  Linear complementary dual (LCD) codes over finite fields are linear codes
satisfying $C\cap C^{\perp}=\{0\}$. We generalize the LCD codes over finite
fields to $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes over the ring
$\mathbf{Z}_2\times(\mathbf{Z}_2+u\mathbf{Z}_2)$. Under suitable conditions,
$\mathbb{Z}_2\mathbb{Z}_2[u]$-linear codes that are
$\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes are characterized. We then prove that
the binary image of a $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD code is a binary LCD
code. Finally, by means of these conditions, we construct new binary LCD codes
using $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes, most of which have better
parameters than current binary LCD codes available.
