Cubic symmetry and magnetic frustration on the $fcc$ spin lattice in K$_2$IrCl$_6$

TL;DR

This study investigates the magnetic properties and exchange interactions in the cubic K$_2$IrCl$_6$ compound, revealing a complex interplay of Heisenberg and Kitaev interactions under high-symmetry conditions, supported by experimental and ab initio results.

Contribution

It provides detailed experimental and theoretical analysis of exchange interactions in K$_2$IrCl$_6$, highlighting its potential as an ideal cubic model for studying magnetic frustration.

Findings

Cubic symmetry persists down to 20 K.

Antiferromagnetic Kitaev exchange is about half of the Heisenberg exchange.

Weak second-neighbor coupling influences magnetic ground state.

Abstract

Cubic crystal structure and regular octahedral environment of Ir$^{4+}$ render antifluorite-type K$_2$IrCl$_6$ a model fcc antiferromagnet with a combination of Heisenberg and Kitaev exchange interactions. High-resolution synchrotron powder diffraction confirms cubic symmetry down to at least 20 K, with a low-energy rotary mode gradually suppressed upon cooling. Using thermodynamic and transport measurements, we estimate the activation energy of $\Delta\simeq 0.7$ eV for charge transport, the antiferromagnetic Curie-Weiss temperature of $\theta_{\rm CW}\simeq -43$ K, and the extrapolated saturation field of $H_s\simeq 87$ T. All these parameters are well reproduced \textit{ab initio} using $U_{\rm eff}=2.2$ eV as the effective Coulomb repulsion parameter. The antiferromagnetic Kitaev exchange term of $K\simeq 5$ K is about one half of the Heisenberg term $J\simeq 13$ K. While this combination removes a large part of the classical ground-state degeneracy, the selection of the unique magnetic ground state additionally requires a weak second-neighbor exchange coupling $J_2\simeq 0.2$ K. Our results suggest that K$_2$IrCl$_6$ may offer the best possible cubic conditions for Ir$^{4+}$ and demonstrates the interplay of geometrical and exchange frustration in a high-symmetry setting.

Tables

Table 1. Table 1: Crystallographic parameters and details of the structure refinement for K 2 IrCl 6 at 300 K and 20 K. The atomic displacement parameters U iso subscript 𝑈 iso U_{\rm iso} are given in Å 2 . The error bars are from the Rietveld refinement. All the crystallographic sites are fully occupied.

$T\left(K\right)$	300 K	20 K
Space group	$Fm\overline{3}m$	$Fm\overline{3}m$
$a=b=c$ (Å)	9.77050(3)	9.66289(3)
$\alpha =\beta =\gamma$	90∘	90∘
$V$(Å3)	932.718(5)	902.238(4)
$R_I/R_P$	0.0400/0.0902	0.0223/0.0928
Atomic parameters
	$x/a=0$	$x/a=0$
Ir	$y/b=0$	$y/b=0$
	$z/c=0$	$z/c=0$
	$U_{\mathrm{iso}}=0.01647\left(9\right)$	$U_{\mathrm{iso}}=0.00163\left(6\right)$
	$x/c=\frac{1}{4}$	$x/a=\frac{1}{4}$
K	$y/b=\frac{1}{4}$	$y/b=\frac{1}{4}$
	$z/c=\frac{1}{4}$	$z/c=\frac{1}{4}$
	$U_{\mathrm{iso}}=0.0387\left(4\right)$	$U_{\mathrm{iso}}=0.0061\left(2\right)$
	$x/a=0.23708\left(10\right)$	$x/a=0.24034\left(12\right)$
Cl	$y/b=0$	$y/b=0$
	$z/c=0$	$z/c=0$
	$U_{\mathrm{iso}}=0.0357\left(3\right)$	$U_{\mathrm{iso}}=0.0057\left(2\right)$

Table 2. Table S1: Hoppings and magnetic interactions calculated for different lattice parameters of K 2 IrCl 6 using U eff = 2.2 subscript 𝑈 eff 2.2 U_{\rm eff}=2.2  eV, J H = 0.3 subscript 𝐽 𝐻 0.3 J_{H}=0.3  eV, and λ = 0.4 𝜆 0.4 \lambda=0.4  eV, as explained in the main text.

$a$ (Å)	$t_1$ (meV)	$t_2$ (meV)	$t_3$ (meV)	$t_4$ (meV)	$J$ (K)	$K$ (K)	$\Gamma$ (K)
11.37	2.77	4.60	$-25.04$	0	0.84	0.47	0.17
10.37	4.30	7.97	$-52.78$	0	4.63	2.04	0.60
10.07	4.78	8.90	$-66.28$	0	7.78	3.20	0.84
9.77	5.42	9.61	$-82.87$	0	12.70	4.99	1.13
9.57	5.55	9.32	$-96.66$	0	18.16	6.77	1.27
9.17	6.67	7.02	$-130.06$	0	34.12	12.32	1.28

Equations

$$\rho (T)=\exp \left(\frac{\Delta }{2k_{B}T}\right),$$

$$\chi ={\chi }_0+\frac{C}{T-{\theta }_{\mathrm{CW}}},$$

$$C_{\mathrm{ph}}(T)=f_D{C}_D({\Theta }_D,T)+\sum _{i=1}^3{g}_i{C}_{E,i}({\Theta }_{E,i},T).$$

$$C_D({\Theta }_D,T)=9R{\left(\frac{T}{{\Theta }_D}\right)}^3{\int }_0^{{\Theta }_D/T}\frac{x^4{e}^x}{(e^x-1{)}^2}dx,$$

$$C_{E,i}({\Theta }_{E,i},T)=3R{\left(\frac{{\Theta }_{E,i}}{T}\right)}^2\frac{\exp ({\Theta }_{E,i/T})}{[\exp ({\Theta }_{E,i/T})-1{]}^2},$$

$$S_{\mathrm{mag}}(T)={\int }_0^T\frac{C_{\mathrm{mag}}}{T^{\prime }}d{T}^{\prime },$$

$$H=\sum _{⟨ij⟩}{\mathbf{S}}_i{\mathbb{J}}_{ij}{\mathbf{S}}_j,$$

$${\mathbb{J}}_{ij}=\left(\begin{array}{ccc}J+{\Gamma }_{xx}& D_z+{\Gamma }_{xy}& -D_y+{\Gamma }_{xz}\\ -D_z+{\Gamma }_{yx}& J+{\Gamma }_{yy}& D_x+{\Gamma }_{yz}\\ D_y+{\Gamma }_{zx}& -D_x+{\Gamma }_{zy}& J+{\Gamma }_{zz}\end{array}\right),$$

$${\mathbb{J}}_{xy}=\left(\begin{array}{ccc}J& \pm \Gamma & 0\\ \pm \Gamma & J& 0\\ 0& 0& J+K\end{array}\right)$$

$${\mathbb{J}}_{xz}=\left(\begin{array}{ccc}J& 0& \pm \Gamma \\ 0& J+K& 0\\ \pm \Gamma & 0& J\end{array}\right)$$

$${\mathbb{J}}_{yz}=\left(\begin{array}{ccc}J+K& 0& 0\\ 0& J& \pm \Gamma \\ 0& \pm \Gamma & J\end{array}\right).$$

$$\mathbb{T}=\left(\begin{array}{ccc}{t}_1& t_2& t_4\\ t_2& t_1& t_4\\ t_4& t_4& t_3\end{array}\right)$$

$$J=\frac{4\mathbb{A}}{9}{\left(2t_1+t_3\right)}^2-\frac{8\mathbb{B}}{9}\left\{9t_4^2+2(t_1-t_3{)}^2\right\}$$

$$K=\frac{8\mathbb{B}}{3}\left\{(t_1-t_3{)}^2+3t_4^2-3t_2^2\right\}$$

$$\Gamma =\frac{8\mathbb{B}}{3}\left\{2t_2(t_1-t_3)+3t_4^2\right\}$$

$$\mathbb{A}=-\frac{1}{3}\left\{\frac{J_H+3(U_{\mathrm{eff}}+3\lambda )}{6J_H^2-U_{\mathrm{eff}}(U_{\mathrm{eff}}+3\lambda )+J_H(U_{\mathrm{eff}}+4\lambda )}\right\}$$

$$\mathbb{B}=\frac{4}{3}\left\{\frac{(3J_H-U_{\mathrm{eff}}-3\lambda )}{(6J_H-2U_{\mathrm{eff}}-3\lambda )}\eta \right\}$$

$$\eta =\frac{J_H}{6J_H^2-J_H(8U_{\mathrm{eff}}+17\lambda )+(2U_{\mathrm{eff}}+3\lambda )(U_{\mathrm{eff}}+3\lambda )}.$$

Full text

I Introduction

Strong spin-orbit coupling is an essential ingredient of correlated insulators with $5d$ transition metals, such as Ir4+ that typically features an octahedral oxygen coordination along with five electrons and one hole in the $t_{2g}$ shell. In the absence of any additional crystal-field splitting, spin-orbit coupling separates the $t_{2g}$ states into the lower-lying $j_{\rm eff}=\frac{3}{2}$ and higher-lying $j_{\rm eff}=\frac{1}{2}$ manifolds, with the latter forming a half-filled band gapped by even moderate electronic correlations Rau et al. (2016). This general scenario has been exemplified in more than a dozen of iridates studied over the last decade Cao and Schlottmann (2018), although the symmetry of Ir4+ is usually lower than cubic, thus leading to crystal-field splittings within the $t_{2g}$ shell Calder et al. (2014); Rossi et al. (2017) or, in cases like CaIrO3 Moretti Sala et al. (2014); Kim et al. (2015) and Sr3CuIrO6 Liu et al. (2012), even to profound deviations from the $j_{\rm eff}=\frac{1}{2}$ scenario.

The quest for cubic systems based on Ir4+ is triggered by interesting predictions for the magnetic interactions that would arise in this setting Rau et al. (2016); Winter et al. (2017). It has been proposed that the combination of the $j_{\rm eff}=\frac{1}{2}$ state and $90^{\circ}$ Ir–O–Ir superexchange leads to the bond-directional (Kitaev) anisotropy of exchange interactions Jackeli and Khaliullin (2009), which, in turn, has broad implications for exotic quantum states and even topological quantum computing Kitaev (2006). This physics is presently explored in the honeycomb iridates A2IrO3 (A = Li, Na) and related materials Winter et al. (2017); Hermanns et al. (2018).

Here, we report on the crystal and electronic structures as well as the magnetic behavior of K2IrCl6, an Ir4+ compound that retains its cubic symmetry and, thus, the ideal octahedral coordination of Ir4+ down to low temperatures. This renders K2IrCl6 an interesting model material that combines the geometrically frustrated (fcc) arrangement of the Ir4+ ions with sizable exchange anisotropy, a rare case among the $5d$ materials. We confirm the frustrated nature of K2IrCl6 experimentally, derive the relevant microscopic parameters, and discuss the extent of exchange anisotropy in this compound.

K2IrCl6 belongs to the K2PtCl6 family of cubic antifluorite-type A2MX6 hexahalides that feature isolated MX6 octahedra arranged on the fcc lattice and separated by alkali-metal cations (Fig. 1, left) Armstrong (1980). The magnetic behavior of K2IrCl6 was reported back in 1950’s Cooke et al. (1959); Griffiths et al. (1959), but, surprisingly, even the exact crystal structure was not determined, and no microscopic information for this compound is available to date. Thermodynamic measurements Cooke et al. (1959); Bailey and Smith (1959); Willemsen et al. (1977); Moses et al. (1979) and neutron diffraction Hutchings and Windsor (1967); Minkiewicz et al. (1968); Lynn et al. (1976) suggest the onset of magnetic order below $T_{N}\simeq 3$ K with the collinear spin structure and a possible field-induced magnetic transition Meschke et al. (2001).

II Methods

Powder samples of K2IrCl6 are commercially available from Alfa Aesar (Ir 39% min) and were used without further purification. Sample quality was confirmed by powder x-ray diffraction (XRD) data collected at the Rigaku MiniFlex diffractometer (CuKα radiation). High-resolution XRD was performed at the MSPD beamline Fauth et al. (2013) of the ALBA synchrotron facility ($\lambda=0.4129$ Å) and at the ID22 beamline of the ESRF ($\lambda=0.35456$ Å). The sample was placed into a thin-wall borosilicate capillary and cooled using the He cryostat. The capillary was spun during the data collection. Diffracted signal was recorded by 14 (ALBA) and 9 (ESRF) point detectors preceded by Si (111) analyzer crystals. Jana2006 software was used for the structure refinement Petr̆íc̆ek et al. (2014). No crystalline impurity phases were detected in the commercial samples from Alfa Aesar within the sensitivity of the synchrotron measurement (about 0.5 wt.%).

Temperature and field dependence of the dc magnetization was measured using the Quantum Design SQUID-VSM magnetometer (MPMS 3). Specific heat was measured in a Quantum Design Physical Properties Measurement System (QD-PPMS) using the relaxation method. The dc electrical resistivity was also measured in the PPMS using the standard four-probe technique. Electrical contacts were attached with the high-conducting silver paste. The two voltage leads were separated by 0.79 mm, and the current was flowing through a cross-section of about 2.5 by 0.45 mm2. The powder was pressed into plate-like samples for the heat-capacity and resistivity measurements.

High-field magnetization data up to 56 T were collected in the Dresden High Magnetic Field Laboratory using a pulsed magnet. Experimental details of the measurement can be found elsewhere Tsirlin et al. (2009). The collected high-field magnetization data were scaled using the magnetization data measured with the SQUID-VSM in static fields up to 7 T.

Density-functional (DFT) band-structure calculations were performed within the FPLO code Koepernik and Eschrig (1999) using the experimental crystal structure determined at 20 K and local density approximation (LDA) for the exchange-correlation potential Perdew and Wang (1992). The $8\times 8\times 8$ $k$-mesh was used for integration over the Brillouin zone. Additionally, we explored changes in the magnetic interactions in K2IrCl6 under the effect of strain that was modeled by changing the cubic lattice parameter and relaxing the Cl position until residual forces were below 0.001 eV/Å.

Strong local correlations were included on the mean-field level via the DFT+$U$ procedure with the on-site Coulomb repulsion $U_{d}$ and Hund’s coupling $J_{d}$ acting on the $5d$-states of Ir atoms, and the atomic-limit flavor of the double-counting correction. Hopping parameters for the $t_{2g}$ states were extracted from the LDA band structure using Wannier functions implemented in FPLO.

III Results

III.1 Crystal structure

Many of the K2PtCl6-type compounds undergo symmetry-lowering transitions upon cooling Rössler and Winter (1977). Therefore, we verified the cubic symmetry of K2IrCl6 using high-resolution synchrotron XRD and also refined the crystal structure, as no structural information was available in the literature. No deviations from the face-centered cubic symmetry are observed down to 20 K (Fig. 2), and the temperature-independent peakwidth of $0.007^{\circ}$ for the 111 reflection suggests excellent crystallinity of the sample. Moreover, the absence of any thermodynamic anomalies below 20 K and down to the magnetic ordering transition (see below) implies that K2IrCl6 should retain its cubic symmetry down to at least $T_{N}\simeq 3$ K.

The lattice shrinks upon cooling, as seen from the refined lattice parameters, compare $a_{\rm 300K}=9.77050(3)$ Å at 300 K to $a_{\rm 20K}=9.66289(3)$ Å at 20 K. This corresponds to a 3% volume reduction. Potential signatures of the symmetry lowering can be seen in the relatively high atomic displacement parameters of K and Cl at 300 K (Table 1). However, both displacements are significantly reduced upon cooling and drop well below 0.01 Å2 at 20 K (Fig. 2), suggesting the presence of a soft phonon mode but no static disorder. This is notably different from another Ir-based fcc antiferromagnet, the metrically cubic perovskite Ba2CeIrO6, where atomic displacements of oxygen remain well above 0.01 Å2 even at 100 K indicating static local disorder Revelli et al. .

Thermal ellipsoid of Cl is stretched along the direction perpendicular to the Ir–Cl bond (Fig. 2). This would be typical for a rotary mode (cooperative rotations of the IrCl6 octahedra), which is indeed common among the A2IrX6 antifluorite compounds Rössler and Winter (1977); Lynn et al. (1978); Armstrong (1980). The reduction in the displacements upon cooling indicates the gradual suppression of such a mode in K2IrCl6, and underpins the absence of local distortions in this cubic compound at low temperatures.

The Ir–Cl distance changes from 2.3164(10) Å at 300 K to 2.3224(11) Å at 20 K. The Cl–Ir–Cl angles are fixed at $90^{\circ}$ by the cubic symmetry, resulting in the regular IrCl6 octahedra.

III.2 Resistivity

K2IrCl6 is a robust insulator. Its resistivity increases upon cooling from 390 K to 300 K and exceeds 2 M$\Omega$ cm at room temperature. This confirms the anticipated Mott-insulating nature of the compound. The inset of Fig. 3 shows that the $T$ dependence of $\rho$ is well described by the activation behavior

[TABLE]

where $\Delta$ is the activation energy and $k_{B}$ is the Boltzmann constant. From the linear fit to the $\ln\rho$ vs $1/T$ curve, we estimate $\Delta\sim 0.7$ eV. The $\Delta$ value for K2IrCl6 is larger than that for the Ir4+ oxides, such as Na2IrO3 ($\Delta\simeq 0.35$ eV) Manni (2014) and La2MgIrO6 ($\Delta\simeq 0.16$ eV) Cao et al. (2013). Although the exact $\Delta$ value for the polycrystalline sample may be affected by grain boundaries, the difference from the Ir4+ oxides appears large enough to conclude that K2IrCl6 demonstrates higher ionicity. This is compatible with the results of our computational analysis presented in Sec. III.5 below.

III.3 Magnetization

The temperature-dependent dc-magnetic susceptibility $\chi=M/H$ measured under the $H=0.1$ T applied magnetic field is shown in Fig. 4a. Upon cooling, the susceptibility curve exhibits a broad maximum at $T_{\rm max}\sim$ 6.0 K implying the onset of short-range spin-spin correlations. Below 6.0 K, the susceptibility decreases down to 2 K with the maximum in the Fisher’s heat capacity $d(\chi T)/dT$ located around 3.1 K, where a magnetic transition was reported in previous studies Cooke et al. (1959); Minkiewicz et al. (1968). The susceptibility data above 100 K have been fitted using the following expression,

[TABLE]

where $\chi_{0}$ is the temperature-independent contribution due to the core diamagnetism ($\chi_{\rm core}$) and Van Vleck paramagnetism ($\chi_{\rm VV}$). The second term represents the Curie-Weiss law with the Curie constant $C$ and Curie-Weiss temperature $\theta_{\rm CW}$. The Curie constant is given by $C=N_{\rm A}\mu_{\rm eff}^{2}/3k_{\rm B}$, where $N_{\rm A}$ is Avogadro’s number, $\mu_{\rm eff}$ is the effective magnetic moment, and $k_{\rm B}$ is the Boltzmann constant.

The least-square fitting with Eq. (2) above 100 K returns $\chi_{0}=-8.53(9)\times 10^{-5}$ emu/mol, $C=0.374(1)$ emu K/mol, and $\theta_{\rm CW}=-42.6(1)$ K. The negative value of $\theta_{\rm CW}$ implies predominant AFM exchange interaction between the Ir4+ ions in K2IrCl6. The frustration parameter, $f=|\theta_{\rm CW}/T_{\rm N}|$, is estimated to be 13.7 and suggests the presence of strong magnetic frustration in K2IrCl6.

The effective magnetic moment, $\mu_{\rm eff}=1.73$ $\mu_{\rm B}$ is in excellent agreement with the expected value $\mu_{\rm eff}=g\sqrt{J(J+1)}\mu_{\rm B}$ using the Landé $g$-factor $g=2$ for the ideal $j_{\rm eff}=\frac{1}{2}$ state expected for Ir4+ in the cubic crystal field. The temperature-independent contribution $\chi_{0}$ includes the Van Vleck part $\chi_{\rm VV}$ and the core part $\chi_{\rm core}$=$-$2$\times$10*-4* emu/mol Bain and Berry (2008). The Van Vleck susceptibility is then evaluated by subtracting $\chi_{\rm core}$ from $\chi_{0}$, resulting in $\chi_{\rm VV}$= 1.1$\times$10*-4* emu/mol very similar to $1.05\times 10^{-4}$ emu/mol reported for Na2IrO3 Mehlawat et al. (2017).

Figure 4(b) shows temperature-dependent susceptibility measured under different magnetic fields. Below $T_{\rm max}$, the susceptibility increases with increasing the field, and the transition at $T_{N}$ is gradually smeared out, but can be still observed in $d(\chi T)/dT$. The ordering temperature $T_{N}$ changes insignificantly. For example, the peak in $d(\chi T)/dT$ shifts from 3.14 K at 0.01 T to 3.07 K at 7 T sup .

Linear field dependence of the magnetization measured at 1.6 K (Fig. 4(b)) persists up to 56 T, the highest field of our experiment. The scaling against the low-field data measured in static fields suggests that at 56 T the magnetization reaches 63.8% of the expected saturation value of $M_{s}=gS\mu_{\rm B}=1.0$ $\mu_{\rm B}$/Ir4+. Linear extrapolation to $M_{s}$ yields the saturation field of $H_{\rm sat}\sim 87$ T.

III.4 Heat capacity

Fig. 5(a) shows the heat capacity of K2IrCl6. A broad hump around 5 K is due to the magnetic contribution, which remains unchanged even in the applied field of 14 T. This field only weakly polarizes the system to produce about 15 % of the maximum magnetization $M_{s}$, thus having no significant effect on the magnetic contribution to the specific heat. Interestingly, neither our magnetization data nor the specific heat reveal any signatures of the field-induced transition reported in Ref. Meschke et al., 2001 at about 5 T. However, these signatures may be weak and not easily resolvable in a polycrystalline sample.

To determine the magnetic contribution to the specific heat, the phonon part $C_{\rm ph}(T)$ was estimated by fitting experimental heat capacity above 35 K with an empirical model that involves a superposition of one Debye-type and three Einstein-type terms as follows Koteswararao et al. (2014):

[TABLE]

The Debye term $C_{D}(\Theta_{D},T)$ is given by

[TABLE]

and the Einstein term $C_{E,i}(\Theta_{E,i},T)$ is given by

[TABLE]

where $R$ is the universal gas constant, $k_{B}$ is the Boltzmann constant, $\Theta_{D}$ and $\Theta_{E}$ are the Debye and Einstein temperatures, respectively. In this combined Debye-Einstein ($D+E$) model, the total number of vibration modes $n=9$ is the total number of atoms in the formula unit.

A stable least-square fit of Eq. (3) to the $C_{p}(T)$ data above $T=35$ K has been obtained by a combination of one Debye term and three Einstein terms with 24% of the total modes contributed by the Debye term [Fig. 5(a)]. During the fitting procedure, the Debye temperature $\Theta_{D}$ was kept fixed to that reported for the iso-structural non-magnetic K2PtCl6 compound, $\Theta_{D}$=91.1(5) K Moses et al. (1979). The fit yields three Einstein temperatures corresponding to the three Einstein terms of the fitted model as $\Theta_{E}$= 459(7), 242(4), and 136(2) K. These Einstein modes can be compared to the relatively flat phonon modes in the isostructural K2OsCl6 in the energy range between 150 and 250 K Sutton et al. (1983), whereas the relatively low (effective) Debye temperature may be caused by the aforementioned soft rotary mode Mintz et al. (1979); Lynn et al. (1978). This interpretation compares favorably to the results of the structure analysis that reveal strong temperature dependence of the atomic displacements parameters for K and Cl (Table 1).

By subtracting $C_{\rm ph}(T)$ from the $C_{p}(T)$ data, we obtain the temperature-dependent magnetic contribution $C_{\rm mag}$ as shown in Fig. 5(b). It reveals a broad maximum around 5 K comparable to $T_{\max}\simeq 6$ K in the magnetic susceptibility, and gradually decreases toward higher temperatures. From $C_{\rm mag}(T)$ data, the total magnetic entropy is estimated as follows

[TABLE]

with the high-temperature limit of 5.50 J mol*-1* K*-1*. The proximity to the theoretical value of $S_{\rm mag}=R\ln(2J+1)=5.76$ J mol*-1* K*-1* validates our analysis.

The maximum value of $C_{\rm mag}$ is about $0.45R$ and comparable to that in non-frustrated square-lattice antiferromagnets Bernu and Misguich (2001). This value gauges the effect of quantum fluctuations, because both low-dimensionality and magnetic frustration tend to impede short-range order and reduce the maximum in $C_{\rm mag}$. Our experimental value for K2IrCl6 suggests that the geometrical and exchange frustration of the three-dimensional fcc lattice in K2IrCl6 cause quantum effects that are as strong as in square-lattice antiferromagnets, but weaker than in frustrated two-dimensional systems, such as triangular antiferromagnets with the maximum value of $C_{\rm mag}$ of $0.22R$ only Bernu and Misguich (2001).

In contrast to previous studies Bailey and Smith (1959); Moses et al. (1979), we do not observe a sharp transition anomaly at $T_{N}$ and rather detect a broad maximum of $C_{\rm mag}$ around this temperature. The sharp anomaly in $d(\chi T)/dT$ at 3.1 K measured on the same sample shows that the absent specific-heat anomaly is not a drawback of sample quality. Indeed, we repeated specific-heat measurement on a different sample, but the anomaly remained broad.

III.5 Electronic structure

We now proceed to the computational analysis. The uncorrelated (LDA+SO) electronic structure of K2IrCl6 shown in Fig. 6 reveals a combination of Ir $t_{2g}$ and Cl $3p$ states at the Fermi level. The $t_{2g}$ bands between $-1$ and $0.5$ eV develop the characteristic two-peak structure that corresponds to the splitting of $t_{2g}$ states into the $j=\frac{1}{2}$ and $j=\frac{3}{2}$ levels.

Correlation effects in the Ir $5d$ shell are taken into account within LSDA+$U$+SO. The on-site Coulomb repulsion $U_{d}=1.7$ eV and Hund’s exchange $J_{d}=0.3$ eV are commonly used for the Ir4+ oxides Winter et al. (2016, 2017). This set of parameters leads to a band gap of 0.5 eV, which is even lower than the activation energy of $\Delta\simeq 0.7$ eV for the electrical transport. The experimental value of $\Delta$ is well reproduced with $U_{d}=2.2$ eV that we choose as the optimal value for K2IrCl6. In Sec. III.6, we show that the very same value of $U_{\rm eff}=U_{d}=2.2$ eV leads to a good agreement with the experimental Curie-Weiss temperature and saturation field and thus can be used for the evaluation of magnetic parameters.

The increased on-site Coulomb repulsion reflects the weaker screening by ligands and the higher ionicity of K2IrCl6 in agreement with the larger $\Delta$, which is unusually high for an Ir4+ compound. Interestingly, this increased ionicity is not immediately visible in the atomic-resolved LDA+SO density of states (DOS), where Cl $3p$ orbitals contribute about 44 % of the total DOS at the Fermi level, an even a larger contribution than 34 % of O $2p$ states in Na2IrO3.

The spin and orbital moments obtained in LSDA+$U$+SO are consistent with the anticipated $j_{\rm eff}=\frac{1}{2}$ state of Ir4+. We find the spin moment of 0.33 $\mu_{B}$ and the orbital moment of $0.6-0.7$ $\mu_{B}$ only weakly dependent on the $U_{d}$ value (Fig. 7, right).

III.6 Microscopic magnetic model

Exchange couplings in Ir4+ compounds are generally anisotropic. The spin Hamiltonian can be written as

[TABLE]

with the sum taken over all pairs of atoms. The $\mathbb{J}_{ij}$’s are exchange tensors of the form:

[TABLE]

where $J$ is the isotropic (Heisenberg) coupling, $D$ is the Dzyaloshinskii-Moriya interaction vector, and $\Gamma$ is the second-rank traceless tensor that describes the symmetric portion of the anisotropic exchange.

The cubic symmetry of the structure leads to the following form of the exchange tensor for three groups of nearest-neighbor interactions,

[TABLE]

[TABLE]

[TABLE]

Here, $\mathbb{J}_{xy}$, $\mathbb{J}_{xz}$, and $\mathbb{J}_{yz}$ stand for exchange tensors of the bonds on the respective faces of the cubic unit cell (Fig. 1, right). All components of the $\Gamma$-tensor are reduced to only two parameters, the diagonal (Kitaev) exchange $K$ and the off-diagonal anisotropy $\Gamma$, whereas Dzyloshinskii-Moriya interaction vanishes, owing to the inversion symmetry of the nearest-neighbor Ir–Ir exchange bonds.

To estimate the $J$, $K$, and $\Gamma$ parameters, we use the perturbation-theory approach detailed in Refs. Rau et al., 2014; Winter et al., 2016. LDA hoppings within the $t_{2g}$ manifold form the hopping matrix

[TABLE]

written in the $d_{yz}-d_{xz}-d_{xy}$ basis, respectively. Magnetic interaction parameters are obtained as Winter et al. (2016)

[TABLE]

using the constants

[TABLE]

Using the spin-orbit coupling $\lambda=0.4$ eV, Hund’s coupling $J_{H}=J_{d}=0.3$ eV, as well as the effective Coulomb repulsion $U_{\rm eff}=2.2$ eV determined in the previous section, we arrive at $J=13$ K, $K=5$ K, and $\Gamma=1$ K. The calculated Curie-Weiss temperature $\Theta=-(3J+K)=-44$ K is in good agreement with the experimental value of $\Theta=-42.6(1)$ K. Moreover, our ab initio $J$ compares favorably to $J\simeq 11.5$ K extracted from electron spin resonance experiments on the magnetically diluted samples Griffiths et al. (1959). The same experiments provide an estimate for the exchange anisotropy $J^{z}-(J^{x}+J^{y})/2\simeq 2$ K Griffiths et al. (1959), which is comparable to our $K$, although one should keep in mind that Ref. Griffiths et al., 1959 assumed the conventional orthorhombic exchange anisotropy instead of the actual Kitaev one.

The ratios of $K/J\simeq 0.38$ and $\Gamma/J\simeq 0.08$ would place K2IrCl6 into the region of collinear antiferromagnetic order with the propagation vector $\mathbf{k}=(1,0,0)$ in the classical phase diagram for the $J-K-\Gamma$ model on the fcc lattice Cook et al. (2015). However, experimental neutron study Hutchings and Windsor (1967) revealed a different flavor of collinear antiferromagnetic order described by $\mathbf{k}=(1,0,\frac{1}{2})$. To resolve this discrepancy, we re-constructed the phase diagram (Fig. 8) using the Luttinger-Tisza method, and recognized that these two states remain degenerate unless a second-neighbor interaction $J_{2}$ is included Ter Haar and Lines (1962); Tahir-Kheli et al. (1966). Whereas a ferromagnetic $J_{2}$ would stabilize the $\mathbf{k}=(1,0,0)$ or type-I order, an antiferromagnetic $J_{2}$ leads to the $\mathbf{k}=(1,0,\frac{1}{2})$ or type-IIIA order 111For both types of order, cubic symmetry allows different choices of the propagation vector. Here, we assume the doubled periodicity along $c$ and, therefore, write $k_{z}=\frac{1}{2}$, but for example $\mathbf{k}=(1,\frac{1}{2},0)$ of Ref. Revelli et al., is equivalent to our $\mathbf{k}=(1,0,\frac{1}{2})$ with the $b$ and $c$ directions swapped.. In our case, we find a weakly antiferromagnetic $J_{2}\simeq 0.2$ K that would lead to the $\mathbf{k}=(1,0,\frac{1}{2})$ order observed experimentally. On the other hand, the $\mathbf{k}=(1,0,0)$ order has been experimentally observed in Ba2CeIrO6 Aczel et al. (2019), where it may be triggered by a weakly ferromagnetic $J_{2}$ or by local deviations from the cubic symmetry.

The second-neighbor interaction $J_{2}$ remains very weak, $J_{2}/J\simeq 0.015$, as expected from the large Ir–Ir distance of nearly 10 Å. This DFT estimate is compatible with the remarkably low Néel temperature $T_{N}/J\simeq 0.24$ that both spin-wave Lines (1963) and Green-function Lines (1964) calculations predict in the region of $J_{2}/J_{1}<0.05$ only.

From the energy difference between the $\mathbf{k}=(1,0,\frac{1}{2})$ state and the fully polarized ferromagnetic state we estimate the saturation field of $H_{s}=(4J+2K)k_{B}/(g\mu_{B}j_{\rm eff})\simeq 92$ T in good agreement with our extrapolated value of 87 T (Fig. 4b). We further explored the stability of the $\mathbf{k}=(1,0,\frac{1}{2})$ order and performed ab initio calculations for different values of the cubic lattice parameter of K2IrCl6 sup . Whereas compression leads to only minor changes in the exchange couplings and shifts the system deeper into the region of collinear order, an expansion of the structure would increase $K/J$. The classical phase boundary is not crossed, though.

The antiferromagnetic Kitaev exchange is somewhat uncommon, as honeycomb iridates and their analogs all show $K<0$ Winter et al. (2017). The origin of $K>0$ can be understood from Eq. (9). In K2IrCl6, $t_{1}=5.4$ meV, $t_{2}=9.6$ meV, $t_{3}=-82.9$ meV, and $t_{4}=0$. Therefore, $K$ is mainly due to $t_{3}$ and antiferromagnetic, in contrast to the honeycomb iridates where large $t_{2}$ leads to $K<0$. As $t_{3}$ contributes to $J$ as well, the antiferromagnetic $K$ is necessarily supplemented by an even larger antiferromagnetic $J$ that occurs in K2IrCl6 indeed. Microscopically, the large-$t_{2}$ regime corresponds to the $90^{\circ}$ Ir–O–Ir superexchange between the edge-sharing IrO6 octahedra, whereas in K2IrCl6 a longer Ir–Cl$\ldots$Cl–Ir superexchange pathway between the disconnected IrCl6 octahedra leads to the large-$t_{3}$ regime caused by the Cl-mediated $d_{xy}-d_{xy}$ hopping. This microscopic mechanism is remarkably similar to the long-range superexchange in V4+ compounds, where only the $d_{xy}$ orbital is magnetic Tsirlin et al. (2011).

IV Discussion and Summary

Using high-resolution synchrotron x-ray diffraction, we verified the cubic symmetry of K2IrCl6 and the regular octahedral environment of Ir4+. From crystallographic point of view, neither global nor local symmetry lowering would be expected in this compound. This is different from Ba2CeIrO6 that likely features local distortions revealed by the abnormally high displacement parameters of oxygen atoms Revelli et al. and from other Ir-based double perovskites, where even the symmetry of the average structure is lower than cubic Aczel et al. (2019). While spectroscopy experiments would be needed to demonstrate the absence of the $t_{2g}$ crystal-field splitting and to ultimately confirm the $j_{\rm eff}=\frac{1}{2}$ state of Ir4+, we note that the results of such experiments may be temperature-dependent. Our data reveal dynamic distortions of the structure driven by the soft rotary mode. At elevated temperatures, this mode renders the system locally and instantaneously non-cubic. It may even be possible to observe not only the $j_{\rm eff}=\frac{1}{2}$ state at low temperatures but also the gradual departure from it upon heating.

Our computational analysis backed by the results of thermodynamic and transport measurements reveals the increased on-site Coulomb repulsion compared to Ir4+ oxides. The magnetic model comprises a sizable antiferromagnetic Kitaev exchange, albeit superimposed on an even larger Heisenberg term, which is unavoidable in this setting, because the leading hopping process is qualitatively different from that in honeycomb iridates with $K<0$ and small $J$.

The large frustration ratio of $\theta/T_{N}=13.7$ indicates a strongly impeded magnetic order, although its full suppression with the formation of a spin liquid appears impossible in the parameter range of K2IrCl6. Neither compressive nor tensile strain moves the system sufficiently close to a classical phase boundary, where long-range order can be destabilized. That said, K2IrCl6 with its robust magnetic order appears to be a good model fcc antiferromagnet. An excellent match between the ab initio results and experiment paves the way to studying excitations of this frustrated antiferromagnet and other properties that can be influenced by the frustration. One of them is the possible structural component in the magnetic ordering transition at $T_{N}$, or even the occurrence of two consecutive transitions at about 2.8 and 3.1 K Moses et al. (1979). Their detailed nature lies beyond the scope of our present study and requires dedicated experiments such as single-crystal neutron diffraction. Here, we only note that the absence of a clear transition anomaly in the specific heat of our polycrystalline samples serves as an additional evidence for the first-order nature of the transition(s) and, thus, for the presence of a structural component. This is not unexpected given the abundance of magnetic frustration and availability of soft phonon modes. Further work in this direction would be interesting.

In summary, we confirmed the cubic symmetry of K2IrCl6 and detected a soft rotary mode that is gradually suppressed upon cooling. This compound is a geometrically frustrated fcc antiferromagnet with the nearest-neighbor Heisenberg exchange $J\simeq 13$ K and Kitaev exchange $K\simeq 5$ K augmented by a weak next-nearest-neighbor coupling $J_{2}\simeq 0.2$ K that stabilizes the $\mathbf{k}=(1,0,\frac{1}{2})$ order. The activation energy of charge transport $\Delta\simeq 0.7$ eV and the effective Coulomb repulsion $U_{\rm eff}=2.2$ eV are higher than in Ir4+ oxides, suggesting an increased ionicity of the chloride. The leading Ir–Ir hopping differs from that in the honeycomb iridates and triggers the antiferromagnetic Kitaev term accompanied by an even stronger Heisenberg one.

Acknowledgements.

We acknowledge the provision of synchrotron beamtime by the ALBA and ESRF and thank Francois Fauth, Alexander Missyul, Oriol Vallcorba, Wilson Mogodi, and Mauro Coduri for their experimental assistance. We are also grateful to Anton Jesche for his help with the magnetization measurements and general advice, and to Philipp Gegenwart for useful comments. AT thanks Anna Efimenko, Liviu Hozoi, Vladimir Hutanu, Jeffrey Lynn, Sergey Zvyagin, and Adam Aczel for various communications on K2IrCl6. The support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL), is acknowledged. The work in Augsburg was supported by the Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of Alexander von Humboldt Foundation. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Projektnummer 107745057 – TRR 80 (Augsburg) and SFB1143 (Dresden-Rossendorf). The work of VGM was supported by the Russian Science Foundation Grant No. 18-12-00185.