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By Andy Zelinski and Steve Pfifer
Introduction Around the 1950s, researchers began studying the
effects of disorder on electronic spectra.
Disorder:
Impurities(Alloys. Initial studies done here)
Defects
Topological (liquid, amorphous material)
P.W. Anderson!
Strong enough disorder can localize all states
Zero conductivity at zero temp, despite non-zero DOS.
Disorder Studies Anderson Localization: Disorder induced metal-insulator
transition. Anderson localized states have an envelope which decays
exponentially at large distances from localization center with localization length ξ.
Standard Model: Hamiltonian of a single particle in a random potential. The random potential simulates the disorder.
A complete understanding is still not reached.
j jxExxxV
dx
xd
m)()()(
)(
202
22
Randomly Distributed
Disordered SystemsCompositional: Lattice points occupied by materials with non-identical potentials
Disordered binary alloys
Translational: An array of identical potentials that are not located in a periodic array
Amorphous
Glassy
Liquid
The Electronic Density of States of a Liquid-Like Random Kronig-Penny Model in the Coherent Potential Approximation. Chin-Yuan Lu and E-Nr Foo, Chinese Journal of Physics, Vol. 16, No. 1, Spring, 1978.
Disorder Studies
Various aspects have been studied over the years…. Here are some of the important contributions.
In most cases, the works are very mathematically involved.
Some different approaches…
DOS for 1-D impurity bands. (Lax, Phillips) Multiple Scattering theory with effective mass
approximation(important simplifications):
Noted that relevant wavelengths in an impurity band ≥ the mean separation between impurities >> lattice constant.
we can neglect the periodic structure of the host altogether, set the host potential = zero and allow impurity atoms to assume random positions on a continuous domain.
These assumptions contributed to successful insight into DOS.
Distribution of Energy Levels (Frisch, Lloyd)
Considered a sequence of atoms positions
Treat each sequence as a single point in an infinite dimensional space Ω. Every possible sequence in Ω carries a probability measure P{ }, and the are “random variables”.
For each sequence, the corresponding sequence of eigenvalues(solutions to (1)) are for
Also for each sequence, let , for -∞< E < ∞,
They proved that exists and is independent of ω!
j jxExxxV
dx
xd
m)()()(
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,...),,(...,101
xxx
)1(
...),(),(21
LELE Lx0
]),(Esatisfy which )',( ofnumber [1
),(m
ELsLEL
EmL
),(lim EL
L
A look at Low Lying Energy Spectrum (Luttinger)
Case 1: V0∞
Wave functions are localized between two potentials( zero probability of finding electron outside of potential)
DOS can be derived by considering probability distribution functions of the cell lengths.
Total length L divided into n cells of random length,
Then and
=CDOS= no. of energy levels below E.
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iL
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)(2i
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s
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...2,1 ,...1 sni
n
i s
iLm
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1 1
2
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)(2)(
DOS, cont
Need to know the probability distribution function of the lengths(all positions equally possible:
DOS for a random system is given by its ensemble average:
-Where aCDOS(E) is just the avg. CDOS per unit cell! Standard deviation is negligible.
)]...(/)!1[(
),.., being lengths cell specific offunction on distributiy Probabilit()...(
211
2121
LLLLLn
LLLLLLP
nn
nn
nn
n
i s
idLdLLLLP
Lm
sEEN ..),...,(
)(2)(
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1 1
2
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)(EaCDOSn
DOS, cont. Finally, the authors proved that this analysis works for
general Vo! (Details beyond scope here)
Case 2. Vo is arbitraty.
If a big cell is in the presence of only small ones, the low lying energy state is localized in the neighborhood of the big cell.
What about structural disorder?
For structural disorder(liquid, amorphous solid), the spatial arrangement of scatterers does not exhibit long range order, but density correlations prevent overlap of atomic potentials. Thus they are said to have short range order.
Short range order modeling approach: Single site approximations-atoms surrounding a given site are represented only in terms of that sites average environment, e.g. depends on the average two-site distribution function.
Modeling Short range order: one might say the probability of finding nearest neighbors separated by a distance, x, is zero for x less than a hard-rod length a and decreases exponentially for x>a.
Liquid Metal Basics From the One electron Hamiltonian
One considers the probability distribution for the nearest neighbors, e.g. p(x)=the probability of finding, at x, the neighbor an atom of a known position.
Tells us the extent to which the average positions of the ions are correlated.
The nearest neighbor distribution functions have been found to be a function of hard rod length.
N
j jxExxxV
dx
xd
m 002
22
)()()()(
2
A detailed look at one model
Random Kronig Penney Model
The Random and Conventional Kronig-Penny Models
Kronig-Penny Model Liquid-Like Random Kronig-Penny Model
The Electronic Density of States of a Liquid-Like Random Kronig-Penny Model in the Coherent Potential Approximation. Chin-Yuan Lu and E-Nr Foo, Chinese Journal of Physics, Vol. 16, No. 1, Spring, 1978.
The Liquid-Like Random Kronig-Penny Model Want to find the electron distribution in a liquid metal.
Metal atoms have same potential.
Potential represented as Dirac delta train of height Vol.
Vol equals a nonzero height Vo when the site is occupied.
Vol is zero when site is unoccupied.
Probability site is occupied is uniformly distributed.
Average Density of States: The Liquid-Like Random Kronig-Penny Model
Ordered KP Model Slightly Disordered KP Model
The Electronic Density of States of a Liquid-Like Random Kronig-Penny Model in the Coherent Potential Approximation. Chin-Yuan Lu and E-Nr Foo, Chinese Journal of Physics, Vol. 16, No. 1, Spring, 1978.
Average Density of States: The Liquid-Like Random Kronig-Penny Model
Partially Disordered KP: Bands start to merge
Almost Totally Random KP: Bands Merge
The Electronic Density of States of a Liquid-Like Random Kronig-Penny Model in the Coherent Potential Approximation. Chin-Yuan Lu and E-Nr Foo, Chinese Journal of Physics, Vol. 16, No. 1, Spring, 1978.
Adapt the Liquid KP Model to Metals with Spatially Varying Brownian Motion
Solid metal at center.
More and more Brownian motion as we move radially from the center.
0
0.2
0.4
0.6
0.8
1
1.2
-39 -35 -31 -27 -23 -19 -15 -11 -7 -3 1 5 9 13 17 21 25 29 33 37
Pro
ba
bil
ity
"c"
Radius, in Units of the Number of Effective Lattice Constants
Probability "c"
Other Random KP ModelsRandom KP Potential
RepresentationsLocalization Lengths
Universality and Scaling Law of Localization Length in One-Dimensional Anderson Localization. Masato Ishikawa and Jun Kondo, Journal of the Physical Society of Japan, Vol. 65 No. 6, June 1996.
Can we suppress localization and create good transport? Prevalent view: disorder induces localization of all eigenstates of
a one-D system.
Beginning in the 1990’s, “engineered disorder” started being considered.
Can suppress localization by: Correlations
Nonlinearity of excitations
Tight binding Hamiltonians suggest the occurrence of disorder correlations: neighbor random parameters not independent within a correlation length.
This short range order leads to new phenomena-the competition between long range disorder and short range correlation causes the appearance of delocalization, long range transport.
Suppression of LocalizationCorrelations allow for extended states.
Model: KP with paired correlated δ-function strengths to present delocalized electronic states!
1. Consider an electron moving in a 1-D potential
introduce paired correlated disorder
takes on only two values, , where only appears in pairs of neighboring sites(dimer model).
2. Corresponding Schrodinger:
3. Discritize Schrodinger via mapping function to relate wave function at three consecutive points.
This form yields a condition for electron to move:
allowed energies.
n nnnxxxV 0 );()(
n ' and '
)()()(2
2
xExnxdx
d
n n
11) ... () ... (
nnn
1sin2
cos EE
EIf = 0, reflection coef. at dimer vanishesCan find this “resonant” energy, where R0
Scattering from lattice with random dimer defects
Back to
We can introduce the transmission and reflection amplitudes through the relation:
where and are the reflection and transmission amplitudes of a system of N scatterers.
• These coefficients can be computed. From , we can get other relevant magnitudes:
Transmission coefficient, Lyapunov coefficient,
Resistance, rate of growth of wavefunciton: inverse of localization length
IDOS,
Can study system by varying: strengths of the scatterers, λ and λ’, the defect concentration, and the length of the system, N.
)()()(2
2
xExnxdx
d
n n
N xif ,
1 xif ,)(
xEi
N
xEi
N
xEi
et
erex
Nr
Nt
Nt
N
1/1NN
)(NN
f
)(N
tf
Some results of analyses. Transmission States with energy close to the “resonant” energy show
good transport properties!
Lyapunov Since transmission 1 , localization lengths of these
states are large. (> system size)
IDOS, DOS Also affected by short range correlated disorder.
There exists a number of electronic states that remain unscattered by dimer defects, e.g. those with > system length.
Most importantly! Sanchez et. al found that the resonance energy in
which T1 is independent of impurity concentration, c.
Suggests that we can modify c to shift Fermi level to match one of these resonances.
This would yield a large conductance peak.
Thus, one can engineer the σ properties of disordered systems.
Conclusion Complete understanding of disordered systems still
not reached.
Some of the one-dimensional models discussed show promise, especially the KP model. Thin GaAs wires, for example are thin enough to show 1-D behavior.