Quantum Dots, Self-Assembled: Calculation of Electronic Structures and Optical Properties


Andrew Williamson Lawrence Livermore National Laboratory

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As evidenced by many of the articles in this encyclopedia, the past decade has witnessed a series of significant advances in the growth of semiconductor nanostructures. These nanostructures range in size from a few atoms up to several million atoms and have been produced by myriad synthesis techniques. In this article, we describe calculations of the electronic and optical properties of self-assembled (so-called Stranski–Krastanow) quantum dots produced by molecular beam epitaxy. Even more dramatic than the advances in the growth of these structures has been the development of extremely sophisticated techniques for measuring the optical, electronic, magnetic, and transport properties of these quantum dot systems. The availability of such high-quality measurements for a wide range of properties presents theorists with the formidable challenge of constructing models that can explain the origins of these experimental observations in terms of the underlying energy states of the quantum dots.

Calculating such energy states in self-assembled semiconductor quantum dots is rendered particularly challenging by a number of factors:

  1. The quantum dots contain a large number of atoms. A typical self-assembled quantum dot has a base of ∼ 300 Å and a height of ∼ 50 Å. Therefore the dot itself may contain ∼ 105 atoms. This dot then needs to be surrounded by a barrier material to isolate it from other dots. Therefore a representative system containing both the dot and barrier typically contains ∼ 106 atoms.

  2. By the nature of the growth process, self-assembled quantum dots are highly strained. For example, in the most common InAs/GaAs material combination, the lattice mismatch is 7%. Therefore an accurate solution of the strain profile in the system is required before the electronic structure can be calculated.

  3. The valence band maximum (VBM) in III–V semiconductor materials is threefold degenerate (in the absence of strain and spin–orbit splitting), therefore any realistic approach must describe at least the band mixing between the three valence band edge states.

  4. In InAs, the bulk band gap is 0.42 eV and the spin–orbit splitting is 0.38 eV, therefore the mixing between valence and conduction band states and split off states also have to be taken into account.

  5. In a zero-dimensional InAs/GaAs quantum dot system, the charge carriers are artificially confined inside the dot, which is typically smaller than the bulk excitonic radius—the “Strong Confinement” regime. This dramatically enhances the Coulomb interaction between charges in the dot and strongly modifies the dielectric screening.