Depletion approximation

• Example: Traditional GaInP/InGaAs/Ge solar cell

Material and device parameters

The table below lists the device and material parameters required for the depletion approximation solver. It is not necessary to set all these parameters explicitly, as some can be calculated from other parameters or have default values. However, there is no reason to assume the default values are reasonable for your material/device!

Parameter	Set location	Solcore label	Units	Default	Calculable?
n-type region SRV	Junction	sn [1]	m s-1	0	No
p-type region SRV	Junction	sp [1]	m s-1	0	No
Permittivity	Junction or material	permittivity	F m-1	none	Yes, from relative_permittivity [2]
Electron diffusion length	p-type material	electron_diffusion_length	m	none	No
Hole diffusion length	or n-type material	hole_diffusion_length	m	none	No
Electron mobility	p-type material	electron_mobility	m2 V-1 s-1	0.94	No [3]
Hole mobility	n-type material	hole_mobility	m2 V-1 s-1	0.05	No [3]
Intrinsic carrier density	material	ni	m-3	none	Yes, from Nc, Nv and band_gap [4]
Donor (n-type) density	n-type material	Nd	m-3	1	No
Acceptor (p-type) density	p-type material	Na	m-3	1	No
Built-in voltage	Junction (optional)	Vbi	V	none	Yes, from Nd, Na and ni [5]
Shunt resistance	Junction (optional)	R_shunt	Ohm m2	1e14	No [6]

Notes:

1. The n/p labels for the surface recombination velocities sn and sp refer to the region, not the carrier type, e.g. sn is the recombination velocity at the n-type surface, which can be the front or rear surface depending on the cell configuration.

2. relative_permittivity is a dimensionless quantity: permittivity = vacuum permittivity * relative permittivity

3. If the material is not in the mobility model database, which calculates the mobility based on the alloy composition (if applicable), temperature, and doping level, the model will default to calculating the mobility for GaAs. Thus if you are using a custom material not included in the mobility database, you should set this explicitly, since there is no reason to assume the GaAs values are reasonable for your material.

4. The intrinsic carrier density ni can be calculated from the conduction and valence band effective density of states (Nc and Nv respectively, units m-3) and band_gap (SI units used by Solcore: J, NOT eV!).

5. In general, you don’t need to set the built-in voltage Vbi since it is calculated from Nd, Na and ni, all of which are already required.

6. An ideal cell has infinitely high shunt resistance (hence the extremely high default value).

Background

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 , the solution to Poisson’s equation will be:

where and 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 and the potential need to be continuous at . is the built-in voltage, which can be expressed in terms of the doping concentration on each side, and , and the intrinsic carrier concentration in the material, :

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 . 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 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

References

[1]

Nelson, J.: The Physics of Solar Cells. Imperial College Press, London; River Edge, NJ: Distributed by World Scientific Pub. Co, (2003)