DOCSLIB.ORG

Fermi Liquid Theory of a Fermi Ring

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fermi Liquid Theory of a Fermi Ring FermiFermi liquidliquid theorytheory ofof aa FermiFermi ringring T. Stauber, N. Peres, F. Guinea, and A. Castro Neto cond-mat/0611468 OverviewOverview Biased bilayer graphene One band approximation: Mexican hat dispersion Coulomb interaction: Self-energy correction Ferromagnetism: Ginzburg-Landau functional Polarizability A) High energies (Non-Fermi liquid behaviour) B) Low energies (Fermi liquid with modified Friedel oscillations) Plasmon dispersion Biased bilayer graphene A2 B2 V t⊥ A1 B1 t V= t⊥ = 3.0 eV Biased bilayer graphene Mexican hat dispersion Mexican hat dispersion a) b) Ek 0.4 , Ek 0.04 k 0.02 0.05 0.05 0 0.044 ky 0.02 0.02 0.4 0 kx 0.02002 0.04 0.04 Fermi ring: k k0 min b kmax Mexican hat dispersion E()= k Δα −2 k λ + 4 k α k 2 = −2πn min,σ 2λ σ α k 2 = +2πn max,σ 2λ σ Coulomb interaction 1 + + Hint = V q() ck+ q,σp c− qσ,,′ p σ′ ck ,σ c ∑∑ 2Ak,, p, σ q σ′ 2πe2 V() q= a) Self-energy orrectionsc to the dispersion ε 0q b) Exchange energy k a = min kmax Self-energy corrections n<nc n>nc 12 -2 nc =1.5x10 cm -->“exactly” filled Mexican hat Ferromagnetism Density fo states: 1 1 ρ() ε = π4 λ ε− min ε Stoner criterion: ε= ε( + 2 πn 2 λ ) F min σ (ρ ε F )U > 1 Ginzburg-Landau functional n= n++ − n m= + n− − n F a=()() n m2 + b4 n… + m Ginzburg-Landau functional 0.01 b(n) a(n) 0 0 A A -0.0001 -0.01 0.0002 -0.02 - -0.03 -0.0003 0 1 2 3 4 5 6 1 1.5 2 2.5 3 3.5 4 11 −2 11 −2 n [in 10 cm ] n [in 10 cm ] with self-energy correction m= n Ginzburg-Landau functional 0 F(m) A -2 -4 n =1 -6 n =5 n =2.5 -8 -10 0 2 4 6 11 −2 m [in 10 cm ] Polarizability “Mexican hat dispersion” 2 E( k )= (| k | −0 k ) High energy regime (b → 0) Asymptotic behaviour: −5 Im P +0. 3 1 2.5 ImP ( qω ,→ ) 2 ω 2 ω k ω 0 1.5 asiparticleQu lifetime: 1 Γ(ε ) ∝ ε | − ε | 0.5 F 0 0 0.5 1 1.5 2 2.5 3 q0 k Low-energy regime (ω ¡ b2) A) Forward scattering −0.5 Im P +0. Asymptotic behaviour 3 2.5 2 b2 ω 1.5 1 0.5 Perfect nesting at q=2b: 0 0 0.5 1 1.5 2 2.5 3 q b Friedel oscillations: cos(br 2 ) δn() r→ r 2 2 B) Backward scattering Low-energy regime (ω ¡ b ) −0.5 Im P +0. −0.5 Im P +0. 3 3 Friedel oscillations: 2.5 2.5 2 2 ω b2 ω b2 1.5 1.5 q→2 k0 + q 1 cos(1 2br ) cos( k0 r 2 ) δn() r → 2 0.5 0.5 r 0 0 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 q b q b Asymptotic behaviour: Perfect nesting: q=2k0 q=2k0-2b q=2k0+2b Example Numerical Numerical Analytical*1.5 Analytical/1.3 30 1,5 q=3b/2 q=2k0-b/2 20 1 -Im P -Im P 10 high-energy0,5 regime 0 0 024 024 2 2 ω/b ω/b 2 2 low-energyω/b regime ω/b 15 q=3b 1 q=2k0-3b 10 -Im P-Im P -Im P-Im P 0,5 5 0 0 0 5 10 15 5 0 5 10 15 20 2 2 ω/b ω/b Plasmon dispersion Random-phase approximation: (q ,ε )ω = 1 V− ( q ) Pω ( q , ) Plasmon dispersion: (ε q ,Ωq )= 0 2 DEG: Bilayer graphene: Summary 9 Biased bilayer graphene at low densities leads to a topologically non-trivial Fermi-surface. 9 At intermediate electronic densities, the one-band approximation becomes unstable. 9 Thegroundstateisunstabletowardferromagnetic ordering. 9 Including self-energy effects leads to a saturation in the magnetisation. 9 At low energies, the system is a Fermi-liquid, albeit with peculiar Friedel oscillations. 9 At high energies or low densities, the system shows non- Fermi liquid behavior. 9 The plasmons show the familiar two-dimensional square- root disperion, but with a larger energy scale..
Load more

Recommended publications
  • Lecture 3: Fermi-Liquid Theory 1 General Considerations Concerning Condensed Matter
    Phys 769 Selected Topics in Condensed Matter Physics Summer 2010 Lecture 3: Fermi-liquid theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 General considerations concerning condensed matter (NB: Ultracold atomic gasses need separate discussion) Assume for simplicity a single atomic species. Then we have a collection of N (typically 1023) nuclei (denoted α,β,...) and (usually) ZN electrons (denoted i,j,...) interacting ∼ via a Hamiltonian Hˆ . To a first approximation, Hˆ is the nonrelativistic limit of the full Dirac Hamiltonian, namely1 ~2 ~2 1 e2 1 Hˆ = 2 2 + NR −2m ∇i − 2M ∇α 2 4πǫ r r α 0 i j Xi X Xij | − | 1 (Ze)2 1 1 Ze2 1 + . (1) 2 4πǫ0 Rα Rβ − 2 4πǫ0 ri Rα Xαβ | − | Xiα | − | For an isolated atom, the relevant energy scale is the Rydberg (R) – Z2R. In addition, there are some relativistic effects which may need to be considered. Most important is the spin-orbit interaction: µ Hˆ = B σ (v V (r )) (2) SO − c2 i · i × ∇ i Xi (µB is the Bohr magneton, vi is the velocity, and V (ri) is the electrostatic potential at 2 3 2 ri as obtained from HˆNR). In an isolated atom this term is o(α R) for H and o(Z α R) for a heavy atom (inner-shell electrons) (produces fine structure). The (electron-electron) magnetic dipole interaction is of the same order as HˆSO. The (electron-nucleus) hyperfine interaction is down relative to Hˆ by a factor µ /µ 10−3, and the nuclear dipole-dipole SO n B ∼ interaction by a factor (µ /µ )2 10−6.
    [Show full text]
  • Coexistence of Superparamagnetism and Ferromagnetism in Co-Doped
    Available online at www.sciencedirect.com ScienceDirect Scripta Materialia 69 (2013) 694–697 www.elsevier.com/locate/scriptamat Coexistence of superparamagnetism and ferromagnetism in Co-doped ZnO nanocrystalline films Qiang Li,a Yuyin Wang,b Lele Fan,b Jiandang Liu,a Wei Konga and Bangjiao Yea,⇑ aState Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, People’s Republic of China bNational Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, People’s Republic of China Received 8 April 2013; revised 5 August 2013; accepted 6 August 2013 Available online 13 August 2013 Pure ZnO and Zn0.95Co0.05O films were deposited on sapphire substrates by pulsed-laser deposition. We confirm that the magnetic behavior is intrinsic property of Co-doped ZnO nanocrystalline films. Oxygen vacancies play an important mediation role in Co–Co ferromagnetic coupling. The thermal irreversibility of field-cooling and zero field-cooling magnetizations reveals the presence of superparamagnetic behavior in our films, while no evidence of metallic Co clusters was detected. The origin of the observed superparamagnetism is discussed. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dilute magnetic semiconductors; Superparamagnetism; Ferromagnetism; Co-doped ZnO Dilute magnetic semiconductors (DMSs) have magnetic properties in this system. In this work, the cor- attracted increasing attention in the past decade due to relation between the structural features and magnetic their promising technological applications in the field properties of Co-doped ZnO films was clearly of spintronic devices [1]. Transition metal (TM)-doped demonstrated. ZnO is one of the most attractive DMS candidates The Zn0.95Co0.05O films were deposited on sapphire because of its outstanding properties, including room- (0001) substrates by pulsed-laser deposition using a temperature ferromagnetism.
    [Show full text]
  • Solid State Physics 2 Lecture 5: Electron Liquid
    Physics 7450: Solid State Physics 2 Lecture 5: Electron liquid Leo Radzihovsky (Dated: 10 March, 2015) Abstract In these lectures, we will study itinerate electron liquid, namely metals. We will begin by re- viewing properties of noninteracting electron gas, developing its Greens functions, analyzing its thermodynamics, Pauli paramagnetism and Landau diamagnetism. We will recall how its thermo- dynamics is qualitatively distinct from that of a Boltzmann and Bose gases. As emphasized by Sommerfeld (1928), these qualitative di↵erence are due to the Pauli principle of electons’ fermionic statistics. We will then include e↵ects of Coulomb interaction, treating it in Hartree and Hartree- Fock approximation, computing the ground state energy and screening. We will then study itinerate Stoner ferromagnetism as well as various response functions, such as compressibility and conduc- tivity, and screening (Thomas-Fermi, Debye). We will then discuss Landau Fermi-liquid theory, which will allow us understand why despite strong electron-electron interactions, nevertheless much of the phenomenology of a Fermi gas extends to a Fermi liquid. We will conclude with discussion of electrons on the lattice, treated within the Hubbard and t-J models and will study transition to a Mott insulator and magnetism 1 I. INTRODUCTION A. Outline electron gas ground state and excitations • thermodynamics • Pauli paramagnetism • Landau diamagnetism • Hartree-Fock theory of interactions: ground state energy • Stoner ferromagnetic instability • response functions • Landau Fermi-liquid theory • electrons on the lattice: Hubbard and t-J models • Mott insulators and magnetism • B. Background In these lectures, we will study itinerate electron liquid, namely metals. In principle a fully quantum mechanical, strongly Coulomb-interacting description is required.
    [Show full text]
  • Unconventional Hund Metal in a Weak Itinerant Ferromagnet
    ARTICLE https://doi.org/10.1038/s41467-020-16868-4 OPEN Unconventional Hund metal in a weak itinerant ferromagnet Xiang Chen1, Igor Krivenko 2, Matthew B. Stone 3, Alexander I. Kolesnikov 3, Thomas Wolf4, ✉ ✉ Dmitry Reznik 5, Kevin S. Bedell6, Frank Lechermann7 & Stephen D. Wilson 1 The physics of weak itinerant ferromagnets is challenging due to their small magnetic moments and the ambiguous role of local interactions governing their electronic properties, 1234567890():,; many of which violate Fermi-liquid theory. While magnetic fluctuations play an important role in the materials’ unusual electronic states, the nature of these fluctuations and the paradigms through which they arise remain debated. Here we use inelastic neutron scattering to study magnetic fluctuations in the canonical weak itinerant ferromagnet MnSi. Data reveal that short-wavelength magnons continue to propagate until a mode crossing predicted for strongly interacting quasiparticles is reached, and the local susceptibility peaks at a coher- ence energy predicted for a correlated Hund metal by first-principles many-body theory. Scattering between electrons and orbital and spin fluctuations in MnSi can be understood at the local level to generate its non-Fermi liquid character. These results provide crucial insight into the role of interorbital Hund’s exchange within the broader class of enigmatic multiband itinerant, weak ferromagnets. 1 Materials Department, University of California, Santa Barbara, CA 93106, USA. 2 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA. 3 Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. 4 Institute for Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.
    [Show full text]
  • Study of Spin Glass and Cluster Ferromagnetism in Rusr2eu1.4Ce0.6Cu2o10-Δ Magneto Superconductor Anuj Kumar, R
    Study of spin glass and cluster ferromagnetism in RuSr2Eu1.4Ce0.6Cu2O10-δ magneto superconductor Anuj Kumar, R. P. Tandon, and V. P. S. Awana Citation: J. Appl. Phys. 110, 043926 (2011); doi: 10.1063/1.3626824 View online: http://dx.doi.org/10.1063/1.3626824 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i4 Published by the American Institute of Physics. Related Articles Annealing effect on the excess conductivity of Cu0.5Tl0.25M0.25Ba2Ca2Cu3O10−δ (M=K, Na, Li, Tl) superconductors J. Appl. Phys. 111, 053914 (2012) Effect of columnar grain boundaries on flux pinning in MgB2 films J. Appl. Phys. 111, 053906 (2012) The scaling analysis on effective activation energy in HgBa2Ca2Cu3O8+δ J. Appl. Phys. 111, 07D709 (2012) Magnetism and superconductivity in the Heusler alloy Pd2YbPb J. Appl. Phys. 111, 07E111 (2012) Micromagnetic analysis of the magnetization dynamics driven by the Oersted field in permalloy nanorings J. Appl. Phys. 111, 07D103 (2012) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 12 Mar 2012 to 14.139.60.97. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions JOURNAL OF APPLIED PHYSICS 110, 043926 (2011) Study of spin glass and cluster ferromagnetism in RuSr2Eu1.4Ce0.6Cu2O10-d magneto superconductor Anuj Kumar,1,2 R. P. Tandon,2 and V. P. S. Awana1,a) 1Quantum Phenomena and Application Division, National Physical Laboratory (CSIR), Dr.
    [Show full text]
  • Landau Effective Interaction Between Quasiparticles in a Bose-Einstein Condensate
    PHYSICAL REVIEW X 8, 031042 (2018) Landau Effective Interaction between Quasiparticles in a Bose-Einstein Condensate A. Camacho-Guardian* and Georg M. Bruun Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark (Received 19 December 2017; revised manuscript received 28 February 2018; published 15 August 2018) Landau’s description of the excitations in a macroscopic system in terms of quasiparticles stands out as one of the highlights in quantum physics. It provides an accurate description of otherwise prohibitively complex many-body systems and has led to the development of several key technologies. In this paper, we investigate theoretically the Landau effective interaction between quasiparticles, so-called Bose polarons, formed by impurity particles immersed in a Bose-Einstein condensate (BEC). In the limit of weak interactions between the impurities and the BEC, we derive rigorous results for the effective interaction. They show that it can be strong even for a weak impurity-boson interaction, if the transferred momentum- energy between the quasiparticles is resonant with a sound mode in the BEC. We then develop a diagrammatic scheme to calculate the effective interaction for arbitrary coupling strengths, which recovers the correct weak-coupling results. Using this scheme, we show that the Landau effective interaction, in general, is significantly stronger than that between quasiparticles in a Fermi gas, mainly because a BEC is more compressible than a Fermi gas. The interaction is particularly large near the unitarity limit of the impurity-boson scattering or when the quasiparticle momentum is close to the threshold for momentum relaxation in the BEC.
    [Show full text]
  • Attractive Fermi Polarons at Nonzero Temperatures with a Finite Impurity
    PHYSICAL REVIEW A 98, 013626 (2018) Attractive Fermi polarons at nonzero temperatures with a finite impurity concentration Hui Hu, Brendan C. Mulkerin, Jia Wang, and Xia-Ji Liu Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria 3122, Australia (Received 29 June 2018; published 25 July 2018) We theoretically investigate how quasiparticle properties of an attractive Fermi polaron are affected by nonzero temperature and finite impurity concentration in three dimensions and in free space. By applying both non- self-consistent and self-consistent many-body T -matrix theories, we calculate the polaron energy (including decay rate), effective mass, and residue, as functions of temperature and impurity concentration. The temperature and concentration dependencies are weak on the BCS side with a negative impurity-medium scattering length. Toward the strong attraction regime across the unitary limit, we find sizable dependencies. In particular, with increasing temperature the effective mass quickly approaches the bare mass and the residue is significantly enhanced. At temperature T ∼ 0.1TF ,whereTF is the Fermi temperature of the background Fermi sea, the residual polaron-polaron interaction seems to become attractive. This leads to a notable down-shift in the polaron energy. We show that, by taking into account the temperature and impurity concentration effects, the measured polaron energy in the first Fermi polaron experiment [Schirotzek et al., Phys.Rev.Lett.102, 230402 (2009)] could be better theoretically explained. DOI: 10.1103/PhysRevA.98.013626 I. INTRODUCTION Experimentally, the first experiment on attractive Fermi polarons was carried out by the Zwierlein group at Mas- Over the past two decades, ultracold atomic gases have pro- sachusetts Institute of Technology (MIT) in 2009 using 6Li vided an ideal platform to understand the intriguing quantum many-body systems [1].
    [Show full text]
  • Electron-Electron Interactions(Pdf)
    Contents 2 Electron-electron interactions 1 2.1 Mean field theory (Hartree-Fock) ................ 3 2.1.1 Validity of Hartree-Fock theory .................. 6 2.1.2 Problem with Hartree-Fock theory ................ 9 2.2 Screening ..................................... 10 2.2.1 Elementary treatment ......................... 10 2.2.2 Kubo formula ............................... 15 2.2.3 Correlation functions .......................... 18 2.2.4 Dielectric constant ............................ 19 2.2.5 Lindhard function ............................ 21 2.2.6 Thomas-Fermi theory ......................... 24 2.2.7 Friedel oscillations ............................ 25 2.2.8 Plasmons ................................... 27 2.3 Fermi liquid theory ............................ 30 2.3.1 Particles and holes ............................ 31 2.3.2 Energy of quasiparticles. ....................... 36 2.3.3 Residual quasiparticle interactions ................ 38 2.3.4 Local energy of a quasiparticle ................... 42 2.3.5 Thermodynamic properties ..................... 44 2.3.6 Quasiparticle relaxation time and transport properties. 46 2.3.7 Effective mass m∗ of quasiparticles ................ 50 0 Reading: 1. Ch. 17, Ashcroft & Mermin 2. Chs. 5& 6, Kittel 3. For a more detailed discussion of Fermi liquid theory, see G. Baym and C. Pethick, Landau Fermi-Liquid Theory : Concepts and Ap- plications, Wiley 1991 2 Electron-electron interactions The electronic structure theory of metals, developed in the 1930’s by Bloch, Bethe, Wilson and others, assumes that electron-electron interac- tions can be neglected, and that solid-state physics consists of computing and filling the electronic bands based on knowldege of crystal symmetry and atomic valence. To a remarkably large extent, this works. In simple compounds, whether a system is an insulator or a metal can be deter- mined reliably by determining the band filling in a noninteracting cal- culation.
    [Show full text]
  • Ferromagnetic State and Phase Transitions
    Ferromagnetic state and phase transitions Yuri Mnyukh 76 Peggy Lane, Farmington, CT, USA, e-mail: [email protected] (Dated: June 20, 2011) Evidence is summarized attesting that the standard exchange field theory of ferromagnetism by Heisenberg has not been successful. It is replaced by the crystal field and a simple assumption that spin orientation is inexorably associated with the orientation of its carrier. It follows at once that both ferromagnetic phase transitions and magnetization must involve a structural rearrangement. The mechanism of structural rearrangements in solids is nucleation and interface propagation. The new approach accounts coherently for ferromagnetic state and its manifestations. 1. Weiss' molecular and Heisenberg's electron Its positive sign led to a collinear ferromagnetism, and exchange fields negative to a collinear antiferromagnetism. Since then it has become accepted that Heisenberg gave a quantum- Generally, ferromagnetics are spin-containing materials mechanical explanation for Weiss' "molecular field": that are (or can be) magnetized and remain magnetized "Only quantum mechanics has brought about in the absence of magnetic field. This definition also explanation of the true nature of ferromagnetism" includes ferrimagnetics, antiferromagnetics, and (Tamm [2]). "Heisenberg has shown that the Weiss' practically unlimited variety of magnetic structures. theory of molecular field can get a simple and The classical Weiss / Heisenberg theory of straightforward explanation in terms of quantum ferromagnetism, taught in the universities and presented mechanics" (Seitz [1]). in many textbooks (e. g., [1-4]), deals basically with the special case of a collinear (parallel and antiparallel) spin arrangement. 2. Inconsistence with the reality The logic behind the theory in question is as follows.
    [Show full text]
  • Non-Fermi Liquids in Oxide Heterostructures
    UC Santa Barbara UC Santa Barbara Previously Published Works Title Non-Fermi liquids in oxide heterostructures Permalink https://escholarship.org/uc/item/1cn238xw Journal Reports on Progress in Physics, 81(6) ISSN 0034-4885 1361-6633 Authors Stemmer, Susanne Allen, S James Publication Date 2018-06-01 DOI 10.1088/1361-6633/aabdfa Peer reviewed eScholarship.org Powered by the California Digital Library University of California Reports on Progress in Physics KEY ISSUES REVIEW Non-Fermi liquids in oxide heterostructures To cite this article: Susanne Stemmer and S James Allen 2018 Rep. Prog. Phys. 81 062502 View the article online for updates and enhancements. This content was downloaded from IP address 128.111.119.159 on 08/05/2018 at 17:09 IOP Reports on Progress in Physics Reports on Progress in Physics Rep. Prog. Phys. Rep. Prog. Phys. 81 (2018) 062502 (12pp) https://doi.org/10.1088/1361-6633/aabdfa 81 Key Issues Review 2018 Non-Fermi liquids in oxide heterostructures © 2018 IOP Publishing Ltd Susanne Stemmer1 and S James Allen2 RPPHAG 1 Materials Department, University of California, Santa Barbara, CA 93106-5050, United States of America 062502 2 Department of Physics, University of California, Santa Barbara, CA 93106-9530, United States of America S Stemmer and S J Allen E-mail: [email protected] Received 18 July 2017, revised 25 January 2018 Accepted for publication 13 April 2018 Published 8 May 2018 Printed in the UK Corresponding Editor Professor Piers Coleman ROP Abstract Understanding the anomalous transport properties of strongly correlated materials is one of the most formidable challenges in condensed matter physics.
    [Show full text]
  • Band Ferromagnetism in Systems of Variable Dimensionality
    JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS Vol. 10, No. 11, November 2008, p.3058 - 3068 Band ferromagnetism in systems of variable dimensionality C. M. TEODORESCU*, G. A. LUNGU National Institute of Materials Physics, Atomistilor 105b, P.O. Box MG-7, RO-77125 Magurele-Ilfov, Romania The Stoner instability of the paramagnetic state, yielding to the occurence of ferromagnetism, is reviewed for electron density of states reflecting changes in the dimensionality of the system. The situations treated are one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) cases and also a special case where the density of states has a parabolic shape near the Fermi level, and is half-filled at equilibrium. We recover the basic results obtained in the original work of E.C. Stoner [Proc. Roy. Soc. London A 165, 372 (1938)]; also we demonstrate that in 1D and 2D case, whenever the Stoner criterion is satisfied, the system evolves spontaneously towards maximum polarization allowed by Hund's rules. For the 3D case, the situation is that: (i) when the Stoner criterion is satisfied, but the ratio between the Hubbard repulsion energy and the Fermi energy is between 4/3 and 3/2, the system evolves towards a ferromagnetic state with incomplete polarization (the polarization parameter is between 0 and 1); (ii) when , the system evolves towards maximum polarization. This situation was also recognized in the original paper of Stoner, but with no further analysis of the obtained polarizations nor comparison with experimental results. We apply the result of calculation in order to predict the Hubbard interaction energy.
    [Show full text]
  • Condensed Matter Option MAGNETISM Handout 1
    Condensed Matter Option MAGNETISM Handout 1 Hilary 2014 Radu Coldea http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option Syllabus The lecture course on Magnetism in Condensed Matter Physics will be given in 7 lectures broken up into three parts as follows: 1. Isolated Ions Magnetic properties become particularly simple if we are able to ignore the interactions between ions. In this case we are able to treat the ions as effectively \isolated" and can discuss diamagnetism and paramagnetism. For the latter phenomenon we revise the derivation of the Brillouin function outlined in the third-year course. Ions in a solid interact with the crystal field and this strongly affects their properties, which can be probed experimentally using magnetic resonance (in particular ESR and NMR). 2. Interactions Now we turn on the interactions! I will discuss what sort of magnetic interactions there might be, including dipolar interactions and the different types of exchange interaction. The interactions lead to various types of ordered magnetic structures which can be measured using neutron diffraction. I will then discuss the mean-field Weiss model of ferromagnetism, antiferromagnetism and ferrimagnetism and also consider the magnetism of metals. 3. Symmetry breaking The concept of broken symmetry is at the heart of condensed matter physics. These lectures aim to explain how the existence of the crystalline order in solids, ferromagnetism and ferroelectricity, are all the result of symmetry breaking. The consequences of breaking symmetry are that systems show some kind of rigidity (in the case of ferromagnetism this is permanent magnetism), low temperature elementary excitations (in the case of ferromagnetism these are spin waves, also known as magnons), and defects (in the case of ferromagnetism these are domain walls).
    [Show full text]