Hidden magnetic order in the triangular-lattice magnet ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$

The manganese tellurate ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ consists of trigonal spin lattices made up of ${\mathrm{Mn}}^{4+}$ ($d^3$, $S=3/2$) ions. The magnetic properties of this compound were characterized by several experimental techniques, which include magnetic susceptibility, specific-heat, dielectric permittivity, electron-spin-resonance, nuclear magnetic resonance (NMR), and neutron powder-diffraction measurements, and by density functional theory calculations. The magnetic susceptibility $\chi \left(T\right)$ demonstrates very unusual behavior. It is described by the Curie-Weiss law at high temperature with Curie-Weiss temperature of $\mathrm{\Theta }=-74\mathrm{K}$ and exhibits no obvious anomaly indicative of a long-range magnetic ordering at low magnetic fields. At high magnetic fields, however, the character of $\chi \left(T\right)$ changes showing a maximum at about 9 K. That this maximum of $\chi \left(T\right)$ reflects the onset of an antiferromagnetic order was confirmed by specific-heat measurements, which exhibit a clear $\lambda$-type anomaly at $T_{\mathrm{N}}\approx 8.5\mathrm{K}$ even at zero magnetic field, and by ${}^7\mathrm{Li}$ NMR and dielectric permittivity measurements. The magnetic structure of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$, determined by neutron powder-diffraction measurements at 1.6 K, is described by the ${120}^{\circ }$ noncollinear spin structure with the propagation vector $\mathit{k}=\left(1/3,1/3,0\right)$. Consistent with this finding, the spin-exchange interactions evaluated for ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ by density functional calculations are dominated by the nearest-neighbor antiferromagnetic exchange within each triangular spin lattice. This spin lattice is strongly spin frustrated with $f=\left|\mathrm{\Theta }\right|/T_{\mathrm{N}}\approx 8$ and exhibits a two-dimensional magnetic character in a broad temperature range above $T_{\mathrm{N}}$.

Figure 1

Polyhedral view of the layered crystal structure of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ ($P\overline{3}1c$ space group): (a) A perspective view of layers. (b) A projection of the layer of magnetic ions along the $c$ direction, in which the ${\mathrm{MnO}}_6$ and ${\mathrm{TeO}}_6$ octahedra alternate sharing their edges. The green and blue octahedra represent the ${\mathrm{MnO}}_6$ and ${\mathrm{TeO}}_6$, respectively. The Li and O atoms are represented by yellow spheres and black balls, respectively.

Figure 2

Refined neutron-diffraction pattern of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ measured on the HRPT diffractometer at $T=300\mathrm{K}$. The red dots represent the experimental data, the black line shows the calculated intensity, the green ticks indicate the position of the Bragg reflections, and the blue line shows the difference between experimental and calculated data, plotted at the bottom for convenience.

Figure 3

The temperature dependence of the magnetic susceptibility $\chi =M/B$ for ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ at $B=0.1\mathrm{T}$ in both FC and ZFC modes (upper panel) and the integrated ESR intensity (green symbols). The inset to this panel represents the $M/B\left(T\right)$ curves for the ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ at various external magnetic fields. The solid red curve represents the Curie-Weiss law. The temperature dependence of the Curie constant $C=\left(\chi \text{–}{\chi }_0\right)\left(T\text{–}\mathrm{\Theta }\right)$ is shown in the lower panel along with the inverse magnetic susceptibility $1/\chi$. The dashed line represents the limiting value $C=1.84\mathrm{emu}/\left(\mathrm{mol}\mathrm{K}\right)$.

Figure 4

The specific heat of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ (filled symbols) and that of nonmagnetic isostructural analog ${\mathrm{Na}}_2{\mathrm{GeTeO}}_6$ (open circles) under zero magnetic field. The upper inset shows the magnetic specific heat $C_{\mathrm{m}}\left(T\right)$ (green symbols) and the magnetic entropy $S_{\mathrm{m}}\left(T\right)$ (violet symbols). The red line is the result of the spin-wave theory. The lower inset shows $C_{\mathrm{p}}\left(T\right)$ under various magnetic fields, where the data were shifted upwards for clarity.

Figure 5

The magnetic phase diagram for ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$. The inset shows the arrangement of magnetic sublattice.

Figure 6

Temperature dependence of the specific heat at zero magnetic field (upper panel), the real part ${\varepsilon }^{\prime }$ of the permittivity recorded at frequency $f=20\mathrm{kHz}$ (middle panel), and the relaxation rate obtained at 116.56 MHz (lower panel), where the yellow area highlights the ordering transition region.

Figure 7

(a) The temperature-dependent ESR spectra of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$. The inset shows a representative ESR spectrum, where the Lorentzian fitting profile is given by the red solid line. (b)The temperature dependence of the effective $g$ factor (green symbols) and the width of the ESR line (navy symbols). The solid orange curve corresponds to the temperature dependence of the linewidth within the framework of classical critical behavior and the magenta dashed line represents the prediction of Eq. (2).

Figure 8

(a) Line shift and (b) linewidth of the ${}^7\mathrm{Li}$ NMR signal at 13.05 MHz as a function of the static magnetic susceptibility. The dashed lines show a linear dependence of the local counterpart from the bulk static susceptibility.

Figure 9

(a) Temperature dependence of the relaxation rate obtained at 13.05 MHz (black solid circles) and 116.56 MHz (red open circles). The solid lines are guide for the eye. The inset shows high-temperature region at 13.05 MHz. (b) A log-log plot of the normalized $\left(1/{}^7{T}_1\right)/\left(1/{}^7{T}_{1\infty }\right)$ against the reduced temperature $|T\text{–}{T}_{\mathrm{N}}^{\mathrm{NMR}}|/T_{\mathrm{N}}^{\mathrm{NMR}}$ for both frequencies $(T_{\mathrm{N}}^{\mathrm{NMR}}=6.7\mathrm{K})$. The dashed line represents Eq. (5) with the critical exponent of $p=1.12$. (c) Example of an inversion recovery measurement on 13.05 MHz at $T=15\mathrm{K}$. Red line shows a fit by single exponent law (see text in Sec. 2).

Figure 10

The plot of ${\left(T_{1}TK\right)}^{-1}$ vs $T$, where ${\left(T_{1}T\right)}^{-1}$ represents the local dynamic susceptibility, and $K$ is the local static susceptibility. The shaded area marks the region where this ratio is a constant, which corresponds to the pure paramagnetic regime.

Figure 11

The ${}^7\mathrm{Li}$ spectrum of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ at 4.2 K at two different frequencies (i.e., external fields). The blue lines correspond to the AF powder profiles as described by Eq. (6), the red dashed lines to the impurity contribution (see the text), and the black dashed lines to the Larmor field $H_{\mathrm{L}}$.

Figure 12

Temperature dependence of the intensity of the NPD maxima measured on the DMC diffractometer at low temperatures. The indices of the most intense magnetic reflections are marked.

Figure 13

Refinement of the neutron-diffraction pattern of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$ measured on the HRPT diffractometer at $T=1.6\mathrm{K}$. The red dots represent the experimental data, the black line shows the calculated intensity, the green ticks indicate the positions of the nuclear and magnetic Bragg reflections, and the blue line shows the difference between experimental and calculated data, plotted at the bottom for convenience. The inset shows a small-angle portion of the neutron-diffraction pattern with an additional blue line showing calculated scattering from a magnetic phase.

Figure 14

Magnetic structure of ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$: (a) An extended projection view on the $ab$ plane of the ${120}^{\circ }$ noncollinear spin arrangements of two adjacent trigonal layers (at $z=0.25$ and 0.75). (b) A perspective view of the ${120}^{\circ }$ noncollinear spin arrangements of two adjacent trigonal layers. The two triangles at $z=0.25$ and 0.75 have a staggered arrangement. The thin black lines denote the crystallographic unit cells formed by the unit vectors a and b. The thick green lines highlight the Shubnikov magnetic unit cell formed by the unit vectors A and B. The relation between the parent crystal unit cell and the magnetic unit cell is determined by $\mathbf{A}=2\mathbf{a}+\mathbf{b}$, $\mathbf{B}=-\mathbf{a}+\mathbf{b}$. [$a=b=5.0114\left(2\right)\AA$, $c=9.4915\left(9\right)\AA$ at $T=1.6\mathrm{K}$].

Figure 15

Spin-exchange paths $J_1$ and $J_2$ in ${\mathrm{Li}}_2{\mathrm{MnTeO}}_6$. The blue circles represent the ${\mathrm{Mn}}^{4+}$ ions.

Figure 16

Spin ordered arrangements of (a) FM, (b) AF1, and (c) AF2 states, where the gray and white circles represent the up and down spin sites, respectively.