  In the last decade rare-earth hexaborides have been investigated for their
fundamental importance in condensed matter physics, and for their applications
in advanced technological fields. Among these compounds, LaB$_6$ has a special
place, being a traditional d-band metal without additional f- bands. In this
paper we investigate the bulk electronic structure of LaB$_6$ using hard x-ray
photoemission spectroscopy, measuring both core-level and angle-resolved
valence-band spectra. By comparing La 3d core level spectra to cluster model
calculations, we identify well-screened peak residing at a lower binding energy
compared to the main poorly-screened peak; the relative intensity between these
peaks depends on how strong the hybridization is between La and B atoms. We
show that the recoil effect, negligible in the soft x-ray regime, becomes
prominent at higher kinetic energies for lighter elements, such as boron, but
is still negligible for heavy elements, such as lanthanum. In addition, we
report the bulk-like band structure of LaB$_6$ determined by hard x-ray
angle-resolved photoemission spectroscopy (HARPES). We interpret HARPES
experimental results by the free-electron final-state calculations and by the
more precise one-step photoemission theory including matrix element and phonon
excitation effects. In addition, we consider the nature and the magnitude of
phonon excitations in HARPES experimental data measured at different
temperatures and excitation energies. We demonstrate that one step theory of
photoemission and HARPES experiments provide, at present, the only approach
capable of probing true bulk-like electronic band structure of rare-earth
hexaborides and strongly correlated materials.
