Large easy-axis anisotropy in the one-dimensional magnet $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$

We present an extensive experimental and theoretical study on the low-temperature magnetic properties of the monoclinic anhydrous alum compound $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$. The magnetic susceptibility reveals strong antiferromagnetic interactions ${\theta }_{CW}=-167\mathrm{K}$ and long-range magnetic order at $T_N=22\mathrm{K}$, in agreement with a recent report. Powder neutron diffraction furthermore shows that the order is collinear, with the moments near the $ac$ plane. Neutron spectroscopy reveals a large excitation gap $\mathrm{\Delta }=15\mathrm{meV}$ in the low-temperature ordered phase, suggesting a much larger easy-axis spin anisotropy than anticipated. However, the large anisotropy justifies the relatively high ordered moment, Néel temperature, and collinear order observed experimentally and is furthermore reproduced in a first-principles calculations by using a new computational scheme. We therefore propose $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$ to host $S=1$ antiferromagnetic chains with large easy-axis anisotropy, which has been theoretically predicted to realize novel excitation continua.

Figure 1

(a) The crystal structure of $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$ as observed along the $\langle 111\rangle$ direction. Here, ${\mathrm{Mo}}^{+4}$ is presented in purple, ${\mathrm{Ba}}^{2+}$ in yellow, and ${\mathrm{P}}^{5+}$ in green. (b) An illustration of the anisotropic triangular arrangement of ${\mathrm{Mo}}^{4+}$ moments and their suggested directions in the magnetic structure of $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$ based on the refinement of NPD data at 1.5 K where the intrachain ($J$) and interchain ($J^{\prime }$) exchange interactions are depicted in green and purple, respectively. The structure was generated using the vesta visualization software [6].

Figure 2

(a) Rietveld refinement of the $\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2$ nuclear structure according to data collected on the backscattering bank of HRPD at 2 K. Observed and calculated points are shown in red and black, respectively, and their difference is shown in blue. The allowed Bragg reflections for all the refined phases are depicted in black $\left[\mathrm{BaMo}{\left({\mathrm{PO}}_4\right)}_2\right]$, purple (${\mathrm{MoO}}_2$), green (Al), and gray (${\mathrm{Ba}}_2{\mathrm{P}}_2{\mathrm{O}}_7$) tick marks. The agreement with the calculated pattern is good, with $R_{wp}=2.55\%$, ${\chi }^2=10.7$. (b) Rietveld refinement of the ${\mathrm{P}}_s\overline{1}$ magnetic structure using 1.5–30 K subtracted data collected on the WISH instrument. Observed data points are shown in red, the calculated pattern in black, their difference in blue, and the magnetic reflections are indicated with black tick marks.

Figure 3

(a) Temperature dependence of the zero-field-cooled dc molar magnetic susceptibility (black circles) collected in a 1000 Oe applied magnetic field. The Curie-Weiss fit shown in red was obtained for a temperature interval of 200 to 300 K and the extracted parameters are $C=0.88\left(2\right)\mathrm{emu}\mathrm{K}{\mathrm{mol}}^{-1}$, ${\theta }_{CW}=-167.0\left(5\right)\mathrm{K}$, and ${\chi }_0=-4.31\left(1\right)\times {10}^{-5}\mathrm{emu}{\mathrm{mol}}^{-1}$. The green line represents the tenth-order high-temperature series expansion fit using the $S=1$ anisotropic triangular lattice model with $J^{\prime }=0.28\mathrm{meV}$, $J=3.91\left(6\right)\mathrm{meV}$, and ${\chi }_0=4.86\left(6\right)\times {10}^{-4}\mathrm{emu}{\mathrm{mol}}^{-1}$. Shown in the inset is the temperature derivative of the magnetic susceptibility as a function of temperature, where a peaked feature is observed at $T_N=22\mathrm{K}$. (b) Temperature dependence of the specific heat $C_p$. The inset shows the magnetic contribution to the specific heat as obtained by subtracting the phonon contribution using a polynomial form $C_p=a_1{T}^3+a_2{T}^5+a_3{T}^7$ fit to the high-temperature data. An anomaly, not observed in the previous work [22], is seen at $T_N=22\mathrm{K}$.

Figure 4

$S(Q,\mathrm{\Delta }E)$ at $T=60\mathrm{K}$ (top) and $T=5\mathrm{K}$ (bottom), along with energy-integrated data for $E_i=50.4\mathrm{meV}$ (red) and $E_i=19\mathrm{meV}$ (blue).

Figure 5

(a) Experimental $S(Q,\mathrm{\Delta }E)$ at 5 K. The orange boxes indicate the integration regions for the three cuts used in the fitting. (b), (c) Simulated $S(Q,\mathrm{\Delta }E)$ using parameter sets 1 (b) and 2 (c) (Table. 2). (d), (e) $\mathrm{\Delta }E$-integrated cuts (open circles) fit to parameter set 1 (d) and set 2 (e) (solid lines), as described in the text, for $Q=1.3$ (blue), $Q=1.5$ (red), and $Q=1.9$ (yellow).

Figure 6

Three-dimensional map of the magnetic anisotropy energy obtained by interpolating a finite number of configurations with different directions of the magnetic moment within the primitive unit cell (one Mo atom) and using the PAW-based vasp code. Panels (a) and (b) show the results for $J_H=0.6\mathrm{eV}$ ($E_{\max }=3.76\mathrm{meV}$) and $J_H=0.8\mathrm{eV}$ ($E_{\max }=2.04\mathrm{meV}$), respectively. The black line is the rotation plane used in Fig. 7. The green arrow labeled “min” is the direction of the energy minimum, whereas ND stands for the experimental spin direction determined from neutron diffraction. A switch between an effective easy-axis to an effective easy-plane anisotropy is observed upon increasing $J_H$.

Figure 7

(a) Anisotropy energy, (b) angular dependence of the orbital moment, and (c) the angle between the spin and orbital moments depending on the spin direction in the $xz$ plane. The experimental spin direction $\mathit{\gamma }\sim (0.70,0,0.72)$ is chosen as zero. The results are obtained by using $J_H=0.6\mathrm{eV}$. Note that the spin moment $M_S=2{\mu }_B$ stays constant within this rotation.