  We study the beam spin asymmetry $A_{LU}^{\sin\phi_h}$ in semi-inclusive
$\pi^0$ electroproduction contributed by the T-odd twist-3 distribution
function $g^\perp(x,k_T^2)$. We calculate this transverse momentum dependent
distribution function for the $u$ and $d$ quarks inside the proton in a
spectator model including the scalar and the axial-vector diquark components.
Using the model results, we estimate the asymmetry $A_{LU}^{\sin\phi_h}$ in the
$e p \rightarrow e^\prime \pi^0 X$ process in which the lepton beam is
longitudinally polarized. The model prediction is compared with the data
measured by the CLAS and the HERMES Collaborations and it is found that our
numerical results agree with experimental data reasonably. Especially, our
results can well describe the CLAS data at the region where the Bjorken $x$ and
the pion transverse momentum is not large. We also make a prediction on the
asymmetry $A_{LU}^{\sin\phi_h}$ in a $\pi^0$ electroproduction at CLAS12 using
the same model calculation.
