  Intrinsic Rashba spin-orbital coupling (SOC) can exist in centrosymmetric
materials with local inversion symmetry breaking. Here we show that such a SOC
can induce topological superconductivity together with an in-plane Zeeman field
in planar Josephson junctions formed by the centrosymmetric materials. A single
Majorana mode can be created at each end of the junction. We demonstrate this
result in a model based on iron-based superconductors. We derive the necessary
Fermi surface condition for the topological planar junction and calculate the
topological phase diagram with respect to the in-plane Zeeman field and the
phase difference between the two superconductors. We provide experimental
characteristics for the topological superconductivity, including the
differential conductance and the Fano factor tomography which can be measured
in the scanning tunneling spectroscopy. Our study reveals that the
centrosymmetric systems with local-inversion-symmetry breaking can serve as new
platforms for the topological planar Josephson junctions, and help to find more
experimentally feasible materials for the topological superconductors.
