Perovskite solar cell: hysteresis from mobile ions

A lead-halide perovskite cell's current–voltage curve depends on how fast you measure it, because the perovskite holds mobile iodide vacancies as well as electrons and holes. Held at 1.2 V, then swept down to 0 V and back, the vacancies take seconds to follow: scanned slowly they keep up, scanned fast they stay frozen, and in between the two sweeps disagree. Watch the loop open and close with the scan rate, and the vacancies pile up in nanometre-thin layers at each face, screening the field the carriers need. Checked against IonMonger, an independent perovskite simulator. (More about the model below.)

About this simulation

The cell is IonMonger's default planar device [1]: TiO₂ (100 nm) | MAPbI₃ (400 nm) | spiro-OMeTAD (200 nm) at 298 K, lit through the TiO₂ with Beer–Lambert absorption, with recombination through traps (SRH) in the perovskite and at each face. The perovskite holds iodide vacancies, 1.6 × 10¹⁹ cm⁻³ over an equal immobile negative background, moving with D = 1.0 × 10⁻¹⁷ m²/s; they can't leave it. That's three species, electrons, holes and vacancies, each moving down its own electrochemical potential. The same device runs in driftlet's test suite against IonMonger's own results at seven scan rates, agreeing to a few parts in 10⁴ in the hysteresis index and within a third of a millivolt along the whole loop; the rings on the right are IonMonger's.

The level diagram uses the species voltage V_i = μ̄_i/(z_iF), a species' electrochemical potential per unit charge. For electrons and holes it's the ordinary band diagram upside down, in volts: the quasi-Fermi levels (thick) and the band edges (thin). The vacancies' level, drawn shifted by 4.5 V (⌇) to sit beside them, is φ itself wherever they're at their background density, so in the bulk its slope is the field: the field that pushes electrons toward the TiO₂ and holes toward the spiro, and that drives the vacancies too. Held at 1.2 V, past the open-circuit voltage, the vacancies settle where they screen out the built-in field. Sweeping down, they lag: their layers still screen the old voltage, so the bulk field points the carriers' way and the current is high. Sweeping back up, the lagging layers now push the wrong way, carriers pile up at the faces and recombine there, and the forward curve sags. Too fast, and the vacancies never move at all: both sweeps see the same frozen layers and agree again.

The two small diagrams zoom on the vacancies at each face, against their background density: layers a nanometre or two thick (the Debye length at this density), piling up at one face and thinning at the other. Most of the voltage drops across the lightly doped transport layers, so these layers stay modest, a few times the background at most. The hysteresis index is (P_rev − P_fwd)/P_rev at maximum power. It peaks where one sweep lasts about as long as the vacancies take to rearrange, a few seconds. The dashed J–V curve is the steady state (vacancies given time to settle at every voltage), the dotted one has them frozen where 1.2 V left them. One dimension makes this a planar cell: no grain boundaries or lateral currents. The vacancies follow Boltzmann statistics, with no limit on how many fit in a layer.

[1] N. E. Courtier, J. M. Cave, J. M. Foster, A. B. Walker and G. Richardson, "How transport layer properties affect perovskite solar cell performance: insights from a coupled charge transport/ion migration model", Energy Environ. Sci. 12, 396 (2019); IonMonger: N. E. Courtier et al., J. Comput. Electron. 18, 1435 (2019), github.com/PerovskiteSCModelling/IonMonger.