Spin-orbit Hamiltonian for organic crystals from first-principles electronic structure and Wannier functions

Spin-orbit coupling in organic crystals is responsible for many spin-relaxation phenomena, going from spin diffusion to intersystem crossing. With the goal of constructing effective spin-orbit Hamiltonians to be used in multiscale approaches to the thermodynamical properties of organic crystals, we present a method that combines density functional theory with the construction of Wannier functions. In particular we show that the spin-orbit Hamiltonian constructed over maximally localized Wannier functions can be computed by direct evaluation of the spin-orbit matrix elements over the Wannier functions constructed in absence of spin-orbit interaction. This eliminates the problem of computing the Wannier functions for almost degenerate bands, a problem always present with the spin-orbit-split bands of organic crystals. Examples of the method are presented for isolated organic molecules, for monodimensional chains of Pb and C atoms and for triarylamine-based one-dimensional single crystals.

Figure 1

Atomic structure of (a) a plumbane molecule, (b) a chain of lead atoms, and (c) a chain of methane molecules. We have also calculated the electronic structure of a chain of C atoms, which is essentially identical to that presented in (b). Color code: Pb = gray, H = light blue, C = yellow.

Figure 2

Band structure of a 1D Pb chain calculated (a) with Siesta and (b) by diagonalizing the Hamiltonian matrix constructed over the MLWFs. Black and red lines are for the bands obtained without and with SO coupling, respectively. The $\sigma , {\sigma }^{*}$, and $\pi$ bands are identified in the picture.

Figure 3

The SO matrix elements of a chain of lead atoms calculated with respect to some representative Wannier functions and plotted as a function of the site index, i.e., of the distance between the Wannier function. (a) and (b) The real and imaginary components, respectively.

Figure 4

Difference $E_{\mathrm{SO}}-E_{\mathrm{NSO}}$ between the band structure of chain of carbon atoms calculated with $E_{\mathrm{SO}}$ and without $E_{\mathrm{NSO}}$, considering SO interaction. The bands are labeled in increasing energy order without taking into account spin degeneracy. For instance the bands ${\psi }_1$ and ${\psi }_2$ are the two spin subbands corresponding to the $\sigma$ band (see Fig. 2 for notation). The left-hand side panels show results for the SO-DFT calculations performed with Siesta, while the right-hand side one, those obtained from the MLWFs.

Figure 5

Difference $E_{\mathrm{SO}}-E_{\mathrm{NSO}}$ between the band structure of chain of methane molecules calculated with $E_{\mathrm{SO}}$ and without $E_{\mathrm{NSO}}$ considering SO interaction. The bands are labeled in increasing energy order without taking into account spin degeneracy. The left-hand side panels show results for the SO-DFT calculations performed with Siesta, while the right-hand side one, those obtained from the MLWFs. The inset shows an isovalue plot of one of the four MLWFs with the red and blue surfaces denoting positive and negative isovalues, respectively. All the MLWFs have similar structure and they resemble those of the isolated methane molecule because of the small intermolecular chemical bonding owing to the large separation.

Figure 6

Structure of the triarylamine molecule (upper picture) and of the triarylamine-based nanowire investigated here. The radicals associated with the triarylamine derivative are ${\mathrm{C}}_8{\mathrm{H}}_{17}$, H, and Cl, respectively. Color code: C = yellow, H = light blue, O = red, N = gray, Cl = green.

Figure 7

Band structure of the 1D triarylamine-based nanowire constructed with the precursor 1 of Ref. [39]. This is plotted over the 1D Brillouin zone ($\mathrm{Z}=\pi /a$ with $a$ the lattice parameter). The Fermi level is marked with a dashed black line and it is placed just above the HOMO-derived valence band (in red). The lower panel is a magnification of the valence band. Note the bandwidth of about 100 meV and the fact that the band has a cosine shape, fingerprint of a single-orbital nearest-neighbor tight-binding-like interaction. Only the HOMO band is considered when constructing the MLWFs.

Figure 8

Plot of ($E_{\text{SO}}-E_{\text{NSO}}$) as a function of k in arbitrary unit over a Brillouin zone for the highest occupied band of a 1D chain of triarylamine derivatives. The blue and the red points correspond to calculations with Siesta and Wannier90, respectively.

Figure 9

SO matrix elements of a triarylamine-based nanowire calculated with respect to the Wannier functions obtained from the HOMO band. (a) and (b) Matrix elements calculated between for same and different spins, respectively.