Fermi level

The Fermi level chemical potential for electrons (or electrochemical potential for electrons) of a body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by µ or EFfor brevity. The Fermi level does not include the work required to remove the electron from wherever it came from.A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics. The Fermi level chemical potential for electrons (or electrochemical potential for electrons) of a body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by µ or EFfor brevity. The Fermi level does not include the work required to remove the electron from wherever it came from.A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics. In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time.The position of the Fermi level in relation to the band energy levels is a crucial factor in determining electrical properties.The Fermi level does not necessarily correspond to an actual energy level (in an insulator the Fermi level lies in the band gap), nor does it require the existence of a band structure.Nonetheless, the Fermi level is a precisely defined thermodynamic quantity, and differences in Fermi level can be measured simply with a voltmeter. Sometimes it is said that electric currents are driven by differences in electrostatic potential (Galvani potential), but this is not exactly true.As a counterexample, multi-material devices such as p–n junctions contain internal electrostatic potential differences at equilibrium, yet without any accompanying net current; if a voltmeter is attached to the junction, one simply measures zero volts.Clearly, the electrostatic potential is not the only factor influencing the flow of charge in a material—Pauli repulsion, carrier concentration gradients, electromagnetic induction, and thermal effects also play an important role. In fact, the quantity called voltage as measured in an electronic circuit has a simple relationship to the chemical potential for electrons (Fermi level).When the leads of a voltmeter are attached to two points in a circuit, the displayed voltage is a measure of the total work transferred when a unit charge is allowed to move from one point to the other.If a simple wire is connected between two points of differing voltage (forming a short circuit), current will flow from positive to negative voltage, converting the available work into heat. The Fermi level of a body expresses the work required to add an electron to it, or equally the work obtained by removing an electron.Therefore, VA − VB, the observed difference in voltage between two points, A and B, in an electronic circuit is exactly related to the corresponding chemical potential difference, µA − µB, in Fermi level by the formula where −e is the electron charge. From the above discussion it can be seen that electrons will move from a body of high µ (low voltage) to low µ (high voltage) if a simple path is provided.This flow of electrons will cause the lower µ to increase (due to charging or other repulsion effects) and likewise cause the higher µ to decrease.Eventually, µ will settle down to the same value in both bodies.This leads to an important fact regarding the equilibrium (off) state of an electronic circuit: This also means that the voltage (measured with a voltmeter) between any two points will be zero, at equilibrium.Note that thermodynamic equilibrium here requires that the circuit be internally connected and not contain any batteries or other power sources, nor any variations in temperature. In the band theory of solids, electrons are considered to occupy a series of bands composed of single-particle energy eigenstates each labelled by ϵ. Although this single particle picture is an approximation, it greatly simplifies the understanding of electronic behaviour and it generally provides correct results when applied correctly.