Flake Growth — Background

Some scientific background

Gold flakes are single crystalline platelets featuring large lateral extensions and at the same time tiny thicknesses (see Fig. 1). The size and shape vary widely and a typical “big” flake can be as larger as 250 µm and 60 nm in thickness while a “smaller” one might be only 40 µm in lateral size (about the diameter of a human hair) and 12 nm thick. This means it consists of 36 layers of gold atoms only.

Fig. 1: Reflection image of some gold flakes on a glass substrate.

These gold flakes have interesting optical, electrical as well as catalytical properties and due to their single-crystallinity are ideally suited for fabricating precise nanostructure down to feature size of 5 nm.

They can be grown with various photo-chemical, electro-chemical or pure chemical recipes. Our favorite one just uses ethylene glycol (a common coolant in cars), gold salt (HAuCl4) and water.

Fig. 2: Main steps of the growing process.

Growing nanocrystals involves various steps (see Fig. 2) — a small overview can be found here — and depending on the crystal structure of the seed very different nanoparticles (spheres, nanorods, bars, …) can be the outcome. For flakes for example it is important to have a seed with at least two stacking faults.

Why this is the case and how many stacking faults are favorable? Why are some flakes very hexagonal while other are more triangular? And how to achieve very large (millimeter size) but still atomically thin flakes?

To answer these questions, I wrote the flake growing simulation.

The algorithm

The main idea is that it is energetically favorable when a new atom is added to a position where it can saturate the most surface bounds. By doing so the surface free energy is minimized, e.g., we are in the kinetic growth regime. In the following images you can see how this basically works:

We take a mini flake and highlight the surface vacancies in different colors depending on their coordination number (number of neighbors).

Then we can list the theoretically possible number of neighbors and count their occurrence in our example:

Furthermore, we must define how likely it is that a new atom randomly added to the flake occupies a certain coordination number. As explained above these probabilities should be higher for higher number of neighbors but their actual values are just educated guesses as too many experimental parameters are simply unknown. (Hopefully, with enough experiments and simulations one might find realistic values for certain conditions.)

Nevertheless, now we can calculate a probability product for each column and put everything into a weighted probability list as shown below:

Now we have to roll the dice to pick on element out of the 1159194 :

The number 37 512 belongs to the section of the list associated with a coordination number of five. This means we will grow an atom at one of the vacancies which have 5 gold atoms as neighbors.

By accident for coordination number 5 there are also 5 possible positions where the atom could land. We have to pick one by rolling the dice again…

…and, our winner is vacancy 3 in list 5 !

Finally, we must examine the coordination number for all positions around the landed atom (some should have changed) and update all the lists in order to play the game again…

That’s basically it. The true difficulties arise when adding stacking faults, managing over 1 billion atoms and trying to still obtain an adequate performance.