Magnetic characterization of the frustrated three-leg ladder compound $\left[{\left(\mathrm{Cu}{\mathrm{Cl}}_2\mathrm{tach}\mathrm{H}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$

We report the magnetic features of a one-dimensional stack of antiferromagnetically coupled equilateral copper(II) triangles. High-field magnetization measurements show that the interaction between the copper triangles is of the same order of magnitude as the intratriangle exchange although only coupled via hydrogen bonds. The infinite chain turns out to be an interesting example of a frustrated cylindrical three-leg ladder with competing intra- and inter-triangle interactions. We demonstrate that the ground state is a spin singlet which is gaped from the triplet excitation.

Figure 1

The figure shows the molecular structure of the cluster framework and the dotted lines represent the hydrogen bonds which connected the triangular units to the triangular antiprismatic chain; the copper(II) ions as large black spheres, the nitrogen atoms as small black spheres, the chloride ions as large grey spheres, the carbon atoms as large white spheres, and the hydrogen atoms as small white spheres.

Figure 2

Schematic structure of the triangular copper chain: The copper ions are placed at the vertices, intratriangle exchange pathways of the related Heisenberg Hamiltonian (1) are drawn by grey lines $\left(J_1\right)$, intertriangle couplings are given by black lines $\left(J_2\right)$.

Figure 3

Magnetic low-field susceptibility ${\chi }_0$ per copper triangle: The solid curve shows the experimental result obtained at $B=0.5\mathrm{T}$ down to temperatures of $2\mathrm{K}$. The dashed curve displays the zero-field susceptibility of an isotropic Heisenberg model, compare Eq. (1), with exchange parameters $J_1=-0.9\mathrm{K}$ and $J_2=-1.95\mathrm{K}$ as given in the figure.

Figure 4

Magnetic low-field susceptibility $T{\chi }_0$ per copper triangle: Again the solid curve shows the experimental result, the dashed curve the theoretical one. The thin horizontal line is the high temperature limit.

Figure 5

Magnetization per copper triangle: Experimental data taken at a cryostat temperature of $0.4\mathrm{K}$ are given by thick solid lines, the two data sets are practically identical. The theoretical estimates using the same parameters as in Fig. 3 are given for 18 and 24 spins, i.e., 6 or 8 triangles. The saturation field is $B_{\mathrm{sat}}=11\mathrm{T}$.

Figure 6

Energy eigenvalues for a finite chain of six triangles using the exchange parameters given in Fig. 3.

Figure 7

Magnetization curves for $T=0$ for various hypothetical couplings of a chain of eight triangles with cyclic boundary condition. The used combinations of $J_1$ and $J_2$ are given in the figure together with the respective line mode.

Figure 8

Finite size extrapolation of the singlet-triplet gap for $\left[{\left(\mathrm{Cu}{\mathrm{Cl}}_2\mathrm{tach}\mathrm{H}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$. The gap for the infinite chain is estimated to be $\Delta ⪆0.4\pm 0.05\mathrm{K}$.

Figure 9

Finite size behavior of the singlet-singlet gap for $\left[{\left(\mathrm{Cu}{\mathrm{Cl}}_2\mathrm{tach}\mathrm{H}\right)}_3\mathrm{Cl}\right]{\mathrm{Cl}}_2$. The two leftmost data points for eight and ten triangles have an uncertainty of about $\pm 0.2\mathrm{K}$ due to the restricted accuracy of the Lanczos procedure for excited states.