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As far as I understand, both quantum dots and nanoparticles are mainly characterised by the fact that all three dimensions are in the nanoscale. Quantum dots are always mentioned to be made from a semiconductor, while nanoparticles can be anything (dielectric, metal, semiconductor).
Is semiconducting nanoparticle always called a quantum dot or is there a stricter definition of both?
There is no strict definition, although as you suspect, quantum dots are a subset of the more generic “nanoparticles”. The reason “quantum dot” is normally referring to a semiconducting system is that it evokes a certain group of physical properties used in a certain way: the band states being quantized, for example, which is useful for engineering the density of states for light emission/absorption. E.g if you want to make a laser, you might start with a semiconducting system with an appropriate band gap and make quantum dots out of it to enhance the internal quantum efficiency. A metallic nanoparticle that’s small enough to result in quantum confinement effects on the conducting electrons could, I guess, be referred to as a quantum dot. But people might be momentarily confused.
