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Physics Letters A 313 (2003) 485–490

www.elsevier.com/locate/pla

About wave localization in two-dimensional random media

Zhen Ye∗, Bikash C. Gupta1

Wave Phenomena Lab and Center of Complex Systems, Department of Physics, National Central University,Chungli, Taiwan 32054, Taiwan, ROC

Received 30 March 2003; received in revised form 21 May 2003; accepted 23 May 2003

Communicated by R. Wu

Abstract

In this Letter, we wish to discussion some basic questions pertinent to the phenomenon of Anderson localization ofwaves in two-dimensional random media. Although a definite answer to the two-dimensional localization is not yea common consensus has been reached based upon the scaling analysis Abraham et al. [Phys. Rev. Lett. 43 (1979is, all waves are localized in two dimensions for any given amount of disorders. This view has been prevalent for moredecades. Here, we explain some recent results and considerations that tend to be contrary to this view or the consequ 2003 Elsevier B.V. All rights reserved.

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The concept of localization was originally introduced by Anderson [1] for electrons in a crystal.the case of a perfectly periodic lattice, except ingaps all the electronic states are extended and areresented by Bloch states. When a sufficient amoundisorders is added to the lattice, for example, inform of random potentials, the electrons may becospatially localized due to the multiple scattering bydisorders. In such a case, the eigenstates are expotially confined in the space [2]. The inception of thecalization concept has opened a new era for the sof electrons in disordered systems, and stimulatetremendous research. The concept of localization

* Corresponding author.E-mail address: zhen@phy.ncu.edu.tw (Z. Ye).

1 Now at Department of Physics, University of Illinois, ChicagIL, 60612, USA.

0375-9601/03/$ – see front matter 2003 Elsevier B.V. All rights reserdoi:10.1016/S0375-9601(03)00854-5

-

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also rendered a great development in many other fisuch as seismology [3], oceanology[4], and randlasers [5], to name just a few. The great efforts hbeen summarized in a number of excellent revie(e.g., [2,6–11])

By a scaling analysis [12], Abraham et al. sugested that there can be no metallic state or meinsulator transition in two dimensions in zero magnefield. In other words, all electrons are always locized in two dimensions (2D). The fact that the eletronic localization is due to the wave nature of eletrons has led to the conjecture that the localizatphenomenon also exists for classical waves, sucacoustic and electromagnetic waves, in random meConsiderable efforts have been subsequently devto the study of classical wave localization in 2D (e.[13–19]). And all predictions for the electronic locaization are believed to hold for classical waves. Thuwas widely accepted that all classical waves are lo

ved.

486 Z. Ye, B.C. Gupta / Physics Letters A 313 (2003) 485–490

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ized in 2D random media. This has been the prevaiview for the past many years. Hereafter we will reto this view as ‘2D conjecture’.

In recent years, however, new experimental resstarted to surface and indicated that unusual metbehavior could exist in 2D random electronic syste[20–22], as reviewed in [23]. Although it has benow widely accepted by the community that tunusual metallic behavior shown in 2D systemscaused by electronic interactions, which havebeen included in the consideration of the previotheory [23], significant disputes remain. For exampit is pointed out in [24] that the previous view on 2localization is apparently incomplete and maybe,the general case, incorrect. Whether all the electrostates in 2D disordered media are localized withelectronic interactions therefore still poses an oquestion [25]. A principle of scientific objectivittowards controversial issues may be: it is not fto simply ignore the work of others, whether thmeans contrary ideas or embarrassing facts [26].these motivate us to consider further the problemlocalization in 2D disordered systems.

In this Letter, we propose that the popular view2D localization may not be universally true. Here wwill restrict to classical waves. For the purpose,will first discuss the ‘2D conjecture’ and brief the curent supporting theory on 2D wave localization [2,1Then point out possible ambiguities in the theory adiscuss the evidence that is in conflict with thesults that have beenthought to support the ‘2D conjecture’. Specially, we take the following steps. (1) Dcuss the previously scaling analysis. (2) Examinepredictions from the current theory that supports‘2D conjecture’, verifying its validity. (3) Examine thprevious experimental and numerical results that clto support the ‘2D conjecture’, checking for their apropriateness. (4) Find the self-conflicting pointsthe current supporting theory, and discuss an appamechanism which is in conflict with the theory.

The idea is that while it is definitely hard to prvide a definite answer to the long standing problin one endeavor, the hope to reaching a final ansmay be raised in finding possible vagueness in theconsiderations for improvements. If there is indevagueness, the application of ‘2D conjecture’ to clsical waves should be at least skeptical. A good thabout two dimensions is: wave scattering in many

t

systems is exactly calculable following the ideasFoldy [27] and Lax [28], and formulations of Twesky [29], thereby allowing for close checks on preous predictions. To be brief, we will only outline thideas. The discussion leading us to the conclusionwaves are not necessarily always localized in 2D wrely on three sources: some published or to be plished results, and some new results. The technicatails will not be presented here.

1. The 2D conjecture and the supporting theory

1.1. The conjecture

In [12], an hypercubic geometry is used for tscaling analysis. In the metallic state, the resistafollows the Ohmic behaviorR ∼ L2−d , whered is thedimension. For a localized state, the resistance grexponentiallyR ∼ eL/L1, whereL1 is the localizationlength. A scaling function is defined asβ = ∂ lnR

∂ lnL .

Then the asymptotic behavior is obtained

(1)β ∼{

lnR, asR→ ∞ (Localized),2− d, asR→ 0 (Ohmic).

From the asymptotic behavior in Eq. (1), one csketch the universal curves ind = 1,2,3 dimensionsThe central assumptions in [12] are (1)β is contin-uous; (2)β is a function ofR and depends on otheparameters such as disorders and length scalethroughR, i.e., the single parameter assumption; a(3) once wave is localized, the increasing samplewould always mean more localization.

The behavior ofβ is plotted in Fig. 1. In the 3Dcase, the curve crosses the horizontal axis, yieldin

Fig. 1. The scaling functionβ vs. lnR from Eq. (1).

Z. Ye, B.C. Gupta / Physics Letters A 313 (2003) 485–490 487

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unstable fixed point (B). Above this point, the wavebecome more and more localized as the sampleincreases. Below the critical point, the system teto follow the Ohmic behavior as the sample sizeenlarged. This fixed point separates the localizednon-localized states. For the two-dimensional casethe Ohmic regimeβ approaches zero as ln(R) → 0.But the perturbation calculation including the wainterference effect shows thatβ is always greater thazero. Therefore, for both one and two dimensions,curves do not cross the horizontal axis, and therthus no fixed point. As the sample size increasesstates move towards the localization regime. Thisto the conclusion that all waves are localized in oand two dimensions.

The above analysis is not rigorous, and the sgle parameter assumption is pointed out to be pobly inappropriate [31]. And the conclusion has bechallenged (see the review [23] and [31]). Recenthrough direct calculation of the conductance withuse of the Kubo formula [24], Tarasov showed ththe difference between 2D and 3D localization issignificant. It is further pointed out recently that tdifferentiation should be made between the prohibpropagation and localization [32]. The scaling analyseems more appropriate for propagation phenomand it has been shown that a prohibited propagadoes not necessarily lead to localization [32]. In adtion, the recent discussion by one of us suggestedthe prevailing ‘2D conjecture’ may involve ambiguoboundary effects [33], and such effects may makeconjecture less problem-free.

1.2. The supporting theory

For the sake of a general reader, now we briereview the existing theory for localization that tento support the ‘2D conjecture’ discussed above.a full account, the reader is referred to the excelbook [8] and Ref. [13]. As wave propagates in randmedia, it experiences multiple scattering, and aresult, the wave loses its phase, leading to the gradecreases of the coherence of the wave in the absof absorption. Meanwhile, diffusive wave is buup as more and more scattering takes place.procedure to obtain the localization state can be brisummarized as follows.

,

le

The quantity,D(B) which is a measure of diffusioof classical waves is called the classical Boltzmdiffusion constant and it may be derived undercoherent potential approximation, and is given as

(2)D(B) ∼ vt l

d,

wherevt is the transport velocity,l is the mean freepath andd is the dimensionality.

As waves scattered along any two reversed paththe backward direction interfere constructively, leaing to the enhanced backscattering effect, which wadd corrections to the diffusion coefficient. In the fietheory approach, such an enhanced backscatterinfect is represented by a set of maximally crossedder diagrams [30]. In the two dimension case, the euation of these diagrams leads to an integrationwhich two cut-off limits have to be introduced to avothe divergence. The correction to the diffusion costant for two-dimensional systems is thus found as

(3)δD ∼ − ln(LM/lm),

whereLM and lm are the two cut-off limits. It isthen interpreted in the previous theory that the cutlimit lm is a measure of the minimum scaling for twaves and is thought to be related to (for exampthe mean free path, whereasLM is a measure of theffective size of the sample. It is rather importato note that the correction in Eq. (3) is not onnegative but diverges asLM approach infinity. Thisis obviously unphysical, since the corrected diffusconstant cannot be negative. To avoid the problemwas suggested thatLM is related to the localizatiorange, or simply the localization length, in such wthat whenLM is equal to the localization lengtdenoted byξ say, the corrected diffusion coefficiebecomes zero:

(4)DR(ξ)=D(B) + δD(ξ)= 0.

The localization lengthξ is subsequently solved fofrom this equation. It is obvious that this equatialways allows a solution.Therefore, a localizationlength can always be found in two dimensions. Sucha backscattering induced absence-of-diffusion mecha-nism supports the ‘2D conjecture’.

488 Z. Ye, B.C. Gupta / Physics Letters A 313 (2003) 485–490

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2. The discussion

2.1. Comparison between the prediction from thecurrent theory which is thought to support the ‘2Dconjecture’ and the exact numerical results for anexactly solvable model

Among many, there are two basic ways to proceFirst, the key parameter obtained from the curr

theory is the localization length. This quantity can abe obtained exactly by numerical computation usthe scheme detailed in [29]. Then we can make a cparison of the two results. We consider the modeacoustic scattering in water with air-cylinders detaiin [35]. The comparison is shown in Fig. 2. The coparison clearly shows that there is a significant dference between the numerically exact results andresults obtained from the theory. Further comparisindicate that the difference between the two resultsignificant not only quantitatively but qualitatively.

Second, one can first evaluate the localizatlength from the theory. Then using an exactly soable model, inspect the spatial distribution of the waenergy density and check whether localizationcurs when the system is larger than the theoreticobtained localization length. Again we consider tmodel of acoustic localization in air-filled cylindein water. From Fig. 2, the theory would predict ththe shortest localization length is aroundka = 0.005which is the nature frequency of an air-cylinder in wter [35]. The numerical results show, however, t

Fig. 2. Localization length (ξ ) is shown as a function of frequenc(ka) for β = 0.001. The dashed curve with circles representexact values obtained numerically while the solid curve is obtaifrom theory. Herek is the wavenumber,a is the radius of thecylinder, andβ is the fraction of area occupied by the cylinders punit area.

no localization occurs around this frequency. Instelocalization appears at somewhat higher frequeranges with the same system size, referring to Fiof [35].

2.2. On the previous experimental and numericalevidence

It was considered that the previous theory2D localization has been tested experimentallybe successful. We find that the claimed succesmainly based on two types of experiments. Onethe indirect method which measures the effects ofenhanced backscattering (e.g., Ref. [36]). This kof measurements actually measure the effect so-caweak localization. It is known that weak localizatiois not directly related to the localization describabove [10].

Asides from few exceptions [14], the other tyof experiments is based on observations of the exnential decay of waves as they propagatethrough dis-ordered media. This is pointed out in Ref. [37]. Acording to the above and the reasons to be disculater, this type of experiments isnot sufficient to dis-cern whether the medium really only has localizstates. Unwanted effects of non-localization origin calso contribute to the exponential decay, making dinterpretation ambiguous. In a conclusion, Sigalaal. [17] pointed out thatthere is no conclusive exper-imental evidence for localization of EM waves in 2D.We mention that there was a report of the observaof microwave localization in two dimensions whentransmitting source is inside disordered media [1However, the diffusion based theory has not beenified against this experimental result. Rather, thisperimental result can be explained by the numersimulation [38,39], following the coherent-phase pture of localization which is to be addressed later.

The same situation may also be said about themerical simulations. Take the models in Refs. [17,as the example. The authors considered the EM wpropagation in random arrays of dielectric cylindeThe localization length is computed from the redution in the transmissionacross the random sampleFollowing their methods, we first compute the tranmission vs. the sample size at various frequenciesthe source and the receiver located at the opposides of a completely random array. We do obse

Z. Ye, B.C. Gupta / Physics Letters A 313 (2003) 485–490 489

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exponential decays, and the decay rates depend ofrequency. As in [17,18], the localization length is esmated from the decay rates. Then as long as the sasize is bigger than this length, we would also expecobserve the same exponential decay in the transsion when the source is put inside the medium. Butfound that the exponential decay disappears for sfrequencies. Here we report one example for breby considering the model in [17]. We use the pameters in [17] in the computation. The dielectric costants of the cylinders and the medium are 10 and 1spectively. The fraction of area occupied by the cylders per unit area, is 0.28. The radiusa of the cylin-ders is 0.38 cm. The lattice constantd of the corre-sponding square lattice array is calculated as 1.28All lengths are scaled by the lattice constantd . As anexample, the results for two frequencies are showFig. 3.

Here we see that the exponential decay14.70 GHz, shown when the transmission is acrthe sample, disappears when the source is movedthe medium. The results suggest that waves are nocalized at this frequency. One may still argue thatnon-localization is due to the fact that the localizatlength is long compared to the sample size in theside’ case. Even if this were the case, the exponedecay shown for the ‘Outside’ case could not be dto the localization effect. Reiterating, it is not sufficieto extract the localization effect by merely computi

Fig. 3. The logarithmic average transmissionT and its fluctuationvs. the sample size for two frequencies. The estimated slops fotransmission are indicated in the figure. The ‘Outside’ and ‘Insrefer respectively to when the transmitting source is located outand inside the medium. The former is the scenario in [17]. Mdetails on the explanation of the ‘Inside’ and ‘Outside’ notatioand about the model are in [34].

ethe transmission reduction across the sample. Thfore, the claim about 2D localization like in [17] manot be sufficient. At 11.75 GHz, we observe thatexponential decay with nearly the same slop holdsboth cases, and reveals the expected localizationhavior. Note here that for the ‘Outside’ case, the finwidth will be an effect. We have done the followinsimulations. With fixed lengths of sample, we plot ttransmission as vs. width. We find when the widthlarge enough, the transmission will saturate to cervalues. Then we plot these values vs. sample lenThe exponential decays observed in Fig. 3 remaindicating that the exponential decays are not causethe finite sample width. A cause of the decay is likethe reflection at the boundary facing the source whhas not been avoided by current experiments normerical simulations.

2.3. The physical picture of localization

It seems that a general picture of localization mbe obtained. For quantum mechanic or acoustic wa(the same argument also holds for EM systemes [3the current can be written as�J ∼ Re[ψ†(−i)∇ψ],whereψ stands for the wave function for quantumechanical systems and for the pressure in acousystems. Writing the field asψ = |ψ|eiθ , the currentbecomes �J ∼ |ψ|2∇θ. It is clear that whenθ isconstant at least by domains while|ψ| = 0, theflow stops, i.e., �J = 0, and the wave is localized ispace, i.e.,|ψ|2 = 0. Obviously the constant phaseθindicates the appearance of a coherence in the sysThis coherent-phase picture has been demonstratesuccessfully not only for two-dimensional media [3but for one and three dimensions as well [40].

The current diffusion-based theory does not suppthe above mechanism. The physical picture oftheory is: waves will undergo a diffusion process whthe system is smaller than the localization size. Assystem increases, the diffusion gradually diminisand finally comes to a complete stop when the sexceeds the localization length. If this picture wevalid, then one would expect a significant changethe spatial distribution of the energy density, from tof diffusion to the exponentially confined envelop,expected from the scaling analysis [12]. This is nevident in the theory and is not supported by numerresults. In contrast, the numerical results in Fig

490 Z. Ye, B.C. Gupta / Physics Letters A 313 (2003) 485–490

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In summary, concerns have been raised aboutprevious claim that all waves are localized in 2D. Evif previous evidence showing non-localized states2D [35] could yet be argued to be due to the finsample size limited by computing facilities, there astill many other reasons for being doubtful about‘2D conjecture’. Some main reasons are corroborahere.

Acknowledgements

The work received support from NSC and NCB.G. is supported by a post doctoral fellowship throuthe grant from NSC. Discussion with various coleagues is appreciated.

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