• ALD modeling;
  • ALD surface coverage dynamics;
  • Nanostructured substrates


We present a general model based on a time-dependent, reaction–diffusion equation to determine the dosing times and coverage profiles in structured substrates during atomic layer deposition (ALD). We first derive expressions comprising a non-linear diffusion–reaction equation coupled to a surface kinetic equation. In their non-dimensional forms, these equations show that coverage dynamics during ALD in nanostructured substrates depend only on two non-dimensional parameters, the Damkoler and precursor excess (number of molecules per surface site in the nanostructure) numbers. Using the assumptions of molecular flow in a circular pore, we derive a general, analytic equation to predict saturation exposure times. To demonstrate the utility of our model, we derive additional expressions incorporating a precursor loss term relevant to predicting exposure times during ozone-based ALD. Because our model makes no assumptions about the diffusion coefficient or sample geometry, it can easily be adapted to describe a broad range of ALD systems such as trenches and vias, anodized alumina, or aerogels under almost any conditions including molecular, viscous, and transition flow regimes