Friedel oscillations,
[
] named after French physicist
Jacques Friedel
Jacques Friedel ForMemRS (; 11 February 1921 – 27 August 2014) was a French physicist and material scientist.
Education
Friedel attended the Cours Hattemer, a private school. He studied at the École Polytechnique from 1944 to 1946, and the É ...
, arise from localized perturbations in a metallic or semiconductor system caused by a defect in the
Fermi gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
or
Fermi liquid
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-bod ...
.
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
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
. In 2020, magnetic Friedel oscillations were observed on a metal surface.
One-dimensional electron gas
As a simple model, consider one-dimensional electron gas in a half-space
. The electrons do not penetrate into the half-space
, so that the boundary condition for the electron wave function is
. The oscillating wave functions that satisfy this condition are
,
where
is the electron wave vector, and
is the length of the one-dimensional box (we use the 'box" normalization here). We consider degenerate electron gas, so that the electrons fill states with energies less than the Fermi energy
. Then, the electron density
is calculated as
,
where summation is taken over all wave vectors less than the
Fermi wave vector , the factor 2 accounts for the spin degeneracy. By transforming the sum over
into the integral we obtain
.
We see that the boundary perturbs the electron density leading to its spatial oscillations with the period
near the boundary. These oscillations decay into the bulk with the decay length also given by
. At
the electron density equals to the unperturbed density of the one-dimensional electron gas
.
Scattering description
The electrons that move through a
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
or
semiconductor
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
behave like
free electrons of a
Fermi gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
with a
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
-like
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
, that is
:
.
Electrons in a metal behave differently than particles in a normal gas because electrons are
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
and they obey
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
. This behaviour means that every k-state in the gas can only be occupied by two electrons with opposite
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
. The occupied states fill a sphere in the
band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
k-space, up to a fixed energy level, the so-called
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
. The radius of the sphere in k-space, ''k''
F, is called the
Fermi wave vector.
If there is a foreign atom embedded in the metal or semiconductor, a so-called
impurity
In chemistry and materials science, impurities are chemical substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound. Firstly, a pure chemical should appear thermodynam ...
, 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 k
i of the electron wave function is scattered to a final state wave vector k
f. Because the electron gas is a
Fermi gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
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 energy ''E''
F 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 energy are scattered resulting in a density modulation around the impurity of the form
:
.
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 Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric ...
. The quantum mechanical description of a perturbation in a one-dimensional Fermi fluid is modelled by the
Tomonaga-Luttinger liquid.
[D. Vieira ''et al''., “Friedel oscillations in one-dimensional metals: From Luttinger’s theorem to the Luttinger liquid”, ''Journal of Magnetism and Magnetic Materials'', vol. 320, pp. 418-420, 2008.]
(arXiv Submission) 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 effect 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
Scanning tunneling microscope, 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.
See also
*
Lindhard theory
In condensed matter physics, Lindhard theoryN. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quant ...
References
External links
* http://gravityandlevity.wordpress.com/2009/06/02/friedel-oscillations-wherein-we-learn-that-the-electron-has-a-size/ - a simple explanation of the phenomenon
{{DEFAULTSORT:Friedel Oscillations
Condensed matter physics