12. Ferromagnetism and AntiFerromagnetismFerromagnetic OrderCurie Point and the Exchange IntegralTemperature Dependence of the Saturation MagnetizationSaturation Magnetization at Absolute ZeroMagnonsQuantization of Spin WavesThermal Excitation of MagnonsNeutron Magnetic ScatteringFerrimagnetic OrderCurie Temperature and Susceptibility of FerrimagnetsIron GarnetsAntiferromagnetic OrderSusceptibility Below the Neel TemperatureAntiferromagnetic MagnonsFerromagnetic DomainsAnisotropy EnergyTransition Region Between DomainsSolitonsOrigin of DomainsCoercivity and HysteresisSingle Domain ParticlesGeomagnetism and BiomagnetismMagnetic Force MicroscopyMagnetic Bubble Domain

Ferromagnetic OrderM = 0M =saturation moment 0Spontaneous moment: M 0 when Hext = 0.M =saturation moment 0M =saturation moment 0M =saturation moment 0

Curie Point and the Exchange IntegralParamagnet N ions with spin S Exchange (internal aligning) Field BEFerromagnetThermal Agitation (Large) (Small)M = Magnetization = average magnetic moment per unit moment.Mean field approximation: indep of T.System in thermal equilibriun at TCurie temperature TC :critical temperature for transitions between para- & ferro-magnetic phasesParamagnetic phase under applied field Ba :Curie-Weiss lawCurie law

Ni, TC = 358C Using ( see Quantum Theory of Paramagnetism, Chap.11 ),we haveFor iron, TC 1000K, g 2, J = S = 1 (Fe ~3d8), N = 1023 cm3 so that 7300.With M 1700, we have BE 107 G = 103 T .Whereas a magnetic ion produces a field at its neighbor B / a3 103 G = 0.1 T. Heisenberg model :J = exchange integralPaul exclusion principle 2 es with parallel spins have lower electrostatic energy.This energy difference is called the exchange energy.Mean field with z n.n. For iron with S = 1, J = 8.3 meVBetter statistics

S = scbccfcckB T / z J0.280.3250.346

Temperature Dependence of the Saturation MagnetizationMean field theory for T < TC , S = 1/2 , and replacing the Curie law with Brillouin expression where t = 1NiT

Experiment:( compared well with spin wave theory)A 7.5 106 deg 3/2 for NiA 3.4 106 deg 3/2 for Fe

Saturation Magnetization at Absolute Zero4 f 73d 63d 73d 8bandferromagsimple ferromag

Cause for non-integral nB : spin-orbit interaction. local magnetization of conduction e. ferrimagnetismKnown simple ferromagnetic insulators (spin all lined up in ground state): CrBr3 , EuO, EuSFerromagnetism that can be described by band electron model:transition metals Fe, Co, Ni Cu: 3d10 4s1 paramagneticNi: 3d8 4s2 ferromagneticT > TCT < TCnB = 0.61n n = 0.54 e/atomProblem 11.6

MagnonsMagnon = Quantum of spin waveNearest neighbor Heisenberg model for atomic chain of N atoms:G.S.: U0 = 2 N J S21st E.S.: U1 = U0 + 8J S2U0 < U < U1Magnon:Total Sz =1Ref: Ziman, Principles of the theory of solids, 2nd ed., 10.11.S =

Terms involving pth spin :Exchange / effective fieldEquation of motion:Cartesian coordinates:Linearization for

real solution:

Long wavelength limit:n.n. interaction only:Neutron scattering: cubic lattices:a = lattice constant

FeCoNiD at 295K [ meV A2]281500364

Quantization of Spin WavesMagnon is a boson :

Thermal Excitation of MagnonsBose-Einstein distribution:Average number of excited magnons at T :ka

Neutron Magnetic ScatteringFeNeutrons interacts with magnetic moments (p, n & e) Mn Pt3bcc, h+k+l = even

Ferrimagnetic OrderMagnetite ( Fe3O4 = FeO Fe2 O3 ):If all spins are aligned, thenper Fe3O4 Observed values is 4.1 (see Table 1) Ferrimagnetism Ferric ( Fe3+ : d 5 ) ion: S = 5/2. Ferrous ( Fe2+ : d 6 ) ion: S = 2. cancelledFerrimagnetism is found in ferrites ( MO Fe2 O3 ), M = Zn, Cd, Fe, Ni, Co, Mg.Most are poor conductors (good as rf transformer cores).spinelMgAl2O4a ~ 8AAll J < 0 (antiferromagnetic) |JAB | > |JAA | , |JBB | AA, BB //, AB anti //.Mg: 8 tetrahedral sitesAl: 16 octahedral sitesO: 32 vertices

U is lowest when ( ferrimagnetism )Forlowest U is 0 Ferrimagnetism is favorable if U < 0, i.e., This condition doesnt rule out other special configurations that may have lower energies.

Curie Temperature and Susceptibility of FerrimagnetsLet = = 0 Mean field approx.:For Ba = 0, a non-zero solution is possible only if For Ba 0,Fe3O4

Iron GarnetsIron Garnets: cubic ferrimagnetic insulators M3Fe5O12 , e.g., YIG ( Y3Fe5O12 ). Ionic states: M+3, Fe+3 (ferric, S = 3/2, L = 0 )In YIG (Y3Fe5O12), Y+3 is diamagnetic.At T = 0, each Fe+3 contributes 5B .However, 5 Fe+3 3 (d-sites) + 2 (a sites) net 1 nB(0) = 5 per formula unit.Mean field at a site (due to d site ions) 1.5104 Md .TC = 559K is due to ad interaction.Fe+3 are in L = 0 state interaction with phonons is weak sharp resonance lines

Antiferromagnetic OrderMnO (NaCl lattice, Mn on fcc)~ fcc, a = 4.43A~ sc, a = 8.85AneutronX-ray:a = 4.43A for both.Spins on single (111) plane are //.Spins on adjacent (111) planes are anti-//.

For an anti-ferromagnet in paramagnetic phase (at T > TN ), See Table 2 for values of TN / .Discrepancy due mostly to 2nd neighbor interactions & sublattice structures.Adding constant to mean field gives Prob 12.3

Susceptibility Below the Neel TemperatureFor T > TN , is independent of direction of Ba .For T

Case // :independent of Ba MnF2In strong fields, spins jump from // to , which has lower U.See D.Wagner,Introduction to the Theory of Magnetism, p.167-70.

Antiferromagnetic MagnonsSetting the linearized version of the equations of motion becomesSettingwe have

Nonzero solution only ifRbMnF3Magnons were observed up to 0.93 TN in MnF3

Ferromagnetic DomainsFor T < TC , observed M of macroscopic specimen are often much less than expected.Reason:Full spin alignment occurs only in small regions called domains.Direction of magnetization in different domains are uncorrelated so that M tends to be 0 when Ha = 0.M = 0Domains also form in antiferromagnets, ferroelastics, superconductors, and sometimes metals undergoing strong dHvA effect.

For weak Ha , volume of domains with M // Ha increases total M increases.As Ha increases, domains begin to merge or disappear, the process becomes irreversible (hysteresis).In strong Ha , M in all domains begin to align with Ha.Hysteresis loop Hc = coercivity Br = Remanence Bs = Saturation induction

Anisotropy EnergyIn ferromagnetic crystals, M tends to align with certain directions (easy axes).This preference is due to the anistropy (magnetocrystalline) energy. bcchexagonalfcc

For Co(hcp): = angle from hexagonal axis. For Fe(bcc):j = direction cosine wrt cubic axes. spin-orbit coupling makes L , S anti-parallel.

Transition Region between DomainsBloch wallFor small , Doing a spin-flip in N steps :For a spin-flip ( = ) between adjacent sites :Let However, larger N increases wall thickness & hence the anisotropy energy.

Let denote energy per unit area of wall, thenFor a cubic crystal with lattice constant a, 300 in Fe 1 erg /cm2 in FeAccurate calculation for a 180 wall in a (100) plane gives Exchange energy for line of N atoms thru wall

Origin of DomainsU = U0 U =U0 / 4 U = 0 U = 0U =U0 / 2 domians of closure

Coercivity and HysteresisCoercivity HC = field requires to reverse B to 0. Softless impurity / straine.g., Fe-Si, Fe-Co-Mn, Ni78Fe22, NiZn, MnZn, metallic glass Fe79B13Si9, amorphous alloysHardhigher straine.g. precipitated phaseAlnico VSi: , anis .Ni78Fe22 : anis 0, m 0.High HC of small grains or powder ( d < 105 cm ): single domains, M by rotation.Rare earth metals in alloy with Mn, Fe, Co, Ni have high K & anis ~ 2K/M. E.g., Fe (Hc ~500G), SmCo5 (Hc ~ 2K/M ~ 290kG ), Nd2 Fe14 B ( B H ~50MGOe )

SupermalloyFe-SiAlnico VSmCo5HC [G]0.0020.560010,000Applicationpulse transformercommercial transformerloud speakermagnet

Single Domain ParticlesDominant application of ferromagnetism: Magnetic recording devices, e.g., hard disks, audio & video tapes.Recording medium are composed of single domain particles / regions.Ideal single domain particle = elongated (acicular) particle with M // long axis.For digital behavior, particle size ~ 10100 nm. First successful recording material is acicular -Fe2O3 with long/short axes ratio ~5:1, HC ~ 200 Oe, length < 1 m. Better material is CrO2 , with axes ratio ~20:1, HC ~ 500 Oe. Superparamagnetism: paramagnetism below TC or TN .Single domain consists of chain of spheres with fixed .M follows Curie-Brillouin-Langevin law if particles are in liquid & free to rotate.

If liquid is then frozen, a permanent M retains after removal of Ha

Geomagnetism and BiomagnetismSedimentary rocks containing single domain particles can carry remanent M that gives a record of earths H at the time of deposition.These can be 500 million yrs old.They provide the strongest support to continental drift theory ( plate tectonics).Brunhes (1906) discovered earths H can reverse direction (explained by standard dynamo theory). Such reversal took place abruptly every 104 to 107 yrs. Single domain particles of Fe2O3 are behind the direction-seeking (magnetotaxis) behavior of biological entities such as migrating birds, bees, and bacteria.Magnetotaxis bacteria with chain of 50 nm particles of Fe2O3

Magnetic Force Microscopy (M F M)Resolution ~ 10-100 nm