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Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Module 5 Sub-atomic – pm-nm Mesoscale – nm-mm 15/02/2011 Magnetic order Macro – mm-mm Intended Learning Outcomes (ILO) (for today’s module) 1. 2. 3. 4. List the various forms of magnetic order in magnetic materials Calculate the room-T magnetization of a given ferromagnet Relate exchange interactions with the ”molecular field” in Weiss models Explain the peak in magnetic susceptibility at the Neel temperature in antiferromagnets Flashback Hˆ spin 2JS1 S2 The spin Hamiltonian A new set of orbitals Superexchange Crystal field splitting J * * (r ) a 1 b (r2 )Hˆ a (r2 )b (r1 )dr1dr2 The exchange integral Ferromagnetism Hˆ 2 Jij Si S j gm B Si B i j B mf i 2 gm B J S ij j j We define an effective field acting upon each spin due to exchange interactions Hˆ gm B Si B B mf i Bmf M In the Weiss model for ferromagnetism, exchange interactions are responsible for the huge “molecular field” that keeps moments aligned. The Hamiltonian now looks just like the paramagnetic Hamiltonian, except there’s a field even with no applied field We relate the molecular field with the “order parameter”, i.e. the magnetization Review Brillouin paramagnetism J mJ exp mJ x mJ m J J J exp m x J=1/2 gJ m B B , x k BT J=5 J m J J M ngJ m B m J ng J m B JBJ (y) M S BJ (y) 2J 1 1 y 2J 1 g m JB BJ (y) coth y coth , y J B 2J 2J 2J 2J k BT meff gJ m B J(J 1) ms gJ m B J J=3/2 3 S(S 1) L(L 1) gJ 2 2J(J 1) The spontaneous magnetization By solving numerically the two equations, we determine the spontaneous magnetization (in zero applied field) at a given temperature M M S BJ (y) gJ m B J(B M ) y k BT T>TC T=TC T<TC BJ (y) J 1 y 3J 2 gJ m B (J 1)M S nmeff T C 3k B 3k B Re-estimate the effective molecular field Bmf=MS if TC is 1000 K and J=S=1/2. The temperature dependence M(T) The case of Nickel (S=1/2) M (TC T )1/2 M 1 Near TC (mean-field critical exponent) Low T (as required by thermodynamics) Estimate the room-T M/Ms of Fe (J=S=3/2, Tc=1043 K) Ferromagnet and applied field T>TC T=TC T<TC Increasing B M B1/ 3 T TC Origin of the molecular field If we assume that exchange interactions are effective over z nearest-neighbours, we find: When L is involved (e.g. 4f ions), only a part of S contributes to the spin Hamiltonian: de Gennes factor 2zJ 2 2 ng m B So that we reveal the proportionality between Tc and the exchange constant 2zJ (J 1) TC J 3k B This is valid when L is quenched (3d ions) and, therefore, J=S J L L+2S=J+S S (gJ-1)J 2zJ(gJ 1)2 ngJ2m 2B 2zJ (J 1)gJ 1 TC J 3k B 2 Antiferromagnetism B M M B M M Neglect those for now (but they are important for a realistic theory) g J m B J M m M M S BJ k BT M M M M M M Staggered magnetization (order parameter) 2 gJ m B (J 1) M S n meff TN 3k B 3k B The magnetic susceptibilities Paramagnet Ferromagnet Antiferromagnet 1 , TC T TC 1 , 0 T 1 , TN T TN AFM with a strong magnetic field E MB cos MBcos AM 2 cos( ) MB arccos 2 2AM cos2 cos2 2 Types of antiferromagnetic order Simple cubic BCC Ferrimagnetism and helical order E 2NS 2 (J1 cos J2 sin 2 ) J1 4J2 cos sin 0 J1 cos 4J2 Ferrimagnets: important technologically for their non-metallic nature and flexible magnetic response Sneak peek B M B m0 (M Hd ) M Shape effects and magnetic domains Wrapping up •Ferromagnetism •Spontaneous magnetization •Ferromagnetic-to-Paramagnetic transition at Tc •Antiferromagnetism •Susceptibilities and Curie-Weiss laws •Ferrimagnetism •Helical order Next lecture: Friday February 18, 8:15, KU room 411D Micromagnetics I (MB)