Aging dynamics across a dynamic crossover line in the three-dimensional short-range Ising spin-glass ${\mathrm{Cu}}_{0.5}{\mathrm{Co}}_{0.5}{\mathrm{Cl}}_2-\mathrm{Fe}{\mathrm{Cl}}_3$ graphite bi-intercalation compound

${\mathrm{Cu}}_{0.5}{\mathrm{Co}}_{0.5}{\mathrm{Cl}}_2-\mathrm{Fe}{\mathrm{Cl}}_3$ graphite bi-intercalation compound is a three-dimensional short-range Ising spin glass with a spin freezing temperature $T_c(=3.92\pm 0.11\mathrm{K})$. The stability of the spin glass phase in the presence of a magnetic field $H$ is examined from (i) the spin freezing temperature $T_f(\omega ,H)$ at which the differential field-cooled (FC) susceptibility $\partial M_{\mathit{FC}}(T,H)∕\partial H$ coincides with the dispersion ${\chi }^{\prime }(\omega ,T,H=0)$ with the angular frequency $\omega$, and (ii) the time dependence of zero-field-cooled (ZFC) susceptibility ${\chi }_{\mathit{ZFC}}$ after a ZFC aging protocol with a wait time $t_W(=1.0\times {10}^4)$. The relaxation rate $S\left(t\right)(=d{\chi }_{\mathit{ZFC}}∕d\ln t)$ exhibits a local maximum at a characteristic time $t_{\mathit{cr}}$, reflecting nonequilibrium aging dynamics. The peak time $t_{\mathrm{cr}}(T,H)$ decreases with increasing $H$ at the fixed temperature $T(2.9\mathrm{K}⩽T<T_c)$. The spin freezing temperature $T_f(\omega ,H)$ provides evidence for the instability of the spin glass phase in thermal equilibrium in a finite magnetic field. Both $t_{\mathit{cr}}(T,H)$ and $T_f(\omega ,H)$ exhibit certain scaling behavior predicted from the droplet picture.

Figure 1

(Color online) Plot of $T_g\left(H\right)\{=T_f(\omega ,H)∕[1+{(2\pi ∕\omega {\tau }_0^{*})}^{-1∕x}]\}$ vs $H$ with $x=10.3$ and ${\tau }_0^{*}=5.27\times {10}^{-6}\sec$, where $\omega (=2\pi f)$ is the angular frequency. $T_f(\omega ,H)$ is defined as a temperature at which $\partial M_{\mathit{FC}}(T,H)∕\partial H$ crosses ${\chi }^{\prime }(\omega ,T,H=0)$ at each $f$. The local minimum temperature of $d\Delta \chi ∕dT$ vs $T$ at $H$ (◆) [denoted as the line $T_{g0}\left(H\right)$ ($d\Delta \chi ∕dT$ vs $T$)] is also shown for comparison. A solid line is a least-squares fitting curve, which is expressed by Eq. (1) with $T_c$, $p$, and $A$ given in the text.

Figure 2

(Color online) Scaling plot of $\ln \left(Y_1\right)$ vs $\ln \left(X_1\right)$ for the data with $T_f(\omega ,H)∕T_c⩽0.9$, where $Y_1={[1-T_f(\omega ,H)∕T_c]}^{a_{\mathit{eff}}}∕H$ and $X_1={(2\pi ∕\omega {\tau }_0^{*})}^{T_f(\omega ,H)∕T_c}$. The best result for the scaling plot is obtained when $a_{\mathit{eff}}=0.55$, where $T_c(=3.92\mathrm{K})$ and ${\tau }_0^{*}(=5.27\times {10}^{-6}\sec )$ are fixed. The least-squares fitting curve $(Y_1\approx {\zeta }^{1∕\delta }{X}_1^{b∕\delta })$ is denoted by a straight line with a slope of $b∕\delta =0.198$ and $\zeta =1.5\times {10}^{-4}(\delta =0.81)$.

Figure 3

(Color online) $t$ dependence of (a) ${\chi }_{\mathit{ZFC}}\left(t\right)$ and (b) $S\left(t\right)$ at various $H(20⩽H⩽300\mathrm{Oe})$. $t_W=1.0\times {10}^4\sec$. $T=3.50\mathrm{K}$.

Figure 4

(Color online) $t$ dependence of (a) ${\chi }_{\mathit{ZFC}}\left(t\right)$ and (b) $S\left(t\right)$ at various $H(150⩽H⩽400\mathrm{Oe})$. $t_W=1.0\times {10}^4\sec$. $T=3.10\mathrm{K}$.

Figure 5

(Color online) (a) $t_{\mathit{cr}}$ vs $H$ at various $T$ and (b) $t_{\mathit{cr}}$ vs $T$ at various $H$. The solid lines are guides to the eyes. $t_W=1.0\times {10}^4\sec$.

Figure 6

(Color online) (a) Scaling plot of $Y$ vs $X$. $X$ and $Y$ are defined by Eqs. (13a, 13b), respectively, where $b∕\delta =0.198$, $b=0.16$, $a_{\mathit{eff}}=0.55$, $\zeta =1.5\times {10}^{-4}$, $T_c=3.92\mathrm{K}$, and ${\tau }_0^{*}=5.27\times {10}^{-6}\sec$. $t_{\mathit{cr}}$ is a time of the peak for $S\left(t\right)$ vs $t$ with $t_W=1.0\times {10}^4\sec$ for $2.9⩽T⩽3.75\mathrm{K}$ and $5⩽H⩽500\mathrm{Oe}$. (b) The same scaling plot of $Y$ vs $X$ for $0⩽X⩽0.6$. The solid lines are denoted by Eq. (14) with $c=0$, 0.15, 0.3, and 0.71, where $\delta =0.81$. Note that the solid line with $c=0.71$ also coincides with that derived from the least-squares fit of the data to Eq. (14) with $c=0.71\pm 0.06$ and $\delta =0.81\pm 0.07$.

Figure 7

(Color online) (a) $H\text{−}T$ line (denoted by the solid lines). The value of $H$ is calculated from Eq. (15) with $t_W=1.0\times {10}^4\sec$ and $X_c$ being varied as a parameter $(0.5⩽X_c⩽1.5)$, where ${\tau }_0^{*}=5.27\times {10}^{-6}\sec$, $\zeta =1.5\times {10}^{-4}$, $\delta =0.81$, $b∕\delta =0.198$, and $a_{\mathit{eff}}=0.55$. The dotted lines denote the least-squares fitting curves of the $H$ vs $T$ relation for each $X_c$ to Eq. (1) with $p\approx 1.5$. (b) $H\text{−}T$ line (denoted by the solid lines). The value of $H$ as a function of $T$ is calculated from Eq. (15) with $X_c=0.28$ and $t_W=100\sec$, where the values of ${\tau }_0^{*}$, $\zeta$, $\delta$, $b∕\delta$, and $a_{\mathit{eff}}$ are the same as those for (a). The line $T_{g0}\left(H\right)$ ($d\Delta \chi ∕dT$ vs $T$) (●) of Fig. 1 is also shown for comparison.