Diploma - Max Planck Institute for Solid State Research

26 4 EuRh 2 Si 2 – semi-localized electrons
Figure 4.3: (a) comparative band structure plot of high-symmetry cuts of SPG 123 (red)
and SPG 139 (green) in the BZ; the corresponding paths have been sketched in (b)+(c);
in (c) additionally the backfolding is sketched: the band structure of the upper green plane is
reflected along the red plane (BZ border of SPG 123) onto the one of the lower green plane
(see the arrow); the band structure along Γ-Z-Γ indicates the reflecting character of the red
plane; since SPG 123 has a cuboid BZ, the bisection can be infinitely repeated resulting in the
mentioned projected band structure
translational invariant quantity – the unit cell – is superior and backfolding does not
occur since the corresponding components of the potential’s Fourier transformation
vanish. But in PE projected band structure occurs, because the initial state is the sum
of all states in the region of the surface (can have different k z ) and the indeterminacy of
k z in the final state due to finite penetrating depth. Therefore, the surface configuration
should consist of projected bulk band structure due to the change in periodicity as
well as of surface states / resonances depending on the respective boundary conditions.
Both parts are important to compare the surface / bulk sensitivity of the measurements.
Speer et. al. [39] made a detailed analysis exemplarily for silver on the topic of emerging
band structure in photoemission. Since semi-bulk and surface have the same space
group, the projection onto the k x × k y plane for the surface BZ will be denoted by a
bar over the respecting high-symmetry points.
Below, the relationship between the BZs of SPG 123 / 139 depicted in fig. 4.2 will be
discussed. A cut at k z = 0 is illustrated denoting the boundary of the BZs inplane as
solid and structures of lower and higher parallel layers by dotted lines. Hence the stacking
of the bulk BZ in the x/y-direction is shifted by (0, 0, ±1/2) for the next-neighbour
BZ, the Z-point is the mid-distance point of the Γ-Γ path for each spatial direction (x,
y, z, -x, -y, -z). Neglecting body-centered symmetry halves the BZ (cf. the 3D image in
fig. 4.2) and each Z-point is mapped onto Γ. These backfolding from SPG 139 to 123
will be exemplarily shown for two high-symmetry directions: the diagonal path Z-X-Γ
(blue) gets mirrored with respect to X reassembling the Γ-M-Γ path in SPG 123 whereat
the folding orientation is given by the arrows. Correspondingly, Z-X maps onto Γ-X.