Anderson localization

In physics, Anderson localization or strong localization refers to the suppression of diffusion in disordered environments if the degree of disorder (concentration of the impurities ) exceeds a certain threshold. The effect is named after Philip Warren Anderson , who in 1958 proposed a simple model for describing such transport processes in the paper Absence of Diffusion in Certain Random Lattices and who predicted the effect.

The Hamilton operator for the Anderson model is:

{\ displaystyle H = \ sum \ limits _ {n, m} t_ {nm} {\ big (} | n \ rangle \ langle m | + | m \ rangle \ langle n | {\ big)} + W \ sum \ limits _ {n} v_ {n} | n \ rangle \ langle n |}

With

${\ displaystyle | n \ rangle}$ the condition at the grid location (see Wannier basis ); the sums run over all grid positions of the -dimensional hypercubic grid ${\ displaystyle n}$ ${\ displaystyle d}$

the hopping matrix element for the hopping process between the grid locations and (and vice versa)

{\ displaystyle t_ {nm}}

{\ displaystyle n}

{\ displaystyle m}

the potential strength

{\ displaystyle W}

the quantity as a random arrangement of the on-site energies. ${\ displaystyle v_ {n} \ in \ left [- {\ tfrac {1} {2}}, {\ tfrac {1} {2}} \ right]}$

In simplified terms, only hopping processes between nearest neighbors are often considered, which then all have the same hopping matrix element; then one recognizes a tight-binding model , i. H. the particle (here no interaction effects , therefore single-particle image ) receives kinetic energy through hopping processes, but has to pay a potential energy dependent on the lattice position (hence on-site energy). In this model, the electron can be localized for two reasons : When the potential becomes very strong and when it is sufficiently disordered.

The Anderson localization describes only single-particle systems or many-particle systems without interaction among the particles. Many-body systems with interacting particles can also develop a localized phase. This process is called many-body localization.

impact

As a result of the Anderson localization, the electrical conductivity and all other variables associated with diffusivity disappear at absolute temperature zero when the threshold mentioned is exceeded ; one therefore speaks of an (Anderson) metal-insulator transition (there is also the Mott metal-insulator transition; this is not caused by disorder, but by electrostatic correlation effects).

In the quantum mechanical localization theory, a particle is considered in a microscopically disordered environment ( random potential ), while in the analogous classical problem, the percolation problem , a macroscopically inhomogeneous system is present. In both cases a phase transition occurs, which is characterized by the existence of a critical energy . ${\ displaystyle E _ {\ mathrm {c}}}$

When dealing with Anderson transitions, the one-electron wave functions are special

for "extended" (i.e. not square integrable , but conductive) ${\ displaystyle E> E _ {\ mathrm {c}}}$

for they drop exponentially (that is, they are “localized”, that is, they can be square-integrated and are non-conductive).

{\ displaystyle E <E _ {\ mathrm {c}}}

Therefore, the electronic transport in a disordered system depends essentially on the position of the Fermi edge relative to : ${\ displaystyle T = 0}$ ${\ displaystyle E _ {\ mathrm {F}}}$ ${\ displaystyle E _ {\ rm {c}}}$

there is a ladder for ${\ displaystyle E _ {\ mathrm {F}}> E _ {\ mathrm {c}}}$
for on the other hand an insulator. ${\ displaystyle E _ {\ mathrm {F}} <E _ {\ mathrm {c}}}$

literature

PW Anderson: Absence of Diffusion in Certain Random Lattices . In: Physical Review . tape 109 , no. 5 , March 1, 1958, p. 1492-1505 , doi : 10.1103 / PhysRev.109.1492 .
Diederik S. Wiersma, Paolo Bartolini, Ad Lagendijk, Roberto Righini: Localization of light in a disordered medium . In: Nature . tape 390 , no. 6661 , December 18, 1997, p. 671-673 , doi : 10.1038 / 37757 .

Individual evidence

^ ^A ^b André Wobst: Phase-space signatures of the Anderson transition . In: Physical Review B . tape 68 , no. 8 , January 1, 2003, doi : 10.1103 / PhysRevB.68.085103 ( aps.org [accessed July 29, 2016]).

[:0-1] A ^b André Wobst: Phase-space signatures of the Anderson transition . In: Physical Review B . tape 68 , no. 8 , January 1, 2003, doi : 10.1103 / PhysRevB.68.085103 ( aps.org [accessed July 29, 2016]).