2FerromagnetismThe atomic moments in these materials exhibit very strong interactions, resulting in a parallel or antiparallel alignment of atomic moments.Exchange forces are very large, equivalent to a field on the order of 1000 Tesla, or approximately a 100 million times the strength of the earth's field.The exchange force is a quantum mechanical phenomenon due to the relative orientation of the spins of two electron.
3FerromagnetismFerromagnetic materials exhibit parallel alignment of moments resulting in large net magnetization even in the absence of a magnetic field.The elements Fe, Ni, and Co and many of their alloys are typical ferromagnetic materials.Two distinct characteristics of ferromagnetic materials are their (1) spontaneous magnetization and the existence of (2) magnetic ordering temperature
5Origin of Ferromagnetism Hund’s ruleFe: [Ar]3d64s2
6Ferromagnetism Ferromagnetic transition metals: Fe, Co, Ni i) Magnitude of Msii) Way to reach MsM (ferro) >> M (para) : 1700 emu/cm3 for Fe >> 10-3 emu/cm3Hs = 50 Oe
7Weiss' AssumptionMolecular field is acting in FM not only above Tc but also below Tc and this field is so strong that it could magnetize the substance to saturation even in the absence of an applied field. → spontaneously magnetized (Self-saturating)Magnetic domain : In demagnetized state, a ferromagnetic material is divided into a number of small regions called domains, each of which is spontaneously magnetized.
8Magnetization process a) Unmagnetized specimen for random orientation of domainsb) – c) Single domain process (motion of domain wall)d) Rotation of the domain along the fieldQuestion: Spontaneous magnetization?Division into domains?
9Magnetic Domain(a) A single-domain sample with a large stray field. (b) A sample split into two domains in order to reduce the magnetostatic energy. (c) A sample divided into four domains. The closure domains at the ends of the sample make the magnetostatic energy zero.
13Magnetic OrderAre ferromagnets already in an ordered state before a magnetic field is applied or is the order by the field?
14Explanation of magnetic order in ferromagnets Weber (1852): The material could already have small atomic magnetic moments within the solid which are randomly aligned in the demagnetized but which became ordered under the action of a magnetic field.Poisson (1983) : The atomic magnetic moments may not exist at all in the demagnetized state but could be induced by a mangetic field.
15Explanation of magnetic order in ferromagnets Ampère (1827): The origin of the atomic moments was suggested that they were due to electrical currents continually circulating within the atom.Ewing (1893): Followed Weber’s idea and interested in explaining hysteresis.
16Weiss domain theoryAtomic magnetic moments were in permanent existence (Weber’s hypothesis)Atomic magnetic moments were ordered even in the demagnetized state. It was the domains only which were randomly aligned in the demagnetized state.The magnetization process consisting of reorienting the domains so that more domains were aligned with field.
17Magnetic DomainIn order to minimize its magnetostatic energy, the magnetic material divides up into magnetic domains.Weiss (1907): concept of magnetic domains. A magnetic material consisted of a number of distinct regions termed ‘domains’ each of which was saturated in a different direction.The concept of domains is able to explain why ferromagnetic materials can be demagnetized even below their Curie temperature.
18Weiss Mean Field Theory What is the origin of the alignment of the atomic magnetic moments?It is the Weiss mean field (later the “molecular field”, further later exchange coupling from quantum mechanics)
19Curie-Weiss LawCurie's law: Individual carrier of magnetic moment (atoms ormolecules) do not interact with one anotherCurie-Weiss law: Under the consideration of interaction between electrons Fictitious internal field Hm (“molecular field”) for interaction: molecular field constant
20Molecular field theory Pierre Weiss introduced molecular field concept.Interaction between magnetic moments Fictitious internal filed: molecular field constantFor > 0, Hm || M
22Curie TemperatureEven though electronic exchange forces in ferromagnets are very large, thermal energy eventually overcomes the exchange and produces a randomizing effect.This occurs at a particular temperature called the Curie temperature (TC).Below the Curie temperature, the ferromagnet is ordered and above it, disordered.The saturation magnetization goes to zero at the Curie temperature.
23Curie temperature Saturation magnetization of Fe, Co, Ni as a function of temperature
24Exchange EnergyExchange force depends on relative orientation of spins of two electrons due to Pauli's exclusion principleWhen two atoms, such as hydrogen atoms, are coming together, there are electrostatic attractive (e-↔p+) and repulsive (e-↔e-, p+↔p+) forces and exchange force.The internal field is produced by interactions between nearest-neighbor dipole moments.The interaction arises from the electrostatic electron-electron interaction, and is called the ”exchange interaction” or exchange force.
25Exchange Energy: Heisenberg Model Si·Sj: spin angular momentumJe : a numerical quantity called exchange integralRelative orientation of two spins determines the energy states.ra/r3dBethe-Slater curve(1) If Jex is positive, Eex is a minimum when the spins are parallel, leading to ferromagnetism(2) If Jex is negative, Eex is a minimum when the spins are antiparallel, leading to antiferromagnetism.
26Band Theory of Ferromagnetism A simple extension of the band theory of paramagnetism by the introduction of an exchange coupling between the electrons.Source of magnetic moments: unpaired electronsIn partially filled energy band, an imbalance of spins leads to a net magnetic moment per atom.
27Band TheoryWhen N atoms come together to form a solid, each level of the free atom must split into N levels.In transition metal elements, the outermost electrons are the 3d and 4s; these electron clouds are the first to overlap as the atoms are brought together, and the corresponding levels are the first to split.
31Anti-ferromagnetismIf the A and B sublattice moments are exactly equal but opposite, the net moment is zero. This type of magnetic ordering is called antiferromagnetism.The clue to antiferromagnetism is the behavior of susceptibility above a critical temperature, called the Néel temperature (TN).Above TN, the susceptibility obeys the Curie-Weiss law for paramagnets but with a negative intercept indicating negative exchange interactions.
32Wess Model on Anti-ferromagnetism Two identical sublattices A and B: While the interaction with the moments on other sublattices with a negative coupling coefficient, interaction with the moments on their own sublattice with a positive coupling coefficientOn the basis of nearest-neighbor interactions, with a negative interaction between nearest neighbors, this leads to simple antiferromagnetism
33Anti-ferromagnetism Electrical Insulator (no free electron) Molecular field theoryBCC crystal (Cr)
34Anti-ferromagnetism TN : Néel temperature T < TN : AF state T > TN : paramagnetic
37FerrimagnetismIn ferrimagnets, the magnetic moments of the A and B sublattices are not equal and result in a net magnetic moment.Ferrimagnetism is therefore similar to ferromagnetism. It exhibits all the hallmarks of ferromagnetic behavior- spontaneous magnetization, Curie temperatures, hysteresis, and remanence. However, ferro- and ferrimagnets have very different magnetic ordering.
38Ferrimagnetism Two groups of ferrites depending on their structure Cubic : General formula : MOFe2O3 where M is a divalent metal ion (Mn, Ni, Fe, Co, Mg, ...) CoOFe2O3 is magnetically hard, but all the other cubic ferrites are magnetically soft. magnetite : Fe3O4 = FeOFe2O3 : oldest ferrite (lodestone, iron ferrite)2. Hexagonal : Barium ferrite (BaO6Fe2O3) is magnetically hard
39Cubic ferrites (Spinel structure) MO·Fe2O3: M = Mn, Ni, Fe, Co, Mg, etc. In the unit cell, total 56 ions (8 M2+ ions, 16 Fe3+ ions, 32 O2- ions) 64 tetrahedral A site / 8 = 8 32 octahedral B site / 2 = 16 Normal Spinel : 8 M2+ in A, 16 Fe3+ in B Inverse Spinel : 8 Fe3+ in A, 8 M Fe3+ in B Intermediate structure : Nor perfectly normal or inverse structure MnO · Fe2O3 (80% on A, 20% on B) MgO · Fe2O3 (10% on A, 90% on B) Most commercial ferrites : a mixed ferrite like (Ni, Zn)O · Fe2O3
40Hexagonal Ferrites MO·6Fe2O3(= BaFe12O19) where M = Ba, Sr Calculated saturation magnetization= 20μB/molecule (experimental) Other oxides BaO·2MO·8Fe2O3 W 2(BaO·2MO·3Fe2O3) Y 3BaO·2MO·12Fe2O3) Z where, M is a divalent ion
41Other Ferrites γ-Fe2O3 : tetragonal (calculated net moment/molecule = 2.5μB↔ 2.39μB experimental)Garnets : 3M2O3・5Fe2O3 (M = Y or RE)Alloys : Mn2Sb, Mn3Ga, Mn3Ge2, Mn3In, FeGe2, FeSe,Cr3As2, CrPt3,RECo5 (RE: Gd, Tb, Dy, Ho, Eu, or Tm)
42FeO·Fe2O3 (Iron ferrite) Crystal structureFeO·Fe2O3 (Iron ferrite)Magnetite is a well known ferrimagnetic material. Indeed, magnetite was considered a ferromagnet until Néel in the 1940's, provided the theoretical framework for understanding ferrimagnetism.Tetrahedral site:Fe ion is surrounded by four oxygensOctahedral site:Fe ion is surrounded by six oxygens
43Magnetite Fe3O4Magnetite (Fe3O4) has a very high Curie temperature (850 °C), but shows complex magnetic behavior. For this reason it seems to be a promising candidate for a high spin polarization degree near 100% even at room temperature.
45Differences with Ferromagnetism Smaller s/0 than that for FeCurie-Weiss behavior above Tc is not obeyed (Non-linear)NiO •Fe2O3 :Expected to have 12 B if ferromagneticExperiment: 2.3 B (56 emu/g) at 0 K
46Spontaneous magnetizations Spontaneous magnetizations of the A and B sublattices and the resultant s
47Kinds of Magnetism Diamagnetism Paramagnetism Non-cooperative (statistical) behaviorFerromagnetismAntiferromagnetismFerrimagnetismCooperative behavior