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“ LOCALIZED EIGENSTATES IN DISORDERED SYSTEMS ”

E. Economou

To cite this version:

E. Economou. “ LOCALIZED EIGENSTATES IN DISORDERED SYSTEMS ”. Journal de Physique

Colloques, 1972, 33 (C3), pp.C3-145-C3-151. �10.1051/jphyscol:1972321�. �jpa-00215055�

JOURNAL DE

PHYSIQUE Colloque C3, supple'ment au no 5-6, Tome 33, Mai-Juin 1972, page C3-145

<< LOCALIZED EIGENSTATES IN DISORDERED SYSTEMS >>

E. N. ECONOMOU

Department of Physics and Center for Advanced Studies, University of Virginia CharIottesville, Virginia 22901 U. S. A.

RCsumk. - Nous etudions une generalisation du modkle de liaison serree #Anderson pour les rkseaux desordonnes. Nous demontrons que le spectre d'knergie consiste en regions d'btats btendus et localises separQs par des knergies critiques appelks bords de mobilit6; la separation est abrupte et dkfinit une transition en accord avec le modkle Mott-CFO. La transition #Anderson a lieu quand deux bords de mobilite adjacents

se

confondent, eliminant ainsi une region d'etats etendus. Nous discutons brikvement plusieurs applications de la thkorie. Celle-ci montre que pour les syst&mes desordonnes unidimensionnels tous les Ctats propres sont localises. Dans ce cas des expressions explicites seront presentees pour la longueur de localisation.

Abstract. -

^A

generalization of Anderson's tight binding model for disordered lattices is studied.

It is shown that the energy spectrum consists of regions of extended and of regions of localized states separated by critical energies of sharp transition termed mobility edges in agreement with the Mott-CFO model. Anderson's transition occurs when two adjacent mobility edges merge eliminating thus a region of extended states. Several applications of the theory will

be

discussed briefly. The theory shows that for 1-D disordered systems all the eigenstates are localized. In this case explicit expressions will be given for the localization length.

I. Introduction.

-

The electronic structure of disor- dered materials is currently under active investigation both because of the technological importance of these materials and the challenging theoretical problems associated with them.

Disordered materials are characterized by the exis- tence of randomness in their positional and/or compo- sitional structure. As a result, this randomness is reflected in the values of the potential V(r) felt by an electron in the system ; V(r) for a given r is a statistical variable. The defining property of a disordered system is the complete lack of long range order ; by that one means that the values V(r,), V(r,) of the potential a t the points r,, r, are statistically independent whenever the distance I r, - r, I is much larger than a charac- teristic length of the system measuring the distance over which short range order exists.

In this paper some recent work is presented aiming a t the discovery of the universal features of disordered systems, i. e., of those features which stem from the lack of long range order and are consequently common to all disordered systems. Mott

[l]

and Cohen et al. [2]

were the first to address themselves to this question.

Their conjecture, known as the Mott-CFO model, will be presented briefly in Sec. 11. In Sec. 111 a theory will be outlined aiming at a first principles quantum mechanical justification of the Mott-CFO conjecture.

Next, in Sec. IV, within the framework of a simple model system, some quantitative results will be obtain- ed which illustrate clearly the general phenomena

occurring in disordered systems and which can serve as partial checks of the general theory. Finally, in Sec. V, the theory is applied to a simple 1-dimensional model and pertinent results are presented.

If. The Mott-CFO conjecture.

-

The content of this paragraph is summarized in figure 1, where the path from perfect (first line) to imperfect crystals (second line) is extrapolated in a <<natural

⁾⁾

way as to yield the Mott-CFO conjecture (third line) for the universal features of disordered systems. The quan- tity n(E) plotted schematically in figure 1 against the energy E is the average density of states per atom. The essential feature of the Mott-CFO model is the assumed existence of critical energies E,, termed mobility edges, at which the nature of the eigenstates changes abruptly from localized to extended. Thus the spectrum is separated by the mobility edges E, in adjacent alter- nating regions each one made up entirely either from localized eigenstates or from extended eigenstates.

Furthermore, it is assumed that the localized eigen- states regions coincide with the tails of the bands or, more generally, with regions where n(E) is small.

The concepts of localized and extended eigenstates in disordered systems need certain clarifications

:

Localized states are not in general associated with single imperfections ; they correspond to an electron bound to a finite cluster of atoms, where the potential deviates from the mean value. Such clusters are always present in disordered systems

;

tEeir magnitude and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972321

C3-146 E. N. ECONOMOU

FIG. 1.

-

Summary of the universal features o f Perfect Crystals, Imperfect Crystals, and Disordered Systems.

shape can be studied using probabilistic considerations.

On the other hand, extended eigenstates are in general associated with clusters of atoms extending to infinity along at least one path. In other words, electrons occupying extended states can escape to infinity although by no means does this imply that all regions of space are accessible to them. Such a definition of the concept of localized or extended eigenstate implies that the mobility p(E) (or conductivity o(E)) is zero when E belongs to a region of localized eigen- states (phonon-assisted hopping conduction is neglect- ed). On the other hand y(E) # 0 when E belongs to a region of extended eigenstates. This justifies the name mobility edge for E,.

U n i v e r s a l F e a t u r e s

n ( E )

Ifin ^{- E}

(a) Bands and Gaps

(b) A1 t e i g e n s t a t e s extended ( c ) l n f i n i t e phase coherence l e n g t h

C o n d u c t i v i t y e i t h e r i n f i n i t e o r z e r o

n

( E l

I d l -E

(a) Bands, Gaps, and d i s c r e t e spectrum

(b) S t a t e s i n t h e bands extended, s t a t e s o f t h e d i s c r e t e spectrum l o c a l i z e d

( c ) F i n i t e phase coherence l e n g t h f o r t h e extended s t a t e s C o n d u c t i v i t y e i t h e r f i n i t e o r z e r o

n ( E )

I

^c

^{E C E~ Ec-E} 1

(a) Bands and, p o s s i b l y , Gaps

( b ) Extended e i s e n s t a t e s (unshaded)

,

l o c a l i zed e i g e n s t a t e s (shaded) separated i n energy by mobi 1 i t y edges Ec (c) F i n i t e phase coherence l e n o t h f o r t h e extended s t a t e

C o n d u c t i v i t y e i t h e r f i n i t e o r z e r o

J Sys tem

P e r f e c t C r y s t a l

I m p e r f e c t C r y s t a l s

D i s o r d e r e d Systems

What happens as the degree of randomness increa- ses

?

The mobility edges E, move inwards into the bands and the regions of extended states shrink. When a certain critical randomness is reached, the two mobility edges merge together and the region of extended states disappears. This is the Anderson transition [3].

Main

C h a r a c t e r i s t i c

P e r f e c t s h o r t and l o n g range o r d e r

S h o r t and l o n g range o r d e r i n t e r r u p t e d o c c a s i o n a l l y because o f i m p u r i t i e s , e t c .

No l o n g range o r d e r

111. Theory of localization

:

the renormalized per- turbation expression (RPE) for the self-energy. -

We consider a system described by a Hamiltonian of the form

(3. l a )

(< LOCALIZED EIGENSTATES I N DISORDERED SYSTEMS >> C3-147

where

and

< n , i I V I m , j > = V n m ; i j , ( 3 . 1 ~ ) with Vn,

; i j

proportional to (1 - an, In (3. lb) and (3. lc) 1 n,

i

> is the i-h atomic orbital centered around n and the points { n ) form a regular lattice. If

i

is allowed to run through a complete set of atomic orbitals, then the Hamiltonian (3.1) is general enough to describe any solid. For the purposes of this work it is assumed that the matrix elements Vnm;

i j

are zero when either one of the quantities I n - ^{m I and}

1

i

- j 1 exceeds a finite number. The results to be presented here can be extended to the more general case where Vnm,

^{i j}

fall to zero

cc

sufficiently

⁾⁾

fast with I n - m I and/or I

i

-

j

1. In any case to neglect matrix elements between sufficiently remote sites or sufficiently remote (in energy) bands is physically a quite reasonable assumption

;

thus our model Hamil- tonian (3.1) is expected to describe realistic solids in spite of the restrictions on the matrix elements Vnm

^{; ii.}

In dealing with a disordered material one should consider the quantities

E ~

and Vn,;

, ~ i j

as random variables following a given distribution law

such that matrix elements referring to remote sites should be statistically independent.

Anderson [3] considered a special case of (3.1)

;

he assumed that there is one orbital per site and V,,, equals a constant number V when nm are nearest neighbors and zero otherwise. He demonstrated 131 that there is a critical value of the width W of the distribution of E,, WC - ^{2 Z V In}

^2,

(Z is the number of nearest neighbors) such that for W

2

W, the eigen- states at the middle of the band (and by inference all eigenstates) are localized. This disappearance of extended eigenstates in a given band is known as Anderson's transition.

Anderson's method was extended and modified recently

by

Economou and Cohen

[4],

who demons- trated within the framework of Andersons' model that a localization function L(E) can be defined such that L(E) > 1

(<

1) corresponds to extended (loca- lized) eigenstates. If W 2 Wc L(E) < l everywhere and consequently we have Anderson's original result.

For W < Wc the energy spectrum can be separated in regions where L(E) < 1 (localized eigenstates) and in regions where L(E) > 1 (extended eigenstates).

The mobility edges Ec are given as solutions of the equation L(E),

=

1. Thus within the framework of Anderson's model the Mott-CFO conjecture has been verified.

The techniques used for Anderson's model can be generalized [5] as to be applicable to the more general and realistic case (3.1) where both diagonal and off-

diagonal randomness is incorporated in the model as described by the distribution (3.2).

One considers the self-energy A0,,(E) defined by

where the matrix element G,;,(E) of the Green's function G(E) = (E

-

A)-' is defined the usual way

A perturbation series expansion in powers of V can be written [5] for A0,,(E) in a way similar to that used in references D] ^{and [4]}

^;

this series can be reorganized as to give the renormalized perturbation expression [3], [4] (RPE) for A0,,(E). One then shows [5]

by following arguments similar to that employed by Economou and Cohen that the RPE for A0,,(E) diverges if E corresponds to extended eigenstates and converges if E corresponds to localized eigenstates or no states at all. Thus the problem of localization reduces to that of studying the convergence properties of the RPE. Following an analysis similar to that of reference [4] one can show [4], [5] that the absolute value of the N-th order term of the RPE for Ao;i(E) behaves like [L(E)]~ and consequently L(E) is a localization function possessing the properties dis- cussed previously. Using the assumption of a com- mon sign of the matrix elements Vnm;ij, as well as that of strong statistical correlation of the terms contributing to the N-th order term, as in reference [4], one can obtain an explicit expression for L(E)

N

- . -

L(E)

=

lim '

^{0 ; a}

N-m [ { z j )

Vonl

; i j l

Gn1;

^jl

VnInz

; j l j z

where the general factors

are defined as

where the superscripts in G denote that in evaluating the matrix element in the state < ^nk

^{jk (}

one should take

= = - a ' = E , ~ - , ; ~ ~ - , = CO,

and the symbol

< > indicates averaging over all the random variables {

E,;

), ( Vnm

^; i j

). The symbol 2 in (3.5) denotes

I n ; j l

summation over all indices n,

^{; jl,}

n,

;

j,, ..., nN

; j,

with the restriction that n,

; j, # o ; i,

n2

; j2 #

o

; i,

n,

; jl,

etc., where n ;

j #

m

; i

means that at least one of the equations n

=

m,

j = i

is violated.

It should be noted that if the quantities Vnmiij do

not possess a common sign L(E) is no longer given

by eq. (3.5). In this case the terms contributing to

the N-th order term of the RPE for A,;,(E) have a random sign and an analysis similar to that of Ander- son [3] as corrected by Thouless [6] is more relevant.

It is then expected that the true localization function would have values between those resulting from eq. (3.5) and those resulting from Anderson's original analysis [3].

Let us now imagine our random system as being created from a perfect crystal through a certain disor- dering procedure. In terms of the matrix elements of the Hamiltonian (3.1) this means that we start with constant

E ~

Vnmiij

; ~

satisfying the relations

and then as disorder is introduced into the system

gnii,

Vnmiij fluctuate around the values ci, Vn-m;ij.

Let T be a proper measure of the degree of randomness in our system such that T

=

0 would correspond to the periodic case. We shall examine the qualitative behavior of L(E) (given by eq. (3.5)) as Tincreases from zero. For T

=

0 one obtains, by a straightforward generalization of the methods used in reference [4], that L,,,(E) > 1 within the bands, with the equality achieved at the band edges. This, of course, is in agreement with the fact that all eigenstates are extended for a periodic system. On the other hand, one can easily show that L(E)

+ 0

for every value of E as the width of the fluctuation of

E,;~

goes to infinity i. e.

as T

+

co. Assuming a continuous behavior of L(E) as a function r we can conclude that for each E there is a critical value of T, rc(E) such that L(E) < 1 when r > r , ( E ) . Introducing Tc

E

max Tc(E) we have that for r ^{> Tc} L(E) is smaller than unity for every E. Hence for r ^> rc all eigenstates become localized. This is Ahderson's transition. For T < T', there will be regions of extended states (L(E) > 1) separated from regions of localized states (L(E) < 1) by mobility edges E, (L(E,)

=

I), in complete agree- ment with the Mott-CFO conjecture.

IV. A simple model for the random binary alloy. - Thus far we were able to justify the Mott-CFO conjec- ture for any disordered system which can be adequa- tely described by a Hamiltonian of the form (3.1).

It would be highly desirable, however, if one could go beyond the qualitative statements of the Mott-CFO conjecture and obtain quantitative results regarding the positions of the mobility edges and Anderson's critical value of the randomness. The function L(E), useful as it was for deriving certain qualitative results, cannot be used for quantitative calculations due to its complexity. It seems inevitable that certain simpli- fications of the general model would be needed as well as certain approximations.

For the time being quantitative results have been obtained only in highly simplified unrealistic models.

Nevertheless, the study of these simple model systems

is extremely useful, since it provides concrete illus- trations of the general phenomena occurring in disor- dered systems. In this section we shall apply the general theory developed in Sec. I11 to a very simple model for a random binary alloy [7], [8]. This simple model provides not only concrete examples of mobility edges, mobility gaps, and Anderson's transitions, but can serve also as a check of the general theory.

The model Hamiltonian assumed to describe a random substitutional alloy A, B, -, is an Anderson type with one orbital I n > at every site n. The off- diagonal matrix elements Vnm are taken as

Vnm

=

V, a positive constant for nearest neighbors ,

= 0,

otherwise .

The diagonal matrix element

E,

are random variables that can take two values eA or

^E,

depending on whether site n is occupied by A or B. The probability of occu- pation by A is x and by B is 1 - x. As degree of ran- domness we can consider the dimensionless parameter 6

=

(E* - zB)/ZV. Even for this simple model a direct exact calculation of L(E) is out of discussion. However, one could show [4] that within the framework of any single site approximation [9] (SSA) [in the SSA average quantities are calculated by replacing each random variable

E,

by a common quantity (E) and thus reducing the problem to a periodic one] one can cal- culate a function F(E) such that

F(E) < 1 implies L(E) < 1 .

(4.1)

From (4.1) we can conclude that those regions of the spectrum for which F(E) < 1 are consisting entirely of localized states. In addition, the mobility edges lie always in the regions where F(E) > 1. As the degree of randomness becomes smaller the mobility edges Ec approach the positions E: where F(E,*)

=

1.

In other words, the function F(E) not only can be used as a reasonable estimate of the localization function L(E) for small randomness, but can also serve for the derivation of inequalities for the positions of the mobi- lity edges. What is more important, the function F(E), in contrast to L(E), is no more difficult to cal- culate than the average density of states. Thus the results of the considerable work devoted to the cal- culation of the latter [9] can be used for the determina- tion of F(E).

The function P(E) for the particular case under consideration was determined by using the coherent potential approximation [9] (CPA) introduced by Soven [lo].

Most of the results obtained by using F(E) as an

estimate of the true localization function L(E), are

presented in figure 2. We have plotted the average

density of states n(E) = < ^no(E) >, the function

F(E), and the so-called fractional A parentage of

states nA(E) as a function of the energy E. The quantity

nA(E) is defined as nA(E)

=

xpA(E)/n(E), where

pA(E) is the conditionally average density of states

pA(E)

=

< no(E) >,,, with the site o restricted to

<< LOCALIZED EIGENSTATES IN DISORDERED SYSTEMS ^)>

contain an A atom. The concentration x of A has been taken as x

=

.I. The curves have been given for four different values of 6 increasing as we go from the bottom to the top section of the figure.

x (per cent)

FIG.

3. - Phase diagram of the alloy model, as obtained from the CPA self-energy. 6~ denotes the critical value of 6 for the appearance of a mobility gap at a given concentration, 6~ the opening of a gap in the density of states, and 6A the Anderson

... transition for the impurity states. The concentration above

which the Anderson transition no longer takes place is

2 0

o denoted xc.

Ec Ec

(ENERGY)

FIG.

2. - Alloy density of states (in arbitrary units) and pro- perties related to localization, calculated by the CPA with x = .l, and 6 = (a) .7, (b) .8, (c) .95, and (d) 1.1. Regions of localized states have been shaded. The heavy line above each density of states indicates the <<localization >> function F(E) satisfying (4.1) ; the dotted line is the A parentage nA(E) defined

in the text. After reference [7].

As can be seen from the sequence of events shown in figure 2 and discussed in detail in reference 171, one can define a critical value of 6, a,, at which the mobility gap first appears, another, 6,, at which a gap in the density of states first appears and a third one, 6,, at which Anderson's transition takes place. All these critical quantities are functions of the concen- tration x, as is shown in figure

3.

We shall discuss first briefly what modifications one would expect if the localization function L(E) were to be used instead of the function P(E). As has been discussed before, we expect only quantitative modifications and always in the direction that given effects occur at smaller randomness. Thus the curves for 6 , and 6 , resulting from L(E) are expected to lie below the corresponding ones shown in figure 3

;

the corrections should be more significant for 6, and 6, since 6, corresponds to a condition of higher disorder where the discrepancy between F(E) and L(E) is more pronounced. Conse- quently, the critical value x, of the concentration is expected to be substantially higher than the one given from the present approach.

Figure 3 can be used also to describe qualitatively the sequence of events associated with the introduction of impurities in a given crystalline material. As the concentration of the impurities increases from zero

an impurity subband is formed consisting initially entirely from localized eigenstates

;

the first extended states appear as we cross the 6, curve and at higher concentration, as we crossed , a, the impurity subband joins the host material's band while a mobility gap is still present. As the concentration increases further this mobility gap disappears (when one crosses the 6, curve).

The quantitative results obtained for this extremely simple model are in full agreement with the Mott- CFO conjecture. Perhaps the most striking success of the whole theory occurs for the limiting case 6

-,

co.

In this case the only way by which an electron ini- tially on an A site can escape to infinity is by finding a path of A sites extending to infinity. Thus the fraction of extended eigenstates in the A-subband as 6

-+

co is given by the probability PA(x) to find A-paths extend- ing to infinity and passing through a given site. This probability is the well known percolation probability first studied by Broadbent and Hammersley [Ill.

PA(x) is exactly zero for x < x, and rises very steeply

for x > x, with the critical concentration x, depend-

ing on the lattice. As can be seen from figure 3, the

present approach reproduces correctly this qualitative

behavior with the critical concentration x, given by

the asymptote of the 6,-curve. Using the methods of

this section we have estimated the fraction of extended

states in the minority subband as 6

-,

co for a simple

cubic lattice and compare this quantity with the per-

colation theory prediction for PA(x) calculated by

Monte Carlo methods. The results are shown in

figure 4. Curve (a) is the prediction of percolation

theory and curve (b) is the prediction of a theory

using F(E) instead of L(E). As was discussed before,

C3-150 E. N. ECONOMOU

I

,r

^{,hK. S.C. ; t(2.61}

with

F ( E ) t CPA ,S.C.(Z=6)

10 20 30 40 50 60

FIG. 4. - The fraction of the extended states in the minority subband (%) as a function of theminority element concentration for 6 = co. Curve (a) has been calculated from percolation theory for a simple cubic lattice. Curve (b) is calculated from the approximate localization function F(E) computed within

the CPA for simple cubic lattice.

if the exact L(E) was used curve (b) would move to the right. Thus the present approach not only reproduces in the 6

+ co

limit the qualitative characteristic of the existence of a critical concentration but gives quanti- tative results consistent with the exact results of the percolation theory. Note the striking similarity of curves (a) and (b) in figure 4. It seems that the use of the exact L(E) would simply shift curve (b) parallel to itself until it will coincide with curve (a). It should be mentioned that other approximate methods of calculating L(E) either failed [12] to reproduce the correct behavior in the 6

-,

co limit or, as in the case of Anderson's estimations [3], have not been checked yet in this respect.

V. A one-dimensionai model.

-

The theory of localization developed in Sec. I11 is based upon the RPE for the quantity A,,(E). One could obtain the same results by considering the RPE for the quantity G,,,,,(E). The analysis based on the RPE for G0,;,,.(E) yields in addition that the localized eigenstate decays in an exponential way and thereby provides the means for calculating localization lengths [13]. Here this method will be presented for a simple 1-D model and some relevant numerical results will be given.

Consider a 1-D lattice of points

n =

0,

1,

-t 2, ...

With each lattice point an atomic orbital 1 n > is associated and the Hamiltonian is assumed to be

where a superscript j in the diagonal matrix element G! denotes that in calculating the matrix element Gi = <

i

I

G

I

i

>

^E,.

should be taken as infinite. Using the general expression for Go, in terms of a complete set of eigenstates I n),

it can be shown that

where

(

n;) are all the eigenstates corresponding to eigenvalue E and having non negligible values around the lattice point 1. Thus if

(

nk) are extended states, Fo,(E) would approach a finite limit as 1

+ a3 ;

if

I nL) are localized then Fo,(E)

-+ 0.

I-+ m

Eq. (5.5) alIows us to define the localization length R(E) for the states

(

nk) as

where R(E) is measured in units of the lattice spacing. Using a statistical analysis similar to that of reference [4] one can show that

R(E)

=

- ¹ ( 5 . 7 ~ )

log ( v ~ ( E ) ) ' where

G(E)

=

exp[< log I G:+ ' ( E ) I >] . ^(5.7b)

From eq. (5.7) one can easily show that for the periodic case

^8, =

0 (n

= 0,

f 1, ...) R(E)

=

oo for - 2 V < E < 2 V rederiving so the well known result of a band of extended states (Bloch states) of total width 4 V.

In the case. where the probability distribution func- tion for the quantities

E,

is a Lorentzian of width r

centered around the origin R(E) can be calculated exactly

< n l Him>

=

8,6,, + ^{V,, (5.1)} R ( E ) = [ l / l o g I ~ + i y + d ( ~ + i ~ ) ~ - l l ] (5.8)

where where

Vnm

=

Vfor nearest neighbors , E = E / ~ V , y = r / 2 V . (5.9)

=

0 otherwise . ( 5 ' 2 )

It is worthwhile to mention that R(E) (eq. (5.8)) is The quantities

E,

are random variables which for finite for every E as long as r

^{# 0}

in agreement with simplicity are assumed independent. The RPE for the the theorem stating that all eigenfunctions of a 1-D matrix element G,,(E) can be written [3] as disordered system are localized.

In figure 5 the quantity R(E) as given by eq. (5.8)

Go,

=

F,,

G,

(5.3a) is plotted for different values of I: F o r small degrees

<< LOCALIZED EIGENSTATES IN DISORDERED SYSTEMS B C3-151

References

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lo3 R

1

lo2

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[I41

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(R.

E.),

PYOC. R. SOC., 1963, A 274, 529.

[15]

MAKINSON

(R.

E.

B.)

and ROBERTS ( A .

P.), Proc.

Phys. SOC., 1962,79,630.

[16]

WILLIAMS

(F. W.)

and MATTHEWS (N.

F.), J., Phys.

Rev., 1969, 180, 864.

of randomness (small

y )

there is a very rapid change of the localization length R from large and rather cons- tant values in the interior of the band to very small values in the tails. This drastic change is reduced as we consider higher values of the degree of random- ness

y

and finally as y approaches

y, (y,

5 l), where y, is the critical value of

^y

for Anderson's transition to occur in the corresponding 3-D case, the localization length remains more or less uniform over the whole spectrum and of the order of magnitude of the atomic spacing.

These results are in agreement with numerical works [14]-[16] indicating that there are some narrow regions in the energy spectrum where the localization

-

-

l r . . . . , . . . . , . . . . ,

0 0.5 1.0 1.5

l l l l . , ~ ~ ~ ~ , ~

-

-

- -

length changes abruptly from small values in the tails of the band to large values in the interior of the band.

A more detailed discussion of this point can be found in reference [13].

r

:

-E

In conclusion it should be mentioned that the loca-

FIG. 5.

-Localization length

R (in

units of the interatomic lization length R(E) can be calculated numerically distance) of the eigenfunctions

in

a

1-D

tight-binding approxi- for different more realistic distributions of the quan- mation with Lorentzian distribution

of

width

y

(in units of

Eb)

tities

E,.

~h~ results of these calculations as well as

of

the sites' energies, versus

the

corresponding eigenenergy

8

(in

units of Eb), where Eb half the bandwidth for the periodic

the detailed derivation of the method outlined in this

case. section will be reported elsewhere.

-

; ^-

:

-

- -

t

-