Coupled spin-lattice dynamics from the tight-binding electronic structure

We developed a method which performs the coupled adiabatic spin and lattice dynamics based on the tight-binding electronic structure model, where the intrinsic magnetic field and ionic forces are calculated from the converged self-consistent electronic structure at every time step. By doing so, this method allows us to explore limits where the physics described by a parameterized spin-lattice Hamiltonian is no longer accurate. We demonstrate how the lattice dynamics is strongly influenced by the underlying magnetic configuration, where disorder is able to induce significant lattice distortions. The presented method requires significantly less computational resources than ab initio methods, such as time-dependent density functional theory (TD-DFT). Compared to parameterized Hamiltonian-based methods, it also describes more accurately the dynamics of the coupled spin and lattice degrees of freedom, which becomes important outside of the regime of small lattice and spin fluctuations.

Figure 1

The spin-lattice parameter ${\mathrm{\Gamma }}_{ijk}^{\mu }$ for the triangular trimer of Cr using the ferromagnetic configuration as reference (top) and the Néel antiferromagnetic configuration as reference (bottom). The indexes $i$, $j$, and $k$ follows the definition of ${\mathrm{\Gamma }}_{ijk}^{\mu }=\frac{\partial {\mathcal{J}}_{ij}}{\partial u_k^{\mu }}$.

Figure 2

Schematic representation of (a) the triangle trimer and (b) the nanowire with 10 atoms, with their respective coordinate system represented by the tripod.

Figure 3

On-site force constant matrix for the rotated magnetic moment in the Fe and Cr triangular trimer, top and bottom, respectively. The $\theta$ refers to the angle being rotated.

Figure 4

Spin-lattice parameters ${\mathrm{\Gamma }}_{ijk}^{\mu }$ (exchange striction) and isotropic spin-spin exchange parameter $J_{ij}$ for the nanochains of Fe (top) and Cr (bottom). On the left panel, atom 5 is taken as the reference atom while varying the $k$-th atom. On the right panel, atom 5 is also taken as reference while the $j\mathrm{th}$ atom is the one varying. Still on the right panel, for the spin-lattice parameter, the $k\mathrm{th}$ atom is also being fixed.

Figure 5

The angle between the magnetic moments of the Fe and Cr triangular trimer as a function of time in femtoseconds, as well as their respective exchange coupling parameters calculated at every time step. In the bottom panel, we show the results when the magnetic moments are considered fixed (lattice dynamics only), for the Cr triangular trimer case. The disordered magnetic moments induce a distortion in the lattice.

Figure 6

Tight-binding spin-lattice dynamics for the triangular trimers of Fe and Cr. For these calculations, both the spin and lattice damping parameters are set to 0.05. On the left (right), we show the Fe (Cr) triangular trimer relaxation process for the atomic positions (top) and magnetic moment length (down). The figures in the middle denote the start configuration (left) and the final configuration (right).

Figure 7

Tight-binding spin-lattice dynamics for the nanochains with ten atoms of Fe and Cr. For these calculations, both the spin and lattice damping parameters are set to 0.5. The figures in the middle denote the start configuration (left) and the final configuration (right). The full lines concern the atoms in the left half of the chain, while the dashed lines stand for the right half. One can note that during the first femtoseconds of simulation, they can be easily distinguished due to the disordered magnetic configuration.

Figure 8

Comparison between the semiclassical (full line) and the tight-binding (dashed line) spin-lattice dynamics. The damping for both lattice and spin were set to 0.1. The parameters to the semiclassical spin-lattice Hamiltonian were calculated from the systems respective ground state, which are FM to the Fe trimer and Néel AFM for the Cr. Each system starts with disordered magnetic moments and in their respective ionic ground state.

Figure 9

The spin-lattice parameter ${\mathrm{\Gamma }}_{ijk}^{\mu }$ for the triangle trimer of Fe.

Figure 10

Schematic representation of the calculated forces using the semiclassical model (blue) and the tight-binding electronic structure (red), for the Fe (left), and Cr (right) triangular trimer. The arrows point to the direction corresponding to the calculated force for each atom. The magnitude of the force is proportional to the size of the arrow.