Chapter 6 Antiferromagnetism and Other Magnetic Ordeer

Home , Antiferromagnetism, Ferrimagnetism

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 2007 1 1 Molecular Field Theory of Antiferromagnetism 2 equal and oppositely-directed magnetic sublattices 2 Weiss coefficients to represent inter- and intra-sublattice interactions.

HAi = n’WMA + nWMB +H

HBi = nWMA + n’WMB +H Magnetization of each sublattice is represented by a Brillouin function, and each falls to zero at the critical temperature TN (Néel temperature)

Sublattice magnetisation

Sublattice magnetisation for antiferromagnet

TCD February 2007 2 Above TN

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 + nW)

TCD February 2007 3 The antiferromagnetic axis along which the sublattice magnetizations lie is determined by magnetocrystalline anisotropy

Response below TN depends on the direction of H relative to this axis. No shape anisotropy (no demagnetizing field)

TCD February 2007 4 Spin Flop

Occurs at Hsf when energies of paralell and perpendicular configurations are equal:

HK is the effective anisotropy ﬁeld

i 1/2 This reduces to Hsf = 2(HKH ) for T << TN Spin Waves General: " n h q ~ q ! M and specific heat ~ Tq/n Antiferromagnet: " h q ~ q ! M and specific heat ~ Tq

TCD February 2007 5 2 Ferrimagnetism Antiferromagnet with 2 unequal sublattices ! YIG (Y3Fe5O12) Iron occupies 2 crystallographic sites one octahedral (16a) & one tetrahedral (24d) with O

! Magnetite(Fe3O4) 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 2007 6 Below TN, magnetisation of each sublattice is zero. Sublattice magnetisation:

Above TN:

Spontaneous magnetisation for nonzero solution in zero ﬁeld

TCD February 2007 7 3 Frustration

TCD February 2007 8 3.1 FCC Antiferromagnets 4 separate Cubic lattices. In zero field

TCD February 2007 9 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 2007 10 4 Amorphous Magnets No crystal structure Generally alloys or compounds of the 3d or 4f elements

TCD February 2007 11 TCD February 2007 12 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 2007 13 Topological disorder ! frustration of the individual superexchange bonds. ! Spins freeze into a random, noncollinear ground state with a high degree of degeneracy, Tf << 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 2007 14 TCD February 2007 15 .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 2007 16 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 Tf , 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 Tf? 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 2007 17

TCD February 2007 18 TCD February 2007 19 TCD February 2007 20 6.2 Critical Behaviour

# Reduced Temperature: = (1-T/Tc)

Critical Exponents:

Ising

Relations:

3D n-vector model

TCD February 2007 21 TCD February 2007 22 6 Magnetic Model Compounds

6.1 Heisenberg, xy and Ising models Heisenberg:

2D ! xy:

Dimensions in which phase 1D ! Ising: transition is possible

Onsager Solution 2D Ising

TCD February 2007 23