3D face generation has achieved high visual quality and 3D consistency thanks to the development of neural radiance fields (NeRF). However, these methods model the whole face as a neural radiance field, which limits the controllability of the local regions. In other words, previous methods struggle to independently control local regions, such as the mouth, nose, and hair. To improve local controllability in NeRF-based face generation, we propose LC-NeRF, which is composed of a Local Region Generators Module (LRGM) and a Spatial-Aware Fusion Module (SAFM), allowing for geometry and texture control of local facial regions. The LRGM models different facial regions as independent neural radiance fields and the SAFM is responsible for merging multiple independent neural radiance fields into a complete representation. Finally, LC-NeRF enables the modification of the latent code associated with each individual generator, thereby allowing precise control over the corresponding local region. Qualitative and quantitative evaluations show that our method provides better local controllability than state-of-the-art 3D-aware face generation methods. A perception study reveals that our method outperforms existing state-of-the-art methods in terms of image quality, face consistency, and editing effects. Furthermore, our method exhibits favorable performance in downstream tasks, including real image editing and text-driven facial image editing.