The whole idea of this post stemmed from a discussion I had with my dad after he listened to a radio interview in which I called graphene a "2D material". He argued that anything with a thickness, even if it is only one atom thick like graphene, is by definition a 3D material. In which case, we'd have to accept that any physical thing that exists on this planet is three dimensional.

As a physical chemist, I am accustomed to referring to different semiconductor nanostructures as one, two or even zero dimensional. And as a stubborn human being, I now feel compelled to explain my reasoning in excruciating depth to at least this small corner of the internet. As a side bonus, quantum dots — which you will soon be convinced are zero dimensional — are beautiful and I love any excuse to show photos like the one above.

We will start with a discussion of the exciton Bohr radius. Now I've already loosely defined "exciton" in my previous post about solar cells (hoorah for well-ordered posts that allow me to link back to my own writing), but I'll expand on that slightly using Figure 1.

On the left side is a simple example of the band structure of a semiconductor. In blue, at the lower end of the energy scale, is the valence band, and it is full of electrons. Above it is the band gap, notable for not only being empty of electrons, but also empty of any energy levels that could possibly contain an electron. On top, in grey, is the conduction band. It too is empty of electrons, but possesses states that could support a sufficiently energetic electron.

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When there is some sort of energetic input (like sunlight) that is of high enough energy to overcome the band gap, an electron gets all excited and jumps from the low energy valence band to the high energy conduction band. What it leaves behind is an empty space, an absence-of-electron, which is not really anything, but which behaves much like a positive charge in a sea of electrons. So we give it a name, a "hole", and treat it like a particle. Now that we've clarified the exact nature of an exciton, defining the exciton Bohr radius is really easy: it's just the average distance between the electron and the hole of an exciton in a particular material.

How does this all relate to dimensions? Well, I'm getting there. First, let's go back to the quantum dots. Quantum dots are really cool. They come in a rainbow of colours, they have an awesome-sounding name, and they do all sorts of interesting things like help treat cancers and improve solar cells. Where do all these interesting properties come from? They stem from what are called Quantum Confinement Effects and ... get ready for the punch line ... these arise when the diameter of the quantum dot is smaller than the exciton Bohr radius in that material. Aha! It's all coming together now.

Figure 2 shows us what happens when things are quantum-ly confined. Now the electrons no longer exist in continous energy bands. Instead, they are confined to discrete ("quantized") levels. The spacing of these levels varies with the size of the quantum dot. That means that the size of the band gap, and thus the necessary energy required to surmount it, also varies with dot diameter. That's why quantum dots can be made in a rainbow of colours — change the diameter of the dot and you change the wavelength of light it absorbs!

Ok, I'll admit, the quantum dot discussion was a bit of a tangent. I am easily distracted by glowing, colourful things. But to conclude this long-winded definition: the dimensionality of a semiconductor materials is the number of dimensions in which the excitons are not confined. I'm sure you will all sleep a little easier tonight with that pressing mystery cleared up. You're welcome.

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Again, this is written from my head and I don't claim to be infallible so I'll offer some suggestions for further reading. In this case, much of the subject matter is pretty general so a good undergraduate chemistry textbook or even wikipedia (quantum dots, excitons) will be decent resources. If you've got $200 to blow and want to know way more about this stuff than I do, there's also this book: Excitons in Low-Dimensional Semiconductors.

Addendum: I had originally written this post to be a generic discussion of dimensionality before realizing that my definition may not apply to metals and insulators (and graphene is a semi-metal! More confusion!). This is actually (shamefully) the very first time this has occurred to me, and now I'm desperately curious to know how dimensions are defined in those materials. Do you know? Tell me in the comments, I mean the "group chat" [heaps scorn on new Kinja]. If not, I'll do some research and maybe write a follow-up post.

Top image, wikipedia.