  We investigated the effect due to perpendicular magnetic field on quantum
wires where spin-orbit interaction (SOI) of electrons is not neglected. Based
on the calculated energy dispersion, the nonlinear ballistic conductance ($G$)
and electron-diffusion thermoelectric power ($S_d$) are calculated as functions
of electron density, temperature and applied bias voltage. A low-temperature
half-step feature in $G$, which was observed experimentally by Quay et al. [see
Nature Physics {\bf 6}, 336 (2010)], as well as a new peak in $S_d$ are
reproduced here in the low density regime. These phenomena are related to the
occurrence of the Zeeman splitting and SOI induced saddle point in the band
structure, where the channel chemical potential lies within an anticrossing gap
between the saddle point of the lower subband and the bottom of the upper
subband. Additionally, side peaks in $G$ far away from the zero bias for the
nonlinear transport, as well as a quadratic bias-voltage dependence of $G$ near
zero voltage, are predicted and discussed.
