Fermi energy

The Fermi energy (including Fermi level or Fermi potential, entourage Fermi level, after Enrico Fermi ) is a physical concept from quantum statistics. It indicates the maximum energy that a particle in a many-body system of identical fermions (called Fermi gas) can have when the system as a whole in its ground state.

All states with energies between the lowest level and the Fermi energy are then fully occupied with particles about no. This is a consequence of the only fermions (e.g., electrons) Ausschließungprinzips force, according to which more than one particle may be located in any condition; for a more detailed explanation, please see Fermi -Dirac statistics. The Fermi energy is indicated as a potential difference with respect to the lowest energy level.

It leads to the system power, then refers to the Fermi energy, that the energy at which the probability of occupation is in thermodynamic equilibrium just ½, see chemical potential.

The Fermi energy makes, for example, in the photoelectric effect on metal surfaces in the form of the work felt: this is the work that needs to be supplied to an electron at the Fermi level at least to get it to blaze out of the metal.

Description

The Fermi level is defined as:

With

  • The Planckian ( divided by ) the quantum of action
  • The particle mass
  • The particle density the number of particles
  • The volume

The Fermi energy is a consequence of quantum physics, in particular quantum statistics. The exact theoretical justification of the concept requires a large number of non - interacting particles. Due to the various interactions of the fermions, the Fermi energy is therefore, strictly speaking, an approximation which everywhere has great significance, where the properties of the system are determined not so much by the interaction of the particles but stronger by mutual exclusion.

The Fermi energy passes for the properties of a Fermi gas, not only in its ground state ( ) plays an important role, but also at higher temperatures, so long as the thermal energy is lower than the Fermi energy:

With

  • Boltzmann constant
  • Absolute temperature.

The Fermi level is then no absolutely sharp boundary more, where the occupation number of the single-particle states from 1 to 0 dB, but something softened: the occupation number falls in an energy range from a few steadily from ( almost ) 1 to ( almost ) 0 from. Such Fermi gases are referred to as degenerate. Each Fermi gas is degenerate if it is not too dilute and the temperature is not too high. The exact function of the occupation number of the energy and the temperature is at the Fermi distribution.

Derivation for a simple example

For this derivation is considered a solid body with an independent electron gas, ie neglecting the electron -electron interaction. Also, you look at it in the ground state, ie, at a temperature of 0 Kelvin. As an approximation for the solid one assumes an infinite, periodic potential and describes the wave function in a cube of edge length L, so that applies to the wave function as a boundary condition:

With the Bloch function as a solution to the stationary Schrödinger equation is obtained for the components of the wave vector as a condition:

Wherein all numbers are and i is the x, y and z component.

For the ground state energy levels are up to the Fermi energy

Completely filled, ie after the Pauli principle, each with a maximum of two electrons ( ie with spin up and down).

From the condition for follows that of a k-space volume

Exactly one state is in a sphere with the radius and the volume ( of the Fermi sphere ) so there are twice as many states or electrons.

If one changes this relationship gradually begins in the Fermi energy, there is the above- mentioned formula:

Fermi level in the semiconductor and insulator

The Fermi level in the semiconductor / insulator is approximately in the middle of the band gap. This results from the Fermi -Dirac statistics. Therein the Fermi energy parameter describes the energy at which an electron state ( if there is one there at this point) is occupied with probability ½ (which is not to be confused with the concept of probability, which is the absolute square of the wave function of an electron in a particular place referred to ).

By doping the Fermi energy can be shifted in the semiconductor:

  • A p -type impurity moves the Fermi level due to the increased number of positive charge carriers ( holes) in the direction of the valence band.
  • An n -type impurity moves the Fermi level due to the increased number of negative charge carriers ( delocalized electrons) in the direction of the conduction band.

Thus, the Fermi energy has an important influence on the electrical properties of a semiconductor and is of enormous importance in the design of electrical components (eg transistors).

Examples

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

  • The fixed work function for conduction electrons in a metal (see photo effect, contact potential, Electrochemical series, sacrificial anode ) is just the energy difference between the Fermi level and the energy of the electron in vacuum.
  • The specific heat of metals is much lower than expected, according to classical physics. For the conduction electrons in it that you have to warm up with, form a degenerate Fermi gas, the need for heating much less energy than a regular gas. The reason is that it is forbidden for the very most of the electrons to absorb energy of the order, because the corresponding higher levels no place is free. Only the relatively very few electrons near the Fermi level can change their energy to these small amounts and thus contribute more in thermal equilibrium. To illustrate how narrow the Fermi level is compared to its distance from the lower band edge, it is also expressed as the Fermi temperature. For most metals is well above its melting point.
  • The electrical conductivity of metals is much greater than to understand with classical physics, because carry most of the electrons neither for current transport in (because they are in pairs flying in opposite directions) yet they stand as a collision partner for the current -carrying electrons near the edge available ( because it lacks unoccupied states into which could be scattered). In addition, the high velocity of the electrons at the Fermi level decreases scattering by lattice defects. This Fermi velocity ( the electron mass ) is for most metals at about half a percent of the speed of light.
  • White dwarf -type stars are stabilized by the degenerate electron gas at a certain radius, because further continued compression, the Fermi energy of the electron gas would rise more than is covered by the gain in gravitational energy. This applies to the Chandrasekhar limit for the mass.
  • White dwarfs or cores of giant stars with a larger mass to explode as a supernova. In the course of the continued compression of the Fermi - gas reaches the proton such a high Fermi energy that they can be converted into the neutron ( slightly heavier ). This opens the door to further and even accelerated compression, up to approximately the density of nuclear matter.
  • The classification of solids according to their electrical conductivity in insulators, semiconductors, metals depends on where the Fermi level lies in relation to the energy bands of electrons. It falls in a band gap is an insulator (broad band gap) or a semiconductor (narrow band gap) before it falls in the middle of a tape is a metal.
  • The variable within a wide range electrical conductivity of semiconductors ( that is, the technical basis of the electronic components ) is largely determined, where the Fermi level in the band gap is located precisely: with an intrinsic semi-conductors in the middle, with a p-type conductor close to the lower edge on top, in an n - ladder.
  • Can two systems exchange particles are similar except their temperatures and their Fermi energies. This creates, for example, in the contact of a p-type semiconductor with an n-type semiconductor diode.
  • The chemical reaction in a mixture of different substances is generally determined by the fact that it leads to the approximation of the chemical potentials of all substances. For a substance whose particles are fermions, the chemical potential is therefore given by the Fermi level.

Evidence

  • Solid State Physics
  • Enrico Fermi
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