Fundamentals of Magnetism - PDF Free Download

Fundamentals of Magnetism Part II Albrecht Jander Oregon State University

Real Magnetic Materials, Bulk Properties M-H Loop M M s M R B-H Loop B B s B R H ci H H c H M S - Saturation magnetization H ci - Intrinsic coercivity M R - Remanent magetization B s - Saturation flux density B R - Remanent flux density H c - Coercive field, coercivity Note: these two figures each present the same information because B=µ o (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 M = NN

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

Atoms without net magnetic moment Diamagnetism M e - H H Diamagnetic substances: Ag -1.0x10-6 Be -1.8x10-6 Au -2.7x10-6 H 2 O* -8.8x10-6 NaCl -14x10-6 Bi -170x10-6 Graphite -160x10-6 Pyrolitic ( ) -450x10-6 graphite ( ) -85x10-6 χ is small and negative Atomic number Atomic density χ = N Aρ W A μ 0 Ze 2 r 2 6m e Orbital radius * 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. M colder hotter H H Susceptibility is small and positive Susceptibility decreases with temperature. Magnetization saturates (at low T or very large H) H µ mh << k T mh > k T 0 B µ 0 B

Langevin Theory of Paramagnetism* Energy of a magnetic moment, m, in a field, H E = µ 0m H = µ 0mH cos( θ ) Independent moments follow Boltzmann statistics P( E) E / kbt µ 0mH cos( θ )/ kbt = e = e 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 p( θ ) 0 e sin( θ ) sin( θ ) dθ Which any fool can see is just: e µ mh cos( θ )/ k T 0 0 = π µ mh cos( θ )/ k T B B The magnetization of a sample with N moments/volume will be M M π µ 0mH cos( θ )/ kbt π e cos( θ 0 = Nm cos( θ ) = Nm cos( θ ) p( θ ) dθ = Nm π 0 µ 0mH cos( θ )/ kbt e sin( ) µ 0mH = kbt Nm coth kbt µ 0mH 0 θ m )sin( θ ) dθ θ dθ Paul Langevin (1872-1946) *Langevin, P., Annales de Chem. et Phys., 5, 70, (1905). H

Langevin paramagnetism: M µ 0mH = kbt Nm coth kbt µ 0mH M 2 Nm = µ 0 3k T B H Which gives Curie s Law: χ = M H = µ 0 2 Nm 3k T B = C T Curie s Law* For µ ο mh<<k B T (most practical situations) L( a) Curie constant 1 1 1 0 1 L( a) = coth( a) a slope = 1/3 1 10 0 10 10 a 10 Curie s Law Pierre Curie (1859-1906) Note: quantization of the magnetic moment requires a correction to the above classical result which assumes all directions are allowed: χ M H 2 2 Ng µ J ( J + 1) µ 3J k T B = = 0 = 2 B C T 1 χ 0 T *Curie, P., Ann. Chem. Phys., 5, 289 (1895)

Pauli Paramagnetism No magnetic field In magnetic field, B Electron energy [J or (ev)] Electron energy [J or (ev)] Fermi level, E f B [T] [#/J-m 3 ] Density of states, D(E) E = 2µ B B Density of states [#/J-m 3 ] Μ = µ B2 BD(E f )

Paramagnetic Substances Paramagnetic susceptibility: Liquid oxygen Sn 0.19x10-6 Al 1.65x10-6 O 2 (gas) 1.9x10-6 W 6.18x10-6 Pt 21.0x10-6 Mn 66.1x10-6

Curie-Weiss* Law Consider interactions among magnetic moments: E = µ 0m ( 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: M Again for µ ο mh<<k B T M µ 0m( H + αm ) = kbt Nm coth kbt µ 0m( H + αm ) 2 Nm = µ 0 ( H +αm ) 3k T Which gives the Curie-Weiss Law: M C C χ = = = H T αc T T C = µ 0 B αnm 3k B 2 T C 1 χ Curie Temperature 0 T C Curie-Weiss Law T Pierre Weiss (1865 1940) *Weiss, P., J. de Phys., 6, 661 (1907)

Weiss Theory* of Ferromagnetism Starting with the Langevin equation with mean field interactions M µ 0m( H + αm ) = kbt Nm coth kbt µ 0m( H + αm ) Set the external field to zero, H=0 M M 0 M µ 0m( αm ) = kbt Nm coth kbt µ 0m( αm ) This can be solved for M graphically: M S 1 3 x mαm k T *Weiss, P., J. de Phys., 6, 661 (1907) x = µ 0 B right side of eq. coth ( x) left side of eq. M kbt = x 2 M µ Nm α 0 0 1 x Langevin function Plot both sides of the equation as a function of x M S = µ 0 M 0 0 T C T Curie Temperature mαm k T Intersection is spontaneous magnetization, M s Note: line with slope proportional to T Asymptote of Langevin function is 1/3 so the only solution for M S is zero when k B T 1 > 2 µ 0Nm α 3 Spontaneous magnetization decreases with temperature and vanishes at: 2 µ Nm T 0 α C = 3k B B

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 M = µ 0m( H + αm ) Nm tanh kbt (c) Measurements Points to remember: Ferromagnets have spontaneous magnetization, M S even in zero field M S goes to zero at the Curie temperature, T C Above T C material is paramagnetic.

Ferromagnetic Materials Material: M s [A/m] T C [ C] Ni 0.49x10 6 354 Fe 1.7x10 6 770 Co 1.4x10 6 1115 Gd 19 Dy -185 NdFeB 1.0x10 6 310 NiFe ~0.8x10 6 447 FeCoAlO 1.9x10 6

Ferromagnetic Breakfast 2 x 10-5 One Flake of Total Cereal Moment (A/m 2 ) 1 0-1 -2-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 Field (T)

Ferromagnetic Exchange Interaction m J 2 E, = µ 0Jm cos( θ ) i+ 1 m i m e - e - e - e - e - e - e - e -

Antiferromagnetic and Ferrimagnetic Materials Negative exchange coupling: Antiferromagnet Ferrimagnet

Crystalline Anisotropy 2 4 Uniaxial: sin sin... E = K θ + K θ + u easy axis 2 θ easy plane Cubic: E K > 0 < 0 u K u 2 2 2 2 2 2 2 2 2 = K1 ( α1α 2 + α 2α3 + α1α3 ) + K2( α1α 2α3 ) +... θ 1 θ 2 θ 3 α i = cos( θ i ) 3 easy axes 4 easy axes K > 0 0 1 K 1 <

Crystalline Anisotropies of Some Magnetic Materials Material Structure K 1 [J/m 3 ] K 2 [J/m 3 ] x10 5 x10 5 Cobalt hcp 4.1 1.5 Iron bcc 0.48-0.1 Nickel fcc -0.045-0.023 SmCo 5 170

Other sources of anisotropy Induced Heat in a field, stress, plastic deformation (e.g., rolling), etc. Exchange anisotropy coupling between FM and AFM materials

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H eeee M s VVVV (θ) H eeee Easy Easy θ 0 180 360 Total energy (Magnetic field along easy axis) E M Hard axis Easy axis 0 180 360 θ * E. Stoner and P. Wohlfarth, A Mechanism of Magnetic Hystereisis in Heterogeneous Alloys, IEEE Trans. Magn., 27, 3475, 1991

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H eeee M s VVVV (θ) H eeee = 0 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M eeee M 0 180 360 θ H eeee

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H eeee M s VVVV (θ) H eeee = 1 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M eeee M H eeee 0 180 360 θ

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H eeee M s VVVV (θ) H eeee = 2 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M M eeee H eeee 0 180 360 θ

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H eeee M s VVVV (θ) H eeee = 3 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M eeee M 0 180 360 θ H eeee

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H haaa M s VVVV (θ) H haaa Easy Easy θ 0 180 360 Total energy (Magnetic field along easy axis) E M Hard axis Easy axis 0 180 360 θ

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H haaa M s VVVV (θ) H haaa = 0 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M haaa M 0 180 360 θ H haaa

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H haaa M s VVVV (θ) H haaa = 1 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M haaa M H haaa 0 180 360 θ

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H haaa M s VVVV (θ) H haaa = 2 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M haaa M H haaa 0 180 360 θ

Anisotropy energy E aaaa = K u Vsss 2 (θ) E Hard Stoner-Wohlfarth Theory* Hard Magnetostatic energy E mm = H haaa M s VVVV (θ) H haaa = 3 Easy Easy 0 180 360 θ Easy axis Total energy (Magnetic field along easy axis) E M haaa M 0 180 360 θ H haaa

Stoner-Wohlfarth Theory Easy axis loop Hard magnet Permanent magnets Data storage 45 loop M H Hard axis Hard axis loop Easy axis Soft magnet Transformers/inductors Flux guides Recording heads Lowest switching field MRAM Anisotropy field H K = 2K u M S

Magnetic Domains

Why do Magnetic Domains Form? H d Large demagnetisation field and energy in external field. Reduced demagnetisation field and external field.

Closure Domains Reduced demagnetisation field and external field. No demagnetisation field and external field.

Domain Configurations Domain walls are the boundary between to regions with different magnetization direction. 180 wall 90 wall M s A ferromagnet with net magnetization less than saturation breaks into domains due to demagnetizing fields. Domains form to reduce the magnetostatic energy. Within a domain, the magnitude of magnetization is the spontaneous magnetization, M s The direction of magnetization varies from domain to domain.

Two Types of Domain Walls Bloch wall magnetization rotates around axis normal to wall M pointing at you! div( M) = 0 nopoles in bulk Felix Bloch (1905-1983) Néel wall Louis Néel (1904-2000) domain wall thickness div( M) 0 distributed poles in bulk

Domain Wall Exchange Energy 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 Exchange energy between adjacent moments: θ 2 2 2 ( θ ) E = cos( ) 1 m, µ 0 + 1 0 m θ µ 0 m i m J J i 2 Summing through wall thickness n= 0.. N E m, m 1 n n+ Nµ Jm 2 2 µ 0Jm π = 2N Summing over wall area: = 0 2 2 ( θ ) + C 2 2 Area µ 0Jm π E exch = + C 2 a 2N 2 + C Using δ=na: δ=na 0 1 2 3 N n 2 2 E exch µ 0Jm π 1 = Area 2a δ Exchange energy favors thick walls. a

Domain Wall Anisotropy Energy Assume linear transition over N lattice points n θ ( n ) = π for n = 0.. N N Anisotropy energy penalty for each moment: a E m i = K u sin 2 ( θ ) a Summing through wall thickness n= 0.. N N 2 3 K Em n Ku sin ( θ ) a dθ = π Summing over wall area: π 0 3 uniaxial anisotropy energy / volume u Na 2 3 π θ 0 δ=na 0 1 2 3 N n Or, E anis = Area 2 a K u Na 2 3 Anisotropy energy favors thin walls. E anis K = u δ Area 2

Domain Wall Thickness Domain wall thickness is determined by a balance between anisotropy and exchange energy: E anis K = u δ Area 2 2 2 E exch µ 0Jm π 1 = Area 2a δ µ m Define exchange stiffness: A 0 J = a Total wall energy is: E π 2 wall Ku A = δ + Area 2 2δ The minimum energy is achieved at: 2 Wall energy E wall E anis E exch δ 0 Wall thickness, δ de dδ wall K 2 π A 2δ 2 = u = 0 2 δ 0 = π A K u Where the wall energy is: E wall = π K u A Area

Single Domain Particles Single domain Multi domain M Demag. Energy r 3 Less Demag. Energy Wall Energy r 2 Below a critical size, domain wall can not form.

λ s Magnetostriction Random crystal orientation: > 0 l l = 3 λ (cos 2 s 2 θ 1 ) 3 M H E.g. Terfenol TbFe 2 : λ s = 1753x10-6 λ s < 0 M H E.g. Ni (fcc): λ s = -34x10-6 Fe (bcc): λ s = -7x10-6

Magnetostrictive Effect on Domain Walls 180 wall 90 wall Magnetostrictive strain energy favors 180 degree walls.

Domain Walls in Thin Films Bloch wall Néel wall Magnetic poles on surface Wall energy (10-3 J/m 2 ) 10 8 6 4 2 0 Néel wall Bloch wall 40 80 120 160 Film Thickness (nm) For A = 10-11 J/m, B s = 1 T and K = 100 J/m 3 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 high remanent magnetization. µ r B B s B R H c H

What Determines? M s depends primarily on composition. H c depends also on : grain size/orientation crystalline anisotropy magnetostriction defects

Soft Iron Inexpensive, high coercivity material µ r ~ 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 B s, T c 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) Commercial iron 5-10 Si-Fe hot rolled 1-3 Si-Fe CRGO 0.3-0.6 Lamination reduces eddy currents: *Di Napoli, A.; Paggi, R.; A model of anisotropic grainoriented steel, IEEE Transactions on Magnetics 19, 1557, (1983)

Ni-Fe Alloys (Permalloys) Zero magnetostriction at 80% Zero anisotropy at 78%

Amorphous Alloys, Metglas (Metallic No crystalline anisotropy Glasses) Ultra-low coercivity and super-high permeability Available in thin ribbons only: Rapid quenching: melt spinning

Some Soft Magnetic Materials Material Composition Relative Permeability µ i µ max H c (ka/m) Iron 100% Fe 150 5000 80 2.15 B s (T) Silicon-iron (non-oriented) Silicon-iron (grain oriented) 96% Fe 4% Si 97% Fe 3% Si 78 Permalloy 78% Ni 22% Fe Hipernik Supermalloy Mumetal Permendur Hiperco Supermendur Metglass (amorphous) 50% Ni 50% Fe 79% Ni, 16% Fe 5% Mo 77% Ni, 16% Fe 5% Cu, 2% Cr 50% Fe 50% Co 64% Fe, 35%Co 0.5% Cr 49% Fe, 49%Co 2% V 1.6%Ni,4.4%Fe, 8.6%Si,82%Co, 3%B 500 7000 40 1.97 1500 40,000 8 2.0 8000 100,000 4 1.08 4000 70,000 4 1.60 100,000 1,000,000 0.16 0.79 20,000 100,000 4 0.65 800 5000 160 2.45 650 10,000 80 2.42-60,000 16 2.40-1,000,000 0.01 0.57

Ferrites Ceramic materials of type MO Fe 2 O 3 where M = transition metal Magnetic properties are not as good as metals. But, high resistivity makes them useful for high frequency applications. Skin depth: δ = 1 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 µ 0 M are plotted on same axis and often in CGS units. These days often in Chinese. µ r

Advances in Permanent Magnet Materials http://www.aacg.bham.ac.uk/magnetic_materials/history.htm

Common Permanent Magnet Materials Material B r (T) H c (ka/m) (BH) max (kj/m 3 ) T c of B r (%/ C) T max ( C) T curie ( C) Cost (relative) Notes SmCo 1.05 730 206-0.04 300 750 Highest (100) NdFeB 1.28 980 320-0.12 150 310 High (50) Hard, brittle Corrodes 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 (1) Injection molded

Thermal Decay of Magnetization

Superparamagnetism For H a =0: E Easy axis ΔE = K u V M t = M 0 e t τ τ = τ 0 e ΔE k BT ΔE 0 180 360 θ τ 0 = attempt period ~ 1ns Superparamagnetic if K u V < k B T

Thermally Activated Reversal What happens over time? E dn 1 2 N (t)e dt 1 E 1 kt 0 2 1 π θ dn 2 1 N (t)e dt 2 E 2 kt dn 1 (t) dt = dn dn 2 1 1 2 dt dt dm(t) dt dn (t) = 2M 1 r dt τ -1 M(t) = 2NM e NM where τ = C e r t r E kt

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 2 2 r e A I m π π ω = = 2 r m L eω = Magnetic moment Angular momentum m e e L m 2 = 2 L = e B m e m 2 = m e e L m = Orbital Spin

The g-factor or gyromagnetic ratio Orbital Spin m L = 2 e m e m L = e m e m e g =1 = g g = 2 L 2 m e e m e =1.76 10 11 SI units: coul kg = 2 A m 2 kg m / sec = rad/sec tesla

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 * From Kittel via Chikazumi s Physics of Magnetism Ferromagnetic materials have g 2 most of magnetic moment is from spin

Einstein-DeHaas Experiment (1915) m L e = g m = g L 2 m e e 2m e m = 2M S Vol

The gyromagnetic constant, γ γ m L = g 2 e m e For most materials: γ = 1.76 10 11 rad / s tesla = 28 GHz tesla

Spin Dynamics: The Landau-Lifschitz Equation Torque on a magnetic moment: T = m B = m µ 0 H m The gyroscope equation: dl = T dt The gyromagnetic relation: m = γ L and M m = / volume Gives: dm dt = γ M µ 0 H

Spin Precession dm dt ω = γµ 0 H M M H f = 2π 0 1 γµ H dm dt dm dt = γ M µ 0 H Magnetization never turns towards H, just precesses at frequency GHz 28 tesla

Landau-Lifshitz Equation with Gilbert Damping dm dt = µ 0 γ α ( M H ) + M M dt α M dm α is the Gilbert damping parameter dm dt M dm dt Equivalent: dm dt = µ 0 γ λ M ( M H ) + [ M ( M H )]

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 M dm dt dm dt demag field Kittel relation: H eff = H ( H + M )

Ferromagnetic Resonance (FMR) AC field Top View M DC field DC field