Multi-junction electrical solver

A complete photovoltaic solar cell can include one or more junctions, metal contacts, optical layers (including anti-reflective coatings and nano-photonic structures) and tunnel junctions. The junctions, in turn, might range from simple PN homojunctions to complex heterojunctions, including multi-quantum well structures. The solvers described so far only calculate the properties of single junction devices. To combine them into a multi-junction device, it is necessary to consider that the individual junctions are electrically connected in series and the potential coupling of light emitted by the wider bandgap junctions into those with smaller bandgap.

No radiative coupling

We first consider the case of no radiative coupling between junctions. This is a good approximation for materials which do not radiate efficiently or radiative materials working at low concentration, when the fraction of radiative recombination compared to non-radiative recombination is low. In this case, the IV curve of each junction can be calculated independently of each other and the current flowing through the MJ structure is limited by the junction with the lowest current at any given voltage. Series resistances defined for each junction are now added together and included as a single term. The operating voltage of each of the junctions is finally back-calculated and added together to get the voltage of the MJ device.

The pseudocode for this solver is:

  1. Calculate the I_j(V) of each junction j in the structure.

  2. Find the current flowing through the MJ device as I_{MJ}(V) = I_j(V), if |I_j(V)| = \min ([|I_1(V)|...|I_N(V)|]).

  3. Calculate the voltage of each junction by interpolating its IV curve at the new current values, V_j(I_{MJ}), and the voltage dropped due to the series resistances, V_{Rs} = R_s I_{MJ}.

  4. Calculate the total voltage at a given current as V_{MJ} = V_{Rs} + \sum_j V_j.

  5. Interpolate the I_{MJ}(V_{MJ}) and the I_{MJ}(V_j) to the desired output voltage values.

The figure shows the simulated IV curve of a 3J solar cell made of Ge/InGaAs/GaAsP. The electrical properties of the three junctions were calculated using the depletion approximation solver. In the dark the voltages of each of the junctions at a given current add together, resulting in the total voltage of the MJ structure. The R_s contribution to the voltage goes in the same direction as those of the junctions. Under illumination the junction producing the lower current (the top junction in this case) limits the overall current of the MJ device. At zero bias, or even at some negative bias, the non-limiting junctions are positively biased, recombining all the photocurrent that cannot be extracted because of the limiting top cell. The contribution of the R_s to the voltage of the MJ device is negative, resulting in a reduction of the fill factor and the overall efficiency of the solar cell.


With radiative coupling

Radiative coupling takes place when the light produced by a high bandgap junction due to radiative recombination is absorbed by a lower bandgap junction, contributing to its photocurrent and changing the operating point. It has been identified in numerous highly radiative materials such as quantum well solar cells and III-V MJ solar cells . It appears as an artefact during the QE measurements of MJ solar cells but it is also an effect that can be exploited to increase the performance of MJ devices and their tolerance to spectral changes, resulting in superior annual energy yield.

The radiative coupling formalism included in Solcore is based on the works by Chan et al. (2014) and Nelson et al. (1997).

For a more detailed description of the radiative coupling calculator refer to the main Solcore paper (open access) and references therein.

An example of the radiative coupling in action can be found here.

Tunnel junctions

Solcore includes partial support for tunnel junctions. They represent an optical loss due to parasitic absorption in the layers, but also an electrical loss. At the moment, there are two models for tunnel junctions. The first one is a simple resistive model, where the tunnel junction is simply modelled as a series resistance. This approximation should be valid in most cases, but will break down if the current is close to or higher than the peak current density of the junction.

The second model is a parametric model, based on the simple formalism described by Sze. In this model, the total current of the tunnel junction will have three components: the tunnel current J_T accounting for band-to-band transport, the excess current J_{ex} related to transport across states inside the forbidden gap, and the diffusion current J_D, which is the usual minority-carrier injection current in PN junctions. The following equations summarise all these components.

J_{TJ} = J_{T} + J_{ex} + J_D

J_{T} = \frac{J_P V}{V_P} \exp{\left( 1 - \frac{V}{V_P} \right) }

J_{ex} =J_V \exp{\left[ C \left( V - V_V \right) \right] }

J_{D} =J_0 \left[  \exp{ \left( \frac{qV}{k_b T}  \right) } - 1  \right]

As illustrated in the next figure, J_P and V_P are the peak current and voltage, J_V and V_V are the valley current and voltages, C is a prefactor of the exponent and J_0 the reverse saturation current. In this simple implementation, these 6 parameters need to be provided as inputs, and can be used as fitting parameters to reproduce experimental data. This allows to correctly account for the break down of the tunnel junction in situations when the current is above the peak current. The code for this example can be found here.


Solcore can also accept external IV data for the tunnel junctions and the implementation of the more rigorous, but still analytic model, described by Louarn et al. (2016) is currently under way in order to relate the tunnel currents with the actual materials and layer structure used in the tunnel junction definition.

Multi-junction IV functions