Spin structures and phase diagrams of the spin-$\frac{5}{2}$ triangular-lattice antiferromagnet Na$_2$BaMn(PO$_4$)$_2$ under magnetic field

TL;DR

This study addresses the spin structures and finite-temperature phase diagram of the classical spin-$\frac{5}{2}$ triangular-lattice antiferromagnet Na$_2$BaMn(PO$_4$)$_2$ under magnetic fields along the $c$-axis and in the $ab$-plane. By combining unpolarized single-crystal neutron diffraction, heat capacity measurements, and Monte Carlo simulations, the authors map field-induced transitions and refine spin structures through representation analysis of the modulation vector $\bm{k}=(\tfrac{1}{3},\tfrac{1}{3},k_z)$. They show that weak, frustrated interlayer exchange generates a nonzero $k_z$ in the field-dependent states and reproduce the main features of the $H$-$T$ phase diagrams with a minimal 3D Heisenberg model that includes easy-axis anisotropy and small interlayer couplings. The results demonstrate that three-dimensional interlayer interactions play a crucial role in stabilizing the rich sequence of Y, UUD, and V-like phases, extending beyond strictly two-dimensional TLAF physics, and motivate further inelastic studies to fully determine the Hamiltonian governing Na$_2$BaMn(PO$_4$)$_2$.

Abstract

We combine single-crystal neutron diffraction studies and Monte Carlo simulations to determine the spin structures and finite-temperature phase diagram of the spin-5/2 triangular-lattice antiferromagnet Na$_2$BaMn(PO$_4$)$_2$ in magnetic field. With the application of a magnetic field in two different directions, namely along the $c$-axis and in the $ab$-plane of the trigonal symmetry, we track the evolution of the spin structure through changes of the magnetic propagation vector. We account for these results with a minimal Heisenberg Hamiltonian that includes easy-axis anisotropy and weak, frustrated interlayer couplings in addition to intralayer exchange. Guided by representation analysis, we refine symmetry-allowed modes to the measured intensities and obtain the spin structures for all field-induced phases, which we compare quantitatively with simulated configurations. Taken together, our measurements and simulations show that frustrated interlayer exchange -- rather than purely 2D physics -- organizes the unexpectedly rich field-induced phases of Na$_2$BaMn(PO$_4$)$_2$.

Figures (8)

• Figure 1: Crystal structure of Na$_2$BaMn(PO$_4$)$_2$ as determined by single crystal X-ray diffraction. View along the (a) in-plane direction, highlighting the layered structure. (b) View along the $c$-axis. (c,d) General view highlighting the connectivity of Mn ions, where (d) shows the Mn--Mn bonds used in theoretical modeling of the spin interactions, in-plane J$_1$ (bond length is $5.37\,$Å), out-of-plane J$_2$ ($7.094\,$Å), and diagonal out-of-plane J$_a$ and J$_b$ (both $8.9\,$Å). Even though the bond length for J$_a$ and J$_b$ is the same, the bonds are not equivalent in the $P\bar{3}$ space group.
• Figure 2: Color-coded intensity plots of single crystal neutron diffraction data of Na$_2$BaMn(PO$_4$)$_2$ collected at 300 and 1200 mK as a function of $\bm{Q} = (1/3, 1/3, Q_{l})$ and applied magnetic field along (a), (c) the $c$-axis and (b), (d) in the $ab$-plane. White horizontal lines represent the identified phase boundaries.
• Figure 3: Representative reciprocal space scans at $\bm{Q} = (1/3, 1/3, Q_{l})$ measured at 600 mK, under different applied fields applied along (a) the $c$-axis and (b) in the $ab$-plane. Lines represent fits with Gaussian functions on top of a constant background.
• Figure 4: Temperature and magnetic field phase diagrams of Na$_2$BaMn(PO$_4$)$_2$ with magnetic field applied along (a) the $c$-axis and (b) in the $ab$-plane. Red and black circles indicate critical fields determined from specific heat (background heat map) and neutron diffraction measurements, respectively. Phase boundaries are marked by dashed gray lines and are guides for the eyes.
• Figure 5: Magnetic structures of Na$_2$BaMn(PO$_4$)$_2$ based on single crystal neutron diffraction measurements under applied field. All structures are determined at 600 mK except the AFM2 phase (e). Each structure is depicted from the directions along the $c$, $a$, and $a^{*}$-axis. Axes coordinates for these views are depicted only in (a) showing $a_1$, $a_2$ and $c$-axis in red, green, blue respectively. (a) At zero field and 600 mK, the spin configuration is incommensurate, co-planar with Y arrangement of spins and components along the $c$-axis and $[120]$ direction. (e) At 1200 mK it is collinear along the $c$-axis, with moments in an up-up-down pattern. When a magnetic field is applied along the $c$-axis, spins gradually align with the field forming (b) alternating YT layers at 1.2 T, (c) the UUD phase at 2.2 T, and (d) the umbrella V arrangement at 3.8 T. For the field applied along the $[1 \bar{1} 0]$ direction the spins also gradually align along the field, with (f) commensurately alternating V-W layers at 1.8 T, (g) commensurately stacked W layers at 3 T, and (h) incommensurately modulated V-W layers at 4.4 T. A detailed description of all phases is given in the main text; numerical values used to render all panels are listed in the Supplementary Material in Table SII Supplemental.