On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe₂ [1st ed. 2019] 978-3-030-29824-1, 978-3-030-29825-8

Table of contents :
Front Matter ....Pages i-xvi
Introduction (Chuan Chen)....Pages 1-14
Excitonic Character of CDW in TiSe2 (Chuan Chen)....Pages 15-29
Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2 (Chuan Chen)....Pages 31-49
Excitonic CDW Fluctuations and Superconductivity (Chuan Chen)....Pages 51-67
Anomalous Quantum Metal Phase in TiSe2 (Chuan Chen)....Pages 69-81
Conclusions (Chuan Chen)....Pages 83-85
Back Matter ....Pages 87-105

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Springer Theses Recognizing Outstanding Ph.D. Research

Chuan Chen

On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe₂

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Chuan Chen

On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 Doctoral Thesis accepted by the National University of Singapore, Singapore, Singapore

123

Author Dr. Chuan Chen Center for Advanced 2D Materials National University of Singapore Singapore, Singapore

Supervisor Prof. Vitor M. Pereira Department of Physics National University of Singapore Singapore, Singapore

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-29824-1 ISBN 978-3-030-29825-8 (eBook) https://doi.org/10.1007/978-3-030-29825-8 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

The interplay of charge density wave (CDW) order and superconductivity (SC) has been a perennial topic of interest in condensed matter physics, especially since it was understood that the underlying physics might unlock the promise of high-temperature SC. Although CDWs—collective macroscopic charge modulations in an otherwise uniform electronic system—come in many guises and across a diverse range of solid-state systems, they all stem from electronic interactions: CDW instabilities are a result of either interactions purely among electrons or between electrons and other degrees of freedom, most frequently phonons. These instabilities, and the associated phase transitions, are frequently controllable by temperature, electronic doping, pressure, or strain. Experimental research on correlated phases such as CDWs has been explosively revived over the past few years by the new ability of exploring them in a number of different realizations of two-dimensional crystals, where carrier densities can be tuned on demand by field effect, without introducing disorder. The nature of the CDW instability in the semimetallic dichalcogenide TiSe2 presents a notable case since it had been conjectured, from as early as 1976, as a candidate excitonic insulator. Despite long-standing theoretical proposals for their existence, unequivocal identification of a material representative of the excitonic insulating state has remained elusive. Recent experiments, some of which appeared while the work for this thesis was being developed, have changed that and reinforce the view that the CDW phase in TiSe2 is a direct manifestation of its intrinsic excitonic character. This doctoral thesis presents an encompassing theoretical analysis of the nature of the CDW in two-dimensional TiSe2 in precisely that context. It makes a compelling case for a view where excitonic physics and excitonic fluctuations likely underpin the entirety of its experimental phase diagram, including the emergence of an experimentally well-established superconducting dome at a critical electron doping. The core of the author’s thesis begins (Chap. 2) by introducing the excitonic mechanism and exploring its consequences on the basis of a realistic parametrization of TiSe2. The most important observation is that the calculated suppression of the CDW transition temperature captures the experimental situation v

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Supervisor’s Foreword

with very good quantitative agreement—with only one free parameter in the theory—while also explaining a number of other known experimental features at this transition. Propelled by these findings—and thereby anchoring the excitonic degrees of freedom as a key premise of the underlying physical picture— the author then studies how SC order might arise in the midst of a CDW phase. This is first done at a phenomenological level, by developing a model of coupled CDW and SC order parameters, and exploring its implications (Chap. 3). Of these, the most significant are the facts that CDW discommensurations that have been experimentally seen in different experiments naturally arise from this model and that the model harbours a dome-shaped SC phase at finite doping that tallies with experiments. A novel and unusual SC phase is also predicted, characterized by spatial non-uniformity and multiple dimensional crossovers that have well defined experimental implications. The same problem is subsequently analysed at the microscopic level, by proposing a specific SC pairing mechanism driven by fluctuations of the excitonic condensate that defines the CDW order (Chap. 4). These fluctuations are shown to mediate an attractive (pairing) interaction among the excitonic quasiparticles with energy scales quite appropriate to explain the experimental SC transition temperatures. This further reinforces the view that SC in TiSe2 is boosted by loss of CDW commensurability, as suggested by the experimental phase diagram, while simultaneously providing a microscopic foundation for the phenomenological model discussed before. Finally, some implications of this interplay between CDW and SC order are discussed in the context of the author’s theoretical contribution to an experimental collaboration that identified an anomalous quantum metallic state in TiSe2 (Chap. 5). This observation adds TiSe2 to an already large set of two-dimensional systems where the existence of such zero-temperature metallic phase demands one to revisit the nature of the superconducting ground state in strictly two-dimensional systems. By tackling different microscopic aspects of both the CDW and SC phases, in particular, their interplay, the work reported here has been able to, for the first time, reproduce the experimental phase diagram with extremely good quantitative agreement. This thesis thus represents an important theoretical counterpart to recent experiments in establishing TiSe2 as an example of a correlated excitonic material. The knowledge and ramifications gathered in the context of this compound will certainly springboard the search of other platforms for correlated exciton physics. The long-predicted excitonic insulator state of matter is no longer as elusive as it had been for decades. Singapore June 2019

Prof. Vitor M. Pereira

Abstract

This thesis presents mainly analytical theoretical studies on the interplay between charge density waves (CDW) and superconductivity (SC) in the actively studied transition-metal dichalcogenide 1T-TiSe2. It begins by re-approaching a year-long debate over the nature of the phase transition to the commensurate CDW (CCDW) state and the role played by the intrinsic tendency towards excitonic condensation in this system. It is shown that an excitonic mechanism is capable of reproducing with very good quantitative agreement the experimental phase diagram for the CCDW transition temperature as a function of carrier density. Combined with an investigation by density-functional perturbation theory of the extent to which phonons contribute to the CDW instability, the overall set of results shows that a combination of electronic correlation and electron-phonon coupling precipitates the CDW transition and, therefore, TiSe2 is indeed an excitonic solid at low temperatures, as suggested by a recent experiment. A Ginzburg-Landau phenomenological theory was subsequently developed to understand the experimentally observed transition from commensurate to incommensurate CDW (ICDW) order with doping or pressure and the emergence of a superconducting dome that coexists with ICDW. By phenomenologically mapping the lock-in energy to the carrier density, one obtains results and a phase diagram where: (i) the CCDW is suppressed and evolves into ICDW with doping or pressure, (ii) SC can coexist with ICDW at low temperatures and a finite range of densities and (iii) SC order first arises within the CDW discommensurations at the near-commensurate transition. These results are in line with the experimental phase diagram. To characterize microscopically the effects of the interplay between CDW and SC, the spectrum of CDW fluctuations beyond mean-field was studied in detail. The effective action for the amplitude and phase modes was obtained, revealing the gapless (Goldstone) character of the phase while the amplitude modes are gapped. The addition of charge carriers to the system fills the CDW-dressed conduction band. Integrating CDW fluctuations out leads to a retarded attractive interaction between the CDW ‘renormalized’ quasiparticles. The result indicates that SC order in TiSe2 can be either driven by fluctuation-induced pairing or, at least, enhanced by vii

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Abstract

CDW fluctuations. This tallies with the experimental observation that the SC dome appears centred at the putative quantum critical point of the CCDW phase. Finally, collaborating with experimental groups, we discovered that, with increasing magnetic field, an anomalous quantum metal (AQM) phase emerges between the SC to normal metal transition. A scaling analysis of the resistance suggests that within the AQM phase there is a crossover from a low field Bose metal regime to a high field vortex quantum creeping regime. Since the onset of SC behaviour in TiSe2 coincides with the disruption of commensurate CDW order through discommensurations, we advance that the development of the SC order parameter is inescapably intertwined with that of the charge density and its fluctuations. This has a direct implication in terms of providing both an intrinsic spatial non-uniformity for the development of the AQM, as well as a natural dissipation channel via phase fluctuations of the CDW.

Publications Related To This Thesis 1. Chuan Chen, Bahadur Singh, Hsin Lin, and Vitor M. Pereira. “Reproduction of the charge density wave phase diagram in 1T-TiSe2 exposes its excitonic character.” Phys. Rev. Lett. 121, 226602 (2018) 2. Chuan Chen, Lei Su, A. H. Castro Neto, and Vitor M. Pereira. “Discommensuration-enhanced superconductivity in the charge density wave phases of transition-metal dichalcogenides.” Phys. Rev. B 99, 121108 (R) (2019) 3. Linjun Li, Chuan Chen, Kenji Watanabe, Takashi Taniguchi, Yi Zheng, Zhuan Xu, Vitor M. Pereira, Kian Ping Loh, and A. H. Castro Neto. “Anomalous quantum metal in a 2D crystalline superconductor with intrinsic electronic non-uniformity.” Nano Lett. 19, 4126 (2019) 4. Chuan Chen, A. H. Castro Neto, and Vitor M. Pereira.“Pairing induced by fluctuations of an excitonic insulator: the case of superconductivity in TiSe2.” (manuscript submitted).

ix

Acknowledgement

Time flies. My doctoral period went past in a way I could never have imagined before it started. During the 4 years in Singapore, I received a lot of help from many people, knowingly or not. Here I want to express my deepest gratitude to them. First and foremost, I want to thank my supervisor Prof. Vitor M. Pereira. At the beginning of my Ph.D., I was really a layman. I still remember how frustrated I felt when Vitor asked me to do some literature research. However, Vitor is very patient and he is good at separating a big problem into small pieces so we can make progresses step by step. The discussions about physics with him are always beneficial, he is serious about the details and he could always find a way out whenever we got stuck on the project. Most importantly, he is a nice man who is willing to listen to everyone’s thoughts and provide as much help as he can. Our research group is like a family. During the group lunch meetings, we shared our opinions about recent news and learned a lot about each others’ culture. I could never forget how impressive it was when I first knew he was taking a class on Chinese and when he said “chi fan ma” before our group lunch. Whenever I got depressed about works and life, Vitor was always a nice guy to talk with and his encouragements helped me calm down and go through the difficult days. There is always too much to learn from him and I feel really honoured and fortunate to be able to be his student. I want to thank Professor Antonio H. Castro Neto, who is a brilliant physicist and a great leader of the Centre for Advanced 2D Materials. The discussions with him are always fruitful and I believe nobody can prevent from being influenced by his optimism and passion about physics and life. He is a great example to me. The coursework in NUS is nice. I learned a lot from the theoretical modules, including Selected Topics in Quantum Field Theory by Prof. Belal E. Baaquie, Advanced Solid State Physics by Prof. Benoit Gremaud and especially the Advanced Statistical Mechanics by Prof. Wang Jian-Sheng, Prof. Wang’s wide knowledge about statistical physics and the good organization of the course were really impressive to me. At the end of my first year, I got a problem on my nose and I experienced a hard time feeling anxious and disappointed. I want to thank Dr. Mark Thong Kim Thye

xi

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in NUH for doing a successful surgery on my sinus and the helpful consultations. I thank Yanwu Gu, Bo Lei and Li Zhang for their encouragement and help during that difficult time. I would like to thank my “Big Brother” Fabio Hipolito for his endless help on LaTeX, Linux, physics and …. Man, you are always “watching” me. As a member of “Ruozhi Bang”, I am grateful to my buddies Chuang-Han Hsu and Lei Su for their support and I will never forget those memorable meals and discussions we had together about physics, life and future. I also want to thank my friends in NUS, including Yifeng Chen, Xingyu Gu, Wen He, Evan Lakasono, Jia Ning Leaw, Jingsi Qiao, Emilia Ridolfi, Zhou Shen, Bahadur Singh, Ho-Kin Tang, Juefan Wang, Zhe Wang, Ziying Wang, Yaze Wu, Kaijian Xiao, Fengyuan Xuan, Indra Yudhistira, Li Zhang and Yige Zhou. Thanks for the happiness you bring to me. Especially, I am in debt to Runrun Xu for her company and love. I acknowledge Centre for Advanced 2D Materials, which provides a comfortable environment for me to do research. I sincerely thank Marilen Buenviaje and Miguel Costa for their IT support, Bernice Kiong, Jia Chyi Lam, Wei Fen Lee, Shirley Sim and Andy Voon for their administrative supports. Last but not least, I have to thank my family. Especially my father Kai Chen, mother Ling Wang, grandpa Wuzhou Chen and grandma Guangzhen Jiao. It’s a fortune to be your kid and I am grateful for your endless love.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Brief Introduction to Charge Density Waves . . . . . . . . . . . 1.2 Overview of the Properties of 1T-TiSe2 . . . . . . . . . . . . . . . 1.2.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 CDW in TiSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Superconductivity in TiSe2 . . . . . . . . . . . . . . . . . . . 1.3 Brief History of the Theoretical Understanding of the CDW Transition in TiSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Electron-Phonon Coupling and Electron-Electron Correlation in TiSe2 . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Signatures of Exitonic Condensation . . . . . . . . . . . . 1.4 Recent Experimental Developments on Two-Dimensional TiSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Excitonic Character of CDW in TiSe2 . . . . . 2.1 Excitonic Instability (EI) of TiSe2 . . . . . . 2.1.1 The EI Hamiltonian . . . . . . . . . . . 2.1.2 Mean Field Approach to the EI . . 2.1.3 Phase Diagram . . . . . . . . . . . . . . 2.2 DFT Study of the Lattice Instability . . . . 2.2.1 Isolating the Effect of Cu Doping . 2.2.2 Phonon Softening with Cu Doping 2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2 . . . . . . . . . . . . . 3.1 The CDW Is Not Alone . . . . . . . . . . . . . . . 3.2 Phenomenological Ginzburg–Landau Theory 3.3 Harmonic Expansion . . . . . . . . . . . . . . . . . . 3.4 CDW Phase Diagram . . . . . . . . . . . . . . . . . 3.5 Coupling Between a Charge Density Wave and Superconductivity . . . . . . . . . . . . . . . . . 3.6 More on Superconductivity . . . . . . . . . . . . . 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Excitonic CDW Fluctuations and Superconductivity . . . . . . 4.1 An Inspiration From the Spin-Bag Mechanism . . . . . . . . 4.2 Simplified Parameterization of the Excitonic Hamiltonian 4.3 Charge Density and Auxiliary Field . . . . . . . . . . . . . . . . 4.4 The Mean Field CDW Solution . . . . . . . . . . . . . . . . . . . 4.5 Effective Action of CDW Fluctuations . . . . . . . . . . . . . . 4.6 Fluctuation-Induced Pairing Between Quasiparticles . . . . 4.7 Exploration of the Joint CDW-Superconductivity Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Mean Field Treatment of CDW with Both Excitons and Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix B: The Lattice Instability Ab initio . . . . . . . . . . . . . . . . . . . . . .

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Appendix C: Effective Action of CDW Fluctuations . . . . . . . . . . . . . . . . .

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5 Anomalous Quantum Metal Phase in TiSe2 . . . . . . . 5.1 A Brief Introduction to the Anomalous Quantum Metal Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Summary of the Experiments . . . . . . . . . . . . . . . 5.3 Theoretical Interpretation and Background . . . . . . 5.3.1 Thermally Activated Flux Flow . . . . . . . . 5.3.2 Bose Metal and Vortex Quantum Creeping 5.3.3 Importance of the CDW Background . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acronyms

AQM ARPES BCS BKT BM BZ CCDW CDW C-IC CuxTiSe2 DC DFPT DFT DOS EI EPC GGA G–L hBN HTSC ICDW M-EELS NbSe2 NC-CDW PLD QCP RPA SC SDW STM

Anomalous quantum metal Angle-resolved photoemission spectroscopy Bardeen–Cooper–Schrieffer Berezinskii-Kosterlitz-Thouless Bose metal Brillouin zone Commensurate charge density wave Charge density wave Commensurate to incommensurate Copper-doped titanium diselenide Discommensuration Density-functional perturbation theory Density-functional theory Density of states Excitonic insulator Electron-phonon coupling Generalized gradient approximation Ginzburg–Landau Hexagonal boron nitride High-temperature superconductor Incommensurate charge density wave Momentum-resolved electron energy-loss spectroscopy Niobium diselenide Near-commensurate charge density wave Periodic lattice distortion Quantum critical point Random-phase approximation Superconductivity or superconducting Spin density wave Scanning tunnelling microscopy

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TAFF TaS2 TiSe2 TMD VASP VQC

Acronyms

Thermally assisted flux flow Tantalum disulfide Titanium diselenide Transition-metal dichalcogenide Vienna ab-initio simulation package Vortex quantum creeping

Chapter 1

Introduction

1.1 Brief Introduction to Charge Density Waves Charge Density Waves (CDW) are one of the most interesting correlated phenomena in condensed matter physics. The very first proposal was given by Peierls [1] and Fröhlich [2] in the 1950s on a 1D free electronic system, now known as the Peierls instability (PI). As illustrated in Fig. 1.1, in a 1D electronic system, due to the electronphonon coupling (EPC), the system has a tendency to undergo a metal-insulator phase transition at low temperature, accompanied by a periodic lattice distortion (PLD). After the phase transition, the electronic band becomes gapped at the Fermi “surface” (two points at ±k F in the 1D case), and the system exhibits a new charge density distribution/wave and a PLD: ρ(r) = ρ0 + ρ cos (Q · r + φ) ,

(1.1)

with wavevector Q = 2k F and φ being a constant phase. When the periodicity of CDW is proportional to the original lattice constant a by a rational number, i.e., 2π m = a, (m, n ∈ R) , Q n the density wave is defined as commensurate charge density wave (CCDW), otherwise it is designated as incommensurate charge density wave (ICDW). In the very early example given by Peierls, the system is at half-filling (see Fig. 1.1) and the CDW wavevector is: π Q = 2k F = , (1.2) a thus it corresponds to a CCDW transition. Moreover, the frequency of the CDWrelated phonon mode ω Q softens as the temperature decreases toward the CDW transition temperature Tc . A weak-coupling approximation yields [3] © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_1

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1 Introduction

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ω2Q = (ω0Q )2 − |g|2 (Q, 0)

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with g being the electron-phonon coupling constant and (Q, 0) the static (bare) charge density susceptibility at the CDW wavevector. After the work of Peierls, there have also been important developments in understanding this problem subsequently followed from the seminal work of Su, Schrieffer and Hegger [4, 5]. Experimentally, a CDW can be identified with different types of measurements: electronic transport, neutron or X-ray scattering, scanning tunnelling microscopy (STM), etc. When a system undergoes a CDW transition with decreasing temperature, the resistivity versus temperature curve would show an anomalous hump around the Tc [6]. Because of the PLD in the CDW phase, there will be new Bragg peaks in the X-ray scattering pattern, and the real space charge modulation with a new periodicity can also be directly observed by STM (Fig. 1.2). Almost 20 years after that theoretical proposal by Peierls and Fröhlich, a CDW transition was first discovered in the so-called Krogmann’s salt K2 Pt(CN)4 Be0.3 ·xH2 O through an X-ray scattering experiment [10]. Later on, it was also observed in many other quasi-1D systems, such as in the charge transfer salt TTF-TCNQ [11], transition metal tri-chalcogenide NbSe3 [12], and blue bronze K0.3 MoO3 [13]. Although the theoretical prediction was made for a 1D system in the very beginning, CDW have been confirmed to exist in many higher dimensional systems as well. Moreover, in addition to EPC, electron-electron interaction can be the driving force for a CDW transition, in which case a CDW can even occur without PLD in many strongly correlated electronic systems, e.g., K0.9 Mo6 O17 [14]. In high-temperature

1.1 Brief Introduction to Charge Density Waves

3

Fig. 1.2 Experimental methods to probe a CDW. a Resistivity versus temperature in TiSe2 [7]. It is clear that there is a hump near the CDW transition temperature. b X-ray scattering pattern in TaS2 [8]. In the CDW phase, an additional Bragg peak at CDW wavevector Q1 can be seen from the scattering pattern. c STM on monolayer NbSe2 [9]. The 3×3 superlattice can be seen explicitly

superconductor (HTSC) cuprates, CDW order has also been observed within certain intervals of the hole doping [15, 16]. However, the microscopic origin of this CDW order is still under debate and it has been suggested that it is intertwined with the superconductivity (SC) [17, 18]. Transition metal dichalcognides (TMDs) are also well-known to support CDW at low temperature, where the microscopic origin of CDW varies from system to system. In the case of 2H-NbSe2 or 2H-TaSe2 , it has been confirmed that they are driven by the strong EPC [19–21]; while in other cases like 1T-TiSe2 , it has been accepted that the electron-electron correlation can also play an important role in the CDW instability [22, 23].

1.2 Overview of the Properties of 1T-TiSe2 1.2.1 Structural Properties 1T-TiSe2 (TiSe2 , in short) is one of the layered TMDs. Each layer has an hexagonal 3 ¯ (P 3m1, 164), which consists of sublayers Bravais lattice with space group D3d in the order Se-Ti-Se within a unit cell where hexagonally arranged Ti atoms are octahedraly coordinated by Se. The first Brillouin zone (BZ) is hexagonal with three high-symmetry points , M, and K, as illustrated in Fig. 1.3c. The 3d and 4s valence electrons from Ti effectively transfer to the 4 p orbitals of the Se atoms, forming ionic bondings. As a result, the valence and conduction bands mainly come from Se 4 p and Ti 3d orbitals.

1.2.2 CDW in TiSe2 From electronic transport [7], Angle-resolved photoemission spectroscopy (ARPES) [25] and X-ray scattering experiments [26], the system was observed to undergo a second-order phase transition to a triple-q commensurate CCDW phase at Tc ∼ 200 K.

4

1 Introduction

(a) (c)

(b)

(d)

(e)

Fig. 1.3 Crystal structure, Brillouin zone, and bare band topology of the TiSe2 monolayer. a 1T phase with octahedral prismatic local structure of Ti and Se atomic layers. b Top view of the TiSe2 monolayer with Cu doping. The black rhombus identifies the 2×2 supercell and red balls show the Cu atoms which lie above the central Ti atoms in the supercell. a1 and a2 are the primitive vectors in the normal phase (1×1). c BZ of the normal (1×1, solid black) and distorted (2×2, dashed blue) phases, where the points marked with ∗ refer to the reduced Brillouin zone. d The Fermi contours and e schematic band structure near E F in the normal phase. The Se-derived valence band has its maximum at while the Ti-derived electron pockets are centered at the M points. Their overlap is quantified by bo and M = Qcdw . This figure is taken from Ref. [24]

This means that the charge density acquires a modulation characterised by the linear combination of three wave vectors: ρ(r) = ρ0 + ρQ(1) (r) + ρQ(2) (r) + ρQ(3) (r) cdw

cdw

cdw

(1.4)

In the CDW phase, the bulk (monolayer) system exhibits a 2×2×2 (2×2) charge density modulation and a PLD with ordering wavevector Q(1) cdw = L ( M) [7, 27, (2,3) 28], with the other two wavevectors Qcdw being related to Q(1) cdw by C 3 rotations. In

1.2 Overview of the Properties of 1T-TiSe2

5

Fig. 1.4 Experimental signatures of the CDW phase in TiSe2 . a Resistivity (ρ) versus temperature (T ) in TiSe2 [7]. The anomalous hump in resistivity signals a CDW transition at Tc ∼ 200 K, the inset shows the derivative of ρ with T . b Phonon softening in TiSe2 [20]. A transverse phonon mode with the CDW wavevector is quenched into a static lattice distortion at 190 K ( ≈ Tc ), indicating a PLD with the same wavevector. c STM on TiSe2 [29]. The system exhibits a 2×2 superlattice within the CDW phase, which can also be inferred from the Fourier transformed plot in the inset: the points with yellow circles are the CDW wavevectors, while the faint outer dots mark the position of the Bragg peaks of the lattice in the normal state

addition, from ARPES experiments, the Se 4 p band repeats with a lower intensity at L (M) point, which is also the zone centre after the zone folding (see an example for the bulk case in Fig. 1.6a).

1.2.3 Superconductivity in TiSe2 In 2006, Morosan et al. discovered that, by doping bulk TiSe2 by Cu intercalation, the CDW phase is suppressed and, at certain doping level (Cu0.04 TiSe2 ), the system can become a superconductor with the SC phase acquiring a dome shape in the phase diagram [30]. Later, Kusmartseva et al. found that TiSe2 can also be SC under pressure [31]. The phase diagrams of these two experiments are qualitatively similar and are shown in Fig. 1.5a, b. In both experiments, the CDW phase transitions were identified through transport measurements, i.e., looking at the hump in resistivity versus temperature curve. It can be seen that the CDW order terminates at a certain doping level/pressure with a finite temperature, which is due to the fact that the hump signature is no longer visible after that point. To have a better understanding of the evolution of CDW order, it is better to probe it from a different perspective. Joe et al. performed X-ray scattering experiments on TiSe2 under pressure. According to their results, at low pressure, the CDW order has a similar suppressed behaviour, which is consistent with Kusmartseva’s result. However, the CDW order doesn’t cease to appear abruptly. Instead, it persists to much higher pressure with a quantum critical point (QCP) relatively far-away from the end point of the SC dome. It thus indicates that the emergence of SC is not due to the quenching of CDW order. More interestingly, it is found that above the SC dome in the phase diagram, the CDW order becomes incommensurate (ICDW) [33]. Later on, the Cu-doped TiSe2 was also

6

1 Introduction

Fig. 1.5 Experimental phase diagram of Cux TiSe2 and pressurised TiSe2 . a Phase diagram of Cux TiSe2 [30]. CDW phase is suppressed with increasing doping and the SC order has a dome shape. b Phase diagram of pressurised TiSe2 [31]. CDW order is suppressed by applying pressure. Superconductivity also appears as a dome in the phase diagram. c Phase diagram of Cux TiSe2 (X-ray) [32]. CDW phase changes to incommensurate near the onset of SC and the SC dome exists entirely inside the ICDW regime. d Phase diagram of pressurised TiSe2 (X-ray) [33]. Above the SC dome, the CDW exhibits an incommensurate nature

revisited by Kogar et al. with a X-ray probing of the CDW phase and qualitatively similar results were found [32]. Near the arising of SC order, CDW also changes to incommensurate and persists to very high doping level. The SC dome exists entirely within the ICDW phase. These experiments thus provide strong indications of the coexistence of CDW and SC orders in the system and make TiSe2 an ideal platform to study this multi-order phenomenon that, although with differences and individual specificities, is an important and appealing scenario due to its prevalence across a number of layered electronic systems, including high temperature superconductors (Fig. 1.4).

1.3 Brief History of the Theoretical Understanding of the CDW Transition in TiSe2

7

1.3 Brief History of the Theoretical Understanding of the CDW Transition in TiSe2 1.3.1 Electronic Structure The nature of the interplay between CDW and SC in TiSe2 is an open problem that has, and continues to, attract considerable interest, both experimentally and theoretically. Since experimentally it is well established that SC emerges within the CDW phase, and that the CDW persists in the SC phase [32–34], understanding the nature of SC requires a robust microscopic understanding of the causes of the CDW in the first place, which is the necessary reference state for the electrons above the Tsc . Since the discovery of the CDW phase transition in TiSe2 , there have been many theoretical studies trying to characterize the underlying driving force. To begin with, one needs to know the electronic band structure. From ARPES experiments, it is difficult to conclude whether the system is a semimetal with a small band overlap [25] or a semiconductor with a small indirect band gap [35]. This difficulty arises from the extremely small occupation of the conduction band in the normal state, combined with the fact that the transition takes place at too high temperatures to allow the necessary resolution in the ARPES spectra. This is illustrated in Fig. 1.6 that show spectra whose interpretation lead the authors to opposing conclusions. On the other hand, according to ab initio density functional theory (DFT) calculations, the system is found to be a semimetal with a band overlap between two hole-like valence bands near the point of the BZ and three electron-like conduction bands near the L (M) point for bulk (monolayer) [36, 37], see Fig. 1.7 A schematic of the Fermi surface is shown in Fig. 1.3d, it is clear that the nesting condition is poor near the Fermi energy. In most of the existing literature, the semimetallic picture of the normal state has been mostly accepted and used as a starting point for the study of CDW transition in TiSe2 [22, 37, 38].

1.3.2 Electron-Phonon Coupling and Electron-Electron Correlation in TiSe2 Given that there is no Fermi surface nesting and the existence of a robust PLD in the CDW phase, it is reasonable to believe that a strong EPC should play the dominant role in the CDW transition. Motizuki et al. found that, after taking into account the EPC, the effective ion-ion interaction can indeed lead to a complete softening of the L− 1 (1) phonon mode [40]. Based on ab initio calculations, Calandra and Mauri were able to show that the pressure dependence of Tc can be well explained by only taking into account the EPC, and the dome shape of the SC order in the phase diagram can also be qualitatively described by EPC using the McMillan formula [41]. However, assumptions regarding the functionals used in these calculations were challenged

8

1 Introduction

Fig. 1.6 ARPES experiments on TiSe2 . a ARPES experiment done by Cercellier et al. [25]. At T = 250 K > Tc , there is a small overlap between valence and conduction bands, which suggests a semimetallic nature of the normal phase. At T = 65 K < Tc in the CDW phase, the valence Se 4 p band folds to L point and a gap opens between topmost valence band and conduction band. b ARPES done by Kidd et al. suggests that the normal phase is a semiconductor with small indirect band gap [35]

[42], while it was also pointed out that the interpretation of DFT results is sensitive to the level at which electronic correlations are taken into consideration [36]. On the other hand, due to the small overlap between the electron and hole pockets separated by Qcdw in the normal state band structure, it is natural to expect that the Coulomb interaction is poorly screened. Thus, electrons from the conduction band and holes from valence band have a tendency to bind together and form excitons, the condensation of which is equivalent to a CDW phase. The instability of a negative gap semiconductor towards exciton condensation is a rich problem in condensed matter physics, and with a long history [43–46]. Based on a Bardeen–Cooper–Schrieffer (BCS) type of approach, Monney et al. showed that the ARPES data can indeed be fit well by an exitonic condensation mechanism [38]. In view of these two aspects, one major and perennial question has been trying to settle whether electrons are the key actors driving the CDW instability with the phonons as spectators (simply adjusting to a new Born–Oppenheimer energy landscape in the CDW phase, thereby generating a PLD as a secondary effect), or if the opposite is true, with the EPC playing the dominant role. However, this question is somehow ill-posed in the sense that the acoustic phonon spectrum can be correctly

1.3 Brief History of the Theoretical Understanding of the CDW Transition in TiSe2

9

Fig. 1.7 The first calculated ab initio band structure of TiSe2 [39]. There is a band overlap between and L points, which indicates that the normal system should be a semimetal

reproduced only after taking EPC into account, thus the lattice and charge degrees of freedom are intimately coupled to each other. Moreover, it has been difficult even to establish prominence of either, in practice, because both scenarios lead to a state with simultaneous CDW and PLD. Hence, formulating the question in an experimentally addressable way has been a major upfront challenge.

1.3.3 Signatures of Exitonic Condensation Significant progress has been made towards pinpointing excitonic signatures of the CDW transition in TiSe2 with the publication of one of the latest experimental developments in a late 2017 Science paper. Using momentum-resolved electron energy-loss spectroscopy (M-EELS), Kogar et al. were able to identify the energymomentum spectrum of an electronic (plasmon) mode, the softening of which was shown to be compatible with the establishment of an excitonic condensation at Tc [47]. Moreover, the excitonic and phonon modes were observed to be hybridized with each other through EPC (see Fig. 1.8), which provide compelling evidence that excitonic condensation indeed coexists with the PLD. Actually, that excitons and electronic interactions are essential at both the qualitative and quantitative levels has been increasingly better documented by a number of modelling refinements. Cercellier, Monney et al. first showed that the excitonic mechanism alone could account for a number of features observed in the evolution of the ARPES spectrum of undoped TiSe2 through the CDW transition [23, 25, 38, 48]. Based on an simplified quasi-1D

10

1 Introduction

Fig. 1.8 The momentum dependence of the soft plasmon mode in TiSe2 [47]. The plasmon behavior is that of the soft mode of a phase transition, demonstrating the condensation of electron-hole pairs at Tc

model, van Wezel et al. found that, within certain ranges of the parameter space of the model, excitonic condensation can actually enhance the PLD and the excitonic order can exist without the lattice distortion [22]. The spectral transfer above Tc is consistent with the ARPES observation of a “renormalised” band structure before CDW transition. Therefore, the outstanding problem to be addressed is not longer whether excitonic physics plays a role or not, but how much it does.

1.4 Recent Experimental Developments on Two-Dimensional TiSe2 Because of their layered structure, many TMDs have been extensively studied in their 2D (monolayer) form. For many compounds exhibiting CDW phases, the CDW order was found to be suppressed when the dimensionality was reduced from 3D to 2D [9, 49]. However, opposite to this tendency, the CDW order was observed to be enhanced in monolayer TiSe2 . According to the ARPES experiment by Chen et al. , it was found that the CDW Tc can actually increase from ∼ 200 K in bulk to ∼ 230 K in the monolayer limit, and the fitting of the order parameter versus temperature indicates a second order phase transition [28]. To study the interplay between CDW and SC orders in 2D limit, Li et al. were able to dope few-layer TiSe2 with an ion-gel gate. Similarly to the bulk case, the CDW order is found to be suppressed with doping and a SC dome can be achieved [34]. The scaling of resistance versus temperature indicates that the SC phase is of Berezinskii–Kosterlitz–Thouless (BKT) type, as expected for 2D superconductors [50]. Moreover, a periodic oscillation of magnetoresistance within the SC phase was

1.4 Recent Experimental Developments on Two-Dimensional TiSe2

11

Fig. 1.9 Experimental results on ion-gel doped few layer TiSe2 [34]. a Phase diagram of iongel doped TiSe2 . b The magnetoresistance for a charge-carrier density of 5.9×1014 cm−2 shows periodic oscillations

discovered, which is reminiscent of the Little–Park physics, where the electronic resistance of a superconducting thin-wall cylinder exhibits a periodic oscillation with the magnetic flux piercing it [51]. It was then inferred that the Cooper pairs might move inside a network matrix formed by the ICDW (CCDW domain walls), which encompasses the CCDW domains. The coexistence of CDW and SC phases and the easy tunability of charge carrier density in 2D TMDs have made them better playgrounds to study the interplay between many-body states than conventional HTSC systems. This thesis is devoted to study the interplay between CDW and SC in two-dimensional TiSe2 theoretically (Fig. 1.9).

1.5 Organization of the Thesis This thesis is organized as follows. In Chap. 2, we introduce our study on the CDW transition of TiSe2 . In the first part of this chapter, the mean field theory of the excitonic instability (EI) along with the main results from this calculation will be thoroughly discussed. In the second part, we will present the results from DFT calculations done by Dr. Bahadur Singh, who is one of the authors of Ref. [24]. Combining the results from both methods, this chapter ends with a discussion of the importance of the exitonic instability in the CDW transition of TiSe2 . Since it has been observed experimentally that TiSe2 can become a superconductor under pressure or doping, we then move on to a phenomenological study on the interplay between CDW and SC. In Chap. 3, we will first establish a phenomenolog-

12

1 Introduction

ical Ginzburg–Laudau (G–L) free energy to characterize the CDW phase transition, where the existence of CCDW and incommensurate CDW (ICDW) orders are confirmed. Then, by proposing an an appropriate coupling between CDW and SC order parameters, it will be shown that the SC phase can be enhanced by CDW fluctuations. We show the main results of the phase diagram and real space profile of the CDW and SC orders in Sect. 3.4. After the phenomenological G–L study, we proceed to a microscopic study of the interplay between CDW and SC in Chap. 4. By expanding the CDW order parameter on top of its mean field configuration, we are able to obtain the effective action for amplitude and phase modes of the CDW (CDW fluctuations). By integrating out the CDW fluctuations, we obtain a retarded attractive interaction between the CDW “renormalized” quasiparticles mediated by the CDW amplitude mode. This chapter ends with a numerical calculation of the SC transition and predicted phase diagram of the simplified model, and the main results of Tsc versus chemical potential will be shown in Fig. 4.4. In Chap. 5, we present an experimental magnetoresistance measurement on iongel gated thin film TiSe2 . An anomalous quantum metal (AQM) phase is found intervening the SC and normal metal phases. Inside the AQM phase, there is a crossover between a Bose metal phase where the resistance scales as a power of magnetic field to a vortex quantum creeping (VQC) regime where the resistance depends on the magnetic field exponentially. The underlying CDW background is conjectured to play an important role for the existence of the AQM phase for providing non-uniformity and a natural channel of dissipation. The main results on the phase diagram are shown in Fig. 5.6.

References 1. Peierls RE (1955) Quantum theory of solids. Oxford University Press, Oxford 2. Fröhlich H (1954) On the theory of superconductivity: the one-dimensional case. Proc R Soc A 223:296–305 3. Mahan GD (2013) Many-particle physics. Springer Science & Business Media, New York 4. Su WP, Schrieffer JR, Heeger AJ (1979) Solitons in polyacetylene. Phys Rev Lett 42:1698–1701 5. Su WP, Schrieffer JR, Heeger AJ (1980) Soliton excitations in polyacetylene. Phys Rev B 22:2099–2111 6. Grüner G (1994) Density waves in solids. Addison-Wesley, Boston 7. Di Salvo FJ, Moncton DE, Waszczak JV (1976) Electronic properties and superlattice formation in the semimetal TiSe2 . Phys Rev B 14:4321–4328 8. Han T-RT, Zhou F, Malliakas CD, Duxbury PM, Mahanti SD, Kanatzidis MG, Ruan C-Y (2015) Exploration of metastability and hidden phases in correlated electron crystals visualized by femtosecond optical doping and electron crystallography. Sci Adv 1:e1400173 9. Ugeda MM, Bradley AJ, Zhang Y, Onishi S, Chen Y, Ruan W, Ojeda-Aristizabal C, Ryu H, Edmonds MT, Tsai H-Z et al (2016) Characterization of collective ground states in single-layer NbSe2 . Nat Phys 12:92 10. Comès R, Lambert M, Launois H, Zeller HR (1973) Evidence for a Peierls distortion or a Kohn anomaly in one-dimensional conductors of the type K2 Pt(CN)4 Br0.3 · x H2 O. Phys Rev B 8:571–575

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11. Denoyer F, Comès F, Garito AF, Heeger AJ (1975) X-ray-diffuse-scattering evidence for a phase transition in tetrathiafulvalene tetracyanoquinodimethane (TTF-TCNQ). Phys Rev Lett 35:445–449 12. Monçeau P, Ong NP, Portis AM, Meerschaut A, Rouxel J (1976) Electric field breakdown of charge-density-wave-induced anomalies in NbSe3 . Phys Rev Lett 37:602–606 13. Pouget JP, Hennion B, Escribe-Filippini C, Sato M (1991) Neutron-scattering investigations of the Kohn anomaly and of the phase and amplitude charge-density-wave excitations of the blue bronze K0.3 MoO3 . Phys Rev B 43:8421–8430 14. Su L, Hsu C-H, Lin H, Pereira VM (2017) Charge density waves and the hidden nesting of purple bronze K0.9 Mo6 O17 . Phys Rev Lett 118:257601 15. Chang J, Blackburn E, Holmes AT, Christensen NB, Larsen J, Mesot J, Liang R, Bonn DA, Hardy WN, Watenphul A et al (2012) Direct observation of competition between superconductivity and charge density wave order in YBa2 Cu3 O6.67 . Nat Phys 8:871 16. Ghiringhelli G, Le Tacon M, Minola M, Blanco-Canosa S, Mazzoli C, Brookes NB, De Luca GM, Frano A, Hawthorn DG, He F et al (2012) Long-range incommensurate charge fluctuations in (Y, Nd)Ba2 Cu3 O6+x . Science 337:821–825 17. Fradkin E, Kivelson SA (2012) High-temperature superconductivity: ineluctable complexity. Nat Phys 8:864 18. Fradkin E, Kivelson SA, Tranquada JM (2015) Colloquium: theory of intertwined orders in high temperature superconductors. Rev Mod Phys 87:457–482 19. Moncton DE, Axe JD, DiSalvo FJ (1975) Study of superlattice formation in 2H -NbSe2 and 2H -TaSe2 by neutron scattering. Phys Rev Lett 34:734–737 20. Weber F, Rosenkranz S, Castellan J-P, Osborn R, Hott R, Heid R, Bohnen K-P, Egami T, Said AH, Reznik D (2011) Extended phonon collapse and the origin of the charge-density wave in 2H-NbSe2 . Phys Rev Lett 107:107403 21. Leroux M, Errea I, Le Tacon M, Souliou S-M, Garbarino G, Cario L, Bosak A, Mauri F, Calandra M, Rodière P (2015) Strong anharmonicity induces quantum melting of charge density wave in 2H − NbSe2 under pressure. Phys Rev B 92:140303 22. van Wezel J, Nahai-Williamson P, Saxena SS (2010) Exciton-phonon-driven charge density wave in TiSe2 . Phys Rev B 81:165109 23. Monney G, Monney C, Hildebrand B, Aebi P, Beck H (2015) Impact of electron-hole correlations on the 1T-TiSe2 electronic structure. Phys Rev Lett 114:086402 24. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T -TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602 25. Cercellier H, Monney C, Clerc F, Battaglia C, Despont L, Garnier MG, Beck H, Aebi P, Patthey L, Berger H, Forró L (2007) Evidence for an excitonic insulator phase in 1T-TiSe2 . Phys Rev Lett 99:146403 26. Holt M, Zschack P, Hong H, Chou MY, Chiang TC (2001) X-ray studies of phonon softening in TiSe2 . Phys Rev Lett 86:3799–3802 27. Fang X-Y, Hong H, Chen P, Chiang TC (2017) X-ray study of the charge-density-wave transition in single-layer TiSe2 . Phys Rev B 95:201409 28. Chen P, Chan YH, Fang XY, Zhang Y, Chou MY, Mo SK, Hussain Z, Fedorov AV, Chiang TC (2015) Charge density wave transition in single-layer titanium diselenide. Nat Commun 6:8943 29. Yan S, Iaia D, Morosan E, Fradkin E, Abbamonte P, Madhavan V (2017) Influence of domain walls in the incommensurate charge density wave state of Cu intercalated 1T-TiSe2 . Phys Rev Lett 118:106405 30. Morosan E, Zandbergen HW, Dennis BS, Bos JWG, Onose Y, Klimczuk T, Ramirez AP, Ong NP, Cava RJ (2006) Superconductivity in Cux TiSe2 . Nat Phys 2:544–550 31. Kusmartseva AF, Sipos B, Berger H, Forró L, Tutiš E (2009) Pressure induced superconductivity in pristine 1T-TiSe2 . Phys Rev Lett 103:236401 32. Kogar A, de La Pena GA, Lee S, Fang Y, Sun SX-L, Lioi DB, Karapetrov G, Finkelstein KD, Ruff JPC, Abbamonte P, Rosenkranz S (2017) Observation of a charge density wave incommensuration near the superconducting dome in Cux TiSe2 . Phys Rev Lett 118:027002

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33. Joe YI, Chen XM, Ghaemi P, Finkelstein KD, de La Peña GA, Gan Y, Lee JCT, Yuan S, Geck J, MacDougall GJ, Chiang TC, Cooper SL, Fradkin E, Abbamonte P (2014) Emergence of charge density wave domain walls above the superconducting dome in 1T-TiSe2 . Nat Phys 10:421–425 34. Li LJ, O’Farrell ECT, Loh KP, Eda G, Özyilmaz B, Neto AC (2015) Controlling many-body states by the electric-field effect in a two-dimensional material. Nature 529:185–189 35. Kidd TE, Miller T, Chou MY, Chiang T-C (2002) Electron-hole coupling and the charge density wave transition in TiSe2 . Phys Rev Lett 88:226402 36. Cazzaniga M, Cercellier H, Holzmann M, Monney C, Aebi P, Onida G, Olevano V (2012) Ab initio many-body effects in TiSe2 : a possible excitonic insulator scenario from GW band-shape renormalization. Phys Rev B 85:195111 37. Singh B, Hsu C-H, Tsai W-F, Pereira VM, Lin H (2017) Stable charge density wave phase in a 1T-TiSe2 monolayer. Phys Rev B 95:245136 38. Monney C, Cercellier H, Clerc F, Battaglia C, Schwier EF, Didiot C, Garnier MG, Beck H, Aebi P, Berger H, Forró L, Patthey L (2009) Spontaneous exciton condensation in 1T-TiSe2 : BCS-like approach. Phys Rev B 79:045116 39. Zunger A, Freeman AJ (1978) Band structure and lattice instability of TiSe2 . Phys Rev B 17:1839–1842 40. Motizuki K, Suzuki N, Yoshida Y, Takaoka Y (1981) Role of electron-lattice interaction in lattice dynamics and lattice instability of 1T-TiSe2 . Solid State Commun 40:995–998 41. Calandra M, Mauri F (2011) Charge-density wave and superconducting dome in TiSe2 from electron-phonon interaction. Phys Rev Lett 106:196406 42. Olevano V, Cazzaniga M, Ferri M, Caramella L, Onida G (2014) Comment on “charge-density wave and superconducting dome in TiSe2 from electron-phonon interaction”. Phys Rev Lett 112:049701 43. Keldysh LV, Kopaev YV (1965) Possible instability of semimetallic state toward Coulomb interaction. Sov Phys Solid State USSR 6:2219 44. Jérome D, Rice TM, Kohn W (1967) Excitonic insulator. Phys Rev 158:462–475 45. Halperin BI, Rice TM (1968) Possible anomalies at a semimetal-semiconductor transistion. Rev Mod Phys 40:755–766 46. Allender D, Bray J, Bardeen J (1973) Model for an exciton mechanism of superconductivity. Phys Rev B 7:1020–1029 47. Kogar A, Rak MS, Vig S, Husain AA, Flicker F, Joe YI, Venema L, MacDougall GJ, Chiang TC, Fradkin E, van Wezel J, Abbamonte P (2017) Signatures of exciton condensation in a transition metal dichalcogenide. Science 358:1314 48. Monney C, Battaglia C, Cercellier H, Aebi P, Beck H (2011) Exciton condensation driving the periodic lattice distortion of 1T-TiSe2 . Phys Rev Lett 106:106404 49. Yang Y, Fang S, Fatemi V, Ruhman J, Navarro-Moratalla E, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P (2018) Enhanced superconductivity upon weakening of charge density wave transport in 2H -TaS2 in the two-dimensional limit. Phys Rev B 98:035203 50. Minnhagen P (1987) The two-dimensional Coulomb gas, vortex unbinding, and superfluidsuperconducting films. Rev Mod Phys 59:1001–1066 51. Little WA, Parks RD (1962) Observation of quantum periodicity in the transition temperature of a superconducting cylinder. Phys Rev Lett 9:9–12

Chapter 2

Excitonic Character of CDW in TiSe2

Since the dependence of Tc on electronic density is well known experimentally, we submit that the predicted density dependence of Tc in a description with and without account of the excitonic mechanism should be different. As a result, it provides a direct, well defined means to quantify the importance of excitonic condensation in the transition to the CDW phase in TiSe2 . In this chapter, we demonstrate that the experimental density dependence in Cux TiSe2 cannot be captured without explicitly accounting for electron-electron interactions and the excitonic instability (EI).

2.1 Excitonic Instability (EI) of TiSe2 2.1.1 The EI Hamiltonian We begin by modelling the valence hole pocket as an isotropic paraboloid centred at  point 2 k 2 + bo εvk ≡ − 2m v and three conduction electron pockets at each M point having anisotropic effective mass 2 (k − Mi )2 2 (k − Mi )2⊥ + εck,i ≡ 2m c,⊥ 2m c,

This chapter is taken and edited from Ref. [1] of which the author of this thesis is the first author. All calculations described were carried out by the author. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_2

15

16

2 Excitonic Character of CDW in TiSe2

as indicated in Fig. 1.3d. In the pristine system, the chemical potential (μ) of TiSe2 is placed near the intersection of the conduction and valence pockets, which is in agreement with the folded DFT bandstructure calculated in an unrelaxed 2×2 superlattice [cf. Fig. 2.4a later], and also in line with transport experiments that reveal both electron and hole carriers in the normal state [2, 3]. The band parameters are extracted by fits to ARPES data in Ref. [4] in the normal state and read m v = 0.63 m e , m c,⊥ = 1.38 m e , m c, = 3.46 m e , bo = 0.1 eV (band overlap). Since the bands strongly renormalize near E F and CDW fluctuations are likely present at T  Tc [5], the fitting takes into account large energy ranges above and below the E F , since high energies remain unaltered in the excitonic phase. With these, our normal state electron density is n e ∼ 4×1013 cm−2 , consistent with the experimental Hall data [2]. The Hamiltonian consists of these 4 “bare” bands and a direct Coulomb interaction between electrons from the valence pocket and three conduction pockets: H≡



† εvk ck,σ ck,σ +

k,σ



† εck,i di,k,σ di,k,σ

k,σ,i

1   † † + Vi,q ck+q,σ di,k  −q,σ  di,k ,σ  ck,σ , N i k,k ,q,σ,σ 

(2.1)

Here, ck,σ (di,k,σ ) are annihilation operators for electrons at the valence (i-th conduction) pocket with momentum k (Mi + k) and spin σ , and N is the number of unit cells of the crystal. Notice that the momentum k for a given electron pocket at Mi represents the momentum measured with respect to Mi . In addition, the chemical potential μ is implicit in the electron and hole dispersions εck/vk which are measured with respect to it.

2.1.2 Mean Field Approach to the EI In order to get the mean field Hamiltonian, we first decouple the Coulomb interaction in the particle-hole channel as † di,k ,σ  = δk+q,k δσ,σ  ck†  ,σ di,k ,σ  ck+q,σ   † + ck+q,σ di,k ,σ  − δk+q,k δσ,σ  ck†  ,σ di,k ,σ  † di,k  −q,σ  ck,σ

=

(2.2a)

† δk+q,k δσ,σ  di,k,σ  ck,σ 

  † † + di,k  −q,σ  ck,σ − δk+q,k δσ,σ  di,k,σ ck,σ 

Neglecting the product of fluctuations, the mean field Hamiltonian thus reads

(2.2b)

2.1 Excitonic Instability (EI) of TiSe2

HMF ≡



17

† † εvk ck,σ ck,σ + εck,i di,k,σ di,k,σ

k,σ,i

−



† † ∗ i,k,σ ck,σ di,k,σ − i,k,σ di,k,σ ck,σ

k,σ,i

+

1  † † Vi,k−k ck,σ di,k,σ di,k  ,σ ck ,σ , N i,σ k,k

(2.3)

where the order parameter i,k,σ ≡

1  † Vi,k−k di,k  ,σ ck ,σ  N k

(2.4)

gives a measure of the Fourier component at wavevector Qcdw of the charge density. Since all terms in Eq. (2.3) are spin diagonal, the spin indices will be suppressed from now on, but are implicitly included in the final result. The theory is formulated in terms of the equation of motion method. Beginning with the definition of normal and anomalous Matsubara Green functions Gv (τ, k) = −T ck (τ )ck† (0),

(2.5a)

† D j,i (τ, k) = −T d j,k (τ )di,k (0), † Fi (τ, k) = −T di,k (τ )ck (0),

(2.5b) (2.5c)

it is straightforward to show that they obey the following equations of motion (in Matsubara frequency (ωn ) space)  (iωn − εvk ) Gv (ωn , k) + i i Fi (ωn , k) = 1,   iωn − εck,i Fi (ωn , k) + i∗ Gv (ωn , k) = 0,   iωn − εck, j D j,i (ωn , k) + ∗j Fi† (ωn , k) = δ j,i ,  (iωn − εvk ) Fi† (ωn , k) + j  j D j,i (ωn , k) = 0.

(2.6a) (2.6b) (2.6c) (2.6d)

There are 16 coupled equations in the set Eq. (2.6) since i, j ∈ {1, 2, 3}, which are to be solved self-consistently to obtain the order parameter i,k =

kB T  + Vi,k−k eiωn 0 Fi† (ωn , k ). N k ,ω

(2.7)

n

In this weak-coupling calculation, the q-dependence in Vi,q is neglected. Combining this with the fact that both the CCDW phase and the original band structure exhibit C3 symmetry, both indices k and i can be dropped for Vi,q , i.e.,  ≡ i,k . According to Eq. (2.6), one can obtain the gap equation of the CDW order parameter

18

2 Excitonic Character of CDW in TiSe2

+

A V kB T  = 0, N k ,ω ||2 B − C

(2.8)

n

with A ≡ (iωn − εc,2 )(iωn − εc,3 ), B ≡ (iωn − εc,2 )(iωn − εc,3 ) + (iωn − εc,1 )(iωn − εc,3 ) + (iωn − εc,1 )(iωn − εc,2 ), C ≡ (iωn − εc,1 )(iωn − εc,2 )(iωn − εc,3 )(iωn − εv ), where all the εc,i and εv are evaluated at k . By solving the self-consistent gap equation, one can identify the CDW transition temperature Tc , and obtain the EI phase diagram as a function of doping and temperature. According to the definition in Eq. (2.4) and the approximations indicated above, the order parameter  is directly related to the amplitude of the CDW at the wavevectors Q(i) cdw . It is worth noting that Eq. (2.8) can also be obtained by minimizing the free energy functional of the order parameter F [], which is usually formulated within a path-integral formalism [6]. In other words, Eq. (2.8) is equivalent to ∂ F [] /∂ = 0, which will be a useful identification below to establish the order of the transition to the excitonic/CDW phase.

2.1.3 Phase Diagram The phase diagram (as a function of doping and temperature) predicted by the solution of Eq. (2.8), along with the experimental result [3], is shown in Fig. 2.1. It is one of our fundamental results. It can be clearly seen that: (i) doping will suppress the CDW phase and the decreasing trend from x = 0 follows very well the experimental behavior until x ≈ 0.038; (ii) the calculation predicts a quantum critical point (QCP) at precisely the doping where the SC phase starts to emerge experimentally (x ≈ 0.04); (iii) the EI transition is of 2nd order until x ≈ 0.038, then changes to 1st order beyond that doping level, which correlates with the onset doping of discommensurations or ICDW observed in recent experiments [7]. Our curve Tc (x) was technically obtained by solving Eq. (2.8) at different chemical potentials (measured from the bottom of the conduction bands); these were then converted to electronic densities and doping level x where, as we explicitly demonstrate ab initio below, x is directly mapped to the number of electrons per unit cell. Examples of calculated Tc curves at different μ can be seen in Fig. 2.2a, which also shows the effect of varying the overlap bo between the bare electron and hole pockets. It is important to highlight that, having set all the bare band parameters (including bo ) from ARPES data as described earlier, our theory of the exitonic instability depends only on one parameter: the coupling constant V . By setting it at 450 meV, we are able to match the highest value of Tc to the experimental value ≈ 220 K reported in Ref. [3] [cf. Fig. 2.1

2.1 Excitonic Instability (EI) of TiSe2

19

Fig. 2.1 Comparison between experimental and theoretical phase diagrams for the CDW transition in Cux TiSe2 . The blue circles represent Tc obtained from the self-consistent solution of Eq. (2.8) which includes only the excitonic mechanism. Its reduction with doping follows the experimental trend with very good qualitative and quantitative agreement up to x ≈ 0.038. At this doping, the transition becomes of 1st order in our calculation which approximately coincides with the experimental observation of ICDW signatures and the onset of the SC dome at x ≈ 0.04 [7]. The strength of the coupling is our only adjustable parameter, and it was fixed at V = 450 meV in order to yield Tc = 220 K without doping. The red diamonds show the variation in the critical smearing parameter (σc ) above which the dynamical phonon instability disappears in the DFPT [8] calculations (note the break in the horizontal axis). σc remains nearly unaltered in the experimentally relevant range of x, and only drops to zero beyond x ≈ 0.2, suggesting that, without the adequate account of the electronic interactions, the phonon calculation predicts the PLD to be stable up to unrealistically high doping. The experimental data for Tc (open squares) and Tsc (open circles) are those reported by Morosan et al. [3]. This figure is taken from Ref. [1]. © 2019 American Physical Society

and the horizontal dashed line in Fig. 2.2a], and this was thus the value used in all our calculations. With V thus fixed at x = 0, the results for Tc at different x follow without any further parameters. Moreover, at x = 0 we have (T = 0) ≈ 25 meV [cf. Fig. 2.2b] which agrees reasonably well with ARPES data of the renormalized bandstructure in the CDW phase [9]. The magnitude of the band overlap is quantitatively important but not qualitatively, as can be seen from Fig. 2.2. When bo is reduced by an order of magnitude, the transition temperature Tc only decreases by 50%. This fact makes sense physically because the EI is governed primarily, not by the number of charge carriers, but by the density of states (DOS) near the region where valence and conduction bands

20

2 Excitonic Character of CDW in TiSe2

intersect. Since the CDW gap opens at that intersection (not at the chemical potential μ), the CDW phase is most favourable when μ coincides with the band intersection. The transition temperature will be suppressed by either electron or hole doping in that case. The intersection point is indicated by the vertical dashed lines in Fig. 2.2a, which correlates with the optimal Tc obtained from the self-consistent solution of the coupled mean field equations. This characteristic of the EI transition can actually be used as an experimental confirmation of the strong excitonic character of the CDW transition. Therefore, one can probe this experimentally by looking at whether an optimal Tc exists with either hole or electron doping, and see if it is related to the band intersection. Figure 2.2b–d show the temperature dependence of the CDW order parameter. At low doping level, it is clear that the transition is of second order, Fig. 2.2b, c, while at higher doping (x  0.038) it changes to first order. The fact that the phase transition changes to 1st order can be understood as an indication of phase separation at x  0.038, or there might be some regions where the fluctuation of commensurate CDW is strong, it is thus reasonable to expect to observe the CDW discommensurations in real space at high doping levels [10]. Since the nesting condition is poor in TiSe2 , the “renormalised” band structure in the CDW phase is only partially gapped. Thus the resistance will be enhanced near transition temperature but the system still holds a metallic transport character at low temperature, which is consistent with previous transport experiments [2, 3]. Moreover, the quasiparticles within the CDW “renormalised” conduction bands might participate the superconducting pairing at high doping levels, which gives rise to a co-existence of CDW and superconductivity, which is a view in agreement with recent experimental findings [11]. The calculations described so far rely only based on electron-electron interaction and indicate that the exitonic mechanism is able to capture quantitatively well the doping dependence of Tc and many other qualitative aspects of CDW phase (Fig. 2.3). Finally, we want to point out that the self-consistent gap equation we get from our mean field treatment is formally the same as the one arising from a model with only EPC included. Actually, it can be shown that Eq. (2.8) is formally unchanged if one includes both electron-electron interaction and EPC with Qcdw in the Hamiltonian at the beginning (more details can be found in Appendix A). However, since it has already been shown that the presence of excitonic condensation will lead to a PLD accompanying it with an displacement amplitude similar to experimental value [12], including both types of effect at the mean field level seems to be unnecessary and it would only increase the number of parameters in the theory. Meanwhile, it is natural to expect that electron-electron interactions should play a prominent role due to the small carrier density. It thus suggests that although excitons and phonons will couple to each other, the CDW transition has a strong excitonic character. In order to assess the importance of electron-electron interaction in the description of the CDW transition, we investigate next the predictions that follow in the absence of it. For that, DFT calculations of phonon softening and TiSe2 band structure with Cu doping were performed, which will be discussed in the next section.

2.2 DFT Study of the Lattice Instability

21

(b)

300

bo bo bo

250

= 0.01 eV = 0.1 eV = 0.13 eV

25 20

Δ (meV)

(a)

15 10 5

(c)

200

Δ (meV)

Tc (K)

0

150

−0.02

Δ (meV)

50

0

0.02

0.04

µ (eV)

0.06

100 150 T (K)

200

250

50

100 150 T (K)

200

250

F [Δ]

15 10

F [Δ]

5 0

(d)

50

25 20

100

0

0

0

20 15

F [Δ]

10 F [Δ]

5

0.08

0

0

20

40

60 80 100 120 140 T (K)

Fig. 2.2 The CDW phase transition in TiSe2 according to the EI alone. a Critical temperature (Tc ) as a function of chemical potential (μ) for different levels of overlap between the electron and hole pockets, bo . Zero in the horizontal scale corresponds to μ coinciding with the bottom of conduction pockets. The vertical dashed lines indicate the energies where the bare electron and hole pockets intersect for each case. It is clear that, while small variations in μ tend to quickly reduce Tc , variations in the band overlap have a less significant effect. b–d Temperature dependence of the excitonic order parameter, (T ), at different doping: b x = 0, c x = 0.003, and d x = 0.042. The top (bottom) insets in (c) and (d) show the behavior of the free energy as a function of order parameter for T > Tc (T < Tc ), and illustrate that the transition becomes of first order in our calculation beyond x  0.038. This figure is taken from Ref. [1]. © 2019 American Physical Society

2.2 DFT Study of the Lattice Instability The DFT calculations were performed by Dr. Bahadur Singh as part of this project’s collaboration and are reported in Ref. [1], of which the author is the first author. As they provide a very important complementary perspective and are important to assess the importance of the excitonic mechanism, I describe here its main results and their connection with the preceding discussion.

2.2.1 Isolating the Effect of Cu Doping The ab initio calculations were done within the DFT [13] framework with the projecting augmented wave method implemented in the Vienna Ab initio Simulation

22

(b)

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

E (eV)

E (eV)

(a)

2 Excitonic Character of CDW in TiSe2

0 −0.1

0 −0.1

−0.2

−0.2

−0.3

−0.3

−0.4 −0.4 −0.3 −0.2 −0.1

0

0.1

0.2

0.3

−0.4 −0.4 −0.3 −0.2 −0.1

0.4

ky (A−1 )

(c)

(d)

0.2

0.1

0.2

0.3

0.4

0.2

0.1

E (eV)

E (eV)

0.1

0

−0.1

−0.2 −0.2

0

ky (A−1 )

0

−0.1

−0.1

0

0.1 −1

ky (A

)

0.2

−0.2 −0.2

−0.1

0

0.1

0.2

ky (A−1 )

Fig. 2.3 Bare and renormalized (folded) bandstructure in the excitonic phase. a and b show, respectively, the bare and the renormalized bands for bo = 0.1 eV. The insets show the corresponding Fermi surfaces with the dashed circle marking the hole pocket. c and d show the equivalent representation at bo = 0.01 eV. The curves are plotted in the reduced Brillouin zone as a function of k y near the folded  point. The chemical potential (μ, dashed gray line at E = 0) is set at the value that corresponds to the highest transition temperature for each case. This figure is taken from Ref. [1]. © 2019 American Physical Society

Package (VASP) [14, 15]. The generalized gradient approximation (GGA) for the exchange-correlation functional was used and spin-orbit coupling was was taken into account. The effects of additional carriers in TiSe2 were investigated with two complementary strategies. First, we explicitly studied the doping induced by Cu by simulating supercells with adsorbed Cu. Subsequently, for the systematic study of the phonon instabilities, electron (hole) doping was considered by adding (removing) electrons to the unit cell, with a neutralizing uniform background charge. The internal atomic positions in the unit cell were relaxed for each doping while keeping the lattice parameter fixed at its undoped value. For the study of Cu-intercalated TiSe2 ,

2.2 DFT Study of the Lattice Instability

23

Cu atoms were placed directly above and below the central Ti in a 2×2 supercell [see Fig. 1.3a–b]. All the atoms were allowed to freely relax inside the unit cell. Interestingly, despite unconstrained, the Cu atoms adsorb onto the surface without bouncing back into the vacuum with an adsorption energy of E ads = 2.456 eV/Cu, where E ads = −[Er el (Cux TiSe2 ) − Er el (TiSe2 ) − E(Cu)]. The structural stability of these Cu-adsorbed TiSe2 monolayers was further scrutinized by calculating the phonon spectrum. The band structure in the normal phase consists of one Se p-derived hole pocket at  point and three Ti d-derived electron pockets at M point, with overlapping between them. Because the pockets are related by Qcdw , in a 2 × 2 superlattice representation, they will fold back to  point of the reduced Brillouin zone as shown in Fig. 2.4(a). The Fermi energy (E F ) is slightly below the intersection between hole and electron pockets, as required by the charge neutrality condition. It should be noted that the band overlap is likely to be slightly enlarged because of the problem with local functionals within DFT. If the ions are fixed, these bands do not interact and resort to their representative primitive Brillouin zone (BZ) as shown in the unfolded band structure in Fig. 2.4(b). When the ions are free to relax, a distorted band structure with PLD was achieved. It is clear that the hybridization of overlapping pockets leads to a lowering of occupied states and a raising of unoccupied states near E F , with a gap emerging (E g = 82 meV and 325 meV within GGA and HSE approximation). Also, there is a restructuring of the band structure near E F , as shown in Fig. 2.4(c)–(d) [see also Fig. B.1]. The maxima of two topmost valence bands move away from  point and form a circle, and results in a inverted Mexican hat shape. Figure 2.4(e) shows the band structure in the reduced BZ of a fully optimized 2 × 2 supercell with two Cu atoms (one above and one below the slab in order to maintain the symmetry). It is obvious that the addition of Cu atoms increases E F and restores the partial overlap between electron and hole pockets. The unfolded band structure in Fig. 2.4 resembles that of Fig. 2.4 with E F shifted to higher energy, which means that the system at this doping level is still a semimetal with a rigid up shift of the Fermi energy. This is further demonstrated by Fig. 2.4(g)–(h), which show that, after Cu doping, the hole pocket shrinks while the electron pockets get enlarged. To highlight, there are two crucial aspects of the effect of Cu doping. First, a careful inspection of the bands in Figs. 2.4(c–e) shows that it does not remove the nontrivial restructuring of the dispersion that occurs at the electron-hole intersection of the pristine monolayer. Figure 2.4(i) emphasizes this observation by placing the undoped and doped bandstructures near E F side by side, and makes explicit the fact that, after Cu doping, E F is rigidly raised by 0.37 eV but the Mexican hat shape is preserved. Second, an analysis of atomic relaxations further reveals that the Cu doping nullifies the large atomic relaxation amplitudes observed in the distorted state of the undoped system and thus suppresses the PLD in TiSe2 (Fig. B.2). These two aspects justify the assumptions related to the density-dependence and bandstructure used in the many-body calculations described in Sect. 2.1.2. In addition, this picture agrees entirely with recent STM measurements showing that the gap in the CDW

24

2 Excitonic Character of CDW in TiSe2

(a)

(c)

(e)

(b)

(d)

(f)

(g)

(h)

(i)

Fig. 2.4 Effect of the PLD and Cu doping on the electronic structure of TiSe2 . Shown are the band structure in the unrelaxed, normal phase (a, b), in the relaxed distorted phase (c, d), and in the relaxed phase with Cu doping (e, f). Each column plots the bands in the 2×2 reduced Brillouin zone of the distorted phase (top) and the corresponding unfolded version in the BZ of the undistorted phase (bottom). The bands obtained without lattice relaxation in (a) have the Ti-derived electron pocket folded into the  point without any restructuring or gap opening; they are restored back to the M point in the unfolded representation (b). The situation is quite different when the lattice is allowed to relax on a 2×2 supercell: in addition to the spontaneous lattice distortion, we observe a finite band gap and the appearance of two back-folded bands at the M point (c, d). Adsorbed Cu atoms electronically dope the system and raise the Fermi level higher in the conduction bands without noticeable disruption to the overall dispersion (e, f). The calculated energy contours at E F before (g) and after (h) show larger (smaller) electron (hole) pockets induced by Cu doping. Panels (e–f) and (g–h) indicate that Cu substitution adds electrons to the system in a way that amounts to a simple increase in E F . This conclusion is further confirmed in panel (i) that shows a side-by-side closeup of (c) and (e). This figure is taken from Ref. [1]

2.2 DFT Study of the Lattice Instability

25

phase of Cux TiSe2 appears below E F and moves to higher binding energies with increasing Cu doping [11]. Finally, it should be noted that the analysis above about Cu doping corresponds to a doping level at 50%, which is extremely high compared with the experimental solubility limit (11%) [3, 16]. Despite such high doping, our results provide clear evidence that the leading effect of Cu doping is to donate charge carriers (electrons) to the electronic band, with a rigid shift of E F . Thus it is natural to expect this simple shift of E F to remain the only band structure of significance at the much smaller doping levels covered in the intercalation experiments (below 10%). Therefore, in order to scrutinize in detail the phonon instability at small doping, we resort to the second doping strategy discussed above (which would otherwise require prohibitively large supercells in the DFT and phonon calculations if done with actual cells reflecting those concentrations).

2.2.2 Phonon Softening with Cu Doping The low-symmetry CDW phase is considered to be the stable configuration of the high symmetry phase in response to a modified Born–Oppenheimer potential. Therefore, soft phonon mode anomalies in the vibrational spectrum of the normal phase can be directly related to the lowering of energy and the PLD that is stabilized at low temperatures. The phonon spectrum was calculated using the density functional perturbation theory (DFPT). Figure 2.5 presents the phonon energy dispersion of undoped TiSe2 in the normal (1 × 1) phase. The qualitative effect of temperature is studied by changing the electron smearing parameter σ , which is normally used as a technical tool in ab-initio calculations to accelerate the convergence (it acquires the physical meaning of electronic temperature only when used in conjunction with finite temperature smearing methods [17–19]). The dependence of soft phonon mode on σ is commonly used to check the qualitative tendency of real phonon spectrum as temperature changes. A soft phonon mode with imaginary frequencies (represented as negative frequencies) for k enclosing M point can be seen at the smallest smearing (σ = 0.1 eV). This is a tell-tale sign of a instability of TiSe2 towards a 2 × 2 PLD below Tc . The fraction of BZ associated with imaginary frequencies shrinks as σ increases and disappears above a critical value σc ∼ 0.4 − 0.5 eV. It is interesting to see that only one acoustic phonon mode is sensitive to the smearing parameter and get softened at small σ while no significant changes occur to other modes, as also seen in experiment [20]. It is also important to notice that the precise relationship between σ and T depends on the smearing strategy used in the calculation. In addition, the true physical temperature includes not only the contribution from electrons but also the one from phonons. Therefore, σc can only be safely used as an indication of whether there is a finite Tc or not for CDW transition, instead of the magnitude of Tc . In other words, one can predict whether a CDW/PLD occurs at a certain doping level from σc . In order to scrutinize the effect of uniformly adding charge carriers to the system, we studied the phonon spectrum with different concentrations of electrons in the unit

26

2 Excitonic Character of CDW in TiSe2

(a)

(b)

(d)

(c)

Fig. 2.5 Calculated phonon modes and the suppression of the PLD in the TiSe2 monolayer with temperature and doping. a Phonon spectrum of the 1×1 normal phase with different electronic smearing parameter σ (undoped). The hardening of the soft phonon at M with increased σ indicates a structural phase transition at a finite critical temperature. b The soft mode along the relevant high-symmetry directions for different σ (see legends) and fixed doping of x = 0.04 electrons per formula unit (e/fu). c Same as (b) but for different electronic doping at fixed σ = 0.01 eV. Legends show x in e/fu. d Evolution of the soft mode frequency at the M point as a function of σ (abscissas) and x (legends). This figure is taken from Ref. [1]. © 2019 American Physical Society

cell. Figure 2.5c shows the evolution of the soft phonon mode at different doping x (measured in electrons per formula unit, FU), at small σ . The imaginary frequencies decrease gradually and disappear at xc ∼ 0.18 − 0.2. To analyse how σc evolves with doping, we calculated the phonon spectrum over a range of σ for each doping. It can be seen from Fig. 2.5b that the imaginary frequency disappears for σ > 0.4 eV with doping level x = 0.04. As this critical smearing parameter is very similar to that of the pristine case, it suggests that the PLD at this doping level is as robust as in the

2.2 DFT Study of the Lattice Instability

27

pristine case, which is different from experiment [3]. The phonon frequency at the M point is shown as a function of doping x and smearing σ , and plotted in Fig. 2.5. It is clear that σc gets suppressed as more electrons are doped into the system and the frequency becomes real and less dependent on σ above a critical doping level xc ∼ 0.18 − 0.2. The evolution of σc with doping is plotted as red diamonds in Fig. 2.1. This conclusion is robust with respect to the smearing methods used in the calculation (see Appendix B.3 for more details).

2.3 Discussion This chapter has described two rather different types of calculations. However, they provide a self-contained theoretical description of the influence of both temperature and doping on the CDW phase transition. The DFT+DFPT calculation studied the lattice instability from both ground state energy and phonon softening sides, and demonstrates that the experimental CDW phase diagram of TiSe2 cannot be described quantitatively. Although it captures the fact that the Tc will be suppressed as the system is doped, it requires a significantly higher doping level to fully remove the CDW phase. On the other hand, the self-consistent mean field calculation of the EI based on the electron-electron interaction predicts a doping dependence of Tc which agrees quantitatively well with experiment. Since the band parameters are fitted from ARPES experiments and the interaction coupling constant V is fixed once and for all in the undoped case, such an agreement with experimental results is significant. Such a good agreement with experiment from a mean field calculation can be understood from three perspectives: (i) in the commensurate CDW phase, both amplitude and phase modes are gapped [21], thus the fluctuations are more difficult to excite; (ii) since the transition temperature Tc is relatively high at low doping level, the quantum fluctuations are expected to damp out [6]; (iii) the Mexican hat feature of the “renormalized” band structure and the fact that the spectral weight transfer affects only relatively small energy scales indicate that the physics should be well described by a weak coupling theory [22]. We attribute the contrast between the results of the two approaches to the fact that, although DFT+DFPT captures the electron-phonon coupling and, at an approximate level, the electron-electron interaction, it does not capture the excitonic physics and thus the EI in TiSe2 . Because of the very low density of charge carriers and the semimetallic nature of the system, DFT calculations cannot describe the phonon softening if the treatment of exchange and correlation do not properly capture this physics. Such a sensitivity to the details of electron interaction has been pointed out by several previous studies which show that the stability of PLD and “renormalized” band structure depend strongly on the type of exchange and correlation functional, the usage of a local or non-local density approximation and quasiparticle corrections [17, 23–26]. As a result, our results suggest that excitons play an essential role in the microscopic description of the CDW/PLD transition, and that the electron-phonon

28

2 Excitonic Character of CDW in TiSe2

coupling is expected to be strongly renormalized by their interplay with the lattice. In this regard, Monney et al. have shown that, given a finite CDW order parameter (T ) which fits the experimental result, combining the exciton condensate with EPC leads to a PLD with correct order of magnitude in the lattice displacement of undoped TiSe2 [12]. Although their calculation is not fully self-consistent and only establishes how the lattice softens in response to the exciton condensate, the result does reinforce the strong influence that the EI can have in the stability of the lattice. Furthermore, incorporating our self-consistent calculation of  in the (T, x) parameter space Fig. 2.2 into such calculations, provides a complete picture of the CDW/PLD in TiSe2 where the driving physics would arise from electron-electron interactions. This would not be surprising in itself because electronic interactions are known to be sufficient to stabilize robust CDW phases even in the absence of a PLD, particularly in reduced dimensions [21, 27–30]. However, it would be incorrect to consider phonons as simple spectators because, on the one hand, self-consistency involving both electronelectron and electron-phonon interaction is expected to further stabilize the PLD; on the other hand, the latest experimental evidence shows that this system carries hybrid electronic and lattice elementary excitations close to Tc [31]. In conclusion, we believe that our results contribute in a clear way towards establishing theoretically the indispensable contribution and celebrated role of the excitonic mechanism in the CDW phase of TiSe2 . In addition, despite the long history of TiSe2 as an interesting and challenging condensed matter system, the theoretical results plotted in Fig. 2.1 constitute the first self-consistent, and self-contained, reproduction of the experimental weakening of the commensurate CDW phase with doping. It is noteworthy how this can be achieved with a single tunable interaction parameter, and at a mean field level. A detailed study of the effects and new implications of CDW fluctuations to superconductivity in TiSe2 is deferred to Chap. 4.

References 1. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T -TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602 2. Di Salvo FJ, Moncton DE, Waszczak JV (1976) Electronic properties and superlattice formation in the semimetal TiSe2 . Phys Rev B 14:4321–4328 3. Morosan E, Zandbergen HW, Dennis BS, Bos JWG, Onose Y, Klimczuk T, Ramirez AP, Ong NP, Cava RJ (2006) Superconductivity in Cux TiSe2 . Nat Phys 2:544–550 4. Chen P, Chan YH, Fang XY, Zhang Y, Chou MY, Mo SK, Hussain Z, Fedorov AV, Chiang TC (2015) Charge density wave transition in single-layer titanium diselenide. Nat Commun 6:8943 5. Monney C, Cercellier H, Clerc F, Battaglia C, Schwier EF, Didiot C, Garnier MG, Beck H, Aebi P, Berger H, Forró L, Patthey L (2009) Spontaneous exciton condensation in 1T-TiSe2 : BCS-like approach. Phys Rev B 79:045116 6. Altland A, Simons BD (2010) Condensed matter field theory. Cambridge University Press, Cambridge 7. Kogar A, de la Pena GA, Lee S, Fang Y, Sun SX-L, Lioi DB, Karapetrov G, Finkelstein KD, Ruff JPC, Abbamonte P, Rosenkranz S (2017) Observation of a charge density wave incommensuration near the superconducting dome in Cux TiSe2 . Phys Rev Lett 118:027002

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8. Baroni S, de Gironcoli S, Dal Corso A, Giannozzi P (2001) Phonons and related crystal properties from density-functional perturbation theory. Rev Mod Phys 73:515–562 9. Cercellier H, Monney C, Clerc F, Battaglia C, Despont L, Garnier MG, Beck H, Aebi P, Patthey L, Berger H, Forró L (2007) Evidence for an excitonic insulator phase in 1T-TiSe2 . Phys Rev Lett 99:146403 10. Li LJ, O’Farrell ECT, Loh KP, Eda G, Özyilmaz B, Castro Neto AH (2015) Controlling manybody states by the electric-field effect in a two-dimensional material. Nature 529:185–189 11. Spera M, Scarfato A, Giannini E, Renner C (2017) Electron hole complementary spatial texturing of the charge order in a doped commensurate charge density wave. arXiv:1710.04096 [cond–mat.str–el] 12. Monney C, Battaglia C, Cercellier H, Aebi P, Beck H (2011) Exciton condensation driving the periodic lattice distortion of 1T-TiSe2 . Phys Rev Lett 106:106404 13. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864–B871 14. Kresse G, Furthmüller J (1996) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 54:11169–11186 15. Kresse G, Joubert D (1999) From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B 59:1758–1775 16. Wu G, Yang HX, Zhao L, Luo XG, Wu T, Wang GY, Chen XH (2007) Transport properties of single-crystalline Cux TiSe2 (0.015  x  0.110). Phys Rev B 76:024513 17. Singh B, Hsu C-H, Tsai W-F, Pereira VM, Lin H (2017) Stable charge density wave phase in a 1T-TiSe2 monolayer. Phys Rev B 95:245136 18. Mermin ND (1965) Thermal properties of the inhomogeneous electron gas. Phys Rev 137:A1441–A1443 19. Duong DL, Burghard M, Schön JC (2015) Ab initio computation of the transition temperature of the charge density wave transition in TiSe2 . Phys Rev B 92:245131 20. Holt M, Zschack P, Hong H, Chou MY, Chiang T-C (2001) X-ray studies of phonon softening in TiSe2 . Phys Rev Lett 86:3799–3802 21. Grüner G (1994) Density waves in solids. Addison-Wesley, Boston 22. Phillips P (2012) Advanced solid state physics, 2nd edn. Cambridge University Press, Cambridge 23. Cazzaniga M, Cercellier H, Holzmann M, Monney C, Aebi P, Onida G, Olevano V (2012) Ab initio many-body effects in TiSe2 : A possible excitonic insulator scenario from GW band-shape renormalization. Phys Rev B 85:195111 24. Olevano V, Cazzaniga M, Ferri M, Caramella L, Onida G (2014) Charge-density wave and superconducting dome in TiSe2 from electron-phonon interaction. Phys Rev Lett 112:049701 25. Calandra M, Mauri F (2011) Charge-density wave and superconducting dome in TiSe2 from electron-phonon interaction. Phys Rev Lett 106:196406 26. Hellgren M, Baima J, Bianco R, Calandra M, Mauri F, Wirtz L (2017) Critical role of the exchange interaction for the electronic structure and charge-density-wave formation in TiSe2 . Phys Rev Lett 119:176401 27. Su L, Hsu C-H, Lin H, Pereira VM (2017) Charge density waves and the hidden nesting of purple bronze K0.9 Mo_6O17 . Phys Rev Lett 118:257601 28. Wise WD, Boyer MC, Chatterjee K, Kondo T, Takeuchi T, Ikuta H, Wang Y, Hudson EW (2008) Charge-density-wave origin of cuprate checkerboard visualized by scanning tunnelling microscopy. Nat Phys 4:696 29. Mou D, Sapkota A, Kung H-H, Krapivin V, Wu Y, Kreyssig A, Zhou X, Goldman AI, Blumberg G, Flint R, Kaminski A, Mou D, Goldman AI, Kreyssig A, Wu Y, Flint R (2016) Discovery of an unconventional charge density wave at the surface of K0.9 Mo6 O17 . Phys Rev Lett 116:196401 30. Chen C-W, Choe J, Morosan E (2016) Charge density waves in strongly correlated electron systems. Rep Prog Phys 79:084505 31. Kogar A, Rak MS, Vig S, Husain AA, Flicker F, Joe YI, Venema L, MacDougall GJ, Chiang TC, Fradkin E, van Wezel J, Abbamonte P (2017) Signatures of exciton condensation in a transition metal dichalcogenide. Science 358:1314

Chapter 3

Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

3.1 The CDW Is Not Alone It is always of great research interest to explore the possibility of SC in a material. In 2006, E. Morosan et al. discovered that, by intercalating Cu atoms in TiSe2 (Cux TiSe2 ), the CDW transition temperature decreases and the system could become a superconductor at doping level x = 0.04. The SC phase appears as a dome shape in the phase diagram as shown in Fig. 1.5 [2]. Their discovery boosted the research interest on this “well-known” TMD material. Since then, plenty of studies have been made from the perspective of both identifying the driving force of the CDW and charactering the properties of the SC phase. Soon after the discovery of SC order in Cux TiSe2 , Kusmartseva et al. found that a similar type of phase diagram containing both CDW and SC orders can also be obtained in pressurized TiSe2 [3]. ARPES measurements reveal that upon Cu doping the (“renormalized”) valence band gets quickly occupied, thus the making SC order mainly contributed by electrons within the conduction bands; a multi-band (between valence and conduction) pairing is hence ruled out [4]. According to the fact that the thermal conductivity does not have a T -linear residual term in the T → 0 limit and varies slowly with magnetic field, Li et al. suggested a s-wave nature for the SC gap in the conduction bands (for the case of Cu0.06 TiSe2 ) [5]. In both the experiments by E. Morosan et al. and Kusmartseva et al., the onset of the CDW phase is determined by the anomalous hump in the resistance-temperature curve. However, at certain pressure or doping level, this feature is no longer visible and thus the CDW Tc –x (x can be either pressure or doping) curve derived from these two early experiments did not allow the scrutiny of the entire range of doping over which the CDW persists, nor the identification of the critical density for its suppression. This motivated other research groups to explore, with different techniques, whether there is a CDW QCP inside the SC dome or very close to the boundary of This chapter is taken and edited from Ref. [1] of which the author of this thesis is the first author. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_3

31

32

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

it, which would indicate that the quenching of CDW order could contribute to the establishment of the SC order. Using high-pressure X-ray scattering, Y. I. Joe et al. were able to suppress the CDW Tc to zero temperature, and found a QCP of the CDW phase which is quite far from the end of the SC dome (more than 1 GPa). Their results suggest that there might be no direct relationship between the vanishing of CDW order and the emergence of the SC phase. However, unexpectedly, an ICDW phase was observed right above the SC dome. This incommensurability is strongest along the c axis and can be understood as an existence of ordered CCDW domains separated by phase-slip domain walls [6]. Such a coincidence between ICDW and SC phase suggests that the SC might be induced by the dynamics of CDW domain walls within which the CDW fluctuations proliferate. The coexistence of ICDW with SC was further confirmed by A. Kogar et al. through an X-ray diffraction experiment in Cu doped TiSe2 . It was found that the ICDW phase appeared roughly at the same doping level as the onset of SC, and ICDW order persisted up to much higher doping than the termination of the SC dome. The entire SC dome is under the ICDW phase in their phase diagram [7]. According to these two X-ray experiments, there should be no doubt of the intimate relationship between ICDW and SC phases in TiSe2 . However, in both experiments, the incommensurability of ICDW phase is most significant along the out-of-plane direction and the in-plane components difficult to observe. So a natural question one can ask is what would happen in the quasi-2D limit where the CDW only has in-plane components. By using the ionic gel electrolyte gating technique, Li et al. performed transport measurements on few-layer (with thickness around 10 nm) TiSe2 [8]. Similar to previous experiments on the bulk system, above a certain charge carrier density (doping level), the system becomes a superconductor and the SC phase appears as a dome shape in the phase diagram. The SC transition was found to be of BKT type, as expected for a 2D system. More interestingly, a periodic oscillation of magnetoresistance was observed inside the SC dome and such behaviour was explained as a Little–Parks effect [9], where the Cooper pairs are confined to move inside closed loops. It was then conjectured that above a certain doping, a network of CDW domain walls emerges in the system and SC starts to nucleate inside this matrix of CDW fluctuations [8]. This interpretation relies on the existence of an ICDW phase (CCDW domains separated by domain walls) in the 2D limit of TiSe2 . It is important to note that, although both applying pressure and doping can be used to suppress the CCDW phase, it has been suggested that they have fundamentally different effects on TiSe2 . One the one hand, a Raman scattering study found that the CDW amplitude mode can be softened by either increasing temperature (towards Tc ) or doping. Moreover, the softening frequency has the same power law scaling with respect to both reduced temperature and reduced doping level [10]. On the other hand, a similar of study of the CDW amplitude modes in TiSe2 under pressure does not find such a critical scaling in the form of the frequency softening with respect to pressure; the softening was even suppressed at low pressure [11]. Consequently, it is reasonable to believe that doping and pressure effects are described by different types of theory, or at least by different terms within one theory.

3.1 The CDW Is Not Alone

33

Following all these experiments, it becomes clear that, in order to explain the SC phase in TiSe2 , one has to establish a theory which is able to predict the existence of an ICDW phase. In this chapter, we will describe a Ginzburg–Landau (G–L) theory that was developed to study the interplay between CCDW, ICDW and SC.

3.2 Phenomenological Ginzburg–Landau Theory In the 1970 s, W. L. McMillan first introduced a G–L approach to describe the free energy across a CDW phase transition in TMDs, e.g., 2H-TaSe2 , in terms of which the commensurate to incommensurate (C-IC) phase transition was first described [12, 13]. Studying a simplified 1D version of the problem, he found that near the lock-in transition from ICDW to CCDW the phase θ (x) of the CDW order parameter has a stair-like structure: the space dependence of the phase contains periodic plateaus (CCDW domains) separated by phase jumps that define domain walls separating adjacent commensurate regions. Inside each domain wall, there is a drastic phase slip of a quantized value, which depends on the commensurability factor of the CCDW phase, and a defect of this type was then designated as a discommensuration (DC) by McMillan [14]. It is interesting to notice that, DCs have been recently observed in TiSe2 in STM experiments performed above the optimal SC transition temperature (Tscmax  4 K) [15, 16], and are implied by inelastic X-ray scattering [7]. These findings suggest that CDW converts from CCDW to ICDW through a near-commensurate charge density wave (NC-CDW) regime characterised by a finite density of DCs, similarly to the cases of 2H-TaSe2 [14] or 1T-TaS2 [17]. In addition, as mentioned before, in both pressure and doping experiments, DCs were also predicted to exist and suggested to be intimately related to the emergence of SC [6, 8]. Inspired by this coincidence, we decided to establish a similar type of G–L theory to describe the interplay of CDW and SC in the TiSe2 . In order to write the G–L free energy, we first define the triple CDW order parameter ψ j ’s according to their relationship to the charge density variation δρ (r) =

3 

eiQ j ·r ψ j (r) + c.c C

(3.1)

j=1

where QCj = G j /2 are the three CCDW wavevectors known experimentally ( M’s). In order to describe the ICDW phase, we introduce three ICDW wavevectors Q Ij and for there introduce q Ij = Q IJ − QCj . In line with experiments, we take Q Ij = (1 + δ)QCj where δ quantifies the incommensurability, and further define q I = |q Ij | = δ|QCj |. In this way, ψi (r) is simply a constant for a homogeneous CCDW phase and proportional to eiq j ·r for a uniform ICDW. The crystal lattice of TiSe2 in the normal (high temperature) phase has four types of symmetry: C3 rotation, mirror about  M’s, inversion and translation by any vector of I

34

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

the Bravais lattice. Under these symmetry operations, the order parameter transforms as, respectively, ψ j (r) = ψ j+1 (C3−1 r)

(3.2a)

ψ j (x, ψ j (r)

(3.2b)

y) = ψ j (−x, y) =

ψ ∗j (−r)

(3.2c)

iQCj ·Rb.l

(3.2d)

ψ j (r) = e

ψ j (r − Rb.l )

Requiring the total free energy to follow these symmetries, one can write down a free energy density in the same spirit of that originally conceived by McMillan to describe the vicinity of the normal to CDW phase transition: f cdw (r ) =A



|ψ j |2 + B

j

  | i j

  ∂  ∂ + q Ij ψ j |2 + C | ψ j |2 + G |ψ j |4 ∂ x, j ∂ x⊥, j j

j

(3.3)

 K   M  3D  ψ j ψ ∗j+1 ψ ∗j+2 + c.c + ψ1 ψ2 ψ3 + ψ1∗ ψ2∗ ψ3∗ − − |ψi ψ j |2 2 2 2  E  2 − ψ j + ψ ∗2 j 2

j

i= j

j

The free energy in Eq. (3.3) consists of standard G–L quadratic and quartic terms (with coupling constants A and G), derivative terms (with coupling constants B and C accounting for the energy cost due to the deviation from the reference ICDW wavevector), which favours an ICDW phase with the wavevectors of the charge density being the Q Ij ’s, a lock-in term (with coupling constant E) which favours a CCDW phase, and terms that capture the interaction between three density waves (with coupling constants D, M and K ). The lock-in term comes from the fact that in a CCDW phase, the periodicity λ of density modulation is commensurate with the underlying lattice constant a, i.e., λ = a m/n (m, n ∈ Z), and the energy of a CCDW phase will depend explicitly on the phase θ of the CDW order parameter [18]. Supposing that a mean field solution of the CDW order parameter takes the form = 0 eiθ , the real space density variation has the form: (3.4) δρ(r) ∝ 2 0 cos (Qcdw · r + θ ) Since the value of θ is also determined by minimizing the energy, the real space configuration of the density wave is thus “locked-in” to the underlying lattice. This lock-in effect breaks the translational symmetry of a density wave and makes it different from an incommensurate phase, where the energy does not depend on the phase of the order parameter and translational symmetry is retained. It should be noted that, generally speaking, there are other types of terms which may also respect the symmetries of the system (e.g., higher order terms beyond the

3.2 Phenomenological Ginzburg–Landau Theory

35

fourth order). However, in this study, our main goal is to establish the minimal model which is essential to capture the features of the CCDW and ICDW transitions, so we use Eq. (3.3) as the CDW part of the free energy ( f cdw ) throughout the remainder of this work.

3.3 Harmonic Expansion In principle, there are two straightforward ways to determine the minimum of the CDW free energy: (3.5) Fcdw = dr f cdw (r). One is to solve the Euler–Lagrangian equation of the order parameter, the other is to numerically search for the minimum directly. The former is more efficient if one can write down the non-linear equations analytically provided that there are not too many solutions, while the later one is more task-oriented because we can minimize the free energy directly. Since it is difficult to numerically minimize the free energy, in this study, we implemented the harmonic expansion method, which was first introduced by Nakanishi et al. in their study of the C-IC transition in another TMD material TaS2 [19–21]. The idea of this method is to first expand the CDW order parameter as a summation of a constant term (corresponds to the CCDW phase) and multiple harmonic terms which are induced by the interaction terms in Eq. (3.3) when we consider ψ j ∝ eiq j ·r : ψ j (r) = j;0 +



  j;lmn exp iq j;lmn · r ,

(3.6)

l,m,n≥0; l·m·n=0

with l, m, n ∈ Z and q j;lmn defined as: q j;lmn = (2l + 1)q j + 2mq j+1 + 2nq j+2 , qj =

ηq Ij .

(3.7a) (3.7b)

In this chapter, the index j always identifies one of the three contributions that make up the modulation in the electronic density and, therefore, always runs over 1, 2, 3. This set of wavevectors defined by all possible combinations of l, m, n spans the higher harmonics of q j;lmn induced by the non-linear terms in f cdw after an integration over real space. The parameter η determines if the solution is a CCDW (η = 0), a uniformly ICDW (η = 1), or in between (NC-CDW). The exact value of q Ij , which is set experimentally, is not very important at this stage. Since in the CDW phase, there is still C3 rotation symmetry, one can eliminate the j-dependence of j;lmn and make the simplifying assumptions j;0 = 0 ∈ R, j;lmn = lmn ∈ R. It is also reasonable to expect that the anisotropy in the derivative terms should not change

36

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

the physics qualitatively, so we set B = C in the following discussion. Combining all these together, one can get the CDW free energy as a function of the harmonic amplitudes lmn ’s and the deviation form commensurability η:    K 4 2 ˜ Fcdw /3 = A + B − E 0 + G + 0 2    

K 2 2 ˜ A + B q˜ lmn + 2 G + | 0 | lmn lmn + 2 l,m,n≥0 

(3.8)

l·m·n=0

− E lmn l  m  n  δl+l  +1,m+m  ,n+n  − D lmn l  m  n  l  m  n  δl+n  +m  ,m+l  +n  ,n+m  +l  − M lmn l  m  n  l  m  n  δl−n  −m  +1,m−l  −n  ,n−m  −l  + G lmn l  m  n  l  m  n  l  m  n  δl−l  +l  −l  ,m−m  +m  −m  ,n−n  +n  −n  + K lmn l  m  n  l  m  n  l  m  n  δl−l  +m  −m  ,m−m  +n  −n  ,n−n  +l  −l  2 with q˜ lmn = 4η2 [(l 2 + m 2 + n 2 ) − (lm + ln + mn)] + 2(2l − m − n)η(η − 1) + 2 (η − 1) and B˜ = B|qiI |2 . The δ-function in Eq. (3.8) is defined as:

δl,m,n =

1 l = m = n, 0 otherwise.

(3.9)

The free energy requires one to specify the values of the constants (A to K ). The harmonic amplitudes and the factor η constitute the variational parameters that determine the equilibrium profile of the charge density at a given temperature. Once the constants are set, to find the absolute minimum of the free energy Eq. (3.8), we begin by fixing η and obtaining the saddle points in the multidimensional space spanned by the real parameters 0 and lmn . The saddle points are determined by numerically solving the Euler–Lagrangian equations for the ’s. The temperature is included in the quadratic coefficient A, which is assumed to vanish linearly at a critical temperature: (3.10) A ≡ t ∝ T − Ticdw with t being the reduced temperature. The result of this step is a curve of the minimum min (η), examples of which are shown in Fig. 3.1 for free energy as a function of η, Fcdw different effective temperatures. The calculations require setting an harmonic cutoff N that restricts the expansion to terms with 0 ≤ l, m, n ≤ N . The number of variational parameters is then given by (N + 1)3 − N 3 + 2, which takes into account the constraint that at least one of l, m, n must be zero (l · m · n = 0), and includes 0 and η. The convergence of the harmonic expansion is relatively fast and we verified that N = 3 yields a good compromise without affecting the accuracy of the results in the range of parameters studied. A typical example of the rapid decay of the higher harmonics is shown in Fig. 3.2b.

3.3 Harmonic Expansion

(a)

(c)

37

(b)

(d)

Fig. 3.1 Minimum of the CDW free energy (in arbitrary units) with respect to η at E = 2.2. a For t = 1.2, the minimum free energy is numerically zero for all η (note the extremely magnified vertical scale to emphasize the threshold of numerical accuracy), indicating a normal state. b At t = 0.3, the minimum of free energy is independent of η and negative. Only the harmonic amplitude 0 is finite (not shown), which defines a CCDW state. c–d For t = −1.2 and t = −2.3, the minimum free energy is obtained at a finite η, which implies an ICDW state. This figure is taken from Ref. [1]. © 2019 American Physical Society

The equilibrium state is identified by the harmonic content of the order parameter for the value of η that yields the absolute minimum of the free energy, as illustrated min (η) = 0 as in Fig. 3.1a, the in Fig. 3.1. There are three possible outcomes: (i) If Fcdw equilibrium corresponds to the normal state, without either CCDW or ICDW. (ii) If min (η) ≤ 0 but constant with η as in Fig. 3.1b, we have a CCDW state, which is Fcdw always confirmed by inspecting that 0 is the only non-zero component of ψ(r). min (η) will have a global minimum at a given η, as in the cases (iii) Otherwise, Fcdw shown in Fig. 3.1c, d, which indicates that the equilibrium state is an ICDW. Under these criteria, we have mapped the phase diagram shown in Fig. 3.3 for the CDW phases in a restricted parameter space relevant for the experiments that probe the C-lC transition, which will be discussed in more detail below. By looking at the tendency of η (with the minimum of f cdw ) with respect to t, the order of CCDWICDW transition can be deduced. As shown in Fig. 3.2, because of a discontinuity in the equilibrium η at t ≈ −1.0, the C-IC transition is of first order.

38

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

(a)

(b)

Fig. 3.2 Representative results from the numerical minimization of the free energy. a Plot of η versus t at a E = 2.2. The discontinuity at t ≈ −1 indicates a first order nature of C-IC phase transition. b The ten largest ’s at (E, t) = (2.2, −1.7). Since 0,0,0 is the largest, it suggests that the CDW phase is incommensurate for these specific parameters. This figure is taken from Ref. [1]. © 2019 American Physical Society

3.4 CDW Phase Diagram As we are only interested in scrutinizing the C-IC transition and presently disregard effects that might arise from induced anisotropy, strain, etc., we map the phase diagram in the E–t plane fixing the remaining parameters to the values A = t,

K = G = 2B(q I )2 = −2D = 2M = 2.

(3.11)

We verified that this choice allows us to concentrate on the vicinity of the C-IC boundary shown in Fig. 3.3, and drive the transition via the lock-in parameter E which, formally, controls the energy gain of having a CCDW. Physically, a smaller E is associated to larger electron densities because: (i) phenomenologically, electron doping reduces the stability of the CCDW state in favor of an ICDW [7, 8, 15]; (ii) microscopically, a theory of the CDW instability in TiSe2 must yield a reduction of the CCDW order parameter with doping in order to match the experimental observations [22–24]; the microscopic lock-in gain, being associated with the condensation energy, is itself determined by the magnitude of the order parameter [18, 25]. For this reason, the horizontal E axis in the figure is reversed so that electron densities increase from left to right, as is usually presented in the experimental phase diagrams [7, 8]. The phase diagram in Fig. 3.3 exhibits the anticipated stability of the CCDW state at large E (low density) and its suppression below a critical and temperaturedependent lock-in parameter: E c (t). Note that the normal-CDW critical temperature, tc (E), is reduced as the system progresses from the CCDW to the ICDW state, in agreement with the experimental trend [7, 8]. Likewise in agreement is the abrupt loss of the CCDW phase indicated by the relatively steep slope of the line E c (t). In light of our definition of t above, the asymptotic tendency tc (E → 0) ≈ 0 means that Tc → Ticdw , suggesting that an IC state is ultimately preferred in the absence

3.4 CDW Phase Diagram

39

Fig. 3.3 Phase diagram of the the G–L model defined in Eq. (3.3), obtained by minimizing Eq. (3.8). When Fcdw < 0, the system is in a CDW state and the CCDW phase corresponds to η = 0. The green line represents the C-IC boundary, E c (t). The red line indicates the boundary of the SC phase including the linear E dependence in the CDW-SC coupling as of Eq. (3.12); it becomes the gray line if as is E-independent. The inset shows the equilibrium η at tc (the transition is first order) and at low temperature. The phase diagram from the G–L model has qualitatively similar shape as the one from experiments (see Fig. 1.5c by Kogar et al. [7]). This figure is taken from Ref. [1]. © 2019 American Physical Society

of lock-in energy. To be more specific, the inset shows the equilibrium value of the parameter η at the critical temperature of the normal-CDW phase transition and at low temperatures: It grows towards η ≈ 1 with decreasing E, implying that the dominant wavevectors contributing to δρ(r) at each equilibrium state increasingly approach the reference ICDW vector Q Ij . Knowledge of η is insufficient to characterize the rich spatial texture of the charge modulation which, following Eq. (3.6), depends on the detailed harmonic content that minimizes Fcdw . Figure 3.4a shows a real-space plot of δρ(r) at the representative point close to the CCDW boundary marked by  in Fig. 3.3. The color density scale shows that there is no single periodicity. In more detail, Fig. 3.4b, c show line cuts of the phase and amplitude of the order parameters ψ j (r) ≡ ϕ j (r)eiθ j (r) along the vertical dashed line in panel (a). The phase θ j (r) displays a stepwise variation with periodic phase slips of π . Seeing that the definition Eq. (3.1) implies that regions where θ j (r) ≈ 0 mod π are commensurate with the Bravais lattice, the spatial pro-

40

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

(a)

(b)

(c)

(d) (e)

Fig. 3.4 a Real space plot of the density profile δρ(r) at E = √ 2.2, t = −1.7 (the point marked with  in the phase diagram of Fig. 3.3). Lengths are in units of 3a/2π , with a the lattice constant of TiSe2 in the normal state. The yellow-dashed lines mark the places where the phase of each CDW order parameter, ψ j (r), jumps by π . b and c respectively show the phase and amplitude of ψ j (r) along the white vertical cut marked in a. d The SC order parameter, (r), in the same region shown in a. e (r) along the vertical cut marked in a. This figure is taken from Ref. [1]. © 2019 American Physical Society

file of the phase reveals that the equilibrium state is characterized by domains of approximately CCDW separated by DCs at which the phase jumps by π . This is the NC-CDW regime which replicates the characteristics of CDW domain walls investigated by STM slightly above Tsc [15, 16]. More generally, adapting the interaction terms in Eq. (3.3) to a commensurability condition QC = νG with ν a rational number (ν = 1/2 for TiSe2 ), one obtains a corresponding domain structure with the phase jumping by 2π ν across the boundary of two neighboring C domains [20, 21, 26–28]. In 1D and phase-only reductions of this problem [ϕ j (r) = const.], the saddle-point condition for Fcdw reduces to the static sine-Gordon equation [26, 27] and it becomes clear that a DC (domain wall) corresponds to its soliton solutions. Even though the general problem of interest to us

3.4 CDW Phase Diagram

41

is two-dimensional, Eq. (3.1) still consists of a linear combination of one-dimensional CDW modulations along the direction of G j . It is thus not surprising that each θ j (r) seen in Fig. 3.4b retains the soliton-like nature characteristic of the 1D solution. The DCs form a 2D network whose periodicity is highlighted by the yellowdashed contours in Fig. 3.4a. They expose a Kagome superlattice overlaying the CCDW, which is the natural effect of superimposing three equivalent 1D-like DC staircases along directions 120◦ apart. For a general √ commensurability fraction ν, the period of the DC network is L = 2π ν/(ηq I ) = 3a/(ηδ), where a is the lattice constant of the crystal in the normal phase. Note that, in addition to the phase, the amplitude of ψi (r) is also significantly modulated. In particular, Fig. 3.4c shows it can drop more than 30% at each DC. The high variational freedom introduced by the expansion Eq. (3.6) permits the CDW to distort from the simple plane wave solution in order to minimize both the lock-in and gradient terms in Eq. (3.3). It can therefore be, simultaneously, as close to a CCDW and an ICDW configuration as possible, which the solution in Fig. 3.4a–c accomplishes by: (i) having domains of nearly flat phase and high amplitude (CCDW) joined by (ii) domain boundaries where, on the one hand, the phase jumps so that on spatial average θ j (r) ≈ q Ij · r and, on the other, the amplitude drops to minimize the cost in deviating from commensurability at those regions.

3.5 Coupling Between a Charge Density Wave and Superconductivity It is natural to expect these DCs to couple strongly with the SC order parameter: On the one hand, the development of a DC superlattice as in Fig. 3.4a introduces new low energy phonons associated with the superlattice [27, 29]. From the perspective of SC pairing due to retarded electron-phonon interactions, the emergence of this DC lattice might enhance any intrinsic pairing tendency already present in the absence of a CDW. On the other hand, DCs are nothing but CDW fluctuations. While both phase and amplitude fluctuations are gapped in the CCDW regime [18], the transition to the NC-CDW state releases them to potentially favor SC through fluctuation-induced pairing. This is the analogue of pairing induced by fluctuations of magnetic order proposed for high-temperature superconductors [30]. As a minimal approach to describe this interplay between the two orders we propose extending the conventional [31] Ginzburg–Landau free energy associated with the SC order parameter, (r), and writing Fsc ≡



dr as (T, ∇ψ j ) ||2 + bs |∇|2 + cs ||4 .

(3.12)

Making as a function of ∇ψ j permits the enhancement of SC by deviations from a CCDW. To lowest order in the interaction and inhomogeneity, as should have the form

42

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

as = a0 − a1



|∇ψ j |2 ,

(3.13)

j

where a0 ∝ T − Tsc is the conventional quadratic coefficient and a1 > 0 so that SC is stabilized within regions of fluctuating CCDW order (we take a1 to be T -independent). This captures phenomenologically both the effect of fluctuationinduced (a0 = const.) and fluctuation-enhanced (a0 ∝ T − Tsc ) pairing, as well as the spatial enhancement of the electronic DOS at DCs [15]. The total free energy now combines Eqs. (3.5) and (3.12), F = Fcdw + Fsc and the coupling in Eq. (3.12) requires one to self-consistently find the saddle point for both ψ j (r) and (r). As in TiSe2 Tcdw  40 K and Tsc  4 K Tcdw [2, 3, 8], the CDW order parameter is well developed when the SC instability appears. Combined with the fact that we are in a time-independent Ginzburg–Landau framework, this justifies solving the two problems independently and tackle Eq. (3.12) given the solution to the CDW alone. In practice, this entails minimizing Fsc in the presence of the static CDW background that minimizes Eq. (3.3). For every point (E, t) in the parameter space of the phase diagram, we replace ψ j (r) by the corresponding equilibrium solution arising from the minimization of Fcdw . Since ψ j (r) is nonuniform in space outside the CCDW phase, this turns the Eqs. (3.12) into non-uniform Ginzburg–Landau problems. Numerically, we solve the Euler–Lagrange equations  for (r) using the CDW texture ( j |∇ψ j (r )|2 ) itself as the initial trial solution which is then relaxed under periodic boundary conditions consistent with the CDW and discommensuration network. A representative outcome of such procedure is shown in Fig. 3.4d for the CDW solution in panel (a).1 The most significant feature is the non-uniformity of (r) that follows the spatial texture of the DC network. The section plotted in Fig. 3.4e shows there is no SC within the CCDW domains [(x1 ) = 0] but only at and near the DCs, and that SC is reinforced when two DCs overlap at the vertices of the Kagome: (x3 ) ≈ 2(x2 ). Other examples of the real-space profile of the equilibrium SC order parameter are shown in Fig. 3.5 for different points in the phase diagram of Fig. 3.3. Most interestingly, it is clear from how ∇ψ j enters the quadratic coefficient as in Eq. (3.12) that, depending on the parameters, the development of SC in the NC-CDW regime might take place in three stages with decreasing temperature, see Fig. 3.6a: (i) it begins at Tsc0d with the nucleation of isolated SC dots at the Kagome vertices; (ii) at Tsc1d  Tsc0d these SC dots have grown and overlap to percolate the system in a connected network, as in Fig. 3.4d; (iii) ultimately, at Tsc2d  Tsc1d the whole system becomes superconducting. (The SC boundaries in the phase diagram correspond to Tsc0d .) The coupling proposed in Eq. (3.12) therefore predicts that, depending on 1 The solution for SC order parameter shown in panel (d) of Fig. 3.4 here is given by replacing a1 → a1 E, as will be discussed later.

3.5 Coupling Between a Charge Density Wave and Superconductivity

43

the temperature, the SC order can have either a 0d, 1d or 2d character. This can be experimentally probed with temperature-dependent local spectroscopy across the SC transition. Also, in the absence of other intrinsic pairing mechanisms this picture predicts that if the localization length of (r) is not much larger than the width of a DC, it is possible that Tsc2d = 0 in the NC-CDW region of the phase diagram. SC would then span the system, at most, through the 1d network [cf. Fig. 3.6a]. The area of SC stability in the phase diagram depends on whether the parameter a1 in Eq. (3.12) varies with E. If it does not, SC persists from the NC-CDW to the ICDW limit at temperatures below the gray line in Fig. 3.3. It remains in the ICDW limit because |∇ψ j | is finite in the ICDW limit, thereby supporting uniform SC. This is a sensible physical outcome from a perspective of fluctuation-induced pairing because phase fluctuations of an ICDW are gapless. In the specific case of doped TiSe2 , however, SC seems to exist only over a dome-shaped portion of the phase diagram, over a finite density range [2, 8]. This phenomenology can be captured by replacing a1 → a1 E in the parameter as , making it depend both implicitly (through ψ j ) and explicitly on the lock-in parameter E. This means that the SC contribution to the free energy now reads explicitly Fsc =

dr

 

  a0 − a1 E j |∇ψ j |2 ||2 + bs |∇|2 + cs ||4 ,

(3.14)

where the constants are chosen as above, a0 = 10t + 60, bs = cs = 1, except that a1 = 500 now. In this case, the effect of the lock-in energy E appears both explicitly, as a prefactor to the interaction, and implicitly, through its effect on ψ j (r). In view of the previously argued negative correlation between E and the electronic density, this amounts to making the coupling to fluctuations weaker at higher densities, which is also physically plausible. Indeed, the phase boundary computed in this way corresponds to the dome-shaped red line in Fig. 3.3: by restricting the SC phase to the NC-CDW region it renders the phase diagram qualitatively correct.

3.6 More on Superconductivity The feasibility of non-uniform percolative SC in the NC-CDW regime is determined by the characteristic width of DCs (w), their separation (L, the size of the C domains), and the SC coherence length ξ (∼12 nm in TiSe2 [32]). It is likely that w  ξ , not sufficient to permit fully developed SC grains in the temperature range Tsc1d < T < Tsc0d where the model predicts nucleation at the vertices of the DC network. But it might be enough to stabilize Cooper pairs within each vertex reducing the problem to that of interacting Bose particles on a lattice with a metallic background (similarly to cold atoms on an optical lattice [33], except that the periodic potential comes here from the DC network and is temperature dependent). Tunneling of non-condensed pairs among this realization of a Josephson junction array would tally with the observation

44

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

(a1)

(b1)

(a2)

(b2)

(c1)

(c2)

(d1)

(d2)

3.6 More on Superconductivity

45

Fig. 3.5 Real space plots of the SC order parameter, (r), computed for the CDW state at specific points (E, t) of the phase diagram shown in Fig. 3.3 of the main text. Each panel in the bottom row shows a vertical section along the line x = 0 of the density plot directly above it. a (E, t) = (2.2, −1.1): right below Tsc0d , SC order nucleates on isolated 0d regions that coincide with the vertices of the Kagome lattice defined by the intersection of DCs. b (E, t) = (2.2, −3): reducing the temperature from Tsc0d , stabilizes the SC state further. Both the amplitude and spatial extent of the SC order parameter increase monotonically. The case shown corresponds to a temperature below the percolation threshold, T < Tsc1d . c (E, t) = (1, −2.4): the linear dependence of the CDWSC coupling constant as with E weakens the SC amplitude when the lock-in energy is reduced. d (E, t) = (0, −2): unlike the previous cases, here (r) has been obtained using an E-independent coupling as , as described in the text in relation to the gray SC boundary line in Fig. 3.3. With no lock-in energy (E = 0), the equilibrium CDW solution approximates a homogeneously IC state (i.e., one without DCs). SC is therefore stabilized over the whole system. This figure is taken from Ref. [1]. © 2019 American Physical Society

[34] of an anomalous-metallic phase in TiSe2 near T = 0 K [35–38], and the analysis of its vortex phases would be similar to that in references [39, 40]. The situation in the range Tsc2d < T < Tsc1d has interesting implications in the presence of a magnetic field, B. First, vortices are naturally pinned by the DC lattice, even in the absence of disorder, and their motion correlated. Second, given the likelihood that w  ξ , vortices would not squeeze within DCs; the supercurrent would instead circulate along the linked network of 1D SC channels [31], as illustrated in Fig. 3.6b. If L  ξ , we may regard this as a microscopic version of SC wire grids which have been extensively studied experimentally [41–44] and theoretically [45–49]. A distinctive feature of these grids are oscillatory dips as a function of B in thermodynamic [41] and transport [43] properties, with period determined by rational fractions, f = φ/φ0 , of the flux through the grid’s elementary plaquette (φ ∼ B L 2 , φ0 ≡ h/2e) [47, 48, 50]. Given that such a networked texture of the SC order parameter is a natural implication of the model studied here, it is tempting to speculate this to be the origin of Little–Parks-type resistance oscillations found in the SC/anomalous-metal phase of TiSe2 near optimum doping [8]. To put this to the test, assume the grid is hexagonal as in Fig. 3.4d ( f = 1/4 [48]) and take the first magnetoresistance dip at B  0.13 T in the experiment √ √ of Li et al. [8]. Hence, L = [φ0 /(2 3B)]1/2  70 nm. We noted above that L = 3a/(Lη) where a = 0.35 nm for TiSe2 [51, 52]. Since η ∼ 1 (Fig. 3.3), it follows that δ ∼ 0.01. In other words, interpreted from the perspective of our model, the experimental resistance oscillations indirectly suggest a CDW incommensurability factor δ ∼ 1% and a typical distance between DCs L ∼ 70 nm. Compellingly, x-ray diffraction does reveal δ ∼ 5–15% in the superconducting dome [7], which is in the same range as in other metallic TMDs [53], and STM finds DCs separated by 10’s of nm at optimum doping above Tsc [15]. Having in mind that Cu intercalation likely pins and disorders the DCs into an irregular network, these estimates seem in noteworthy agreement with experiments. Experimental evidence that SC emerges with the suppression of the CCDW in the NC-CDW regime is increasingly better documented across a large range of 2H and 1T TMDs [54]. These span both good metals and semimetals, as well as a num-

46

3 Phenomenological Model of Coexisting CDW and Superconductivity in TiSe2

Fig. 3.6 a Schematic of the distinct non-uniform SC regimes spatially correlated with the DC network: nucleation and expansion of the SC order parameter (Tsc1d < T ≤ Tsc0d ), percolation (Tsc2d < T ≤ Tsc1d ), and finite everywhere. See Fig. 3.5 for actually calculated textures. b Illustration of how the connectivity in the percolation regime constrains the vortex structure, with impact in the magnetic response. This figure is taken from Ref. [1]. © 2019 American Physical Society

ber of distinct commensurability conditions. Our model straightforwardly extends to these cases, each corresponding to a particular region of the general phase diagram in Fig. 3.3. It thus provides a definite and universal phenomenological foundation to further explore the interplay between these two coexisting orders and their fluctuations.

3.7 Summary In this chapter, we have established a phenomenological G–L theory to describe the interplay between CCDW, ICDW and SC in TiSe2 . After fixing most of the parameters in the free energy and allowing only the reduced temperature t and lockin energy E to vary, the phase diagram predicted by our theory is able to capture all the phases that have been observed experimentally. Different from cases of the TaS2 and TaSe2 , decreasing the temperature in a certain lock-in energy interval, the sequence of phases predicted by our theory is NormalCCDW-ICDW [55, 56]. At low temperatures and with decreasing lock-in energy, the system undergoes a first order phase transition from CCDW to ICDW and the early stages of the ICDW phase is characterised by the NC-CDW regime. Within the NC-CDW, DCs form a network structure encompassing the CCDW domains. Inside the each DC, the CDW phase exhibits a rapid slip of π and the amplitude goes through a minimum. By considering a simple form of coupling between CDW and SC order parameters, the SC phase is predicted to exist inside the ICDW regime and has a dome shape. Moreover, the SC order starts to nucleate inside DCs and also forms a network structure. Such a network structure of SC order is exactly what has been conjectured from the Little–Parks oscillations in the ion-gel doping experiment with few-layer TiSe2 [8]. After mapping the lock-in energy to the doping level/charge carrier density, the phase diagram predicted by our G–L theory is qualitatively the

3.7 Summary

47

same as the one from that experiment. In addition, the estimated discommensurability from the network structure is also at the same order of magnitude as that expected from the Cu doping experiment [7]. There are several things not included in our phenomenological theory: (i) We are mainly focusing on the 2D limit and thus do not include the interlayer coupling, which might be important for explaining the effect of pressure in bulk TiSe2 . (ii) The dynamics of the CDW and SC are not studied in our theory; perhaps one can try to develop a dynamical Landau theory in the future. (iii) So far, our theory is purely phenomenological. It will be of great interest to establish a fully microscopic theory and derive the parameters in the G–L free energy from there. Despite these simplifications, all the predictions from our G–L theory exhibit excellent consistency with existing experiments, and we believe this phenomenological study paves a way for the future microscopic studies of this problem. The natural follow up to this phenomenological approach is to establish a fully microscopic theory. Since the formalism and the qualitative aspects of the C-IC transition are relatively well established in the literature, it is of particular interest to develop the theory for how CDW fluctuations can induce superconducting pairing from a microscopic point of view. In addition to providing the microscopic mechanisms and values for the parameters in the G–L free energy, such a microscopic theory would also establish the regimes of validity of the phenomenological theory. In view of this, we have looked in detail at the elementary amplitude and phase excitations of the CDW in the microscopic framework discussed in Chap. 2, where the CDW arises as a result of excitonic instability. The next chapter reports our findings in this context, and the results ultimately reinforce the phenomenological coupling introduced here between CDW and SC order parameters.

References 1. Chen C, Su L, Neto AHC, Pereira VM (2019) Discommensuration-driven superconductivity in the charge density wave phases of transition-metal dichalcogenides. Phys Rev B 99:121108 2. Morosan E, Zandbergen HW, Dennis BS, Bos JWG, Onose Y, Klimczuk T, Ramirez AP, Ong NP, Cava RJ (2006) Superconductivity in Cux TiSe2 . Nat Phys 2:544–550 3. Kusmartseva AF, Sipos B, Berger H, Forró L, Tutiš E (2009) Pressure induced superconductivity in pristine 1T-TiSe2 . Phys Rev Lett 103:236401 4. Zhao JF, Ou HW, Wu G, Xie BP, Zhang Y, Shen DW, Wei J, Yang LX, Dong JK, Arita M, Namatame H, Taniguchi M, Chen XH, Feng DL (2007) Evolution of the electronic structure of 1T-Cu x TiSe2 . Phys Rev Lett 99:146401 5. Li SY, Wu G, Chen XH, Taillefer L (2007) Single-gap s-wave superconductivity near the charge-density-wave quantum critical point in Cux TiSe2 . Phys Rev Lett 99:107001 6. Joe YI, Chen XM, Ghaemi P, Finkelstein KD, de La Peña GA, Gan Y, Lee JCT, Yuan S, Geck J, MacDougall GJ, Chiang TC, Cooper SL, Fradkin E, Abbamonte P (2014) Emergence of charge density wave domain walls above the superconducting dome in 1T-TiSe2 . Nat Phys 10:421–425 7. Kogar A, de La Pena GA, Lee S, Fang Y, Sun SX-L, Lioi DB, Karapetrov G, Finkelstein KD, Ruff JPC, Abbamonte P, Rosenkranz S (2017) Observation of a charge density wave incommensuration near the superconducting dome in Cux TiSe2 . Phys Rev Lett 118:027002

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8. Li LJ, O’Farrell ECT, Loh KP, Eda G, Özyilmaz B, Castro Neto AH (2015) Controlling manybody states by the electric-field effect in a two-dimensional material. Nature 529:185–189 9. Little WA, Parks RD (1962) Observation of quantum periodicity in the transition temperature of a superconducting cylinder. Phys Rev Lett 9:9–12 10. Barath H, Kim M, Karpus JF, Cooper SL, Abbamonte P, Fradkin E, Morosan E, Cava RJ (2008) Quantum and classical mode softening near the charge-density-wave-superconductor transition of Cux TiSe2 . Phys Rev Lett 100:106402 11. Snow CS, Karpus JF, Cooper SL, Kidd TE, Chiang T-C (2003) Quantum melting of the chargedensity-wave state in 1T-TiSe2 . Phys Rev Lett 91:136402 12. McMillan WL (1975) Landau theory of charge-density waves in transition-metal dichalcogenides. Phys Rev B 12:1187–1196 13. McMillan WL (1975) Time-dependent Landau theory of charge-density waves in transitionmetal dichalcogenides. Phys Rev B 12:1197–1199 14. McMillan WL (1976) Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition. Phys Rev B 14:1496–1502 15. Yan S, Iaia D, Morosan E, Fradkin E, Abbamonte P, Madhavan V (2017) Influence of domain walls in the incommensurate charge density wave state of Cu intercalated 1T-TiSe2 . Phys Rev Lett 118:106405 16. Novello AM, Spera M, Scarfato A, Ubaldini A, Giannini E, Bowler DR, Renner Ch (2017) Stripe and short range order in the charge density wave of 1T-Cu x TiSe2 . Phys Rev Lett 118:017002 17. Thomson RE, Burk B, Zettl A, Clarke J (1994) Scanning tunneling microscopy of the chargedensity-wave structure in 1T-TaS2 . Phys Rev B 49:16899 18. Grüner G (1994) Density waves in solids. Addison-Wesley, Boston 19. Nakanishi K, Takater H, Yamada Y, Shiba H (1977) The nearly commensurate phase and effect of harmonics on the successive phase transition in 1T-TaS2 . J Phys Soc Jpn 43:1509–1517 20. Nakanishi K, Shiba H (1977) Domain-like incommensurate charge-density-wave states and the first-order incommensurate-commensurate transitions in layered tantalum dichalcogenides. I. 1T-polytype. J Phys Soc Jpn 43:1839–1847 21. Nakanishi K, Shiba H (1978) Domain-like incommensurate charge-density-wave states and the first-order incommensurate-commensurate transitions in layered tantalum dichalcogenides. II. 2H-polytype. J Phys Soc Jpn 44:1465–1473 22. Monney C, Schwier EF, Garnier MG, Mariotti N, Didiot C, Cercellier H, Marcus J, Berger H, Titov AN, Beck H, Aebi P (2010) Probing the exciton condensate phase in 1T-TiSe2 with photoemission. New J Phys 12:125019 23. Van Wezel J, Nahai-Williamson P, Saxena SS (2011) Exciton-phonon interactions and superconductivity bordering charge order in TiSe2 . Phys Rev B 83:024502 24. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T -TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602 25. Lee PAA, Rice TMM, Anderson PWW (1974) Conductivity from charge or spin density waves. Solid State Commun 14:703 26. Bak P, Emery VJ (1976) Theory of the structural phase transformations in tetrathiafulvalenetetracyanoquinodimethane (TTF-TCNQ). Phys Rev Lett 36:978 27. McMillan WL (1977) Collective modes of a charge-density wave near the lock-in transition. Phys Rev B 16:4655–4658 28. Jacobs AE, Walker MB (1980) Phenomenological theory of charge-density-wave states in trigonal-prismatic, transition-metal dichalcogenides. Phys Rev B 21:4132–4148 29. Nakanishi K, Shiba H (1978) Domain-like incommensurate charge-density-wave states and collective modes. J Phys Soc Jpn 45:1147 30. Scalapino DJ (1999) Superconductivity and spin fluctuations. J Low Temp Phys 117:179 31. Tinkham M (1996) Introduction to superconductivity, 2nd edn. McGraw Hill Inc., New York 32. Morosan E, Li L, Ong NP, Cava RJ (2007) Anisotropic properties of the layered superconductor Cu0.07 TiSe2 . Phys Rev B 75:104505

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Chapter 4

Excitonic CDW Fluctuations and Superconductivity

In Chap. 3, we built a G-L theory to describe the interplay between CDW and SC. Through a simple form of the coupling between them, we found that SC can be enhanced by CDW fluctuations. However, the underlying physical mechanism for SC is still obscure within such a phenomenological theory. On the other hand, we have shown in Chap. 2 that the normal to CCDW transition seems to be very well described microscopically by the phenomenon of excitonic condensation. The meanfield treatment used to tackle that problem leaves open the role and nature of CDW fluctuations in the context of this excitonic picture. Inspired by a model developed in the context of HTSC cuprates, this chapter bridges the microscopic model discussed in Chap. 2 with the coupling of its fluctuations to superconductivity described in the previous chapter. We show that the CDW fluctuations can indeed mediate a BCS type of pairing in the system.

4.1 An Inspiration From the Spin-Bag Mechanism In the late 1980s, soon after the discovery of HTSC cuprates, Schreiffer, Wen and Zhang proposed a so-called spin-bag mechanism for the origin of SC pairing in these systems [1, 2]. According to the fact that the spin density wave (SDW) order persists for a finite range of doping close to the SC dome, and the SDW wavevector Qsdw (which is exactly the nesting vector for a 2D single band Hubbard model at half filling) remains commensurate throughout this process, they proposed a weak-coupling theory (the band width W is larger than the on-site Hubbard interaction strength U ) for the SC pairing based on a single band Hubbard model. In the hole-doped regime, the quasiparticles in the normal phase (with SDW order) are spin- 21 fermions, corresponding to holes surrounded by regions with reduced SDW order; their energy spectrum is the SDW-dressed band structure obtained from a weak-coupling mean-field © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_4

51

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4 Excitonic CDW Fluctuations and Superconductivity

calculation based on Fermi surface nesting. The spin wave spectrum was calculated in the random-phase approximation (RPA), and they were able to show at the time that an attractive interaction between quasiparticles can be mediated by the collective SDW amplitude mode. However, through extensive experimental studies, more and more exotic phenomena have been discovered later, e.g., dynamical spectral weight transfer on large energy scale [3], the appearance of a strange metal phase [4] and CDW phase [5], etc, all of which indicate that cuprates cannot be described by such a simple weak-coupling theory. Although the spin-bag mechanism has been ruled out for the case of cuprate, the idea of a fluctuation-mediated pairing between dressed quasiparticles can still be grafted to the CDW systems. In fact, there are several experimental facts indicating that such type of mechanism might account for the SC pairing in TiSe2 . First of all, the band dispersion mainly changes near the Fermi level after the CCDW transition and the topmost CDW-renormalized valence band exhibits an inverted Mexican hat shape (see Fig. 2.3). Since there is no large-scale spectral weight transfer throughout the CDW phase transition, it is reasonable to suppose that it can be well described by a weak-coupling mechanism [6, 7]. Secondly, STM studies of Cu-doped TiSe2 have been observed that the CDW gap persists even at high doping levels and shifts to lower energy, which is consistent with the DFT calculation discussed in Sect. 2.2 [7, 8]. Finally, a thermal conductivity measurement indicates that the CDW valence (Se 4 p) band is far below the Fermi level and SC is mainly contributed from the CDWrenormalized conduction (Ti 3d) bands. Combining all these with the fact that the SC dome emerges inside the ICDW phase for Cu doped TiSe2 [9], it is thus reasonable to propose the following picture for SC pairing in Cux TiSe2 : When the electron density increases, the long-range CCDW order is globally suppressed but still remains locally, so the charge carriers are the quasiparticles with CDW-renormalized energy dispersion. The fluctuations of CDW order might induce an attractive interaction between these new quasiparticles. In the remainder of this chapter, using the path integral formalism, we will show that CDW amplitude fluctuations indeed can mediate an attractive interaction between the CDW-dressed quasiparticles. Subsequently, we explore the SC phase diagram that is induced by this fluctuation-mediated pairing without the need to introduce any new microscopic parameter.

4.2 Simplified Parameterization of the Excitonic Hamiltonian For the purpose of allowing a direct quantitative comparison between the calculated doping dependence of CDW transition temperature Tc and the experimental values, the calculations of Chap. 2 were based on a realistic parameterisations of the hole pocket at the  point and the three electron pockets at M [see Fig. 1.3], which were extracted from ARPES data as described there. Here, on the other hand, the primary goal is to characterize the spectrum of excitonic fluctuations, which motivates the

4.2 Simplified Parameterization of the Excitonic Hamiltonian

53

following two simplifications. To begin with, we ignore the ellipticity of the electron pockets and shall consider them to be parabolic and isotropic. This does not produce any qualitative difference, and is expected to have only a minor quantitative effect. Secondly, as we are interested in the excitonic fluctuations associated with each electron-hole channel, we will consider only one electron and one hole pocket. This effective band structure that we now use to describe the reference (normal, noninteracting) state is represented in Fig. 4.1. The non-interacting part of the simplified model consists thus of a hole-like valence band (centred at the  point of the BZ) and an electron-like conduction band. Both have parabolic dispersion and are separated in the BZ by a wavevector Qcdw . To simplify the notation, we denote the (annihilation) field operator for the σ , while the one for the conduction band valence band with momentum k, spin σ as cL,k σ with momentum Qcdw + k, spin σ as cR,k . Assuming perfect nesting between the two pockets, i.e., εk+Qcdw = −εk (the chemical potential μ is equal to 0 here) with k ∈ L (L stands for the region near the valence pocket), the system has a CDW instability with an inter-valley interaction. With the electronic interaction, the Hamiltonian of the simplified model has the following form: H=

  U    †σ σ †σ σ †σ †σ  σ  σ εk cL,k cL,k − cR,k cR,k − cR,k+q cL,k cL,k  cR,k  +q . N q∼0 k∈L,σ k  ∈L,σ  k∈L,σ 

(4.1) We formulate now the problem within a path integral approach. To do so we first introduce the Grassmann field for the fermions:    σ  †σ σ σ cR,k → ψ¯ R,k , (4.2a) , cR,k , ψR,k     †σ σ σ σ cL,k → ψ¯ L,k , cL,k , ψL,k . (4.2b) By a Hubbard–Stratonovich transformation on the interaction term [10, 11], we introduce the auxiliary field  (CDW order parameter), and the action of the system can be written as  σ   β    ∂τ − εk N  ∗ ψR,k 0 σ σ ¯ ¯ ψR,k ψL,k + dτ q q S= σ + ε ψ 0 ∂ U q∼Q τ k L,k 0 k∈L,σ cdw     σ σ σ σ ψ¯ R,k+q (4.3) − ψL,k Qcdw +q + ψ¯ L,k ψR,k+q ∗Qcdw +q , q∼0 k∈L,σ

with τ being the imaginary time.

54

4 Excitonic CDW Fluctuations and Superconductivity

(a)

(b)

Fig. 4.1 Band structure of simplified model for TiSe2 . a The “bare” band structure in the original (unfolded) BZ consists of a hole-like valence band (red) and an electron-like conduction band (blue). The two bands have identical parabolic behaviour and are perfectly nested to each other, the nesting wavevector is Qcdw . b The mean field CDW-renormalized band. The gap between conduction and valence band is 20 , with 0 being the mean field value of the CDW order parameter. The conduction band has a Mexican hat shape while the valence band is an inversion of it, as expected

4.3 Charge Density and Auxiliary Field The real space electron charge density is defined as (the spin index has been dropped here since it is the same for both cases): ρ R = c†R c R     1  −ik·R  † † cL,k + e−iQcdw ·R cR,k = e eik ·R cL,k  + eiQcdw ·R cR,k  N k∈L k  ∈L ⎡  1  iq·R   † † = ⎣ cL,k cL,k+q + cR,k e cR,k+q N q∼0 +





k∈L

ei(q+Qcdw )·R

q∼0



† cL,k cR,k+q + e−i(q+Qcdw )·R

k∈L

 k  ∈L

⎤ † ⎦ (4.4) cR,k  +q cL,k 

Meanwhile, according to the saddle-point condition involving the auxiliary field, δSeff [∗ , ] = 0, δ(∗ , )

(4.5)

with Seff [∗ , ] defined as Seff [∗ , ] = − ln



¯

¯ ψ)e−S[ψ,ψ, D(ψ,

∗

,]

,

one can get the self-consistent relationship between the (saddle-point value of) auxiliary field and the expectation value of the charge density:

4.3 Charge Density and Auxiliary Field

Qcdw +q = ∗Qcdw +q =

55

U  σ σ ψ¯ ψ  ∝ ρQcdw +q  N k∈L,σ L,k R,k+q

U  σ ψ¯ ψ σ  ∝ ρ−(Qcdw +q)  N k∈L,σ R,k+q L,k

(4.6a) (4.6b)

The expectation value of the real space electron density can be written as ρ R  =



eiq·R ρq  + α

q∼0

=

ρ Rslow 

  ei(q+Qcdw )·R Qcdw +q + e−i(q+Qcdw )·R ∗Qcdw +q q∼0

  + α  R + ∗R .

(4.7)

Here the ρ Rslow  is the part of the charge density consisting of small wavevector components (which is a constant in this case), the α is simply a constant factor involved in the proportional relationship in Eq. (4.6). It is clear that the additional contribution of charge density (with Fourier component near Qcdw ) are proportional to the auxiliary field. Because of this relation, we will simply call  the CDW order parameter from now on. The Fourier transformation of  to real space is defined through  R = eiQcdw ·R



eiq·R Qcdw +q

(4.8a)

q∼0

= eiQcdw ·R (0 + ϕ(R))eiθ(R) ,

(4.8b)

where 0 is the mean field value of the auxiliary field, which will be explained in detail in the next section. ϕ(R) and θ (R) are the amplitude and phase fluctuations of CDW order on top of 0 . By replacing the  R with Eq. (4.8b), the electron density in Eq. (4.6) can be recast into a more familiar form: ρ R  = ρ Rslow  + 2α(0 + ϕ(R)) cos (Qcdw · R + θ (R))

(4.9)

By direct comparison with Eq. (1.1), it becomes evident that the ϕ(R) enters in the amplitude of the CDW and θ (R) is the phase degree of freedom corresponding to a sliding of CDW.

4.4 The Mean Field CDW Solution In the mean field approach to the CDW phase with wavevector Qcdw , the only nonzero Fourier component of the order parameter is Qcdw = 0 . Then, by replacing Qcdw +q = 0 δq,0 in Eq. (4.3), and Fourier transform the imaginary time to Matsubara frequency, one can obtain the CDW mean field action:

56

4 Excitonic CDW Fluctuations and Superconductivity

SMF =

 σ     −iωn − εk N ψR,k −0 σ σ ψ¯ R,k ψ¯ L,k + β 20 . σ −0 −iωn + εk ψL,k U

(4.10)

k∈L,ωn

After integrating out the fermionic fields, the effective action for 0 can be deduced to be  1 −SMF ¯ F [0 ] = − ln D(ψ, ψ)e β   N 2  = 20 − ln ωn2 + εk2 + 20 eiωn 0+ U β k∈L,ωn     N 2 4 β Ek , (4.11) = 0 − ln 2 cosh U β 2 k∈L

with Ek =



εk2 + 20

and a factor of 2 has been included in the second line of Eq. (4.11), which comes from the spin degrees of freedom. From the saddle point condition on 0 , i.e., ∂F [0 ]/∂0 = 0, one can finally arrive at the self-consistent mean field gap equation:   β E k 0 N 0 = tanh . (4.12) U 2 Ek k∈L The (mean field) CDW-renormalized band dispersion is described by ±E k , which shows that a CDW gap with value of 20 opens at the Fermi level. The conduction band has a Mexican hat shape while the valence band is simply an inversion of it, as depicted in Fig. 4.1b. The corresponding Grassmann fields for each band are denoted σ σ (conduction) and χ−,k (valence), and are related to the original fermion fields as χ+,k by a canonical transformation: σ σ σ = cos(θk )χ+,k + sin(θk )χ−,k , ψL,k

(4.13a)

σ ψR,k

(4.13b)

=

σ − sin(θk )χ+,k

+

σ cos(θk )χ−,k ,

with the transformation coefficients given by  (1 − εk /E k )/2,  cos(θk ) = (1 + εk /E k )/2. sin(θk ) =

(4.14a) (4.14b)

At this stage we have a CDW for the undoped system. In order to have a superconductor, we need to add free charge carriers, which can be either electrons or holes. Since for the case of TiSe2 Cu doping has been studied extensively in experiments, we will consider the case of donating electrons into the system in this study. Compared

4.4 The Mean Field CDW Solution

57

with the mean-field calculation based on realistic bands in Chap. 2, the simplified model captures the basic feature of the CDW-renormalized energy dispersion, e.g., Mexican hat shape. Because in real TiSe2 , the band is only partially gapped in the CDW phase, so there are remaining mobile electrons after CDW transition which can participate in SC pairing, it is thus conjectured to be easier for the emergence of the SC order.

4.5 Effective Action of CDW Fluctuations Having obtained the CDW “renormalized” quasiparticle band structure, the next step is to look for the energy dispersion of CDW fluctuations. In this section, we will achieve that by deriving the effective action for the CDW amplitude and phase modes Seff [ϕ, θ ]. Firstly, it is useful to define a slowly varying field φ(R) which is also a measure of the deviation of CDW order from the mean field configuration: (R) = eiQcdw ·R [0 + φ(R)]

(4.15)

Assuming the phase fluctuation θ to be small, we can get the relationship between φ, θ and ϕ modes according to Eq. (4.9), 1   φq ≈ ϕq − 0 θ 2 q + i0 θq . 2

(4.16)

Then, one can separate the fluctuation of the order parameter from its mean filed value in the action,      ψR,k  ,ωm −1 ¯ ¯ S= ψR,k,ωn ψL,k,ωn −(G 0 )k,k  ;ωn ,ωm + Mk,k  ;ωn ,ωm ψL,k  ,ωm  k,k ∈L ωn ,ωm

+

N U

 dτ

1  [0 + ϕ(R, τ )]2 , N R

(4.17)

where the mean field Green function G 0 and coupling matrix between fermions and CDW fluctuations, M, are defined through  −0 −iωn − εk  ;ω ,ω  δω ,ω , ) = δ −(G −1 k,k k,k n m n m 0 −0 −iωn + εn  0 −φk−k  ,ωn −ωm Mk,k  ;ωn ,ωm = . −φk∗ −k,ωm −ωn 0

(4.18a) (4.18b)

In order to get the effective action for thefluctuations,one needs to integrate out the ¯ ψ), which gives Det −G −1 fermionic fields (ψ, 0 + M . Using the identity Det(A) =

58

4 Excitonic CDW Fluctuations and Superconductivity

eTr ln(A) , one arrives at     −1 exp Tr ln(−G −1 0 ) + M = exp Tr ln(−G 0 ) + Tr ln(1 − G 0 M)

∞ 1 n Tr [(G 0 M) ] . = const. × exp − n n=1

(4.19)

It should be noted that the spin index has been dropped here since it is the same for both cases, and one just needs to double all the contributions from the integration ¯ ψ). Since we are only interested in the low energy effective action for CDW over (ψ, amplitude and phase modes, we truncate the Taylor expansion at second order and replace φ by θ and ϕ according to Eq. (4.16). Here we simply write down the final form of the effective action:    Seff [ϕ, θ ] = dτ dr ρθ (∂τ θ )2 + Jθ (∇θ )2 + ρϕ (∂τ ρ)2 + Jϕ (∇ϕ)2 + m 2 ϕ 2 . (4.20) The details of the derivation and expressions for the various coefficients are discussed in Appendix C. As expected, the phase mode is gapless (Goldstone mode) while the amplitude mode is gapped.

4.6 Fluctuation-Induced Pairing Between Quasiparticles Since we have obtained the effective action for the CDW amplitude and phase modes, we can use it as the starting point and consider the interaction between CDW “renormalized” quasiparticles mediated by them. From Eq. (4.17), one can read the coupling between fermions and CDW fluctuations has the form ⎛ ⎞ ⎜   ⎟ σ σ ∗ σ σ ψ¯ R,k,ω ψ¯ L,k,ω Scoup = − ψL,k− ψR,k+ ⎝φ p,ω p p,ωn −ω p + φ p,ω p p,ωn +ω p ⎠ . n n p,ω p

k,ωn σ

k,ωn σ

(4.21) Using Eq. (4.16), one can replace the φ field with ϕ and θ , and rewrite Scoup as: Scoup = −



ϕ p,ω p

p,ω p

+ θ p,ω p i0

 k,ωn σ

 k,ωn σ

σ σ ∗ ψ¯ R,k,ω ψL,k− p,ωn −ω p + ϕ p,ω p n



σ σ ψ¯ L,k,ω ψR,k+ p,ωn +ω p n

k,ωn σ σ σ ∗ ψL,k− ψ¯ R,k,ω p,ωn −ω p − θ p,ω p i0 n



σ σ ψR,k+ ψ¯ L,k,ω p,ωn +ω p . n

k,ωn σ

(4.22)

4.6 Fluctuation-Induced Pairing Between Quasiparticles

59

Here, the coupling between fermionic fields and θ 2 has been dropped since we are assuming θ to be small. Now we can combine Eqs. (4.20) and (4.22) and integrate out the amplitude and phase modes. The integration over the amplitude mode ϕ contributes ⎛

 1 1 1 2 βV 2 ρϕ ω p + Jϕ p2 + m 2 ( p,ω p )-half    σ σ σ ¯ σ ψL,k− ψ¯ R,k,ω p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p n exp ⎝

k,k  ωn ,ωm σ,σ 

σ σ σ ¯σ + ψ¯ L,k  ,ω ψR,k  − p,ω −ω ψR,k,ω ψL,k+ p,ω +ω m m p n n p σ σ σ ¯σ + ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n 



 σ σ σ ¯ σ +ψ¯ L,k,ω ψR,k+ p,ωn +ω p ψL,k  ,ωm ψR,k  − p,ωm −ω p n ⎛ 1 1 1 = exp ⎝ 2 βV p,ω 2 ρϕ ω p + Jϕ p2 + m 2 p    σ σ σ ¯ σ ψ¯ R,k,ωn ψL,k− p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p k,k  ωn ,ωm σ,σ 

1 σ σ σ ¯ σ + ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n 2 1 σ σ σ σ ¯ ¯ + ψL,k,ωn ψR,k+ p,ωn +ω p ψL,k  ,ωm ψR,k  − p,ωm −ω p , 2

(4.23)

so the interaction mediated by the amplitude modes is: 1 1 1 2 βV p,ω 2 ρϕ ω p + Jϕ p2 + m 2 p    σ σ σ ¯ σ ψ¯ R,k,ω ψL,k− p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p n

Sint-ϕ = −

k,k  ωn ,ωm σ,σ 

1 σ σ σ ¯ σ + ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n 2 1 σ σ σ ¯ σ + ψ¯ L,k,ω ψR,k+ p,ωn +ω p ψL,k  ,ωm ψR,k  − p,ωm −ω p n 2 Meanwhile, the integration over the phase field θ gives ⎛ exp ⎝

1 βV

 ( p,ω p )−h. f

20 1 2 ρθ ω2p + Jθ p2

(4.24)

60

4 Excitonic CDW Fluctuations and Superconductivity

   k,k  ωn ,ωm σ,σ 

σ σ σ ¯ σ ψL,k− ψ¯ R,k,ω p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p n

σ σ σ ¯σ + ψ¯ L,k  ,ω ψR,k  − p,ω −ω ψR,k,ω ψL,k+ p,ω +ω m m p n n p 



σ σ σ ¯ σ − ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n

 σ σ σ σ ¯ L,k − ψ¯ L,k,ω ψ ψ  ,ω ψR,k  − p,ω −ω R,k+ p,ω +ω n n p m m p ⎛ 1 1 20 = exp ⎝ 2 βV p,ω 2 ρθ ω p + Jθ p2 p    σ σ σ ¯ ¯ σ ψR,k,ωn ψL,k− p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p k,k  ωn ,ωm σ,σ 

1 σ σ σ ¯ σ − ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n 2 1 σ σ σ σ ¯ − ψ¯ L,k,ωn ψR,k+ ψ ψ   p,ωn +ω p L,k ,ωm R,k − p,ωm −ω p , 2

(4.25)

and the θ -mediated interaction reads 20 1 1 βV p,ω 2 ρθ ω2p + Jθ p2 p    σ σ σ ¯ σ ψ¯ R,k,ω ψL,k− p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p n

Sint-θ = −

k,k  ωn ,ωm σ,σ 

1 σ σ σ ¯ σ − ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n 2 1 σ σ σ ¯ σ − ψ¯ L,k,ω ψR,k+ p,ωn +ω p ψL,k  ,ωm ψR,k  − p,ωm −ω p . n 2

(4.26)

One should recall at this point that the Grassmann fields ψ describe the original elections and holes in the two pockets that characterize the reference band structure is the normal state. As discussed in Sect. 4.4, they are related to the excitonic (CDWrenormalized) quasi particles by the unitary transformation in Eq. (4.13). Since we are interested in the interaction between those CDW-renormalized quasiparticles, we ¯ ψ) with those of the can replace the Grassmann fields of the original fermions (ψ, new quasiparticles (χ¯ , χ ). Because we are interested in the electron-doped case, and the inter-band processes correspond to a relatively large energy scale (CDW gap is equal to 20 ), we only focus on the intra-conduction-band interaction. Picking up only the BCS type of pairing, i.e., (k ↑, −k ↓) → (k  ↑, −k  ↓), from the amplitude mode mediated interaction in Eq. (4.24), one can get for each term in Eq. (4.26) the following: σ σ σ ¯ σ ψ¯ R,k,ω ψL,k− p,ωn −ω p ψL,k  ,ωm ψR,k  + p,ωm +ω p n

4.6 Fluctuation-Induced Pairing Between Quasiparticles ↑

↓

↓

61

↑

= ψ¯ R,k,ωn ψ¯ L,−k,ωm ψR,−k  ,ωm +ω p ψL,k  ,ωn −ω p + (↑↔↓) =

1 ↑ ↓ ↓ ↑ sin(2θk ) sin(2θk  )χ¯ +,k,ωn χ¯ +,−k,ωm χ+,−k  ,ωm +ω p χ+,k  ,ωn −ω p + (↑↔↓) (4.27a) 4

σ σ σ ¯ σ ψ¯ R,k,ω ψL,k− p,ωn −ω p ψR,k  ,ωn ψL,k  + p,ωn +ω p n ↑ ↓ ↓ ↑ = ψ¯ R,k,ωn ψ¯ R,−k,ωn ψL,−k  ,ωn +ω p ψL,k  ,ωn −ω p + (↑↔↓) ↑

↓

↓

↑

= sin2 (θk ) cos2 (θk  ) χ¯ +,k,ωn χ¯ +,−k,ωm χ+,−k  ,ωm +ω p χ+,k  ,ωn −ω p + (↑↔↓) (4.27b) σ σ σ ¯ σ ψR,k+ ψ¯ L,k,ω p,ωn +ω p ψL,k  ,ωm ψR,k  − p,ωm −ω p n ↑ ↓ ↓ ↑ = ψ¯ L,k,ωn ψ¯ L,−k,ωm ψR,−k  ,ωm −ω p ψR,k  ,ωn +ω p + (↑↔↓) ↑

↓

↓

↑

= cos2 (θk ) sin2 (θk  ) χ¯ +,k,ωn χ¯ +,−k,ωm χ+,−k  ,ωm +ω p χ+,k  ,ωn −ω p + (↑↔↓) (4.27c) At low doping levels, since εk ≈ 0, the coefficients within the canonical transformation can be approximated as   1 1− 2   1 1+ cos (θk ) = 2 sin (θk ) =

εk Ek εk Ek

1/2 1/2

1 ≈√ , 2 1 ≈√ . 2

(4.28a) (4.28b)

Combining all the terms from Eq. (4.27), one obtains a BCS-type pairing arising from the fluctuations of the CDW amplitude mode SBCS-ϕ = −

1   ↑ ↓ ↓ ↑ Dk−k  ,ω p χ¯ +,k,ωn χ¯ +,−k,ωm χ+,−k  ,ωm +ω p χ+,k  ,ωn −ω p . 2βV ω  p

k,k ωn ,ωm

(4.29) The Dk−k  ,ω p here is essentially the Green function of the amplitude modes Dk−k  ,ω p =

ρϕ ω2p

1 . + Jϕ (k − k )2 + m 2

(4.30)

On the other hand, due to the negative sign before two of the contributions in Eq. (4.26), one can show that the phase mode does not contribute to a BCS type of pairing between quasiparticles, so only the CDW amplitude mode can lead to pairing. Now one can perform the Fourier transform to convert the interaction to (imaginary) time:

62

4 Excitonic CDW Fluctuations and Superconductivity

1  ↑ ↓ ↓ ↑ Dk−k  ,ω p χ¯ +,k,ωn χ¯ +,−k,ωm χ+,−k  ,ωm +ω p χ+,k  ,ωn −ω p β ω ω ,ω p n m  ↑ ↓ ↓ ↑ = − dτ1 · · · dτ4 χ¯ +,k (τ1 ) χ¯ +,−k (τ2 ) χ+,−k  (τ3 ) χ+,k  (τ4 )

−

1  iωn (τ1 −τ4 ) iωm (τ2 −τ3 ) 1 Dk−k  ,ω p eiω p (τ4 −τ3 ) 2 e e β ω β ω ,ω p n m  1 ↑ ↓ ↓ ↑ = − dτ1 dτ2 χ¯ +,k (τ1 ) χ¯ +,−k (τ2 ) χ+,−k  (τ2 ) χ+,k  (τ1 ) Dk−k  ,ω p eiω p (τ1 −τ2 ) β ω ×

p

(4.31) In order to do the summation over the Matsubara frequencies ω p , we implement the standard trick for Matsubara summations, which is to introduce an auxiliary (complex) function and convert the summation into a contour integral. Here we separate the two possible cases of τ1 − τ2 : (i) τ1 − τ2 > 0 In this case, we choose the auxiliary function as the Bosonic distribution function n B (z) = eβz1−1 : 1 1 eiω p (τ1 −τ2 ) β ω ω2p + 2ϕ,k−k  p  1 1 = dz n B (z) 2 e z(τ1 −τ2 ) 2πi C ϕ,k−k  − z 2  1  n B (k−k  ) ek−k  (τ1 −τ2 ) − n B (−k−k  ) e−k−k  (τ1 −τ2 ) = 2k−k  1 β→∞ −−−→ e−k−k  (τ1 −τ2 ) . (4.32) 2k−k  (ii) τ1 − τ2 < 0 In this case, to make the contour integral converge, we choose the auxiliary βz function as n B (z) + 1 = eβze +1 : 1 1 eiω p (τ1 −τ2 ) β ω ω2p + 2ϕ,k−k  p  1 1 eβz = dz βz e z(τ1 −τ2 ) 2 2πi C e − 1 ϕ,k−k  − z 2  βk−k  e 1 e−βk−k  k−k  (τ1 −τ2 ) −k−k  (τ1 −τ2 ) e e = − −β  k−k − 1 2k−k  eβk−k  − 1 e 1 β→∞ −−−→ ek−k  (τ1 −τ2 ) . (4.33) 2k−k 

4.6 Fluctuation-Induced Pairing Between Quasiparticles

63

In these expressions  p is simply the energy dispersion of the CDW amplitude modes [cf. Eq. (4.30)]: Jϕ p2 + m 2 . (4.34) 2p = ρϕ Finally, one can write down the amplitude mode mediated pairing interaction between the excitonic quasiparticles in imaginary time as −

1 2Vρϕ

 dτ1 dτ2

 k,k 

1 ↑ ↓ ↓ ↑ e−k−k  |τ1 −τ2 | χ¯ +,k (τ1 ) χ¯ +,−k (τ2 ) χ+,−k  (τ2 ) χ+,k  (τ1 ) 2k−k 

(4.35) It is a retarded attractive interaction in the Cooper channel, which qualitatively has the same form as the phonon mediated interaction in the BCS problem [12].

4.7 Exploration of the Joint CDW-Superconductivity Phase Diagram In order to have an idea of stability of the SC phase as a function of carrier density, we performed a numerical mean field calculation of the SC order parameter and transition temperature on a specific parameterization for the simplified model. Before implementing the mean field calculation, one needs to approximate the retarded interaction in Eq. (4.35) by an instantaneous one to get a time-independent Hamiltonian, which can be done by dropping the frequency dependence of the amplitude mode’s Green function in Eq. (4.31). By further neglecting the momentum dependence of D p,ω p , an instantaneous attractive interaction can be obtained, which has the form 1  † † χ χ χ−k  ,↓ χk  ,↑ . (4.36) Hint = − g N k,k  k,↑ −k,↓ The coupling constant g = 1/2a 2 m 2 , with a being the lattice constant of the system. Such an attractive interaction would favour an s-wave superconducting pairing between quasiparticles. It is important to note that, as this pairing is due only to the effect of excitonic fluctuations, the effective coupling constant is determined by the initial parameters of the problem. In other words, there are no new parameters to quantify the strength of this attractive interaction. Recalling that we saw in Chap. 2 that, after fixing the normal state band structure to ARPES, our theory relies only on a single constant V (the attractive interaction), this means that our approach is capable of generating both the CDW and SC transition lines as a function of doping with a single free parameter in the theory. It should be noted that this way of obtaining the instantaneous interaction will definitely overestimate the interaction strength, because the D p,ω p is larger than it should be in Eq. (4.31). There are two energy cutoffs involved in the SC pairing, the first is due to the amplitude mode

64

4 Excitonic CDW Fluctuations and Superconductivity

Fig. 4.2 CDW order parameter as a function of temperature. The CDW transition in the pristine system is of second order

30 25 20 15 10 5 0

0

50

100

150

200

frequency, which in our calculation is approximated as the mass of the amplitude  mode, i.e., m 2 /ρϕ ; the other one comes from the approximation that we used for the coefficients in the canonical transformation Eq. (4.28), and in our calculation we set |εk | ≤ 0 . In the numerical calculation, the parabolic band has the form 2 k 2 εbo − μ, + 2m c 2 2 k 2 εbo = −εk − μ = − − μ, 2m c 2

εv,k = εk − μ = −

(4.37a)

εc,k

(4.37b)

where we use band parameters appropriate for the bare band structure of TiSe2 m c = 0.63 m e , band overlap εbo = 0.2 eV, and interaction strength U = 0.1 eV. The pristine system (μ = 0) undergoes a second order phase transition to the CDW phase at low temperatures with Tc ≈ 200 K (see Fig. 4.2), the mean field value of the CDW order parameter is 0 ≈ 0.03 eV at zero temperature. From a numerical calculation, the SC pairing coupling constant g ∼ U . The SC order parameter  as a function of temperature is shown in Fig. 4.3a, which reveals a second order nature of the SC phase transition, as expected. The predicted phase diagram of the simplified model is shown in Fig. 4.4. The CDW transition is of second order with a small amount of doped electrons (blue diamonds, obtained from mean-field calculation) and becomes first order at higher doping levels (blue dashed line), which is consistent with the results based on a realistic band dispersion of TiSe2 shown in Chap. 2. The red diamonds indicate the SC transition temperature Tsc obtained from mean-field calculation with the assumption that the CDW order parameter being fixed ( = 30 meV). As one can see from Fig. 4.4, when the chemical potential starts to increase from the bottom of the CDW-renormalized conduction band (μ = 30 meV), Tsc increases first, then decreases and becomes suppressed at relatively high doping levels. The red dashed line is inferred by extending the calculated data. Since in realistic TiSe2 the CDW-renormalized band structure is only partially gapped, it is natural to expect that the remaining mobile electrons would

4.7 Exploration of the Joint CDW-Superconductivity Phase Diagram

(a)

65

(b)

Fig. 4.3 SC order parameter as a function of temperature. a The SC transition is of second order. As the chemical potential increases from the bottom of the CDW-renormalized conduction band (μ = 0.03 eV), the SC phase is favoured first and then suppressed. b CDW-renormalized band structure (along x-direction) for pristine system. The dashed lines indicate two chemical potential values corresponding to low (μ = 0.03 eV) and relatively high doping levels (μ = 0.054 eV)

Fig. 4.4 Phase diagram inferred from the simplified model. The blue diamonds indicate the second order (low doping) CDW transition temperature Tcdw and red diamonds stand for the SC transition temperature Tsc , Tcdw has been obtained by solving Eq. (4.12) at different μ, and Tsc corresponds to the mean field solution of the pairing interaction Eq. (4.36). The blue dashed line stands for the regime where the CDW transition becomes first order. When the chemical potential increases from the bottom of the CDW-renormalized conduction band, the SC order is initially enhanced at low doping, but is ultimately suppressed above a critical doping. This curve of Tsc vs doping is shaped like a dome. The red dashed line is conjectured by continuing the calculated ones. The changing of CDW transition to first order suggests that in real space CDW domains are separated by CDW fluctuations (DCs). Combined with the results from the G-L theory in Chap. 3, we thus conjecture that the SC dome is surrounded by NC-CDW and ICDW, as suggested by the experimental phase diagram [9]

participate SC pairing before the chemical potential reaching the bottom of the CDWrenormalized conduction band. As has been discussed in Chap. 2, the fact that the CDW transition changes to first order can be understood as an indication of phase

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separation. We thus expect that, although there is no true long range order (predicted from mean-field theory) above the blue dashed lines, there are CDW domains separated by domain walls (DCs). Combined with the result obtained from G-L theory in Chap. 3, we conjecture that the SC dome is surrounded by the NC-CDW and ICDW regimes. It should be noted that the Tsc obtained from our calculation is overestimated, which is due to the approximations we made of simplifying and using an instantaneous interaction, as has been discussed above. In addition, recall that our simplified representation of the election-hole pockets lead to perfect nesting and the opening of a full CDW gap which is only partial predicted from the realistic treatment of the band structure in Chap. 2. Hence, the current approximations create a stronger renormalization of the electronic quasiparticles and are thus also expected to slightly overestimate the fluctuations. These considerations lead to the conclusion that, if one uses this realistic band structure and does not suppress the retardation in the pairing interaction, the values of Tsc would come down and approach the experimental ones in TiSe2 . Finally, the values of Tsc reported in Fig. 4.4 are obtained in a mean field calculation which itself overestimates the critical temperatures. Considering all these aspects, we believe that the fluctuation-induced pairing can indeed be the dominant mechanism responsible for the appearance of SC order in TiSe2 in a strong inter-dependence with the loss of commensurability of the underlying CDW.

4.8 Summary In this chapter, we have studied the interplay between CDW fluctuations and SC order microscopically based on a simplified model for the band structure of TiSe2 in the normal state. Due to a perfect nesting between the two (electron and hole) pockets, the system exhibits a CDW instability at low temperatures and opens a gap (20 ) at the Fermi surface, which is confirmed by a mean field calculation. After obtaining the mean field value of the CDW order parameter, we expanded the order parameter on top of it and introduced the CDW amplitude and phase modes (CDW fluctuations). In order to study pairing mediated by these fluctuations of the charge density, we first derived the effective action of the amplitude and phase modes Seff [ϕ, θ ]. It was found that the amplitude mode is gapped while the phase mode is gapless, which corresponds to a Goldstone mode. It should be noted that the inter-valley interaction in our simplified model does not capture the umklapp effect which is due to electronphonon coupling in a CCDW phase and can give rise to a gapped phase mode, thus our model is suitable for an electronic correlation induced CCDW or electron-phonon coupling induced ICDW phase. In the electron-doped case, the additional electrons “feel” the (mean-field) CDW background and are described as new quasiparticles with CDW-dressed (conduction) band dispersion. By integrating out the amplitude and phase modes based on Seff [ϕ, θ ] and their coupling to fermions in the Cooper channel, we found that the phase mode does not contribute to any pairing, while the amplitude mode can mediate a retarded attractive interaction between the CDW

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quasiparticles. Finally, a numerical calculation on a specific parameterization of the simplified model confirms the existence of a second order SC phase transition. As the system is doped (chemical potential μ starts to increase from the bottom of conduction band), the SC order gets enhanced at first and then gets suppressed, resulting in a dome shape of the SC phase in the phase diagram. Although the effect of doping on the CDW order has not been studied selfconsistently in this work, in the case of TiSe2 it has been confirmed that the CDW gap persists up to doping levels where SC order exists [9]. Thus, it is reasonable to believe the qualitative results from our simplified model are correct. The results from the microscopic study are also in line with the G-L study described in Chap. 3, where it has been shown that the SC order is enhanced by CDW fluctuations (the discommensurations). To conclude, our microscopic study indicates that, the CDW fluctuations can induce, or at least enhance, the SC pairing in TiSe2 .

References 1. Schrieffer JR, Wen X-G, Zhang S-C (1988) Spin-bag mechanism of high-temperature superconductivity. Phys. Rev. Lett. 60:944–947 2. Schrieffer JR, Wen XG, Zhang SC (1989) Dynamic spin fluctuations and the bag mechanism of high-Tc superconductivity. Phys Rev B 39:11663–11679 3. Peets DC, Hawthorn DG, Shen KM, Kim Y-J, Ellis DS, Zhang H, Komiya S, Ando Y, Sawatzky GA, Liang R, Bonn DA, Hardy WN (2009) X-ray absorption spectra reveal the inapplicability of the single-band hubbard model to overdoped cuprate superconductors. Phys Rev Lett 103:087402 4. Batlogg B, Takagi H, Kao H-L, Kwo J (1993) Charge dynamics in metallic CuO2 layers. Electronic properties of high-Tc superconductors. Springer, Berlin, pp 5–12 5. Chang J, Blackburn E, Holmes AT, Christensen NB, Larsen J, Mesot J, Liang R, Bonn DA, Hardy WN, Watenphul A et al (2012) Direct observation of competition between superconductivity and charge density wave order in YBa2 Cu3 O6.67 . Nat Phys 8:871 6. Cercellier H, Monney C, Clerc F, Battaglia C, Despont L, Garnier MG, Beck H, Aebi P, Patthey L, Berger H, Forró L (2007) Evidence for an excitonic insulator phase in 1T-TiSe2 . Phys Rev Lett 99:146403 7. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T -TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602 8. Novello AM, Spera M, Scarfato A, Ubaldini A, Giannini E, Bowler DR, Renner Ch (2017) Stripe and Short Range Order in the Charge Density Wave of 1T-Cux TiSe2 . Phys Rev Lett 118:017002 9. Kogar A, de la Pena GA, Lee S, Fang Y, Sun SX-L, Lioi DB, Karapetrov G, Finkelstein KD, Ruff JPC, Abbamonte P, Rosenkranz S (2017) Observation of a charge density wave incommensuration near the superconducting dome in Cux TiSe2 . Phys Rev Lett 118:027002 10. Stratonovich RL (1957) On a method of calculating quantum distribution functions. Dokl Phys 2:416 11. Hubbard J (1959) Calculation of partition functions. Phys Rev Lett 3:77–78 12. Bruus H, Flensberg K (2004) Many-body quantum theory in condensed matter physics: an introduction. Oxford University Press, Oxford

Chapter 5

Anomalous Quantum Metal Phase in TiSe2

5.1 A Brief Introduction to the Anomalous Quantum Metal Phase The superconductor-insulator transition in low dimensional (D ≤ 2) electronic systems has been extensively studied since the late 1980s. In the early stages of this research area, it was found experimentally that, by either tuning the thickness of a Bi film deposited Ge [2] or changing the external magnetic field in InOx [3], the system can actually undergo a superconductor insulator transition, apparently without an intermediate metallic phase (see Fig. 5.1). At zero temperature (T → 0), systems with large thickness (small magnetic field) show zero resistance, which indicates a superconducting ground state, while for small thickness (high magnetic field), an insulator with infinite resistance is achieved. Moreover, in the thickness-tuning experiment on Ge, right at the critical point, the value of the resistivity is universal and equals to quantum of resistance for charge 2e Cooper pairs (ρ Q ≡ h/4e2 ) [2]. From the theoretical side, this superconductor-insulator transition has been regarded as a paradigm for quantum phase transitions (QPT) [4]. Such a direct superconductor-insulator transition can be captured by a Josephson junction array Hamiltonian [5] (or Bose–Hubbard model which can be mapping to Josephson junction array at integer fillings [6]):   nˆ 2j − J cos(θˆi − θˆ j ) (5.1) H = EC j

i, j

Since the Cooper pairs are always retained and pair breaking processes are not included, this model is also commonly referred to as a phase-only model (the amplitude of the SC order parameter || is not affected). Due to the canonical conjugate relation between particle number and phase This chapter is taken and edited from Ref. [1] of which the author of this thesis is the second author overall, and first theoretical one in the author list. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_5

69

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5 Anomalous Quantum Metal Phase in TiSe2

Fig. 5.1 Experimental results on the superconductor-insulator transition. a Evolution of the temperature dependence of the sheet resistance R(T ) with thickness for a Bi film deposited onto Ge [2]. b Logarithmic plots of the resistance transitions at zero field (filled circle) and nonzero field (open symbols) for a film with Tc = 0.29 K [3]

[nˆ i , θˆ j ] = iδi, j ,

(5.2)

the system favours a bosonic insulating ground state when the effective coupling g = E C /J is large [6]. In this state, the Cooper pairs (bosons) are localized and there is a well defined particle number at each SC grain (eigenstate of nˆ j ) and the phase is randomized. On the other hand, in the limit where g ∼ 0, the phase coherence term dominates (ground state minimizes the cos(θˆi − θˆ j ) term) and there is no cost for charging a grain. Cooper pairs can then tunnel “freely” between the grains and a phase coherent SC ground state is obtained. Right at the critical point gc , it has been shown that the resistivity should be proportional to ρ Q , with a coefficient not necessarily being equal to one, even in the case where long-range Coulomb interaction between SC grains is included [5, 7, 8]. So it seemed that the experiments could be very well described by this theory. However, such an agreement didn’t last for a long time. By changing the thickness of Ga deposited on alumina, the low temperature (T → 0) levelling of resistivity does not go to infinity immediately after the SC phase. Instead, the resistivity continuously increases as the sample’s thickness decreases and diverges (insulator) for sufficiently thin limit (see Fig. 5.2). In other words, an intervening metallic phase exists between the SC and insulating phases. Moreover, in the metallic phase near the critical point (critical thickness gc ) of the SC to metal transition, the resistivity turns on as a power

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Fig. 5.2 Experimental observation of an AQM. Resistance versus temperature at different magnetic fields [10]. The resistance saturates at low temperatures and increases gradually from zero after the quenching of the SC phase

of g − gc [9]. Similar behaviour is also found in the magnetic field tuned systems. As the temperature is lowered, the resistivity starts to saturate from a thermally activated flux flow regime (TAFF). For a certain range of magnetic field (H < Hc2 ), the resistivity in the zero temperature limit is also finite and much smaller than ρ Q [10–12]. To understand this peculiar metallic phase, the first thing one should figure out is the nature of the charge carriers, i.e., whether they are bosons (Cooper pairs) or fermions (electrons). From tunnelling experiments, the critical field (Hc ) for the metal to insulator transition is found to be proportional to the amplitude of the SC order parameter (||). Because such a relationship also holds for the field at which Cooper pairs start to fall apart into electrons (Hc2 ), it indicates that Hc ≈ Hc2 [13]. Within the intervening metallic phase, since H Hc2 , it is reasonable to believe that bosons (Cooper pairs) carry the current. So the problem has been to construct a theory which predicts a bosonic metallic phase. Several models have been proposed to capture the metallic phase between the SC and insulator states. Since the basic excitations are Cooper pairs, some theories are based on Josephson junction array embedded on a normal metal where, by design, a SC to metal transition is predicted and the failure of SC phase simply unmasks the metallic background [14]. In the absence of the host metal, the normal electrons provide the dissipation within such hybrid models. However, the metallic phase only exists at the separatrix between SC and insulator transition instead of persisting for a finite range of the external driving parameter (field or disorder) [15]. In other models, Coulomb interactions between superconducting grains were included. It was proposed that the system can become a superconductor when the grains exceed some critical size and the shortrange Coulomb interaction is crucial near the QCP for the SC to metal phase [16]. On the other hand, scaling arguments demonstrate that the short-range Coulomb interaction is irrelevant near the corresponding QCP [17, 18]. Meanwhile, it has been suggested that dissipation might play an important role for the existence of metallic phase [19, 20]. However, it was shown that the conventional way of treating the dissipation in a Josephson junction array cannot give rise to

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a metallic phase at T = 0 [21, 22]. All these facts motivated Philip Phillips and collaborators to establish a quantum-phase glass model where the dissipation is selfgenerated [18]. To capture the SC and glassy phase, the Josephson coupling Ji j is allowed to vary with some probability distribution supporting a non-zero (positive) mean value. The randomized value of Ji j is due to the scattering of Cooper pairs by disorder [23]. In the glassy phase, the phase at each site has a given direction although the global phase coherence is missing. The glass order parameter couples to the bosonic excitations from the SC order and serves as the origin of dissipation. More interestingly, the calculation of the conductivity through Kubo formula shows that, near the transition from SC to the metallic phase, resistivity scales as a power law with the coupling constant g − gc . This phase glass model is thus a good candidate for explaining the intervening metallic phase, which is also designated as Bose metal by Philip Phillips et al. [24–27] or anomalous quantum metal (AQM) by Kapitulnik et al. [28]. Although the superconductor-insulator transition and SC to quantum metal transition has been studied extensively in thin-film superconductors, it has not been probed in a crystalline 2D superconductor with coexisting and fluctuating quantum orders. This chapter describes my theoretical contribution to a collaboration that has studied the superconducting phase in 2D films of ion-gel gated TiSe2 under magnetic fields. The experiments were carried out by Dr. Linjun Li who is the first author of reference [1]. Even though the emphasis in this thesis is on the theoretical interpretation and modelling of these experiments, for completeness, I provide a brief overview of the experiments in the following section. The theoretical modelling and analysis of the experiments is done in the subsequent sections.

5.2 Summary of the Experiments Transport measurements were carried out in top-gated TiSe2 electrical double-layer transistors using an ion-gel solution. As illustrated in Fig. 5.3a, b, while the charge distribution in the gel is spatially uniform at zero gate (panel (a)), at finite gate voltages the electric field promotes charge separation and accumulation in surface charge layers that substantially contribute to increase the nominal carrier number in the target nanosheet of TiSe2 . This permits the large field-effect doping necessary to drive the system into the superconducting regime at low temperatures and, as a result, allows one to map the phase diagram of TiSe2 as a function of the three key parameters shown in Fig. 5.6. As having the ion gel in direct contact with the TiSe2 flake can lead to detrimental chemical reactions or ion intercalation, we separated them by encapsulating the whole device with an atomically thin spacer (1–2 layers) of crystalline hexagonal boron nitride. This is an important advantage in studying the possible existence of quantum phase transitions because it removes sample variability and the concomitant variation in the level of electronic disorder. A schematic of these devices is shown in Fig. 5.3 (hBN sheet is not shown for keeping simplicity) and

5.2 Summary of the Experiments

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Fig. 5.3 Two-dimensional superconductivity in a gated 1T-TiSe2 nanosheet device. a and b illustrate the process and consequences of ion gel gating (to keep simplicity, hBN layer is not shown). Without gate (a) the ion gel is not polarized and TiSe2 undergoes only one phase transition as a function of temperature to its intrinsic, homogeneous, CCDW phase. A finite voltage applied to the top gate (b) polarizes the ion-gel which, in turn, dopes TiSe2 . In a finite interval about optimal doping and intermediate temperatures, the CDW phase breaks into CCDW domains separated by a network of DCs [29]. Superconductivity emerges upon further lowering of temperature in this region. Panel c displays the measured I–V curves near optimal doping (n = 4 × 1014 cm−2 ) and different temperatures. The transition temperature (TBKT ) is defined as that when V ∝ I 3 . The inset shows the resistive transition at zero field, where Tc = 2.3 K is defined as the temperature at which the resistance drops to 90% of the normal state value. Reprinted (adapted) with permission from Nano Letters 2019 19 (6), 4126–4133. Copyright 2019 American Chemical Society

additional details can be found in Ref. [29], which established this as a dependable and versatile strategy to map the phase diagram as a function of density using one single sample. The zero-field superconducting transition is of the BKT type which places our devices in the two-dimensional regime. Figure 5.3c shows the nonlinear I –V characteristics at different temperatures, when the carrier density in the device is tuned to n ≈ 4 × 1014 cm−2 , near optimal doping. The distinctive power law behavior, V ∝ I α , immediately below the critical current is characteristic of the BKT transition where the critical temperature TBKT is identified as that corresponding to α = 3. We obtain TBKT ≈ 1.6 K, indicated by the dashed line in the figure. In addition, we define another temperature, Tc , for the onset of superconducting correlations as the temperature at which the resistance drops to 90% of the normal state value. For the near optimal doping densities used in Fig. 5.3c, we have Tc ≈ 2.3 K, which is considerably above the BKT transition. On the one hand, a broad temperature separation between the onset of pairing at Tc and the development of quasi long-range phase coherence at TBKT [(Tc − TBKT /TBKT ) 0.44] in a clean system is expected as a result of the largely enhanced thermal fluctuations in this 2D crystal. On the other hand, such a temperature separation could also be a signature of inhomogeneous superconductivity, in which case the transition width is proportional to the normal state resistance (Rn ) [30]. In this context, we note that the device has Rn 500 / at this doping, far from the resistance quantum, ρ Q ≡ h/4e2 6.4 k, and also much

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smaller than in typical MoGe and Ta thin films [10]. The k f l value can be estimated as k f l = h/(2e2 )/Rn ≈ 26, much larger than the Ioffe-Regel limit (k f l ≈ 1), showing that the normal state is in the clean regime. Therefore, we attribute the relatively large separation between Tc and TBKT here to strong thermal fluctuations in the phase of the superconducting order parameter, rather than extrinsic inhomogeneity. In addition, being in the clean regime enables the observation of the quantum metallic state in TiSe2 at low temperatures, similarly to recent reports in crystalline bilayer NbSe2 [31] and ZrNCl [32].

5.3 Theoretical Interpretation and Background 5.3.1 Thermally Activated Flux Flow Figure 5.4 summarizes the magnetic field dependence of the sheet resistance near optimal doping. Panel (a) shows the resistance as a function of both temperature and the perpendicular external field. If we define a temperature-dependent upper critical field, Hc2 (T ), as the threshold at which the resistance crosses 90% of the normal state value, we see that it displays the typical mean-field-like diagram (dashed line in Fig. 5.4a). Constant-field traces are plotted in panel (b) as a function of reciprocal temperature and show the hallmarks of the AQM state in the extrapolated T ≈ 0 limit (the lowest achievable temperature in our experiments is 0.25 K): the saturation of

Fig. 5.4 Magnetic field and temperature dependence of the resistance. a Density plot of the sheet resistance (R S ) as a function of perpendicular magnetic field and temperature near optimal doping (n = 4 × 1014 cm−2 ). The line labeled Hc2 (T ) marks the onset of resistance drop with respect to the normal state. b The same data shown in a, except that RS is now plotted against 1/T for different fields. The resistance drop is characterized by two distinct temperature regimes: at intermediate temperatures, below Tc , the resistance displays thermally activated behavior, as emphasized by the dashed lines in the figure. Below a field-dependent crossover temperature (Ta ), R S plateaus at finite values establishing the presence of a metallic state at zero temperature. Panel c displays the semi-logarithmic plot of U/k B versus H , where U is the thermal activation energy derived from the slope of the dotted lines in b. The data can be fit with a dependence U (H ) = U0 ln(H0 /H ) (solid line). Reprinted (adapted) with permission from Nano Letters 2019 19 (6), 4126–4133. Copyright 2019 American Chemical Society

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the sheet resistance, R S (T ≈ 0), at finite values much smaller than Rn , combined with the systematic increase of R S (T ≈ 0) with increasing field up to Hc2 (0). There is a giant magnetoresistance in this regime, with the resistance at saturation spanning more than two orders of magnitude below Rn . The dissipative regime that takes place between the onset of resistance saturation and the normal state shows behavior characteristic of TAFF: R S (T, H ) = R0 exp[−U (H )/k B T ], which is highlighted by the dashed lines in Fig. 5.4b. The activation barrier is seen in Fig. 5.4c to vary with magnetic field as U (H ) = U0 ln(H0 /H ), which is typical of collective flux creeping [11, 33], and our fit yields U0 = 4.8 K and H0 = 0.30 T. In order to mark the crossover between the TAFF and the AQM regimes for a given field, we define Ta as the temperature at which the dashed line (TAFF) intersects the solid horizontal one (saturation); its dependence on magnetic field is shown in Fig. 5.6, where we see that Ta decreases with increasing field.

5.3.2 Bose Metal and Vortex Quantum Creeping From Fig. 5.4, one can see an AQM state exists at T = 0 K, which has challenged the microscopic understanding of the possible ground states in this problem. As we have mentioned in Sect. 5.1, Wu and Phillips [27, 34] have explored the possible emergence of a Bose metal in the so-called phase glass at finite fields. In particular, they predicted the resistance at the zero-temperature SC to quantum metal transition to scale with field as (5.3) R(H ) ∝ (H − HC0 ) y , where HC0 is the critical field for the SC to quantum metal transition and y is derived from the power-law divergence of the superconducting coherence length [27]; Wu and Phillips specifically predict y = 2. This scaling has been observed recently in the 2D superconducting bilayer NbSe2 [31] and, in the following, we describe how this is also observed at the SC to quantum metal transition in our case of TiSe2 at small fields. Figure 5.5a reports the field dependence of the longitudinal resistance at temperatures below Ta (i.e., in the saturation region of Fig. 5.4c, outside the TAFF regime). There is an interval approximately between 0.01 and 0.2 T where it behaves as R(H ) ∝ (H − Hc0 ) y (below 0.01 T, the data falls to noise level), as highlighted by the straight lines superimposed on each experimental curve. The exponent y that best fits the data in this interval of magnetic field is indicated in the inset; it approaches y ≈ 2 at our lowest temperature, which tallies with the calculations of Wu and Phillips [27]. The fitting also gives us HC0 ≈ 0 T at the lowest temperature, which is in accordance with the fact that the R(H ) traces in Fig. 5.5a have an upward-positive curvature as H → 0, indicative of the finite resistance at zero field all the way down to, and including, T = 0.25 K.

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Fig. 5.5 Magnetic field scaling of the resistance in the anomalous metallic state. a In a field range Hc0 < H < Hc , where Hc < Hc2 is a crossover field that depends weakly on temperature (see Fig. 5.6a), the sheet resistance at fixed T can be described by the dependence R S ∝ (H − Hc0 ) y characteristic of the phase glass model in the low field limit. The inset reports the exponent y obtained by fitting the curves at the different temperatures shown in the main graph. Panel b shows that, at higher magnetic fields, Hc < H < Hc2 , the measured resistance can be satisfactorily fit by the expression in equation (2) predicted for dissipation due to quantum creeping of vortices. c The magnetic field dependence of the I –V curves at T = 0.25 K. The inset magnifies the behavior near the origin at small fields where no hysteresis is observed. Reprinted (adapted) with permission from Nano Letters 2019 19 (6), 4126–4133. Copyright 2019 American Chemical Society

This scaling is no longer capable of accurately describing the measured traces above H ≈ 0.2 T and up to Hc2 , as is clear already in Fig. 5.5a from the negative curvature of the data in that field range. Instead, we found that the resistance is best described at these fields by the dependence proposed for dissipation in superconductors due to quantum tunneling of vortices (quantum creeping) at T < Ta [19, 32]: 

Cπ  H − HC κ ∝ exp Rn e 2 H κ R(H ) ∝ R Q , 1−κ

 (5.4a) (5.4b)

where C is a dimensionless constant. Figure 5.5b shows the best fit to this field dependence. The AQM in our TiSe2 sample is thus exhibiting a crossover at H ∼ Hc between the regimes described by Eqs. (5.3) and (5.4). The obtained crossover field, Hc , is essentially independent of temperature (Fig. 5.6a), as expected from a purely quantum-mechanical mechanism. Such specific crossover has been predicted by Das and Doniach, albeit in the context of strongly disordered superconductors where the AQM gives way to an insulating state at high field [35]. Nevertheless, despite our TiSe2 sample being in the clean limit, one expects the same physics underlying the field-driven crossover between the two regimes to be at play here. Namely, whereas in the phase glass picture the SC to quantum metal transition and the scaling Eq. (5.3) is primarily a result of gauge field fluctuations [27, 35], these are overcome by zero-point quantum fluctuations at higher fields. In this case, the resistance becomes “activated” with magnetic field which is the parameter controlling the strength of quantum fluctuations [35]. While the field-dependence Eq. (5.3) has been reported

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recently in NbSe2 [31] and the behavior Eq. (5.4) has been separately seen in ZrNCl [32], the observation of the crossover in a single sample is, to the best of our knowledge, so far unique to gate-doped TiSe2 .

5.3.3 Importance of the CDW Background Our observations related to this TiSe2 device with density tuned near optimal doping are globally summarized in the H –T phase diagram of Fig. 5.6a. The normal state is separated from the phases exhibiting superconducting correlations at low T and low H by the mean-field-type Hc2 (T ) line. At zero field, superconducting correlations begin developing at Tc ≈ 2.3 K, but vortex-antivortex excitations remain unbound down to TBKT ≈ 1.6 K, at which temperature the BKT transition takes place and a true superconducting state (R S = 0) is expected in a prefect 2D superconductor. At finite but small fields, the system transitions from the normal state to the TAFF regime at Tc (H ) with decreasing temperature, followed by a crossover to the AQM regime below Ta . In the portion of the phase diagram below the line Ta (H ) it remains

Fig. 5.6 Phase diagram of two-dimensional superconducting 1T-TiSe2 . a The field- temperature phase diagram when the carrier density is tuned to near optimal doping (n = 4.0 × 1014 cm−2 ). Thermally assisted flux flow (TAFF) exists between the lines labeled Hc2 (T ) and Ta (H ) while the anomalous quantum metal (AQM) is observed below the Ta (H ) line. With increasing field in the region T ≈ 0, the system transitions from a dirty Bose metal (BM) regime to vortex quantum creeping (VQC) around Hc , before reaching the normal state (Normal) for fields higher than Hc2 (T ). The scaling exponent zν obtained for different temperature ranges is also indicated. Its systematic increase from the clean to quantum percolation regimes with decreasing temperature is indicative of spatial inhomogeneity likely attributed to the underlying CDW order, and the dominant role of quantum fluctuations at the lowest temperatures (see text). b Phase diagram extended along the density axis summarizing the transport behavior for the under- and over-doped cases. The AQM is prevalent as T → 0, and the different regimes have a dome-like dependence on carrier density. Reprinted (adapted) with permission from Nano Letters 2019 19 (6), 4126–4133. Copyright 2019 American Chemical Society

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metallic as T → 0, although the extrapolated sheet resistance can be more than two orders of magnitude smaller than Rn Fig. 5.4b; this giant positive magnetoresistance is in correspondence with the prevalent behavior of the AQM in a variety of other 2D superconductors having Rn R Q [28]. With increasing field, the resistance of the AQM displays the crossover discussed above which, although first predicted in reference [35], has remained unreported and can be an indication that, in TiSe2 , there is a more substantial enhancement of quantum fluctuations by the external field. There are two important issues that we should now briefly address: dissipation and spatial non-uniformity. In relation to the first, one must ponder whether the AQM might be simply an effect arising from the presence of the ionic gate placed at a subnanometric separation from the current channel, thus acting as a potential source of dissipation. However, controlled experiments in MoGex indicate that the close proximity of a metallic gate tends to act against the metallic state [36]. This, combined with the similarity in the behavior seen here in TiSe2 to that reported in other ionicgated crystalline superconductors [32], supports intrinsic dissipation channels. At the level of microscopic models, metallic states have certainly been obtained when dirtybosons are coupled ad-hoc to gapless degrees of freedom. Nonetheless, Phillips and collaborators have shown that the Bose-metallic state does not necessarily require external dissipation but only a lack of phase coherence [34, 35] (plus interactions) and, moreover, have shown that dissipation can in fact be self-generated by coupling to the gapless excitations of the phase glass [27] (it is in the context of this model that their specific prediction y = 2 for the scaling in Eq. (5.3) arises). For the case under consideration, however, it is important to recall that the superconducting dome of TiSe2 arises from (and coexists with) a well-established CDW order. Crucially, the fact that superconductivity emerges only when doping causes a commensurate to near-commensurate CDW transition [37–39] is unlikely to be a coincidence. As we have discussed in previous chapters, the presence of CDW phase fluctuations can simultaneously contribute to enhance the pairing (by fluctuation-induced pairing, thereby explaining the coincidence in the loss of CDW commensuration with the superconducting dome) as well as provide a natural, intrinsic dissipation channel for the preformed Cooper pairs, necessary to stabilize the Bose metal. This brings us to the issue of spatial non-uniformity. Irrespective of whether CDW fluctuations conspire or not to promote superconducting pairing and dissipation, their presence unavoidably implies an intrinsic non-uniformity of the electronic system. In fact, as has also been shown in Chap. 3, the mean-field-level solution of the C-IC transition in this class of dichalcogenides consists, in the vicinity of the transition, of a 2D superlattice structure of finite-sized CCDW domains separated by DCs in the form of phase slips of the CDW order parameter [40]. As this state of affairs remains below the superconducting Tc , we conjecture that the spatial non-uniformity of the underlying electronic system in the normal state translates into the non-uniform development of a superconducting order parameter [41]. Ironically, despite its clean crystalline nature, this would render the microscopic situation in TiSe2 somewhat

5.3 Theoretical Interpretation and Background

79

similar to that of granular superconducting films, although for very different reasons and with important differences: (i) the inhomogeneity is intrinsic, rather than extrinsic; (ii) it is not caused by static disorder, but rather a combination of a largescale mean-field (static) super-periodicity of CCDW domains with dynamical CDW fluctuations. In this situation, a natural starting point is to consider a network of superconducting domains Josephson-coupled to each other, which indeed has been the common launch pad to most attempts to model the Bose metal microscopically. Since the CDW does not gap the electronic spectrum and TiSe2 is thus a relatively good metal in the non-superconducting NC state, it is reasonable to assume that, in the above “granular” picture, the superconducting domains are embedded in a metallic matrix alive with CDW fluctuations (indirect evidence for this metallic background is provided by the presence of a zero-bias conductance peak in these devices [41]). The conditions for the development of an AQM phase in this scenario have been studied in detail by Spivak and collaborators [42] (although without any coexisting CDW order).

5.4 Summary In conclusion, detailed magnetotransport measurements on an ion-gel gated 2D device have been made to explore the nature of superconductivity in TiSe2 sheets as a function of magnetic field and deviations of density from optimal doping. Having a BN spacer and a single-crystal sample ensures a clean device. Our key observations are summarized in the 3-parameter phase diagram of Fig. 5.6. Most notably, between the lowest temperatures and Ta ∼ 0.7 − 1.0 K, transport is dominated by an anomalous quantum-metallic phase at all finite fields. The giant positive magnetoresistance in this phase displays a crossover between the two regimes proposed within the phase glass picture [16, 27, 34, 35], which are designated “Bose metal” and “vortex quantum creeping” in the phase diagrams of Fig. 5.6. Since the onset of SC behavior in TiSe2 coincides with the disruption of commensurate CDW order through DCs, we advance that the development of the SC order parameter is inescapably intertwined with that of the charge density and its fluctuations. This has a direct implication in terms of providing both an intrinsic spatial non-uniformity for the development of the AQM, as well as a natural dissipation channel via phase fluctuations of the CDW. The presence of an additional quantum-fluctuating order parameter might explain what seems to be a persistence of the AQM in the absence of magnetic field. These findings and the gate-tunability of TiSe2 open the door to exploring, in a controlled way, the fate of superconductivity in 2D in the presence of competing or coexisting orders.

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References 1. Li L, Chen C, Watanabe K, Taniguchi T, Zheng Y, Xu Z, Pereira VM, Loh KP, Castro Neto AH (2019) Anomalous quantum metal in a 2D crystalline superconductor with electronic phase nonuniformity. Nano Lett 19(6):4126–4133. PMID:31082262 2. Haviland DB, Liu Y, Goldman AM (1989) Onset of superconductivity in the two-dimensional limit. Phys Rev Lett 62:2180–2183 3. Hebard AF, Paalanen MA (1990) Magnetic-field-tuned superconductor-insulator transition in two-dimensional films. Phys Rev Lett 65:927–930 4. Goldman AM (2011) Superconductor-insulator transitions. BCS: 50 years. World Scientific, Singapore, pp 255–275 5. Fisher MPA (1990) Quantum phase transitions in disordered two-dimensional superconductors. Phys Rev Lett 65:923–926 6. Phillips P (2012) Advanced solid state physics, 2nd edn. Cambridge University Press, Cambridge 7. Wen X-G, Zee A (1990) Universal conductance at the superconductor-insulator transition. Int J Mod Phys B 4:437–445 8. Herbut IF (1998) Finite temperature transport at the superconductor-insulator transition in disordered systems. Phys Rev Lett 81:3916–3919 9. Christiansen C, Hernandez LM, Goldman AM (2002) Evidence of collective charge behavior in the insulating state of ultrathin films of superconducting metals. Phys Rev Lett 88:037004 10. Yazdani A, Kapitulnik A (1995) Superconducting-insulating transition in two-dimensional a-MoGe thin films. Phys Rev Lett 74:3037–3040 11. Ephron D, Yazdani A, Kapitulnik A, Beasley MR (1996) Observation of quantum dissipation in the vortex state of a highly disordered superconducting thin film. Phys Rev Lett 76:1529–1532 12. Mason N, Kapitulnik A (2001) True superconductivity in a two-dimensional superconductinginsulating system. Phys Rev B 64:060504 13. Hsu S-Y, Chervenak JA, Valles JM Jr (1995) Magnetic field enhanced order parameter amplitude fluctuations in ultrathin films near the superconductor-insulator transition. Phys Rev Lett 75:132–135 14. Spivak B, Zyuzin A, Hruska M (2001) Quantum superconductor-metal transition. Phys Rev B 64:132502 15. Wagenblast K-H, van Otterlo A, Schön G, Zimányi GT (1997) New universality class at the superconductor-insulator transition. Phys Rev Lett 78:1779–1782 16. Das D, Doniach S (1999) Existence of a Bose metal at T = 0. Phys Rev B 60:1261–1275 17. Cardy J (1996) Scaling and renormalization in statistical physics. Lecture notes in physics. Cambridge University Press, Cambridge 18. Phillips P, Dalidovich D (2002) Short-range interactions and a Bose metal phase in two dimensions. Phys Rev B 65:081101 19. Shimshoni E, Auerbach A, Kapitulnik A (1998) Transport through quantum melts. Phys Rev Lett 80:3352–3355 20. Kapitulnik A, Mason N, Kivelson SA, Chakravarty S (2001) Effects of dissipation on quantum phase transitions. Phys Rev B 63:125322 21. Dalidovich D, Phillips P (2000) Fluctuation conductivity in insulator-superconductor transitions with dissipation. Phys Rev Lett 84:737–740 22. Ng TK, Lee DKK (2001) Possibility of a metallic phase in granular superconducting films. Phys Rev B 63:144509 23. Spivak BI, Kivelson SA (1991) Negative local superfluid densities: the difference between dirty superconductors and dirty Bose liquids. Phys Rev B 43:3740–3743 24. Dalidovich D, Phillips P (1999) Landau theory of bicriticality in a random quantum rotor system. Phys Rev B 59:11925–11935 25. Phillips P, Dalidovich D (2003) Absence of phase stiffness in the quantum rotor phase glass. Phys Rev B 68:104427

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26. Phillips P, Dalidovich D (2003) The elusive Bose metal. Science 302:243–247 27. Wu J, Phillips P (2006) Vortex glass is a metal: unified theory of the magnetic-field and disordertuned Bose metals. Phys Rev B 73:214507 28. Kapitulnik A, Kivelson SA, Spivak B (2017) Anomalous metals – failed superconductors. arXiv:1712.07215 [cond–mat.supr–con] 29. Chervenak JA, Valles JM (2000) Absence of a zero-temperature vortex solid phase in strongly disordered superconducting Bi films. Phys Rev B 61:R9245–R9248 30. Beasley MR, Mooij JE, Orlando TP (1979) Possibility of vortex-antivortex pair dissociation in two-dimensional superconductors. Phys Rev Lett 42:1165–1168 31. Tsen AW, Hunt B, Kim YD, Yuan ZJ, Jia S, Cava RJ, Hone J, Kim P, Dean CR, Pasupathy AN (2016) Nature of the quantum metal in a two-dimensional crystalline superconductor. Nat Phys 12:208–212 32. Saito Y, Kasahara Y, Ye J, Iwasa Y, Nojima T (2015) Metallic ground state in an ion-gated two-dimensional superconductor. Science 350:409–413 33. Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, Vinokur VM (1994) Vortices in hightemperature superconductors. Rev Mod Phys 66:1125–1388 34. Dalidovich D, Phillips P (2002) Phase glass is a Bose metal: a new conducting state in two dimensions. Phys Rev Lett 89:027001 35. Das D, Doniach S (2001) Bose metal: gauge-field fluctuations and scaling for field-tuned quantum phase transitions. Phys Rev B 64:134511 36. Mason N, Kapitulnik A (2002) Superconductor-insulator transition in a capacitively coupled dissipative environment. Phys Rev B 65:220505 37. Joe YI, Chen XM, Ghaemi P, Finkelstein KD, de La Peña GA, Gan Y, Lee JCT, Yuan S, Geck J, MacDougall GJ, Chiang TC, Cooper SL, Fradkin E, Abbamonte P (2014) Emergence of charge density wave domain walls above the superconducting dome in 1T-TiSe2 . Nat Phys 10:421–425 38. Yan S, Iaia D, Morosan E, Fradkin E, Abbamonte P, Madhavan V (2017) Influence of domain walls in the incommensurate charge density wave state of Cu intercalated 1T-TiSe2 . Phys Rev Lett 118:106405 39. Kogar A, de la Pena GA, Lee S, Fang Y, Sun SX-L, Lioi DB, Karapetrov G, Finkelstein KD, Ruff JPC, Abbamonte P, Rosenkranz S (2017) Observation of a charge density wave incommensuration near the superconducting dome in Cux TiSe2 . Phys Rev Lett 118:027002 40. Nakanishi K, Shiba H (1977) Domain-like incommensurate charge-density-wave states and the first-order incommensurate-commensurate transitions in layered tantalum dichalcogenides. I. 1T-polytype. J Phys Soc Jpn 43:1839–1847 41. Li LJ, O’Farrell ECT, Loh KP, Eda G, Özyilmaz B, Castro Neto AH (2015) Controlling manybody states by the electric-field effect in a two-dimensional material. Nature 529:185–189 42. Spivak B, Oreto P, Kivelson SA (2008) Theory of quantum metal to superconductor transitions in highly conducting systems. Phys Rev B 77:214523

Chapter 6

Conclusions

In this thesis, we have discussed the nature of CDW and SC in TiSe2 . The calculations have been directed to the 2D (monolayer) form of this crystal but, by virtue of the strongly layered structure of its bulk form, our conclusions are expected to apply equally well to the 3D case. The journey began with revisiting the long-standing problem of the nature of the CCDW transition in TiSe2 . As described in Chap. 2, a mean field calculation based on the mechanism of excitonic instability can capture the doping dependence of the CDW transition temperature quantitatively well while the DFT calculation (including mainly EPC) overestimates the critical doping by an order of magnitude (see Fig. 2.1). Combining the results from both sides, it was concluded that the excitonic physics plays an indispensable role in the phase transition, which has also been confirmed by a recent experiment where a softened plasmon mode was discovered. Of course, in order to have a more fully quantitative description of the role played by excitons, a self-consistent calculation including both electrons and phonons at the same level is in demand. After the exploration of the CCDW problem, motivated by recent pressure and doping experiments where ICDW and SC phases were discovered, we moved on to study the interplay between CDW and SC. In Chap. 3, we established a phenomenological G-L theory to describe these two phases in monolayer TiSe2 . By introducing a coupling between CDW fluctuations (DCs) and SC order, and phenomenologically mapping the lock-in energy to the carrier density, our model is able to predict a phase diagram which includes CCDW, ICDW and SC phase. The SC order emerges with a dome shape inside the ICDW phase and nucleates first inside the CDW DCs. All the predictions made from our theory are in line with existing experiments. In the future, to have an understanding of the pressure effect in the bulk system, one might need to extend the theory to include the interlayer coupling. To understand the interplay between CDW and SC at a more microscopic level, we performed a field-theoretical calculation on a simplified two-band model in Chap. 4. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8_6

83

84

6 Conclusions

The expansion of the CDW order parameter around its mean field solution gave rise to the effective action for CDW fluctuations (amplitude and phase modes), where the amplitude mode is gapped while the phase mode is gapless (Goldstone mode). After integrating over the CDW fluctuations, it was found that the amplitude mode can mediate a retarded attractive interaction/pairing between the CDW-dressed quasiparticles. A numerical mean field calculation of the SC transition temperature with a specific parameterization of the model shows that the SC order has a dome shape in the phase diagram, which is consistent with experiments. Our microscopic study indicates that, indeed, the CDW fluctuations can induce or at least enhance the SC phase (for example, they can boost the underlying pairing instability expected to be present independently of fluctuations because of conventional EPC). Of course, in order to accurately explain the case of TiSe2 , more complicated calculation including all three electron pockets is needed. Finally, in Chap. 5 we described a theory-experiment collaboration that allowed the discovery and characterization of an AQM phase intervening the SC to normal metal transition in thin-film TiSe2 . As the magnetic field increases, there is a crossover between the so called Bose metal phase, where the resistivity scales as a power of magnetic field and the VQC regime where the resistivity exhibits an exponential dependence on magnetic field. The existence of the AQM phase is in correspondence with the picture that SC emerges within CDW DCs that percolate and enclose the CCDW domains, in the sense that the CDW background can provide both an intrinsic spatial non-uniformity and dissipation for the development of AQM. These findings and the gate-tunability of TiSe2 open the door to exploring, in a controlled way, the fate of superconductivity in 2D in the presence of competing or coexisting orders. The interplay between multiple coexisting or competing orders is always one of the most fascinating and challenging problems in condensed matter physics. At the beginning of this big project, I really had no idea of where to start and what we could contribute to this field. However, as the saying goes, “You never know what you can do till you try.” We climbed step by step on this mountain and were able to obtain a multi-perspective and self-contained picture regarding the nature of CDW, SC and their interplay in TiSe2 . Despite its first experiments dating back to 1976, the electronic properties of TiSe2 remain a lively source of new discoveries still today. Most importantly, in the very first 1976 experiment that identified the CDW instability in TiSe2 Di Salvo et al. [1] advanced the possibility that this system could be a realization of the excitonic insulator that had been predicted by Jérome, Kohn and Rice [2, 3]. This view remained dormant for most of the time until the early 2000’s with the developments related to high resolution ARPES data. But, even thereafter, this view has encountered some “resistance” to widespread acceptance. The work reported here shows that, not only is the excitonic mechanism capable of accounting for the CDW instability, but also that the underlying fluctuations are likely to play a key rule in providing the “glue” that stabilizes the SC dome at the point is the phase diagram where the long scale commensurability of the CDW is lost. For these reasons, I believe the work reported in this thesis makes considerable contributions to the understanding of this problem.

References

85

References 1. Di Salvo FJ, Moncton DE, Waszczak JV (1976) Electronic properties and superlattice formation in the semimetal TiSe2 . Phys Rev B 14:4321–4328 2. Kohn W (1967) Excitonic phases. Phys Rev Lett 19:439–442 3. Jérome D, Rice TM, Kohn W (1967) Excitonic insulator. Phys Rev 158:462–475

Appendix A

Mean Field Treatment of CDW with Both Excitons and Phonons1

In Sect. 2.1.2, we have performed the mean field calculation of CDW transition in TiSe2 with only electron-electron interactions (excitonic effect) taken into account. However, because of the mean field formalism, one might realise that mean field theory would give rise to the same type of gap equation if, instead of having electronelectron interaction, one starts only with EPC. Actually, there has been theoretical work where both types of effects were included [2–4]. However, they are either not fully self-consistent [2, 3] or were based on a simplified quasi-1D model [4]. A fully self-consistent calculation based on an exact model is in demand. Here we will show that for a Hamiltonian including both electron-electron interaction and EPC, the gap equation one gets from mean field theory has the same form as we had in Eq. (2.8). In order to describe both kinds of effect, one can simply add the free phonon part and EPC into the Hamiltonian we had in Eq. (2.1). Supposing the EPC has the form   1  † † + h.c, di,k+q ck aQi +q + a−Q He-p = √ i −q N i k,q

(A.1)

the mean field Hamiltonian will be of the form H = H1 + H2  ωq aq† aq H1 =

(A.2) (A.3)

q



      1  † † † † + c a gk,Qi di,k ck aQi + a−Q d + a i,k −Q i k Q i i N i,k  † † H2 = εvk ck,σ ck,σ + εk,i di,k,σ di,k,σ (A.4) +

k,σ,i 1 Part

of this chapter is taken and edited from Ref. [1] of which the author of this thesis is the first author. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8

87

88

Appendix A: Mean Field Treatment of CDW with Both Excitons and Phonons

−



 †  †

∗ i,k,σ + δi ck,σ di,k,σ − i,k,σ + δi di,k,σ ck,σ . k,σ,i

Assuming the order parameters are real and do not depend on the direction of the CDW and momentum k, one can get the following relations for both order parameters: ≡

1   †  V di,k ck , N k

(A.5)

 1 g 2   †   †  g2 1  . (A.6) ck di,k + di,k ck = 2 δi = − √ g aQ† + a−Q = N ωQ k ωQ V N Comparing the

H2 here with the mean field Hamiltonian we had in Eq. (2.3), it can be seen that 1 + 2g 2 /ωQ V i plays the role of i before. It is interesting that the plus sign here indicates both effects collaborate. It can be easily show that the gap equation of the new Hamiltonian is of the form   2g 2  AV 1 +  ωQ V kB T + =0   2 2 N k,ω 1 + ω2gQ V 2 B − C n

(A.7)

where α = 1 + 2g 2 /ωQ V , V  = αV , and  = α. The gap equation above can be recast as k B T  AV   = 0, (A.8)  + N k,ω 2 B − C n

which is exactly the same as Eq. (2.8) with the following identifications: V  ↔ V,

 ↔ .

(A.9)

Because of the simple linear relationship between V,  and V  ,  , the solution of Eq. (A.8) gives rise to a unique solution for . Correspondingly, the transition temperature is also the same in both cases if V  here is equal to the value of V in the main text. To conclude, the results based on an Hamiltonian containing only electronelectron interaction can also be regarded as the results of a system with both effects included, with a simple rescaling of the coupling constants.

Appendix A: Mean Field Treatment of CDW with Both Excitons and Phonons

89

References 1. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T-TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602. 2. Monney C, Battaglia C, Cercellier H, Aebi P, Beck H (2011) Exciton condensation driving the periodic lattice distortion of 1T-TiSe2 . Phys Rev Lett 106:106404. 3. Monney C, Monney G, Aebi P, Beck H (2012) Electron-hole instability in 1T-TiSe2 . New J Phys 14:075026. 4. van Wezel J, Nahai-Williamson P, Saxena SS (2010) Exciton-phonon-driven charge density wave in TiSe2 . Phys Rev B 81:165109.

Appendix B

The Lattice Instability Ab initio2

B.1

Renormalized Band Structure: Mexican Hat Features

In Fig. B.1 we show a close-up of the restructured bands in undoped TiSe2 , whose ground state we determine to be the 2×2 PLD with wavevector Qcdw after full relaxation of the ions in the unit cell [see also Fig. 2.4 in the main text]. The top valence bands are seen to lose their −k 2 parabolic dispersion and develop the shape of an inverted Mexican hat. More specifically, the top of the valence band moves from k = 0 Å−1 to lie at k = 0.0785 Å−1 . A 3D rendition of this band shows that the maximum defines a circle of diameter 0.157 Å−1 centered at k = 0, as marked by the blue circle in Fig. B.1. This is additionally supported by the fact that the energy dispersion along two orthogonal directions flattens exactly on this circle. Both perspectives establish the inverted Mexican hat shape of the band dispersion in the distorted phase (ground state). As discussed earlier by Kohn [2] and emphasized by Cazzaniga et al. with DFT+GW calculations for bulk TiSe2 [3], this shape is typical of the excitonic phases.

B.2

Phonon Hardening with Cu Doping

As discussed in the main text as well in Ref. [4], the low-temperature state of an undoped TiSe2 monolayer is the CDW phase with an accompanying PLD. Our DFT and DFPT calculations demonstrate that Cu adsorption (which corresponds to intercalation in bulk systems) suppresses the PLD and eventually stabilizes a 1×1 undistorted structure at zero temperature. In the main text, this transition as a function of doping is established by studying the density beyond which the dynamical phonon

2 Part

of this chapter is taken and edited from Ref. [1] of which the author of this thesis is the first author. © Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8

91

92

Appendix B: The Lattice Instability Ab initio

(a)

(b)

Fig. B.1 Inverted Mexican hat shape of energy dispersion of undoped TiSe2 . a Electronic band structure of undoped TiSe2 monolayers with the PLD. b Zoom-in structure of top valence band in the E − k x − k y space within the reduced Brillouin zone. Black dashed lines show the band structure along two orthogonal directions and blue ring highlights the valence band maximum in the reduced Brillouin zone. This figure is taken from Ref. [1]. © 2019 American Physical Society

Fig. B.2 Phonon band dispersion (left) and phonon density of states (right) of a Cu-doped 2×2 TiSe2 monolayer at low smearing parameter. The absence of imaginary frequencies in the reduced Brillouin zone confirms the dynamical stability in the presence of the Cu adsorbates. This figure is taken from Ref. [1]. © 2019 American Physical Society

instability disappears at T = 0 (cf. Fig. 2.5); as explained and justified there, these calculations are done by adding additional electrons to the unit cell. Here, we wish to demonstrate than this conclusion holds when the phonon spectrum is computed including the Cu atoms explicitly in the unit cell from the outset. Figure B.2 summarizes the phonon spectrum and corresponding density of states

Appendix B: The Lattice Instability Ab initio

93

obtained under such conditions, with two Cu atoms per 2×2 supercell (one adsorbed above and the other symmetrically below the TiSe2 monolayer, as in Fig. 1.3b of the main text). We highlight that this phonon spectrum has been calculated using the same structure that is employed to determine the band structure shown in Fig. 2.4e of the main text. Figure B.2 does not show any imaginary frequencies nor soft acoustic branch, thereby confirming the dynamical stability of the Cu doped monolayer at low temperature. This small simulation cell corresponds to a 50% Cu content (x = 0.5), it tallies with the evolution of σc shown in Figs. 2.1 and 2.5 (main text) based on the electron doping approach that predicts the lattice to be stable for x  0.20. It also reinforces the validity of the latter approach for numerical expediency in treating the experimentally relevant doping levels of 0 < x < 11 which, to explore with actual Cu atoms in the supercell, would require extremely large supercells, prohibitive for both the DFT electronic structure and DFPT phonon calculations.

B.3 Robustness of the PLD and CDW Transition with Smearing Function In electronic structure calculations, the smearing function is routinely used to decide how to set the partial occupancies for each wavefunction. For a particular smearing method, σ determines the width of smearing. An optimal choice for smearing function and σ depends not only on improved convergence but also on the system and properties of interest. However, all methods should converge to the ground state in the limit σ → 0. In order to verify the robustness of our conclusions regarding the evolution of the dynamical phonon instability and PLD with doping, we calculated the phonon dispersions of undoped TiSe2 using two independent strategies: the Methfessel–Paxton (MP) smearing and Fermi–Dirac (FD) smearing methods. The resulting phonon spectra at different σ are shown in Fig. B.3. It is evident that both approaches correctly predict the freezing of the longitudinal acoustic mode at the experimentally correct Qcdw , and the existence of a threshold σc above which this instability suppressed. Note, however, that the magnitude of σc varies for different smearing strategies, and reflects the fact that σ is not the physical temperature, and should only be used to explain qualitative trends in the lattice structure as a function of temperature. These results are consistent with earlier calculations done for bulk TiSe2 [5]. One of our central conclusions in the main text is that relying only on the phonons calculated within DFT+DFPT to predict the critical doping (xc ) above which the CDW/PLD is no longer stable, leads us to values of xc that overshoot the experimental threshold by about one order of magnitude. Crucially, this result is also independent of the smearing method used, as we show in Fig. B.4: both MP and FD smearing predict the PLD to remain as the ground state at least up to x = 0.16. Conversely, this implies, by extrapolation, that xc  0.20 if xc is extracted from the criterion σc (x) = 0.

94

Appendix B: The Lattice Instability Ab initio

(a)

(b)

Fig. B.3 Phonon dispersion of the 1×1 TiSe2 monolayer (normal phase) calculated with different smearing parameter (σ ). a Methfessel–Paxton smearing and b Fermi–Dirac smearing. Imaginary frequencies are represented as negative values. This figure is taken from Ref. [1]. © 2019 American Physical Society

Fig. B.4 The critical smearing parameter (σc ) as a function of doping x according to the two different smearing strategies. The values of σc are obtained from studying the phonon spectrum for different σ as in Fig. B.3. All phonon frequencies are real (no dynamical instability) for σ > σc . The red points were obtained with Methfessel–Paxton and the blue using Fermi–Dirac smearing (black lines are guides). Note how, despite having different magnitudes (see text), the two strategies agree in the qualitative prediction that the phonon instability persists up to very large values of x. This figure is taken from Ref. [1]. © 2019 American Physical Society

Seeing that there is complete consistency among the calculated phonon dispersion and phonon instabilities with these two smearing methods, we chose to present in the main text the results obtained with the Methfessel–Paxton smearing because of its improved convergence and accuracy over Fermi–Dirac smearing.

Appendix B: The Lattice Instability Ab initio

95

References 1. Chen C, Singh B, Lin H, Pereira VM (2018) Reproduction of the charge density wave phase diagram in 1T-TiSe2 exposes its excitonic character. Phys Rev Lett 121:226602. 2. Kohn W (1967) Excitonic phases. Phys Rev Lett 19:439–442. 3. Cazzaniga M, Cercellier H, Holzmann M, Monney C, Aebi P, Onida G, Olevano V (2012) Ab initio many-body effects in TiSe2 : a possible excitonic insulator scenario from GW band-shape renormalization. Phys Rev B 85:195111. 4. Singh B, Hsu C-H, Tsai W-F, Pereira VM, Lin H (2017) Stable charge density wave phase in a 1T-TiSe2 monolayer. Phys Rev B 95:245136. 5. Duong DL, Burghard M, Schön JC (2015) Ab initio computation of the transition temperature of the charge density wave transition in TiSe2 . Phys Rev B 92:245131.

Appendix C

Effective Action of CDW Fluctuations

In this chapter, we will show explicitly the derivation for the effective action of CDW amplitude and phase modes Seff [ϕ, θ ].

C.1

Expansion for the Mass Term

In this section, we look for the mass term for both CDW amplitude and phase modes. As mentioned in Sect. 4.5, the phase mode is massless while the amplitude mode is gapped, which will be shown explicitly by the end of this section. To avoid readers’ confusion when reading the details of the derivation, here we want to first clarify the convention for Fourier transformation that we used throughout this study. For both amplitude and phase modes, the Fourier transformation is defined as the following:  e−iωn τ +iq·R ϕωn ,q , (C.1a) ϕ(τ, R) = ωn ,q

ϕωn ,q =

1 βN

dτ



eiωn τ −iq·R ϕ(τ, R).

(C.1b)

R

In the first order expansion from Eq. (4.19), keeping the second order of φ and θ (since we are expanding around a saddle point of the effective action for CDW order parameter, the summation of all the first order terms of θ and ϕ should vanish), one obtains: 

 0 ∗ φ0,0 + φ0,0 (iωn + E k ) (iωn − E k ) k,ω n

=

 k,ωn

−

2 20 θ 0,0 (iωn + E k ) (iωn − E k )

© Springer Nature Switzerland AG 2019 C. Chen, On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe2 , Springer Theses, https://doi.org/10.1007/978-3-030-29825-8

97

98

Appendix C: Effective Action of CDW Fluctuations



1 β Ek 2  θ 0,0 tanh 2E k 2 k  1 β dτ = c0 θ (τ, R)2 Nβ R 1 = dτ dr c0 × θ (τ, r )2 , V 

 1 β Ek . c0 = tanh 2E k 2 k =



β

(C.2)

(C.3)

Notice that in the final step of Eq. (C.2), we made a continuous approximation: θ (τ, R) → θ (τ, r ) and V stands for the volume of the system, V = N a 2 , with a being the lattice constant. In the second order, there is: 1  G 0,k;ωn Mk,k  ;ωn ,ωm G 0,k  ;ωm Mk  ,k;ωm ,ωn 2  k,k ωn ,ωm

=

1  (iωn − εk ) (iωm + εk  ) ∗ φk−k  ,ω −ω φk−k  ,ωn −ωm n m  ) (iωm − E k  ) 2 + E − E + E (iω ) (iω ) (iω n k n k m k  k,k ωn ,ωm

(iωn + εk ) (iωm − εk  ) φ ∗ φk  −k,ωm −ωn (iωn + E k ) (iωn − E k ) (iωm + E k  ) (iωm − E k  ) k −k,ωm −ωn

∗ 20 φ  + φ ∗   (iωn + E k ) (iωn − E k ) (iωm + E k ) (iωm − E k ) k−k ,ωn −ωm k −k,ωm −ωn  (C.4) + φk−k  ,ωn −ωm φk  −k,ωm −ωn . +

Since we are currently focusing on the mass term, we can ignore the momentum and frequency difference in the coefficient. The first term in Eq. (C.4) can be written as: β  1   (iωm − εk  ) (iωm + εk  ) ∗ φ p,ω p φ p,ω p = c1 × φ ∗p,ω p φ p,ω p (C.5) 2 2 2 p,ω k  ,ω (iωm + E k  ) (iωm − E k  ) 2 p,ω p p m

with the coefficient c1 being c1 =

 k

[1 − 2 f (E k )]

20 1 − 3 2E k 4E k

 −β

20 f (E k ) (1 − f (E k )) 2E k2

(C.6)

it can be shown that the contribution from the second term is the same as the first one, so from the first two terms of Eq. (C.4), one can obtain

Appendix C: Effective Action of CDW Fluctuations

β



c1 × φ ∗p,ω p φ p,ω p =β

p,ω p



99

  ∗ c1 × ϕ ∗p,ω p ϕ p,ω p + 20 θ p,ω θ p p,ω p

p,ω p

1 = V





dτ dr c1 × ϕ (τ, r )2 + 20 θ (τ, r )2

(C.7)

The last two terms contribute:   1 20 ∗ ∗ φ φ + φ φ p,ω − p,−ω p p p,ω − p,−ω p p 2 p,ω k  ,ω (iωm + E k  )2 (iωm − E k  )2 p m   β  = c2 × φ ∗p,ω p φ−∗ p,−ω p + φ p,ω p φ− p,−ω p 2 p,ω p   β  ∗ = c2 × 2 ϕ ∗p,ω p ϕ p,ω p − 20 θ p,ω θ p,ω p p 2 p,ω p    ∗ =β c2 ϕ ∗p,ω p ϕ p,ω p − 20 θ p,ω θ p p,ω p p,ω p

=

1 V





dτ dr c2 × ϕ (τ, r )2 − 20 θ (τ, r )2 ,

(C.8)

with the coefficient c2 being c2 =



[1 − 2 f (E k )]

k

20 20 − β f (E k ) (1 − f (E k )) . 4E k3 2E k2

(C.9)

Recalling that there is also a free part of the action of the order parameter (introduced when one do the H-S transformation): N 1  N  ∗ q q = (C.10) dτ dτ (0 + ϕ(R, τ ))2 U q U N R using the mean field gap Eq. (4.12), one can replace the N /U 

 1 N β Ek = = 2c0 tanh U Ek 2 k

(C.11)

thus the Eq. (C.10) can be rewritten as: dτ 2c0 ×

 p

ϕ ∗p (τ ) ϕ p (τ ) =

1 V

dτ dr 2c0 × ϕ (τ, r )2 .

Combining Eqs. (C.2), (C.7), (C.8), (C.12) together, one can obtain:

(C.12)

100

Appendix C: Effective Action of CDW Fluctuations

1 V



  dτ dr 2 (c0 + c1 + c2 ) ϕ (τ, r )2 + 220 (c0 + c1 − c2 ) θ (τ, r )2 .

(C.13)

Here the factor 2 comes from the spin degree of freedom, and it is straightforward to show c0 + c1 − c2 = 0. Finally, one can arrive at the mass term of the amplitude and phase modes dτ dr m 2 × ϕ (τ, r )2

(C.14a)

2 (c0 + c1 + c2 ) V

(C.14b)

m2 =

C.2 Expansion for the Spatial Derivative (Momentum) Term In the previous section, we derived the mass term, which is given by neglecting the momentum and frequency difference in the two G 0 ’s of Eq. (C.4). In this section, we derive the momentum dependence of amplitude and phase modes. This can be done by expanding the p ≡ k − k  to second order while neglecting the frequency difference. Since εk  = −2 k 2 /2m c + bo , we can rewrite: εk =εk  + x E k2

=E k2

x =−

+ 2εk  x + x

(C.15a) 2

(C.15b)

   2 k ·p− p mc 2m c 2

2

(C.15c)

so the first term of Eq. (C.4) can be rewritten as: 1 (iωm − εk  − x) (iωm + εk  )    φ ∗ φ p,ω p 2 2 p,ω k  ,ω (iωm ) − E k2 − 2εk  x − x 2 (iωm )2 − E k2 p,ω p p

(C.16)

m

after an expansion over x, it gives (the zero order term is dropped since it’s just the mass term): 

 iωm + εk  (iωm )2 − 2iωm εk  + 2 k2 − E k2 − x  3 (iωm )2 − E k2 

iωm + εk  (iωm )3 − 3 (iωm )2 εk  − iωm E k2 + 4iωm εk2 + 3E k2 εk  − 4εk3 x2 + · · · +  4 2 2 (iωm ) − E k 

(C.17)

Appendix C: Effective Action of CDW Fluctuations

101

after the summation over Matsubara frequency ωm , one can show that the first term in Eq. (C.17) is: 



  β Ek 2 sinh E k + εk2 2β 2 E k2 − 3 − 2β 2 E k4

 3 2 4E k5 eβ Ek + 1 k

 



 3β E k β Ek + E k2 − 3εk2 sinh − 2β E k E k2 − 3εk2 cosh (C.18) 2 2

−β



εk

e

3β E k 2

which is an odd function of εk . Since the density of state for εk is a constant, it is reasonable to set this term to zero. The second term of Eq. (C.17) is c1,k =

e2β Ek

4 (Ak + Bk + Ck ) 12E k7 eβ Ek + 1

(C.19)

with Ak = 3E k4 [sinh(β E k ) (cosh (β E k ) + 1) + β E k (β E k sinh (β E k ) − cosh (β E k ) − 1)] (C.20a) Bk = E k2 k2 [21 sinh(β E k ) (cosh(β E k ) + 1) + β E k (β E k (−9 sinh (β E k ) − 2β E k (cosh (β E k ) − 2)) − 21 (cosh (β E k ) + 1))]

(C.20b) Ck =

k4 [2β E k

(15 (cosh (β E k ) + 1) + β E k (6 sinh (β E k ) + β E k (cosh (β E k ) − 2)))

− 15 (2 sinh (β E k ) + sinh (2β E k ))]

(C.20c)

keeping terms up to second order in momentum p, one can obtain:   β   4 k  2 c  p2 φ ∗p,ω p φ p,ω p 2 3 1,k 2 p,ω m c k

(C.21)

p

It can be shown that the result from the second term in Eq. (C.4) is the same, thus combining the first two terms of Eq. (C.4) contribute to the momentum term as βα1



p2 φ ∗p,ω p φ p,ω p

(C.22a)

p,ω p

   4 k  2 c1,k  α1 = m 2c 3 k

(C.22b)

The third term of Eq. (C.4) gives 20 1    φ∗ φ∗ 2 p,ω k  ,ω (iωm )2 − E k2 − 2 k  x − x 2 (iωm )2 − E k2 p,ω p − p,−ω p p m

(C.23)

102

Appendix C: Effective Action of CDW Fluctuations

expanding x to the second order, it gives:   20 (iωm )2 − E k2 + 4εk2 2 x + ···  3 x +  4 (iωm )2 − E k2 (iωm )2 − E k2 220 εk 

(C.24)

since the first term is an odd function of εk , it should be zero. After frequency summation over the second term, it results in: c2,k =

20 e2β Ek

4 (Dk + E k ) 24E k7 eβ Ek + 1

(C.25)

with  Dk = 6E k2 β E k (β E k sinh (β E k ) + 3 cosh (β E k ) + 3)  − 3 sinh(β E k ) (cosh (β E k ) + 1)  E k = k2 30 (2 sinh (β E k ) + sinh (2β E k ))

(C.26a)

+ 4β E k (−15 (cosh (β E k ) + 1) − β E k (6 sinh (β E k ) + β E k (cosh (β E k ) − 2)))



(C.26b) keeping terms up to second order in momentum p, it results in:   β   4 k  2 c  p2 φ ∗p,ω p φ−∗ p,−ω p 2 3 2,k 2 p,ω m c k

(C.27)

p

the expansion for the last term of Eq. (C.4) is the same, so the last two terms of Eq. (C.4) gives:  β  2 ∗ p φ p,ω p φ−∗ p,−ω p + φ p,ω p φ− p,−ω p 2 p,ω p    4 k  2 α2 = c2,k  . m 2c 3 k α2 ×

(C.28a)

(C.28b)

Combining Eqs. (C.22a) and (C.28a), one can finally obtain the spatial derivative (momentum) term β



 ∗ p2 2 (α1 + α2 ) ϕ ∗p,ω p ϕ p,ω p + 220 (α1 − α2 ) θ p,ω θ p,ω p p

p,ω p



=

  dτ d x Jϕ (∇r ϕ (τ, r ))2 + Jθ (∇r θ (τ, r ))2

here the additional factor of 2 comes from the spin degree of freedom, with

(C.29)

Appendix C: Effective Action of CDW Fluctuations

103

1 2 (α1 + α2 ) V 1 Jθ = 2 (α1 − α2 ) 20 V

Jϕ =

C.3

(C.30) (C.31)

Expansion for the Stiffness Term

In this section, we will expand the frequency ω p ≡ ωn − ωm in the two G 0 ’s of Eq. (C.4) up to second order while neglecting the momentum difference. From the first term in Eq. (C.4): 1 (iωm + z − εk  ) (iωm + εk  ) φ ∗ φ p,ω p 2 p,ω k  ,ω (iωm + z + E k  ) (iωm + z − E k  ) (iωm + E k  ) (iωm − E k  ) p,ω p p

m

(C.32) after expanding to second order in z (z = iω p ), it gives:   (iωm + εk ) (iωm )2 + E k2 − 2iωm εk x −  3 (iωm )2 − E k2

   (iωm + εk ) (iωm )3 − εk E k2 + 3 (iωm )2 + 3iωm E k2 2 x . +  4 (iωm )2 − E k2

(C.33)

The frequency summation over the first term above gives

β

sech2



β Ek 2



(sinh [β E k ] − β E k ) 8E k3

εk ,

(C.34)

which is an odd function of εk and therefore should be zero after integrating out εk . The second order gives  

2 e2β Ek 3 3 5 2

4 6β E k + 8β E k − 3 sinh(2β E k ) k + E k 5 β E 24E k e k + 1 



 + 6 sinh(β E k ) k2 + E k2 β 2 E k2 − 1 + k2 6β E k − 8β 3 E k3



+ 2β E k cosh(β E k ) 3E k2 − 2β 2 E k4 + k2 3 + 2β 2 E k2 (C.35)

−a1,k =

which leads to

  β   a1,k  ω2p φ ∗p,ω p φ p,ω p 2 p,ω  k p

(C.36)

104

Appendix C: Effective Action of CDW Fluctuations

since the contribution from the second term in Eq. (C.4) is the same, the first two terms of Eq. (C.4) contribute βζ1



ω2p φ ∗p,ω p φ p,ω p

(C.37a)

p,ω p

ζ1 =



a1,k 

(C.37b)

k

The third term of Eq. (C.4) gives:     20 3 (iωm )2 + E k2 2 1 2iωm 20 ∗ ∗ −  z+   z φ p,ω p φ− p,−ω p . 2 2 2 3 2 4 2 p,ω k  ,ω (iωm ) − E k  (iωm ) − E k  p m (C.38) Because the first term is an odd function of iωm , it vanishes after frequency summation. The second order term gives a2,k =

 20 e2β E k 4 −3 (2 sinh (β E k ) + sinh (2β E k ))

5 β E k 24E k e +1

 + 2β E k (3 (cosh (β E k ) + 1) + β E k (3 sinh (β E k ) + 2β E k (cosh (β E k ) − 2)))

(C.39) and thus leads to

  β   a2,k  ω2p φ ∗p,ω p φ−∗ p,−ω p 2 p,ω  k

(C.40)

p

the last term of Eq. (C.4) will give the same result after expansion over z, so the sum of these two terms give  β  2 ∗ ζ2 ω p φ p,ω p φ−∗ p,−ω p + φ p,ω p φ− p,−ω p 2 p,ω

(C.41)

p

ζ2 =



a2,k 

(C.42)

k

Combining Eqs. (C.37a) and (C.41), one can obtain the stiffness part of the effective action (an additional factor of 2 has been included due to the spin degree of freedom): β



 ∗ ω2p 2 (ζ1 + ζ2 ) ϕ ∗p,ω p ϕ p,ω p + 220 (ζ1 − ζ2 ) θ p,ω θ p,ω p p

p,ω p



=

dτ dr ρϕ (∂τ ϕ (τ, r ))2 + ρθ (∂τ θ (τ, r ))2

(C.43)

Appendix C: Effective Action of CDW Fluctuations

105

with 1 2 (ζ1 + ζ2 ) V 1 ρθ = 220 (ζ1 − ζ2 ) V

ρϕ =

C.4

(C.44a) (C.44b)

Effective Action for CDW Amplitude and Phase Modes

Combining Eqs. (C.14a), (C.29) and (C.43), one can write down the effective action of CDW fluctuations   S[ϕ, θ ] = dτ dr ρϕ (∂τ ϕ (τ, r ))2 + Jϕ (∇r ϕ (τ, r ))2 + m 2 ϕ (τ, r )2   (C.45) + ρθ (∂τ θ (τ, r ))2 + Jθ (∇r θ (τ, r ))2 When written in momentum-frequency space, it has the form S[ϕ, θ ] =βV





∗

 ϕ ω p , p ρϕ ω2p + Jϕ p2 + m 2 ϕ ω p , p p,ω p



∗



+ θ ω p , p ρθ ω2p + Jθ p2 θ ω p , p

(C.46)