Abstract
Disorder effects on three-dimensional second-order topological insulators (3DSOTIs) are investigated numerically and analytically. The study is based on a tight-binding Hamiltonian for non-interacting electrons on a cubic lattice with a reflection symmetry that supports a 3DSOTI in the absence of disorder. Interestingly, unlike the disorder effects on a topological trivial system that can only be either a diffusive metal (DM) or an Anderson insulator (AI), disorders can sequentially induce four phases of 3DSOTIs, three-dimensional first-order topologicalinsulators (3DFOTIs), DMs and AIs. At a weak disorder when the on-site random potential of strength W is below a low critical value W_c1 at which the gap of surface states closes while the bulk sates are still gapped, the system is a disordered 3DSOTI characterized by a constant density of states and a quantized integer conductance of e²/h through its chiral hinge states. The gap of the bulk states closes at a higher critical disorder W_c2, and the system is a disordered 3DFOTI in a lower intermediate disorder between W_c1 and W_c2 in which electron conduction is through the topological surface states. The system becomes a DM in a higher intermediate disorder between W_c2 and W_c3 above which the states at the Fermi level are localized. It undergoes a normal three-dimension metal-to-insulator transition at W_c3 and becomes the conventional AI for W>W_c3. The self-consistent Born approximation allows one to see how the density of bulk states and the Dirac mass are modified by the on-site disorders.