Spin spirals

Spin spirals are magnetic structures in which the direction of the magnetization rotates continuously from one unit cell to the next. In VASP they are modeled with a generalization of the Bloch condition, which captures incommensurate magnetic order in the primitive cell and avoids the need for large supercells.

This page describes the underlying formalism. For step-by-step instructions on setting up, running, and analyzing a spin-spiral calculation, see Spin-spiral calculations.

Generalized Bloch condition

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

[math]\displaystyle{ \left[ \begin{array}{c} \Psi^{\uparrow}_{\bf k}(\bf r) \\ \Psi^{\downarrow}_{\bf k}(\bf r) \end{array} \right] = \left( \begin{array}{cc} e^{-i\bf q \cdot \bf R / 2} & 0\\ 0 & e^{+i\bf q \cdot \bf R / 2} \end{array}\right) \left[ \begin{array}{c} \Psi^{\uparrow}_{\bf k}(\bf r-R) \\ \Psi^{\downarrow}_{\bf k}(\bf r-R) \end{array} \right], }[/math]

i.e., from one unit cell to the next the up- and down-spinors pick up an additional phase factor of [math]\displaystyle{ \exp(-i{\bf q}\cdot {\bf R}/2) }[/math] and [math]\displaystyle{ \exp(+i{\bf q}\cdot {\bf R}/2) }[/math], 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 lattice, and is specified by means of the QSPIRAL tag.

Magnetization density

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

[math]\displaystyle{ {\bf m} ({\bf r} + {\bf R})= \left( \begin{array}{c} m_x({\bf r}) \cos({\bf q} \cdot {\bf R}) - m_y({\bf r}) \sin({\bf q} \cdot {\bf R}) \\ m_x({\bf r}) \sin({\bf q} \cdot {\bf R}) + m_y({\bf r}) \cos({\bf q} \cdot {\bf R}) \\ m_z({\bf r}) \end{array} \right) }[/math]

The components of the magnetization in the xy-plane rotate about the spin-spiral propagation vector q, while the out-of-plane component [math]\displaystyle{ m_z }[/math] retains the usual cell periodicity. This is depicted schematically below:

Mind: The generalized Bloch condition does not mean that the magnetization density may not have contributions along the z-direction; these are simply unaffected by it. To keep the magnetization density from developing a component along z, set LZEROZ = .TRUE., which sets [math]\displaystyle{ m_z({\bf r}) = 0 }[/math] at each step of the electronic minimization.

Modified Hamiltonian

The generalized Bloch condition redefines the Bloch functions as follows:

[math]\displaystyle{ \Psi^{\uparrow}_{\bf k}(\bf r) = \sum _{\bf G} \rm C^{\uparrow}_{\bf k \bf G} e^{i(\bf k + \bf G -\frac{\bf q}{2})\cdot \bf r} }[/math]

[math]\displaystyle{ \Psi^{\downarrow}_{\bf k}(\bf r) = \sum _{\bf G} \rm C^{\downarrow}_{\bf k \bf G} e^{i(\bf k + \bf G +\frac{\bf q}{2})\cdot \bf r} }[/math]

This changes the Hamiltonian only minimally:

[math]\displaystyle{ \left( \begin{array}{cc} H^{\uparrow\uparrow} & V^{\uparrow\downarrow}_{\rm xc} \\ V^{\downarrow\uparrow}_{\rm xc} & H^{\downarrow\downarrow} \end{array}\right) \rightarrow \left( \begin{array}{cc} H^{\uparrow\uparrow} & V^{\uparrow\downarrow}_{\rm xc} e^{-i\bf q \cdot \bf r} \\ V^{\downarrow\uparrow}_{\rm xc}e^{+i\bf q \cdot \bf r} & H^{\downarrow\downarrow} \end{array}\right), }[/math]

where in [math]\displaystyle{ H^{\uparrow\uparrow} }[/math] and [math]\displaystyle{ H^{\downarrow\downarrow} }[/math] the kinetic energy of a plane-wave component changes to:

[math]\displaystyle{ H^{\uparrow\uparrow}:\qquad |{\bf k} + {\bf G}|^2 \rightarrow |{\bf k} + {\bf G} - {\bf q} /2|^2 }[/math]

[math]\displaystyle{ H^{\downarrow\downarrow}:\qquad |{\bf k} + {\bf G}|^2 \rightarrow |{\bf k} + {\bf G} + {\bf q} /2|^2 }[/math]

Because the only change to the off-diagonal exchange-correlation potential is a phase modulation, a spin-spiral calculation has approximately the same computational cost as a standard noncollinear calculation of the primitive cell.

Basis-set considerations

Because the two spinor components are shifted in reciprocal space by [math]\displaystyle{ \pm{\bf q}/2 }[/math], the plane-wave cutoff must be chosen carefully. The cutoff of the basis set of the individual spinor components is specified by means of the ENINI tag, while ENMAX must be chosen large enough that the plane-wave components of both spinors have a kinetic energy below ENMAX. This is the case when

[math]\displaystyle{ \mathtt{ENMAX} \geq \frac{\hbar^2}{2m}\left( G_{\rm ini} + |q| \right)^2, \qquad G_{\rm ini}=\sqrt{\frac{2m}{\hbar^2}\mathtt{ENINI}}. }[/math]

In practice it is more than sufficient to set ENMAX to ENINI + 100 eV. The practical settings, and the runtime warning VASP prints when ENMAX is too small, are described in Spin-spiral calculations.

Symmetry

The introduction of a spin spiral generally lowers the symmetry of the system, and VASP cannot currently account for the presence of a spin spiral in its symmetry analysis. For this reason the use of symmetry has to be switched off completely (ISYM = -1) in spin-spiral calculations.

Related tags and articles

Spin-spiral calculations

LNONCOLLINEAR, LSPIRAL, QSPIRAL, LZEROZ, ENINI, ENMAX, ISYM, MAGMOM