Spin-orbit coupling driven orbital-selective doping effect in ${\mathrm{Sr}}_2{\mathrm{Ru}}_{1-x}{\mathrm{Ir}}_x{\mathrm{O}}_4$

Orbital-selective phenomena in mutliorbital systems have received much attention due to their uniqueness as well as possible connections to other phenomena. As orbital-selectiveness is mostly related to the crystal structure, finding a new control parameter other than structure would be of significant importance. Here we report discovery of an orbital-selective doping effect in ${\mathrm{Sr}}_2{\mathrm{Ru}}_{1-x}{\mathrm{Ir}}_x{\mathrm{O}}_4$ (SRIO). Our systematic electronic structure study of SRIO reveals an anomalous orbital-selective doping effect and concomitant Lifshitz transitions (LTs) in the $\gamma$ band. With the help of a tight-binding calculation, we find that the orbital-selective doping effect is due to variation in the spin-orbit coupling (SOC) strength. Our findings not only elucidate the mechanism of LTs in the $\gamma$ band in SRIO but may also open new avenues for novel SOC-controlled orbital-selective phenomena.

Figure 1

Fermi surface (FS) maps for SRIO. (a)–(c) Experimental FS map for (a) $x=0.0$, (b) $0.2$, and (c) $0.5$, and (d)–(f) corresponding schematic band structures. The solid black squares mark the bulk Brillouin zone (BZ) at each doping level, and dashed black ones are the $\sqrt{2}\times \sqrt{2}$ reduced BZ due to surface octahedral rotation (OR). At $x=0.5$, there is bulk OR distortion which leads to a $\sqrt{2}\times \sqrt{2}$ reduced BZ as the surface OR does in (a) and (b). The blue, green, and red lines in panels (d)–(f) indicate $\alpha$, $\beta$, and $\gamma$ bands, respectively. Solid (dashed) lines correspond to bulk (surface) FS pockets. For an easier understanding, symbols of the $\sqrt{2}\times \sqrt{2}$ reduced BZ are used for high symmetry points as denoted in (a).

Figure 2

(a) Experimentally (filled square) and theoretically (empty square) obtained electron occupation numbers as a function of the Ir concentration ($x$) for $\alpha$, $\beta$, and $\gamma$ bands. Due to the difficulty in tracking the $\gamma$ electron occupation ($n_{\gamma }$), $n_{\gamma }$ is estimated from $n_{tot}-n_{\alpha }-n_{\beta }$, and $n_{\gamma }$ is set to be 2.0 for $x\ge 0.5$. The theoretical electron numbers are calculated by assuming a rigid-band-like shift in our tight-binding (TB) calculation [Fig. 3]. The black arrows indicate the change of the experimental value between $x=0.0$ and $0.5$. (b–d) Band dispersions along the $\mathrm{\Gamma }-{\mathrm{\Gamma }}^{\prime }$ high symmetry line for (b) $x=0.0$, (c) $0.2$, and (d) $0.4$. The colored dashed lines are extrapolated (quadratic) band dispersions for the bands. The energy difference between the top of $\alpha$ and $\beta$ bands ($\mathrm{\Delta }$) is expected to be proportional to the SOC strength since $\alpha$ ($\beta$) band corresponds to $J_{\mathrm{eff}}=3/2$ ($J_{\mathrm{eff}}=1/2$) near the Fermi energy. (e) $x$-dependent $\mathrm{\Delta }$ for $x\le 0.5$.

Figure 3

(a)–(c) SOC-dependent electronic structures from TB calculations for (a) $\lambda =0$, (b) 130, and (c) 300 meV, and (e)–(g) corresponding FS maps. $\mathrm{\Delta }$, the splitting between $\alpha$ and $\beta$ bands at $\mathrm{\Gamma }$, is defined in Fig. 2. (h) FS map of (c) ($\lambda =300\mathrm{meV}$) but with 0.54 electron doping which is taken from the $n_{\gamma }$ in Fig. 2. (d), (i) OR effect is further considered in addition to $\lambda =300\mathrm{meV}$ and 0.54 electron doping. To show the SOC-driven orbital mixing effect in panels (a)–(d), the band characters are color coded with $d_{xy}$ (red), $d_{zx}$ (green), and $d_{yz}$ (blue) projection. The TB calculation results in panels (f) and (i) closely resemble the experimentally obtained FS maps of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ [Fig. 1] and ${\mathrm{Sr}}_2{\mathrm{Ru}}_{0.5}{\mathrm{Ir}}_{0.5}{\mathrm{O}}_4$ [Fig. 1], respectively.

Figure 4

Ir concentration dependent evolution of the $\gamma$ band. The upper panel displays the $x$-dependent topology of the $\gamma$ band. The corresponding electron occupation number $n_{\gamma }$ is shown in the middle panel. Based on the TB calculation result in Fig. 3, the change in the electron number is quantitatively attributed to contributions from electron doping, SOC, and OR. On the bar in the lower panel, the doping range is categorized in terms of the number of FSs.