Valley degeneration

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Valley degeneracy in solid state physics is the appearance of degenerate states that belong to wave vectors in different crystal directions. The associated degeneracy factor is often referred to as. If the surfaces of constant energy (similar to the Fermi surface ) near the energy minimum in a diagram illustrating the surroundings of the energy minima as sinks (show Valleys ). The degeneracy factor is the number of complete environments in the first Brillouin zone . ${\displaystyle g_{v}}$ ${\displaystyle g_{v}}$

The degeneracy can be described in the quantum mechanical analysis as a pseudo-spin and taken into account in the Dirac equation by the Pauli matrices .

Examples

Gallium arsenide

The energetic minimum of the conduction band structure of gallium arsenide lies at the wave vector and the surfaces of constant energy around the minimum form spherical surfaces. There is only a minimum in the first Brillouin zone, hence the degeneracy factor . ${\displaystyle {\vec {k}}={\vec {0}}}$ ${\displaystyle g_{v}=1}$

silicon

In silicon , the energy minimum is in the direction. Due to the crystal symmetry, the six directions in the band structure are equivalent to the Miller indices . Thus there are six minima in the first Brillouin zone and a degeneracy factor of . ${\displaystyle (100)}$ ${\displaystyle \{(100),({\bar {1}}00),(010),(0{\bar {1}}0),(001),(00{\bar {1}})\}}$ ${\displaystyle g_{v}=6}$

Germanium

For germanium , the energy minima in the direction and their equivalents with the Miller indices occur a total of eight times. Since these minima are at the edge of the first Brillouin zone, they are only counted proportionally. By shifting around reciprocal lattice vectors, the surroundings of the degenerate states can be generated on the surroundings of four different states in the first Brillouin zone. Hence the degeneracy factor . ${\displaystyle (111)}$ ${\displaystyle \{({\bar {1}}11),(1{\bar {1}}1),(11{\bar {1}}),({\bar {1}}{\bar {1}}1),({\bar {1}}1{\bar {1}}),(1{\bar {1}}{\bar {1}})({\bar {1}}{\bar {1}}{\bar {1}})\}}$ ${\displaystyle g_{v}=4}$

Graph

Graph has a so-called two-valley structure , that is . ${\displaystyle g_{v}=2}$

Individual evidence

1. ^ SM Sze: Physics of Semiconductor Devices . Second Edition Wiley-Interscience, p. 14th