Terahertz photoconductive antenna (PCA) is an important device for generating ultrabroadband terahertz radiations, being applicable in various scenarios. However, the metallic electrodes in PCAs, a pair of coplanar strip lines (CSL), always produce horizontal electrode modes in a broad THz band, thus resulting in low directivity in the vertical direction. Here, we introduce spoof surface plasmon polariton (SSPP) structures to suppress horizontal electrode modes in a broad band. The suppression principles are accounted to both the forbidden band of the fundamental SSPP mode and the orthogonality between source and higher-order SSPP modes. In the SSPP-modified PCA, we achieve around 2 dBi higher directivity in the vertical direction compared to a typical CSL PCA. Unlike the narrow bands inheriting from conventional metamaterial resonators, the relative operational band of the SSPP-modified PCA is as broad as 48%. This planar SSPP structure is compatible with the well-developed micro fabrication technologies. Thus, our scheme can be combined with the semiconductor material engineering and plasmonic nanoscale structures for further increasing THz output power.