Contents - Glossary - Study Aids - 1 2 3 4 5 6 7 8 9

In this section:

- Introduction
- Energy-wavenumber
(
*E-k*) diagram of silicon - Detailed parameters for Ge, Si and GaAs
- Density of states mass
- Conductivity mass
- Short list of parameters for Ge, Si and GaAs

Most semiconductors can be described as having one band minimum at *k* =
0 as well as several equivalent anisotropic band minima at *k* ¹ 0. In addition there are three band maxima of interest
which are close to the valence band edge.

Two of the three band maxima occur at 0 eV. These bands are refered to as the
light and heavy hole bands with a light hole mass of
*m _{lh}*

Name | Symbol | Germanium | Silicon | Gallium Arsenide |
---|---|---|---|---|

Band minimum at k = 0 |
||||

Minimum energy | E_{g,direct} [eV] |
0.8 | 3.2 | 1.424 |

Effective mass | m_{e}^{*}/m_{0} |
0.041 | ?0.2? | 0.067 |

Band minimum not at k = 0 |
||||

Minimum energy | E_{g,indirect} [eV] |
0.66 | 1.12 | 1.734 |

Longitudinal effective mass | m_{e,l}^{*}/m_{0} |
1.64 | 0.98 | 1.98 |

Transverse effective mass | m_{e,t}^{*}/m_{0} |
0.082 | 0.19 | 0.37 |

Wavenumber at minimum | k [1/nm] |
xxx | xxx | xxx |

Longitudinal direction | (111) | (100) | (111) | |

Heavy hole valence band maximum at E = k = 0 |
||||

Effective mass | m_{hh}^{*}/m_{0} |
0.28 | 0.49 | 0.45 |

Light hole valence band maximum at k = 0 |
||||

Effective mass | m_{lh}^{*}/m_{0} |
0.044 | 0.16 | 0.082 |

Split-off hole valence band maximum at k = 0 |
||||

Split-off band valence band energy | E_{v,so} [eV] |
-0.028 | -0.044 | -0.34 |

Effective mass | m_{h,so}^{*}/m_{0} |
0.084 | 0.29 | 0.154 |

- (f24a)

- (24b)

for instance for a single band minimum described by a longitudinal mass and two transverse masses the effective mass for density of states calculations is the geometric mean of the three masses. Including the fact that there are several equivalent minima at the same energy one obtains the effective mass for density of states calculations from:

- (f65)

where *M*_{c} is the number of equivalent band minima. For
silicon one obtains:

As the conductivity of a material is inversionally proportional to the effective masses, one finds that the conductivity due to multiple band maxima or minima is proportional to the sum of the inverse of the individual masses, multiplied by the density of carriers in each band, as each maximum or minimum adds to the overall conductivity. For anisotropic minima containing one longitudinal and two transverse effective masses one has to sum over the effective masses in the different minima along the equivalent directions. The resulting effective mass for bands which have ellipsoidal constant energy surfaces is given by:

- (f66)

provided the material has an isotropic conductivity as is the case for cubic
materials. For instance electrons in the *X* minima of silicon have an
effective conductivity mass given by:

Name | Symbol | Germanium | Silicon | Gallium Arsenide |
---|---|---|---|---|

Smallest energy bandgap at 300 K | E_{g} (eV) |
0.66 | 1.12 | 1.424 |

Effective mass for density of states calculations |
||||

Electrons | m_{e}^{*}_{,dos}/m_{0} |
0.56 | 1.08 | 0.067 |

Holes | m_{h}^{*}_{,dos}/m_{0} |
0.29 | 0.57/0.81^{1} |
0.47 |

Effective mass for conductivity calculations |
||||

Electrons | m_{e}^{*}_{,cond}/m_{0} |
0.12 | 0.26 | 0.067 |

Holes | m_{h}^{*}_{,cond}/m_{0} |
0.21 | 0.36/0.386^{1} |
0.34 |

© Bart J. Van Zeghbroeck, 1997