Physics:Quantum Fermi surfaces: Difference between revisions
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In | '''Fermi surfaces''' is a Book I topic in the Quantum Collection. In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at absolute zero temperature. Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of electronic energy bands. The existence of a Fermi surface follows directly from the Pauli exclusion principle, which allows only one electron per quantum state. The study of Fermi surfaces is called fermiology. In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at absolute zero temperature. Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of electronic energy bands. | ||
Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of | |||
The study of Fermi surfaces is called | |||
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== Theory == | == Theory == | ||
For an ideal | For an ideal Fermi gas, the occupation of quantum states is governed by the Fermi–Dirac distribution: | ||
<math display="block">\langle n_i\rangle = \frac{1}{e^{(\epsilon_i-\mu)/k_{\rm B}T}+1}</math> | <math display="block">\langle n_i\rangle = \frac{1}{e^{(\epsilon_i-\mu)/k_{\rm B}T}+1}</math> | ||
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\end{cases}</math> | \end{cases}</math> | ||
All states below the | All states below the Fermi energy are filled, while all above are empty. In momentum space, these occupied states form a sphere of radius {{math|''k''<sub>F</sub>}}, whose boundary is the Fermi surface. | ||
For a free electron gas: | For a free electron gas: | ||
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</div> | </div> | ||
In real materials, Fermi surfaces can be highly complex. For example, [[graphite]] exhibits both electron and hole pockets due to multiple bands crossing the Fermi level. In many metals, the Fermi surface extends beyond the first | In real materials, Fermi surfaces can be highly complex. For example, [[graphite]] exhibits both electron and hole pockets due to multiple bands crossing the Fermi level. In many metals, the Fermi surface extends beyond the first Brillouin zone and is folded back into it using the reduced-zone scheme. | ||
Materials in which the Fermi level lies inside a [[band gap]] (such as semiconductors and insulators) do not have a Fermi surface. | Materials in which the Fermi level lies inside a [[band gap]] (such as semiconductors and insulators) do not have a Fermi surface. | ||
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* stability of low-temperature phases | * stability of low-temperature phases | ||
Systems with a high density of states at the Fermi level often become unstable and develop new ground states such as | Systems with a high density of states at the Fermi level often become unstable and develop new ground states such as superconductivity, ferromagnetism, or spin density waves. | ||
At finite temperatures, the sharp boundary of the Fermi surface becomes slightly blurred due to thermal excitations. | At finite temperatures, the sharp boundary of the Fermi surface becomes slightly blurred due to thermal excitations. | ||
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Fermi surfaces can be measured experimentally using several techniques: | Fermi surfaces can be measured experimentally using several techniques: | ||
* | * de Haas–van Alphen effect | ||
* | * Shubnikov–de Haas effect | ||
* | * ARPES | ||
These methods rely on quantum oscillations or direct measurement of electron energies in momentum space. | These methods rely on quantum oscillations or direct measurement of electron energies in momentum space. | ||
A key result by | A key result by Lars Onsager relates oscillation periods in magnetic fields to the cross-sectional area of the Fermi surface: | ||
<math display="block">A_{\perp} = \frac{2 \pi e \Delta H}{\hbar c}</math> | <math display="block">A_{\perp} = \frac{2 \pi e \Delta H}{\hbar c}</math> | ||
Another method is | Another method is ACAR, which measures electron momentum distributions through positron annihilation. | ||
== See also == | == See also == | ||
Revision as of 08:17, 20 May 2026
Fermi surfaces is a Book I topic in the Quantum Collection. In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at absolute zero temperature. Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of electronic energy bands. The existence of a Fermi surface follows directly from the Pauli exclusion principle, which allows only one electron per quantum state. The study of Fermi surfaces is called fermiology. In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at absolute zero temperature. Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of electronic energy bands.
Theory
For an ideal Fermi gas, the occupation of quantum states is governed by the Fermi–Dirac distribution:
At zero temperature (T → 0), this simplifies to:
All states below the Fermi energy are filled, while all above are empty. In momentum space, these occupied states form a sphere of radius kF, whose boundary is the Fermi surface.
For a free electron gas:
The shape of the Fermi surface determines how electrons respond to electric, magnetic, and thermal fields. Therefore, many physical properties of metals—such as conductivity—are controlled by states near the Fermi surface.
In real materials, Fermi surfaces can be highly complex. For example, graphite exhibits both electron and hole pockets due to multiple bands crossing the Fermi level. In many metals, the Fermi surface extends beyond the first Brillouin zone and is folded back into it using the reduced-zone scheme.
Materials in which the Fermi level lies inside a band gap (such as semiconductors and insulators) do not have a Fermi surface.
Physical significance
The Fermi surface plays a central role in determining:
- electrical conductivity
- thermal conductivity
- magnetic properties
- stability of low-temperature phases
Systems with a high density of states at the Fermi level often become unstable and develop new ground states such as superconductivity, ferromagnetism, or spin density waves.
At finite temperatures, the sharp boundary of the Fermi surface becomes slightly blurred due to thermal excitations.
Experimental determination
Fermi surfaces can be measured experimentally using several techniques:
- de Haas–van Alphen effect
- Shubnikov–de Haas effect
- ARPES
These methods rely on quantum oscillations or direct measurement of electron energies in momentum space.
A key result by Lars Onsager relates oscillation periods in magnetic fields to the cross-sectional area of the Fermi surface:
Another method is ACAR, which measures electron momentum distributions through positron annihilation.
See also
Table of contents (198 articles)
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References
Source attribution: Physics:Quantum Fermi surfaces