Spin-Orbit Coupling, Spin Relaxation, and Spin Diffusion in Organic Solids

We develop a systematic approach of quantifying spin-orbit coupling (SOC) and a rigorous theory of carrier spin relaxation caused by the SOC in disordered organic solids. The SOC mixes up and down spin in the polaron states and can be characterized by an admixture parameter ${\gamma }^2$. This mixing effects spin flips as polarons hop from one molecule to another. The spin relaxation time is ${\tau }_{\mathrm{sf}}={\overline{R}}^2/(16{\gamma }^{2}D)$, and the spin diffusion length is $L_s=\overline{R}/4|\gamma |$, where $\overline{R}$ is the mean polaron hopping distance and $D$ the carrier diffusion constant. The SOC in tris-(8-hydroxyquinoline) aluminum (${\mathrm{Alq}}_3$) is particularly strong due to the orthogonal arrangement of the three ligands. The theory quantitatively explains the temperature-dependent spin diffusion in ${\mathrm{Alq}}_3$ from recent muon measurements.

Figure 1

Admixture ${\gamma }^2$ as a function of torsion angle $\theta$ in biphenyl. Green (light gray) and red (dark gray) lines are for the electron and hole polarons, respectively. Solid (dashed) lines describe ${\gamma }^2$ (${\gamma }_{\uparrow \downarrow }^2$). The inset shows the energy splitting between the HOMO and $\mathrm{HOMO}+(-)1$ for the electron (hole) polaron.

Figure 2

(a) Logarithm of spin polarization as a function of $x$ with spin injected at the plane of $x=0$ in a $32\times 32\times 32$ cubic lattice by numerically solving Eq. (11). The four lines, from bottom to top, correspond to $T=80$, 40, 20, and 10 K, respectively. (b) Spin diffusion length as a function of temperature for ${\mathrm{Alq}}_3$. Circles are experimental values. (c)–(f) are the distributions of hopping distance at $T=80$, 40, 20, and 10 K, respectively.