Magnetic order in the $S=1∕2$ two-dimensional molecular antiferromagnet copper pyrazine perchlorate $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$

We present an investigation of magnetic ordering in the two-dimensional $S=1∕2$ quantum magnet $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ using specific heat and zero-field muon-spin relaxation $\left({\mu }^{+}\mathrm{SR}\right)$. The magnetic contribution to the specific heat is consistent with an exchange strength of $17.7\left(3\right)\mathrm{K}$. We find unambiguous evidence for a transition to a state of three-dimensional long-range order below a critical temperature $T_N=4.21\left(1\right)\mathrm{K}$ using ${\mu }^{+}\mathrm{SR}$ even though there is no feature in the specific heat at that temperature. The absence of a specific heat anomaly at $T_N$ is consistent with recent theoretical predictions. The ratio of $T_N∕J=0.24$ corresponds to a ratio of intralayer to interlayer exchange constants of $\mid J^{\prime }∕J\mid =6.8\times {10}^{-4}$, indicative of excellent two-dimensional isolation. The scaled magnetic specific heat of $\left[\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right]\mathrm{B}{\mathrm{F}}_4$, a compound with an analogous structure, is very similar to that of $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ although both differ slightly from the predicted value for an ideal 2D $S=1∕2$ Heisenberg antiferromagnet.

Figure 1

Specific heats of $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ (☉) and $\left[\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right]\mathrm{B}{\mathrm{F}}_4$ (×) as functions of temperature. The solid lines correspond to the best-fit predictions for the specific heats based on the parameters given in the text. The dashed lines represent the estimated lattice specific heats for $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ (short dashes) and $\left[\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right]\mathrm{B}{\mathrm{F}}_4$ (long dashes). The arrows mark the temperatures of the magnetic ordering transitions for the two compounds as determined by muon-spin relaxation studies. Note the absence of anomalies in the specific heats at those temperatures.

Figure 2

(a) Magnetic specific heats of $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ (☉) and $\left[\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right]\mathrm{B}{\mathrm{F}}_4$ (×) as functions of temperature with the solid lines corresponding to the best-fit predictions for the magnetic specific heats based on the parameters given in the text. The dashed lines (short dashes for $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ and long dashes for $\left[\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right]\mathrm{B}{\mathrm{F}}_4$) mark the estimated lattice contributions that were subtracted from the data sets in Fig. 1 to produce the magnetic specific heats. (b) The reduced magnetic specific heats $(C_{\mathrm{mag}}∕R)$ of $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ (☉) and $\left\{\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2\left(\mathrm{H}{\mathrm{F}}_2\right)\right\}\mathrm{B}{\mathrm{F}}_4$ (×) as functions of the reduced temperature $T∕J$ with the solid line corresponding to the theoretical prediction (Ref. 3) for the magnetic specific heat of the 2DSLQHA.

Figure 3

(a) ZF ${\mu }^{+}\mathrm{SR}$ spectra in $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$ measured at $T=0.35\mathrm{K}$ and $5.02\mathrm{K}$. At temperatures $T⩾4.27\mathrm{K}$, Eq. (4) describes the data while the oscillations measured below this temperature are fitted to Eq. (5).

Figure 4

Temperature evolution of the three magnetic oscillation frequencies in $\mathrm{Cu}{\left(\mathrm{Pz}\right)}_2{\left(\mathrm{Cl}{\mathrm{O}}_4\right)}_2$, extracted from fits to Eq. (5). The solid lines are fits to the functional form ${\nu }_i\left(T\right)={\nu }_i\left(0\right){[1-{(T∕T_N)}^{\alpha }]}^{\beta }$ (see main text).