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 -6 Atomic density Au -2.7x10 -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<<kBT (most practical situations) slope = 1/3 2 Nm 1 1 M = µ H 10 0 10 0 10 a 10 Pierre Curie 3kBT Curie constant (1859-1906) • Which gives Curie’s Law: Curie’s Law M Nm2 C χ = = µ0 = H 3kBT T 1 Note: quantization of the magnetic χ moment requires a correction to the above classical result which assumes all directions are allowed: 0 M Ng 2µ 2 J (J +1) C T χ = = µ B = 0 2 H 3J kBT T *Curie, P., Ann. Chem. Phys., 5, 289 (1895) Pauli Paramagnetism 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: -6 Sn 0.19x10 -6 Al 1.65x10 -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<<kBT Nm2 M = µ0 (H +αM ) 3kB T 1 χ • Which gives the Curie-Weiss Law: M C C χ = = = H T −αC T −T C 0 T C T αNm2 T = µ C 0 Curie Temperature 3kB *Weiss, P., J. de Phys., 6, 661 (1907) Weiss Theory* of Ferromagnetism • Starting with the Langevin equation with “mean field” interactions M µ m(H +αM ) k T 0 M = Nmcoth 0 − B M S kBT µ0m(H +αM ) • Set the external field to zero, H=0 T 0 C µ0m(αM ) kBT M = Nmcoth − T kBT µ0m(αM ) Curie Temperature • 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 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 푁푒푒 Easy Easy Easy axis M 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 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 0 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 1 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 2 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E M 푀푒푒 θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 3 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 = 1 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 0 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 푒푒 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푉푉푠 휃 = 0 푁푒푒 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀푒푒 M θ 푁푒푒 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 푁ℎ푎푎푎 Easy Easy Easy axis M 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 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 = 0 푁ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ 푁ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 = 1 푁ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ 푁ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 = 2 푁ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ 푁ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃 = 3 푁ℎ푎푎푎 Easy Easy Easy axis θ 0 180 360 Total energy (Magnetic field along easy axis) E 푀ℎ푎푎푎 M θ 푁ℎ푎푎푎 0 180 360 Stoner-Wohlfarth Theory* Anisotropy energy Magnetostatic energy = ( ) = ( ) 2 E푎푎푎푎 푢 푚푎 ℎ푎푎푎 푎 퐸 퐾Hard푉 푠푠푠 휃 Hard 퐸 푁 푀 푉푠푠푠 휃