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 ()gμ 2 ≅≅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 H = λMfor FM χχE

2 JJ(1)+ C MHH=+( ), where = CT / is PM susceptibility χμ= ng() ≡ χ pE p pJB3kT T MC C ⇒ = =≡ (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) CngSSB (1)+ 3 μ Ms ～1700 G, HE ～10 T. Temperature dependence of magnetization μ N ⎛⎞gJHJBμ MgJ= JBBJ ⎜⎟ V ⎝⎠kT 21JJ++⎛⎞ 21 1 ⎛⎞ x where BxJ ( )≡− coth⎜⎟x coth ⎜⎟ 2222JJJJ⎝⎠ ⎝⎠ λ == tanh xJ (for 1/2)

HMH≈ (ext neglected) μμμλM ∴ Mn= tanh (≡ gJ ) kT JB m Mm⎛⎞kTμ or m ≡=μ tanh ⎜⎟t ≡ 2 n tn⎝ λ ⎠

−2x μ At low T, use tanhx 1− 2e →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.

NNμμ μ HJSSS=−∑∑pp ⋅()−+11 + p =−μ p ⋅Bp pp=11= ħS is the classical where =−gS =emc / 2 pBpB()angular momentum J μ B ≡−()S +S effective B field (exchange field) pppgμ −1 +1 B dS p torque =×=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 ⎪ =−−JS()2 Syy S S y ⎪ dt pp−+11 p ⇒ ⎨ y ⎪ dS p xx x =−JS()2 S − S − S ⎩⎪ dt pp−+11 p x i() pka−−ωω t y i () pka t let Suepp== ; Sve , ka ⎛⎞⎛⎞ω iJω 2 Skau(1− cos ) then ⎜⎟⎜⎟= 0 ⎝⎠⎝⎠−−2(1cos)JS ka iω v ⇒= 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 ωωV ⎛⎞ 1/2 D()= 22⎜⎟ 4 JSa ⎝⎠π 3/2 1/2 VkT⎛⎞∞ x ∴ ndxk = ∑ 22⎜⎟∫0 x k 4 ⎝⎠JSa e −1 π 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 Considerλλ a AFM consists of 2 FM sublattices A, B. H A =−MMBA; H B =− Use separate Curie consts CC , for sublattices A, B • For identical sublattices, AB χ ⎧ C 22CT− λ C2 2 C MHM=−A ()λ ==, T = λC ⎪ AB 2 2 N ⎪ T TC− ()λ T +TN ⎨ ⎪ CB 2C MHMBA=−()λ Experiment: = ⎩⎪ T χT +θ λ ⎛⎞⎛⎞⎛⎞TCMCAA A →=⎜⎟⎜⎟⎜⎟λ H ⎝⎠⎝⎠⎝⎠CTMBB CB There is non-zero solution at H = 0 only if det= 0 ⇒ T = C C 1/2 N ()A B λ

At TT> N , MM+ χ = AB H

()2CCTAB+ − () CC AB = 22 TT− N λ • 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/