1. Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Module 6 18/02/2011 Micromagnetism I Mesoscale – nm-  m Reference material: Blundell, section 6.7 Coey, chapter 7 These lecture notes

2. Intended Learning Outcomes (ILO) (for today’s module) 1.Explain why and how magnetic domains form 2.Estimate the domain wall width 3.Calculate demagnetizing fields in simple geometries 4.Describe superparamagnetism in simple terms 5.List Brown’s equation in micromagnetics 6.Explain how hysteresis arises in a simple Stoner-Wolfharth model

3. Flashback

4. Edge effects and consequences This is a bit misleading

5. Dipoles Two interacting dipoles Dipole field Dipolar energy Torque Dipole vector potential 11 22 H12H12 H21H21 Zeeman energy

6. Energy of magnetized bodies This is to avoid double-counting d2d2 d1d1 Each dipole (magnetic moment) within a magnetized body interacts with each and every other. The sum of all that is the “self energy” of a magnetized body. Recognize this? It’s the dipole field “density”. The demagnetization field

7. For spheres, ellipsoids, and a few other shapes, the demag field is uniform throughout the shape. In general, the demag field is highly non-uniform. =+ BMH

8. Demag field for uniformly magnetized objects Introducing the characteristic function D(r), with value 1 inside the object, and 0 outside, we disentangle shape effects and get a convenient expression for the demag field. Representation of the demag field for a uniformly magnetized tetrahedron

9. Demag energy and demag factors Demag field as a result of a tensor operation on the magnetization Demag factors The demag tensor (a function of position) The demag energy: a 2-form involving the three demag factors along main axes and the magnetization direction cosines This is valid for any shape, provided its magnetization is uniform. NxNx NyNy NzNz

10. Domain walls Large dipolar energy, no exchange energy Snaller dipolar energy, some exchange energy Idem Bloch walls: bulk, thick objects Neel walls: thin films, thin objects Cross-over between dipolar and domain wall energies for a sphere (idealized model)

11. Wall width The strong commercial magnet NdFeB has K=4.3e6 J/m 3, and A=7.3e-12 J/m. Estimate the domain wall width in this material. The domain wall energy is proportional to the area

12. Magnetocrystalline anisotropy The crystal structure creates anisotropy: some directions are more responsive (“easier to magnetize”) to applied fields than others. Consider a sphere of radius R magnetized along some easy axis u with anisotropy constant K u =4.53e5 J/m 3 (value for Co). If the magnetization flips to –u, the energy remains the same (up and down states are degenerate). But, in order to rotate from +u to –u, the magnetization has to go through a high energy state, i.e. when M points perpendicular to u. Suppose that the temperature is such that k B T is of the same order of the energy barrier separating the degenerate states. What happens? Uniaxial Cubic M u

13. Stoner-Wolfharth x y M H The direction of M at any given applied field Single-domain hysteresis is a consequence of anisotropy (shape or magnetocrystalline).

14. Brown’s equations The whole set of equations provides a full description of the energy landscape of a micromagnetic system (such as the one shown above) and drives its evolution towards the ground state of minimum energy

15. Sneak peek Micromagnetic simulations LLG equation Magnetodynamics and evolution Searching for ground states

16. Wrapping up Next lecture: Tuesday February 22, 13:15, KU (A9) Micromagnetism II (MB) Magnetic domains Bloch and Neel walls, and wall widths Dipolar/magnetostatic/demag energy Demagnetization fields and factors Stoner-Wolfharth hysteresis Magnetocrystalline anisotropy Brown’s equations Please remember to: Install OOMMF on your laptop Familiarize a little bit with it Bring your laptop to class on Tuesday, February 22