Closed-form expression for α(k) in the non-constant capacitance model

Determine a closed-form expression for the function α(k) that enforces linear growth of the scaling matrix entries S(n) with respect to n in the proposed non-constant capacitance extension of the Constant Interaction Model for a single-dot device. Specifically, derive α(k) such that the recurrence S(n+2) = (n+2) / ( (2 α(k) (k+2) / (n + k + 2)) + n / S(n) ) with boundary conditions S(0) = 1 and S(1) = 1 yields S(n) increasing linearly in n, consistent with the target spacing Δv_{n+1} = |e| / (2 C_DG [ (n+2)/S(n+2) − n/S(n) ]) ∝ (k+2)/(k+n+2).

Background

To account for experimentally observed charging-energy variations, the paper proposes an extension of the Constant Interaction Model in which the dot-dot and dot-gate capacitance matrices depend on the dot occupation via a diagonal scaling matrix S(n). In the single-dot case, matching the observed power-law spacing between charging transitions leads to a recurrence for S(n) in terms of a smoothing parameter k and an unknown scaling function α(k).

The authors impose the constraint that S(n) should increase linearly with n, but report that they could not find a closed-form solution for α(k), and therefore fitted α(k) empirically for 1 < k < 100 to achieve linearity up to n ≤ 1000. A closed-form α(k) would formalize this extension and remove reliance on empirical fitting.