(3) Ferromagnetism (additional)

Seoul National UniversitySuperconductors and Magnetic Materials Lab. Seoul National UniversitySuperconductors and Magnetic Materials Lab.

(3) Ferromagnetism (additional)

▶ The Slater-Pauling curve (see Fig. 16 in Chap. 21 of Kittel)

Magnetic moments per atom of 3d metals and their alloys (from Mn to Cu) from the premises of the itinerant electron theory. - Interpretation: in terms of the rigid band model. μtot = [5 - (n - x - 5)]μB = [10 - (n - x)]μB- Max. moments : at a point between Fe and Co (n ~ 8.3). - 3d and 4s electrons are responsible for the magnetic properties of these metals and alloys.

Observations

- When n > 8.3, good agreement between theory and experimental data for alloys of adjacent elementsIf 3d band is just filled, μtot = 0 (e.g., Ni-60 at% Cu alloy, see Fig. 12 & 13 in Chap. 21 of Kittel)However, For n > 8.3, marked disagreement between theory and experimental data for alloys of nonadjacent elements

cf) saturation magnetization/atom = μtote.g., n = 9.5 for Co - 50 at% Ni , (μtot)theory ~ (μtot)exp ↔ n = 9.5 for Mn - 83 at% Ni , (μtot)theory < (μtot)exp

- For n < 8.3, no agreement between theory and experimental data: (μtot)exp decreases as n decreases.

- Ni-based alloys with non-transition elements (e.g., Cu, Zn, Al, Si, etc.) (see Fig. 15 in Chap. 21 of Kittel)Magnetization decreases at a rate proportional to the valence of the solute. The solute atom seems to contribute all its valence electrons to the 3d band of the alloy : the larger the valence, the more rapidly does the band fill up, and hence the more rapidis the decrease in magnetization. → good agreement with the band theory

- Fe- and Co-rich alloys : The solute atom appears to act as a simple diluent; the magnetization decreases as though the magnetic atoms were being replaced by atoms of zero moment → inexplicable in terms of the band theory



The Slater-Pauling curve

Transition elements

Sc : 4S23d1

Ti : 4S23d2

V : 4S23d3

Cr : 4S13d5

Mn : 4S23d5

Fe : 4S23d6

Co : 4S23d7

Ni : 4S23d8

Cu : 4S13d10

Zn : 4S23d10


Fig. 15 in Chap. 21 of KittelSaturation magnetization of nikel alloys in Bohr magnetons per atom as a function of

the atomic percent of solute element




▶ Superexchange

(see Fig. 4.5 in O’Handley)

The superexchange interaction is a consequence of the different symmetry of p and d states.

-One of the three p orbitals represents two electron states (spin up and down) whose spatial wave functions have 360°(single fold) rotational symmetry.

- One of the five d orbitals represents two sates that have 180° rotational symmetry.

- Thus a p state has spin up and down electrons concentrated in lobes 180° apart while a d state has opposite spin concentrations show every 90° rotation about the iron.

-The above symmetry considerations, plus the fact that electron hopping conserves spin orientation, produce the antiferromagnetic spin coupling between nearest neighbor transition metal ions.

- Applications : transition metal monooxides of the rocksalt structure, complex spinel, garnet, perovskiteoxides



▶ Indirect ExchangeThe conduction electrons mediate the exchange interaction.

The coupling of nuclear spins to s-electron spins by a contact interaction. :

RKKY (Ruderman, Kittel, Kasuya, Yoshida) model

Kondo effect : a consequence of the magnetic ion-conduction electron interaction

(see Fig. 18 in Chap. 21 of Kittel)



▶ Magnon : a quantized spin wave.(see Fig. 8 and Fig. 9 in Chap. 15 of Kittel and Fig. 3.21 in O’Handley)

- a good explanation for magnetization at low temperature : Bloch T3/2 law



▶ Magnon : a quantized spin wave.(see Fig. 8 and Fig. 9 in Chap. 15 of Kittel

and Fig. 3.21 in O’Handley)

-a good explanation for magnetization

at low temperature : Bloch T3/2 law

Fig. 3.21 in O’Handley


▶ Characteristics-Small positive susceptibility at all temperatures - Antiparallel ordering of spins - Zero net moment (no spontaneous moment) below the ordering temperature, called Neel temperature TN

▶ The spin arrangementConfirmed by neutron diffraction and magnetic measurements In case of MnO : The spins in a single (111) plane are parallel, but those in adjacent (111) planes are antiparallel. (see Fig. 21 in Chap. 15 of Kittel)

▶ Interpretation on the basis of the Weiss model

Two ways(i) 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 sublatticewith a positive coupling coefficient

(ii) On the basis of nearest-neighbor interactions, with a negative interaction between nearest neighbors, this leads to simple antiferromagnetism

(4) Antiferromagnetism



▶ The susceptibility in the paramagnetic region T > TN

Mean(or molecular) field approximation for two identical sublattices A and B HeA = - γMB and HeB = - γMA where γ is mean field coefficient

Assuming Curie-law χ = M/ρH = C/T or MT = ρCH

For each sublattice: MAT = ρC(H - γMB) and MBT = ρC(H - γMA)

Then (MA + MB)T = 2ρCH - ρCγ(MA+ MB) MT = 2ρCH - ρCγM

M(T + ρCγ) = 2ρCH

χ = M/ρH =

χ = Curie-Weiss law in AF

where θ = ρCγ = TN (He is opposite to the direction of H in paramagnetic region)

γρCTC

+2

)(22

θθ −−=

+ TC

TC


▶The susceptibility in the antiferromagnetic region T < TNWhen H = 0,

M = MA + MB = 0 and MA = - MBAt a temperature infinitesimally below TN, we may assume M is still proportional to the total field because saturation effects are unimportant near TN. At T = TN and H = 0, MATN = ρC(H - γMB) = -ρCγMBThus ρCγ = θ = - (MA/MB)TN = TN

For any temperature and field Where HeA = - γMB = γMA

Therefore, the fractional spontaneous magnetization of the A sublattice is given by

Similarly, (where, m = gJμB)

Although the net magnetization is zero below TN, H can produce a small positive M. In most antiferromagnets, the orbital contribution is almost entirely quenched. The spin axis usually coincides with an important crystallographic direction.


)(xBMM

JOA

A =kT

HHmx eAo )( +=μ

)()(kT

MmB

kTmHB

MM SAo

JeAo

JOA

SA γμμ==

)()(kT

MmB

kTmHB

MM SBo

JeBo

JOB

SB γμμ==


(4) Ferrimagnetism

▶CharacteristicsSimilarity to ferromagnetics Difference from ferromagnetics-A substantial spontaneous magnetization Ms - Imperfect antiferromagnetic arrangement -Self-saturated domains of the atomic moment - Magnetic saturation and hysteresis - Temperature dependence of Ms-T > Tc, paramagnetic - A marked deviation from the Curie-Weiss law

in the paramagnetic region

▶Structure of Ferrites (ref. J. Smit and H. P. J. Wijn, Ferrites, New York, Wiley, 1959)

Cubic ferrites (Spinel structure) :MO·Fe2O3, where 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 BInverse Spinel : 8 Fe3+ in A, 8 M2+ + 8 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


(4) Ferrimagnetism

Structure of a cubic ferrite (Fig. 6.2 in Cullity)


(4) Ferrimagnetism

▶Structure of Ferrites (ref. J. Smit and H. P. J. Wijn, Ferrites, New York, Wiley, 1959)

Hexagonal 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

Other ferrimagnetsγ-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)


(4) Ferrimagnetism

▶Saturation MagnetizationCalculation of the saturation magnetization of a ferrite at 0 K By considering the spin-only contribution of magnetic ion, the distribution of ions, and negative exchange interaction between A and B sites.

Ferrite Mn Fe Co Ni Cu Zn Calculated 5 4 3 2 1 0 Measured 4.6 4.1 3.7 2.3 1.3 0

(in μB per molecule)

- The discrepancies are generally ascribed to incomplete quenching of the orbital moments and/or incomplete inverse structure. - Ionic distribution and net moment per molecule of typical ferrites (see Table 6.3 in Cullity). - Zn ferrite (ZnO·Fe2O3) : Zn2+ ions of zero moment in A sites, no AB exchange interaction and thus a negative BBinteraction, leading to on net moment. - Mixed ferrites : the addition of the nonmagnetic Zn2+ ion increases the Ms. (see Fig. 4.7 in O'Handley)


▶Saturation Magnetization

Table 6.3 in Cullity Fig. 4.7 in O'Handley

(4) Ferrimagnetism


(4) Ferrimagnetism

▶ Exchange Interactions- Exchange interactions between 3d cations in ionic solids: antiferromagnetic superexchange coupling. - The exchange is mediated by an intervening anion p orbital. When the three ions are collinear, the interaction is the strongest. - In the spinel structure, A-O-B bond angle = 125°, B-O-B bond angle = 90°, no A-O-A bond: the strongest superexchange interaction is the 125° A-B coupling.

▶ Site PreferenceFactors affecting site selection: ionic valence, cation size, crystal electric field(CEF) (see Fig. 4.8 in O'Handley) - Magnetite(Fe3O4 = FeO·Fe2O3) : Fe3+ has a much stronger A site preference than Fe2+→ inverse spinel- Divalent transition metal substitutions for Fe

Fe3O4 = Fe2+O2-·(Fe3+)2(O2-)3→ T2+O2-(Fe3+)2(O2-)3For T = Cr, Fe, Co, Ni, Cu ; Fe3+in A, T2+ & Fe3+ in BFor ZnO·Fe2O3 ; Zn2+ in A & Fe3+ in BFor (T2+)1-x(Zn2+)x(Fe3+)2O4 ; (Zn2+)x(Fe)1-x in A, (T2+)1-x(Fe3+)1+x in BSaturation magnetization (net moment)/molecule(or formula unit: FU) = (10x)μB + μT(1-x)


▶Temperature Dependence of Magnetization in Ferrites(see Fig. 4.9, Fig. 4.11 in O’Handley)

(4) Ferrimagnetism

(3) Ferromagnetism (additional)

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