Tasks

Monday 05.10 1: Tasks: Read: Kittel, Chapter: Diamagnetism & Hand-in exercise sheet Tuesday 06.10 Magnetism 1: Paramagnetism Tasks: Read: Kittel, Chapter: Diamagnetism & Paramagnetism Monday 12.10 Magnetism 2: Tasks: Read: Kittel, Chapter: Ferromagnetism & Hand-in exercise sheet Tuesday 13.10 Guest lecture: Christof Aegerter. Tasks: Monday 19.10 Magnetism 2: Antiferromagnetism & Magnetic excitations Tasks: Read: Kittel, Chapter: Ferromagnetism & Antiferromagnetism Tuesday 20.10 Magnetism Summary + Tasks: Student presentations (Charles and Alexandra) Monday 26.10 interactions 1 Tasks: Read: Kittel – Fermi Surfaces and Metals Tuesday 27.10 Electron interactions 2 Monday 02.11 Quantum Oscillations Tuesday 03.11 Angle-Resolved Photoemission Spectroscopy Organization – Nov & Dec

Monday 9.11 Tuesday 10.11 Student Presentation: Reza – Magnetic excitations David – spin Monday 16.11 Tuesday 17.11 Student Presentation: Jens – Vortex, Charge order & SC Lorena – Skyrmions Monday 23.11 Tuesday 24.11 Student Presentation: Ron – Magic angle graphene & SC NN – Room temperature SC Monday 30.11 Guest Lecture: Fabian Natterer Tuesday 01.12 Guest Lecture: Thomas Greber Monday 07.12 Guest Lecture: Marta Gibert Tuesday 08.12 Guest Lecture: Marc Janoschek Monday 14.12 Recap + Student presentations Tuesday 15.12 Recap + Exam Prep. Exam date: Friday 29th of January (last Friday in January) LITERATURE CLUB - II

(1) Quantum Oscillations B. Ramshaw et al., Science 348, 317 (2015)

(2) Fermi Custers et al., Nature 424, 524, (2003) N. Doiron-Leyraud et al., Physical Review B 80, 214531 (2009)

(3) Unconventional Science 336, 1554-1557 (2012) – Penetration depth @ QCP Nature 11, 17–20 (2015) – SC fluctuations in URu2Si2 (4) Superconductivity without Nature 394, 39 (1998) – Pressure induced superconductivity Nature 450, 1177 (2007) – Review article (5) What is the evidence p-wave superconductivity? Starting point: Rev. Mod. Phys. 75, 657 (2003) liv

Electron Paramagnetic Resonance 386 Exchange Narrowing 386 Zero-field Splitting 386 Principle of Maser Action 386 Three-Level Maser 388 Lasers 389 Kittel Reading for Summary 390 next w Problems 391

CHAPTER 14: , , AND 393 Dielectric Function of the Electron 395 Definitions of the Dielectric Function 395 Optics 396 Dispersion Relation for Electromagnetic Waves 397 Transverse Optical Modes in a Plasma 398 Transparency of Metals in the Ultraviolet 398 Longitudinal Plasma Oscillations 398 Plasmons 401 Electrostatic Screening 403 Screened Coulomb Potential 406 Pseudopotential Component U(O) 407 Mott Metal- Transition 407 Screening and Phonons in Metals 409 Polaritons 410 LST Relation 414 Electron-Electron Interaction 417 Fermi Liquid 417 Electron-Electron Collisions 417 Electron- Interaction: Polarons 420 Peierls Instability of Linear Metals 422 Summary 424 Problems 424

CHAPTER 15: OPTICAL PROCESSES AND 427 Optical Reflectance 429 Kramers-Kronig Relations 430 Mathematical Note 432 Magnetism - Overview

This week Next week Isolated magnetic Interacting magnetic moments moments

Reading tasks Reading tasks Kittel: Kittel: Chapter: Diamagnetism & Chapter: Ferromagnetism & Paramagnetism Antiferromagnetism (b) Measurement of spin waves

Spin wave dispersions can be measured using inelastic neutron scattering. In such an experiment the magnitude of the incident neutron wave vector ki is not equal to the magnitude of the scattered neutron 2 2 2 2 wave vector kf . The energy of the neutron also changes from Ei =¯h ki /2mn to Ef =¯h kf /2mn because the neutron produces an excitation in the sample of energy ¯hω and wave vector Q. Conservation of energy and momentumFerromagnetic implies that : E E =¯hω (38) i − f Neutron Spectroscopyk k = Q, (39) i − f so that a measurement of ki, kf , Ei and Ef allows a determination of ω and Q. Brockhouse 1960

Shared Nobel prize 1994 Spin wave evergy vs. momentum in an alloy of Co0.92Fe0.08 obtained at room temperature (Sinclair and Brockhouse 1960).

Spin wave dispersion relations in ferromagnetic Gadolinium at 78 K (TC = 300 K) along different directions in the Brillouin zone (there are 2 modes as there as 2 atoms in the unit cell). The energy extrapolates to a quadratic form q 2 near Γ as expected for a ferromagnet. ∼ | |

28 Phonons in Sr RuO Time-of-flight spectrometry 2 4

Initial alignment scans revealed nicely “c-axis” phonons.

https://www.helmholtz-berlin.de/forschung/zukunftsprojekte/neat2_en.html Antiferromagnetic magnons

Materials

Christensen et al, PNAS 104 15264 (2007) Antiferromagnetic magnons week ending PRL 105, 247001 (2010) PHYSICAL REVIEW LETTERS 10 DECEMBER 2010 400 a La4002CuO4 40 )

350 1

300 − 30

300 f.u. 1 − 250 200 20 eV 2 B µ

200 ( ) ω , Energy (meV) Energy (meV)

10 Q 150 ( 100 ′′ χ 100 0 0 50 20 c d 1 e Wavevector( h, k) (r.l.u.) 1.2 400 )

1 300 − 15 1 0.5 M

f.u. 200 2 B SW µ 0.8 I/I

) ( ) 100 10 Γ Q Energy (meV) ( X 0 0.6 0 SW I 0 0.5 1 1 5 0.5 1 0.4 k 00 h 0.5

0 0.2 (3/4,1/4)(1/2,1/2) (1/2,0) (3/4,1/4) (1,0) (1/2,0) (3/4,1/4)(1/2,1/2) (1/2,0) (3/4,1/4) (1,0) (1/2,0) Wave vector ( h, k ) (r.l.u.) Wave vector ( h, k ) (r.l.u.)

FIG. 2 (color online). q dependence of the magnetic excitations in La CuO . (a) One- dispersion (T 10 K) along lines in 2 4 ¼ (c, inset). Symbols indicate Ei: 160 meV (h), 240 meV ( ), and 450 meV ( ). The line is a SWT fit based on Eq. (1). (b) Measured q;! . Dashed circle highlights the anomalous4 scattering near 1=2; 0 . An !-dependent background determined Christensen et al, PNAS 104, 15264 (2007) Headings et al., PRL 00ð Þ 105, 247001 (2010) ð Þ near 1; 0 has been subtracted. (c) One-magnon intensity. Line is a fit to SWT with renormalization@ factor Zd 0:4 0:04. (d) One- magnonð intensityÞ divided by SWT prediction. (e) SWT dispersion (color indicates SW intensity). ¼ Æ

In general terms, our results show that at the q continuum and (ii) the q dependence to the intensity of 1=2; 0 position the spin waves are more strongly coupled¼ the SW pole. We estimate the total observed moment toð otherÞ excitations than at q 1=4; 1=4 . This coupling squared (including the Bragg peak) is M2 1:9 ¼ð Þ 2 h i ¼ Æ provides a decay process and therefore damps the spin 0:3B. The continuum scattering accounts for about 29% wave, reducing the peak height and producing the tail. of the observed inelastic response. The total moment sum 2 2 2 The question is, What are these other excitations? An rule [15] for S 1=2 implies M g BS S 1 2 ¼ h i ¼ ð þ Þ¼ interesting possibility is that the continuum is a manifes- 3B. We consider two reasons why we fail to observe 2 tation of high-energy spinon proposed in the full fluctuating moment of the Cu þ ion. First, our theoretical models of the cuprates [1–3,13,19–21]. These experiment is limited in energy range to about 450 meV; assume that Ne´el order coexists with additional spin cor- thus, there may be significant spectral weight outside the relations with the magnetic state supporting both low- energy window of the present experiment. Raman scatter- energy SW fluctuations of the Ne´el order parameter as ing [22] and optical absorption [23] spectra show excita- well as distinct high-energy spin-1=2 spinon excitations tions up to about 750 meV. Recent RIXS measurements created above a finite [20,21]. Spinons are S also show high-energy features [24] which appear to be 1=2 quasiparticles which can move in a strongly fluctuating¼ magnetic in origin. The second reason why we may fail to background. The anomaly we observe at 1=2; 0 may be see the full fluctuating moment may be covalency effects ð Þ explained naturally in a model where spinons exist at high [25,26]. The Cu dx2 y2 and O px orbitals hybridize to yield energies and have a d-wave dispersion [20,21] with min- the Wannier orbitalÀ relevant to superexchange. This will ima in energy at q 1=4; 1=4 and 1=4; 1=4 . Under lead to a reduction in the measured response. However, the ¼ ðÆ Þ ð Æ Þ these circumstances, the lower boundary of the two-spinon (at most) 36% reduction observed in La2CuO4 is substan- continuum is lowest in energy at 1=2; 0 and significantly tially less than the 60% reduction recently reported in the ð Þ higher at 1=4; 1=4 . This provides a mechanism for the cuprate chain compound Sr2CuO3 [26]. spin wavesð at 1=Þ 2; 0 to decay into spinons [with Our results have general implications for the cuprates. 1=4; 1=4 ] andð those atÞ 1=4; 1=4 to be stable. Firstly, they show that the collective magnetic excitations ð TheÆ newÞ features in theð collectiveÞ magnetic excitations of the cuprate parent compounds cannot be fully described observed in the present study are (i) a q-dependent in terms of the simple SW excitations of a Ne´el ordered 247001-3 Resonant inelastic x-ray scattering

http://www.esrf.eu/news/spotlight/spotlight140/index_html Antiferromagnetic magnons

Bi2201

NBCO

CCO

Nature Physics (2017) doi:10.1038/nphys4248 Antiferromagnetic magnons week ending PRL 108, 177003 (2012) PHYSICAL REVIEW LETTERS 27 APRIL 2012

in the strong SOC limit, on which a novel platform for high 0.25 (a) data temperature superconductivity (HTSC) may be designed. fit In the last few years, RIXS has become a powerful tool 0.20 week ending PRL 108, 177003 (2012) PHYSICAL REVIEW LETTERS 27 APRIL 2012 to studySr2 magneticIrO4 excitations [11]. We report measurement

1.0 ) of(a) single magnons using hard x(b) rays, which has comple- (c) V 0.15

e

(

e-h continuum

mentary advantages over soft x rays, as detailed later on. (opticallyy allowed) (π,0) g

0.8 r

The RIXS measurements were performed at the 9-1D and e n 0.10

(0,0) E 30-ID) beam line of the Advanced Photon Source. A hori-

t i spin-orbit

n 0.6

u E

.

zontal scattering geometry was used with the -polarized n

b e

r 0.05 (optically r a ( /2, /2) π π g

(

y forbidden)

incidenty photons. A spherical diced Si(844) analyzer was

t (

i

0.4 e

s V

n ) used.e The overall energy and(π,π momentum) resolution of the

t

n I 0.00 RIXS spectrometer at the Ir L3 edge ( 11:2 keV) was 1  0.2 (π,π) (b) 130 meV and 0:032 A À , respectively. (π,π)  Æ (π,0) magnons As shown(0,0) in Fig. 1(b), Sr2IrO4 has a canted antiferro- 0.0 ) (0,0)

t

i

magnetic (AF) structure [(π8/2,],π/2) with TN 240 K [12]. n

u

1.0 0.8 0.6 0.4 0.2 0.0 (π/2,π/2)(π,0) (π,π)(π/2,π/2) (0,0) (π,0) .

Although the ‘‘internal’’ structure of a single magnetic b Energy (eV) Momentum q Intensityr

a

( moment in Sr2IrO4, composed of orbital and spin, is dras- (d) y

t

i

tically different from that of pure spins in La2CuO4,a s

n

e parent insulator for cuprate superconductors, the two com- t

n pounds share apparently similar magnetic structure. I Figures 2(a) and 2(b) show the dispersion and intensity, respectively,PRL 108, 177003 (2012) of the single magnon extracted by fitting the energy distribution curves shown in Fig. 3(a) [13]. We (π/2,π/2) (π,0) (π,π) (π/2,π/2) (0,0) (π,0) Momentum q FIG. 3highlight (color online). three (a) Energy important loss spectra observations. recorded at T 15 First, K, well belownot only the TN the240 K [8,12], along a path in the constant L 34 plane. The path was chosen to avoid the magnetic Bragg¼ peaks, which appear at two of the four corners of the unfolded unit cell¼ (blackdispersion square) shown but in the also inset the (where momentum the same conventions dependence as in Fig. 2 are of used). the (b) ImageFIG. plot 2 of (color the data shownonline). in (a). (a) Blue dots with error bars show the (c) Schematicintensity of the show three representative striking features similarities in the data. to (d) those A real space observed description in of the spin-orbitsingle exciton magnon mode. dispersion extracted by fitting the energy loss superimposedthe cuprates on top of (for a continuum instance, generated in La by2CuO particle-4) by inelasticby magnons neutron as that experiencedcurves by a doped shown hole in [28 Fig.]. It is3(b) [13]. The magnons disperse up to hole excitationsscattering across [14 the]. This Mott gap provides [24] (estimated confidence to be thatwell-known the observed that the dispersion of205 a doped meV holeat in cuprates; 0 and 110 meV at =2; =2 . The solid  ð Þ ð Þ 0:4mode eV from is optical indeed spectroscopy a singlemagnon [25]). This excitation is sche- has [15 a minimum–18]. Using at =2; =2purple[29], i.e., line at the shows AF magnetic the best fit to the data with J 60, J0 20,  ð Þ ¼ ¼À matically shown in Fig. 3(c). Taking the second derivative Brillouin zone boundary. SinceandSrJ2IrO4 has15 a meV similar. (b) mag- Momentum dependence of the intensities of thehard raw data x-ray deemphasizes RIXS the allows intensity mapping arising from of the annetic entire order Brillouin [8], it can be understood00 by analogy that the showing¼ diverging intensity at ;  and vanishing intensity at particle-holezone within continuum only and a revealsfew degrees a clear dispersiveof 90 scatteringdispersion geometry of the spin-orbit exciton should also have its ð Þ featureso above that 0.4 the eV, spectrum as shown in reveals Fig. 4(a) the. The intrinsic energy dynamicalminimum at  struc-=2; =2 . (0,0). The inset shows the Brillouin zone of the undistorted scale of this excitation coincides with the known energy The overallð bandwidthÞ is determinedtetragonal by (I4 the=mmm parameters) unit cell (black square) and the magnetic tural factor with minimal RIXS matrix element effects. scale of spin-orbit coupling in Sr2IrO4 (SO 0:5 eV)[7], involved in the hopping process,cell (blue which diamond), is depicted and in the notation follows the convention for and thusSecond, we assign the it to measured intrasite excitations magnon of a hole dispersion across relationFig. 4(d) in strongly the hole picture. It involves moving an excited the tetragonal unit cell, as, for instance, in La2CuO4. the spin-orbit split levels in the t2g manifold, i.e., from the hole to a neighboring site, which happens in two steps. J supports1=2 level the to one theories of J predicting3=2 quartet levels that theFirst, superexhange the excited hole in site i hops to a neighboring site j eff ¼ eff ¼ [7,13interactions,15] [see Fig. 4(d) of]. WeJ refer to1= such2 moments an excitation on( at3=2 squareprocess), lattice generating an intermediate state with energy eff ¼ in the case of Sr2CuO2Cl2 [19]. However, here we do not as a ‘‘spin-orbitwith corner-sharing exciton’’; see Fig. octahedra4(d) [26]. are governedU0, which by is a the SU(2) Coulomb repulsion between two holes at a The dispersion of the spin-orbit exciton with a bandwidth site in two different spin-orbitalpursue quantum this levels. path Then, because, the as we show below, another kind invariant Hamiltonian with AF coupling [2,10]. Third, of at least 0.3 eV implies that this local excitation can other hole in site j hops backof magnetic to site i (t1=2 modeprocess). in Sr2IrO4, which is not present in propagatethe coherently magnon through mode the inlattice.Sr Our2IrO model4 has of aThus, bandwidth the energy of scale ofcuprates, the dispersion may is affect set by the magnon dispersion. the spin-orbit exciton starts from a recognition that the 2t1=2t3=2=U0, which is of the order of the magnetic ex- 200 meV, as compared to 300 meV in La2CuO4 [14] Characterizing the magnon mode is important because it hopping process is formally analogous to the problem of a change couplings. In fact, these processes lead to the holeand propagatingSr2CuO in the2Cl background2 [19], ofwhich AF ordered is consistent mo- superexchange with energy interactionsstrongly responsible renormalizesfor the magnetic the dispersion of a doped hole or ments,scales which has of been hopping extensivelyt and studied on-site in the context Coulombordering, energy but hereU theyin involveelectron both the and ground is believed state and to provide a pairing mechanism for of cuprate HTSC [27]. Although the spin-orbit exciton does excited states of Ir ions. not carrySr2IrO a charge,4 being its hopping smaller creates by a trailroughly of misaligned50% thanTechnically, those reported the spin-orbitHTSC. exciton hopping We now can be show de- that Sr2IrO4 supports an exciton spinsfor and thus the is cuprates subject to the [10 same,20 kind,21 of]. renormalization scribed by the following Hamiltonianmode, which [13]: gives access to the dynamics of a particle For a quantitative description, we have fitted the magnon propagating in the background of AF ordered moments 177003-3 dispersion using a phenomenological J-J0-J00 model [22]. even in an undoped case. Figures 3(a) and 3(b) show the Here, the J, J0, and J00 correspond to the first, second, and energy loss spectra along high symmetry directions. No third nearest neighbors, respectively. In this model, the corrections to the raw data, such as normalization or sub- downward dispersion along the magnetic Brillouin zone traction of the elastic contaminations, have been made. from ; 0 to =2; =2 is accounted for by a ferromag- Another virtue of using hard x ray is that, by working in ð Þ ð Þ netic J0 [14,22]. We find J 60, J0 20, and J00 the vicinity of 90 scattering geometry, elastic (Thompson) 15 meV. The nearest-neighbor¼ interaction¼À J is smaller¼ scattering can be strongly suppressed. In addition to the than that found in cuprates by roughly 50%. The fit can low-energy magnon branch ( 0:2 eV), we observe high- be improved by including higher-order terms from longer- energy excitations with strong momentum dependence range interactions, which were also found to be important in the energy range of 0:4 0:8 eV. This mode is 

177003-2 Resonant inelastic x-ray scattering (RIXS)

K-edge: 1s L-edge: 2s or 2p M-edge: 3s …

http://physics.usask.ca/~chang/homepage /xray/xray.html Rev. Mod. Phys. 83, 705 (2011) Resonant inelastic x-ray scattering (RIXS)

http://physics.usask.ca/~chang/homepage Rev. Mod. Phys. 83, 705 (2011) /xray/xray.html Resonant inelastic scattering @ PSI RIXS- Example

M Moretti Sala et al 2011 New J. Phys. 13 http://www.nature.com/nphys/journal/vaop/ 043026 ncurrent/pdf/nphys4248.pdf Literature Club

Probing of anti-ferromagnetic spin excitations

Phys. Rev. Lett. 105, 247001 (2010) – Inelastic Neutron Scattering (INS) – La2CuO4

Nature Physics 13, 633 (2007) – Inelastic Neutron Scattering (INS) – Ca2RuO4

Phys. Rev. Lett. 108, 177003 (2012) – Resonant Inelastic X-ray scattering (RIXS) – Sr2IrO4

Nat. Comm. 10, 786 (2019) – Resonant Inelastic X-ray scattering (RIXS) – La2CuO4

Skyrmions

Science 323, 915-919 (2009) – Skyrmions in reciprocal space (MnSi)

Nature 465, 901–904 (2010) – Skymions in real space (Fe0.5Co0.5Si) Lorena 17.11.2020 Mott-insulators

The Nobel Prize in Physics 1977 1/3 Nevill F. Mott https://history.aip.org/phn/11806013.html Metal

Mott Insulator

https://www.nanowerk.com/news/newsid=17393.php Mott-insulators: Resistivity

M. Imada et al., Rev. Mod. Phys. 70, 1040 (1998) Mott-insulator: Filling condition

M. Imada et al., Rev. Mod. Phys. 70, 1040 (1998) Mott-insulator: Filling condition Mott-insulator: Filling condition

Hydrogen Case

Electron interaction U is fixed

Lattice constant: a

t = probability for hopping M. Imada et al., Rev. Mod. Phys. 70, 1040 (1998) Mott-insulator: Filling condition

M. Imada et al., Rev. Mod. Phys. 70, 1040 (1998) Mott-insulator: Filling condition

J. Phys. Soc. Japan 75 083710 (2006) Quantum Criticality

http://www.u-tokyo.ac.jp/en/utokyo-research/research- news/anomalous-critical-state-of--near-a-metal- Nature Physics 4, 170 (2008) insulator-boundary/ PHYSICAL REVIEW B 79, 195106 ͑2009͒

Bandwidth-controlled in ␬-(BEDT-TTF)2Cu[N(CN)2]BrxCl1−x: Optical studies of correlated carriers

Michael Dumm, Daniel Faltermeier, Natalia Drichko, and Martin Dressel 1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart Germany

Cécile Mézière and Patrick Batail Laboratoire CIMMA, UMR 6200 CNRS-Université d’Angers, Bât. K, UFR Sciences, 2 Boulevard Lavoisier, F-49045 Angers, France ͑Received 19 February 2009; published 6 May 2009͒

In the two-dimensional organic charge-transfer salts ␬-͑BEDT-TTF͒2Cu͓N͑CN͒2͔BrxCl1−x a systematic variation in the Br content from x=0 to 0.9 allows us to tune the Mott transition by increasing the bandwidth. At temperatures below 50 K, an energy gap develops in the Cl-rich samples and grows to approximately 1000 cm−1 for T 0. With increasing Br concentration spectral weight shifts into the gap region and eventually → fills it up completely. As the samples with x=0.73, 0.85, and 0.9 become metallic at low temperatures, a Drude-type response develops due to the coherent quasiparticles. Here, the scattering rate shows a ␻2 dependence and the effective mass of the carriers is enhanced in agreement with the predictions for a Fermi liquid. These typical signatures of strong electron-electron interactions are more pronounced for com- positions close to the critical value xc Ϸ0.7, where the metal-to-insulator transition occurs.

DOI: 10.1103/PhysRevB.79.195106 PACS number͑s͒: 71.30.ϩh, 71.10.Hf, 74.70.Kn, 74.25.Gz

I. INTRODUCTION cooled below 50 K. The application of external pressure shifts the compound across the boundary. It becomes One of the most intriguing issues in condensed-matter metallic and even undergoes a superconducting transition, physics is the transition from a metal to an insulator driven very similar to the Br analog at ambient pressure. Recently, by electronic correlations. Why does an electron in a crystal the critical behavior in the vicinity of the metal-insulator change from itinerant to localized behavior when a control transition and the critical endpoint was thoroughly investi- parameter such as magnetic field, or pressure is var- Mottgated by- dcinsulator starting point for SC measurements under external pressure and mag- ied? For a system with a half-filled conductance band this netic field.12–14 In the present study we gradually substitute problem is known as the Mott transition, one of the central Cl by the isovalent Br in the anion layers and obtain the problems of strongly correlated electrons.1 While studies of the influence of electron-electron interactions in materials Cu[N(CN)2]Br I3 with open d and f electron shells have a long history,2–4 only Cu[N(CN)2]Cl Cu(NCS)2 recently it was realized that also in molecular conductors ͑where the charges originate from molecular orbitals͒ elec- 5–10 tronic correlations are very significant. In transition-metal κ-(BEDT-TTF)2X , the Coulomb interaction is crucial for any under- 100 Mott standing of the unconventional metallic and superconducting insulator bad metal properties, as well as the vicinity to magnetic order; this is most pronounced in the underdoped cuprates.4 However, in Fermi liquid metal 10 many regards organic conductors turn out to be superior antiferro- magnetic model systems to study certain effects of electron-electron insulator superconductor interaction since their properties can be more easily tuned by Temperature (K) ͑physical or chemical͒ pressure. Varying U/t, where U is 1 on-site repulsion and t is a transfer integral, opens the road to Pressure the bandwidth-controlled Mott transition. U / t b1 / p The family of half-filled organic conductors PHYSICAL REVIEW B 79, 195106 ͑2009͒ FIG. 1. Color online Schematic phase diagram of Bandwidth-controlled Mott transition in ␬-(BEDT-TTF)͑ Cu[N(CN) ]Br Cl ͒: Optical studies ␬-͑BEDT-TTF͒2X ͓anions X=Cu͑CN͒3, Cu͓N͑CN͒2͔Cl, 2 2 x 1−x ␬-͑ofBEDT-TTF correlated carriers͒2X. The on-site Coulomb repulsion with respect Cu͓N͑CN͒2͔Br, Cu͑SCN͒2, and I3͔͑Ref. 11͒ has a particu- Michael Dumm, Danielto the Faltermeier, hopping Natalia Drichko, integral and Martin DresselU/t can be tuned either by external 1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart Germany lary rich phase diagram as a function of pressure and tem- pressure or modifying the anions X. The arrows indicate Cécile Mézière and Patrick Batail perature as depicted in Fig. 1. The abscissaLaboratoire of this CIMMA, phase UMR 6200 CNRS-Universitéthe d’Angers, approximate Bât. K, UFR Sciences, 2 Boulevard position Lavoisier, F-49045 Angers, of France ␬-phase salts with diagram can be interpreted as the variation in the relative ͑Received 19 February 2009; published 6 May 2009͒ In the two-dimensional organicX charge-transfer=Cu͓N͑ saltsCN␬-͑BEDT-TTF͒2͔Cl,͒2Cu Cu͓N͑CN͓N͒2͔Br͑CNxCl1−x ͒a2 systematic͔Br, Cu͑NCS͒2, and I3 at ambient variation in the Br content from x=0 to 0.9 allows us to tune the Mott transition by increasing the bandwidth. Coulomb interaction U/t. Similar to external pressure,At temperatures below a 50 K, anpressure, energy gap develops respectively. in the Cl-rich samples and The grows to bandwidth-controlled approximately 1000 cm−1 for T 0. With increasing Br concentration spectral weight shifts into the gap region and eventually → variation in anions also changes the bandwidth and,fills it up completely. thus, As the samplesbetween with x=0.73, the 0.85, and insulator 0.9 become metallic and at low temperatures, the Fermi a liquid/superconductor can Drude-type response develops due to the coherent quasiparticles. Here, the quasiparticle scattering rate shows reduces U/t. For large values of U/t the half-filleda ␻2 dependence system and the effectivebe mass of the explored carriers is enhanced in by agreement gradually with the predictions for a replacing Cl by Br in Fermi liquid. These typical signatures of strong electron-electron interactions are more pronounced for com- becomes a Mott insulator. This behavior is observedpositions close to inthe critical value␬x-c͑ϷBEDT-TTF0.7, where the metal-to-insulator͒2Cu transition͓N͑CN occurs.͒2͔BrxCl1−x. Here c and a are the lattice DOI: 10.1103/PhysRevB.79.195106 PACS number s : 71.30.ϩh, 71.10.Hf, 74.70.Kn, 74.25.Gz ␬-͑BEDT-TTF͒2Cu͓N͑CN͒2͔Cl: at ambient conditions the parameters; b1͑ ͒ and p indicate intradimer and interdimer transfer narrow-gap semiconductor gradually gets insulatingI. whenINTRODUCTION integrals, respectively.cooled below 50 K. The application of external pressure shifts the compound across the phase boundary. It becomes One of the most intriguing issues in condensed-matter metallic and even undergoes a superconducting transition, physics is the transition from a metal to an insulator driven very similar to the Br analog at ambient pressure. Recently, 1098-0121/2009/79͑19͒/195106͑11͒ by electronic correlations. Why195106-1 does an electron in a crystal the critical behavior in the vicinity©2009 of the metal-insulator The American Physical Society change from itinerant to localized behavior when a control transition and the critical endpoint was thoroughly investi- parameter such as magnetic field, doping or pressure is var- gated by dc measurements under external pressure and mag- ied? For a system with a half-filled conductance band this netic field.12–14 In the present study we gradually substitute problem is known as the Mott transition, one of the central Cl by the isovalent Br in the anion layers and obtain the problems of strongly correlated electrons.1 While studies of the influence of electron-electron interactions in materials Cu[N(CN)2]Br I3 with open d and f electron shells have a long history,2–4 only Cu[N(CN)2]Cl Cu(NCS)2 recently it was realized that also in molecular conductors ͑where the charges originate from molecular orbitals͒ elec- 5–10 semiconductor tronic correlations are very significant. In transition-metal κ-(BEDT-TTF)2X oxides, the Coulomb interaction is crucial for any under- 100 Mott standing of the unconventional metallic and superconducting insulator bad metal properties, as well as the vicinity to magnetic order; this is most pronounced in the underdoped cuprates.4 However, in Fermi liquid metal 10 many regards organic conductors turn out to be superior antiferro- magnetic model systems to study certain effects of electron-electron insulator superconductor interaction since their properties can be more easily tuned by Temperature (K) ͑physical or chemical͒ pressure. Varying U/t, where U is 1 on-site repulsion and t is a transfer integral, opens the road to Pressure the bandwidth-controlled Mott transition. U / t b1 / p The family of half-filled organic conductors FIG. 1. ͑Color online͒ Schematic phase diagram of ␬-͑BEDT-TTF͒ X ͓anions X=Cu͑CN͒ , Cu͓N͑CN͒ ͔Cl, 2 3 2 ␬-͑BEDT-TTF͒ X. The on-site Coulomb repulsion with respect Cu N CN Br, Cu SCN , and I Ref. 11 has a particu- 2 ͓ ͑ ͒2͔ ͑ ͒2 3͔͑ ͒ to the hopping integral U/t can be tuned either by external lary rich phase diagram as a function of pressure and tem- pressure or modifying the anions X. The arrows indicate perature as depicted in Fig. 1. The abscissa of this phase the approximate position of ␬-phase salts with diagram can be interpreted as the variation in the relative X=Cu͓N͑CN͒2͔Cl, Cu͓N͑CN͒2͔Br, Cu͑NCS͒2, and I3 at ambient Coulomb interaction U/t. Similar to external pressure, a pressure, respectively. The bandwidth-controlled phase transition variation in anions also changes the bandwidth and, thus, between the insulator and the Fermi liquid/superconductor can reduces U/t. For large values of U/t the half-filled system be explored by gradually replacing Cl by Br in becomes a Mott insulator. This behavior is observed in ␬-͑BEDT-TTF͒2Cu͓N͑CN͒2͔BrxCl1−x. Here c and a are the lattice ␬-͑BEDT-TTF͒2Cu͓N͑CN͒2͔Cl: at ambient conditions the parameters; b1 and p indicate intradimer and interdimer transfer narrow-gap semiconductor gradually gets insulating when integrals, respectively.

1098-0121/2009/79͑19͒/195106͑11͒ 195106-1 ©2009 The American Physical Society Mott-insulator starting point for SC Example: Ca2RuO4

Physical Review Letters 93, 146401 (2004) Example: Ca2RuO4

ARPES Calculation Band structure Band structure

Band Structure Calculation Ingredient: Crystal Lattice

Denys Sutter & Fabio Cossalter et al., Nature Communications 8, 15176 (2017) Combine band – Mott insulator

Crystal field splitting

Ca dxy

dxz dyz Compression Elongation Apical oxygen week ending PRL 101, 076402 (2008) PHYSICAL REVIEW LETTERS 15 AUGUST 2008

 ! was obtained by using Kramers-Kronig (KK) trans- one remaining electron in the Jeff 1=2 band, the system formation.ð Þ The validity of KK analysis was checked by is effectively reduced to a half-filled¼J 1=2 single band eff ¼ independent ellipsometry measurements between 0.6 and system [Fig. 1(c)]. The Jeff 1=2 spin-orbit integratedweek ending PRL6.4 eV.101, XAS076402 spectra (2008) were obtained at 80PHYSICAL K under vacuum REVIEWstates form LETTERS a narrow band so¼ that even small15U AUGUSTopens a 2008 of 5 10 10 Torr at the Beamline 2A of the Pohang Light Mott gap, making it a J 1=2 Mott insulator [Fig. 1(d)]. À eff ¼ Source! Âwas with obtainedÁh by0:1 using eV. Kramers-Kronig (KK) trans- Theone narrow remaining band width electron is due in theto reducedJeff hopping1=2 band, elements the system ð Þ ¼ is effectively reduced to a half-filled¼J 1=2 single band formation.Here we The propose validity a schematic of KK model analysis for wasemergence checked of a by of the Jeff 1=2 states with isotropic orbitaleff and mixed independent ellipsometry measurements between 0.6 and system [Fig.¼ 1(c)]. The J 1=2 spin-orbit¼ integrated novel Mott ground state by a large SO coupling energy SO spin characters. The formationeff of the Jeff bands due to the states form a narrow band so¼ that even small U opens a 6.4as shown eV. XAS in Fig. spectra1. Under wereobtained the Oh symmetry at 80 K under the 5d vacuumstates large SO explains why Sr2IrO4 (SO 0:4 eV) is insulat- 10  ofare5 split10 intoÀ Torrt andat thee Beamlineorbital states 2A by of the Pohang crystal field Light ingMott while gap,Sr makingRhO ( it a Jeff0:151 eV=2)Mott is metallic. insulator [Fig. 1(d)].  2g g 2 4 SO  ¼ Sourceenergy with 10Dq.Áh In general,0:1 eV4.d and 5d TMOs have suffi- TheThe narrowJeff band band formation width is is due well to justified reduced in hopping the LDA elements and ¼ cientlyHere we large propose 10Dq a toschematic yield a modelt5 low-spin for emergence state for of a LDAof theUJeffcalculations1=2 states on Sr with2IrO isotropic4 with and orbital without and in- mixed 2g þ novel Mott ground state by a large SO coupling energy  cludingspin characters. the SO¼ coupling The formation presented of in the Fig.J 2bands. The due LDA to the Sr2IrO4, and thus the system would become a metal withSO eff as shown in Fig. 1. Under the O symmetry the 5d states resultlarge [Fig. 2(a)explains] yields why a metalSr IrO with( a wide0t:24g eVband) is as insulat- in partially filled wide t2g band [Fig.h1(a)]. An unrealistically SO 2 4 SO  are split into t and e orbital states by the crystal field Fig.ing1(a) while, andSr theRhO Fermi( surface0:15 (FS) eV is) nearly is metallic. identical to large U W 2couldg leadg to a typical spin S 1=2 Mott 2 4 SO  energyinsulator 10Dq. [Fig. 1(b) In general,]. However,4d aand reasonable5d TMOsU¼cannot have lead suffi- thatThe of SrJeff2RhOband4 [ formation12,13]. The is well FS, justified composed in the of LDA one- and 5 dimensional yz and zx bands, is represented by holelike cientlyto an insulating large 10Dq state toas seen yield from a t thelow-spin fact that Sr stateRhO for LDA U calculations on Sr2IrO4 with and without in- 2g 2 4 and þ sheets and an electronlike sheet centered at SrisIrO a normal, and thus metal. the As system the SO would coupling become is a taken metal into with cludingX the SO coupling presentedM in Fig. 2. The LDA 2 4 À, X, and M points, respectively [12]. As the SO coupling account, the t states effectively correspond to the orbital result [Fig. 2(a)] yields a metal with a wide t2g band as in partially filled2g wide t2g band [Fig. 1(a)]. An unrealistically is included [Fig. 2(b)], the FS becomes rounded but retains angular momentum L 1 states with zx Fig. 1(a), and the Fermi surface (FS) isweek nearly ending identical to largePRLU 101,W076402could (2008) lead to a typicalPHYSICALm spinl 1S 1=2 Mott REVIEW LETTERS 15 AUGUST 2008 ¼ ¼Æ ¼ Çðj i Æ the overall topology. Despite small variations in the FS i yz =p2 and xy . In the strong SO¼ coupling that of Sr2RhO4 [12,13]. The FS, composed of one- insulator [Fig. 1(b)ml].0 However, a reasonable U cannot lead topology, the band structure changes remarkably: Two j iÞ ¼ ¼ j i dimensional yz and zx bands, is represented by holelike tolimit, an insulating theffiffiffi t2g band state splits as seen into effective from thetotal fact angular that Sr2RhO mo- 4 narrow bands crossing EF are split off from the rest due Novel Jeff 1=2 Mott State Induced by Relativistic and Spin-OrbitX sheets and Coupling an electronlike in Sr 2IrOM sheet4 centered at ismentum a normalJeff metal.1=2 doublet As¼ the and SOJeff coupling3=2 quartet is taken bands into account,[Fig. 1(c) the][17t¼].states Note effectivelythat the J correspond¼1=2 is energetically to the orbital À, X, and M points, respectively [12]. As the SO coupling B. J. Kim,21gHosub Jin,1 S. J. Moon,eff ¼2 J.-Y. Kim,3 B.-G. Park,4 C.(a)is S. LDA included Leem,5 Jaejun [Fig. 2(b) Yu,1],T. the W. FS Noh, becomes2 C. Kim, rounded5 S.-J. Oh, but1 retains angularhigher than momentum the Jeff L 3=12,states seemingly with against3,4, the Hund’szx 6 6 7 ¼ J.-H. Park,ml 1* V. Durairaj, G. Cao,the overalland E. Rotenberg topology. Despite small variations in the FS rule, since the J ¼1=2 is branched off¼Æ from¼ theÇðjJ i Æ E-E i yz =p2 and eff 1 xy . In the strong SO coupling5=2 M ml ¼0 School of Physics and Astronomy, Seoul Nationaltopology, University, the Seoul band 151-747, structure Korea changes remarkably: Two ¼2 F (j5diÞ ) manifold due to¼ thej largei crystal field as depicted in (eV) limit,5=2 the t bandReCOE splits into & Schooleffective of Physicstotal and angular Astronomy, mo- SeoulX NationalΓ University, Seoul 151-747, Korea ffiffiffi 2g 3 narrow bands crossing EF are split off from the rest due Fig. 1(e). As a result,Pohang with Accelerator the filled Laboratory,Jeff 3 Pohang=2 band University and of Science and Technology, Pohang 790-784, Korea mentum J 41=2 doublet and J 3¼=2 quartet bands eff ¼ eSSC & Department ofeff Physics,¼ Pohang University of Science and Technology, Pohang 790-784, Korea [Fig. 1(c)][17]. Note that the5InstituteJeff of1 Physics=2 is andenergetically Applied Physics, Yonsei(a) LDA University, Seoul, Korea 6 ¼ M X higher than the JeffDepartment3=2, seemingly of Physics and against Astronomy, the Hund’s University of Kentucky, Lexington,Γ Kentucky 40506, USA Γ 7 ¼ (b) LDA+SO

Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 96720, USA E-E rule, since the Jeff 1=2 is branched off from the J5=2 M ¼ (Received 23 January 2008; published 15 August 2008) F E-E

(5d ) manifold due to the large crystal field as depicted in (eV) 5=2 M ΓβM X α β We investigated the electronic structure of 5d transition-metal XSr2IrO4 using angle-resolved F Fig. 1(e). As a result, with the filled Jeff 3=2 band and (eV) photoemission, optical conductivity,¼ x-ray absorption measurements,X Γ and first-principles band calcula- tions. The system was found to be well described by novel effective total angular momentum Jeff states, in which the relativistic spin-orbit coupling is fully taken into account under a largeΓ crystalM field. DespiteX Γ (b) LDA+SO delocalized Ir 5d states, the Jeff states form such narrow bands that even a smallΓ correlationM energy leadsX to Γ (c) LDA+SO+U the Jeff 1=2 Mott ground state with unique electronic and magnetic behaviors, suggesting a new class of E-E ¼ M βM Jeff quantum spin driven correlated-electron phenomena.

α β E- M X F β (eV) X αΓ X µ DOI: 10.1103/PhysRevLett.101.076402 PACS numbers: 71.30.+h, 71.20. b, 78.70.Dm, 79.60. i (eV) X Γ À À

Mott physics based on the Hubbard Hamiltonian, which Sr IrO , however, is unexpectedly an insulator with weak 2 4 Γ M X Γ is at the root of various noble physical phenomena such as ferromagnetism(c) LDA+SO+U [8].Γ At this point,M it is naturalX to considerΓ metal-insulator transitions, magnetic spin orders, high TC the(d) spin-orbitLDA+U (SO) coupling as a candidate responsible for superconductivity, colossal magneto-resistance, and quan- the insulating nature since its energy is much larger than M E- β E-E tum criticality, has been adopted to explain electrical and that inM 3d andα 4dX systems. Recent band calculations µ (eV) F

X Γ (eV) magnetic properties of various materials in the last several showedX thatΓ the electronic states near EF can be modified decades [1–5]. Great success has been achieved in 3d considerably by the SO coupling in 5d systems, and sug- transition-metal oxides (TMOs), in which the localized gested a new possibility of the Mott instability [14]. It indi- 3d states yield strongly correlated narrow bands with a cates that the correlation effectsM can be importantX even in Γ Γ M X Γ Γ large on-site Coulomb repulsion U and a small5 5 band width 5d (d)TMOs LDA+U when combined with strong SO coupling. FIG. 1. Schematic energy diagrams for the 5d (t2g) configu- rationW. (a) As without predicted, SO and mostU, stoichiometric (b) with an unrealistically3d TMOs are large anti-U FIG.In 2 this (color Letter, online). we show Theoretical formation Fermi of new surfaces quantum and state band butferromagnetic no SO, (c) with (AFM) SO but Mott no insulatorsU, and (d) [5 with]. On SO the and otherU. dispersionsbands withM in effective (a) LDA, total (b) angularLDA momentumSO, (c) LDAJeff inSO5d U E-E F

hand, 4d and 5d TMOs were considered as weakly- electron systems under a large crystalþ field, in whichþ theþ (eV) Possible optical transitions A and B are indicated by arrows. (2 eV),X and (d)Γ LDA U. In (c), the left panel shows topology correlated wide band systems with largely reduced U due SO coupling is fullyþ taken into account, and also report for (e) 5d level splittings by the crystal field and SO coupling. of valence band maxima (EB 0:2 eV) instead of the FS. to delocalized 4d and 5d states [6]. Anomalous insulating the first time manifestation¼ of a novel J 1=2 Mott 076402-2 eff ¼ behaviors were recently reported in some 4d and 5d TMOs ground state realized in Sr2IrO4 by using ARPES, optical Γ M X Γ [7–10], and the importance of correlation effects5 5 was rec- conductivity, and x-ray absorption spectroscopy (XAS) FIG. 1. Schematic energy diagrams for the 5d (t2g) configu- ognized in 4d TMOs such as Ca2RuO4 and Y2Ru2O7, and first-principles band calculations. This new Mott rationwhich (a) without were interpreted SO and U, as (b) Mott with insulators an unrealistically near the border large U groundFIG. 2 state (color exhibits online). novel Theoreticalelectronic and Fermi magnetic surfaces behav- and band but no SO, (c) with SO but no U, and (d) with SO and U. dispersions in (a) LDA, (b) LDA SO, (c) LDA SO U line of the Mott criteria, i.e., U W [7]. However, as 5d ior, for example, spin-orbit integrated narrowþ bands andþ an þ Possiblestates optical are spatially transitions more A extended and B areand indicatedU is expected by arrows. to be exotic(2 eV), orbital and dominated (d) LDA localU. In magnetic (c), the left moment, panel suggest- shows topology (e) 5d level splittings by the crystal field and SO coupling. of valence band maximaþ (E 0:2 eV) instead of the FS. further reduced, insulating behaviors in 5d TMOs such as ing a new class of the Jeff quantumB ¼ spin driven correlated- Sr2IrO4 and Cd2Os2O7 have been puzzling [8,9]. 076402-2electron phenomena. Sr2IrO4 crystallizes in the K2NiF4 structure as La2CuO4 Single crystals of Sr2IrO4 were grown by flux method and its 4d counterpart Sr2RhO4 [8,11]. Considering its odd [15]. ARPES spectra were obtained at 100 K from cleaved number of electrons per unit formula (5d5), one expects a surfaces in situ under vacuum of 1 10 11 Torr at the  À metallic state in a naı¨ve band picture. Indeed Sr2RhO4 beamline 7.0.1 of the Advanced Light Source with h# (4d5) is a Fermi liquid metal. Its Fermi surface (FS) mea- 85 eV and ÁE 30 meV. The chemical potential " was¼ ¼ sured by the angle-resolved photoemission spectroscopy referred to EF of electrically connected Au. The band (ARPES) agrees well with the band calculation results calculations were performed by using first-principles [12,13]. Since both systems have identical atomic arrange- density-functional-theory codes with LDA and LDA U ments with nearly the same lattice constants and bond methods [16]. The optical reflectivity R ! was measuredþ angles [8,11], one expects almost the same FS topology. at 100 K between 5 meV and 30 eV andð theÞ conductivity

0031-9007=08=101(7)=076402(4) 076402-1 Ó 2008 The American Physical Society