Perovskite solar cells are notorious for exhibiting transient behaviour not seen in conventional inorganic semiconductor devices. Significant inroads have been made into understanding this fact in terms of rapid ion migration, now a well-established property of the prototype photovoltaic perovskite MAPbI$_3$ and strongly implicated in the newer mixed compositions. Here we study the manifestations of ion migration in frequency-domain small-signal measurements, focusing on the popular technique of Electrical Impedance Spectroscopy (EIS). We provide new interpretations for a variety of previously puzzling features, including giant photo-induced low-frequency capacitance and negative capacitance in a variety of forms. We show that these apparently strange measurements can be rationalized by the splitting of AC current into two components, one associated with charge-storage, and the other with the quasi-steady-state recombination current of electrons and holes. The latter contribution to the capacitance can take either a positive or a negative sign, and is potentially very large when slow, voltage-sensitive processes such as ion migration are at play. Using numerical drift-diffusion semiconductor models, we show that giant photo-induced capacitance, inductive loop features, and low-frequency negative capacitance all emerge naturally as consequences of ion migration via its coupling to quasi-steady-state electron and hole currents. In doing so, we unify the understanding of EIS measurements with the comparably well-developed theory of rate dependent current-voltage (I-V) measurements in perovskite cells. Comparing the two techniques, we argue that EIS is more suitable for quantifying I-V hysteresis than conventional methods based on I-V sweeps, and demonstrate this application on a variety of cell types.
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