Localization (physics)

In the area of condensed matter , localization means that the amplitude of a wave is not homogeneous in the entire space, but is concentrated in one position. At this position the amplitude has a maximum , with increasing distance from this position it decreases exponentially . This behavior is found in disordered systems such as amorphous materials .

Explanation

The cause of localization is the constructive interference of waves that are scattered multiple times in disordered systems . In an ordered system, the scattering centers are arranged periodically . Therefore the wave can be described as a Bloch function , i. H. as a plane wave with a periodically varying amplitude. A plane wave is spatially homogeneous and has no preferred locations. In a disordered system, however, such a Bloch approach is not useful due to the lack of periodicity. On the contrary: if one assumes that a wave propagates from a certain point A in space, then it follows that it is scattered in any direction at the non-periodically arranged scattering centers. The repeated scattering at other scattering centers can now lead to the formation of closed paths, so that the scattered wave arrives at its starting point A again . Waves can run through this closed path in the opposite direction. In doing so, they each experience the same change in their phase , which means that they can constructively interfere with one another at point A. This constructive interference leads to an increased likelihood that waves will return to point A instead of propagating away from it. Increasing the disorder leads to a higher density of the scattering centers, so that the probability of the waves propagating on such closed paths becomes greater and greater, as a result of which the conductivity decreases. From a certain degree of disorder , the waves can only move on such paths, i. That is, all states are localized and the conductivity at very low temperatures is zero, in contrast to the usual metallic behavior. Since in this case the amplitude of the waves is concentrated around a certain point in space, one speaks of localization. As mentioned above, the result is that the waves do not have a spatially extended character, but rather have a maximum amplitude at one position. As the distance from this position increases, the amplitude continues to decrease. ${\ displaystyle W_ {C}}$

This complete localization only from a certain degree of disorder is only found in three-dimensional disordered systems, where the phenomenon is called Anderson localization , while in one and two-dimensional systems all states are always localized, even with very weak disorder. Furthermore, the critical disorder strength depends on the frequency from. This means that waves with different frequencies or different energy values have different critical disorders from which they are localized. This effect leads to the formation of so-called mobility edges . The states outside of these characteristic energies of the more or less broad so-called " energy bands " responsible for electrical conductivity are localized, while the states inside are "extended". With increasing disorder, this area becomes narrower and narrower until it finally disappears completely. ${\ displaystyle W_ {C}}$ ${\ displaystyle W_ {C}}$ ${\ displaystyle W_ {C}}$

Localization due to disorder can only take place if the waves are scattered within their coherence length ( also referred to as free path ). With the wavelength of the electrons, results (with weak scattering it would be against ). The specified identity defines the so-called Ioffe rule criterion, which states that in a strongly scattering medium a wave must completely carry out at least one oscillation during scattering . (Due to the use of the size and the (phenomenological!) Image of the fixed electron wavelength instead of a superposition of different waves, the specified condition is only of limited value in a mathematically sound theory of the phenomena discussed.) ${\ displaystyle l}$ ${\ displaystyle l}$ ${\ displaystyle \ lambda \ ,,}$ ${\ displaystyle 2 \ pi \, l _ {\, {\ text {Ioffe rule}}} \, \ equiv \, \ lambda \,}$ ${\ displaystyle l \ gg \ lambda \,}$ ${\ displaystyle l}$

In an in-depth presentation (see below), the phenomena are treated analogously to the theory of phase transitions and critical phenomena with so-called scale concepts.

Mathematics (point spectrum vs. continuous spectrum)

The electrons form in the named systems - both with plane waves and with the localized eigenfunctions - a dense system of states with a spectrum of so-called " unreal " (i.e. not square integrable ) or "actual" (i.e. square integrable) eigenfunctions, the is continuous in normal behavior (so-called continuous spectrum, improper eigenfunctions). From the improper eigenfunctions, since they cannot be square-integrated, one must, as usual, form square-integrable wave packets through superposition . In the localization case, however, it is a point spectrum , i.e. H. the eigenfunctions are square-integrable from the outset.

Scale concepts (localization length)

The problem of the electrical conductivity of the system is closely related to the concept of localization of the wave functions :

It depends very much on whether a first characteristic energy of the charge carriers, the so-called Fermi energy, lies in the range of the conductive (i.e. the continuous or delocalized ) states of the system or in the range of the localized states. Usually, i.e. H. if the disorder is weak, the former is the case; then the free path l is very large compared to the wavelength of the charge carriers. Furthermore, l is inversely proportional to the charge carrier concentration, i.e. it decreases with increasing disorder. I.e. a second characteristic energy of the system, the so-called. localization edge thus is weak disorder, for example below of the case of very strong disorder, however, above . In the latter case, the states are “localized”; d. H. the wave function decreases with increasing distance r from the localization center, for example, according to the law , exponentially on the scale of a so-called localization length.This diverges when the Fermi energy and localization edge approach, with a so-called critical exponent, i.e. according to the law of the exact value von is less interesting compared to the fact that it is universal , i.e. always has the same value for completely different three-dimensional systems. ${\ displaystyle E_ {F} \ ,,}$ ${\ displaystyle \ lambda}$ ${\ displaystyle E_ {c} \ ,,}$ ${\ displaystyle E_ {F}}$ ${\ displaystyle \ psi}$ ${\ displaystyle | \ psi | \, \ sim \ exp (-r / \ xi) \ ,,}$ ${\ displaystyle \ xi.}$ ${\ displaystyle \ nu \ ,,}$ ${\ displaystyle \ xi \ sim | E_ {F} -E_ {c} | ^ {- \ nu}.}$ ${\ displaystyle \ nu}$

In addition, the term localization length itself is of course very clear.

In the conductive area, there is also a characteristic length that is less clearly defined, but is also referred to as and for which the same critical exponent also applies. Now the conductivity is defined, namely by the approach whereby the pre-factor can be expressed by natural constants. Again, however, is no longer applies (localized states), but applies ("extended" states). ${\ displaystyle \ xi}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ xi}$ ${\ displaystyle \ sigma}$ ${\ displaystyle \ sigma \ sim 1 / \ xi \ ,,}$ ${\ displaystyle \ xi \ sim | E_ {F} -E_ {c} | ^ {- \ nu} \ ,,}$ ${\ displaystyle E_ {F} \, <\, E_ {c} \,}$ ${\ displaystyle E_ {F} \,> \, E_ {c} \,}$

Electron localization

The idea of localizing electrons in disordered semiconductors was first discussed by the American physicist Philip Warren Anderson . He found that electrons cannot move freely in such systems, which makes a conductive material an insulator .

Anderson vs. Mott's metal-insulator transition

The so-called Mott's metal-insulator transition (named after Sir Nevill Mott ) must be distinguished from the Anderson metal-insulator transition , a single-particle effect , which is induced by “disorder” . This is a many-particle effect ; In Mott's case, the charge carriers are localized without any disorder, solely through a particularly strong mutual repulsion of the charge carriers. Scaling effects do not play a role here, since the phase transition is now discontinuous.

Light localization

The physicist Sajeev John discussed the idea of localizing light in special structures in an important paper . Similar to electrons in semiconductors, it can be achieved in such systems that light can no longer propagate, but rather is bound to individual positions. This was proven experimentally by a group around DS Wiersma.

literature

David Thouless , Introduction to localization . In: Physics Reports . Volume 67, 1980, Les Houches Lectures

References and footnotes

↑ There is also the case of destructive interference, e.g. B. with dominant so-called. Spin-orbit interaction instead of potential scattering. In this case the conductivity would increase with increasing disorder .
↑ The frequency ν and the energy E of a quantum mechanical system are known to be proportional to each other: E = hν , with Planck's constant h .
^ AF Ioffe and AR rule: Non-crystalline, amorphous, and liquid electronic semiconductors . In: Prog. Semiconduct . Volume 4, 1960, p. 237
↑ In the case of weak disorder, the conductivity is proportional to the reciprocal value of the free path, but in the case of severe disorder, the usual formulas cannot be used and must be calculated using the scale concepts given below. ${\ displaystyle \ sigma}$ ${\ displaystyle \ sigma}$
^ W. Gebhardt and U. Krey, Phase transitions and critical phenomena - an introduction , Vieweg, Braunschweig 1980, ISBN 3-528-08422-7
↑ PW Anderson: Absence of diffusion in certain random lattices . In: Phys. Rev . Volume 109, 1958, p. 1492
^ S. John: Strong localization of photons in certain disordered dielectric superlattices . In: Phys. Rev. Lett . Volume 58, 1987, p. 2486
^ DS Wiersma, P. Bartolini, A. Lagendijk and R. Righini: Localization of light in a disordered medium . In: Nature . Volume 390, 1997, p. 671

[1] There is also the case of destructive interference, e.g. B. with dominant so-called. Spin-orbit interaction instead of potential scattering. In this case the conductivity would increase with increasing disorder .

[2] The frequency ν and the energy E of a quantum mechanical system are known to be proportional to each other: E = hν , with Planck's constant h .

[3] AF Ioffe and AR rule: Non-crystalline, amorphous, and liquid electronic semiconductors . In: Prog. Semiconduct . Volume 4, 1960, p. 237

[4] In the case of weak disorder, the conductivity is proportional to the reciprocal value of the free path, but in the case of severe disorder, the usual formulas cannot be used and must be calculated using the scale concepts given below. ${\ displaystyle \ sigma}$ ${\ displaystyle \ sigma}$

[5] W. Gebhardt and U. Krey, Phase transitions and critical phenomena - an introduction , Vieweg, Braunschweig 1980, ISBN 3-528-08422-7

[6] PW Anderson: Absence of diffusion in certain random lattices . In: Phys. Rev . Volume 109, 1958, p. 1492

[7] S. John: Strong localization of photons in certain disordered dielectric superlattices . In: Phys. Rev. Lett . Volume 58, 1987, p. 2486

[8] DS Wiersma, P. Bartolini, A. Lagendijk and R. Righini: Localization of light in a disordered medium . In: Nature . Volume 390, 1997, p. 671