Anderson Localization: Localization by...
Transcript of 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
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Delocalization in perfect crystals:
Electronic states
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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
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Electronic states in perfect crystals
Bloch states are extended
Density of states: D(E ) = dNdE
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Disorder induced localization:
Intuitive explanations
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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
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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
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Randomly scattered water waves
regularly distributed scatterers: waves are spreading all over the surface
randomly distributed scatterers: waves remain localized in some areas
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Disorder induced localization:
Theoretical concepts
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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
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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)
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Compositional disorder: Anderson model
Tight binding Hamiltonian:
H =∑jν
Ejν |jν〉〈jν|+∑jν,kµ
Vjν,kµ|jν〉〈kµ|
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
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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!
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Localization in 1 dimension
Schrödinger equation for 1 dimensional tight-binding model:∑jν
Ejν |jν〉〈jν|+∑jν,kµ
Vjν,kµ|jν〉〈kµ|
|ψ〉 = E |ψ〉
ansatz: |ψ〉 =∑
jν ajν |jν〉
〈nν|H|ψ〉 = E 〈nν|ψ〉
⇒ Enν anν +∑kµ
Vnν,kµ akµ = E anν
consider only one atomic orbital, only next neighbor coupling V = 1
⇒ En an + an+1 + an−1 = E an
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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!
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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!
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Green's functions:
Disordered systems of arbitrarydimension
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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
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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 >
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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) >
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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!
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Disorder and Transport
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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
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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
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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
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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
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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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