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. The challenge is now to describe magnetism from first principles using ab-initio simulations.

In VASP, the electronic spin can be treated either within a so-called spin-polarized calculation (ISPIN=2) or a noncollinear calculation (LNONCOLLINEAR=T)^[1]. The aim is to solve the Kohn-Sham (KS) equations including the spin degree of freedom to yield spin-dependent KS orbitals, thus fulfilling quantum mechanical statistics. In this context it is important to choose the pseudopotential such that the electrons that give rise to magnetism are treated as valence electrons. The electron–electron interaction is included based on the selected exchange-correlation functional. For a general introduction to magnetism in the context of density-functional theory (DFT), we recommend the book Theory of itinerant electron magnetism by Jürgen Kübler ^[2].

Spin-polarized calculation

In spin-polarized calculations, there is a distinct spin-up and spin-down charge density

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

similar to the well-known Stoner model. As in standard DFT, one electron moves in an effective potential, but the effective potential ${\displaystyle v_\sigma^{eff}({\bf r})}$ 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 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 ${\displaystyle \psi_{\uparrow n}}$. 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. It is useful in order to describe collinear magnetic systems without spin-orbit coupling. Also see MAGMOM, LORBIT.

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. The magnetization (CHGCAR, PROCAR) and on-site magnetic moments (see MAGMOM, LORBIT) live in spinor space as defined by SAXIS.

Advanced methods

Some methods below can be applied in the context of both spin-polarized and noncollinear calculations.

Constrained magnetic moments

See I_CONSTRAINED_M.

Spin spirals

Spin spirals, or spin waves, are a magnetic order that can be described in terms of the generalized Bloch theorem. It states that the spin-up and spin-down components, and thus the magnetization, are rotated based on a certain spin-spiral propagation vector ${\displaystyle \mathbf{q}}$. If the corresponding wavelength is a multiple of the lattice vector of the crystal unit cell, the spin wave is commensurate. While the method can be used for both commensurate and incommensurable spin spirals, in practice it is more useful in case of incommensurable spin spirals. This is because spin spirals cannot be combined with spin orbit coupling and often systems with commensurate spin waves exhibit strong SOC, while in many compounds with incommensurable spin spirals SOC can be neglected. The main tag is LSPIRAL.

To learn how to apply the method read more on spin spirals.

Spin-orbit coupling

Spin-orbit coupling (SOC) is supported as of VASP.4.5 and later described in Ref. ^[3]. The main tag is LSORBIT, which automatically sets LNONCOLLINEAR and requires using vasp_ncl. SOC couples the spin degrees of freedom with the lattice degrees of freedom, see SAXIS.

Nuclear magnetic resonance

See LCHIMAG.

External magnetic field

Apply a constant external magnetic field (Zeeman-like term): BEXT

References