Energy Levels of a Semiconductor Quantum Dot

As many quantum effects are more pronounced in semiconductors compared to metals, attention will now be focused on the case of a semiconducting material. The changes that occur in the properties of a free electron gas change when the dimensions of the solid are reduced were described in Section 2.4. Although the model of the free electron gas does not include the nature of the solid, from a macroscopic point of view it is necessary to distinguish between metals, semiconductors, and insulators [15], Whilst the model of a free electron gas describes relatively well the case of electrons in the conduction band of metals, the electrons in an insulating material are only poorly described by the free electron model. In 

If the box is a sphere of diameter A, then the Schrodinger equation can be solved by introducing spherical coordinates and by separating the equation in a radial part, and also in a part that contains the angular momentum [52, 53]. The lowest energy level (with angular momentum = 0) is then  

This term may be quite significant, because the average distance between an electron and a hole in a QD dot can be small [55-59]. Thus, it is possible to estimate the size-dependent energy gap of a spherical semiconductor QD, as [54—59]  

The size-dependent energy gap may serve as a useful tool when designing materials with well-controlled optical properties, and a much more detailed analysis of this topic is available [62]. Before describing the physical consequences of the size-dependent band gap on the optical and electronic properties, however, a brief overview will be provided of how QDs may be fabricated in practice. 

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