MSE 7025 Magnetic Materials (and Spintronics) Lecture 4: Category of Magnetism Chi-Feng Pai cfpai@ntu.edu.tw
Course Outline Time Table Week Date Lecture 1 Feb 24 Introduction 2 March 2 Magnetic units and basic E&M 3 March 9 Magnetization: From classical to quantum 4 March 16 No class (APS March Meeting, Baltimore) 5 March 23 Category of magnetism 6 March 30 From atom to atoms: Interactions I (oxides) 7 April 6 From atom to atoms: Interactions II (metals) 8 April 13 Magnetic anisotropy 9 April 20 Mid-term exam 10 April 27 Domain and domain walls
Course Outline Time Table Week Date Lecture 11 May 4 Magnetization process (SW or Kondorsky) 12 May 11 Characterization: VSM, MOKE 13 May 18 Characterization: FMR 14 May 25 Transport measurements in materials I: Hall effect 15 June 1 Transport measurements in materials II: MR 16 June 8 MRAM: TMR and spin transfer torque 17 June 15 Guest lecture (TBA) 18 June 22 Final exam
Hund s Rules So, how do we determine the ground state? For a given atom with multiple electrons, the total orbital angular momentum L and spin angular momentum S can have (2l+1)(2s+1) combinations. (j) Blundell, Magnetism in Condensed Matter (2001)
Hund s Rules Coulomb interaction Spin-orbit interaction Note: Apply strictly to atoms, loosely to localized orbitals in solids, not at all to free electrons Blundell, Magnetism in Condensed Matter (2001)
Hund s Rules So, how do we determine the ground state? Blundell, Magnetism in Condensed Matter (2001)
Hund s Rules Why Hund s rules are important? Blundell, Magnetism in Condensed Matter (2001)
Hund s Rules Why Hund s rules are important?
Hund s Rules Why Hund s rules are important? eff eff gb j( j 1) gm z z j B Blundell, Magnetism in Condensed Matter (2001)
Magnetization M and magnetic susceptibility χ Magnetic susceptibility Ferromagnetism 0 0
Magnetization M and magnetic susceptibility χ Magnetic susceptibility Hummel, Electronic Properties of Materials (2000)
Magnetization M and magnetic susceptibility χ Magnetic susceptibility (T-dependence) Diamagnetism
Category of magnetism Different behaviors of susceptibility 1 Coey, Magnetism and Magnetic Materials (2009)
Category of magnetism Diamagnetism Lenz law C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Paramagnetism Ferromagnetic material above T ORD (T c ) becomes paramagnetic C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Ferromagnetism M s J 0 Ferromagnetic material below T ORD (T c ) C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Antiferromagnetism J 0 p Antiferromagnetic material below T ORD (T N ) C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Ferrimagnetism ( Inverse spinel) M s C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Ferrimagnetism M s (tetrahedral) (octahedral) J 0 C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)
Category of magnetism Ferrimagnetism Iron garnets, R 3 Fe 2 (FeO 4 ) 3 Yttrium Iron Garnet (YIG), Y 3 Fe 2 (FeO 4 ) 3 Fruit-fly for magnetic studies Charles Kittel
Category of magnetism Ferro, antiferro, and ferrimagnetism Exchange interaction J 0 J 0 J 0
Category of magnetism Magnetic states of different elements at room temp.
Classical diamagnetism Diamagnetism Diamagnetism is present in all matter, but is often obscured by paramagnetism or ferromagnetism. Classically can be viewed as a manifestation of Lenz law. However, the classical picture is not accurate (Bohr-van Leeuwen theorem). Still, let s look at the classical (orbital) picture of diamagnetism, which is related to the Larmor precession LI dl Id I Larmor 2 e 2 2 e e 2 2 d dl m e x y B x y B 2m e 2m e 4me
Diamagnetism Classical diamagnetism 2 e 2 2 e e 2 2 d dl m e x y B x y B 2m e 2m e 4me 2 2 x y x y z r 3 3 2 2 2 2 2 2 e r dia d 6m 2 2 M H M B n B dia 0 0 v dia e n v B N / V n e v 2 2 0 6m e r
Quantum diamagnetism Diamagnetism Consider the Hamiltonian operator (total energy operator) and momentum operator Hˆ e 2 p A p i 2m e ˆ ie e H A A A 2m 2m 2m 2 2 2 2 e e e B A Consider magnetic field only along z-direction yb xb A,,0 A 0 2 2
Quantum diamagnetism Diamagnetism Consider the Hamiltonian operator (total energy operator) and momentum operator Hˆ e 2 p A p i 2m e Lz i x y y x ˆ ie B e B H x y x y 2me 2me y x 2 8me 2 2 2 2 2 2 Kinetic energy (Orbital) paramagnetism Diamagnetism e B e B Edia x y r 8m 12m 2 2 2 2 2 2 2 e e
Quantum diamagnetism Diamagnetism Consider the Hamiltonian operator (total energy operator) and momentum operator e B e B Edia x y r 8m 12m 2 2 2 2 2 2 2 e e E B dia 2 eb r 2 6m e dia n e v 2 2 0 r 2 Zeff r 6me (indep. of T)
Classical paramagnetism Paramagnetism Consider an ensemble of atoms (moments) Statistical approach E U μ B Bcos P exp E kt B (probability) cos z cos exp Bcos k T d exp B cos k T d B B dsindd (solid angle)
Classical paramagnetism Paramagnetism Consider an ensemble of atoms (moments) Statistical approach z 1 1 1 1 cos exp B cos k T sind exp B cos k T sind cos exp B cos k T d cos exp B cos k T d cos x exp exp sx dx sx dx B B B B s B kbt x cos
Classical paramagnetism Paramagnetism Consider an ensemble of atoms (moments) Statistical approach 1 1 s s s s s e e xexp sx dx s e e e e 1 z 1 s s exp sx dx 1 z coth s L( s) s (Langevin function) At low field and/or high T B s 0 L( s) s / 3 3 kt B
Classical paramagnetism Curie s law Paramagnetism 2 v 0 para M H 0nv z B 3kBT n C T Langevin function
Quantum paramagnetism Paramagnetism Discrete possible states Consider the total angular momentum quantum number j gm z j B E U μ B gm B j B M n g m n g m j j j v j B v B j m j m exp gm B / k T j j j B B exp gm B / k T j B B
Quantum paramagnetism Paramagnetism Discrete possible states Consider the total angular momentum quantum number j M n g j B ( x) M B ( x) v B j s j x g jb / k T B B s v B M n g j (Saturation magnetization) Brillouin function 2 j 1 2 j 1 1 x Bj ( x) coth x coth 2 j 2 j 2 j 2 j
Paramagnetism Quantum paramagnetism Discrete possible states Consider the total angular momentum quantum number j Brillouin function 2 j 1 2 j 1 1 x Bj ( x) coth x coth 2 j 2 j 2 j 2 j j j 1/ 2 B ( x) L( x) j Langevin function B ( ) tanh( ) j1/2 x x (spin-1/2 system)
Paramagnetism Quantum paramagnetism Discrete possible states Consider the total angular momentum quantum number j Brillouin function 2 j 1 2 j 1 1 x Bj ( x) coth x coth 2 j 2 j 2 j 2 j j1 3 Bj ( x) O x when x 1 3 j x 1 1 j x j gb jb M nv gb j nv gb 3 j 3k T B
Quantum paramagnetism Paramagnetism Discrete possible states Consider the total angular momentum quantum number j M n v 1 2 j j g B 3kT B B 2 eff para-j v 0 1 2 n j j g 3kT B B n 2 v 0 para 3kBT C T (Curie s law) eff gb j( j 1)
Quantum paramagnetism Discrete possible states Paramagnetism Consider the total angular momentum quantum number j Curie constant C for quantum scenario C T C 2 2 1 n j j g n 3k 3k v 0 B v 0 eff B B
Quantum paramagnetism Brillouin function Paramagnetism when x 1 M n g j s v B when x 1
Quantum paramagnetism Paramagnetism when x 1 when x 1 M n g j s v B
Pauli paramagnetism Free electron model Band structure (Fermi energy) Paramagnetism http://nptel.ac.in/courses/115103038/module2/lec7/
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field HE M 2000 (Remember Hw2?) applied E applied M H H H H M C M T H M applied M C C H T C T T applied c
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field HE M 2000 (Remember Hw2?) Connection to exchange interaction and total Hamiltonian Hˆ J S S g S B g S B B ij i j B i B i i, j i i B B H H H M mf 0 E 0 mf
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field M C C H T C T T applied c
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field C C T C T T c Curie Temperature T c T c C nv 3k 2 0 eff B
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field Again, borrow from theory of quantum paramagnetism M n g j B ( x) M B ( x) M tanh x M n g j s v B v B j s j s x g jb / k T g j H M / k T B B B 0 applied B j 1/ 2 1 kbt 1 kbt M x Happlied x Happlied gbj0 B0 j 1/ 2
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field Again, borrow from theory of quantum paramagnetism Find M(x) by plotting these two functions! M / Ms M / M tanh x s 1 kt B M / M s x H nvb B0 applied tanh x H applied n v B
Curie-Weiss law Ferromagnetism Weiss internal field, Weiss exchange field, Weiss molecular field Again, borrow from theory of quantum paramagnetism M M M / Ms / s
Curie-Weiss law Ferromagnetism Curie temperature of various ferromagnetic materials