  The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown
to be global well-posed for the initial date in $H^{s}$ provided
$s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below
$L^2$, we construct an almost conserved quantity using multilinear correction
term following the $I$-method introduced by Colliander, Keel, Staffilani,
Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle
the resonant interactions which are significant due to the multidimensional and
multilinear setting of the problem. The proof relies upon the bilinear
Strichartz estimate and the nonlinear Loomis-Whitney inequality.
