Tasks Monday 05.10 Magnetism 1: Diamagnetism Tasks: Read: Kittel, Chapter: Diamagnetism & Paramagnetism Hand-in exerCise sheet Tuesday 06.10 Magnetism 1: Paramagnetism Tasks: Read: Kittel, Chapter: Diamagnetism & Paramagnetism Monday 12.10 Magnetism 2: Ferromagnetism Tasks: Read: Kittel, Chapter: Ferromagnetism & Antiferromagnetism 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 EleCtron 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 liquid 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 Liquids Custers et al., Nature 424, 524, (2003) N. Doiron-Leyraud et al., Physical ReView B 80, 214531 (2009) (3) Unconventional superconductivity Science 336, 1554-1557 (2012) – Penetration depth @ QCP Nature Physics 11, 17–20 (2015) – SC fluctuations in URu2Si2 (4) Superconductivity without phonons 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: PLASMONS, POLARITONS, AND POLARONS 393 Dielectric Function of the Electron Gas 395 Definitions of the Dielectric Function 395 Plasma 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-Insulator 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-Phonon Interaction: Polarons 420 Peierls Instability of Linear Metals 422 Summary 424 Problems 424 CHAPTER 15: OPTICAL PROCESSES AND EXCITONS 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 magnons: 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-magnon dispersion (T 10 K) along lines in 2 4 ¼ (c, inset). Symbols indicate Ei: 160 meV (h), 240 meV ( ), and 450 meV ( ). The solid 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 quasiparticles 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 energy gap [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].