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Transverse Hamiltonian terms

Quantum tunneling splitting

MQT is only possible in the presence of off-diagonal terms in the Hamiltonian (see eq.(1)). When the spin system is at resonance, Hz = kD/gB, opposite spin-projections at both sides of the anisotropy barrier are energy degenerated (see figs. 1C and 3). A transverse term in the Hamiltonian breaks this degeneracy and generates a so called tunnel splitting, , between symmetric and anti-symmetric superposition states of the eigen-states, |m, of the diagonal Hamiltonian, Hdiag,

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where m = -S,…, S, and . MQT probability is thus determined by the tunnel splitting and governs the magnetic relaxation of SMMs at low temperatures.

Transverse interactions

In SMMs these transverse terms have different origins. One possibility is to apply a magnetic field perpendicular to the (easy) z-axis of the molecules, Hx (or Hy), which generates a Zeeman transverse energy term due to the coupling with the transverse components of the spin, gnBSxHx. Applied transverse fields are very important tools in the study of MQT in SMMs due, in part, to the high power dependence of the tunnel splitting on the magnitude of the transverse field, , leading to variations of twelve orders of magnitude with transverse fields of a few Tesla. Tunning the tunnel splitting magnitude with an external transverse field allows the study of MQT with many different measurement techniques; from dc and low-frequency (~Hz-kHz) susceptibility measurements up to high-frequency (~MHz-THz) electro paramagnetic resonance (EPR), infrared spectroscopy or neutron scattering experiments, to mention a few.

Transverse magnetic field components are also generated by hyperfine and dipolar interactions. In this case, magnetic field distributions generate a tunnel splitting distribution along the molecules of the crystal. This should be taken into account in the possible applications of these systems in technology devices, because tunnel splitting distributions lead to different MQT behaviors for different molecules within a single crystal. Therefore, one of the most peculiar characteristics of SMMs, the mono-dispersion (identical molecules), can be altered. The understanding of interactions in SMMs is one of the aims of this group and the focus will be on the synthesis and study of materials with no dipolar interactions (anti-ferromagnetic SMMs) and without significant hyperfine interactions (magnetic ions with no nuclear spin). Different methods of measurement will be used to study these phenomena in new SMMs, such as hole digging procedures in which a digging magnetic field with a given direction or application time is used to dig a hole in the distribution. This will allow us to probe the structure of internal  magnetic interactions. Moreover, tilts of the easy axes of the molecules can also generate transverse field components if a longitudinal field is present in the experiment. In fact del Barco and collaborators at NYU and Steve Hill and collaborators at UF have demonstrated the presence of distributions of easy-axis tilts in three different SMMs.

Transverse anisotropy

Another origin of transverse terms in the Hamiltonian common among the SMMs known to date is transverse molecular anisotropy. This anisotropy is imposed by the symmetry of the molecules. For example, a molecule with rhombic site symmetry (i.e. Fe8) generates a second-order transverse anisotropy term in the spin Hamiltonian, E(Sx2-Sy2), coupling spin states, m, differing by 2. Other examples are Mn12, Mn4 or Ni4 SMMs which have tetragonal symmetry, and consequently, have a fourth-order transverse anisotropy term in the Hamiltonian, C(S+2+S-2). Transverse anisotropies also break the degeneracy of spin states at resonance and allow MQT to occur, leading to a zero-field tunnel splitting. Moreover, the distortion of the anisotropy barrier due to these transverse terms leads to different MQT behaviors in the presence of transverse fields, depending on the relative orientation of the field with respect to the transverse anisotropy axes (see figure on the right). For example, for a given magnitude of an external transverse field, the tunnel splitting is bigger when the transverse field is applied along the direction of a medium axis than along a hard axis, leading to oscillations of the MQT probability as a function of the angle of orientation of the transverse field. Moreover, transverse anisotropy terms generate MQT selection rules, imposing restricted k-resonances for which the tunnel relaxation is only possible. Interestingly, for certain directions of application of a transverse field, MQT can be suppressed for several field magnitudes, leading to a fundamental quantum interference effect known as Berry phase. This has been observed in several SMMs. Berry phase interference in SMMs is very sensitive to the symmetry of the molecules and defects that modify or distort the molecular symmetry and could be used to tune MQT for applications of SMMs in electronic devices in information technology.

Recent studies carried out by del Barco and collaborators at NYU have shown that disorder in the environment of the magnetic molecules changes the symmetry of the molecules affecting the MQT properties. This finding has generated important advances in the understanding of the origin of MQT in SMMs, explaining, among others, the observance of MQT relaxation in the absence of transverse magnetic fields and the absence of the quantum selection rules imposed by the symmetry of the molecules (see research/symmetry for more information.
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