Wigner crystal
Encyclopedia
A Wigner crystal is the solid (crystalline) phase of electron
s first predicted by Eugene Wigner in 1934. A gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a b.c.c. (body-centered cubic) lattice in 3D, a triangular lattice in 2D and a evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed . A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner-crystalization that has led to observation of insulating behaviours in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization
.
There is a single dimensionless parameter characterising the state of a uniform electron gas at zero temperature, the so-called Wigner-Seitz radius rs = a / ab, where a is the average inter-particle spacing and ab is the Bohr radius
. Quantum Monte Carlo
simulations indicate that the uniform electron gas crystallizes at rs = 106 in 3D and roughly rs = 35 in 2D.
For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: G = e2 / (kB Ta). The Wigner transition occurs at G = 170 in 3D and G = 125 in 2D. It is believed that ions, such as those of iron, form a Wigner crystal in the interiors of white dwarf
stars.
More generally, a Wigner crystal phase can also refer to a crystal phase occurring in non-electronic systems at low density. In contrast, most crystals melt as the density is lowered. Examples seen in the laboratory are charged colloids or charged plastic spheres.
In practice, it is difficult to experimentally realize a Wigner crystal because quantum mechanical fluctuations overpower the Coulomb repulsion and quickly cause disorder. Low electron density is needed. One notable example occurs in quantum dot
s with low electron densities or high magnetic fields where electrons will spontaneously localize in some situations, forming a so-called rotating "Wigner molecule", a crystalline-like state adapted to the finite size of the quantum dot.
Another experimental realisation of the Wigner crystal occurs in single electron transistors with very low currents, where a 1D Wigner crystal will form. The current due to each electron can be directly detected experimentally.
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s first predicted by Eugene Wigner in 1934. A gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a b.c.c. (body-centered cubic) lattice in 3D, a triangular lattice in 2D and a evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed . A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner-crystalization that has led to observation of insulating behaviours in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization
Anderson localization
In condensed matter physics, Anderson localization, also known as strong localization, is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W...
.
There is a single dimensionless parameter characterising the state of a uniform electron gas at zero temperature, the so-called Wigner-Seitz radius rs = a / ab, where a is the average inter-particle spacing and ab is the Bohr radius
Bohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...
. Quantum Monte Carlo
Quantum Monte Carlo
Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem. They use, in one way or another, the Monte Carlo method to handle the many-dimensional integrals that arise...
simulations indicate that the uniform electron gas crystallizes at rs = 106 in 3D and roughly rs = 35 in 2D.
For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: G = e2 / (kB Ta). The Wigner transition occurs at G = 170 in 3D and G = 125 in 2D. It is believed that ions, such as those of iron, form a Wigner crystal in the interiors of white dwarf
White dwarf
A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
stars.
More generally, a Wigner crystal phase can also refer to a crystal phase occurring in non-electronic systems at low density. In contrast, most crystals melt as the density is lowered. Examples seen in the laboratory are charged colloids or charged plastic spheres.
In practice, it is difficult to experimentally realize a Wigner crystal because quantum mechanical fluctuations overpower the Coulomb repulsion and quickly cause disorder. Low electron density is needed. One notable example occurs in quantum dot
Quantum dot
A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...
s with low electron densities or high magnetic fields where electrons will spontaneously localize in some situations, forming a so-called rotating "Wigner molecule", a crystalline-like state adapted to the finite size of the quantum dot.
Another experimental realisation of the Wigner crystal occurs in single electron transistors with very low currents, where a 1D Wigner crystal will form. The current due to each electron can be directly detected experimentally.