Realization of random-field Ising ferromagnetism in a molecular magnet

The longitudinal magnetic susceptibility of single crystals of the molecular magnet ${\text{Mn}}_{12}$-acetate obeys a Curie-Weiss law, indicating a transition to a ferromagnetic phase at $\sim 0.9\text{K}$. With increasing magnetic field applied transverse to the easy axis, a marked change is observed in the temperature dependence of the susceptibility, and the suppression of ferromagnetism is considerably more rapid than predicted by mean-field theory for an ordered single crystal. Our results can instead be fit by a Hamiltonian for a random-field Ising ferromagnet in a transverse magnetic field, where the randomness derives from the intrinsic distribution of locally tilted magnetic easy axes known to exist in ${\text{Mn}}_{12}$-acetate crystals, suggesting that ${\text{Mn}}_{12}$-acetate is a realization of the random-field Ising model in which the random field may be tuned by a field applied transverse to the easy axis.

Figure 1

(Color online) Longitudinal magnetization as a function of the longitudinal-field swept at $6.7\times {10}^{-3}\text{T}/\text{s}$ for the indicated transverse magnetic fields at $T=2.0\text{K}$. Inset: schematic of the experiment setup, showing the relative positions of sample, Hall sensor and applied magnetic field.

Figure 2

(Color online) Determination of the blocking temperature. Insets: Magnetization as a function of the longitudinal-field swept at the indicated rates for $H_{\perp }=2\text{T}$ at $T=2.15\text{K}$ and $T=2.40\text{K}$. Main panel: Blocking temperatures for three longitudinal-field sweep rates as a function of $H_{\perp }$.

Figure 3

(Color online) Temperature and field dependence of the inverse susceptibility of ${\text{Mn}}_{12}$-ac. Main panel: filled symbols denote the inverse susceptibility of a single crystal of ${\text{Mn}}_{12}$-ac as a function of temperature in different transverse fields $H_{\perp }$ as labeled. The solid lines are the result of mean-field calculations for a hypothetical system with no tilts. The dashed lines are obtained from mean-field calculations incorporating the effects of random tilt angles, as discussed in Sec. 4B of the main text with root mean square tilt angle $1.2°$. The dotted lines present theoretical results for a different distribution with the same mean square tilt angle but in which 25% of the sites are not tilted. Inset: Symbols represent the difference $\left[{\chi }^{-1}\left(H_{\perp }\right)-{\chi }^{-1}\left(H_{\perp }=0\right)\right]$ versus $H_{\perp }^2$. The dashed, dotted and dash-dotted lines are calculated using the model of Sec. 4B but with different root mean square tilt angles as indicated. The solid line displays results for the pure case.

Figure 4

(Color online) Effect of easy axis tilts on the transition temperature. In zero field a perfectly ordered crystal and a crystal in which there are easy axis tilts (e.g., the double-line red spins) will order at nearly the same temperature (the small tilts do not greatly modify the interaction between spins, which depends on the longitudinal component of the magnetic moment). In an applied transverse field, the spins of misaligned molecules experience a field along their Ising axis. When this field is comparable to the exchange field these spins are frozen (double-line red spins) and do not order. This leads to an effective dilution of the spins, a decrease in the susceptibility and a reduction in the transition temperature. It also increases the random field on the other sites in the crystal.

Figure 5

(Color online) The Curie-Weiss and the ferromagnetic transition temperatures as a function of transverse field. The intercepts $T_{CW}$ (squares) are obtained from the straight-line portion of the data curves in Fig. 3. The dotted and dash-dotted lines are mean-field $T_{CW}$ results for the pure and random case, respectively. The light and heavy solid lines are mean-field transition temperatures, $T_C$, calculated for the pure and random case, respectively. In the random case at higher field, the detailed structure depends on the specific details of the distribution of random fields. Here we plot $T_C$ using the distribution obtained by Park[19] where the parameters ($\theta =1.2°$ and $\varphi =0$) are the same as those used to fit the data in Fig. 3, and residual randomness is introduced as described in the text. For this assumed distribution and these parameters, $T_C$ drops discontinuously to zero at 4.5 T.