Anderson localization
Encyclopedia
In condensed matter physics
, Anderson localization, also known as strong localization, is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was 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 waves, acoustic waves, quantum waves, spin waves, etc. This phenomenon is to be distinguished from weak localization
, which is the precursor effect of Anderson localization (see below), and from Mott localization
, named after Sir Nevill Mott, where the transition from metallic to isolating behaviour is not due to disorder, but to a strong mutual Coulomb repulsion of electrons.
where the Hamiltonian
H is given by
with Ej random and independent, and interaction V(r) falling of as r-2 at infinity. For example, one may take Ej uniformly distributed in [−W, +W], and
Starting with ψ0 localised at the origin, one is interested in how fast the probability distribution |ψt|2 diffuses. Anderson's analysis shows the following:
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 disorder-induced metal-insulator transition
(MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field and in the absence of spin-orbit coupling. Much further work has subsequently supported these scaling arguments both analytically and numerically (Brandes et al., 2003; see Further Reading). 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 either decreasing system size for fixed disorder or increasing 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)
s can operate using this phenomenon.
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
, Anderson localization, also known as strong localization, is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first one to suggest the possibility of electron localization inside a semiconductor, provided that the degree of randomness of the impurities or defects
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
is sufficiently large.
Anderson localization is a general wave phenomenon that applies to the transport of electromagnetic waves, acoustic waves, quantum waves, spin waves, etc. This phenomenon is to be distinguished from weak localization
Weak localization
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....
, which is the precursor effect of Anderson localization (see below), and from Mott localization
Mott transition
A Mott transition is a metal-nonmetal transition in condensed matter. Due to electric field screening the potential energy becomes much sharper peaked around the equilibrium position of the atom and electrons become localized and can no longer conduct a current.-Conceptual explanation:In a...
, named after Sir Nevill Mott, where the transition from metallic to isolating behaviour is not due to disorder, but to a strong mutual Coulomb repulsion of electrons.
Introduction
In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Zd is given by the Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
where the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
H is given by
with Ej random and independent, and interaction V(r) falling of as r-2 at infinity. For example, one may take Ej uniformly distributed in [−W, +W], and
Starting with ψ0 localised at the origin, one is interested in how fast the probability distribution |ψt|2 diffuses. Anderson's analysis shows the following:
- if d is 1 or 2 and W is arbitrary, or if d ≥ 3 and W/ħ is sufficiently large, then the probability distribution remains localized:
- uniformly in t. This phenomenon is called Anderson localization.
- if d ≥ 3 and W/ħ is small,
- where D is the diffusion constant.
Analysis
The phenomenon of Anderson localization, particularly that of weak localization, 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 disordered medium.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 disorder-induced metal-insulator transition
Metal-insulator transition
Metal-insulator transitions are transitions from a metal to an insulator...
(MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field and in the absence of spin-orbit coupling. Much further work has subsequently supported these scaling arguments both analytically and numerically (Brandes et al., 2003; see Further Reading). 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 either decreasing system size for fixed disorder or increasing disorder for fixed system size.
Most numerical approaches to the localization problem use the standard tight-binding Anderson Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
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
Transfer-matrix method
In physics and mathematics, the transfer-matrix method is a general technique for solving problems in statistical mechanics.The basic idea is to write the partition function in the form...
(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)
Experimental evidence
Two reports of Anderson localization of light in 3D random media exist up to date (Wiersma et al., 1997 and Storzer et al., 2006; see Further Reading), 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., 2006). 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). Random laserRandom laser
A random laser is a laser that uses a highly disordered gain medium. A random laser uses no optical cavity but the remaining principles of operation remain the same as for a conventional laser...
s can operate using this phenomenon.
External links
- Fifty years of Anderson localization Physics Today, August 2009.
- Example of an electronic eigenstate at the MIT in a system with 1367631 atoms Each cube indicates by its size the probability to find the electron at the given position. The color scale denotes the position of the cubes along the axis into the plane
- Anderson localization of elastic waves
- Popular scientific article on the first experimental observation of Anderson localization in matter waves
