Chapter 6 Antiferromagnetism and Other Magnetic Ordeer

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1 Chapter 6 Antiferromagnetism and Other Magnetic Ordeer 6.1 Mean Field Theory of Antiferromagnetism 6.2 Ferrimagnets 6.3 Frustration 6.4 Amorphous Magnets 6.5 Spin Glasses 6.6 Magnetic Model Compounds TCD February

2 1 Molecular Field Theory of Antiferromagnetism 2 equal and oppositely-directed magnetic sublattices 2 Weiss coefficients to represent inter- and intra-sublattice interactions. H Ai = n W M A + n W M B +H HBi = nwm A + n W M B +H Magnetization of each sublattice is represented by a Brillouin function, and each falls to zero at the critical temperature T N (Néel temperature) Sublattice magnetisation Sublattice magnetisation for antiferromagnet TCD February

3 Above T N The condition for the appearance of spontaneous sublattice magnetization is that these equations have a nonzero solution in zero applied field Curie Weiss C = 2C, P = C (n W + n W ) TCD February

4 The antiferromagnetic axis along which the sublattice magnetizations lie is determined by magnetocrystalline anisotropy Response below T N depends on the direction of H relative to this axis. No shape anisotropy (no demagnetizing field) TCD February

5 Spin Flop Occurs at H sf when energies of paralell and perpendicular configurations are equal: H K is the effective anisotropy field This reduces to H sf = 2(H K H i ) 1/2 for T << T N Spin Waves General: h q ~ q n M and specific heat ~ T q/n Antiferromagnet: h q ~ q M and specific heat ~ T q TCD February

6 2 Ferrimagnetism Antiferromagnet with 2 unequal sublattices YIG (Y 3 Fe 5 O 12 ) Iron occupies 2 crystallographic sites one octahedral (16a) & one tetrahedral (24d) with O Magnetite(Fe 3 O 4 ) Iron again occupies 2 crystallographic sites one tetrahedral (8a A site) & one octahedral (16d B site) 3 Weiss Coefficients to account for inter- and intra-sublattice interaction TCD February

7 Below T N, magnetisation of each sublattice is zero. Sublattice magnetisation: Above T N : Spontaneous magnetisation for nonzero solution in zero field TCD February

8 3 Frustration a b TCD February

9 3.1 FCC Antiferromagnets 4 separate Cubic lattices. In zero field TCD February

10 3.2 Helimagnets Layer structure with ferromagnetic layers coupled ferromagnetically to the neighbouring layers, but antiferromagnetically to the next-neighbour layers helical spin structure. minimised for Helicial structure if TCD February

11 4 Amorphous Magnets No crystal structure Generally alloys or compounds of the 3d or 4f elements TCD February

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13 4.1 One Network Structures Differences to regular ferromagnet are small No magnetocrystalline anisotropy field and no general easy axis of magnetisation Find local easy axes random local anisotropy insufficient to pin the magnetization in 3d alloys but may pin it in 4f alloys. Handrich Model: TCD February

14 Topological disorder frustration of the individual superexchange bonds. Spins freeze into a random, noncollinear ground state with a high degree of degeneracy, T f << p Spin correlations are negative an nearest neighbour distance, but average to zero on larger scale Speromagnetism Intermediate situation for broad exchange distribution centred at net positive value. Locally tendency towards a net magnetization, but the ferromagnetic axis wanders under the influence of local balance of exchange Asperomagnetism Ferromagnet Asperomagnet Speromagnet TCD February

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16 .4.2 Two Network Structures May be possible to distinguish two magnetic subnetworks on a chemical basis. Generally 3d & 4f atoms Strongly ferromagnetic d-d exchange (ferromagnetic 3d subnetwork) 3d - 4f interactions tend to align subnetwork spins antiparallel TCD February

17 5 Spin Glasses Dilute alloys with magnetic impurity atoms in a nonmagnetic matrix Positions of the magnetic atoms are essentially random Scaling: C/x = f(h/x,m/x);m/x = g(h/x;m/x) Below T f, a small remnant magnetization is observed Marked difference in the field-cooled and zero-field cooled response in small fields. Magnetisation measured in small applied field for ferromagnet, antiferromagnet and spin glass. Dashed line = zero-field cooled behaviour. Is there a phase transition at T f? Order parameter? Mean field approach is correct in 6-D, with 3-D bheing the lower critical dimension where the transition is marginal, and stabilized by anisotropy TCD February

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21 6.2 Critical Behaviour Reduced Temperature: = (1-T/T c ) Critical Exponents: Ising Relations: 3D n-vector model TCD February

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23 6 Magnetic Model Compounds 6.1 Heisenberg, xy and Ising models Heisenberg: 2D xy: 1D Ising: Dimensions in which phase transition is possible Onsager Solution 2D Ising TCD February