Magnetic properties of the triangular lattice magnets $A_4{B}^{\prime }{B}_2{\mathrm{O}}_{12}$ ($A=\mathrm{Ba}$, Sr, La; $B^{\prime }=\mathrm{Co}$, Ni, Mn; $B=\mathrm{W}$, Re)

The geometrically frustrated two-dimensional triangular lattice magnets $A{}_4{B}^{\prime }B{}_2{\mathrm{O}}_{12}$ ($A=\mathrm{Ba}$, Sr, La; $B^{\prime }=\mathrm{Co}$, Ni, Mn; $B=\mathrm{W}$, Re) have been studied by x-ray diffraction, ac and dc susceptibilities, powder neutron diffraction, and specific-heat measurements. The results reveal the following: (i) The samples containing ${\mathrm{Co}}^{2+}$ (effective spin-1/2) and ${\mathrm{Ni}}^{2+}$ (spin-1) ions with small spin numbers exhibit ferromagnetic (FM) behavior and ordering, respectively, while the sample containing ${\mathrm{Mn}}^{2+}$ (spin-5/2) ions with a large spin number exhibits antiferromagnetic (AFM) ordering. We ascribe these spin-number-manipulated ground states to the competition between the AFM $B^{\prime }$-O-O-$B^{\prime }$ and FM $B^{\prime }$-O-$B$-O-$B^{\prime }$ superexchange interactions. (ii) The chemical pressure introduced into the Co-containing samples through the replacement of different-size ions on the $A$ site finely tunes the FM behavior temperature of the system. This effect is not simply governed by the lattice parameters and requires more precise structural measurements to elucidate.

Figure 1

(a) Crystalline structure of $A{}_4\mathrm{Co}B{}_2{\mathrm{O}}_{12}$ with the space group R-3mH. (b) Staggered pattern of triangular ${\mathrm{Co}}^{2+}$ planes along the $c$ axis.

Figure 2

(a)–(e) Powder x-ray diffraction patterns (red crosses) and Rietveld refinements (black line) for five $A{}_4\mathrm{Co}B{}_2{\mathrm{O}}_{12}$ compounds. The blue line at the bottom of each panel is the difference curve. The black ticks are the reflection positions.

Figure 3

(a)–(e) The inverse dc susceptibility of the $A{}_4\mathrm{Co}B{}_2{\mathrm{O}}_{12}$ compounds. The solid and dashed lines are Curie-Weiss fittings of the low-temperature and high-temperature regimes, respectively. Insets: dc magnetization taken at 1.8 K. The saturation magnetization of the ${\mathrm{Co}}^{2+}$ ion is extrapolated using a linear fit to account for the Van Vleck paramagnetic contribution.

Figure 4

(a)–(e) The real part of the ac susceptibility ${\chi }_{\text{AC}}$ for cobalt containing $A{}_4{B}^{\prime }B{}_2{\mathrm{O}}_{12}$ compounds measured from 0.3 to 2.0 K under different dc magnetic fields. Excitation fields of 2 Oe at low ac frequencies were used.

Figure 5

For ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{NiW}}_2{\mathrm{O}}_{12}$, (a) the Rietveld refinement of the NPD pattern measured at room temperature using a neutron wavelength of $\lambda =1.5405\AA$. (b) The temperature dependence of $C_{\text{P}}$ measured at different dc fields. Inset: the field dependence of $T_{\text{C}}$. (c) The inverse dc susceptibility. The solid line is the linear fitting. Inset: the temperature dependence of dc susceptibility measured under different fields. (d) The dc magnetization curve measured at 2 K.

Figure 6

NPD of ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{NiW}}_2{\mathrm{O}}_{12}$ taken at 2 and 20 K. The overlap of magnetic Bragg peaks with lattice Bragg peaks indicates FM long-range ordering.

Figure 7

For ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{MnW}}_2{\mathrm{O}}_{12}$, (a) the Rietveld refinement of the NPD pattern taken at room temperature. (b) The temperature dependence of $C_{\text{P}}$ measured at zero field. (c) The inverse dc susceptibility. The solid line is the linear fitting.

Figure 8

(a) The temperature dependence of ${\chi }_{\text{AC}}$ for ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{MnW}}_2{\mathrm{O}}_{12}$ under applied dc fields. (b) The dc field dependence of ${\chi }_{\text{AC}}$ at various temperatures. Inset: the enlargement of the data around 4 T. (c) The derivative of the field dependence of ${\chi }_{\text{AC}}$ at different temperatures.

Figure 9

Magnetic phase diagram of ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{MnW}}_2{\mathrm{O}}_{12}$. Transition temperatures were found from the temperature derivative (red solid squares), the field derivative (blue open squares) of ${\chi }_{\text{AC}}$, and zero-field $C_{\text{P}}$ measurements (green triangle).

Figure 10

(a) Pathways for FM $B^{\prime }$-O-$B$-O-$B^{\prime }$ (black dashed line) and AFM $B^{\prime }$-O-O-$B^{\prime }$ (green solid line) superexchange interactions. (b) Energy levels of $4f$ orbitals in cubic symmetry, with the highest energy levels having the same symmetry as $p$ orbitals. (c) Orbital diagram of FM $B^{\prime }$-O-$B$-O-$B^{\prime }$ interaction with $f$ orbitals represented by symmetrically similar $p$ orbitals.

Figure 11

Temperature dependence of lattice parameters $a$ and $c$ for ${\mathrm{Ba}}_2{\mathrm{La}}_2{\mathrm{CoW}}_2{\mathrm{O}}_{24}$. No structural distortion is seen upon the transition from high spin ($S=3/2$) to low spin ($S=1/2$) of the ${\mathrm{Co}}^{2+}$ ions.