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Magneti sm 101 : All Ab out Spins Jeffrey E. Shield Department of Mechanical Engineering Nebraska Center for Materials and Nanoscience UiUniversit y of fNb Nebras ka Applications of Magnets Hard (Permanent) Magnets Power generation (DC motors) Hybrid cars All-electric airplanes/ships Microsurgical tools Cordless power tools Wind generators Consumer Electronics Computers (Disk drive voice coil motors) Speakers Magnetic Resonance Imaging Soft Transformer cores Electromagnets Semi-hard Data storage The Markets Soft Market Hard Market Magnets Share Magnets Share 1980 $1.89 B 70% $0.81 B 30% 1990 $2.69 B 57.4% $2.0 B 42.6% 2000 $3.55 B 44.2% $4.48 B 55.8% 2010 $5.0 B 28.7% $12.44 B 71.3% th Source: Y. Luo, 18 Workshop on High Performance Magnets and Their Applications, Annecy, France, 2004 Magnetism Basics Maggpnetism arises from electron spins • electrons rotate about nucleus, creating “magnetic moments” • Sometimes, all of the moments cancel each other, resulting in “diamagnetism” • Here, we will only worry about magnetic moments from at/toms/compoun dthtdtds that do not cance l • Specifically, UNPAIRED INNER SHELL ELECTRONS Magnetism Basics • Current (moving electric charge) causes magnetic field Courtesy Todd Zimmerman Magnetism Basics Electrons are “spinning” charges→Have magnetic field • Sometimes, all of the moments cancel each other, resulting in “diamagnetism” •Her e, w e will onl y w orr y about m agn eti c m om en ts fr om atom s/com poun ds that do not cancel • Specifically, UNPAIRED INNER SHELL ELECTRONS Courtesy Todd Zimmerman Magnetism Basics N Atom with a magnetic moment Atom (transition metal, lanthanide, actinide) S Transition metals Lanthanides Actinides Magnetism Basics Periodic array of atoms --“Crystal” Magnetic moments point in random directions PARAMAGNETIC Magnetic moments align parallel FERROMAGNETIC Magnetic moments align antiparallel ANTIFERROMAGNETIC Structure has two magnetic species FERRIMAGNETIC Spontaneous Magnetization Periodic array of atoms --“Crystal” Magnetic moments point in random directions PARAMAGNETIC Magnetic moments align parallel FERROMAGNETIC Magnetic moments align antiparallel ANTIFERROMAGNETIC Structure has two magnetic species FERRIMAGNETIC Ferromagnetic Elements Paramagnets M H H M H H M H Ferromagnets M ? But ferromagnets are not “spontaneously” magnetized H --Think about any steel—screwdriver, automobile body, etc. Flux lines Flux outside a magnet costs energy “Magnetostatic Energy” 2 Ems = ½NdMs Where Nd depends on the shape of the magnet Ferromagnets The material “self divides” into “MAGNETIC DOMAINS ” --regions with spins pointing in a common direction Flux lines Increase in magnetization due to domain growth M H Ferromagnets Domain Walls: Thin and Thick δ = domain wall width = π(A/K)1/2 A = exchange stiffness ~ 10-13 J/m K = Anisotropy constant = 103 – 107 J/m3 So δ ~ 0.1 – 100 nm Ferromagnets Domain walls can be eliminated by reducing the size of the crystal --Competition between Magnetostatic energy (volume dependent) and domain wall energy (surface area) γ = domain wall energy = 4(AK)1/2 2 Ems(single domain)= μoMs V/6 Ems(multidomain) = 0.5Ems(single domain) Ems(multidomain) = Edw 1/2 2 Resulting in, for a sphere, RSD = 36(AK) /μoMs RSD ~ few nm to 1 μm Anisotropy Anisotropy—Answers the question “How easy is it to rotate a spin away from its preferred direction?” θ From 1) Shape 2) Crystal (“Magnetocrystalline”) 3) Stress Shape Anisotropy ⌠ E = -μo H dM ms ⌡ 2 Ems = (μo/2) NdM Since Hd = - NdM is the self de-magnetizing field M a θ c 2 2 2 Ems = (μo/2) [(Mcos θ) NcM +(Msin θ) Na] 2 2 2 Ems = (μo/2) [M Nc +(Na -Nc)M sin θ] 2 Ks = (μo/2) (Na -Nc)M Na andNd Nc are dtiiftlddemagnetizing factors along a and c 4 3 When a = c (sphere), Ks = 0; Max Ks~10 J/m Magnetocrystalline Anisotropy For cubic structures 2 2 2 2 2 2 2 2 2 E = Ko + K1(α1 α2 + α2 α3 + α1 α3 ) + K2(α1 α2 α3 ) Since Ko is angle-independent and K2 is small 2 2 2 2 2 2 E = K1(α1 α2 + α2 α3 + α1 α3 ) For uniaxial structures (hexagonal and tetragonal) 2 4 E = Ko + K1sin θ + K2sin θ Since Ko is angle-independent and K2 is small 2 E = K1sin θ 3 7 3 K1 = 10 -10 J/m Magnetization Easy axis M Easy axis Hard axis H HA HA = anisotropy field field necessary to saturate in the hard direction HA = 2K/μοMs Anisotropy and Length Scales LOW K (“SOFT” MAGNETS) Large domain wall width Small single domain limit SSamall HA HIGH K (“HARD MAGNETS”) Small domain wall width Large single domain limit Large HA Magnetic Properties: Boring Definitions Saturation Magnetization The maximum obtainable magnetization in a material in response to an applie d magne tic fie ld Remanent Magnetization (Remanence—Mr)) The magnetization “remaining” in a material after the applied field has been removed Coercivity (Hc) The magnetic field, applied in the opposite direction, 1 necessary to make the magnetization zero Remanence Anisotropy 0.5 The resistance of a magnetic moment Coercivity s M to point in an unfavorable direction // r 0 M -50 -40 -30 -20 -10 0 10 20 30 40 50 -0.5 -1 Applied Field (kOe) 21 “Hard” v. “Soft” Magnets Coercivity This (arbitrarily) separates hard (high coercivity) and soft (low coercivity) magnetic materials Hard 1 Hc > 2000 Oe 0.5 Soft s M // H < 100 Oe r 0 c M -50 -40 -30 -20 -10 0 10 20 30 40 50 -0.5 -1 Applied Field (kOe) 22 Hard (()gPermanent) Magnets Energy Product The amount of energy stored in a permanent magnet, given by the largest rectangle that can fit in the second quadrant of a B-H graph Defining property for most applications of permanent magnets November 16, 2007 23 Full Hysteresis Loops M Hysteresis loop of a uniililiaxial single- domain particle H along its easy axis IDEAL LOOP! Mr = Ms Hci = HA = 2K/μoMs 2 (BH)max = μoMs /4 Full Hysteresis Loops But a real system has many single-domain particles AthtthdlitdAssume that these are randomly oriented with respect to their easy axis Stoner-Wohlfarth Model Mr = ½Ms Hci = 0.48HA 2 (BH)max = μoMs /16 Stoner-Wohlfarth Systems • Dilute systems of nanocrystalline hard magnetic particles imbedded in a non -magnetic matrix approach the Stoner-Wohlfarth coercivity Nd-Fe-B Er. Girt, K.M. Krishnan, G. Thomas, E. Girt and Z. Altounian, J. Magn. Magn. MtMater. 231, 219 (2001). FePt in C Yingfan Xu, M. L. Yan, J. Zhou, and D. J. Sellmyy,er, J. A ppl. Ph ys. 97, 10J320 (2005) 5 nm FePt C “Real” Magnets • In a real material, we want a high density of magnetic grains • But then, interactions take over! •Hci << 0.48HA •Two basic interactions: •Exchange Interactions--these are short-range 1/2 • lex = (A/K) Exchange-Spring Interactions • Spins across interfaces influence each other ¾ Interaction length ½ ª lex~ (Α/Κ) Interface Exchange-Spring Interactions • Spins across interfaces influence each other ¾ Interaction length ½ ª lex~ (Α/Κ) • Near-interface regions thus have spins influenced by neighboring grains Interface lex Exchange-Bias Interactions Interactions between a ferromagnetic (FM) and antiferromagnetic (AFM) phase Interface FM Phase AFM Phase H The FM spins are “pinned” at the interface ⇒ A (()small) reverse field will not reverse the ferroma gnet “Real” Magnets • •Two basic interactions: •Exchange Interactions--these are short-range 1/2 • lex = δ = π(A/K) •Dipolar Interactions--these are long-range and depend on strength of the dipole • ∝1/r3 The reversal looks kind of like: Strong dipolar, weak exchange Strong dipolar, strong exchange Permanent Magnets 40 ) ee 30 20 vity Hc (kO Hc vity ii 10 Coerc 0 0246 %Additive y (C) Percent Cx in or Sm y 12Co88 32 “Real” Magnets • Soft Magnetic Materials • Low Coercivity • desire strong interactions • Transformers, AC motors or flux-directors (e.g., write- heads) •Hard Magnetic Materials (aka Permanent Magnets) • Hig h Coerc iv ity • Desire weaker interactions • DC motors, flux sources (()gMRI), magnetic recordin g Magnetic Materials: Summary • Important Properties • CiitRCoercivity, Remanence, SttiESaturation, Energy PdtProduct • Important Parameters • Anisotropy • Important Length Scales •RSD = single domain size ~ nanometers to microns • lex= exchange length ~ a few nanometers • δ = domain wall width ~ a few nanometers • Important Features • Microstructure/nanostructure So what happens when Important Length Scales meet up with the scale of the Important Features Magnetic Materials: Practical Examples Low K (soft) materials Characteristics Low K •Large δ (()domain wall width) •Small RSD So,,yy relatively easy reduce microstructure/nanostructure scale so that these converge Nanocrystalline Materials Hci ∝d6 ∝d-1 0 dsp lex Grain Size Superparamagnetic Limit: dsp ~ 25kT/K Nanocrystalline Materials Grain Boundary Grain Boundary Reduction in Grain Size lex lex Resulting in the anisotropy to be “averaged” over all possible orientations “Random Anisotropy”—Lowers K (remember that Hc ∝ K) Material Properties Typical Fe-Si alloys (widely used commercially) Hc ~ 0.08 Oe Nanocrystalline Materials (Nanoperm or Finemet) Hc ~ 0.006 Oe Magnetic Materials: Practical Examples • High K (hard) materials + Low K (soft) materials o“Nanocomposite” magnets Exchange-Spring Interactions Two-phase Mixtures of Hard and Soft Magnetic Phases • With no exchange coupling, the soft phase reverses easily in an applied magnetic field Interface HdPhHard Phase SfPhSoft Phase Exchange-Spring Interactions Two-phase Mixtures of Hard and Soft Magnetic Phases • With no exchange coupling, the soft phase reverses easily in an applied magnetic field Applied Field Interface HdPhHard Phase SfPhSoft Phase Exchange-Spring Interactions Two-phase Mixtures of Hard and Soft Magnetic
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