  In this paper, we study one-Lee weight and two-Lee weight codes over
$\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee
weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a
complete classification of one-Lee weight
$\mathbb{Z}_2\mathbb{Z}_2[u]$-additive formally self-dual codes is obtained.
The structure of two-Lee weight projective $\mathbb{Z}_2\mathbb{Z}_2[u]$ codes
is determined. Some optimal binary linear codes are obtained directly from
one-Lee weight and two-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive
codes via the extended Gray map.
