^{1,2}, Robert Colby

^{1,2}, Isaac H. Wildeson

^{2,3}, David A. Ewoldt

^{1,2}, Timothy D. Sands

^{1,2,3}, Eric A. Stach

^{1,2}and R. Edwin García

^{1,2,a)}

### Abstract

The effect of image forces in GaN pyramidal nanorod structures is investigated to develop dislocation-free light emitting diodes (LEDs). A model based on the eigenstrain method and nonlocal stress is developed to demonstrate that the pyramidal nanorod efficiently ejects dislocations out of the structure. Two possible regimes of filtering behavior are found: (1) cap-dominated and (2) base-dominated. The cap-dominated regime is shown to be the more effective filtering mechanism. Optimal ranges of fabrication parameters that favor a dislocation-free LED are predicted and corroborated by resorting to available experimental evidence. The filtering probability is summarized as a function of practical processing parameters: the nanorod radius and height. The results suggest an optimal nanorod geometry with a radius of (26 nm) and a height of (65 nm), in which is the magnitude of the Burgers vector for the GaN system studied. A filtering probability of greater than 95% is predicted for the optimal geometry.

The authors would like to acknowledge the support by the Department of Energy under Award No. DE-FC26-06NT42862. R.C. and E.A.S. would like to acknowledge additional funding by NSF under Grant No. NSF-DMR 0606395. I.H.W. would like to acknowledge funding from the NDSEG research fellowship supported by the U.S. Department of Defense.

I. INTRODUCTION

II. THEORETICAL FRAMEWORK

III. NUMERICAL IMPLEMENTATION

IV. MODEL VALIDATION

V. ANALYZED GEOMETRY

VI. RESULTS AND DISCUSSION

VII. SUMMARY AND CONCLUSIONS

### Key Topics

- Nanorods
- 43.0
- Dislocations
- 26.0
- Nanostructures
- 22.0
- III-V semiconductors
- 13.0
- Light emitting diodes
- 11.0

## Figures

TEM images of a dislocation found in a GaN nanorod structure that turns toward the side wall and results in a dislocation-free pyramidal cap (figure modified from Ref. 10 ). [(a) and (b)] A filtered dislocation is shown in weak beam dark field images tilted with , viewed at two angles separated by 60°. (c) Bright field image of the same dislocation and nanorod along with the schematic of the cross-section used in the three-dimensional simulations. The radius of the nanorod is while the height is . The distance between the dislocation line and the central axis of the nanorod is . The pyramidal nanorod observed here corresponds to , , and .

TEM images of a dislocation found in a GaN nanorod structure that turns toward the side wall and results in a dislocation-free pyramidal cap (figure modified from Ref. 10 ). [(a) and (b)] A filtered dislocation is shown in weak beam dark field images tilted with , viewed at two angles separated by 60°. (c) Bright field image of the same dislocation and nanorod along with the schematic of the cross-section used in the three-dimensional simulations. The radius of the nanorod is while the height is . The distance between the dislocation line and the central axis of the nanorod is . The pyramidal nanorod observed here corresponds to , , and .

Comparison of analytical and numerical shear stress, , for a screw dislocation. The solid curve corresponds to the analytical solution, △ to the large numerical domain with , and ○ to the small numerical domain with .

Comparison of analytical and numerical shear stress, , for a screw dislocation. The solid curve corresponds to the analytical solution, △ to the large numerical domain with , and ○ to the small numerical domain with .

Schematic deconvolution of integration path [dotted line in (a)] decomposed into two segments [(b) and (c)] to calculate the force on the dislocation, (a) shows a cross-section from the three-dimensional nanorod structure, (b) corresponds to the configuration with a oriented slip plane (shaded area) with a positive Burgers vector, and (c) corresponds to a configuration with a oriented slip plane (shaded area) with a negative Burgers vector. In this configuration, the line integral avoids the slip plane.

Schematic deconvolution of integration path [dotted line in (a)] decomposed into two segments [(b) and (c)] to calculate the force on the dislocation, (a) shows a cross-section from the three-dimensional nanorod structure, (b) corresponds to the configuration with a oriented slip plane (shaded area) with a positive Burgers vector, and (c) corresponds to a configuration with a oriented slip plane (shaded area) with a negative Burgers vector. In this configuration, the line integral avoids the slip plane.

Force density acting along a dislocation line in a pyramidal nanorod (◻), and in a nanopillar (○)-In both cases, (40 nm), (100 nm) and . The Peierls–Nabarro force density for GaN, , is shown as a dashed line for reference. The force density at the cap region for the pyramidal nanorod is greater than the lattice resistance, while the dislocation in the nanopillar is metastable.

Force density acting along a dislocation line in a pyramidal nanorod (◻), and in a nanopillar (○)-In both cases, (40 nm), (100 nm) and . The Peierls–Nabarro force density for GaN, , is shown as a dashed line for reference. The force density at the cap region for the pyramidal nanorod is greater than the lattice resistance, while the dislocation in the nanopillar is metastable.

Effect of nanorod radius on the force density along the dislocation line. Nanorod height is set to (100 nm). (a) corresponds to (60 nm), (b) corresponds to (25 nm), and (c) corresponds to (10 nm). In each plot, △ corresponds to , ◇ to , ◻ to , and ○ to .

Effect of nanorod radius on the force density along the dislocation line. Nanorod height is set to (100 nm). (a) corresponds to (60 nm), (b) corresponds to (25 nm), and (c) corresponds to (10 nm). In each plot, △ corresponds to , ◇ to , ◻ to , and ○ to .

Effect of nanorod radius on the effective force density, . Nanorod height is set to (100 nm). In each plot, △ corresponds to , ◇ to , ◻ to , and ○ to . Error bars represent the standard deviation of force density along each dislocation line.

Effect of nanorod radius on the effective force density, . Nanorod height is set to (100 nm). In each plot, △ corresponds to , ◇ to , ◻ to , and ○ to . Error bars represent the standard deviation of force density along each dislocation line.

Effect of nanorod height, , on the force density along the dislocation lines for (25 nm) and . The dashed line shows , the Peierls–Nabarro force density for GaN. The symbol of ● corresponds to (10 nm), to (50 nm), and × to (100 nm).

Effect of nanorod height, , on the force density along the dislocation lines for (25 nm) and . The dashed line shows , the Peierls–Nabarro force density for GaN. The symbol of ● corresponds to (10 nm), to (50 nm), and × to (100 nm).

Effect of nanorod height, , on the effective force density. Nanorod radius is set to (25 nm). The dashed line shows the Peierls–Nabarro force density for GaN, . In the plot, △ corresponds to and ○ to . Error bars represent the standard deviation of force density along the dislocation.

Effect of nanorod height, , on the effective force density. Nanorod radius is set to (25 nm). The dashed line shows the Peierls–Nabarro force density for GaN, . In the plot, △ corresponds to and ○ to . Error bars represent the standard deviation of force density along the dislocation.

Effective force density isocontour maps as a function of normalized nanorod radius, , and height, , for (a) (b) and (c) . The isocontour lines represent the magnitude of the effective force density, . The regions outlined by dotted lines correspond to nanostructures where , while the regions outlined by dashed lines correspond to geometries with cap-dominated behavior. The overlap of the dotted and dashed region, , corresponds to the predicted window of nanorod geometries for dislocation-free LEDs. Region corresponds to cap-dominated behavior and . Finally, region corresponds to base-dominated behavior and .

Effective force density isocontour maps as a function of normalized nanorod radius, , and height, , for (a) (b) and (c) . The isocontour lines represent the magnitude of the effective force density, . The regions outlined by dotted lines correspond to nanostructures where , while the regions outlined by dashed lines correspond to geometries with cap-dominated behavior. The overlap of the dotted and dashed region, , corresponds to the predicted window of nanorod geometries for dislocation-free LEDs. Region corresponds to cap-dominated behavior and . Finally, region corresponds to base-dominated behavior and .

Dislocation filtering probability maps, (a) illustrates the pore arrangement of the mask. is the pore radius, is the pore spacing, and is the critical radius for effective dislocation filtering, (b) corresponds to the map of dislocation filtering probability for pore filtering, . (c) corresponds to the map of dislocation filtering probability for an individual, isolated pyramidal nanorod, . Region and region indicate parameter ranges for low filtering probabilities . (d) corresponds to the map of total probability for pore-plus-nanopyramid filtering, .

Dislocation filtering probability maps, (a) illustrates the pore arrangement of the mask. is the pore radius, is the pore spacing, and is the critical radius for effective dislocation filtering, (b) corresponds to the map of dislocation filtering probability for pore filtering, . (c) corresponds to the map of dislocation filtering probability for an individual, isolated pyramidal nanorod, . Region and region indicate parameter ranges for low filtering probabilities . (d) corresponds to the map of total probability for pore-plus-nanopyramid filtering, .

Force density acting along a dislocation line located off-center in different directions for a pyramidal nanorod with , , and . In each plot, ○ corresponds to along , ◇ to along , and ◻ to a direction in between.

Force density acting along a dislocation line located off-center in different directions for a pyramidal nanorod with , , and . In each plot, ○ corresponds to along , ◇ to along , and ◻ to a direction in between.

## Tables

Article metrics loading...

Full text loading...

Commenting has been disabled for this content