Spin spirals: Difference between revisions

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:<math>
:<math>
G_{\rm ini}=\sqrt{\frac{2m}{\hbar^2}\mathtt{ENMAX}}
G_{\rm ini}=\sqrt{\frac{2m}{\hbar^2}\mathtt{ENINI}}
</math>
</math>

Revision as of 13:50, 6 July 2018

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