To increase the amount of lithium ions that can be received and released in and from a positive electrode active material to achieve high capacity and high energy density of a secondary battery. A composite material of crystallites of LiMn2O4 (crystallites with a spinel crystal structure) and crystallites of Li2MnO3 (crystallites with a layered rock-salt crystal structure) is used as a positive electrode active material. The lithium manganese oxide composite has high structural stability and high capacity.