Weak localization is a physical effect that occurs in disordered electronic systems at low temperatures. The effect manifests itself as a positive additive to the resistivity of a metal or semiconductor .
The effect has a quantum-mechanical nature and is associated with the interference of electron waves. In a random electronic system, the electron motion is not ballistic (that is, without scattering), but diffusive, when the electron experiences many collisions with impurities and other defects of the crystal lattice .
The resistivity of the system is related to the probability of an electron passing from one point to another. Classical physics suggests that total probability is the sum of the probabilities of all paths connecting two points. However, quantum mechanics tells us that in order to find the full probability, we must add up the quantum-mechanical amplitudes of the paths, and not the probabilities themselves. Therefore, the correct (quantum-mechanical) probability formula for an electron to move from point A to point B includes the classical part (individual probabilities of propagating paths) and the set of interference terms (product of amplitudes corresponding to different paths). The usual formula for the conductivity of a metal (the so-called Drude formula ) corresponds to the first of the mentioned methods of adding probabilities, while weak localization corresponds to the last quantum terms averaged over realizations of the disorder or, in other words, over the positions of scattering centers.
It can be shown that the main contribution to weak localization is made by self-intersecting trajectories in which the electron can propagate clockwise and counterclockwise around the loop. Due to the identical length of these two paths, the phases of the wave functions change by the same amount, which leads to their reduction in the interference terms, therefore, the interference term becomes a real value and does not disappear when averaged over the realization of the disorder. Since it is much more likely to find self-intersecting trajectories in low-dimensional systems, the effect of weak localization is manifested much stronger in two-dimensional systems and quantum wiring. Weak localization is characterized by a phase failure length parameter , that is, the length at which the wave function loses information about the initial phase, for example, due to inelastic scattering.
In the case of a quasi-one-dimensional system, the change in conductivity due to the weak localization effect can be written in the form [1]
- {\ displaystyle \ Delta \ sigma = -s {\ frac {e ^ {2}} {h}} {\ frac {1} {S}} (L _ {\ phi} -l_ {e}),}
where S is the cross-sectional area of the wire, s = 2 is the spin degeneracy, e is the elementary charge, h is the constant bar, - the mean free path of the electron. It is clear that the latter cannot be less than the phase coherence length since the particle trajectory can be closed only after scattering. For two-dimensional systems
And for three-dimensional systems, the effect of weak localization is the smallest
Notes
- ↑ Eric Akkermans, Gilles Montambaux. Mesoscopic Physics of Electrons and Photons. - 1 ed .. - Cambridge, UK: Cambridge University Press, 2007. - 606 p. - ISBN 978-0521855129 .