  Ensemble-based quantum memories are key to developing multiplexed quantum
repeaters, able to overcome the intrinsic rate limitation imposed by finite
communication times over long distances. Rare-earth ion doped crystals are main
candidates for highly multimode quantum memories, where time, frequency and
spatial multiplexing can be exploited to store multiple modes. In this context
the atomic frequency comb (AFC) quantum memory provides large temporal
multimode capacity, which can readily be combined with multiplexing in
frequency and space. In this article, we derive theoretical formulas for
quantifying the temporal multimode capacity of AFC-based memories, for both
optical memories with fixed storage time and spin-wave memories with longer
storage times and on-demand read out. The temporal multimode capacity is
expressed in key memory parameters, such as AFC bandwidth, fixed-delay storage
time, memory efficiency, and control field Rabi frequency. Current experiments
in europium- and praseodymium-doped Y$_2$SiO$_5$ are analyzed within this
theoretical framework, and prospects for higher temporal capacity in these
materials are considered. In addition we consider the possibility of spectral
and spatial multiplexing to further increase the mode capacity, with examples
given for both rare earh ions.
