Geometric-phase-effect tunnel-splitting oscillations in single-molecule magnets with fourth-order anisotropy induced by orthorhombic distortion

We analyze the interference between tunneling paths that occurs for a spin system with both fourth-order and second-order transverse anisotropy. Using an instanton approach, we find that as the strength of the second-order transverse anisotropy is increased, the tunnel splitting is modulated, with zeros occurring periodically. This effect results from the interference of four tunneling paths connecting easy-axis spin orientations and occurs in the absence of any magnetic field. Copyright c © EPLA, 2009 Geometric-phase effects play an important role in spin dynamics. Notably, a geometric phase lies at the heart of spin-parity effects, such as Kramers degeneracy [1–3]. Tunneling of a spin (or magnetic particle) between degenerate orientations can be modulated by such geometricphase effects via the interference between multiple tunneling paths, with complete suppression (or “quenching”) of the tunnel splitting occurring when tunneling paths destructively interfere. Garg has considered the case in which the tunneling of a spin with biaxial anisotropy can be modulated by the application of a magnetic field along the hard axis [4,5]. Specifically, he studied the Hamiltonian H=D(J− Ĵ z )+E(Ĵ x − Ĵ y )−hxĴx, (1) for spin J and anisotropy constants D>E > 0 and hx ≡ gμBHx. He showed periodic quenching of the tunnel splitting as a function of magnetic field Hx due to geometric-phase interference between tunneling paths. Several extensions and variants of this problem have been studied theoretically [5–18]. In a ground-breaking experiment, Wernsdorfer and Sessoli [19] found clear evidence of this effect in the single-molecule magnet Fe8, which is reasonably well described by eq. (1). (a)Present address: Department of Physics, University of Colorado Boulder, CO 80309, USA. (b)E-mail: [email protected] The case originally studied by Garg involved no magnetic field along the longitudinal (z) direction. Tunneling can occur between the metastable ground state |J〉 and an excited state | −J +n〉 when the applied field brings them into resonance, i.e., when hz ≡ gμBHz = nD, n= 0, 1, 2, . . . , 2J − 1 [20]. In Fe8, it was discovered that at these resonance longitudinal fields, a transverse field along the hard axis continued to produce interference effects [19]. Thus, at certain values of hz and hx, destructive interference occurs, suppressing tunneling. These so-called diabolical points in hz-hx parameter space have been studied theoretically by several authors using different approaches [5–7,10,11,15,17]. The Mn12 single-molecule magnet has a fourfold transverse magnetic anisotropy and, like Fe8, displays resonant tunneling between two easy-axis orientations [20]. One variant of this molecule, Mn12-tBuAc, appears to display the fourfold symmetry with high accuracy [21–23]. Here we consider the interference that occurs between tunneling paths in such a system and how that interference can be modified by the presence of a second-order transverse anisotropy perturbation. Such a perturbation could potentially be induced by the application of uniaxial pressure to a sample of Mn12. We find that the tunnel splitting is periodically quenched as a function of the strength of the perturbation. This interference effect takes place in the absence of any magnetic field. In addition, we find a