Ferromagnetism
Technische Universität Graz Institute of Solid State Physics Ferromagnetism elow 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 J S S g S i, j Mean field approximation H MF Si Ji, S g i i, j i j i i Exchange energy sums over the neighbors of spin i 1 MF Ji, S g Looks like a magnetic field MF magnetization N M g S V eliminate <S>
Mean field theory MF V Ng 2 2 zjm z is the number of nearest neighbors In mean field, the energy of the spins is 1 E g ( MF a ) 2 We calculated the populations of the spins in the paramagnetism section
Spin populations N1 exp( / kt) N exp( / k T) exp( / k T) N2 exp( / kt) N exp( / k T) exp( / k T) M ( N N ) 1 2 exp( / kt ) exp( / kt ) N exp( / kt) exp( / kt) N tanh kt
Mean field theory M 1 N g MF a g tanh 2 V 2kT For zero applied field M Tc M M S tanh T M s N z M S g and Tc J 2V 4k M s = saturation magnetization T c = Curie temperature
Mean field theory M Tc M M S tanh T M s m tanh m t Experimental points for Ni.
Ferromagnetism Material Curie temp. (K) Co 1388 Fe 1043 FeOFe 2 O 3 858 NiOFe 2 O 3 858 CuOFe 2 O 3 728 MgOFe 2 O 3 713 Mni 630 Ni 627 MnSb 587 MnOFe 2 O 3 573 Y 3 Fe 5 O 12 560 CrO 2 386 MnAs 318 Gd 292 Dy 88 EuO 69 Electrical insulator Nd 2 Fe 14 353 M s = 10 M s (Fe) Sm 2 Co 17 700 rare earth magnets
Curie - Weiss law M 1 N g MF a g tanh 2 V 2kT MF V Ng 2 2 zjm Above T c we can expand the hyperbolic tangent tanh( x) x for x 1 1 2 2 N V M g zjm 2 2 4 VkT Ng a Solve for M Curie Weiss Law M 2 2 g N a 4Vk T T dm C dh T T c c T c z 4k J Critical fluctuations near T c
Ferromagnets are paramagnetic above T c Ferromagnetic Paramagnetic Critical fluctuations near T c.
Magnetic ordering Ferromagnetism Ferrimagnetism Antiferromagnetism Helimagnetism All ordered magnetic states have excitations called magnons
Ferrimagnets Magnetite Fe 3 O 4 (Magneteisen) Ferrites MO. Fe 2 O 3 M = Fe, Zn, Cd, Ni, Cu, Co, Mg Two sublattices A and. MgAl 2 O 4 Spinel crystal structure XY 2 O 4 8 tetrahedral sites A (surrounded by 4 O) 5 16 octahedral sites (surrounded by 6 O) 9 per unit cell
Ferrimagnets Magnetite Fe 3 O 4 Ferrites MO. Fe 2 O 3 M = Fe, Zn, Cd, Ni, Cu, Co, Mg Exchange integrals J AA, J A, and J are all negative (antiparallel preferred) J A > J AA, J
Mean field theory Heisenberg Hamiltonian 1 1 J S J S H J S S g S i, j i j i i MF, A i, A i, AA A g g i, j Exchange energy Mean field approximation 1 1 J S J S MF, i, A A i, g g N M g S V A A N M g S V
Mean field theory The spins can take on two energies. These energies are different on the A sites and because the A spins see a different environment as the spins. 1 EA g ( MF, A a) 2 1 E g ( MF, a) 2 M A Calculate the average magnetization with oltzmann factors: MF, A a N tanh kt M MF, a N tanh kt M M A 0 AM 0 AAM A ac M s, A tanh kt 0 AM A 0 M a M s, tanh kt
Ferrimagnetism gauss = 10-4 T oersted = 10-4 /4x10-7 A/ D. Gignoux, magnetic properties of Metallic systems Kittel
Antiferromagnetism Negative exchange energy J A < 0. At low temperatures, below the Neel temperature T N, 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. 0 M A M C T a Curie-Weiss temperature
Antiferromagnetism Average spontaneous magnetization is zero at all temperatures.
from Kittel