The number of available electronic states per energy interval in a material.The density of states (DOS) is a fundamental quantity in solid-state physics that counts how many electronic states are available at each energy level. It is a function that peaks where many states exist (e.g., in flat bands) and vanishes in band gaps. The DOS determines how many electrons can occupy a given energy range and directly influences a material's electrical, thermal, and optical behavior.
In a crystal, electrons occupy discrete energy bands. The DOS is computed by counting, for each energy E, the number of quantum states with energy between E and E+dE. It depends on the dispersion relation (how energy varies with momentum). For example, in a 2D electron gas, the DOS is constant per unit area, while in 1D it has sharp peaks called van Hove singularities.
The DOS is essential for interpreting scanning tunneling spectroscopy, designing thermoelectric materials, and predicting the performance of transistors and quantum well lasers.
In graphene, the density of states vanishes linearly at the Dirac point, making it a zero-gap semiconductor with unusual electronic behavior.
| \Gamma_{i\to f} | variable | Transition rate from initial state i to final state f — transition rate per second |
| \psi_i | variable | Initial quantum state — wave function |
| \psi_f | variable | Final quantum state — wave function |
| H' | variable | Perturbation Hamiltonian — perturbation hamiltonian joule |
| \rho(E_f) | variable | Density of final states at energy E_f — density of states per joule |
| \hbar | constant | Reduced Planck constant, relating energy to angular frequency — reduced planck constant = 1.054571817e-34 joule second |
| \langle \psi_f | H' | operator | Matrix element of the perturbation between initial and final states — matrix element |
| J | variable | joint density of states — density of states per joule per cubic metre |
| E | variable | photon energy — energy joule |
| E_c | variable | conduction band energy — energy joule |
| E_v | variable | valence band energy — energy joule |
| \mathbf{k} | variable | crystal momentum — wave vector per metre |
| \int_{\text{BZ}} d^3 | operator | integration over the first Brillouin zone — integral over brillouin zone |
| \delta | operator | selects states satisfying energy conservation — dirac delta function |
| g(ω) | variable | vibrational density of states — density of states second per radian |
| ω | variable | angular frequency — angular frequency radian per second |
| T | variable | temperature — temperature kelvin |
| ħ | constant | Reduced Planck constant, quantum of angular momentum — reduced planck constant = 1.054571817e-34 joule second |
| k | constant | Relates energy to temperature in statistical mechanics — boltzmann constant = 1.380649e-23 joule per kelvin |
