Photovoltage in c-Si/mc-Si Solar Cells: Grain Size, Base Thickness, and Illumination Mode: A Mathcad-Based Analytical Model

Available online www.ejaet.com European Journal of Advances in Engineering and Technology, 2025, 12(10):37-46 Research Article ISSN: 2394 - 658X 37 Photovoltage in c-Si/mc-Si Solar Cells: Grain Size, Base Thickness, and Illumination Mode: A Mathcad-Based Analytical Model Youssou GNING1, Aly TOURE1, Dimitry DIASSY1, Mamadou Lamine SAMB1*, Moussa TOURE1, Ahmed MOHAMED-YAHYA2 1Department of Physics and Chemistry, University Iba Der Thiam of Thies, Thies, Senegal 2Applied Research Unit for Renewable Energies, University of Nouakchott, Nouakchott, Mauritania *[email protected] _____________________________________________________________________________________________ ABSTRACT We present a three-dimensional diffusion–recombination model that yields closed-form expressions for the photovoltage Vph of crystalline-silicon (c-Si/mc-Si) solar cells under front, rear, and dual illumination. The framework solves the minority-carrier continuity equation in (x,y,z) with finite lateral dimensions (gx, gy), explicitly retaining geometric eigenmodes and boundary conditions, and parameterizes microstructure through a grain size g and device geometry through the base thickness H. We introduce carrier-collection velocity (CCV) at the junction as the operating-point knob connecting transport to the diode relation for voltage. The analysis clarifies three regimes: i. a low-CCV extraction-limited regime (open-circuit-like for current), ii. a transition regime where the slope of Vph versus CCV diminishes, and iii. a high-CCV efficient-collection regime (short-circuit-like current plateau), in which Vph is lowest for a given generation level. Parametric results show that Vph decreases with increasing CCV, consistent with the reduction of steady-state excess density and quasi-Fermi-level splitting. Larger grains systematically increase Vph by lowering grainboundary recombination, while thinner bases shorten transport paths and curb bulk SRH losses. Front illumination is weakly sensitive to H (generation near the junction); rear illumination shows a strong dependence on H (longer diffusion paths); and dual illumination raises the voltage baseline and mitigates thickness penalties by shortening average collection distances. At fixed CCV near open-circuit, the voltage generally increases monotonically with grain size across illumination modes; in dual illumination we observe “blockwise” grouping by H, consistent with enhanced generation and reduced path lengths. Design guidance follows directly: prioritize large grains (g≳0.01 cm), moderate H (≈ 150 µm) unless superior passivation is available, and dual illumination when applicable. Low-J0, low-SRV passivating contacts lift the entire Vph-CCV characteristic. The model’s closed-form nature enables rapid exploration of geometry, microstructure, operation couplings and provides physically transparent targets for bifacial c-Si optimization. Keywords: bifacial photovoltaics; photovoltage; carrier-collection velocity (CCV); grain size; base thickness; analytical modeling. _____________________________________________________________________________________________ INTRODUCTION Bifacial crystalline-silicon (c-Si/mc-Si) photovoltaics have moved from niche to mainstream, driven by their superior energy yield and competitive LCOE when integrated with modern trackers and optimized site albedo [1]. Accurate prediction and interpretation of device-level observables, particularly the photovoltage under front, rear, and dual illumination, require coupling optical irradiance models with charge-transport and recombination physics. Recent techno-economic and field assessments quantify bifacial gains and underscore the need for electrical models that resolve front/rear generation asymmetry and extraction bottlenecks [1,2]. On the optical side, practical and computationally efficient schemes for rear-side irradiance (now standard in bankable rating methods) enable robust links between geometry, albedo, and backside flux [3,4]. Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 38 Electrically, open-circuit photovoltage 𝑉𝑜𝑐 and its evolution with operating point hinge on the balance between photogeneration and recombination currents (𝐽0 terms). Progress in passivating contacts has lowered 𝐽0 and stabilized performance under outdoor stress, directly boosting 𝑉𝑜𝑐 in both mono and bifacial cells [5–7]. At the material scale, modern views of mc-Si microstructure, grain boundaries, dislocations, and impurity complexes, relate structural descriptors to recombination activity, informing grain-engineering strategies that raise carrier lifetime and improve voltage at a given base thickness [8]. Finally, refined intrinsic recombination parameterizations (Auger and radiative) are essential to avoid systematic bias when extrapolating 𝑉𝑜𝑐 trends across injection regimes and doping ranges [9]. In this work we adopt a 3D diffusion–recombination framework and derive closed-form photovoltage expressions for front, rear, and dual illumination. We emphasize the coupled roles of lateral dimensions, grain size, and base thickness, and we benchmark qualitative trends against the contemporary understanding summarized above. PHYSICAL MODEL Description of the Simulated Solar Cell We consider a 3D columnar domain centered on a single grain of polycrystalline silicon. The device is bifacial and comprises four regions: a thin, highly doped n+ emitter (≈ 0.5 − 1 µ𝑚; 1017–1019 𝑐𝑚−3) contacted by a metal grid; the space-charge region (𝑝−𝑛 junction) that separates carriers; a lightly doped p-type base (1015–1017 𝑐𝑚−3) governing generation, diffusion and recombination with thickness 𝐻∈[100,400] 𝜇𝑚; and a rear p+ BSF that redirects minority carriers toward the junction to limit deep recombination. This layout reflects modern c-Si/bifacial practice and passivating-contact stacks for high collection efficiency [10,11]. Figure 1: Simulated solar cell model Modeling assumptions: the emitter’s direct contribution to the photocurrent is small compared with the base; only the junction field is retained in the quasi-neutral regions. The junction is at 𝑧=0 (origin of the (𝑥,𝑦) axes); 𝐻 is swept from 100 to 400 µ𝑚. Carrier transport and recombination follow the drift–diffusion + SRH framework [12,13]. The minority-carrier continuity equation in the base becomes ∂2 ∂x2δ(x,y,z)+∂2 ∂y2δ(x,y,z)+∂2 ∂z2δ(x,y,z) − δ(x,y,z) L2 =−G(z) 𝐷 (3) with δ(x,y,z) the excess minority-carrier density, G(z) the generation rate D the diffusion coefficient and 𝐿 the diffusion length. We represent the depth-dependent generation by a compact multi-exponential fit to AM1.5: G(z) = n∑aie−biz 3 i=1 (4) where n is the concentration factor (in “suns”) and {ai, 𝑏i} are fit coefficients capturing the wavelength-integrated absorption [14]. A separated-variables solution is sought under grain-boundary (GB) recombination at the lateral edges. The general form is: δ(x,y,z)=∑ ∑ Zj,k(z)cosCjxcosCky kj (5) with lateral wavenumbers 𝐶𝑗, 𝐶𝑘 fixed by GB boundary conditions W H O z y x Heavily doped p-region ( ) Emitter ( ) Base (p) Junction Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 39 ∂ ∂xδ(x,y,z)]x=±gx 2=∓ Sg 2D∗δ(±gx 2,y,z) (6) ∂ ∂yδ(x,y,z)]y=±gy 2=∓ Sg 2D∗δ(x,±gy 2,z) (7) Where 𝑆𝑔 is the GB recombination velocity. These yield two transcendental relations tan(Cjgx 2)= Sg 2D Cj (8) tan(Ckgy 2)= Sg 2DCk (9) solved graphically or numerically [13,15]. Substituting (5) into (3) gives, for each mode (𝑗,𝑘), 𝑑2 𝑑z2Zj,k(z)−Zj,k(z) (𝐿𝑗,𝑘)2=−G(z) 𝐷𝑗,𝑘 (10) with effective modal parameters 1 Lj,k²=Cj2+Ck 2+1 L² (11) 1 Dj,k =16sin(Cjgx 2)sin(Ckgy 2) D[Cjgx+sin(Cjgx)][Ckgy+sin(Ckgy)] (12) so Lj,k and Dj,k act as effective diffusion length and coefficient [13,15]. A particular solution based on (4) leads to Zj,k(z)=Aj,ksinh z Lj,k +Bj,kcosh z Lj,k +∑kie−biz 3 i=1 (13) with: ki=− n Dj,k aiLj,k² bi²Lj,k²−1 (14) The constants Aj,k and Bj,k follow from the front and rear surface conditions 𝐷∂ ∂zδ(x,y,z)=Sfδ(x,y,z) pour z=0 (15) D∂ ∂zδ(x,y,z)=−Sbδ(x,y,z) pour z=H (16) Where 𝑆𝑓=𝑆𝑓0 +𝑆𝑓𝑗 combines the intrinsic junction loss (e.g. shunt-related) and the operating-point-dependent flux imposed by the external circuit; 𝑆𝑏 is the rear SRV. Modern extractions of 𝑆𝑅𝑉/𝐽0 motivate these boundary terms [16]. Expression of the photovoltage (in V) The photovoltage under illumination mode m ∈{Front, Rear, Dual} is computed from the junction-plane excess: Vphm=VTln(1+ 1 nO∫ ∫ δm(x,y,0)dxdy gy 2 −gy 2 gx 2 −gx 2) (17) with VT=𝑘𝑇/𝑞 and nO=ni2/ Nb (intrinsic density 𝑛𝑖, base doping Nb) [12]. Front-side Illumination Under this illumination, absorption occurs primarily near the upper surface, i.e., in the vicinity of the junction, which facilitates strong carrier collection. The corresponding photovoltage is given by: Vphfront =VTln(1+ 1 nO∑ ∑ (Bfrontj,k+∑ki 3 i=1 ) kj ×4sin(Cjgx 2)sin(Ck(B)gy 2) CjCk(B) ) (18) with: 𝐵front,j,k: mode amplitude coefficients for front-side illumination 𝑘𝑖 : constant derived from the analytical solution, associated with generation and recombination mechanisms. Rear-side Illumination When the device is illuminated from the rear, carriers are produced farther from the depletion region and must diffuse across the base, which penalizes collection. The expression is given by: Vphrear =VTln(1+ 1 nO∑ ∑ (Bearrj,k +∑kie−biH 3 i=1 ) kj ×4sin(Cjgx 2)sin(Ck(B)gy 2) CjCk(B) ) (19) where: 𝐵rear,j,k: mode amplitude coefficients for rear-side illumination, 𝑏𝑖: constant derived from the analytical solution, associated with generation and recombination mechanisms. 𝑒−𝑏𝑖𝐻: exponential attenuation term representing carrier recombination in deeper regions. Dual-face Illumination With dual-side incidence, the two carrier streams overlap within the base; recombination and lateral transport couple their effects, so the photovoltage departs from a simple sum. The expression is: Vphdual =VTln(1+ 1 nO∑ ∑ (Bdualj,k+∑ki(1+e−biH) 3 i=1 ) kj ×4sin(Cjgx 2)sin(Ck(B)gy 2) CjCk(B) ) (20) Where: Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 40 𝐵dual,j,k: mode amplitude coefficients under dual-face illumination, (1−𝑒−𝑏𝑖𝐻): contribution of photogenerated carriers throughout the full base thickness. RESULTS AND DISCUSION Influence of carrier collector velocity 𝐂𝐂𝐕 on the photovoltage of a solar cell illuminated from the front, the rear, and on both sides, for different grain sizes Front-side illumination. Figures 2 and 3 plot the photovoltage Vph under front-side illumination as a function of the CCV, for several grain sizes and two base thicknesses (H=150 μm and H=300 μm). Figure 1 : Photovoltage of a 150μm (left) and a 300μm (right) PV cell under front illumination as a function of carrier collection velocity for various grain sizes In all cases, Vph decreases as CCV increases, consistent with the diode relation Vph ∝ln(1+Jph/J0): faster extraction lowers the excess carrier density at the junction and reduces the quasi-Fermi-level splitting [17]. The sensitivity of Vph to CCV also reflects surface recombination and contact quality (which control J0 and effective SRV); improved passivation/contact stacks shift the curves upward and delay the drop of Vph with CCV [11,16]. Regarding grain size, larger grains yield higher Vph owing to reduced grain-boundary recombination and longer effective diffusion paths before loss [8,18]. In our data, the highest values occur for g=0.015 cm, with a maximum Vph ≈0.30 V at H=150 μm. Finally, the small gap between the H=150 and 300 μm curves under front illumination is consistent with near-junction generation: once carriers are created close to the depletion region, increasing H has only a modest influence on Vph at a given CCV [17]. Rear-side illumination. Figures 3 plot the photovoltage 𝑉𝑝ℎ under rear-side illumination as a function of the CCV, for several grain sizes and two base thicknesses (𝐻=150 𝜇𝑚 and 𝐻=300 𝜇𝑚). Figure 2 : Photovoltage of a 150μm and a 300μm PV cell under rear illumination as a function of carrier collection velocity for various grain sizes 101102103104105106107 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Carrier collector velocity (cm/s) Photovoltage (V) Front side illumination Base thickness = 150 m 101102103104105106107 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Carrier collector velocity (cm/s) Photovoltage (V) Front side illumination Base thickness = 300 m 101102103104105106107 0,05 0,10 0,15 0,20 0,25 0,30 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Photovoltage (V) Carrier collector velocity (cm/s) Rear side illumination Base thickness = 150 m 101102103104105106107 0,00 0,05 0,10 0,15 0,20 0,25 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Carrier collector velocity (cm/s) Photovoltage (V) Rear side illumination Base thickness = 300 m Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 41 As in the front-side case, Vph decreases with increasing CCV, reflecting the reduction of excess carrier density at the junction and the consequent drop in quasi-Fermi-level splitting [17]. However, the dependence on base thickness is much stronger from the rear: carriers are generated farther from the junction and must traverse the base, so a larger H increases transit time and exposure to bulk SRH and grain-boundary recombination, thereby lowering Vph at a given CCV [16-18]. The grain-size trend remains robust: larger grains (lower boundary density) yield higher Vph across the CCV range [18]. In our data, the upper envelope reaches about Vph≈0.30 V for g=0.015 cm at H= 150 μm. Finally, rear-surface passivation/contacts (which set SRV and J0) are critical under rear incidence; improved stacks shift the curves upward and mitigate the loss in thick bases [16,17]. These behaviors are consistent with recent assessments of bifacial silicon operation, where rear-side generation accentuates transport-andrecombination limitations through the base [10]. Dual-side illumination. Figure 4 show the photovoltage Vph under dual-side illumination as a function of the CCV, for several grain sizes and two base thicknesses (H=150 μm and H=300 μm). Figure 3: Photovoltage of a 150μm and a 300μm PV cell under dual illumination as a function of carrier collection velocity for various grain sizes As expected for bifacial operation, illuminating both faces increases the overall generation and shifts the Vph −CCV curves upward compared with single-face cases [10,19]. At fixed irradiance and grain size, Vph still decreases with increasing CCV, consistent with the diode relation Vph ∝ln(1+Jph/J0): faster extraction reduces the steady-state excess carrier density at the junction and hence the quasi-Fermi-level splitting [17]. The absolute Vph is, however, higher under dual illumination due to the larger Vph input; this effect is more pronounced when surface passivation and contacts are optimized (lower SRV and J0), which lifts the curves and mitigates losses across the CCV range [11,16]. The grain-size trend remains robust: larger grains (fewer boundaries) reduce recombination and yield higher Vph [18]. In our data, a maximum Vph ≈0.35 V is observed for g=0.015 cm at H=150 μm. The smaller gap between the H=150 and H=300 μm curves compared with rear-only illumination is consistent with shorter average collection distances provided by dual incidence [10,19]. Complementary view (grain-size sweep at fixed CCV). The figures below report the variation of Vph versus grain size at CCV = 30.2 cm·s−1 for multiple base thicknesses and illumination modes, illustrating the monotonic rise of Vph with grain size and the additional benefit of dual illumination [10,18,19]. Influence of grain size on the photovoltage of a solar cell illuminated from the front, the rear, and on both sides, for different base thickness Figures 5, 6, and 7 display the photovoltage 𝑉𝑝ℎ as a function of grain size 𝑔 for front, rear, and dual illumination, respectively, at several base thicknesses 𝐻. The carrier-collection velocity is fixed at 𝐶𝐶𝑉 =30,2 𝑐𝑚.𝑠−1, i.e., near open-circuit, so these trends closely reflect the open-circuit voltage 𝑉𝑜𝑐 behavior via Vph ≈Voc ∝ln(1+Jph/J0 ) [17]. 101102103104105106107 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Carrier collector velocity (cm/s) Photovoltage (V) Dual illumination Base thickness = 150 m 101102103104105106107 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,002 cm 0,005 cm 0,01 cm 0,012 cm 0,014 cm 0,015 cm Carrier collector velocity (cm/s) Photovoltage (V) Dual illumination Base thickness = 300 m Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 42 Figure 5: Photovoltage as a function of grain size for a PV cell with different base thicknesses under front-side illumination. Figure 6: Photovoltage as a function of grain size for a PV cell with different base thicknesses under rear-side illumination Figure 7: Photovoltage as a function of grain size for a PV cell with different base thicknesses under dual Two levers govern the evolution: i. any increase in Jph (e.g., dual illumination) or decrease in J0 (better passivation/contacts) raises Vph [11,16,17]; ii. larger grains reduce grain-boundary recombination, effectively lowering recombination losses and lifting Vph across H [8,18]. Front illumination (Fig. 5). The curves for different H are nearly superposed: generation occurs close to the junction, so thickness changes have little impact on Vph at fixed CCV. The dominant trend is the monotonic rise of Vph with grain size, driven by the reduced boundary density [8,17,18]. Rear illumination (Fig. 6). The curves separate with thickness: a larger H lengthens transport paths and exposure to bulk SRH and grain-boundary recombination, thus lowering Vph at a given g. For a fixed H, Vph increases with grain size, and the thinner base (e.g., H=150 μm) yields the highest voltages, especially for g∈[0, 0.012] cm [8,16-18]. Dual illumination (Fig. 7). The behavior is intermediate between front and rear: dual incidence boosts Jph, lifting all curves, while the dependence on H is weaker than rear-only because average collection distances shorten when both sides generate carriers. The grain-size benefit remains clear and is enhanced by good surface passivation and passivating contacts that suppress J0 [10,11,16]. Influence of CCV on photovoltage for different base thicknesses and fixed grain size Figures 8–9 show the photovoltage 𝑉𝑝ℎ 𝐶𝐶𝑉 for front, rear, and dual illumination, respectively, at several base thicknesses 𝐻, with a fixed grain size 𝑔=0.01 𝑐𝑚. 0,004 0,008 0,012 0,016 0,020 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 Grain size (cm) Photovoltage (V) Front side illumination Carrier collection velocity 30,2 cm/s 100m 150m 200m 250m 300m 350m 400m 0,004 0,008 0,012 0,016 0,020 0,12 0,16 0,20 0,24 0,28 0,32 Grain size (cm) Rear side illumination Carrier collector velocity 30,2 cm/s 100m 150m 200m 250m 300m 350m 400m Photovoltage (V) 0,004 0,008 0,012 0,016 0,020 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 Grain size (cm) Photovoltage (V) Dual illumination Carrier collector velocity 30,2 cm/s 100m 150m 200m 250m 300m 350m 400m Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 43 Figure 8: Photovoltage as a function of carrier collection velocity for PV cells of varying base thickness under front illumination and a constant grain Figure 9: Photovoltage as a function of carrier collection velocity for PV cells of varying base thickness under rear illumination and a constant grain size. Figure 10: Photovoltage as a function of carrier collection velocity for PV cells of varying base thickness under dual illumination and a constant grain size. As a reminder, low CCV corresponds to an extraction-limited regime (open-circuit-like for current), whereas high CCV indicates efficient extraction (short-circuit-like, with a current plateau) [17]. Consistent with Vph ≈Voc ∝ ln(1+Jph/J0), Vph decreases as CCV increases, because faster extraction lowers the steady-state excess carrier density at the junction and the quasi-Fermi-level splitting [17]. The magnitude and CCV sensitivity of Vph are set by surface recombination and contact quality (SRV, J0), improved passivation/contact stacks generally lift the curves and delay the drop with CCV [11,16]. By illumination mode, the trends match the transport distance and recombination exposure. Front illumination (Fig. 8): the curves are nearly superposed across H, since generation occurs near the junction; thickness plays a minor role at fixed CCV. Rear illumination (Fig. 9): curves separate with H, for H=100 −150 μm the behavior remains close, but thicker bases show a clear Vph reduction at the same CCV due to longer paths and higher bulk/GB recombination [17,18,20]. Dual illumination (Fig. 10): we observe blockwise superposition: a first group for H=100 − 200 μm with Vph ≈ 0.28 V and a second for H=250 − 400 μm with Vph ≈0.325 V, reflecting the higher Jph and shorter average collection distances of bifacial operation [10,20]. Absolute variation of photovoltage over the grain-size range as a function of base thickness Figures 11, 12, and 13 report the absolute variation of photovoltage over the grain-size range, ΔVph =Vph(gmax )−Vph(gmin ) (21) under front, rear, and dual illumination, respectively, as a function of base thickness 𝐻. The junction 𝐶𝐶𝑉 is fixed at 30.2 𝑐𝑚 𝑠−1, i.e., near an open-circuit-like regime for the voltage (slow extraction, larger excess carrier density) [1]. 101102103104105106107 0,10 0,15 0,20 0,25 0,30 0,35 Carrier collector velocity (cm/s) Photovoltage (V) 150m 200m 250m 300m 350m 400m Front side illumination g = 0,01 cm 101102103104105106107 0,00 0,05 0,10 0,15 0,20 0,25 0,30 Carrier collector velocity (cm/s) Photovoltage (V) 100m 150m 200m 250m 300m 350m 400m Rear side illumination g = 0,01 cm 101102103104105106107 0,10 0,15 0,20 0,25 0,30 Carrier collector velocity (cm/s) Photovoltage (V) Dual illumination g = 0,01 cm 100m 150m 200m 250m 300m 350m 400m Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 44 Figure 11: Absolute variation of the Photovoltage over the grain size range under front illumination as a function of base thickness Figure12: Absolute variation of the Photovoltage over the grain size range under rear illumination as a function of base thickness Figure 43: Absolute variation of the Photovoltage over the grain size range under dual illumination as a function of base thickness Physical reading. • Role of 𝐻: Increasing 𝐻 lengthens transport paths and raises exposure to bulk SRH and grain-boundary recombination. Since 𝑉𝑝ℎ ≈Voc ∝ln(1+Jph/J0), any thickness-induced increase of J0 (via higher SRV or deeper recombination) compresses the voltage spread between largeand small-grain cases, tending to reduce 𝛥𝑉𝑝ℎ [11,16,17]. • Front illumination (Fig. 11). Generation close to the junction makes 𝑉𝑝ℎ only weakly dependent on 𝐻; residual surface/contact losses dominate. In this regime, 𝛥𝑉𝑝ℎ often decreases with 𝐻 as added thickness brings little benefit to large grains but increases parasitic losses [17]. • Rear illumination (Fig. 12). Carriers originate far from the junction; thicker bases penalize small grains more (higher boundary density), which can increase 𝛥𝑉𝑝ℎ over part of the range (contrast effect) before very large 𝐻 suppresses both endpoints [8,17,18]. • Dual illumination (Fig. 13). Bifacial generation shortens average collection distances and mitigates thickness penalties; accordingly, 𝛥𝑉𝑝ℎ typically decreases with 𝐻, but more slowly than under rear-only operation [10,16]. Design implication. To preserve a sizable 𝛥𝑉𝑝ℎ near open-circuit: i. limit 𝐻 (e.g., 150 𝜇𝑚 vs. 300−400 𝜇𝑚) for front/dual operation; ii. in rear operation, note that thicker bases may temporarily widen 𝛥𝑉𝑝ℎ (contrast) but ultimately lower absolute voltages; and iii. combine large grains (reduced GB recombination) with low-𝐽0, low-SRV passivating contacts to lift both 𝑉𝑝ℎ(𝑔𝑚𝑎𝑥 ) and the overall curve [10,11,16]. 150 200 250 300 350 400 0,1090 0,1092 0,1094 0,1096 0,1098 0,1100 Base thickness (m) Front side illumination Carrier collector velocity 30,2 cm/s Absolute variation of the photovoltage over the grain size range (A/cm²) 150 200 250 300 350 400 0,116 0,118 0,120 0,122 0,124 Absolute variation of the photovoltage over the grain size range (A/cm²) Rear side illumination Carrier collector velocity 30,2 cm/s Base thickness (m) 150 200 250 300 350 400 0,1090 0,1095 0,1100 0,1105 0,1110 Rear side illumination Carrier collector velocity 30,2 cm/s Base thickness (m) Absolute variation of the photovoltage over the grain size range (A/cm²) Gning Y et al Euro. J. Adv. Engg. Tech., 2025, 12(10):37-46 45 CONCLUSION This work established closed-form expressions for the photovoltage under front, rear, and dual illumination in a 3D diffusion–recombination framework. Across operating conditions, the carrier-collection velocity (𝐶𝐶𝑉) emerges as the primary knob controlling the quasi-Fermi-level splitting at the junction: 𝑉𝑝ℎ decreases as 𝐶𝐶𝑉 increases, transitioning from an extraction-limited (open-circuit-like) to an efficient-collection (short-circuit-like) regime. Microstructure and geometry co-govern the voltage: larger grains (𝑔) reduce grain-boundary recombination and consistently raise 𝑉𝑝ℎ, while thinner bases (𝐻) shorten transport paths and curb bulk losses. Illumination mode sets the transport distance and thus the sensitivity to 𝐻: front illumination is weakly thickness-dependent, rear illumination is strongly thickness-dependent, and dual illumination lifts the voltage baseline while mitigating the thickness penalty. Design guidance. • Favor large grains (𝑔≳0.01 𝑐𝑚) to suppress grain-boundary recombination. • Use moderate 𝐻 (e.g., 150 µ𝑚) to limit bulk SRH losses; thicker bases demand superior passivation. • Implement low-𝐽0, low-SRV passivating contacts; improvements here lift all 𝑉𝑝ℎ 𝐶𝐶𝑉 curves. • Prefer dual illumination when available; it raises 𝑉𝑝ℎ and reduces thickness sensitivity, especially at low 𝐶𝐶𝑉. Scope & outlook. The analysis assumes uniform material properties, 𝑔𝑥=𝑔𝑦 domains, and constant temperature. Extending the model to include lateral non-uniformities, realistic optics/albedo for bifacial operation, series resistance, and temperature-dependent recombination would enable direct prediction of full 𝐽−𝑉 characteristics under field conditions. 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