Weak ferromagnetic order breaking the threefold rotational symmetry of the underlying kagome lattice in $\mathrm{CdC}{\mathrm{u}}_3{\left(\mathrm{OH}\right)}_6{\left({\mathrm{NO}}_3\right)}_2·{\mathrm{H}}_2\mathrm{O}$

Novel magnetic phases are expected to occur in highly frustrated spin systems. Here, we study the structurally perfect kagome antiferromagnet $\mathrm{CdC}{\mathrm{u}}_3{\left(\mathrm{OH}\right)}_6{\left({\mathrm{NO}}_3\right)}_2·{\mathrm{H}}_2\mathrm{O}$ by magnetization, magnetic torque, and heat capacity measurements using single crystals. An antiferromagnetic order accompanied by a small spontaneous magnetization that surprisingly is confined in the kagome plane sets in at $T_{\mathrm{N}}\sim 4\mathrm{K}$, well below the nearest-neighbor exchange interaction $J/k_{\mathrm{B}}=45\mathrm{K}$. This suggests that a unique “$\mathbf{q}=0$” type ${120}^{\circ }$ spin structure with “negative” (downward) vector chirality, which breaks the underlying threefold rotational symmetry of the kagome lattice and thus allows a spin canting within the plane, is exceptionally realized in this compound rather than a common one with “positive” (upward) vector chirality. The origin is discussed in terms of the Dzyaloshinskii-Moriya interaction.

Figure 1

Three coplanar ${120}^{\circ }$ spin structures. A spin on each lattice point is represented by an arrow, and the direction of the vector chirality on each triangle is shown by “$+$” (up) or “$-$” (down). The two $\mathbf{q}=0$ type structures, “positive” vector chirality (PVC) and “negative” vector chirality (NVC) structures, are shown in (a) and (b), respectively, and the $\surd 3\times \surd 3$ structure with $\mathbf{q}=(1/31/3)$, a staggered vector-chirality (SVC) structure, in (c). The numbers around the lower-left triangle of (a) refer to those in Eq. (1).

Figure 2

Crystal structures of $\mathrm{CdC}{\mathrm{u}}_3{\left(\mathrm{OH}\right)}_6{\left({\mathrm{NO}}_3\right)}_2·{\mathrm{H}}_2\mathrm{O}$ (CdK) (a) and the kagome layer viewed along the [001] direction (b) [21]. In (b), the arrangement of $d(x^2-y^2)$ orbitals is depicted by the red lobes. The nearest-neighbor magnetic exchange interactions are shown by the thick lines. $D_z$ and $D_p$ in the pair of triangles represent the out-of-plane and in-plane components of the DM vector, respectively. The arrows on the pair of triangles represent the rotational direction in the cross products of the DM interactions, which is always counterclockwise.

Figure 3

(a) Crystal growth by the hydrothermal transport method and grown crystals. (b) Crystal of CdK used in the present experiments.

Figure 4

Temperature dependence of magnetic susceptibilities from the one single crystal of CdK, the photograph of which is shown in Fig. 3. The measurements were carried out upon cooling in a magnetic field of 1 T applied along the $a$ or $c$ axis. The dashed lines on the data show fits to calculations by the high-temperature series expansion, which yields $J/k_{\mathrm{B}}=45.44\left(3\right)\mathrm{K}$. The inset shows magnetizations measured upon heating after zero-field cooling and then upon cooling below 10 K in a low field of 0.01 T.

Figure 5

Temperature dependence of heat capacity divided by temperature at zero magnetic field.

Figure 6

Magnetization processes measured at 2 K. Blue, green, and red marks are data for $B\parallel c, a-b$, and $a$, respectively. The inset expands the small field range of $-0.02\sim 0.02\mathrm{T}$.

Figure 7

Magnetic torque τ at various temperatures measured by rotating one crystal of CdK around the $c$ axis in a magnetic field of 10 T. The rotation angle $\varphi$ is set to zero at $B\parallel a$. The solid line on each data set is a fit to Eq. (2). Easy axes appear with ${60}^{\circ }$ interval at $a, a+b,...,$ as indicated at the top of the panel. The inset depicts a kagome net with thus determined local easy axes shown by the broken lines.

Figure 8

Phase diagram for the ground state of the classical kagome antiferromagnet with the nearest-neighbor antiferromagnetic interaction $J$ and DM interaction [36, 37]. Negative and positive $D_z$ tend to stabilize the PVC and NVC spin structures, respectively. In PVC, every spin is canted upward and downward by positive and negative $D_p$, respectively, keeping the threefold rotational symmetry. This causes an out-of-plane net magnetization shown by the thick arrow. In NVC, spins are mostly within the plane and the threefold rotational symmetry is broken. The dashed lines on the triangle represent $\langle 100\rangle$ easy axes. When one of the three spins on the triangle is pinned along the easy axis, the other two spins are not, which can generate a small tilting and cause an in-plane net magnetization.

Figure 9

Schematic representation of basis vectors for the space group $P-3m1$ with $\mathbf{k}=\left(0,0,0\right)$. A spin on each lattice point is represented by an arrow when it is confined in the plane, and by a circle when it is perpendicular to the plane. In ${\psi }_9$ one large downward spin and two small upward spins in the triangle cancel with each other.