# Absorption and Emission Spectra of Quantum Dots

For one of my next labs, I will be measuring the absorbtion and emission spectra of a tolulene solution with quantum dots in it. The gist of the idea is that I will end up obtaining a graph similar to this one:

However, from this graph, the size of the quantum dots apparently can be estimated, and maybe the distribution of sizes of these quantum dots. Are there any textbooks/papers that provide a theoretical prediction for this graph? That way I can work backwards from the data and find more information about the dots in the solution.

Let me first point out that there exist many different kinds of quantum dots - depending on how they are fabricated and, by extension, what they serve to - notably, colloidal quantum dots are often used for their optical properties, whereas the QDs fabricated by split-gate techniques are studied in the context of quantum transport - see also threads What happens to an electron-hole in a quantum dot when you irradiate it with photons? and How come we can grow nanowires? Semiconductors.

However, from this graph, the size of the quantum dots apparently can be estimated, and maybe the distribution of sizes of these quantum dots.

The absorption spectrum of a semicondictor/insulator usually begins at energies of photons larger than the gap, $$\hbar\omega\geq E_g$$, with a few excitonic picks below this threshold and continuous absorption above it. That is, for above-the-gap frequencies, the photon energy is divided as $$\hbar\omega=E_g+\frac{\hbar^2k_e^2}{2m_e}+\frac{\hbar^2k_h^2}{2m_h}$$

Quantum dot can be seen as a confining potential acting on both the electron and the hole, which means that instead of continuum of states, the states of both electrons and holes are quantized. If we denote the energies of the electron and hole states in their respective confining potentials as by $$\epsilon_i^{e}$$ and $$\epsilon_i^h$$, then the absorption spectrum is discrete even above the band gap, and the energy conservation gives us $$\hbar\omega = E_g + \epsilon_i^{e} + \epsilon_j^h$$ this discreteness is one of the reasons why the quantum dots are often referred-to as artificial atoms. ($$E_g$$ in this equation corresponds to that of the material from which QD is made, and may differ from that of the host material.)

Since the magnitude of the energies $$\epsilon_i^{e}$$ and $$\epsilon_i^h$$ depends on the shape of the confinement potential, i.e., how big is the dot - one can in principle estimate the size of the dot from the position of the absorption and the emission peaks, but one needs also to know the material properties of the dot and host material.

Are there any textbooks/papers that provide a theoretical prediction for this graph?

Since QDs are a subject that has been studied for several decades by now, there exist multiple textbooks and reviews about them. Admittedly, the later literature is too biased towards quantum computation, so for simple discussion of optical properties one likely has to turn to the older books and reviews. Due to my personal heavily theoretical bias, I cannot recommend something directly answering the question, but there might be some reference in this well-known review on the optics of semiconductors. There are also a couple of references at the bottom of this article, which features and discusses the figure similar to the one given in the OP. Finally, by simply googling for the quantum dots optical properties I encountered this book: Quantum Dots - Optics, Electron Transport and Future Applications.