SOLID STATE PHYSICS PART III Magnetic Properties of Solids
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Arxiv:Cond-Mat/0106646 29 Jun 2001 Landau Diamagnetism Revisited
Landau Diamagnetism Revisited Sushanta Dattagupta†,**, Arun M. Jayannavar‡ and Narendra Kumar# †S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, India ‡Institute of Physics, Sachivalaya Marg, Bhubaneswar 751 005, India #Raman Research Institute, Bangalore 560 080, India The problem of diamagnetism, solved by Landau, continues to pose fascinating issues which have relevance even today. These issues relate to inherent quantum nature of the problem, the role of boundary and dissipation, the meaning of thermodynamic limits, and above all, the quantum–classical crossover occasioned by environment- induced decoherence. The Landau diamagnetism provides a unique paradigm for discussing these issues, the significance of which is far-reaching. Our central result connects the mean orbital magnetic moment, a thermodynamic property, with the electrical resistivity, which characterizes transport properties of material. In this communication, we wish to draw the attention of Peierls to term diamagnetism as one of the surprises in the reader to certain enigmatic issues concerning dia- theoretical physics4. Landau’s pathbreaking result also magnetism. Indeed, diamagnetism can be used as a demonstrated that the calculation of diamagnetic suscep- prototype phenomenon to illustrate the essential role of tibility did indeed require an explicit quantum treatment. quantum mechanics, surface–boundary, dissipation and Turning to the classical domain, two of the present nonequilibrium statistical mechanics itself. authors had worried, some years ago, about the issue: Diamagnetism is a material property that characterizes does the BV theorem survive dissipation5? This is a natu- the response of an ensemble of charged particles (more ral question to ask as dissipation is a ubiquitous property specifically, electrons) to an applied magnetic field. -
Magnetism, Free Electrons and Interactions
Magnetism Magnets Zero external field Finite external field • Types of magnetic systems • Pauli paramagnetism in metals Paramagnets • Landau diamagnetism in metals • Larmor diamagnetism in insulators Diamagnets • Ferromagnetism of electron gas • Spin Hamiltonian Ferromagnets • Mean field approach • Curie transition Antiferromagnets Ferrimagnets …… … Pauli paramagnetism Pauli paramagnetism Let us first look at magnetic properties of a free electron gas. ε =−p2 /2meBmc= /2 ε =+p2 /2meBmc= /2 ↑ ↓G Electron are spin-1/2 particles #of majority spins: dp3 NV= f()ε In external magnetic field B – Zeeman splitting #of minority spins: ↑,,↓ ∫ (2π= )3 ↑ ↓ 2 = 2 = ε↑ =−p /2meBmc /2 ε↓ =+p /2meBmc /2 Magnetization (magnetic moment per unit volume): - minority spins e= M =−()NN : aligned along the field and proportional Fermi level ↑ ↓ 2Vmc to B in low fields χ - magnetic succeptibility - majority spins M = χB χ > 0 - paramagnetism Pauli succeptibility Landau quantization G 2 2 A free electron in magnetic field: B & zˆ ε↑ =−p /2meBmc= /2 ε↓ =+p /2meBmc= /2 G 2 µ+eB= /2 mc 2 G VgeB= Schrödinger equation: = ⎛⎞ieA ABxAA===;0 NN−= gd()εε ≈ V −∇+=⎜⎟ψ εψ yxz ↑↓ ∫ 2mc= 22µ−eB= /2 mc mc ⎝⎠ B=1T corresponds toeBmc= /1=× K k provided m is free electrons’s mass Solutions: labeled by two indices nk, B G z For any fields, eBmc=/ µ ψ nk(r )= exp( ik y y+− ik z z )ϕ n ( x= ck y / eB ) Magnetic succeptibility: ϕn - wave functions of a harmonic oscillator 22 2 Energies: ε =+==kmeBmcn/2 ( / )( + 1/2) - strongly degenerate!! ⎛⎞e= nk z χP = ⎜⎟g ⎝⎠2mc We “quantized” momenta transverse to the field (Landau levels) 1 Landau diamagnetism Electrons in metals A free electron in magnetic field: moves along spiral trajectories We know that there are diamagnetic metals. -
Canted Ferrimagnetism and Giant Coercivity in the Non-Stoichiometric
Canted ferrimagnetism and giant coercivity in the non-stoichiometric double perovskite La2Ni1.19Os0.81O6 Hai L. Feng1, Manfred Reehuis2, Peter Adler1, Zhiwei Hu1, Michael Nicklas1, Andreas Hoser2, Shih-Chang Weng3, Claudia Felser1, Martin Jansen1 1Max Planck Institute for Chemical Physics of Solids, Dresden, D-01187, Germany 2Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, D-14109, Germany 3National Synchrotron Radiation Research Center (NSRRC), Hsinchu, 30076, Taiwan Abstract: The non-stoichiometric double perovskite oxide La2Ni1.19Os0.81O6 was synthesized by solid state reaction and its crystal and magnetic structures were investigated by powder x-ray and neutron diffraction. La2Ni1.19Os0.81O6 crystallizes in the monoclinic double perovskite structure (general formula A2BB’O6) with space group P21/n, where the B site is fully occupied by Ni and the B’ site by 19 % Ni and 81 % Os atoms. Using x-ray absorption spectroscopy an Os4.5+ oxidation state was established, suggesting presence of about 50 % 5+ 3 4+ 4 paramagnetic Os (5d , S = 3/2) and 50 % non-magnetic Os (5d , Jeff = 0) ions at the B’ sites. Magnetization and neutron diffraction measurements on La2Ni1.19Os0.81O6 provide evidence for a ferrimagnetic transition at 125 K. The analysis of the neutron data suggests a canted ferrimagnetic spin structure with collinear Ni2+ spin chains extending along the c axis but a non-collinear spin alignment within the ab plane. The magnetization curve of La2Ni1.19Os0.81O6 features a hysteresis with a very high coercive field, HC = 41 kOe, at T = 5 K, which is explained in terms of large magnetocrystalline anisotropy due to the presence of Os ions together with atomic disorder. -
Magnetism, Magnetic Properties, Magnetochemistry
Magnetism, Magnetic Properties, Magnetochemistry 1 Magnetism All matter is electronic Positive/negative charges - bound by Coulombic forces Result of electric field E between charges, electric dipole Electric and magnetic fields = the electromagnetic interaction (Oersted, Maxwell) Electric field = electric +/ charges, electric dipole Magnetic field ??No source?? No magnetic charges, N-S No magnetic monopole Magnetic field = motion of electric charges (electric current, atomic motions) Magnetic dipole – magnetic moment = i A [A m2] 2 Electromagnetic Fields 3 Magnetism Magnetic field = motion of electric charges • Macro - electric current • Micro - spin + orbital momentum Ampère 1822 Poisson model Magnetic dipole – magnetic (dipole) moment [A m2] i A 4 Ampere model Magnetism Microscopic explanation of source of magnetism = Fundamental quantum magnets Unpaired electrons = spins (Bohr 1913) Atomic building blocks (protons, neutrons and electrons = fermions) possess an intrinsic magnetic moment Relativistic quantum theory (P. Dirac 1928) SPIN (quantum property ~ rotation of charged particles) Spin (½ for all fermions) gives rise to a magnetic moment 5 Atomic Motions of Electric Charges The origins for the magnetic moment of a free atom Motions of Electric Charges: 1) The spins of the electrons S. Unpaired spins give a paramagnetic contribution. Paired spins give a diamagnetic contribution. 2) The orbital angular momentum L of the electrons about the nucleus, degenerate orbitals, paramagnetic contribution. The change in the orbital moment -
Density of States Information from Low Temperature Specific Heat
JOURNAL OF RESE ARC H of th e National Bureau of Standards - A. Physics and Chemistry Val. 74A, No.3, May-June 1970 Density of States I nformation from Low Temperature Specific Heat Measurements* Paul A. Beck and Helmut Claus University of Illinois, Urbana (October 10, 1969) The c a lcul ati on of one -electron d ensit y of s tate va lues from the coeffi cient y of the te rm of the low te mperature specifi c heat lin ear in te mperature is compli cated by many- body effects. In parti c ul ar, the electron-p honon inte raction may enhance the measured y as muc h as tw ofo ld. The e nha nce me nt fa ctor can be eva luat ed in the case of supe rconducting metals and a ll oys. In the presence of magneti c mo ments, add it ional complicati ons arise. A magneti c contribution to the measured y was ide ntifi e d in the case of dilute all oys and a lso of concentrated a lJ oys wh e re parasiti c antife rromagnetis m is s upe rim posed on a n over-a ll fe rromagneti c orde r. No me thod has as ye t bee n de vised to e valu ate this magne ti c part of y. T he separati on of the te mpera ture- li near term of the s pec ifi c heat may itself be co mpli cated by the a ppearance of a s pecific heat a no ma ly due to magneti c cluste rs in s upe rpa ramagneti c or we ak ly ferromagneti c a ll oys. -
Unusual Quantum Criticality in Metals and Insulators T. Senthil (MIT)
Unusual quantum criticality in metals and insulators T. Senthil (MIT) T. Senthil, ``Critical fermi surfaces and non-fermi liquid metals”, PR B, June 08 T. Senthil, ``Theory of a continuous Mott transition in two dimensions”, PR B, July 08 D. Podolsky, A. Paramekanti, Y.B. Kim, and T. Senthil, ``Mott transition between a spin liquid insulator and a metal in three dimensions”, PRL, May 09 T. Senthil and P. A. Lee, ``Coherence and pairing in a doped Mott insulator: Application to the cuprates”, PRL, Aug 09 Precursors: T. Senthil, Annals of Physics, ’06, T. Senthil. M. Vojta, S. Sachdev, PR B, ‘04 Saturday, October 22, 2011 High Tc cuprates: doped Mott insulators Many interesting phenomena on doping the Mott insulator: Loss of antiferromagnetism High Tc superconductivity Pseudogaps, non-fermi liquid regimes , etc. Stripes, nematics, and other broken symmetries This talk: focus on one (among many) fundamental question. How does a Fermi surface emerge when a Mott insulator changes into a metal? Saturday, October 22, 2011 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator? Evolution from Mott insulator to overdoped metal : emergence of large Fermi surface with area set by usual Luttinger count. Mott insulator: No Fermi surface Overdoped metal: Large Fermi surface ADMR, quantum oscillations (Hussey), ARPES (Damascelli,….) Saturday, October 22, 2011 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator? Large gapless Fermi surface present even in optimal doped strange metal albeit without Landau quasiparticles . Mott insulator: No Fermi surface Saturday, October 22, 2011 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator? Large gapless Fermi surface present also in optimal doped strange metal albeit without Landau quasiparticles . -
An Overview of Representational Analysis and Magnetic Space Groups
Magnetic Symmetry: an overview of Representational Analysis and Magnetic Space groups Stuart Calder Neutron Scattering Division Oak Ridge National Laboratory ORNL is managed by UT-Battelle, LLC for the US Department of Energy Overview Aim: Introduce concepts and tools to describe and determine magnetic structures • Basic description of magnetic structures and propagation vector • What are the ways to describe magnetic structures properly and to access the underlying physics? – Representational analysis – Magnetic space groups (Shubnikov groups) 2 Magnetic Symmetry: an overview of Representational Analysis and Magnetic Space groups Brief History of magnetic structures • ~500 BC: Ferromagnetism documented Sinan, in Greece, India, used in China ~200 BC • 1932 Neel proposes antiferromagnetism • 1943: First neutron experiments come out of WW2 Manhatten project at ORNL • 1951: Antiferromagnetism measured in MnO and Ferrimagnetism in Fe3O4 at ORNL by Shull and Wollan with neutron scattering • 1950-60: Shubnikov and Bertaut develop methods for magnetic structure description • Present/Future: - Powerful and accessible experimental and software tools available - Spintronic devices and Quantum Information Science 3 Magnetic Symmetry: an overview of Representational Analysis and Magnetic Space groups Intrinsic magnetic moments (spins) in ions • Consider an ion with unpaired electrons • Hund’s rule: maximize S/J m=gJJ (rare earths) m=gsS (transtion metals) core 2+ Ni has a localized magnetic moment of 2µB Ni2+ • Magnetic moment (or spin) is a classical -
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. -
Rich Magnetic Quantization Phenomena in AA Bilayer Silicene
www.nature.com/scientificreports OPEN Rich Magnetic Quantization Phenomena in AA Bilayer Silicene Po-Hsin Shih1, Thi-Nga Do2,3, Godfrey Gumbs4,5, Danhong Huang6, Hai Duong Pham1 & Ming-Fa Lin1,7,8 Received: 13 June 2019 The rich magneto-electronic properties of AA-bottom-top (bt) bilayer silicene are investigated using Accepted: 27 August 2019 a generalized tight-binding model. The electronic structure exhibits two pairs of oscillatory energy Published: xx xx xxxx bands for which the lowest conduction and highest valence states of the low-lying pair are shifted away from the K point. The quantized Landau levels (LLs) are classifed into various separated groups by the localization behaviors of their spatial distributions. The LLs in the vicinity of the Fermi energy do not present simple wave function modes. This behavior is quite diferent from other two-dimensional systems. The geometry symmetry, intralayer and interlayer atomic interactions, and the efect of a perpendicular magnetic feld are responsible for the peculiar LL energy spectra in AA-bt bilayer silicene. This work provides a better understanding of the diverse magnetic quantization phenomena in 2D condensed-matter materials. Silicene, an isostructure to graphene, is purely made of silicon atoms through both the sp2 and sp3 bondings. So far, silicene systems have been successfully synthesized by the epitaxial growth on various substrate surfaces. Monolayer silicene with diferent sizes of unit cells has been produced on several substrates, such as Si(111) 1,2 3,4 5 6 33× -Ag template , Ag(111) (4 × 4) , Ir(111) ( 33× ) and ZrB2(0001) (2 × 2) . -
Chapter 6 Antiferromagnetism and Other Magnetic Ordeer
Chapter 6 Antiferromagnetism and Other Magnetic Ordeer 6.1 Mean Field Theory of Antiferromagnetism 6.2 Ferrimagnets 6.3 Frustration 6.4 Amorphous Magnets 6.5 Spin Glasses 6.6 Magnetic Model Compounds TCD February 2007 1 1 Molecular Field Theory of Antiferromagnetism 2 equal and oppositely-directed magnetic sublattices 2 Weiss coefficients to represent inter- and intra-sublattice interactions. HAi = n’WMA + nWMB +H HBi = nWMA + n’WMB +H Magnetization of each sublattice is represented by a Brillouin function, and each falls to zero at the critical temperature TN (Néel temperature) Sublattice magnetisation Sublattice magnetisation for antiferromagnet TCD February 2007 2 Above TN The condition for the appearance of spontaneous sublattice magnetization is that these equations have a nonzero solution in zero applied field Curie Weiss ! C = 2C’, P = C’(n’W + nW) TCD February 2007 3 The antiferromagnetic axis along which the sublattice magnetizations lie is determined by magnetocrystalline anisotropy Response below TN depends on the direction of H relative to this axis. No shape anisotropy (no demagnetizing field) TCD February 2007 4 Spin Flop Occurs at Hsf when energies of paralell and perpendicular configurations are equal: HK is the effective anisotropy field i 1/2 This reduces to Hsf = 2(HKH ) for T << TN Spin Waves General: " n h q ~ q ! M and specific heat ~ Tq/n Antiferromagnet: " h q ~ q ! M and specific heat ~ Tq TCD February 2007 5 2 Ferrimagnetism Antiferromagnet with 2 unequal sublattices ! YIG (Y3Fe5O12) Iron occupies 2 crystallographic sites one octahedral (16a) & one tetrahedral (24d) with O ! Magnetite(Fe3O4) Iron again occupies 2 crystallographic sites one tetrahedral (8a – A site) & one octahedral (16d – B site) 3 Weiss Coefficients to account for inter- and intra-sublattice interaction TCD February 2007 6 Below TN, magnetisation of each sublattice is zero. -
Phys 446: Solid State Physics / Optical Properties Lattice Vibrations
Solid State Physics Lecture 5 Last week: Phys 446: (Ch. 3) • Phonons Solid State Physics / Optical Properties • Today: Einstein and Debye models for thermal capacity Lattice vibrations: Thermal conductivity Thermal, acoustic, and optical properties HW2 discussion Fall 2007 Lecture 5 Andrei Sirenko, NJIT 1 2 Material to be included in the test •Factors affecting the diffraction amplitude: Oct. 12th 2007 Atomic scattering factor (form factor): f = n(r)ei∆k⋅rl d 3r reflects distribution of electronic cloud. a ∫ r • Crystalline structures. 0 sin()∆k ⋅r In case of spherical distribution f = 4πr 2n(r) dr 7 crystal systems and 14 Bravais lattices a ∫ n 0 ∆k ⋅r • Crystallographic directions dhkl = 2 2 2 1 2 ⎛ h k l ⎞ 2πi(hu j +kv j +lw j ) and Miller indices ⎜ + + ⎟ •Structure factor F = f e ⎜ a2 b2 c2 ⎟ ∑ aj ⎝ ⎠ j • Definition of reciprocal lattice vectors: •Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law): σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • What is Brillouin zone kl kl 2 2 C •Elastic wave equation: ∂ u C ∂ u eff • Bragg formula: 2d·sinθ = mλ ; ∆k = G = eff x sound velocity v = ∂t 2 ρ ∂x2 ρ 3 4 • Lattice vibrations: acoustic and optical branches Summary of the Last Lecture In three-dimensional lattice with s atoms per unit cell there are Elastic properties – crystal is considered as continuous anisotropic 3s phonon branches: 3 acoustic, 3s - 3 optical medium • Phonon - the quantum of lattice vibration. Elastic stiffness and compliance tensors relate the strain and the Energy ħω; momentum ħq stress in a linear region (small displacements, harmonic potential) • Concept of the phonon density of states Hooke's law: σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • Einstein and Debye models for lattice heat capacity. -
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.