Unraveling the nature of ferrimagnetism and associated exchange interactions in distorted honeycomb ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$

Ferrimagnetism in orthorhombic ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$ below its Néel temperature, $T_{FN}\sim 76\mathrm{K}$ is reported to result from two inequivalent ${\mathrm{Ni}}^{2+}$ ions having different magnetic moments. However, a clear understanding of the temperature variation of its magnetization $\left[M\right(T$)] for $T>T_{FN}$ and $T<T_{FN}$ in terms of a single set of exchange parameters is still lacking. In this work, experimental results obtained from a detailed analysis of the temperature and magnetic field dependence of magnetization $\left[M\right(T,H)$], ac-magnetic susceptibility [${\chi }_{\text{ac}}(f,T,H$)], and heat-capacity [$C_P(T,H)$] measurements are combined with theoretical analysis to provide new insights into the nature of ferrimagnetism in ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$. X-ray diffraction/Rietveld analysis of the prepared sample yielded the structural parameters of the orthorhombic crystal in agreement with previous studies, whereas x-ray photoelectron spectroscopy confirmed the ${\mathrm{Ni}}^{2+}$ and ${\mathrm{Nb}}^{5+}$ electronic states in ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$. Analysis of ${\chi }_{\text{ac}}\left(T\right)$ shows the paramagnetic-to-ferrimagnetic transition occurs at 76.5 K ($T_{FN}$), which increases with applied field $H$ as $T_{FN}\propto H^{0.35}$ due to the coupling of the ferromagnetic component with $H$. For $T>T_{FN}$, the ${\chi }_{dc}$ versus $T$ data are fitted to the Néel's expression for ferrimagnets, yielding the $g$-factors for the two ${\mathrm{Ni}}^{2+}$ ions as $g_A=2.47$ and $g_B=2.10$. Also, the antiferromagnetic molecular field constants between the $A$ and $B$ sublattices were evaluated as $N_{AA}=26.31, N_{BB}=8.59$, and $N_{AB}=43.06,$ which, in turn, yield the antiferromagnetic exchange parameters: $J_{AA}/{\mathrm{k}}_B=4.27$ K, $J_{BB}/{\mathrm{k}}_B=1.40$ K, and $J_{AB}/{\mathrm{k}}_B=6.98$ K. For $T<T_{FN}$, the $M$ versus $T$ data clearly show the magnetic compensation point at $T_{\text{COM}}\sim 33$ K. The mathematical model presented here using the magnitudes of $N_{AA}, N_{BB}$, and $N_{AB}$ correctly predicts the position of $T_{\text{COM}}$ as well the temperature variation of $M$ both above and below $T_{\text{COM}}$. The data of $C_P(T$) versus $T$ shows a $\lambda$-type anomaly across $T_{FN}$. After subtracting the lattice contribution, the $C_P(T$) data are fitted to $C_P=A{(T-T_N)}^{(-\alpha )}$ yielding the critical exponent $\alpha =0.14\left(0.12\right)$ for $T<T_{FN}(T>T_{FN}),$ which is a characteristic of second-order phase transition. Magnetic entropy changes determined from the $M\text{−}H$ isotherms shows that the applied field $H$ enhances the magnetic ordering for $T>T_{FN}$ and $T<T_{\text{COM}}$, but for $T_{COM}<T<T_{FN}$, the spin disorder increases with the increase in $H$. The temperature variation of the measured coercivity $H_C(T$) and remanence $M_R(T$) from 1.9 K to $T_{FN}$ initially show a decreasing trend, becoming zero at $T_{\text{COM}}$, then followed by an increase and eventually becoming zero again at $T_{FN}$.

Figure 1

Schematic model of the distorted honeycomb structure of ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$ containing (a) $A$ layer and two $BB$ layers stacked vertically along the $c$-axis along with superexchange path $J_{AB}$ projected in $ac$-plane. (b) and (c) The superexchange paths $J_{BB}$ and $J_{AA}$ are projected on the $\mathit{ab}$-plane in the $B$-layer and the $A$-layer, respectively. (d) The distorted honeycomb structure is represented in $BB$-layer.

Figure 2

Experimental x-ray diffraction pattern and the corresponding Rietveld refinement (RR) of the bulk polycrystalline ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$ system of space group $Pbcn$ where the RR was performed using the FullProf suite. The Miller indices of all the lines are listed along with their positions (green vertical lines). The pink line at the bottom represents the difference between the experimental and simulated patterns.

Figure 3

Electronic structure determined by the x-ray photoelectron spectroscopy (XPS) survey scan showing the (a) photoelectron intensity measured in counts per second as a function of the binding energy (eV). The individual core-level binding energy spectra of Nb, Ni, and O are represented in (b), (c), and (d), respectively. Here the solid lines are the mathematical fits to the experimental data (solid circles) using the Tougard baseline correction method.

Figure 4

The core-level x-ray photoelectron spectra of Ni $3p$ in the lower binding energy region for the bulk polycrystalline sample of ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$.

Figure 5

Plotted here are the temperature variations (74 K to 79 K) of the ac-magnetic susceptibilities (a) ${\chi }^{\prime }$ ($T$) and (b) ${\chi }^{\prime \prime }$ ($T$). The insets in both figures show the zoomed view for three typical frequencies, $f$ = 0.17 Hz, 17.1 Hz, and 1202 Hz, showing peak shifts to higher temperatures with increase in $f$.

Figure 6

Effect of applied magnetic field $H$ (50 Oe to 800 Oe) on the temperature variation of the ac susceptibility ${\chi }^{\prime }$ shows that with increase in $H$, the broad hump near 75 K disappears and the peak near 76.6 K moves to higher temperatures.

Figure 7

Magnetic field variation of the ferrimagnetic Néel temperature $T_{FN}(H$). Circles represent the experimental data points and the dotted line is fit to the equation: $T_{FN}(H$) = $T_{FN}$(0)$+{AH}^n$ with the fitted parameters listed therein.

Figure 8

Temperature dependence of measured magnetization $M\left(T\right)$ in $H=500$ Oe recorded under zero-field-cooled warming (ZFCW) and field-cooled warming (FCW) conditions. Notable temperatures where major changes are observed are marked for clarity including the compensation temperature $T_{\text{COM}}\sim 33$ K. For ZFCW, the peak at 76.6 K represents the ferrimagnetic ordering temperature $T_{FN}$.

Figure 9

Temperature dependence of the inverse paramagnetic susceptibility ($\chi$- ${\chi }_0{)}^{-1}$ after correcting for the temperature-independent term ${\chi }_0=-5\times {10}^{-5}$ emu/mol.Oe discussed in the text. The solid line represents best fit to the Néel's expression for ferrimagnets ([Eq. (1)] with the magnitudes of the fitted parameters listed inside the figure.

Figure 10

Measured hysteresis loop variations of magnetization vs. magnetic field $H$ under ZFC condition for four selected temperatures. The inset shown in bottom panel shows nonzero coercivity ($H_C$) and remanence ($M_R$) at 10 K and 65 K, but which are zero at $T_{\text{COM}}=33$ K and $T_{FN}\sim 76\mathrm{K}$.

Figure 11

Temperature dependence of coercivity $H_C$ and remanence $M_R$ determined from the $M$ vs. $H$ hysteresis loops at different temperatures under ZFC conditions, both going to zero at the compensation temperature $T_{\text{COM}}=33$ K.

Figure 12

Temperature dependence of magnetization obtained from both (a) experiment and (b) theory using Eq. (1) for T $>T_{FN}$ and Eq. (12) for $T<T_{FN}$. For $T>T_{FN}, H=90$ kOe is used in the calculations for $T>T_{FN}$ to match the calculated $M$ with that for $T<T_{FN}$, the latter assuming zero coercivity. Note that curves in (a) and (b) have similar features in that with the decrease in temperature, a sharp rise on approach to $T_{FN}$ is observed followed by a drop to negative values and then the observation of compensation temperature $T_{\text{COM}}$.

Figure 13

(a) Plots of $M$ vs. $T$ for $H\left(\text{kOe}\right)=0.5$, 1, 5, 20, 40 60, 80, and 90. (b) Computed plots of $d\left(\chi T\right)/dT$ vs. $T$ for the same $H$ values where $\chi =M/H$ was used; the inset shows the zoomed view for $H=90$ kOe near the compensation temperature.

Figure 14

(a) Temperature dependence of the measured specific heat [$C_P(T$)] under both zero-field ($H=0$ Oe) and high field ($H=90$ kOe) conditions. The individual contributions of lattice specific heat ($C_{Ph}$) and magnetic specific heat ($C_M$) are also shown in the same panel which was extracted from the Debye model yielding Debye temperature ${\mathrm{\Theta }}_D=353$ K. (b) Temperature variation of $C_M/T$, for both $H=0$ and 90 kOe.

Figure 15

Thermal variation of the magnetic entropy $S_M(T$) determined from the specific heat $C_P(T$) vs. $T$ data for $H=0$ Oe and 90 kOe.

Figure 16

Temperature variation of the magnetic specific heat $C_M$ ($H=0$) near $T_{FN}$ of ${\mathrm{Ni}}_4{\mathrm{Nb}}_2{\mathrm{O}}_9$ system fitted to the expression: $C_M$ = $A|T-T_{FN}{|}^{-\alpha }$ using the double logarithmic plot of $C_M$ vs. $|T-T_{FN}|$. The solid lines are linear fits in the limited temperature range close to $T_{FN}$ yielding $\alpha$ for $T>T_{FN}$ and $T<T_{FN}$.

Figure 17

Thermal variation of the magnetic entropy change, $-\mathrm{\Delta }{S}_M\left(T\right)$ for a field change of $\mathrm{\Delta }H=90$ kOe estimated both from heat capacity data and isothermal magnetization plots recorded at various temperatures below the $T_{FN}$ employing the Maxwell's expression.