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Density of States Derivation

Density of States Derivation The Density of States gives the number of allowed electron (or hole) States per volume at a given energy. It can be derived from basic quantum mechanics. Electron Wavefunction The position of an electron is described by a wavefunction zyx,, . The probability of finding the electron at a specific point (x,y,z) is given by 2,,zyx , where total probability dxdydzzyxspaceall 2,, is normalized to one. Particle in a Box The electrons at the bottom of a conduction band (and holes at the top of the valence band) behave approximately like free particles (with an effective mass) trapped in a box. We will consider here conduction band electrons, but the result for holes is similar. For our parabolic conduction band: 22*2ckEEm For electrons in a rectangular volume Lx by Ly by Lz with an infinite confining potential ((U(x,y,z)=0 inside the box and outside), the electron wavefunction must go to zero on the boundaries, and takes the form of a harmonic function within the region.)

The resulting density of states for a quantum well is a staircase, as below in red. Further restriction of the semiconductor dimensionality to 1-D (quantum wire) and 0-D (quantum dot) results in more and more confined density of states functions. Density of states for 0-D through 3-D regions.

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