  Narrow bandgap semiconductor nanostructures have been explored for
realization of topological superconducting quantum devices in which Majorana
states can be created and employed for constructing topological qubits.
However, a prerequisite to achieve the topological phase transition in these
nanostructures is application of a magnetic field, which could complicate the
technology development towards topological quantum computing. Here we
demonstrate that a topological phase transition can be achieved in a narrow
bandgap semiconductor nanolayer under application of a perpendicular electric
field. Based on full band structure calculations, it is shown that the
topological phase transition occurs at an electric-field induced band inversion
and is accompanied by a sharp change of the $\mathbb{Z}_{2}$ invariant at the
critical field. We also demonstrate that the nontrivial topological phase is
manifested by the quantum spin Hall edge states in a band-inverted nanolayer
Hall-bar structure. We present the phase diagram of the nanolayer in the space
of layer thickness and electric field strength, and discuss the optimal
conditions to achieve a large topological bandgap in the electric-field induced
topological phase of a semiconductor nanolayer.
