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The silicon solar cell with series-connected vertical junction is studied with different lamella widths—the expression of the ac recombination velocity of the excess minority carrier at the back surface is established. Spectroscopy technique reveals dominated impact of the lamella widths of the base.

The techniques of characterization of solar cell for quality control [

The phenomenological parameters [

1) in the bulk [

2) on the surfaces [^{+}) [

The solar cell is placed under different operating modes [

The solar cell can be maintained under different experimental conditions while varying: temperature [

To achieve low cost solar concentrator cell, vertical multi-junction (VMJ) cells have been manufactured [

In our study, the structure of the series-connected vertical junction solar cell [

^{+}) and base (p) we have the space charge region (SCR), called the junction. And at the back side of each base region, there is a high doping layer (p^{+}) giving rise to a back surface field (BSF), which induced the back surface recombination velocity (Sb) (

In this series-connected architecture, each solar cell unitis separated on both sides by metal contacts [

The continuity equation at which the density of minority charge carriers in excess obeyed δ ( x , t ) at the position x in the base, in an instant t, is given by [

D ( ω ) ⋅ ∂ 2 δ ( x , t ) ∂ x 2 − δ ( x , t ) τ = − G ( z , t ) + ∂ δ ( x , t ) ∂ t (1)

The ac component of the excess minority density is in the following form:

δ ( x , t ) = δ ( x ) ⋅ e j ω t (2)

j is the complex notation.

With δ ( x ) is the steady state minority carrier density position dependent.

The expression of the ac generation rate G ( z , t ) of the minority carrier at depth z, is given by [

G ( z , t ) = g ( ω , α , z ) ⋅ e J ω ⋅ t (3)

with:

g ( ω , α , z ) = K ( ω , α ) ⋅ exp ( − α ⋅ z ) (4)

and

K ( ω , α ) = − [ α ⋅ I ( λ ) ⋅ ( 1 − R ( λ ) ) D ( ω ) ⋅ [ α 2 − 1 L ( ω ) 2 ] ] (5)

I (λ) is the intensity of the monochromatic illumination of wavelength λ. α is the absorption coefficient of the monochromatic light incident on the cell and R (λ) its reflectance coefficient.

D (ω) and L (ω) are respectively, the excess ac minority carrier diffusion coefficient and diffusion length in the base subjected to illumination in frequency modulation (ω).

L (ω) and D (ω) ac expressions are giving by [

D ( ω ) = D ⋅ [ 1 1 + ( ω ⋅ τ ) 2 − j ⋅ ω ⋅ τ 1 + ( ω ⋅ τ ) 2 ] (6)

L ( ω ) = D ⋅ [ τ 1 + ( ω ⋅ τ ) 2 − j ⋅ ω ⋅ τ 2 1 + ( ω ⋅ τ ) 2 ] (7)

where D denotes the diffusion constant and τ the bulk lifetime in steady state.

By substituting Equation (1) together with Equation (2) and Equation (3), leads to:

∂ 2 δ ( x , ω ) ∂ x 2 − δ ( x , ω ) L ( ω ) 2 + g ( ω , α , z ) D ( ω ) = 0 (8)

Thus the resolution of Equation (6) gives the excess minority carrier density in the base through the following expression:

δ ( x , ω , α , z ) = A cosh ( x L ( ω ) ) + B sinh ( x L ( ω ) ) + K ( ω , α ) ⋅ exp ( − α ⋅ z ) (9)

with coefficients A et B are deduced from the boundary conditions:

1) At the junction (x = 0), the expression of the photocurrent J p h ( 0 , ω , α , z ) [

q ⋅ D ( ω ) ∂ δ ( x , ω , α , z ) ∂ x | x = 0 = q ⋅ S f ⋅ δ ( 0 , ω , α , z ) = J p h ( 0 , ω , α , z ) (10)

2) On the back side in the base at x = H.

∂ δ ( x , ω , α , z ) ∂ x | x = H = − S b D ( ω ) δ ( H , ω , α , z ) (11)

Sf and Sb are respectively the recombination velocities of the excess minority carrier at the junction and at the back surface. The recombination velocity Sf reflects the charge carrier velocity of passage at the junction, in order to participate in the photocurrent. It is then imposed, by the external load which fixes the solar cell operating point [^{+} layer, which generates an electric field for throwing back the charge carrier toward the junction [

After calculation, the following expression of the ac excess minority carrier density is obtained by:

δ ( x , ω , α , z ) = A ( H , ω , α , z , S f , S b ) cosh ( x L ( ω ) ) + B ( H , ω , α , z , S f , S b ) sinh ( x L ( ω ) ) + K ( ω , α ) ⋅ exp ( − α ⋅ z ) (12)

The excess minority carrier recombination velocity at the back surface is deduced from the resolution of the following equation [

∂ J p h ( H , ω , α , z , S f , S b ) ∂ S f = 0 (13)

Solving Equation (13) leads to two solutions [

S b ( ω , H ) = − D ( ω ) L ( ω ) ⋅ tanh ( H L ( ω ) ) (14)

Sb in complex form (real and imaginary components) is presented by analogy of the effect of Maxwell-Wagner-Sillars (MWS) model [

S b ( ω , H ) = S b ′ ( ω , H ) + J ⋅ S b ″ ( ω , H ) (15)

We define the ac phase as following equation:

tan ( ϕ ( ω , H ) ) = S b ″ ( ω , H ) S b ′ ( ω , H ) (16)

S b a m p l ( ω , H ) and ϕ ( ω , H ) correspond to the amplitude and phase component of Sb.

We represent in

The large (H) thicknesses give weak oscillation periods. Thus whatever the lamella width, the oscillation is around a fixed (Sb0) value of the excess minority carrier recombination velocity.

For high range frequencyvalues (ωτ > 1) we observe a damped sinusoid (aperiodic response), to then give the appearance of an exponential growth. For large lamella width, these curves tend towards an asymptote (Sb0 = 4184 cm/s), whatever the frequency.

The amplitude and the phase spectra of the recombination velocity (Sb) are given respectively in

On the frequency axis, the region corresponding to the frequencies below 10^{4} rd/s (i.e. ω τ ≪ 1 ), constitutes the steady state.

In this zone the amplitude of Sb_{Ampl} believes with the thickness H. The phase remains constant and obviously equal to zero.

In high frequency region 10^{4} rd/s < ω, tarts the dynamic regime i.e. ω τ ≫ 1 ), showing a sinusoid of amplitude (Sb_{Ampl}), oscillating around Sb0, the asymptotical recombination velocity, with periods T_{Sb} decreasing with H (see

The phase spectrum shows regular sinusoids with constant amplitudes f_{ampl}, for each given H lamella widths, but decreases with H. The period Tf_{ampl} of these oscillations decreases with the lamella thickness H (see

The circles obtained have for their center Sb0, on the axis of the reals. The radius of the circles increases when the lamella thickness H decreases. According to the spectroscopy techniques [

H (cm) | 0.013 | 0.015 | 0.017 | 0.02 |
---|---|---|---|---|

(SbAmpl) (10^{5} rad/s) | 10.070^{ } | 8.776^{ } | 7.700^{ } | 6.472^{ } |

T (SbAmpl) (10^{−}^{6} s) | 6.240 | 7.160 | 8.160 | 9.700 |

Sb (Ampl) (cm/s) | 4573 | 4422 | 4329 | 4254 |

H(cm) | 0.013 | 0.015 | 0.017 | 0.02 |
---|---|---|---|---|

f (Ampl) | 0.089^{ } | 0.055^{ } | 0.034 | 0.017 |

ω_{f} (Ampl) (10^{5} rad/s) | 10.120^{ } | 8.584^{ } | 7.952^{ } | 6.604^{ } |

T_{f} (10^{−6} s) | 6.200 | 7.320 | 7.902 | 9.515 |

phenomenon (Lh), both decrease with the lamella thickness H. The ac equivalent circuit of Sb, suggests that, the capacitor and the inductor are associated in parallel and connected in series with a resistance (obtained for large frequency) [

The series-connected vertical multi-junction silicon solar cell was studied under frequency modulated illumination and yielded the determination of the ac back surface recombination velocity of the excess minority carrier. It is expressed as dependent of both lamella thickness and illumination frequency. The excess minority carrier recombination is investigated through the Bode diagrams of its amplitude and phase. The study also showed through the Nyquist diagram, the capacitive, inductive and resistive responses of the ac recombination velocity Sb, as well as the effect of the thickness of the lamella based solar cell.

The authors declare no conflicts of interest regarding the publication of this paper.

Gueye, M., Diallo, H.L., Moustapha, A.K.M., Traore, Y., Diatta, I. and Sissoko, G. (2018) Ac Recombination Velocity in a Lamella Silicon Solar Cell. World Journal of Condensed Matter Physics, 8, 185-196. https://doi.org/10.4236/wjcmp.2018.84013