Fermi Level - Some Other Complications

Some Other Complications

• Charging effects. In cases where the "charging effects" due to a single electron are non-negligible, the above definition does not quite work. For example, consider an capacitor made of two identical parallel-plates. If the capacitor is uncharged, the Fermi level is the same on both sides, so one might think that it should take no energy to move an electron from one plate to the other. But when the electron has been moved, the capacitor has become (slightly) charged, so this does take a slight amount of energy. In a normal capacitor, this is negligible, but in a nano-scale capacitor it can be more important. This can be dealt with, for example, by saying that the Fermi level is N times the energy required to move 1/Nth of an energy to the reference level (where N is a very large number), or by saying that the Fermi level is the energy required to move an electron to the reference level, not counting the energy stored in electrostatic fields.

• Non-equilibrium effects. In many cases, the occupancy of electronic states is not described by the Fermi-Dirac distribution, because the electron distribution is not in local thermodynamic equilibrium. For example, when light is shining on a semiconductor, there is no value of the Fermi-Dirac distribution function f(E) that describes the actual occupancy of electronic states. (The light shifts electrons to more energetic levels in a characteristic way.) In such cases, there is no Fermi level. Sometimes, the occupancy of conduction-band states can be described by putting the Fermi level in a certain position relative to the band structure, whereas the occupancy of valence-band states can be described putting the Fermi level in a different position relative to the band structure. In this case, the two different Fermi levels are called quasi-Fermi levels, as in the case described earlier. In other situations, such as immediately after a high-energy laser pulse, one cannot even define quasi-Fermi levels; the electrons and holes are simply said to be "non-thermalized".

• Fermi level equilibration. Any material or device in thermodynamic equilibrium will have a constant Fermi level everywhere in the device. Neither the electrostatic potential energy nor the internal chemical potential on their own need to be constant in the device, but the Fermi level (their sum) does. For example, in an equilibrium p-n junction, the electrostatic potential energy is higher on the n-type side than the p-type side (this is associated with the so-called "built-in field"), but this is precisely offset by the equal-and-opposite change in internal chemical potential.

The Fermi level can vary (or not exist at all) in any non-equilibrium situation, such as:

• Under an applied voltage,
• Under illumination from a light-source with a different temperature, such as the sun (this allows for photovoltaics),
• When the temperature is not constant within the device (this allows for thermocouples, for example),
• When the device has been altered, but has not had enough time to re-equilibrate (this allows for pyroelectricity, for example).