Magnetic transition in K${}_4$Cu${}_4$OCl${}_{10}$: A model system of three-dimensional spin-$\frac{1}{2}$ tetrahedra

Isolated spin tetrahedral systems with weak intertetrahedral couplings, as have been reported for Cu${}_2$Te${}_2$O${}_5$$X_2$ ($X$ $=$ Cl, Br) and the related compound Cu${}_4$Te${}_5$O${}_{12}$Cl${}_4$, have received much attention recently because they represent an interesting class of magnets that consist of weakly coupled magnetic clusters and, in particular, they can directly demonstrate the interplay of intertetrahedral couplings with built-in tetrahedral frustration. However, there is much debate about the structural low dimensionality of the Cu-Te-O-Cl(Br) compounds and its effect on the magnetism of the material. Here, we present a model spin tetrahedral system K${}_4$Cu${}_4$OCl${}_{10}$, with almost isotropic magnetic coupling within the tetrahedron and three-dimensional connection of the tetrahedra. The system enters a spin-singlet state with a susceptibility maximum at $T_{\max }$ $=$ 11 K, and then enters an antiferromagnetic order at $T_{\mathrm{N}}$ $=$ 4.4 K. The ratio $T_{\mathrm{N}}$/$T_{\max }$ $=$ 0.40 is close to the $T_{\mathrm{N}}$/$T_{\max }$ $=$ 0.38 for Cu${}_2$Te${}_2$O${}_5$Br${}_2$, which is viewed as an indicator of closeness to quantum criticality. Evidence in muon-spin rotation or relaxation suggests an incommensurate ordering. This work shows that the previously revealed ground state in anisotropically structured Cu${}_2$Te${}_2$O${}_5$$X_2$ compounds also exists in an isotropic spin tetrahedral system.

Figure 1

Crystal structures of spin tetrahedral systems of (a) Cu${}_2$Te${}_2$O${}_5$Cl${}_2$ and (b) K${}_4$Cu${}_4$OCl${}_{10}$. Ions are represented by colored balls, with red for Cu${}^{2+}$, blue for O${}^{-2}$, green for halogen ions, and gray and purple for Te${}^{4+}$ and K${}^{+}$, respectively. In panel (c) the local environment for the Cu${}^{2+}$ spins is illustrated more clearly.

Figure 2

Temperature dependence of magnetic susceptibility χ per mol Cu measured at 1 T for K${}_4$Cu${}_4$OCl${}_{10}$. The thick dashed line is a curve fit for noninteracting spin tetrahedra and the thick gray line is that containing a Curie-Weiss component, as described in the text (data at temperatures above 6 K are used for fitting). For reference, a curve fit (thin solid line) using the data above 50 K to the Curie-Weiss formula is also plotted. The inset shows an enlarged plot of the low-temperature part.

Figure 3

(a) Specific heat per mol formula for K${}_4$Cu${}_4$OCl${}_{10}$ (left axis) and the roughly estimated magnetic entropy (solid line, right axis). The dashed line is an assumed lattice contribution to the specific heat. The inset plot shows the zero-field $C$/$T$ depicting the broad anomaly around 11 K; the thin line is to guide to eyes to help to see the small, broad anomaly. (b) Specific-heat data under external fields showing the decrease of $T_{\mathrm{N}}$ with increasing field. The specific-heat data are shifted upward to clearly show the $T_{\mathrm{N}}$ decrease.

Figure 4

Zero-field μSR asymmetry spectra at typical temperatures for K${}_4$Cu${}_4$OCl${}_{10}$ obtained from the RIKEN-RAL beam line. The solid lines on the back of high-temperature data are curves fits as described in the text. The inset shows the estimated nuclear field distribution Δ.

Figure 5

Fourier transform of asymmetry spectra showing broad-peak features of the muon spin precession frequencies in K${}_4$Cu${}_4$OCl${}_{10}$.

Figure 6

Examples of curve fits (thick line) using (a) a zero-order Bessel function and (b) a modified Bessel function as defined in Eq. (3) and (c) Eq. (4) for a fictitious commensurate order for the zero-field μSR asymmetry spectrum at 2 K (TRIUMF).

Figure 7

Temperature dependence of muon spin procession frequencies in K${}_4$Cu${}_4$OCl${}_{10}$.