Fermi Surface · aiwiki.page

A Fermi surface is a set of points in momentum space at which the energy needed to add or remove a particle vanishes, measured relative to the chemical potential. For noninteracting fermions at absolute zero, it separates occupied from unoccupied single-particle states. In condensed matter physics, its shape and size are central to understanding the low-energy behavior of electrons in metals. It is not a physical surface inside a material: it is a boundary in the space of momentum labels. (damtp.cam.ac.uk)

Definition and occupation

For a noninteracting system with band energies $\varepsilon_n(\mathbf{k})$ , the Fermi surface is defined by

\varepsilon_n(\mathbf{k})=\mu_0,

where $n$ labels the band, $\mathbf{k}$ is the wave vector, and $\mu_0$ is the zero-temperature chemical potential, conventionally identified with the Fermi energy $E_F$ . The Fermi–Dirac distribution

f(\varepsilon)= \frac{1}{\exp[(\varepsilon-\mu)/(k_BT)]+1}

becomes a step at zero temperature: states below $\mu_0$ are occupied and states above it are empty. Here $T$ is temperature and $k_B$ is the Boltzmann constant. This filling follows from the Pauli exclusion principle. (damtp.cam.ac.uk)

The occupied region is called the Fermi sea, whereas the Fermi surface is its boundary. At nonzero temperature, occupation changes smoothly across an energy interval of order $k_BT$ . The term “Fermi surface” nevertheless remains useful for the corresponding energy contour when thermal broadening is small. (damtp.cam.ac.uk)

Free-electron example

For nonrelativistic free electrons of mass $m$ ,

\varepsilon(\mathbf{k})=\frac{\hbar^2|\mathbf{k}|^2}{2m},

where $\hbar$ is the reduced Planck constant. The occupied states form a ball, and its boundary is a sphere of radius $k_F$ . For an unpolarized, three-dimensional gas with two spin states and electron number density $n_e$ ,

k_F=(3\pi^2n_e)^{1/3}, \qquad E_F=\frac{\hbar^2k_F^2}{2m}.

The corresponding momentum is $p_F=\hbar k_F$ . These expressions count the available momentum states, including their spin degeneracy. (damtp.cam.ac.uk)

More generally, a regular Fermi surface has dimension $d-1$ in a $d$ -dimensional system: a surface in three dimensions, a contour in two dimensions, and isolated Fermi points in one dimension. Special band-touching situations can produce lower-dimensional zero-energy sets. (ocw.mit.edu)

Fermi surfaces in crystals

In a crystal, electronic band structure replaces the free-particle dispersion. The wave vector labels crystal momentum and is equivalent modulo a reciprocal lattice vector. Fermi surfaces are therefore commonly drawn inside the first Brillouin zone, with appropriate boundary identifications. Pieces separated in a drawing may connect across those boundaries. (davidtong.org)

A crystal can have several Fermi-surface sheets belonging to different bands. Common geometries include nearly spherical pockets, elongated pockets, approximately cylindrical sheets, and sheets that cross zone boundaries. An electron pocket encloses occupied states near a band minimum; a hole pocket encloses unoccupied states near a band maximum, which can be described using holes. (davidtong.org)

Within ordinary band theory, partially filled dispersive bands generally produce metallic behavior. If the chemical potential lies in a band gap, no band crosses it and there is no electronic Fermi surface. This describes a band insulator, but does not exhaust the possible effects of interactions or special band crossings. (davidtong.org)

Why the surface controls low-energy behavior

Electrons far below the Fermi energy cannot easily move into nearby states because those states are already occupied. Low-energy excitations consequently involve states close to the Fermi surface. In a conventional metallic regime, this gives an electronic heat capacity proportional to temperature at sufficiently low temperature, rather than the temperature-independent classical-gas result. (damtp.cam.ac.uk)

For a band or quasiparticle dispersion, the Fermi velocity is

\mathbf{v}_{F,n} =\left.\frac{1}{\hbar}\nabla_{\mathbf{k}} \varepsilon_n(\mathbf{k})\right|_{\mathrm{FS}}.

It is normal to a regular constant-energy surface. Variations in velocity, band curvature, and scattering rates help determine directional differences in electrical transport and response to a magnetic field. Thus, the Fermi surface is more than a count of carriers: its local geometry also matters. (ocw.mit.edu)

Interacting systems and Luttinger’s theorem

The occupied–empty boundary is exact for independent particles, but interactions modify that picture. In Landau Fermi-liquid theory, low-energy excitations are long-lived quasiparticles. At zero temperature, the electron momentum distribution retains a discontinuity at the Fermi surface, although its height need not equal one. The surface can remain sharply defined despite a redistributed occupation away from it. (doi.org)

In a paper published on August 15, 1960, J. M. Luttinger related this discontinuity to a Fermi surface for interacting fermions and established a constraint on its enclosed momentum-space volume. Under the conventional Fermi-liquid assumptions, Luttinger’s theorem relates that volume to particle density: interactions may change the shape without freely changing the volume at fixed density. In crystals, the counting must account for filled bands and degeneracies. This is not an unrestricted statement about every strongly interacting phase. (doi.org)

A Fermi-surface-like singularity does not automatically imply conventional quasiparticles. For example, interacting one-dimensional systems can have well-defined Fermi momenta while their low-energy excitations are collective rather than Landau quasiparticles. (physics.aps.org)

Topology and nesting

A Lifshitz transition changes the connectivity or number of Fermi-surface components as a control parameter varies. Examples include the appearance of a pocket or the joining of previously separate contours. Such changes can result from shifting the chemical potential or modifying the band dispersion. Here topology refers to the connectivity of the momentum-space set, not necessarily to a topological classification of electronic wave functions. (arxiv.org)

Fermi-surface nesting occurs when substantial portions of a Fermi surface can be mapped onto one another by translation through a common wave vector $\mathbf{Q}$ . It can enhance electronic response at that wave vector. However, geometric nesting alone does not establish the occurrence or wave vector of a charge-density wave: interactions, electron–phonon coupling, and the full electronic susceptibility also matter. (triqs.github.io)

Experimental determination

Angle-resolved photoemission spectroscopy (ARPES) measures the energies and emission directions of electrons released by light. These measurements provide access to the occupied electronic spectral function and allow identification of band crossings near the chemical potential. Surface sensitivity, photoemission matrix elements, and uncertainty in momentum perpendicular to the surface complicate interpretation, particularly for three-dimensional materials. (arpes.stanford.edu)

Quantum oscillations provide a complementary probe. In conventional metallic cases, electronic properties oscillate approximately periodically in inverse magnetic field as quantized orbital levels cross the chemical potential. The Onsager relation connects the oscillation frequency $F$ to an extremal Fermi-surface cross-sectional area $A_{\mathrm{ext}}$ , perpendicular to the field:

F=\frac{\hbar}{2\pi e}A_{\mathrm{ext}},

where $e$ is the positive elementary charge. Rotating the field provides different cross sections, constraining the three-dimensional geometry. Oscillations in magnetization are associated with the de Haas–van Alphen effect; resistance oscillations with the Shubnikov–de Haas effect. (davidtong.org)

Experimental contours and theoretical Fermi surfaces need not coincide without qualification. Thermal broadening, finite lifetimes, limited resolution, and strongly momentum-dependent spectral weight can obscure portions of a surface. ARPES and quantum oscillations measure different aspects of electronic structure, so their interpretation requires the relevant spectral or orbital model. (arpes.stanford.edu)

References

Solid State Physics — David Tongdavidtong.org
730 Physics for Solid State Applications, Lecture 21ocw.mit.edu
Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermionsdoi.org
Topological approach to Luttinger's theorem and the Fermi surface of a Kondo latticearxiv.org
In a tight spot, spin and charge separatephysics.aps.org
Tunable Fermi surface topology and Lifshitz transition in bilayer graphenearxiv.org
Fermions on the square lattice & perfect nestingtriqs.github.io
Fermi surface nesting and the origin of Charge Density Waves in metalsarxiv.org
Angle-resolved Photoemission Spectroscopy — Shen Laboratoryarpes.stanford.edu
Laser ARPES — Shen Laboratoryarpes.stanford.edu
Fermi pockets and quantum oscillations of the Hall coefficient in high temperature superconductorsarxiv.org