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  • Print publication year: 2016
  • Online publication date: February 2019

16 - Reduced-dimensional systems and nanostructures

from Part IV - Many-body effects, superconductivity, magnetism, and lower-dimensional systems


The electronic, transport, optical, and other properties of a material often undergo drastic changes, and new phenomena emerge, when the system is reduced in size such that quantum confinement of electrons (i.e. the wavelength of the electron is comparable to the confining structure) occurs in one or more dimensions. These reduced-dimensional systems and nanostructures have physical properties and phenomena that are of basic interest and are also potentially useful in applications. Because many of their properties are fundamentally derived from restricted geometry, the behaviors of these reduced dimensional systems and nanostructures are tunable by changing their size, and they are strongly influenced by considerations such as quantum confinement, enhanced many-electron interactions, reduced number of degrees of freedom, and symmetry effects. In this chapter, we shall discuss some basic elements of the electronic, transport, and optical properties of reduced-dimensional systems. We consider several systems, including semiconductor two-dimensional electron gas (2DEG) systems, quantum dots, graphene, carbon nanotubes, atomically thin quasi two-dimensional (2D) crystals, and molecular junctions, illustrating that small is different.

Density of states and optical properties

Electrons confined in a semiconductor quantum well or at a metal-oxide–1h are fragments of semiconductor crystals consisting of hundreds to many thousands of atoms with surface states eliminated by passivating adsorbates, or enclosure in a material that has a larger bandgap, is another class of quantum dots. Semiconductor wires of several nanometers in diameter can also be grown. Examples of other fascinating nanostructures that have been synthesized and behave like reduced-dimensional systems even at room temperature include graphene, carbon nanotubes, atomically thin quasi-2D materials consisting of mono- and few-layer Van der Waals crystals (such as hexagonal BN, transition metal dichalcogenides, etc.), and structures derived from these systems, such as nanoribbons.

The electron density of states in a reduced-dimensional system has distinct characteristics, i.e. van Hove singularities of different nature than for three dimensional systems, leading to many of its characteristic properties. Let us idealize the carriers in a semiconductor quantum well, e.g. a thin layer of GaAs of thickness L sandwiched between two AlAs crystals, as free electrons with an effective mass m *. The larger bandgap of AlAs effectively forms a potential confining the lower-energy carriers (both electrons and holes) to move within the GaAs layer, as illustrated in Fig. 16.1 for the states in the conduction band of GaAs.