Three-dimensional electronic structures and the metal-insulator transition in Ruddlesden-Popper iridates

In this study, we systematically investigate three-dimensional (3D) momentum ($\hslash k$)-resolved electronic structures of Ruddlesden-Popper-type iridium oxides ${\mathrm{Sr}}_{n+1}{\mathrm{Ir}}_n{\mathrm{O}}_{3n+1}$ using soft-x-ray (SX) angle-resolved photoemission spectroscopy (ARPES). Our results provide direct evidence of an insulator-to-metal transition that occurs upon increasing the dimensionality of the ${\mathrm{IrO}}_2$-plane structure. This transition occurs when the spin-orbit-coupled $j_{\mathrm{eff}}=1/2$ band changes its behavior in the dispersion relation and moves across the Fermi energy. In addition, an emerging band along the $\mathrm{\Gamma }$(0,0,0)-$\mathrm{R}(\pi ,\pi ,\pi$) direction is found to play a crucial role in the metallic characteristics of ${\mathrm{SrIrO}}_3$. By scanning the photon energy over 350 eV, we reveal the 3D Fermi surface in ${\mathrm{SrIrO}}_3$ and $k_z$-dependent oscillations of photoelectron intensity in ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$. In contrast to previously reported results obtained using low-energy photons, folded bands derived from lattice distortions and/or magnetic ordering make significantly weak (but finite) contributions to the $k$-resolved photoemission spectrum. At the first glance, this leads to the ambiguous result that the observed $k$-space topology is consistent with the unfolded Brillouin zone (BZ) picture derived from a nonrealistic simple square or cubic Ir lattice. Through careful analysis, we determine that a superposition of the folded and unfolded band structures has been observed in the ARPES spectra obtained using photons in both ultraviolet and SX regions. To corroborate the physics deduced using low-energy ARPES studies, we propose to utilize SX-ARPES as a powerful complementary technique, as this method surveys more than one whole BZ and provides a panoramic view of electronic structures.

Figure 1

Constant-energy surfaces in the $\mathit{k}$ space obtained using SX-ARPES. The first BZs of (a) Sr113, (b) Sr327, and (c) Sr214 for RP-type crystal structures without consideration of the ${\mathrm{IrO}}_6$ octahedral rotation are shown. Note that the lattice constants $a$ and $c$ are omitted from the coordinate points. The constant-energy surfaces in the $k_{\left|\right|}-k_z$ and $k_x-k_y$ planes for Sr113 [(d) and (g)], Sr327 [(e) and (h)], and Sr214 [(f) and (i)] are displayed. Solid red lines indicate the cross-sectional view of the BZs illustrated in (a)–(c). Dashed curves in (d) show the cross-sectional FS discussed in Sec. 3b. The surfaces in (d)–(i) were obtained by integrating the momentum-distribution curves around a certain $E_B$ over an energy window of $\pm 100$ meV [35]. Their surface topology is qualitatively similar to that with smaller energy windows under the present experimental conditions.

Figure 2

$j_{\mathrm{eff}}=1/2$ band width depending on the dimensionality of the ${\mathrm{IrO}}_2$-plane structure. (a) Energy bands seen in the SX-ARPES spectra of Sr214 and the schematic view of their characters assigned by calculations (see Fig. 6 for details). The $j_{\mathrm{eff}}=1/2$ band dispersions along the (0,0,0)-($\pi$,0,0) line are shown for (b) Sr113, (c) Sr327, and (d) Sr214. Open circles indicate the peak positions estimated from the momentum- (energy-)distribution curves of Sr113 (Sr327 and Sr214). The error bar represents statistical variability and the variation of the peak position between the original and the second-derivative images.

Figure 3

Energy-band dispersions along high-symmetry lines in (a) Sr113, (b) Sr327, and (c) Sr214. To yield a better view of the dispersions, the second-derivative images are shown. The color of the images corresponds to that of the BZ planes [Figs. 1–1], in which the high-symmetry line is included. The arrows in (a) indicate the positions where the observed bands go across $E_F$ [see also (d)]. The LDA+SOC band structure of cubic Sr113 is also shown in (a). Solid orange lines indicate the $j_{\mathrm{eff}}=1/2$ band. (d) Momentum-distribution curves of Sr113 near $E_F$ along the high-symmetry lines, where the $j_{\mathrm{eff}}=1/2$ band goes across the $E_F$. Peak positions for the momentum-distribution curves, estimated using a peak-fitting analysis, are also shown by red bars [33]. (e) and (f) Enlarged band dispersions in the vicinity of $E_F$. Open circles in red indicate the peak positions in either the momentum- or energy-distribution curve. Calculated $j_{\mathrm{eff}}=1/2$ bands are shown by solid orange lines. (g) $W/2z$ value corresponding to an averaged hopping integral $t$ in the $j_{\mathrm{eff}}=1/2$ band, where $z$ stands for the number of nearest-neighbor Ir atoms. Here, $W$ for Sr214 and Sr327 is defined as $2w$, where $w$ is the $j_{\mathrm{eff}}=1/2$ band width in the occupied state. The dashed line divides the insulating and metallic states.

Figure 4

FS topology of Sr113. (a) Constant-energy surface (including the FS) in the $\mathrm{R}(\pi ,\pi ,\pi$)-$\mathrm{M}(\pi ,0,\pi$)-X(0,0,$\pi$)-M(0,$\pi ,\pi$) plane. Open circles in red indicate the $k_F$ points estimated from the ARPES spectra [33]. (b) Schematic image of a holelike FS for a simple cubic structure is shown. The cross-sectional views of the FS are also illustrated by the dashed line in (a) and Fig. 1. Electron reservoirs just below $E_F$ are displayed in (c) and (d).

Figure 5

Bilayer splitting and matrix element effect in Sr327. (a) Energy-distribution curves along the X-X line in the $k_z$ direction. (b) Transition probability along the $k_z$ axis, $M_z^2$, induced by photoexcitation for a bilayer-coupled system. The photoelectron intensity along the X-X line from Fig. 1 is also shown. (c) Schematic illustration of the bilayer-coupled wave functions.

Figure 6

(a) Theoretical energy spectrum based on the three-orbital Hubbard model with consideration for the unfolded BZ of Sr214. (b)–(d) Spectra projected onto the atomic $j_{\mathrm{eff}}=1/2$, 3/2 ($j_z=\pm 1/2$), and 3/2 ($j_z=\pm 3/2$) bases, respectively. The parameters and their values for the calculation are identical to those used in the previous study (i.e., $U=1.44$ eV, $U^{\prime }=1.008$ eV, $J=0.216$ eV, and $\lambda =0.432$ eV in Ref. [46]). (e) Experimental energy-band dispersion along $\mathrm{\Gamma }$(0,0,0)-$\mathrm{X}(\pi ,\pi$,0) line. (f) Constant-energy surface at $E_B=5.2$ eV. These images were taken at $\mathrm{h}\nu =760$ eV. In (f), unfolded and folded BZs are drown as solid and dotted lines, respectively.

Figure 7

Metal-insulator transition in RP iridates. (a) $k$-integrated spectra of Sr113 $(n=\infty )$, Sr327 ($n=2)$, and Sr214 ($n=1)$ obtained by summing up the SX-ARPES spectra in both $k_x-k_y$ and $k_{\left|\right|}-k_z$ planes shown in Figs. 1–1. (b) Temperature dependence of the SX-ARPES spectrum at the M point ($\pi$,0,0). (c) The structural dependence of the electronic nature and its temperature variation, where $z$ stands for the number of nearest-neighbor Ir atoms. Open circles (cross marks) indicate the temperatures at which the Fermi cutoff was (not) observed.