Electrical transport and magnetic properties of the triangular-lattice compound Zr₂NiP₂

Figure 1(c) shows the powder XRD pattern measured at 300 K and the final Rietveld refinement. Except the possible nonmagnetic ZrO₂ (weight fraction $\sim$ 3%), no other impurity phases are visible in the samples of Zr₂NiP₂. The grain size is estimated to be larger than 100 nm, according to the narrow full width at half maximum $\sim$ 0.09^∘ 111The instrumental resolution is about 0.05^∘. observed at 2$\theta$ = 34.5^∘ (see Figure 1(c)) [23]. The detailed crystallographic parameters and atomic positions are listed in Table 1. The refinement result suggests that the title compound has a hexagonal structure and the $P$6₃/$mmc$ space group with two formulae per unit cell, as previously reported in reference [22]. Moreover, our refined crystal structure shows quantitative agreement with the previous result [22]. Ni atoms occupy the 2b Wyckoff position and form a geometrically perfect triangular lattice (see Figure 1(b)). Along the $c$ axis, the triangular layers of Ni atoms are stacked in an “AA”-type stacking fashion, and well separated by the double layers of ZrP₆ octahedra. The interlayer Ni-Ni distance is $c$/2 = 6.4430(2) Å, much larger than the intralayer Ni-Ni distance of $a$ = 3.7717(1) Å. The Ni-Zr distances $\sim$ 2.8 Å are reasonably short, which may have considerable influence on the magnetism of Zr₂NiP₂. All of these indicate a quasi-two-dimensional nature of the system of Zr₂NiP₂. Moreover, no antisite mixing was reported [22] due to the large chemical difference among Zr, Ni, and P atoms, and thus the structural disorder should have marginal effects in Zr₂NiP₂.

Figure 2(a) and (b) show the electrical resistivity ($\rho$) measured on two independent as-grown samples of Zr₂NiP₂ at $\mu_{0}H$ = 0 and 9 T down to 2 K, and no significant sample dependence is observed. $\rho$ decreases with the decrease of temperature, and gradually reaches a minimum below $\sim$ 25 K. The measured temperature dependence can be well described by the BGM law [28, 29],

where $\rho_{0}$ is the temperature-independent residual resistivity, the second term accounts for the electron-phonon scattering, and the $T^{3}$ term is due to the Mott-type $s$-$d$ electron scattering. All the least-squares fitting parameters are listed in Table 2. The resulted $\rho_{0}$, Debye temperature $\Theta_{\mathrm{D}}$, and electron-phonon coefficient $R$ of Zr₂NiP₂ show typically metallic features [30, 31]. In comparison, the coefficient $K$ that is directly related to the strength of the $s$-$d$ interaction is about one order of magnitude smaller than those of many other metals [30, 31].

Figure 2(c) shows the magnetoresistance measured on Zr₂NiP₂ at 9 T, MR(9 T) = [$\rho$(9 T)$-\rho$(0 T)]/$\rho$(0 T), which also exhibits the common positive values of metals, MR $\sim$ several percent [32, 33]. As presented in Figure 2(c), the temperature evolution of MR(9 T) roughly follows that of 1/[$\rho$(0 T)]² in the case of Zr₂NiP₂. To understand this observation, we consider both holes and electrons ($n_{1}$, $q_{1}$, $\mu_{1}$, and $n_{2}$, $q_{2}$, $\mu_{2}$, respectively),

where the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are applied perpendicular and parallel to the $z$ axis, respectively. For simplicity, we assume the carrier density $n_{1}$ = $n_{2}$ = $n$, zero-field carrier mobility $\mu_{1}$ = $-\mu_{2}$ = $\mu$, and charge $q_{1}$ = $-q_{2}$ = $e$, so that the Hall term is negligible and current density $\mathbf{j}$ is parallel to $\mathbf{E}$. Thereby, the magnetoresistance is given as, MR($B$) = $B^{2}\mu^{2}$ = $B^{2}$/[2$ne\rho$(0 T)]², in agreement with the experimental observation.

The electrical transport characterizations (Figure 2) consistently suggest that Zr₂NiP₂ is a typical metal without any evident transition down to 2 K. Furthermore, both $\rho$ and low MR imply a nonmagnetic state of Ni in Zr₂NiP₂. To confirm this, we conducted the magnetic property measurements (see below).

Neither clear anomaly/kink nor splitting between the ZFC and FC susceptibilities is observed (see Figure 3(a)), which also precludes the existence of magnetic transition in Zr₂NiP₂ down to 1.8 K. Above $\sim$ 10 K, the magnetic susceptibility is negative, $\chi$ $<$ 0, thus suggesting a diamagnetism. In contrast, the susceptibility exhibits a profound Curie-like tail at low temperatures, which is most likely to originate from a paramagnetic impurity. To confirm this, we fit the experimental magnetization ($M$) by a combination of a Brillouin function for free $s$ = 1 Ni spins (impurity) and a linear magnetization for the bulk system,

where $f$ is the fraction of free spins. When the $g$ factor is fixed to the expected value $g$ = 2 with negligible spin-orbit coupling, we obtain $f$ = 0.25% and $\chi_{\mathrm{bulk}}$ = $-$0.0052, $-$0.0043, $-$0.0037, $-$0.0034, $-$0.0037 cm³/mol-Ni at 1.8, 3, 5 10, 50 K, respectively, with the least-squares standard deviation $\sigma_{\mathrm{LS}}$ = 0.0002$\mu_{\mathrm{B}}$/Ni (see Figure 3(c)). Moreover, we also treat $g$ as an adjustable parameter, the goodness of fit gets better ($\sigma_{\mathrm{LS}}$ = 0.00004$\mu_{\mathrm{B}}$/Ni), and we find $g$ = 3.3, $f$ = 0.09%, and $\chi_{\mathrm{bulk}}$ = $-$0.0026, $-$0.0027, $-$0.0029, $-$0.0032, $-$0.0037 cm³/mol-Ni at 1.8, 3, 5 10, 50 K, respectively (see Figure 3(d)). Both combined fits suggest that the bulk system of Zr₂NiP₂ remains diamagnetic/nonmagnetic down to 1.8 K, i.e., $\chi_{\mathrm{bulk}}$ $<$ 0. The fitted $\chi_{\mathrm{bulk}}$($T$) is roughly consistent with the total susceptibility measured at high temperatures, but doesn’t exhibit a profound Curie-like tail at low temperatures (see Figure 3(a)).

No obvious magnetic hysteresis is observed, thus precluding ferromagnetic freezing down to 1.8 K (Figure 3(b)). The diamagnetism is confirmed by $M$ measured above $\sim$ 10 K, as d$M$/d$H$ $<$ 0. Below $\sim$ 10 K, the paramagnetism of tiny impurities (i.e. d$M$/d$H$ $>$ 0) dominates at $H$ $<$ $H_{\mathrm{p}}$, whereas the bulk diamagnetism dominates at $H$ $>$ $H_{\mathrm{p}}$ after the polarization of the impurity spins. At $\mu_{0}H_{\mathrm{p}}$ = 2-3 T and 1.8 $\leq$ $T$ $\leq$ 10 K, $M$ exhibits a broad hump with the maximum magnetization, $M_{0}$ $\leq$ 0.002$\mu_{\mathrm{B}}$/Ni, suggesting that most of Ni atoms are in the nonmagnetic state in our samples of Zr₂NiP₂.

To understand the observed metallic and nonmagnetic characters, we calculated the density of states (DOS) by DFT+$U$ for Zr₂NiP₂. Figure 4 shows the total and partial DOS of Zr₂NiP₂ calculated with $U$ = 0 and 5 eV, respectively. Several dispersive bands passing over the Fermi energy ($E_{\mathrm{F}}$ = 0 eV) are clearly observed with significant DOS (see Figure 4(b), (d), and (e)), suggesting a typically metallic character of Zr₂NiP₂. The DOS at $E$ $\geq$ $E_{\mathrm{F}}$ is dominantly contributed by Zr atoms, which accounts for the metallic nature.

The core diamagnetic contribution to the magnetic susceptibility can be estimated as $\chi_{\mathrm{D}}$ $\sim$ $-$MW$\times$2$\pi\times$10^-6 cm³/mol (SI unit) [34] $\sim$ $-$2$\times$10^-3 cm³/mol-Ni that is close to the experimental value $\chi_{\mathrm{bulk}}$ $\sim$ $-$3$\times$10^-3 cm³/mol-Ni (see Figure 3(a)), where MW = 303 is the molecular weight of Zr₂NiP₂. Therefore, it is reasonable to estimate that the Pauli susceptibility ($\chi_{\mathrm{Pauli}}$) should be less than $\sim$ 3$\times$10^-3 cm³/mol-Ni. We can obtain the DOS at the Fermi level $D$($E_{\mathrm{F}}$) = $\chi_{\mathrm{Pauli}}$/($N_{\mathrm{A}}\mu_{0}\mu_{\mathrm{B}}^{2}$) $<$ 7 States/eV, which does not contradict with our DFT results. Alternately, the Pauli susceptibility can be estimated as $\chi_{\mathrm{Pauli}}$ = $N_{\mathrm{A}}\mu_{0}\mu_{\mathrm{B}}^{2}$$D$($E_{\mathrm{F}}$) = 2$\times$10^-4 cm³/mol-Ni, where $D$($E_{\mathrm{F}}$) $\sim$ 0.5 States/eV is calculated by DFT (see Figure 4(b) and (d)). Usually, it is very difficult to precisely measure such a small Pauli susceptibility.

In the DFT+$U$ simulations, we considered both the initial antiferromagnetic and ferromagnetic structures for Ni atoms, $|\uparrow\downarrow\rangle$ and $|\uparrow\uparrow\rangle$ with a large magnetic moment $m$ = 2$\mu_{\mathrm{B}}$, but finally got the same nonmagnetic state with $m$ $<$ 0.01$\mu_{\mathrm{B}}$ after optimizations. Moreover, the spin-up and spin-down DOS are almost symmetric (see Figure 4), which naturally explains the non-magnetism observed in Zr₂NiP₂ (see above) [35, 36]. The number of states (occupancy) integrated the Ni DOS (Figure 4(a)) over $-$15 $\leq$ $E$ $\leq$ 0 eV (Fermi energy $E_{\mathrm{F}}$ = 0 eV) is calculated to be 9.5, and thus the Ni valence may be $\sim$ +0.5 in Zr₂NiP₂. The band structure projected to the Ni-3$d$ states is shown in Figure 4(f), most of the weight is below the Fermi level, suggesting Ni tends to have the nonmagnetic electronic configuration $d^{10}$.

There exists no essential difference between the DFT results calculated with $U$ = 0 and 5 eV. Furthermore, we also performed the exactly similar DFT+$U$ simulations for the other members of the family, Ln₂NiX₂ with Ln = Y, In, Lu, Zr and X = P, As, and obtained similar electrical and magnetic properties, i.e., metallic nature and nonmagnetic state of Ni. These results imply that the metallic and nonmagnetic ground state of Zr₂NiP₂ is fairly stable.