Magnetic quantum tunneling in ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ studied by ac susceptibility: Temperature and magnetic-field dependence of the spin-relaxation time

The geometrically frustrated spin-chain compound ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ was recently found to exhibit properties suggesting the phenomenon of quantum tunneling of the magnetization (QTM). The present paper addresses this issue by investigating the temperature and magnetic-field dependence of the characteristic spin-relaxation time $\tau$. This study is based on the analysis of the out-of-phase ac magnetic susceptibility of single crystals. The overall behavior is found to support the occurrence of QTM in ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$: (i) saturation of $\tau$ at low $T$; (ii) observation of identical $\tau$ values in the magnetic fields corresponding to the steps in the $M$ vs $H$ curve at low $T$; (iii) existence of a dip centered at zero field in the $\tau$ vs $H$ curve at low $T$. These QTM-like features found in ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ are compared to those previously reported in other materials such as single-molecular magnets.

Figure 1

${\chi }^{\prime }$ data in zero field, recorded at different frequencies: $0.01\mathrm{Hz}$ (squares), $0.1\mathrm{Hz}$ (circles), $1\mathrm{Hz}$ (triangles up), $10\mathrm{Hz}$ (triangles down), $100\mathrm{Hz}$ (diamonds), $1000\mathrm{Hz}$ (stars).

Figure 2

${\chi }^{″}$ data in zero field. (a) ${\chi }^{″}\left(T\right)$ curves recorded at different frequencies: $0.01\mathrm{Hz}$ (squares), $0.1\mathrm{Hz}$ (circles), $1\mathrm{Hz}$ (triangles up), $10\mathrm{Hz}$ (triangles down), $100\mathrm{Hz}$ (diamonds), $1000\mathrm{Hz}$ (stars). (b) ${\chi }^{″}\left(f\right)$ curves recorded at different temperatures: $2.25\mathrm{K}$ (squares), $3.5\mathrm{K}$ (circles), $6\mathrm{K}$ (triangles up), $8\mathrm{K}$ (triangles down), $10\mathrm{K}$ (diamonds), $11.5\mathrm{K}$ (stars), $13\mathrm{K}$ (triangles right).

Figure 3

${\chi }^{″}\left(f\right)$ curves at three temperatures in zero field. At each temperature, curves were recorded after a waiting time equal to zero (squares), $2\mathrm{h}$ (circles), and $4\mathrm{h}$ (diamonds).

Figure 4

Temperature dependence of the characteristic spin-relaxation time in zero field, which was derived from the location of the peaks in ${\chi }^{″}\left(T\right)$ or ${\chi }^{″}\left(f\right)$. The squares come from ${\chi }^{″}\left(T\right)$ curves such as those of Fig. 2a $[\tau \left(T_{\max }\right)=1∕2\pi f]$; the circles come from ${\chi }^{″}\left(f\right)$ curves such as those of Fig. 2b $[\tau \left(T\right)=1∕2\pi f_{\max }]$; and the diamonds from ${\chi }^{″}\left(f\right)$ curves recorded after either 0, 2, or $4\mathrm{h}$, such as in Fig. 3. The solid line shows the Arrhenius regime, while the dashed line corresponds to the quantum regime associated with a constant relaxation time ${\tau }_S\sim 0.4\mathrm{s}$.

Figure 5

${\chi }^{″}\left(f\right)$ in zero field at $2.25\mathrm{K}$. The solid line is a fit to the data assuming the existence of a distribution function $g\left(\tau \right)$ in the $\tau$ values (see text). A normalized plot of this function is shown in the inset.

Figure 6

Top panel: Normalized frequency dependence of ${\chi }^{″}$ in zero field at $10\mathrm{K}$ (solid triangles), $7\mathrm{K}$ (open circles), and $5\mathrm{K}$ (solid squares). Bottom panel: ${\chi }^{″}\left(f\right)$ curve in zero field at $T=7\mathrm{K}$. The solid line is the theoretical curve derived from a fit to the data for $f>f_{\max }$.

Figure 7

${\chi }^{″}\left(f\right)$ curves at $2\mathrm{K}$ in a series of magnetic fields applied along the $c$ axis. The inset shows rescaled curves for $0\mathrm{T}$ (stars), $1.2\mathrm{T}$ (triangles up), and $2.3\mathrm{T}$ (triangles down).

Figure 8

Field dependence of $\tau$ at $T=2\mathrm{K}$. The data were derived from the location of the maximum in ${\chi }^{″}\left(f\right)$ curves $[\tau \left(H\right)=1∕2\pi f_{\max }\left(H\right)]$.