  We investigate a cross-correlation between the weak gravitational lensing
field of the large-scale structure, $\kappa$, and the secondary temperature
fluctuation field, $\Delta$, of cosmic microwave background (CMB) induced by
Thomson scattering of CMB photons off the ionized medium in the mildly
nonlinear structure. The cross-correlation is expected to observationally
unveil the biasing relation between the dark matter and ionized medium
distributions in the large-scale structure. We develop a formalism for
calculating the cross-correlation function and its angular power spectrum based
on the small angle approximation. As a result, we find that leading
contribution to the cross-correlation comes from the secondary CMB fluctuation
field induced by the quadratic Doppler effects of the bulk velocity field $v$
of ionized medium. The magnitude of the cross-correlation can be estimated as
$\kaco{\kappa\Delta}\sim 10^{-10}$ on small angular scales ($l\simgt 1000$)
under the currently favored cold CDM model of the structure formation and the
simplest scenario of homogeneous reionization after a given redshift of $z_{\rm
ion}\simgt 5$. Interestingly, we find that the density-modulated quadratic
Doppler effect of $O(\delta_e v^2)$ produces the cross-correlation with the
weak lensing field that has {\em linear} dependence on the electron density
fluctuation field $\delta_e$ or equivalently on the biasing relation between
the dark matter and electron distributions. In other words, the angular power
spectrum could be both positive or negative, depending on the positive biasing
or antibiasing, respectively. Detection of the cross-correlation thus offers a
new opportunity to observationally understand the reionization history of
intergalactic medium connected to the dark matter clustering.
