Electron & holes: realstic band structure in semiconductor
This section draws the realstic band structure of the semicondfuctor & explains it in details.
Realistic band structures in semiconductors:
- Figure (a) shows the band structures for Si and GaAs viewed along two major directions.
- While the shape is parabolic near the band edges, there are significant non-parabolicities at higher energies.
- The energies are plotted along the high symmetry  and  directions in the crystal.
- The k = 0 point is denoted as .When we go along the  direction, we reach a valley near X, while we reach the L valley along the  direction
- Since the energies are plotted along different directions, the curves do not look symmetric.
- The valence band maximum in most semiconductors is at the point.
- It has three branches:
- The heavy hole band with the smallest curvature,
- A light hole band with a larger curvature, and
- A split-off band at a different energy.
- We notice that for GaAs the conduction band minimum and the valence band maximum are both at k = 0, therefore it is direct band gap.
- Silicon, on the other hand, has 6 equivalent conduction minima at X along the 6 equivalent (100) directions, therefore, it is indirect.
- Figure (b) shows the constant energy surface for electrons in one of the six conduction bands for Si.
- The way to relate these surfaces to the band structures shown in Fig.(a) is to consider a certain value of energy, and determine all the k vectors in 3 dimensions for which we get this energy.
- We find that for Si we have 6 cigar-shaped ellipsoidal equi-energy surfaces near the conduction band minima along the six equivalent X-directions with a longitudinal effective mass, ml along the major axis, and two transverse effective masses, mi along the minor axes.
- For GaAs, the conduction band is more or less spherical for low energies.
- On the other hand, we have warped spherical surfaces in the valence band.