This paper investigates theoretically the behavior of the space charge region of a silicon solar cell and its associated capacitance under the effect of an external electric field. The purpose of this work is to show that under illumination the solar cell’s space charge region width varies with both operating point and the external induced electric field and how the solar cell capacitance varies with the space charge region width. Based on a 1D modelling of the quasi-neutral p-base, the space charge region width is determined and the associated capacitance is calculated taking into account the external electric field and the junction dynamic velocity. Based on the above calculations and simulations conducted with Mathcad, we confirmed the linear dependence of the inverse capacitance with space charge region width for thin space charge region and we exhibit an exponential dependence for large space charge region.
The p-n junction remains the core of semiconductor devices; the device performance and behavior are related directly to those of this p-n junction. Among many parameters that influence the operation of the p-n junction, the injection level plays a very important role [
The space charge region and the total capacitance can also characterize the p-n junction [
[
The aim of this paper is to analyze the behavior of the space charge region and the capacitance of a forward biased silicon solar cell on the one hand, and, on the other, the relationship between the capacitance and the space charge region width for various polarization induced electric fields.
When illuminated, three major phenomenon occur inside the solar cell: carrier generation, recombination and drift/diffusion. Given that the emitter thickness is typically less than 1 µm with a high doping density contrary to that of the base region (thickness up to 300 µm and doping about one hundred less than that of the emitter), the contribution of the emitter is very less than that of the base. We then neglect the emitter region and consider only the base region of the solar cell. Regarding to the space charge region width xE + xB (
We also assume a quasi-neutral base with low injection condition, no lateral effect and the solar cell forward biased in steady state. Then, the main transport mechanism is described by a one-dimension diffusion of minority carriers (electrons) written as [
∂ 2 δ ( x ) ∂ x 2 + μ ⋅ E D ⋅ ∂ δ ( x ) ∂ x − δ ( x ) L 2 + G ( x ) D = 0 (1)
Let us consider L E = μ ⋅ E ⋅ τ and taking into account that L 2 = τ ⋅ D , Equation (1) is rewritten as:
∂ 2 δ ( x ) ∂ x 2 + L E L ⋅ ∂ δ ( x ) ∂ x − δ ( x ) L 2 = − G ( x ) D (2)
δ ( x ) is the excess minority carrier density, L is their diffusion length and D their diffusion coefficient.
µ is the excess minority carrier mobility and E the external induced electric field.
Carriers generation rate G(x) at the depth x in the base is in the form [
G ( x ) = ∑ i = 1 3 a i ⋅ exp ( − b i ⋅ x ) (3)
ai and bi are tabulated values obtained from AM1.5 solar irradiance and the dependence of the absorption coefficient on the illumination wavelength [
The excess minority carrier density δ(x) is then given by:
δ ( x ) = exp ( β ⋅ x ) ⋅ [ A ⋅ cosh ( α ⋅ x ) + B ⋅ sinh ( α ⋅ x ) ] + ∑ i = 1 3 K i ⋅ exp ( − b i ⋅ x ) (4)
with α = ( L E 2 + 4 ⋅ L 2 ) 1 2 2 ⋅ L 2 , β = − L E 2 ⋅ L 2 and K i = − a i ⋅ L 2 D ⋅ [ L 2 ⋅ b i 2 − L E ⋅ b i − 1 ]
By help of the following boundary conditions (Equation (5) and Equation (6)), coefficients A and B are evaluated easily:
- at the junction (x = 0):
∂ δ ( x ) ∂ x | x = 0 = S f D ⋅ δ ( 0 ) (5)
- at the backside (x = H):
∂ δ ( x ) ∂ x | x = H = − S b D ⋅ δ ( H ) (6)
H is the base thickness, Sb is the back surface velocity that traduce carrier lost at the backside of the cell; Sf is the junction dynamic velocity describing how carrier flow through the junction and is directly related to the external load conditions [
We now derive the capacitance C of the solar cell as follow [
C = d Q d V p h (7)
Q is the total amount of charge across the junction and Vph the photovoltage across the junction; we have Q = q δ ( x = 0 ) with q the elementary charge.
The photovoltage result from the Boltzmann relation as:
V p h = V t ⋅ ln ( N b ⋅ δ ( 0 ) n i 2 + 1 ) (8)
Vt is the thermal voltage, ni the intrinsic carrier concentration and Nb the base doping density.
Replacing Q by its expression and rearranging the capacitance equation, we obtain:
C = q ⋅ d δ ( x = 0 ) d S f ⋅ 1 d V p h d S f (9)
Finally, replacing Vph by the corresponding expression and doing some calculations we obtain the capacitance as [
C = q ⋅ n i 2 N b V t + q δ ( 0 ) V t (10)
Based on the above mathematical formulation, we performed simulation using programs written in Mathcad software with various induced fields through different forward biased voltages and various external load conditions through the dynamic junction velocity.
The relative excess minority carrier density is the ratio of the minority carrier density and the corresponding maximum value for a given operating conditions.
This figure shows that the excess minority carrier density increase with the depth x in the base until a maximum situated at a certain depth x0 in the base. Above x0, the excess minority carrier density decrease very quickly. Since illumination comes from the front side and taking into account the “Beer-Lambert” law, there are more carrier generated near the front side. Due to the presence of the junction, the photogenerated carrier in the neighborhood of the junction flow through the junction, giving rise to the observed maximum at x0.
For increasing electric field, this maximum is left shifted leading to a space charge region width reduction; this reduction corresponds to an increase of the carrier stored in the base because there is less and less carrier flowing through the junction. This increase of the carrier stored in the base lead to an increasing capacitance. That is, the solar cell capacitance depends directly on the induced electric field.
We now present the relative excess minority carrier density versus the depth x in the base for various junction dynamic velocities (
This figure shows that the maximum of relative excess minority carrier density is also shifted according to the junction dynamic velocity but now the maximum is right shifted contrary to that obtained with the induced electric field. This behavior could be well explained by the fact that when the junction dynamic velocity Sf increase, carrier flow through the junction increase also. There are less and less free carrier in the neighborhood of the junction leading to more depleted region that is, a widening of the space charge region. The decrease in excess minority carrier density in the base and especially near the junction is
associated to a decreasing capacitance since there is less and less free carrier stored in the base.
From the maximum of the excess minority carrier density, we have extracted the corresponding depth xSCR in the base given to be the space charge region width.
As stated previously, there is a widening of the space charge region with increasing junction dynamic velocity. Near short circuit, carrier flow through the junction is very important and these carriers come mainly from the neighborhood of the junction, emptying this region and thus leading to a roll back of the maximum of the excess minority carrier density deeply in the base. That is, the more the solar cell operate near short circuit, the more the space charge region is wide.
For increasing induced electric field, more and more carriers in the bulk of the base are drifted to the neighborhood of the junction so that they can flow through the junction. Then the bulk is emptied as seen on
Since charge variation in the solar cell is related to a variation of the solar cell capacitance and given that the charge variation is also related to the space charge region width, then the capacitance is directly related to the space charge region width. To illustrate this, we have plotted in
We can observe an increasing inverse capacitance with space charge region width and this increase is more marked when the space charge region is the largest for a given operating conditions. We can see in
small space charge region width as previously demonstrated [
For larger space charge region, this behavior is fitted as an exponential grow (
1 C = y 0 + A 1 ⋅ exp ( x − x 0 t 1 ) + A 2 ⋅ exp ( x − x 0 t 2 ) (11)
Finally, we can write that:
{ 1 C = a ⋅ x S C R + b if x < x t h r e s 1 C = y 0 + A 1 ⋅ exp ( x − x 0 t 1 ) + A 2 ⋅ exp ( x − x 0 t 2 ) if x > x t h r e s (12)
xthres is a certain threshold depending on both operating conditions and base doping density; a, A1, A2, x0, y0, t1 and t2 are adjustments coefficients presented in
| Linear fit parameters | |
|---|---|
| Intercept (b) | 2141.1187 |
| Slope (a) | 5.96156E6 |
| Exponential grow fit parameters | |
| y0 | 39,860.75428 |
| x0 | 0.00613 |
| A1 | 0.01518 |
|---|---|
| t1 | 7.53284E−5 |
| A2 | 21,702.42063 |
| t2 | 4.68509E−4 |
The behavior of the excess minority carrier density in the base of a silicon solar cell was investigated with respect to induced electric field and junction dynamic velocity. We showed that the space charge region width is directly related to both induced electric field and junction dynamic velocity. This dependence was then studied and we proved that the space charge region behaves as a plane capacitor only for thinner space charge region. For larger space charge region, based on a theoretical fit of the 1/C-xSCR curve we pointed out a nonlinear relation (exponential growth) between the space charge region width and the inverse of the capacitance.
Future work is on the one hand experimental to validate the theoretical model used and, on the other hand, to consider 3D modeling for poly or multi-crystalline solar cells by taking into account the effects of grain boundaries.
The authors declare no conflicts of interest regarding the publication of this paper.
Zouma, B., Barro, F.I. and Honadia, P.A.A. (2021) Silicon Solar Cell Space Charge Region and Capacitance Behavior under Electric Field. Energy and Power Engineering, 13, 41-50. https://doi.org/10.4236/epe.2021.131003