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Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Gauss's Law in Dielectrics

Gauss's Law in Dielectrics

Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...

Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...

Electric Field of Two Equal and Opposite Charges

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Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral. However, the centers of the positive and negative charges do not always coincide. In such a scenario, the electric field of an atom may not be zero.
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Magnetic Vector Potential

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In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic field ensures that the magnetic field can be expressed as the curl of a vector potential. This potential is known as the magnetic vector potential.
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Related Experiment Video

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Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities

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Non-Abelian gauge potentials in graphene bilayers.

P San-Jose¹, J González, F Guinea

¹Instituto de Estructura de la Materia (IEM-CSIC), Serrano 123, 28006 Madrid, Spain.

Physical Review Letters

|September 26, 2012

Summary

Spatial modulations in twisted bilayer graphene create non-Abelian gauge potentials. These potentials lead to charge accumulation and flat zero-energy bands, confining states within moiré patterns at specific periods.

Area of Science:

Condensed Matter Physics
Materials Science
Quantum Mechanics

Background:

Graphene bilayers exhibit unique electronic properties influenced by interlayer coupling.
Spatial modulations, like those from twisting or shearing, significantly alter these properties.
Understanding these modulations is key to designing novel electronic devices.

Purpose of the Study:

To investigate the impact of spatial modulations in interlayer hopping on graphene bilayer physics.
To elucidate the role of non-Abelian gauge potentials in governing the electronic behavior.
To explore the formation and spatial confinement of zero-energy bands.

Main Methods:

Theoretical analysis of single-particle physics in modulated graphene bilayers.

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Derivation of low-energy electronic theory incorporating non-Abelian gauge potentials.

Examination of the relationship between pattern period (L) and electronic band structure.

Main Results:

Spatial modulations induce a non-Abelian gauge potential due to interlayer coupling.
Charge accumulation and recurrent formation of zero-energy bands are observed as pattern period L increases.
Specific values of L lead to spatial confinement of zero-energy states within moiré patterns.

Conclusions:

The electronic physics of modulated graphene bilayers is governed by emergent non-Abelian gauge potentials.
The recurrence of flat zero-energy bands is directly linked to the non-Abelian nature of these potentials.
These findings offer insights into controlling electronic states in moiré materials.