In stochastic processes, Anderson localization, also known as strong localization, is the absence of diffusion of waves in a random medium. This phenomenon is named after the American physicist P. W. Anderson, who is the first one to suggest the possibility of electron localization inside a semiconductor, provided that the degree of randomness of the impurities or defects is sufficiently large. Anderson localization is a general wave phenomenon that applies to the transport of electromagnetic wave, acoustic wave, quantum wave and spin wave, etc. This phenomenon is to be distinguished from weak localization, which is the precursor effect of Anderson localization. This phenomenon finds its origin in the wave interference between multiple-scattering paths. In the strong scattering limit, the severe interferences can completely halt the waves inside the random medium.
Localized states have been predicted but never observed to easily exist inside bandgaps upon structural disorders in periodic structures.
For non-interacting electrons, a highly successful approach was put forward in 1979 by Abrahams et al. This scaling hypothesis of localization suggests that a metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field B and in the absence of spin-orbit coupling. Much further work has subsequently supported these scaling arguments both analytically and numerically. In 1D and 2D, the same hypothesis shows that there are no extended states and thus no MIT. However, since 2 is the lower critical dimension of the localization problem, the 2D case is in a sense close to 3D: states are only marginally localized for weak disorder and a small magnetic field or spin-orbit coupling can lead to the existence of extended states and thus an MIT. Consequently, the localization lengths of a 2D system with potential disorder can be quite large so that in numerical approaches one can always find a localization-delocalization transition when decreasing either system size for fixed disorder or disorder for fixed system size.
Most numerical approaches to the localization problem use the standard tight-binding Anderson Hamiltonian with onsite potential disorder. Characteristics of the electronic eigenstates are then investigated by studies of participation numbers obtained by exact diagonalization, multifractal properties, level statistics and many others. Especially fruitful is the transfer-matrix method (TMM) which allows a direct computation of the localization lengths and further validates the scaling hypothesis by a numerical proof of the existence of a one-parameter scaling function. Direct numerical solution of Maxwell equations to demonstrate Anderson localization of light has been implemented (Conti and Fratalocchi, 2008).
Two reports of Anderson localization of light in 3D random media exist up to date (Wiersma et al., 1997 and Storzer et al., 2006), even though absorption complicates interpretation of experimental results (Scheffold et al., 1999). Anderson localization can also be observed in a perturbed periodic potential where the transverse localization of light is caused by random fluctuations on a photonic lattice. Experimental realizations of transverse localization were reported for a 2D lattice (Schwartz et al., 2007) and a 1D lattice (Lahini et al., 2008). It has also been observed by localization of a Bose-Einstein condensate in a 1D disordered optical potential (Billy et al., 2008; Roati et al., 2008). Anderson localization of elastic waves in a 3D disordered medium has been reported (Hu et al., 2008). The observation of the MIT has been reported in a 3D model with atomic matter waves (Chabé et al., 2008).
Further reading
- Anderson, P. W. (1958). "Absence of Diffusion in Certain Random Lattices". Phys. Rev. 109: 1492–1505. doi:.
- Wiersma, Diederik S.; et al. (1997). "Localization of light in a disordered medium". Nature 390 (6661): 671–673. doi:.
- Störzer, Martin; et al. (2006). "Observation of the critical regime near Anderson localization of light". Phys. Rev. Lett. 96: 063904. doi:.
- Scheffold, Frank; et al. (1999). "Localization or classical diffusion of light?". Nature 398: 206–207. doi:.
- Schwartz, T.; et al. (2007). "Transport and Anderson Localization in disordered two-dimensional Photonic Lattices". Nature 446: 52–55. doi:.
- Lahini, Y.; et al. (2008). "Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices". Phys. Rev. Lett. 100: 013906. doi:.
- Billy, Juliette; et al. (2008). "Direct observation of Anderson localization of matter waves in a controlled disorder". Nature 453 (7197): 891–894. doi:.
- Roati, Giacomo; et al. (2008). "Anderson localization of a non-interacting Bose-Einstein condensate". Nature 453 (7197): 895–898. doi:.
- Ludlam, J. J.; et al. (2005). "Universal features of localized eigenstates in disordered systems". Journal of Physics: Condensed Matter 17: L321–L327. doi:.
- Conti, C; A. Fratalocchi (2008). "Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals". Nature Physics 4: 794–798. doi:.
- Hu, Hefei; et al. (2008). "Localization of ultrasound in a three-dimensional elastic network". Nature Physics 4: 945. doi:.
- Chabé, J.; et al. (2008). "Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves". Phys. Rev. Lett. 101: 255702. doi:.
External links
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