Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism Ferromagnetism. Molecular field theory Exchange interaction

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1 1 Lecture contents Magnetic properties Diamagnetism and paramagnetism Atomic paramagnetism Ferromagnetism Molecular field theory Exchange interaction NNSE 58 EM Lecture #1

2 [SI] M magnetization or magnetic dipole density Diamagnetic ~ Paramagnetic ~ +1-5 Ferromagnetic spontaneous magnetization, large H M H M R R H 1 Magnetic properties of materials From Cusack, 1963 NNSE 58 EM Lecture #1

3 Diamagnetism (classical) Arises from Lentz s law: when magnetic flux changes in a circuit, a current is induced which opposes the change of flux 3 Orbiting electron creates magnetic dipole (circulating current) charge I q period In magnetic field, Lorentz s force is added to centrifugal force And corresponding change of rotational frequency If change in orbital motion is small ( ) The energy associated with this frequency is ohr magneton The change in frequency can be associated with induced magnetic dipole moment: IA 1 m R F q R H E q R v q m m q H m q H m H q m T qr 4m H Am 4 H = H Lentz NNSE 58 EM Lecture #1

4 Diamagnetism (classical) contd. 4 Small magnetic field-induced magnetic dipole moment: Now we can apply the result to spherical closed-shell atom Averaging over 3D gives mean square radial distance Sum over all Z electrons in the atom Sum over all atoms in a unit volume, density N, to obtain magnetization Finally susceptibility All atoms and ions display diamagnetic response Almost independent of temperature Molar susceptibility is often used to describe magnetism of atoms (should be multiplied by molar volume to obtain dimensionless susceptibility) M H q ZN 6m R R m 3 R M qr 4m Larmor or Langevin diamagnetic susceptibility H x y R 1 x y z R 3 q ZN 6m H R Molar susceptibilities of some atoms and ions (x1-6 cm 3 /mole) From urns, 199 NNSE 58 EM Lecture #1

5 Contrary to diamagnetism, paramagnetism arises from non-zero magnetic moments: Free electron (Pauli) spin paramagnetism Langevin atomic paramagnetism Paramagnetism An electron has an intrinsic magnetic dipole moment associated with its spin S, equal to ohr magneton: We can expect that the magnetic dipoles will rotate towards low-energy state ( U from to ) The fraction of electrons with magnetic moments parallel to magnetic field exceeds the anti-parallel fraction by H For n free electrons, the magnetization ut we need to take band structure into account! M g s =.3 q S m q m T n H 5 Am 4 For =1 T (H = 8x1 5 A/m ) U 58 ev.67 K Field alignment is weak! NNSE 58 EM Lecture #1

6 Paramagnetism of free spins H M n Magnetization is ~1 times higher than observed in real materials 6 In a band only a thermal fraction of electrons contributes to paramagnetism (compare to transport) E F Energy vs. density of states In a magnetic field before the spins reorient In equilibrium Magnetization is M n H F Similar to transport, more accurate averaging over the distribution function gives susceptibility 3 n F For example, for Na From urns, 199 NNSE 58 EM Lecture #1

7 Langevin atomic paramagnetism 7 Similar to free spins, if an atom has a magnetic moment, it can align along the magnetic field Magnetization of a material with atomic density N is (averaging included) 1 H M N 3 And susceptibility Atom with orbital, spin and total angular momenta, L,S, and = L+S, will have magnetic moment 3 N C T g L S Curie law for paramagnetics With Curie constant C N 3k With Lindé g-factor Complications g 1 S L =.3 Quantum mechanical averaging of m Ions Quenching of orbital momentum in the crystal field (Stark splitting of L+1 degeneracy ) NNSE 58 EM Lecture #1

8 Quantum mechanical averaging over (+1) projections With rillouin function: Atomic paramagnetism - Quantum theory g m g me M N Ng g m y N y e y ( y) coth y coth M sat Magnetic moment vs. H/T 8 with y k T If magnetic energy is small compared to thermal energy, y << 1, rillouin function gives ( y 1) y g k T 1 3 This results in classic susceptibility 3 N with quantum averaged g ( 1) NNSE 58 EM Lecture #1

9 Ground states of ions predicted by Hund s rules Magnetic moments of ions S 1 L Values of magnetic moments of 4f and 3d ions in insulating compounds 9 From urns, 199 NNSE 58 EM Lecture #1

10 Ferromagnetism Molecular field theory 1 Spontaneous magnetization occurs in some (Ferromagnetic) materials composed of atoms with unfilled shells For some reason magnetic moments are aligned even at relatively high temperature Values of Curie temperatures and spontaneous magnetism (at K in Gauss) for a few ferromagnetic materials Hypothesis: magnetic order is due to strong local magnetic field (Weiss ective field) at the site of each dipole with a constant M Consider a collection of N identical atoms per unit volume, with total angular momentum, and use QM treatment of atomic paramagnetism loc M M y Msat N sat a with y M g M loc a a k T k T k T From urns, 199 NNSE 58 EM Lecture #1

11 Now let s find the spontaneous magnetization ( a = ) M M y sat Solving equation against y : Ferromagnetism Molecular field theory Solution of equation with rillouin function with y y y M sat M 11 Depending on temperature spontaneous magnetization can be either M= or finite At low y 1 ( y 1) y 3 We can find critical Curie temperature: C M sat 1 3 N T g C 1 3k From urns, 199 NNSE 58 EM Lecture #1

12 Ferromagnetism Molecular field theory 1 At temperatures T > T c, there is no spontaneous magnetization, and we can find temperature dependence of magnetizatiob ( a > ) M M y Solving for M: with M Then susceptibility of ferromagnet in paramagnetic region sat Some estimation for iron: y g M a C a T C with Curie constant C T 8 3 g ; 1; N m C 1.77 K T C 1 N M M sat y a M 3 3k T C N 3k Reciprocal susceptibility vs. temperature for nickel T C TC 143 K; M 17 Gauss; 1 Gauss=1 T Huge! From urns, 199 NNSE 58 EM Lecture #1

13 Ferromagnetism Heisenberg exchange interaction 13 What is the reason for so high local magnetic field? Exchange interaction Consider two electrons on two atoms. We need to find the energy difference = exchange integral ex ) : Singlet H s s ex ex 1 Triplet 3 s1 s s 1 s Their wavefunction is antisymmetric due to Pauli exclusion principle Usually the interaction between the space and spin parts is small, and the variables can be separated : Antiparallel spins give antisymmetric spin wavefunction, etc. We can construct wavefunctions for singlet and triplet states with correct symmetry: r, s, r, s r, s, r, s space spin r, s, r, s r, r g s, s Wavefunction Singlet Triplet Total Antisym. Antysym. Spin part Antisym. Symmetric Space part Symmetric Antisym. x, x r r r r S 1 a 1 b a b 1 1 x, x r r r r T 1 a 1 b a b 1 1 NNSE 58 EM Lecture #1

14 Ferromagnetism Heisenberg exchange interaction-contd 14 The energy shift of singlet and triplet states can be calculated from perturbation theory: E V V V V V S S S T T T E V V V V V V r, r The energy difference between the singlet and triplet states E E 4 V 4 r r V r r 1 e r r r r ab a b1 1 S T exc 1 a 1 b a b 1 e r r r r 4 r r r r 4 r r r r a 1 b a b 1 a 1 b a b 1 a 1 b a b 1 ra rab r1 At small a-b distance At large a-b distance exc exc and singlet state is favorable and triplet state is favorable TW if the electrons are on the same atom, the ion interaction change is zero, exc and antiparallel spins are favorable = Hund s rule NNSE 58 EM Lecture #1

15 Ferromagnetism Heisenberg exchange interaction contd. Exchange interaction can be ferromagnetic or antiferromagnetic depending on interatomic distance Exchange interaction is electrostatic (strong) in nature To correlate it with molecular field theory, we can write: H s s g s ex ex i j i loc i, j i z s g ex loc Number of nearest neighbors Exchange integral vs. interatomic distance r d average radius of 4d electron 15 From Christman, 1988 For Fe: ex g zs M g n 11 z mev Electrostatic interaction easily accounts for this value NNSE 58 EM Lecture #1

16 Energy is minimized by ordering spins into domains Net moment, M, would cause external field, increase energy Magnetic domains cancel so that M = Natural ferromagnetism does not produce net magnetic field To magnetize a ferromagnet, impose H Domain walls move to align M and H Defects impede domain wall motion Magnetization (M r ) retained when H removed Magnetic properties Ferromagnetic materials M s = saturation magnetization (All spins aligned with field) M r = remanent magnetization (Useful moment of permanent magnet) H c = coercive force (Field required to erase moment) Area inside curve = magnetic hysteresis (Governs energy lost in magnetic cycle) Ferromagnetic material is always locally saturated Hysteresis loop of a ferromagnetic material 16 NNSE 58 EM Lecture #1

17 Core magnetism materials with spontaneously ordered magnetic dipoles 17 High temperature: - Spins disordered paramagnetism Low Temperature (T < Tc) - Spins align = ferromagnetism Elements: Fe, Ni, Co, Gd, Dy Alloys and compounds: AlNiCo, FeCrCo, SmCo 5, Fe 14 Nd - Like spins alternate = antiferromagnetism (RbMnF 3 ) - Unlike spins alternate = ferrimagnetism Compounds: Fe 3 O 4 (lodestone, magnetite), CrO 3, SrFe O 3, other ferrites and garnets NNSE 58 EM Lecture #1

18 Other types of exchange interaction 18 Superexchange (transition metal oxides) Can be ferromagnetic or anfiferromagnetic depending upon the energy of delocalization of the p-electrons on M 1 and M Ordering temperature up to 9 K in ferrites (NiFe O K) Sign mostly negative, though ferromagnetics are known: EuO (T c =69K) or Crr 3 (T c =37K) RKKY interaction (Ruderman-Kittel-Kasuya- Yosida) - Indirect exchange over relatively large distances trough spin of conduction electrons (4f metals) Interaction oscillates with (k F R), Fermi wavevector determines the wavelength of oscillations The interaction is of the same order for all rare earths, but ordering temperatures vary due to magnetic moment: 19K for Nd, 89 K for Gd) k F R Mn O - Mn RKKY interaction NNSE 58 EM Lecture #1

19 Magnetic materils 19 Magnetic induction field is the same in a From Goldberg, 6 NNSE 58 EM Lecture #1

20 Magnetic units V s m 1 Wb T m 1 T Wb 1H A T m A From Tremolet de Lacheisserie, 5 NNSE 58 EM Lecture #1

21 Physical constants 1 From Tremolet de Lacheisserie, 5 NNSE 58 EM Lecture #1