Frustrated magnetic interactions in an $S=3/2$ bilayer honeycomb lattice compound ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}$(${\mathrm{NO}}_3$)

An inelastic neutron scattering study has been performed in an $S=3/2$ bilayer honeycomb lattice compound ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$ at ambient and high magnetic fields. Relatively broad and monotonically dispersive magnetic excitations were observed at ambient field, where no long-range magnetic order exists. In the magnetic-field-induced long-range ordered state at 10 T, the magnetic dispersions become slightly more intense, albeit still broad as in the disordered state, and two excitation gaps, probably originating from an easy-plane magnetic anisotropy and intrabilayer interactions, develop. Analyzing the magnetic dispersions using the linear spin-wave theory, we estimated the intraplane and intrabilayer magnetic interactions, which are almost consistent with those determined by ab initio density functional theory calculations [M. Alaei et al., Phys. Rev. B 96, 140404(R) (2017)], except the third and fourth neighbor intrabilayer interactions. Most importantly, as predicted by the theory, there is no significant frustration in the honeycomb plane but frustrating intrabilayer interactions probably give rise to the disordered ground state.

Figure 1

Schematic structure of the honeycomb bilayer in ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$. The open circles represent the ${\mathrm{Mn}}^{4+}$ moments. Magnetic interactions in a honeycomb plane $J_1$ (bold solid line), $J_2$ (dashed line), and $J_3$ (long dashed dotted line) (a) and intrabilayer interactions $J_{1c}$ (bold solid line), $J_{2c}$ (dashed line), $J_{3c}$ (long dashed dotted line), and $J_{4c}$ (long dashed double-dotted line) (b), which are used to calculate the spin-wave dispersions.

Figure 2

Color contour maps of the inelastic neutron scattering intensity $S\left(\right|Q|,E)$ for ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$ powder measured with $E_{\mathrm{i}}=20.0$ meV on ARCS at ambient field and at $T=5$ (a), 30 (b), 60 (c), 80 (d), 160 (e), and 250 K (f).

Figure 3

Color contour maps of $S\left(\right|Q|,E)$ for ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$ powder measured with $E_{\mathrm{i}}=25.3$ meV on DCS at $T=2.1$ K and at $H=0$ (a) and 10 T (b). Energy cuts of the excitations at $H=0$ and 10 T integrated over the range of $1.3\le Q\le 1.7$ Å${}^{-1}$ (c) and $0.85\le Q\le 1$ Å${}^{-1}$ (d). $Q$ cuts of the excitations at $H=0$ and 10 T integrated over the range of $2\le E\le 4$ meV (e) and $4\le E\le 6$ meV (f). Solid lines in (d) are fits with a Lorentzian function. Solid lines in (e) and (f) are fits with a Gaussian function.

Figure 4

Low energy magnetic excitations in ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$ powder measured at $T=2.1$ K with $E_{\mathrm{i}}=6.3$ meV on DCS. Color contour maps of $S\left(\right|Q|,E)$ at $H=0$ (a) and 10 T (b). (c) Energy cut of the excitations integrated over the range of $1.4\le Q\le 1.65$ Å${}^{-1}$ at $H=0$ and 10 T. $Q$ cut of the excitations at $H=0$ and 10 T integrated over the range of $0.8\le E\le 1.8$ meV (d), $1.8\le E\le 2.5$ meV (e), and $2.5\le E\le 3.2$ meV (f). Solid lines in (c) are guides to the eye. Solid lines in (d), (e), and (f) are fits with a Gaussian function.

Figure 5

Summary of the calculated magnetic excitations using the linear spin-wave theory with magnetic field (10 T) perpendicular to the spin direction. (a), (e), and (i) Powder averaged inelastic neutron scattering intensity $S\left(\right|Q|,E)$. (b), (f), and (j) Low-energy and low-$Q$ region of $S\left(\right|Q|,E)$. (c), (g), and (k) Magnetic dispersions along the high symmetric directions in the reciprocal space. (d), (h), and (l) $Q$ dependence of the scattering intensities $S\left(\right|Q|,E)$ integrated over the energy range of $8\le E\le 11$ meV. The experimental values measured on ARCS at 0 T and 5 K and on DCS at 0 and 10 T and 2.1 K are also shown. Panels (a)–(d) are the results with theoretically predicted interactions $J_2=0.084J_1, J_3=0.112J_1, J_{1c}=0.28J_1, J_{2c}=0.10J_1, J_{3c}=0.047J_1$, and $J_{4c}=0.084J_1$. $J_1=2.64$ meV is used to adjust the energy scale of the experimental data. Panels (e)–(h) are the results with $J_1=2.55$ meV, $J_2=0.084J_1, J_3=0.112J_1, J_{1c}=0.28J_1$, and $J_{2c}=0.094J_1$. Note that $J_{2c}$ is fixed at a slightly smaller value to keep the ground state stable with the spin correlations same as those observed. Panels (i)–(l) are the results with $J_1=3.3$ meV, $J_2=0.14J_1, J_3=0.088J_1, J_{1c}=0.29J_1, J_{2c}=0.09J_1$, and $D=0.012$ meV.

Figure 6

Summary of the calculated magnetic excitations in the magnetic field of 10 T using the linear spin-wave theory. (a), (d), (g), and (j) Powder averaged inelastic neutron scattering intensity $S\left(\right|Q|,E)$. (b), (e), (h), and (k) Low-energy and low-$Q$ region of $S\left(\right|Q|,E)$. (c), (f), (i), and (l) Magnetic dispersions along the high symmetric directions in the reciprocal space. Panels (a)–(f) are the results with $J_1=2.64$ meV, $J_2=0.084J_1, J_3=0.112J_1, J_{1c}=0.28J_1, J_{2c}=0.10J_1, J_{3c}=0.047J_1$, and $J_{4c}=0.084J_1$. The ratios of the interactions theoretically predicted [29] are used. [(a)–(c) Magnetic field perpendicular to the spins. (d)–(f) Magnetic field parallel to the spins.] Panels (g)–(l) are the results with $J_1=2.55$ meV, $J_2=0.084J_1, J_3=0.112J_1, J_{1c}=0.28J_1$, and $J_{2c}=0.093J_1$. [(g)–(i) Magnetic field perpendicular to the spins. (j)–(l) Magnetic field parallel to the spins.] The sharp and vertically elongated spots seen at $\sim 1.2$ meV in (e) and (k) originate from the sharp edge of the dispersion at $\mathrm{\Gamma }$ point, generated by the longitudinal magnetic field. Due to the almost flat dispersion along the $c$ axis, many spots are present at the same energy.