Density of States - Introduction

Introduction

In quantum mechanical (QM) systems, not all waves, or wave-like particles, are allowed to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. The most familiar systems, like neutronium in neutron stars and free electron gases in metals (examples of degenerate matter and a Fermi gas), have a 3-dimensional Euclidean topology. Less familiar systems, like 2-dimensional electron gases (2DEG) in graphite layers and the Quantum Hall effect system in MOSFET type devices, have a 2-dimensional Euclidean topology. Even less familiar are Carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. The topological properties of the system have a major impact on the properties of the density of states. Systems with 1D and 2D topologies are likely to become more common, assuming developments in nanotechnology and materials science proceed. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels.

For example, the density of states for electrons in a semiconductor is shown in red in Fig. 4 (in section 5). For electrons at the conduction band edge, very few states are available for the electron to occupy. As the electron increases in energy, the electron density of states increases and more states become available for occupation. However, because there are no states available for electrons to occupy within the bandgap, electrons at the conduction band edge must lose at least of energy in order to transition to another available mode. The density of states can be calculated for electron, photon, or phonon in QM systems. The DOS is usually represented by one of the symbols g, ρ, D, n, or N, and can be given as a function of either energy or wave vector k. To convert between energy and wave vector, the specific relation between E and k must be known.