# Misfit Dislocations in Heterojunctions

| Calculating the critical thickness of a layer with lattice constant a1 on top of a substrate with lattice constant a2 can become rather involved, if all components contributing to the elastic energy are taken into account.
|  | In particular, you may want to consider the anisotropy of the situation, the effect of a finite thickness of the top layer, the real geometry with respect to the dislocations (their line energy depends on this and that, and they may be split into partial dislocations).
|  | Then, after arriving at a formula, you may chose to make all kinds of approximations.
|  | In the backbone part of the script we had a simple formula (taken from a paper of the very well known scientist Sir Peter Hirsch) which you can find in the link (together with some comments):
dcrit = | = | b 8 · p · f · (1 + n) | · ln | e · dcrit r0
--- | --- | --- | --- | ---
|  | With b = Burgers vector of the dislocations; usually somewhat smaller than a lattice constant, f = misfit parameter = (a1 - a2)/a1, n = Poisson ration » 0,4, e = really e = base of natural logarithms r0 = inner core radius of the dislocation; again in the order of lattice constant.
| Lets look at some other approaches
|  | A formula taking into account most everything going back to J. W. Matthews and A.E. Blakeslee (1974) , who pioneered this field of research, is
dcrit | = | b · (1 – n ) · cos2Q 8 · p · (1 + n) · f · cosl | · | æ ç è | ln | æ è | dcrit b | ö ø | + 1 | ö ÷ ø
|  | with Q = angle between the dislocation line and its Burgers vector, l = angle between the slip direction and that line in the interface plane that is normal to the line of intersection between the slip plane and the interface.
|  | For simple systems (Q = 90o and l = 0o), we have
dcrit | = | b 8 · p · f · (1 + n) | · | æ ç è | ln | æ è | dcrit r0 | ö ø | + 1 | ö ÷ ø
|  | And that is Sir Peters equation if you insert ln(e) for the 1 in the ln term.
| While Sir Peter used the simple approximation
dcrit | » | b 6f
--- | --- | ---
|  | the comparison with the (computer-generated) correct functional dependence suggests
dcrit | » | b 9f
--- | --- | ---
|  | which is a bit different!
| A plot of the full formula and the approximation looks like this:
| Similar curves are contained in the books of Mayer and Lau or Tu, Mayer and Feldmann; they supposedly use the same equation but show rather different results.
|  | Well, somewhere should be a mistake (maybe I made one?). In any case, it nicely demonstrates the point made in the backbone section: Do not blindly believe a theory. In case of doubt, try it out.

With frame

5.3.3 Real Heterojunctions

© H. Föll (Semiconductors - Script)