We elucidate the flexoelectricity of semiconductors in the high strain gradient regime, the underlying mechanism of which is less understood. By using the generalized Bloch theorem, we uncover a strong flexoelectric-like effect in bent thinfilms of Si and Ge due to a high-strain-gradient-induced band gap closure. We show that an unusual type-II band alignment is formed between the compressed and elongated sides of the bent film. Therefore, upon the band gap closure, electrons transfer from the compressed side to the elongated side to reach the thermodynamic equilibrium, leading to a pronounced change of polarization along the film thickness dimension. The obtained transverse flexoelectric coefficients are unexpectedly high with a quadratic dependence on the film thickness. This new mechanism is extendable to other semiconductor materials with moderate energy gaps. Our findings have important implications for the future applications of flexoelectricity in semiconductor materials.