3D single object tracking is a key issue for robotics. In this paper, we propose a transformer module called Point-Track-Transformer (PTT) for point cloud-based 3D single object tracking. PTT module contains three blocks for feature embedding, position encoding, and self-attention feature computation. Feature embedding aims to place features closer in the embedding space if they have similar semantic information. Position encoding is used to encode coordinates of point clouds into high dimension distinguishable features. Self-attention generates refined attention features by computing attention weights. Besides, we embed the PTT module into the open-source state-of-the-art method P2B to construct PTT-Net. Experiments on the KITTI dataset reveal that our PTT-Net surpasses the state-of-the-art by a noticeable margin (~10%). Additionally, PTT-Net could achieve real-time performance (~40FPS) on NVIDIA 1080Ti GPU. Our code is open-sourced for the robotics community at https://github.com/shanjiayao/PTT.