Ag | AgNO₃ | Ag: concentration polarisation and saturation
10 mM silver nitrate between two silver electrodes 20 µm apart, driven by a square wave of cell voltage: the current's burst and sag as salt piles up at one electrode and depletes at the other, and its saturation at a limiting current however hard the cell is driven. Vary the voltage and the half-period. (More about the model below; the impedance demo is the same cell's small-signal side.) Further down, a transistor saturates for the same reason.
About this simulation
The electrolyte is strictly neutral. Ag⁺ is plated at one electrode and stripped at the other; NO₃⁻ can't cross either. So each reversal starts with a burst of current limited only by the solution's resistance; then salt piles up at one electrode and depletes at the other, and the current sags toward its steady value, (4FD₊c/L) tanh(FV/4RT) (dashed), which can never exceed the limiting current 4FD₊c/L. Half-periods shorter than the diffusion time, (L/2)²/D ≈ 60 ms here, reverse the salt before it has piled up, and the current stays near ohmic.
The steady current is distinctly nonlinear. Below about 4RT/F ≈ 0.1 V it rises in proportion to the voltage, but beyond that it saturates: the salt at the depleting electrode is nearly gone, and no more voltage can draw Ag⁺ to it any faster than it diffuses there. The extra voltage goes into the burst at each reversal instead, which only the solution's resistance limits, because the salt hasn't had time to move.
The level diagram uses the species voltage Vi = μ̄i/(ziF): a species' electrochemical potential per unit charge, in volts, so electrons and ions share one axis. Each ion moves down its own μ̄i, so its current flows down slopes in its own Vi (thick; Ag⁺ moves downhill, NO₃⁻ uphill); the thin line under it, the standard level V°i, is where Vi would sit at the reference concentration. Ag⁺'s voltage slopes from one electrode's to the other's, driving the current. NO₃⁻'s is level once it settles, since nitrate carries no current; meanwhile its standard level is sloped, and that slope is the salt's concentration gradient.
The same ceiling in a transistor
A thin-film transistor: an undoped semiconductor film 10 nm thick and 10 µm long between a source and a drain, with a gate along it through 1 mF/m² of dielectric (35 nm of SiO₂). The gate draws electrons into the film, and the drain pulls them along it. Raise the drain voltage and the current climbs, then stops climbing: it saturates.
The film is thin and the channel long, so each slice of it sees the gate straight across: the gate holds a charge C(VG − VFB − φ) per area against it, and the film's electrons balance it (the gradual-channel approximation). That makes this a 1D problem along the channel, the gate a capacitance spread along the film. The band diagram is the familiar one, electron energy up. The gate's Fermi level is flat, since no current flows into it, and it holds the band edge Ec through the capacitance, much as the blocked nitrate's flat level holds the salt in the cell above. The current flows down the electrons' quasi-Fermi level EFn, which the drain pulls down. Where EFn falls away below Ec, the electrons thin out, until at the drain they all but vanish: the channel pinches off. There Ec flattens onto a ceiling set by the gate's level, 0.3 eV above it (the flat-band voltage, the offset between the two materials), the gate rather than the electrons now holding it, and any more drain voltage drops across that sliver while the current hardly changes. The dashed curve is the charge-sheet model's current, I/W = (μ/L)[(σs² − σd²)/(2C) + VT(σs − σd)], with σ the gate charge at each end; it's exact for this model, from below threshold to saturation.
That's where 1D stops. Near pinch-off the field in a real transistor turns two-dimensional, and the saturated current keeps creeping up as the pinch-off point moves toward the source (channel-length modulation, absent here). A silicon MOSFET also has a body under its channel, whose depletion charge this film doesn't have. The model is the textbook long-channel one, not a device simulation.