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{{Short description|Abstract boundary in condensed matter physics}}
{{Quantum book backlink|Condensed matter and solid-state physics}}

In [[Physics:Condensed matter physics|condensed matter physics]], the '''Fermi surface''' is the surface in [[Physics:Reciprocal lattice|reciprocal space]] which separates occupied electron states from unoccupied electron states at absolute zero temperature.<ref name="Dugdale2016">{{cite journal|last1=Dugdale|first1=S B|title=Life on the edge: a beginner's guide to the Fermi surface|journal=Physica Scripta|volume=91|issue=5|year=2016|article-number=053009|doi=10.1088/0031-8949/91/5/053009|doi-access=free|bibcode=2016PhyS...91e3009D|hdl=1983/18576e8a-c769-424d-8ac2-1c52ef80700e|hdl-access=free}}</ref>

Its shape is determined by the periodicity and symmetry of the crystal lattice and by the occupation of [[Physics:Electronic band structure|electronic energy bands]]. The existence of a Fermi surface follows directly from the [[Physics:Pauli exclusion principle|Pauli exclusion principle]], which allows only one electron per quantum state.<ref>{{cite book |first1=N. |last1=Ashcroft |first2=N. D. |last2=Mermin |title=Solid-State Physics |year=1976 |publisher=Holt, Rinehart and Winston |isbn=0-03-083993-9 }}</ref><ref>{{cite book |first=W. A. |last=Harrison |title=Electronic Structure and the Properties of Solids |date=1989 |publisher=Courier Corporation |isbn=0-486-66021-4 }}</ref><ref>{{cite book |first=J. M. |last=Ziman |title=Electrons in Metals: A Short Guide to the Fermi Surface |publisher=Taylor & Francis |year=1963 }}</ref>

The study of Fermi surfaces is called '''fermiology'''.
<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:400px; text-align:center;">
[[File:fermi_yellow.png|400px]]
<div style="font-size:90%; margin-top:6px;">Fermi surface and electron momentum density of copper measured using 2D ACAR techniques.</div>
</div>

== Theory ==

For an ideal [[Physics:Fermi gas|Fermi gas]], the occupation of quantum states is governed by the [[Physics:Fermi–Dirac statistics|Fermi–Dirac distribution]]:
<math display="block">\langle n_i\rangle = \frac{1}{e^{(\epsilon_i-\mu)/k_{\rm B}T}+1}</math>

At zero temperature ({{math|''T'' → 0}}), this simplifies to:
<math display="block">\langle n_i\rangle =
\begin{cases}
1 & (\epsilon_i < \mu) \\
0 & (\epsilon_i > \mu)
\end{cases}</math>

All states below the [[Physics:Fermi energy|Fermi energy]] are filled, while all above are empty. In [[Physics:Momentum space|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:
<math display="block">k_{\rm F} = \frac{\sqrt{2 m E_{\rm F}}}{\hbar}</math>

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.

<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:360px; text-align:center;">
[[File:graphiteFS.png|340px]]
<div style="font-size:90%; margin-top:6px;">Fermi surface of graphite showing anisotropic electron and hole pockets in the Brillouin zone.</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 [[Physics:Brillouin zone|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 [[Physics:Superconductor|superconductivity]], [[Physics:Ferromagnet|ferromagnetism]], or [[Physics:Spin density wave|spin density waves]].

At finite temperatures, the sharp boundary of the Fermi surface becomes slightly blurred due to thermal excitations.

== Experimental determination ==

<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:360px; text-align:center;">
[[File:Fermi surface of BSCCO exp.jpg|340px]]
<div style="font-size:90%; margin-top:6px;">Fermi surface of a cuprate superconductor measured using angle-resolved photoemission spectroscopy (ARPES).</div>
</div>

Fermi surfaces can be measured experimentally using several techniques:

* [[Physics:De Haas–van Alphen effect|de Haas–van Alphen effect]]
* [[Physics:Shubnikov–de Haas effect|Shubnikov–de Haas effect]]
* [[Physics:Angle-resolved photoemission spectroscopy|ARPES]]

These methods rely on quantum oscillations or direct measurement of electron energies in momentum space.

A key result by [[Biography:Lars Onsager|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>

Another method is [[Physics:Angular Correlation of Electron Positron Annihilation Radiation|ACAR]], which measures electron momentum distributions through positron annihilation.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Fermi surface|1}}

Physics:Quantum Fermi surfaces - Revision history

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