Observation of local atomic displacements intrinsic to the double zigzag chain structure of 1$T$-$M$Te₂ ($M$ = V, Nb, Ta).

Single crystal samples of 1$T$-$M$Te₂ ($M$ = V, Nb, Ta) were synthesized by a conventional solid phase reaction method. The mixture of the constituent elements in their amphoteric ratios was vacuum-sealed and sintered at 1000 $\tccentigrade$ for $M$ = Nb and Ta, and at 850 $\tccentigrade$ for $M$ = V for 15 hours. The obtained samples are basically powders, but some of them contain tiny single crystals of several tens of micrometers on a side, which were used for diffraction experiments using synchrotron radiation x-rays. X-ray diffraction (XRD) experiments were conducted using the BL02B1 beamline at SPring-8 at an x-ray energy of 40 keV. The typical dimensions of the 1$T$-$M$Te₂ ($M$ = V, Nb, Ta) single crystal used for the XRD experiment were 20 × 20 × 20 $\mu$m³. A He-gas-blowing device was employed to cool the sample to 100 K. A 2D CdTe PILATUS detector was utilized with the diffractometer. The CrysAlisPro program was used to integrate the diffraction profiles. Diffraction intensity averaging and refinement of structural parameters were performed using Jana2006 program [29]. Crystal structure was visualized by using VESTA [30]. The obtained powder diffraction data were indexed using Conograph [31], and the analysis was performed using Rietan-FP [32].

Fig. 1(c) and (d) show single-crystal XRD patterns of 1$T$-TaTe₂ in the high temperature (300 K) and low temperature (125 K) phases. Although the Bragg peaks remain sharp, the presence of a merohedral domain sharing a 20$\bar{1}$ plane, as shown in Fig. 1(e), is confirmed. In the following, we show the results of the analysis in which only one domain component is extracted. The diffraction pattern changes below the phase transition around 170 K, and the superlattice peak shown by the circle is observed at low temperatures.

The obtained crystal structures are shown in Figs. 1(a) and (b), both of which are consistent with the previously reported structures [25, 28, 33]. Details of the structure analysis results are summarized in the Appendix of this paper. It is noteworthy that the Ta-Ta distance constituting the linear trimer changes significantly between the high and low temperature phases. In the high-temperature phase, the linear trimer is composed of equally spaced Ta2-Ta1-Ta2 arrays, whereas in the low-temperature phase, the displacement of Ta1b ions is accompanied by a large difference in the distance between adjacent Ta-Ta inside the Ta2a-Ta1b-Ta2b array. This indicates that the Ta2a-Ta1b-Ta2b array does not form a linear trimer in the low-temperature phase, but rather transforms into a Ta2a-Ta1b dimer and isolated Ta2b ion. Although the contraction of the Ta-Ta distance upon heptamerization is greatest between Ta1a and Ta1b, the Ta1a-Ta1b distance of $\sim$3.34 Å in the low temperature phase is still much longer than the Ta2a-Ta1b distance of $\sim$3.18 Å, indicating that the Ta2a-Ta1b bond is more essential. The large change in the Ta1a-Ta1b distance associated with the phase transition should be a side effect of the $b$ axis displacement of Ta1b to form dimers with two Ta2a simultaneously.

Structural analysis revealed that the atomic positions of the high-temperature phase are almost the same as those previously reported [25], and the anisotropic ADP at the Ta1 site also shows an anomalous elongation in the double zigzag chain direction as shown in the inset of Fig. 2(a), as indicated in a previous report [28]. Fig. 2(a) shows that the $U_{22}$ parameter has an unusually large value compared to $U_{11}$ and $U_{33}$. It is important to note that the $U_{11}$ and $U_{33}$ parameters are zero when extrapolated toward 0 K, but the $U_{22}$ parameter clearly reaches a finite value. This indicates that the anomalous increase in $U_{22}$ is not simply anisotropic strong thermal oscillation, but rather a local distortion at the Ta1 site. This is consistent with the possibility of dynamic disorder discussed in the previous report [28].

In order to clarify the local displacement of the ions at the Ta1 site, we performed a structural analysis using a split-site model. This is an analytical method that examines the change in the confidence factor $R$ with the change in distance $r$, assuming that the Ta1 site ions are not at the central position, but two ions with occupancy 0.5 exist at positions +$r$ and -$r$ along the $b$ axis, as shown in the inset of Fig. 2(b). If the atomic displacement as represented by this model does not actually occur, the $R$ value will show a minimum at $r$ = 0. If it occurs, the $R$ value will be a minimum at a finite $r$. As shown in Fig. 2(b), the analysis shows that the $R$ value varies significantly with $r$, reaching a minimum at about $r$ = 0.13 Å. This indicates that there is an intrinsic local atomic displacement in the high-temperature phase of 1$T$-TaTe₂, where the local atomic displacement $r$ value that minimizes the $R$ value is almost independent of temperature, and consequently the Ta1-Ta2 bond that splits into two types, as shown in Fig. 2(c). It seems strange that such local distortion is independent of temperature, but it should be noted that similar behavior has been observed at high temperatures in AlV₂O₄ [14] and Li₂RuO₃ [15] where molecular formation occurs at low temperature.

Will similar local atomic displacements appear in 1$T$-NbTe₂ and 1$T$-VTe₂, where the double zigzag chain structure is maintained down to low temperatures? The anisotropic ADPs of the high-temperature phases of these two materials are shown in the inset of Fig. 3(a). Although a larger $U_{22}$ is realized compared to $U_{11}$ and $U_{33}$, its value is smaller than that of 1$T$-TaTe₂. As summarized in Appendix, the values of lattice constants and bond lengths are close between 1$T$-TaTe₂, 1$T$-NbTe₂ and 1$T$-VTe₂. Thus, the difference in magnitude of $U_{22}$ indicates a clear difference in local structure between these telluriums. Also, as shown in Fig. 3(a), the $U_{22}$ parameter decreases in a temperature-dependent manner to zero at 0 K. Fig. 3(b) shows the results of the split-site model analysis for these two materials: 1$T$-NbTe₂ exhibits a minimum $R$ value at finite $r$, but the value of $r$ that minimizes $R$ at 300 K is about 0.09 Å, which is smaller than that of 1$T$-TaTe₂. The change of $R$ with $r$ is also very small compared to 1$T$-TaTe₂. This trend is more pronounced for 1$T$-VTe₂, where the $R$ value is almost constant over a wide $r$ range. As shown in Fig. 3(c), the local atomic displacements, if any, of 1$T$-NbTe₂ and 1$T$-VTe₂ are much smaller than those of 1$T$-TaTe₂ and decrease upon cooling to zero at 0 K.

These experimental results raise two questions. The first question is what is the origin of these local atomic displacements. In the high-temperature phase of 1$T$-TaTe₂, we can predict that two dimer patterns with equal lattice energy can be realized, as shown in the right side of Fig. 4(a), which are thermally fluctuating, resulting in a linear trimer in the average structure, as shown in the left side of Fig. 4(a). In the three-center, two-electron bonding state derived from the linear trimer, three orbitals are formed: a bonding orbital, a non-bonding orbital, and an anti-bonding orbital, with two electrons stored in the bonding orbitals, which are energetically stabilized, as shown in the left energy scheme of Fig. 4(a). When local atomic displacement occurs, the linear trimer is transformed into a dimer-isolated atom pair, which should result in the formation of a bonding orbital and an anti-bonding orbital, derived from the dimer, and an isolated atomic orbital, as shown in the right energy scheme of Fig. 4(a). The three energetically equally spaced orbitals formed by the linear trimer and dimer-isolated atom pairs are similar at first glance. However, if the actual interatomic distances of the dimer are sufficiently closer than those of the linear trimer, the bonding orbitals should be $\Delta$ lower in energy for the dimer. If the energy difference $\Delta$ is larger than the energy loss due to lattice distortion, $\varepsilon$, the local dimer is expected to stabilize. Since the energies between the two dimer patterns are equal, we expect the two dimer patterns to appear thermally fluctuating, as shown in Fig. 4(a). The transition process between the two dimer patterns is via a trimer state, and the low energy of the bonding orbital in this trimer state should facilitate the transition between the two patterns. The long-range ordering of this local distortion at low temperatures leads to the formation of heptamer clusters, as shown in Fig. 1(b).

Another question is why different behaviors appear for 1$T$-TaTe₂, 1$T$-NbTe₂, and 1$T$-VTe₂. As shown in Fig. 3(c), local atomic displacements are largest for Ta(5$d$) and small for Nb(4$d$) and V(3$d$). In 1$T$-TaTe₂ with large 5$d$ orbitals, the electronic energy gain $\Delta$ is expected to be the largest because the orbitals overlap more during dimerization than in 1$T$-NbTe₂ and 1$T$-VTe₂. If $\varepsilon$ $<$ $\Delta$ is realized at 1$T$-TaTe₂, the overall energy of the system including lattice energy with respect to atomic positions will be as shown in Fig. 4(b). Since atomic displacement and the associated local dimerization stabilize the system, and this trend does not change even when the temperature is lowered, the atomic displacement $r$ does not show any temperature dependence, and the local distortion is expected to survive until just above the phase transition. On the other hand, for 4$d$ Nb and 3$d$ V, the orbital overlap is small and the energy is not very stable due to local dimerization. Therefore, $\varepsilon$ $\geq$ $\Delta$ is realized and the energy of the whole system including lattice energy has a flat shape with respect to $r$. The energy drops to a low central position when the temperature is lowered. Because of these differences in energy schemes, it is thought that among the three tellurides, only 1$T$-TaTe₂ produces large local distortions, which are maintained at low temperatures and develop into heptamers. From the above discussion, it seems likely that in 1$T$-NbTe₂ and 1$T$-VTe₂, when the distance between adjacent transition metals is closer, the orbital overlap becomes larger and dimer (heptamer) states such as 1$T$-TaTe₂ are formed. It should be noted that it has been reported that applying pressure to 1$T$-NbTe₂ induces a structural evolution from the trimeric to the dimeric structure [34], confirming the validity of this argument.

An argument similar to the local distortion of 1$T$-TaTe₂ might be applied to other material systems that exhibit molecular formation at low temperatures, such as the spinel compound AlV₂O₄. It has been argued that vanadium in AlV₂O₄ spontaneously forms a heptamer molecule at about 700 K. Based on the Curie paramagnetic component in the low-temperature phase by magnetization measurements, the heptamer has been revealed to be composed of 9 bonds by 18 electrons [6]. However, recent structural analysis of the split-site model on synchrotron XRD data has revealed that the heptamer is actually composed of trimer/tetramer pairs [14]. The two structures are compared in Fig. 25 of the paper by D.I. Khomskii and S.V. Streltsov [35]. The nature of the difference between the two types of molecular structures proposed for AlV₂O₄ can be interpreted as whether the three bonds connecting the upper and lower triangular trimer are linear trimers consisting of three-center two-electron bonds or pairs of dimers and isolated atoms. This is very similar to the present case. An interesting difference is that such local distortions appear in the low-temperature ordered phase in AlV₂O₄, whereas they appear only in the high-temperature phase in 1$T$-TaTe₂. This indicates that unconventional orbital ordering states that appear in the low-temperature phase can universally appear as local distortions in the high-temperature phase, even in systems like 1$T$-TaTe₂, where there is no short-range order in the low-temperature ordered phase.

The presence of local distortion in the high-temperature phase not only has important implications for the mechanism of phase transitions, but is also important for understanding the thermodynamics of the high-temperature phase. For example, while high thermal conductivity is generally realized in the metallic state due to itinerant electrons, it has been observed that in CuIr₂S₄, which undergoes insulating transition with molecular formation at low temperatures, the thermal conductivity of the high temperature metallic phase is lower than that of the insulating low temperature phase. It is argued that local distortion suppresses phonon thermal conduction [36, 37, 38]. In addition, the multiple degrees of freedom of electrons are expected to play an important role in the entropy changes associated with molecular formation [8, 5]. The presence of local distortion will be a new factor, not previously considered, to the concept of electron degrees of freedom in the high-temperature phase. In this respect, the finding of differences in local structure among the systematics of $M$ = V, Nb, and Ta in this study should correspond to providing an attractive stage for discussing the role of local distortion on physical properties through comparison.