Magnetic interactions in $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$: Quantum spin-2 system on a spatially anisotropic two-dimensional triangular lattice

The dc magnetization and the electron spin resonance (ESR) measurements have been performed on $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$ polycrystalline sample, a quantum spin system on a frustrated two-dimensional (2D) triangular lattice with spatially anisotropic Heisenberg exchange. The former measurements reveal a realization of the high-spin state $(S=2)$ on magnetic ${\mathrm{Mn}}^{3+}$ sites. From the susceptibility curve, we have determined the nearest-neighbor antiferromagnetic exchange coupling constants (namely, in the preferably coupled spin chains, $J_1∕k_B=65\mathrm{K}$, and perpendicular to them, $J_2∕J_1=0.44$) by employing the finite-temperature Lanczos method. The dominant magnetic anisotropy term of the single-ion type, $D∕k_B=-4.1\mathrm{K}$, which establishes an easy-axis direction, is evaluated from the ESR linewidth. We find that the temperature dependence of the linewidth reflects the onset of short-range static spin correlations below room temperature.

Figure 1

(Color online) (a) Crystal structure of monoclinic $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$ as viewed along the (010) direction. The purple (small), yellow (medium), and blue (large) spheres denote manganese, sodium, and oxygen ions, respectively. (b) Arrangement of edge-sharing $\mathrm{Mn}{\mathrm{O}}_6$ octahedra in the $\mathit{ab}$ crystal plane and the associated spatially anisotropic triangular lattice of ${\mathrm{Mn}}^{3+}$ $S=2$ spins. (c) The finite $m\times n$ cluster used in the FTLM calculations (periodic boundary conditions) with intrachain $J_1$ and interchain $J_2$ exchange.

Figure 2

(Color online) (a) The temperature dependence of the magnetic susceptibility $({\chi }_0=M∕H)$ measured at $1\mathrm{kOe}$ (circles) and the corresponding numerical fit on $N=6\times 2$ site cluster (solid line) determining the magnetic exchange parameters, $J_1∕k_B=65\mathrm{K}$ and $J_2∕k_B=29\mathrm{K}$. The fit corresponds to temperatures above $T_N=45\mathrm{K}$, which is emphasized with the vertical line. The inset shows the low-temperature part. (b) The finite-size analysis of the susceptibility calculated by FTLM on various $N=m\times n$ clusters, with $m=2-6$ and $N⩽12$ (lines). We also show a classical Monte Carlo result on $16\times 16$ site cluster (squares) to emphasize the quantum effects in a given spin-2 system. All curves are calculated with the same exchange parameters as in (a). The inset shows the temperature derivative of the measured susceptibility.

Figure 3

(Color online) The temperature evolution of the derivative $X$-band ESR spectra in $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$ (circles) and the corresponding fits (solid lines), given by Eq. (3).

Figure 4

(Color online) The experimental (circles) and numerically predicted (line) temperature dependence of the ESR linewidth in $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$ at $9.40\mathrm{GHz}$. The FTLM calculation was performed on $N=6\times 2$ site cluster with the same exchange parameters as in Fig. 2a and $\mid D\mid ∕k_B=4.1\mathrm{K}$. The inset shows the temperature changes in the dispersion-to-absorption ratio.

Figure 5

(Color online) The room-temperature derivative ESR spectrum of $\alpha \text{−}\mathrm{Na}\mathrm{Mn}{\mathrm{O}}_2$ measured at (a) $9.40\mathrm{GHz}$ and (b) $211.2\mathrm{GHz}$. The solid lines are Lorentzian fits [Eq. (3)].

Figure 6

(Color online) (a) The room-temperature frequency dependence of the linewidth and (b) the frequency-field dependence, determining the $g$ factor $g=1.993$.

Figure 7

The temperature dependence of the ESR linewidth (at $9.40\mathrm{GHz}$) multiplied by the susceptibility and temperature.