Accurate and efficient spin-phonon coupling and spin dynamics calculations for molecular solids

Abstract

Molecular materials are poised to play a significant role in the development future opto-electronic and quantum technologies. A crucial aspect of these areas is the role of spin-phonon coupling and how it facilitates energy-transfer processes such as intersystem crossing, quantum decoherence, and magnetic relaxation. Thus, it is of significant interest to be able to accurately calculate molecular spin-phonon coupling and spin dynamics in the condensed phase. Here we examine the various approximations inherent in spin-phonon coupling and spin dynamics calculations on molecular solids by performing a case study on a single-molecule magnet. Three key results are: i) finite crystalline slab calculations should be avoided; ii) the phonon spectrum in reciprocal space should be sampled as densely as possible; and iii) phonon linewidths, as calculated by periodic density-functional theory, are likely overestimated at low temperature, but are not essential to obtain accurate magnetic relaxation rates provided point ii is adhered to. Calculations using this approach are facilitated by the open-source packages we have developed, which enable cost-effective and accurate spin-phonon coupling calculations on molecular solids with quantitative accuracy.

Spin–phonon coupling and magnetic relaxation in single-molecule magnets

“…32 However, we have recently shown that there is little impact on the spin dynamics when either fixed, approximated, or calculated linewidths are used, provided the BZ is integrated with a sufficiently dense q-point grid. 33 4 Electronic structure of lanthanide complexes Now we turn to the spin states of the molecule, where herein we focus on the low-lying electronic states of Ln complexes. For most cases, the trivalent Ln 3+ oxidation state is most relevant, with a well-isolated ground configuration 4f n 5s 2 5p 6 ; the 6s and 5d may become relevant depending on the oxidation state and the ligands surrounding the metal.…”

“… 32 However, we have recently shown that there is little impact on the spin dynamics when either fixed, approximated, or calculated linewidths are used, provided the BZ is integrated with a sufficiently dense q -point grid. 33 …”

“…Moreover, magnetic relaxation is typically much slower than the underlying phonon dynamics, thus allowing for a clear separation of system and bath timescales; we have recently confirmed this separation of timescales by calculating the phonon lifetimes for a molecular crystal of a Dy( iii ) SMM. 33 These additional constraints simplify the problem considerably, allowing one to treat the phonons as a weakly coupled bath with no memory (on the timescale of the spin system). Under these assumptions, commonly referred to as the Born–Markov approximation, the dynamics of the reduced density matrix is described by the Redfield equation: 44,56 where the square brackets denote the commutator ([ Â , B̂ ] = ÂB̂ − B̂Â ), and and are the Redfield superoperators arising from the first- and second-order spin–phonon coupling in eqn (16) .…”

“…87 We have recently shown that the Madelung potential can be closely-approximated in a consistent manner by employing a spherical cluster of unit cells embedded in a conductor-like reaction field to screen the non-physical surface charges that arise from finite-size unit-cell expansions. 33 …”

“…87 We have recently shown that the Madelung potential can be closely-approximated in a consistent manner by employing a spherical cluster of unit cells embedded in a conductorlike reaction field to screen the non-physical surface charges that arise from finite-size unit-cell expansions. 33 The third step, computation of the spin-phonon coupling parameters, requires knowledge of the electronic response to nuclear distortions which is encoded in derivatives of the CFPs as presented in Section 5. The most obvious method to obtain these parameters is by computing the electronic structure and CFPs at distorted geometries along the normal mode coordinates and fitting the parameters to a polynomial (Fig.…”