We analyze the performance of a cellular network, where Poisson point process distributed half-duplex (HD) downlink (DL) and uplink (UL) users are served by multiple full-duplex (FD) base stations (BSs). To address the surge in interference in the network due to the simultaneous operation in time and frequency of the FD BSs, we (a) adopt a self-interference cancellation scheme at each BS, and (b) apply linear interference alignment in each cell to cancel the intra-cell interference. Further, to better capture the distribution of the FD BSs, we model the BSs as a Matérn hard-core point process, in which a minimum distance is imposed between points. The performance of both UL and DL users is analyzed by deriving general expressions and closed-form approximations for the outage probability and throughput. Next, simulations are carried out for both macro and micro cell environments under both FD and HD operations with respect to various network parameters. Our results reveal several fundamental characteristics and the necessary conditions required for the successful deployment of such networks.INDEX TERMS Full-duplex communication, interference alignment, Matérn hard-core point process, stochastic geometry.