springer theses recognizing outstanding ph.d. research ...fullerides conducted by dr. yusuke nomura...

Springer ThesesRecognizing Outstanding Ph.D. Research

Ab Initio Studies on Superconductivity in Alkali-Doped Fullerides

Yusuke Nomura

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Yusuke Nomura

Ab Initio Studieson Superconductivityin Alkali-Doped FulleridesDoctoral Thesis accepted byThe University of Tokyo, Tokyo, Japan

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AuthorDr. Yusuke NomuraDepartment of Applied PhysicsThe University of TokyoTokyoJapan

SupervisorsDr. Ryotaro AritaCenter of Emergent Matter ScienceRIKENSaitamaJapan

Prof. Masatoshi ImadaDepartment of Applied PhysicsThe University of TokyoTokyoJapan

ISSN 2190-5053 ISSN 2190-5061 (electronic)Springer ThesesISBN 978-981-10-1441-3 ISBN 978-981-10-1442-0 (eBook)DOI 10.1007/978-981-10-1442-0

Library of Congress Control Number: 2016945997

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Supervisor’s Foreword 1

This book describes a theoretical study on superconductivity in alkali-dopedfullerides conducted by Dr. Yusuke Nomura during his doctoral program at theDepartment of Applied Physics, the University of Tokyo. Superconductivity is afascinating macroscopic quantum phenomenon observed in various metals at lowtemperatures. Its quantitative description from first principles is of great interest, buthas been a significant challenge. One reason is that the typical energy scale ofelectronic structure treated in standard ab initio calculation is larger by far than thatof superconductivity, e.g., its transition temperature (Tc). Thus we need a reliablemultiscale scheme, which spans the scale of 1 K to 105 K, and the development ofsuch methods is indeed an exciting frontier in computational physics.

Another problem is that the mechanism of superconductivity is not fullyunderstood and still remains a mystery. While the celebrated BCS theory hassucceeded in explaining the basic properties of conventional phonon-mediatedsuperconductors, general consensus on the origin of superconductivity in the high-Tc cuprates, iron-based superconductors, and other exotic superconductors has yetto be achieved. Interestingly, Tc’s of such unconventional materials are often higherthan those of conventional superconductors and their pairing mechanism has beenlong a central topic in condensed matter physics.

Alkali-doped fullerides are one of such intriguing unconventional supercon-ductors, having the highest Tc among molecular solids. In the phase diagram, anisotropic s-wave superconductivity resides next to a Mott insulating phase. Thissituation is more surprising than that of d-wave superconductivity in the cuprates inthat we naively expect that local Coulomb correlations causing the Mott transitionseverely suppress onsite s-wave pairing.

In the first stage of his work, Dr. Nomura developed an ab initio scheme toderive low-energy Hamiltonians for electron–phonon coupled systems. In ful-lerides, the energy scale of lattice vibration is extremely high, so that both theelectronic correlations and electron–phonon interactions play a crucial role inthe superconductivity. However, it has not been established how to incorporate thephonon degrees of freedom into the multi-scale ab initio scheme for correlated

v

electron systems. Dr. Nomura’s method works very effectively and successfully forfullerides, and he determined all the parameters in the low-energy Hamiltonian fromfirst principles. Then he solved the resulting Hamiltonian by exploiting cutting-edgetechnology of many-body theory. He revealed that nontrivial physics emerges when“Mott meets BCS”, and reproduced the experimental phase diagram with anaccuracy of a few K.

The success achieved in the present study demonstrates that we have indeedtaken the first step toward predictive calculation for unconventional superconduc-tors. I believe that this book will stimulate and encourage many readers who areinterested in and aiming at materials design of novel superconductors.

Saitama Dr. Ryotaro AritaMarch 2016

vi Supervisor’s Foreword 1

Supervisor’s Foreword 2

Superconductivity is one of the most fascinating phenomena in condensed matterphysics. At the frontier of theoretical studies on understanding superconductivity,quantitative predictability has recently been sought for from a first-principles basis.For conventional superconductors describable within the framework of the BCStheory, indeed the superconducting density functional theory has achieved a mea-sure of success with agreement of critical temperatures between the experimentsand theories within about 10 % accuracy for several compounds. However,applications of this method to the unconventional superconductivity emerging fromstrong electron correlations result in failure with underestimates of the criticaltemperatures. Not only in typical unconventional superconductors such ascopper-oxide and iron-based superconductors, but also in others such as thealkaline-doped fullerene, A3C60, the discrepancies are substantial.

On the other hand, studies on understanding and predicting strongly correlatedelectron materials by applying first-principles methods have rapidly been devel-oped, thanks to the success of the hierarchical (multiscale) ab initio scheme forcorrelated electrons (MACE). In this hierarchical scheme in the energy space,low-energy effective Hamiltonians are derived from global electronic structures,where the electrons far from the Fermi level (high-energy electrons) are success-fully described by the conventional methods such as the density functional theoryand a perturbative many-body method called the GW approximation. After renor-malizing effects of high-energy electrons into the electrons near the Fermi level à larenormalization group spirit, the derived low-energy effective Hamiltonians aresolved by refined low-energy solvers beyond the limitation of the density functionaland GW theories. The dynamical mean-field theory is one such solver.

Based on these trends, in this book Dr. Yusuke Nomura has applied thedynamical mean-field theory to the effective low-energy model for A3C60, afterextending the scheme by taking into account electron–phonon interactions. Hismajor achievement is the precise reproduction of the superconductivity in agree-ment with the experimental phase diagram and elucidation of the mechanism of thesuperconductivity.

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After reviewing the history of studies on A3C60 in the introduction, thetheoretical framework MACE applied in this book is formulated in Chap. 2.Notably, the existing MACE scheme is extended to incorporate the electron–pho-non interaction, which is Dr. Nomura’s original contribution. This extension hasbeen made possible by developing a constrained density-functional perturbationtheory.

In Chap. 3, the constrained density-functional perturbation theory is applied toA3C60 and by combining it with the MACE, a low-energy effective Hamiltonian isderived after taking into account the electron–phonon interaction. The derivedlow-energy Hamiltonian is solved using the dynamical mean-field theory. Theresult shows quantitative agreement with the known experimental phase diagramincluding the superconducting and the Mott insulating phases.

Dr. Nomura next analyzes the obtained numerical results to extract the mecha-nism of superconductivity and the Mott insulating phase. In the derived effectiveHamiltonian for A3C60, three bands constructed from the Wannier orbitalsspreading in a C60 molecule are half filled in total. In addition, due to the largespatial extension of the molecule, the screened exchange interaction is very small.Then, when the electron–phonon interaction is further considered on top of it, thesign of the effective Hund’s rule interaction becomes negative owing to the con-tribution from the electron–phonon interaction. Consequently, it has an unusual andremarkable property that the effective onsite intraorbital Coulomb interactionU becomes smaller than the interorbital interaction U’. This “negative Hund’s rulecoupling” accounts for the low-spin state of the Mott insulating phase in theexperiments. Furthermore, the experimental facts on dynamical fluctuations of theJahn–Teller distortion are naturally explained from the strong tendencies for orbitalpolarizations.

Superconductivity is also well reproduced in terms of alkaline-element andpressure dependences. To understand the mechanism, Dr. Nomura shows numericalresults that the superconductivity disappears when the interorbital interaction largerthan the intraorbital interactions are reversed in addition to the removal of thepair-hopping term contained in the effective Hamiltonian. This indicates that theU smaller than U’ with a low-spin state generates the effective attraction betweenthe electron on the same orbital and it is strengthened by the pair-hopping term inthe form of the kinetic energy gain of the pair. Nevertheless, the energy scale of theattraction is very small. However, because the kinetic energy is largely renormal-ized by the electron correlation effect, a small attraction efficiently contributes tosuperconductivity. The mechanism of the superconductivity is then successfullyunderstood from the simultaneous and crucial contributions of both the electron–phonon interaction and strong electron correlations.

viii Supervisor’s Foreword 2

http://dx.doi.org/10.1007/978-981-10-1442-0_2http://dx.doi.org/10.1007/978-981-10-1442-0_3

I believe the findings by Dr. Nomura described in detail in this book largelycontribute to the understanding of the superconducting mechanism in the dopedfullerene and more generally contribute to the understanding of the physics of theunconventional superconductors through the developed methodological framework.

Tokyo Prof. Masatoshi ImadaMay 2016

Supervisor’s Foreword 2 ix

Preface

The superconductivity in the A3C60 systems with A = K, Rb, Cs (alkali-dopedfullerides) has been intensively studied since its discovery in 1991. The maximumsuperconducting transition temperature Tc of � 40 K is the highest among themolecular superconductors. The superconductivity was seemingly well understoodby the conventional phonon mechanism, in which the intramolecular Hg modesplayed a main role: Experimentally, the conventional s-wave pairing symmetry andthe positive correlation between the superconducting transition temperature and thelattice constant supported this scenario. Theoretically, it has been claimed in theliterature that the total electron phonon coupling k � 0.5–1 and the high phononfrequencies on the order of � 0.1 eV produce a high transition temperature com-parable to the experimental ones.

However, the recent success in synthesizing fcc/A15 Cs3C60 brought to light asevere contradiction with the conventional scenario. They are both Mott insulatorsat ambient pressure, and the superconductivity is realized only when the latticeconstant is shrunken by applying pressure. The critical temperature Tc as a functionof the lattice constant shows a dome-like shape for both the A15 and fcc systems.These features cannot be explained by the conventional Migdal–Eliashberg theory.In fact, the existence of the superconducting phase in the vicinity of the Mottinsulating phase indicates that the electron correlation might be essential.Furthermore, the observed low-spin state and the dynamical Jahn–Teller effect inthe insulating phase revealed a substantial role for the electron–phonon interactions.Therefore, in order to understand the pairing mechanism, the Mott transition, andthe low-spin state in the Mott insulating phase in a comprehensive manner, it isnecessary to elucidate a nontrivial interplay between the electron correlations andthe electron–phonon interactions.

In this thesis, we aim to obtain a unified description of the phase diagram.Especially, we try to answer why the s-wave superconductivity is stabilized in thevicinity of the Mott insulating phase in contrast to a naive expectation that thestrong electron correlations are incompatible with the s-wave pairing. While it wasproposed that a new type of phonon-mediated superconductivity distinct from the

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BCS (Bardeen–Cooper–Schrieffer) superconductivity emerges near the Mott tran-sition by cooperation of the Jahn–Teller phonons and the strong correlations, toobtain conclusive statements, we need arguments as unbiased as possible.

Another goal of the thesis is the non-empirical calculation of the transitiontemperatures of the alkali-doped fullerides. Historically, the Tc calculation cruciallyrelies on empirical parameters, such as the Coulomb pseudopotential. While recentprogress has enabled fully ab initio Tc calculations in high accuracy for the con-ventional superconductors, there is still no reliable way to predict Tc for uncon-ventional superconductors.

To achieve the goals, we derive, from first-principles calculations, effectivelow-energy Hamiltonians for the fcc A3C60 systems. The derived Hamiltoniansconsist of the electron transfer, the Coulomb interaction, the electron–phononinteraction, and the phonon one-body terms. By analyzing them accurately with amodel calculation technique, we study the low-energy phenomena. This schemerequires only the chemical composition and the crystal structure, which enablesquantitative studies without employing any empirical parameters.

Because a previous study has evaluated the electron transfer and the Coulombinteraction parameters, in this thesis we focus on the derivation of thephonon-related terms. To this end, we formulate a novel ab initio scheme, con-strained density functional perturbation theory (cDFPT). In the cDFPT, partiallyrenormalized phonon frequencies and electron–phonon couplings are calculatedwith excluding the low-energy-subspace renormalization effects, which are used asthe parameters in the low-energy models. The partial renormalization allows us totake into account the effects of high-energy bands and to avoid the double countingof the low-energy-subspace renormalization effects, which are to be consideredwhen the models are analyzed.

We apply the cDFPT to the alkali-doped fullerides. Then, the static part of thephonon-mediated negative exchange interactions Jph(x = 0) is estimated to be Jph(x= 0) � −51 meV. We find that the magnitude of the negative Jph(x = 0) is largerthan that of the positive Hund’s coupling JH � 34 meV. It means that, effectively,negative exchange and pair-hopping interactions are realized in the A3C60 systems,while the amounts of the interactions (�−17 meV) are tiny compared to theHubbard repulsion U � 1 eV. Furthermore, we see that, due to thephonon-mediated attractions, the effective interorbital repulsion U’eff becomesslightly larger (� 5 %) than the effective intraorbital repulsion Ueff.

To analyze the Hamiltonians with the unusual form of the intramolecularinteractions, we adopt the extended dynamical mean-field theory (extended DMFT)with employing the continuous-time quantum Monte Carlo method based on thestrong coupling expansion as an impurity-model solver. The extended DMFT is oneof the most powerful methods to study strongly correlated materials in threedimensions, which can accurately treat the local phonon dynamics and thedynamical screening effects originating from the long-range Coulomb interactions,on top of the local electron correlations.

We perform the extended DMFT analysis of the derived ab initio models anddraw a theoretical phase diagram as a function of lattice constant and temperature.

xii Preface

We obtain the paramagnetic metal, the superconducting phase, and the paramag-netic Mott insulator, which will reproduce the experimentally observed phases. As aconsequence of the effective negative exchange interaction, the low-spin state isrealized in the insulating phase. Remarkably, the agreement is not only at aqualitative level but also at a quantitative level. In particular, the calculationsindicate the maximum Tc of � 28 K, in good agreement with the experimentalresult (� 35 K).

As for the pairing mechanism, we identify two crucial factors. One is the singletpair generation by U’eff > Ueff, and the other is the tunneling of the pairs by thenegative pair-hopping interactions (the Suhl–Kondo mechanism). The inequalityU’eff > Ueff and the negative pair-hopping term originate from the phonon-mediatedattractions, thus the superconductivity essentially relies on the phonons. However,this superconductivity differs from the conventional ones in that the strong electroncorrelations also play an important role: The pair formation is originally inefficientbecause the difference between U’eff and Ueff is very small (� 0.03−0.04 eV)compared to the typical kinetic energy � 0.5 eV, and becomes efficient only whenthe electronic kinetic energy is suppressed by the correlations. As a result, we seethe increase of Tc with the increase of the correlation strength. These considerationslead to a conclusion that the alkali-doped fullerides are unconventional supercon-ductors, whose essence is the unusual synergy between the strong electron corre-lations and phonons.

Paris Yusuke NomuraMarch 2016

Preface xiii

Parts of this thesis have been published in the following journal articles:

• Yusuke Nomura, Shiro Sakai, Massimo Capone, and Ryotaro Arita“Exotic s-wave superconductivity in alkali-doped fullerides”J. Phys.: Condens. Matter 28, 153001 (2016)

• Yusuke Nomura and Ryotaro Arita“Ab initio downfolding for electron-phonon-coupled systems: Constraineddensity-functional perturbation theory”Phys. Rev. B 92, 245108 (2015)Editor’s suggestion

• Karim Steiner, Yusuke Nomura, and Philipp Werner“Double-expansion impurity solver for multiorbital models with dynamicallyscreened U and J”Phys. Rev. B 92, 115123 (2015)

• Yusuke Nomura, Shiro Sakai, Massimo Capone, and Ryotaro Arita“Unified understanding of superconductivity and Mott transition in alkali-dopedfullerides from first principles”Science Advances 1, e1500568 (2015)

• Yusuke Nomura, Kazuma Nakamura, and Ryotaro Arita“Effect of Electron-Phonon Interactions on Orbital Fluctuations in Iron-BasedSuperconductors”Phys. Rev. Lett. 112, 027002 (2014)

• Yusuke Nomura, Kazuma Nakamura, and Ryotaro Arita“Ab initio derivation of electronic low-energy models for C60 and aromaticcompounds”Phys. Rev. B 85, 155452 (2012)

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Acknowledgments

I would like to express my sincere gratitude to Dr. Ryotaro Arita andProf. Masatoshi Imada for their supervision during my doctoral program, for fruitfuldiscussions, and for their critical reading of the manuscript. I am especially gratefulto Dr. Arita: The daily communications with him and his continual encouragementsince my master’s course were essential to accomplish the present work. I alsothank Prof. Atsushi Fujimori for supervising me through Advanced LeadingGraduate Course for Photon Science (ALPS) and for useful suggestions andcomments on my studies from an experimental point of view.

I am indebted to Prof. Kazuma Nakamura, who patiently taught me the basicsof the electronic structure calculations. I acknowledge the helpful advice fromDr. Ryosuke Akashi, especially on phonon calculations. As for model calculationtechniques, I am particularly thankful to Dr. Shiro Sakai for his instructions on thedynamical mean-field theory (DMFT). In preparing the calculation codes,I appreciate their kindness to provide me with various useful computational codes.Some of the subroutines employed in the calculation were also provided byProf. Yoshihide Yoshimoto and Dr. Yoshiro Nohara. Without constructive dis-cussions with them, the present achievement would not have been possible.

I acknowledge excellent collaboration with Prof. Massimo Capone on thepresent topic. I also thank him and his group members for their hospitality duringmy stay in Trieste in 2014.

I am grateful to the referees of the dissertation defense, Profs. Masatoshi Imada,Yoshihiro Iwasa, Atsushi Oshiyama, Atsushi Fujimori, and Dr. Ryotaro Arita forreviewing the thesis and for their valuable comments and discussions about thestudy. In correcting grammatical errors in the manuscript, various comments fromDr. Shiro Sakai were very helpful.

I wish to thank Prof. Philipp Werner, Thomas Ayral, Yuta Murakami,Dr. Hiroshi Shinaoka, Dr. Nicolaus Parragh, and Prof. Giorgio Sangiovanni forfruitful discussions on the practical implementation of the DMFT code. In partic-ular, I appreciate Thomas Ayral, Yuta Murakami, and Dr. Nicolaus Parragh for

xvii

providing their numerical results for the benchmark of my codes. I would liketo thank Profs. Yuichi Kasahara, Yoshihiro Iwasa, and Pawel Wzietek for fruitfuldiscussions on the experimental results for alkali-doped fullerides. I acknowledgethe kindness of Prof. Yuichi Kasahara to provide the experimental phase diagramand to allow me to use it in the thesis.

I am grateful to Profs. Silke Biermann, Giorgio Sangiovanni, Karsten Held,Adolfo Eguiluz, and their group members for their hospitality and for variousdiscussions when I visited their groups. The experience in the foreign instituteshighly stimulated my motivation. The discussions and communications with thepresent and former members in the Arita group, Dr. Takashi Koretsune, Dr. MichitoSuzuki, Dr. Masayuki Ochi, Wataru Sano, Takahiro Kurosu, Hideyuki Miyahara,and Tatsuya Tomiuchi, were also encouraging. I also thank the present and formermembers in the Imada and Motome groups. I learned many things through the jointseminar conducted every Friday and the communications with them.

This work was supported by a Grant-in-Aid for JSPS Fellows (12J08652). Someof the calculations were performed at the Supercomputer Center, ISSP, theUniversity of Tokyo. I am thankful to Prof. Naoto Nagaosa for providing excellentcomputational resources at RIKEN and Dr. Wataru Koshibae for taking care of theRIKEN clusters.

Last but not least, I would like to express my gratitude to my parents, brother,and grandparents for their support and encouragement in all my life.

xviii Acknowledgments

Contents

1 Introduction to Superconductivity in Alkali-Doped Fullerides . . . . . 11.1 Superconductivity in Alkali-Doped Fullerides . . . . . . . . . . . . . . 1

1.1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Properties of Alkali-Doped Fullerides Revealed by

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Electronic Structure and Electronic Correlations . . . . . . . . 91.1.4 Electron-Phonon Interactions and Phonon Frequencies . . . 121.1.5 On Applicability of Conventional Mechanism . . . . . . . . . 161.1.6 Unconventional Mechanisms . . . . . . . . . . . . . . . . . . . . . 19

1.2 Aim of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Methods: Ab Initio Downfolding and Model-CalculationTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 Multi-energy-scale Ab Initio Scheme for Correlated

Electrons (MACE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.2 Low-Energy Effective Hamiltonian . . . . . . . . . . . . . . . . . 32

2.2 Ab Initio Downfolding for Electron-Phonon CoupledSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . 352.2.2 Maximally Localized Wannier Function . . . . . . . . . . . . . 382.2.3 Constrained Random Phase Approximation . . . . . . . . . . . 392.2.4 Density-Functional Perturbation Theory . . . . . . . . . . . . . 412.2.5 Constrained Density-Functional Perturbation Theory . . . . . 53

2.3 Analysis of Low-Energy Hamiltonian . . . . . . . . . . . . . . . . . . . . 572.3.1 Dynamical Mean-Field Theory . . . . . . . . . . . . . . . . . . . . 582.3.2 Extended Dynamical Mean-Field Theory . . . . . . . . . . . . . 622.3.3 Impurity Solver: Continuous-Time Quantum Monte

Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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2.3.4 Simulation of Superconducting State Within ExtendDMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.4 Combining Model Derivation and Model Analysis . . . . . . . . . . . 902.4.1 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.4.2 Overview of Whole Scheme . . . . . . . . . . . . . . . . . . . . . 95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3 Application of cDFPT to Alkali-Doped Fullerides . . . . . . . . . . . . . . 1013.1 Calculated Materials and Calculation Conditions . . . . . . . . . . . . 1013.2 cDFPT Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.2.1 Partially Renormalized Phonon Frequencies . . . . . . . . . . . 1033.2.2 Effective Onsite Interactions Mediated by Phonons . . . . . . 1043.2.3 Dynamical Structure of Onsite Interaction Including

Coulomb and Phonon Contributions Along RealFrequency Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.3 Comparison Between Partially Renormalized and FullyRenormalized Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.1 Difference in Frequencies . . . . . . . . . . . . . . . . . . . . . . . 1083.3.2 Difference in Phonon-Mediated Interactions . . . . . . . . . . . 109

3.4 Smallness of Electron-Phonon Vertex Correctionin Downfolding Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4 Analysis of Low-Energy Hamiltonians with Extended DMFT . . . . . 1194.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.2 Frequency Dependence of Effective Onsite Interaction . . . . . . . . 1204.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3.1 Comparison Between Theory and Experiment . . . . . . . . . 1214.3.2 Accuracy of Phase Boundaries . . . . . . . . . . . . . . . . . . . . 122

4.4 Metal-Insulator Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.4.1 Physical Quantities at 40 K . . . . . . . . . . . . . . . . . . . . . . 1234.4.2 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.5 Nature of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5.1 Gap Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5.2 Pairing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5.3 Possible Explanations on Origin of Dome-Shaped Tc . . . . 131

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.1 Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.2 Future Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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Chapter 1Introduction to Superconductivityin Alkali-Doped Fullerides

1.1 Superconductivity in Alkali-Doped Fullerides

1.1.1 Historical Background

The fullerene is a generic term for a closed hollow cluster composed of carbonatoms. This new type of the carbon allotropes other than the well-known diamondand graphite has come to draw strong attention since 1985 [1], when Kroto et al.discovered the first fullerene molecules, namely, C60 fullerene (hollow sphere) andC70 fullerene (hollow ellipsoid) [2].1 It is interesting to note, however, that there hadexisted several independent predictions of the existence of the fullerene, though theyhad not been widely recognized at the time of the discovery (see e.g., Refs. [4–9]).

One of the most intriguing properties of the fullerenes is their characteristic shapes:Especially, the C60 molecule is highly symmetric and has the symmetry of the icosa-hedral group, which is the point group with the largest possible symmetry opera-tions [10]. Its shape (Fig. 1.1) resembles a soccer ball and the geodesic domes, anarchitecture designed by Buckminster Fuller. Therefore, C60 was named “buckmin-sterfullerene” [2] and the shortened name “fullerene” is now used as a generic termfor the hollow carbon clusters.

The success of the production of C60 solids (fullerites) by Krätschmer et al. in1990 [12] opened up a way to study C60 in the context of the condensed matter physics.Since then, enormous number of solid-state experiments have been performed and avariety of remarkable physical properties have been revealed. Among them, super-conductivity is, in particular, of great interest. The superconductivity was first foundin potassium-doped C60 with a superconducting transition temperature Tc = 18 K(Hebard et al. 1991 [13]), soon after the discovery of metallic behavior in alkali-doped C60 compounds (fullerides) by Haddon et al. in 1991 [14]. Before a year hadpassed, several different groups also succeeded in observing superconductivity in

1In 1996, Nobel Prize in chemistry was awarded to Curl, Kroto, and Smalley for the discovery [3].

© Springer Science+Business Media Singapore 2016Y. Nomura, Ab Initio Studies on Superconductivity in Alkali-Doped Fullerides,Springer Theses, DOI 10.1007/978-981-10-1442-0_1

1

2 1 Introduction to Superconductivity in Alkali-Doped Fullerides

Fig. 1.1 The C60 moleculedrawn by VESTA [11]

alkali-doped fullerides [15–18] and Tc reached 33 K in RbCs2C60 [17]. In 1995, Tcof ∼40 K in Cs3C60 under pressure was reported by Palstra et al. [19], however, due tothe small superconducting volume fraction, the composition of the superconductingphase had not been identified. Only recently, thanks to the advancement in experi-mental techniques, A15 Cs3C60 and fcc Cs3C60 were shown to be superconductingwith the maximum Tc’s of 38 K [20–22] and 35 K [22, 23], respectively.

At that time (in 1990s), Tc of the alkali-doped fullerides was the second high-est after the cuprates.2 The discovery of a new type of superconductors with sucha high Tc had attracted strong attention and there had been much effort to under-stand the superconducting mechanism. It had been found that the Bardeen-Cooper-Schrieffer(BCS)-like mechanism [24] is compatible with several experimental resultsincluding the full gap s-wave pairing symmetry [25–35], the singlet pairing indicatedby the loss of the spin susceptibility in the superconducting state [36],3 the existenceof the Hebel-Slichter peak [39] in the nuclear magnetic resonance (NMR) [29, 36]and the muon spin relaxation (μSR) [30] measurements, and the positive correlationbetween the lattice constant and Tc [18, 40–44]. According to the McMillan-Allen-Dynes formula [45–47], the dimensionless electron-phonon coupling constant λ ∼0.5–1 and the high phonon frequencies of the intramolecular vibration modes on theorder of ∼0.1 eV give a high transition temperature comparable to the experimen-tal ones [44, 48]. Then, in late 1990s, the mechanism was widely believed to bethe conventional phonon-mediated one, while there was no conclusive experimen-tal/theoretical evidence [44, 48].

On the other hand, the importance of the electron correlations was indicated sincethe early stage: The intramolecular Coulomb repulsion was estimated to be ∼1.6 eVby the Auger spectroscopy of C60 solid [49], which is larger than the typical band-width ∼0.5 eV. Indeed, several C60 compounds were shown to be insulating [50–57].However, their structure typically differed from the cubic fcc structure [50–57] andsome materials have a different electron filling from that of the fcc A3C60 system(A = alkali atoms) [54–57]. Then the several effects, e.g., the lift of the orbital

2Even at present, it is the highest among the molecular superconductors.3In the case of Sr2RuO4, which is widely believed to be a spin-triplet superconductor, the spinsusceptibility remains unchanged across the transition into the superconducting state [37, 38].

1.1 Superconductivity in Alkali-Doped Fullerides 3

degeneracy and the filling difference, might be important to describe the insulatingbehavior. For example, the insulating behaviors in A2C60 and A4C60 are naturallyexplained by considering the interplay between the Jahn-Teller nature of the C60molecule and the electron filling [58–60]. It was also argued that the difference incrystal structure changes the structure of the electron transfer and hence the kineticenergy scale, which makes a difference in the tendency towards localization [61].Therefore, it is dangerous to discuss the physics on the same footing between the fccA3C60 systems and the others. As far as the cubic fcc A3C60 system is concerned, ametallic behavior had been observed experimentally. Theoretically, the orbital degen-eracy had been considered to play an important role to weaken the correlation effectsand to avoid the Mott localization [62, 63]. Therefore, many works had not takeninto account the correlation effects seriously to explain the superconductivity in thefcc A3C60 systems.

However, the recent success of the synthesis of fcc Cs3C60 [23] has renewed ourunderstanding. It is a Mott insulator at ambient pressure and the superconductiv-ity appears only under pressure [22, 23]. Across the metal-insulator transition, nostructural change is observed [23]. The transition temperature of Cs3C60 as a func-tion of the lattice constant was smoothly connected to those of Rb3C60 and K3C60.Therefore, it provides a unique playground for systematically studying the supercon-ductivity with a broad parameter range from very strongly correlated region to lesscorrelated region. The existence of the superconductivity in the vicinity of the Mottinsulating phase and the observed dome-shaped Tc [23] can not be explained withinthe conventional BCS picture. For some reason(s), the s-wave superconductivity isresistant to the strong Coulomb interactions, or even the electron correlations mighthelp the superconductivity. Furthermore, the traces of the phonons observed in theMott insulating phase, such as the low-spin state [23] and the dynamical Jahn-Tellereffect [64, 65], revealed that there is some non-trivial interplay between the elec-tron correlations and the electron-phonon interactions: If they simply compete witheach other, it should be difficult to detect the signature of the phonons in the Mottphase, where the Coulomb interactions are dominant. These facts suggest that, tounderstand the pairing mechanism, it is important to consider both the Coulomb andelectron-phonon interactions and to elucidate the relation between them. Althoughthere exist several studies on the superconductivity in this direction [66–73], we stilllack global consensus. This is partly because there are no realistic and unbiasedcalculations which reflect the material dependence of the electronic structure, theelectron correlations, and the electron-phonon interactions.

1.1.2 Properties of Alkali-Doped Fullerides Revealedby Experiments

While C60 solid is a semiconductor, the superconductivity appears by electron dop-ing. Among various known superconducting C60 compounds, A3C60 (A = K, Rb,Cs) systems are of great interest since they have the highest Tc [74] and allow the


systematic study without an unnecessary disturbance such as the structural change.Hereafter, we restrict ourselves to the A3C60 systems. In this section, we review theexperimental results.

1.1.2.1 K3C60 and Rb3C60

The superconductivity in K3C60 (Tc = 19 K) and Rb3C60 (Tc = 29 K), which con-dense into fcc lattice (see Fig. 1.3a for the crystal structure), has been known formore than 20 years and has been intensively studied. The main focus of the studies iswhether the superconductivity is explicable within the conventional BCS-like pictureor not.

• Pairing symmetry and superconducting energy gapPairing symmetry is an important clue to understand the superconducting mech-anism since it reflects the structure of the pairing interaction: While an attractivepairing interaction favors full-gapped s-wave pairing symmetry, a sign change inthe gap functions indicates the existence of a repulsive pairing interaction. Forexample, in the case of the cuprates [75], the observed d-wave pairing symme-try [76, 77] strongly suggests a non-phonon mechanism. On the other hand, inK3C60 and Rb3C60, the full s-wave gap, which is expected from the phonon-mechanism scenario, has been observed by many different experiments using theSTM [25, 26], the NMR [27–29], the μSR [30], the infrared spectroscopy [31–33],the tunneling measurement [33], and the photoemission spectroscopy [34, 35].Another indication for the pairing mechanism comes from the Hebel-Slichterpeak [39]. The Hebel-Slichter peak is a peak of the relaxation rate in e.g., theNMR measurements, which occurs slightly below Tc. Since it is characteristic ofthe conventional s-wave superconductors, its existence is sometimes used as apowerful evidence for the BCS-like mechanism [78]. While the early NMR mea-surements on K3C60 and Rb3C60 did not observe it [27], later experiments by theNMR [29] and the μSR [30] showed the existence of the Hebel-Slichter peak forK3C60 and Rb3C60, respectively.The ratio between the superconducting energy gap Δ and Tc is also an importantquantity. The BCS theory predicts the universal value of 2Δ/Tc = 3.53 [24] andthe deviation to a larger value can be understood as the strong-coupling effects [79].There exist a lot of experimental estimates on the ratio [25–35], however, they didnot obtain a consistent value. We list the values below.For K3C60, the obtained values are 5.3 [26], 3.0 [27], 3.4 ± 0.2 [28], 4.31 [29],3.44 [32], and ∼3.53 [35].For Rb3C60, the obtained values are 5.3 [25], 5.2 [26], 4.5 [27], 3.6 [30], 3–5 [31],3.45 [32], 4.2 [33], 4.1 [34], and ∼3.53 [35].

• Isotope effectThe change of Tc by varying the mass of atoms M is known as the isotope effect,whose exponent α is defined as Tc ∝ M−α . In the BCS theory [24], the supercon-ducting transition temperature is given by the formula


Tc ∼ ωexp[− 1N (0)V

](1.1)

with ω, N (0), V being the typical phonon frequency, the density of states at theFermi level, the effective pairing attraction between the electrons, respectively. If asystem consists of a single type of atoms, ω scales as ω ∝ M−0.5, while N (0) andV do not depend on the mass M . Therefore, within the BCS framework, Tc shouldbehave as Tc ∝ M−0.5, i.e., α = 0.5. If we take into account the strong couplingeffects, the isotope exponent α generally becomes smaller than 0.5 [79].Similarly to the results of the gap-Tc ratio, experimentally derived α values arescattered. The obtained values for the carbon isotope exponent are1.45 ± 0.3 [28], 1.3 ± 0.3 [80], and 0.3 ± 0.06 [81] for K3C60,1.4 ± 0.5 [82], 0.37 ± 0.05 [83], 2.1 ± 0.35 [80], and 0.21 ± 0.012 [84] forRb3C60.If we focus on the measurements using 99 % enriched 13C60 samples [81, 84](the others used incompletely enriched samples), the values (0.3 ± 0.06 for K3C60and 0.21 ± 0.012 for Rb3C60) seem to be compatible with the Migdal-Eliashbergtheory [85–92].

• Coherence lengthIt was shown that the fcc A3C60 systems have a rather short coherence length. Theestimated values are31 Å [93], 26 Å [94], and 45 Å [95] for K3C60,24 Å [93] and 20 Å [43] for Rb3C60.Since the distances between the neighboring C60 molecules (at ambient pressure)in fcc K3C60 and Rb3C60 are a ∼ 10.1 Å and a ∼ 10.2 Å, respectively [96], thelisted values are about a few times larger than the C60-C60 distance.

• Positive correlation between the lattice constant and the transition temperatureThe lattice parameter can be modified by changing the chemical species of thedopant atoms (chemical pressure) and/or adding pressure. It was shown that thechange in dopant atoms leads to a positive correlation between the lattice constantand Tc [18]. Furthermore, the pressure dependence of Tc in K3C60 [40–42] and inRb3C60 [42, 43] also revealed the positive correlation (see Fig. 1.2).The relationship can be naturally understood within the BCS framework [17, 18]:When the lattice constant is increased, the transfer integral of the electrons betweenthe C60 molecules decreases. It leads to the decrease in the bandwidth and theincrease in the density of states at the Fermi level. According to the BCS transition-temperature equation (1.1), higher density of states at the Fermi level gives ahigher Tc.

• Properties of the normal stateAboveTc, fcc K3C60 and Rb3C60 show a metallic behavior: The resistivity increaseswith increasing temperature [97–101]. There exists the Drude peak in the opticalconductivity [32, 102]. The photoemission studies observed finite density of states


at the Fermi level [103–106].4 The electron spin resonance measurements [108,109] and the NMR measurements [27, 36, 78] showed that the susceptibility in thenormal state is Pauli-like (nearly temperature independent). While the signatureof the electron correlation was not clear in the NMR results [78], the transferof the spectral weight to the incoherent part in the photoemission data [103–106] was ascribed to the strong correlations or the effects of the electron-phononinteractions. The temperature dependence of the resistivity supports a metallicbehavior as described above. However, the resistivity data indicate a ‘bad metal’behavior in that the deduced mean free path is apparently smaller than the nearestneighbor C60-C60 distance [110].

Overall, while there is some uncertainty in several experiments, many of them arecompatible with or even support the BCS-like mechanism. Especially, the positivecorrelation between Tc and the lattice constant is considered as a representative evi-dence of the conventional superconductivity [74], which, however, has been stronglychallenged by the appearance of fcc Cs3C60 [22, 23] and A15 Cs3C60 [20–22].

Fig. 1.2 Superconducting transition temperature Tc of alkali-doped fullerides as a function of thevolume VC603− occupied per C60

3− anion. VC603− is given by VC603− = a3/4 with the lattice constanta. Filled (open) circles represent the data at ambient pressure (under pressure). Reproduced fromIwasa and Takenobu, J. Phys.: Condens. Matter (2003). doi:10.1088/0953-8984/15/13/2025 ©IOPPublishing. Reproduced by permission of IOP Publishing. All rights reserved

4There exists inconsistency among the photoemission measurements. For example, in Ref. [107],the pseudogap behavior was observed around the Fermi level.5Y Iwasa and T Takenobu, Superconductivity, Mott-Hubbard states, and molecular orbital order inintercalated fullerides. J. Phys.: Condens. Matter 15, R495-R519. Published 24 March 2003.

http://dx.doi.org/10.1088/0953-8984/15/13/202


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Fig. 1.3 a,b Crystal structure of a fcc A3C60 (A = K, Rb, Cs) and b A15 Cs3C60. Drawn byVESTA [11]. The positions of the dopant are depicted as the green spheres. In reality, in fcc A3C60,there exists the disorder in the orientations of the C60 molecules. However, it is neglected in the panela for clarity. c, dThe experimental phase diagram of c fcc A3C60 and dA15 Cs3C60. The data pointsare taken from Refs. [23, 111] and [21] for c and d, respectively. Tc and TN are superconductingtransition temperature and Néel temperature, respectively. SC and AFI denote the superconductingphase and the antiferromagnetic insulating phase, respectively. In the phase diagram of fcc A3C60[panel c], while not shown explicitly, there is an anomalous metallic region near the Mott phase,which is dubbed “Jahn-Teller metal” [111] (see the main text for detail). Adapted from Nomura etal., Ref. [115]

1.1.2.2 Cs3C60

The superconductivity in Cs3C60 has been observed either in the fcc structure or theA15 structure [20, 23]. In the A15 structure, the C60 molecules are located at thebcc sites, however, due to the difference in the orientation between the neighboringC60 molecules, the unit cell contains two C60 molecules [Fig. 1.3b]. The orientationsof the neighboring C60 molecules are related with each other by a 90◦ rotation [20].Since the number of C603− anions in the conventional cell is different between thefcc polymorph (four) and the A15 polymorph (two), in order to compare the phasediagram between the two polymorphs, it is convenient to introduce a volume occupiedper C603− anion in solids, VC603− , instead of the lattice constant.


Figure 1.3c, d show the experimental phase diagrams as a function of the tem-perature and VC603− for the fcc [111] and A15 [21] systems, respectively. They areremarkably similar [20–23]: At ambient pressure, both systems are insulators. Byapplying pressure and thus decreasing VC603− , the superconductivity emerges througha first order transition. Across the transition, no structural change is observed. Tc’sshow dome-like shapes as a function of the pressure or VC603− . The values of Tc arealso similar: At optimal pressure where Tc takes a maximum, Tc = 35 K for the fccstructure [23] and Tc = 38 K for the A15 structure [20, 21].

However, the fcc and A15 systems differ in the magnetic property. While the low-spin state is observed in both polymorphs in the insulating region [21–23, 112], theNéel temperature TN for the A15 samples ∼46 K [21] is drastically higher than thatof the fcc samples ∼2 K [23]. In the most recent experiment [113], it was found thatfcc Cs3C60 is not a pure antiferromagnetic insulator, but the glass-like magneticallydisordered state and the antiferromagnetically ordered state coexist spatially in thesample at low temperature. The supression of TN and the spin-glass behavior inthe fcc polymorphs are ascribed to the geometrical frustration effects of the fcclattice [113, 114], and disorder in the effective super-exchange interaction betweenthe spins due to the merohedral disorder (the disorder in the orientation of C603−anions) [113]. On the other hand, the A15 structure is based on the bcc lattice, andis less frustrated, which results in the antiferromagnetic long-range order with thewave vector q = ( 12 , 12 , 12 ) at 46 K [112].

The seminal discoveries have stimulated subsequent studies to understand thenature of the insulating phase, the metal-insulator transition, and the superconduc-tivity [64, 65, 116–119], which have elucidated a nontrivial relationship among theelectron correlations, the electron-phonon interactions, the magnetism, and the super-conductivity. Ihara et al. [116] investigated the spin-dynamics around the metal-insulator transition by the NMR. Near the transition on the metallic side, theyobserved the development of the spin fluctuations beyond the values expected fromthe smooth extrapolation from the data in the small VC603− region. They also found thatthe critical pressure has a considerable temperature dependence. Klupp et al. [64,65] carefully examined the infrared (IR) spectrum and found the signature of thedynamical Jahn-Teller effect in the insulating phase (dynamical distortion of C603−).On the other hand, a more recent NMR experiment [117] suggests the freezingof the Jahn-Teller dynamics and the development of the orbital glass state as thetemperature is lowered. These IR and NMR measurements give the following indi-cations [117]: At room temperature, the time-scale of the Jahn-Teller distortion isin between the IR time scale of 10−11 s and the NMR time scale of 10−5 s. TheJahn-Teller dynamics gradually slows down with the decrease of the temperatureand eventually becomes static within the NMR time scale. Kawasaki et al. [118]observed a phase separation into the Mott insulating phase and the metallic phasethrough the NMR measurements. Basing on its analysis, they argued that the mag-netism and the superconductivity are competing orders. Wzietek et al. [119] andPotočnik et al. [120] independently performed the NMR experiments and estimatedthe size of the superconducting energy gap Δ in A15 Cs3C60 and fcc Cs3C60, respec-tively. They both obtained a similar VC603− dependence of the gap Δ: The gap size


successively increases as VC603− increases even in the region close to the Mott transi-tion, while Tc decreases in this region. The ratio 2Δ/kBTc takes a value close to theBCS value of 3.53 in the region away from the Mott transition, however, as the Motttransition approaches, it becomes larger than the BCS value [120]. Zadik et al. [111]identified an anomalous metallic region close to the metal-insulator boundary, whichthey call “Jahn-Teller metal”. In this region, the IR spectrum is similar to that ofthe Mott insulating phase [111]. This result can be interpreted in terms of a slowingdown of the dynamical distortion of the C60 molecules when VC603− increases. Whenthe Mott localization is approached, the distortion timescale becomes eventually solong that the IR experiment probes the system in a distorted state on its characteristictimescale.

1.1.3 Electronic Structure and Electronic Correlations

1.1.3.1 Electronic Structure of Undoped C60 Solid

C60 solid has a three-dimensional crystal structure [121–125] because of the ratherisotropic shape of the C60 molecule. The centers of the molecules form an fcc lattice.At high temperature, the molecules are rotationally disordered and the correspondingcrystal symmetry is Fm3̄ (fcc structure). Below ∼ 250 K, the system takes the simple-cubic structure with Pa3̄ symmetry, in which four molecules at the fcc positions inthe conventional cell take orientations different from each other [121–125].

Since undoped C60 is a molecular solid, a good starting point to understand itselectronic structure is to consider a molecular limit. In the molecular limit, manymolecular orbitals are degenerate due to the highly symmetric shape of the molecule:the C60 molecule has the fivefold-degenerate HOMO, threefold-degenerate LUMO,and three-fold degenerate LUMO+1 orbitals, which are called hu , t1u , and t1g orbitals,respectively, according to their symmetry properties (left panel of Fig. 1.4a). In solids,they acquire a dispersion by electron transfers between the molecules. However, dueto the smallness of the transfer integral, the bandwidth of each band becomes small(typically ∼ 0.5 eV) and hence, there is almost no overlap between the bands of thedifferent molecular orbitals (left panel of Fig. 1.4a). Then, C60 solid becomes a bandinsulator in which the LUMO t1u band is empty and the HOMO hu band is fullyoccupied [Fig. 1.4(b)].

1.1.3.2 Electronic Structure and Electron Correlations of A3C60

Among the A3C60 families, K3C60 and Rb3C60 take the fcc structure [127, 128].In Cs3C60, both the A15 and fcc structures are realized. In the fcc systems, thereexists a disorder in the orientations of the C60 molecules (merohedral disorder). TheC60 molecules almost randomly take one of two orientations which are related by


(a) (b)

Fig. 1.4 a Left panel: Energy levels of the C60 molecular orbitals. The arrows indicate the opticallyallowed excitation processes whose energy cost is less than 6 eV. Right panel: Band structure offcc C60 solid assuming Fm3̄ symmetry (one molecule per unit cell). In reality, at low temperature,C60 solid takes Pa3̄ structure (four molecules per unit cell). b Enlarged band structure around theenergy gap. Reprinted with permission from Saito and Oshiyama, Ref. [126]. Copyright 1991 bythe American Physical Society

90◦ rotation about [100] direction (Fm3̄m structure).6 Although some calculationstook into account the effects of the merohedral disorder [130–132], most of thecalculations for fcc A3C60 have been performed with assuming a fixed orientation(Fm3̄ structure).

The systematic studies on the electronic structures of the fcc and A15 systemswere done by Nomura et al. [133]. In both the fcc and A15 systems, each alkali atomdonates about one electron into the t1u band [134–136]. As a result, the t1u bandbecomes half-filled [see Fig. 1.5a, b for the band structures of fcc K3C60 at ambi-ent pressure and A15 Cs3C60 under pressure, respectively]. As the lattice constantincreases, the t1u bandwidth becomes smaller, which can be clearly seen in Fig. 1.5d,e. Note that, due to the inability of the density functional theory (DFT) to describethe Mott physics, the band structure calculations based on the DFT predict a metallicbehavior even for expanded systems such as fcc Cs3C60 and A15 Cs3C60 at ambientpressure, in which the insulating behaviors are observed experimentally.7

They also derived the one-body Hamiltonian for the subspace consisting of the t1uorbitals and estimated the effective Coulomb interactions between the t1u electronsusing the techniques of the maximally localized Wannier function [138–140] andthe constrained random phase approximation [141]. Reflecting the molecular natureof the solids, the maximally localized Wannier orbitals (MLWO’s) constructed from

6It is not completely random since a short range correlation has been observed: neighboring buck-yballs tend to have different orientations [129].7Giovanetti and Capone obtained the antiferromagnetic insulating phase in A15 Cs3C60 from thespin-polarized DFT calculations with employing the hybrid functional [137].


(a) (b)

(c)

(d) (e)

Fig. 1.5 Band structures for a fcc K3C60 at ambient pressure and b A15 Cs3C60 under pressure.The Wannier interpolated band dispersions are depicted as blue dotted curves. c px -like maximallylocalized Wannier orbital for A15 Cs3C60. Density of states for the t1u bands in the d fcc systemsand e A15 Cs3C60 systems. For fcc Cs3C60 and A15 Cs3C60, the calculations are done for severaldifferent volumes occupied per C603− anion, which are labelled as V opt.PSC (762 Å3), VMIT (784Å3), and VAFI (804 Å3) for fcc Cs3C60, and V

high PSC (751 Å

3), V opt. PSC (774 Å3), VMIT (791 Å3),

and VAFI (818 Å3) for A15 Cs3C60, respectively. Adapted with permission from Nomura et al.,Ref. [133]. Copyright 2012 by the American Physical Society

the t1u bands are well localized on one molecule (Fig. 1.5c). They obtained threeMLWO’s (px -, py-, and pz-like orbitals) per molecule and constructed the tight-binding Hamiltonian in the Wannier basis. There, each site corresponds to eachmolecule and the hopping parameters are obtained by calculating the transfer integralbetween the MLWO’s. The interpolated band dispersions derived by the Hamiltonian(blue dotted curves in Fig. 1.5a, b) well reproduce the original t1u band dispersions[red curves in Fig. 1.5a, b].

The calculated effective Coulomb interactions between the t1u electrons are listedin Table 1.1. There are no orbital dependence in these interactions because of the highsymmetry of the MLWO’s. The intraorbital Coulomb repulsionU ’s are ∼1 eV, whichare slightly smaller than the previous estimates ∼ 1–1.5 eV [49, 142–144]. Com-pared to U ’s, the Hund’s coupling JH’s are very small ∼ 0.035 eV. The smallness ofthe Hund’s coupling can be ascribed to the molecular-orbital nature of the MLWO’s:In the case of atomic-orbital-like MLWO, JH tends to be as large as ∼ 0.5 eVfor e.g., the 3d electrons in transition metal oxides [145], while, in the case of the


Table 1.1 The volume occupied per C3−60 anion, the DFT bandwidth, and the effective Coulombinteractions between the t1u electrons in various A3C60 systems. The listed values are taken fromRef. [133]. U , U ′, JH, and V denote the intramolecular intraorbital interaction, the intramolecularinterorbital interaction, the Hund’s coupling, and the nearest-neighbor intermolecular repulsion,respectively

Volume/C3−60(Å3)

bandwidth(eV)

U (eV) U ′ (eV) JH (meV) V (eV)

fcc K3C60 722 0.50 0.82 0.76 31 0.24–0.25

fcc Rb3C60 750 0.45 0.92 0.85 34 0.26–0.27

fcc Cs3C60 762 0.43 0.94 0.87 35 0.27–0.28

fcc Cs3C60 784 0.38 1.02 0.94 35 0.28–0.29

fcc Cs3C60 804 0.34 1.07 1.00 36 0.30

A15 Cs3C60 751 0.74 0.93 0.87 30 0.30

A15 Cs3C60 774 0.66 1.02 0.95 36 0.31

A15 Cs3C60 791 0.61 1.07 0.99 36 0.32

A15 Cs3C60 818 0 54 1.14 1.06 37 0.34

molecular-orbital-like MLWO, the wave functions are delocalized in space and thusthe exchange Coulomb integral becomes small. The nearest-neighbor intermolecu-lar Coulomb repulsions V ∼ 0.25–0.30 eV are not negligible considering the ratioV/U ∼ 0.28.

One of the useful measures for the electron correlation strength is the ratio (U −V )/W with W being the bandwidth. In Fig. 1.6, a plot of the correlation strength(U − V )/W is superimposed on the phase diagram. We see that the alkali-dopedfullerides can be categorized into strongly correlated systems since (U − V )/W � 1and that the correlation strength increases with the increase in the volume per C603−anion. Furthermore, one can see that there is a positive correlation between thecorrelation strength and Tc, while Tc eventually goes down and the system goes intothe insulating phase with the further increase in the correlation strength.

1.1.4 Electron-Phonon Interactions and Phonon Frequencies

The phonon modes of A3C60 that may couple to the t1u electrons can be classifiedinto the libration modes, the intermolecular modes, the optical modes involving alkalications and C603− anions, and the intramolecular modes (see Fig. 1.7 for a schematicpicture) [146]. It has been shown that, among them, the couplings of the libration,intermolecular, and alkali-ion modes to the t1u electrons are small compared to thoseof the intramolecular modes [44, 147–152]. Therefore, in this section, we mainlyfocus on the properties of the intramolecular phonons.

Since the C60 molecule is highly symmetric, only limited types of the intramole-cular phonon modes can couple to the t1u electrons for symmetry reasons. If we


(a) (b)

Fig. 1.6 The material dependence of the correlation strength (Ū − V̄ )/W is depicted on the phasediagram for a fcc A3C60 (A = K, Rb, Cs) and b A15 Cs3C60. Ū and V̄ are the average of theintramolecular intraorbital Coulomb repulsion and the nearest-neighbor intermolecular Coulombrepulsion, respectively. In the case of fcc A3C60 and A15 Cs3C60, the three MLWO’s per mole-cule are equivalent, therefore Ū = U . Reprinted with permission from Nomura et al., Ref. [133].Copyright 2012 by the American Physical Society

Fig. 1.7 Schematic picture of the phonon modes of A3C60 (A = K, Rb, Cs). From the low frequencyside, a the libration modes, b the intermolecular modes, c the optical modes between the alkali-cation and the C603− anions, d, e the intramolecular Hg modes are depicted. Reproduced withpermission from Hebard, Ref. [146]. Copyright 1992, American Institute of Physics

consider the ideal icosahedral symmetry of the isolated C60 molecule, there exist60 × 3 − 6 = 174 intramolecular vibrational modes (the subtracted six modes cor-respond to the translations and the rotations of the molecule), which are classified intotwo Ag modes, one Au mode, three T1g modes, four T1u modes, four T3g modes, fiveT3u modes, six Gg modes, six Gu modes, eight Hg modes, and seven Hu modes [110,


153–155]. The Ag(u) modes have no degeneracy, while the T1g(u) and T3g(u) modes, theGg(u) modes, and the Hg(u) modes are three-fold, four-fold, and five-fold degenerate,respectively [110, 153–155]. Among the intramolecular modes, only the phononmodes with the Ag and Hg symmetries have finite electron-phonon couplings tothe t1u electrons [156, 157]. When we write the electron-phonon coupling Hamil-tonian as

Ĥel−ph =3∑

i, j=1

∑σ

∑ν

gνi j ĉ†iσ ĉ jσ

(b̂†ν + b̂ν

)(1.2)

with the creation (annihilation) operators for electrons with the i th orbital and thespin σ ĉ†iσ (ĉiσ ), the creation (annihilation) operators for phonons with the νth modeb̂†ν (b̂ν), and the electron-phonon coupling g

νi j , the matrix elements of g

ν shouldbecome [156–158]

g(Ag) ∝⎛⎝ 1 0 00 1 0

0 0 1

⎞⎠ (1.3)

for the Ag modes and

g(Hg) ∝⎧⎨⎩

⎛⎝−1 0 00 −1 0

0 0 2

⎞⎠ ,

⎛⎝

√3 0 0

0 −√3 00 0 0

⎞⎠ ,

⎛⎝ 0

√3 0√

3 0 00 0 0

⎞⎠ ,

⎛⎝ 0 0

√3

0 0 0√3 0 0

⎞⎠ ,

⎛⎝ 0 0 00 0 √3

0√

3 0

⎞⎠

⎫⎬⎭ (1.4)

for the five-fold degenerate Hg modes, respectively. While the Ag modes couple tothe total density of the t1u electrons N̂ = n̂1 + n̂2 + n̂3 with n̂i = ∑σ ĉ†iσ ĉiσ , the Hgmodes do not since the traces of the g(Hg) matrices are zero. The Hg modes split theenergy levels of the t1u orbitals, therefore they are classified as Jahn-Teller modes.

Reflecting the solid C-C bond and the lightness of the carbon atoms, the intramole-cular Ag and Hg modes have rather high frequencies. Experimentally, they fall intothe range of 0.03–0.2 eV [159–165]. Theoretically calculated phonon frequenciesalso fall into this range [148, 156, 165–179]. The relatively high phonon frequenciesωph ∼ 0.1 eV compared to the bandwidth of t1u band W ∼ 0.5 eV indicate that theMigdal theorem [86], which states that the vertex correction for the electron-phononcoupling is negligible if ωph � W , is no more valid.

The electron-phonon coupling constant λ has also been estimated experimen-tally [101, 161–165, 180] and theoretically [148, 156, 165–176, 181]. The theo-retical values lie in the range λ = 0.3–0.9, while the experimental ones are a bitlarger λ = 0.5–1.2. If we restrict our attention to the recent studies, both the exper-iments [164, 165] and the calculations using the hybrid functional [165, 174] and


the GW approximation [175] give a similar value of λ/N (0) ∼ 100 meV. There, thehybrid functional and the GW approximation were found to give a larger electron-phonon coupling than that of the local density approximation [165, 174, 175]. Akashiand Arita [176] studied the material dependence of λ among fcc K3C60, Rb3C60, andCs3C60. They found that while the conventional formula

λ = 2NkN (0)

∑qν

∑knn′

∣∣gνn′n(k,q)∣∣2ωqν

δ(εnk)δ(εn′k+q) (1.5)

gives similar couplings for all the three (0.562, 0.570, and 0.603 for K3C60, Rb3C60,and Cs3C60, respectively), a more sophisticated formula to explicitly treat the energyexchange via the phonons [182, 183]

λ = 2NkN (0)

∑qν

∑knn′

∣∣gνn′n(k,q)∣∣2ω2qν

[f (εnk) − f (εn′k+q)

]δ(εn′k+q − εnk − ωqν)

(1.6)

results in a significant material dependence (0.489, 0.542, and 0.652 for K3C60,Rb3C60, and Cs3C60, respectively). Here, gνn′n(k,q) denotes the electron-phonon-coupling matrix element of the process where the νth phonon mode with the wavevector q scatters the electron from the Bloch state ψnk into the Bloch state ψn′k+q.ωqν is the phonon frequency for the mode (qν), and the εnk is the Kohn-Shameigenenergy for the Bloch state ψnk. Nk and N (0) are the number of k-mesh pointsand the electronic density of states at the Fermi level, respectively. f is the Fermidistribution function. The difference in the results between the conventional andrefined formulae comes from the high phonon frequencies and the two peak structureof the electronic density of states: The phonons can connect the electronic statesresiding in the different peaks.

Fig. 1.8 Tc as a function ofμ∗ calculated by theMcMillan-Allen-Dynesformula [Eq. (1.7)] withωln = 0.1 eV = 1160.4 K.We show the results forλ = 0.4, 0.6, 0.8 and 1.0.The gray region shows therange where theexperimental Tc’s distribute


1.1.5 On Applicability of Conventional Mechanism

1.1.5.1 Arguments Based on Migdal-Eliashberg Theory: Phonons

According to the McMillan-Allen-Dynes formula [45–47], Tc is given by

Tc = ωln1.2

exp

[− 1.04(1 + λ)

λ − μ∗(1 + 0.62λ)]

, (1.7)

where ωln is the logarithmic average of phonon frequencies and μ∗ is the Coulombpseudopotential [89, 92, 184]. Since the intramolecular phonon modes have highfrequencies ∼0.1 eV, even if λ is not so large, Tc comparable to the experimental onecan be expected. Figure 1.8 shows Tc curves for reveal λ as a function of μ∗ calculatedby the McMillan-Allen-Dynes formula [Eq. (1.7)] with ωln = 0.1 eV = 1160.4 K.Considering that the theoretical [148, 156, 166–176, 181] and experimental [101,161–165, 180] estimates of λ range from 0.3 to 1.2, one can say that the experimen-tal Tc’s are reasonably explained by the conventional phonon mechanism [85–92].Indeed, at the early stage of the studies, the intramolecular-phonon-mediated pairingmechanism was proposed based on the above-mentioned argument [156, 166–168].

There also exist studies which claimed the importance of the phonons other thanthe intramolecular ones. Zhang et al. [185] proposed that the alkali-ion optical modesmay strongly couple to the t1u electrons and give an effective attractive interactionat low-energy. However, later, it was shown that the metallic screening would sig-nificantly suppress the coupling of alkali phonons [148, 186]. Mazin et al. [187]analyzed the experimental data by the strong-coupling theory with using modelEliashberg functions. Then, they found that various experimental results are consis-tently explained if the low-frequency intermolecular modes and the high-frequencyintramolecular modes are both relevant to the superconductivity.

1.1.5.2 Arguments Based on Migdal-Eliashberg Theory: CoulombPeudopotential

Another important problem is how large the Coulomb pseudopotential μ∗ is [48,186, 188]. The Coulomb pseudopotential μ∗ is a parameter to describe the effects ofCoulomb interaction on the s-wave superconductivity. Due to the difference betweenthe electronic energy scale and the phonon one, low-energy electrons with phononenergy scale will feel a renormalized Coulomb interaction as a result of the retardationeffect [89, 92, 184]. Such a renormalization effect is incorporated in μ∗. Though μ∗ isa key parameter in the McMillan-Allen-Dynes formula [Eq. (1.7)], useful methodsto estimate it have not been established [189]. Traditionally, the formula [89, 92,184]


μ∗ = μ1 + μ log

(Eelωph

) (1.8)

has often been employed. Here, μ is a dimensionless Coulomb potential calculatedas the product of the Fermi-surface-averaged screened Coulomb interaction and thedensity of states at the Fermi level, and Eel (ωph) is a typical electronic (phonon)energy scale. The formula can be derived by projecting out the high-energy electronicstates via the summation of the ladder diagram, in which a constant density of statesand a momentum- and frequency-independent Coulomb interaction are assumed [89,92, 184].

As for the estimate of μ∗ in the alkali-doped fullerides, several problems werepointed out [48, 186, 188]. For example, there exists uncertainty in the appropriatevalue for the typical electronic energy scale Eel. Varma et al. [156] and Schlüteret al. [166] assumed that Eel is on the order of 10 eV based on the fact that theladders of narrow bands spread out over several tens of eV while there are gaps onthe order of 1 eV between the bands. In this case, μ∗ becomes small enough to allowthe BCS-like superconductivity. On the contrary, Anderson [190] and Chakravartyet al. [191] argued that Eel should be on the order of the narrow bandwidth ∼0.5 eV, which leads to almost unrenormalized Coulomb pseudopotential μ∗ ∼ μ.Then, it is difficult to explain the high transition temperature of the fullerides by theconventional phonon mechanism. Another problem is whether the traditional formula[Eq. (1.8)] is applicable or not. In the fullerides, the validity of the assumptions of theconstant density of states and the momentum- and frequency-independent Coulombmatrix elements is highly questionable [48, 186, 188]. Since the traditional formulaessentially relies on these assumptions, it might be inappropriate to apply it for thecalculation of μ∗.

Gunnarsson and Zwicknagl [186], and Gunnarsson et al. [188] studied a simplemodel to answer these problems. They introduced a Hubbard-like two-band model

H =∑i

∑σ

(ε1n̂

σ1i + ε2n̂σ2i

)− t

∑〈i, j〉

∑σ

(ĉσ†1i ĉ

σ1 j + ĉσ†2i ĉσ2 j

)

+ U∑i

∑(lσ)


summing up ladder diagrams, which leads to the Coulomb pseudopotential

μ∗ ∼ N (0)[U scr − (U

scr12 )

2

2Δε

]∼ N (0)

[1

2N (0)− U

212

2Δε

]� 0.5. (1.10)

In deriving this formula, they utilized the fact that, within the random phaseapproximation (RPA), the screened Coulomb interactions U scr and U scr12 are givenby U scr ∼ U/[1 + 2N (0)U] ∼ 1/[2N (0)] and U scr12 ∼ U12/[1 + 2U12/Δε] ∼ U12,respectively. However, in this parameter setting, a correct treatment is to first trace outthe high-lying levels, which yields an effective single band model with the Coulombrepulsion

Ueff ∼ U − U212

2Δε∼ U. (1.11)

One finds that the effect of the high-lying band is very small. Then, within the RPA,μ∗ is given by

μ∗ ∼ N (0) Ueff1 + 2N (0)Ueff ∼ 0.5. (1.12)

Therefore, the traditional approach overestimates the renormalization effect comingfrom the high-lying band and gives an incorrect result. Basing on these considera-tions, they suggested that the conventional theory for μ∗ can not simply be appliedto the alkali-doped fullerides and that the retardation effects might be inefficient inthe fullerides.

1.1.5.3 Reinvestigation of Conventional Mechanism by SCDFT

Akashi and Arita [176] examined the validity of phonon mechanism by means of thedensity functional theory for superconductors (SCDFT) [192–195]. The SCDFT is apowerful method to calculate Tc with assuming the phonon-mechanism from purelyfirst principles calculations; i.e., it is free from any empirical parameters such as theCoulomb pseudopotential μ∗. Since it has succeeded in reproducing the experimentalTc’s of conventional superconductors such as simple metals [195] and MgB2 [196]with deviations less than a few K, it can be used as a “Litmus paper” to examinewhether a superconductor is of conventional type or not [176, 197]: If the transitiontemperature calculated by the SCDFT agrees well with the experimental one, it willgive a strong support for the phonon mechanism. Otherwise, it strongly suggeststhat the phonon mechanism alone can not explain the superconductivity. In the caseof the alkali-doped fullerides, the calculated Tc’s are 7.5, 9.0, and 15.7 K for fccK3C60 (ambient pressure), Rb3C60 (ambient pressure), and Cs3C60 (under pressureof 7 kbar), respectively [176], which are much lower than the experimental Tc’s (19,


29, and 35 K). Therefore, they concluded that the alkali-doped fullerides are not asimple phonon-mediated superconductor.

1.1.6 Unconventional Mechanisms

1.1.6.1 Purely Electronic Mechanisms

Baskaran and Tosatti [198], Chakravarty and Kivelson [199], and Chakravarty etal. [200] proposed that an attraction, which drives superconductivity, may appear bya purely electronic mechanism. They argued that, as a consequence of the correlationeffects, π electrons on a C60 molecule form the resonating valence bond (RVB)state [201–204]. Baskaran and Tosatti [198] considered that the singlet nature isstable even in a charged C60 molecule. They pointed out that this tendency towardthe singlet formation can be parametrized by an effective negative exchange coupling.The negative exchange interaction serves as an attractive scattering channel for theCooper pairs. Chakravarty and Kivelson [199], and Chakravarty et al. [200] statedthat, if the excitation energy of the spinon (spin-1/2 and charge-0 excitaion) is largeenough, it should be energetically favored to add an even number of electrons intoa neutral C60 molecule, which can avoid the spinon excitation, than to add an oddnumber of electrons, which is inevitably accompanied by the spinon excitation. Asa result, an effective attraction arises in the state with odd number of extra electrons.Friedberg et al. [205] suggested the possibility to form “boson” band involving t1u andt1g electrons in the fullerides. They discussed that the Bose-Einstein condensationof the boson degrees of freedom might be responsible for the superconductivity.

1.1.6.2 Mechanisms Related to Suhl-Kondo Mechanism

Rice et al. [206], Asai and Kawaguchi [169], and Suzuki et al. [207] pointed out thatthe pair-hopping interaction plays an important role via the Suhl-Kondo-like mech-anism [208, 209]. Rice et al. [206] found that the superconductivity is enhancedregardless of the sign of the interband scattering, while a negative interband scat-tering is more efficient. Asai and Kawaguchi [169] studied the origin of the inter-band pair-transfer interaction. They argued that the nonadiabatic couplings, whichare neglected in the Born-Oppenheimer approximation [210], give a positive pair-transfer interaction. Suzuki et al. [207] took into account the phonon-mediated attrac-tion in the antiadiabatic limit on top of the Coulomb interaction, and showed that,based on the BCS-type mean-field argument, the superconductivity can occur ifVintra + 2K < 0 with the intraorbital interaction Vintra and pair-hopping interactionK . Here, the intraorbital interaction Vintra does not need to be attractive; even if it isrepulsive, the superconductivity emerges if the attractive pair-hopping interaction Ksurpasses it.


1.1.6.3 Mechanisms Involving Both Coulomb Interactionsand Electron-Phonon Interactions

Takada [211] took into account the effects of the plasmon on top of those of phonons.Furthermore, he studied the vertex corrections both for the plasmon part and thephonon part. He found that the inclusion of the vertex corrections results in theincrease in Tc for the phonon-mediated mechanism, while Tc for the plasmon-mediated mechanism is suppressed by the vertex corrections. By considering boththe plasmon and intramolecular-phonon contributions, he calculated Tc’s for severalsystems with different lattice constants, which were in good agreement with theexperimental ones.

Capone et al. [66–70] assumed that, as a result of the competition between thepositive Hund’s coupling and the negative phonon-mediated exchange interaction,an effectively small negative interaction |J | ∼ 20 meV is realized in the system.They considered it in antiadiabatic limit (|J | is static in this limit) and analyzed thenegative-J three-orbital [66–68, 70] and two-orbital [69] Hubbard models definedon the Bethe lattice by means of the dynamical mean-field theory (DMFT) [212].Figure 1.9 shows thus calculated anomalous self-energy at ω = 0 (ΔSC) of the half-filled three-orbital model as a function of the Hubbard U for |J |/W = 0.02, 0.05,0.1, and 0.2. They found that, when |J | is very small compared to the bandwidthW (|J | � 0.05W ), the s-wave superconductivity appears in the vicinity of the Motttransition. On the other hand, if |J | � 0.1W , ΔSC takes a maximum value at U = 0.

Fig. 1.9 Anomalous self-energy at ω = 0 (≡ ΔSC) as a function of HubbardU obtained by DMFTanalysis of negative-J three-orbital Hubbard model at half-filling. The semicircular density of states(Bethe lattice) is employed in the calculations. Figure shows ΔSC’s for |J |/W = 0.02, 0.05, 0.1,and 0.2. The positive values of J in the figure means that the exchange interaction is negativeaccording to the notation used by Capone et al. [70]. Reprinted with permission from Capone et al.,Ref. [70]. Copyright 2009 by the American Physical Society


Although both superconducting regions show s-wave character, the origins of themare totally different. The latter one is understood by the BCS mechanism, in which thebare attraction − 103 |J | drives the superconductivity.8 The repulsiveU competes withthis attraction and suppresses the superconductivity. On the other hand, the formerone is a new type of phonon-mediated superconductivity which crucially relies on thestrong correlation, which they call strongly correlated superconductivity. In this case,the Coulomb repulsion suppresses the charge fluctuation and the onsite occupationN tends to be fixed at N = 3 (half-filling). Then the Coulomb repulsion between thequasip

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