Organic bilayer solar cell: excitons

In an organic semiconductor, light doesn't make free electrons and holes: it makes excitons, bound pairs that are electrically neutral. They wander by diffusion, whatever the field, for about ten nanometres before they decay, and they split into an electron and a hole only at an interface between a donor and an acceptor. So a third species, beside the electrons and holes, decides how much current the cell gives: a thicker donor absorbs more light, but excitons made far from the interface never reach it. Vary the bias, the donor's thickness, the excitons' diffusion length and the light, and compare with theory. (More about the model below.)

About this simulation

The cell is anode | donor | acceptor | cathode, with light entering through the anode. The donor carries holes and excitons, the acceptor electrons; the numbers are illustrative, of the order of the classic copper phthalocyanine/C₆₀ cell: the donor's HOMO 5.2 eV below the vacuum level, the acceptor's LUMO 4.0 eV, an exciton energy of 1.7 eV, α = 10⁵ cm⁻¹, and contacts with work functions 5.0 and 4.2 eV. The levels are aligned on the vacuum level, an estimate stated as such. Only the donor absorbs here (a real acceptor absorbs too), and the anode blocks excitons.

Three species, each with its own electrochemical potential. Light makes excitons, photon → X, which decay, X → 0, in τ = 1 ns, or reach the interface and split there, X → h⁺(donor) + e⁻(acceptor), downhill by the 0.5 eV between the exciton's energy and the interface's charge-transfer gap E_CT = 1.2 eV (from the acceptor's LUMO to the donor's HOMO). The reverse of that, and the interface's recombination e⁻ + h⁺ → 0, are the same reactions run backward, so detailed balance holds exactly. An exciton, being neutral, has no level on a band diagram; its chemical potential μ_X is what the splitting reaction compares with E_Fn − E_Fp, the quasi-Fermi levels' splitting across the interface. Its concentration tells the same story here, so that's what's drawn.

The band diagram is the familiar one, electron energy up: the band edges E_c and E_v (thin: the LUMO and the HOMO) and the quasi-Fermi levels E_Fn and E_Fp (thick), one Fermi level wherever the carriers are in equilibrium with each other. A slope in a quasi-Fermi level is what drives its carriers' current. Underneath, driftlet works in electrochemical potentials: electrons are a species of charge −1 with μ̄_e⁻ = E_Fn per particle, holes one of +1 with μ̄_h⁺ = −E_Fp, which is exactly the usual drift–diffusion (van Roosbroeck) model, and generation and recombination are the reaction e⁻ + h⁺ ⇌ 0. (The same lines in volts and turned over are species voltages, which earn their place where ions share the diagram, as in the electrochemistry and perovskite demos.)

The checks. An exciton made at depth x reaches the interface with probability η(x) = cosh(x/L)/(cosh(d/L) + (D/vL) sinh(d/L)), from diffusion with decay (L = √(Dτ)), a blocking anode, and the interface taking excitons at the velocity v its splitting reaction gives; so J_sc = qΦ∫αe^−αxη(x)dx, which the thickness plot draws against simulated points. At open circuit no current flows in either layer, the quasi-Fermi levels are flat out to the interface, and there the charge-transfer recombination balances the splitting: V_oc = V_T ln(J_ph/J₀ + 1), with J₀ = F k₀ e^{−(1−α)E_CT/kT} by detailed balance, whatever the layers' transport. A bilayer like this is honestly one-dimensional; the bulk heterojunctions that solve the diffusion-length problem, by mixing donor and acceptor on a 10 nm scale, are not.