Thermal evolution of quasi-one-dimensional spin correlations within the anisotropic triangular lattice of $\alpha -{\mathrm{NaMnO}}_2$

Magnetic order on the spatially anisotropic triangular lattice of $\alpha -{\mathrm{NaMnO}}_2$ is studied via neutron diffraction measurements. The transition into a commensurate, collinear antiferromagnetic ground state with $\mathbf{k}=(0.5,0.5,0)$ was found to occur below $T_{\mathrm{N}}=22$ K. Above this temperature, the transition is preceded by the formation of a coexisting, short-range ordered, incommensurate state below $T_{\mathrm{IC}}=45$ K whose two-dimensional propagation vector evolves toward $\mathbf{k}=(0.5,0.5)$ as the temperature approaches $T_{\mathrm{N}}$. At high temperatures ($T>T_{\mathrm{IC}}$), quasielastic scattering reveals one-dimensional spin correlations along the nearest-neighbor Mn-Mn “chain direction” of the ${\mathrm{MnO}}_6$ planes. Our data are consistent with the predictions of a mean-field model of Ising-like spins on an anisotropic triangular lattice, as well as the predominantly one-dimensional Heisenberg spin Hamiltonian reported for this material.

Figure 1

Summary of the $\alpha -{\mathrm{NaMnO}}_2$ structure and defect types. (a) The defect-free $\mathit{ab}$ and $\mathit{ac}$ planes, with the chemical unit cells for both outlined in black. The $\mathit{ab}$-plane schematic shows a single layer of Mn atoms with nearest neighbors connected by solid orange lines and next-nearest neighbors connected by dashed black lines. ${\mathrm{MnO}}_6$ octahedra are shaded in gray in the $\mathit{ac}$-plane schematic. The orientation of the $\mathit{ac}$ plane was made to highlight the direction in which defects form—breaking symmetry along the $\left[-1,0,1\right]$ direction—and matches the orientation of the structures in panels (b) and (c). The black rectangle in (b) defines the repeating unit of a layer which can stack to form the defect-free structure in (a), the stacking fault in (b), or the twin boundary in (c).

Figure 2

Neutron data and simulations/refinements of the $\alpha -{\mathrm{NaMnO}}_2$ structure using the program FAULTS. (a) The refinement using a defect-free model as discussed in the text. (b) The refinement allowing the defect percentages to refine. The insets in (a) and (b) show the high $2\theta$ range, which highlights the differences between the defect-free fit and the defect-refined fit. The asterisk (*) symbol denotes a Bragg peak associated with ${\mathrm{Mn}}_3{\mathrm{O}}_4$ intergrowth. (c) Data from the CORELLI experiment, showing the $(H,2,L)$ scattering plane. (d) A simulation of the $(H,0,L)$ scattering plane using the refined percentage of stacking defects found from the fit presented in (b).

Figure 3

Slices of scattering planes from the neutron spectrometer, CORELLI, at different temperatures. Panels (a)–(e) show slices of the ($\mathit{H}$, $\mathit{K}$, 0.25) plane with 0.15 $<$ $\mathit{L}$ (r.l.u.) $<$ 0.35. Panels (f)–(j) show slices of the (0.25, $\mathit{K}$, $\mathit{L}$) plane with 0.15 $<$ $\mathit{H}$ (r.l.u.) $<$ 0.35. Maxima along the diffuse scattering rods correspond to a different set of diffuse scattering rods along $[\overline{1},0,1]$, which cut through the planes at these points.

Figure 4

Temperature dependence of the scattering in the $(H,K,0)$ plane about the magnetic ordering wave vector. Panels (a), (b), and (c) were taken at 5 K, 30 K, and 50 K, respectively. (d)–(g) $H$ scans at select temperatures, where $K$ is either at the commensurate position, $K=K_{\mathrm{C}}=0.5$ r.l.u., or the incommensurate position $K=K_{\mathrm{IC}}\left(T\right)$. Both $K_{\mathrm{C}}$ and $K_{\mathrm{IC}}$ are indicated in the insets of each panel, which are $K$ scans taken at $H=0.5$ r.l.u.. Only one peak, located at $K_{\mathrm{C}}$, was observed for the $T=5$ K data shown in panel (d). $L=0$ r.l.u. for all figures, and solid black lines are Voigt function fits.

Figure 5

Temperature-dependent parameters extracted by fitting $H$, $K$, and $L$ scans at the commensurate (C) and/or incommensurate (IC) Bragg peak positions. Panel (a) shows the center positions of the incommensurate peak, with respect to the commensurate position, along $\mathit{K}$, where each data point has a different fixed $H=H_{\mathrm{IC}}\left(T\right)$, which is the center of the incommensurate peak along $\mathit{H}$. (b) The integrated area of $H$ scans taken either at $K=K_{\mathrm{C}}=0.5$ r.l.u. (orange circles) or at $K=K_{\mathrm{IC}}\left(T\right)$ (blue squares). The inset is a schematic showing the direction and location of each scan. (c) The correlation length along $H$ for the C (${\xi }_{H,\mathrm{C}}$) and IC (${\xi }_{H,\mathrm{IC}}$) peaks, as well as along $L$ for the C peak (${\xi }_{L,\mathrm{C}}$), as a function of temperature. The inset shows the correlation lengths along $K$ of the IC peak, ${\xi }_{K,\mathrm{IC}}$, as a function of temperature, where $H=H_{\mathrm{IC}}\left(T\right)$.

Figure 6

Tracking of the incommensurate scattering at the magnetic zone boundary as a function of temperature. The green filled squares in (a) reflect the tracking of the diffuse critical scattering at the zone boundary and represent the peak of the diffuse scattering along $\mathit{K}$ at the zone boundary (ZB), where $\mathit{H}=0.25$ r.l.u.. The open blue squares are the same data from Fig. 5 near the magnetic zone center. The filled green squares in panel (b) show the integrated area of the diffuse scattering peak at the ZB, and the open gray diamonds represent the correlation length of the diffuse scattering at the ZB. These data were taken with $\mathrm{open}-{80}^{\prime }-{80}^{\prime }-\mathrm{open}$ collimation.

Figure 7

The center position of the incommensurate peak, with respect to the commensurate position, along $\mathit{H}$. Each data point has a different fixed $K=0.5+{\delta }_K\left(T\right)$, which is the center of the incommensurate peak along $\mathit{K}$.

Figure 8

Schematic of the ${\mathrm{Mn}}_3{\mathrm{O}}_4$ (MO) intergrowth with $\alpha -{\mathrm{NaMnO}}_2$ (NMO). (a) The $ac$ plane of both phases shown meeting at the boundary of intergrowth, highlighted in gray. The chemical unit cells of each phase are outlined with solid black lines, and the extension of the MO unit cell into the NMO unit cell is depicted as a dashed black line. (b) Comparison of two parallel planes: the $ab$ plane of $\alpha -{\mathrm{NaMnO}}_2$ and the ${\mathrm{Mn}}_3{\mathrm{O}}_4$ plane spanned by ${\mathbf{b}}_{\mathrm{MO}}$ and ${[1,0,-1]}_{\mathrm{MO}}$.

Figure 9

Bulk magnetization measurements taken on single crystals of $\alpha -{\mathrm{NaMnO}}_2$. The data show $T=2$ K magnetic field sweeps with $\mathbf{H}\parallel b$ axis of $\alpha -{\mathrm{NaMnO}}_2$ and ${\mathrm{Mn}}_3{\mathrm{O}}_4$. The dashed orange line is a fit to the high field data between $8.5<H\left(\mathrm{T}\right)<9$ used for extracting the mass percentage of ${\mathrm{Mn}}_3{\mathrm{O}}_4$ intergrowth in the sample as described in the text.

Figure 10

Reciprocal space maps for the ${\mathrm{Mn}}_3{\mathrm{O}}_4$ magnetic unit cell and $\alpha -{\mathrm{NaMnO}}_2$ chemical unit cell, orientated so that they reflect the real space intergrowth direction. Panels (a) and (b) assume the a-domain magnetic structure for ${\mathrm{Mn}}_3{\mathrm{O}}_4$, and (c) and (d) assume the b-domain magnetic structure for ${\mathrm{Mn}}_3{\mathrm{O}}_4$. Panels (a) and (c) show the $a^{*}{c}^{*}$ plane for both ${\mathrm{Mn}}_3{\mathrm{O}}_4$ and $\alpha -{\mathrm{NaMnO}}_2$, where the $b^{*}$ axis is going into the page for both phases. Panels (b) and (d) show the $a^{*}{b}^{*}$ plane for $\alpha -{\mathrm{NaMnO}}_2$ and the plane which is parallel in ${\mathrm{Mn}}_3{\mathrm{O}}_4$: either the ${(2H,K,-H)}_{\mathrm{MO}}$ for the a domain or the ${(H,K,-H)}_{\mathrm{MO}}$ for the b domain. Integral miller index positions for $\alpha -{\mathrm{NaMnO}}_2$ are defined by the intersections of red lines and integral miller index positions for ${\mathrm{Mn}}_3{\mathrm{O}}_4$ are defined by the intersections of blue lines. Allowed Bragg peaks are marked according to the legend at the top of the figure. The black arrows represent the direction in which magnetic Bragg peaks shift for ${\mathrm{Mn}}_3{\mathrm{O}}_4$ when in its IC phase, as reported in Ref. [51].

Figure 11

Neutron data showing the presence of ${\mathrm{Mn}}_3{\mathrm{O}}_4$ magnetic impurity within $\alpha -{\mathrm{NaMnO}}_2$. (a) Triple-axis neutron data of integrated $\mathit{K}$ scans centered at $\mathbf{Q}={(0,1,0)}_{\mathrm{NMO}}$. This momentum corresponds to the ${(0,2,0)}_{\mathrm{MO}}$ in the a domain, and ${(0,4,0)}_{\mathrm{MO}}$ in the b domain, of ${\mathrm{Mn}}_3{\mathrm{O}}_4$. The solid red line is a power-law fit to the data yielding the critical exponent, $\beta$, and ordering temperature, $T_{\mathrm{C}}$, as indicted. (b) Neutron powder diffraction data taken at $T=5$ K of a crushed single crystal of $\alpha -{\mathrm{NaMnO}}_2$. The nuclear and magnetic structure refinement using the Rietveld method is shown as a solid red line. The positions of Bragg peaks are marked with ticks underneath the spectra. Going from top to bottom they represent the ${\mathrm{Mn}}_3{\mathrm{O}}_4$ nuclear, ${\mathrm{Mn}}_3{\mathrm{O}}_4$ magnetic, $\alpha -{\mathrm{NaMnO}}_2$ nuclear, and $\alpha -{\mathrm{NaMnO}}_2$ magnetic Bragg peak positions.