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Fermi Level

"Fermi level" is the term used to describe the top of the collection ofelectron energy levels at absolute zero temperature. This conceptcomes from Fermi-Dirac statistics. Electrons are fermionsand by thePauli exclusion principlecannot existin identical energy states. So atabsolute zero they pack into the lowest available energy states and

build up a "Fermi sea" of electron energy states. The Fermi level is thesurface of that sea at absolute zero where no electrons will haveenough energy to rise above the surface. The concept of the Fermienergy is a crucially important concept for the understanding of theelectrical and thermal properties of solids. Both ordinary electrical andthermal processes involve energies of a small fraction of an electronvolt. But the Fermi energies of metals are on the order of electronvolts. This implies that the vast majority of the electrons cannotreceive energy from those processes because there are no availableenergy states for them to go to within a fraction of an electron volt of

their present energy. Limited to a tiny depth of energy, theseinteractions are limited to "ripples on the Fermi sea".

At higher temperatures a certainfraction, characterized by the Fermifunction, will exist above the Fermilevel. The Fermi level plays animportant role in theband theory ofsolids. In doped semiconductors,p-typeandn-type, the Fermi level is shifted by

the impurities, illustrated by theirbandgaps. The Fermi level is referred to asthe electron chemical potential in othercontexts.

In metals, the Fermi energy gives usinformation about the velocities of theelectrons which participate in ordinaryelectrical conduction. The amount ofenergy which can be given to an

electron in such conduction processes ison the order of micro-electron volts (seecopper wire example

), so only thoseelectrons very close to the Fermi energy

Index

Semiconductorconcepts

Semiconductorsfor electronics

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can participate. The Fermi velocityof

these conduction electrons can becalculated from the Fermi energy.

Table

This speed is a part of the microscopicOhm's Lawfor electrical conduction.For a metal, the density of conductionelectronscan be implied from the Fermienergy.

The Fermi energy also plays an important role in understanding themystery of why electrons do not contribute significantly to the specificheat of solids at ordinary temperatures, while they are dominantcontributors to thermal conductivity and electrical conductivity. Sinceonly a tiny fraction of the electrons in a metal are within the thermal

energy kT of the Fermi energy, they are "frozen out" of the heatcapacity by the Pauli principle. At very low temperatures, the electronspecific heatbecomes significant.

Fermi energies for metals

Table of Fermi energies

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Fermi Function

The Fermi function f(E) gives the probability that a given available electronenergy state will be occupied at a given temperature. The Fermi function comesfrom Fermi-Dirac statisticsand has the form

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The basic nature of this function dictates that at ordinary temperatures, most ofthe levels up to the Fermi levelEFare filled, and relatively few electrons have

energies above the Fermi level. The Fermi level is on the order of electron volts(e.g., 7 eV for copper), whereas the thermal energykT is only about 0.026 eV

at 300K. If you put those numbers into the Fermi function at ordinarytemperatures, you find that its value is essentially 1 up to the Fermil level, andrapidly approaches zero above it.

The illustration below shows the implications of the Fermi function for theelectrical conductivity of a semiconductor. Theband theory of solidsgives the

picture that there is a sizable gap between the Fermi level and the conductionband of the semiconductor. At higher temperatures, a larger fraction of theelectrons can bridge this gap and participate in electrical conduction.

Note that although the Fermi function has a finite value in the gap, there is noelectron population at those energies (that's what you mean by a gap). The

population depends upon the product of the Fermi function and the electrondensity of states. So in the gap there are no electrons because the density ofstatesis zero. In the conduction band at 0K, there are no electrons even thoughthere are plenty of available states, but the Fermi function is zero. At high

Index

Semiconductorconcepts

Semiconductorsfor electronics

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html#c5http://hyperphysics.phy-astr.gsu.edu/hbase/solids/sselcn.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlhttp://-/?-http://hyperphysics.phy-astr.gsu.edu/hbase/solids/intrin.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disene.html#c2http://hyperphysics.phy-astr.gsu.edu/hbase/solids/semcn.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/eqpar.html#c2

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temperatures, both the density of states and the Fermi function have finitevalues in the conduction band, so there is a finite conducting population.

Fermi-Dirac distribution as a function of temperature

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Density of Energy States

The Fermi functiongives the probability of occupying an available energystate, but this must be factored by the number of available energy states todetermine how many electrons would reach the conduction band.Thisdensity of states is the electron density of states, but there are differences inits implications for conductors and semiconductors. For the conductor, thedensity of states can be considered to start at the bottom of the valence bandand fill up to the Fermi level, but since the conduction band and valence

band overlap, the Fermi level is in the conduction band so there are plenty ofelectrons available for conduction. In the case of the semiconductor, thedensity of states is of the same form, but the density of states for conduction

electrons begins at the top of the gap.

Index

Semiconductorconcepts

Semiconductors

for electronics

ReferenceSimpson

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Electron energy density function

Sec 4.7

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Population of Conduction Band for a

Semiconductor

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