Magnetic susceptibilities in a family of $S=\frac{1}{2}$ kagome antiferromagnets

Hexagonal antiferromagnets ${\text{Cs}}_2{\text{Cu}}_{3}M{\text{F}}_{12}$ ($M=\text{Zr}$, Hf, and Sn) have uniform kagome lattices of ${\text{Cu}}^{2+}$ with $S=1/2$, whereas ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ has a $2a\times 2a$ enlarged cell as compared to the uniform kagome lattice. The crystal data of ${\text{Cs}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ synthesized in the present work are reported. We performed magnetic-susceptibility measurements on this family of kagome antiferromagnet using single crystals. In the ${\text{Cs}}_2{\text{Cu}}_{3}M{\text{F}}_{12}$ systems, structural phase transitions were observed at $T_t=225$, 172, and 185 K for $M=\text{Zr}$, Hf, and Sn, respectively. The magnetic susceptibilities observed for $T>T_t$ are almost perfectly described using theoretical results obtained by exact diagonalization for the 24-site kagome cluster with $J/k_B=244$, 266, and 240 K, respectively. Magnetic ordering accompanied by the weak ferromagnetic moment occurs at $T_N=23.5$, 24.5, and 20.0 K, respectively. The origins of the weak ferromagnetic moment should be ascribed to the lattice distortion that breaks the hexagonal symmetry of the exchange network for $T<T_t$ and the Dzyaloshinsky-Moriya interaction. ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ is magnetically described as a modified kagome antiferromagnet with four types of neighboring exchange interactions. Neither structural nor magnetic phase transition was observed in ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$. Its magnetic ground state was found to be a spin singlet with a triplet gap. Using exact diagonalization for a 12-site kagome cluster, we analyzed the magnetic susceptibility and evaluated individual exchange interactions. The causes leading to the different ground states in ${\text{Cs}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ and ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ are discussed.

Figure 1

(Color online) (a) Crystal structure of ${\text{Cs}}_2{\text{Cu}}_{3}M{\text{F}}_{12}$ ($M=\text{Zr}$, Hf, and Sn) viewed along the [1, 1, 0] direction and (b) its projection onto the $c$ plane. Shaded octahedra represent ${\text{CuF}}_6$ octahedra. Dotted lines denote the unit cell.

Figure 2

(Color online) Crystal structures of (a) ${\text{Cs}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ and (b) ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ viewed along the $c$ axis, where fluorine ions located outside the kagome layer are omitted. Dashed lines denote the unit cell.

Figure 3

(Color online) Temperature dependences of magnetic susceptibilities of (a) ${\text{Cs}}_2{\text{Cu}}_3{\text{ZrF}}_{12}$, (b) ${\text{Cs}}_2{\text{Cu}}_3{\text{HfF}}_{12}$, (c) ${\text{Cs}}_2{\text{Cu}}_3{\text{SnF}}_{12}$, and (d) ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ measured at $H=1\text{T}$ for $H\parallel c$ and $H\perp c$. Arrows in (a)–(c) denote structural and magnetic phase-transition temperatures $T_t$ and $T_N$. Dashed lines denote the fits using the theoretical susceptibilities for the uniform HKAF obtained from exact diagonalization for the 24-site kagome cluster (Ref. 29). The magnetic parameters used for the fits for Cs compounds are given in Table and those for ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ are shown in the text. Insets of (a) and (d) are low-temperature susceptibilities. Solid lines in (d) denote the results obtained by exact diagonalization for a 12-site kagome cluster with the exchange parameters $J_1/k_B=234\text{K}$, $J_2/k_B=211\text{K}$, $J_3/k_B=187\text{K}$, and $J_4/k_B=108\text{K}$ and $g_{\parallel }=2.44$ and $g_{\perp }=2.15$. The dotted line in the inset of (d) denotes the fit using Eq. (3) with $\Delta /k_B=21\text{K}$.

Figure 4

(Color online) Arrangement of the $\mathit{D}$ vectors for $T>T_t$ in ${\text{Cs}}_2{\text{Cu}}_{3}M{\text{F}}_{12}$ systems. (a) The $c$ axis component $D^{\parallel }$ and (b) the $c$ plane component $D_{\perp }$. Symbols $\odot$ and $\otimes$ in (a) and arrows in (b) denote the local positive directions of parallel and perpendicular components $D^{\parallel }$ and $D^{\perp }$, respectively. Large arrows in (a) denote the $q=0$ spin structure stabilized when $D^{\parallel }>0$.

Figure 5

(Color online) Anisotropic exchange network composed of $J_1$ (solid lines) and $J_2$ (dashed lines), which results from orthorhombic distortion of the kagome lattice. Large arrows represent the $q=0$ spin structure composed of three sublattice spins ${\mathit{S}}_1$, ${\mathit{S}}_2$, and ${\mathit{S}}_3$. $\theta$ denotes the angle between the $c$ plane component of $-{\mathit{S}}_1$ and that of ${\mathit{S}}_2$ or ${\mathit{S}}_3$.

Figure 6

(Color online) Exchange network in the kagome layer of ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$. Pinwheels composed of $J_1$, $J_2$, and $J_4$ interactions are connected by triangles of $J_3$ interactions.

Figure 7

(Color online) Temperature dependence of magnetic susceptibilities per site obtained by exact diagonalizations for 12-site kagome cluster (a) for $J_4/J_1=0.5$, 0.75, 1, and 1.5 with $J_1=J_2=J_3$, (b) for $J_2/J_1=0.5$, 1, 1.5, and 2 with $J_1=J_3=J_4$, and (c) for $J_3/J_1=0.5$, 1, 1.5, and 2 with $J_1=J_2=J_4$. Temperature is scaled using the average of the exchange interactions $J_{\text{av}}$. The thick solid line in (a) denotes the susceptibilities for the uniform HKAF obtained by exact diagonalization for the 24-site kagome cluster (Ref. 29).

Figure 8

(Color online) Magnetization curves of ${\text{Rb}}_2{\text{Cu}}_3{\text{SnF}}_{12}$ measured at $T=1.3\text{K}$ for $H\parallel c$ and $H\perp c$. Arrows indicate the critical field $H_c$. Inset shows $dM/dH$ vs $H$ for $H\parallel c$.