Classical localization

Physica A 167 (1990) 215-230 North-Holland

CLASSICAL LOCALIZATION E.N. E C O N O M O U Department of Physics, University of Crete, Heraklion, Crete, Greece and Research Center of Crete, P. 0. Box 1527, Heraklion, Crete, Greece

The problem of the classical wave equation is mapped to the electronic Schr6dinger equation. Thus various calculational techniques developed for the latter can be employed to study the question of localization of classical waves. Results are presented for a random binary system.

I. Introduction The question of localized eigenstates in disordered media has been studied extensively in connection with electronic transport properties in condensed matter [1-8]. The question of localization of classical waves has attracted attention only during the last seven years [9-22] although there is an extensive literature dealing with various aspects of classical wave propagation in r a n d o m media [23-27]. Indeed classical waves, such as light waves, microwaves, acoustic waves, seismic waves, ultrasound waves, etc. are of great interest for their own sake. F u r t h e r m o r e , classical waves offer "clean" conditions allowing tests of localization theory against experimental data, without the usual electronic complications (energy not fixed, finite t e m p e r a t u r e effects, e l e c t r o n electron interactions, e l e c t r o n - p h o n o n interactions, etc.). One then wonders why localization theory has not been applied mainly to classical waves instead of electron waves. The main reason for this is the fact that classical waves are much m o r e difficult to localize than electronic waves. Indeed, it is not yet proved beyond doubt that classical wave localization does exist, although recently [28-35] strong evidence supporting the existence of classical wave localization has b e e n produced. Thus the outstanding p r o b l e m in classical wave localization is to find the optimal conditions for its realization. To be m o r e concrete, one usually considers a binary composite m e d i u m consisting of two lossless materials, 1 and 2. Material 2 with real constant phase velocity c 2 is the "slow" material, while 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

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E . N . E c o n o m o u / Classical localization

material 1 with real constant phase velocity c 1 (C 1 ~ C2) is the "fast" material. The ratio m = c 1 / c 2/> 1 is called the contrast; the quantity /x = c J2 c 22 = m 2 = e2/e 1/> 1 is also used; el, 2 is the dielectric constant of material 1, 2 respectively (e 2 >I 81 > 0). Besides m (or/x), the random medium is characterized by x, the volume fraction occupied by medium 2, and by the particular geometric characteristics determining the way the two materials are mixed together. E.g. one may consider spheres of material 2 randomly placed within material 1, or vice versa; the radius a of the spheres may be fixed or may be a random variable characterized by its average value ~ and its standard deviation o-~ (assuming Gaussian distribution). Besides spheres, other shapes such as cubes may also be considered in which case one is faced with orientation disorder as well. It must be stressed that the geometric characteristics may prove very important. E.g. the spheres do not form a connected infinite channel (percolation channel) even for very high volume fraction x, while a completely random mixture (of all shapes and sizes for both material 1 and material 2) form a percolation channel of material 2 for x ~> 0.2. Besides - the contrast, m ( o r / x = m2), - t h e volume fraction of material 2, x, - and the geometric characteristics, another important parameter is - t h e frequency to of the wave. The question one faces is to find in this multiparameter space the regions corresponding to localization. To examine this problem we first start by reviewing briefly the relevant work in the electronic case (section 2). Then in section 3 we show how the classical wave problem can be mapped to the electronic one. In the next section 4 we generalize some of the techniques used in the electronic problem as to be applicable to the classical wave case. In sections 5 and 6 we examine a periodic problem for which very accurate results can be obtained and which, at least in some limit, is relevant to the question of classical wave (CW) localization. In section 7 we present theoretical results and we compare them with experimental data. Finally in the last section we present our conclusions.

2. B r i e f r e v i e w o f e l e c t r o n i c l o c a l i z a t i o n

Ordinary transport theory for 3D systems is based on the implicit assumption that disorder modifies the phase ~0(r) of the unperturbed eigenfunctions A e i k ° r so that (e~(~)-,~(o)) = ei~ e-~/2~

(2.1)

E.N. Economou / Classical localization

217

In other words the unperturbed wave vector k 0 is renormalized to k and at the same time acquires an imaginary part i / 2 l , where l is the phase coherence length (for isotropic scatterers 1 coincides with the transport mean free path [4]), i.e. k o - - - > k + i / 2 l . As long as kl>>l the effect of disorder on the amplitude A of the unperturbed eigenfunctions is negligible. (For anisotropic systems it is more appropriate to replace kl by SI 2, where S is the surface of constant energy in k-space. For isotropic systems Sl 2 = 4xrk212). However, as kl approaches unity the amplitude A develops large fluctuations of ever increasing extent and magnitude. The extent as well as the magnitude of the larger fluctuations is characterized by a length ~, which at the critical point a~ (i.e. as kl---> a+~) blows up as as kl---> a+~ .

~--> b / ( k l - a¢) ~ ,

(2.2)

For kl < a~, the eigenfunctions become localized, i.e. their amplitude decays exponentially on the average for large distances, , e -'/x

A(r)

,

as

r---> ~ ,

(2.3)

r ~

where the localization length A blows up at the critical point in the same way as ~, i.e., A---> b / ( a c - k l ) ~ ,

as kl--~ a~ .

(2.4)

The best way to define and calculate ~ is by considering a quasi-one dimensional system (i.e. a wire) of square cross section M x M (in units of lattice spacing a) and calculate the largest localization length AM for this system of M 2 coupled chains [36-38]. For kl > a c, one finds numerically that (2.5)

AM/M =f(M/~),

where f ( x ) seems to be a universal function [39] #t of its argument as long as M > > a , l . Moreover, f(x)-->0.6 for x--->0 and as x--->~, f ( x ) - - - > c x , 1 / c = 4.82_+0.4. The universality of f ( x ) provides strong support for the one parameter scaling hypothesis [5]. Now following Anderson's analysis [40] E c o n o m o u et al. [41] have shown that the resistance R of our wire, for AM ~> M (i.e. for M ~> ~), is given by R = ~ ~ps L

"Ps ~

- 1 ,

(2.6)

• 1Some doubts about the universality off(x) has recently been raised by B. Kramer et al. (see ref [39]).

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E.N. Economou / Classical localization

where y = 1.13 according to Anderson [40] or y---0.99---0.1 according to ref. [41] and & is a slowly varying function of the resistance with the following limits [40]: pS = ½ as R--+0 and & = 1/1.764 as R--+ m. Taking the large L = M limit of eq. (2.6) one obtains the 3D dc conductivity o- as e2 (0.066 +

b)

(2.7)

,

where the coefficient 0.066 is accurate to about 10%, while the coefficient b is more uncertain and seems to vary from b = 0.05 for weak disorder to about half this value at the critical point. (It must be pointed out that strictly speaking the above derivation of eq. (2.7) is not valid at the critical point, since then the inequality M >> ~ is violated). It follows from the above discussion that numerical determination of h M versus M allows us to obtain ~:, h and or. E c o n o m o u et al. [42, 43] have shown that the above results of localization theory for ~, h are analogous to those obtained in a single effective potential well, where the quantity ~ corresponds to the scattering length of the scattering solution [43] and h corresponds to the decay length of the bound solution [42]. By fitting the results obtained by the potential well analogy (PWA) to those obtained by the numerical determination of h M versus M, E c o n o m o u et al. [41-43] have obtained the following simple empirical interpolation formulae: 2.72l - x2F(x) - '

(2.8)

a -

2.2 + 14.12x 2 1 -- X 2

o" =

o'oF(x)

(2.9)

l,

(2.10)

,

where

1

e2

° ' ° - 12,rr3 h

(2.11)

Sl

is the weak disorder limit for the dc conductivity and F(x)

=

X2(X2 -- 1) X2(X---5~--{)+ 6 '

X = kl/a¢,

a c = 0.844.

(2.12)

It must be pointed out that the above formulae assume that u = 1 in eqs. (2.2) and (2.4); furthermore eq. (2.10) gives that the first correction to ¢r for weak disorder goes as 1 / l 3, while Kirkpatrick and Belitz [44] have obtained that the first correction is - ~ o ~ r h / E r , i.e. independent of l, and the coherent potential approximation gives [4] that the first correction is proportional to 1/1.

E.N. Economou / Classical localization

219

This discrepancy can be easily corrected by fitting the numerical AM versus M data by introducing an F(x, y) where y = ka. It must be stressed that the plethora of results [36-38, 45-47] based on the reliable AM versus M method as well as the PWA results [41-43, 45-47] based on eqs. (2.8)-(2.12) have been obtained for a tight-binding lattice model, where the Hamiltonian matrix element < n l H I m > equals e n for n = m, equals V for n, m nearest neighbours and is zero otherwise. The quantities {en} are independent random variables of binary or Gaussian or rectangular distribution. Each state In > is an atomic-like orbital centered around lattice site n. However such a lattice model does not contain the full richness of a continuum model, because a lattice point with just one orbital associated with it, does not correspond to a finite region of space. This can be seen from the fact that the lattice point n can sustain at most one bound state, no matter how large the difference l en - < Em :>l is, while a potential well can sustain as many bound states as we wish, if its depth is large enough. Another equivalent way of seeing this point is by thinking in terms of the wavelength: for a lattice model to approximate the continuum the wavelength must be much larger than the lattice spacing. On the other hand in the continuum system that we discussed in the introduction, the wavelength may become comparable to or smaller than the characteristic geometric dimension, e.g. the radius of each sphere. Thus for a lattice model to approximate our composite binary continuum system, we must introduce another length scale, the correlation length d, much larger than the lattice spacing a so that ka ~ 2rr ~ kd. Over the length d the quantities en have the same value. We return to this point in section 4.

3. Mapping CW to electron waves

Consider now the composite binary system described in the introduction. The potential felt by an electron located in material 1 is E A and in material 2 is E B (8-= E A - E a / > 0 ) ; x is again the volume fraction of material 2. For an electron of energy E Schr6dinger's equation has the form 2m V2~ + ~ ( E - V)4, :- 0 ,

(3.1)

with {E-E E - V=

a EB

for material 1 , for material 2 .

(3.2)

The corresponding scalar classical equation has the form (.0

V2u+ ~

2

u =0

(3.3)

E.N. Economou / Classical localization

220 with

for material 1,

"--5

1

Cl

~=

(3.4)

1

for material 2.

C2

T h e q u e s t i o n is h o w t o m a p a g i v e n p o i n t in t h e 8, E __

2

2

to the l~(=cl/c2) , immediately that

0,)2

E A electronic plane

C W p l a n e . B y i n s p e c t i o n o f e q s . ( 3 . 1 ) to ( 3 . 4 ) w e s e e

2

0.) 2m -~-g ¢:> - ~ - ( E - E A )

(3.5)

I

and 2

0.) 2m --7- ¢¢' ~ (E - EB) ,

(3.6)

C2

f r o m w h i c h it f o l l o w s t h a t 8

/x = 1 + EE -- ~

'

(3.7)

X
"'\

?

~-'~// 0

:!c

/,'l

'lCo " E-EA

Fig. 1. Schematic diagram for a composite binary random electronic system A 1-x Bx; E is the energy and 8 = E A - E B ~>0, where E A (EB) is the bottom of the spectrum of pure A (B) material. The line OB corresponds to E = E B. The expected mobility edge trajectory (MET) separating localized states to its left from extended states to its right, is also shown schematically either for x < x c (heavy solid line) or for x > xc (heavy dashed line); x c is the critical percolation concentration for the B material. The slope of the line OC equals /.~ - 1 and OC o determines elto2/c 2, where /x eE/e1 (see text). =

E . N . E c o n o m o u / Classical localization

2

2mc2~

= -V-

(E -

EA)

221

(3.8)

.

Eqs. (3.7), (3.8) achieved the desired mapping. The to2-axis is mapped, according to eq. (3.7), to a straight line passing through the origin of the 6, E - E A plane with slope equal to/x - 1. The value of to 2 along this straight line is given essentially by the corresponding E - E A line, according to eq. (3.8). It is now clear why it is more difficult to have CW localization rather than electronic localization. The entire physical CW plane/~, to 2 is mapped into the positive 6, positive E - E A quadrant of the electronic plane. This means that CW localization is equivalent to electronic for energies E larger than the higher among the two values of the potential. In other words, if one draws the line that connects all the critical points (for which kl = ac) in the 6, E - E A plane (the so-called mobility edge trajectory (MET)), this line (which for low 6 starts to the left of the 6-axis) must bend over and cross to the right of the 6-axis, as shown in fig. 1.

4. Generalization of the electronic localization techniques The tight binding AM versus M method needs to be generalized as was discussed in section 2. For this purpose the wire is divided in cubes each containing N s sites (N = 2, 3, 4). Within each cube all the sites are either A (with probability 1 - x) or B (with probability x). Numerical limitations are imposed by the fact that M can hardly exceed 12, while at the same time one should try to fit as many and as large as possible cubes within the cross section of the wire (in our calculations the number of cubes within each cross section is

(M/N)2). The approximate PWA technique requires as inputs the renormalized wave vector k and the mean free path l. A very efficient way to obtain both k and I is the so called coherent potential approximation (CPA), which introduces an effective, complex, frequently dependent dielectric constant e¢ or, equivalently, an effective propagation constant q, such that

1,2 q=

ee]

i = k + 2~ "

(4.1)

The quantity q (or ee) is determined by the condition that the scattering resulting, when a spherical region of the effective medium is replaced by the true random medium, be equal to zero on the average. This condition is easily satisfied for a simple lattice model because the cross section is isotropic.

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E . N . E c o n o m o u / Classical localization

However in the continuum the scattering amplitude is not isotropic, thus a question arises of what to put equal to zero. We have decided [28-30] to set the so-called average total cross section equal to zero. We obtained thus k, v = t o ~ k , l, and the weak scattering value for the diffusion coefficient Do, D O = ~vl.

(4.2)

Then using eqs. (2.5), (2.8)-(2.10) we calculated AM, sc, h, the mobility edge trajectory (MET), and the diffusion coefficient D, D = DoF(x ) .

(4.3)

5. Scalar CW band structure

We start with the observation that the physical origin of band gaps, Bragg diffraction and localization is the same i.e. self reinforcing multiple scattering. To be more specific consider identical spheres of material 2 arranged periodically, e.g. in an fcc lattice. A scalar wave propagating in this periodic medium may exhibit gaps of forbidden frequencies. Now we gradually introduce [48] some disordering process regarding the positions of the spheres. For weak disorder, it is well known that the bands will be practically unaffected by the disorder, while tails of localized states will be developed in the gaps. Thus, in this case of weak positional disorder, as described above, the regions of localized states will practically coincide with the positions of the gaps. Let us point out also that at very high x (approaching the close-packed limit) there is very little room to move the spheres around out of their periodic position; thus, at high x we are close to the above-mentioned weak positional disorder. We also point out that John and co-workers [49] have proposed the periodic arrangement of the spheres as a starting point followed by a weak disordering process in order to achieve optical localization at relatively low x. Their calculations, based on the Korringa-Kohn-Rostoker method [50], gave an optimum x around 0.12, in disagreement with earlier estimations. We have studied [48] systematically, using standard APW [51], fcc and bcc muffin-tin structures [51]. Within the muffin-tin the electronic potential is - 3 , while in the interstitial region the potential is zero. We have determined the position of the gaps in the 3, E plane and then, using the mapping of section 3, we have found under which conditions frequency gaps appear. Recently Yablonovitch and Gmitter [32] have shown experimentally that a band gap for electromagnetic waves appears in the microwave region for an fcc periodic structure consisting of overlapping air spheres (of volume fraction

E.N. Economou

/ Classical localization

223

86%) embedded in a dielectric of e = 12.25. In this case the connected pieces of dielectric should be viewed as the scattering centers rather than the air spheres.

6. Photonic band structure

Standard band structure techniques, such as APW [51] or KKR [50], have been designed for the scalar case and are not directly applicable to the electromagnetic case. We report here briefly some preliminary results [52] concerning the generalization of the APW method as to be applicable to electromagnetic fields. The generalization is based on the two Hertz-Debye potentials [53, 54] in terms of which the electric and the magnetic fields can be expressed. The Hertz-Debye potentials satisfy the wave equations. However, through the boundary conditions on the fields, the two Debye-Hertz potentials are coupled together. The Debye-Hertz potentials H 1 and /I2 are written as linear combinations of APW functions,

//1 = E E (axiYli + ayiI"ti)AilA?pu ,

(6.1)

H 2 = ~ • (-ayiYti + axiff"u)Ai2At~ou,

(6.2)

i

l

i

1

where Al=it_l

2l+1

(6.3)

t(l + 1) ' ~bl(kir)

r>~,

r

q~ti =

Ct(kiot) ~,((to/c)v~z r) ~bt((to/c)x/-~2 a) r

Yti = P}I)(cos Oi)cos 9i,

/'1

I7"1i = P}a)(cos Oi) sin q~i,

(6.4) r
k~ '

r>a,

(6.7a)

~1 1 e2 k2 ,

r< ~ ,

(6.7b)

A l l ~-

¢

Ai2-

oJki .

(6.8)

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E.N. Economou / Classical localization

The function 6l(x)=-xjt(x), where Jt is the spherical Bessel function, and k i = k + Ki, where k is in the first Brillouin zone and K i is a vector of the reciprocal lattice. The angles 0~ ~, are those of the vector k~. Note that the A P W basic functions do not satisfy all four continuity equations at [he interfaces. Eqs. (6.1) and (6.2) only guarantee that H 2 and e l i I are continuous everywhere. By a standard minimization procedure (which should take into account the discontinuities of H l and a l I 2 / a r ) one obtains a linear system for the coefficients axi, ay i of the form

J

(Fxixja~j + F~iy/ayj)

= 0,

(6.9)

(Fyixjaxj + Fyiyjayj) = O,

(6.10)

J

where the F's are complicated expressions to be presented elsewhere [52]. We see from eqs. (6.9) and (6.10) that the photonic A P W equations form a 2n x 2n system, where n is the number of reciprocal lattice vectors kept in the summation. Note that the factor of 2 is due to the two polarizations of the electromagnetic field which are coupled together doubling the size of the system and violating the symmetric character of the A P W electronic matrix. Furthermore, the expressions for the F's are considerably more complicated in the photonic case rather than in the electronic case.

7. Results

In fig. 2 we present [28] results for the M E T based on the generalized AM versus M numerical method (solid line) as well as C P A - P W A results (dashed line) for x = 10%. The reliable numerical results indicate clearly that M E T crosses to the right of the g-axis for ~ ~ 4 and 8 ~ 10 (in units of h2/2mot2). Thus scalar CW localization does exist in our lattice model. It has been argued [28-30] that localization is easier in a continuum model of spheres than in our lattice model of cubes. This is so because (i) The cubes can be easily joined together to form other shapes with different geometric characteristics and, consequently, different resonant energies; this is not so for the spheres. (ii) The lattice model exhibits a gradual transition (over a lattice spacing) from "material" 1 to "material" 2 in contrast to the continuum model, which shows an abrupt transition, thus increasing the scattering cross section. It must be pointed out that the energies at which the M E T crosses to the right of the g-axis are associated with the values of 8 for which a new bound

E.N. Economou / Classical localization I

~

I

I

225

I

x

=

0.10

f %

211 \ I

\ -~\"-.

15

5_

57

iS,

I 15

/, 5

EM

B i t, I ----~ I 1 0 1 2 E-EA Fig. 2. Mobility edge trajectory obtained from numerical results (solid line) and CPA results (dashed line) for the scalar case with x = 0.10. In the insert the CPA mobility edge trajectory for the electromagnetic wave case for x = 0.1 is shown. The units of 6 and E - E A are equal to hE/2ma 2. The accuracy of the correspondence of our tight-binding model with the continuum model breaks down for 6 > 10 (solid line). -2

-1

0

state just appears in a potential well of d e p t h & A t these values section exhibits an infinite r e s o n a n c e located at E - E g = 0. A s 6 starting f r o m o n e of these critical values the r e s o n a n c e m o v e s inside and tends to be less p r o n o u n c e d . T h e critical values of 6 for a potential well are given as solutions of the e q u a t i o n

jt_l(k2 a) = 0 ,

11> 1 ,

the cross decreases the b a n d spherical

(7.1)

where 6 = h2k~/2m, k 2 = t o / C 2 and Jt is the spherical Bessel function corres p o n d i n g to angular m o m e n t u m l. F o r l = 0, the critical values are given by

ka = (2n + 1 ) w / 2 ,

l= 0.

(7.2)

Thus for each l there are infinitely m a n y critical values of 8 (or ka). In table I the first four critical values of 6 (and of 2a/A2) for each o f l = 0 (s), 1 (p), 2 (d) and 3 (f) are given; A2 = 2~rc2/to is the wavelength inside the sphere and 2 a is the d i a m e t e r o f the sphere. N o t e that resonances a p p e a r w h e n the d i a m e t e r is a b o u t equal to half an integer times the w a v e l e n g t h for the sphere material. N o t e also the almost d e g e n e r a c y o f the critical 6 of I and o r d e r n with the 6 of l + 2 and o r d e r n - 1. T h e s e resonances in the optical case have b e e n well k n o w n for o v e r eighty years as Mie resonances [53, 54]. In the lattice cube case the crossings o f the M E T is again related with the r e s o n a n c e energies o f a

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E . N . Economou / Classical localization

Table I Values of 8 (in units o f h2/2ma 2) and 2 a / A 2 (8 = (2~rct/h2)2) for resonances to appear at E - EA = 0, i.e. /z = % for scattering by a single sphere. The numbers in parentheses are the 8-values and the ones without are the 2a/h 2 values. Order 1

2 3 4

l 0 (s)

1 (p)

0.5

1

(2.467) 1.5 (22.21) 2.5 (61.69) 3.5 (120.90)

2 (d) 1.43

3 (f) 1.83

(9.87) (20.19) (33.22) 2 2.46 2.90 (39.48) (59.68) (82.72) 3 3.47 3.92 (88.83) (118.90) (151.85) 4 4.48 4.94 (157.91) (197.86) (240.70)

single cubic potential well. The critical values of 8 foi" the cube are quite close but not identical to those of a sphere of equal volume. The close correspondence shown in fig. 2 between the crossings of the M E T as calculated by the generalized AM versus M method and the C P A - P W A method, as well as the easier localization in the continuum, lend further credibility, to the C P A - P W A method, which has the advantage that it can be used both in the scalar and electromagnetic case. Furthermore, the C P A - P W A technique allows us to obtain many other quantities of direct physical interest; it allows also the possibility of checking the resonance interpretation. Indeed, by excluding from t h e / - s u m m a t i o n in the total cross section particular values of 1 we have shown that the corresponding structure in the M E T disappears. For low values of x the lowest resonance (the sl for the scalar case and the p l for electromagnetic case) seems to dominate by pushing the M E T structure to the far right, lowering thus the m i n i m u m / z required for C W localization. The optimum value of x for the lowest structure seems to be x ---0.13 --- 0.03 for the lattice model according to the AM versus M numerical method and the corresponding minimum value of ~ = 14---2. For the C P A - P W A method in the continuum system the corresponding values are x---0.2 and /z = 7. As x increases beyond the optimum value the lowest structure recedes to the left and higher structures become more pronounced. These structures also recede to the left as x passes through corresponding optimum values. The C P A - P W A for the electromagnetic case shows a similar behaviour (the l = 0 component and the corresponding structures are missing) with a general tendency for easier localization and higher values of optimum x.

E.N. Economou / Classical localization

227

We have used our CPA-PWA method to calculate quantities measured by Drake and Genack [31], most notably the diffusion coefficient. The agreement between theory and experiment is rather impressive [29], given the fact that essentially there are no adjustable parameters (with the exception of x which is not exactly known in the Drake and Genack experiment). The CPA-PWA method predicts that slight variations in the Drake and Genack experiment (e.g. lower dispersion in the sphere diameters or slightly lower x) could produce localization around the pl resonance, i.e. for vacuum wavelength about 1.5 times the average sphere diameter. The scalar case besides the numerical AMversus M method has been checked by using standard APW techniques for a periodic placement of spheres. Some of our results are summarized in fig. 3 (see ref. [48]), where a surprising perseverence of the p3 resonance for very high values of x appears. This behavior is quite similar with our CPA-PWA results for the electromagnetic case. In fig. 4 we plot the threshold value o f / x versus x for each structure labeled by the corresponding resonance(s). We see that the optimum value of x is again 0.12 +-0.03 (for the sl resonance) and the corresponding value of g is about 4. The p3 resonance could give gaps for /~ ~>5 for a wide range of x values (0.5 < x < 0.75). Preliminary calculations for the system studied by Yablonovich and Gmitter [32] seem to indicate that the gap they have observed is due to the " p l " resonance by the dielectric between the spheres. A characteristic geometric dimension of the dielectric is about 6 mm, while the wavelength inside the dielectric which produces the gap is 6.12 mm.

70

'~

60~~d2+s3 SO~

/

= ° 4 0 ~ ~p2

/

/

:oOF x=ot,,

~"sl_ 0

2 4

X= 0.40

_ . , ........ 6 8 10 12 14 E

i

0

,

. . . . . . . . . . . . . .

2 4

6 8 10 12 14 E

0

S_ 2 4

6 8 10 12 14 E

Fig. 3. Positions of the gaps (solid horizontal lines) in an fcc muffin-tin periodic potential ( - 8 inside each sphere, zero in the interstitial region). The quantity x is the volume fraction occupied by the spheres. The energy E and the depth 8 are measured in units of h2/2rna 2, where a is the muffin-tin radius. The thin line passing through the ends of the horizontal lines is a guide to the eye, roughly indicating the trajectory(ies) of the band edges for positive E. The symbols indicate the dominant resonant scatterings responsible for the corresponding gap (see text).

228

E . N . E c o n o m o u / Classical localization 20 18 16

14

ii ¢2 ~i~ \ ~,

~3 II

is3+d2

12 10

'i,

8

\

I

," ~

6 4 2

i

i

20 pl ~ 2 + d l 18~"

i

i

~1

i

i

i

!

:k ,o

i

!!

L, ~\

\~

5i/I

oL ',W-"2 o

o.,

o.2 0.3 d.4 d.s o'.6 d~

d.8 o'.9 1.o

x Fig. 4. Each curve represents the threshold value of the ratio ~ = e2/e 1 for which a frequency gap of a specific type just opens up, plotted against the volume fraction x. The type of the gap (and consequently the corresponding curve) is characterized by the dominant resonance (see text and fig. 1) responsible for its existence.

8. Conclusions

1) The generalized AM versus M method [28], the scalar periodic case [48], and the recent experimental work by Yablonovich and Gmitter [32] strongly suggest that classical localization is possible. Optimum conditions seem to be x = 0.15 (although much higher values of x for the sphere case appear to be favorable). Values of/x -- e2/e 1 ~> 10 seem to be more than sufficient to produce localization (although in the sphere case much lower values of /x may still produce localization). The favorable frequency(ies) to appear to be such that the corresponding wavelength 2"trCE/to should be comparable to the geometric dimension of each scatterer. The scatterers are the slow propagation velocity material. 2) Optical localization was almost achieved experimentally by Drake and Genack [31]. Their data is in reasonably good agreement with the results of the PPA-PWA calculations [29].

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3) CPA-PWA calculations are in fair agreement with the results for the scalar periodic case. 4) CPA-PWA shows that it is easier to localize electromagnetic rather than scaler waves, Hovever, there is no independent verification of this prediction. 5) One favorable regime for localization is x ~ 0 . 1 5 and frequency corresponding to the lower resonance. But CPA-PWA and periodic calculations supported by the experiment of Drake and Genack [31] show that localization can also be achieved (for the sphere case) for high values of x (0.5 ~< x ~< 0.75) and higher resonances (d2, p2, p3). 6) The threshold value of /x may be as low as four. For /~ ~> 10 it seems certain that classical wave localization can be achieved.

References [1] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [2] N.F. Mott and E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed. (Clarendon, Oxford, 1979). [3] J.M. Ziman, Models of Disorder (Cambridge Univ. Press, London, 1979). [4] E.N. Economou, Green's Functions in Quantum Physics, 2nd ed. (Springer, Berlin, 1983), and references therein. [5] E. Abrahams. P.W. Anderson, D.C. LicciardeUo and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [6] P. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 291. [7] B. Souillard, Waves and Electrons in Non-Homogeneous Media, in: Proc. Summer school Chance and Matter, Les Houches, 1986 (North-Holland, Amsterdam, 1987). [8] D.J. Thouless, in: Ill-Condensed Matter, R. Balian, R. Maynard and G. Toulouse, eds. (North-Holland, Amsterdam, 1979). [9] S. John, Phys. Rev. Lett. 53 (1983) 2169. [10] P.W. Anderson, Phil. Mag. B 52 (1985) 505. [11] S. John, Comments Cond. Mat. Phys. 14 (1988) 193. [12] Y. Kuga and A. Ishimaru, J. Opt. Soc. Am. A 1 (1984) 831. [13] M.P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55 (1985) 2692; M.P. Van Albada et al., Phys. Rev. Lett. 58 (1987) 361. [14] P.E. Wolf and G. Maret, Phys. Rev. Lett. 55 (1985) 2696. [15] S. Etemad, R. Thompson and M.J. Andrejo, Phys. Rev. Lett. 57 (1986) 574; S. Etemad et al., Phys. Rev. 59 (1987) 1420. [16] A.Z. Genack, Phys. Rev. Lett. 58 (1987) 2043. [17] M. Kaveh, M. Rosenbluh, I. Edrei and I. Freund, Phys. Rev. Lett. 57 (1986) 2049; M. Rosenbluh et al., Phys. Rev. A 35 (1987),4458. [18] Ping Sheng and Z.Q. Zhang, Phys. Rev. Lett. 57 (1986) 1879. [19] K. Arya, Z.B. Su and J.L. Birman, Phys. Rev. Lett. 57 (1986) 2725. [20] C.A. Condat and T.R. Kirkpatrick, Phys. Rev. Lett. 58 (1987) 226. [21] Shanjin He, J.D. Maynard, Phys. Rev. Lett. 57 (1986) 3171. [22] N. Garcia and A.Z. Genack, Phys. Rev. Lett. 63 (1989) 1678. [23] M. Minnaert, Phil. Mag. 16 (1933) 235. [24] L. Brekhovskikh and Yu. Lysanov, Fundamentals of Ocean Acoustics (Springer, Berlin, 1982).

230 [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

E . N . E c o n o m o u / Classical localization

M. Strasberg, J. Acoust. Soc. Am. 28 (1956) 20. E. Silberman, J. Acoust. Soc. Am. 29 (1957) 925. R.L. Weaver, J. Acoust. Soc. Am. 85 (1989) 1005. C.M. Soukoulis, E.N. Economou, G.S. Grest and M.H. Cohen, Phys. Rev. Lett. 62 (1989) 575. E.N. Economou and C.M. Soukoulis, Phys. Rev. B 40 (1989) 7977. E.N. Economou and C.M. Soukoulis, in: Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989). J.M. Drake and A.Z. Genack, Phys. Rev. Lett. 63 (1989) 259; see also K.M. Yoo, K. Arya, G.C. Tang, J.L. Birman and R.R. Alfano, Phys. Rev. A 39 (1989) 3728. E. Yablonovitch and T.J. Gmitter, Phys. Rev. Lett. 63 (1989) 1950. R.L. Weaver, Wave Motion, to appear. D. Sornette and B Souillard, preprint. M. Bolzons, P. Devillard, Ft. Dunlop, E. Guazzelli, O. Parodi and B. Souillard, preprint. J.L. Pichard and G. Sarma, J. Phys. C 14 (1981) L127, L617. A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47 (1981) 1546; 49 (1982) 695; Z. Phys. B 53 (1983) 1. C.M. Soukoulis, I. Webman, G.S. Grest and E.N. Economou, Phys. Rev. B 26 (1982) 1838. B. Kramer, K. Broderix, A. MacKinnon and M. Schreiber, Physica A 167 (1990) 163, these Proceedings. P.W. Anderson, Phys. Rev. B 23 (1981) 4828. E.N. Economou, C.M. Soukoulis and A.D. Zdetsis, Phys. Rev. B 31 (1985) 6483. E.N. Economou, C.M. Soukoulis and A.D. Zdetsis, Phys. Rev. B 30 (1984) 1686. E.N. Economou, Phys. Rev. B 31 (1985) 7710. T.R. Kirkpatrick and D. Belitz, Phys. Rev. B 34 (1986) 2168. A.D. Zdetsis, C.M. Soukoulis, E.N. Economou and Gary S. Grest, Phys. Rev. B 32 (1985) 7811. C.M. Soukoulis, A.D. Zdetsis and E.N. Economou, Phys. Rev. B 34 (1986) 2253. C.M. Soukoulis, E.N. Economou and G.S. Grest, Phys. Rev. B 36 (1987) 8649. E.N. Economou and A.D. Zdetsis, Phys. Rev. B 40 (1989) 1334. S. John, Phys. Rev. Lett. 58 (1987) 2486; S. John and R. Rangavajan, Phys. Rev. B 38 (1988) 10101. J. Korringa, Physica 13 (1947) 292; W. Kohn and N. Rostoker, Phys. Rev. 94 (1954) 1111. See e . g . D . A . Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum, New York, 1986). M. Sigalas and E.N. Economou, unpublished. See e.g.M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1975). M. Kerker, The Scattering of Light (Academic Press, New York, 1969).