Dislocation correlations in GaN epitaxial films revealed by EBSD and XRD

Correlations between dislocations in crystals reduce the elastic energy via screening of the strain by the surrounding dislocations. We study the correlations of threading dislocations in GaN epitaxial films with dislocation densities of $5\times10^{8}$ cm$^{-2}$ and $1.8\times10^{10}$ cm$^{-2}$ by X-ray diffraction (XRD) in reciprocal space and by high-resolution electron backscatter diffraction (HR-EBSD) in real space, where the strain is derived from a cross-correlation analysis of the Kikuchi patterns. The measured XRD curves and HR-EBSD strain and rotation maps are compared with Monte Carlo simulations within one and the same model for the dislocation distributions. The screening of the dislocation strains is modeled by creating pairs of dislocations with opposite Burgers vectors, with the mean distance between dislocations in a pair equal to the screening distance. The pairs overlap and cannot be distinguished as separate dipoles. The HR-EBSD-measured autocorrelation functions of the strain and rotation components follow the expected logarithmic law for distances smaller than the screening distances and become zero for larger distances, which is confirmed by the Monte Carlo simulations. The kink in the plot of the autocorrelation function allows a robust and accurate determination of the screening distance without making any simulation or fit. Screening distances of 2 $μ$m and 0.3 $μ$m are obtained for the samples with low and high dislocation densities, respectively. The dislocation strain is thus screened by only 4 neighboring dislocations. In addition, an anisotropic resolution of the HR-EBSD measurements is observed and quantified. In this version, an error in the processing of the HR-EBSD maps of the Si wafer is specified.

💡 Research Summary

This paper investigates the spatial correlations of threading dislocations in GaN epitaxial films by combining X‑ray diffraction (XRD) and high‑resolution electron backscatter diffraction (HR‑EBSD). Two GaN samples with markedly different dislocation densities—5 × 10⁸ cm⁻² (low) and 1.8 × 10¹⁰ cm⁻² (high)—are examined, together with a low‑density free‑standing GaN reference (≈6 × 10⁵ cm⁻²) and a Si(001) wafer for calibration.

The authors adopt a single statistical model for the dislocation arrangement: dislocations are generated in random opposite‑Burgers‑vector pairs. The distance between the two members of a pair, R, is drawn from a log‑normal distribution whose mean defines the “screening distance”. When R exceeds the mean inter‑dislocation spacing (ρ⁻¹/²), the pairs overlap and cannot be distinguished as isolated dipoles—exactly the situation in real GaN films with high densities. The model includes all relevant GaN threading dislocation types (a‑type edge, c‑type screw, and a + c mixed) with equal probability.

XRD measurements are performed in a skew‑geometry double‑crystal setup using Cu Kα₁ radiation. The diffracted intensity I(q) is computed as the probability density of the distortion component q = K̂_out·∇(Q·U), where U(r) is the total displacement field from all dislocations, Q the diffraction vector, and K̂_out the direction of the diffracted beam. The experimental line profiles display a central Gaussian core and q⁻³ tails. The transition (“kink”) between these regimes directly yields the screening distance R without any fitting. For the low‑density sample R≈2 µm; for the high‑density sample R≈0.3 µm. These values correspond to screening by roughly four neighboring dislocations, confirming that only a few nearby lines dominate strain compensation.

HR‑EBSD is carried out in a Zeiss Ultra 55 SEM at 15 kV, 6 nA, with the sample tilted 70° to the incident beam. Kikuchi patterns are recorded at 470 × 470 px resolution and processed by cross‑correlation (CrossCourt software) to extract sub‑pixel shifts. Using GaN elastic constants, the full strain tensor (ε₁₁…ε₃₃, ε₁₂, ε₁₃, ε₂₃) and rotation tensor (ω₁₂, ω₁₃, ω₂₃) are mapped over areas of 2 × 2 µm² (20 nm step) and 10 × 10 µm² (50 nm step).

From these maps the strain‑strain and rotation‑rotation autocorrelation functions C(r)=⟨ε_i(r₀)ε_i(r₀+r)⟩ are calculated. For distances r < R the functions follow a logarithmic dependence (∝ ln r), reflecting the 1/r decay of a single dislocation’s strain field. For r > R the correlations vanish, producing a flat tail. Plotting C(r) versus ln r yields a characteristic hook‑shaped curve; the kink precisely marks the screening distance. The HR‑EBSD derived R values match those from XRD and the Monte Carlo simulations, confirming the robustness of the “fit‑free” method.

Monte Carlo simulations are performed for both XRD profiles and HR‑EBSD maps using the same dislocation‑pair model. The simulated XRD line shapes and HR‑EBSD correlation functions reproduce the experimental data with excellent agreement, demonstrating that a single statistical description suffices for both reciprocal‑space and real‑space techniques.

An additional finding concerns the anisotropic spatial resolution of HR‑EBSD. Because the electron beam is inclined, the resolution along the beam direction is roughly twice worse than perpendicular to it. This anisotropy is quantified by comparing correlation functions along orthogonal directions. Moreover, measurements on the strain‑free Si wafer reveal a random‑noise background in shear strain components that should be zero; the width of these distributions provides an empirical estimate of the HR‑EBSD strain accuracy (≈10⁻⁴).

In summary, the paper establishes that (i) dislocation strain screening in GaN can be accurately quantified by a simple pair‑screening model, (ii) the screening distance can be extracted directly from the kink in strain‑strain correlation functions without any fitting, and (iii) HR‑EBSD, despite its anisotropic resolution, yields consistent real‑space information that complements XRD. The methodology is applicable to other III‑nitride and wide‑bandgap semiconductor systems where threading dislocations dominate the defect landscape, offering a powerful tool for assessing defect‑induced strain and its impact on device performance.