Unified theory of ultrafast femtosecond and conventional picosecond magnetic dynamics in the distorted three-center magnetic cluster Ni${}_3$Na${}_2$

With the use of a full ab initio method, based on quantum chemical wave function methods with the inclusion of spin-orbit coupling and a static external magnetic field, we investigate the laser-induced ultrafast magnetic switching in the three-magnetic-center cluster Ni${}_3$Na${}_2$ under various static distortions. Thus we (i) establish spin dynamics as an unprecedentedly accurate sensor of the bond length between metallic centers, and (ii) derive a unified picture of ultrafast (femtosecond) and slow (ps, phonon-magnon) spin dynamics in molecular magnets by combining electron-spin and electron-lattice coupling. The resulting unexpectedly rich set of magnetic phases is exploited in order to provide time scales of phonon-magnon mediated spin dynamics in magnetic molecules.

Figure 1

(a) The geometry structure of Ni${}_3$Na${}_2$ with the atoms and the interatomic distances. Stretching and shrinking is understood with respect to the original value of 3.6 Å. (b) The five magnetic phases due to the different interatomic distances (see text). The arrows indicate the spin direction when the spin density is localized on the respective atoms.

Figure 2

Phonon energies as a function of the equilibrium interatomic distances $d1$ (a), $d2$ (b), $d3$ (c), and $d4$ (d). For the definitions of $d1$, $d2$, $d3$, and $d4$ see Fig. 1a. 1, 2, and 3 stand for the three normal modes along the $x$ axis and 4 stands for one normal mode along the $y$ axis. The red, dashed lines indicate the critical distances at which the magnetic phase transitions occur (see text). The insets in (a), (b), and (c) show the energy gap at the critical distances as a function of the phonon energy (center of the gap).

Figure 3

Fidelities (filled symbols, upper panels) and pulse widths (FWHM) (empty symbols, lower panels) of the ultrafast spin flips at Ni1 (blue squares), Ni2 (violet circles), and Ni3 (green triangles) as a function of the equilibrium interatomic distances $d1$ (a), $d2$ (b), $d3$ (c), and $d4$ (d). The initial and final states of spin flip are listed for Ni1, Ni2, and Ni3 atoms also in blue, violet, and green, respectively. For the definitions of $d1$, $d2$, $d3$, and $d4$ see Fig. 1a. The red, dashed lines indicate the critical distances at which the magnetic phase transitions occur (see text).

Figure 4

Phonon (filled symbols) and electron (empty symbols) energies, for phononic vibrations along the $x$ axis (labeled 1, 2, and 3) and along the $y$ axis (labeled 4) vs the interatomic distances $d1$ (a), $d2$ (b), $d3$ (c), and $d4$ (d) in the Ni${}_3$Na${}_2$ cluster. The electron energies are labeled as 5. See also the Appendix.

Figure 5

Unified illustration of the two microscopic mechanisms for the slow (picosecond) interphase, and the ultrafast (femtosecond) intraphase spin-flip process. $|e_1\rangle$ stands for one or more spin-mixed electronic states (due to SOC), while $|e_2\rangle$ stands for one or more phonon-mixed electronic states. Both processes start from an initial state $|i\rangle$ and end to a final state with different spin orientation ($|f_1\rangle$ or $|f_2\rangle$) via photon (ultrafast) or phonon (slow) scattering.