Unidirectional spin density wave state in metallic (Sr1−xLax)2IrO4

Materials that exhibit both strong spin–orbit coupling and electron correlation effects are predicted to host numerous new electronic states. One prominent example is the Jeff = 1/2 Mott state in Sr2IrO4, where introducing carriers is predicted to manifest high temperature superconductivity analogous to the S = 1/2 Mott state of La2CuO4. While bulk superconductivity currently remains elusive, anomalous quasiparticle behaviors paralleling those in the cuprates such as pseudogap formation and the formation of a d-wave gap are observed upon electron-doping Sr2IrO4. Here we establish a magnetic parallel between electron-doped Sr2IrO4 and hole-doped La2CuO4 by unveiling a spin density wave state in electron-doped Sr2IrO4. Our magnetic resonant X-ray scattering data reveal the presence of an incommensurate magnetic state reminiscent of the diagonal spin density wave state observed in the monolayer cuprate (La1−xSrx)2CuO4. This link supports the conjecture that the quenched Mott phases in electron-doped Sr2IrO4 and hole-doped La2CuO4 support common competing electronic phases.

shows a schematic depicting the scattering geometry employed in the experiment, using conventional notations for the photon polarization vectors [1]. A vertical scattering plane of (H, 0, L) or (0, K, L) was employed, and the incoming beam was horizontally σ polarized. Magnetic scattering was isolated in the polarization rotated σ − π channel, and unless otherwise specified, data were collected in σ − π scattering channel at temperature 10 K.
Once the vertical scattering plane (H, 0, L) was defined, only the (0, 1, 4N + 2) or (1, 0, 4N ) (N = integer) type magnetic reflections are resolvable [2][3][4]. (1, 0, 4N + 2) and (0, 1, 4N ) type magnetic reflections arising from the other magnetic domain (with rotated moments and modified interplane phasing) are hidden due to the azimuthal dependence of the scattering intensity. Correspondingly, when the sample is rotated by 90 • with respect to the crystal's c-axis ( Fig. 3f and Supplementary Figures 2(b-f )), then the (0, K, L) scattering plane is accessed and only those previously silent (1, 0, 4N + 2) or (0, 1, 4N ) type magnetic peaks from the other domain set will contribute. In short, only one domain of the crystal will contribute to scattering either in the (H, 0, L) or (0, K, L) vertical scattering planes as a result of the azimuthal dependence of the magnetic intensity. Finally, the incident X-ray beam has a large horizontal divergence perpendicular to the scattering plane. As a result, the instrumental resolution is broadened along this direction (e.g. broad along K for the (H, 0, L) scattering plane), as shown in Figs. 3(a-b) and Figs. 4(a-b).

Supplementary Note 2: Single crystal refinement
As a further consistency check, single crystal X-ray diffraction measurements and structural refinements were performed on select samples used in synchrotron experiments. Crystals were mounted on a glass fiber and transferred to a Bruker Kappa APEX II diffractometer with a Mo K α source. The APEX2 [5] program was used to determine the unit cell parameters and data collection was performed using 10 sec / frame and 0.5 deg./ frame Omega scan. Data were collected at room temperature and refined using the SHELXTL [6] program. The similar R factors, Goodness of Fit (GoF) and comparable atomic displacement parameters (ADP) indicate consistent crystal qualities of doped samples when compared to the parent system.  Supplementary Figure 2a for comparison. Both the commensurate and incommensurate magnetic peaks disappear at the same temperature, which is estimated to be T AF = 188 ± 15K. Supplementary Figures 2(b-f ) show data obtained after the sample was rotated counter-clockwise by 90 • with respect to the crystal c-axis. Following this rotation, the scattering plane becomes (0,K,L), and Supplementary Figure 2b shows the H, K map of scattering intensities collected about the (1, 0, 10) magnetic zone center at the temperature 10 K. It should be noted that for the parent magnetic structure, the (1, 0, 10) peak intensity is interpreted as arising from the second allowed magnetic domain in the crystal. After the sample rotation, the resolution along K (estimated to be 0.0005 r.l.u.) becomes much better than that along H (estimated to be 0.0023 r.l.u.) rendering any small incommensurate splitting along H difficult to resolve. The K scan in Fig. 3f shows only one peak, demonstrating no equivalent splitting along K.
A remaining question concerns the degree to which the unidirectional incommensurate splitting along H is resolvable in the rotated (1, 0, 10) zone. Cuts taken through the resolution broadened peak (indicated by lines marked #1 and #2 in Supplementary Figure 2b) are plotted in Supplementary Figures 2(c-d). These cuts suggest the presence of two weakly resolved components split along H. This implies that the incommensurate splitting is still present along H after the sample's rotation, yet it is largely blurred by the instrumental resolution.
The observation of weakly resolvable incommensurate splitting in the 90 • rotated zone is further supported by H scans collected through the equivalent (1, 0, 14) magnetic zone at a range of different energies plotted in Supplementary Figures 2(e-f ). As discussed in the main manuscript text, the resonance energies of the commensurate and incommensurate magnetic peaks of this sample are shifted from one another by approximately 1.5 eV. By tuning to each resonance energy individually and performing H scans (denoted by lines in Supplementary Figure 2e), subtle shifts between convolved components forming the single peak can potentially be resolved. Supplementary Figure 2f shows the resulting scans at energy E = 11.217 keV, where the right shoulder of the broad peak is promoted, and at energy E = 11.209 keV where the left shoulder of the peak is enhanced. This subtle offset in the spectral weight of the peak with changing energy suggests the presence of the lower energy incommensurate peaks convolved within the resolution broadened peak.
A rough estimation of the splitting between the convolved peaks in this rotated zone shows the separation between two components along H to be about 0.005 r.l.u., close to the measured splitting of 0.0045 r.l.u. in the (0, K, L) zone. This combined analysis strongly suggests that the incommensurate peaks split along H are still present in the scattering data after the sample is rotated by 90 • . This is consistent with the incommensurate state being unidirectional and occurring only along H direction, regardless of the magnetic zones chosen or the relative orientation of the sample.