Memory and aging effect in hierarchical spin orderings of the stage-2 $\mathrm{Co}{\mathrm{Cl}}_2$ graphite intercalation compound

Stage-2 $\mathrm{Co}{\mathrm{Cl}}_2$ graphite intercalation compound undergoes two magnetic phase transitions at $T_{cl}$ $(=7.0\mathrm{K})$ and $T_{cu}$ $(=8.9\mathrm{K})$. The aging dynamics of this compound is studied near $T_{cl}$ and $T_{cu}$. The intermediate state between $T_{cl}$ and $T_{cu}$ is characterized by a spin-glass phase extending over ferromagnetic islands. A genuine thermoremnant magnetization (TRM) measurement indicates that the memory of the specific spin configurations imprinted at temperatures between $T_{cl}$ and $T_{cu}$ during the field-cooled (FC) aging protocol can be recalled when the system is reheated at a constant heating rate. The zero-field cooled (ZFC) and TRM magnetization is examined in a series of heating and reheating processes. The magnetization shows both characteristic memory and rejuvenation effects. The time $\left(t\right)$ dependence of the relaxation rate $S_{\mathrm{ZFC}}\left(t\right)=(1∕H)\mathrm{d}{M}_{\mathrm{ZFC}}\left(t\right)∕\mathrm{d}\ln t$ after the ZFC aging protocol with a wait time $t_w$ exhibits peaks at two different characteristic times in the intermediate state between $T_{cl}$ and $T_{cu}$. This result indicates the coexistence of two types of ordered domains. The observed aging and memory effects are discussed in terms of the droplet model.

Figure 1

(Color online) (a) $T$ dependence of $M_{\mathrm{ZFC}}$, $M_{\mathrm{FC}}$, $M_{\mathrm{TRM}}$, and $\Delta M$ $(=M_{\mathrm{FC}}-M_{\mathrm{ZFC}})$ for stage-2 $\mathrm{Co}{\mathrm{Cl}}_2$ GIC. $H=1\mathrm{Oe}$. (b) $T$ dependence of $M_{\mathrm{TRM}}$ at $H_c=1.0$ and $0.15\mathrm{Oe}$ and $\Delta M$ at $H=H_c=1$ and $0.15\mathrm{Oe}$. $T_{cu}=8.9\mathrm{K}$.

Figure 2

(Color online) $T$ dependence of $M_{\mathrm{TRM}}(T;T_s,t_s)$ (closed circles) and of $M_{\mathrm{TRM}}^{\mathrm{ref}}\left(T\right)$ (open circles). $M_{\mathrm{TRM}}(T;T_s,t_s)$ is measured with increasing $T$ at $H=0$ after the FC aging protocol at $H_c=1\mathrm{Oe}$ with a stop-wait at $T_s$ for $t_s=1.0\times {10}^4\mathrm{s}$. $M_{\mathrm{TRM}}^{\mathrm{ref}}$ is measured with increasing $T$ after the FC aging protocol at $H_c=1\mathrm{Oe}$ without such a stop-wait protocol. (a) $T_s=6.6\mathrm{K}$, (b) $7.2\mathrm{K}$, (c) $8.0\mathrm{K}$, and (d) $8.5\mathrm{K}$.

Figure 3

(Color online) $T$ dependence of $\Delta M_{\mathrm{TRM}}(T;T_s,t_s)$ $[=M_{\mathrm{TRM}}(T;T_s,t_s)-M_{\mathrm{TRM}}^{\mathrm{ref}}\left(T\right)]$. $t_s=1.0\times {10}^4\mathrm{s}$. $6.4⩽T_s⩽9.1\mathrm{K}$. $H_c=1\mathrm{Oe}$. (a) $6.6⩽T⩽7.4\mathrm{K}$. (b) $7.6⩽T⩽8.3\mathrm{K}$. (c) $8.5⩽T⩽9.1\mathrm{K}$.

Figure 4

(Color online) $T_s$ dependence of the peak height of $\Delta M_{\mathrm{TRM}}(T;T_s,t_s)$ denoted by ${\left(\Delta M_{\mathrm{TRM}}\right)}_{\max }$. The curve of $\Delta M_{\mathrm{TRM}}$ vs $T$ shows a peak at $T_s$. $T_s$ dependence of the full-width at half-maximum $(=\Delta T_{\mathrm{FWHM}})$ for the peak of the curve of $\Delta M_{\mathrm{TRM}}$ vs $T$ located at $T=T_s$.

Figure 5

(Color online) $t$ dependence of $S_{\mathrm{ZFC}}\left(t\right)$ at various $t_w$ ($t_{w1}=2.0\times {10}^3\mathrm{s}$, $t_{w2}=1.0\times {10}^4\mathrm{s}$, and $t_{w3}=3.0\times {10}^4\mathrm{s}$). $H=1\mathrm{Oe}$. (a) $T=7.0\mathrm{K}$, (b) $7.5\mathrm{K}$, (c) $8.0\mathrm{K}$, and (d) $8.5\mathrm{K}$. The lines denote the least-squares fits to a SER given by Eq. (2) for $S_{\mathrm{ZFC}}$ vs $t$ in the vicinity of the peak time $t_{cr}$. The fitting parameters $\tau$ and $n$ for each $T$ are shown in Figs. 6a, 6b. The inset of (a) shows the relation between $t_{\mathrm{cr}}$ and $t_w$.

Figure 6

(Color online) $T$ dependence of the fitting parameters $\tau$ and $n$. $H=1\mathrm{Oe}$. (a) $\tau$ vs $T$ and (b) $n$ vs $T$ at various $t_w$ [$t_w=t_{w1}$ (▾), $t_{w2}$ (◻), $t_{w3}$ (●, ▴, 엯, ▵)]. For $t_w=t_{w3}$, two SER peaks are observed at $t<3.0\times {10}^3\mathrm{s}$ (▴, ▵) and $t>1.0\times {10}^4\mathrm{s}$ (●, 엯) above $8.0\mathrm{K}$. $\tau$ is equal to the peak time $t_{cr}$ of the curve $S_{\mathrm{ZFC}}\left(t\right)$ vs $t$ at the fixed $T$. In order to check the reproducibility of the data at $t_w=t_{w3}$, the measurements were carried out twice independently: $n$ and $\tau$ (엯, ▵) for the first measurement and $\tau$ (●, ▴) for the second measurement. The data of Figs. 5a, 5b correspond to those of the second measurement. The arrows indicate the location of $T_{cl}$ and $T_{cu}$.

Figure 7

(Color online) (a) TRM case. Repeated processes of heating from $T_i=3.0\mathrm{K}$ to $T_r$ (U-turn temperature), cooling from $T_r$ to $T_i$, and reheating from $T_i$ K to $T_{r+1}$ $(>T_r)$ in the absence of $H$, after the FC aging protocol (quenching from $50\mathrm{K}$ to $T_i$ at $H=H_c=1\mathrm{Oe}$ and annealing at $T_i$ for $1.0\times {10}^2\mathrm{s}$). (b) ZFC case. Repeated processes of heating from $T_i=3.0\mathrm{K}$ to $T_r$ (U-turn temperature), cooling from $T_r$ to $T_i$, and reheating from $T_i$ to $T_{r+1}$ $(>T_r)$ in the presence of $H$ $(=1\mathrm{Oe})$, after the ZFC aging protocol (quenching from $50\mathrm{K}$ to $T_i$ at $H=0$ and aging at $T_i$ for $1.0\times {10}^2\mathrm{s}$).

Figure 8

(Color online) (a) and (b) $T$ dependence of $M_{\mathrm{TRM}}$ measured at $H=0$ in a series of heating (closed circles) and cooling (open circles) processes [see Fig. 7a and text for detail] after the FC aging protocol of the system from 50 to $T_i=3.0\mathrm{K}$ in the presence of $H_c$ $(=1\mathrm{Oe})$. Note that the data of $M_{\mathrm{TRM}}$ shown here are corrected one by the subtraction of the data of $M_{\mathrm{TRM}}$ vs $T$ obtained with decreasing $T$ from $T_f$ $(=12\mathrm{K})$ to $2.0\mathrm{K}$, which may be affected by the remanent magnetic field $(\approx 3\mathrm{m}\mathrm{Oe})$.

Figure 9

(Color online) $T$ dependence of $M_{\mathrm{TRM}}$ at $H=0$ which is measured with decreasing $T$ from $T_r$ to $T_i=3.0\mathrm{K}$ [$M_{\mathrm{TRM}}(T\downarrow )$, open circles] and subsequently measured with increasing $T$ from $T_i$ to $T_r$ [$M_{\mathrm{TRM}}(T\uparrow )$, closed circles]. (a) $T_r=7.0\mathrm{K}$, (b) $7.4\mathrm{K}$, (c) $7.8\mathrm{K}$, (d) $8.2\mathrm{K}$, (e) $8.6\mathrm{K}$, and (f) $8.9\mathrm{K}$.

Figure 10

(Color online) (a) $T_r$ dependence of $M_r$ and $M_i$. $M_r$ and $M_i$ are the value of $M_{\mathrm{TRM}}$ at $T=T_r$ and $T_i$ $(=3.0\mathrm{K})$, respectively, which are obtained in the measurement of $M_{\mathrm{TRM}}$ with decreasing $T$ from $T_r$ to $T_i$ [see Fig. 8a]. (b) $T_r$ dependence of $\mathrm{d}{M}_i∕\mathrm{d}{T}_r$. (c) $T_r$ dependence of $\Delta M_{ir}^{\mathrm{TRM}}$ $(=M_i-M_r)$.

Figure 11

(Color online) (a) $T$ dependence of $\Delta M_{\mathrm{TRM}}$ $[=M_{\mathrm{TRM}}(T\uparrow )-M_{\mathrm{TRM}}(T\downarrow )]$ for $6.0\mathrm{K}⩽T⩽T_r$. $T_r=7.8$, 8.2, 8.6, and $8.9\mathrm{K}$. (b) $T$ dependence of $\Delta M_{\mathrm{ZFC}}$ $[=M_{\mathrm{ZFC}}(T\uparrow )-M_{\mathrm{ZFC}}(T\downarrow )]$ for $6.0\mathrm{K}⩽T⩽T_r$. $T_r=7.8$, 8.2, 8.6, and $9.0\mathrm{K}$.

Figure 12

(Color online) (a) and (b) $T$ dependence of $M_{\mathrm{ZFC}}$ measured at $H=1\mathrm{Oe}$ in a series of heating (closed circles) and cooling (open circles) processes [see Fig. 7b and the text for detail], after the ZFC cooling of the system from $50\text{to}3.0\mathrm{K}$ in the absence of $H$.

Figure 13

(Color online) $T$ dependence of $M_{\mathrm{ZFC}}$ at $H=1\mathrm{Oe}$ measured with decreasing $T$ from $T_r$ to $T_i$ $(=3.0\mathrm{K})$ [$M_{\mathrm{ZFC}}(T\downarrow )$, open circles] and subsequently measured with increasing $T$ from $T_i$ to $T_r$ [$M_{\mathrm{ZFC}}(T\uparrow )$, closed circles]. (a) $T_r=7.8$, (b) 8.2, (c) 8.6, and (d) $8.9\mathrm{K}$.

Figure 14

(Color online) (a) $T_r$ dependence of $M_r$ and $M_i$. $M_r$ and $M_i$ are the value of $M_{\mathrm{ZFC}}$ at $T=T_r$ and $T_i$ $(=3.0\mathrm{K})$, respectively, which are obtained in the measurement of $M_{\mathrm{ZFC}}$ with decreasing $T$ from $T_r$ to $T_i$ [see Fig. 12a]. (b) $T_r$ dependence of $\mathrm{d}{M}_i∕\mathrm{d}{T}_r$. (c) $T_r$ dependence of $\Delta M_{ir}^{\mathrm{ZFC}}$ $(=M_i-M_r)$.

Figure 15

(Color online) (a) and (b) $T$ dependence of $M_{\mathrm{FC}}^{\mathrm{IS}}(T\downarrow )$ (엯) and $M_{\mathrm{FC}}^{\mathrm{IS}}(T\uparrow )$ (●) observed in the following FC aging protocol. The system was quenched from $50\text{to}12\mathrm{K}$ in the presence of $H$$(=1\mathrm{Oe})$. $M_{\mathrm{FC}}^{\mathrm{IS}}(T\downarrow )$ was measured with decreasing $T$ from $12\text{to}1.9\mathrm{K}$ but with intermittent stops at $T=8.5$, 6.5, and $4.5\mathrm{K}$ for a wait time $t_w=3.0\times {10}^4\mathrm{s}$. The field is cut off during each stop. $M_{\mathrm{FC}}^{\mathrm{IS}}(T\uparrow )$ was measured at $H=1\mathrm{Oe}$ with increasing $T$ after the above cooling process. The $T$ dependence of $M_{\mathrm{FC}}^{\mathrm{ref}}$ (▴) and $M_{\mathrm{TRM}}^{\mathrm{ref}}$ (▵) are also shown as reference curves. These are measured after the FC aging protocol without intermittent stop (reference curves). (c) $T$ dependence of the difference $\Delta M_{\mathrm{FC}}^{\mathrm{IS}}$ $[=M_{\mathrm{FC}}^{\mathrm{IS}}(T\downarrow )-M_{\mathrm{FC}}^{\mathrm{IS}}(T\uparrow )]$.