Fermi energy

The Fermi energy (also Fermi level or Fermi potential, closest environment Fermi edge; after Enrico Fermi ) is a physical term from quantum statistics . It indicates the highest energy that a particle can have in a many-body system of similar fermions (a Fermi gas ) when the system as a whole is in its ground state .

At absolute zero , all states between the lowest possible level and the Fermi energy are fully occupied, none above. This is a consequence of the Pauli principle, which only applies to fermions (e.g. electrons ), according to which no more than one particle can be in any state; for a more detailed explanation see Fermi-Dirac statistics . The states are thus filled with the state of lowest energy. For the ground state, the Fermi energy separates the occupied from the unoccupied states. The Fermi energy is given as the energy difference to the lowest possible energy level.

If one adds energy to the system, then the Fermi energy denotes the energy at which the occupation probability is just 50% in thermodynamic equilibrium , see chemical potential .

The Fermi energy makes itself z. B. noticeable in the photoelectric effect on metal surfaces in the form of the work function, i.e. the work that must at least be supplied to an electron at the Fermi edge in order to knock it out of the metal.

Some authors refer to the Fermi energy as the energy difference that the highest energetic occupied state has above the single-particle ground state, while the Fermi level can refer to any zero point and is used in particular for . ${\displaystyle T=0}$ ${\displaystyle T>0}$

description

The following applies to the Fermi energy in a gas consisting of non-interacting fermions with mass and energy dispersion relation ${\displaystyle E_{\mathrm {F} }}$ ${\displaystyle m}$ ${\displaystyle E(k)=\hbar ^{2}k^{2}/(2m)}$

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m}}(3{\pi }^{2}n)^{\frac {2}{3}}}$

With

• the reduced Planck quantum of action (the Planck quantum of action divided by ), ${\displaystyle \hbar }$ ${\displaystyle 2\pi }$ ${\displaystyle h}$
• the particle mass ${\displaystyle m,}$
• the particle density ${\displaystyle n={\frac {N}{V}},}$
  • the number of particles ${\displaystyle N,}$
  • the volume and ${\displaystyle V}$
• the wave vector ${\displaystyle {\vec {k}}.}$

The Fermi energy is a consequence of quantum physics , especially quantum statistics. The exact theoretical justification of the term requires a large number of non - interacting particles. Due to the diverse interactions of the fermions, the Fermi energy is therefore, strictly speaking, an approximation that is of great importance wherever the properties of the system are determined not so much by the interaction of the particles but more by the mutual exclusion.

Fermi distribution
for different temperatures

The Fermi energy plays an important role for the properties of a Fermi gas not only in its ground state ( ), but also at higher temperatures as long as the thermal energy is significantly lower than the Fermi energy: ${\displaystyle T=0\,\mathrm {K} }$ ${\displaystyle k_{\mathrm {B} }T}$

${\displaystyle k_{\mathrm {B} }T\ll E_{\mathrm {F} }}$

With

• the Boltzmann constant and ${\displaystyle k_{\mathrm {B} }}$
• the absolute temperature ${\displaystyle T.}$

The Fermi edge is then no longer an absolutely sharp limit, where the occupation number of the single-particle states jumps from 1 to 0, but is somewhat softened: The occupation number falls steadily from (almost) 1 to (almost) 0 in an energy range of a few . Such Fermi gases are called degenerate . Every Fermi gas is degenerate if it is not too diluted and the temperature is not too high. The exact dependence of the occupation number on the energy and the temperature is described by the Fermi distribution . ${\displaystyle k_{\mathrm {B} }T}$

It is true that for weakly interacting fermionic systems it is no longer true that all states which are energetically below the Fermi energy are occupied and all above are unoccupied, but here too the Fermi energy is still of great importance. Excited states with energy just above or below the Fermi energy are then so long-lived that they are still well defined as particles (lifetime , we speak of quasiparticles or holes). “Well defined as a particle” is to be understood here as meaning that these excited states near the Fermi edge, which are not eigenstates of the Hamilton operator with electron-electron interaction (hence the finite lifetime), are approximately identified with the eigenstates of the interaction-free Hamilton operator can. All of this is described in the Fermi fluid theory. From this theory it becomes clear that the states with energies close to the Fermi edge z. B. for transport phenomena such as electrical or thermal conductivity are essential and why simple theories that completely neglect the electron-electron interaction, such as the Drude and Sommerfeld theory, sometimes deliver acceptable results for real materials (mostly only for materials without strong interaction or complicated band structures ). ${\displaystyle E}$ ${\displaystyle \tau \propto 1/[(E-E_{\mathrm {F} })^{2}+(\pi k_{\mathrm {B} }T)^{2}]}$

Derivation for a simple example

For this derivation one considers a solid with an independent electron gas , thus neglecting the electron-electron interaction. It is also viewed in its basic state, i.e. at a temperature of 0  Kelvin . As an approximation for the solid body, one takes an infinite, periodic potential and describes the wave function in a cube of edge length , so that the wave function is a boundary condition ${\displaystyle L}$

${\displaystyle \Psi (x+L,y,z)=\Psi (x,y,z)}$(analogous for and ) ${\displaystyle y}$ ${\displaystyle z}$

applies. With the Bloch function as the solution for the stationary Schrödinger equation , the conditions for the components of the wavenumber vector are obtained

${\displaystyle k_{i}={\frac {2\pi n_{i}}{L}},}$

with whole numbers , where stands for the -, - or - component. ${\displaystyle n_{i}}$ ${\displaystyle i}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

For the basic state, the energy levels up to the Fermi energy are with ${\displaystyle E_{\mathrm {F} }}$

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}k_{\mathrm {F} }^{2}}{2m}}}$

completely filled, i.e. according to the Pauli principle with a maximum of two electrons each with opposite spins . Here is the wave number belonging to the Fermi surface . ${\displaystyle k_{\mathrm {F} }}$

From the condition for it follows that in a -space-volume of ${\displaystyle k_{i}}$ ${\displaystyle k}$

${\displaystyle V_{0}=\left({\frac {2\pi }{L}}\right)^{3}={\frac {(2\pi )^{3}}{V}}}$

there is exactly one state, in a sphere with radius and volume (the Fermi sphere ) there are states, i.e. H. there are twice as many electrons. ${\displaystyle k_{\mathrm {F} }}$ ${\displaystyle V_{\mathrm {F} }=(4\pi /3){k_{\mathrm {F} }}^{3}}$ ${\displaystyle V_{\mathrm {F} }/V_{0}}$

If you rearrange this relationship and use it in the Fermi energy, you get the formula mentioned at the beginning ${\displaystyle k_{\mathrm {F} }}$

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m}}(3{\pi }^{2}n)^{\frac {2}{3}}.}$

Fermi energy in the semiconductor and insulator

The Fermi energy in the semiconductor / insulator is roughly in the middle of the band gap . This results from the Fermi-Dirac statistics . In it, the parameter Fermi energy describes the energy at which an electron state (if there were one at this point) is occupied with probability ½ (which should not be confused with the term probability of being, which is the absolute square of the wave function of an electron at a certain location designated).

The Fermi energy in the semiconductor can be shifted by doping :

• a doping moves the Fermi energy, due to the increased number of positive charge carriers (holes) in the direction of the valence band . ${\displaystyle p}$
• a doping moves the Fermi energy, due to the increased number of negative charge carriers (electrons localized de-), in the direction of the conduction band . ${\displaystyle n}$

The Fermi energy thus has an important influence on the electrical properties of a semiconductor and is of enormous importance in the design of electrical components (e.g. transistors ).

Examples

The Fermi energy helps in many areas of physics to describe phenomena that have no classical meaning.

• The fixed work function when conduction electrons in a metal (s. Photoelectric effect , contact potential , Electrochemical Series , sacrificial anode ) is just the difference in energy between the Fermi level and the energy of the electron in a vacuum.
• The specific heat of the metals is much lower than expected from classical physics. Because the conduction electrons in it, which have to be heated up, form a degenerate Fermi gas that needs much less energy for heating than a normal gas. The reason is that it is forbidden for the vast majority of electrons to absorb energies of this magnitude , because there is no free space at the corresponding higher levels. Only very few electrons (relative to the total amount of electrons) near the Fermi edge can change their energy by these small amounts and therefore contribute to the thermal equilibrium. To make it clear how narrow the Fermi edge is in comparison to its distance from the lower band edge, this is also expressed as the Fermi temperature . For most metals it is well above their melting point. ${\displaystyle k_{\mathrm {B} }T\,}$ ${\displaystyle T_{\mathrm {F} }=E_{\mathrm {F} }/k_{\mathrm {B} }}$
• The electrical conductivity of metals is much greater than can be understood from classical physics, because most electrons neither contribute to the transport of electricity (because they fly in pairs in opposite directions) nor are they available as collision partners for the electrons carrying the current near the edge (because there are no unoccupied states into which scattering could take place). In addition, the high speed of the electrons at the Fermi edge reduces the scattering of lattice disturbances. This Fermi speed (with the electron mass ) is about half a percent of the speed of light for most metals . ${\displaystyle \textstyle v_{\mathrm {F} }={\sqrt {\frac {2E_{\mathrm {F} }}{m_{\mathrm {e} }}}}}$ ${\displaystyle m_{\mathrm {e} }}$
• White dwarf- type stars are stabilized by the degenerate electron gas at a certain radius, because with continued compression, the Fermi energy of the electron gas would rise more than is covered by the gain in gravitational energy . This is true for the crowd up to the Chandrasekhar limit .
• White dwarfs or cores of giant stars with greater mass explode than supernovae . In the course of the continued compression, the Fermi gas of the protons reaches such a high Fermi energy that they can transform into the (somewhat heavier) neutrons . This opens up the possibility of further and even accelerated compression, for example up to the density of the nuclear matter.
• The division of solid materials into insulators , semiconductors and metals according to their electrical conductivity depends on where the Fermi level lies in relation to the energy bands of the electrons. If it falls into a band gap , if it is an insulator (wide band gap) or a semiconductor (narrow band gap), if it falls in the middle of a band, it is a metal.
• The widely variable electrical conductivity of semiconductors (i.e. the technical basis of electronic components) is largely determined by the exact location of the Fermi energy in the band gap: in the case of an intrinsic semiconductor in the middle, in the case of a p-conductor close to the bottom Edge, with an n-conductor at the top.
• If two systems can exchange particles, then not only their temperatures but also their Fermi energies adjust themselves. So z. B. in contact of a p-semiconductor with an n-semiconductor a diode .
• The chemical reaction in a mixture of different substances is generally determined by the fact that it leads to the equalization of the chemical potentials of all substances. For a substance whose particles are fermions, the chemical potential is given by the Fermi level.

proof

1. ↑ Enrico Fermi: On the quantization of the monatomic ideal gas. In: Journal of Physics. Vol. 36, 1926, pp. 902-912, DOI: 10.1007 / BF01400221 .
2. ↑ For example, the superconductivity in the BCS theory is explained by the fact that the energy of the electron gas in the "normal" ground state can still be lowered by, as a result of an attractive electron-electron interaction, which is mediated by the crystal lattice , under energy gain Cooper - Form pairs .
3. ↑ Sometimes , unlike in this article, the term Fermi energy is only used for systems at , while Fermi level can also be used at. This distinction is not common and is not made here. ${\displaystyle T=0\,\mathrm {K} }$ ${\displaystyle T>0}$
4. ^ Gabriele Giuliani, Giovanni Vignale:  Quantum Theory of the Electron Liquid. Cambridge University Press, 2005.