Critical behavior of two molecular magnets probed by complementary experiments

The fast developing field of molecule-based magnets involving organic and coordination chemistry provides the physicist with a multitude of novel compounds of unprecedented structure. The magnetic structure of ${\text{Cu}}_4\left(\text{tetren}\right){\left[\text{W}{\left(\text{CN}\right)}_8\right]}_4$ $\left(\mathbf{1}\right)$ was shown to consist of weakly coupled double layers. By contrast, in the structurally similar compound ${\text{Cu}}_{2+x}{\text{Cu}}_4{\left[\text{W}{\left(\text{CN}\right)}_8\right]}_4$ $\left(\mathbf{2}\right)$ the free spaces between the double layers are filled with paramagnetic copper(II) ions leading to a unique magnetic network. Both compounds exhibit the transition to a magnetically ordered phase at $T_c\approx 33\text{K}$ and $T_c\approx 40\text{K}$, respectively. The critical behavior of $\mathbf{1}$ and $\mathbf{2}$ is investigated using complementary methods: ac magnetometry, relaxation calorimetry, and muon spin-rotation spectroscopy. Apart from $\alpha$, $\beta$, and $\gamma$, critical exponents $\kappa$ and ${\kappa }^{\prime }$ describing the combined scaling of excess entropy and order parameter are determined for both compounds. This type of scaling is verified for $\mathbf{1}$, the system revealing the signatures of the Berezinskii-Kosterlitz-Thouless transition. For $\mathbf{2}$ their values imply that the system is close to the universality class of the three-dimensional Heisenberg model. The relatively small value of exponent $\gamma =1.05$ for $\mathbf{2}$ indicates the presence of noncollinearity in the spin arrangement. Exponents $\kappa$ and ${\kappa }^{\prime }$ for $\mathbf{2}$ are also found consistent with noncollinear models. The shift of the heat-capacity anomaly toward higher temperatures with increasing applied field indicates the presence of ferromagnetic interactions in $\mathbf{2}$.

Figure 1

(Color online) Projection of the crystal structure of $\mathbf{1}$ onto the $ab$ crystallographic plane. Alternating arrangement of Cu(II) and W(V) ions forms anionic double layers. The space between them is filled with ${\text{tetrenH}}_5$ countercations and water molecules (not shown for clarity).

Figure 2

(Color online) Projection of the crystal structure of $\mathbf{2}$ onto the $bc$ crystallographic plane. The system of Cu(II)-W(V) double layers is similar as in $\mathbf{1}$. The space between them is filled with additional paramagnetic Cu(II) ions and water molecules, of which only oxygen atoms are shown.

Figure 3

(Color online) Top panel: powder-diffraction patterns for $\mathbf{2}$ recorded at 2, 35, and 45 K. Above considerable background due to incoherent scattering well-defined Bragg peaks are present. Bottom panel: comparison of the diffraction patterns detected at 45 K (above magnetic transition) and 2 K (below magnetic transition). No additional Bragg peaks and no intensity enhancement of the existing peaks are observed.

Figure 4

(Color online) Real part of ac magnetic susceptibility for $\mathbf{1}$ and $\mathbf{2}$. For both compounds sharp anomalies at $T_c=33.04\text{K}$ and $T_c=39.9\text{K}$, respectively, mark the transition to the magnetically ordered phase. Higher transition temperature of $\mathbf{2}$ is attributed to the presence of additional Cu(II) ions exchange coupled with the spin centers constituting the double-layer sheets.

Figure 5

(Color online) Classical scaling analysis of the ac susceptibility signals for $\mathbf{1}$ and $\mathbf{2}$. The value of the gamma exponent of the quasi-2D spin network (compound $\mathbf{1}$) is enhanced as compared to that of compound $\mathbf{2}$ displaying the 3D connectivity of spin centers.

Figure 6

(Color online) Scaling analysis of the ac susceptibility signals for $\mathbf{1}$ and $\mathbf{2}$. The intercepts of the best-fit curves (solid lines) with both axes yield $T_c$ and $1/\gamma$. The value of the gamma exponent of the quasi-2D spin network (compound $\mathbf{1}$) is enhanced as compared to that of compound $\mathbf{2}$ displaying the 3D connectivity of spin centers.

Figure 7

(Color online) Total heat capacity at zero magnetic field for compounds $\mathbf{1}$ and $\mathbf{2}$ measured on powder samples. For clarity the heat capacity of $\mathbf{1}$ has been shifted by $100\text{J}{\text{K}}^{-1}{\text{mol}}^{-1}$. The $\lambda$-shaped anomalies signal the presence of a magnetic continuous phase transition. Solid lines are the baselines used to extract the magnetic contribution to the heat capacity (see text).

Figure 8

(Color online) Magnetic contribution to the total heat capacity of $\mathbf{2}$ for an array of applied field values. The anomaly is suppressed and shifted toward higher temperatures with increasing applied field implying a ferromagnetic character of exchange interactions.

Figure 9

(Color online) Scaling plots for the zero-field magnetic heat capacity of $\mathbf{1}$ (circles) and $\mathbf{2}$ (squares). To avoid overlap of the data the heat-capacity values of $\mathbf{1}$ were multiplied by factor 1.5. The positive values of the $\alpha$ exponent suggest the presence of noncollinearity in spin arrangement.

Figure 10

(Color online) Magnetic contribution to the total heat capacity of $\mathbf{1}$ for an array of applied field values. The magnetic field is applied in the direction perpendicular to the double layers. The anomaly is suppressed and shifts toward lower temperatures with increasing applied field. This behavior is due to the presence of the strong easy-plane anisotropy.

Figure 11

(Color online) Time dependence of the muon asymmetry function observed in the vicinity of the transition point of $\mathbf{2}$ under zero field. The onset of spontaneous oscillations in the relaxation signal marks the transition to an ordered phase. Solid lines show the fits to the relaxation function given by Eq. (8) where a single precession frequency was assumed.

Figure 12

(Color online) Temperature dependence of the local magnetic fields as inferred from the $\mu \text{SR}$ experiment for $\mathbf{1}$ and $\mathbf{2}$. Solid lines show the fits to the phenomenological law given in Eq. (9). For $\mathbf{1}$ two local fields were detected whereas for $\mathbf{2}$ only one local field was observed.

Figure 13

(Color online) $\Delta S/Q^2$ vs $\frac{T_c-T}{T_c}$ in log-log plot for $\mathbf{1}$ and $\mathbf{2}$. For both compounds the experimental points tend to align while approaching the transition temperature, which signals the algebraic scaling behavior. Solid lines show the corresponding linear fits.

Figure 14

(Color online) The log-log plot of the excess entropy as function of the square of order parameter for $\mathbf{1}$ and $\mathbf{2}$. For both compounds the alignment of the data points while approaching the transition temperature is clearly visible. Solid lines show the corresponding linear fits.