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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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19.01.2015 Fermi level and Fermi function
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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
HyperPhysics*****Condensed Matter *****Electricity and
Magnetism
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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
HyperPhysics*****Condensed Matter *****Electricity and Magnetism R NaveGo Back
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
HyperPhysics*****Condensed Matter *****Electricity and MagnetismR
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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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