Fundamentals of Magnetism
Part II
Albrecht Jander Oregon State University Real Magnetic Materials, Bulk Properties
M-H Loop B-H Loop M B
Bs Ms BR MR
H H Hci Hc
MS - Saturation magnetization Bs - Saturation flux density Hci - Intrinsic coercivity BR - Remanent flux density MR - Remanent magetization Hc - Coercive field, coercivity
Note: these two figures each present the same information because B=µo(H+M) Minor Loops M
H
M M
DC demagnetize AC demagnetize
H H Effect of Demag. Fields on Hysteresis Loops
Measured loop Actual material loop
Measured M Measured M
Applied H Internal H
=
푀 푁푁 Types of Magnetism
Diamagnetism – no atomic magnetic moments Paramagnetism – non-interacting atomic moments Ferromagnetism – coupled moments Antiferromagnetism – oppositely coupled moments Ferrimagnetism – oppositely coupled moments of different magnitude Diamagnetism M • Atoms without net magnetic moment
H e-
H is small and negative Diamagnetic substances: -6 휒 Ag -1.0x10 Atomic number -6 Be -1.8x10 Orbital radius Au -2.7x10-6 Atomic density -6 H2O* -8.8x10 = 6 2 2 NaCl -14x10-6 푁퐴휌 휇0푍푒 푟 휒 − Bi -170x10-6 푊퐴 푚푒 Graphite -160x10-6 Pyrolitic (┴) -450x10-6 graphite (║) -85x10-6 * E.g. humans, frogs, strawberries, etc. See: http://www.hfml.ru.nl/froglev.html Paramagnetism
• Consider a collection of atoms with independent magnetic moments. • Two competing forces: spins try to align to magnetic field but are randomized by thermal motion.
H H
M µ0mH << kBT µ0mH > kBT colder hotter • Susceptibility is small and positive H • Susceptibility decreases with temperature. • Magnetization saturates (at low T or very large H) Langevin Theory of Paramagnetism* H • Energy of a magnetic moment, m, in a field, H E = −µ0m⋅ H = −µ0mH cos(θ ) θ m • Independent moments follow Boltzmann statistics − µ θ P(E) = e E / kBT = e 0mH cos( )/ kBT • The “density of states” on the sphere is dn = 2πr 2 sin(θ )dθ • So the probability of a moment pointing in direction θ to θ+dθ is µ θ e 0mH cos( )/ kBT sin(θ ) p(θ ) = π µ θ ∫ e 0mH cos( )/ kBT sin(θ )dθ 0 Paul Langevin N (1872-1946) • The magnetization of a sample with moments/volumeπ will be µ θ 0mH cos( )/ kBT θ θ θ π ∫ e cos( )sin( )d = θ = θ θ θ = 0 M Nm cos( ) Nm∫ cos( ) p( )d Nm π µ θ 0 ∫ e 0mH cos( )/ kBT sin(θ )dθ • Which any fool can see is just: 0 µ0mH kBT M = Nmcoth − kBT µ0mH *Langevin, P., Annales de Chem. et Phys., 5, 70, (1905). Curie’s Law* 1 • Langevin paramagnetism: L(a) = coth(a) − a 1 1 µ0mH kBT M = Nmcoth − kBT µ0mH L( a) 0
• For µοmH<
No magnetic field In magnetic field, B
Electron energy Electron energy [J or (eV)] [J or (eV)]
Fermi level, Ef B [T]
∆E = 2µBB
[#/J-m3] [#/J-m3] Density of states, D(E) Density of states
2 Μ = µB BD(Ef ) Paramagnetic Substances
Liquid oxygen Paramagnetic susceptibility: Sn 0.19x10-6 Al 1.65x10-6 -6 O2 (gas) 1.9x10 W 6.18x10-6 Pt 21.0x10-6 Mn 66.1x10-6
Curie-Weiss* Law • Consider interactions among magnetic moments: E = −µ0m⋅(H +αM ) where M represents the average orientation of the surrounding magnetic moments (“Mean field theory”) and α is the strength of the interactions
• The Langevin equation then becomes: µ0m(H +αM ) kBT Pierre Weiss M = Nmcoth − (1865–1940) kBT µ0m(H +αM ) Curie-Weiss Law • Again for µοmH< • This can be solved for M graphically: Plot both sides of the equation as a function of µ mαM x = 0 M S kBT 1 right side of eq. Intersection is spontaneous magnetization, M x 1 s M 3 coth(x)− x Note: line with slope proportional to T M Langevin function 0 Asymptote of Langevin function is 1/3 so left side of eq. the only solution for MS is zero when M k T kBT 1 = B x > 2 µ 2α M 0 µ0 Nm α 0 Nm 3 Spontaneous magnetization decreases µ mαM = 0 with temperature and vanishes at: x 2 kBT µ0 Nm α TC = *Weiss, P., J. de Phys., 6, 661 (1907) 3kB Weiss Theory of Ferromagnetism • Fits data very well! • Again, corrections need to be made for quantization of magnetic moments (a) Classical result using Langevin function (b) Quantum result using Brilloun function µ0m(H +αM ) M = Nm tanh kBT (c) Measurements Points to remember: • Ferromagnets have spontaneous magnetization, MS even in zero field • MS goes to zero at the Curie temperature, TC • Above TC material is paramagnetic. Ferromagnetic Materials Material: Ms [A/m] TC [°C] Ni 0.49x106 354 Fe 1.7x106 770 Co 1.4x106 1115 Gd 19 Dy -185 NdFeB 1.0x106 310 NiFe ~0.8x106 447 FeCoAlO 1.9x106 Ferromagnetic Breakfast -5 One Flake of Total Cereal x 10 2 1 ) 2 0 Moment (A/m -1 -2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Field (T) Ferromagnetic Exchange Interaction J m E = −µ m2 cos(∆θ ) mi ,mi+1 0 J - e e- e- e- e- e- e- e- Antiferromagnetic and Ferrimagnetic Materials Negative exchange coupling: Antiferromagnet Ferrimagnet Crystalline Anisotropy θ 2 4 • Uniaxial: E = Ku sin θ + K2 sin θ +... easy axis easy plane Ku > 0 Ku < 0 2 2 2 2 2 2 2 2 2 θ1 • Cubic: E = K1(α1 α 2 +α 2α3 +α1 α3 ) + K2 (α1 α 2α3 ) +... θ θ3 2 αi = cos(θi ) 4 easy axes 3 easy axes K1 > 0 K1 < 0 Crystalline Anisotropies of Some Magnetic Materials 3 3 Material Structure K1 [J/m ] K2 [J/m ] x105 x105 Cobalt hcp 4.1 1.5 Iron bcc 0.48 -0.1 Nickel fcc -0.045 -0.023 SmCo5 170 Other sources of anisotropy Induced • Heat in a field, stress, plastic deformation (e.g., rolling), etc. Exchange anisotropy • coupling between FM and AFM materials Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 �푒𝑒푒 Easy Easy M Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E Hard axis θ 0 180 360 * E. Stoner and P. Wohlfarth, “A Mechanism of Magnetic Hystereisis in Heterogeneous Alloys”, IEEE Trans. Magn., 27, 3475, 1991 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 0 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 1 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 2 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E M 푀푒𝑒푒 θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 3 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 1 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 0 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� 푒𝑒푒 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉푉푉� 휃 = 0 �푒𝑒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒𝑒푒 M θ �푒𝑒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 �ℎ푎푎푎 Easy Easy M Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E Hard axis θ 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 0 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 1 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 2 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 3 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 2 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚� ℎ푎푎푎 � 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 � 푀 푉𝑉� 휃 = 0 �ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ �ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory Easy axis Easy axis loop • “Hard” magnet Hard axis • Permanent magnets 푀 • Data storage Hard axis loop • “Soft” magnet • Transformers/inductors • Flux guides � • Recording heads 45° loop • Lowest switching field • MRAM Anisotropy field 2 = 퐾푢 �퐾 푀푆 Magnetic Domains Why do Magnetic Domains Form? Hd Large demagnetisation field Reduced demagnetisation and energy in external field. field and external field. Closure Domains Reduced demagnetisation No demagnetisation field and field and external field. external field. Domain Configurations Domain walls are the boundary between to regions with different magnetization direction. • A ferromagnet with net magnetization 90° wall less than saturation breaks into domains due to demagnetizing fields. 180° wall • Domains form to reduce the magnetostatic energy. • Within a domain, the magnitude of Ms magnetization is the “spontaneous magnetization”, Ms • The direction of magnetization varies from domain to domain. Two Types of Domain Walls •Bloch wall magnetization rotates around axis normal to wall Felix Bloch (1905-1983) div(M) = 0 ⇒ nopoles in bulk M pointing at you ! •Néel wall Louis Néel (1904-2000) domain wall thickness div(M) ≠ 0 ⇒ distributed poles in bulk Domain Wall Exchange Energy a • Assume linear transition over N lattice points n θ (n) = π for n = 0..N N • Angle between adjacent moments is π ∆θ = N π δ=Na • Exchange energy between adjacent moments:θ (∆θ )2 E = −µ Jm2 cos(∆θ ) ≅ −µ Jm2 1− 0 mi ,mi+1 0 0 2 0 1 2 3… N n • Summing through wall thickness 2 2 δ Nµ Jm (∆θ ) • Using =Na: E = 0 + C ∑ mn ,mn+1 n=0..N 2 2 2 Eexch µ0 Jm π 1 2 2 = µ0 Jm π δ = + C Area 2a 2N • Summing over wall area: • Exchange energy favors thick walls. Area µ Jm2π 2 E = 0 + C exch a2 2N Domain Wall Anisotropy Energy a • Assume linear transition over N lattice points n θ (n) = π for n = 0..N N • Anisotropy energy penalty for each moment: E = K sin 2 (θ )a3 mi u π δ=Na uniaxial anisotropy energy / volume θ • Summing through wall thickness π 3 0 N 2 3 K Na 0 1 2 3… N E ≅ K sin (θ )a dθ = u ∑ mn ∫ u n n=0..N π 0 2 • Summing over wall area: Area K Na3 E = − u anis a2 2 Or, • Anisotropy energy favors thin walls. E K anis = u δ Area 2 Domain Wall Thickness • Domain wall thickness is determined by a balance between anisotropy and exchange energy: 2 2 Eanis Ku E µ Jm π 1 = δ exch = 0 Area 2 Area 2a δ 2 µ0 Jm • Define exchange stiffness: A = Ewall a Eanis • Total wall energy is: Wall energy Wall Eexch 2 Ewall Ku π A = δ + δ Area 2 2δ 0 • The minimum energy is achieved at: Wall thickness, δ dE K π 2 A A wall = u − = ⇒ δ = π 2 0 0 dδ 2 2δ Ku • Where the wall energy is: Ewall = π Ku A Area Single Domain Particles Single domain Multi domain M Demag. Energy r3 Less Demag. Energy ∝ Wall Energy r2 ∝ Below a critical size, domain wall can not form. Magnetostriction Random crystal orientation: ∆l 3 1 = λ (cos2 θ − ) l 2 s 3 λs > 0 → M E.g. “Terfenol” -6 TbFe2: λs = 1753x10 H λs < 0 E.g. -6 → M Ni (fcc): λs = -34x10 -6 Fe (bcc): λs= -7x10 H Magnetostrictive Effect on Domain Walls 180° wall 90° wall • Magnetostrictive strain energy favors 180 degree walls. •BlochDomain wall Walls in Thin Films ) 10 2 8 J/m 3 - Néel wall Magnetic poles on surface 6 4 Bloch wall 2 •Néel wall 0 Wall energy energy (10 Wall 40 80 120 160 Film Thickness (nm) -11 3 For A = 10 J/m, Bs = 1 T and K = 100 J/m from O’Handley, Modern Magnetic Materials Magnetic poles in bulk Domain Configuration • Balance of energy between: – Magnetostatic self energy (demag. field -> try to reduce magnetic poles) – Domain wall energy (exchange and anisotropy -> reduce wall area) – Strain energy (magnetostriction -> favor 180° walls) – Magnetostatic energy in applied field (E=-m·B -> favors alignment with field) Domains in Fe film MFM images From: http://physics.unl.edu/~shliou/ResearchActivity/research3.htm Barkhausen Noise M Heinrich Barkhausen (1881-1956) H H Barkhausen, Two phenomena uncovered with help of the new amplifiers, Z. Phys, 1919 Domains in NiFe patterned film MFM images from http://www.philsciletters.org/3rd%20week%20June/Visualizing%20objects%20a t%20nanometer%20scales.htm Stripe domains in CoFeB thin film with Perpendicular Anisotropy -0.11 mT -0.06 mT 0.01 mT 25 µm Polar Kerr images. 0.06 mT 0.11 mT 54 Flower Domains in YIG MFM Image From: http://www.nanoworld.org/russian/s_11.html Magnetic Materials Magnetic materials are typically classified as “hard” or “soft” depending on the intended application. Soft magnets have low coercivity and high permeability. Hard magnets have high coercivity and µr high remanent magnetization. B Bs BR Hc H What Determines? Ms depends primarily on composition. Hc depends also on : • grain size/orientation • crystalline anisotropy • magnetostriction • defects Soft Iron • Inexpensive, high coercivity material • µr ~ 1000 - 5000 • Used in DC/low frequency applications Silicon Steel (“Electrical Steel”) • Most common in power (50/60 Hz) applications • Adding Si, Al increases resistivity, permeability and reduces coercivity. • Bad effects: reduces Bs, Tc and increases brittleness • Higher resistivity and lower coercivity -> less losses • Higher permeability -> better coupling • Si limited to 3-4%, material gets too brittle. Littmann, M.; Iron and silicon-iron alloys, IEEE Transactions on Magnetics, 7, 48 - 60 (1971) Fe,Si,Al Cold-rolled Grain-Oriented Steel (CRGO) •Reduces hysteresis further by orienting grains: * Losses (W/kg) Lamination reduces eddy currents: Commercial iron 5-10 Si-Fe hot rolled 1-3 Si-Fe CRGO 0.3-0.6 *Di Napoli, A.; Paggi, R.; A model of anisotropic grain- oriented steel, IEEE Transactions on Magnetics 19, 1557, (1983) Ni-Fe Alloys (Permalloys) Zero magnetostriction at 80% Zero anisotropy at 78% Amorphous Alloys, “Metglas” (Metallic Glasses) • No crystalline anisotropy • Rapid quenching: melt spinning • Ultra-low coercivity and super-high permeability • Available in thin ribbons only: Some Soft Magnetic Materials Material Composition Relative Permeability Hc Bs µi µmax (kA/m) (T) Iron 100% Fe 150 5000 80 2.15 Silicon-iron 96% Fe 500 7000 40 1.97 (non-oriented) 4% Si Silicon-iron 97% Fe 1500 40,000 8 2.0 (grain oriented) 3% Si 78 Permalloy 78% Ni 8000 100,000 4 1.08 22% Fe Hipernik 50% Ni 4000 70,000 4 1.60 50% Fe Supermalloy 79% Ni, 16% Fe 100,000 1,000,000 0.16 0.79 5% Mo Mumetal 77% Ni, 16% Fe 20,000 100,000 4 0.65 5% Cu, 2% Cr Permendur 50% Fe 800 5000 160 2.45 50% Co Hiperco 64% Fe, 35%Co 650 10,000 80 2.42 0.5% Cr Supermendur 49% Fe, 49%Co - 60,000 16 2.40 2% V Metglass 1.6%Ni,4.4%Fe, - 1,000,000 0.01 0.57 (amorphous) 8.6%Si,82%Co, 3%B Ferrites Ceramic materials of type MO·Fe2O3 where M = transition metal Magnetic properties are not as good as metals. But, high resistivity makes them useful for high frequency applications. Skin depth: 1 δ = µσπf = 8.6 mm for Cu at 60 Hz = 3 mm for Si-Fe at 60 Hz. Hard Magnetic Materials • Permanent magnet materials. • Manufacturers typically provide data in the form of second quadrant B-H response or “demagnetisation curves” • Often B and µ0M are plotted on same axis and often in CGS units. • These days often in Chinese. µr Advances in Permanent Magnet Materials http://www.aacg.bham.ac.uk/magnetic_materials/history.htm Common Permanent Magnet Materials Material Br Hc (BH)max Tc of Br Tmax Tcurie Cost Notes (T) (kA/m) (kJ/m3) (%/°C) (°C) (°C) (relative) SmCo 1.05 730 206 -0.04 300 750 Highest Hard, brittle (100) NdFeB 1.28 980 320 -0.12 150 310 High Corrodes (50) easily Alnico 1.25 50 45 -0.02 540 860 Medium (30) Ferrite 0.39 250 28 -0.20 300 460 Low (5) Bonded 0.25 160 11 -0.19 100 - Low Injection (1) molded Thermal Decay of Magnetization Superparamagnetism Easy axis = For H =0: a Δ퐸 =퐾푢푉 E −푡⁄휏 푀 =푡 푀0푒 −Δ퐸⁄푘퐵푇 휏 휏0푒 θ = attempt period ~ 1ns 0 Δ퐸 180 360 휏0 “Superparamagnetic” if < 퐾푢푉 푘퐵푇 Thermally Activated Reversal What happens over time? ∆ E − 1 dN → 1 2 ∝ N (t)e kT E 1 → → dt → → → → → → → ∆ E 1 2 → dN − → → → → 2 2 1 ∝ N (t)e kT θ dt 2 0 π dN (t) dN dN 1 = 2 → 1 − 1 → 2 dt dt dt dM(t) dN (t) = 2M 1 dt r dt t ∆E − τ -1 kT M(t) = 2NM re − NM r where τ = C e Spin Dynamics The response of magnetization to time-varying fields is affected by the angular momentum associated with spin or orbital magnetic moment. Consider this when: • Frequencies above ~ 1 GHz • Pulses shorter than ~ 1 nsec Magnetic Moment and Angular Momentum Are Linked Orbital Spin Angular momentum L = m ωr 2 L = e 2 Magnetic moment − eω 2 e m = I A = πr mB = 2π 2me m − e m − e = = L 2m e L me The g-factor or “gyromagnetic ratio” Orbital Spin m − e m − e = = L 2me L me m − e g =1 = g g = 2 L 2me SI units: e 11 coul A ⋅m2 rad/sec = ⋅ = = 1.76 10 2 kg kg ⋅m / sec tesla me g-factors Substance g * ------Iron 1.93 Cobalt 1.85 Nickel 1.84-1.92 Magnetite 1.93 Heusler alloy 2.00 Permalloy 1.9 Supermalloy 1.91 Ferromagnetic materials have g ≈ 2 ⇒ most of magnetic moment is from spin * From Kittel via Chikazumi’s Physics of Magnetism Einstein-DeHaas Experiment (1915) m − e − e = g ∆m = g ∆L ∆m = 2M Vol L 2m S e 2me The gyromagnetic “constant”, γ m − e γ ≡ = g L 2me For most materials: rad / s GHz γ =1.76×1011 = 28 tesla tesla Spin Dynamics: The Landau-Lifschitz Equation Torque on a magnetic moment: m T = m× B = m× µ0 H The gyroscope equation: dL = T dt The gyromagnetic relation: m = −γ L Gives: dM and = −µ γ M × H dt 0 M = m / volume Spin Precession dM = −µ γ M × H dt 0 dM dt dM dt M M Magnetization never turns towards H, just precesses at H GHz frequency 28 1 tesla ω = γµ0 H f = 2π γµ0 H Landau-Lifshitz Equation with Gilbert Damping dM α dM = −µ γ × + × 0 (M H ) M dt M dt α is the Gilbert damping parameter dM α M × dt dM dt M dM λ Equivalent: = −µ γ ′(M × H )+ [M × (M × H )] dt 0 M Reversal with Damped Precession Upon reversal of field, magnetization vector follows a complex path. H α <<1 α =1 α >>1 Precessional Switching Reversal in a Thin Film • M tries to precess out of film plane • Demagnetizing field pushes back • Demag. field increases effective field H dM M dt dM demag field dt Kittel relation: H eff = H (H + M ) Ferromagnetic Resonance (FMR) Top View AC field M DC field DC field