Realization of $Z_{2}$ Topological Metal in Single-Crystalline Nickel Deficient NiV₂Se₄

Single-crystal x-ray diffraction (SXRD) data were processed by Eval15 [22] and SADABS [23]. All diffraction maxima could be indexed by a single unit cell with lattice parameters $a=6.1453(3)$ Å, $b=3.4327(2)$ Å, $c=11.5901(4)$ Å and $\beta=91.619(3)$ deg, similar to the published unit cell of NiV₂Se₄ [24]. Crystal structure refinements were done with Jana2006 [25] with the monoclinic space group $I2/m$. Table 1 shows the structural analysis for three different models on basis of the occupancy of the Ni site, in order to determine whether there are vacancies on the Ni site or there exists Ni/V disorder.

Initially, model I was tested where the Ni site is completely filled. The composition is NiV₂Se₄ and it leads to a good fit of the diffraction data with $R=0.0317$. In model II, a Vanadium atom (V_Ni) was introduced on the Ni site, such that the sum of the occupancies remains 1, Occ[Ni] + Occ[V_Ni] = 1. Refinement of Occ[V_Ni] then leads to a significant improvement of the fit, with $R=0.0213$. However, the chemical composition derived from model II (Table 1) is not in agreement with the composition
Ni_0.925(9)V_2.054(7)Se₄ found by an electron probe micro-analyzer (EPMA). Such a large difference in the measured compositions between SXRD and EPMA led us to believe that the partially filled site of Ni is not occupied by V. Therefore, model II is not suitable. Model III involves vacancies at the Ni site. Refinement of the occupancy of Ni results in an even better fit to the data, with $R=0.0209$. Moreover the chemical composition of model III is in agreement with that obtained by EPMA, with a difference of about 7% deemed acceptable. One can also notice the standard uncertainty (su.) in the refined occupancy of Ni in model III is four times smaller than in model II. Our findings from SXRD and EPMA suggest that there is 15% vacancies on the Ni site. Table 2 provides the atomic coordinates and atomic displacement parameters (ADPs) of model III.

From the measured diffraction data at 298 K and at 100 K, reciprocal layers were reconstructed for $0kl$, $hk0$ and $h0l$ (Figure S1, S2 and S3 in the Supplemental Material [26]). They demonstrate that, upon cooling the crystal down to 100 K, there are no superlattice reflections observed and there is no change to the lattice, indicating the absence of a CDW phase transition. Table 3 shows the crystallographic information of the crystal at 298 and 100 K, respectively, as based on model III.

For details regarding data processing of the 100 K data refer Section S1 in the supplemental material [26]. The absence of the CDW transition is substantiated by our band structure calculations described in Section 2.2.

We now discuss the stability and nontrivial state of NiV₂Se₄ with regards to the calculated band structure (Figure 2).

These results also apply to Ni_0.85V₂Se₄, because the major effect of the introduction of vacancies is a shift of the Fermi level, which is presented in Section S3 of the Supporting Information. The results on Ni-deficient material show a 50 meV downward shift of the Fermi level as compared to pristine NiV₂Se₄. The phonon dispersion curves and associated phonon density of states (PDOS) are shown in Figure 2(a). The absence of imaginary phonon frequency in the full BZ ensures that NiV₂Se₄ is dynamically stable. Moreover, the absence of a soft phonon mode indicates the absence of a structural instability or a periodic lattice distortion, as it would be expected in CDW materials, in agreement with the experiments. The PDOS suggests that Ni and Se atoms contribute to low-lying phonon modes, whereas the V and Se atoms dominate the high-frequency phonon modes.

The bulk band structure along the high-symmetry directions in the BZ of NiV₂Se₄ is shown in Figure 2(b) without spin-orbit coupling (SOC). It is metallic, where both the valence and conduction bands cross the Fermi level. Notably, the valence and conduction bands cross along $\Gamma$–$C$, $M$–$D$, $D$–$A$ directions at generic $\mathbf{k}$ points close to the Fermi level, as indicated by red circles. The crossing points are stable against band hybridization to realize nodal line crossings. The band structure in presence of SOC is shown in Figure 2(c). Although the system remains metallic, the band crossing points are now gapped, separating the valence and conduction bands locally at each $\mathbf{k}$ point. The orbital resolved band structure in Figure 2(d) shows that these band crossing points are composed of Ni–$d$, V–$d$, and Se–$p$ orbitals. The existence of local band gap allows the computation of the $Z_{2}$ invariants $(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ in a manner similar to $Z_{2}$ topological insulators. Since NiV₂Se₄ crystal respects inversion symmetry, we can calculate the $Z_{2}$ invariants $(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ from the parity eigenvalues of the valence bands at each time-reversal-invariant momentum (TRIM) point [28]. The product of the parity eigenvalues of the occupied bands at the eight TRIM points is shown in Figure 2(e). The calculated $Z_{2}$ of $(1;110)$ indicates that NiV₂Se₄ is $Z_{2}$ topological metal. The topological phases are immune to the effects of a Hubbard $U$ for the Ni and V atoms, as discussed in Section S4 of the Supporting Information.

The nontrivial state is further demonstrated by calculations of the surface electronic structure. They reveal a single Dirac cone topological state present within the energy gap of the bulk band structure (see Section S3 of the Supplemental Material for details [26]). Figure 2(f)-(h) show the calculated Fermi surface of NiV₂Se₄. Owing to the multiband crossings at the Fermi level, the Fermi surface consists of both electron and hole bands which are shown separately in Figure 2(f) and 2(g), respectively. The metallic behavior is consistent with the experimental measurements.

The electrical resistivity, $\rho(T)$, of Ni_0.85V₂Se₄ decreases upon cooling from room temperature down to 1.5 K (Figure 3).

In particular, a broad hump is observed between 150–200 K, which also agrees with Bouchard et al. (1966) [21], who first reported an anomaly at about 160 K in $\rho(T)$ of a polycrystalline sample of NiV₂Se₄. They suggested that the observed anomaly may be due to either a crystallographic or a magnetic transition. However, we find that the broad anomaly between 150–200 K cannot be ascribed to a structural transition, since our diffraction studies did not find any evidence for it (Section 2.1). Explanations for these features could be the development of spin fluctuations or CDW fluctuations as opposed to long-range magnetic order or CDW ordering. The latter involve structural distortions, like in CuV₂S₄ [9], BaFe₂Al₉ [29], and $R_{2}$Ir₃Si₅ ($R$ = Lu, Er) [30, 31].

The metallic behavior of Ni_0.85V₂Se₄ is consistent with the properties of polycrystalline materials of the defect-NiAs structure [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Below the broad hump, $\rho(T)$ exhibits an unusual $T^{3/2}$ dependence for 15–150 K. Such a behavior is generally observed in spin glasses and amorphous ferromagnets, where a $T^{3/2}$ dependence in $\rho(T)$ is the result of a diffusive motion of the charge carriers [32, 33]. Most theories suggest an explanation of the anomalous resistivity by an inhomogeneous magnetic state, through comparison with the resistivity of spin glasses [34]. Since Ni_0.85V₂Se₄ is a single crystal, the diffusive carrier motion has to be intrinsic somewhat similar to the behavior observed in MnSi under pressure [35].

It is well established that the electronic properties of metals on the border of magnetic phase transitions at low temperatures are often found to exhibit temperature dependencies that differ from the predictions of Fermi liquid theory. Early attempts to explore such non-Fermi-liquid behavior have been based on mean-field treatments of the effects of enhanced magnetic fluctuations, as in the self-consistent renormalization (SCR) model [36]. The SCR model seems to explain the temperature dependence of the resistivity of Ni₃Al at high pressures, where its ferromagnetism is suppressed, leading to a negligible magnetic correlation vector $\kappa(T)$. In the idealized limit $\kappa\rightarrow 0$ and $T\rightarrow 0$, the SCR model predicts that in 3D, the quasiparticle scattering rate $(\tau_{qp})^{-1}$ varies linearly with the quasiparticle excitation energy, rather than quadratically as in the standard Fermi-liquid picture. This form of $(\tau_{qp})^{-1}$ is similar to that of the marginal Fermi-liquid model [37], which is normally associated with a linear temperature dependence of the resistivity. However, at the border of ferromagnetism, the relevant fluctuations responsible for quasiparticle scattering are of long wavelength and, thus, are ineffective in reducing the current. This leads to a transport relaxation rate $(\tau_{tr})^{-1}$ that differs from $(\tau_{qp})^{-1}$ and is not linear in $T$. Instead, it varies as $T^{5/3}$, which was observed in high pressure studies on Ni₃Al.

For a metal on the border of metallic antiferromagnetism in three dimensions, the SCR model predicts $\rho(T)$ to vary as $T^{3/2}$ in the idealized limit $\kappa\rightarrow 0$, $T\rightarrow 0$, where $\kappa(T)$ now stands for the correlation wave vector for the staggered magnetization. This simple model for the scattering from antiferromagnetic fluctuations assumes that the scattering rate can be averaged over the Fermi surface. Within the SCR model, this procedure would seem to require the presence of a sufficient level of defects, the latter which might be provided by the chemical disorder due to vacancies at the A-site in Ni_0.85V₂Se₄. This site vacancy is supposed to inhibit the short circuiting caused by the carriers on the cold spots of the Fermi surface, i.e., regions far from the hot spots connected by the antiferromagnetic ordering wave vector and, thus, strongly affected by spin-fluctuation scattering. Such a behavior has been observed in NiSSe [38].

In the case of single crystalline Ni_0.85V₂Se₄, the temperature dependence of the electrical resistivity between 15 and 150 K could be fitted by (upper inset (b) of Figure 3)

where $\rho_{0,1}$ is the residual resistivity and the second term is the contribution from scattering proposed by the SCR model. It is of interest to see that below 15 K, $\rho(T)$ can be described by (lower inset (c) of Figure 3)

which suggests that the anomalous temperature dependence ceases to exist below 15 K. The second term in Equation 2 arises from the scattering of electrons due to localized spin fluctuations [39].

The temperature dependence of the magnetic susceptibility clearly shows paramagnetic behavior (Figure 4).

However, it does deviate from a simple Curie-Weiss behaviour, as discussed below. Magnetization measurements performed under field cooled (FC; not shown) and zero-field cooled (ZFC; Figure 4) conditions produced identical results, implying that spin glass phenomena can be ruled out in the case of Ni_0.85V₂Se₄. The high temperature (150–300 K) data can be described by a modified Curie-Weiss law,

where $\chi_{0}$ includes the diamagnetic and temperature-independent paramagnetic contributions. From the value of the Curie constant $C$, the effective magnetic moment is estimated as $\mu_{eff}=0.57\,(1)\;\mu_{B}$ per formula unit. Usually, the Ni ion does not have a magnetic moment in metallic compounds of the type AT₂X₄, since all valence electrons of Ni are completely delocalized in fully ordered crystal structures. Apparently, this scenario breaks down in Ni_0.85V₂Se₄, which again might be related to the vacancies on the Ni site. The Weiss temperature is found to be $\theta=4.8\,(1)$ K, indicative of weak antiferromagnetic exchange interactions. The inverse susceptibility (Figure 4) shows small deviations from linear behavior below 150 K.

Isothermal magnetization (M–H) curves have been measured at selected temperatures within the range 2–300 K up to magnetic fields of $\pm 7$ T (Figure 5).

Hysteresis is absent at all temperatures, demonstrating that Ni_0.85V₂Se₄ does not undergo a magnetic phase transition, but remains in the paramagnetic state down to 2 K. At the lowest measured temperatures, the M–H dependence is S-shaped instead of linear at higher temperatures. This might be related to the fact that lower temperatures allow to reach higher values of H/T, thus bringing the M–H relation outside the linear regime. However, attempts failed to bring all data onto a universal H/T dependence. This seems to imply that the crystal does not behave like an usual paramagnet below 15 K.

The specific heat, $C_{p}(T)$, decreases on cooling (Figure 6).

The absence of sharp features and the presence of a broad, weak anomaly at $T$ = 150–170 K rule out a possible structural phase transition, which is in agreement with our single-crystal x-ray diffraction data. However, the presence of this anomaly in the specific heat data is in agreement with the anomalies observed in resistivity data. The origin of this anomaly could be attributed to spin or CDW fluctuations. It is important to specify here, that the anomalous temperature dependence of the resistivity ($T^{3/2}$) starts below this transition. The low temperature data can be fitted with the model [40],

The first term is the contribution due to conduction electrons, the second term is the contribution from the lattice, and the third term is the contribution from spin fluctuations due to disorder. This model was successfully employed to explain a similar anomaly in the specific heat of Fe_1-xV_2+xAl in another context [40]. The fit yields $\gamma=104.0\,(1)$ mJ/mol K² and the Debye temperature $\theta_{D}=252.9\,(1)$ K. The existence of magnetic fluctuations below 15 K has been corroborated by the isothermal magnetization data, as explained earlier. The presence of such magnetic fluctuations seems to change the anomalous temperature dependence of resistivity behavior below 15 K.