Bipolar membranes and pn junctions

The previous topic ended with a fixed background charge forcing the standard-state ladder to step across a boundary, the Donnan potential. Put two such regions back to back, with fixed charge of opposite sign on each side, and you have built one of the most satisfying objects in this subject: a device that is at once a semiconductor pn-junction diode and an electrochemical bipolar membrane. Drawn as ViV_i diagrams, the two share one construction and one set of rules: depletion, generation, recombination.

Functional twins

A pn junction joins two doped semiconductors: an nn-type region, with fixed positive donors and mobile electrons, against a pp-type region, with fixed negative acceptors and mobile holes. A bipolar membrane joins two fixed-charge polymer films in the very same arrangement. An anion-exchange membrane carries fixed positive groups and conducts mobile anions (here OH−\mathrm{OH}^-), making it the electrochemical nn-type; a cation-exchange membrane carries fixed negative groups and conducts mobile cations (here H+\mathrm{H}^+), the electrochemical pp-type.[1]

pn junction bipolar membrane
mobile positive carrier hole h+\mathrm{h}^+ cation, e.g. H+\mathrm{H}^+
mobile negative carrier electron e−\mathrm{e}^- anion, e.g. OH−\mathrm{OH}^-
fixed background charge ionized dopants ND+N_{\mathrm{D}}^+, NA−N_{\mathrm{A}}^- bound charges on the membrane
the two halves nn-type and pp-type anion- and cation-exchange
pairing reaction e−+h+⇌∅\mathrm{e}^- + \mathrm{h}^+ \rightleftharpoons \varnothing H++OH−⇌H2O\mathrm{H}^+ + \mathrm{OH}^- \rightleftharpoons \mathrm{H_2O}
junction potential step built-in potential Donnan potential

At equilibrium

At equilibrium every mobile carrier's ViV_i runs flat across the whole device. The pairing reaction sets the spacing between the two carrier levels, and flatness carries it across the device. For the diode, e−+h+⇌∅\mathrm{e}^- + \mathrm{h}^+ \rightleftharpoons \varnothing gives Ve−=Vh+V_{\mathrm{e}^-} = V_{\mathrm{h}^+}, the single Fermi level. For the membrane, H++OH−⇌H2O\mathrm{H}^+ + \mathrm{OH}^- \rightleftharpoons \mathrm{H_2O} gives VH+−VOH−=μH2O/F=−2.457 VV_{\mathrm{H}^+} - V_{\mathrm{OH}^-} = \mu_{\mathrm{H_2O}}/F = -2.457~\mathrm{V}, the same fixed offset we met for autoionized water. The only thing distinguishing the two cases is the formation energy of the product, zero for the electron-hole pair, −237 kJ/mol-237~\mathrm{kJ/mol} for water, and it is the one visible difference between the twins on the diagram: the diode's two carrier levels coincide into a single Fermi line, while the membrane's sit a fixed 2.457 V2.457~\mathrm{V} apart.

The carrier voltages stay flat, but the standard-state ladder cannot. The fixed charge differs on the two sides, so the ladder must rest at a different offset on each side to keep its bulk neutral, precisely as it did across the Donnan membrane. The transition between the two offsets is the junction potential step (built-in potential to the diode, Donnan potential to the membrane), and it falls across a narrow junction region where the mobile carriers have been swept aside, leaving only the bare fixed charge. This is the depletion zone. In the membrane the fixed-charge density is so high that this zone is squeezed down to a few nanometres, far thinner than its semiconductor counterpart (tens of nanometres to a micron, depending on doping), but it plays exactly the same role.

pn junction
pn ⇌⇌Vh+∘V_{\mathrm{h}^{+}}^\circVe−=Vh+V_{\mathrm{e}^-}{=}V_{\mathrm{h}^+}Ve−∘V_{\mathrm{e}^{-}}^\circ−1.2−1.0−0.8−0.6−0.4−0.20.00.20.40.60.81.01.21.4Species Voltage (V)
bipolar membrane
CEMAEM ⇌⇌VOH−V_{\mathrm{OH}^{-}}VOH−∘V_{\mathrm{OH}^{-}}^\circVH+∘V_{\mathrm{H}^{+}}^\circVH+V_{\mathrm{H}^{+}}−2.6−2.4−2.2−2.0−1.8−1.6−1.4−1.2−1.0−0.8−0.6−0.4−0.20.0Species Voltage (V)

Equilibrium, side by side: a pn junction and a bipolar membrane as ViV_i diagrams. The mobile carrier voltages run flat (held a fixed distance apart by the pairing reaction), while the Vi∘V^\circ_i ladder bends through the depletion zone, carrying the junction potential step. (Junction widths schematic, not to scale.) The two panels share the same voltage scale, so the twins' one visible difference stands out: the diode's carrier levels merge, the membrane's sit 2.457 V2.457\ \mathrm{V} apart.

Under bias

Reverse-bias the device and the mobile carriers are pulled away from the junction, widening the depletion zone until almost none remain there to carry the current. Current can continue only if fresh carriers are created at the junction itself. In the diode, thermal or optical energy generates electron-hole pairs, ∅→e−+h+\varnothing \rightarrow \mathrm{e}^- + \mathrm{h}^+. In the membrane, the intense junction field splits water, H2O→H++OH−\mathrm{H_2O} \rightarrow \mathrm{H}^+ + \mathrm{OH}^-, which is exactly what an industrial bipolar membrane is built to do: manufacture acid and base from water and electricity. On the diagram the two carrier voltages split apart at the junction, and the slope of that split sweeps the new carriers out to their respective sides.

Forward-bias it instead, and carriers are injected toward the junction from both sides, where they meet and annihilate. Electrons and holes recombine, e−+h+→∅\mathrm{e}^- + \mathrm{h}^+ \rightarrow \varnothing, and the energy they give up emerges as light in an LED or otherwise as heat; protons and hydroxide ions neutralize, H++OH−→H2O\mathrm{H}^+ + \mathrm{OH}^- \rightarrow \mathrm{H_2O}, releasing theirs as heat.

pn junction
pn Vh+∘V_{\mathrm{h}^{+}}^\circVe−=Vh+V_{\mathrm{e}^-}{=}V_{\mathrm{h}^+}Ve−∘V_{\mathrm{e}^{-}}^\circ−2.0−1.5−1.0−0.50.00.51.0Species Voltage (V)
bipolar membrane
CEMAEM ⇌VOH−V_{\mathrm{OH}^{-}}VOH−∘V_{\mathrm{OH}^{-}}^\circVH+∘V_{\mathrm{H}^{+}}^\circVH+V_{\mathrm{H}^{+}}−3.5−3.0−2.5−2.0−1.5−1.0−0.50.00.5Species Voltage (V)

Under bias (a slider sweeps from reverse to forward). Reverse bias widens the depletion zone and splits the carrier voltages, driving pair generation in the diode or water splitting in the membrane; forward bias injects carriers inward to recombine or neutralize at the junction.

Takeaways

A silicon diode and a water-splitting membrane are functional twins, one junction built from two different sets of carriers. Both join a region of one fixed-charge sign to a region of the other; both hold their mobile carriers a fixed distance apart at equilibrium; both bend the standard-state ladder through a depleted junction; and both answer a reverse bias by creating carrier pairs and a forward bias by destroying them.

We have leaned all along on that depleted junction, the place where neutrality finally breaks and the ladder is free to bend. What sets its width, and the shape of the bend? Those are questions of electrostatics, which is the piece we put in place next.

NEXT TOPIC: Basic electrostatics

  1. These fixed-charge media go by many names depending on the field (ionomers and polyelectrolytes to a materials chemist, permselective or ion-exchange membranes to an engineer), but physically each is just a medium carrying a built-in static charge, the direct counterpart of a semiconductor dopant. The dopant picture is an idealization, though: a real membrane is its own medium, with its own Vi∘V^\circ_i ladder set by how it solvates and sterically packs each ion (the same per-medium ladder as the ITIES of the previous topic), and the counter-ions may even bind locally to the fixed groups. We treat it here as simply doped water for the sake of the analogy. ↩︎