# NUSOD Blog

Connecting Theory and Practice in Optoelectronics

## How to calculate the piezoelectric polarization of semi-polar InGaN?

GaN-based light-emitting devices suffer from strong polarization of the InGaN light-emitting layers when grown along the wurtzite c-axis. That is why researchers worldwide work on growing these devices in different, so-called semi-polar crystal directions that promise less polarization. These promises are typically derived from theoretical polarization models. The most cited model was published by Romanov et al. in 2006 (link). It gives analytical formulas for the piezoelectric polarization as function of the angle between growth direction and c-axis. However, when I tried to use this model, I realized that the given formulas do not produce the results plotted in the paper (see figure).

Luckily, two of our NUSOD 2015 authors (link) just published a more general model for InGaN quantum dot polarization (link). They found perfect agreement with Romanov’s model, but only after making several corrections to his equation (18). First, they replaced the second piezoelectric coefficients e33 by e31, which obviously was a typo, because this produced the results plotted in the paper (blue line). Second, they introduced factor 2 in front of all e15 coefficients, as required for the transition from Voigt notation to Cartesian notation. This generated the green line in the figure and full agreement between both models.

Maybe these corrections are already discussed in the literature. If you, Dear Reader, are aware of such a paper, please let me know.

I think, the above two improvements of the Romanov’s model are identified correctly. I know that many people simulating semipolar LEDs have come to the same conclusion using the general approach suggested by Romanov and deriving the final equations by themselves. Personally, I used the compliance matrix to get more compact expressions for strain and electric polarization in semipolar LED structures but within the same Romanov’s approach. The results are indentical to those obtained the the stiffness matrix, accounting for e31 and the factor 2.

Funato, J. Appl. Phys. 107, 123501 (2010), compare Fig. (6), have shown that the approch used by Romanov is not correct at all. I don’t complety agree with that paper, because in my opinion the correct approach is to project the primitive translation vectors (and not the reciprocal ones) of the epilayer and the substrate on the interface, as Park, Phys. Rev. B 59, 4725 (1996) did. A discussion of the different approaches can be also found in Shen et al. Phys. Status Solidi C 7, No. 10, 2378–2381 (2010), who claimed that Romanov’s approach is the correct one, although they did not remark on the mistakes in the paper. So the question which is the correct approach is not answered yet. If someone knows more recent papers let me know.