1. Magnetic Materials in PWSCF Tutorial session, Thursday September 29 2005 1.Spin polarized systems: The collinear magnetic case Exercise: Electronic structure and DOS of Ni 2.Non-collinear magnetism: extension of the formalism Exercise: Magnetism in bcc-iron 3.Non-collinear magnetism: Constraints using a penalty functional Exercise: Angle-dependence of magnetism in bcc-iron Ralph Gebauer

2. Spin polarized systems So far in this course: we “neglected” spin of the electrons: Implicitly, we have assumed that spin-up and spin-down wavefunctions are the same: Advantage: we need only N el /2 orbitals

3. Spin polarized systems For the case of spin-polarized systems, we drop this constraint and Use N orbitals:

4. Spin polarized systems Look at how the total energy depends on the two charge densities: Calculate the functional derivative in order to obtain the Hamiltonian:

5. We can also write the exchange-correlation energy as. Spin polarized systems Since we have: The Hamiltonian looks like:

6. Spin polarized systems

7. Exercise: Use the spin-polarized formalism to calculate The electronic structure of Ni

8. Non-collinear magnetism The exchange-correlation energy depends only on the amount of magnetization, not on its direction in space. The axis for the spin-up and spin-down projection is completely arbitrary and not related to the system’s geometry. This axis is UNIQUE all over space (collinear magnetism). It is possible to generalize the concept of magnetization to complex, non-collinear magnetic geometries.

9. Non-collinear magnetism We introduce two-component spinors: With these spinors, we define the vector magnetization as: The Pauli matrices are defined as:

10. Non-collinear magnetism As before, the exchange-correlation energy depends only on the modulus of m. The generalized Schrödinger equation reads now:

11. Non-collinear magnetism Exercise: Use the non-collinear program to calculate the magnetic structure of bcc-iron.

12. Non-collinear magnetism It is possible to constrain the magnetic moments of some atoms to some fixed value to a fixed direction. This can be done using a penalty functional: Which results in an additional term in the Hamiltonian:

13. Non-collinear magnetism For λ→∞, the magnetic moment takes the value m fixed. In that case, B pen is finite. Therefore the penalty energy E pen →0.

14. Non-collinear magnetism Exercise: Use the non-collinear program to constrain the magnetic moments in bcc-iron, and plot a curve of the total energy as a function of the angle.