1. Models of Ferromagnetism Ion Ivan

2. Contents: 1.Models of ferromagnetism: Weiss and Heisenberg 2.Magnetic domains

3. Langevin Theory * ignore the fact that magnetic moments can point only along certain directions because of quantization The probability of having angle between θ and θ+dθ at temperature T is proportional to the fraction of shaded area and the Boltzmann factor The average moment n the number of magnetic moments per unit volume Curie’s law * Magnetism in condensed matter, Sthephen Blundell

4. In 1907, Weiss developed a theory of effective fields Magnetic moments in ferromagnetic material aligned in an internal (Weiss) field: HwHw H (applied) H W = wM w=Weiss or molecular field coefficient * Fizica Solidului, Ion Munteanu Weiss Theory of Ferromagnetism *

5. -average magnetization If H ext = 0 x At T=T c 1 0 1 M/M s T/T c At T c, spontaneous magnetization disappears and material become paramagnetic

6. Central for understanding magnetic interactions in solids Arises from Coulomb electrostatic interaction and the Pauli exclusion principle The Exchange Interaction Coulomb repulsion energy high Coulomb repulsion energy lowered

7. The Exchange Interaction Consider two electrons in an atom: + r1r1 r2r2 12 Ze e-e- e-e- r 12 Hamiltonian:

8. Pauli principle One orbital aproximation * Because of the indistinguishability of electrons If the alectron are in different states this would conflict with the indistinguishability of electrons because it is possible to know with certainty that electron 1 si in state a and electron 2 is in state b If consider the spin of electron Total wave function must be antisymmetrical * Solid state electronics (Shyh Wang), Qunatum mechanics for chemists (David O. Hayward )

9. Singlet state S = 0 m s = 0 Triplet state S=1, m s = 1,0,-1 Using one electron approximation: singlet triplet

10. Using one electron approximation: singlet triplet Coulomb repulsion = 2K 12 Exchange terms =2 J 12 If J 12 is positive Lowest energy state is for triplet, with

11. The energies of the parallel and antiparalel spin pairs differ by -2J 12 The coupling energy between spins of neighboring atoms If J > 0, is mininum if If J < 0, is mininum if ferromagnetism antiferomagnetism

12. Magnetic Domains * Why do domains occur? Magnetostatic energy Magnetostrictive energy Magnetocrystalline energy Competition between Magnetostatic energy To minimise the total magnetic energy the magnetostatic energy must be minimised. This can be achieved by decreasing the external demagnetising field by dividing the material into domains Magnetocrystalline energy There is an energy difference associated with magnetisation along the hard and easy axes which is given by the difference in the areas under (M,H) curves. This energy can be minimised by forming domains such that their magnetisations point along the easy crystallographic directions. * http://www.msm.cam.ac.uk/doitpoms//tlplib/ferromagnetic/index.php

13. Magnetostrictive energy Magnetostriction: when a ferromagnetic material is magnetised it changes length An increase in length along the direction of magnetisation is positive magnetostriction (e.g. in Fe), and a decrease in length is negative magnetostriction (e.g. in Ni). Domain walls * : The tranzition layer wich separates adjacent magnetic domains The width of domain walls is controlled by the balance of two energy contributions: Exchange energy Anisotropy energy * Fundamentals of magnetism, Mathias Getzlaff

14. When neighboring spins make small angles with each other If a is lattice constant, the exchange energy stored per unit area of tranzition region In turning away from the easy axys the magnetization must increase its anisotropy Energy per unit area: KNa, K is anisotropy constant. The total energy per unit area The tickness of tranzition region The first term favors a large number N with spins involved in the domain wall whereas the second term favors a small number. The energy minimum can be determined by setting the first derivative to zero: Domain Wall Width