Slow magnetic dynamics and hysteresis loops of the bulk ferromagnet Co${}_7$(TeO${}_3$)${}_4$Br${}_6$

Magnetic dynamics of a bulk ferromagnet, a new single crystalline compound Co${}_7$(Te$$O${}_3$)${}_4$Br${}_6$, was studied by ac susceptibility and related techniques. Very large Arrhenius activation energy of 17.2 meV (201 K) and long attempt time ($2\times {10}^{-4}$ s) span the full spectrum of magnetic dynamics inside a convenient frequency window, offering a rare opportunity for general studies of magnetic dynamics. Within the experimental window, the ac susceptibility data build almost ideally semicircular Cole-Cole plots. A comprehensive study of experimental dynamic hysteresis loops of the compound is presented and interpreted within a simple thermal-activation-assisted spin-lattice relaxation model for spin reversal. Quantitative agreement between the experimental results and the model’s prediction for dynamic coercive field is achieved by assuming the central physical quantity, the Debye relaxation rate, to depend on frequency, as well as on the applied field strength and sample temperature. Crossover between minor to major hysteresis loops is carefully analyzed. Low-frequency limitations of the model, relying on domain-wall pinning effects, are experimentally detected and appropriately discussed.

Figure 1

Dynamic hysteresis loops of $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ taken at several temperatures within the ferromagnetic transition. Numbers adjacent to the loops designate the temperatures. A small hysteresis asymmetry (a shift to the left) is commented in Sec. 4.

Figure 2

Frequency dependence of ${\chi }^{\prime }$ (open symbols) and ${\chi }^{\prime \prime }$ (solid symbols) of $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ just below $T_c=27.2$ K (main panel). Solid lines are fits to the standard expressions for the relaxing complex susceptibility (Ref. 9) [Eq. (3)] with the addition of a prefactor in the formula for ${\chi }^{\prime \prime }$ to account for its asymmetry. Dynamic hysteresis and dissipation scans were measured in the tail area characterized by ${\chi }^{\prime \prime }>{\chi }^{\prime }$. The inset shows the same data from the main panel but plotted in the ${\chi }^{\prime },{\chi }^{\prime \prime }$ diagram; the data are now collapsed on the single Cole-Cole semicircle.

Figure 3

Temperature dependence of the mean relaxation time (main panel) obeys the Arrhenius-type thermal activation with approximately temperature-independent activation energy $E_a$ (inset). $E_a$ is unusually big due to strong magnetocrystalline anisotropy $K$ resulting from the strong single-ion anisotropy energy.

Figure 4

Plot of the Arrhenius thermal activation $\tau (T,E_a,{\tau }_0)={\tau }_0\text{exp}(E_a/k_{B}T)$ at fixed temperature $T=26$ K in its dependence on the attempt time ${\tau }_0$ and the activation energy $E_a$. Thick black curves define approximately the experimental window involved with the present ac susceptibility study. Intrinsic parameters of $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ (values of $E_a$ and ${\tau }_0$) at $T=26$ K impose magnetic dynamics to take place in the position of the black dot, i.e., by coincidence, within the experimental window of readily observable area on the ln $\tau ({\tau }_0,E_a)$ surface. By further cooling, magnetic dynamics slows down [the ln($\tau$) surface rapidly elevates] and the system’s dynamics status (black dot) rapidly escapes from the experimental window area.

Figure 5

Dissipation scans (see text), shown at different temperatures, measure the rate of spin reversals as the ramping dc field progressively ejects, and then nucleates back, the domain walls. Field sequence is indicated by horizontal arrows in (a). The points ascribed to the domain walls’ nucleation ($N$) and propagation ($P$) are indicated by the inclined arrows. A smooth background dissipation is attributed to incomplete spin reversal. The virgin curve dissipations are shown in (d).

Figure 6

An example of dynamic hysteresis measurement on $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ in the time domain. Magnetic field pulse $H_{\text{ac}}\left(t\right)$ and the induced voltage $V_{\text{ind}}\left(t\right)$ (circles) are measured and recorded directly while magnetization $\left[M\right(t)$ (triangles)] is obtained by numerical integration of $V_{\text{ind}}\left(t\right)$. The phase shift between $H_{\text{ac}}\left(t\right)$ and $M\left(t\right)$ brings a direct information on the involved magnetic dynamics (see text). All quantities are plotted scaled by their maximal values.

Figure 7

Experimental, induction-type dynamic hysteresis on a 2.9-mg single crystal taken with sinusoidal magnetic field pulses varying in amplitudes (5–100 Oe RMS). Calibration to absolute units was performed by the use of dc susceptibility results. The pulse widths were $t=34$ ms and $t=0.59$ s, shown in panels (a), (c), (d), and (b), respectively. At higher frequencies (29 Hz), the loops (generic ellipses) are characteristically horizontally aligned [panel (c)], reflecting fulfillment of the condition ${\chi }^{\prime \prime }\gg {\chi }^{\prime }$ (see text). At lower temperatures, the domain-wall propagation does not reach the sample boundaries within the available field amplitudes. All aspects of magnetic dynamics are revealed in the set of 29-Hz hysteresis loops at different temperatures, shown in (d), including the collapse of order parameter for $T$ passing through $T_c$. The loops are also characterized by small asymmetry, commented in the text.

Figure 8

Coercive field factor $h(\omega ,\tau )\equiv \omega \tau /\sqrt{1+{\omega }^2{\tau }^2}$ for three representative values of $\tau$, effective in thermal activation of $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ at approximately 24, 26.5, and 27 K, have been selected for the plots. Inset: Magnetization vs applied field from Eq. (1), normalized to respective amplitude values $H_0,M_0$, for several chosen values of the phase-shift parameter $\phi$ (numbers adjacent to each loop).

Figure 9

Temperature dependence of $H_c$ as determined from the dynamical hysteresis curves collected during a slow temperature drift [such as those shown in Fig. 7d]. Some sample dependence has been observed in the $H_c\left(T\right)$ measured on different samples. The solid line is a guide to the eye.

Figure 10

Dynamic hysteresis on $m=2.4$ mg $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ sample at 26.5 K in sequence of applied field amplitudes $H_0$(10–200 Oe, 10 Oe steps), at the two measuring frequencies 10 and 60 Hz. To save space, the sets of hysteresis loops taken at a number of other frequencies are not shown.

Figure 11

Dependence of $H_c$ on frequency $f$ at fixed $H_0$. The experimental $H_c$ points (circles) are derived from measurements partially shown in Fig. 10 (for two frequencies only). Thin solid lines represent the predictions of the model, Eq. (4), treating $\tau$ as a fit variable depending on $H_0$ only. Thick solid lines represent fits to Eq. (4) treating $\tau$ as a function of $H_0$ and $f$ in the form of a simple product $\tau (f,H_0)\equiv {\tau }_f\left(f\right){\tau }_h\left(H_0\right)$. Domain-wall pinning takes an important role in the triangle area.

Figure 12

Plots of the single-variable functions ${\tau }_f\left(f\right)$ and ${\tau }_h\left(H_0\right)$ used in heuristic expression for $\tau$, $\tau (f,H_0)={\tau }_f\left(f\right){\tau }_h\left(H_0\right)$. These functions are used to interpret the experimental $H_c\left(f\right)$ and $H_c\left(H_0\right)$ points (Figs. 11 and 13, respectively).

Figure 13

Coercive field $H_c$ as a function of applied field amplitudes $H_0$ for three measuring frequencies. The plots are derived from measurements shown in Fig. 10. Solid lines represent the $H_c$ values calculated from Eq. (4) using the same $\tau (f,H_0)$ functional dependence involved in the thick-line fit specified in Fig. 11, with the parameters listed in Fig. 12.

Figure 14

The dc SQUID hysteresis of $\text{Co}{}_7\text{(Te}\text{O}{}_3\text{)}{}_4\text{Br}{}_6$ at 20 K. Field cycle time took approximately 20 hours.