  We develop a first-principles electron-transport simulator based on the
Lippmann--Schwinger (LS) equation within the framework of the real-space
finite-difference scheme. In our fully real-space based LS (grid LS) method,
the ratio expression technique for the scattering wave functions and the
Green's function elements of the reference system is employed to avoid
numerical collapse. Furthermore, we present analytical expressions and/or
prominent calculation procedures for the retarded Green's function, which are
utilized in the grid LS approach. In order to demonstrate the performance of
the grid LS method, we simulate the electron-transport properties of the
semiconductor/oxide interfaces sandwiched between semi-infinite metal
electrodes. The results confirm that the leakage current through the
(001)Si/SiO$_2$ model becomes much larger when the dangling-bond (DB) state is
induced by a defect in the oxygen layer while that through the (001)Ge/GeO$_2$
model is insensitive to the DB state.
