Fermi energy

The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature This article requires a basic knowledge of quantum mechanics.

1 Introduction
- 1.1 Context
- 1.2 Advanced context
2 Illustration of the concept for a one dimensional square well
3 The three-dimensional case
4 Typical fermi energies
- 4.1 White dwarfs
- 4.2 Nucleus
5 Fermi level
6 Quantum mechanics
7 Pinning of Fermi level
8 Free electron gas
9 See also
10 References
11 External links

Introduction

Context

In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. To determine the lowest energy a system of fermions can have, we first group the states in sets with equal energy and order these sets by increasing energy. Starting with an empty system, we then add particles one at a time, consecutively filling up the unoccupied quantum states with lowest-energy. When all the particles have been put in, the Fermi energy is the energy of the highest occupied state. What this means is that even if we have extracted all possible energy from a metal by cooling it down to near absolute zero temperature (0 kelvins), the electrons in the metal are still moving around, the fastest ones would be moving at a velocity that corresponds to a kinetic energy equal to the Fermi energy. The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic This is the Fermi velocity. The Fermi energy is one of the important concepts of condensed matter physics. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. It is used, for example, to describe metals, insulators, and semiconductors. Insulator may refer to Insulator (genetics Insulator (electrical Thermal insulation Building A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that It is a very important quantity in the physics of superconductors, in the physics of quantum liquids like low temperature helium (both normal ³He and superfluid ⁴He), and it is quite important to nuclear physics and to understand the stability of white dwarf stars against gravitational collapse. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical Nuclear physics is the field of Physics that studies the building blocks and interactions of Atomic nuclei. A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity.

Advanced context

The Fermi energy (E_F) of a system of non-interacting fermions is the increase in the ground state energy when exactly one particle is added to the system. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Quantum mechanics, a stationary state is an Eigenstate of a Hamiltonian, or in other words a state of definite energy In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός It can also be interpreted as the maximum energy of an individual fermion in this ground state. The chemical potential at zero temperature is equal to the Fermi energy. In Thermodynamics and Chemistry, chemical potential, symbolized by μ, is a term introduced by the American engineer chemist and mathematical

Illustration of the concept for a one dimensional square well

The one dimensional infinite square well is a model for a one dimensional box. In Physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting of a single particle inside It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labeled by a single quantum number n and the energies are given by

$E_n = \frac{\hbar^2 \pi^2}{2 m L^2} n^2 \,$ .

Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1/2. In Quantum mechanics, spin is an intrinsic property of all elementary particles. Then only two particles can have the same energy i. e. two particles can have the energy of $E_1=\frac{\hbar^2 \pi^2}{2 m L^2}$ , or two particles can have energy $E 2 = 4 E 1$ and so forth. The reason that two particles can have the same energy is that a spin-1/2 particle can have a spin of 1/2 (spin up) or a spin of -1/2 (spin down), leading to two states for each energy level. When we look at the total energy of this system, the configuration for which the total energy is lowest (the ground state), is the configuration where all the energy levels up to n=N/2 are occupied and all the higher levels are empty. The Fermi energy is therefore

$E_f=E_{N/2}=\frac{\hbar^2 \pi^2}{2 m L^2} (N/2)^2 \,$ .

The three-dimensional case

The three-dimensional isotropic case is known as the fermi sphere. Isotropy is uniformity in all directions Precise definitions depend on the subject area

Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). In Physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a problem consisting of a single particle inside This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers n_x, n_y, and n_z. The single particle energies are

$E_{n_x,n_y,n_z} = \frac{\hbar^2 \pi^2}{2m L^2} \left( n_x^2 + n_y^2 + n_z^2\right) \,$

n_x, n_y, n_z are positive integers.

There are multiple states with the same energy, for example $E 100 = E 010 = E 001$ . Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case for N is large.

If we introduce a vector $\vec{n}=\{n_x,n_y,n_z\}$ then each quantum state corresponds to a point in 'n-space' with Energy

$E_{\vec{n}} = \frac{\hbar^2 \pi^2}{2m L^2} |\vec{n}|^2 \,$

The number of states with energy less than E_f is equal to the number of states that lie within a sphere of radius $|\vec{n}_f|$ in the region of n-space where n_x, n_y, n_z are positive. In the ground state this number equals the number of fermions in the system.

$N =2\times\frac{1}{8}\times\frac{4}{3} \pi n_f^3 \,$

The free fermions that occupy the lowest energy states form a sphere in momentum space. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The surface of this sphere is the Fermi surface. In Condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal electrical magnetic and optical properties of Metals

the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We find

$n_f=\left(\frac{3 N}{\pi}\right)^{1/3}$

so the Fermi energy is given by

$E_f =\frac{\hbar^2 \pi^2}{2m L^2} n_f^2$

$= \frac{\hbar^2 \pi^2}{2m L^2} \left( \frac{3 N}{\pi} \right)^{2/3}$

Which results in a relationship between the fermi energy and the number of particles per volume (when we replace L² with V^2/3):

$E_f = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3} \,$

The total energy of a fermi sphere of $N 0$ fermions is given by

$E = {\int_0}^{N_0} E_f(N) dN = {3\over 5} N_0 E_f$

Typical fermi energies

White dwarfs

Stars known as White dwarfs have mass comparable to our Sun, but have a radius about 100 times smaller. A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. The Sun (Sol is the Star at the center of the Solar System. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. Degenerate matter is matter which has sufficiently high Density that the dominant contribution to its Pressure rises from the Pauli Exclusion In Solid-state physics, the free electron model is a simple model for the behaviour of Valence electrons in a Crystal structure of a Metallic The number density of electrons in a White dwarf are on the order of 10³⁶ electrons/m³. This means their fermi energy is:

$E_f = \frac{\hbar^2}{2m_e} \left( \frac{3 \pi^2 (10^{36})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 3 \times 10^5 \ \mathrm{eV} \,$

Nucleus

Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:

$R = \left(1.25 \times 10^{-15} \mathrm{m} \right) \times A^{1/3}$

where A is the number of nucleons. The size of an atomic nucleus is of the order of 10^{-15} metres In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton.

The number density of nucleons in a nucleus is therefore:

$n = \frac{A}{\begin{matrix} \frac{4}{3} \end{matrix} \pi R^3 } \approx 1.2 \times 10^{44} \ \mathrm{m}^{-3}$

Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus, and vice versa. This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive

So the fermi energy of a nucleus is about:

$E_f = \frac{\hbar^2}{2m_p} \left( \frac{3 \pi^2 (6 \times 10^{43})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 30 \times 10^6 \ \mathrm{eV} = 30 \ \mathrm{MeV}$

The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the fermi energy usually given is 38 MeV. The size of an atomic nucleus is of the order of 10^{-15} metres

Fermi level

The Fermi level is the highest occupied energy level at absolute zero, that is, all energy levels up to the Fermi level are occupied by electrons. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 [1] In this state (at 0 K), the average energy of an electron is given by:

$E_{av} = \frac{3}{5} E_f$

where $E f$ is the Fermi energy. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic

The Fermi momentum is the momentum of fermions at the Fermi surface. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal electrical magnetic and optical properties of Metals The Fermi momentum is given by:

$p_F = \sqrt{2 m_e E_f}$

where $m e$ is the mass of the electron.

This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In more general cases, one must consider the Fermi energy.

The Fermi velocity is the velocity of fermions at the Fermi surface. It is defined by:

$V_f = \sqrt{\frac{2 E_f}{m_e}}$

where $m e$ is the mass of the electron.

Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by:

$T_f = \frac{E_f}{k}$

where k is the Boltzmann constant. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics

Quantum mechanics

According to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. In Mathematics, a half-integer is a Number of the form n + 1/2 where n is an Integer. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Consequently, fermions obey Fermi-Dirac statistics. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO); however, within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV.

Pinning of Fermi level

When the energy density of surface states is very high (>10¹²/cm²), the position of the Fermi level is determined by the neutral level of the Surface states [2] and becomes independent of Work Function [3] variations.

Free electron gas

In the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labeled according to their momentum. A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" or "crystal momentum" (see Bloch wave). In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. The Momentum space associated with a particle is a vector space in which every point {k_x k_y k_z} corresponds to a possible value of the Momentum vector \vec{k} In Condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal electrical magnetic and optical properties of Metals For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. Electrical conductivity or specific conductivity is a measure of a material's ability to conduct an Electric current. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.

The Fermi energy of the free electron gas is related to the chemical potential by the equation

$\mu = E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$

where E_F is the Fermi energy, k is the Boltzmann constant and T is temperature. In Thermodynamics and Chemistry, chemical potential, symbolized by μ, is a term introduced by the American engineer chemist and mathematical Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E_F/k. The characteristic temperature is on the order of 10⁵ K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that

References

Kroemer, Herbert; Kittel, Charles (1980). A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " A Quasi Fermi level is a term used in Quantum mechanics and especially in Solid state physics to describe the change in Charge carrier Energy Thermal Physics (2nd ed. ). W. H. Freeman Company. ISBN 0-7167-1088-9.
Table of fermi energies, velocities, and temperatures for various elements.
a discussion of fermi gases and fermi temperatures.

External links

Dictionary

-noun

(physics) The highest energy of a particle that is part of a many-particle system of identical fermions in its ground state.

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