More Fermì Surface Shapes* Consider a divalent atom, such as...

Name: Problem 3, Chapter 16: The Oxford Solid State Basics
Uploaded: 2022-01-21T18:08:48-08:00
Duration: 10 min 43 s
Channel: Hafiz Shahzaib
Description: More Fermì Surface Shapes* Consider a divalent atom, such as Ca or Sr, that forms an fce lattice (with a single atom basis). In the absence of a periodic potential, would the Fermi surface touch the Brillouin zone boundary? What fraction of the states in the first Brillouin zone remain empty?

More Fermì Surface Shapes* Consider a divalent atom, such as Ca or Sr, that forms an fce lattice (with a single atom basis). In the absence of a periodic potential, would the Fermi surface touch the Brillouin zone boundary? What fraction of the states in the first Brillouin zone remain empty?

More Fermì Surface Shapes*
Consider a divalent atom, such as Ca or Sr, that forms an fce lattice (with a single atom basis). In the absence of a periodic potential, would the Fermi surface touch the Brillouin zone boundary? What fraction of the states in the first Brillouin zone remain empty?

Key Concepts

Electron Filling and Band Occupation

In crystals the number of electrons per unit cell determines how the available electronic states (which, when counted with spin degeneracy, are located within the Brillouin zone) are occupied. For instance, in a divalent metal with one atom per cell, the free electron sphere that encloses exactly the right number of k–states may extend beyond the geometric boundaries of the Brillouin zone. Upon folding these states back into the zone, some regions near the zone boundary remain unoccupied, thus defining both the shape of the Fermi surface in the reduced zone scheme and the fraction of empty states.

Free Electron Model

The free electron model treats conduction electrons as noninteracting particles moving in a constant (or zero) periodic potential. This simplification leads to a spherical Fermi surface that is independent of the detailed ionic potential, highlighting the role of electron density in setting the Fermi energy.

Zone Folding

Zone folding refers to the mapping of electronic states from the extended (infinite) k–space back into the first Brillouin zone due to the periodicity of the lattice. When the free electron Fermi sphere is not commensurate with the shape of the Brillouin zone, parts of the spherical surface are folded back, producing a nonuniform filling inside the zone.

Fermi Surface

The Fermi surface is the boundary in momentum space separating occupied from unoccupied electron states at absolute zero. In the free–electron picture it is a sphere determined solely by the electron density, and its geometric properties (size and shape) underpin many electronic properties of metals.

Brillouin Zone

The Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice and represents the set of unique wavevectors due to lattice periodicity. It is central to understanding electronic band structure because it defines the region within which all distinct electron states are contained.

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