Friedel oscillations
Encyclopedia
Friedel Oscillations arise from localized perturbations in a metallic or semiconductor system caused by a defect in the Fermi gas
or Fermi Liquid
. Friedel Oscillations are a quantum mechanical analog to electric charge screening
of charged species in a pool of ions. Whereas electrical charge screening utilizes a point entity treatment to describe the make-up of the ion pool, Friedel Oscillations describing fermions in a Fermi fluid or Fermi gas require a quasi-particle or a scattering treatment. Such oscillations depict a characteristic exponential decay in the fermionic density near the perturbation followed by an ongoing sinusoidal decay resembling sin(x)/x.
or semiconductor
behave like a free electrons
of a Fermi gas
with wave function
Electrons in a metal behave differently than particles in a normal gas because electrons are Fermions and they obey Fermi–Dirac statistics. This means that every k-state in the gas can only be occupied by two electrons with opposite spin. The occupied states fill a sphere in the band structure
k-space up to a fixed energy level, the so called Fermi level
EF. The radius of the sphere in k-space, kF, is called the Fermi wave vector.
If there is a strange atom embedded in the metal or semiconductor, a so called impurity
, the electrons that move freely through the solid are scattered by the deviating potential of the impurity. During the scattering process the initial state wave vector ki of electron wave function is scattered to a final state wave vector kf. Because the electron gas is a Fermi gas
only electrons with energies near the Fermi level can participate in the scattering process because there must be empty final states for the scattered states to jump to. Electrons that are too far below the Fermi level EF can't jump to unoccupied states. The states around the Fermi level that can be scattered occupy a limited range of k-values or wavelengths. So only electrons within a limited wavelength range near the Fermi level are scattered resulting in a density modulation around the impurity like
where kF is the Fermi wave vector.
The quantum mechanical description of a perturbation in a one-dimensional fermionic fluid is modelled by the Tomanaga-Luttinger liquid
. The fermions in the fluid that take part in the screening cannot be considered as a point entity but a wave-vector is required to describe them. Charge density away from the perturbation is not a continuum but fermions arrange themselves at discrete spaces away from the perturbation. This is the cause of the circular ripples around the impurity.
N.B. Where classically near the charged perturbation an overwhelming number of oppositely charged particles can be observed, in the quantum mechanical scenario of Friedel Oscillations periodic arrangements of oppositely charged fermions followed by spaces with same charged regions.
In the figure to the right, a 2-dimensional Friedel Oscillations has been illustrated with an STM
image of a clean surface. As the image is taken on a surface, the regions of low electron density leave the atomic nuclei ‘exposed’ which result in a net positive charge.
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
or Fermi Liquid
Fermi liquid
Fermi liquid theory is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interaction between the particles of the many-body system does not need to be small...
. Friedel Oscillations are a quantum mechanical analog to electric charge screening
Electric field screening
Screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases and conduction electrons in semiconductors and metals....
of charged species in a pool of ions. Whereas electrical charge screening utilizes a point entity treatment to describe the make-up of the ion pool, Friedel Oscillations describing fermions in a Fermi fluid or Fermi gas require a quasi-particle or a scattering treatment. Such oscillations depict a characteristic exponential decay in the fermionic density near the perturbation followed by an ongoing sinusoidal decay resembling sin(x)/x.
Scattering Description
The electrons that move through a metalMetal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
or semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
behave like a free electrons
Free electron model
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 solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it...
of a Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
with wave function
Electrons in a metal behave differently than particles in a normal gas because electrons are Fermions and they obey Fermi–Dirac statistics. This means that every k-state in the gas can only be occupied by two electrons with opposite spin. The occupied states fill a sphere in the band structure
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
k-space up to a fixed energy level, the so called Fermi level
Fermi level
The Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...
EF. The radius of the sphere in k-space, kF, is called the Fermi wave vector.
If there is a strange atom embedded in the metal or semiconductor, a so called impurity
Impurity
Impurities are substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound.Impurities are either naturally occurring or added during synthesis of a chemical or commercial product...
, the electrons that move freely through the solid are scattered by the deviating potential of the impurity. During the scattering process the initial state wave vector ki of electron wave function is scattered to a final state wave vector kf. Because the electron gas is a Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
only electrons with energies near the Fermi level can participate in the scattering process because there must be empty final states for the scattered states to jump to. Electrons that are too far below the Fermi level EF can't jump to unoccupied states. The states around the Fermi level that can be scattered occupy a limited range of k-values or wavelengths. So only electrons within a limited wavelength range near the Fermi level are scattered resulting in a density modulation around the impurity like
where kF is the Fermi wave vector.
“Hand-Waving” Qualitative Description
In the classic scenario of electric charge screening, a dampening in the electric field is observed in a mobile charge carrying fluid upon the presence of a charged object. Since electric charge screening considers the mobile charges in the fluid as point entities, the concentration of these charges with respect to distance away from the point decreases exponentially. This phenomenon is governed by Poisson–Boltzmann equation.The quantum mechanical description of a perturbation in a one-dimensional fermionic fluid is modelled by the Tomanaga-Luttinger liquid
Luttinger liquid
A Tomonaga-Luttinger liquid, more often referred to as simply a Luttinger liquid, is a theoretical model describing interacting electrons in a one-dimensional conductor...
. The fermions in the fluid that take part in the screening cannot be considered as a point entity but a wave-vector is required to describe them. Charge density away from the perturbation is not a continuum but fermions arrange themselves at discrete spaces away from the perturbation. This is the cause of the circular ripples around the impurity.
N.B. Where classically near the charged perturbation an overwhelming number of oppositely charged particles can be observed, in the quantum mechanical scenario of Friedel Oscillations periodic arrangements of oppositely charged fermions followed by spaces with same charged regions.
In the figure to the right, a 2-dimensional Friedel Oscillations has been illustrated with an STM
Scanning tunneling microscope
A scanning tunneling microscope is an instrument for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer , the Nobel Prize in Physics in 1986. For an STM, good resolution is considered to be 0.1 nm lateral resolution and...
image of a clean surface. As the image is taken on a surface, the regions of low electron density leave the atomic nuclei ‘exposed’ which result in a net positive charge.
See also
- http://gravityandlevity.wordpress.com/2009/06/02/friedel-oscillations-wherein-we-learn-that-the-electron-has-a-size/ - a simple explanation of the phenomenon