Pressure-induced formation of rhodium zigzag chains in the honeycomb rhodate Li$_2$RhO$_3$

TL;DR

This study investigates how pressure affects the crystal structure of Li₂RhO₃, revealing the formation of rhodium zigzag chains at high pressure, which is supported by experimental diffraction data and theoretical calculations.

Contribution

It demonstrates pressure-induced formation of rhodium zigzag chains in Li₂RhO₃ and explains the structural transition through anisotropic pressure effects.

Findings

Zigzag chains form at pressures above 14 GPa.

Structural phase mixture occurs between 6.5 GPa and 14 GPa.

The zigzag pattern is reproducible under anisotropic pressure conditions.

Abstract

We use powder x-ray diffraction to study the effect of pressure on the crystal structure of the honeycomb rhodate Li$_2$RhO$_3$. We observe low-pressure ($P$$<$$P_{c1}$ = 6.5 GPa) and high-pressure ($P$$>$$P_{c2}$ = 14 GPa) regions corresponding to the monoclinic $C2/m$ symmetry, while a phase mixture is observed at intermediate pressures. At $P$$>$$P_{c2}$, the honeycomb structure becomes distorted and features short Rh--Rh bonds forming zigzag chains stretched along the crystallographic $a$ direction. This is in contrast to dimerized patterns observed in triclinic high-pressure polymorphs of $\alpha$-Li$_2$IrO$_3$ and $\alpha$-RuCl$_3$. Density-functional theory calculations at various pressure conditions reveal that the observed rhodium zigzag-chain pattern is not expected under hydrostatic pressure but can be reproduced by assuming anisotropic pressure conditions.

Tables

Table 1. Table 1: Bulk moduli B 0 , V subscript 𝐵 0 𝑉 B_{0,V} and B 0 , r subscript 𝐵 0 𝑟 B_{0,r} with r = a , b , c 𝑟 𝑎 𝑏 𝑐 r=a,b,c in the low-pressure ( P 𝑃 P < < 6.5 GPa) and high-pressure ( P 𝑃 P > > 14 GPa) phases, as obtained from fitting the volume V 𝑉 V and lattice parameters r 𝑟 r with a MOS, with B 0 ′ superscript subscript 𝐵 0 ′ B_{0}^{\prime} set to 4.

	GPa	$P>15$ GPa
$V_0$ (Å3)	216.90(3)	212.21(16)
$B_{0,V}$ (GPa)	100.4(4)	118.6(9)
$B_{0,a}$ (GPa)	105.7(5)	155.6(15)
$B_{0,b}$ (GPa)	100.2(6)	122.2(12)
$B_{0,c}$ (GPa)	94.8(7)	93.9(8)

Table 2. Table 2: Structural parameters for the low-pressure phase at ambient pressure and for the high-pressure phase at 25.2 GPa. At ambient pressure, the lattice parameters are a = 5.11126 ​ ( 10 ) 𝑎 5.11126 10 a=5.11126(10)  Å, b = 8.83473 ​ ( 16 ) 𝑏 8.83473 16 b=8.83473(16)  Å, c = 5.10034 ​ ( 11 ) 𝑐 5.10034 11 c=5.10034(11)  Å, β = 109.6105 ​ ( 18 ) ∘ 𝛽 109.6105 superscript 18 \beta=109.6105(18)^{\circ} , V = 216.955 ​ ( 8 ) 𝑉 216.955 8 V=216.955(8)  Å 3 , and at 25.2 GPa a = 4.7732 ​ ( 5 ) 𝑎 4.7732 5 a=4.7732(5)  Å, b = 8.3980 ​ ( 7 ) 𝑏 8.3980 7 b=8.3980(7)  Å, c = 4.8027 ​ ( 3 ) 𝑐 4.8027 3 c=4.8027(3)  Å, β = 109.034 ​ ( 11 ) ∘ 𝛽 109.034 superscript 11 \beta=109.034(11)^{\circ} , V = 181.99 ​ ( 3 ) 𝑉 181.99 3 V=181.99(3)  Å 3 . The isotropic atomic displacement parameters U iso subscript 𝑈 iso U_{\rm iso} were fixed to 0.005 Å 2 for all atomic positions, except for the Rh(1)/Li(1) one.

		low-pressure phase (0 GPa)					high-pressure phase (25.2 GPa)
Atom	Site	$x$	$y$	$z$	Occupancy	$U_{\text{iso}}$(Å2)	$x$	$y$	$z$	Occupancy	$U_{\text{iso}}$(Å2)
Rh(1)	$4g$	0	0.3311(2)	0	0.864(3)	0.0029(3)	0	0.3225(5)	0	0.899(7)	0.0047(11)
Li(1)	$4g$	0	0.3311(2)	0	0.136(3)	0.0029(3)	0	0.3225(5)	0	0.101(7)	0.0047(11)
Li(2)	$2a$	0	0	0	0.728(3)	0.005	0	0	0	0.798(7)	0.005
Rh(2)	$2a$	0	0	0	0.273(3)	0.005	0	0	0	0.202(7)	0.005
Li(3)	$4h$	0	0.820(3)	0.5	1	0.005	0	0.808(8)	0.5	1	0.005
Li(4)	$2d$	0	0.5	0.5	1	0.005	0	0.5	0.5	1	0.005
O(1)	$8j$	0.252(17)	0.3209(7)	0.7631(10)	1	0.005	0.271(3)	0.3332(16)	0.754(3)	1	0.005
O(2)	$4i$	0.274(2)	0	0.7726(19)	1	0.005	0.287(4)	0	0.774(4)	1	0.005

Equations

$$V(p)$$

$$r(p)$$

Full text

I Introduction

In recent years, $4d$ and $5d$ transition-metal compounds were intensively studied due to their extremely rich physics. In comparison to $3d$ compounds, where the electronic correlation $U$ dominates over the spin-orbit coupling constant $\lambda\textsubscript{SOC}$ and Hund’s coupling $J\textsubscript{H}$, spin-orbit coupling (SOC) becomes more and more important for $4d$ and $5d$ transition-metal compounds, whereas the strength of electronic correlations decreases. The actual physics of these compounds thereby depends on a delicate balance between $U$, $\lambda\textsubscript{SOC}$ and $J\textsubscript{H}$, as well as the crystal structure. The class of layered honeycomb-type $4d$ and $5d$ transition-metal compounds, such as $A_{2}M$O3 ($A$= Li, Na and $M$= Ir, Rh) and $\alpha$-RuCl3, is especially interesting in this regard, as this delicate balance of parameters was discussed in terms of Kitaev physics and possible spin-liquid state Kitaev (2006); Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Choi et al. (2012); Plumb et al. (2014); Chun et al. (2015); Winter et al. (2017); Luo et al. (2013); Khuntia et al. (2017). However, in Na2IrO3, $\alpha$-Li2IrO3, and $\alpha$-RuCl3 the quantum spin liquid ground state is not realized, since these materials were found to order magnetically at low temperatures Choi et al. (2012); Sears et al. (2015); Banerjee et al. (2016); Williams et al. (2016).

As for Li2RhO3, its magnetic ground state is still under debate. No long-range magnetic order could be found down to $\approx$0.5 K, but instead at small magnetic fields spin freezing was observed below 6-7 K Luo et al. (2013); Khuntia et al. (2017), although it is suspected that the majority of magnetic moments form a fluctuating liquid-like state Khuntia et al. (2017). Whether this partial spin freezing is due to a proximity to the Kitaev quantum spin liquid ground state or due to unavoidable defects (anti-site disorder and/or stacking faults) is still unclear Khuntia et al. (2017); Katakuri et al. (2015). However, ab initio and effective-model calculations showed that Li2RhO3 bears similar electronic structure to the iridates Mazin et al. (2013) and hosts anisotropic Kitaev interaction terms of the same magnitude as in $5d$ iridates Katakuri et al. (2015). According to electrical resistivity measurements, Li2RhO3 is insulating at ambient pressure Luo et al. (2013); Mazin et al. (2013).

Another interesting aspect of Li2RhO3 is its behavior under pressure, where honeycomb iridates Hermann et al. (2018); Clancy et al. (2018) and $\alpha$-RuCl3 Bastien et al. (2018); Biesner et al. (2018) become dimerized and, consequently, non-magnetic. Previously Hermann et al. (2018), we showed that the size of the central ion, the strength of the spin-orbit-coupling, electronic correlations, and Hund’s coupling all act against the dimerization. In comparison to $\upalpha$-Li2IrO3, the $\lambda\textsubscript{SOC}$ in Li2RhO3 is expected to be lower, while the electronic correlations should be enhanced in Rh*+4* compared to Ir4+, as screening by oxygen orbitals is reduced. Thereby, one generally expects a higher transition pressure in Li2RhO3 and a larger pressure range for tuning the putative Kitaev magnetism of this compound. Here, we show that this is the case, but also that the pressure-induced transformations are very different from the dimerization observed in honeycomb iridates.

II Methods

II.1 Experimental Details

A powder sample of Li2RhO3 was prepared by a solid-state reaction of Li2CO3 and Rh in oxygen flow at $850^{\circ}$ C with several intermediate re-grindings. The sample quality was confirmed by laboratory x-ray diffraction (XRD) using the Rigaku MiniFlex diffractometer (CuKα radiation, Bragg-Brentano geometry). This synthesis procedure yields samples with the best structural order achieved so far Manni (2014), although stacking faults are still present. Their concentration is discussed in Sec. III.1 below.

Li2RhO3 powder was loaded into a diamond anvil cell (DAC) for pressure generation, and helium was used as pressure transmitting medium. The powder x-ray diffraction patterns were obtained using synchrotron radiation at the beamline ID15B at the European Synchrotron Radiation Facility (ESRF), Grenoble at room temperature. The wavelength of the radiation was 0.411267 Å, and the patterns were obtained in the 2$\theta$-range between 2 and 33∘. The pressure in the DAC was determined in situ by the ruby luminescence method. The resulting patterns were analyzed by Rietveld refinements using the Jana2006 software Petříček et al. (2014). The quality of the fit is gauged by the weighted structure factor $R_{\omega}$, as defined in Ref. Young, 2002 and by the commonly used weighted profile factor $R_{\omega p}=(\sum_{i}w_{i}(y^{\prime}_{i}(\text{\rm obs})-y^{\prime}_{i}(\text{\rm calc}))^{2})/(\sum_{i}w_{i}y^{\prime}_{i}(\text{\rm obs})^{2})$, where $y^{\prime}_{i}$ are intensities corrected for background. The absorption correction for a cylindrical sample was calculated to be lower than the value one (using Ref. von Dreele et al. (2013)), so no absorption correction was applied. The isotropic atomic displacement parameters $U_{\rm iso}$ were fixed to the value of 0.005 Å2 for all atomic positions, except for Rh(1) and Li(1).

II.2 Computational details

Structural optimizations were performed under different pressure conditions by using projector-augmented planewave Blöchl (1994) method based on density functional theory (DFT), as implemented in the VASP package Kresse and Hafner (1993). Calculations were done within the generalized gradient approximation (GGA), GGA+U Dudarev et al. (1998), and GGA+SOC+U (including spin-orbit coupling effects for Rh). The value of the on-site Coulomb parameter $U$ was chosen based on the reproducibility of the experimental structure, as will be shown below. The cutoff for the wavefunction was set at 650 eV. K-point meshes of size $8\times 6\times 8$ were used for all the structural optimizations.

We performed two types of structural optimizations: (i) allowing relaxation of both lattice parameters and atomic positions under fixed hydrostatic pressure; we refer to this as “full relaxation”, and (ii) keeping the lattice parameters fixed according to given pressure conditions and allowing only the relaxation of the atomic positions. In both cases, the system is allowed to relax until the total force acting on the system was less than 0.005 eV/Å. At each pressure value, several different initial magnetic configurations were considered: (a) ferromagnetic (FM), (b) zigzag antiferromagnetic (AFM), (c) Neél AFM, (d) stripy AFM, and (e) nonmagnetic (NM) (see Fig. 1).

The analysis of the electronic properties was done with the Full Potential Local Orbital (FPLO) basis Koepernik and Eschrig (1999).

III Results and Discussion

III.1 Experimental results

The x-ray powder diffraction patterns at ambient pressure and at the highest studied pressure (25.2 GPa) together with the corresponding fits from the Rietveld refinement are displayed in Fig. 2. Both refinements were performed within the monoclinic unit cell with the $C2/m$ symmetry. The same crystal symmetry is found for the closely related honeycomb iridates O’Malley et al. (2008); Freund et al. (2016); Choi et al. (2012). In the refinements, stacking faults associated with shifts between successive LiRh2 layers were taken into account, as observed in $\alpha$-Li2IrO3 O’Malley et al. (2008); Freund et al. (2016) and other Li${}_{2}M$O3 ($M$ = Mn, Pt, Ru) compounds O’Malley et al. (2008); Bréger et al. (2005); Casas-Cabanas et al. (2007). Stacking faults affect the intensity and lineshape of several peaks and lead to an additional intensity between the (020) and (110) peaks as marked by the dashed red arrows in the insets of Fig. 2. The presence of stacking faults was taken into account by introducing the Li/Rh mixing for the Rh(1)/Li(1) and Li(2)/Rh(2) sites while constraining the overall stoichiometry to Li2RhO3. This reproduces the peak intensity but not their shape O’Malley et al. (2008).

The lattice parameters as a function of pressure, as obtained by the Rietveld fits of the x-ray powder diffraction diagrams, are depicted in Fig. 3. Up to the critical pressure $P_{c1}$=6.5 GPa, the lattice parameters $a$, $b^{\prime}=b/\sqrt{3}$, and $c$ decrease monotonically with increasing pressure in a very similar manner. The $c/a$ value, shown in the inset of Fig. 3(a), reveals that the strongest pressure-induced effect occurs for the lattice parameter $c$. The monoclinic angle $\beta$ decreases slightly but monotonically within this pressure range.

Above $P_{c1}$, a second phase with the same $C2/m$ symmetry appears and gets more pronounced with increasing pressure. Above the critical pressure $P_{c2}$=14 GPa, this second phase is dominant and the high-pressure diffractograms can be well described by a single phase with the $C2/m$ symmetry. There are only traces of the low-pressure phase found in the diffraction patterns above $P_{c2}$ and up to the highest studied pressure, marked with black arrows in the inset of Fig. 2(b). Most importantly, we can rule out a symmetry lowering above $P_{c2}$, as such a symmetry lowering would induce peak splittings, for example for the (021) and the (111) diffraction peaks. These peaks are observed at 7.7∘ and 9.0∘ in the inset of Fig. 2(b) and are obviously not split. Thus, both the low-pressure (P$$<$$P_{c1}) and high-pressure (P$$>$$P_{c2}) phases in Li2RhO3 have the $C2/m$ symmetry. This result is in contrast to the recent findings for $\alpha$-Li2IrO3, where a pressure-induced structural phase transition with symmetry lowering from monoclinic to the triclinic symmetry caused by the Ir–Ir dimerization occurs at 3.8 GPa Hermann et al. (2018). Analogously, the monoclinic to triclinic symmetry lowering with the Ru-Ru dimerization is observed in $\alpha$-RuCl3 at $P\approx 1$ GPa Bastien et al. (2018).

The refinement of the diffraction patterns for the intermediate pressure range, P_{c1}$$<$$P$$<$$P_{c2}, with a phase mixture of the low-pressure and high-pressure phases did not yield stable fits, as many of the peaks of the two phases are broad and overlapping. Since the refinement with only one phase does not reproduce the actual peak shape, we marked this range with open symbols in Figs. 3, 5, 6 and 7.

Between $P_{c1}$ and $P_{c2}$, the lattice parameter $a$ decreases drastically by about 3 %, while there is only a slight but abrupt increase in the lattice parameter $b$, and $c$ follows the pressure-induced monotonic decrease as observed below $P_{c1}$ [see Fig. 3(a)]. The abrupt decrease in the $a$ parameter is also revealed by the abrupt increase in the $c$/$a$ ratio. Accordingly, the most pronounced pressure-induced change happens along the $a$ lattice direction, as will be discussed in more detail later. The monoclinic angle $\beta$ abruptly decreases above $P_{c1}$, and above $P_{c2}$ it monotonically increases with increasing pressure [see Fig. 3(b)]. The kink in the pressure evolution of $\beta$ in the pressure range 10-12 GPa, i.e., in the intermediate phase, is not discussed here, because the phase mixture affects the refinements in this pressure range.

The pressure dependencies of the volume $V$ and the lattice parameters $r$ ($r=a,b,c$) were fitted separately for the low- and high-pressure phases, neglecting the intermediate regime, with a second-order Murnaghan equation of state (MOS) Murnaghan (1944), to obtain the bulk moduli $B_{0,V}$ and $B_{0,r}$ according to:

[TABLE]

with $B^{\prime}$ fixed to 4. The results are summarized in Table 1. The bulk modulus $B_{0,V}$ of the low- and high-pressure phases amounts to 100.4(4) GPa and 118.6(9) GPa, respectively. This means that the material is less compressible in the high-pressure phase. In the low-pressure phase (P$$<$$P_{c1}), the contribution of the $c$ direction to the bulk modulus is the lowest with $B_{0,c}$=94.8(7) GPa, as already indicated by the pressure dependence of the $c$/$a$ ratio [inset of Fig. 3 (a)]. Thus, the material is most compressible along the $c$ direction. The largest contribution to the bulk modulus is attributed to the $a$ crystal direction, with $B_{0,a}$=105.7(5) GPa.

In the high-pressure phase ($P>P_{c2}$), the contribution $B_{0,c}$ of the $c$ direction remains low and is even slightly decreased as compared to the low-pressure phase. Most interestingly, the contribution $B_{0,a}$ is strongly increased to 155.6(15) GPa in the high-pressure phase, while $B_{0,b}$ is much less increased, i.e., to 122.2(12) GPa. Hence, the honeycomb layers along the $ab$-plane become less compressible in the high-pressure phase, whereby the pressure-induced hardening has the strongest effect along the $a$ direction.

For a more detailed discussion, the atomic parameters of the refinement are shown in Table 2. The partial exchange of Li and Rh accounts for the stacking faults, as described in Ref. O’Malley et al., 2008. Since three Rh atoms are required to change place with one Li atom in order to mimic one stacking fault, each 5.5(1) unit cells one stacking fault occurs at ambient pressure. This value is very similar to previous reports in Li2RhO3 Luo et al. (2013); Todorova and Jansen (2011) and slightly higher than in $\upalpha$-Li2IrO3 and Li2PtO3 O’Malley et al. (2008). On the other hand, different studies of Na2IrO3 Ye et al. (2012); Choi et al. (2012) reported the concentrations of stacking faults that are either larger or smaller than in Li2RhO3. The number of unit cells per stacking fault increases monotonically with increasing pressure and reaches 7.4(5) at 25.2 GPa, as shown in Fig. 4, i.e., the number of stacking faults is slightly reduced by external pressure. The parameters for the oxygen positions are changed in the high-pressure phase as compared to the low-pressure phase, thus affecting the RhO6 octahedra. The most interesting change, though, is observed for the $y$-parameter of Rh(1) that determines the Rh-Rh distances in the honeycomb network (see Table 2).

To evaluate this behavior further, we compare the pressure evolution of the three Rh-Rh bond lengths in the $ab$-plane, namely, the $Z1$ bond and the two degenerate $X1/Y1$ bonds as depicted in Fig. 5(b). At ambient pressure, the $Z1$ bond length amounts to 2.985(3) Å, while the $X1/Y1$ bond length is 2.9296(13) Å [see Fig. 5(a)], leading to a slightly distorted honeycomb. The corresponding bond disproportionation $l_{l}/l_{s}$, with $l_{l}$ and $l_{s}$ being the long and short bonds of the hexagonal Rh network, respectively, amounts to $l_{l}/l_{s}$=1.02. For the high-pressure phase, both $X1$ and $Y1$ bonds are drastically reduced by $\approx$0.15 Å, while the $Z1$ bond is increased by the same amount [Fig. 5(a)]. Hence, the bond disproportionation increased to $l_{l}/l_{s}$=1.11 at 25 GPa. The $X1/Y1$ bond length of $\approx$2.7 Å above $P_{c2}$ is close to but still larger than the interatomic distances in metallic rhodium ($d$=2.69 ÅBale (1958)). We thus conclude that external pressure introduces zigzag chains of rhodium atoms along the $a$ direction, as illustrated in Fig. 5(b). A similar structure but with closer bond lengths is found for 5% Na doped crystals (Li0.95Na0.05)2RuO3 at ambient pressure Mehlawat and Singh (2017). On the other hand, pure Li2RuO3 is dimerized at ambient pressure with an armchair pattern of the short Ru–Ru bonds (see Fig. 1(e) for illustration) Mehlawat and Singh (2017); Kimber et al. (2014).

Next, we consider the pressure-induced changes in the RhO6 octahedra. To this end, we define various Rh–O bond lengths and Rh–O–Rh bond angles that are responsible for the direct metal-to-metal and indirect oxygen-mediated contributions. The octahedra possess a two-fold rotational C2-axis which is indicated by the arrow in Fig. 6(c). There are three unique Rh–O bonds labeled $R_{1}$, $R_{2}$, $R_{3}$ and two unique Rh–O–Rh angles $\delta_{1}$ and $\delta_{2}$, where $\delta_{1}$ ($\delta_{2}$) involves two Rh atoms connected via the $X1/Y1$-bond ($Z1$-bond) [see Fig. 6(c)]. The pressure dependence of the various bonds and bond angles is depicted in Figs. 6(a) and (b), respectively. At ambient pressure, the largest Rh–O bond length is found for the apical oxygen atom, thus the RhO6 octahedra show a tetragonal distortion with axial elongation. In the low-pressure phase (P$$<$$P_{c1}), the bond length $R_{1}$ is pressure-independent, whereas $R_{2}$ and $R_{3}$ slightly decrease under pressure.

At $P_{c2}$, the length $R_{1}$ is increased as compared to the low-pressure phase, whereas $R_{2}$ and $R_{3}$ are decreased. Upon further compression, $R_{1}$ decreases, $R_{3}$ seems to be unaffected, and $R_{2}$ shows a small anomaly at 15-20 GPa that may be significant, as the changes exceed the error bars.

The formation of zigzag chains is predominantly due to a change of Rh-O-Rh angles as described in the following. The pressure dependence of the Rh–O–Rh bond angles $\delta_{1}$ and $\delta_{2}$ is shown in Fig. 6(b). At ambient pressure, the values of $\delta_{1}$ and $\delta_{2}$ amount to 94.0(3)∘ and 96.4(4)∘, respectively. While $\delta_{2}$ is independent of pressure in the low-pressure phase, $\delta_{1}$ decreases by increasing pressure. When entering the high-pressure phase above $P_{c2}$ the bond angle $\delta_{1}$ is strongly decreased to the value 87*∘*. Interestingly, the onset of the intermediate phase at $P_{c1}$ appears at a pressure, when $\delta_{1}$ approaches 90∘, which is a distinct angle for the contributions of the ligand-mediated hopping to the hopping parameters, as discussed in more detail in Refs. Winter et al., 2016 and Winter et al., 2017. The strong pressure-induced decrease in the angle $\delta_{1}$ between $P_{c1}$ and $P_{c2}$ confirms the formation of Rh zig-zag chains along the $a$ direction. Consistently, the bond angle $\delta_{2}$ is strongly increased, as the $Z_{1}$ bond length is increased [see Fig. 5(a)]. Again an anomaly is observed for the Rh–O–Rh bond angles between 15-20 GPa, which is directly related to the anomaly for the Rh–O distances and thereby has the same origin.

The electronic states of Li2RhO3 are affected by the distortion of the RhO6 octahedra. Therefore, we followed the pressure dependence of the octahedral distortion using the bond-length distortion $\Delta_{\text{oct}}$ and the bond-angle distortion $\sigma_{\text{oct}}^{2}$ Ertl et al. (2002); Kim et al. (2004); Hogan et al. (2017). The bond-length distortion is defined as $\Delta_{\text{oct}}$=$\frac{1}{6}\sum_{i=1}^{6}[(d_{i}-d_{\text{av}})/d_{\text{av}}]^{2}$, where $d_{i}$ is an individual Rh–O bond length and $d_{\text{av}}$ the average Rh–O bond length in the RhO6 octahedron. The bond-angle distortion is calculated according to $\sigma_{\text{oct}}^{2}$=$\frac{1}{11}\sum_{i=1}^{12}(\alpha_{i}-90)^{2}$, where $\alpha_{i}$ is an individual O–Rh–O bond angle. At ambient pressure, the distortion parameters are $\Delta_{\text{oct}}$=3.4(11)$\times$10*-4* and $\sigma^{2}=9.6(6)$ comparable to the results in Ref. Todorova and Jansen (2011) [$\Delta_{\text{oct}}$=1.2(6)$\times$10*-4* and $\sigma^{2}$=9.6(5)], although the Rh position seems to be fixed in that report. Comparison of our refinement to previous ones Mazin et al. (2013); Luo et al. (2013) is not straightforward since in those studies some oxygen parameters were fixed or calculated. The distortion parameters for Li2RhO3 reported in our study are comparable to the ones of the related materials $\alpha$-Li2IrO3 and Li2PtO3 O’Malley et al. (2008); Freund et al. (2016). For Na2IrO3 the $\Delta_{\text{oct}}$ value is about one magnitude smaller, while the bond-angle distortion $\sigma^{2}$ is nearly doubled Ye et al. (2012); Choi et al. (2012). A comparison to the octahedral distortions in dimerized Li2RuO3 is difficult, since the reported values determined by various studies are not consistent. For example, the $\Delta_{\text{oct}}$ values between 1.4$\times$10*-4* and 24$\times$10*-4* have been reported, and the values for $\sigma^{2}$ range between 4.7 and 54 Wang et al. (2014); Kobayashi et al. (1995); Mehlawat and Singh (2017).

In the low-pressure phase, the bond-length distortion only slightly increases with increasing pressure, whereas the bond-angle distortion decreases (Fig. 7). At the critical pressure $P_{c2}$, both parameters $\Delta_{\text{oct}}$ and $\sigma_{\text{oct}}^{2}$ are drastically enhanced compared to the low-pressure range. Such an enhanced distortion was also reported in Ref. Wang et al. (2014) for dimerized Li2RuO3 compared to the non-dimerized samples. It is therefore likely that the enhancement of $\Delta_{\text{oct}}$ and $\sigma_{\text{oct}}^{2}$ at $P_{c2}$ is caused by the lattice strain due to the formation of the Rh–Rh zigzag chains in Li2RhO3.

Of further interest is the tetragonal distortion (elongation or compression along the $z$-direction) of the octahedra, as this would cause a splitting of the Rh $t_{2g}$ states. As a measure of the tetragonal distortion we define the parameter $\Delta_{\text{tetr}}$ as the deviation of the apical Rh–O bond length $R_{1}$ from the average Rh–O bond length $d_{\text{av}}$ according to $\Delta_{\text{tetr}}=(R_{1}-d_{\text{av}})/d_{\text{av}}$ Kim et al. (2004). For positive (negative) nonzero values of $\Delta_{\text{tetr}}$ the octahedra are elongated (compressed) along the apical bond direction. Such a distortion can be explained by a cooperative first-order Jahn-Teller effect neglecting stress on the system Gehring and Gehring (1975); Kaplan and Vekhter (1995). The Jahn-Teller effect is expected to be weak but nonzero in a $d^{5}$ configuration. The pressure dependence of $\Delta_{\text{tetr}}$ is depicted in Fig. 7. We note that the tetragonal distortion at ambient pressure amounts to $\Delta_{\text{tetr}}$=0.023(4), which is comparable to the value $\Delta_{\text{tetr}}$=0.015(3) given in Ref. Todorova and Jansen (2011). In the low-pressure phase, $\Delta_{\text{tetr}}$ increases slightly but steadily upon compression, i.e., the elongation increases.

Between the low- and high-pressure phases, $\Delta_{\text{tetr}}$ is nearly doubled, before it decreases upon further compression above $P_{c2}$. While the tetragonal distortion in the low-pressure phase is comparable to that in Li${}_{2}M$O3 ($M$= Ir, Pt), it is much more pronounced than in Na2IrO3, where $\Delta_{\text{tetr}}$ is close to zero. We thus conclude that the lattice strain rather than the Jahn-Teller effect is the driving force for the distortion of the octahedra in the honeycomb lattices. The tetragonal distortion in the high-pressure phase of Li2RhO3 is comparable to the tetragonal distortion in the perovskites Sr2RhO4 and Sr2RuO4, where the bond angle distortion is zero Huang et al. (1994); Vogt and Buttrey (1996).

III.2 Computational results

A question that remains open is why Li2RhO3 retains the monoclinic $C2/m$ symmetry and shows the zigzag-chain pattern of short Rh–Rh bonds under pressure, whereas $\alpha$-Li2IrO3 Hermann et al. (2018); Clancy et al. (2018) and $\alpha$-RuCl3 Bastien et al. (2018) become triclinic following the formation of metal-metal dimers.

In previous studies, the experimentally observed dimerization pattern in $\alpha$-Li2IrO3 and $\alpha$-RuCl3 was identified by DFT calculations within the GGA+SOC+$U$ scheme Hermann et al. (2018), as a consequence of a complex interplay of SOC, magnetism, correlation, and covalent bonding. Following this knowledge, we performed first full relaxations of Li2RhO3 as a function of hydrostatic pressure with and without SOC. As initial guess for the geometrical optimization at each pressure, we considered two structures. The experimental low-pressure ‘undistorted’ structure at 5 GPa and the experimental high-pressure ‘zigzag chain’ structure at 25.2 GPa. In our notation, we assume a structure to be undistorted when the corresponding bond disproportionation $l_{l}/l_{s}<1.04$. Moreover, for each of these initial geometries, we considered five different spin configurations, as explained in Section II.2.

Test calculations performed at 25 GPa reveal that after relaxation the structure becomes dimerized, regardless of the initial configuration. At a given pressure, the energetics of the various different configurations are obtained by comparing the corresponding enthalpies. Due to the underbinding problem of GGA (relaxed interatomic distances are longer than their experimental counterparts), the volume corresponding to 2 GPa reproduces the experimental volume at ambient pressure. This has been corrected by systematically subtracting $\Delta P=2$ GPa from all simulated pressure values.

The value of Hubbard correlation $U=1.5$ eV was chosen such that at 5 GPa (within GGA+SOC+$U$ scheme): (a) the optimized lowest-enthalpy magnetic configuration corresponds to $C2/m$ symmetry and reproduces the experimental value of $l_{l}/l_{s}$ = 1.02, and (b) the nonmagnetic configuration is dimerized (though parallel $X1/Y1$ type). The latter confirms dimerization at finite pressure for Li2RhO3  as magnetism is known to work against dimerization by pushing the transition pressure to a higher value Hermann et al. (2018).

At a pressure P_{c}^{\rm noSOC}$$\approx11 GPa, we find that Li2RhO3 undergoes a phase transition from a homogeneous to a dimerized phase with bond disproportionation $l_{l}/l_{s}$=1.146 within the GGA+$U$ scheme (not shown here). Upon dimerization, Li2RhO3 becomes nonmagnetic. However, there are a few discrepancies with the experimental structures: (i) below $P_{c}^{\rm noSOC}$, in the homogeneous structure, the shorter bond corresponds to the $Z1$ bond, rather than to the $X1$ and $Y1$ bonds as observed in experiments, and (ii) the dimerized phase does not have the $C2/m$ symmetry, rather it has the triclinic ($P{\overline{1}}$) symmetry, similar to $\alpha$-RuCl3 Bastien et al. (2018) and $\alpha$-Li2IrO3 Hermann et al. (2018); Clancy et al. (2018). The inclusion of SOC reproduces the shorter bond to be $X1/Y1$ in the homogeneous phase (below $P_{c}^{\rm SOC}$) and shifts the transition pressure $P{{}_{cm}}$ to 27 GPa. However, the high-pressure structure still becomes triclinic with $l_{l}/l_{s}$=1.15 (see Fig. 8).

The above results show that the experimental zigzag pattern of the short Rh–Rh bonds cannot be obtained from hydrostatic pressure simulations. We therefore proceed by simulating uniaxial pressure with the $b$ and $c$ parameters fixed to their experimental values at 25.5 GPa and the $b/a$ ratio varied systematically (Fig. 9). This approach yields the zigzag-chain structure observed experimentally. However, we had to increase the $b/a$ ratio to 1.95, in order to reproduce the ratio $l_{l}/l_{s}$ between the long and short bonds.

For obtaining the pressure evolution of Rh-Rh bonds under uniaxial condition, we next repeated the above calculations but this time with $b/a=$ 1.95, while fixing the $b$ and $c$ parameters to their experimental values at the corresponding pressure. By comparing the results of the hydrostatic and uniaxial pressure simulations, we conclude (see Fig. 10) that the evolution of Li2RhO3 up to $P_{c1}$ is compatible with hydrostatic pressure conditions, whereas at higher pressures the system progressively moves toward the behavior expected under uniaxial pressure. The uniaxial pressure accounts for the formation of zigzag chains instead of dimers, although it does not fully account for the evolution of the longer Rh–Rh bonds that evolve smoothly in the simulation but show a step-like anomaly experimentally (Fig. 5).

At ambient condition, Li2RhO3  is an insulator as shown in Ref. Mazin et al., 2013. Our calculated density of states (DOS) for the experimentally obtained structures at 25.2 GPa (Fig. 11) show that unlike other dimerized phases in $\alpha$-Li2IrO3 and $\alpha$-RuCl3, in Li2RhO3 the degeneracy between $yz$ and $xz$ orbitals of Rh $d$ states does not get lifted as the symmetry remains the same. Moreover, the system probably becomes metallic under pressure due to the formation of zigzag chains, which provide new hopping pathways.

The origin of the uniaxial-like pressure conditions requires further investigation. Experimental pressure conditions in a DAC with helium as pressure-transmitting medium are expected to be hydrostatic. Therefore, we consider the nature of the Li2RhO3 sample as a more plausible reason. In particular, stacking faults that occur, on average, at every 6-7 layers could act as a local strain and affect the evolution of the structure under pressure. Our data show that the concentration of stacking faults in Li2RhO3 is higher than in the polycrystalline samples of $\alpha$-Li2IrO3 and in single crystals of $\alpha$-Li2IrO3 that were used in our previous study Hermann et al. (2018). Interestingly, Na2IrO3 shows different pressure evolution of the crystal structure in powders Xi et al. (2018) and single crystals Hermann et al. (2017). Given the proclivity of Na2IrO3 to the formation of stacking faults, a similar mechanism may be operative there and deserves further systematic investigation.

IV Conclusion

In contrast to $\alpha$-Li2IrO3 and $\alpha$-RuCl3, where a dimerized triclinic phase is stabilized under pressure, Li2RhO3 retains its ambient-pressure monoclinic symmetry and develops zigzag chains of short Rh–Rh bonds. This structural phase transition is not abrupt, since traces of the low-pressure phase can still be found even at the highest pressure of 25.2 GPa, but above $\approx$14 GPa the high-pressure phase is dominant. Our density-functional calculations suggest that such a behavior is not anticipated in Li2RhO3 under hydrostatic pressure, where conventional dimerization should occur. On the other hand, uniaxial pressure may explain the experimental observations and promote the formation of zigzag chains instead of dimers.

Acknowledgements.

We thank the ESRF, Grenoble, for the provision of beamtime at ID15B. DKh, RV, AAT, and PG acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG), Germany, through TRR 80, SPP 1666, TRR 49, and SFB 1238. AJ acknowledges support from the DFG through Grant No. JE 748/1. SB thanks Stephen Winter for helpful discussions. AAT acknowledges financial support from the Federal Ministry for Education and Research via the Sofja-Kovalevskaya Award of Alexander von Humboldt Foundation, Germany.