Dept of Phys Ferromagnetism and antiferromagnetism • ferromagnetism (FM) M.C. Chang • exchange interaction, Heisenberg model • spin wave, magnon • antiferromagnetism (AFM) • ferromagnetic domains • nanomagnetic particles Magnetic order: NdFeB 釹鐵硼 (1982) Nature, April 2011 Ferromagnetic insulator (no itinerant electron) •FM is not from magnetic dipole-dipole interaction, nor the SO interaction. It is a result of electrostatic interaction! • Estimate of order: 1 Dipole-dipole Ummmrmr=⋅−⋅⋅⋅⎡⎤3()()ˆˆ r3 ⎣⎦12 1 2 2 ()gμ ≅≅B 10−4 eV (∼ 1 K) for r ≅2A r3 • Because of the electrostatic interaction, some prefers ↑↑, some prefers↑↓(for example, H2). • Effective interaction between a pair of spinful ions ⎧−3/ 4 for singlet • J is called the exchange coupling const. SS12⋅=⎨ (for 2-e system, the GND state must be a singlet) ⎩ 1/ 4 for triplet ∴ Heisenberg wrote • FM has J>0, AFM has J<0 1 ⎧E • The tendency for an ion to align the spins UEESSEE=−() − ⋅ + (3) + → s st12 s t ⎨ of nearby ions is called an exchange field 4 ⎩Et HE (or molecular field, usually much =−JS12 ⋅ S +constant (Heisenberg model) stronger than applied field.) • Weiss mean field HE = λMfor FM 2 JJ(1)+ C MHH=+χχ( ), where = CT / is PM susceptibility χμ= ng() ≡ pE p pJB3kT T MC C ⇒ χ= =≡ (Curie-Weiss law, for TT> c only ) HTC−−λ TTC ～ ～ ～ TkT3 For iron, Tc 1000 K, g 2, S 1 λ= cBc= 22 ∴λ～5000 (no unit in cgs) CngSSμB (1)+ 3 Ms ～1700 G, HE ～10 T. Temperature dependence of magnetization N ⎛⎞gJHJBμ MgJ= JBμ BJ ⎜⎟ V ⎝⎠kT 21JJ++⎛⎞ 21 1 ⎛⎞ x where BxJ ( )≡− coth⎜⎟x coth ⎜⎟ 2222JJJJ⎝⎠ ⎝⎠ ==tanh xJ (for 1/2) HMH≈ λ (ext neglected) μλM ∴ Mn= μμtanh (≡ gJμ ) kT JB m Mm⎛⎞kT or m ≡=tanh ⎜⎟t ≡ 2 nμ tn⎝ μ λ ⎠ At low T, use tanhx 1− 2e−2x →ΔM ≡MMT(0) − ( ) −2/λμnkT2 M ()T ≈ 2neμ M (0) Does not agree with experiment, which is ～ T3/2. Explained later using spin wave excitation. Spin wave in 1-dim FM (classical approach) N Heisenberg model HJSSJ= −⋅2(0)∑ pp+1 > p=1 2 • Ground state energy E0 =－2NJS • Excited state: Flip 1 spin costs 8JS2. But there is a cheaper way to create excited state. NN HJSSS=−∑∑pp ⋅()−+11 + p =−μ p ⋅Bp pp=11= ħS is the classical where μμ=−gS μ =emc / 2 pBpB()angular momentum J B ≡−S +S effective B field (exchange field) pppgμ ()−1 +1 B dS torque p =×=μ BJSS×+× SS dt pp() pp−+1 pp1 Dispersion of spin wave zxyz assume SSSSSpppp≈<<; , , xy neglect nonlinear terms in SSpp, ε x (k) ⎧ dS p yy y ⎪ =−−JS()2 Spp S−+11 S p ⎪ dt ⇒ ⎨ y ⎪ dS p xx x =−JS()2 Spp − S−+11 − S p ⎩⎪ dt x i() pka−−ωω t y i () pka t let Suepp==; Sve , ka ⎛⎞⎛⎞iJω 2 Skau(1− cos ) then ⎜⎟⎜⎟= 0 ⎝⎠⎝⎠−−2(1cos)JS ka iω v ⇒=ω 2(1cos)JS − ka ∝ k 2 at long wave length 磁振子 • Quantized spin wave is called magnon ( ∈ boson) • magnon energy ε kk=+(n 1/2) ω k • magnons, like phonons, can interact with each other if nonlinear spin interaction is included. Thermal excitations of magnons gμB ⎛⎞ M ()TNSnT=−⎜⎟∑ k () (one magnon reduces spin by 1) V ⎝⎠k Number of magnons being excited, ndDn= ωω() ∑ kk∫ k 1 nk = (boson) eωk /kT −1 DOS in 3-dimension, 3/2 V ⎛⎞ 1/2 D()ωω= 22⎜⎟ 4π ⎝⎠JSa 3/2 1/2 VkT⎛⎞∞ x ∴ ndxk = ∑ 22⎜⎟∫0 x k 4π ⎝⎠JSa e −1 0.0587(4π 2 ) n ΔM ∑ k =∝k T 3/2 (Bloch T 3/2 law) MNS(0) FM in Fe, Co, Ni (with itinerant electrons) Cu, nonmagnetic ℓ=0 Ni, magnetic T > Tc T < Tc • ferromagnetism (FM) • antiferromagnetism (AFM) • susceptibilities • ferrimagnetism • ferromagnetic domains • nanomagnetic particles Antiferromagnetism (predicted by Neel, 1936) • many AFM are transition metal oxides. • net magnetization is zero, not easy to show that it’s a AFM. First confirmed by Shull at 1949 using neutron scattering. MnO, transition temperature=610 K Neel temperature Neel 1970 Shull 1994 T-dependence of susceptibility for T > TN Consider a AFM consists of 2 FM sublattices A, B. H A =−λλMMBA; H B =− Use separate Curie consts CCAB, for sublattices A, B • For identical sublattices, 2 ⎧ CA 22CT− λ C 2 C MHMAB=−()λ χ ==, TN = λC ⎪ T TC2 − λ 2 T +T ⎨ () N ⎪ CB 2C MHMBA=−()λ Experiment: χ = ⎩⎪ T T +θ ⎛⎞⎛⎞⎛⎞TCMCλ AA A →=⎜⎟⎜⎟⎜⎟H ⎝⎠⎝⎠⎝⎠λCTMBB CB There is non-zero solution at H = 0 only if det= 0 1/2 ⇒ TN = λ ()CACB At TT> N , MM+ χ = AB H ()2CCTAB+ − λ () CC AB = 22 TT− N • Susceptibility for T < TN (Kittel, p343) M = χ⊥ H H flexible M = χ// H rigid • Dispersion relation for AFM spin wave (see Kittel, p344 for details) Linear dispersion at small k (Cf, FM spin wave) Ferrimagnetic materials 磁鐵礦 Hematite 赤鐵礦 Magnetite (Fe3O4 or FeO．Fe2O3) • Curie temperature 585 C • belong to a more general class of ferrite MO．Fe2O3 (M=Fe, Co, Ni, Cu, Mg…) 磁性氧化物 Iron garnet 鐵石榴石 e.g., • Yttrium iron garnet (YIG)Y3Fe2(FeO4)3, or Y3Fe5O12 釔鐵石榴石 is a ferrimagnetic material with Curie temperature 550 K. • YIG has high degree of Faraday effect, high Q factor in microwave frequencies, low absorption of infrared wavelengths up to 600 nm … etc (wiki) • ferromagnetism (FM) • antiferromagnetism (AFM) • ferromagnetic domains • nanomagnetic particles Magnetic domains (proposed by Weiss 1906) Why not all spins be parallel to reduce the exchange energy? → it would cost “stray field” energy magnetic energy B2 / 8π Little leaking field ≈106 erg/cm3 • Magnetization and domains Transition between domain walls • Bloch wall Why not just → Would cost too much exchange energy (not so in ferroelectric materials) • Neel wall Domain wall dynamics • domain wall motion induced by current •… Race-track memory (Parkin, IBM) Hysteresis Easy/hard axis From hyperphysics The first single molecule magnet (1980): Mn12-acetate (s=10) Can be described by Stoner–Wohlfarth model (T=0) From W. Wernsdorfer’s pdf 磁流體 Single domain particle: ferrofluid, magnetic data storage … • superparamagnetism 超順磁性 (T≠0, small enough single domain particle) ferrofluid 趨磁性 • Magnetotaxsis bacteria (Phototaxis - 趨光性) Magnetospirillum magnetotacticum http://www.calpoly.edu/~rfrankel/mtbphoto.html supplementary The zoo of magnetoresistance (first discovered by Lord Kelvin, 1857) • GMR (giant MR, Fert and Grünberg 1988) 巨磁阻 • CMR (colossal MR, Jonker and can Santen 1950’s) 龐磁阻 • EMR (extraordinary MR, Solin, 2000) 異常磁阻 • TMR (tunneling MR, Julliere, 1975) 穿隧磁阻 •… Soh and Aeppli, Nature (2002) Getzlaff and Mathias - Fundamentals of Magnetism, p.259 Giant MR （of multi-layer magnetic materials) supplementary (Gruenberg JAP; Fert PRL, 1988) • In 1988, GMR was discovered • In 1996, GMR reading heads were commercialized • Since 2000: Virtually all writing heads are GMR heads Solin, ScientificAmerican,2004 2007 A. Fert and P. Grünberg supplementary GMR read head IBM’s demo Fig from http://www.stoner.leeds.ac.uk/research/.