  Recently, a new spin-1/2 fermionic quantum field with mass dimension one in
four dimensions--Elko field $\lambda$ was introduced as a candidate of dark
matter. In this paper, we investigate the localization of 5D Elko {spinors} on
Minkowski branes by presenting the equation of {the} Elko KK modes. For the 5D
free massless Elko field, the zero mode can be localized on Randall-Sundrum
thin brane but can not be localized on the majority of thick branes. There {do}
not exist bound massive KK modes on all these branes. If the 5D mass term is
introduced, there will exist bound Elko zero mode in Randall-Sundrum brane
model. And when we introduce the Yukawa type coupling $\eta \phi^2 \mathop
\lambda\limits^\neg\lambda$ with $\phi$ the background scalar field, the Elko
zero mode can be localized on some special thick branes with a particular
coupling constant $\eta$. Nevertheless, the massive KK modes still can not be
localized on these branes. These results are very different from that of the
conventional Dirac spinor {field} and the scalar field.

High Energy Physics - Theory;General Relativity and Quantum Cosmology