Pronounced $\frac{2}{3}$ magnetization plateau in a frustrated $S=1$ isolated spin-triangle compound: Interplay between Heisenberg and biquadratic exchange interactions

We report the synthesis and characterization of a new quantum magnet [2-[Bis(2-hydroxybenzyl) aminomethyl]pyridine]Ni(II)-trimer (BHAP-${\mathrm{Ni}}_3$) in single-crystalline form. Our combined experimental and theoretical investigations reveal an exotic spin state that stabilizes a robust 2/3 magnetization plateau between 7 and 20 T in an external magnetic field. AC-susceptibility measurements show the absence of any magnetic order/glassy state down to 60 mK. The magnetic ground state is disordered and specific-heat measurements reveal the gapped nature of the spin excitations. Most interestingly, our theoretical modeling suggests that the 2/3 magnetization plateau emerges due to the interplay between antiferromagnetic Heisenberg and biquadratic exchange interactions within nearly isolated spin $S=1$ triangles.

Figure 1

(a) A perspective view of the ${\mathrm{Ni}}^{2+}$ (red spheres) spin triangle along with the exchange interactions (yellow lines) in BHAP-${\mathrm{Ni}}_3$. Each Ni ion is situated in a distorted octahedral environment (gray shadow) formed by four O (green spheres) and two N (light blue spheres) ions. (b) Distribution of isolated spin triangles in the lattice, light (dark) red circles denote Ni ions in the lower (upper) layer along the $a$ direction. (c) Zero-field-cooled magnetization as a function of temperature. The magnetic field was applied both parallel ($M_{\parallel }$) and perpendicular ($M_{\perp }$) to the triangle plane. (d) Temperature dependence of the real part of the ac susceptibility measured without and at several magnetic fields applied within the triangle plane ($H_{\parallel }$).

Figure 2

Isothermal field dependence of the magnetization ($M_{\parallel }/M_{\mathrm{S}}$) measured at $T=360\mathrm{mK}$ in a pulsed magnetic field up to 40 T as well as the theoretical magnetization curve showing the 2/3-magnetization plateau. Inset: $M\left(H\right)$ curves obtained at 1.9 K using a vibrating sample magnetometer.

Figure 3

(a) Effect of a small biquadratic exchange term $K$ on the magnetization curve for the simplified model of a spin-1 triangle with isotropic spin exchange $J$. Ground-state compositions at the plateaus ($A,B,C$) are discussed in the text. (b) Illustration of the Ni triangle with two oxygen atoms O1 and O2 (red) not being shared between nickel atoms Ni1 and Ni3 (gray). Twisted ring exchange involving these four atoms might be responsible for the biquadratic interaction term in the spin model. (c) Sketch of the Dzyaloshinskii-Moriya vectors ${\mathbf{d}}_{ij}$ (blue) for the spin model.

Figure 4

(a) Temperature dependence of specific heat ($C$) as obtained experimentally. In the inset, $lnC$ ($C$ in J/mol K) is plotted against $1/T$ down to 360 mK showing exponential behavior. The red line is the linear fit to the data which allows us to extract the spin gap $\mathrm{\Delta }$. (b) Magnetic contribution to the specific heat ($C_{\mathrm{mag}}$) calculated using the model with the parameters as described in the text (same as in Fig. 2). Inset shows $ln{C}_{\mathrm{mag}}$ ($C_{\mathrm{mag}}$ in J/mol K) vs $1/T$ plot to show the similar exponential behavior with spin gap as of experiment.

Figure 5

Top panel: Platelike tiny BHAP-${\mathrm{Ni}}_3$ single crystals. Bottom panel: Perspective view of BHAP-${\mathrm{Ni}}_3$ crystal structure obtained by solving the single-crystal x-ray diffraction data, showing 50% thermal ellipsoids for all nonhydrogen atoms at 293 K. Solvents are omitted for clarity.

Figure 6

Top panel: Temperature variation of inverse dc magnetic susceptibility $(1/\chi =H/M)$ for powder sample. Inset shows Curie-Weiss fit to the powder data. Bottom panel: Angular dependence of magnetic moment ($m$) in the ($b,c$) plane measured at 1.8 K using single crystal.

Figure 7

Magnetization as a function of applied magnetic field for both the VSM and the pulsed field measurements at a temperature of $\sim 2\mathrm{K}$.

Figure 8

Calculated $\mathrm{GGA}+U$ ($U_{\mathrm{eff}}=5\mathrm{eV}$) spin polarized density of states (DOS) projected onto relevant orbitals of different atoms in the unit cell is shown. The Fermi energy was set at the zero in the energy scale. The Ni-$d$ integrated DOS suggest that below the Fermi energy all five $d$ states in the majority spin channel and three $d$ states in the minority are filled, while the remaining two states in the minority $d$ states are empty.

Figure 9

Construction of the DM vectors (a) of the Ni triangle (gray spheres). (b) Two Ni atoms are connected via three oxygen atoms (red spheres), which mediate the superexchange and are considered for the DM interactions. Each Ni-O-Ni triangle contributes a DM component (green arrows) perpendicular to it resulting in the DM vector (blue). (c) View along the Ni-Ni axes indicated in (a): The three O atoms in the left and center panel deviate from a threefold symmetric arrangement (dashed lines) and lead to a finite net DM vector. In the right panel, only two O atoms are shared between the Ni atoms.

Figure 10

(a) Experimentally obtained specific heat ($C$) data plotted as a function of temperature. Measurements were performed both in zero field and with different field up to 9 T. (b) Theoretically obtained magnetic contributions of the specific heat ($C_{\mathrm{mag}}$) both in zero field and in the presence of magnetic field.

Figure 11

Temperature dependence of the magnetic properties of the model. (a) Calculated magnetization as a function of temperature for an external magnetic field strength of ${\mu }_{0}H=0.1\mathrm{T}$. (b) In-plane susceptibility as a function of temperature for different magnetic field strengths. The curves have been shifted for better visibility.

Figure 12

Field dependence of the magnetic properties of the model. (a) In- and out-of-plane magnetization at $T=1.9\mathrm{K}$. The inset shows the derivative $\mathrm{d}M/\mathrm{d}H$ and the dashed lines correspond to the field strength for which $M/M_{\mathrm{sat}}=1/3$. (b) Effect of the strength of the biquadratic exchange term on the in-plane magnetization. All parameters of the model except of the biquadratic exchange $K/J$ are as described in the paper.

Figure 13

Properties of the alternative model: Magnetization curve as a function of applied field strength (a) and temperature (b) for in-plane and perpendicular magnetic field direction. (c) In-plane susceptibility and (d) specific heat data as a function of temperature for different magnetic field strengths.