ANALYTICAL CARRIER-COLLECTION RETENTION AND DEGRADATION MARGIN IN A SERIES VERTICAL-JUNCTION SILICON SOLAR CELL

This work develops an analytical theory of carrier-collection retention during diffusion-length degradation in a series vertical-junction silicon solar cell. Starting from the reduced collection law C(h;f,b), degradation is parameterized by γ=L/L₀ under the reference assumption of lifetime-driven degradation, with constant diffusion coefficient D, thickness H, and interface velocities Sf and Sb. The induced trajectory is (h,f,b)=(h₀/γ,γf₀,γb₀). An exact retention law is obtained. It is shown that ℛ_C(1)=1 and that ℛ_C(γ) tends linearly to zero as γ→0⁺, which guarantees the existence of a retention threshold for every 0<η<1. The directional derivative satisfies γ d ln C_γ/dγ=−𝒮_h+𝒮_f+𝒮_b. Its exact factorization reveals a boundary separating an M regime of locally monotonic loss from an NM regime of transient compensation. A reduction to two invariants, a=h₀f₀ and d=h₀b₀, reduces every trajectory to a one-dimensional function and shows that any possible compensation is necessarily transient. A materialized numerical campaign in the b₀=1 section, complemented by three representative trajectories, illustrates the strong dependence of the end-of-life margin on the initial regime. For η=0.90, the representative cases yield margins of approximately 10.89%, 56.34%, and 99.53%. A generic conversion to irradiation fluence is finally established from L⁻²=L₀⁻²+k_lφ. The results explicitly distinguish diffusion-length degradation, transient compensation of the reduced model, and improvement in overall photovoltaic performance, which is not claimed.