Impurities play a pivotal role in semiconductors. One part in a million of phosphorous in silicon alters the conductivity of the latter by several orders of magnitude. Indeed, the information age is possible only because of the unique role of shallow impurities in semiconductors. Although work in semiconductor nanostructures (SN) has been in progress for the past two decades, the role of impurities in them has been only sketchily studied. We outline theoretical approaches to the electronic structure of shallow impurities in SN and discuss their limitations. We find that shallow levels undergo a SHADES (SHAllow-DEep-Shallow) transition as the SN size is decreased. This occurs because of the combined effect of quantum confinement and reduced dielectric constant in SN. Level splitting is pronounced and this can perhaps be probed by ESR and ENDOR techniques. Finally, we suggest that a perusal of literature on (semiconductor) cluster calculations carried out 30 years ago would be useful.

The Coulomb blockade (CB) in quantum dots (QDs) is by now well documented. It has been used to guide the fabrication of single electron transistors. Even the most sophisticated techniques for synthesizing QDs (e.g. MOCVD/MBE) result in an assembly in which a certain amount of disorder is inevitable. On the other hand, theoretical approaches to CB limit themselves to an analysis of a single QD. In the present work we consider two types of disorders: (i) size disorder; e.g. QDs have a distribution of sizes which could be unimodal or bimodal in nature. (ii) Potential disorder with the confining potential assuming a variety of shapes depending on growth condition and external fields. We assume a Gaussian distribution in disorder in both size and potential and employ a simplified mean field theory. To do this we rely on the scaling laws for the CB (also termed as Hubbard 𝑈) obtained for an isolated QD [1]. We analyze the distribution in the Hubbard 𝑈 as a consequence of disorder and observe that Coulomb blockade is partially suppressed by the disorder. Further, the distribution in 𝑈 is a skewed Gaussian with enhanced broadening.

The effective mass of electrons in low-dimensional semiconductors is position-dependent. The standard kinetic energy operator of quantum mechanics for this position-dependent mass is non-Hermitian and needs to be modified. This is achieved by imposing the BenDaniel–Duke (BDD) boundary condition. We have investigated the role of this boundary condition for semiconductor quantum dots (QDs) in one, two and three dimensions. In these systems the effective mass m i inside the dot of size R is different from the mass m o outside. Hence a crucial factor in determining the electronic spectrum is the mass discontinuity factor $\beta = m_{i} /m_{o}$ . We have proposed a novel quantum scale, 𝜎, which is a dimensionless parameter proportional to $\beta{2}R^{2}V_{0}$ , where $V_{0}$ represents the barrier height. We show both by numerical calculations and asymptotic analysis that the ground state energy and the surface charge density, $(\rho(R))$, can be large and dependent on 𝜎. We also show that the dependence of the ground state energy on the size of the dot is infraquadratic. We also study the system in the presence of magnetic field 𝐵. The BDD condition introduces a magnetic length-dependent term $(\sqrt{\hbar /eB})$ into 𝜎 and hence the ground state energy. We demonstrate that the significance of BDD condition is pronounced at large 𝑅 and large magnetic fields. In many cases the results using the BDD condition is significantly different from the non-Hermitian treatment of the problem.