AC Back Surface Recombination Velocity as Applied to Optimize the Base Thickness under Temperature of an (n⁺-p-p⁺) Bifacial Silicon Solar Cell, Back Illuminated by a Light with Long Wavelength

The bifacial silicon solar cell, placed at temperature ( T) and illuminated from the back side by monochromatic light in frequency modulation ( ω), is studied from the frequency dynamic diffusion equation, relative to the density of excess minority carriers in the base. The expressions of the dynamic recombination velocities of the minority carriers on the rear side of the base Sb1( D( ω, T); H) and Sb2( α, D( ω, T); H), are analyzed as a function of the dynamic diffusion coefficient ( D( ω, T)), the absorption coefficient ( α( λ)) and the thickness of the base ( H). Thus their graphic representation makes it possible to go up, to the base optimum thickness (Hopt(ω, T)), for different temperature values and frequency ranges of modulation of monochromatic light, of strong penetration. The base optimum thickness (Hopt(ω, T)) decreases with temperature, regardless of the frequency range and allows the realization of the solar cell with few material (Si).

• Bifacial Silicon Solar Cell
• Absorption Coefficient
• Frequency
• Temperature
• Recombination Velocity
• Optimum Thickness

1. Introduction

This work aims to determine the optimum thickness of the base [1] [2] [3] of the bifacial silicon solar cell (n⁺/p/p⁺) [4] [5] [6] [7] [8] . The base (p) of the bifacial silicon solar cell (n⁺/p/p⁺) maintained at temperature (T), is studied under monochromatic illumination (λ) in frequency modulation (ω).

The monochromatic light chosen, induces an absorption coefficient (α(λ) = 6.02 cm⁻¹) [9] [10] and penetrates deep into the silicon material. The frequency dynamic diffusion equation relating to the density of the excess charge minority carriers in the base of the solar cell is solved. The boundaries of this zone, which are the junction (n⁺/p) in x = 0 and the rear face (p/p⁺) in x = H, impose conditions that use the recombination velocities of minority carriers, respectively (Sf) [11] [12] [13] [14] [15] and (Sb) [16] - [24] . Taking into account the dynamic diffusion coefficient (D(ω, T)) related to the temperature (T) [25] [26] [27] and the frequency (ω) of modulation [28] - [33] , as well as the recombination velocities (Sf) at the junction and (Sb) at the rear side, the expression of the dynamic photocurrent density of the minority charge carriers Jph(Sf, Sb, α, H, D(ω, T)) is established and represented graphically as a function of (Sf), for different temperatures and frequency zones.

From this representation, the expressions Sb(α, H, D(ω, T)) and Sb(H, D(ω, T)) of the dynamic rate of recombination of the minority charge carriers on the rear side of the base, are deduced. The graphic technique of representation [34] - [44] of these expressions as a function of the base thickness (H), makes it possible to determine the optimum thickness (Hopt), for given temperature values and for different frequency zones. The optimum thickness (Hopt(T)) is represented by curves as function of temperature, for each frequency range studied.

The modeling analysis of these curves, through the effects of temperature (thermal agitation), frequency (relaxation time) and a low absorption coefficient (deep penetration), shows the possibility of developing the bifacial solar cell by a reduction in thickness, according to the conditions of use while producing an optimal photocurrent.

2. Theory

The structure of the n⁺-p-p⁺ bifacial silicon solar cell at (T) temperature and back illuminated with monochromatic light in frequency, is represented by Figure 1.

The excess minority carriers’ density δ(x, t) generated in the base of the solar cell, under monochromatic illumination, is governed by the following continuity equation [31] [32] [33] :

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

The expression of the excess minority carriers’ density is written, according to the space coordinates (x) and the time t, as:

δ ( x , t ) = δ ( x ) ⋅ e − j ω t (2)

Ac Carrier generation rate G(x, t) is given by the relationship:

G ( x , t ) = g ( x ) ⋅ e − j ω t (3)

With:

g ( x ) = α ( λ ) ⋅ I 0 ( λ ) ⋅ ( 1 − R ( λ ) ) ⋅ e − α ( λ ) ⋅ ( H − x ) (4)

I₀ is the incident photon flux on the rear (p⁺), at the solar cell base depth (x = H), while α(λ) and R(λ) are respectively the optical [9] [10] absorption and reflection coefficients of Si material D(ω, t) is the complex diffusion coefficient of excess minority carrier in the base under T temperature. Its expression is given by the relationship [22] [23] [31] [32] [33] :

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

D(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 is the Boltzmann constant:

K b = 1.38 × 10 − 23   m 2 ⋅ Kg ⋅ S − 1 ⋅ K − 1

The mobility coefficient [25] [26] [45] for electrons µ(T), is expressed according to the temperature and is given by:

μ ( T ) = 1.43 × 10 19 T − 2.42 (7)

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

∂ 2 δ ( x ) ∂ x 2 − δ ( x ) L 2 ( ω , T ) = − g ( x ) D ( ω , T ) (8)

L(ω, T) is the complex diffusion length of excess minority carriers’ in the base given by:

L ( ω , T ) = D ( ω , T ) τ 1 + j ω τ (9)

The solution is expressed as following:

δ ( x , λ , ω , T ) = A ⋅ cosh [ x L ( ω , T ) ] + B ⋅ sinh [ x L ( ω , T ) ] + K ⋅ e − α ( λ ) ⋅ ( H − x ) (10)

With:

K = − α ( λ ) ⋅ I 0 ⋅ ( 1 − R ) ⋅ [ L ( ω , T ) ] 2 D ( ω , T ) [ L ( ω , T ) 2 ⋅ α ( λ ) 2 − 1 ] and Ψ ˙ = ∂ Ψ ∂ C : C ˙ + ∂ Ψ ∂ θ θ ˙ . (11)

Coefficients A and B are determined through the boundary conditions:

• At the junction (x = 0)

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

• On the back side in the base (x = H)

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

(Sf) and (Sb) are respectively the recombination velocities of the excess minority carriers [46] at the junction and at the back surface. The recombination velocity Sf is imposed by the external load which fixes the solar cell operating point [11] [12] [47] [48] [49] [50] . At low value, it becomes intrinsic component which represents the carrier losses associated with the shunt resistor in the solar cell electrical equivalent model under open circuit operation [7] [15] [50] . The minority carrier recombination velocity (Sb) on the back surface is the consequence of the electric field created by the junction (p/p⁺) which rejects the carrier toward the junction (n⁺/p) [51] [52] .

3. Results and Discussions

3.1. Ac Diffusion Coefficient and Frequency Domains

Previous works [18] [19] [29] [30] [53] [54] [55] have shown that the dynamic diffusion coefficient of minority carriers in the base of the solar cell is of constant amplitude at low frequencies (steady state: ω < ω_c, ω_c is the cut-off frequency) and it decreases at high frequencies (dynamic regime: ω < ω_c). The amplitude of the diffusion coefficient of minority carriers decreases very rapidly with frequency. The higher the frequency, the lower the relaxation time of the minority carriers, then this produces as a consequence a greater probability of recombination of minority carriers, hence the drastic decrease of the diffusion coefficient, which corresponds to the third frequency interval. For the rest of the study, three frequencies from these intervals will be taken into account.

3.2. Photocurrent

Ac photocurrent density at the junction is obtained from ac minority carriers’ density in the base and δ(x, Sf, Sb, H, ω, T, α(λ)) is given by the following expression:

J p h ( S f , S b , H , ω , T , α ( λ ) ) = q D ( ω , T ) ∂ δ ( x , S f , S b , H , ω , T , α ( λ ) ) ∂ x | x = 0 (14)

where q is the elementary electron charge.

Figures 2-4 show ac photocurrent versus junction surface recombination velocity respectively, for different frequency values (ω = 10³ rad∙s⁻¹, ω = 10⁵ rad∙s⁻¹, ω = 10⁶ rad∙s⁻¹).

The amplitude of the density of the dynamic photocurrent (J_ph), as function of the recombination rate (Sf) at the junction, for different temperature values, is represented by the Figures 2-4, corresponding to the three frequency domains, shows three zones. At low values of (Sf), the amplitude of the photocurrent density is zero, regardless of temperature, for the three frequency domains and corresponds to the open circuit operation of the solar cell. The amplitude of the photocurrent density increases rapidly with (Sf), and reaches an asymptotic value corresponding to the short-circuit operation of the solar cell. The amplitude of the short-circuit photocurrent density (J_ph,sc) decreases with temperature and frequency. Indeed, the density profile of minority carriers at depth in the base of the short-circuited solar cell is modified with temperature (amplitude reduction) [56] and with frequency (by reducing the relaxation time) [18] [19] [20] [21] [29] [30] as suggested throughout Equations (5), (6) and (9).

3.3. Back Surface Recombination Velocity Determination and Base Thickness Optimization with Both, Temperature and Frequency

The plot of ac photocurrent density amplitude versus minority carriers’ recombination velocity at the junction shows a bearing sets up, for very large Sf, and corresponds to short-circuit current density (J_phsc). Then in this zone, we can write [12] [13] [14] [55] :

∂ J p h ( S f , S b , H , ω , T , α ( λ ) ) ∂ S f | S f ≥ 10 5 cm ⋅ s − 1 = 0 (15)

The solution of Equation (15) leads to the ac recombination velocity in the back surface expressions given by Equations (16) and (17):

S b 1 ( ω , H , T ) = − D ( ω , T ) L ( ω , T ) ⋅ tanh ( H L ( ω , T ) ) (16)

S b 2 ( ω , H , T , λ ) = D ( ω , T ) L ( ω , T ) ⋅ [ α ( λ ) ⋅ L ( ω , T ) − ( sinh ( H L ( ω , T ) ) + α ( λ ) ⋅ L ( ω , T ) ⋅ cosh ( H L ( ω , T ) ) ) exp ( − α ( λ ) ⋅ H ) exp ( − α ( λ ) ⋅ H ) ⋅ ( cosh ( H L ( ω , T ) ) + α ( λ ) ⋅ L ( ω , T ) ⋅ sinh ( H L ( ω , T ) ) ) − 1 ] (17)

Figures 5-7 representation two expressions of back surface recombination velocity versus solar cell base thickness, for different temperature values, respectively in the three frequency domains. Tables 1-3, give the results obtained, for the solar cell base optimum thickness again temperature, in the three frequency domains.

The optimum thickness is extracted from Figure 4, Figure 5 and Figure 9 for given temperatures and for the three frequency domains. The results are presented in the Tables 1-3.

Deduced from Tables 1-3, Figure 8 and Figure 9 are respectively the plots of Hopt again temperature and diffusion coefficient for frequency domains.

Table 1

Optimum thickness obtained, for different temperatures and for the frequency range (ω = 10 3 rad∙s −1 )

T (K)	200	215	230	250	265	280	300	315
D (cm2/s)	66.65	60.14	54.65	48.55	44.69	41.33	37.48	34.97
Hop (cm)	0.0108	0.0104	0.0101	0.0098	0.0094	0.0092	0.0089	0.0086

Table 2

Optimum thickness obtained, for different temperatures and for the frequency range (ω = 10 5 rad∙s −1 )

T (K)	200	215	230	250	265	280	300	315
D (cm2/s)	33.3278	30.0751	27.3285	24.2770	22.3491	20.6683	18.7395	17.4851
Hop (cm)	0.0089	0.0086	0.0083	0.0080	0.0078	0.0076	0.0074	0.0072

Table 3

Optimum thickness obtained, for different temperatures and for the frequency range (ω = 10 6 rad∙s −1 )

T (K)	200	215	230	250	265	280	300	315
D(cm2/s)	6.63	5.98	5.44	4.83	4.45	4.11	3.73	3.48
Hop (cm)	0.0072	0.007	0.0068	0.0066	0.0065	0.0064	0.0063	0.0062

4. Discussions

The modeling of the curves gave the mathematical correlations of the optimum thickness of the base Hopt(T) and Hopt(D), respectively as a function of the temperature (Figure 8) and the diffusion coefficient of the minority carriers of the base (Figure 9), according to the three frequency regimes:

1) Frequency regime: ω ≤ 10 3 rad / s

H o p ( c m ) ( T ) = − 1.8 × 10 − 5 × T + 0.014 (18)

H o p ( c m ) ( D ) = 6.9 × 10 − 5 × D + 0.0063 (19)

2) Frequency regime: 10 3 rad / s < ω ≤ 10 5 rad / s

H o p ( c m ) ( T ) = − 1.4 × 10 − 5 × T + 0.012 (20)

H o p ( c m ) ( D ) = 1.1 × 10 − 4 × D + 0.0055 (21)

3) Frequency regime: 10 5 rad / s < ω

H o p ( c m ) ( T ) = 4.9 × 10 − 8 × T 2 − 3.4 × 10 − 5 × T + 0.012 (22)

H o p ( c m ) ( D ) = 3.1 × 10 − 4 × D + 0.0051 (23)

The results through Figure 8, show that, regardless of the frequency range of modulation of the incident light, the optimum thickness decreases with temperature (Equations (18), (20) and (22)). The increase in the density of the photogenerated carriers with temperature causes the maximum density of the minority carriers to be reduced towards the junction (p/p⁺) which is the illuminated surface [56] . Obtaining a large photocurrent then imposes a reduction in the thickness of the base. On the other hand, the increase in the diffusion coefficient of the minority charge carriers consequently leads to the increase in the optimum thickness (Figure 9), which is modeled by Equations (19), (21) and (23). The optimum thickness of the base is large at low frequencies corresponding to the static regime (ωτ = 1) whatever the temperature (Figure 8). In this interval of low frequencies, the relaxation time of the minority carriers is important (large values of D in Equation (5)), allowing a large distance of travel (Einstein relation). A large optimum thickness of the base is then obtained [43] [57] [58] (Figure 9). On the other hand, at high frequency values, the relaxation time (ωτ ? 1) of minority carriers is reduced, as well as the coefficient (Equation (5)) and the scattering length (Equation (9)) of minorities corresponding to a short travel distance, then the optimum thickness [57] [58] of the base needed to collect the charge carriers is small, regardless of the temperature. Previous work on solar cells, monofacial, bifacial or multi-vertical junctions, corroborates these results, showing the reduction of the optimum thickness of the base of the silicon solar cell under illumination, according to the front (n⁺) [34] [58] or rear (p⁺) [38] [43] [57] , while it is subjected to:

­ Monochromatic light with constant flux [59] and absorption coefficient α(λ)).

­ Monochromatic light (α(λ)) in frequency modulation (ω) [22] [23] [43] [57] [58] .

­ Constant B magnetic field [60] [61] .

­ Irradiation by charged particles [43] [62] .

­ Temperature variation [22] [63] .

­ Combination of several parameters leads to a situation of resonance of minority carriers’ diffusion coefficient in the base of the solar cell under magnetic field:

1) In frequency [64] .

2) In temperature, under front [36] [37] , rear [39] [41] , or vertical illumination [65] .

3) And under irradiation of charged particles, in frequency [66] .

4) Frequency and temperature [40] [42] .

The analysis of the physical mechanisms [1] [24] , through the phenomological parameters of the material [67] , would lead to the development of optimized and economical solar cells [68] [69] , taking into account above all the external physical factors, which constitute the operating environment.

5. Conclusions

The study of the density of the minority charge carriers in the base of the bifacial solar cell (n⁺/p/p⁺) silicon, under a given temperature and under monochromatic illumination in frequency modulation by the back side (p⁺), showed the effect of the frequency, by the folding of the maximum density towards the junction (p/p⁺) and the increase of the density of the photogenerated carriers with temperature. The graphic technique for the determination of the optimum thickness of the base of the bifacial solar cell (n⁺/p/p⁺) at temperature (T) and subjected to monochromatic illumination (α(λ)) in frequency modulation (ω) by the back side, was used. The analysis of the dynamic photocurrent density curve as a function of the recombination velocity of minority carriers at junction (Sf), the dynamic recombination velocity expressions of minority carriers on the back side (Sb(α, H, D(ω, T))), and (Sb(H, D(ω, T)) were deduced.

The graphical representation of the expressions of the dynamic recombination velocity of the minority carriers on the back side as a function of the thickness (H), made it possible to extract the base optimum thickness (Hopt), for each frequency domain and for different temperature values. The representation of the optimum thickness (Hopt(T)) gives a curve profile that is modeled by a function decreasing with temperature (T), regardless of the frequency of modulation of the incident light.

The variations of the dynamic coefficient (D(ω, T)) of diffusion of the charge minorities are obtained by the variation of the frequency of modulation of the monochromatic light incident on the rear side (p⁺). Thus the base optimum thickness Hopt decreases sharply with the frequency regime whatever the temperature (T).

Acknowledgements

In memory of the deceased professors emeritus, Professor Gerard W. COHEN SOLAL (Montpellier University-France), Professor Michel RODOT (CNRS/ Meudon Bellevue, France), Professor Yembila Abdoulaye TOGUYENI and Professor Sié Faustin SIB, both at Joseph Ki ZERBO University ( Ouagadougou, Burkina Faso).

Conflicts of Interest

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

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