# Nonlocal diffusion currents at the nanoscale

## TL;DR

This paper derives a nonlocal diffusion equation for impurity currents at the nanoscale, revealing short-time nonlocal effects and proposing experimental tests, with implications for nanostructured materials.

## Contribution

It introduces a generalized nonlocal diffusion model that accounts for boundaries and variable defect concentrations, extending existing pair mechanism theories.

## Key findings

- Nonlocal effects are significant at short diffusion times.
- Predictions align with local theories at large times.
- Experimental verification of uphill diffusion and impurity segregation profiles is feasible.

## Abstract

An integro-differential expression for the diffusion current of the impurities diffusing by the mechanism of bound impurity-defect pairs has been derived. The ensuing nonlocal diffusion equation generalizes the existing theories of diffusion by the pair mechanism in unbounded systems with homogeneous defect distributions on the systems with boundaries and variable defect concentration. It has been established that the nonlocality manifests itself only at short diffusion times while at large times, in particular, under the stationary conditions, the predictions of the nonlocal theory would coincide with existing approaches based on local diffusion currents. Possibilities of experimental verification of the nonlocal theory have been suggested. In particular, the explanation of the uphill diffusion in silicon by the pair drag can be tested within slightly modified conventional experimental setups. Also, it has been shown that in the nonlocal theory the impurity segregation profile caused by the drag should differ at the initial stage of evolution from the predictions of the local theories. Because the segregation influences mechanical, corrosive, and electronic properties of materials, the nonlocal character of the pair diffusion may have important implications for nanostructured materials.