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Curie-Weiss Law – Weiss Internal Field, Weiss Exchange Field, Weiss Molecular Field

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Curie-Weiss Law – Weiss Internal Field, Weiss Exchange Field, Weiss Molecular Field MSE 7025 Magnetic Materials (and Spintronics) Lecture 4: Category of Magnetism Chi-Feng Pai [email protected] 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 g B j( j 1) z zgm 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 TORD (Tc) becomes paramagnetic C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism • Ferromagnetism M s J 0 Ferromagnetic material below TORD (Tc) C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism • Antiferromagnetism J 0 p Antiferromagnetic material below TORD (TN) 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 (tetrahedral) (octahedral) M s J 0 C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism • Ferrimagnetism – Iron garnets, R3Fe2(FeO4)3 – Yttrium Iron Garnet (YIG), Y3Fe2(FeO4)3 – “Fruit-fly for magnetic studies” Charles Kittel Category of magnetism • Ferro, antiferro, and ferrimagnetism Exchange interaction J 0 J 0 Category of magnetism • Magnetic states of different elements at room temp. Diamagnetism • Classical 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 ddLII Larmor e e e2 d dL m x2 y 2 B x 2 y 2 B e 2me 2 m e 4 m e Diamagnetism • Classical diamagnetism 22 x2 y 2 x 2 y 2 z 2 r 2 33 er22 dia dB 6me 22 nv e0 r e M H M B n e B e2 d dL dia m x200 y 2 v dia B 6m x 2 y 2 B e e 2me 2 m e 4 m e nv N/ V Diamagnetism • Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator… pA e 2 Hˆ p i 2me 22ie e HAAAˆ 22 2me 2 m e 2 m e BA Consider magnetic field only along z-direction yB xB A , ,0 A 0 22 Diamagnetism • Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator… pA e 2 Lz i x y Hˆ p i yx 2me 2 2 2 ˆ 2ie B e B 2 2 H x y x y 2me 2 m e y x 2 8 m e Kinetic energy (Orbital) paramagnetism Diamagnetism 2 2 2 2 e B2 2 e B 2 Edia x y r 8mmee 12 Diamagnetism • Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator… e2 B 2 e 2 B 2 E x2 y 2 r 2 dia 8mmee 12 E eB2 dia r 2 Bm6 e 22 nv e0 r 2 dia Zreff (indep. of T) 6me Paramagnetism • Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach E EUB μ B cos P exp(probability) kTB cos exp B cos k T d B z cos expB cos k T d B dsin d d (solid angle) Paramagnetism • Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach cos exp B cos k T sin d B z expB cos k T sin d B cos exp B cos k T d cos B expB cos k T d cos B 1 xexp sx dx 1 1 s B kB T expsx dx 1 x cos Paramagnetism • Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach 1 xexp sx dx s es e s e s e s 1 z 1 ss expsx dx s e e 1 1 z coths L ( s ) (Langevin function) s At low field and/or high T B s 0 L( s ) s / 3 3kTB Paramagnetism • Classical paramagnetism – Curie’s law 2 nv0 C para M H 0 n v z B 3kB T T Langevin function Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j z gm j B E U μ B gmjB B j mjexp gm j B B / k B T mjj M nv g m j B n v g B j expgmj B B / k B T mjj Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j M nv g B j B j()() x M s B j x (Saturation x gBB jB/ k T Ms n v g B j magnetization) Brillouin function 2j 1 2 j 1 1 x Bj ( x ) coth x coth 2j 2 j 2 j 2 j Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j Brillouin function j j 1/ 2 2j 1 2 j 1 1 x B( x ) coth x coth B()() x Lj x Bj 1/2 ( x ) tanh( x ) j 2j 2 j 2 j 2 j Langevin function (spin-1/2 system) Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j Brillouin function jx1 3 Bj () x O x when x 1 3 j 2j 1 2 j 1 1 x Bj ( x ) coth x coth 2j j11 x 2 j 2 j j g 2B j jB M nv g B j n v g B 33j kB T Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j 2 j j1 gB B Mn v 2 3kTB eff 2 2 nvB0 j j1 g nv0 C para para-J 3k T T 3kTB B (Curie’s law) eff gB j( j 1) Paramagnetism • Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j – Curie constant C for quantum scenario C T 2 n j j1 g n 2 C vB0 v 0 eff 33kkBB Paramagnetism • Quantum paramagnetism – Brillouin function when x 1 Ms n v g B j when x 1 Paramagnetism • Quantum paramagnetism when x 1 Ms n v g B j when x 1 Paramagnetism • Pauli paramagnetism – Free electron model – Band structure (Fermi energy) http://nptel.ac.in/courses/115103038/module2/lec7/ Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field 2000 HME (Remember Hw2?) MHHHHM applied E applied CM THMapplied MCC HTCTTapplied c Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field 2000 HME (Remember Hw2?) – Connection to “exchange interaction” and total Hamiltonian ˆ H JijSSSBSBB i j g B i g B i mf i, j i i BBHHHMmf 0 E 0 Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field MCC HTCTTapplied c Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field CC TCTT c • Curie Temperature Tc 2 nv 0 eff TCc 3kB Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism MngjBxMBxMv B j( ) s j ( ) s tanh x Ms n v g B j xgjBkTgjBBBB// 0 H applied MkT 11kBB T k T M x Happlied x H applied gj BB00j 1/ 2 Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism – Find M(x) by plotting these two functions! MM/ s tanhx M/ Ms tanh x 1 kTB M/ Ms x Happlied nv B B 0 Happlied nvB Ferromagnetism • Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism MM/ s Ferromagnetism • Curie-Weiss law – Curie temperature of various ferromagnetic materials .
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