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Inclined threading dislocations (TDs) piercing the oriented free surface of a crystal are currently observed after growth of oriented thin films on substrates. Up to date the unique way to treat their anisotropic elastic properties nearby the free surface region is to use the integral formalism, which assumes no dislocation core size and needs numerical double integrations. In a first stage of the work, a new and alternative approach to the integral formalism is developed using double Fourier series and the concept of a finite core size, which is often observed in high-resolution transmission electron microscopy. In a second stage, the integral formalism and the Fourier series approaches are applied to the important case of a TD piercing the basal free surface of a hexagonal crystal. For this particular geometry, easy-to-use expressions are derived and compared to a third approach previously known for a plate-like crystal. Finally, the numerical interest and the convergence of these approaches are tested using the basal free surface of the GaN compound, in particular for TDs with Burgers vectors

Epitaxial growth of semiconductor materials offers opportunities in combining and modifying structural and electronic parameters [

A description of the elastic field of inclined dislocations lying in an isotropic plate was recently presented from a Fourier calculation method [

Cartesian frames and symbols attached to (a) an isolated inclined threading dislocation (TD) oriented by

The first step of the calculation is to consider a biperiodic series of largely spaced and identical emerging TDs. Its stress field can be written as the sum

Let us first consider the calculation of

In the plane

According to [

Let us now consider the calculation of

Since this stress field should also be periodic, it can be expressed formally as a double Fourier series. As shown in [

The total stress field is finally obtained from the addition of expressions (

As a consequence, three parameters are required to evaluate the elastic field around an isolated TD: the period

For the case of a hexagonal crystal,

Constants

The elastic field described by the above Fourier approach can be compared with that calculated with the integral formalism [

In the integral formalism, the relaxation stress field of a TD dislocation is calculated for convenience in the frame

For the particular orientation

When

The Burgers vector of an infinitesimal dislocation has the following simple form:

The relaxed displacement field

As an example, for the edge case with

The displacement field regarding a mixed dislocation with

(i) Edge dislocation

(ii) Screw dislocation

Gallium nitride is chosen to perform numerical applications because of its technological interest for electronic devices that can work under high power density and at relatively high temperature [

The thickness of the GaN layer is assumed to be large enough to neglect the elastic interaction of the TD with the GaN/substrate interface. Different anisotropic elastic constants _{3}mc) with lattice parameters

Elastic constants of GaN. In anisotropic elasticity, the equilibrium angle

(GPa) | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

Refs. | AS | A∞ | IS | I∞ | ||||||

[ | 367 | 135 | 103 | 405 | 95 | 113, 0.27 | 2.6° | 3.8° | 4.8° | 11.6° |

[ | 388 | 154 | 84 | 458 | 85 | 112, 0.28 | 2.5° | 3.6° | 4.9° | 11.6° |

[ | 329 | 109 | 80 | 357 | 91 | 107, 0.25 | 2.4° | 3.6° | 4.5° | 11.6° |

[ | 390 | 145 | 106 | 398 | 105 | 120, 0.26 | 2.8° | 4.4° | 4.8° | 11.6° |

[ | 374 | 106 | 70 | 379 | 101 | 121, 0.22 | 2.0° | 3.0° | 4.2° | 11.6° |

[ | 359 | 129 | 92 | 389 | 98 | 113, 0.26 | 2.4° | 3.7° | 4.7° | 11.6° |

[ | 334 | 132 | 99 | 372 | 86 | 100, 0.28 | 2.6° | 3.8° | 4.9° | 11.6° |

Since the isotropic approximation is sometimes used in literature, column 7 indicates the calculated Young modulus

Other TDs in GaN are sometimes directed along the

Case of the

For

Orientation

Case of the (

Figure

The dashed curve denoted “Iso semi-∞” is calculated from the Yoffe-Shaibani-Hazzledine solution [

In this work, the elastic field of an inclined TD has been investigated using different approaches. The first one, based on Fourier series and the concept of extended dislocation core, is developed in the present work from an appropriate combination of partial results obtained in [

When the anisotropic constants tend to those of an isotropic crystal, we verified numerically that the three above approaches converge to the same stress field, namely, that obtained from the Yoffe-Shaibani-Hazzledine expressions [

For a dislocation normal to the basal free surface of a hexagonal crystal

Of course, it is expected that the atomic structure of the dislocation core could influence the elastic field at distance from the core lower than a few Burgers vector moduli. According to our experience the assumption of a square section of the dislocation core has no sensible influence on the stress field beyond a few nanometers from the dislocation line provided that the core size

The following misprints can be noticed in Appendix

In expression

In expression

In [

The parameters

The authors declare that there are no conflicts of interest regarding the publication of the paper.

_{x}Ga

_{1−x}N alloys

_{2}Cu (