Single-molecule magnets are molecules which exhibit bulk magnetic properties. This makes them hot candidates in the development of small data storage units and other electronic devices. To be useful in this context, however, it is mandatory that the magnetic moment of the single-molecule magnet be stable over a significant amount of time. In other words, the magnetic moment should relax as slowly as possible after polarization through an external magnetic field for a single-molecule magnet of commercial interest. At high temperature, one expects the relaxation time to follow an activated Arrhenius-type behavior, i.e. the relaxation time increases exponentially with increasing effective relaxation barrier.
Following this argumentation, efforts have been devoted to the design of single-molecule magnets with large effective relaxation barrier. However, a consistent body of experiments now reports relaxation times at high temperature which are significantly shorter than one would expect from the corresponding, spectroscopically determined large effective relaxation barrier.
Researchers from the University of Florence in Italy and from the Advanced Materials and Bioengineering Research Centre and from the Centre for Research on Adaptive Nanostructures and Nanodevices of Trinity College in Dublin now unraveled this mystery.
One possible explanation for the observed shortness of the relaxation time is a phonon-mediated spin relaxation mechanism. In its simplest form, known as Orbach relaxation, the spin system absorbs a phonon, leading to an excitation of the spin system and resultant relaxation to its ground state or to the corresponding spin-flipped state which is quasi-degenerate with the ground state. Such a phonon-mediated spin relaxation would, however, pose strict requirements on the absorbed phonon. Namely, the absorbed phonon needs to be resonant with the described spin transition and the absorbed phonon needs to have non-vanishing spin-phonon coupling. Such an event would thus be rare.
But what about more complicated phonon-mediated spin relaxation mechanisms? In a more realistic scenario, the spin system should be coupled to a bath of anharmonic phonons, taking into account the fact that phonons have a finite life time.
Lead by the idea that a more realistic scenario is needed to explain the observed relaxation times, the researchers reformulated the basic concepts of spin-phonon relaxation in a quantum-mechanical formalism. They thus provide a theoretical framework that is general and fully amenable to ab initio methods and as such is free from previously adopted approximations. The presented framework requires the calculation of the full phonon spectrum, the spin Hamiltonian and the spin-phonon coupling. These quantities are fed into the master equations that govern the spin dynamics. These can be solved numerically, thus providing a full quantitative analysis of the underlying spin-phonon dynamics.
The researchers demonstrate their framework on a real single-molecule magnet, namely [(tpaPh)Fe]-, an S=2 system. Using ORCA, spin Hamiltonian and spin-phonon couplings are determined on the complete active space self-consistent field (CASSCF) level of theory. The relaxation time at different temperatures is determined through fitting the time evolution of the magnetization obtained from solving the Redfield master equation at the respective temperature to an exponential decay. Indeed the authors find no single exponential dependence of the relaxation time over the entire temperature range. This is a clear indication for a relaxation through non-resonant spin-phonon relaxation channels. The extracted effective relaxation barrier is in excellent agreement with its experimental counterpart and much smaller than the calculated excitation energy. Thus, phonon-induced under-barrier relaxation does drastically reduce the spin lifetime of [(tpaPh)Fe]-, the single-molecule magnet studied.
Thus, the authors demonstrated that phonon dissipation, when treated at a more realistic level, enables off-resonance spin relaxation. Magnitude and temperature dependence of this spin relaxation are determined by the electronic and vibrational details of the specific single-molecule magnet under study. In the case of [(tpaPh)Fe]-, different relaxation mechanisms are dominant in different energy ranges.
Several design strategies for reducing spin relaxation in single-molecule magnets can be deduced from this study. For example, the direct relaxation between two quasi-degenerate ground states can be decelerated either by increasing the smallest phonon frequency or by reducing the spin-phonon coupling. This translates into either designing more rigid small-molecule magnets or by employing the usual quantum-tunnneling reduction methods such as using Kramer ions of high axial symmetry.