Pressure-dependent magnetism of the Kitaev candidate Li2RhO3

Introduction

Honeycomb magnets with dominant Kitaev interactions are predicted to realize a spin-liquid state with emergent topological order and exotic excitations^1,2. Material realizations of this scenario are usually searched for among the low-spin d⁵ compounds following the initial proposal by Jackeli and Khaliullin³. Whereas several honeycomb iridates and Ru³⁺ halides have been extensively studied experimentally^4,5,6, rhodates remain relatively less explored despite the fact that Rh⁴⁺ is isoelectronic to Ir⁴⁺ and well suited for realizing Kitaev exchange.

Lithium rhodate, Li₂RhO₃, features a slightly deformed honeycomb lattice of the Rh⁴⁺ ions⁷ (see Fig. 1). Its silver analog, Ag₃LiRh₂O₆, can be prepared by an ion-exchange reaction⁸. Despite the rather similar structures of the honeycomb layer, these two compounds feature very different magnetic properties. Whereas the electronic state of Rh⁴⁺ in Li₂RhO₃ should be close to ({j}_{{rm{eff}}}=frac{1}{2}), akin to the typical Ir⁴⁺ compounds⁴, a departure from the ({j}_{{rm{eff}}}=frac{1}{2}) state has been detected in Ag₃LiRh₂O₆ by x-ray spectroscopy⁸. Li₂RhO₃ evades long-range magnetic order, but reveals a change in spin dynamics associated with spin freezing below 6 K^9,10, as confirmed by nuclear magnetic resonance and muon spectroscopies¹¹. By contrast, Ag₃LiRh₂O₆ develops long-range antiferromagnetic order below 90 K⁸, which is the highest Néel temperature among the d⁵ honeycomb magnets reported to date. The differences between Li₂RhO₃ and Ag₃LiRh₂O₆ are far more drastic than between the corresponding iridates, α-Li₂IrO₃ and Ag₃LiIr₂O₆, that reveal rather similar magnetic behavior with the long-range magnetic order below 15 K¹² and 8 K¹³, respectively. This comparison suggests that rhodates may be more tunable by (hydrostatic or chemical) pressure compared to the iridates.

Pressure-dependent magnetism of the Kitaev candidate Li2RhO3 — **Fig. 1: Structural details of Li₂RhO₃.**

In the following, we explore this possibility and study pressure evolution of Li₂RhO₃. X-ray diffraction experiments on this compound revealed the structural phase transition with the formation of linear Rh⁴⁺ chains above 6.5 GPa at room temperature¹⁴. It means that Li₂RhO₃ offers a broader pressure window for tuning magnetism of the Rh⁴⁺ honeycombs than different polymorphs of Li₂IrO₃¹⁵ that become nonmagnetic upon the structural dimerization transition already at 3.5–4.0 GPa at room temperature^{16,17,18,19,20} and at even lower pressures of 1.0–1.5 GPa on cooling^21,22. Here, we probe Li₂RhO₃ using magnetization measurements under pressure and follow the evolution of individual magnetic couplings in this material using ab initio and cluster many-body calculations. Previous quantum-chemistry studies reported Kitaev exchange as one of the leading terms in the Li₂RhO₃ spin Hamiltonian at ambient pressure²³.

Results

Magnetization under pressure

Figure 2a shows the inverse magnetic susceptibility H/M of Li₂RhO₃ measured as a function of temperature under various pressures. A temperature-independent term χ₀, which stands for the residual part of the background from the pressure cell, has been subtracted for each pressure, respectively. At all pressures, the susceptibility monotonically increases upon cooling, similar to the ambient-pressure behavior reported in the literature^9,11. The high-temperature part of magnetic susceptibility can be fitted with the Curie-Weiss law (solid lines), χ-χ₀ = C/(T − θ), where C is the Curie constant and θ is the Curie-Weiss temperature.

**Fig. 2: Curie-Weiss analysis of Li₂RhO₃ under pressure.**

From the fits to the data between 150 K and 300 K, we find that the Curie constant and the associated paramagnetic effective moment weakly decrease with pressure, whereas the Curie-Weiss temperature increases and even changes sign from negative to positive (Fig. 2), indicating the growth of ferromagnetic interactions. At ambient pressure, the paramagnetic effective moment is around 2.4 μ_B in agreement with the previous studies¹¹. It decreases with pressure and at 2 GPa reaches 1.8 μ_B, which is close to 1.73 μ_B expected for the ({j}_{{rm{eff}}}=frac{1}{2}) state of Rh⁴⁺.

At low temperatures, the smooth evolution of the magnetic susceptibility is consistent with the absence of any magnetic transition. The spin freezing reported below 6 K at ambient pressure^9,11 can be tracked by the bifurcation of the magnetic susceptibilities measured under field-cooled (FC) and zero-field-cooled (ZFC) conditions in low applied fields. Such measurements become quite challenging under pressure because of the weak signal in low magnetic fields. Nevertheless, it was possible to track the FC/ZFC susceptibilities of Li₂RhO₃ in the applied field of 0.1 T up to 3.46 GPa (Fig. 3). The spin-glass like behavior of Li₂RhO₃ remains almost unchanged upon compression. The bifurcation of the FC/ZFC susceptibilities is seen at around 5.0 K at all pressures.

**Fig. 3: Temperature-dependent dc magnetic susceptibility M(T) of Li₂RhO₃ measured at various pressures in a magnetic field of 0.1 T for run No. 3.**

Microscopic magnetic model

Figure 4 shows the density of states for Li₂RhO₃ calculated on the full-relativistic (PBE+SO) level. Typically for a transition-metal oxide, the bands near the Fermi level are dominated by the d-states that are split into the t_2g and e_g manifolds by the octahedral crystal field. The t_2g − e_g splitting is about 3.0 eV compared to the splitting of 2.0 eV in α-RuBr₃²⁴ and 2.2 eV in α-RuCl₃²⁵, in agreement with the higher negative charge of O²⁻.

**Fig. 4: Full-relativistic (PBE+SO) density of states for Li₂RhO₃ calculated without spin polarization at ambient pressure.**

From the orbital energies obtained via the Wannier fit, we extract a small noncubic crystal-field splitting of only 13 meV, which is well below the spin-orbit coupling constant for Rh⁴⁺. However, the t_2g states do not show the splitting into ({j}_{{rm{eff}}}=frac{3}{2}) and ({j}_{{rm{eff}}}=frac{1}{2}) bands known from the Ir⁴⁺ compounds. There is instead a three-peak structure reminiscent of Na₂IrO₃²⁶, with the peaks at -1.0, -0.7, and 0.0 eV. This splitting into three sub-bands is a fingerprint of the dominant off-diagonal hopping (t₂) that takes place between, e.g., the d_yz and d_xz orbitals and gives rise to the large Kitaev coupling²⁷. Our direct calculation of the exchange tensor confirms this assessment.

We define the exchange tensors in the Kitaev coordinate frame as

$${mathbb{J}}=left(begin{array}{ccc}J&Gamma &{Gamma }^{{prime} }\ Gamma &J&{Gamma }^{{prime} }\ {Gamma }^{{prime} }&{Gamma }^{{prime} }&J+Kend{array}right)$$

(1)

with four independent parameters (J, K, Γ, and ({Gamma }^{{prime} })). Technically, the X– and Y– bonds have a lower symmetry with six independent parameters²⁸, but our calculations show that the approximate form given by Eq. (1) is sufficiently accurate for these bonds as well, so it will be used in the following. The X– and Y– bonds are symmetry-equivalent but different from the Z-bonds. This difference is also relatively small at all pressures, as can be seen in the Supplementary Information. Therefore, in the rest of this work we will discuss the couplings averaged over the X-, Y-, and Z-bonds of the honeycomb lattice. We label these averaged couplings as (bar{J}), (bar{K}), (bar{Gamma }), etc.

Figure 5 shows such averaged exchange couplings as a function of pressure. Li₂RhO₃ is dominated by the ferromagnetic nearest-neighbor Kitaev term K₁ that decreases in magnitude upon compression. The off-diagonal anisotropy Γ₁ increases under pressure, with the ({bar{Gamma }}_{1}/| {bar{K}}_{1}|) ratio changing from 0.34 at 0 GPa to 0.60 at 4.5 GPa. The nearest-neighbor Heisenberg coupling is weakly ferromagnetic and also increases in magnitude upon compression.

**Fig. 5: Pressure evolution of the averaged magnetic couplings in Li₂RhO₃ calculated by exact cluster diagonalization (ED) and second-order perturbation theory (ESOPT).**

These pressure-induced changes in the nearest-neighbor couplings are well in line with the structural changes upon compression. Indeed, the Rh–O–Rh bond angles systematically decrease, leading to a reduction in the off-diagonal hopping t₂ and the weakening of the Kitaev term relative to the other terms of the exchange tensor²⁸. This mechanism appears to be generic for the Kitaev magnets that all show the reduction in the bond angles under hydrostatic pressure and the gradual suppression of the Kitaev term^17,22,24. The evolution of Li₂RhO₃ can also be followed on the phase diagram of the J₁ − K₁ − Γ₁ model (Fig. 6) where pressure systematically shifts the system away from the Kitaev limit located at φ = 3π/2.

**Fig. 6: Li₂RhO₃ placed onto the classical phase diagram of the J₁ − K₁ − Γ₁ model⁴⁷ with ({Gamma }^{{prime} }/A=-0.10).**

Turning to the smaller terms in the spin Hamiltonian, we note that the off-diagonal anisotropy ({Gamma }^{{prime} }) is below 1 meV at all pressures. Its negative sign should increase the proclivity of Li₂RhO₃ for zigzag order, as shown in the phase diagram of Fig. 6. However, the main term stabilizing the zigzag order is believed to be the antiferromagnetic third-neighbor coupling J₃ that has been estimated at about 1−3 meV in α-RuCl₃^4,29,30,31 and 2−6 meV in Na₂IrO₃^28,32,33,34. Surprisingly, the ({bar{J}}_{3}) of Li₂RhO₃ is quite small, about 0.15 meV according to our cluster many-body calculations. This suppression of ({bar{J}}_{3}) may be a result of the smaller spatial extent of the Rh 4d orbitals compared to the Ir 5d orbitals of the iridates, and of the O 2p orbitals compared to the Cl 3p orbitals of α-RuCl₃. Interestingly, ({bar{J}}_{3}) obtained by the cluster many-body calculations is much lower than in the superexchange model. It means that the hoppings to the e_g orbitals yield ferromagnetic contributions that are strong enough to compensate for antiferromagnetic contributions from the intra-t_2g hoppings.

Discussion

Li₂RhO₃ remains a forgotten sibling of the much better known Kitaev iridates and Ru³⁺ halides. Although the first reports of its magnetic properties^9,10 even preceded the discovery of α-RuCl₃ as a Kitaev candidate, relatively little is known about its microscopic regime. The quantum-chemistry study of Li₂RhO₃ demonstrated a strong Kitaev coupling but also reported an unusually strong spatial anisotropy with ({J}_{1}^{Z}=-10.2) meV having a different sign than ({J}_{1}^{X}=2.4) meV²³.

Our study advocates a more conventional microscopic scenario where spatial anisotropy plays only a minor role, and the couplings on the X/Y– and Z-bonds are qualitatively and quantitatively similar to each other. Moreover, the position of Li₂RhO₃ on the phase diagram of the extended Kitaev (J₁ − K₁ − Γ₁) model should resemble that of the Ru³⁺ halides, with the leading ferromagnetic Kitaev term K₁ < 0 and the subleading off-diagonal anisotropy Γ₁ > 0. The nearest-neighbor Heisenberg exchange, J₁ < 0, is relatively small at ambient pressure but becomes increasingly more important upon compression. This trend is corroborated by our magnetization measurements. Indeed, the powder-averaged Curie-Weiss temperature can be calculated as²⁸

$$theta =-(3{bar{J}}_{1}+{bar{K}}_{1})/4{k}_{B}$$

(2)

and does not depend on ({bar{Gamma }}_{1}). Whereas ({bar{K}}_{1}) decreases in magnitude with pressure, the leading trend is determined by the enhancement of ferromagnetic ({bar{J}}_{1}) that causes the increase in θ upon compression (Fig. 5g). Despite this good qualitative agreement, we note that the slope of the calculated pressure dependence is much lower compared to the experiment. Similar discrepancies have been reported in other Kitaev materials and ascribed to deviations from the simple Curie-Weiss law caused by the temperature dependence of the paramagnetic effective moment³⁵.

Although Li₂RhO₃ does not approach the Kitaev limit, the combination of ({bar{K}}_{1} < 0) and ({bar{Gamma }}_{1} > 0) as the dominant terms in the spin Hamiltonian is a precondition for the strong frustration that has been comprehensively studied in the context of the K₁ − Γ₁ model on the honeycomb lattice³⁶. This model connects the limits of the Kitaev spin liquid at large K₁ and classical spin liquid at large Γ₁, whereas the intermediate region is often described as correlated paramagnet³⁶. With Li₂RhO₃ lying close to the K₁ − Γ₁ line on the phase diagram of the ({J}_{1}-{K}_{1}-{Gamma }_{1}-{Gamma }_{1}^{{prime} }) model, at least at ambient pressure, it is not surprising that this material evades long-range magnetic order. Experimentally, we find spin freezing below 5 K that persists at least up to 3.46 GPa. Interestingly, the freezing temperature almost does not change, whereas individual exchange couplings are clearly affected by pressure. This observation suggests that spin freezing is driven by an extrinsic energy scale and may be associated with the structural disorder arising from stacking faults^10,16. The high-pressure x-ray diffraction study of Li₂RhO₃ shows that the concentration of stacking faults is almost unchanged within the pressure range of our study¹⁴.

Although stacking faults are unavoidable in almost all Kitaev candidates because of their layered nature^12,13,37,38, this structural disorder will usually not preclude magnetic ordering. Na₂IrO₃, α-RuCl₃, and α-RuBr₃ all show zigzag magnetic order, which is stabilized by ({Gamma }_{1}^{{prime} }) and J₃. Whereas ({Gamma }_{1}^{{prime} }) is a comparably minor term across the whole family of the existing Kitaev materials, Li₂RhO₃ stands apart from the other Kitaev candidates in that its J₃ is unusually small. One can then interpret the absence of magnetic order in Li₂RhO₃ as the joint effect of the frustration caused by K₁ − Γ₁ and the weakness of J₃. Additionally, the existing Li₂RhO₃ samples¹⁴ show about twice higher concentration of stacking faults compared to Na₂IrO₃³². This increased amount of structural disorder should increase the proclivity for spin freezing.

Finally, we note that our study supports the general trend of tuning Kitaev candidates away from the Kitaev limit by hydrostatic pressure¹⁵. One would then expect negative pressure to enhance the Kitaev term, reduce Γ₁/∣K₁∣, and bring the materials closer to the Kitaev limit. In this context, it is somewhat surprising that Ag₃LiRh₂O₆, the expanded version of the Li₂RhO₃ structure, not only shows magnetic ordering, but also features the highest Néel temperature among all Kitaev candidates reported to date⁸. The negative pressure effects in honeycomb rhodates may be nontrivial and clearly deserve a further dedicated investigation.

Methods

Sample synthesis and characterization

Polycrystalline samples of Li₂RhO₃ used in this work were previously characterized in refs. ^11,14. Magnetization under pressure was measured with the same method as in ref. ³⁹ using the gasket with the sample chamber diameter of 0.9 mm that can reach pressures up to 2 GPa. In order to reach higher pressures, the gasket was pre-indented, and pressures up to 3.46 GPa could be reached in run No. 3. A piece of Pb served to determine pressure from the temperature of its superconducting transition. Daphne oil 7373 was used as pressure-transmitting medium.

DFT and cluster many-body calculations

Density-functional band-structure calculations were performed in the FPLO code⁴⁰ using the Perdew-Burke-Ernzerhof (PBE) version of the exchange-correlation potential⁴¹. Hopping parameters were obtained by the built-in Wannierization procedure of FPLO⁴². Previous studies reported only the lattice parameters of Li₂RhO₃ as a function of pressure¹⁴, whereas oxygen positions had a large uncertainty due to limitations of the x-ray powder diffraction data. Therefore, we chose to fix the lattice parameters to their experimental values at each pressure and relaxed the atomic positions using DFT+U calculations in FPLO, performing force optimization until the residual forces were less than 1 × 10⁻³ eV/Å. At ambient pressure, we obtained the Rh–O distances (R₁, R₂, R₃ in Fig. 1a) of 2.0395/1.9713/2.0379 Å, the Rh–Rh distances (Z, XY) of 2.9812/2.9315 Å, and the Rh–O–Rh bond angles (labeled as δ₁, δ₂) of 93.91/94.01°, which are in a good agreement with the experimental values⁷ of 2.033(6)/2.001(7)/2.029(3)/Å (Rh–O1/Rh–O1/Rh–O2), 2.950(4)/2.954(5) Å (Rh–Rh), and 94.16/93.29° (Rh–O1–Rh/Rh–O2–Rh). We also note that the experimental structural data for Li₂RhO₃ feature a weak Li/Rh site mixing⁷. This site mixing is caused by the interlayer disorder, such as stacking faults, whereas individual honeycomb layers are well-ordered⁸. Therefore, we used the fully ordered structural model in our calculations.

Magnetic couplings in Li₂RhO₃ are defined by the general spin Hamiltonian,

$${mathcal{H}}=sum _{langle ijrangle }{{bf{S}}}_{i}{{mathbb{J}}}_{ij}{{bf{S}}}_{j}$$

(3)

where ({{mathbb{J}}}_{ij}) is the exchange tensor for the respective bond, and the summation is over bonds. The ({{mathbb{J}}}_{ij}) components were determined by two complementary approaches. In the superexchange model, the hoppings within the t_2g manifold of the scalar-relativistic band structure are used to calculate the exchange couplings as described in ref. ²⁸. Weak crystal-field splittings within the t_2g manifold and virtual processes involving the e_g states are neglected in this method.

Cluster many-body calculations allow a comprehensive treatment of the microscopic processes that underlie the exchange couplings. In order to calculate exchange couplings, we start by deriving the electronic Hamiltonian in terms of Rh 4d orbitals Wannier basis⁴². We obtain this Hamiltoninan by performing fully relativistic (PBE+SO) calculations using a k– grid of 12 × 12 × 12 and retaining the translational symmetry of the system. The obtained electronic Hamiltonian is exactly diagonalized on two-site clusters to get the low-energy eigenstates. These eigenstates are then projected to pure spin states by following the des Cloizeaux effective Hamiltonian method⁴³ to obtain the intermediate states which are finally orthonormalized by employing the symmetric (Löwden) approach⁴⁴. The advantage of the above described procedure is that it preserves all the symmetries and includes the effects of the non-cubic crystal-field splitting and nominally empty e_g orbitals that are neglected in the superexchange model.

In both cases, we used the same parameters of U = 2.58 eV and J_H = 0.29 eV for the on-site Coulomb repulsion and Hund’s coupling, respectively, as determined for α-RuCl₃⁴⁵. The spin-orbit coupling λ = 0.15 eV was used in the superexchange model.