Ferromagnetism Institute of Solid State Physics Technische Universität Graz Ferromagnetism

Below a critical temperature (called the Curie temperature) a magnetization spontaneously appears in a ferromagnet even in the absence of a magnetic field.

Iron, nickel, and cobalt are ferromagnetic.

Ferromagnetism overcomes the magnetic dipole-dipole interactions. Is arises from the Coulomb interactions of the electrons. The energy that is gained when the spins align is called the exchange energy. Mean field theory (Molekularfeldtheorie)

  Heisenberg Hamiltonian H  JSSij, i  j g BBS i ij, i Exchange energy Mean field approximation    H SJS g B MF i i, B i 

 sums over the neighbors of spin i Looks like a magnetic field B  1  MF BJS MF i, g B

 N  M  gS magnetization B V eliminate Mean field theory

 V  BzJMMF  22 Ng  B

z is the number of nearest neighbors In mean field, the energy of the spins is

1 Eg ()BB  2 B MF a

We calculated the populations of the spins in the paramagnetism section Spin populations

NBkTexp( / ) 1  B NBkTBkTexp( /BB ) exp( / ) NBkTexp( / ) 2  B NBkTBkTexp( /BB ) exp( / )

MNN()  12 exp(B /kT ) exp( B / kT )  N BB  exp(B /kTBB ) exp( B / kT )  B   N tanh  kTB Mean field theory



1 N gB BMFa  B Mg B tanh  22VkTB

For zero applied field

Tc M MM S tanh  TMs

Nz MgSB and TJ c 24VkB

Ms = saturation magnetization Tc = Curie temperature Mean field theory

Tc M MM S tanh  TMs

m m  tanh  t Experimental points for Ni. Ferromagnetism

Material Curie temp. (K)

Co 1388 Fe 1043

FeOFe2O3 858 NiOFe2O3 858 CuOFe2O3 728 MgOFe2O3 713 MnBi 630 Ni 627 MnSb 587

MnOFe2O3 573 Y3Fe5O12 560 CrO2 386 MnAs 318 Gd 292 Dy 88 EuO 69 Electrical insulator

Nd2Fe14B 353 Ms = 10 Ms(Fe) Sm2Co17 700 rare earth magnets Curie - Weiss law

   1 N gB BMFa  B V Mg tanh BzJMMF  B  Ng 22 22VkTB B

Above Tc we can expand the hyperbolic tangent tanh(xx ) for x 1

1 22 NV Mg B 22zJM Ba 4 VkBB T Ng Solve for M

22 gN B Ba z M  TJc  4VkB T Tc 4k B dM C Curie Weiss Law   dH T Tc

Critical fluctuations near Tc Ferromagnets are paramagnetic above Tc

Ferromagnetic

Paramagnetic

Critical fluctuations near Tc.

Magnetic ordering

Ferromagnetism

Ferrimagnetism

Antiferromagnetism

Helimagnetism

All ordered magnetic states have excitations called magnons Ferrimagnets

Magnetite Fe3O4 (Magneteisen) . Ferrites MO Fe2O3 M = Fe, Zn, Cd, Ni, Cu, Co, Mg

MgAl2O4 Two sublattices A and B.

Spinel crystal structure XY2O4

8 tetrahedral sites A (surrounded by 4 O) 5B 

16 octahedral sites B (surrounded by 6 O) 9B  per unit cell Ferrimagnets

Magnetite Fe3O4

. Ferrites MO Fe2O3 M = Fe, Zn, Cd, Ni, Cu, Co, Mg

Exchange integrals JAA, JAB, and JBB are all negative (antiparallel preferred)

|JAB| > |JAA|,|JBB| Mean field theory

  Heisenberg Hamiltonian H  JSSij, i  j  g B B S i ij, i Exchange energy Mean field approximation  11  BJSJSMFA,, iAB B iAA , A ggBB

 11  BJSJSMFB,, iAB A iBB , B ggBB

  N  N  M A  gS BA M  gS V B BBV Mean field theory

The spins can take on two energies. These energies are different on the A sites and B because the A spins see a different environment1 as the B spins. 1 Eg ()BB  Eg ()BB  A 2 BMFAa, B 2 BMFBa,

Calculate the average magnetization with Boltzmann factors:   BB    BB  MN tanh MF, A a MN tanh MF, B a A kT B kT B B

00 A BBMMB AAA ac MMAsA , tanh  kTB

00 A BAM  BBBMB a MMBsB , tanh  kTB -4 Ferrimagnetism gauss = 10 T oersted = 10-4/4x10-7 A/

Kittel D. Gignoux, magnetic properties of Metallic systems Antiferromagnetism

Negative exchange energy JAB < 0.

At low temperatures, below the Neel temperature TN, the spins are aligned antiparallel and the macroscopic magnetization is zero.

Spin ordering can be observed by neutron scattering.

At high temperature antiferromagnets become paramagnetic. The macroscopic magnetization is zero and the spins are disordered in zero field.   MMAB C Curie-Weiss 0  Ba T   temperature Antiferromagnetism

Average spontaneous magnetization is zero at all temperatures. from Kittel