Geometric-Phase Interference in a $\mathrm{M}{\mathrm{n}}_{12}$ Single-Molecule Magnet with Fourfold Rotational Symmetry

We study the magnetic relaxation rate $\Gamma$ of the single-molecule magnet ${\mathrm{Mn}}_{12}\mathrm{\text{−}}t\mathrm{BuAc}$ as a function of the magnetic field component $H_T$ transverse to the molecule’s easy axis. When the spin is near a magnetic quantum tunneling resonance, we find that $\Gamma$ increases abruptly at certain values of $H_T$. These increases are observed just beyond values of $H_T$ at which a geometric-phase interference effect suppresses tunneling between two excited energy levels. The effect is washed out by rotating $H_T$ away from the spin’s hard axis, thereby suppressing the interference effect. Detailed numerical calculations of $\Gamma$ using the known spin Hamiltonian accurately reproduce the observed behavior. These results are the first experimental evidence for geometric-phase interference in a single-molecule magnet with true fourfold symmetry.

Figure 1

Measured magnetic relaxation rate as a function of longitudinal field near the zero-field resonance for several values of transverse field ($H_T$) from 1.0 to 6.8 kOe in 200 Oe increments. The data were taken from sample $A$ at 3.10 K. The green boxes indicate regions where curves for different values of $H_T$ tend to bunch together. The left inset shows the classical energy landscape in a spherical polar plot for a spin described by Eq. (1). The $z$ axis is the easy axis (energy minima) while the $x$ and $y$ axes are the hard axes (maxima). The value of $C$ has been greatly exaggerated to make the fourfold symmetry evident. The right inset shows a schematic of the apparatus, illustrating control of angles $\theta$ and $\varphi$. $\widehat{\mathbf{n}}$ is normal to detector plane; $\widehat{\mathbf{i}}$ is a hard axis direction.

Figure 2

(a) Relaxation rate as a function of transverse field at $H_z=0$ and $H_z=-400\mathrm{Oe}$, taken from the data shown in Fig. 1. The solid curves are the results of simulations of the relaxation rate using Eq. (2) with $\varphi =7°$, as described in the text. The inset shows the double-well potential for the $n=0$ tunneling resonance. (b) Calculated tunnel splitting for energy-level pairs $m=\pm 2$, $m=\pm 3$, and $m=\pm 4$, as indicated, for $\varphi =0$ (solid lines) and $\varphi =45°$ (dashed lines). The dotted horizontal line marks a threshold splitting at which the system makes a transition from one dominant pair of tunneling levels to another. The dotted vertical lines show the transverse field at which this transition occurs and its correspondence to the observed rapid increases in the relaxation rate. Inset: Probability current diagram for high-lying energy levels at $H_T=4.8\mathrm{kOe}$. The numerical labels indicate the expectation values of $S_z$ for the corresponding energy level; the opacity of the arrows indicates the magnitude of the associated current.

Figure 3

Relaxation rate from sample $B$ as a function of transverse field for several different values of ${\varphi }^{\prime }$ for $H_z=-500\mathrm{Oe}$. The inset shows the raw relaxation rate while the main figure shows the same data after dividing by an exponential function to enhance clarity.

Figure 4

(a) Relaxation rate (after dividing by an exponential to enhance presentation) as a function of $H_T$ for $H_z$ near the $n=1$ resonance. Data was taken from sample $B$ with experimental azimuthal angle ${\varphi }^{\prime }=2°$ and $H_z$ set to $1\mathrm{kOe}$ less than the peak of the resonance. (b) Simulations of relaxation rate as a function of transverse field using Eqs. (2, 3, 4) with $\varphi =7°$. The inset shows the double-well potential for the $n=1$ resonance. (c) Tunnel splitting calculations with $\varphi =0°$ for the $m=-2$, 3 and $m=-3$, 4 energy-level pairs, as indicated. The dotted horizontal line represents a tunneling threshold while the dotted vertical lines correspond to the transverse fields at which abrupt transitions in relaxation rate occur.