Spin glass dynamics at the mesoscale

The mesoscale allows a new probe of spin glass dynamics. Because the spin glass lower critical dimension $d_l>2$, the growth of the correlation length $\xi (t,T)$ can change the nature of the spin glass state at a crossover time $t_{\text{co}}$ when $\xi (t_{\text{co}},T)=\ell$, a minimum characteristic sample length (e.g., film thickness for thin films and crystallite size for bulk samples). For thin films, and times $t<t_{\text{co}}$ such that $\xi (t,T)<\ell$, conventional three-dimensional dynamics is observed. When $t>t_{\text{co}}$, a crossover to $d=2$ behavior takes place. The parallel correlation length, associated with a $T_g=0$ transition, increases in time from the saturated value of the perpendicular correlation length $\ell$ to an equilibrium value of the parallel correlation length proportional to $T^{-\nu }$. This results in a pancakelike correlated state, with a thickness $\ell$ and a temperature-dependent in-plane radius that increases with decreasing temperature. Activated dynamics is associated with these states. Measurements on Cu:Mn thin films are analyzed quantitatively within this framework. We extract a temperature-dependent activation energy from a fit to the frequency dependence of the dynamic susceptibility. The extrapolated temperature at which the activation energy would become large is close to the extrapolated glass transition temperature from ac susceptibility measurements. All known relevant experimental data are consistent with this approach. For polycrystalline materials, there is a distribution of length scales $\mathcal{P}\left(\ell \right)$. For sufficiently broad distributions, a logarithmic time dependence is derived for the time decay of the thermoremanent magnetization $M_{\text{TRM}}(t,T)$ using an approach originally derived by Ma. Properties dependent upon an effective waiting time $t_w^{\mathrm{eff}}$ are derived that are consistent with experiment, and further measurements are suggested.

Figure 1

The temperatures for which $\chi \left(\tau \right)$ peaks are extracted from the plots of $\chi \left(\tau \right)$ versus temperature at different observation times $\tau$, where $\tau ={\omega }^{-1}$, from Figs. 1(a) in Refs. [10, 11], for Cu:Mn (13.5 at.%) films of 30 and 20 Å, respectively. The points at the higher peak temperatures (shorter $\tau$) are less certain because of the broader experimental curves in this time regime.

Figure 2

The temperature-dependent activation energy of the highest barrier ${\Delta }_{\text{max}}\left(T\right)$ from Eq. (9) for the 30 Å Cu:Mn (13.5 at.%) film, plotted against the peak temperature for $\chi \left(\tau \right)$, using the data in Fig. 1. To guide the eye, a best fit to a linear dependence is plotted on the same graph. It is seen that there is a significant departure from a linear relationship between the values of ${\Delta }_{\text{max}}\left(T\right)$ and the peak temperatures for $\chi \left(\tau \right)$ at the larger values of ${\Delta }_{\text{max}}\left(T\right)$ (lower peak temperatures).

Figure 3

The temperature-dependent activation energy of the highest barrier ${\Delta }_{\text{max}}\left(T\right)$ from Eq. (9) for the 20 Å Cu:Mn (13.5 at.%) film, plotted against the peak temperature for $\chi \left(\tau \right)$, using the data in Fig. 1. To guide the eye, a best fit to a linear dependence is plotted on the same graph. Some curvature could be imputed to the experimental points, but the data are too uncertain to claim anything else than a linear relationship between the values of ${\Delta }_{\text{max}}\left(T\right)$ and the peak temperatures for $\chi \left(\tau \right)$.

Figure 4

Plots of $\delta {\Delta }_{\text{max}}\left(T\right)/\delta T$ against ${\Delta }_{\text{max}}\left(T\right)$ extracted (a) from the data points for the 30 Å film from Fig. 2 and (b) from the data points for the 20 Å film from Fig. 3. The solid line for the 30 Å film in Fig. 4(a) is a fit of the 30 Å data to an exponential dependence of $\partial {\Delta }_{\text{max}}\left(T\right)/\partial T$ on ${\Delta }_{\text{max}}\left(T\right)$ [Eq. (10)] following the reasoning of Ref. [20]. The scatter for the 20 Å film in Fig. 4(b) obviates any fit.

Figure 5

Integration of the exponential form for the dependence of $\partial {\Delta }_{\text{max}}\left(T\right)/\partial T$ on ${\Delta }_{\text{max}}\left(T\right)$ for 30 Å film, fitted to the experimental data (solid line portion), as a function of $(T-T^{*})/T_g$, over the full range of ${\Delta }_{\text{max}}\left(T\right)/\left(k_B{T}_g\right)$ and $(T-T^{*})/T_g$ using Eq. (11). The $\times$ denotes the value of ${\Delta }_{\text{max}}$ at quench from Eq. (4).

Figure 6

Reproduction of Fig. 1 from Ref. [27]. Figure 6(a) shows the decay curves for $t_w^{\mathrm{eff}}$ equal to 7, 17.5, 25, 27, 35, 44, and 110 s. In the inset, the relaxation curves $S\left(t\right)$ for each curve are used to determine $t_w^{\mathrm{eff}}$. Figure 6(b) displays the subtraction of the $t_w^{\mathrm{eff}}=7$ s TRM decay curve from the other TRM curves. The $M_{\text{TRM}}\left(t\right)$ curve for $t_w^{\mathrm{eff}}=17.5$ s begins to overlap the $M_{\text{TRM}}\left(t\right)$ curve for $t_w^{\mathrm{eff}}=7$ s at a time $t_{\text{ov}}\approx 6000$ s, while the curve for $t_w^{\mathrm{eff}}=44$ s overlaps at a $t_{\text{ov}}\sim {10}^5$ s, and by extrapolation the curve for $t_w^{\mathrm{eff}}=110$ s overlaps at a time $t_{\text{ov}}\sim {10}^9$ s. The inset displays the relaxation curve $S\left(t\right)$ for the $t_w^{\mathrm{eff}}=7$ s curve. The curve becomes horizontal at $t\sim {10}^3$ s, remaining flat over two orders of magnitude in time, indicating a crossover to a logarithmic time dependence for the decay of $M_{\text{TRM}}\left(t\right)$.

Figure 7

A logarithmic plot of the time for the $M_{\text{TRM}}\left(t\right)$ curves to overlap the $M_{\text{TRM}}\left(t\right)$ curve for $t_w^{\mathrm{eff}}=7$ s, as a function of their respective $t_w^{\mathrm{eff}}$.