Panoramic depth estimation has become a hot topic in 3D reconstruction techniques with its omnidirectional spatial field of view. However, panoramic RGB-D datasets are difficult to obtain due to the lack of panoramic RGB-D cameras, thus limiting the practicality of supervised panoramic depth estimation. Self-supervised learning based on RGB stereo image pairs has the potential to overcome this limitation due to its low dependence on datasets. In this work, we propose the SPDET, an edge-aware self-supervised panoramic depth estimation network that combines the transformer with a spherical geometry feature. Specifically, we first introduce the panoramic geometry feature to construct our panoramic transformer and reconstruct high-quality depth maps. Furthermore, we introduce the pre-filtered depth-image-based rendering method to synthesize the novel view image for self-supervision. Meanwhile, we design an edge-aware loss function to improve the self-supervised depth estimation for panorama images. Finally, we demonstrate the effectiveness of our SPDET with a series of comparison and ablation experiments while achieving the state-of-the-art self-supervised monocular panoramic depth estimation. Our code and models are available at https://github.com/zcq15/SPDET.