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The ac recombination velocity of the excess minority carriers, in the back surface of a silicon solar cell with a vertical junction connected in series, is developed through Einstein’s law giving the diffusion coefficient of minority carriers according to temperature, through mobility. The frequency spectrum of both, amplitude and phase, are produced for the diffusion coefficient and the recombination velocity in the rear face, in order to identify the parameters of equivalent electric models.

Vertical multi-junctions (VMJ) silicon solar cells have an architecture that is an alternative for collecting minority carriers with low-diffusion length [

Two types of VMJ solar cells are developed, by a succession of npp or pnn junctions. The VMJ-P has connections in parallel, between bases and connections between emitters. Thus a base type (p) is surrounded by two emitters allowing the collection of minority carriers at close range, leading to the increase of photocurrent [

The VMJ is designed to operate under light concentration to generate more minority carriers, thereby increasing voltage or current production. In this situation, temperature is an important factor that influences the operating performance of the solar cell, through the physical mechanisms that are important to study [

The structure of the serially connected vertical multi-junctions silicon solar cell, under monochromatic illumination in frequency modulation, is given by

The unit of the solar cell extracted from the series representation, is a npp+ structure, the base of which is studied by variation in the temperature T (

The continuity equation relating to the excess minority carriers density δ ( x , z , T , t ) in the base at temperature T, and under monochromatic illumination in frequency modulation, is given by the relationship [

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

The density of photogenerated carriers is written according to the space coordinates (x, z) and the time t as:

δ ( x , z , t ) = δ ( x , z ) exp ( − j ω t ) (2)

The minority carriers generation rate at depth z in the base and at any point of absciss x, under the modulation frequency ω of the incident wave, is given by the relationship:

G ( z , ω , t ) = g ( z ) exp ( − j ω t ) (3)

where g(z) is the steady stateminority carriers generation rate at the z depth induced in the base by a monochromatic light of incident flow ϕ ( λ ) , respectively with monochromatic absorption and reflecting coefficients α ( λ ) and R ( λ ) . It is then written by the following relation:

g ( z ) = α ( λ ) ⋅ ( 1 − R ( λ ) ) ⋅ ϕ ( λ ) ⋅ exp ( − α ( λ ) ⋅ z ) (4)

With τ the excess minority carrier lifetime in the base.

D ( ω , T ) is the complex diffusion coefficient of excess minority carrier in the base at T-temperature. Its expression is given by the relationship [

D ( ω , T ) = D ( T ) ( 1 − j ⋅ ω 2 τ 2 ) 1 + ( ω τ ) 2 (5)

D 0 ( T ) is the temperature-dependent diffusion coefficient given by Einstein’s relationship [

D ( T ) = μ ( T ) ⋅ K b ⋅ T q (6)

T is the temperature in Kelvin, K_{b} the Boltzmann constant:

K b = 1.38 × 10 − 23 m 2 ⋅ kg ⋅ S − 2 ⋅ K − 1 (7)

The minority carrier mobility coefficient [

μ ( T ) = 1.43 × 10 9 T − 2.42 cm 2 ⋅ V − 1 ⋅ s − 1 (8)

By replacing the Equations (2) and (3) in the Equation (1), the continuity equation for the excess minority carriers density in the base is reduced to the following relationship:

∂ 2 δ ( x , z , ω , T ) ∂ x 2 − δ ( x , z , ω , T ) L 2 ( ω , T ) = − G ( x , z ) D ( ω , T ) (9)

L ( ω , T ) is the complex diffusion length of excess minority carrier in the base; ilest donné par :

L ( ω , T ) = D ( ω , T ) τ 1 + j ⋅ ω ⋅ τ (10)

(ω, T) is the ac minority carriers diffusion coefficient in the base under the influence of temperature and the minority carrier lifetime in the base.

Thus the solution of the Equation (9) is given by the following expression of the ac density of minority carriers:

δ ( x , ω , T , z ) = A cosh ( x L ( ω , T ) ) + B sinh ( x L ( ω , T ) ) + L 2 ( ω , T ) D ( ω , T ) ⋅ α t ( 1 − R ( λ ) ) ⋅ ϕ ( λ ) ⋅ exp ( α t ⋅ z ) (11)

Coefficients A and B are determined from conditions at the base space boundaries, i.e. at the junction (x = 0) and in the rear (x = H) and are expressed by:

1) At, x = 0, at the junction emitter-base (n/p) surface

D ( ω , T ) ⋅ ∂ δ ( x , z , T , ω ) ∂ x | x = 0 = S f ⋅ δ ( x , z , T , ω ) | x = 0 (12)

2) At, x = H, the back surface (p/p+)

D ( ω , T ) ⋅ ∂ δ ( x , z , ω ) ∂ x | x = H = − S b ⋅ δ ( x , z , T , ω ) | x = H (13)

where S f is the junction surface recombination velocity [

S f = S f O + S f j (14)

S f O , defines the lost electrical charges velocity at the junction surface and is related to shunt resistance in establishing the electric model equivalent to the illuminated solar cell [

S f j is the velocity of the flow of electrical charges that crosses through the external charge and defines the solar cell operating point [

Sb is the excess minority carrier recombination velocity at the rear surface of the solar cell’s base (p/p+) [

The amplitude and phase of the diffusion coefficient under different temperatures, are represented versus frequency, through the

For a given temperature, the diffusion coefficient is maximum and virtually constant when the frequency is low. Indeed, in a quasi-static regime the diffusion of minority carriers is not influenced by the frequency which explains the level observed. On the other hand, in a dynamic frequency regime, repeated arousals lead to a problem of relaxation of the solar cell which is a blocking factor for the diffusion of minority carriers. In addition, an increase in temperature decreases the diffusion of minority carriers. The diffusion is more sensitive to temperature in a quasi-static regime.

In a dynamic frequency regime, the problem of relaxation in the solar cell, blocks the diffusion of the minority carriers which gives a negative phase of the diffusion coefficient. The Nyquist diagram is shown in

We find that the radius of the semicircles decreases according to the temperature with a shift from the center of the circles to the origin of the axes. The semicircle indicates a resistor in parallel with a capacitor, so gives rise to a single time constant. The deformation of the semicircles, corresponds to a time constant, time dependent. The exploitation of the half-circle radius allows to determine electrical parameters characteristic of the equivalent electric model.

The density of photocurrent at the junction is obtained from the density of minority carriers in the base and is given by the following expression:

J P h ( ω , T , S f , S b ) = q ⋅ D ( ω , T ) ⋅ ∂ δ ( x , ω , T , S f , S b ) ∂ x | x = 0 (15)

where q is the elementary electron charge.

The representation of photocurrent density according to the junction recombination velocity of minority carriers shows that, for very large Sf, a bearing sets up and corresponds to the short-circuit current density (Jphsc). So in this junction recombination velocity interval, we can write [

∂ J p h ( ω , T , S f , S b ) ∂ S f = 0 (16)

The solution of this equation leads to expressions of the ac recombination velocity in the back surface, given by:

S b 1 ( ω , T ) = D ( ω , T ) ⋅ sinh ( H L ( ω , T ) ) L ( ω , T ) ⋅ [ cosh ( H L ( ω , T ) ) − 1 ] (17)

S b 2 ( ω , T ) = − D ( ω , T ) L ( ω , T ) ⋅ tanh ( H L ( ω , T ) ) (18)

Previous studies have looked at the second solution given to the Equation (18). Our study will consider this second solution, whose module and phase are represented versus the logarithm of the modulation frequency by the

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

S b ( ω , T ) = S b ′ ( ω , T ) + J ⋅ S b ″ ( ω , T ) (19)

We define the ac phase for a given temperature, as following equation:

tan ( ϕ ( ω , T ) ) = S b ″ ( ω , T ) S b ′ ( ω , T ) (20)

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

At low frequencies (≤10^{4} rad/s), the stationary regime is observed and gives constant amplitudes for each T. These amplitudes decrease with the temperature T. Beyond the frequency ( ≫ 10 4 rad/s), the cut-off frequency (ω_{c}, sb(T)) is determined for each temperature. It is noted that the cut-off frequency decreases with the temperature T, as does the amplitude (Sb_{ampl}) and on the other hand the frequency (ω_{Sb}(T)) of oscillations increases, in the part corresponding to the dynamic regime(See

The phase is represented versus the logarithm of the modulation frequency. The part corresponding to the dynamic regime shows sinusoidal oscillations between positive and negative values of the phase, amplitude (Φampl) that decreases with the temperature T and the frequency of oscillations (ω_{Φ}), which on the other hand increases with the temperature T (See

T (K) | 250 | 300 | 350 | 400 | 450 | 500 | 550 | 600 |
---|---|---|---|---|---|---|---|---|

ω_{c} (rad/s) | 10^{4.34} | 10^{4.38} | 10^{4.33} | 10^{4.24} | 10^{4.23} | 10^{4.24 } | 10^{4.06 } | 10^{4.18 } |

Sb (T) (cm/s) | 1837.1 | 1813.1 | 1583.9 | 1523.8 | 1377 | 1295.5 | 1240 | 1156.3 |

Sb, ampl (cm/s) | 2711.3 | 2118.7 | 1940.9 | 1650.4 | 1547.4 | 1414.5 | 1280.5 | 1218.6 |

ω (rad/s) | 10^{5.14} | 10^{5.00} | 10^{5.04} | 10^{4.91} | 10^{4.96} | 10^{4.93} | 10^{4.82} | 10^{4.87 } |

T (K) | 250 | 300 | 350 | 400 | 450 | 500 | 550 | 600 |
---|---|---|---|---|---|---|---|---|

Φ ampl | 0.20 | 0.15 | 0.11 | 0.08 | 0.06 | 0.04 | 0.03 | 0.02 |

ω_{Φ} (rad/s) | 10^{4.85 } | 10^{4.79 } | 10^{4.74 } | 10^{4.70} | 10^{4.67} | 10^{4.64} | 10^{4.61} | 10^{4.57} |

T (K) | 250 | 300 | 350 | 400 | 450 | 500 | 550 | 600 |
---|---|---|---|---|---|---|---|---|

ImSb_{max} (cm/s) | 459.96 | 293.63 | 195.11 | 132.75 | 92.77 | 65.36 | 47.41 | 34.63 |

T (K) | 250 | 300 | 350 | 400 | 450 | 500 | 550 | 600 |
---|---|---|---|---|---|---|---|---|

Re (Sb (cm/s)) | 2712 | 2252 | 1941 | 1717 | 1547 | 1415 | 1307 | 1219 |

1 R e ( S b ( cm / s ) ) ⋅ 10 − 5 | 36.873 | 44.405 | 51.52 | 58.241 | 64.641 | 70.671 | 76.511 | 82.034 |

in 1/T [

The negative phase of the ac Sb recombination velocity and the determination of electrical parameters, with the Bode and Nyquist diagrams, characterizing Sb, allow to determine the equivalent electric model [

The solar cell’s ac back surface (p/p+) recombination velocity that controls the recombination of the excess minority carrier has been determined. Thus, the spectroscopy method allowed the study of the Bode and Nyquist diagrams and extracted certain electrical parameters characterizing the equivalent electric model. The effect of temperature on back surface recombination velocity was explained by umklapp processes.

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

Traore, Y., Thiam, N., Thiame, M., Thiam, A., Ba, M.L., Diouf, M.S., Diatta, I., Mballo, O., Sow, E.H., Wade, M. and Sissoko, G. (2019) AC Recombination Velocity in the Back Surface of a Lamella Silicon Solar Cell under Temperature. Journal of Modern Physics, 10, 1235-1246. https://doi.org/10.4236/jmp.2019.1010082