Switching Molecular Conformation with the Torque on a Single Magnetic Moment

For the endohedral fullerene molecule ${\mathrm{HoLu}}_2\mathrm{N}@{\mathrm{C}}_{80}$, it is shown that the endohedral ${\mathrm{HoLu}}_2\mathrm{N}$ unit may be oriented in a magnetic field. The Ho magnetic moment is fixed in the strong ligand field and aligns along the holmium-nitrogen axis. The torque of a magnetic field on the Ho magnetic moment leads to a hopping bias of the endohedral unit inclining to an orientation parallel to the externally applied field. This endohedral cluster distribution remains frozen below the onset of thermally induced rotation of the endohedral units. We derive an analytical statistical model for the description of the effect that scales below 7 T with the square of the external field strength, and that allows us to resolve the freezing temperature of the endohedral hopping motion. The freezing temperature is around 55 K and depends on the cooling rate, which in turn determines an activation energy for the hopping motion of 185 meV and a prefactor of $1.8\times {10}^{14}{\mathrm{s}}^{-1}$. For ${\mathrm{TbSc}}_2\mathrm{N}@{\mathrm{C}}_{80}$ we find the same behavior with a 3.5% higher freezing temperature.

Figure 1

(a) Ball-and-stick model of ${\mathrm{HoLu}}_2\mathrm{N}@{\mathrm{C}}_{80}$. A $\overrightarrow{B}$ field exerts a torque $\overrightarrow{T}$ on the Ho magnetic moment $\overrightarrow{\mu }$. (b) The energy barrier $E_A$ separates preferred orientations separated by $\mathrm{\Delta }\theta$, while the Zeeman energy imposes a bias for magnetic moments parallel to the $\overrightarrow{B}$ field. (c) (top) Zeeman splitting and (bottom) angular density distribution dependence on the polar angle. The equilibrium angular density distribution for $E_Z=0$ (dashed line) and $k_{B}T/E_Z=1/2$ (solid line) are shown. The inset is the polar diagram of the two distributions. (d) Effective magnetic moment in thermal equilibrium vs $k_{B}T/E_Z$ for freely rotating endohedral units (green) and for frozen, isotropically distributed magnetic moments (black). The magnetization of in-field cooling should lie within the green and the black curves in the shaded area.

Figure 2

(a) Temperature dependence of the magnetic moment $m$ for zero-field cooled (black) and in-field cooled (red) ${\mathrm{HoLu}}_2\mathrm{N}@{\mathrm{C}}_{80}$. (b) Ratio of the magnetic moment for in-field and zero-field cooled $m_i/m_z$ for different externally applied magnetic fields (7 T, 5 T, 3 T). (c) $m_{i0}/m_{z0}/2$ extrapolated to 0 K. From the curvature of the parabola the Zeeman energy:$k_B{T}_F$ ratio is inferred. Inset: Prediction for higher magnetic fields. Heating and cooling rate $\pm \beta =5\times {10}^{-2}\mathrm{K}/\mathrm{s}\mathrm{K}/\mathrm{s}$.

Figure 3

Magnetization $m$ vs externally applied magnetic field ${\mu }_{0}H$ of ${\mathrm{HoLu}}_2\mathrm{N}@{\mathrm{C}}_{80}$ at 1.8 K (raw data). Both zero-field (black squares) and in-field 7 T (red circles) cooled data were fitted (solid lines) simultaneously with a model of isotropically (randomly) oriented Ho magnetic moment and an equilibrium distribution with a freezing temperature $T_F$. The fits yield 9.55 ${\mu }_B$ for the Ho magnetic moment and a $T_F$ of 52.3 K for $B=7\mathrm{T}$ and $\beta =5\times {10}^{-4}\mathrm{K}/\mathrm{s}$. Grey diamonds are the difference between zero-field and in-field cooled. The solid line is the difference of the corresponding fits.

Figure 4

(a) Ratio of magnetic moment for in-field and zero-field cooled $m_i/m_z$ at different temperature sweep rates $\beta$ for ${\mathrm{HoLu}}_2$ (full symbols) and ${\mathrm{TbSc}}_2$ (open symbols). (b) Freezing temperatures $T_F(\beta )$ [with corresponding symbols and colors from (a)] determined from fits of magnetization curves of the in-field cooled samples. The solid line is the best fit of first order kinetics for ${\mathrm{HoLu}}_2$—with a barrier height $E_A/k_B=(2145\pm 20)\mathrm{K}$ and an attempt frequency $\nu =1.8\times {10}^{14\pm 0.15}{\mathrm{s}}^{-1}$.