Phonon bottleneck effect leads to observation of quantum tunneling of the magnetization and butterfly hysteresis loops in ${\left({\mathrm{Et}}_4\mathrm{N}\right)}_3{\mathrm{Fe}}_2{\mathrm{F}}_9$

A detailed investigation of the unusual dynamics of the magnetization of ${\left({\mathrm{Et}}_4\mathrm{N}\right)}_3{\mathrm{Fe}}_2{\mathrm{F}}_9$ $\left({\mathrm{Fe}}_2\right)$, containing isolated ${\left[{\mathrm{Fe}}_2{\mathrm{F}}_9\right]}^{3-}$ dimers, is presented and discussed. ${\mathrm{Fe}}_2$ possesses an $S=5$ ground state with an energy barrier of $2.40\mathrm{K}$ due to an axial anisotropy. Poor thermal contact between sample and bath leads to a phonon bottleneck situation, giving rise to butterfly-shaped hysteresis loops below $5\mathrm{K}$ concomitant with slow decay of the magnetization for magnetic fields $H_z$ applied along the $\mathrm{Fe}\mathrm{Fe}$ axis. The butterfly curves are reproduced using a microscopic model based on the interaction of the spins with resonant phonons. The phonon bottleneck allows for the observation of resonant quantum tunneling of the magnetization at $1.8\mathrm{K}$, far above the blocking temperature for spin-phonon relaxation. The latter relaxation is probed by ac magnetic susceptibility experiments at various temperatures and bias fields $H_{\mathrm{DC}}$. At $H_{\mathrm{DC}}=0$, no out-of-phase signal is detected, indicating that at $T⩾1.8\mathrm{K}$ ${\mathrm{Fe}}_2$ does not behave as a single-molecule magnet. At $H_{\mathrm{DC}}=1\mathrm{kG}$, relaxation is observed, occurring over the barrier of the thermally accessible $S=4$ first excited state that forms a combined system with the $S=5$ state.

Figure 1

Double-well potential of the $S=5$ ground state of ${\mathrm{Fe}}_2$ with the energy barrier $\Delta =\mid D_S{S}^2\mid =2.40\mathrm{K}$. Full and broken arrows indicate spin-phonon and QTM transitions, respectively.

Figure 2

Dependence of the relaxation time ${\tau }_2$ on $H_z$ at $T=1.8\mathrm{K}$ with error $\left(3\sigma \right)$. The dotted line is a guide to the eyes.

Figure 3

Hysteresis of the magnetization of ${\mathrm{Fe}}_2$ single crystals (open circles: sample $\mathbf{A}$; asterisks: sample $\mathbf{B}$) at $1.8\mathrm{K}$ obtained with $\Gamma \approx 3\mathrm{G}∕\mathrm{s}$. The arrows indicate the direction of the measurement. The solid lines are calculated using the resonant-phonon model (Sec. 4C). The dashed line represents the thermal equilibrium curve calculated with $J=-2.23\mathrm{K}$, $D=-0.215\mathrm{K}$, and $g=2.00$ (Ref. 14). The inset highlights the region close to $H_z=0$.

Figure 4

Positive wings of the butterfly hysteresis of the magnetization of a single crystal (sample $\mathbf{A}$) for $\mathbf{H}\parallel c$ orientation at different temperatures as indicated with $\Gamma \approx 3\mathrm{G}∕\mathrm{s}$. The solid lines are calculated using the resonant phonon model (Sec. 4C).

Figure 5

Magnetization data (sample $\mathbf{A}$) at $1.8\mathrm{K}$ obtained as follows: after saturation at $+30\mathrm{kG}$ the field was quickly $(\Gamma \approx 150\mathrm{G}∕\mathrm{s})$ reduced to the given $H_z$ values including negative ones, and the magnetization was measured within $50\mathrm{s}$ after reaching $H_z$. The solid line is the adiabatic curve calculated using the resonant phonon model described in the text with $H_{\mathrm{eff},x}=350\mathrm{G}$, while the dashed line represents the thermal equilibrium curve.

Figure 6

Out-of-phase ac susceptibility ${\chi }^{″}$ vs $T$ at $H_{\mathrm{DC}}=1\mathrm{kG}$ (top) and $5\mathrm{kG}$ (bottom) for a polycrystalline sample $\left(\mathbf{C}\right)$ of ${\mathrm{Fe}}_2$ in a $1\mathrm{G}$ ac field oscillating with the indicated frequencies $\nu$.

Figure 7

Temperature dependence of the ac ${\chi }^{″}$ relaxation rate $1∕\tau =2\pi \nu$ shown as an Arrhenius plot ($\ln 1∕\tau$ vs $1∕T_{\max }$) as extracted from the data in Fig. 6. The solid line represents a least squares fit of the $H_{\mathrm{DC}}=1\mathrm{kG}$ data to the Arrhenius law [Eq. (1)], yielding the prefactor $1∕{\tau }_0$ and the kinetic energy barrier $\Delta E$ as indicated. The inset shows the same data on a linear scale. Note the linear dependence of $1∕\tau$ on $T_{\max }$ for the $H_{\mathrm{DC}}=5\mathrm{kG}$ data.

Figure 8

Out-of-phase ac susceptibility $\chi ″$ vs $T$ for various values of $H_{\mathrm{DC}}\text{for}$ a polycrystalline sample $\mathbf{C}$ of ${\mathrm{Fe}}_2$ in a $1\mathrm{G}$ ac field oscillating at $997\mathrm{Hz}$. The inset shows the field dependences of the integrated ${\chi }^{″}$ intensity and of $T_{\max }$.

Figure 9

Dependence of the 11 energy levels $\mid m⟩$ defined by ${\mathcal{H}}_{\mathrm{eff}}^{\mathrm{grot}}$ [Eq. (10)] on $H_z$, arising from the resonant phonons ($H_{\mathrm{eff},x}=350\mathrm{G}$). The inset highlights the anticrossing at $H_z=0$.

Figure 10

Double-well potential for a bias field of $H_{\mathrm{DC}}=1\mathrm{kG}$. The arrows indicate individual $\Delta S=\Delta M\pm 1$ spin-phonon transitions within the combined $S=5$, $S=4$ spin system and illustrate a likely scenario for spin-phonon relaxation.

Figure 11

Argand or Cole-Cole plots of ${\chi }^{″}$ vs ${\chi }^{\prime }$ at $H_{\mathrm{DC}}=1$ (a) and $5\mathrm{kG}$ (b) at fixed temperatures (∗, $3.0\mathrm{K}$; ●, $4.0\mathrm{K}$; 엯, $5.0\mathrm{K}$). For clarity the data are shifted along the $x$ axis. For a single relaxation process the data points should lie on a semicircle (solid lines) at each temperature.