Pattern formation and directional and spatial ordering of edge dislocations in bulk GaN: Microphotoluminescence spectra and continuum elastic calculations

We present a detailed microphotoluminescence study of the long-range strain fields surrounding threading dislocations as well as their interaction and pattern formation in GaN bulk crystals. The stress patterns are detected by tiny energy shifts of the near-band-edge spectral lines and show a dipolelike stress state around the dislocation core of edge- or mixed-type dislocations, with an angular orientation coinciding with high-symmetry crystal directions. We detect tens of micrometers long linear lineups of edge dislocations with sizable effects on the local strain states and photoluminescence peak positions. With continuum elastic strain simulations, we demonstrate that the observed patterns of threading dislocations are energetically favorable. We calculated a binding energy per threading dislocation length of $50\mathrm{meV}∕\mu \mathrm{m}$ against gliding of an edge dislocation within a lineup of edge dislocations of parallel Burgers vector orientation.

Figure 1

Microphotoluminescence area scan across the (0001) surface of a bulk GaN sample. (a) Integrated near-band-edge intensity and photoluminescence line shift, represented by the first momentum [(b) in grayscale and (c) as contour plot] of the spectrum.

Figure 2

(Color online) Spectra along one strain dipole induced by an edge- or mixed-type dislocation. (a) shows a contour plot of the photoluminescence spectra along the energy shift dipole main axis of the dislocation marked with a black rectangle in Fig. 1. Spectra at four particular positions are drawn explicitly in (b): A, far away from the dislocation and thus undisturbed; B and D, with maximum energy shift; and C, at the dislocation core. The marks represent the measured data points, whereas the solid lines are derived from a multiline fit of the spectra with three Lorentzian curves.

Figure 3

(Color online) (a) illustrates the possible angular orientation of the Burgers vectors (dark arrows: $⟨11\overline{2}0⟩$) of perfect edge dislocations along the $c$ crystal axis in wurtzite GaN. Dots symbolize the hexagonal lattice sites. The grayscale image schematically shows one possible $⟨\overline{1}100⟩$ orientation (light arrow) of the resulting strain dipole (not in scale with the lattice points). In (b) the correlation of crystal planes (lines), Burgers vectors (arrow), and strain dipole (grayscale image, not to scale with crystal planes) is visualized. (c) shows a plan-view HRTEM image of such a dislocation with a $1∕3\left[11\overline{2}0\right]$ edge component (determined by a Burgers circuit).

Figure 4

(Color online) Histogram of the angular dispersion of the stress-related dipoles induced by edge dislocations. The ticks at the outer circle mark the directions of the 126 evaluated dislocation dipoles. The dark and light angular histogram accounts for 84 completely freestanding (undisturbed) dislocations and 42 dislocations which might be slightly influenced by their neighbors, respectively. The full angular circle is divided in 30° sectors, whereby the numbers outside each sector give the total number of dislocations within. For better visualization, the shaded sectors are centered around the $⟨11\overline{2}0⟩$ directions, where dislocations should be absent, and the unshaded sectors are centered around the $⟨1\overline{1}00⟩$ directions (lines), in which one expects the dislocation dipoles.

Figure 5

(a) and (c) show the integrated near-band-edge intensity of area scans at different regions on the investigated sample. The adjacent contour plots [(b) and (d)] depict the energy deviations of the near-band-edge PL from the same region, where every contour level refers to $0.05\mathrm{meV}$ energy shift.

Figure 6

Measured [(a) and (c)] and calculated [(b) and (d)] energy shifts of the donor bound exciton caused by arrays of edge dislocations. The pattern shown in (a) corresponds to the lineup in the lower left corner of Fig. 5c; the one shown in (b) is from a different scan area not included in Fig. 5.

Figure 7

Cut along the line connecting the dislocations in Fig. 6. (a)–(d) correspond to Figs. 6a, 6b, 6c, 6d, respectively. The dotted lines mark the position of the individual dislocations used to simulate the energy shift patterns.

Figure 8

(a)–(d) show linear cuts through the PL energy landscape along dislocation lineups. (a) and (b) correspond to the dominant lineups in Figs. 5b, 5d, respectively. The dotted vertical lines with the attached dislocation marks indicate the assumed position of the single edge dislocations judging from the PL intensity. (c) and (d) show similar cuts through different dislocation lineups (not shown in Fig. 5) and (e) displays cuts through three freestanding and thus undisturbed dislocations along the main strain-dipole axis for comparison.

Figure 9

(a) Calculated potential of an edge dislocation in the strain field of an edge dislocation of the same Burgers vector orientation placed in the center. (b) Potential along the vertical dashed line showing the local minimum for gliding. (c) Binding energy for gliding as function of the distance of the gliding plane (dashed line) from the fixed edge dislocation in the center.

Figure 10

(a) Hydrostatic strain field for a pattern of four fixed and one movable edge dislocations of identical Burgers vector orientation. (b) Potential as function of the position of the movable dislocation. (c) Binding energy for gliding along the vertical dashed line in the center. (d) Potential along the horizontal line through the four fixed edge dislocations.