$S=\frac{1}{2}$ triangular-lattice antiferromagnets ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{O}_9$ and ${\mathrm{CsCuCl}}_3$: Role of spin-orbit coupling, crystalline electric field effect, and Dzyaloshinskii-Moriya interaction

We have performed the detailed investigations of the magnetization of the $S=\frac{1}{2}$ triangular-lattice antiferromagnets ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ and ${\mathrm{CsCuCl}}_3$ with a ${120}^{\circ }$ spin structure in the $ab$ plane. In ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$, the magnetic susceptibility ($\chi$) exhibits a broad maximum above the Néel temperature ($T_{\mathrm{N}}$) as is expected in the low-dimensional antiferromagnet (AFM). In ${\mathrm{CsCuCl}}_3, \chi$ exhibits a continuous increase down to $T_{\mathrm{N}}$ as if it is the three-dimensional AFM. This is induced by the strong ferromagnetic (FM) interaction along the $c$ axis. The magnetic phase diagrams are also very different. Although the transition field from the umbrella to the 2-1-coplanar phase ($H_{u-c}$) for $H\parallel c$ is almost independent of temperature in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$, it shows a considerable decrease with increasing temperature in ${\mathrm{CsCuCl}}_3$. The temperature independent $H_{u-c}$ in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ originates from the magnetic anisotropy from the van Vleck contribution, which does not depend so much on the temperature. The temperature dependent $H_{u-c}$ in ${\mathrm{CsCuCl}}_3$ originates from the magnetic anisotropy from the Dzyaloshinskii-Moriya (DM) interaction, which decreases with increasing temperature. For $H\parallel ab$, the clear transition from the Y-coplanar to the up-up-down ($uud$) phase was observed in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ but not in ${\mathrm{CsCuCl}}_3$. While the reentrant behavior of $T_{\mathrm{N}}$ originating from the thermal and quantum spin fluctuations is observed in both compounds, it is pronounced in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ but small in ${\mathrm{CsCuCl}}_3$. These differences originate from the existence or nonexistence of the DM interaction. The DM interaction in ${\mathrm{CsCuCl}}_3$ suppresses those fluctuations in the $ab$ plane, leading to the less pronounced reentrant behavior of $T_{\mathrm{N}}$ and the broad crossover in place of the phase transition. We analyzed the anisotropic magnetization of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ in the paramagnetic region by the mean field calculation. The spin-orbit (SO) coupling, the uniaxial crystalline electric field, and the isotropic exchange interaction were taken into account. We could estimate the anisotropy ratio of the exchange interaction $J/J_z=1.19$ and $g_{\perp }/g_{\parallel }=1.13$ with $g_{\perp }=4.45$ and $g_{\parallel }=3.95$ in the ground state. We emphasized that although the isotropic exchange interaction was used, the above anisotropies at low temperatures are induced simultaneously through the SO coupling and the uniaxial crystalline electric field and they are closely associated with each other.

Figure 1

(a) Temperature dependence of the magnetic susceptibility of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for $H\parallel ab$ and $H\parallel c$ measured at $H=0.5\mathrm{T}$. The inset exhibits the temperature dependence of the inverse magnetic susceptibility. The dotted lines are drawn to estimate the paramagnetic Curie temperature (${\theta }_{\mathrm{p}}$). (b) Magnetic susceptibilities in an expanded scale below 15 K.

Figure 2

Temperature dependence of the magnetic susceptibility of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for (a) $H\parallel ab$ and (c) $H\parallel c$ at various magnetic fields up to 5 T. The arrows in (a) indicate the transition temperature. Magnetic phase diagrams for (b) $H\parallel ab$ and (d) $H\parallel c$ obtained from the magnetic susceptibility data.

Figure 3

Magnetization curves of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ at $T=1.5\mathrm{K}$ for $H\parallel ab$ and $H\parallel c$.

Figure 4

Temperature dependence of the magnetization of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for (a) $H\parallel ab$ and (c) $H\parallel c$ at various magnetic fields up to 14.8 T. Magnetization curves of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for (b) $H\parallel ab$ and (d) $H\parallel c$ at various temperatures.

Figure 5

(a) Magnetic phase diagram of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for $H\parallel ab$. Green line indicates the transition field from $uud$ to coplanar state, which is cited from Ref. [36]. Along a black dashed line, the thermal and quantum spin fluctuation works most effectively, where $T_{\mathrm{N}}$ is the highest. (b) Magnetic phase diagram for $H\parallel c$. In the hatched area, $T_{\mathrm{N}}$ increases with increasing magnetic field, which might be due to the larger effect of the thermal and quantum spin fluctuation than the Zeeman effect.

Figure 6

(a) Temperature dependence of the magnetic susceptibility of ${\mathrm{CsCuCl}}_3$ for $H\parallel b^{*}$ and $H\parallel c$. The inset shows the temperature dependence of the inverse magnetic susceptibility. The dotted lines in the inset are drawn to estimate the paramagnetic Curie temperature. (b) Magnetic susceptibilities in an expanded scale below $T_{\mathrm{N}}$.

Figure 7

Temperature dependence of the magnetic susceptibility of ${\mathrm{CsCuCl}}_3$ for (a) $H\parallel b^{*}$ and (c) $H\parallel c$ at various magnetic fields up to 5 T. The origin of the vertical axis is shifted in each $\chi -T\mathrm{curve}$. Magnetic phase diagrams for (b) $H\parallel b^{*}$ and (d) $H\parallel c$ obtained by the temperature dependence of the magnetic susceptibility.

Figure 8

Magnetization curves of ${\mathrm{CsCuCl}}_3$ for $H\parallel b^{*}$ and $H\parallel c$ at $T=1.5\mathrm{K}$.

Figure 9

Temperature dependence of the magnetization of ${\mathrm{CsCuCl}}_3$ for (a) $H\parallel b^{*}$ and (c) $H\parallel c$ at various magnetic fields up to 14.8 T. Magnetization curves of ${\mathrm{CsCuCl}}_3$ for (b) $H\parallel b^{*}$ and (d) $H\parallel c$ at various temperatures. The origin of the vertical axis is shifted in each $M-H$ curve.

Figure 10

(a) Magnetic phase diagram of ${\mathrm{CsCuCl}}_3$ for $H\parallel b^{*}$. Red and blue circles are plotted from the temperature dependence of magnetization and green squares are from magnetic field dependence of magnetization. See the text in details. (b) Magnetic phase diagram for $H\parallel c$. In the hatched area, $T_{\mathrm{N}}$ increases with increasing magnetic field, which might be due to the larger effect of the thermal and quantum spin fluctuation than the Zeeman effect.

Figure 11

$\delta /{\lambda }^{{}^{\prime }}$ dependencies of (a) $|c_i|$ ($i=1$, 2, 3), (b) $p, q, p_l, q_l$, (c) $g_{\parallel }$ and $g_{\perp }$ for $k=1$, and (d) anisotropy ratios of $J/J_z$ and $g_{\perp }/g_{\parallel }$. Dashed lines in (b), (c), and (d) indicate the results for $k=0.9$. See the text in details.

Figure 12

Three-sublattice model with ${120}^{\circ }$ spin structure in the $xy$ plane at $H=0$. The antiferromagnetic exchange interaction works between A-B, B-C, and C-A. The magnetic field is applied along the $z$ axis (left) and the $x$ axis (right). $z$ and $x$ axes correspond to $c$ and $a$ axes in ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$, respectively.

Figure 13

Relationship between $g_{\parallel }$ and $g_{\perp }$. Solid line is that for $k=1$. Red circle and black plus symbol indicate $g_{\perp }$ and $g_{\parallel }$ for $\delta =0.5{\lambda }^{{}^{\prime }}$ and 0 when $k=0.95$.

Figure 14

Temperature dependence of (a) magnetic susceptibility and (b) inverse magnetic susceptibility of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ for $H\parallel ab$ and $H\parallel c$. Dotted lines are the calculated results for the three-sublattice model for $H\parallel x$ and $H\parallel z$ by using the parameters of $\delta =0.5{\lambda }^{{}^{\prime }}, k=0.95$, and $J_{\mathrm{ex}}=-18$ K. See the text in details.

Figure 15

(a) Magnetization curves for $H\parallel x$ and $H\parallel z$ in the three-sublattice model with $\delta =0.5{\lambda }^{{}^{\prime }}$ and $J_{\mathrm{ex}}=-18$ K at $T=0\mathrm{K}$ calculated by the mean field approximation. The black dotted line indicates the calculated one with $\delta =0$ for $H\parallel z$. (b) Temperature dependence of the inverse magnetic susceptibilities for $H\parallel x$ and $H\parallel z$ in the three-sublattice model with $\delta =0.5{\lambda }^{{}^{\prime }}$. For $H\parallel z$, the results with $J_{\mathrm{ex}}=-18$ K, $-9$ K, and 0 and for $H\parallel z$, that with $J_{\mathrm{ex}}=0$ K are shown. Black solid line indicates the result with $\delta =0$ for $H\parallel x$. The black dotted lines are drawn to estimate the paramagnetic Curie temperature ${\theta }_{\mathrm{p}}$. See the text for details.

Figure 16

Schematic magnetic phase diagrams of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ (red line) and ${\mathrm{CsCuCl}}_3$ (green line) for $H\parallel c$. See the text in details.