Let the density and its respective radius being known, then a critical beam angle shall be found!

It was happened when I read a paper from V. Consonni et al published in Physical Review B (or you can find it in here) about the shadowing effect part, in the page 4. V. Consonni et al gave a mathematical description (modelling) for the growth rate of self-induced GaN nanowires, and extracted several important parameters (for instance effective diffusion length on the sidewalls and substrate surface, desorption rate, driving forces for the diffusion of gallium adatoms to the nanowire top) based on the finding of equations from the fitting with the experimental results, which are as a function of growth time, gallium rate and growth temperature.

So, I found this “magic” at page 4, on the sub-part of “Theoretical modelling of the NW axial growth rate for the self-induced approach”, namely “shadowing effects”. Beforehand, I agreed with them regarding the geometrical considerations in the molecular beam epitaxy chamber determining the final form of nanowires which are heavily influenced by the incident angle of the gallium effusion cell and nitrogen plasma source with respect to the substrate (read here). It is important also to realize that the incident angle of these sources will be having an impact in the shadowing of the grown nanowires after they reach certain height.

To my surprise, V. Consonni et al provided three examples, just straight answer, of how shadowing effects took a role in affecting structural morphology of the nanowire (density, radius, spacing, height) and the respective angle of gallium beam. In all cases, they assumed nanowire to have regular square area. The first case is nanowire having density, radius, spacing of 100 {\mu}m^{-2}, 30 nm and 51 nm, respectively. They found out that with an angle of 21 degree, the gallium impinged on a nanowire with height of 133 nm. The second example is nanowire with density and radius of 100 {\mu}m^{-2} and 30 nm, a critical gallium beam angle of 49 degree was found. What… The last example, nanowire with radius and gallium beam angle of 30 nm and 21 degree, the density was deduced to be 200 {\mu}m^{-2}. I just don’t understand that straight answer.. It is just BOOM! Just let the radius and density to be known, then gallium beam angle will be discovered by some calculations which I do not understand.

To elucidate this matter, I make simple self-explanation. I used plain figure, based on the first case, which I think the most reasonable example I want to approach due to the more data in it. The objective is simple, before shadowing plays role, I want to find the maximum gallium beam angle and density, while radius, spacing and height are known. The figure has same scale measurement as with nm:

Nanowires configuration based on the first example
Nanowires configuration based on the first example

Let’s consider that gallium beam angle comes to the nanowires in a single line (of course in reality, the nanowires are received bunch of gallium flux on them). The consideration of “just before” shadowing effect means that this effect does not occur, i.e. maximum height of nanowires are found to be around 133 nm. The figure below gives an illustrative idea of how the maximum gallium beam angle “just before” the shadowing effect.

Maximum gallium beam angle with respect to the nanowire before shadowing effect
Maximum gallium beam angle with respect to the nanowire before shadowing effect

If I made a larger angle, than shadowing effect would take place. This is the maximum angle before shadowing. We can calculate that angle by doing a little hack. We can consider those figure with this:

The alternative approach for angle calculation
The alternative approach for angle calculation

I doubt the angle will be 21 degree. With \tan \theta = \frac{111 nm}{133 nm}, the maximum gallium beam angle is 39.84 degree.

The density of the nanowire itself is lower with my calculation. Since 1 {\mu}m^{-2} is equal with 10^{-6} nm^{-2}. We have 1 nanowire every 111 nm, meaning that in 2000 x 500 nm^{-2}, we have only 81 nanowire for 1 {\mu}m^{-2}.

See! The protractor also shows around 39 degree
See! The protractor also shows around 39 degree

It might be that V. Consonni et al have lower maximum limit to ensure that shadowing effect really will not be occurred, while my illustration really push the maximum angle of gallium beam angle. Using my approximation, with gallium beam angle of 21 degree, the maximum height of nanowire can be around 192 nm.

Maximum height of nanowire with gallium incident angle of 21 degree
Maximum height of nanowire with gallium incident angle of 21 degree

Ok, I will leave the shadowing part here.

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