Magnetic phase diagrams from non-collinear canonical band theory

A canonical band theory of non-collinear magnetism is developed and applied to the close packed fcc and bcc crystal structures. This is a parameter-free theory where the crystal and magnetic symmetry and exchange splitting uniquely determine the electronic bands. In this way, we are able to construct phase diagrams of magnetic order for the fcc and bcc lattices. Several examples of non-collinear magnetism are seen to be canonical in origin, in particular, that of $\gamma \text{−}\mathrm{Fe}$. In this approach, the determination of magnetic stability results solely from changes in kinetic energy due to spin hybridization, and on this basis we are able to analyze the microscopic reasons behind the occurrence of non-collinear magnetism in the elemental itinerant magnets.

Figure 1

(Color online) Canonical phase diagram for $\Gamma XW$ spin spirals in the fcc structure. Indicated by symbols are the regions of stability of different spiral structures. The lines beginning at $(n,m)=(7.2,0.0)$ and $(n,m)=(2.0,0.0)$ indicate the stability of the spiral ${\mathbf{q}}_{\Gamma X}=[0,0,0.5]$. The smaller size X’s indicate points discussed in the text.

Figure 2

(Color online) Canonical phase diagram for $\Gamma HP$ spin spirals in the bcc structure. Indicated by symbols are the regions of stability of different spiral structures. The line beginning at $(n,m)=(5.8,0.0)$ indicates the stability of the spiral ${\mathbf{q}}_{\Gamma X}=[0,0,0.5]$. X’s indicate points discussed in the text.

Figure 3

Canonical spin spiral spectra for the symmetry path $\Gamma XWL\Gamma$ in the fcc Brillouin zone. Shown are spectra for (a) $n=8.8$, $m=1.1$, (b) $n=8.8$, $m=0.2$, (c) $n=8.0$, $m=0.8$, (d) $n=7.0$, $m=1.8$, (e) $n=5.0$, $m=3.3$, and (f) the region around the $X$ point for $n=6.5$, $m=0.8$. The top scale refers to graphs (a)–(e), and the bottom right to graph (f).

Figure 4

(Color online) Canonical density of states (DOS) for (a) AFM-1 and (b) spin spiral ${\mathbf{q}}_{\Gamma X}=[0,0,0.5]$ structures. Broken line indicates the FM DOS for the same exchange splitting. In each case, the inset graph shows the difference between the AFM-1 or spiral DOS and the FM DOS. Vertical lines indicate a region where the total AFM-1 or spiral DOS is lower than the FM DOS.

Figure 5

(Color online) Canonical density of states (DOS) for the spin spiral ${\mathbf{q}}_{\Gamma X}=[0,0,0.3]$ structure and, indicated by the broken lines, the FM DOS with identical exchange splitting. The inset shows the difference between the ${\mathbf{q}}_{\Gamma X}=[0,0,0.3]$ and FM DOS.

Figure 6

Canonical band structure for the (a) high-spin AFM-1 and (b) low-spin ${\mathbf{q}}_{\Gamma X}=[0,0,0.3]$ spin spiral structures. In each case, the corresponding FM band structure is indicated by the broken lines.