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

We present a three-dimensional diffusion–recombination model that yields closed-form expressions for the photovoltage V_ph 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 (g_x, g_y), 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 V_ph versus CCV diminishes, and

  iii.     a high-CCV efficient-collection regime (short-circuit-like current plateau), in which V_ph is lowest for a given generation level.

Parametric results show that V_ph decreases with increasing CCV, consistent with the reduction of steady-state excess density and quasi-Fermi-level splitting. Larger grains systematically increase V_ph by lowering grain-boundary 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 V_ph-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.