Ferromagnetism and antiferromagnetism ferromagnetism (FM) antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles

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1 Dept of Phys Ferromagnetism and antiferromagnetism ferromagnetism (FM) exchange interaction, Heisenberg model spin wave, magnon antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles M.C. Chang Magnetic order:

2 NdFeB 釹鐵硼 Nature, April 2011

3 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 3 r Dipole-dipole U = m m 3( m rˆ) ( m rˆ) ( gμ ) ev ( 1 K) for 2A B r 3 r Because of the electrostatic interaction, some prefers, some prefers (for example, H2).

4 Effective interaction between a pair of spinful ions 3/ 4 for singlet S1 S2 = 1/ 4 for triplet Heisenberg wrote 1 Es U = ( Es Et) S1 S2 + ( Es + 3 Et) 4 Et = JS1 S2 +constant (Heisenberg model) J is called the exchange coupling const. (for 2-e system, the GND state must be a singlet) FM has J>0, AFM has J<0 The tendency for an ion to align the spins of nearby ions is called an exchange field H E (or molecular field, usually much stronger than applied field.) Weiss mean field H E = λm for FM M = χp( H + HE), where χp= C/ T is PM susceptibility ( ) 2 J ( J + 1) C χp = n gjμb 3kT T M C C χ= = (Curie-Weiss law, for T > Tc only) H T Cλ T T C T 3k T λ= = C ng S( S + 1) c B c 2 2 μb For iron, T c ~1000 K, g~2, S~1 λ~5000 (no unit in cgs) M s ~1700 G, H E ~10 3 T.

5 Temperature dependence of magnetization N gjμbjh M = gjμbjbj V kt 2J + 1 2J x where BJ ( x) coth x coth 2J 2J 2J 2J = tanh x (for J = 1/2) H λm ( H neglected) μλm M = nμtanh ( μ gjμbj ) kt or m M nμ ext m tanh t kt nμ λ = t 2 m At low T, M ( T ) M (0) ΔM M(0) M( T) 2 2 n / kt μ 2n e λ μ Does not agree with experiment, which is ~ T 3/2. Explained later using spin wave excitation.

6 Spin wave in 1-dim FM N Heisenberg model H = 2 J S p S p+ 1 ( J > 0) p= 1 Ground state energy E 0 =-2NJS 2 Excited state: Flip 1 spin costs 8JS 2. But there is a cheaper way to create excited state. Classical approach, H = J S S + S = B N N p ( p 1 p+ 1) μ p p= 1 p= 1 where μp = gμbsp μb = / 2 J Bp ( S p 1 + S p+ 1) gμb ds p = μ B = J S S S S dt ( e mc) ( ) p p p p p p p ħs is the classical angular momentum effective B field (exchange field)

7 Dispersion of spin wave assume ds dt ds dt z x y z S S; S, S << S, p p p p neglect nonlinear terms in, S ( p p 1 p+ 1) = JS 2 S S S, ( p p 1 p+ 1) = JS 2 S S S. x i( pka ωt ) y i( pka ωt ) let Sp = ue ; Sp = ve, i ω 2 JS(1 cos ka) u then = 0 2 JS(1 cos ka) i ω v S x p x p y y y y p x x x ω = 2 JS(1 cos ka) k 2 y p at long wave length Quantized spin wave is called magnon ( boson) magnon energy ε ( n 1/2) = + ω k k k magnons, like phonons, can interact with each other if nonlinear spin interaction is included

8 Thermal excitations of magnons k n = dωd( ω) n k 1 nk = ωk / kt e 1 DOS in 3-dimension, 3/2 1 = 2 2 D( ω) ω 4π JSa 1 kt nk = k 4π JSa ΔM M(0) M NS n = Number of magnons being excited, n k k = NS k k 1/ T k 3/2 1/2 3/2 3/2 x dx x e (4 π ) (Bloch T law)

9 FM in Fe, Co, Ni (Itinerant electrons) Cu, nonmagnetic Ni, magnetic T > T c T < T c

10 ferromagnetism (FM) antiferromagnetism (AFM) susceptibilities ferrimagnetism ferromagnetic domains nanomagnetic particles

11 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 1970 Shull 1994

12 T-dependence of susceptibility for T > T N Consider a AFM consists of 2 FM sublattices A, B. H = λm ; H = λm A B B A Use separate Curie consts C, C for sublattices A, B MT= C( H λm ); A A B MT= C( H λm ). B B A T λca MA CA = H λcb T MB CB There is non-zero solution when H = 0 only if det = 0 T = λ ( C C ) 1/2 N A B A B At T > T, χ = For identical sublattices, χ M A N + M H B 2 2 2λ 2 = = CT C C T T T ( λc) 2C Experiment: χ = T + θ N, ( ) ( C + C ) T 2λ C C = A B A B 2 2 T TN T N = λc

13 Susceptibility for T < T N (Kittel, p343) M H = χ M = χ H // Dispersion relation for AFM spin wave (Kittel, p344) Linear dispersion at small k (Cf, FM spin wave)

14 磁鐵礦 Ferrimagnetic materials Magnetite (Fe 3 O 4 or FeO.Fe 2 O 3 ) Curie temperature 585 C Hematite 赤鐵礦 belong to a more general class of ferrite MO.Fe 2 O 3 (M=Fe, Co, Ni, Cu, Mg ) 磁性氧化物 cancel e.g., Iron garnet 鐵石榴石 Yttrium iron garnet (YIG) Y 3 Fe 2 (FeO 4 ) 3, or Y 3 Fe 5 O 12 is a ferrimagnetic material with Curie temperature 550 K. 釔鐵石榴石 YIG has high degree of Faraday effect, high Q factor in microwave frequencies, low absorption of infrared wavelengths up to 600 nm etc (wiki)

15 ferromagnetism (FM) antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles

16 Magnetic domains (proposed by Weiss 1906) Why not all spins be parallel to reduce the exchange energy? it would cost stray field energy Magnetization and domains 2 magnetic energy B /8π erg/cm Little leaking field

17 Transition between domain walls Bloch wall Why not just Would cost too much exchange energy (not so in ferroelectric materials) Domain wall dynamics Neel wall domain wall motion induced by spin current Race-track memory (IBM)

18 Hysteresis Easy/hard axis From hyperphysics

19 Single domain particle: ferrofluid, magnetic data storage superparamagnetism 趨磁性 Magnetotaxsis bacteria

20 supplementary The zoo of magnetoresistance (first discovered by Lord Kelvin, 1857) GMR (giant MR) CMR (colossal MR) TMR (tunneling MR, Julliere, Phys. Lett. 1975) EMR (extraordinary MR, Solin, 2000) Soh and Aeppli, Nature (2002) Getzlaff and Mathias - Fundamentals of Magnetism, p.259

21 supplementary Giant MR (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 A. Fert and P. Grünberg 2007 Solin, Scientific American, 2004

22 GMR read head supplementary IBM s demo Fig from