Mixed membership problem for undirected network has been well studied in network analysis recent years. However, the more general case of mixed membership for directed network in which nodes can belong to multiple communities remains a challenge. Here, we propose an interpretable and identifiable model: directed mixed membership stochastic blockmodel (DiMMSB) for directed mixed membership networks. DiMMSB allows that row nodes and column nodes of the adjacency matrix can be different and these nodes may have distinct community structure in a directed network. We also develop an efficient spectral algorithm called DiSP designed based on simplex structures inherent in the left and right singular vectors of the population adjacency matrix to estimate the mixed memberships for both row nodes and column nodes in a directed network. We show that DiSP is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vectors of each row node and each column node using delicate spectral analysis. Numerical results on computer-generated directed mixed membership networks support our theoretical findings and show that our DiSP outperforms its competitor in both error rates and run-time. Applications of DiSP to real-world directed networks demonstrate the advantages of DiSP in studying the asymmetric structure of directed networks.