Phenomenological-Velocity-Indexed Transition-Curvature Engineering for Sequential Charged-Particle Lithography: Activation, Depletion, and Completion

Sequential positive exposure can suppress weak background but provides only limited spatial narrowing, because the inverse variances of its positive transition kernels add and because saturation of the individual positive gates degrades peak curvature at any dose high enough to meet a site-yield requirement. This work develops a stronger charged-particle architecture in which particle velocity, energy, and species index transition-specific activation, depletion, completion, and leakage responses. A first exposure creates an intermediate state, a spatially nodal second exposure selectively deactivates that intermediate outside the desired site, and a third exposure completes the surviving state. The completed-state width is governed by a transition-curvature matrix, giving w=22ln2/λmin(HC), so that localization below 0.5nm requires λmin(HC)>22.18nm−2. Two results are established. First, a common scalar phenomenological-velocity gain cannot narrow a normalized positive kernel, and for a saturating positive gate the peak curvature carries a factor A0/(eA0−1) that is strictly decreasing: increasing gain on a positive channel reduces curvature, whereas gain multiplying a nodal depletion action increases it linearly. At the yield-required action A0≈6, the benchmark 3.0 and 2.2nm positive kernels retain only 0.0047nm−2 of peak curvature, and their simulated completed-state width is 8.73nm rather than the 4.18nm zero-dose limit. Second, for a locally quadratic node the neighbor-rejection constraint requires κ1DHD,0≥55.26nm−2, binding 2.49× harder than the 22.18nm−2 curvature constraint; the depletion requirement is therefore set by contrast, not by width. All of these results depend only on the calibrated transition factors and are identical whether those factors are indexed by energy or by velocity; for a fixed particle species the two formulations are the same model. A direct numerical solution of the ordered response model reproduces the predicted width to within 1% once the depletion term dominates and confirms that the design gain delivers the specified 10−3 neighbor error. For a representative 99%-yield site process, the analysis further requires approximately six effective events per intended transition, total direct-leakage fraction below 1.7×10−4, intermediate survival strictly exceeding the yield target, a single log-loss budget shared between intermediate survival, positive-gate incompleteness, and depletion at the node, which at seven events per channel leaves 3.2×10−3 for node loss and requires the node floor to lie below 4.7×10−4 of the neighbor-distance action; node placement within about 10pm; and negligible diffusion. The architecture is physically plausible as an atomic-site proof of principle in an immobile discrete material, and is not presented as a demonstrated conventional-resist process.