Carrier generation, recombination and drift/diffusion are the three major phenomena that occur inside a solar cell under illumination. If we consider only the base region of the cell, neglecting the emitter since its contribution is very low in comparison to that of the base and also assume a quasi-neutral p-type base (QNB) in low injection condition and no lateral effect, then, the principal transport mechanism remain a one dimensional diffusion of minority carriers (electrons).

In steady state we have:

The solution of Equation (5) can be written as:

Coefficients A₁ and B₁ are determined considering the two following boundary conditions:

- At the junction ():

With S_f being the junction dynamic velocity corresponding to the flow of carrier through the junction.

- In the middle of the base (x = 0):

Giving these two boundary conditions and Equation (6) and after some calculations, the final expression of the minority carrier density is written as:

Based on the excess minority carrier density, we can derive both photocurrent density J_n and photovoltage respectively as:

with q the elementary charge ; the factor 2 comes from the presence of the two junctions on both sides of the base. For the photovoltage, thanks to Boltzmann’s law, we have:

with n_i the intrinsic concentration and V_T the thermal voltage:

where k is the Boltzmann’s constant and T the temperature.

When the solar cell is illuminated, it can deliver the power P given by:

The power loss in the intrinsic diode is neglected given it is negligible compared to the whole power.

The measurement of the quantum efficiency of solar cells is generally used to quantify recombination losses. The internal quantum efficiency R_Q of a solar cell is defined as the fraction of collected electron/hole pairs per photon of wavelength that enters the solar cell. The commonly applied analysis uses a plot of the inverse quantum efficiency versus the inverse of the absorption; in this study, it is the dependence on the base doping that is analyzed and how it is related to the conversion efficiency.

The internal quantum efficiency is expressed as:

We present on Figure 2 the excess minority carrier density versus base doping density for various illumination wavelengths.

Figure 2 shows that excess minority carrier density in the base slightly increases firstly until a base doping density of about 10¹⁷ cm⁻³. Above that doping density, carrier density decreases very significantly. Indeed, below 10¹⁷ cm⁻³, an increase of base doping leads to a little increase in base doping but above 10¹⁷ cm⁻³ recombination processes becomes more and more predominant, leading to a decrease of the carrier density as observed.

In the following figures, we plotted the photocurrent density versus base doping density for various illumination wavelengths. In Figure 3 we considered wavelength range of 0.66 µm to 0.78 µm and in Figure 4 we considered wavelength range of 0.8 µm to 0.92 µm.

These figures show clearly that photocurrent density is more interesting while the base doping density is of the order of 10¹⁷ cm⁻³ and below. These curves also show that there is a limit to base doping density as we can observe a very marked decrease of the photocurrent densities near and above 10¹⁷ cm⁻³. Given that recombination activities (centers) are directly related to base doping density that is why photocurrent density is so sensitive to base doping density.

Below 0.8 µm, photocurrent density increases with illumination wavelength contrary to wavelengths above 0.8 µm; this could be explained by the illumina-

tion spectrum with lower wavelengths associated to higher energy, generating more carriers in the base of the solar cell.

The maximum output power is computed versus base doping density and the results are presented in Figure 5.

We can observe an increase of the maximum output power for lower base doping density until 10¹⁶ cm⁻³, and above that value, we have a very marked decrease as for the excess minority carrier density. This confirm the fact that there is an optimum doping density for the base and above that doping density the performance of the solar cell decreases significantly.

This result has been predicted previously but the difficulty is that the exact profile of the maximum power versus base doping density was not known. This blank is now filled for vertical parallel junction solar cells.

The internal quantum efficiency of the VMJ cell has also been evaluated for various illumination wavelengths and different base doping densities on Figure 6.

This figure shows that VMJ solar cell’s conversion efficiency increases with illumination wavelength between 0.3 µm to a certain threshold l_s from which it began decreasing. This is due to the manner light is absorbed in the semiconductor material by Beer-Lambert law and the dependence of the absorption coefficient to the illumination wavelength. We can also observe the shifting effect of base doping density on the threshold point with a net decrease of the conversion efficiency maximum and shrinkage of the conversion efficiency bandwidth.

We then see the importance of doping density on the VMJ solar cell performance.

We want to know now how the maximum output power is related to the internal quantum efficiency; we then plotted the maximum output power versus internal quantum efficiency in Figure 7.

This figure shows that the output power increases with increasing internal quantum efficiency. Given that internal quantum efficiency is the fraction of collected electron/hole pairs per photon of wavelength that enters the solar cell, an increase in IQE mean that there are more and more collected electron/hole pairs and more collected electron/hole pairs also means more generation and then more power.