Spin-orbit coupling and its effects in organic solids

We present a detailed analysis of spin-orbit coupling (SOC) in pi-conjugated organic materials and its effects on spin characteristics including the spin-relaxation time, spin-diffusion length, and g factor. While pi electrons are responsible for low-energy electrical and optical processes in pi-conjugated organic solids, sigma electrons must be explicitly included to properly describe the SOC. The SOC mixes up- and down-spin states and, in the context of spintronics, can be quantified by an admixture parameter in the electron and hole polaron states in pi-conjugated organics. Molecular geometry fluctuations such as ring torsion, which are common in soft organic materials and may depend on sample preparation, are found to have a strong effect on the spin mixing. The SOC-induced spin mixing leads to spin flips as polarons hop from one molecule to another, giving rise to spin relaxation and diffusion, which are examined by the time-dependent perturbation theory and density-matrix theory. The spin-relaxation rate is found to be proportional to the carrier hopping rate, or equivalently, carrier mobility. The spin-diffusion length depends on the spin mixing and hopping distance but is insensitive to the carrier mobility. An applied electric field causes spin drift and gives rise to upstream and downstream spin-diffusion lengths in the hopping-conduction regime. The SOC influences the g factor of the polaron state and makes it deviate from the free-electron value. The deviation is due to the mixing of different orbitals in the polaron state, which does not include the spin mixing within the same orbital, and therefore underestimates the SOC strength. In particular, the g factor is not sensitive to the molecular geometry fluctuations, where the spin mixing within the same orbital is dominant. The SOCs in tris-(8-hydroxyquinoline) aluminum (Alq(3)) and in copper phthalocyanine (CuPc) are particularly strong, due to the orthogonal arrangement of the three ligands in the former and Cu 3d orbitals in the latter. The theory quantitatively explains the recent measured spin-diffusion lengths in Alq(3) from muon spin rotation and in CuPc from spin-polarized two-photon photoemission.