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We establish an invariance principle and a large deviation principle for a biased random walk
. The scaling limit in the invariance principle is not a d-dimensional Brownian motion. For the large deviation principle, its rate function is different from that of a drifted random walk, as may be expected, though the reflected biased random walk evolves like the drifted random walk in the interior of the first quadrant and almost surely visits coordinate planes finitely many times.
Operations on biological living cells and molecular devices have driven
research towards implementation of high-aspect-ratio nano-needles. However,
current nano-needle fabrication is complicated to control the sizes and
angles. In this work, we develop a simple method to fabricate repeatable and
integrated circuit (IC)-compatible sharp silicon nano-needles based on boron
etch-stop in tetramethyl ammonium hydroxide (TMAH) solutions, and the needle
angles can be accurately controlled. An analytical model is proposed to
efficiently predict the needle sizes and explain the etching evolution of
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