Weakly coupled alternating $S=1/2$ chains in the distorted honeycomb lattice compound Na₂Cu₂TeO₆

Introduction. Spin-1/2 chains with alternating AF and FM couplings are known to exhibit gapped excitations and exponentially decaying correlations [1, 2], which are different from those of the spin-1/2 Bethe chains with uniform couplings [3] but more similar to the spectrum of integer-spin Haldane chains [4, 4]. Assuming alternating couplings of $J_{1}$ (AF) and $J_{2}=\beta J_{1}$ ($\beta<1$), the Hamiltonian of an alternating chain can be written as $\mathcal{H}=\sum_{j}J_{1}\bm{S}_{2j-1}\cdot\bm{S}_{2j}+\sum_{j}J_{2}\bm{S}_{2j}\cdot\bm{S}_{2j+1}$. In the special case of $\beta=0$, an alternating chain can be viewed as disconnected spin dimers, and the local singlet-triplet (triplon) excitations over the dimers account for the excitation gap in the spectrum. Non-zero $J_{2}$ couplings will introduce dispersion for the triplon excitations. At $\beta=1$, one recovers the gapless character of a Bethe chain. In the whole range of $-\infty<\beta<1$, theoretical calculations have revealed a hidden string order that is protected by the $\rm{Z_{2}}\times\rm{Z_{2}}$ global rotation symmetry [6, 7], showing that the gapped ground state of alternating chains is a symmetry-protected topological state of the same type as that of Haldane chains [8]. Considering the prospect of Haldane chains as the resource ground state for measurement-based quantum computation [9], the additional degrees of freedom in $S=1/2$ alternating chains may be important to explore further flexibility in qubit operations.

Experimental realizations of alternating AF-FM chains are limited to a few compounds, including CuNb₂O₆ [10], DMACuCl₃ [11], Na₃Cu₂SbO₆ [12], and BaCu₂V₂O₈ [13] where the exchange couplings have been accurately determined through neutron scattering. Recently, a new candidate compound, Na₂Cu₂TeO₆, was proposed [14, 15, 16, 17, 18]. Similar to Na₃Cu₂SbO₆, the Cu²⁺ ions ($S=1/2$) in Na₂Cu₂TeO₆ form a distorted honeycomb lattice in the $ab$ plane, which are separated by the Na⁺ layers along the $c$ axis (see Fig. 1). Magnetic susceptibility measurements on a powder sample of Na₂Cu₂TeO₆ revealed a spin gap $\Delta\sim 127$ K [14], which has been attributed to strong AF couplings $J_{1}$ in the DFT calculations [14, 16, 17, 18]. However, controversy remains as to the sign of the intrachain coupling $J_{2}$: magnetic susceptibility data can be fitted equally well by the FM or AF $J_{2}$ model [15], and this ambiguity is not resolved by contradictory DFT results that support either AF [14, 16] or FM [17, 18] $J_{2}$ couplings. The magnitude of $J_{3}$ also remains to be determined, as comparable $J_{2}$ and $J_{3}$ couplings might invalidate a spin chain scenario.

To establish whether Na₂Cu₂TeO₆ represents a rare realization of the alternating AF-FM spin chains, here we perform inelastic neutron scattering (INS) experiments on a single crystal sample of Na₂Cu₂TeO₆ to determine the exchange coupling strengths. We confirm the spin gap originates from the dominant AF coupling $J_{1}$ (22.8 meV), and the interchain coupling $J_{3}$ is found to be much weaker (1.3 meV). Most importantly, we reveal the intrachain coupling $J_{2}$ to be FM with a strength ($-8.7$ meV) that is much higher than $J_{3}$, thus establishing Na₂Cu₂TeO₆ as a weakly-coupled alternating AF-FM chain compound. Through the DFT calculations, the alternating signs of the interchain couplings can be attributed to their different exchange paths.

Experimental Details. Polycrystalline Na₂Cu₂TeO₆ was utilized as a source material for crystal growth in a flux based on TeO₂ and Na₂CO₃. Details for the polycrystal synthesis and characterization can be found in the Supplemental Materials [19]. Powders of these materials in a ratio of 2(Na₂Cu₂TeO₆):1(Na₂CO₃):4(TeO₂) were mixed and loaded into a Pt crucible that was covered with a Pt lid. The crucible was heated rapidly to 900 ^∘C where the melt was homogenized for 12 h in air. The furnace was then cooled at 2 ^∘C/h to 500 ^∘C at which point it was turned off to cool. The translucent green crystals were recovered by boiling the product-filled crucible in a hot aqueous solution of potassium hydroxide, followed by additional rinsing in deionized water.

INS experiments on Na₂Cu₂TeO₆ were performed on the fine-resolution Fermi chopper spectrometer SEQUOIA at the SNS of the ORNL. A single crystal (mass $\sim$140 mg) was aligned with the (001) vector vertical. A closed cycle refrigerator (CCR) was employed to reach temperatures $T$ down to 5 K. The incident neutron energy was $E_{i}=60$ meV, and the Fermi chopper frequency of 180 Hz and 420 Hz was selected in the high-intensity and high-resolution modes, respectively. Data were acquired by rotating the sample in 1^∘ steps, covering a total range of 200^∘ at 5 K and 100^∘ at higher temperatures. Data reductions and projections were performed using MANTID [20] and HORACE softwares [21].

First-principles calculations using the projector augmented wave (PAW) method were performed based on the DFT as implemented in the Vienna ab initio Simulation Package (VASP) code [22, 23, 24]. The generalized gradient approximation (GGA) and the revised Perdew-Burke-Ernzerhof (PBEsol) function were used to treat the electron exchange-correlation potential [25, 26]. Based on the ab initio ground-state wave function, the Cu-site centered Wannier functions (WFs) with orbital $d_{x^{2}-y^{2}}$ were constructed using the WANNIER90 code [27, 28].

Results. As shown in Fig. 1(a), the large separation of $c=5.92$ Å between the honeycomb layers indicates likely negligible interlayer couplings. This is immediately confirmed in our single-crystal INS experiment. Figure 2(a) plots a representative slice of the measured spectrum along the $Q_{l}$ direction that is perpendicular to the $ab$ plane. A single excitation mode is observed at $\sim 18$ meV, which is flat within the instrument resolution (1.6 meV at the elastic line). In contrast, strong dispersion in the energy range of $E=$ [17, 28] meV are observed in the $ab$ plane as summarized in Fig. 3(a-d), thus confirming all the related couplings to be within the honeycomb layers.

The lack of dispersion out of the $ab$ plane allows us to integrate data along $Q_{l}$ for better statistics. Fig. 2b plots the scattering intensity integrated in the range of $Q_{l}$ in [-2, 2] (r.l.u.) and $E$ in [17, 28] meV. Along the $Q_{k}$ direction, intensity is strongly modulated with a periodicity of $\sim 1.5$ (r.l.u.). For dimer systems, it is established that intensity of the triplon excitations is modulated by the dimer structure factor $S(\bm{Q})\propto[1-\cos(\bm{Q}\cdot\bm{r})]$, where $\bm{r}$ is the vector that connects the two spin sites within the dimer [29, 30, 31]. Therefore, the modulation along $Q_{k}$ indicates that dimers in Na₂Cu₂TeO₆ are forming along the $b$ axis, and its periodicity tells the bond distance $r$ within the dimers. As shown in Fig. 2(c), the model that assumes dimers forming over the $J_{1}$ bonds with a distance of $r_{1}=2b/3$ accurately describes the intensity modulation, thus confirming the $J_{1}$ couplings to be AF and dominant as proposed by the DFT calculations [15, 16, 17, 18].

Figure 3(a-d) summarizes the dispersion along $Q_{h}$ and $Q_{k}$ in the $ab$ plane. Along the chain direction ($Q_{k}$), the triplon band reaches its lowest energy of $\sim 18$ meV at $Q_{k}=N+1/2$ with integer $N$, indicating the FM character of the interdimer $J_{2}$ couplings [32, 11]. Different from the conventional chain compounds, the top of the dispersion varies at successive integer $Q_{k}$ positions, which might arise from the interchain couplings. As compared in Fig. 3(c) and (d), although the triplon band looks flat along ($Q_{h}$ 0.5 0) at the bottom of the band, dispersion with a bandwidth of $\sim 3$ meV is observed along ($Q_{h}$ 1 0) at the top of the band, which suggests weak but non-negligible $J_{3}$ couplings.

The triplon dispersion in dimer systems can often be analyzed through the random phase approximation [33, 29, 34, 35, 36]. Under this approximation, the dispersion relation can be written as

where $R(T)$ describes the population difference between the singlet and triplet states [29, 34] and for Na₂Cu₂TeO₆ can be approximated by 1 at $T=5$ K due to the large excitation gap. $\mathcal{J}(\bm{Q})$ is the Fourier sum of interactions beyond dimer exchange:

Experimental dispersion values were extracted from Gaussian fits to constant $Q$ scans at 140 points throughout the measured reciprocal space volume, which were then fitted by the dispersion relation in Eq. (1). The fitted coupling strengths are $J_{1}=22.78(2)$ meV, $J_{2}=-8.73(4)$ meV, and $J_{3}=1.34(3)$ meV. The calculated spectra are summarized in Fig. 3(e-h) for comparison with the experimental data. The FM character of the $J_{2}$ couplings is thus unambiguously established, and the weakness of the interchain $J_{3}$ justifies the description of Na₂Cu₂TeO₆ as a weakly-coupled alternating AF-FM chain compound.

In Haldane chains the existence of a spin gap is known to protect the spin entanglement [37]. Na₂Cu₂TeO₆ can serve as a model system to test that the same ideas are valid for alternating spin chains. In long-range ordered magnets, the excitation gap often decreases at higher $T$ due to reduced ordering moments. As a contrast, the gap of Haldane chains increases with $T$ due to the reduction in the coherent length and the consequent finite-size effect [38, 39]. The $E$ scans at $\bm{Q}=$ (1, 0.5, 0) shown in Fig. 4(a) indeed reveal an increased gap at elevated $T$. We parameterize the lineshape using a Gaussian function that is independently fit at each temperature, and the value of $\Delta$ is the centroid of the Gaussian peak. As summarized in Fig. 4(b), in a large range below $\sim 150$ K, the fitted gap size $\Delta$ and full width at half maximum (FWHM) $\Gamma$ exhibit an activated behavior that are characteristic of Haldane chains [40, 41]:

where $\alpha$, $\gamma$, and $\Gamma_{0}$ are fitting parameters, $\Delta_{0}$ is the gap size at 0 K and is fixed at 18.0 meV. The validity of the activated behavior further confirms the similarity between the $S=1/2$ alternating AF-FM chains and the integer-spin Haldane chains, thus revealing the robustness of the topological ground state against weak interchain couplings.

Discussion. The emergence of spin chains in Na₂Cu₂TeO₆ can be ascribed to the distortion of the honeycomb lattice. As shown in Fig. 1(b), the exchange paths of $J_{3}$ involve the longest Cu-O bonds b3 of the distorted octahedra. Therefore, the $J_{3}$ couplings are expected to be weak as the unoccupied Cu $3d_{x^{2}-y^{2}}$ orbitals disfavor the elongated bond direction, which reduces the electron hopping between the chains.

The sign of the interchain couplings $J_{2}$ is more subtle, and different scenarios exist in the previous DFT calculations [14, 16, 17, 18]. As summarized in the Supplemental Materials, our DFT calculations confirm the alternating FM and AF intrachain couplings in agreement with the experiment. The contrasting $J_{1}$ and $J_{2}$ couplings can be understood through the Wannier functions (WFs) as plotted in Fig. 5(a). Due to the contributions from the O $2p$ states, the WFs overlap directly over the $J_{1}$ paths but are almost orthogonal over the $J_{2}$ path. Therefore, the $J_{1}$ path, in spite of its longer distance, develops a stronger coupling than that over the $J_{2}$ path. Based on the WFs, the signs of the couplings can be understood through the Goodenough-Kanamori-Anderson rules [42, 43, 44, 45] as shown in Fig. 5(b). For the $J_{1}$ couplings, the Cu-O$\cdots$O-Cu super-superexchange leads to an AF interaction between the Cu²⁺ spins. While for the $J_{2}$ couplings, the interaction become FM as the angle of ∠Cu-O-Cu is close to $\sim 90^{\circ}$, which means a pair of orthogonal O $2p$ orbitals with parallel spins are involved in the virtual electron hopping.

Conclusions. Neutron scattering experiments have been performed on the honeycomb-lattice compound Na₂Cu₂TeO₆ to study its spin correlations. A triplon excitation mode was observed, which exhibits strong dispersion along the chain but weak dispersion perpendicular to the chain. Under the random phase approximation, the intrachain couplings were found to be alternating AF and FM, and a weak interchain coupling was also established. The emergence of spin chains in Na₂Cu₂TeO₆ was ascribed to the distortion of the honeycomb lattice, and the alternating intrachain couplings were understood through the DFT calculations. Our works establish the existence of weakly-coupled alternating AF-FM spin-1/2 chains in Na₂Cu₂TeO₆ and reveal a robust gapped ground state that is similar to that of the integer-spin Haldane chains.

Acknowledgments. We acknowledge helpful discussions with Jyong-Hao Chen, Tong Chen, and Tao Hong. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. This research used resources at the Spallation Neutron Source (SNS), a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory (ORNL).

Weakly-coupled alternating $S=1/2$ chains in the distorted honeycomb-lattice compound Na₂Cu₂TeO₆
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