  In this work, we consider the following generalized derivative nonlinear
Schr\"odinger equation \begin{align*}
  i\partial_t u+\partial_{xx} u +i |u|^{2\sigma}\partial_x u=0, \quad (t,x)\in
\mathbb R\times \mathbb R. \end{align*} We prove that when $\sigma\ge 2$, the
solution is global and scattering when the initial data is small in
$H^s(\mathbb R)$, $\frac 12\leq s\leq1$. Moreover, we show that when
$0<\sigma<2$, there exist a class of solitary wave solutions $\{\phi_c\}$
satisfying $$ \|\phi_c\|_{H^1(\mathbb R)}\to 0, $$ when $c$ tends to some
endpoint, which is against the small data scattering statement. Therefore, in
this model, the exponent $\sigma\ge2$ is optimal for small data scattering. We
remark that this exponent is larger than the short range exponent and the
Strauss exponent.
