Time Dependence of the freezing temperature for thin film spin glasses

There have been many measurements of the dependence of the ``freezing temperature", $T_f$, on the thickness $\mathcal {L}$ of thin film spin glasses. $T_f$ decreases with decreasing $\mathcal {L}$, but never vanishes. This contribution suggests that the dependence of $T_f$ on $\mathcal {L}$ is a time dependent relationship. Because the lower critical dimension of a spin glass, ${d_\ell}\approx 2.5$, when the spin glass correlation length $\xi(t, T)$ grows to $\mathcal {L}$, the spin glass dimensionality crosses over from $d = 3$ to $d = 2$. What remains are spin glass correlations for length scales $\leq \mathcal {L}$. The time dependence of the magnetization dynamics are then activated, with activation energy equal to a largest barrier ${\Delta_{\text {max}}}({\mathcal {L}})$, and an associated activation time $\tau$. For measurements at time scales such that $\xi(t, T) < \mathcal {L}$, the effective dimension $d = 3$, and the characteristic cusp and knee of a spin glass is observed. For experimental time scales greater than $\tau$, with $\xi(t, T) \approx \mathcal {L}$, the zero-field cooled magnetization has grown to the field-cooled value of the magnetization, leading to the identification of $T_f$. Quantitative agreement with experiment is exhibited.