By Andy Zelinski and Steve Pfifer

Jul 16, 2015

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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)()()(

)(

202

22

,...),,(...,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.

j jxExxxV

dx

xd

m)()()(

)(

202

22

iL

2

222

)(2i

L

s

mE

...2,1 ,...1 sni

n

i s

iLm

sEEN

1 1

2

222

)(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)(

121

1 1

2

222

)(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

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.

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