Models of Ferromagnetism

1. Models of Ferromagnetism Ion Ivan

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

3. ignore the fact that magneticmoments can point only along certain directions because of quantization Langevin Theory* The probability of havingangle between θand θ+dθat temperature T is proportionalto the fraction of shaded area and the Boltzmann factor The average moment nthe number of magnetic moments per unit volume Curie’s law * Magnetism in condensed matter, Sthephen Blundell

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

5. -average magnetization x If Hext= 0 M/Ms 1 At Tc, spontaneous magnetization disappears and material become paramagnetic At T=Tc 0 1 T/Tc

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

7. r12 1 2 e- e- r1 r2 + Ze The Exchange Interaction Consider two electrons in an atom: Hamiltonian:

8. One orbital aproximation* Because of the indistinguishability of electrons Pauli principle this would conflict with theindistinguishability 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 the alectron are in different states 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. ms= 0 Triplet state S=1, ms= 1,0,-1 Singlet state S = 0 Using one electron approximation: triplet singlet

10. Using one electron approximation: singlet triplet Coulomb repulsion = 2K12 Exchange terms =2 J12 Lowest energy state is for triplet, with If J12is positive

11. The energies of the parallel and antiparalel spin pairs differ by -2J12 The coupling energy between spins of neighboring atoms If J > 0, is mininum if ferromagnetism antiferomagnetism is mininum if If J < 0,

12. Magnetic Domains* Magnetostatic energy Magnetostrictive energy Magnetocrystalline energy Competition between Why do domains occur? 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 Exchange energy The width of domain walls is controlled by the balance of two energy contributions: Anisotropy energy *Fundamentals of magnetism, Mathias Getzlaff

14. Domain Wall Width When neighboring spins make small angles with each other If a is lattice constant, the exchange energy stored per unit area of tranzition region The tickness 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 first term favors a large number N with spins involved in the domain wallwhereas the second term favors a small number. The energy minimum can bedetermined by setting the first derivative to zero: