Electronic structure trends in the Sr${}_{n+1}$Ru${}_n$O${}_{3n+1}$ family ($n=1,2,3$)

The identification of electronic states and the analysis of their evolution with $n$ is key to understanding $n$-layered ruthenates. To this end, we combine polarization-dependent O 1$s$ x-ray absorption spectroscopy, high-purity Sr${}_{n+1}$Ru${}_n$O${}_{3n+1}$ ($n=1,2,3$) single crystals, and ab initio and many-body calculations. We find that the energy splitting between the empty $x^2-y^2$ and $3z^2-1$ state is considerably smaller than previously suggested and that, remarkably, Sr bands are essential to understanding the spectra. At low energy, we identify the main difference among the materials with a substantial rearrangement of $t_{2g}$ orbital occupations with increasing $n$. This rearrangement is controlled by the interplay of Coulomb repulsion, dimensionality, and changes in the $t_{2g}$ crystal field.

Figure 1

Left to right: Crystal structure of Sr${}_2$RuO${}_4$, Sr${}_3$Ru${}_2$O${}_7$, and Sr${}_4$Ru${}_3$O${}_{10}$.[1] The tetragonal (Sr${}_2$RuO${}_4$) and pseudotetragonal[2] (Sr${}_3$Ru${}_2$O${}_7$ and Sr${}_4$Ru${}_3$O${}_{10}$) axes are indicated by $x$, $y$, and $z$.

Figure 2

X-ray-diffraction patterns of a cleaved $(001)$ surface of Sr${}_2$RuO${}_4$, Sr${}_3$Ru${}_2$O${}_7$, and Sr${}_4$Ru${}_3$O${}_{10}$ crystals, grown with the flux-feeding floating-zone method. The spectra show the absence of any spurious phase and all the peaks are indexed as $(00l)$ peaks of the Sr-based Ruddelsden-Popper ruthenates structures.

Figure 3

XAS spectra ($T=115$ K) at the O $K$ edge for (a) Sr${}_2$RuO${}_4$, (b) Sr${}_2$Ru${}_3$O${}_7$, and (c) Sr${}_4$Ru${}_3$O${}_{10}$. The spectra are shown for several values of ${\theta }_{\mathrm{inc}}$, the angle between $c$ and the incoming x rays. The cleaved surface is perpendicular to the $c$ axis (Fig. 1). $A$, $B$, and $C$ label the most relevant features.

Figure 4

Top: XAS experimental da (open squares) and LDA results (filled curves) for ${\theta }_{\mathrm{inc}}=0^{°}$ and ${70}^{°}$. The energy zero is the absorption threshold ($E_0$); the normalization is the number of atoms per cell. The gray lines show the apical oxygen contribution. Bottom: $t_{2g}$ density of states (states/eV) per Ru site. For Sr${}_4$Ru${}_3$O${}_{10}$ both central Ru (c) and external Ru (e) contributions are shown. Energy zero is denoted by ${\varepsilon }_F$.

Figure 5

Sr${}_2$RuO${}_4$: Sr $d+s$ and Ru $e_g$ contributions to the XAS intensity (Fig. 4). The dark gray curves show ${\mathrm{O}}_P$XAS (with a 1.3-eV shift) and the light gray curves show ${\mathrm{O}}_A$XAS. $E_0=0$. Left: O $x$ and $y$. Right: O $z$.