Silicon solar cell under light

A silicon solar cell, a thin n⁺ emitter on a 60 µm p base, lit by one colour at a time: its current–voltage curve and its spectral response, both checked against textbook theory, and where the light makes electron–hole pairs and the quasi-Fermi levels split. Blue light is absorbed within a fraction of a micron, in the emitter; infrared goes deep into the base, or straight through. Vary the bias, the wavelength, the light and the base's lifetime. (More about the model below.)


  
About this simulation

  
Light is a reservoir: the reaction photon → e⁻ + h⁺, its rate falling off as the light is absorbed
    (Beer–Lambert), G(x) = Φα e−αx, with silicon's absorption coefficient α at each wavelength from
    M. A. Green, Sol. Energy Mater. Sol. Cells 92, 1305 (2008). The photons' μ is their energy
    ħω, the most it can be: the reverse reaction, a pair recombining into the light, is then negligible while
    the quasi-Fermi levels split by much less than ħω. There's no reflection; every photon enters the front,
    and what reaches the back leaves.

  
The band diagram is the familiar one, electron energy up: the band edges
    Ec and Ev (thin) and the quasi-Fermi levels EFn and EFp
    (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 μ̄e⁻ = EFn per particle, holes
    one of +1 with μ̄h⁺ = −EFp, 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.) Under light the quasi-Fermi levels split even at zero bias; at open circuit they split by
    qVoc and no current flows.

  
A pair is collected if its minority carrier reaches the junction before it recombines or reaches a
    contact, which is a sink here (no back-surface field, no front passivation). That chance η(x) is 1 in the
    depletion layer and falls off as sinh toward each contact, over a diffusion length: Ln in the
    base, Lp in the heavily doped emitter, where the lifetime is a thousand times shorter. So blue
    light, absorbed in the emitter, is mostly lost to the front contact; red light, absorbed within a
    diffusion length of the junction, is collected; and infrared is barely absorbed at all. The middle plot
    draws G(x) and the part collected, G(x)η(x), whose area is Jsc/q. The spectral response plots
    Jsc/qΦ, the external quantum efficiency, at short circuit (so the bias doesn't enter) and the
    present light (which, at low injection, doesn't matter either: the response is linear), simulated at
    each colour against
    ∫αe−αxη(x)dx, the generalisation of Gärtner's formula to this cell (from the depletion
    approximation). The readout checks Voc = VT ln(Jsc/J₀ + 1), with J₀ from
    the dark curve, and the dashed J–V curve is the dark curve shifted by Jsc, the superposition
    principle. Under the plots, the kit's check() audits the open-circuit state: where every
    pair goes (nothing is collected, so each one recombines, in the base or at a contact, which is a sink
    for both carriers), and the same cell on a grid twice as fine, whose Voc and charges must
    agree to 1%.