Anderson Localization: Localization by...

Anderson Localization:

Localization by Disorder

Tobias Binninger

Seminar: Quantendynamik in mesoskopischen Systemen

Albert-Ludwigs-Universität Freiburg

June 8, 2010

Tobias Binninger (Uni Freiburg) Localization by Disorder June 8, 2010 1 / 29

Outline

1 Delocalization in perfect crystals

2 Disorder induced localization: Intuitive explanationsAmorphous SiRandomly scattered water waves

3 Disorder induced localization: Theoretical conceptsBasic de�nitionsLocalization in 1 dimension: Transfer matrixGreen's Functions

4 Disorder and TransportAnderson transitionHopping TransportWeak Localization

5 Summary


Delocalization in perfect crystals:

Electronic states


Electronic states in perfect crystals

Translation operator:(TR ψ)(r) = ψ(r + R)

Perfect crystal: Translational symmetry of the potential

⇒ [H,TR ] = 0 ∀R ∈ Lattice

Bloch Theorem

There exists a basis of Hamiltonian eigenstates of the form:

ψn,k(r) = e ik·r un,k(r)

un,k(r + R) = un,k(r)

Bloch states are delocalized


Electronic states in perfect crystals

Bloch states are extended

Density of states: D(E ) = dNdE


Disorder induced localization:

Intuitive explanations


Localization in amorphous Si

Tight binding model:

Electronic states of thecrystal: linear combinationof atomic orbitals (LCAO)located at each lattice site

Sharp atomic energy levelsare broadened⇒ energy bands

Band width ∝ overlapintegral of neighboringatomic orbitals

Crystalline Si:interatomic distance r0⇒ sharp band gap


Amorphous Si: interatomic distance statistically distributed aroundaverage value:

band gap is softened

spectral density of states (DOS) forms tails reaching into band gap

value of DOS re�ects the probability of having the correspondingbinding length

small DOS in tail region: seldomly realized binding con�gurations⇒ corresponding states are largely separated and don't overlap⇒ localized states in band tails


Randomly scattered water waves

regularly distributed scatterers: waves are spreading all over the surface

randomly distributed scatterers: waves remain localized in some areas


Disorder induced localization:

Theoretical concepts


Models of disorder

Types of disorder:

Structural disorder:disorder in lattice structure(e.g. glassy and amorphoussystems)

Compositional disorder:di�erent kinds of atomsrandomly distributed onlattice sites(e.g. impurities, dopedsemiconductors, alloys)

Figure: Binary alloy

Note: statistical nature of disorder ⇒ statistical ensembles of systems


Compositional disorder: Anderson model

Tight binding Hamiltonian for electrons:

H =∑jν

Ejν |jν〉〈jν|+∑jν,kµ

Vjν,kµ|jν〉〈kµ|

|jν〉: Atomic orbital with quantum number ν located at lattice site j

diagonal elements Ejν : Energy of state |jν〉 (expectation value)

nondiagonal elements Vjν,kµ:matrix elements responsible for �hopping� between di�erent sites

Anderson model:

statistically distributed site energies Ejν

statistically independent site energies:

P({Ejν}) =∏jν

p(Ejν)


Compositional disorder: Anderson model

Tight binding Hamiltonian:

H =∑jν



Distribution functions for site energies

1. binary alloy made up of two-components AxB1−x :

p(Ejν) = x δ(Ejν − EA) + (1− x) δ(Ejν − EB)

2. Anderson model: continuous distribution of site energies:

p(Ejν) =1

WΘ(

1

2W − |Ejν |)

width W : magnitude of disorder


De�nitions of Localization

1. Asymptotic behaviour of wavefunction: Localization length λ:

ψ(r) = f (r) e−r/λ

2. �Diameter� R of wavefunction:

3. Transport related quantities: Transmission probability from r to r ′:

t(r , r ′;E ) = 〈|〈r |G (E )|r ′〉|2〉

G (E ) = (E − H)−1

Localization length λ: exponential decay length of transmissionprobability:

2

λ= − lim

|r−r ′|→∞

log t(r , r ′;E )

|r − r ′|Tobias Binninger (Uni Freiburg) Localization by Disorder June 8, 2010 14 / 29

Localization in 1 dimension:

Transfer matrix formalism

All eigenstates in a one-dimensional disordered lattice are localized!


Transfer matrix

En an + an+1 + an−1 = E an

Solution for initial values a0, a1 (semi-in�nite lattice):(an+1

an

)= TnTn−1 · · ·T1

(a1a0

)Transfer matrices Tn:

Tn =

(E − En −1

1 0

)En random variables ⇒ Tn random matrices!


Fürstenberg's theorem for products of random matrices:

limn→∞

1

nlog ‖Tn · · ·T1

−→a0‖ = γ > 0

Consider �nite system with N lattice sites:localization length (exponential decay length) of solution with energy E:

λ(E ,N) := − 1

Nlog |a0(E ,N)aN(E ,N)|

for N →∞: λ can be related with γ (no riorous proof!):

0 < γ = λ <∞

All eigenstates in a one-dimensional disordered lattice are localized!


Green's functions:

Disordered systems of arbitrarydimension


Green's functions

Hamiltonian H with discrete eigenvalues {E1,E2, ...} and eigenbasis{|φn〉}n∈NGreen's Operator:

G (z) :=1

z − H=∑n

|φn〉〈φn|z − En

for z ∈ C\{E1,E2, ...}

Green's function:G (r , r ′; z) := 〈r |G (z)|r ′〉

Green's function

contains information about system and its Hamiltonian:

eigenvalues En are poles of Green's function

eigenfunctions can be deduced from the residues at the correspondingpoles


Green's functions and disordered solids

From Green's function for a disordered system we can calculate:

Density of states D(E )(for Hamiltonian with continuous spectrum spec(H) ⊆ C):

D(E ) = − 1

2π iTr[G+(E )− G−(E )]

G±(E ) := limη→0+

G (E ± iη)

Localization length λ:

1

λ= − lim

|r−r ′|→∞

< log |G (r , r ′;E )| >|r − r ′|

DC electrical conductivity σ:

σ =2e2

hlimη→0+

4η2∫

dr r2 < |G+(r ;E )|2 >


Calculating Green's Functions

Consider disorder as perturbation:

H = H0 + H1

G0(z) = (z − H0)−1

G (z) = (z − H)−1 = (z − H0 − H1)−1 = (1− G0(z)H1)−1 G0(z)

= G0 + G0 H1 G0 + G0 H1 G0 H1 G0 + · · ·= G0 + G0 H1 G

= G0 + G H1 G0

Di�erent methods to evaluate the ensemble average of theperturbation series: < G (z) >


Anderson Model: Theoretical Results

Dimensionality of the system d ≤ 2

All eigenstates are localized, no matter how weak the disorder!

Dimensionality of the system d = 3

DOS forms tails of the band consisting of localized states

Interior of the band corresponds to extended states

Critical energies Ec separating localized from extended states:Mobility edges

Note: No rigorous proof of these results!


Disorder and Transport


Anderson transition

Only extended states contribute to transport/conductivity signi�cantly

Position of mobility edge depends on magnitude of disorder

At critical magnitude of disorder:Metal-Insulator Phase Transition


Hopping transport

Current transport by electrons with energy E ≈ EF

for EF < Ec : Current transport by localized electrons:Electrons tunnel/hop from one localized state to the other

Tunnel probability i → j : p(rij ,Ej ,Ei ) ∝ e−αrij e−(Ej−Ei )/kT

Mott's T−14 -law for conductivity:

σ(T ) ∝ e−(T0/T )1/4


Weak Localization

for EF > Ec : Fermi-energy within region of extended states:Metallic regime

disorder leads to positive interference of certain terms in perturbationexpansion of conductivity:enhanced backscattering (weak localization)⇒ reduced conductivity

Magnetic �eld destroys positive interference and weak localization


Summary

Perfect crystal: Delocalized electronic eigenstates

1-D and 2-D disordered systems: All eigenstates are localized!

3-D disordered systems:Energy band forms tails of localized states: Mobility edges

Anderson transition:Metal-Insulator phase transition at critical disorder

Hopping transport by localized electrons:

Mott's T−14 -law for conductivity

σ(T ) ∝ e−(T0/T )1/4

Weak localization:Reduced conductivity in metallic regime due to disorder enhancedbackscattering


Bibliography

Ibach/Lüth: Festkörperphysik, Springer

Ziman: Models of disorder, Cambridge Univ. Press

Economou: Green's Functions in Quantum Physics, Springer

Kramer and MacKinnon: Localization: theory and experiment,Rep. Prog. Phys. 56 (1993) 1469-1564

Ishii: Progr. Theor. Phys. Suppl. 53, 77 (1973)


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