The paper reported a theoretical study on the photoconductivity of a bifacial silicon solar cell under polychromatic illumination and a constant magnetic field effect. By use of the continuity equation in the base of the solar cell maintained in a constant temperature at 300 K, an expression of the excess minority carriers’ density was determined according to the applied magnetic field, the base depth and the junction recombination velocity. From the expression of the minority carriers’ density, the photoconductivity of the solar cell was deduced and which allowed us to predict some recombination phenomena, the use of such solar cell in optoelectronics. The profile of the photoconductivity also permitted us to utilize a linear model in order to determine an electrical capacitance that varied with magnetic field.
The photoconductivity is one of the important characteristic quantities of solar cells or semiconductors [
We consider an n+-p-p+ type of a bifacial silicon solar cell [
By illuminating the solar cell, through the emitter considered as transparent to incident light because of its very small thickness (1 μm), generation and recombination of electron-hole pairs occur in the base. The excess photo-generated minority carriers (electrons) in the base, are subjected to the electrical force
q E , magnetic force q v ∧ B and frictional force − m τ v during their diffusion
in the crystal lattice. By use of the basic low, one obtains Equation (1):
m d v d t = q E + q v ∧ B − m τ v (1)
where q and τ are respectively the elementary charge and the average lifetime of electrons in the base; v is the electron velocity; E the electric field resulting
from the base polarization.
In quasi-steady-state, the time derivative of v vanishes [
q E + q v ∧ B − m τ v = 0 (2)
Let’s pose:
E = E x i + E y j + E z k (3)
v = v x i + v y j + v z k and B = B j (4)
Substituting Equations (3) and (4) in Equation (2), and rearranging, we obtain the velocity components:
v x = μ o ( 1 + μ o 2 B 2 ) E x − μ o 2 B ( 1 + μ o 2 B 2 ) E z (5)
v y = μ o E y (6)
v z = μ o 2 B ( 1 + μ o 2 B 2 ) E x + μ o ( 1 + μ o 2 B 2 ) E z (7)
In matrix writing, we get:
( v x v y v z ) = ( μ o ( 1 + μ o 2 B 2 ) 0 − μ o 2 B ( 1 + μ o 2 B 2 ) 0 μ o 0 μ o 2 B ( 1 + μ o 2 B 2 ) 0 μ o ( 1 + μ o 2 B 2 ) ) ( E x E y E z ) (8)
Furthermore the current density is the phoconductivity tensor times the electric field as given by Equation (9):
J = ( σ ) E (9)
where:
( σ ) = q δ n μ o ( 1 + μ o 2 B 2 ) ( 1 0 − μ o B 0 ( 1 + μ o 2 B 2 ) 0 μ o B 0 1 ) (10)
The parameter μ o = q τ m is the initial mobility of the electrons in the base
without magnetic field; δ n is the excess minority carriers’ density.
To simplify our study, a one dimension along the x axis is considered. The expression of the photoconductivity then becomes:
σ p h = q μ o δ n ( 1 + μ o 2 B 2 ) (11)
In Equation (11), the excess minority carriers’ density is necessarily required. Thus, in steady-state, the evolution of the minority carriers’ density is governed by the continuity equation:
D n ∂ 2 δ n ∂ x 2 − δ n τ = − G (12)
where Dn is the minority carriers’ diffusion and its expression can be written as:
D n = D o 1 + μ o 2 B 2 (13)
where Do is the minority carriers’ diffusion coefficient without magnetic field; G is the minority carriers’ generation rate [
G = η ∑ i = 1 3 a i e − b i x (14)
with η the solar number, ai and bi are coefficients deduced from modelling of the generation rate considered over all solar radiation spectrum under AM 1.5.
The solution of Equation (12) is given by:
δ n = A e − x L n + C e x L n − ∑ i = 1 3 a i L n 2 e − b i x D n ( b i 2 L n 2 − 1 ) (15)
with:
L n = D n τ et ( b i 2 L n 2 − 1 ) ≠ 0 (16)
where Ln is the minority carriers diffusion length in the base.
The coefficients A and C are determinated by the following boundary conditions [
- at the junction (x = 0)
D n ⋅ ∂ δ n ∂ x | x = 0 = S f ⋅ δ n | x = 0 (17)
- at the rear side (x = H)
D n ⋅ ∂ δ n ∂ x | x = H = − S b ⋅ δ n | x = H (18)
where Sf and Sb are respectively minority carrier junction and back surface recombination velocities. The recombination velocity Sf is the sum of the junction recombination velocity due to the external load that defines the solar cell operating point and the intrinsic junction recombination velocity that is an effective recombination velocity at the emitter-base interface.
We note that, for each curve data, the photoconductivity decreases with the increase of the base depth. In the vicinity of the junction, with a defined operating
point around Sf = 2 × 102 cm・s−1, the photoconductivity is maximum for each curves since the density of the minority carriers is also maximum. But, when the base depth increases, the absorption of the incident photons is weak; accordingly the low minority carriers photogeneration and their recombination in volume, leads to a decrease of the photoconductivity as predicted by the Beer-Lambert law. With an applied magnetic field, the photoconductivity decreases because of the magneto-photoconductivity phenomenon and the deflected minority carriers towards the surface. For an optimal use of such solar cell in optoelectronics, laser or detector, it is necessary to choose a base thickness less or equal to 2 × 10−3 cm (≈20 µm) for a better incident photons absorption and then a decrease of recombination in volume. To highlight the magneto-photoconductivity, we present the behavior of the photoconductivity with the applied magnetic field.
In
For each curve, the behavior of the photoconductivity is the same. In the interval of the magnetic field [0 T; 10−4 T], the photoconductivity is constant and stays at its maximum value for a given curve. Here, the effect of the magnetic field is weak on the photo generated minority carriers. The magnetic force that acts on minority carriers, is so weak that there is no significant deflected carriers that could be recombined in surface area or in the bulk of the base. In contrast in
the interval [10−4 T; 10−2 T], the photoconductivity decreases markedly because of the strong magnetic force which acts on the minority carriers trajectories that are more and more incurvated. The increase of the magnetic field while decreasing photoconductivity, induced change in the solar cell properties by creating degenerated levels states.
We now show the behavior of the photoconductivity with the operating point of the solar cell.
We present in
We note that, for each curve in
in short circuit situation [5 × 105 cm・s−1; 7 × 107 cm・s−1]. The photo created minority carriers have very high kinetic energy to cross the junction. There are no more significant stored minority carriers and the photoconductivity remains constant but is lower than those obtained in open circuit situation. For all these above cases, an applied magnetic field causes a decrease in the magnitude of the photoconductivity. Therefore, for an application in optoelectronics, laser or detector, we have to select the solar cell operating point before its short-circuit situation. In order to determine the solar cell electrical parameters, in the interval [2 × 102 cm・s−1; 5 × 105 cm・s−1], we can apply the linear model of this part of the photoconductivity curve since there is a straight line observed. Thus, we obtain a relationship between the photoconductivity and the junction recombination velocity as:
σ p h = − C ⋅ S f + σ p h o (19)
where σ p h o is the photoconductivity in open circuit situation; − C is the slope of the linear part and C being the electrical capacitance.
In
We note a decrease of both electrical capacitance and photoconductivity in open circuit situation with the increase of the applied magnetic field. The decrease of the diffusion capacitance is linked to the space charge region (in the vicinity of the junction) increase since there is no significant stored minority carriers and the occurring of the magnetoresistance effect on the solar cell.
| B (T) | 0 | 5 × 10−4 | 10−3 | 1.5 × 10−3 | 2 × 10−3 |
|---|---|---|---|---|---|
| σ p h o (10−4 Ω−1・cm−1) | 7.62 | 5.21 | 2.50 | 1.2 | 0.6 |
| C (pF cm−2) | 894 | 555 | 210 | 73 | 25 |
A steady-state of the photoconductivity of a bifacial silicon solar cell has been presented. The Drüde model and the continuity equation permitted to determine the photoconductivity according to the base depth, the applied magnetic field and the junction recombination velocity. The effects of the depth, the magnetic field and the junction recombination velocity, have enabled us to determine an optimum base depth and to be in open circuit situation for having more photoconductivity. By using a linear model of the photoconductivity, we have determined an electrical capacitance, which decreases with the applied magnetic field. The method of the steady photoconductivity has been used to characterize a thin film silicon solar cell capacitance. For the future works, few parameters, such series and shunt resistances, quantum efficiency, spectral response will be considered in 1D or 3D, taking into account of the grain boundaries recombination. This method will be utilized to heterojunctions solar cells in transient regime, frequency modulation with or without magnetic field effect.
Diao, A., Wade, M., Thiame, M. and Sissoko, G. (2017) Bifacial Silicon Solar Cell Steady Photoconductivity under Constant Magnetic Field and Junction Recombination Velocity Effects. Journal of Modern Physics, 8, 2200-2208. https://doi.org/10.4236/jmp.2017.814135