Fermi surface
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In
Theory
Consider a spin-less ideal Fermi gas of particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy is given by[7]
where
- is the mean occupation number of the th state
- is the kinetic energy of the th state
- is the chemical potential (at zero temperature, this is the maximum kinetic energy the particle can have, i.e. Fermi energy )
- is the absolute temperature
- is the Boltzmann constant
Suppose we consider the limit . Then we have,
By the Pauli exclusion principle, no two fermions can be in the same state. Additionally, at zero temperature the enthalpy of the electrons must be minimal, meaning that they cannot change state. If, for a particle in some state, there existed an unoccupied lower state that it could occupy, then the energy difference between those states would give the electron an additional enthalpy. Hence, the enthalpy of the electron would not be minimal. Therefore, at zero temperature all the lowest energy states must be saturated. For a large ensemble the Fermi level will be approximately equal to the chemical potential of the system, and hence every state below this energy must be occupied. Thus, particles fill up all energy levels below the Fermi level at absolute zero, which is equivalent to saying that is the energy level below which there are exactly states.
In
The linear response of a metal to an electric, magnetic, or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy. In reciprocal space, the Fermi surface of an ideal Fermi gas is a sphere of radius
- ,
determined by the valence electron concentration where is the
Materials with complex crystal structures can have quite intricate Fermi surfaces. Figure 2 illustrates the
The state occupancy of fermions like electrons is governed by Fermi–Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.
Experimental determination
Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields , for example the
.
Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface. Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a mean free path. Therefore, dHvA and SdH experiments are usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic Field Laboratory in France, the Tsukuba Magnet Laboratory in Japan or the National High Magnetic Field Laboratory in the United States.
The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see
With
See also
- Fermi energy
- Brillouin zone
- Fermi surface of superconducting cuprates
- Kelvin probe force microscope
- Luttinger's theorem
References
- ISSN 0031-8949.
- ISBN 0-03-083993-9.
- ISBN 0-486-66021-4.
- ^ VRML Fermi Surface Database
- OCLC 541173.
- S2CID 119246268.
- ISBN 978-0-07-051800-1.
- ^ K. Huang, Statistical Mechanics (2000), p. 244
External links
- Experimental Fermi surfaces of some superconducting cuprates and strontium ruthenates in "Angle-resolved photoemission spectroscopy of the cuprate superconductors (Review Article)" (2002)
- Experimental Fermi surfaces of some transition metal dichalcogenides, ruthenates, and iron-based superconductors in "ARPES experiment in fermiology of quasi-2D metals (Review Article)" (2014)
- Dugdale, S. B. (2016-01-01). "Life on the edge: a beginner's guide to the Fermi surface". Physica Scripta. 91 (5): 053009. ISSN 1402-4896.