Electronic states in $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ and $1∕1$ ${\mathrm{Cd}}_6\mathrm{Ca}$: Relativistic, correlation, and structural effects

The electronic structure of the rational approximants $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ and $1∕1$ ${\mathrm{Cd}}_6\mathrm{Ca}$ to the stable icosahedral ${\mathrm{Cd}}_{5.7}\mathrm{Yb}$ and ${\mathrm{Cd}}_{5.7}\mathrm{Ca}$ quasicrystals is studied by the full-potential linear augmented plane-wave method. A comparison is made between several structural models. We show that the (relativistic) spin-orbit (SO) interaction and electronic correlations that are not described by the usual local density approximation are essential for an accurate description of the electronic structure. In particular, we show that the SO interaction is responsible for a splitting of the Cd $4d$ and Yb $4f$ peaks, and that the experimental peak positions can be reproduced by including a Hubbard $U$ term in the Hamiltonian [$U\left(\mathrm{Cd}\right)=5.6\mathrm{eV}$, $U\left(\mathrm{Yb}\right)=3.1\mathrm{eV}$]. Our results show very good agreement with a photoemission (PE) spectrum of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ [R. Tamura, Y. Murao, S. Takeuchi, T. Kiss, T. Yokoya, and S. Shin, Phys. Rev. B 65, 224207 (2002)] and a $350\mathrm{eV}$ PE spectrum of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Ca}$, which we present in this paper.

Figure 1

(Color online) Unit cell of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ (spacegroup I23). The Cd (small) and Yb atoms (large) form a shell structure. In the center Cd atoms are located on four vertices of a cube. The next shell (the dodecahedral cavity) consists of Cd atoms placed on the vertices of a dodecahedron. The third shell is a Cd icosidodecahedron composed of triangles and pentagons. Yb atoms are located at the centers of the pentagons. The remaining Cd atoms and Cd atoms of the icosidodecahedra in neighboring unit cells (not shown) form a defect triacontahedron (Ref. 8).

Figure 2

(Color online) Models of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ differ from each other in the positions of the Cd atoms inside the dodecahedral cavity (Fig. 1). These positions and the unit cells are schematically indicated for models with symmetries (a) $I23$, (b) $Pn\overline{3}$, (c) $I222$, (d) $C2$, and (e) $Imm2$ (see text).

Figure 3

Electronic DOS of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ (spacegroup I23). (a) Original model (Ref. 6). (b)-(d) Relaxed model (Ref. 22). In (c) and (d) the SO interaction is included. (d) was calculated with the $\mathrm{LDA}+U$ method. All DOS’s were convoluted with a Gaussian with a full width at half maximum of $30\mathrm{meV}$. The DOS in (d) is not as smooth as the other DOS’s, because fewer $\mathbf{k}$ points were used to calculate it.

Figure 4

(Color online) Comparison of the calculated DOS of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Yb}$ with a PE spectrum (Ref. 19). The DOS was convoluted with a Gaussian with a full width at half maximum of $200\mathrm{meV}$.

Figure 5

(Color online) Comparison of the calculated DOS of $1∕1$ ${\mathrm{Cd}}_6\mathrm{Ca}$ with a PE spectrum (see text). The DOS was convoluted with a Gaussian with a full width at half maximum of $400\mathrm{meV}$.