Introduction to Quantum Dots

In 1969, Esaki and Tsu proposed to use a semiconductor superlattice based on a periodic structure of alternating layers of semiconductor materials with wide and narrow band gaps [1].

From then on, the research in quantum nanostructures is thriving.

The wave function of electrons is going to change when they are confined to dimensions comparable with their wavelength. Low threshold considerations dictate the transition from bulk to quantum well to quantum wire to quantum dot systems [2]. Figure 1.1 shows that confining electrons to small structures causes the continuous bulk bands to split up into discrete levels [3]. A quantum well is a potential well that confines electrons, which were originally free to move in three dimensions, to two dimensions, forcing them to occupy a planar region. The effects of quantum confinement take place when the quantum well thickness becomes comparable at the de Broglie wavelength of the carriers, leading to energy levels called energy subbands. For the quantum wire, the electrons in two dimensions are quantized and in one dimension electrons are free to move. And finally, confining electrons to three dimensions forms a quantum dot (QD).

Being quasi-zero-dimensional nanostructures, QDs have a discrete density of states and wave function properties. The confinement can be due to electrostatic potentials, the presence of an interface between different semiconductor materials, the presence of the semiconductor surface, or a combination of above. In contrast to atoms, the energy spectrum of a QD can be engineered by controlling the geometrical size, shape, and the strength of the confinement potential. Such dots are sometimes called "artificial atoms". In a crystal, the periodic atomic potential leads to band formation. But in a superlattice, an artificial, human-made periodical potential causes the formation of small bands. Thus, the utilization of QDs is one of the most promising technologies for applications in optoelectronic devices and lasers.

Figure 1.1 Semiconductor structures and corresponding electronic density of states near the edge of electronic bands. (a) Bulk, (b) quantum well, (c) quantum wire, and (d) quantum dot [3].

1.1.1. The Formation of Self-assembled Quantum Dots

In the equilibrium theory of heteroepitaxial growth, three growth modes are traditionally distinguished [4]:

(1) Frank-van der Merwe (FM mode):

If the sum of the epilayer surface energy and the interface energy is lower than the energy of the substrate surface, the FM mode occurs. It may be described as two-dimensional layer-by-layer growth.

(2) Volmer-Weber (VW mode):

If the sum of the epilayer surface energy and the interface energy is higher than the energy of the substrate surface, the VW mode occurs. It may be described as three-dimensional island growth.

(3) Stranski-Krastanow (SK mode):

For a strained epilayer with small interface energy, the deposited layer initially grows

strain energy in the deposited layer increases. This layer can lower its energy by forming three-dimensional isolated islands in order to release the strain. The SK growth mode leads to a two-dimensional wetting layer with three-dimensional islands on top.

These growth modes are deduced from equilibrium considerations of the energy balance between the surface energy and the interface energy for lattice-matched systems. However, for heteroepitaxial growth of highly strained structures, the elastic strain energy associated with epitaxial lattice mismatch must be considered.

The SK growth mode is favorable for a layer with a lattice constant that differs considerably from that of the substrate, because the islands allow a relaxation of the strain energy. It also prefers to less differences between the epilayer surface energy and the substrate surface energy [5]. The SK growth mode may occur in systems where the formation of a two-dimensional wetting layer is favorable during the deposition of the first few monolayers (ML) of the film [6]. With the increasing layer thickness, the strain energy of two-dimensional wetting layer increases. Above the critical thickness, the onset of three-dimensional island formation on the substrate occurs, mainly because an island offers the possibility of relaxation of elastic strain at its free surface [7]. As a result of formation of such structures in nanometer size, what is called QDs, carriers in inlands are confined in three dimensions, as shown in Figures 1.2 and 1.3.

There has been considerable interest in using SK growth to deposit coherent islands, known as self-assembled QDs. Take InAs/GaAs QDs as an example. The lattice constant of InAs (a_InAs= 6.06 Å) is larger than GaAs (a_GaAs= 5.65 Å) and so has a compressive strain in the lateral direction after island formation. If the size of island exceeds a critical thickness that mainly depends on the misfit between the wetting layer and substrate, it is required for the generation of the misfit dislocations. However, if coherency is maintained within the critical condition, defect free and highly uniform QDs can result from the SK mode.

Figure 1.2 Schematic diagrams of three growth modes for heteroepitaxial systems and evolution of growth. (a) FM mode, (b) VW mode, and (c) SK mode [3].

Figure 1.3 Schematic diagrams of strains in the three growth modes for heteroepitaxial systems that can be distinguished. (a) FM mode, (b) VW mode, and (c) SK mode [7].

Figure 1.4 Room temperature bandgap energy versus lattice constant of common elemental and binary compound semiconductors [8].

1.1.2. The Characteristics and Applications of Quantum Dots

QD is a nanostructure that confines the motion of conduction band electrons, valence band holes, or excitons in all three spatial directions. Hence, a QD has a discrete quantized energy spectrum. The corresponding wave functions are spatially localized within the QD, but extend over many periods of the crystal lattice.

Using the structure of QD as an active region in a semiconductor laser is expected to exhibit superior lasing properties such as higher efficiency of electron confinement, lower threshold current density, lower transparency current density, higher differential gain, higher characteristic temperature, higher temperature stability, and wide modulation bandwdith, and so forth. Therefore, semiconductor of QDs has important potential applications in optoelectronic devices.

QDs are particularly significant for optoelectronic applications due to their theoretically high quantum yield. As a result, they have superior transport and optical properties, and are being studied for use in single electron transistor, quantum dot lasers, amplifiers, and biological sensors. According to recently researches, the emission spectra for QD lasing in these applications are 1.3~1.6 μm.