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

###### Abstract

Temperature-dependent electronic and magnetic properties are reported for a
$Z_{2}$ topological metal single-crystalline nickel-deficient NiV2Se4. It is
found to crystallize in the monoclinic Cr3S4 structure type with space group
$I2/m$. From single-crystal x-ray diffraction, we find that there are
vacancies on the Ni site, resulting in the composition Ni0.85V2Se4 in
agreement with our electron-probe microanalysis. The electrical resistivity
shows metallic behavior with a broad anomaly around 150–200 K that is also
observed in the heat capacity data. This anomaly indicates a change of state
of the material below 150 K. We believe that this anomaly could be due to spin
fluctuations or charge-density-wave (CDW) fluctuations, where the lack of
long-range order is caused by vacancies at the Ni site of Ni0.85V2Se4.
Although we fail to observe any structural distortion in this crystal down to
1.5 K, its electronic and thermal properties are anomalous. The observation of
non-linear temperature dependence of resistivity as well as an enhanced value
of the Sommerfeld coefficient $\gamma=104.0\,(1)$ mJ/mol$\cdot$K2 suggests
strong electron-electron correlations in this material. The first-principles
calculations performed for NiV2Se4, which are also applicable to Ni0.85V2Se4,
classify this material as a topological metal with $Z_{2}=(1;110)$ and
coexisting electron and hole pockets at the Fermi level. The phonon spectrum
lacks any soft phonon mode, consistent with the absence of periodic lattice
distortion in the present experiments.

###### keywords:

Charge-density-wave, Topological metal, Vacancies, Non Fermi liquid

Sitaram Ramakrishnan* Shidaling Matteppanavar Andreas Schönleber Bikash Patra
Birender Singh Arumugam Thamizhavel Bahadur Singh* Srinivasan Ramakrishnan*
Sander van Smaalen*

S. Ramakrishnan
Department of Quantum Matter, AdSE, Hiroshima University, Higashi-Hiroshima
739-8530, Japan
Email Address<EMAIL_ADDRESS>
S. Matteppanavar, B. Patra, B. Singh, B. Singh, A. Thamizhavel, S.
Ramakrishnan
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India
Email Address<EMAIL_ADDRESS><EMAIL_ADDRESS>
A. Schönleber, S. van Smaalen
Laboratory of Crystallography, University of Bayreuth, 95447 Bayreuth, Germany
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Topological metals such as Dirac and Weyl semimetals represent exotic quantum
materials with non-trivial band crossings at the charge neutrality point in
the bulk band structure that lead to peculiar quasi-particle excitations and
physical properties. There has been a surge of interest in realizing these
topological metals, owing to their novelty for fundamental science and device
applications [1, 2]. Here, we show that nickel-deficient Ni0.85V2Se4 is a
topological metal with $Z_{2}=(1;110)$ based on first-principles calculations.
Our bulk measurements on single-crystalline Ni0.85V2Se4 show unusual quasi-
particle excitations. The presence of vacancies is a common feature of
compounds AT2X4 (A and T are transition metals; X is chalcogenide), which
display exotic electronic and magnetic properties.

Cubic thiospinels of the type AT2S4 (A = Cu, Ni; T = V, Rh, Ir) have attracted
attention due to the unusual ground states exhibited by them [3, 4, 5, 6, 7,
8]. For instance, superconductivity (e.g. in CuRh2S4) [3], magnetic order in
insulators (CuIr2S4) [4, 5] and charge density waves (CDWs; CuV2S4) [6, 7]
have been found to exist in this class of compounds. It is worthwhile to
recall here that the CDW in CuV2S4 was suppressed in as-grown crystals,
because of the presence of defects [9]. In addition, structural studies have
shown that phase transitions are accompanied by charge ordering and spin
dimerization [8]. The occurrence of spin dimerization in a three-dimensional
compound, like CuIr2S4, is unique and not understood at present.

Apart from these cubic spinel compounds, there exists more than 50
chalcogenides with the same stoichiometry AT2X4 (A = Y, Cr, Fe, Co, Ni; T =
Ti, Y, Cr; X = S, Se, Te), but crystallizing in the monoclinic defect NiAs
structure type (Figure 1) [10], where very little investigations have been
made.

Figure 1: Crystal structure of Ni0.85V2Se4 in a perspective view along
${\mathbf{b}}$. The shortest distances between metal atoms form an unbounded
network on the $(1\,0\,1)$ plane; $d$(Ni–V) = 2.988 (1) Å and $d$(V–V) = 3.033
(1) Å at room temperature. The next shortest metal distance is $b=3.4327$ Å
along $[0,1,0]$.

When we consider the band structures of compounds with the defect NiAs
structure, we find that only octahedral site ions (both Ni and other
transition metal) are involved. Unlike spinels, all metal ions are on
octahedral sites and, therefore, the $t_{2g}$ levels have the lowest energy.
The metallic behavior for the AT2X4 compounds can be ascribed to partially
filled bands formed as a result of T–T overlap. The NiAs structure is also
known as the Cr3S4 structure type (space group $I2/m$) with two nonequivalent
crystallographic sites for Cr. Chromium atoms occupy Wyckoff positions 2a and
4i, with two and four atoms per unit cell, respectively. These sites can be
occupied by different elements [11, 12, 13, 14, 15, 16, 17, 18, 19].

A study on polycrystalline NiV2S4 has indicated a CDW transition at 165 K
[20], while another study on NiV2Se4 reported a small anomaly in the
resistivity around 160 K [21]. Using the vapor transport method, we have
obtained a Ni-deficient single crystal of composition Ni0.85V2Se4, which has
the same crystal structure as NiV2Se4. Band structure calculations reveal that
it is a $Z_{2}$ topological metal. Our experimental studies on the physical
properties of Ni0.85V2Se4 single crystal unfold an unusual electronic ground
state of this material with exotic properties.

##  2 Results and discussion

###  2.1 Evidence of vacancies and absence of lattice distortion.

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.

Table 1: Summary of results of refinements against SXRD data at 298K for three structure models, differing in the treatment of the occupancy of the Ni site. Model | I | II | III
---|---|---|---
Composition | NiV2Se4 | Ni0.4178(143)V2.5822Se4 | Ni0.8524(35)V2Se4
Occ[Ni] | 1 | 0.4178(143) | 0.8524(35)
Occ[V] | 1 | 1 | 1
Occ[VNi] | - | 0.5822 | -
Unique reflections (obs/all) | 578/685 | 578/685 | 578/685
No. of parameters | 24 | 25 | 25
$R_{F}$(obs) | 0.0317 | 0.0213 | 0.0209
$wR_{F}$(all) | 0.0454 | 0.0253 | 0.0255
GoF (obs/all) | 2.65/2.45 | 1.46/1.37 | 1.43/1.34
$\Delta\rho_{min}$, $\Delta\rho_{max}$(e Å${}^{-3})$ | -2.87, 1.49 | -1.07, 1.30 | -1.08, 1.28

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 (VNi) was introduced on the Ni site,
such that the sum of the occupancies remains 1, Occ[Ni] + Occ[VNi] = 1.
Refinement of Occ[VNi] 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
Ni0.925(9)V2.054(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.

Table 2:  Structural parameters for the monoclinic structure of crystal A of Ni0.85V2Se4 at 298 K. Given are the fractional coordinates $x$, $y$, $z$ of the atoms, their anisotropic displacement parameters (ADPs) $U_{ij}$ $(i,j=1,2,3)$ and the equivalent isotropic displacement parameter $U^{eq}_{iso}$. All atomic sites, except Ni, are fully occupied. Atom | Occ | $x$ | $y$ | $z$ | $U_{11}$ | $U_{22}$ | $U_{33}$ | $U_{12}$ | $U_{13}$ | $U_{23}$ | $U^{eq}_{iso}$
---|---|---|---|---|---|---|---|---|---|---|---
Ni | 0.8524(35) | 0 | 0 | 0 | 0.0158(3) | 0.0019(4) | 0.0017(4) | 0 | 0.0011(3) | 0 | 0.0175(2)
V | 1 | -0.0451(1) | 0 | 0.2556(1) | 0.0133(2) | 0.0021(3) | 0.0153(3) | 0 | -0.0006(2) | 0 | 0.0166(2)
Se1 | 1 | 0.3373(1) | 0 | 0.3641(1) | 0.0119(2) | 0.0146(1) | 0.0176(2) | 0 | -0.0003(1) | 0 | 0.0015(1)
Se2 | 1 | 0.3381(1) | 0 | 0.8924(1) | 0.0138(1) | 0.0153(2) | 0.0163(2) | 0 | -0.0004(1) | 0 | 0.0152(1)

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.

Table 3: Crystallographic data of crystal A of Ni0.85V2Se4 at 298 K and 100 K. Notice that space group $I2/m$ is a non-standard setting of space group No. 12 with standard setting $C2/m$ [27]. Temperature (K) | 298 | 100
---|---|---
Crystal system | Monoclinic | Monoclinic
Space group | $I2/m$ | $I2/m$
Space group no. | 12 | 12
$a$ (Å) | 6.1453(3) | 6.1314(2)
$b$ (Å) | 3.4327(2) | 3.4174(3)
$c$ (Å) | 11.5901(4) | 11.5462(3)
$\beta$ (deg) | 91.619(3) | 91.741(3)
Volume (Å3) | 244.45(3) | 241.83(3)
$Z$ | 2 | 2
Wavelength (Å) | Ag-K$\alpha$ | Ag-K$\alpha$
Detector distance (mm) | 100 | 100
$2\theta$-offset (deg) | 0 | 0, 30
Rotation per image (deg) | 1 | 1
$(\sin(\theta)/\lambda)_{max}$ (Å-1) | 0.683589 | 1.009913
Absorption, $\mu$ (mm-1) | 19.271 | 19.469
Tmin, Tmax | 0.5796, 0.8622 | 0.6018 0.8627
Criterion of observability | $I>3\sigma(I)$ | $I>3\sigma(I)$
Number of reflections |  |
measured | 4410 | 8970
unique (obs/all) | 451/470 | 841/1024
$R_{int}$ (obs/all) | 0.0312/0.0321 | 0.0407/0.0415
Composition | Ni0.8524(35)V2Se4 | Ni0.8488(25)V2Se4
No. of parameters | 25 | 25
Occ[Ni] | 0.8524(35) | 0.8488(25)
$R_{F}$ (obs) | 0.0209 | 0.0196
$wR_{F}$ (all) | 0.0255 | 0.0246
GoF (obs/all) | 1.43/1.34 | 1.32/1.21
$\Delta\rho_{min}$, $\Delta\rho_{max}$(e Å-3) | -1.08, 1.28 | -1.27, 1.71

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.

###  2.2 Structural stability and electronic structure

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

Figure 2: (a) Calculated phonon dispersion and total and partial phonon
density of states of bulk NiV2Se4. The low-frequency phonons mainly arise from
Ni and Se atoms whereas high-frequency phonons arise from V and Se atoms. Bulk
band structure (b) without and (c) with spin-orbit coupling (SOC). The nodal
band crossings along high-symmetry directions are highlighted with red circles
in (b). These crossing points are gapped in (c), leading to a local band gap
between valence (blue lines) and conduction (black lines) bands. (d) Orbital
resolved band structure with SOC. Blue, green, and red markers identify Ni
$d$, V $d$, and Se $p$ states, respectively. (e) Parity eigenvalues of the
occupied bands at eight time-reversal invariant momentum points. (f) and (g)
Calculated individual Fermi pockets and (h) the Fermi surface of NiV2Se4.

These results also apply to Ni0.85V2Se4, 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.

###  2.3 Electrical resistivity

The electrical resistivity, $\rho(T)$, of Ni0.85V2Se4 decreases upon cooling
from room temperature down to 1.5 K (Figure 3).

Figure 3: Electrical resistivity of Ni0.85V2Se4. Curve (a) $\rho(T)$ for the
whole measured range of temperatures 1.5–300 K. Curve (b), upper inset,
provides an expanded view of 15–150 K; the solid line is a fit to a $T^{3/2}$
dependence (Equation 1), with $\rho_{0,1}=231.600\,(1)\;\mu\Omega\,$cm and
$a_{1}=0.038\,(1)\;\mu\Omega\,$cm/K3/2. Curve (c), lower inset, gives an
expanded view of 1.5–15 K; the solid line is a fit to a $T^{2}$ dependence
(Equation 2), with $\rho_{0,2}=229.982\,(1)\;\mu\Omega\,$cm and
$a_{2}=0.017\,(1)\;\mu\Omega\,$cm/K2.

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 Ni0.85V2Se4 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 Ni0.85V2Se4 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 Ni0.85V2Se4. 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 Ni0.85V2Se4, the temperature dependence of
the electrical resistivity between 15 and 150 K could be fitted by (upper
inset (b) of Figure 3)

$\rho(T)=\rho_{0,1}+a_{1}\,T^{3/2}\,,$ (1)

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)

$\rho(T)=\rho_{0,2}+a_{2}\,T^{2}\,,$ (2)

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].

###  2.4 Magnetic susceptiblity

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

Figure 4: Temperature dependence of the magnetic susceptibility of
Ni0.85V2Se4. Data measured in a field of 5 T under zero-field cooled (ZFC)
conditions. A measurement under field cooled (FC) conditions produced data
coinciding with those shown for ZFC conditions. The inset provides the inverse
susceptibility; the solid line is a fit to the modified Curie-Weiss law
(Equation 3), with $\chi_{0}=1.01\,(1)\times 10^{-3}\,$emu/mol,
$C=0.040\,(1)\,$emu/mol K, $\mu_{eff}=0.57\,(1)\,\mu_{B}$ and
$\theta=4.8\,(1)\,$K.

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
Ni0.85V2Se4. The high temperature (150–300 K) data can be described by a
modified Curie-Weiss law,

$\chi=\chi_{0}+C/(T+\theta)\,.$ (3)

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 Ni0.85V2Se4, 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).

Figure 5: Magnetization (M) in dependence on the applied magnetic field (H) at
selected temperatures.

Hysteresis is absent at all temperatures, demonstrating that Ni0.85V2Se4 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.

###  2.5 Specific heat

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

Figure 6: Temperature dependence of the specific heat $C_{p}$. (c) displays
$C_{p}(T)$ for temperatures 2–300 K. The upper inset (a) provides an expanded
view of the region of the anomaly around 160 K. The lower inset (b) displays
$C_{p}/T$ vs $T^{2}$ at low temperatures 2–12 K. Contributions from strong
spin fluctuations are demonstrated by a fit (solid line) of Equation 4 to the
data, resulting in $\gamma=104.0\,(1)$ mJ mol-1 K-2, $\beta=0.83\,(1)$ mJ
mol-1 K-4, $\theta_{D}=252.9\,(1)$ K, $a=20.1\,(1)$ mJ mol-1 K-2 and
$T^{*}=0.86\,(1)$ K.

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],

$C_{p}/T=\gamma+\beta\,T^{2}+a\,\ln(T^{*}/T)\,.$ (4)

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
Fe1-xV2+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.

##  3 Conclusions

The perfectly ordered crystal structure of NiV2Se4 does not have a magnetic
moment on Ni. However, our attempted growth of single crystals of NiV2Se4 has
resulted in Ni-deficient crystals, where the crystallographic Ni site has 15%
vacancies, but full chemical order is preserved at the V and the two Se sites
of the AT2Se4 monoclinic structure type. This observation is in agreement with
reports in the literature of crystallization attempts, where part or all of
the transition metal A can be occupied by defects or V atoms [21, 41].

Single crystalline Ni0.85V2Se4 is found to be a $Z_{2}$ topological metal,
that exhibits a cross-over in its physical properties at temperatures around
150 K. All properties, including low-temperature x-ray diffraction data, are
in agreement with the absence of phase transitions down to $T=1.5$ K in the
absence of a magnetic field. The temperature dependence of the electrical
resistivity of Ni0.85V2Se4 is anomalous in the temperature range from 15–150
K, while the presence of magnetic correlations below 15 K probably leads to T2
dependence of resistivity. It is interesting to compare the present results to
the well established $Z_{2}$ topological metal $R$V6Sn6 ($R$ = Gd and Y) [42],
containing a kagome network of V ions coordinated by Sn and separated by
triangular lattice planes of rare-earth ions. The Gd spins contribute to the
observed magnetism of GdV6Sn6 while YV6Sn6 is in a paramagnetic state. Unlike
the case of YV6Sn6, minor site disorder of Ni in Ni0.85V2Se4 is responsible
for its anomalous low-temperature properties. The broad hump in the
resistivity may be related to the anomaly in the specific heat in the
temperature range 150–160 K. However, the exact nature of the crossover is not
understood at present. These features combine with the gradual build-up of
antiferromagnetic fluctuations upon cooling. Isothermal magnetization curves
confirm the absence of bulk magnetic order down to 2 K, but attempts have
failed to bring all data onto a universal H/T dependence. This seems to imply
that the crystal does not behave like a usual paramagnet below 15 K, and it
suggests a build-up of antiferromagnetic correlations. We believe that
Ni0.85V2Se4 is a $Z_{2}$ is a topological metal that displays unusual
electronic properties due to significant electron correlations. However, to
understand this system, one needs more investigations, notably on the nature
of the anomalous behavior around 160 K and Fermi surface studies using angle-
resolved photoemission spectroscopy.

##  4 Experimental and computational section

Crystal growth and characterization by EPMA
Attempts to synthesize single-crystalline NiV2Se4 resulted in the formation of
Ni0.85V2Se4 prepared by solid-state reaction of the elements in evacuated
quartz-glass tubes at a temperature of $T=1153$ K, employing stoichiometric
amounts of the elements obtained from Alfa Aesar: nickel (99.996% purity),
vanadium (99.5%), and selenium (99.999%). Single crystals have been grown by
vapor transport in evacuated quartz-glass tubes, employing iodine as transport
agent and a temperature gradient of $T=1033\,/\,973$ K over 200 mm. The
chemical composition has been determined as Ni0.925(9)V2.054(7)Se4 by an
electron probe micro-analyzer (EPMA). These values are the average over 57
points measured on a a flat, as-grown surface sliced off from the large single
crystal used for the physical-properties measurements. The finding agrees with
the results of x-ray diffraction (see Section 2.1).

Temperature dependent Single-crystal x-ray diffraction (SXRD)
SXRD has been measured on a mar345dtb image plate diffractometer (marXperts,
Germany) with Ag-K$\alpha$ radiation from a rotating anode generator.
Initially, several crystals from the batch were tested, all showing that they
were Ni deficient similar to what has been found with EPMA. For the final
reported measurement a crystal of dimensions $0.1\times 0.1\times 0.1$ mm3 was
crushed from the large crystal which was used for physical properties
measurements.

A complete set of diffraction images has been measured at temperatures 298 K
and 100 K, employing an exposure time such that the strongest Bragg’s
reflection was close to the saturation of the detector. A second run was
measured with an offset of 30 deg in $2\theta$ for obtaining data at high
resolution, and employing an eight times longer exposure time, resulting in
overexposed strong reflections but allowing for detection of weak scattering
effects. Superlattice reflections could not be detected, in agreement with the
absence of a CDW at 100 K. Details are given in Section S1 and S2 of the
Supplemental Material [26].

Physical property measurements
The electrical resistivity $\rho(T)$ has been measured on cooling from 300
down to 1.5 K in a standard four-probe configuration, employing a cryostat and
a LR-700 (Linear Research, USA) bridge with 5 mA current of small AC frequency
of 16 Hz. Electrical contacts have been made using silver paste and gold wires
($40\;\mu$m diameter). The magnetic susceptibility $\chi(T)$ has been measured
for temperatures 2–300 K, using a commercial SQUID magnetometer (MPMS5 by
Quantum Design, USA). Measurements have been repeated in both zero-field-
cooled (ZFC) and field-cooled (FC) conditions for magnetic fields of different
strength. The isothermal magnetization has been measured for magnetic fields
from ${-}7$ to +7 Tesla at a few selected temperatures. The heat capacity
$C_{p}(T)$ has been measured from 2 to 300 K by the thermal relaxation method
using a Physical property measuring system (PPMS, Quantum Design, USA).

Density functional theory calculations
Electronic structure calculations were performed using the Vienna ab initio
simulation package (VASP) [43] with the projector augmented (PAW) [44] wave
method. Generalized gradient approximation of Perdew, Burke and Ernzerhof
(PBE) [45] was used to include the exchange-correlation effects. An energy
cutoff of 380 eV was used for the plane wave basis and a $10\times 10\times
10$ $\Gamma$ centered k-mesh was employed for the Brillouin zone (BZ)
sampling. Experimental lattice parameters with fully optimized ionic positions
were considered in our computations. We optimized atomic positions until the
residual forces on each atom were less than 0.0001 eV/Å. The material-specific
tight-binding Hamiltonian [46, 47] was generated to compute the topological
properties using the WannierTools [48] package. The Xcrysden program was used
to visualize the Fermi surface [49]. The phonon spectrum was obtained using
the frozen phonon method as implemented in the Phonopy [50] code with a
$2\times 2\times 2$ supercell.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

Single crystals have been grown by Kerstin Küspert at the Laboratory of
Crystallography in Bayreuth. We thank Detlef Krauße of the Bavarian
Geoinstitute (BGI) in Bayreuth for performing the electron microprobe
experiments. This research has been funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation)–265092781. SM thanks
the Tata Institute of Fundamental Research Mumbai for providing a postdoctoral
fellowship and VGST CESEM GRD-852. The work at TIFR is supported by the
Department of Atomic Energy of the Government of India under Project No.
12-R&D-TFR-5.10-0100.

Conflict of interest

The authors declare no conflict of interest

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Table of Contents

Growth of single crystals of nominal NiV2Se4 has produced nickel-deficient
Ni0.85V2Se4. Vacancies at the nickel site explain the absence of a charge-
density-wave (CDW) transition, while the temperature dependencies of physical
properties exhibit broad anomalies, indicative of spin or CDW fluctuations.
Density-Functional-Theory (DFT) calculations reveal that Ni0.85V2Se4 is a
$Z_{2}$ topological metal.