˛H ı/1=2

(4.18)

„ where ˛H D ˛ C h; h D 2eıH and ı D 4mT : It physically means that a bulk superconductor thus behaves like an array of one-dimensional rods parallel to the field with the number of rods per unit area given by eH=; the Landau degeneracy factor for particles of charge 2e: The fluctuation of specific heat is then proportional to the external magnetic field H and becomes one dimensional in nature, diverging within free-fluctuation theory as ˛3=2 . For a thin film with thickness d , the kz degree of freedom is suppressed and final result has a form (two-dimensional system) [424] C eH ˇ : (4.19) D C  d.˛H /2 2

4.1.3.2 Two-Band Approach Using above presented effective mass approach [420] near Tc (2.12), we have the following expression for fluctuation of specific heat in two-band superconductors 

where

C C

 D TB

C /SB . C

.1 C

˛H 3=2 / .1 ˛H

C

4m1 „2

"1 .""1 2eH / „c „2 2eH C˛ .T / 2 4m2 „c

;

(4.20)

/1=2

„ ˛1 .T / 2 2 2eH 2eH . 4m2 C ""1 /" C ""1  "1 „c ; ˛H D 2 „cT . „ 2eH C ˛2 .T // 2

(4.21)

4m2 „c

and ˛SB D

.˛1 .T /˛2 .T /  "2 / „2 2eH . 4m 2 „c

C ˛2 .T //

C

„eH : 2m1 T

(4.22)

As followed from detail analysis of these expressions [425], multiband character of superconductivity leads to enhancement of fluctuations in presence of external magnetic field. Such conclusions confirm by experimental study in two-band superconductor MgB2 [421].

4.2 The Diamagnetic Susceptibility T >Tc

147

4.1.3.3 Inclusion of High-Order Terms Next step in the calculation of specific heat jump is using Hartree approximation (for detail analysis of post-Guassian fluctuations, see paragraph 4.5) [426], which allow us for. Calculations in single-band GL theory leads results: C 1 D ; C .1 C x/

(4.23)

   .8/3=2 2 .˛ H ı/3=2  2=3 : 1 ; y D ˛H ı ; yDx x D 8 .1 C x/ .ˇh/ x

(4.24)

where was introduced notations:

In framework of two-band approximation near upper critical field Hc2 ; we use  effective band expressions [420]. As a result in expressions (4.23) and (4.24) ˛ H replaced by .8/3=2  ; (4.25) ˛ H D .˛HSB C %H /ı .ˇh/ where 2e . %D „cTc

„2 ˛1 .T / 4m2

C ""1 /"2 C ""1  "21 2eH „c

„ 2eH . 4m C ˛2 .T // 2 „c 2

:

(4.26)

Next step in calculations related with inclusion screening effects [427]. Final C has expression for specific heat fluctuations in single-band superconductors C a form C 1Cx D : (4.27) 2 C .1 C 5x C x  2x.1 C x/ ln.1 C 1=x// Generalization of Bray results to the case of two-band superconductors leads renormalization of parameter y D y SB C %H 1=3 ; where y SB D x 2=3 .1  x2  2 ln.1 C 1=x//: As followed from (4.26) and (4.27), in general case of two-band superconductors manifestation of fluctuations on specific heat jump in external magnetic field is determined by the parameters " , "1 and their product ""1 . In the case of MgB2 for the fitting of experimental data, we take ""1 < 0 and this argument seems reasonable to explain [420] experimental data in [421].

4.2 The Diamagnetic Susceptibility T >Tc The diamagnetic susceptibility of the bulk sample is determined by the thermodynamic relation [207, 416] 1 @2 F D : (4.28) V @H 2

148

Fluctuation Effects in Anisotropic and Multiband Superconductors

Using Eqs. (4.14)–(4.17) for the fluctuation part of the free energy, after integration we obtain Z X  T  2eT X T D V ln Ffl D T dkz ln 2 „c „2 k 2 1 " C ˛ .2/ q „!0 .n C 2 / C 4mz C ˛ q n (4.29) After using the Poission summation formula [368] and (4.29) susceptibility of the single-band isotropic superconductors [428, 429] has a form SB D 

.2e/2 Tc : 24c 2 .4ma/1=2 .T  Tc /1=2

(4.30)

As followed from last expression, susceptibility of single-band isotropic superconductor at vanishing small magnetic fields H ! 0 is negative and diverges at T ! Tc : The behavior expected in arbitrary fields H was first obtained by Prange [430]. He found that M 2./1=2 T p D g.x/; (4.31) 3=2 H ˆ0 where g.x/ is a universal function of the variable 2m˛ D xD „eH



dHc2 dT

 Tc

T  Tc ; H

(4.32)

Prange show that as x ! 1 .H D Hc2 .T //; g.x/ diverges as .x C 1/1=2 : For fixed field, this implies that M diverges as (T  Tc2 .H //1=2 as the temperature is reduced. Here, Tc2 .H / is defined as temperature at which Hc2 .T / becomes equal to the applied field as temperature is varied. Inclusion of two-band effects for isotropic superconductors near Hc2 using effective band approximation [420] at small magnetic fields H ! 0 gives result [425] TB Tc D SB Tc1

   1=2 4m1 ""1 1=2 "2 1C 2 1C : „ ˛2 .T / ˛1 .T /˛1 .T /  "2

(4.33)

As followed from the (4.33), fluctuation part of magnetic susceptibility for twoband superconductors is greater than for single-band superconductors. Increasing of fluctuation part is determined by the interaction parameters " and "1 : Strength of 2 enhancement depends by the ratio TTc1c > 1 and by the factor 1 C ˛1 .T /˛"1 .T /"2 ; which also is greater than 1. Product of parameters ""1 < 0 for MgB2 and has inconsiderable influence on final result. For the analysis of the behavior of diamagnetic susceptibility in arbitrary fields H of two-band superconductors, it is useful to use results of Chap. 2. As shown in paragraph 2.1.1, in the case of two-band superconductors, upper critical field near Tc reveals positive curvature. It means that

4.3 Fluctuation of Conductivity Near Tc

149

m = –Mfl / H1/2 Tc [emu / cm3 (Oe)1/2 K]

1E-3

1E-4

1E-5

1E-6

4.7 * 10–7

1E-7 Tc(0) 1E-8 –0.005 0.000

0.005

0.010

0.015

ε = T/Tc(0) – 1

Fig. 4.2 Enhancement of magnetic susceptibility in two-band superconductors. Empty squares correspond to MgB2 ; while full cycles corresponds to optimally doped YBaCuO. The value of scaled magnetization in three-dimensional single-band case corresponds to 4:7  107



dHc2 dT



TB < Tc

dHc2 dT

SB (4.34) Tc

and as a result at fixed temperatuıre and magnetic field variable x decreased, while the value of function g.x/ increased. There are experimental results about enhancement of magnetic susceptibility in two-band superconductors in the limit of vanishing small magnetic fields H ! 0 and an arbitrary magnetic field [431] (Fig. 4.2).

4.3 Fluctuation of Conductivity Near Tc The fluctuations of the order parameter above critical temperature can also contribute to the electrical conductivity, which also called as paraconductivity. Average current in the absence of external magnetic field using GL theory can be written as paraconductivity j.r/ D i

e„  h‰ r‰  ‰r‰  i: 2m

(4.35)

150

Fluctuation Effects in Anisotropic and Multiband Superconductors

Using wave decomposition (4.15), we have j.q/ D

e„V X .2p C q/ 2m p

 p

pCq :

(4.36)

We use Kubo formula [432] for the calculation of electrical conductivity 1 .q/ D 2T

Z1 dthjq .t/jq .0/i:

(4.37)

1

Substituting (4.33) into (4.34) with (4.15) and limiting ourselves to the q=0, we can get following expression V 2 „2 .2e/2 ij .0/ D 32m2 T

Z1 X ˇ dt ki kj ˇh

 k

.t/

 k

ˇ2 .0/iˇ :

(4.38)

k

1

For the calculation of correlation function h k .t/ k .0/i time-dependent GL equations for the single-band case will be used [298] (see paragraph 2.6.1). For single-band superconductors solution of GL equations has a form with the relaxation parameter  and relaxation time k k

.0/et =k ;

(4.39)

1 4m : „2 k 2 C  2

(4.40)

.t/ D

where k D

k

Calculations of integrals leads to result [433] 8   ˆ 1  e 2 ˛mTc 1=2 ˆ ˆ ; 3D case; < 8 „ Tc  T D ˆ 1 e 2 Tc ˆ ˆ : ; 2D case; 16 „d Tc  T

(4.41)

Equations (4.10) and (4.12) for effective single-band approach has been used to calculate the fluctuation conductivity in two-band superconductors [425]. The results of calculations performed using this modified Aslamazov–Larkin formula (4.38) are presented in the Fig. 4.3 in comparison with experimental data on the fluctuation conductivity in MgB2 . For a comparative analysis, the results obtained in a single-band approximation [433] for various temperature regimes are also presented by straight lines (Fig. 4.3). As can be seen from the figure, (4.41) derived with allowance for the two-band character of superconductors provides a better agreement with the experimental data for MgB2 [434]. In the case of a single-band GL theory (4.41), the Aslamazov–Larkin formula for the fluctuation conductivity has the following structure: (Tc  T / ; where is a parameter that takes various

4.4 Fluctuation Effects in Layered Superconductors

151

Fig. 4.3 Three-dimensional character of fluctuations of conductivity in MgB2 ; full squares correspond to experimental data, triangles to TB GL calculations for the fitting parameters of MgB2 : Straight lines correspond to Aslamazov–Larkin approach

values in different regimes (e.g., D 0.5 for three-dimensional fluctuations and D 1 for the two-dimensional case). In the Fig. 4.3, this behavior corresponds to the straight lines with different slopes. Approaching the critical temperature leads to the passage from the two-to three-dimensional case, and the slope of the straight line decreases by half. An analysis of data in the Fig. 4.3 shows that (4.41) obtained in this paragraph using the two-band model allows the regimes with different dimensions to be combined and provides a better description of the experimental points than does the single-band approximation.

4.4 Fluctuation Effects in Layered Superconductors For investigation of fluctuation effects in layered superconductors with weak coupling between planes, we use Lawrence–Doniach functional for free energy [236]

FLD D

XZ n

8 ˇ ˇ2 9  2 ˇ ˇ ˆ < ˛ j‰n .r/j2 C ˇ j‰n .r/j4 C „ ˇ rab  2i e A ‰n .r/ˇ > = ˇ ˇ 2 2 4mab „c d r ; ˆ > : ; 2 Ct j‰n .r/  ‰nC1 .r/j (4.42)

152

Fluctuation Effects in Anisotropic and Multiband Superconductors

where ‰n .r/ is order parameter of the nth superconducting layer. Near critical temperature Lawrence–Doniach model is reduced to the usual GL functional 2 with an effective mass along c direction: M D 4t„d 2 ; where d is the interlayer

„ distance. The superconducting coherence length along c is c2 .0/ D 4M˛ and the dimensionless anisotropy parameter of the Lawrence–Doniach model, which determines the crossover from the three-dimensional to the two-dimensional regime „2 is r D 2c2 .0/=d 2 D 2M˛d 2 [435]. The fluctuational contribution to the free energy in the Gauss approximation (without j‰n .r/j4 / may be written as [416, 432] 2

FLD D T ln

Y T ; E C ˛

(4.43)

where E are the eigenvalue of the equation  2 2i e rab  A ‰n .r/ C t.‰nC1 .r/ C ‰n1 .r/  2‰n .r// D E ‰n .r/: „c (4.44) If take into account Landau quantization in the perpendicular direction to layers, then spectrum En;kz ; we may write „2 4mab

En;kz

„eH D mc

  1 nC C ˛r.1  cos kz d /: 2

(4.45)

The fluctuation contribution to the free energy in the quasi-two-dimensional case [436]

FLD

VTh D 2 2 4 ab .0/d

Z dz

1 X nD0



ln

T ; ˛0 f.2n C 1/h C r.1  cos z/ C g

(4.46)

0 where h D Hc2H.0/ is the dimensionless magnetic field, Hc2 .0/ D 2mc˛ being e„ the linear extrapolation of Hc2 at zero temperature. The magnetic moment can be @F calculated as M D  @H : Using (4.46) in three regime, we can get for the M in the limit =2h << 1 [416, 436, 437]

  ! 1 X 2 3=2 1 1=2 2 3=2 M D .n C 1/  n  n C ; 3=2 2 3 2 2 ab .0/c Hc2 .0/ nD0 3 p 3TH 1=2 2

(4.47)

which in agreement with [427] result and has H 1=2 dependence. For the twodimensional case for high fields (=2h << 1/ M D

1 X

T 3=2

2 2 ab .0/dHc2 .0/ nD0

! nC1 nC1 n ln C ln 1 : n n C 12

(4.48)

4.4 Fluctuation Effects in Layered Superconductors

153

As followed from above presented calculations, fluctuational part of magnetization of layered superconductors is the function of temperature and magnetic field. There is good agreement with experimental data for YBa2 Cu3 O7x compounds [438,439]. Calculations of fluctuation of specific heat using above expression for FLD in a weak magnetic field leads to result [416] C D

1 ; p .r C /

1 2 4 ab .0/d

(4.49)

which contains both two dimensional and three dimensional regimes: 8 ˆ ˆ <

1 1 ; r << ; 2 4 ab .0/ r C D ˆ 1 ˆ : p ; r << h; r

(4.50)

4.4.1 Influence Phase Fluctuations on Critical Temperature in Layered Superconductors For the calculation of fluctuation effects on critical temperature Tc ; fluctuations of order parameter modulus can be neglected [440]. To study the effects of order parameter phase fluctuation on the critical temperature Tc we will start from the free energy functional FΠ (4.42) for quasi two dimensional superconductors [236] F Π D NS2d

XZ j

8 9 < „2  @ 2 = X  j C W? .1  cos j .r/  j Cg .r/ ; d2 r : 8m @r ; gD˙1

(4.51) ˇ ˇ where j .r/ is the phase of the order parameter j D ˇj ˇ exp.i j .r// in the 2 plane j with coordinate r D .x; y/: W? D t is the Josephson energy and Ns2d .T / is the two-dimensional concentration of superconducting electrons defined as NS2d .T /

D

NS2d .0/

1

T .2/

Tco

!

p2 D F2 2„

1

T .2/

Tco

! :

(4.52)

In the expression (4.51) contribution of the modulus of the order parameter is neglected. The mean value of the order parameter is defined by the following expression: R st Π D cos j exp. FkT / hcos j i D : (4.53) R Fst Π D exp. kT /

154

Fluctuation Effects in Anisotropic and Multiband Superconductors

Accurate calculation of the path integral (4.53) with free energy functional (4.51) is not possible. At Tc (4.53) with free energy functional (4.51) has a nonzero solution, which may define the transition temperature Tc : To calculate the integral (4.53), we will use the mean field approximation by replacing the free energy functional (4.51) by the following expression ˆ

F Π D

)  @ j 2 d r C W? hcos i cos. / j @r Z .2d / D F0  Ns W? hcos i d2 r cos .r/;

NS2d

P

(

Z

2

„2 8m



(4.54)

where F0P Œ  is the free energy functional of two-dimensional superconductor and W? D tends to zero. g W? .g/: At the vicinity of Tc , the order parameter ˆ Therefore, the second part of the free energy potential F Œ  may be chosen as a small parameter. Substituting (4.54) into (4.53) instead Fst Œ  and after transformations, we obtain the following equation for Tc : 1D

NS2d .T /W? kTc

Z d2 rhcos .0/ cos .r/i0 ;

(4.55)

where h.....i0 indicates an averaging by means of the free energy functional F0 Π for a single superconducting layer. The correlation function hcos .0/ cos .r/i0 has been calculated in [441] and has a form:

hcos .0/ cos .r/i0 D

8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

 2

2d r

 1 E

kT T F .1 .2/ / Tc0

ˆ 6 1 ˆ kT ˆ 6 ˆ  ˆ ˆ exp 4  ˆ ˆ : EF 1 

; if r > 2d

  T .2/ Tc0

r 2d

2

3 7 7; if r < 2d ; 5 (4.56)

„vF is the coherence length inside the superconducting plane. Equawhere 2d D .0/ tion (4.55) may be solved for Tc with correlator (4.56), if following condition is satisfied kTc 1   < 1: (4.57) 2 Tc EF 1  .2/ Tc0

It means that

Tc < T < Tco.2/ ;

(4.58)

4.5 Post-Gaussian Fluctuations in Superconductors

where Tc is defined as

155

1 1 1 D C :  .2d / kTc 2EF kTc0

(4.59)

Tc is the temperature above which the interlayer phase coherence length is destroyed. Substituting (4.56) into (4.55) under a condition (4.57), we obtain the following expression for the critical temperature Tc : 2 !1=2 3 2 N .2d / .0/W? 22d 1 1 1 41  1 C 5: (4.60) D .2d /  2 Tc EF 22d N .2d / .0/W? Tc0 „ For a small value of the tunneling integral W? < 2m 2 , the last equation takes the 2d following form: Tc (4.61) Tc D h i:  T 2m 2 1  2c „22d . EtF /2 2

In the opposite case W? > sion [442]

„2 2 2m2d

; critical temperature givens by the expres.2d /

Tc D h

Tc 1C

Tc2d kF 2d t

i:

(4.62)

As followed from (4.61) and (4.62), the critical temperature Tc increases with increasing of the tunneling integral t and approaches Tc 2d in the interval Tc 
4.5 Post-Gaussian Fluctuations in Superconductors In above the presented consideration of fluctuation effects, in effective Hamiltionian (4.2) we are neglecting the quartic term in the free energy functional (2.1). Implicit in most of the early work is the assumption that fluctuations do not interact; that is, only Gaussian fluctuations are considered [416, 422]. Although this assumption breaks down in the critical region, it at least captures the qualitative aspects of the fluctuations in zero magnetic field. Interaction between the fluctuations are important near Tc2 .H /; and that these interactions remove the nonanalyticities

156

Fluctuation Effects in Anisotropic and Multiband Superconductors

present in zero magnetic field. The simplest approximation is to treat the quartic term using a self-consistent Hartree approximation [426, 427, 445] and this is implemented by replacing the quartic term j‰i j 4 by the 2h j‰i j 2 i j‰i j 2 : Whence, the new renormalized coefficient of the linear term in the (2.1) is given by 

˛ D ˛ C ˇhj‰i j 2 i:

(4.63)

Then for dimensionless reduced temperature H in layered superconductors [446, 447] is a true equation:

H D



H

when D

N 1 1 .2 2  1/ s X  h ; 4 2 d ƒT nD0 . H C 2hn/1=2 .1 C d 2 . H C 2hn//1=2 ˛.T / ˛.T D0/

(4.64)

is the reduced temperature, H D C h; with h D

H Hc2 .0/ s 2c .0/ is

ˆ2 ƒT = 16 2 k0 B T

a

dimensionless magnetic field, is the thermal length, d D a dimensionless interplanar spacing, c .0/ zero temperature coherence length along c axis with anisotropy parameter  D abc .0/ : In the limit of low magnetic field, in .0/ three-dimensional superconductors is obtained [447] 

 .2 2  1/ s . /1=2 1h 1   I R D C ƒ;

R D C 4 2 d ƒT 2 

(4.65)

in two-dimensional case 

R D C



.2 2  1/ s h 

 ln  : 8 2 d ƒT

(4.66)

It means that Hartree approximation for small magnetic field prevents the order parameter susceptibility from diverging, thereby preventing a transition to a phase with conventional long-range order.  In the limit of high magnetic field ( H << h/ in three-dimensional case, H is determined by the contribution from the lowest Landau level .n D 0/ W 

H D H 

.2 2  1/ s 4 2 d ƒT

h 

1 

2 1=2 . /1=2 .1 C d H /

:

(4.67)

In papers [446, 447], influence of critical fluctuations on transport properties of superconductors was calculated. In strong magnetic field region for the scaling functions of the excess conductance are given by  2D .H / D

T H

1=2

  T  Tc .H / : F2D A .TH /1=2

(4.68)

4.6 GL Theory for Layered Superconductors with Small Coherence Length

157

for two-dimensional superconductors and  3D .H / D

T2 H

1=3

  T  Tc .H / : F3D B .TH /3=2

(4.69)

for three-dimensional superconductors. In last equations F2D ; F3D are unspecified scaling functions, A and B are field- and temperature-independent coefficients. The scaling functions F2D ; F3D have the asymptotic forms F2D ; F3D  x for large negative values of x: For large positive values of x, F2D  x 1 ; F3D  x 1=2 , which reproduces the Gaussian result. Experimental results for nonmagnetic borocarbides [448] and MgB2 [449] reveals three-dimensional character of critical fluctuations. Experimental results for cuprate superconductor YBa2 Cu3 O7x [450] reveal twodimensional behavior of critical fluctuations.

4.6 GL Theory for Layered Superconductors with Small Coherence Length As followed from Chap. 1, one of the distinct features of a class of cuprate compounds is the smallness of the superconducting coherence length extrapolated to zero temperature c .0/ in contrast to conventional superconductors. Properties of such superconductors can be considered on the basis of generalized GL theory, which intended to be used in the critical region near Tc [451, 452]. Such theory is similar to generalized ‰-theory if superfluidity of helium-II near the -point [453]. For free energy in this approach is true following expression expression 



C0 Tc 2  ln  2 ( Z b0 g0 „2 C dV a0  4=3 j‰j2 C  2=3 j‰j4 C j‰j6 C 2 3 4mk

F D Fn0 C

ˇ  ˇ2 ) ˇ ˇ 2i e ˇ rk  Ak ‰ ˇˇ : ˇ „c

Variation in respect to ‰  leads to the following GL equation

(4.70)

 2 n o 2i e rk  Ak ‰ C a0  4=3 C b0  2=3 j‰j2 C g0 j‰j4 ‰ D 0: „c (4.71) Equation (4.71) differs from ordinary GL equations by temperature-changed coefficients and by the presence of the term ‰ 5 : Another two equations of generalized GL theory for vector-potential A and for current density and corresponding remains the same as in ordinary GL theory. From (4.64) and (4.65), for temperature dependence of equilibrium value of superconducting state parameters: 

„2 4mk

158

Fluctuation Effects in Anisotropic and Multiband Superconductors

p a0 0 0 2 ‰eq D ‰eq I .‰eq / D .1  M0 /; b0   2 a0 M0 C D .1  M0 / 1 C ; b0 Tc 3 where M0 D

g0 0 4 .‰ / : a0 eq

(4.72) (4.73)

(4.73a)

Temperature dependence of critical magnetic fields also changes. Upper critical field Hc2 can be calculated as ˆ0 /  4=3 ; (4.74) Hc2 D 2 2 while lower critical field Hc1 can be written as Hc1 D

ˆ0 ln  /  1=3 ln  1 : 42

(4.75)

However, thermodynamic magnetic field Hcm remains linear near Tc  Hcm D

 1=3  M0 4a02 .1  M0 / 1 C : b0 3

(4.76)

As followed from expression (4.74), near Tc in the region critical fluctuations, upper 2 critical field reveals a positive curvature ddTH2c2 . At the same time at low temperatures d2 Hc2 dT 2

reveal negative curvature [451,452] and as a result there is a point of changing of sign of curvature.

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•

Index

1.5 type, 28, 84, 93

ab initio, 20 ab plane, 3, 17, 48 Abrikosov, 70, 126 Acoustic, 116 Acoustical phonons, 107 Adaptive grid, 81 Adiabatic, 108, 110, 137 Al- doped, 104 Allen–Dynes formula, 20 Anderson, 76, 126 Angular, 28, 55–57, 59, 60, 83, 89 Anisotropic, 23, 62, 107 Anisotropy, 2, 5, 14, 24, 50, 52, 57, 68, 77, 98, 106, 118 Anisotropy parameter, 23, 28, 49, 50, 52, 59, 88, 121, 132, 152, 156 Annihilation operator, 6 Anomaly, 4, 9, 24, 88 Anti-bonding, 15 Antidot, 89 Antiferromagnetic, 86 Antiferromagnetism, 2, 7, 12, 24, 86 Arbitrary, 149 ARPES, 8, 9, 36 Aslamazov-Larkin, 150 Attractive, 125 Average current, 149 Averaged, 75, 104, 113, 154 Averaged phonon frequency, 102 Averaging, 40, 99, 111 Band structure, 13, 17, 24, 26 BCS, 2, 6, 15, 27, 45, 64, 74, 87, 95, 98, 119, 125, 128

Berezinski-Kosterlitz-Thouless, 155 Bessel, 55 Bipolaron, 8, 20 Bogoliubov, 87 Bogoliubov–de Gennes, 66 Bogoliubov-Anderson-Goldstone, 138 Bonding, 15 Bor, 88, 116 Borocarbides, 10, 87, 98 Bose–Einstein Condensation, 7, 20 Bound state, 7 Boundary, 29, 52, 80 Boundary conditions, 157 Bragg, 87 Brillouin, 25 Bulk, 51

c-axis, 17, 48 Carbon doped, 104 Carrier density, 3 Charge density, 17 Chemical potential, 108, 127, 133 Chu, 113 Clean, 88 Clean-limit, 14 Coherence, 74, 84 Coherence length, 46, 91, 142, 152, 154, 156, 157 Collective, 140 Condensation, 134 Condensation energy, 15 Continuum, 114 Conventional, 2, 15, 143 Cooper, 19, 20, 46, 92, 97 Coordination number, 22 Core, 15

I. Askerzade, Unconventional Superconductors, Springer Series in Materials Science 153, DOI 10.1007/978-3-642-22652-6, © Springer-Verlag Berlin Heidelberg 2012

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174 Correlation function, 150, 154 Coulomb, 20, 63, 95, 102, 111, 113, 115, 119 Coulomb repulsion, 96, 98 Creation operator, 6 Critical current, 15 Critical current density, 45, 47 Critical fluctuations, 156, 157 Critical level spacing, 135 Critical temperature, 1, 2, 10, 13, 15, 22, 128, 137 Critical velocity, 46 Cross, 107, 109 Crystal structure, 2, 13, 22, 23 CuO planes, 2, 23, 113, 118 Cuprate, 1, 2, 19, 23, 42, 111, 118, 157 Current density, 81 Cut off, 97, 109, 128, 142 Cut off momentum, 136

D-wave, 95, 122 d-wave, 7, 9, 62 Debye, 10, 119, 137 Defects, 155 Density of states, 119, 124, 127 Depairing, 51 Diamagnetic, 87 Diamagnetic susceptibility, 147 Dielectric, 112, 118 Differential conductance, 12 Diffusion, 76 Diffusivity, 77 Dirty, 15, 28 Disorder, 31 Dispersion, 87, 99, 137 Dopant concentration, 130 Doped, 4, 104 Doping, 18–20, 95, 127, 128 Dot, 89, 92

Effective, 48, 51, 62, 63, 76, 86, 107, 124, 136, 138, 143, 147, 155 Effective exchange integral, 7 Effective frequency, 100 Effective mass, 29, 33, 46, 57, 64, 98, 152 Eigenvalue, 69, 152 Eilenberger, 52, 76, 85 Einstein, 100 Electron correlation, 134, 137 Electron density, 143 Electron–phonon, 9, 20 Electron–phonon coupling, 10 Electron–phonon interaction, 2

Index Electron–phonon mechanism, 6 Electron–phonon pairing, 20 Electron-ion, 96 Electron-phonon, 87, 96, 97, 136 Electron-phonon coupling, 131 Eliashberg, 6, 20, 27, 95, 96, 103, 108, 118, 119, 121, 136 Energy spectrum, 100, 107, 109, 112 Enhancement, 145 Equilibrium, 157 Excess conductance, 156 Exchange interaction, 23, 86 External pressure, 105

FeAs planes, 23 Fermi, 5, 14, 17, 25, 26, 33, 63, 76, 87, 97, 99, 100, 103, 107, 108, 111, 112, 116, 121, 124, 131, 137, 143 Fermi level, 10, 13, 136 Fermi Liquid, 5, 8 Fermi surface, 11 Fermi velocity, 17 Ferrell-Prange, 52 Ferromagnetism, 2 Field dependent, 51 Field independent, 62 Film, 39, 42, 47, 81, 89 First-principle calculations, 24 Fluctuation, 141, 142, 146, 150, 151 Fluctuation of specific heat, 153 Fluctuational, 143, 145, 153 Fluctuations, 8, 11, 33, 75, 144, 153, 155 Flux quantization, 89 Fourer, 112, 141, 145 Fourfold, 67 Fourier, 55 Fractional, 89, 92 Fractional states, 92 Frank–Condon, 9 Free energy, 35, 152, 154 Functional, 28, 48, 53, 86

Gauge invariance, 80 Gauss, 152 Gaussian, 155, 157 Generalized GL theory, 157 Gibbs, 70, 87 GL, 8, 27, 29, 34, 51, 61, 63, 85, 87, 132, 145, 147, 150 GL parameter, 38, 42, 70, 80 GL-like, 28, 71 Gorkov, 63, 126

Index Grains, 15, 134 Granularity, 20 Green, 8, 76, 96, 97, 108, 123, 131, 136 Ground state, 7

Haas–van Alphen, 11 Haas–van Alphen frequency, 14 Haas-van Alphen, 26, 118 Hall conductivity, 14 Hamiltonian, 6, 124 Hampshire, 87 Harmonic, 69, 100, 107 Hartree, 147, 156 Heavy fermion, 86 Helical, 87 Helicoidal, 87, 88 Helium-II, 157 Hexagonal, 82 High magnetic field, 156 Hole density, 73 Hole-doped, 9, 124 Hole-like, 26 Hubbard, 6

I type, 42, 84, 93 II type, 42, 84, 93 Impurity, 124, 126 Incommensurate, 12 Integral equations, 100 Interaction, 84, 96, 122, 135, 144, 147, 148, 155 Interband, 18, 29, 44, 77, 105, 126 Intergradient, 32, 52, 59, 69 Intervortex, 84 Intraband, 18, 52, 77 Isotope, 9, 15, 20 Iteration, 98, 101

Josephson coupling, 84, 139, 155 Josephson current, 9 Josephson junction, 9

Kinetic equations, 118 Korringa behavior, 11 Kramer–Pesch, 15

Lambda point, 157 Landau, 88, 145 Landau degeneracy factor, 146

175 Landau quantization, 152 Larmor, 145 Lattice, 22, 25 Lawrence-Doniach, 141, 152 Layered, 48, 50, 95, 111 LDA, 26 Leggett, 16, 113, 138 Linear extrapolation, 152 Little-Parks, 28, 43 Logarithmical, 101 Logarithmically averaged, 104 London, 68 London equations, 37 London penetration depth, 37, 39, 46, 69, 84 Long wavelength, 116, 142 Long-range, 84 Longitudinal, 99 Loss function, 83 Low energy process, 9 Low frequency, 118 Low magnetic field, 156 Lower critical field, 38, 87, 88, 158 Lowest Landau level, 156 LSCO, 3

Magnesium diboride, 16, 27, 98 Magnetic, 54, 87 Magnetic field ratio, 87 Magnetic ordering, 88 Magnetization, 41 Magnetoconductance, 13 Manifestation, 147 Marginal, 8 Mass anisotropy, 48, 59 Mass operator, 139 Mass ratio, 33 Mass tensor, 51, 56 Mathieu, 49, 50 Matrix element, 96, 98 Matsubara, 76, 107, 119 Maxwell, 52 Mc Millan, 98, 109, 113 Mean field, 73, 154, 155 Mean value, 153 Meissner, 36, 69, 82, 84, 91 Mesoscopic, 28, 42, 88–90 Metal-insulator transition, 21 Microwave conductivity, 16 Migdal, 7, 96, 110, 136 Minimization, 29, 70, 124 Mixed state, 82 Mixed type, 93 Momentum, 100, 109

176 Momentum transfer, 108 Mott-Hubbard, 21 Multiband, 15, 76, 81, 124, 126, 146

Nambu, 124 Nano-size, 95 Nanosize two-band superconductivity, 135 Neel, 7, 86 Negative contribution, 136 Negative curvature, 31, 158 Nesting, 14 Non phonon, 127 Non-adiabatic effect, 2 Non-singular, 67 Nonadiabacity, 95 Nonadiabatic, 103, 106, 109, 136, 137 Nondiagonal, 107 Nonlinear behavior, 42 Nonmagnetic, 75, 126 Nonmagnetic borocarbides, 11, 31, 51, 106 Nonparabolic, 43 Nonsingular, 82 Normalized frequency, 124 Nuclear spin-lattice relaxation, 15 Numerical, 92, 103, 116 Numerical modeling, 80

Odd parity, 135 Optical conductivity, 9 Optical phonons, 107 Optimal doping, 5 Order parameter, 7, 29, 71, 89, 141, 153 Oscillations, 43, 116 Overdoped, 5, 75 Overlapping energy bands, 131 Oxypnictide, 19, 98

Paraconductivity, 149 Paramagnetic, 36, 51, 87 Paramagnon, 7 Parity, 135 Partition function, 142 Path integral, 138, 142, 154 Pauli, 96 Pauli spin matrix, 124 Peak, 74 Pellets, 36 Penetration depth, 54 Phase, 19, 29, 131, 155 Phase coherence, 155 Phase diagram, 21

Index Phase fluctuations, 138, 153 Phase transition, 2 Phenomenogical, 8 Phonon, 99 Phonon frequency, 103 Pi band, 17 Pinning, 20 Plasma frequency, 116 Plasmon, 116, 118 Plasmon spectrum, 115 Poission, 148 Polarization operator, 112 Polycrystalline, 21, 47 Positive curvature, 31, 36, 41, 42, 47, 51, 65, 131, 148, 158 Post Gaussian, 141, 147 Prange, 148 Pressure, 104 Projection operator, 7 Pseudogap, 5, 7, 8 Pseudopotential, 102 Psi theory, 157 Pure superconductors, 126

Quadro-layer, 13 Quantization, 42 Quantum, 88 Quantum interference, 42 Quantum numbers, 145 Quantum oscillation, 10 Quantum-phase fluctuations, 73 Quasi-long-range, 155 Quasi-two-dimensional, 24, 25, 107, 111, 120, 152

Rare-earth elements, 3, 19 Ratio, 51, 59, 60, 87, 136 Reentrant superconductivity, 12 Relaxation parameter, 150 Relaxation time, 126, 150 Renormalization parameter, 107 Repulsive, 125, 139 Rietmann, 120 RKKY exchange interaction, 12 Room temperature, 1 Ruthger, 45

s-wave, 9, 28, 111 Scaling function, 157 Scattering rate, 77 Schrodinger, 145

Index Screened, 112 Sheet, 77 Sheets, 17 Short-range, 84 Sigma band, 15, 17 Single crystal, 47 Single-band, 15, 30, 34, 41–43, 45, 111 Singular, 67, 82, 101, 127 Singularity, 101 Size dependence, 134 Slope, 41, 151 Small, 141, 148, 149, 154, 157 Space-time, 82 Specific heat, 74, 88, 144, 147 Specific heat jump, 45, 120, 121, 128, 146 Spectrum, 145, 152 Spin, 88 Spin fluctuations, 20 Spin glass, 9 Spin waves, 88 Spin-fermion, 7 Spin-lattice relaxation, 15 Spin-orbit scattering, 51 Spin-singlet pairing, 63 Strenght, 148 Strong, 7, 87, 121, 156 Strong coupling, 7, 107 Structure of vortex, 67 Superfluid, 46 Superfluid density, 15, 73 Superfluidity, 157 Superlattice, 111, 118 Surface, 34, 85 Surface critical field, 36

t-J model, 7 Temperature dependence, 122, 157 Temperature independent, 157 Temperature-changed, 157 Tetragonal, 21, 23 Thermal, 20 Thermodynamic, 5, 44, 138, 147, 158 Thickness, 116 Thin film, 41

177 Three-dimensional, 13, 25, 151, 157 Time dependent, 77 Time dependent GL equations, 150 Time-dependent, 28, 80, 82 Topological, 155 Transition, 5 Transport, 156 Transverse, 99 Trilayer, 13, 118 Tunneling integral, 98, 116, 155 Two-band, 27, 29, 30, 48, 89, 95, 122, 130, 140, 143, 148, 150 Two-band layered, 48 Two-dimensional, 53, 109, 114, 151, 152, 154, 157

Underdoped, 5, 9 Unit cell, 99 Upper critical field, 14, 29, 30, 32, 41, 49, 58, 130, 131, 158 Uzadel, 57, 76

Van Hove, 95 Van Hove singularity, 27 Variable carrier density, 129, 132 Variational procedure, 7 Vector potential, 157 Velocity, 17, 46 Vertex, 96, 107–109, 111, 136 Viscosity, 83 Vortex, 15, 28, 41, 53, 56, 66, 68–71, 82, 84, 85, 88, 90 Vortex nucleation, 81 Vortex-pattern, 82

Weak coupling, 107, 136, 151

YBCO, 3, 20, 114

Zubarev, 101