What is Fermi Factor
With the Fermi energy can you the highest energy one Particle in one system of Fermions at atemperature of zeroKelvin () to calculate. In this article we will explain what the Fermi energy exactly describes and how you like them to calculate and derive can. You will also learn how to use her materials, in particular Metals, semiconductors and Isolators, characterize can and what at Temperatures higher as 0 Kelvin happens. At the end of the article we will also look at a specific one example view by using the Fermi energyexplicit for copper to calculate.
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Fermi energy explained in simple terms
The Fermi energy gives the energy of highest occupied energy levels in the Basic state of a system Fermions at.
Are you looking at the absolute zero (0 Kelvin) then all energy levels are up to Fermi energy fully occupied and the higher energy levels are vacant. A system is in the basic state when it is in the state lowest possible energy is located.
Fermi energy formula
The Fermi energy lets out for a gas non-interacting fermions calculate with the formula
It represents the reduced Planck quantum of action, the Mass of the fermion and the Fermi wave vector
The Particle density is the quotient from the Particle number and the volume. Using the Fermi energy you can also do the Fermi speed just calculate with
The Fermi speed is the speed of a fermion at which the kinetic energy equal to the Fermi energy corresponds to
With the Fermi wave vector can also be called the Fermi wavelength express
Fermions, such as electrons, both propagate as wave as well as Particle out. The Fermi wavelength is the wavelength of the electrons, which they at the Fermi energy have. In addition, the so-called Fermi temperature to calculate
in which the Boltzmann constant designated.
Fermi energy derivation
In this section we will use the formula for thatFermi energy derive. We assume a square Potential box with the volume out. In addition, the Fermions as non-interacting particles approached and the system is in the ground state at a temperature of 0 Kelvin. Because one of a infinite, periodic potential goes out, applies to the Wave function
By solving the stationary Schrödinger equation with the Bloch function is then obtained as a condition for the Wavenumber vector
(1) With .
From particle physics, there is also the following relationship between the frequency and the energy known
In addition, you also know that the Wavenumber proportional to pulse is
If one uses these relationships, one obtains for them kinetic energy the following formula
If you put in this formula for the Wave vector the Fermi wave vector one, then the result is the Fermi energy
All energy levels are then in the basic state up to this Fermi energy with each two spins occupied. These occupied states are in the Fermi sphere with the volume
The volume, which contains exactly one state, has in the reciprocal space a volume of
If you divide the volume of the Fermi sphere by volume , then you get the number the Electrons in the Potential box
The factor 2 comes from there each state has two electrons can accommodate. With this, the Electron density determine
If you model this equation around
and plug this into formula (2), then you can use the Fermi energy also with the Particle density express
Fermi energy semiconductor, insulator, metal
With the Fermi energy you can now explain why some materials are conductive and some are not. Is the Fermi energy in the Band gap between the fully occupied valence band and the empty conduction band so, for example, (sufficiently strong) thermal excitation, Electrons in that Conduction band reach. This becomes the material conductive and in this case one speaks of one Semiconductor. Is the Band gap however too big, so that the electrons get this by excitation not bridge can, then the material not conductive and one speaks of one Insulator. A metal is a conductive material, since the Fermi energy in the Conduction band is located and thus the Line band partially occupied is. And partially occupied bands are precisely the prerequisite for a material to be conductive.
Fermi edge at higher temperatures
The states are at a temperature of 0 Kelvin just until Fermi level occupied. In this context one speaks of a Fermi edge, because the Fermi distribution For a Edge having. The Fermi distribution gives the probability with which a state with the energy at a certain temperature is busy
Here is the chemical potential and for applies . To energy the Fermi edge are so all states occupied and the states higher energy are all vacant. If the temperature is starting from 0 Kelvin elevated, so it gives way Fermi edge more and more up and so will conditions above that Fermi level occupied. This can be seen in the following illustration.
It must, however, apply that the thermal energy is much smaller than that Fermi energy
Calculate Fermi energy example
With the above formula, the Fermi energy calculate various elements, such as the Fermi energy sodium or the Fermi energy copper. Here we want the example Fermi energy copper to calculate. The free electrons in copper have at a density of , the Mass of electrons is given by and the reduced Planck quantum of action With . So you get the Fermi energy copper
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