Spin Hall effect in disordered organic solids

We study the spin Hall effect (SHE) in disordered $\pi$-conjugated organic solids, where individual molecules are oriented randomly and electrical conduction is via carrier hopping. The SHE, which arises from interference between direct ($i\to j$) and indirect ($i\to k\to j$) hoppings in a triad consisting of three molecules $i$, $j$ and $k$, is found to be proportional to $\lambda ({\mathbf n}_i\times {\mathbf n}_j+{\mathbf n}_j\times {\mathbf n}_k + {\mathbf n}_k\times {\mathbf n}_i)$, where $\lambda$ is the spin admixture of $\pi$ electrons due to the spin-orbit coupling and ${\mathbf n}_i$ is the orientation vector of molecule $i$. Electrical conductivity $\sigma_{qq}$ ($q=x,y,z$) and spin-Hall conductivity $\sigma_{\rm sh}$ are computed by numerically solving the mater equations of a system containing $32\times32\times32$ molecules and summing over contributions from all triads in the system. The obtained value of spin Hall angle, $\Theta_{\rm sh}\equiv \sigma_{\rm sh}/\sigma_{qq}$, is consistent with experimental data in PEDOT:PSS, with a predicted temperature dependence as $\log \Theta_{\rm sh} \sim T^{-1/4}$.