A first-principles linear response theory for open quantum systems and its application to Orbach and direct magnetic relaxation in Ln-based coordination polymers

Abstract

Single-Molecule Magnets (SMMs) exhibit slow magnetic relaxation as a result of axial magnetic anisotropy inhibiting spin-phonon transitions. In order to establish a direct link between physical observables and the microscopic theory of magnetic relaxation, we here develop and numerically implement a first-principles linear-response theory for open quantum systems that provides access to the complex a.c. magnetic susceptibility in the presence of an oscillating a.c. magnetic field. Once combined with density functional theory and multiconfigurational electronic structure simulations, this formalism is applied in a fully first-principles fashion to three cyanido-bridged Ln/Y-based coordination polymers with general formula {Ln$^{III}_x$ Y$^{III}_{1-x}$ [Co(CN)$_6$]}, where Ln = Yb (1), Tb (2), and Dy (3). The method is able to reproduce the low-temperature direct relaxation process and its field dependence, as well as the high-temperature Orbach relaxation regime for all the investigated compounds. These results demonstrate the feasibility of ab initio simulations of magnetic a.c.susceptibility in lanthanide-based SMMs and support the potential of further development of ab initio open quantum systems methods towards the completion of a magnetization dynamics theory.

Figures (12)

• Figure 1: Experimental crystal structure of the anhydrous Ln$^{\mathrm{III}}$/Y$^{\mathrm{III}}$-[$\mathrm{Co}^{\mathrm{III}}(\mathrm{CN})_6$]$^{3-}$ three-dimensional framework: (a) view of the $3\times3\times2$ supercell along the crystallographic $c$ axis, with the unit cell highlighted; (b) geometry of the Ln/Y center surrounded by six cyanido ligands; (c) overlap between the experimental and optimized asymmetric units for the Y-based framework, shown in deep blue and yellow, respectively. In panel (a), the $\mathrm{Co}^{\mathrm{III}}_{\mathrm{LS}}$ label emphasizes the diamagnetic character of the low-spin cobalt(III) centers.
• Figure 2: Scheme of usual experimental workflow for extracting magnetic relaxation times from experiment (a) with typical $\chi'(\nu)$ and $\chi"(\nu)$ traces of Debye model \ref{['eq:debye-model']} compared to HN model \ref{['eq:hn-model']} with $\alpha$ = 0.2, $\beta$ = 0.8 (both for the same $\tau$ = 0.02 s) and the corresponding Cole-Cole plot (c) showing the effect of broadening phenomenological HN parameters.
• Figure 3: Temperature-variable Ab initio I (a) and II (c,d) a.c. magnetic characteristics of 1 under H$_{\mathrm{d.c}}$ = 3000 Oe applied along the main magnetic X-axis for temperatures in the 1.8–3.2 K range, with ab initio data as points and solid lines representing the best-fit curves of the Havriliak–Negami model from \ref{['eq:hn-model']}, with the obtained magnetic relaxation times $\tau$ (b) and the fitted model \ref{['eq:tau-model']} compared to the experimental data (black circles), where the colored dashed lines indicate the contribution of each relaxation path employed. The best-fit parameters are gathered in Table S9.
• Figure 4: Temperature-variable Ab initio III (a-c) magnetic characteristics of 1 under H$_{\mathrm{d.c}}$ = 3000 Oe applied along the main magnetic $X$, $Y$, and $Z$ axes for temperatures in the 1.8–3.2 K range, with ab initio data as points and solid lines representing the best-fit curves of the Havriliak–Negami model from \ref{['eq:hn-model']}, together with the obtained magnetic relaxation times $\tau$ compared to the experimental and those computed with one-phonon Secular-Markov approximation \ref{['eq:one-ph-sec-markow-red']} presented in panel (d).
• Figure 5: Temperature-variable experimental (a,c) magnetic characteristics of 1 under H$_{\mathrm{d.c}}$ = 3000 Oe compared to the corresponding Ab initio III simulation (b,d) averaged along $X$, $Y$, and Z magnetic axes for temperatures in the 1.8–3.2 K range, with experimental and ab initio data as points and solid lines representing the best-fit curves of the Havriliak–Negami model from \ref{['eq:hn-model']}.