Magnetism and Magnetic Materials K. Inomata

Transcription

1 Magnetism and Magnetic Materials K. Inomata 1. Origin of magnetism 1.1 Magnetism of free atoms and ions 1.2 Magnetism for localized electrons 1.3 Itinerant electron magnetism 2. Magnetic properties of ferromagnets 2.1 Magnetic anisotropy 2.2 Magnetoelastic effects 2.3 Magnetic domain walls and domains 2.4 Magnetization process 3. Ferromagnetic materials 3.1 Soft magnetic materials 3.2 Hard magnetic materials 3.3 Magnetic recording materials 4. Magnetic and tranport properties on advanced magnetic materials 4.1 Magnetism in small structures 4.2 Surface and thin-film magnetism 4.3 Electron transport in ferromagnets

2 1. 1. Magnetism of free atoms and ions

3 Three forces in natural world: Universal gravitation, Nuclear force, Magnetic force Magnetic force is one of the oldest physical phenomena that human knows. The origin of the word Magnet History of magnetism Magnetic ores (Magnetite, Fe 3 O 4 ) were found in Magnesia, the name of a region of the ancient Middle East, in what in now Turkey. The Magnet was also found in China in BC 7. Magnet =, In Japanese,. = loving stone, love, treat tenderly, Attraction or friendship, similar to magnet History on magnetism 1. Discovery of functionality of the magnet Loadstone compass : China (BC 3) 2. Discovery of electromagnetic field Oersted (Denmark, 1820): Current gives a force on a loadstone compass Establishment of Electromagnetism Maxwell s equations 3. Relation between the substance and magnetism Ampere : Magnetism is based on the special molecular field (1821) Langivin : Angular momentum by electron orbital motion (1905) Dirac : Discovery of spin angular momentum (1928) Heisenberg: The origin of ferromagnetism is in the exchange interaction (1933)

4 Origin of Atomic Moment What is the microscopic origin of magnetism in materials? Magnetic moment of a free electron Consider an electron which rounds an atomic nucleus with a radius r and an angular velocity ω : Magnetic moment by orbital motion, μ l is defined as -e μ l = μ 0 A i ( 1.1 ) A = π r 2 μ 0 = H/m : Permeability in vacuum r θ ω i = -e = -e ω / 2 : Current ( 1.2 ) A = π r 2 By inserting A π r 2 and i = - ω/2 to (1.1), we obtain μ l μ 0 π r 2 -e ω 2 - eμ 0 ωr /2 ( 1.3 ) While, angular momentum of an electron is given by P l m r ω ( 1.4 ) Thus, magnetic moment by an electron orbital motion μ - (μ 0 e /2 m) P l ( 1.5 )

5 According to the quantum mechanics electron orbital motion around an atomic nucleus is quantized. So, orbital angular momentum is given by using the orbital angular momentum number l P l h l ( 1.6 ) h = h/2π J s Thus, magnetic moment due to the orbital motion of an electron is μ = μ 0 (-e h / 2 m l = - μ B ( 1.7 ) μ B μ 0 e h / 2 m = [A m 2 ] [Wb m] : Bohr magneton On the other hand, magnetic moment by electron spin is given using spin angular momentum number s from the Dirack equation s = - (μ 0 e/m) P s - (μ 0 e/m) h s ( 1.8 ) = - 2 μ B s where P s = h s, s = 1/2. Thus, the total magnetic moment by an electron is μ = μ μ = - ( l 2 s) μ B = - g μ B ( 1.9 ) = l + s : total angular momentum g: g factor g = 2 for l = 0

6 Magnetic moment of an atom with multi-electrons 1) Pauli principle Only two electrons can occupy one energy state including spin 2) Hund rule : Rule for electron occupancy Total S is maximum Total L is maximum Ex) 3d 6 : Fe 2+ ion n = 3 Total L = 2, S = 2 l = 2 L = Σ m l = 0 m l = Principal quantum number n Orbital angular momentum quantum number l Magnetic quantum number m l Spin quantum number m s l = 0, 1, 2..n-1 m l = -1, - l + 1, - l +2,, l m s = +1/2, - 1/2

7 3) Russel-Sanders coupling How do the spin and orbital angular momenta combine to form the total angular momentum of an atom? Spin-orbit interaction: ls j j L S S = s j L = j ( 1.10 ) If spin-spin and orbit-orbit couplings are strongest and spin-orbit interaction is not so strong, l -e +Ze L H s is antiparallel to l s J = L + S = j j : Russel-Sanders coupling J = (L + S), (L + S 1), ( L + S 2),.. L S > 0 : L is antiparallel to S for less than half electrons < 0 : L is parallel to S for more than half electrons Magnetic moment μ μ = - μ B L 2 S - g μ B J S J g = [3J (J +1) +S (S+1) L (L+1) / [2J (J+1)] Lande s g factor ( 1.11 ) Effective magnetic moment μ eff = g μ B JJ+ ( 1) ( 1.12 )

8 Practice problems Nd 4f 5s 5p m l L = 6, S = 3/2 J = L S = 9/2 g = 8/11 μ eff = 3.62 μ Β 2 Tb 3+ 4f 9 5s 2 5p 6 m l L = 5, S = 5/2 J = L + S = 15/2 g = 4/3 μ eff = μ Β

9 Effective magnetic moment: theory and experiment μ eff = g μ B JJ+ ( 1) 5s, 5p μ eff 3 + ion Metals 4f Rare earth metals follow the Russel- Sanders coupling Modified calculation J is a good quntum number. Hund rule Ferromagnetic 3d metals Fe Co Ni Number of 4f electrons

10 Quenching of the orbital angular momentum In 3d series of elements J = S has been found experimentally as shown in the table. Electrons in such as 3d series cannot draw a definite orbit due to the interactions between electrons. This leads to the negligible orbital contribution to the magnetic moment. L = 0 L = 0 Isolated atom Solid Thus, the magnetic moment in 3d magnetic materials is mainly caused by electron spin.

11 j-j coupling There is other way of coupling to form the total angular momentum of an atom. j-j coupling The orbital and spin angular momenta are dependent on each other. It assumes there is a strong spin - orbit ( l - s ) coupling for each electron and that this coupling is stronger than the l - l or s - s coupling between electrons. J = Σ j i = Σ ( l i + s i ) ( 1.13 ) This form of coupling is more applicable to very heavy atoms.

12 2. Magnetism in materials The various different types of magnetic materials are traditionally classified according to their bulk susceptibility. 2.1 Diamagnetism Closed shell of electrons: S = 0 Weak magnetism Negative magnetic susceptivility : = I/H ~ Lenz law Changing magnetic field H induces an electric field whose current generates a magnetic field that opposes the change in the first H field. This field induces the magnetization I Cu Ag Al 2 O 3 C (graphite) H H -e Electric field is induced = - 1 : Perfect diamagnetism = Superconductor r E = - dφ/dt = AdB/dt = - μ 0 AH/dt Φ: magnetic flux B: Flux density A = πr 2 Current i = E / R induces opposite magnetic field

13 2.2 Paramagnetim: Localized model is small and positive, typically ~ μ Atomic magnetic moment μ = - (L + 2S ) μ B No interaction between atomic moments Zeeman energy: E = - μ H = - μ B (L + 2S ) H = - g μ B J H ( 2.1 ) H Random orientation of moment For H // z, E = g μ B J z H = g μ B M J H = E M ( 2.2 ) where M J is the eigen value of J z Probability for occupying energy E M is determined by Boltzmann statictics P(M J ) exp (-E M /k B T) Σ exp(-e M /k B T) Magnetization ( Magnetic moment μ per unit cell ) ( 2.3 ) J z M J = -J H = 0 M J = +J I = Nμ =Ngμ B Σ P(M J )M J = Ngμ B JB J (x) ( 2.4 ) B J (x) = [(2J + 1)/2J] coth [(2J + 1 ) /2J] x (1/2J ) coth (x/2j) : Brillouin function x = gjμ B H/k B T ( 2.5 )

14 High temperature limit B J (x) = [(J + 1)/3J] x lim x 0 I = Ng 2 μ B2 J(J + 1)/3 k B T= [ Nμ eff2 /3k B T] H 1/ ( 2.6 ) Eq.( 2.7 ) High field limit = I/H = C/T : Curie law C = Nμ eff2 /3k B ( 2.7 ) Slope = 1/C T B J (x) = 1 ( H ) lim x I I s =Ngμ B J, I s = N μ s μ s = gμ B J N: Number of atoms per unit volume Saturation ( 2.8 ) Brillouin function Slope = C H / T I s = N μ s μ s = gμ B J H

15 Experiments obey The Brillouin function

16 2.3 Ferromagnetism: Localized model H Weiss theory Molecular field H w = w I Total magnetic field H T = H + w I E = - gμ B M J ( H + w I ) ( 2.8 ) H w I = Nμ =Ngμ B Σ P(M J )M J = Ngμ B JB J (x) = I 0 B J (x) I 0 = Ngμ B J x = gjμ B (H + w I ) / k B T When H = 0 I / I 0 = ktx/ N(gμ B J) 2 w I/I 0 =B J (x) ( 2.9 ) ( 2.10 ) ( 2.10 ) ( 2.9 ) I/I 0 T > Θ in ( 2.10 ) ( 2.9 ) T < Θ in ( 2.10 ) High temperature limit; x << 1 B J (x) = (J + 1 ) x / 3 J k B T/ N(gμ B J) 2 w = (J + 1 )/3 J, T = N(gμ B ) 2 J ( J + 1 )w / 3k B = C w = Θ Θ: Paramagnetic Curie temperature Θ = Nμ eff2 w/3k B ( 2.12 ) C = Nμ 2 eff /3k B, x ( 2.11 )

17 Temperature dependence of spontaneous magnetization I/I 0 T/Θ

18 T > Curie temperature ( ) Θ For x << 1, B J (x) = ( J + 1 ) x /3 J ( 2.9 ) I = Ngμ B J ( J + 1 ) x / 3 = ( C/T ) ( H + w I ) I = C H / ( T-Cw ) = I / H = C / ( T- Θ ), Θ = C w Curie-Weiss law I/I 0 1 / Θ T Θ Molecular field approximation

19 Estimation of w Ex) Fe Θ= 1063 K, μ = 2.2 μ B, N = /m 3 J = 1 w = 3kΘJ/(J + 1 ) N μ 2 = [A m / Wb] Molecular field H m = w I = wnμ = [ A/m ] = Oe = 11 [MOe] This is quite large. The w is based on the quantum mechanical exchange interaction -JS 1 S 2 J is exchange integral

20 Summary: Types of Magnetism Diamagnetism S = 0, Orbital magnetic moment only Paramagnetism S = 0 and no atomic interaction Ferromagnetism S = 0 and parallel coupling -J S 1 S 2, J > 0 J:Exchange Integral Antiferromagnetism S = 0 and antiparallel coupling -J S 1 S 2, J < 0 Ferrimagnetism S = 0, antiparallel coupling and different atomic moments -J S 1 S 2, J < 0 S 1 = S 2