Density Of StatesEdit

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Density of states is a central quantity in quantum mechanics and solid-state physics. It counts how many quantum states are available to electrons per interval of energy in a material. In crystalline solids, the density of states (often abbreviated as DOS) is closely tied to the electronic band structure and plays a decisive role in determining electrical conductivity, optical response, and thermodynamic properties. DOS can be specified per unit energy and per unit volume, or it can be resolved according to spin or other quantum numbers. In many contexts, the notation g(E) or D(E) is used for the density of states as a function of energy E. See also band structure and Fermi energy for related concepts.

Definitions and fundamental ideas

The DOS g(E) is defined so that g(E) dE counts the number of single-particle states with energies between E and E + dE. It is related to the distribution of energy eigenvalues in the system, which in a perfect crystal is determined by the solutions of the single-particle Schrödinger equation in the periodic potential of the lattice.
In thermal equilibrium, the occupancy of states at energy E is governed by the Fermi-Dirac distribution f(E). The product g(E) f(E) describes how many states at energy E are filled on average, and integrating over all energies yields the total number of electrons.
In many practical problems, one also distinguishes the total density of states from the partial or projected DOS, which accounts for contributions from specific atomic orbitals or crystal momenta. The local density of states (LDOS) is particularly important for spatially resolved probes such as scanning tunneling microscopy, and is conceptually related to the spectral function in many-body theory. See local density of states for more.

The DOS is profoundly influenced by dimensionality and the nature of the electronic dispersion. In a free-electron model, electrons are treated as non-interacting particles moving in a uniform background, and the DOS can be obtained by counting states in momentum space. For crystalline solids, the periodic potential creates energy bands, and the DOS reflects the distribution of these bands in energy and momentum space.

Models and analytic results

Free electron gas (three-dimensional): For a spin-degenerate system, the 3D DOS per unit volume behaves as g(E) ∝ sqrt(E). A common explicit form is g(E) = (1/2π^2) (2m/ħ^2)^{3/2} sqrt(E), where m is the electron mass and ħ is the reduced Planck constant. This result underpins much of the early intuition about metals and is modified by real-band structure in actual materials. See free electron model and Fermi energy for context.
Two-dimensional electron gas: In two dimensions, the DOS is essentially constant with energy (for a parabolic band), aside from spin degeneracy. This has important consequences for the physics of quantum wells, heterostructures, and certain two-dimensional materials. See two-dimensional electron gas and band structure for related material.
One-dimensional systems: In 1D, the DOS typically diverges as 1/sqrt(E) near the bottom of a band, reflecting the van Hove singularities that arise at critical points in the dispersion relation. This leads to pronounced features in measurements of the electronic structure. See van Hove singularity for a detailed discussion.
Tight-binding and real materials: In many solids, especially transition metals and semiconductors, the DOS is computed from a detailed band structure obtained from first-principles methods or tight-binding models. Features such as peaks, plateaus, and gaps in the DOS encode the underlying crystal chemistry and bonding. See tight-binding and band theory for more.

In real materials, interactions among electrons (and with phonons, impurities, and other excitations) modify the simple non-interacting DOS. The concept of a non-interacting DOS remains useful as a starting point, and many-body techniques provide corrections, such as the introduction of a self-energy that reshapes the effective DOS. The spectral function A(k, E) and the local density of states LDOS are related tools in this broader framework, connecting to experimental probes like photoemission spectroscopy and tunneling measurements. See spectral function and photoemission spectroscopy for further reading.

Calculating and interpreting the DOS

Analytical calculations are possible in idealized models (free electron gas, tight-binding lattices, particle-in-a-box geometries). These give insight into how dimensionality and dispersion shape g(E) and how degeneracies or symmetry dictate features of the DOS. See Fermi-Dirac distribution and Bloch's theorem for foundational background.
Numerical methods are essential for real materials. Density functional theory (DFT) provides a practical framework for computing the electronic structure, from which the DOS can be extracted. Many-body methods go beyond DFT to address correlation effects. See density functional theory and many-body theory for context.
Experimental probes connect to the DOS in different ways. Photoemission spectroscopy measures the spectral function related to the occupied part of the DOS, while scanning tunneling spectroscopy with a sharp tip probes the LDOS at the sample surface. See photoemission spectroscopy and scanning tunneling microscopy for details.

Local density of states and spatially resolved information

The LDOS ρ(r, E) extends the concept of DOS to depend on position. It is especially important near defects, interfaces, or surfaces where the electronic structure can differ significantly from the bulk. Spatial variations in the LDOS influence surface reactivity, catalytic activity, and nanoscale electronic transport. The LDOS is closely connected to the tunneling current in scanning probe experiments, and it provides a window into how electronic states are distributed in real space. See local density of states and scanning tunneling microscopy for discussion.

Controversies and debates

Interacting systems and the meaning of DOS: In strongly correlated materials, the single-particle DOS is not the whole story, because many-body states and collective excitations can dominate transport and optical responses. While the non-interacting DOS remains a useful reference, interpreting measurements often requires going beyond a simple picture and using Green’s functions, self-energies, or dynamical mean-field theory. See many-body theory and Green's function for more.
Non-equilibrium and time-dependent scenarios: In time-dependent or driven systems, the conventional energy-resolved DOS can be less straightforward to define, since energy is not strictly conserved in the same sense. Researchers handle this with generalized spectral functions and transient formulations, which can lead to debates about the best definitions and practical usefulness in different regimes. See non-equilibrium Green's function for an advanced treatment.
Surface and interface effects: The DOS at surfaces or heterostructures can differ substantially from the bulk DOS due to broken symmetry and altered bonding environments. Reconciling surface-sensitive measurements with bulk calculations can be nuanced, especially in complex materials. See surface states and band structure for related topics.
Definitions and measurement conventions: Different communities sometimes adopt subtly different conventions for density-of-states-related quantities (per unit energy, per unit energy per unit volume, etc.). Clear specification of units, degeneracies, and normalization is essential to avoid misinterpretation when comparing theory and experiment. See normalization (physics) for general discussion.