  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.
