Energia Fermi

8/9/2019 Energia Fermi

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19.01.2015 Fermi level and Fermi function

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Fermi Level

"Fermi level" is the term used to describe the top of the collection of electron

energy levels at absolute zero temperature. This concept comes from Fermi-

Dirac statistics. Electrons are fermionsand by the Pauli exclusion principle

cannot exist in identical energy states. So at absolute zero they pack into the

lowest available energy states and build up a "Fermi sea" of electron energy

states. The Fermi level is the surface of that sea at absolute zero where no

electrons will have enough energy to rise above the surface. The concept of

the Fermi energy is a crucially important concept for the understanding of the

electrical and thermal properties of solids. Both ordinary electrical and

thermal processes involve energies of a small fraction of an electron volt. But

the Fermi energies of metals are on the order of electron volts. This implies

that the vast majority of the electrons cannot receive energy from those

processes because there are no available energy states for them to go to

within a fraction of an electron volt of their present energy. Limited to a tinydepth of energy, these interactions are limited to "ripples on the Fermi sea".

At higher temperatures a certain fraction,

characterized by the Fermi function, will

exist above the Fermi level. The Fermi level

plays an important role in theband theory of

solids. In doped semiconductors,p-type

andn-type, the Fermi level is shifted by the

impurities, illustrated by theirband gaps.

The Fermi level is referred to as the electron

chemical potential in other contexts.

In metals, the Fermi energy gives us

information about the velocities of the

electrons which participate in ordinary

electrical conduction. The amount of energy

which can be given to an electron in such

conduction processes is on the order of

micro-electron volts (see copper wire

example), so only those electrons very closeto the Fermi energy can participate. The

Fermi velocityof these conduction electrons

can be calculated from the Fermi energy.

Table

This speed is a part of the microscopic

Ohm's Lawfor electrical conduction. For a

Index

Semiconductor

concepts

Semiconductors

for electronics
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metal, the density of conduction electrons

can be implied from the Fermi energy.

The Fermi energy also plays an important role in understanding the mystery

of why electrons do not contribute significantly to the specific heat of solids at

ordinary temperatures, while they are dominant contributors to thermal

conductivity and electrical conductivity. Since only 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 heat capacity by the Pauli principle. At very low

temperatures, the electron specific 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 electron energystate will be occupied at a given temperature. The Fermi function comes from Fermi-

Dirac statisticsand has the form

The basic nature of this function dictates that at ordinary temperatures, most of the

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 ordinary temperatures, you find that its value

is essentially 1 up to the Fermil level, and rapidly approaches zero above it.

The illustration below shows the implications of the Fermi function for the electrical

conductivity of a semiconductor. Theband theory of solidsgives the picture that

there is a sizable gap between the Fermi level and the conduction band of the

semiconductor. At higher temperatures, a larger fraction of the electrons can bridge Index
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this gap and participate in electrical conduction.

Note that although the Fermi function has a finite value in the gap, there is no electron

population at those energies (that's what you mean by a gap). The population

depends upon the product of the Fermi function and the electron density of states. So

in the gap there are no electrons because the density of states is zero. In the

conduction band at 0K, there are no electrons even though there are plenty of

available states, but the Fermi function is zero. At high temperatures, both the density

of states and the Fermi function have finite values in the conduction band, so there is

a finite conducting population.

Fermi-Dirac distribution as a function of temperature

Semiconductor

concepts

Semiconductors

for electronics

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Density of Energy States

The Fermi functiongives the probability of occupying an available energy state, but

this must be factored by the number of available energy states to determine how

many electrons would reach the conduction band.This density of states is the

electron density of states, but there are differences in its implications for

conductors and semiconductors. For the conductor, the density of states can be
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considered to start at the bottom of the valence band and 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 of electrons available for conduction. In the

case of the semiconductor, the density of states is of the same form, but the

density of states for conduction electrons begins at the top of the gap.

Electron energy density function

Index

Semiconductor

concepts

Semiconductors

for electronics

Reference

Simpson

Sec 4.7

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Population of Conduction Band for a

Semiconductor
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The population of conduction electrons for a semiconductor is given by

Show

where

For a semiconductor with bandgap eV (1.1 eV for Si, 0.72 eV for Ge)

at temperature K = C

the conduction electron population is x10^ electrons/m3.

You could use this calculation to verify that the conduction electron population Ncbin

germanium doubles for about a 13 degree rise in temperature. For silicon, N cb

doubles for about an 8 degree rise in temperature. Because of the larger band gap,

there will be fewer conduction electrons in silicon than germanium for any given

temperature.

Index

Semiconductor

concepts

Semiconductors

for electronics

Reference

Simpson

Sec 4.7

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Energia Fermi