Electronic band structure and Fermi surface of ferromagnetic Tb: Experiment and theory

We have investigated the bulk valence-band structure of Tb metal in the ferromagnetic phase by angle-resolved photoelectron spectroscopy and full-potential-linearized-augmented-plane-wave calculations. The experiments were performed at undulator beamline 7.0.1 of the Advanced Light Source using a three-axis rotatable low-temperature goniometer and a display-type photoelectron spectrometer that give access to a large region of momentum space. The results of our calculations, which make use of recent progress in the theoretical description of the magnetic properties of $4f$ metals, are in remarkably good agreement with experiment. This can be best seen from a comparison of the electronic structure in high-symmetry directions, at critical points, on Fermi contours, and at band crossings with the Fermi level. To our knowledge, the present work represents the most detailed combined experimental and theoretical study of the electronic structure of a magnetic lanthanide metal to date.

Figure 1

Hexagonal BZ with high-symmetry points and crystallographic directions (left-hand side) and orientation of $\mathbf{k}$-space directions with respect to the BZ (right-hand side). High-symmetry directions are indicated by dotted lines; the $\Gamma A$ direction is parallel to the [0001] direction of the crystal. The $k_z$ axis is parallel to $\Gamma A$, which is along normal emission for PE from a (0001)-oriented hcp crystal.

Figure 2

(Color) Comparison of (a),(b) the experimental electronic structure of Tb metal at $25\mathrm{K}$ with (c) the results of our calculations for the $\Gamma M$, $MK$, and $K\Gamma$ high-symmetry lines. The top panels in (a) and (b) show the PE data as measured. Dark gray tones represent high PE intensity; note that the gray levels in the sections $\Gamma M$, $MK$, and $K\Gamma$ are scaled individually, to compensate for matrix-element effects. The bottom panels serve as guides to the eye and reflect the experimental positions of the bands [same symbols as in (c)]. PE can distinguish between (a) states with ${\Delta }_2$-like symmetry at $h\nu \simeq 105\mathrm{eV}$ and (b) states with ${\Delta }_1$-like symmetry at $h\nu \simeq 155\mathrm{eV}$ (for details, see text). The majority part of the $d$-like surface state is clearly visible in the symmetry band gap around $\Gamma$ both in (a) and (b) (green dashed lines in the bottom panels), reflecting a good quality of the sample. The calculated bands in (c) are in very good agreement with the experimental results in (a),(b). The surface state is absent in the bulk calculations. Red color corresponds to ${\Delta }_2$-like (bands 2 and 3) and blue color to ${\Delta }_1$-like symmetry (bands 1 and 4), filled (open) symbols represent majority (minority) electronic states. Bands that are not involved in the formation of the Fermi surface are represented by plus symbols (+). The area between $E_F$ and the short-dashed line in (c) marks the experimental window.

Figure 3

(Color online) Comparison of (a) the experimental band-dispersion curves measured at $h\nu \simeq 85\mathrm{eV}$ with (b) the results of theory in $AL$, $LH$, and $HA$ high-symmetry directions. Bands $1∕2$ and $3∕4$ are degenerate due to backfolding at the $ALH$ plane. The top panel in (a) shows the data as measured, where dark gray tones correspond to high PE intensities. The bottom panel in (a) displays the experimental band positions as guides to the eye; the same notation as in (b) is used. Green (gray) dashed lines indicate the location of the surface state and surface resonance (see text). In (b) filled (open) symbols represent majority (minority) bands; electronic states above $E_F$ are marked with a plus symbol (+). The area between $E_F$ and the dashed line reflects the experimental window.

Figure 4

(Color) Calculated bands in (a) $\Gamma A$, (b) $KH$, and (c) $ML$ high-symmetry directions of the BZ perpendicular to the (0001) surface (note the different energy scales). Bands $1∕2$ and $3∕4$ are degenerate on the $ALH$ plane. The degeneracy is lifted when the bands disperse toward the $\Gamma MK$ plane, with the exception of $1∕2\uparrow$ and $1∕2\downarrow$, which stay degenerate along $KH$. Blue color represents ${\Delta }_1$-like symmetry (bands 1 and 4) and red color ${\Delta }_2$-like symmetry (bands 2 and 3); filled (open) symbols represent majority (minority) electronic states. Bands that are not involved in the formation of the Fermi surface are marked with a plus symbol (+).

Figure 5

Valence band spectra of Tb at $T=25\mathrm{K}$, measured with $h\nu \simeq 105\mathrm{eV}$ (a) at the ${\Delta }_2$-like $\Gamma$ point and (b) half-way along $\Gamma M$ [see vertical arrow in Fig. 2a]. The solid lines through the data points and the subspectra represent the results of least-squares-fit analyses. In (a), an additional peak $S$ was added to describe the surface state.

Figure 6

(Color) Fermi contours in the $\Gamma MK$ plane, i.e., parallel to the sample surface: (a) Experimental data as measured at $h\nu \simeq 105\mathrm{eV}$, revealing bands 2 and 3 with ${\Delta }_2$-like symmetry [marked in red in (b)]. Bright colors correspond to high PE intensities. (b) Calculated Fermi contours; ${\Delta }_1$-like bands are colored blue, ${\Delta }_2$-like bands are colored red. Around $K$, bands of both symmetries ($1\downarrow$ and $2\downarrow$) cross $E_F$ (marked in green color). The agreement between experiment and theory is excellent.

Figure 7

(Color) Fermi contours in the $\Gamma ALM$ plane, i.e., perpendicular to the sample surface: (a) Data as measured as a function of photon energy (ordinate) and emission angle relative to the surface normal (abscissa); bright colors correspond to high PE intensities. In the present experiments, it is difficult to resolve Fermi contours in the region from ${\Gamma }_1{M}_1$ to $A_2{L}_2$, where the photon energy is passing through the Tb $4d\to 4f$ giant resonance, with a variation of the PE intensity by several orders of magnitude. To obtain the Fermi contours as shown in (b), autocorrelation operations combined with symmetries of the hcp lattice were applied on the as-measured data (for details see text). ${\Delta }_2$-like contours of bands $3\uparrow$ and $3\downarrow$ as well as the ${\Delta }_1$-like band $4\uparrow$ are clearly observed. The white lines represent guides to the eyes and highlight the FS features. (c) Theoretical results for the Fermi surface of Tb metal, where the white lines on the left-hand side of the panel represent the experimental Fermi contours. The agreement along $\Gamma M$ for bands $3\uparrow$ and $3\downarrow$ is remarkably good, while some deviations show up toward $AL$. Note that in (b) and (c) the scales on abscissa and ordinate differ by a factor of 2.