  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 $e^2/h$ 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.
