Retardance of lab grown diamond substrates as a function of thickness: momentum-drift random walk model

Abstract

This work studies the correlation between mean retardance and thickness of diamond substrates grown homoepitaxially via microwave plasma-enhanced chemical vapor deposition (MPCVD). We measure the retardance of a diamond substrate in two orientations: perpendicular and parallel to the growth direction. Our experimental results demonstrate that the correlation between mean retardance and thickness differs for these orientations. When measured perpendicular to the growth direction, the mean retardance is approximately proportional to the square root of the substrate thickness. In contrast, when measured parallel to the growth direction, we observe a generally higher mean retardance and an approximately linear correlation with thickness. This anisotropy arises not from differences in stress magnitude but from differences in the interlayer correlation of the principal stress axes, as evidenced by correlation coefficients between the azimuth angles of consecutive layers in the diamond crystal. To simulate the integrated retardance of diamond wafers, we propose a two-dimensional random walk model with momentum drift, which captures the diamond crystal tendency to preserve the azimuth angle across the samples. By optimizing the momentum factor, we show that the model can closely match experimental data. The momentum factor is found higher along the growth direction, which is consistent with the calculated correlation coefficients. Furthermore, both the model and experiments indicate that retardance-to-thickness ratios of thin samples converge toward similar base retardances in both orientations. These findings establish a quantitative framework for interpreting birefringence in diamond substrates, with implications for material selection and development in thermal management, quantum sensing, high-power electronics, and optical applications.