Spin spiral calculations

Warning

This tutorial is work in progress

In this tutorial we employ the Generalized Bloch’s Theorem approach to calculate the spin spiral ground-state 2. In this approach we can choose any wave vector of the spin spiral \(q\), and rotate the spin degrees through the periodic boundary conditions accordingly. This rotation can be included in Bloch’s theorem by applying a combined translation and spin rotation to the wave function at the boundaries. Then we get the generalized Bloch’s theorem,

\[\phi(\mathbf{k}, \mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} [e^{-i\mathbf{q} \cdot \mathbf{r}/2} F(\mathbf{k}, \mathbf{r}), e^{i\mathbf{q} \cdot \mathbf{r}/2} G(\mathbf{k}, \mathbf{r})]^T\]

With two new spin Bloch functions \(F(\mathbf{k}, \mathbf{r})\) and \(G(\mathbf{k}, \mathbf{r})\) replacing the regular Bloch function \(u(\mathbf{k}, \mathbf{r})\). There are some limitations associated with these wave functions, because the spin structure should decouple from the lattice such that the density matrix is invariant under the spin rotation. In order for this to be the case, we can only apply spin orbit coupling perturbatively, and not as part of the self consistent calculation. Furthermore, with the density being invariant under the this spin rotation, so will also the z-component of the magnetization. This can be understood by looking at the magnetization density \(\tilde{\rho} = I_2\rho + \sigma\cdot\m\) under the spin spiral rotation, where one sees that the entire diagonal is left invariant. Thus we are limited to spiral structures which have magnetization vectors

\[\hat{e} = [\cos(\mathbf{q} \cdot \mathbf{r}), \sin(\mathbf{q} \cdot \mathbf{r}), 0]^T\]

which are called flat spin spirals, because they always rotate in the \(xy\)-plane. However, there is nothing special about the \(xy\)-plane, since spin-orbit is neglected at this stage, the spin spiral is invariant under any global rotation. The reward is that we can simulate any incommensurate spin spiral of this type in the principle unit cell. Additional care does need to be taken when taking structures with multiple magnetic atoms within the unit cell. This is because we only modify the boundary condition of the self-consistent calculation; the magnetization within the unit cell handled as a regular non-collinear magnetization density. For example, with two magnetic atoms in the unit cell, such as Cr2I6, one could consider parallel, anti- parallel or any canted alignment between the two Cr atoms on top of the spin spiral structure. In practice, canted order or ferrimagnetic order can be found self- consistently however finding anti-ferromagnetic order from a ferromagnetic starting point seem unlikely. Thus in order to find most spin spiral structures, one should in run calculations with both collinear starting structures.

Ground state of FCC Fe

At high temperatures, elementary iron has a phase transition to the iron allotrope \({\gamma}-Fe\) which has a FCC lattice. The spin structure of \({\gamma}-Fe\) was measured by stabilizing the phase at lower temperatures using Co. 1 They found a spin spiral ground state with wave vector \(q_{exp}=\frac{1}{5}XW = \frac{2\pi}{a}(1, 0, 1/10)\) at an atomic volume of \({\Omega}=11.44 Å^3\).

DFT simulations of \({\gamma}-Fe\) have found this system to be extremely sensitive to the lattice parameter. In fact we do not find the experimental spin spiral at the experimental volume. Instead we construct a FCC crystal with a slightly smaller unit cell of \({\Omega}=10.72 Å^3\). The following script fe_sgs.py (Warning, requires HPC resources) will construct the Fe FCC lattice and calculate the spin spiral ground-states with q along the high symmetry axis in the reciprocal lattice.

from ase.build import bulk
from gpaw.new.ase_interface import GPAW

# Construct bulk iron with FCC lattice
atoms = bulk('Fe', 'fcc', a=3.50)  # V = 10.72
# atoms = bulk('Fe', 'fcc', a=3.577)  # V = 11.44

# Align the magnetic moment in the xy-plane
magmoms = [[1, 0, 0]]
ecut = 600
k = 14

# Construct list of q-vectors
path = atoms.cell.bandpath('GXW', npoints=31)

results = []
for i, q_c in enumerate(path.kpts):
    # Spin-spiral calculations require non-collinear calculations
    # without symmetry or spin-orbit coupling
    calc = GPAW(mode={'name': 'pw',
                      'ecut': ecut,
                      'qspiral': q_c},
                xc='LDA',
                symmetry='off',
                magmoms=magmoms,
                kpts=(k, k, k),
                txt=f'gs-{i:02}.txt')
    atoms.calc = calc
    atoms.get_potential_energy()
    atoms.calc.write(f'gs-{i:02}.gpw')

As a result we find a spectrum with two local minimum, one of which match the experimentally measured spin spiral. Since only one atom is present in the unit cell, we do not need to worry about any magnetic structure inside the unit cell.

../../../_images/e-spiral.png
../../../_images/m-spiral.png

(see plot.py)

Calculating the energy of the spin spiral ground state could be done using a (2, 1, 10) supercell of the iron lattice in a standard non-collinear ground state calculation. It would however be difficult to verify the local minimum since wave vectors close to the minimum are incommensurate with the unit cell, and so a huge supercell would be required.

1
  1. Tsunoda 1989 J. Phys.: Condens. Matter 1 10427

2

K. Knöpfle, L. M. Sandratskii, and J. Kübler Spin spiral ground state of γ-iron, Phys. Rev. B 62, 5564 – Published 1 September 2000 doi: 10.1103/PhysRevB.62.5564