Fermi-Surface Topology and Helical Antiferromagnetism in Heavy Lanthanide Metals

Detailed angle-resolved photoemission studies of Tb and Dy metal in the paramagnetic phase provide direct experimental proof of the presence of nesting features in the Fermi surfaces (FS) of these heavy lanthanide (Ln) metals. The observations clearly support the hypothesis that nesting of the FS in the paramagnetic phase is responsible for the development of helical antiferromagnetic ordering in heavy Ln metals. They also show that magnetic exchange splitting of the electronic states is responsible for the disappearance of FS nesting in the ferromagnetic phases.

Figure 1

(a) Schematics of the FS of Tb metal in the PM phase, derived from the data of this work. The hexagonal BZ boundaries are indicated by solid lines, with the high-symmetry points marked. FS nesting is present around the $L$ point, with the nesting vector ${\mathbf{q}}_{\mathbf{0}}$, connecting almost parallel sections of the FS. (b) Regions of high PE intensity at $E=E_F$ for Tb metal, reflecting the FS at 240 K in the PM phase; for details, see text. Note that high (low) PE intensity corresponds to light gray (dark gray) tones (i.e., the contrast is opposite of that in Fig. 2).

Figure 2

Fermi contours in the $\Gamma LM$ plane of the BZ for (a),(d) Gd metal (see Ref. 9), (b),(e) Tb metal, and (c),(f) Dy metal in the PM phases (top row), and FM phases (bottom row), respectively; lines serve as a guide to the eyes. The darkness of the gray tones is proportional to the PE intensity. In the PM phase, the contour of (a) Gd ($T=300\mathrm{K}$) crosses the zone boundary close to the center of the $AL$ line, whereas for (b) Tb ($T=240\mathrm{K}$) and (c) Dy ($T=200\mathrm{K}$) it disperses towards $L$ crossing the zone boundary on the $ML$ line, close to $L$. This behavior in the case of Tb and Dy leads to the formation of nesting sheets of the FS around $L$, with the nesting vector ${\mathbf{q}}_{\mathbf{0}}$, indicated by arrows close to $L$ in (b) and (c) (note that for clarity the data were mirrored around the $\Gamma M$, $\Gamma A$, and $AL$ lines). The exchange-split Fermi contours are shown in the FM phases in the same BZ plane for (d) Gd at 50 K, as well as (e) Tb and (f) Dy metal, both at 25 K. Note the very similar FS of the three metals in the FM phases, with the small differences caused by different $\Delta E_{\mathrm{ex}}$ and slightly different dispersions.

Figure 3

Schematics of the influence of magnetic exchange splitting on the valence band that causes FS nesting in Tb and Dy metal. (a) In the PM phase, the band is unoccupied at $L$ and crosses $E_F$ on the $ML$ line at a distance of $|{\mathbf{q}}_{\mathbf{0}}|/2$ from $L$. (b) Exchange splitting in the FM phase lowers the energy of the majority subband (solid line) and increases that of the minority subband (dashed line). Hence, the crossings with $E_F$ are shifted against each other, leading to a disappearance of nesting (see text).

Figure 4

Regions of high PE intensity at $E=E_F$ for Tb metal at 25 K, reflecting the FS in the FM phase. Substantial differences to the PM phase [Fig. 1b] are obvious, caused by a separation of majority and minority FS sheets by magnetic exchange splitting. In particular, FS nesting around $L$ has disappeared. Note that the sections of the $\downarrow$ sheet close to the $MLK$ zone boundary cannot be seen in the 3D plot due to the weak PE signal; its location is visible in the cuts presented in Fig. 2e. The contrast is the same as that in Fig. 1b.