Role of disorder in electronic and magnetic properties of ${\mathrm{Ag}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$

The nature of magnetism in the intercalated honeycomb iridate ${\mathrm{Ag}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ has been a subject of recent intensive debate, where the absence or presence of antiferromagnetic order has been reported to be related to possible structural disorder effects and an enhanced Ir-O hybridization and itinerancy with respect to the parent $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ has been suggested as the origin of distinct x-ray spectroscopy features. In the present work we investigate the microscopic nature of the electronic and magnetic properties of ${\mathrm{Ag}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ via a combination of density functional theory combined with exact diagonalization of ab initio derived models for various experimental and theoretical structures. We evaluate two possible scenarios, the itinerant quasimolecular framework (QMO) on the one hand, and the localized relativistic $j_{\mathrm{eff}}=1/2$ and $j_{\mathrm{eff}}=3/2$ picture on the other hand, and find that the second description is still viable for this system. We further calculate resonant inelastic x-ray scattering spectra and show that agreement with experimental observations can be obtained if the presence of Ag vacancies leading to changes in Ir filling and structural disorder is assumed. Finally, we show that the experimentally observed antiferromagnetic spiral magnetic order is reproduced by our ab initio derived magnetic models.

Figure 1

Crystal structure of ${\mathrm{Ag}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ in the (a) $ac$ plane and (b) $ab$ plane. Ir, Ag, and O are displayed as magenta, dark blue, and yellow balls. Red, green, and blue bonds show the three different types of bonds, $X, Y$, and $Z$, respectively. The green arrows in (b) indicate the stripy magnetic configuration used in the calculations. $l_l$ denotes the length of the long bond ($X$ and $Y$) and $l_s$ the length of the short bond ($Z$) for the structure $S_3$ (see the main text). (c) and (d) show two types of Ag vacancies considered (marked with circles) on $Z$-bond Ir-Ir clusters corresponding to ${\mathrm{Ir}}^{4.5+}$ and ${\mathrm{Ir}}^{5+}$, respectively. $x,y,z$ are the Cartesian coordinates for $d$ orbitals.

Figure 2

Band structure and partial density of states for the relaxed structure $S_2$ within (a) GGA, (b) $\text{GGA}+\text{SO}$, and (c) $\text{GGA}+\text{SO}+U$, respectively, obtained with the LAPW basis [46]. In the $\text{GGA}+\text{SO}+U$ calculation we considered a stripy magnetization as shown in Fig. 1 with doubling of the unit cell.

Figure 3

Geometry of nearest-neighbor hopping integrals (a) $t_1$, (b) $t_2$, (c) $t_3$, and (d) $t_4$ for the $Z$ bond [31]. Both $t_2$ and $t_3$ include contributions of oxygen assisted hopping and direct hopping. $t_2$ is dominated by oxygen assisted hopping while $t_3$ is dominated by direct hopping. In ${\mathrm{Ag}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ the oxygen assisted part of $t_3$ is strongly affected by Ag-O hybridization.

Figure 4

(a)–(c) Nonrelativistic GGA density of states projected onto quasimolecular orbitals and (d)–(f) relativistic $\text{GGA}+\text{SO}$ density of states projected onto the relativistic $j_{\mathrm{eff}}$ basis for structures $S_1\text{−}{S}_3$.

Figure 5

Spectral weight of various states [Eq. (11)] obtained from performing (a) one-site and (b) two-site cluster calculations for the experimental structure $S_3$. $P_1$ indicates the ground state, $P_2$ is a local exciton, $P_3$ are multiple excitons, while $P_4$ are all the projections including $d^4\text{−}{d}^6$. (c) Schematic diagrams of the lowest-energy subspaces ${\mathcal{B}}_1, {\mathcal{B}}_2, {\mathcal{B}}_3$, and ${\mathcal{B}}_4$ as defined in the main text.

Figure 6

(a) JDOS and (b) RIXS results obtained from one-site and two-site cluster calculations (see main text) for the experimental structure $S_3$. The experimental data from Ref. [38] are shown with a black dashed line where A, B, C, D, E indicate the experimental observed peaks.

Figure 7

(a) JDOS and (b) RIXS results obtained from two-site cluster ($Z$-bond) calculations for (i) the structure $S_3$ with Ir occupancies $d^5\text{−}{d}^5$, (ii) the structure $S_4$ with Ir occupancies $d^5\text{−}{d}^5$, (iii) the structure $S_4$ with Ir occupancies $d^{4.5}\text{−}{d}^{4.5}$, and (iv) the structure $S_5$ with Ir occupancies $d^4\text{−}{d}^4$.

Figure 8

(a) Spectral weight [Eq. (11)] of various states obtained from two-site cluster calculations for the $S_4$ structure with Ir occupancies $d^4\text{−}{d}^4$. $P_1^{\prime }$ indicates the ground states and $P_2^{\prime }$ is local excitons for $d^4$. (b) Schematic diagrams of the lowest-energy subspaces ${\mathcal{B}}_i^{\prime }$ ($i=1,2$) for $d^4\text{−}{d}^4$.

Figure 9

(a) JDOS for $S_2$ obtained from DFT $[S_2$(DFT)] and from two-site ED $[S_2$(ED)] in comparison to the ED results for $S_1$ and $S_3$. (b) RIXS results for $S_1\text{−}{S}_3$ structures.