Depletion approximation

The depletion approximation provides an analytical - or semi-analytical - solution to the Poisson-drift-diffusion equations described in the previous section applied to simple PN homojunction solar cells. Historically, it has been used extensively to model solar cells and it is still valid, to a large extent, for traditional PN junctions. More importantly, it requires less input parameters than the PDD solver and these can be easily related to macroscopic measurable quantities, like mobility or diffusion lengths. The DA model is based on the assumption that around the junction between the P and N regions, there are no free carriers and therefore all the electric field is due to the fixed, ionized dopants. This “depletion” of free carriers reaches a certain depth towards the N and P sides; beyond this region, free and fixed carriers of opposite charges balance and the regions are neutral. Under these conditions, Poisson’s equation decouples from the drift and diffusion equations and it can be solved analytically for each region. For example, for a PN junction with the interface between the two regions at z=0, the solution to Poisson’s equation will be:

\phi(z) =
        0  & \mbox{if } z < -w_p \\
        \frac{qN_a}{2\epsilon_s}(z+w_p)^2  & \mbox{if } -w_p < z < 0 \\
        -\frac{qN_d}{2\epsilon_s}(z-w_n)^2 + V_{bi} & \mbox{if } 0 < z < w_n  \\
        V_{bi} & \mbox{if } w_n < z

where w_n and w_p are the extensions of the depletion region towards the N and P sides, respectively, and can be found by the requirement that the electric field F and the potential \phi need to be continuous at z=0. V_{bi} is the built-in voltage, which can be expressed in terms of the doping concentration on each side, N_d and N_a, and the intrinsic carrier concentration in the material, n_i^2:

V_{bi} = \frac{k_bT}{q} \ln \left(\frac{N_dN_a}{n_i^2} \right)

Another consequence of the depletion approximation is that the quasi-Fermi level energies are constant throughout the corresponding neutral regions and also constant in the depletion region, where their separation is equal to the external bias qV. Based on these assumptions, the drift-diffusion equations simplify and an analytical expression can be found for the dependence of the recombination and generation currents on the applied voltage. A full derivation of these expressions is included in Nelson (2003) ([1]).

Solcore’s implementation of the depletion approximation includes two modifications to the basic equations. The first one is allowing for an intrinsic region to be included between the P and N regions to form a PIN junction. For low injection conditions (low illumination or low bias) this situation can be treated as described before, simply considering that the depletion region is now widened by the thickness of the intrinsic region. Currently, no low doping level is allowed for this region.

The second modification is related to the generation profile, which in the equations provided by Nelson is given by the BL law which has an explicit dependence on z and results in analytic expressions for the current densities. In Solcore, we integrate the expressions for the drift-diffusion equations under the depletion approximation numerically or by using the Green’s function method to allow for an arbitrary generation profile calculated with any of the optical solvers. It should be noted that although the equations are integrated numerically this will not be a self-consistent solution of the Poisson-drift-diffusion equations, as is achieved by the PDD solver.

Detailed balance functions