A stochastic simulation of the dislocation-mediated etching of porous GaN distributed Bragg reflectors

1. Problem Statement

The electrochemical porosification of Gallium Nitride (GaN) is a critical process for fabricating Distributed Bragg Reflectors (DBRs) used in optoelectronic devices (e.g., LEDs, lasers). In these structures, alternating layers of highly n-doped GaN and non-intentionally doped (NID) GaN are stacked. The etching process is conductivity-selective, meaning only the n-doped layers are expected to etch, while NID layers remain intact.

However, a fundamental challenge exists: how does the etchant access the buried n-doped layers when the surface is an insulating NID layer?

• Traditional Model ("Kebab Model"): It was previously hypothesized that Threading Dislocations (TDs) act as continuous vertical pipelines (nanopipes) allowing etchant to flow straight down through every layer, creating a "kebab" of porosity along each dislocation.
• Observed Reality: Recent experimental data (using FIB tomography and STEM) revealed that the "kebab" model is insufficient. Instead, etching often follows a "cascade" mechanism: etchant travels down a TD, moves laterally through a doped layer, intersects a different TD, and continues downward. This results in complex, non-uniform pore structures where TDs can be active in some layers and inactive in others.
• Gap: The physical mechanisms controlling the transition between "kebab" and "cascade" morphologies, and how applied voltage influences these pathways, are not fully understood. There is a lack of predictive models to correlate experimental chronoamperometry (current-time) data with the resulting 3D microstructures.

2. Methodology

The authors developed a stochastic simulation to model the electrochemical etching process and compared its outputs against experimental data from GaN DBRs grown on Silicon substrates.

• Simulation Architecture:

  • Built in Python using the numba library for just-in-time compilation to ensure efficiency.
  • Grid Model: The DBR structure is represented as a 2D array of pixels (5 nm × 5 nm).
  • Components: Pixels are assigned as Etchant, NID GaN, Si-doped GaN, or Threading Dislocation (TD).
  • TD Modeling: TDs are modeled as vertical lines (1 pixel wide, 500 nm spacing) penetrating the stack.
  • Etching Logic:
    • NID pixels have a 0% probability of etching.
    • Doped pixels and TD pixels have finite probabilities of etching: $P_{doped}$ (probability of etching the doped layer) and $P_{disloc}$ (probability of etching the TD).
    • Cascade Mechanism: The algorithm allows etchant to move laterally. If a lateral path intersects an inactive TD, that TD can become active, creating a "cascade" pathway.
    • Constraints: Neighboring pores are prevented from merging to mimic the non-intersecting domains observed in microscopy.
  • Current Calculation: Each etched pixel generates a unit of charge, allowing the simulation to generate chronoamperometry curves (Current vs. Iteration/Time).

• Experimental Validation:

  • Samples: Three samples were grown via MOVPE:
    • Sample A: Standard DBR with uniform layer thicknesses (63 nm doped / 51 nm NID).
    • Sample B & C: DBRs with varying doped layer thicknesses (increasing or decreasing through the stack) to test the model's robustness.
  • Etching: Performed in oxalic acid at voltages ranging from 6 V to 12 V without lithographic trenches.
  • Characterization: Chronoamperometry was recorded during etching. Post-etch microstructures were analyzed via SEM and FIB tomography (referencing previous data from Thornley et al.).

3. Key Contributions

1. Development of a Stochastic Cascade Model: The authors created the first simulation capable of reproducing the "cascade" morphology where etching pathways switch between TDs laterally, moving beyond the simplistic "kebab" model.
2. Correlation of Voltage and Probability: The study establishes a direct link between experimental applied bias and simulation parameters. Increasing the applied voltage in experiments corresponds to increasing the probability of etching the doped layers ($P_{doped}$) in the simulation.
3. Quantification of Morphology: The simulation successfully predicts the ratio of "K5" structures (TDs active in all 5 layers) to "C" structures (cascades) as a function of etching conditions, matching trends observed in FIB tomography.
4. Extension to Non-Uniform Structures: The model was validated against samples with non-uniform layer thicknesses (Samples B and C), demonstrating its applicability beyond standard periodic DBRs.
5. Insight into Trench Etching: The simulation was adapted to model etching from lithographically defined trenches, suggesting that even in trench-based approaches, TD-mediated cascades eventually dominate the pore morphology away from the trench sidewall.

4. Key Results

• Chronoamperometry Matching:
  • The simulation reproduces the characteristic oscillating current peaks seen in experiments, where each peak roughly corresponds to the etching of a doped layer.
  • Voltage Effect: Lower voltages (low $P_{doped}$) result in damped oscillations and lower currents, while higher voltages (high $P_{doped}$) produce distinct, high-amplitude peaks.
  • Discrepancy: The simulation does not capture the initial high current spike seen in experiments (attributed to side-wall etching in real samples, which the simulation excludes).
• Morphological Evolution:
  • Low Voltage (Low $P_{doped}$): The simulation shows a mix of short "kebab" segments and frequent cascades. The etch front is non-uniform, leading to irregular current peaks.
  • High Voltage (High $P_{doped}$): The simulation predicts a higher proportion of "K5" (full kebab) structures and fewer cascades. This matches experimental FIB data showing that higher voltages favor more continuous vertical etching.
  • Layer-by-Layer vs. Cascade: At high probabilities, etching appears more "layer-by-layer." At low probabilities, lateral etching competes with vertical propagation, causing the etch front to become irregular and oscillations to die out.
• Variable Thickness Samples:
  • The simulation successfully predicted the chronoamperometry trends for Samples B and C.
  • It correctly identified that samples with thicker layers near the surface (Sample B) show decreasing peak amplitudes over time, while samples with thinner surface layers (Sample C) show more stable peaks.
  • Limitation: The simulation slightly overestimates the etching of deep layers compared to experiments, suggesting that in reality, etchant access to deep layers is more difficult than the model assumes.
• Pore Morphology:
  • The simulation reproduces the "fine, tortuous" pore structures seen at low voltages.
  • Limitation: The simulation fails to reproduce the "conical pore" expansion observed in NID layers at high voltages (where the etchant widens the NID layer around the TD), as the model strictly prohibits etching in NID regions away from the TD core.

5. Significance

• Mechanistic Insight: The study provides strong evidence that the electrochemical etching of GaN is a stochastic process governed by local probabilities rather than a deterministic, uniform flow. It validates the "cascade" hypothesis as a dominant mechanism in TD-mediated etching.
• Process Optimization: By linking applied voltage to the probability of etching ($P_{doped}$), the simulation offers a tool for engineers to predict and control the porosity and reflectivity of DBRs without needing to perform extensive trial-and-error etching.
• Broader Applicability: The model's ability to handle non-uniform layer thicknesses and alternative etching geometries (trenches) suggests it can be a versatile tool for designing complex nitride photonic devices.
• Future Physics: The work highlights that while the macroscopic behavior can be modeled stochastically, the microscopic physics (e.g., Zener tunneling, trap-assisted tunneling at TD cores in NID layers) requires further investigation to fully explain the probability variations with voltage.

In conclusion, this paper bridges the gap between experimental observation and theoretical modeling in GaN electrochemical etching, providing a robust stochastic framework that explains the complex interplay between threading dislocations, applied bias, and resulting 3D pore morphologies.