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Ferromagnetism Institute of Solid State Physics Technische Universität Graz Ferromagnetism

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Ferromagnetism Institute of Solid State Physics Technische Universität Graz Ferromagnetism 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 MFSJS i i, g B B i sums over the neighbors of spin i Looks like a magnetic field B 1 MF BJSMF i, g B N M gS magnetization B V eliminate <S> 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 22VkTB 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 22VkTB 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) 5B 16 octahedral sites B (surrounded by 6 O) 9B 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 ggBB 11 BJSJSMFB,, iAB A iBB , B ggBB 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: BBMF, A a BBMF, B a MN tanh MN tanh A kT B kT B B 00ABBMMB AAA ac MMAsA , tanh kTB 00ABAM BBBMB a MMBsB , tanh kTB -4 Ferrimagnetism gauss = 10 T oersted = 10-4/4x10-7 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.
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