Atomic-precision advanced manufacturing enables unique
silicon
quantum electronics built on quantum dots fabricated from small numbers
of phosphorus dopants. The number of dopant atoms comprising a dot
plays a central role in determining the behavior of charge and spin
confined to the dots and thus overall device performance. In this
work, we use both theoretical and experimental techniques to explore
the combined impact of lithographic variation and stochastic kinetics
on the number of P incorporations in quantum dots made using these
techniques and how this variation changes as a function of the size
of the dot. Using a kinetic model of PH3 dissociation augmented
with novel reaction barriers, we demonstrate that for a 2 × 3
silicon dimer window the probability that no donor incorporates goes
to zero, allowing for certainty in the placement of at least one donor.
However, this still comes with some uncertainty in the precise number
of incorporated donors (either one or two), and this variability may
still impact certain applications. We also examine the impact of the
size of the initial lithographic window, finding that the incorporation
fraction saturates to δ-layer-like coverage as the circumference-to-area
ratio decreases. We predict that this incorporation fraction depends
strongly on the dosage of the precursor and that the standard deviation
of the number of incorporations scales as ∼√n, as would be expected for a sequence of largely independent
incorporation events. Finally, we characterize an array of 36 experimentally
prepared multidonor 3 × 3 nm lithographic windows with scanning
tunneling microscopy, measuring the fidelity of the lithography to
the desired array and the final location of PH
x
fragments within these lithographic windows. We use our kinetic
model to examine the expected variability due to the observed lithographic
error, predicting a negligible impact on incorporation statistics.
We find good agreement between our model and the inferred incorporation
locations in these windows from scanning tunneling microscope measurements.