## Orbital, Fermi-Dirac distribution, DOS >[!Note] > It's just a quick summary, more about this content will be discussed in other pages. See the reverse links for more information. ### Orbital We call orbital the solution of the wave function for a system of a single electron. (i.e., neglected electron-electron interaction in [[Secular equation|LCAO]]) ### Fermi-Dirac distribution $f(\epsilon)=\frac{1}{e^{\frac{\epsilon-\mu}{kT}}+1}$ >[!Note] >$\mu$ is the Fermi energy. >And distributions have similar form. Fermi-Dirac: +1; Maxwell-Boltzmann: 0; Bose-Einstein: -1. ### DOS 3D system (bulk), $DOS \propto \sqrt{E}$. 2D system (2DEG), $DOS \propto C$. 1D system (quantum wire), $DOS \propto \frac{1}{\sqrt{E}}$. 0D system (quantum dot), $DOS \propto 2\delta E$. ![[Drawing 2023-09-28 20.40.11.excalidraw.svg]] >[!Info] >For 3D DOS, the derivation could be done by considering the original definition and apply k.p theory. This is done in [[Band structure near band extrema, k.p theory]].