Length scales and roughness on a growing solid surface: A review

Surfaces of grown solids such as electrochemically deposited materials, need to be smooth and require a method for characterizing the roughness. Theoretical understanding behind formation of surface roughness has been reviewed. It is argued that there is a length scale characteristic of the process that is used to grow the deposit on the substrate. Above such a length scale dendrites form due to interfacial instability but the roughness below such a length scale can be characterized by the solution to the Kardar–Parisi–Zhang equation under certain conditions. Atomic force microscopy measurements for substrates prepared for microelectronics, often fall in this latter category. Diffusion of the reactants in the solution plays a key role in determining the above critical length scale. In particular, the critical length scale becomes infinite when the rate of deposition is limited by the surface reaction which is generally not expected in electrochemical deposition except at low currents and low substrate dimensions. Comparison with experimental observations is made that indicate that at least the direction of the above supposition is correct. For contrast, some discussion of other deposition techniques, such as chemical vapor deposition where the rate controlling step is expected to be reaction limited, is provided. Issues related to the use of additives, multiple reactions, epitaxial growth are briefly discussed.