Category:Magnetism: Difference between revisions

There are three ingredients to obtain magnetism.

• The first is the electronic spin. The origin of spin is explained by Dirac theory, that is the relativistic quantum theory of an electron.
• The second ingredient is quantum mechanical statistics. The Bohr–Van Leeuwen theorem states that a classical particle that follows Boltzmann statistics can never give rise to magnetism, even if the particle carries charge and spin. Therefore, in addition to spin, proper quantum mechanical statistics are necessary to explain magnetism.
• And finally, there is one more necessary ingredient: electron–electron interaction. Only the specific details of the interaction between electrons leads to a finite magnetization, otherwise any material would be magnetic.

In summary, magnetism is a collective, quantum electrodynamic phenomenon.

In VASP, the spin degrees of freedom can be treated either within a so-called spin-polarized calculation (ISPIN=2) or a noncollinear calculation (LNONCOLLINEAR=T).

Spin-polarized calculation

Similar to the Stoner model, there is a distinct spin-up and spin-down charge density

${\displaystyle n_\sigma(\mathbf{r}) = \sum_{n=1}^N |\psi_{\sigma n}(\mathbf{r})|^2. }$

As in standard DFT, one electron moves in an effective potential, but the effective potential of a spin-polarized calculation (ISPIN=2) has an additional spin index

${\displaystyle v_\sigma^{eff}({\bf r})=v^{ext}({\bf r}) + v^{H}({\bf r}) + v_\sigma^{xc}({\bf r}).}$

The exchange-correlation potential is

${\displaystyle v_\sigma^{xc}=\frac{\delta E_{xc}[n_\uparrow,n_\downarrow]}{\delta n_{\sigma}}.}$

The spin-dependent effective potential enters the Kohn-Sham (KS) equations

${\displaystyle \left[-{\mathbf \nabla}^2_n + v^{eff}_\uparrow({\mathbf r}_n)\right]\psi_{\uparrow n}= \varepsilon_{n} \psi_{\uparrow n}}$

${\displaystyle \left[-{\mathbf \nabla}^2_n + v^{eff}_\downarrow({\mathbf r}_n)\right]\psi_{\downarrow n}= \varepsilon_{n} \psi_{\downarrow n}}$

and leads to spin-dependent solutions for the KS orbitals. Finally, the spin-up and spin-down KS orbitals are used to update the spin-up and spin-down charge densities until self-consistency is reached. Note that, the spin species only couple to one another through the exchange-correlation potential where both, the spin-up and spin-down charge densities enter as an argument.

Spin-polarized calculations are enabled by setting ISPIN=2 in the INCAR file and executing vasp_std.

Noncollinear calculation

For noncollinear magnetism (LNONCOLLINEAR) and spin-orbit coupling (LSORBIT), the Hohenberg-Kohn-Sham DFT is extended, once again, by introducing an additional spin index. We introduce the spin-density matrix

${\displaystyle n_{\sigma'\sigma}(\mathbf{r}) = \sum_{n=1}^N \psi_{\sigma' n}(\mathbf{r})\psi^*_{\sigma n}(\mathbf{r}).}$

Accordingly, also the potential becomes a 2x2 matrix

${\displaystyle v_{\sigma'\sigma}^{eff}({\bf r})=v^{ext}_{\sigma'\sigma}({\bf r}) + \delta_{\sigma'\sigma} v^{H}({\bf r}) + v_{\sigma'\sigma}^{xc}({\bf r})}$

and the KS orbitals in the KS equations are two-component spinors

${\displaystyle \sum_{\sigma} \left[-{\mathbf \delta_{\sigma'\sigma} \nabla}^2_n + v^{eff}_{\sigma'\sigma}({\mathbf r}_n)\right]\psi_{\sigma n}= \varepsilon_{\sigma' n} \psi_{\sigma' n}.}$

Note that in the noncollinear case, the KS equations do not decouple for spin-up and spin-down because the potential has off-diagonal elements that couple the two spin species. This is the SCF loop of noncollinear spin-density functional theory (SDFT).

As VASP needs to treat many quantities as matrices instead of arrays, you need to use the vasp_ncl executable for these calculations in addition to setting LNONCOLLINEAR=T and/or LSORBIT=T for spin-orbit coupling.

How to

• Spin-orbit coupling: LSORBIT.
• Noncollinear magnetism: LNONCOLLINEAR.
• Spin spirals: Spin spirals.
• Apply a constant external magnetic field (Zeeman-like term): BEXT

What can one do when convergence is bad:

• Start from charge density of non-spin-polarized calculation using ISTART=0 (or remove the WAVECAR file) and ICHARG=1.
• Use linear mixing by setting BMIX=0.0001 and BMIX_MAG=0.0001.
• Mix slowly, i.e., reduce AMIX and AMIX_MAG.
• REDUCE MAXMIX, the number of steps stored in the Broyden mixer (default MAXMIX=45).
• Restart from partially converged results (stop a calculation after say 20 steps and restart from the WAVECAR file).
• Use constraints to stabilize the magnetic configuration.
• Pray.