Microscopic evidence of a quantum magnetization process in the $S=\frac{1}{2}$ triangular-lattice Heisenberg-like antiferromagnet ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$

We report results of the neutron scattering investigations of frustrated quantum magnet ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ in magnetic fields up to 25.9 T. Contrary to other materials, ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ exhibits properties typical of an ideal $S=\frac{1}{2}$ triangular lattice Heisenberg antiferromagnet, making it a perfect model system for testing theoretical predictions. In this work, we looked into the magnetization process in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ on a microscopic scale with a magnetic field applied in plane. A sequence of magnetic phase transitions, including the new high-field phase at 22.5 T reported recently has been followed at low temperatures as a function of field and modeled using the large-size cluster mean-field plus scaling method. Showing good agreement with the model, our results bridge the theory and the experiment providing microscopic information about the high-field spin ordering in $S=\frac{1}{2}$ triangular lattice Heisenberg-like antiferromagnet ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$.

Figure 1

[(a)–(e)] Spins in a magnetic unit cell (view along the $c$-axis) of the calculated ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ spin configurations in magnetic fields applied along the reciprocal ${\mathbf{a}}_{\mathrm{mag}}^{*}$ direction. The magnetic unit cell is defined as $\sqrt{3}{a}_n\times \sqrt{3}{a}_n\times c$ and its relation with the crystallographic one is shown in (f) [27]. Green- (sites $A,B,C)$ and red-colored (sites $A^{\prime },B^{\prime },C^{\prime })$ spins correspond to adjacent layers along the $c$ axis. (a) and (b) show the zero field ${120}^{\circ }$ structure and the UUD structure $\left(0.3H_s<H<0.47H_s\right)$. (c)–(e) display the co-planar $V$ phase $\left(0.47H_s<H<0.7H_s\right)$ and the proposed high-field co-planar $V^{\prime }$ and $\mathrm{\Psi }$ phases $\left(0.7H_s<H<H_s\right)$. (f) shows the crystallographic and magnetic unit cells as well as the field direction and the Cartesian coordinate system applied in the theoretical calculations.

Figure 2

Schematic presentation (top view) of the high-field scattering setup. The picture includes the HFM and the EXED detectors, sample orientation and scattering geometry for the forward- and back-scattering detectors displayed for a single wavelength (wave vector $\mathbf{k})$.

Figure 3

Refinement of the nuclear structure of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ at 1.5 K: calculated vs observed intensities for the collected nuclear peaks as obtained using fullprof. The straight line shows ${\mathrm{F}}_{\mathrm{obs}}^2={\mathrm{F}}_{\mathrm{calc}}^2$. The insert displays low ${\mathrm{F}}_{\mathrm{obs}}^2$ part of the plot.

Figure 4

Measured (square and circle symbols) and calculated (solid lines) magnetic intensities as a function of magnetic field for the transitions between the ${120}^{\circ }$, UUD, $V$, and $V^{\prime }$ phases. (a) The magnetic contribution to the intensity of (0, 0, $-2)$ as a function of field. Two experimental datasets (1 and 2) are shown. The theoretical calculation (solid blue) and square of magnetization (dashed green, refer to right axis) are also presented. The inset displays the dataset 2 and its first derivative across the phase transition at 22.5 T. (b) Experimental and theoretical curves of the magnetic contribution to $\left(-1,1,-2\right)$ as a function of field. The inset shows a sketch of the experiment geometry. (c) The field dependence of the intensity of the $(\frac{1}{3},-\frac{2}{3},-3)$ and $(-\frac{2}{3},\frac{1}{3},-3)$ reflections. The inset visualizes a first derivative of the averaged experimental and theoretical datasets across the phase transition at 22.5 T. For all the datasets, the results have been scaled by a constant to allow the comparison between the experiment and the theory.

Figure 5

(a) Cluster of $N_C=36$ spins embedded in neighboring two layers with the six-sublattice structure. (b) List of the clusters used in the $\mathrm{CMF}+\mathrm{S}$ analysis with scaling parameter $\lambda$.

Figure 6

(a) Theoretically simulated ground-state magnetization curves for $N_C=15$, 21, 36, and $\infty$ (from bottom to top) when $H\parallel ab$. The curves apart from the bottom one are vertically shifted by 0.05, 0.1, and 0.2, respectively, for clarity. [(b) and (c)] Sublattice magnetic moments ${\mathit{m}}_{\mu }\left(\mu =A,B,C\right)$ calculated by $\mathrm{CMF}+\mathrm{S}\left(\lambda \to 1\right)$ when $H\parallel ab$. The field-perpendicular components $m_{\mu }^z$ are zero. The values of ${\mathit{m}}_{\mu }$ for $\mu =A^{\prime },B^{\prime },C^{\prime }$ are obtained through the relations with $A,B,C$ as displayed in Fig. 1.

Figure 7

Theoretical calculations for the field $H\sim {12}^{\circ }$ off [1,1,0] towards [0,0,1] for the transitions between the ${120}^{\circ }$, UUD, $V$, and $V^{\prime }$ phases. (a) Calculated ground-state magnetization curves for $N_C=15$, 21, 36, and $\infty$ (from bottom to top). The curves apart from the bottom one are vertically shifted for clarity by 0.05, 0.1, and 0.2, respectively. [(b)–(d)] Sublattice magnetic moments ${\mathit{m}}_{\mu }\left(\mu =A,B,C\right)$ calculated by $\mathrm{CMF}+\mathrm{S}\left(\lambda \to 1\right)$. The values of ${\mathit{m}}_{\mu }$ for $\mu =A^{\prime },B^{\prime },C^{\prime }$ are obtained through the relations with $A,B,C$ displayed in Figs. 1–1 while the coordinate system is depicted in Fig. 1.

Figure 8

Integrated intensities of the magnetic $(\frac{1}{3},-\frac{2}{3},-3)$ and $(-\frac{2}{3},\frac{1}{3},-3)$ reflections (a) calculated for the $V,V^{\prime }$, and $\mathrm{\Psi }$ phases across the $H_{c3}$ transition and (b) calculated for ${\omega }_s=0^{\circ }$ and ${12}^{\circ }$ in the entire field range.