Height distribution of equipotential lines in a region confined by a rough conducting boundary

TL;DR

This study analyzes how the height distributions of equipotential lines evolve in a region with a rough cathode boundary and a flat anode, revealing how morphology depends on interface growth parameters and approaches a uniform electric field.

Contribution

It introduces a detailed analysis of equipotential line height distributions in a region with a rough boundary generated by a KPZ-class growth model, linking morphology to growth parameters.

Findings

Skewness increases as lines approach the flat anode.

Equipotential lines converge to a delta distribution in a uniform field.

Morphology varies with the KPZ growth parameter p.

Abstract

This work considers the behavior of the height distributions of the equipotential lines in a region confined by two interfaces: a cathode with an irregular interface and a distant flat anode. Both boundaries, which are maintained at distinct and constant potential values, are assumed to be conductors. The morphology of the cathode interface results from the deposit of $2 \times 10^{4}$ monolayers that are produced using a single competitive growth model based on the rules of the Restricted Solid on Solid and Ballistic Deposition models, both of which belong to the Kadar-Parisi-Zhang (KPZ) universality class. At each time step, these rules are selected with probability $p$ and $q = 1 - p$. For several irregular profiles that depend on $p$, a family of equipotential lines is evaluated. The lines are characterized by the skewness and kurtosis of the height distribution. The results indicate that the skewness of the equipotential line increases when they approach the flat anode, and this increase has a non-trivial convergence to a delta distribution that characterizes the equipotential line in a uniform electric field. The morphology of the equipotential lines is discussed; the discussion emphasizes their features for different ranges of $p$ that correspond to positive, null and negative values of the coefficient of the non-linear term in the KPZ equation.