Magnetic order in single crystals of Na3Co2SbO6 with a honeycomb arrangement of 3d$^7$ Co$^{2+}$ ions

TL;DR

This study synthesizes and characterizes Na$_3$Co$_2$SbO$_6$ single crystals, revealing a monoclinic structure with anisotropic magnetic properties, a zigzag antiferromagnetic order below 5 K, and insights into its electronic ground state.

Contribution

The paper provides detailed structural, magnetic, and theoretical analysis of Na$_3$Co$_2$SbO$_6$, highlighting its zigzag magnetic order and complex electronic interactions.

Findings

Na$_3$Co$_2$SbO$_6$ has a monoclinic structure with layered Co$_2$SbO$_6$ sheets.

The compound exhibits anisotropic magnetic susceptibility and a transition to antiferromagnetic order below 5 K.

Neutron diffraction confirms zigzag magnetic order with moments aligned along the b-axis.

Abstract

We have synthesized single crystals of Na$_3$Co$_2$SbO$_6$ and characterized the structure and magnetic order by measuring anisotropic magnetic properties, heat capacity, x-ray and neutron single crystal diffraction. Magnetic properties and specific heat of polycrystalline Na$_3$Co$_2$SbO$_6$ were also measured for comparison. Na$_3$Co$_2$SbO$_6$ crystallizes in a monoclinic structure (space group $C2/m$) with [Co$_2$SbO$_6$]$^{3-}$ layers separated by Na$^+$ ions. The temperature dependence of magnetic susceptibility shows significant anisotropic behavior in the whole temperature range 2\,K-350\,K investigated in this work. An effective moment of about 5.5\,$\mu_B$/Co$^{2+}$ from a Curie-Weiss fitting of the magnetic susceptibility is larger than the spin only value and signals significant orbital contribution. Na$_3$Co$_2$SbO$_6$ single crystal undergoes a transition into a long-range antiferromagnetically ordered state below $T_N$=5\,K. Neutron single crystal diffraction confirmed the zigzag magnetic structure with a propagation vector k\,=\,(0.5, 0.5, 0). The ordered moment is found to be 0.9\,$\mu_B$ at 4\,K and align along the crystallographic \textit{b}-axis. Density functional theory calculations suggest that the experimentally observed zigzag order is energetically competing with the Neel order. It is also found that the covalency between Co $d$ and O $p$ is quite strong and competes with the local spin-orbit coupling, suggesting a $J_{eff}$=1/2 ground state may not be realized in this compound.

Tables

Table 1. Table 1: Selected crystallographic data and refinement parameters for single crystal x-ray diffraction study at room temperature.

Formula		Na3Co2SbO6
Space Group		C2/m
$a$ (Å)		5.3707(3)
$b$ (Å)		9.2891(6)
$c$ (Å)		5.6533(4)
$\alpha$		90.00
$\beta$		108.566(6)
$\gamma$		90.00
$V$ (Å3)		267.36(3)
$Z$		2
Crystal size (mm)		$0.03\times 0.06\times 0.08$
Temperature (K)		297
$\theta$ Range (∘)		4.34 - 32.07
$\mu$ (mm-1)		41.773
$\lambda$ (Å)		(Mo K$\alpha$) 0.71073
Collected reflections		1713
Unique reflections		444
$R_{int}$		0.0119
$\sigma I/I$		0.0099
$h$		$-7\le h\le 7$
$k$		$-13\le k\le 9$
$l$		$-7\le l\le 7$

Table 2. Table 2: The structure parameters of Na 3 Co 2 SbO 6 measured at 297 K by single crystal x-ray diffraction. The space group is C ​ 2 / m 𝐶 2 𝑚 C2/m . The lattice parameters are a 𝑎 a =5.3707(3)Å, b 𝑏 b =9.2891(6)Å, c 𝑐 c =5.6533(4)Å, α 𝛼 \alpha =90 o , β 𝛽 \beta =108.566(6) o , γ 𝛾 \gamma =90 o . R f ​ a ​ c ​ t ​ o ​ r subscript 𝑅 𝑓 𝑎 𝑐 𝑡 𝑜 𝑟 R_{factor} =1.80 %. χ 2 superscript 𝜒 2 \chi^{2} =1.36. U 𝑈 U have units of Å 2 . Co/Sb antisite defects are included in the refinement to account for the intensity change from diffuse scattering.

atom	$x$	$y$	$z$	$Occ.$	$U_{11}$	$U_{22}$	$U_{33}$	$U_{23}$	$U_{13}$	$U_{12}$
Co1	0.00000(0)	0.66635(5)	0.00000(0)	0.922(1)	0.0039(2)	0.0011(1)	0.0055(2)	0	0.0019(2)	0
Sb11	0.00000(0)	0.66635(5)	0.00000(0)	0.078(1)	0.0039(2)	0.0011(1)	0.0055(2)	0	0.0019(2)	0
Sb1	0.00000(0)	0.00000(0)	0.00000(0)	0.844(1)	0.0023(2)	0.0006(1)	0.0055(2)	0	0.0013(2)	0
Co11	0.00000(0)	0.00000(0)	0.00000(0)	0.156(1)	0.0023(2)	0.0006(1)	0.0055(2)	0	0.0013(2)	0
O1	0.27299(44)	0.34125(25)	0.79466(48)	1	0.0083(9)	0.0025(3)	0.0057(9)	-0.0004(4)	0.0010(7)	-0.0007(4)
O2	0.25106(66)	0.50000(0)	0.20436(68)	1	0.0098(13)	0.0031(4)	0.0053(12)	0	0.0029(10)	0
Na1	0.00000(0)	0.50000(0)	0.50000(0)	0.920(3)	0.0066(11)	0.0019(4)	0.0064(11)	0	0.0005(8)	0
Na2	0.50000(0)	0.32755(21)	0.50000(0)	0.980(4)	0.0075(8)	0.0023(3)	0.0066(8)	0	0.0027(6)	0

Table 3. Table 3: The density functional theory results of the ordered moments, spin S 𝑆 S and angular momentum L 𝐿 L both in unit of μ B subscript 𝜇 𝐵 \mu_{B} , out-of-plane exchange J ⟂ z , y superscript subscript 𝐽 perpendicular-to 𝑧 𝑦 J_{\perp}^{z,y} in meV, and single-site anisotropy K 𝐾 K in meV.

$U_{eff}$ (eV)	Néel		zigzag		$J_{⟂}^z{S}^2$	$J_{⟂}^y{S}^2$	$K{S}^2$
	$2S$	$L$	$2S$	$L$
4.4	2.72	0.27	2.69	0.23	0.1	0.11	3.5
1.0	2.57	0.26	2.58	0.19	0.23	0.16	40.2

Full text

I Introduction

$4d$/$5d$ transition metal compounds such as A2IrO3 (A=Li, Na) and $\alpha$-RuCl3 have been intensively studied in recent years for possible exotic quantum spin liquid state.Takayama et al. (2015); Singh and Gegenwart (2010); Plumb et al. (2014) These compounds have a quasi-two-dimensional structure with weak coupling between layers of honeycomb arranged transition metal ions. The strong spin-orbit coupling in these 4d/5d transition metal ions leads to the effective pseudospin ($J_{eff}=1/2$) electronic state. The in-plane honeycomb arrangement of transition metal ions enables the orbital-dependent hopping processes giving rise to the Kitaev-type interactions. However, all these compounds show a long range magnetic order at low temperatures due to the Heisenberg exchange interactions. Therefore, various approaches have been employed to suppress the long range magnetic order, such as high magnetic fieldsSears et al. (2017); Zheng et al. (2017); Baek et al. (2017); Leahy et al. (2017) and chemical substitutionKitagawa et al. (2018); Lampen-Kelley et al. (2017); Koitzsch et al. (2017); Do et al. (2018). Recently, Liu et al.(Liu and Khaliullin, 2018) and Sano et al.(Sano et al., 2018) proposed that the Kitaev model can be realized in materials based on $d^{7}$ ions with an electronic configuration of $t_{2g}^{5}e_{g}^{2}$ (S=3/2, L=1), which also host the pseudospin-1/2 magnetism. One interesting feature for these $d^{7}$ candidates is that in the magnetic honeycomb plane the ferromagnetic interactions from $e_{g}$ spins compensate the antiferromagnetic interactions resulting from $t_{2g}$ electrons. This provides a new approach to tuning the relative magnitude of Heisenburg and Kitaev interactions for the realization of a spin liquid ground state.

One candidate material proposed by Liu et al.(Liu and Khaliullin, 2018) is Na3Co2SbO6. The monoclinic structure (space group C2/m)Zvereva et al. (2016); Wong et al. (2016) is shown in Fig. 1. In the magneto-active layer, Na3Co2SbO6 has a honeycomb structure of edge-sharing CoO6 octahedra with one SbO6 octahedron occupying the center of each honeycomb. These layers are separated by Na. Previous studiesWong et al. (2016); Viciu et al. (2007) on the magnetic properties and specific heat of polycrystalline samples suggest that Na3Co2SbO6 orders antiferromagnetically below $T_{N}$=8.3 K or 4.4 K which might result from the varying sodium stoichiometry. The Curie-Weiss fitting of the high temperature magnetic susceptibility gives a nearly zero Weiss constant and an effective moment around 5 $\mu_{B}$/Co2+ larger than the expected spin-only value, indicating a significant orbital contribution. Neutron powder diffraction measurementWong et al. (2016) suggests a partially frustrated ‘zigzag’ ordering with a propagation vector k = (0.5, 0.5, 0) and a magnetic moment of 1.79(4) $\mu_{B}$ at 1.5 K aligning nearly normal to the plane of the layer along the crystallographic c-axis.

Considering the quasi-two-dimensional structure of Na3Co2SbO6, anisotropic physical properties are expected which can be learned from investigating single crystals. To the best of our knowledge, single crystals of Na3Co2SbO6 have not been available for a detailed study of the intrinsic and anisotropic properties. In this work, we report the growth of millimeter-sized Na3Co2SbO6 single crystals and the characterization of structure and physical properties. Single crystal x-ray diffraction measurements confirm the monoclinic C2/m structure and observe diffuse scattering along L due to the stacking faults in the as-grown crystals. Magnetic measurements suggest significant anisotropic behavior in the whole temperature range 2 - 350 K. Na3Co2SbO6 single crystal orders antiferromagnetically below 5 K. Neutron single crystal diffraction confirms the zigzag magnetic structure with a propagation vector k = (0.5, 0.5, 0) as reported in a previous neutron powder diffraction study. However, different from that for the polycrystalline samples, the ordered moment in our single crystals is determined to align along the crystallographic b-axis. Our density functional theory (DFT) calculations show that the experimentally found zigzag ordering is energetically competing with the Néel ordering and suggest that strong covalency should be considered when understanding the electronic state of Na3Co2SbO6.

II Experimental details

Polycrystalline Na3Co2SbO6 was synthesized by conventional solid state reaction. The starting materials Na2CO3 (99.995%, Alfa Aesar), Co3O4 (99.7%, Alfa Aesar), and Sb2O3 (99.9%, Alfa Aesar) were mixed in the molar ratio of 1.65:0.67:0.5. A 10% excess Na2CO3 was added to compensate the evaporation loss during sintering. The homogeneous powder was pelletized and then placed in an alumina crucible with a lid and heated to 800∘C in 20 hrs and held at this temperature for 48 hrs. The resulting product was ground and pelletized again and heated to 900∘C and held there for 48 hrs before cooling to room temperature. Room temperature x-ray powder diffraction confirmed that the sample is single phase after the second sintering. The refinement of all powder diffraction patterns in this work is performed using the monoclinic C2/m structure following ref[14; 15] and our single crystal x-ray diffraction study. The lattice parameters from Rietveld refinement are a=5.3557(8) Å, b=9.2701(10) Å, c=5.6486(6) Å, and $\beta$=108.40(2)∘. The lattice parameters are comparable to those reported by Wong et al.Wong et al. (2016)

A dense pellet of Na3Zn2SbO6 was also synthesized using a similar procedure. The temperature dependence of specific heat of Na3Zn2SbO6 was used as the phonon reference when estimating the entropy release across magnetic order of Na3Co2SbO6.

To grow single crystals of Na3Co2SbO6, the above fired pellets were reground together with 30wt% excess Na2CO3. After a thorough mixing, the mixture was pelletized and placed in a Pt crucible with a lid (see Fig. 2). The growth ampoule was then put in a box furnace and heated to 1100∘C in 10 hrs, held there for 48 hrs, and then cooled to 800∘C in 24 hrs. The furnace was then shut down to cool to room temperature.

Some crystals were ground into fine powder for x-ray powder diffraction. Room temperature X-ray powder diffraction patterns were collected on a PANalytical X’Pert Pro MPD with a Cu-Kα,1 incident-beam monochromator. Elemental analysis confirmed the atomic ratio of Na, Co, and Sb, which was carried out on as-grown crystals using a Hitachi-TM3000 microscope equipped with a Bruker Quantax 70 EDS system.

The magnetic properties were measured with a Magnetic Property Measurement System (Quantum Design, QD) in the temperature interval 2 K $\leq$ x $\leq$ 350 K. Specific heat was measured with a QD Physical Property Measurement System (PPMS) in the temperature interval 1.90 K $\leq$ x $\leq$ 250 K. We tried to measure electrical resistivity of Na3Co2SbO6 crystals; however, the total resistance is too large and no reliable resistivity data could be obtained.

X-ray single crystal diffraction was performed to determine the crystal structure using a Rigaku single crystal X-ray diffractometer at SNS, ORNL. Neutron single crystal diffraction was performed to determine the magnetic structure at the HB-3A four-circle diffractometer at the High Flux Isotope Reactor of the Oak Ridge National Laboratory. A neutron wavelength of 1.003 ${\AA}$ was used from a bent perfect Si-331 monochromator.Chakoumakos et al. (2011) The neutron diffraction data refinements are based on 85 magnetic and nuclear reflections with the program Fullprof.

III Results and discussion

III.1 X-ray single crystal diffraction

Plate-like single crystals with a regular hexagon shape were found on the inner surface of the Pt crucible as illustrated in Fig. 2. After crystal growth, the starting pellet was porous and some voids were observed. These observations suggest that the crystal growth occurs with the aid of Na2CO3 vapor. As shown in Fig. 3(a), the as-grown hexagon-shaped crystals are typically 1-2 mm long for each edge. Room temperature x-ray powder diffraction of pulverized single crystals observed no impurity. Rietveld refinement of the diffraction pattern using monoclinic C2/m gives the lattice parameters of a=5.3708(3) Å, b=9.2631(5) Å, c=5.6485(4) Å, and $\beta$=108.50(2)∘. Compared to the polycrystalline samples used for the crystal growth (see experimental details), Na3Co2SbO6 single crystals have a larger a-lattice. As discussed later, this a-lattice seems to be a good indicator of the magnetic ordering temperature.

To further confirm the monoclinic symmetry of our single crystals, we performed x-ray single crystal diffraction at room temperature. The x-ray diffraction pattern shown in Fig. 3(a))shows well-shaped spots close to trigonal symmetry. Automatic indexing of our single crystal diffraction data indeed tends to return a hexagonal unit cell but relative intensities of the reflections indicated a slight preference for monoclinic symmetry. In the reported crystal structure of Na3Co2SbO6 the individual Na layers and the individual Co2SbO6 slabs have very nearly six-fold symmetry, and it is the manner in which the layers are displaced relative to one another in the stacking sequence that results in an overall monoclinic symmetry.Zvereva et al. (2016) Diffuse spot shapes along the stacking directing indicate stacking faults are prevalent in the crystals, which may be partly responsible for the ambiguity in symmetry. The stacking faults can be modeled approximately by allowing Co and Sb to mix on their respective sites, as described below. We checked all the integrated intensities in trigonal symmetry and also monoclinic symmetry within WinGX. P6*/mmm* and C2*/m* give comparable merging R-factors. However, a careful investigation of the Bragg peak intensities shown in Fig. 3(b) suggests that the trigonal symmetry is not satisfied. Our analysis concluded space group symmetry is C2*/m* which gives the smallest merging R-factor of 0.012.

A summary of single crystal x-ray diffraction data and refinement parameters for Na3Co2SbO6 crystals is provided in Table I. Altogether 444 unique peaks were obtained at room temperature with Rint= 1.2$\%$. The refinement was performed with Fullprof suite and it is well converged with $R_{factor}$=1.80 % and $\chi^{2}$=1.36. It is worth mentioning that the refinement was performed with $\sim$10% Co/Sb antisite disorder, i.e. Co and Sb switch positions. Including Co/Sb antisite disorder can significantly improve the fitting as this accounts for the peak intensity suppression by the stacking faults. In an octahedral environment, the ionic radius of 0.745 Å for Co2+ is larger than 0.60 Å for Sb5+.Shannon (1976) We cannot rule out the possibility of Co/Sb antisite disorder. However, considering the observation of stacking faults and the fact that sliding the honeycomb layers along any honeycomb bond direction leads to Co/Sb on top of Sb/Co sites, we tend to look at the amount of antisite defects as an indicator for the amount of stacking faults. Table II shows the atomic coordinates, occupancy, and anisotropic displacement parameters. The refinement suggests Na sites are not fully occupied. This Na nonstoichiometry might account for the variation of $T_{N}$ of different Na3Co2SbO6 samples.

The formation of stacking fault may be an intrinsic feature of Na3Co2SbO6 or come from the crystal growth. As suggested by Zvereva et al.,Zvereva et al. (2016) the monoclinic and the trigonal structures are very similar in both metrics and lattice energy, which leads to the natural formation of stacking faults. On the other hand, the growth technique employed in this work may also facilitate the formation of stacking faults. The observation of Na3Co2SbO6 crystals on the upper part of the Pt crucible signals a vapor transport growth mechanism. Since the Pt crucible is just loosely covered by a Pt lid, all factors affecting the gas flow inside of the Pt crucible and the evaporation of Na2CO3, such as the temperature profile, or the amount of Na2CO3, can affect the growth kinetics and thus formation of lattice defects.

III.2 Magnetic properties

Figure 4 shows the temperature dependence of the magnetic susceptibility measured in a magnetic field of 10 kOe applied either perpendicular,$\chi_{\perp}$, or parallel, $\chi_{\parallel}$, to the honeycomb plane. The anomaly around 5 K in both $\chi_{\parallel}$ and $\chi_{\perp}$ curves signals the long range magnetic order, which is further confirmed by specific heat and neutron single crystal diffraction measurements. The magnetic susceptibility is quite anisotropic with $\chi_{\parallel}$ larger than $\chi_{\perp}$ in the whole temperature range investigated 2 - 350 K. At room temperature, $\chi_{\parallel}\approx$1.2$\chi_{\perp}$. This difference increases with decreasing temperature to $\chi_{\parallel}\approx$6$\chi_{\perp}$ around 5 K. Below 5 K, $\chi_{\perp}$ shows little temperature dependence while $\chi_{\parallel}$ decreases quickly upon further cooling. The different temperature dependence of magnetic susceptibility below 5 K suggests that the spins are aligned in ab plane. This is confirmed by neutron single crystal diffraction measurements presented later.

Figure 4(a) also shows the inverse magnetic susceptibility. The dashed lines highlight the deviation from a linear temperature dependence for 1/$\chi_{\perp}$ below $\sim$150 K and for 1/$\chi_{\parallel}$ above $\sim$250 K. The latter is similar to that observed for polycrystalline sample.Viciu et al. (2007) We then fit the $\chi$ data in the temperature range 150 K$\leq$T$\leq$350 K using $\chi$(T)=$\chi_{0}$+C/(T-$\theta$), where $\chi_{0}$ is the temperature independent term, C is the Curie constant and $\theta$ the Weiss constant. The fitting of $\chi_{\perp}$ gives $\chi_{0}$=-2$\times$10*-4emu/mol, $\theta_{c}$=-170 K and $\mu_{eff}$(c)=5.60 $\mu_{B}$/Co2+. For $\chi_{\parallel}$, the fitting suggests $\chi_{0}$=-5$\times$10-3emu/mol, $\theta_{ab}$=-4 K and $\mu_{eff}$(ab)=5.48 $\mu_{B}$/Co2+*, respectively. The large difference between $\theta_{ab}$ and $\theta_{c}$ is surprising. As presented later, our density functional theory calculations suggest these anomalies result from the single ion anisotropy and the antiferromagnetic interlayer coupling in Na3Co2SbO6. The strong single ion anisotropy can lead to abnormal effective Weiss temperatures, which makes the interpretation of the Weiss constant more complicated.Johnston (2017)

The Co2+ ions in an octahedral crystal field have an electronic configuration of t${}_{2g}^{5}e_{g}^{2}$. An effective moment of 3.87$\mu_{B}$/Co2+ is expected for S=3/2 (assuming g=2) from the spin-only contribution. The experimental values of $\mu_{eff}$(c)=5.60 $\mu_{B}$/Co2+ and $\mu_{eff}$(ab)=5.42 $\mu_{B}$/Co2+ are larger than the expected spin-only value, which signals significant orbital contribution. The above experimental values are also much larger than $\mu_{eff}=3.75\mu_{B}$/Co2+ expected for the pseudospin-1/2 case proposed theoretically.(Liu and Khaliullin, 2018; Sano et al., 2018) This theoretical value is determined by $\mu_{eff}=g_{J}\sqrt{J(J+1)}$ with $g_{J}=-\frac{3}{2}g_{L}+\frac{5}{3}g_{S}$, where $g_{L,S}$ are the $g$ factor for the effective orbital moment and the spin moment, respectively, given by $g_{L}=-\frac{2}{3}$ and $g_{S}=2$.(Khomskii, 2014)

The powder average value of magnetic susceptibility was obtained as 1/3$\chi_{\perp}$+2/3$\chi_{\parallel}$ and plotted in Fig. 4(b). Also plotted in Fig. 4(b) is $\chi(T)$ of polycrystalline Na3Co2SbO6 used for crystal growth in this study. The powder average is close to $\chi(T)$ of the polycrystalline sample but they do not overlap with some difference especially at low temperatures. The reason is that the polycrystalline sample orders magnetically at $T_{N}$=7.8 K where a cusp in $\chi(T)$ curve was observed. This $T_{N}$ is close to 8.3 K reported by Wong et alWong et al. (2016) for their polycrystalline samples. The magnetic ordering temperature of our single crystals of 4.8 K is similar to 4.4 K reported by Viciu et al.Viciu et al. (2007) for their polycrystalline samples.

All samples synthesized in different groups have comparable b-lattice, c-lattice, and $\beta$ angle, but can be classified into two groups following their a-lattice: a=5.3557(8) Å of our polycrystalline sample is comparable to a=5.3565(1) Å of Wong’s polycrystalline sample; a=5.3708(3) Å for our single crystals is comparable to 5.3681(2) Å of Viciu’s polycrystalline samples. The comparison suggests that samples with a larger a-lattice have a lower TN and a-lattice can thus be a good indicator of the Neel temperature.

The variation of $T_{N}$ can result from Na nonstoichiometry as proposed by Wong et al.Wong et al. (2016) Our single crystal diffraction suggests some Na deficiency. It is worth mentioning that the Na content can be purposely controlled by chemical deintercalation.Chou et al. (2004); Van Bruggen et al. (1980) This might be employed to suppress the long range magnetic order of Na3Co2SbO6. The effects of Na content on the magnetism and structure deserve further investigation. One more factor that might affect TN is the stacking fault. As evidenced by the x-ray single crystal diffraction measurements, our single crystals have stacking faults. A disturbed stacking is expected to suppress the long range magnetic order of Na3Co2SbO6. However, we want to mention that in $\alpha$-RuCl3 the stacking fault induces magnetic order at temperatures higher than $T_{N}$=7 K.Cao et al. (2016)

Figure 5 shows the field dependence of magnetization for polycrystalline sample and single crystals. While a nearly linear field dependence was observed when the field is applied long c-axis, the magnetization of polycrystalline samples and single crystals with field applied in ab-plane tends to saturate at high fields even at 30 K. At 2 K, the magnetization in an applied field of 60 kOe is 2.25 $\mu_{B}$/Co2+ for our polycrystalline sample, slightly larger than 2 $\mu_{B}$/Co2+ for our single crystals. Both values are comparable to those reported previouslyViciu et al. (2007); Wong et al. (2016), however, much larger than 0.70 $\mu_{B}$/Co2+ when the external magnetic field is applied along the c-axis. The field derivative, $dM/dH$, shows a maximum around 10 kOe for polycrystalline samples and 9 kOe for single crystals with field applied in ab-plane, signaling the field induced spin-flop transition.Viciu et al. (2007); Wong et al. (2016) However, this feature is absent for single crystals when the magnetic fields are applied along the c-axis. This anisotropic behavior suggests that the magnetic moments are aligned in the ab-plane. This is consistent with the anisotropic temperature dependence of magnetic susceptibility shown in Fig. 4 and our neutron single crystal diffraction measurements further confirm that the moment aligns along the crystallographic b-axis.

III.3 Specific heat

Figure 6 shows the temperature dependence of the specific heat of both polycrystalline pellet and single crystals. A lambda-type anomaly signals the occurrence of a long range magnetic order of Co2+ around 5 K in Na3Co2SbO6 single crystals and around 8 K for the polycrystalline sample. The magnetic ordering temperature obtained from specific heat data agrees with that determined from magnetic susceptibility shown in Fig. 4 and neutron single crystal diffraction presented later in Fig. 8. In order to estimate the entropy release across the magnetic order, specific heat of Na3Zn2SbO6 was measured and used as the phonon reference. The entropy release is 6.7J/mol K and 7.7J/mol K for the single crystal and the polycrystalline pellet, respectively. The entropy release is much lower than the theoretical spin-only value of 23 J/mol K for Co2+ but larger than 5.76 J/mol K expected for the pseudospin-1/2 case.

Figure 7(a) and (b) show the field dependence of magnetic specific heat obtained by subtracting the specific heat of Na3Zn2SbO6 from that of Na3Co2SbO6 measured in different magnetic fields. We monitored and plotted in Fig. 7(c) the field dependence of $T_{max}$ which is defined as the temperature where the magnetic specific heat has a maximum. Below $\sim$6 kOe, $T_{max}$ of polycrystalline sample decreases from 6.60 K in zero magnetic field to 6.52 K in a field of 4 kOe. Above $\sim$6 kOe, Tmax increases rapidly with increasing fields as shown in Figure 7(c). For the single crystal sample, $T_{max}$ has a larger shift to higher temperatures when the magnetic fields are applied in ab-plane than perpendicular to the plane. This field dependence of magnetic order is quite different from that of $\alpha$-RuCl3 where an in-plane field suppresses the long range magnetic order.Sears et al. (2017); Zheng et al. (2017); Baek et al. (2017); Leahy et al. (2017)

III.4 Neutron single crystal diffraction

To further confirm the magnetic order and to determine the magnetic structure, we performed a neutron single crystal diffraction study using one piece of single crystal of about 1 mg. Survey of magnetic propagation vectors at 4 K returns (1/2, 1/2, 0). Figure. 8(a) shows the evolution with temperature of the intensity of the (0.5 0.5 -1) reflection. The temperature dependence suggests $T_{N}$=5.5 K. The best refinement suggests that the Co moments of 0.90(3) $\mu_{B}$/Co at 4 K are in the honeycomb layers and along the b-axis. The fitting quality is displayed in Fig. 8(b) by plotting the observed squared structure factors versus the calculated ones. The zigzag magnetic arrangement in ab-plane is shown in Fig. 8(c). The neighboring layers with the same in-plane zigzag arrangement stack on top of each other along the crystallographic c-axis.

We noticed that the moment direction along the c-axis is proposed based on a neutron powder diffraction study.Wong et al. (2016) Compared to our single crystals with $T_{N}$=5 K, the polycrystalline sample investigated by Wong et al. has a magnetic order at 8.3 K and a smaller a-lattice. It would be interesting to further investigate the effects of Na content on the magnetic structure.

III.5 DFT calculations

In order to understand the magnetic behavior of Na3Co2SbO6, we performed the density functional theory (DFT) calculations. We used the generalized gradient approximation and projector augmented wave (PAW) approach Blöchl (1994) as implemented in the Vienna ab initio simulation package (VASP)Kresse and Joubert (1999); Kresse et al. (1996). For Co, Sb and O, standard potentials were used (Co, Sb and O in the VASP distribution), and for Na a potential in which $p$ states are treated as valence states is used (Napv). As an energy cutoff, we chose 550 eV. To account for correlation effects the local $U$ is included on the Co $d$ states Dudarev (1998). Here, several values of $U_{eff}=U-J_{H}$ are considered including $U_{eff}=4.4$ eV Chen et al. (2011). The SOC is also turned on.

First, we checked the magnetic ground state. We use the structural parameters obtained in our single crystal x-ray diffraction measurements, but the unit cell is doubled along $a$ and $b$ directions to accommodate the experimental zigzag magnetic ordering with the $2\times 2\times 2$ $k$-point grid. In addition to the zigzag state, we also considered Néel ordered state. With $U_{eff}=4.4$ eV, the Néel ordered state is found to be more stable than the zigzag state by 0.10 eV per Co. The relative energy between the two magnetic states is reversed with reducing $U_{eff}$; at $U_{eff}=1$ eV the zigzag state is more stable than the Néel state by $0.01$ eV per Co. Figure 9 shows the total density of state (DOS) and partial DOS projected on Co $d$ states computed with $U_{eff}=4.4$ eV. The Néel state (a) and the zigzag state (b) give nearly identical DOSs with an excitation gap $\approx 2.5$ eV. The gap value is reduced to $\approx 0.8$ eV at $U_{eff}=1$ eV. Thus, the difference between the two magnetic states is rather subtle.

In the presence of trigonal distortion and spin-orbit coupling the ground state doublet is given, in the basis of $|Lz,Sz\rangle$, by: $|\pm 1\rangle$=c1$|\mp 1,\pm 3/2\rangle$+c2$|0,\pm 1/2\rangle$+c3$|\pm 1,\mp 1/2\rangle$, where the coefficients c1, c2, and c3 are real numbers and are determined by $\delta/\lambda$. $\delta$ describes the trigonal crystal field and $\lambda$ the spin-orbit coupling.Osaki and Ury~̂u (1976); Lines (1963) Our DFT calculations show that the spin moment $S$ and the orbital moment $L$ are parallel (see table 3, $S$ and $L$ have the same sign). While this is consistent with the $J_{eff}=1/2$ picture as proposed in (Liu and Khaliullin, 2018; Sano et al., 2018), the ordered spin moment is rather large, comparable to the one expected for the high-spin $S_{z}=3/2$ state, and the reduction due to the mixing with $S_{z}=\pm 1/2$ states seems to be small. This implies that the atomic SOC is competing with the trigonal crystal field in determining the ground state. From our DFT results, we suspect that the covalencey plays an important role to determine the electronic state of Na3Co2SbO6. As seen in the density of states in Fig. 9, Co $t_{2g}$ states (mainly majority spins, but minority spins as well) and O $p$ states are strongly hybridized at $-6\lesssim E\lesssim-1$ (eV) below the Fermi level. Such strong covalency would suppress the mixing between the $S_{z}=3/2$ state and the $S_{z}=\pm 1/2$ states by competing with the local SOC. However, it would be necessary to treat correlation effects more accurately going beyond the DFT framework.

Next, we focus on the on-site spin anisotropy and the interlayer exchange coupling that could provide some insight into the anisotropic magnetic susceptibility. Those interactions could be modeled by $H=\sum_{xyzx^{\prime}y^{\prime}}(J_{\perp}^{z}S_{xyz}^{z}S_{x^{\prime}y^{\prime}z+1}^{z}+J_{\perp}^{y}S_{xyz}^{y}S_{x^{\prime}y^{\prime}z+1}^{y})-K\sum_{xyz}(|S_{xyz}^{z}|^{2}-|S_{xyz}^{y}|^{2})+E_{0}$. Here, $J_{\perp}^{z(y)}$ represents the average exchange coupling for the $z(y)$ component of the spins between neighboring two layers, $K$ is the single-site spin anisotropy, and $E_{0}$ is the energy offset appearing in the DFT calculation. These parameter values can be estimated by comparing the energy of the following 4 magnetic states: ferromagnetic states and layered antiferromagnetic states, which have in-plane ferromagnetic spin alignments and out-of-plane antiferromagnetic alignments, with the spin orientation either along the $y$ direction or the $z$ direction. Considering these magnetic orderings, DFT calculations are carried out using the experimental structure doubled along the $b$ direction and $4\times 4\times 2$ $k$-point grid with the other settings mentioned previously. While the actual magnetic ground state depends on $U_{eff}$, the sign of interlayer exchange and the single-site anisotropy turned out to be independent of the value of $U_{eff}$; the out-of-plane exchange is antiferromagnetic, and the anisotopy is uniaxial (table 3). Note that this does not contradict with the ground state magnetic ordering, either Néel at large $U_{eff}$ or zigzag at small $U_{eff}$, that has ferromagnetic spin alignment between two neighboring layers. These magnetic states are mainly stabilized by in-plane magnetic interactions. While the precise form remains to be determined within the current DFT study,(Liu and Khaliullin, 2018; Sano et al., 2018) in-plane interactions compete with $J_{\perp}$ and $K$ and are expected to have some frustration arising from either Kitaev and $\Gamma$ terms or longer-range exchange. Thus, the trace of magnetic interactions is expected to become small due to the cancellation between different terms.

IV Summary

In summary, we have reported the crystal growth, structure, and magnetic order of Na3Co2SbO6 single crystals. Single crystal x-ray diffraction measurements confirm the monoclinic C2/m structure and find diffuse scattering along L due to the stacking faults in the as-grown crystals. Magnetic measurements suggest significant anisotropic behavior in the whole temperature range 2 K-350 K. Na3Co2SbO6 single crystal orders magnetically below 5 K into the zigzag magnetic structure with a propagation vector k = (0.5, 0.5, 0). The ordered moment is found to be 0.9 $\mu_{B}$ at 4 K and align along the crystallographic b-axis. Our DFT calculations showed that the experimentally found zigzag ordering is energetically competing with the Néel ordering. The zigzag ordering is only stable at relatively small $U_{eff}$ ($\lesssim 1$ eV). The spin moment and the orbital moment on a Co ion are found to be parallel irrespective of the magnetic ordering. While this is consistent with the proposed $J_{eff}=1/2$ picture, the ordered spin moment is comparable to the one from a $S_{z}=3/2$ state, and the influence of the SOC appears to be small. This might indicate that the covalency plays an important role to determine the electronic state of Na3Co2SbO6. Thus, it is necessary to go beyond the standard DFT approach to describe the electronic property of Na3Co2SbO6.

V Acknowledgments

The authors thank A. Banerjee, A. F. May, and B. C. Sales for discussions. Work at ORNL was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering (JQY, QZ, and MAM) and by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences, Division of Materials Sciences and Engineering (SO). The x-ray and neutron single crystal diffraction work at ORNL was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U. S. Department of Energy(YW and HC). HDZ thanks the support from NSF-DMR-1350002.

This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/ downloads/doe-public-access-plan).

VI references