Spin spirals

From VASP Wiki

Generalized Bloch condition

Spin spirals may be conveniently modeled using a generalization of the Bloch condition (set LNONCOLLINEAR=.TRUE. and LSPIRAL=.TRUE.):

i.e., from one unit cell to the next the up- and down-spinors pick up an additional phase factor of and , respectively, where R is a lattice vector of the crystalline lattice, and q is the so-called spin-spiral propagation vector.

The spin-spiral propagation vector is commonly chosen to lie within the first Brillouin zone of the reciprocal space lattice, and has to be specified by means of the QSPIRAL-tag.

The generalized Bloch condition above gives rise to the following behavior of the magnetization density:

This is schematically depicted in the figure at the top of this page: the components of the magnization in the xy-plane rotate about the spin-spiral propagation vector q.

Basis set considerations

The generalized Bloch condition redefines the Bloch functions as follows:

This changes the Hamiltonian only minimally:

where in and the kinetic energy of a plane wave component changes to:

In the case of spin-spiral calculations the cutoff energy of the basis set of the individual spinor components is specified by means of the ENINI-tag.

Additionally one needs to set ENMAX appropriately: ENMAX needs to be chosen larger than ENINI, and large enough so that the plane wave components of both the up-spinors as well as the components of the down-spinor all have a kinetic energy smaller than ENMAX. This is the case when:

where

In most cases it is more than sufficient to set ENMAX=ENINI+100.

To judge whether ENMAX is chosen large enough one will always get a warning at runtime, e.g.

 ----------------------------------------------------------------------------- 
|                                                                             |
|           W    W    AA    RRRRR   N    N  II  N    N   GGGG   !!!           |
|           W    W   A  A   R    R  NN   N  II  NN   N  G    G  !!!           |
|           W    W  A    A  R    R  N N  N  II  N N  N  G       !!!           |
|           W WW W  AAAAAA  RRRRR   N  N N  II  N  N N  G  GGG   !            |
|           WW  WW  A    A  R   R   N   NN  II  N   NN  G    G                |
|           W    W  A    A  R    R  N    N  II  N    N   GGGG   !!!           |
|                                                                             |
|      To represent the spin spiral you requested, with a kinetic             |
|      energy cutoff of ENINI=  300.00 eV, choose ENMAX >  331.21 eV          |
|      Currently ENMAX=  400.00 eV                                            |
|                                                                             |
 -----------------------------------------------------------------------------