Weak localization
Encyclopedia
Weak localization is a physical effect which occurs in disordered electronic systems at very low temperatures. The effect manifests itself as a positive correction to the resistivity
of a metal
or semiconductor
.
The effect is quantum-mechanical in nature and has the following origin: In a disordered electronic system, the electron
motion is diffusive rather than ballistic. That is, an electron does not move along a straight line, but experiences a series of random scatterings off impurities which results in a random walk.
The resistivity
of the system is related to the probability of an electron to propagate between two given points in space. Classical physics assumes that the total probability is just the sum of the probabilities of the paths connecting the two points. However quantum mechanics
tells us that to find the total probability we have to sum up the quantum-mechanical amplitudes of the paths rather than the probabilities themselves. Therefore, the correct (quantum-mechanical) formula for the probability for an electron to move from a point A to a point B includes the classical part (individual probabilities of diffusive paths) and a number of interference terms (products of the amplitudes corresponding to different paths). These interference terms effectively make it more likely that a carrier will "wander around in a circle" than it would otherwise, which leads to an increase in the net resistivity. The usual formula for the conductivity of a metal (the so-called Drude formula) corresponds to the former classical terms, while the weak localization correction corresponds to the latter quantum interference terms averaged over disorder realizations.
The weak localization correction can be shown to come mostly from quantum interference between self-crossing paths in which an electron can propagate in the clock-wise and counter-clockwise direction around a loop. Due to the identical length of the two paths along a loop, the quantum phases cancel each other exactly and these (otherwise random in sign) quantum interference terms survive disorder averaging. Since it is much more likely to find a self-crossing trajectory in low dimensions, the weak localization effect manifests itself much stronger in low-dimensional systems (films and wires).
In this equation
is -1 for weak localization and +1/2 for weak anti-localization. 
is roughly the magnetic field required to disrupt localization effects, and is equal to 
where 
is the coherence length.
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
of a metal
Metal
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...
.
The effect is quantum-mechanical in nature and has the following origin: In a disordered electronic system, the electron
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...
motion is diffusive rather than ballistic. That is, an electron does not move along a straight line, but experiences a series of random scatterings off impurities which results in a random walk.
The resistivity
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
of the system is related to the probability of an electron to propagate between two given points in space. Classical physics assumes that the total probability is just the sum of the probabilities of the paths connecting the two points. However quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
tells us that to find the total probability we have to sum up the quantum-mechanical amplitudes of the paths rather than the probabilities themselves. Therefore, the correct (quantum-mechanical) formula for the probability for an electron to move from a point A to a point B includes the classical part (individual probabilities of diffusive paths) and a number of interference terms (products of the amplitudes corresponding to different paths). These interference terms effectively make it more likely that a carrier will "wander around in a circle" than it would otherwise, which leads to an increase in the net resistivity. The usual formula for the conductivity of a metal (the so-called Drude formula) corresponds to the former classical terms, while the weak localization correction corresponds to the latter quantum interference terms averaged over disorder realizations.
The weak localization correction can be shown to come mostly from quantum interference between self-crossing paths in which an electron can propagate in the clock-wise and counter-clockwise direction around a loop. Due to the identical length of the two paths along a loop, the quantum phases cancel each other exactly and these (otherwise random in sign) quantum interference terms survive disorder averaging. Since it is much more likely to find a self-crossing trajectory in low dimensions, the weak localization effect manifests itself much stronger in low-dimensional systems (films and wires).
Weak anti-localization
In a system with spin-orbit coupling the spin of a carrier is coupled to its momentum. The spin of the carrier rotates as it goes around a self-intersecting path, and the direction of this rotation is opposite for the two directions about the loop. Because of this, the two paths around the loop interfere destructively which leads to a lower net resistivity.In two dimensions
In two dimensions both weak localization and weak anti-localization can be described by the Hikami-Larkin-Nagaoka equation:In this equation
is -1 for weak localization and +1/2 for weak anti-localization. is roughly the magnetic field required to disrupt localization effects, and is equal to where is the coherence length.