Spin-Orbit Coupling in AlGaN/AlN/GaN Heterostructures with a Polarization Induced Two-Dimensional Electron Gas

Spin-orbit coupling is investigated by weak antilocalization and Shubnikov-de Haas measurements in wurtzite Al$_{x}$Ga$_{1-x}$N/AlN/GaN heterostructures with a polarization induced two dimensional electron gas. By employing the persistent photoconductivity effect and by using five different heterostructures with different Al compositions, we cover a carrier density range extending from $0.8\times 10^{12}\mbox{ cm}^{-2}$ to $10.6\times 10^{12}\mbox{ cm}^{-2}$. We determine electron splitting energies for different carrier densities by analyzing the weak antilocalization measurements using the Iordanskii, Lyanda-Geller, and Pikus theory. We find the spin splitting energies do not scale linearly with the Fermi wavevector $k_{F}$ at high carrier densities. By fitting the spin splitting energies to a form $E_{SS}$=2($\alpha k_{F}+\gamma k_{F}^{3})$ we extract linear and cubic spin-orbit coupling parameters $\alpha $=5.13$\times $10$^{-13}$ eV m and $\gamma $=1.2$\times $10$^{-31}$eV m$^{3}$, respectively. The cubic spin-orbit coupling parameter is purely due to the bulk inversion asymmetry of the wurtzite crystal and has not been previously measured for the GaN system.