Quantum spin tunneling

A simple single spin Hamiltonian that describes quantum spin tunneling for a spin ${\displaystyle S}$ is given by:

${\textstyle {\hat {H}}=D{\hat {S}}_{z}^{2}+E({\hat {S}}_{x}^{2}-{\hat {S}}_{y}^{2})}$ [1]

where D and E are parameters that determine the magnetic anisotropy, and ${\displaystyle {\hat {S}}_{x},{\hat {S}}_{y},{\hat {S}}_{z}}$ are spin matrices of dimension ${\displaystyle 2S+1}$ . It is customary to take z as the easy axis so that D<0 and |D|>> E. For E=0, this Hamiltonian commutes with ${\displaystyle {\hat {S}}_{z}}$ , so that we can write the eigenvalues as ${\displaystyle E(S_{z})=DS_{z}^{2}}$, where ${\displaystyle S_{z}}$ takes the 2S+1 values in the list (S, S-1, ...., -S)

describes a set of doublets, with ${\displaystyle E=0}$ and ${\displaystyle D<0}$ . In the case of integer spins the second term of the Hamiltonian results in the splitting of the otherwise degenerate ground state doublet. In this case, the zero field quantum spin tunneling splitting is given by:

${\displaystyle \Delta \propto E\left({\frac {E}{D}}\right)^{(S-1)}}$

From this result, it is apparent that, given that E/D is much smaller than 1 by construction, the quantum spin tunnelling splitting becomes suppressed in the limit of large spin S, i.e., as we move from the atomic scale towards the macroscopic world. The magnitude of the quantum spin tunnelling splitting can be modulated by application of a magnetic field along the transverse hard axis direction (in the case of Hamiltonian [1], with D<0 and E>0, the x axis). The modulation of the quantum spin tunnelling splitting results in oscillations of its magnitude,[6] including specific values of the transverse field at which the splitting vanishes. This accidental degeneracies are known as diabolic points.

Quantum tunneling of the magnetization was reported in 1996 for a crystal of Mn₁₂ac molecules with S=10.[7] Quoting Thomas and coworkers,[7] "in an applied magnetic field, the magnetization shows hysteresis loops with a distinct 'staircase' structure: the steps occur at values of the applied field where the energies of different collective spin states of the manganese clusters coincide. At these special values of the field, relaxation from one spin state to another is enhanced above the thermally activated rate by the action of resonant quantum-mechanical tunneling". Quantum tunneling of the magnetization was reported in ferritin[8] present in horse spleen proteins[9]

A direct measurement of the quantum spin tunneling splitting energy can be achieved using single spin scanning tunneling inelastic spectroscopy, that permits to measure the spin excitations of individual atoms on surfaces.[10] Using this technique, the spin excitation spectrum of an individual integer spin was obtained by Hirjibehedin et al.[11] for a S=2 single Fe atom on a surface of Cu₂N/Cu(100), that made it possible to measure a quantum spin tunneling splitting of 0.2 meV. Using the same technique other groups measured the spin excitations of S=1 Fe phthalocyanine molecule on a copper surface[12] and a S=1 Fe atom on InSb,[13] both of which had a quantum spin tunneling splitting of the ${\displaystyle S_{z}=\pm 1}$ doublet larger than 1 meV.

In the case of molecular magnets with large S and small E/D ratio, indirect measurement techniques are required to infer the value of the quantum spin tunneling splitting. For instance, modeling time dependent magnetization measurements of a crystal of Fe₈ molecular magnets with the Landau-Zener formula, Wernsdorfer and Sessoli [14] inferred tunneling splittings in the range of 10⁻⁷ Kelvin.