  We theoretically investigate the engineering of two-dimensional second-order
topological insulators with corner states by coupling two first-order
topological insulators. We find that the interlayer coupling between two
topological insulators with opposite topological invariants results in the
formation of edge-state gaps, which are essential for the emergence of the
corner states. Using the effective Hamiltonian framework, We elucidate that the
formation of topological corner states requires either the preservation of
symmetry in the crystal system or effective mass countersigns for neighboring
edge states. Our proposed strategy for inducing corner state through interlayer
coupling is versatile and applicable to both $\mathbb{Z}_2$ topological
insulators and quantum anomalous Hall effects. We demonstrate this approach
using several representative models including the seminal Kane-Mele model, the
Bernevig-Hughes-Zhang model, and the Rashba graphene model to explicitly
exhibit the formation of corner states via interlater coupling. Moreover, we
also observe that the stacking of the coupled $\mathbb{Z}_2$ topological
insulating systems results in the formation of the time-reversal invariant
three-dimensional second-order nodal ring semimetals. Remarkably, the
three-dimensional system from the stacking of the Bernevig-Hughes-Zhang model
can be transformed into second-order Dirac semimetals, characterized by
one-dimensional hinge Fermi arcs. Our strategy of engineering second-order
topological phases via simple interlayer coupling promises to advance the
exploration of higher-order topological insulators in two-dimensional spinful
systems.
