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# Band gap formation and Anderson localization in disordered photonic materials with structural correlations

Edited by Salvatore Torquato, Princeton University, Princeton, NJ, and accepted by Editorial Board Member Pablo G. Debenedetti July 18, 2017 (received for review March 28, 2017)

## Significance

It has been shown recently that disordered dielectrics can support a photonic band gap in the presence of structural correlations. This finding is surprising, because light transport in disordered media has long been exclusively associated with photon diffusion and Anderson localization. Currently, there exists no picture that may allow the classification of optical transport depending on the structural properties. Here, we make an important step toward solving this fundamental problem. Based on numerical simulations of transport statistics, we identify all relevant regimes in a 2D system composed of silicon rods: transparency, photon diffusion, classical Anderson localization, band gap, and a pseudogap tunneling regime. We summarize our findings in a transport phase diagram that organizes optical transport properties in disordered media.

## Abstract

Disordered dielectric materials with structural correlations show unconventional optical behavior: They can be transparent to long-wavelength radiation, while at the same time have isotropic band gaps in another frequency range. This phenomenon raises fundamental questions concerning photon transport through disordered media. While optical transparency in these materials is robust against recurrent multiple scattering, little is known about other transport regimes like diffusive multiple scattering or Anderson localization. Here, we investigate band gaps, and we report Anderson localization in 2D disordered dielectric structures using numerical simulations of the density of states and optical transport statistics. The disordered structures are designed with different levels of positional correlation encoded by the degree of stealthiness

- disordered photonics
- photonic band gap materials
- Anderson localization of light
- mesoscopic wave transport
- hyperuniform structures

Light propagation through a dielectric medium is determined by the spatial distribution of the material. Photons scatter at local variations of the refractive index. For a periodically organized system, interference dominates light transport and is responsible for optical phenomena in opal gems and photonic crystals (1). In random media, transport becomes diffusive through successive scattering events. The characteristic length scale over which isotropic diffusion takes place is the transport mean free path. Materials thicker than the mean free path appear cloudy or white. However, when scattering centers are locally correlated, diffraction effects can be significant. The description of light transport then becomes a challenging problem with many applications, such as the transparency of the cornea to visible light (2), the strong wavelength dependence of the optical thickness of colloidal suspensions (3) and amorphous photonic structures (4, 5), and structural colors in biology (6). Critical opalescence and the relatively large electrical conductivity of disordered liquid metals (7) are closely related phenomena.

In the weak scattering limit, photon transport is diffusive and can be described by a local collective scattering approximation, which states that the mean free path **1**.

For quasi-one-dimensional (Q1D; waveguides) and 2D uncorrelated or fully random media with

The opposite limit of a periodically repeating structure, a photonic crystal with sharp Bragg peaks in the structure factor and a full band gap in three dimensions, was discussed in the late 1980s in the pioneering works of Yablonovitch and John (16, 17). As pointed out by John at that time, perturbative introduction of disorder in a crystal can induce strongly localized states in the photonic band gap (PBG), leading to a pseudogap in the photon density of states (DOS), in analogy with electronic pseudogaps in amorphous semiconductors. Strong Anderson localization might then be accessible in the pseudogap frequency range, and there should be a cross-over between these two transport regimes as the structure factor evolves between the two extreme limits of a structureless random medium and Bragg-peaked shape typical of a full band gap photonic crystal (17). However, 30 years after John’s proposal, an experimental realization or a theoretical description of the statistical properties of photon transport in the cross-over regime from weakly to strongly correlated disordered media remains poorly understood, both in three and two dimensions.

For correlated but fully disordered media, a situation clearly distinct from disordered crystals, attention over the last years focused on the emergence of unexpected optical properties like optical transparency (18, 19) and PBGs in 2D and 3D high-refractive-index disordered materials (12, 13, 20, 21). In particular, the concept of stealthy hyperuniformity as a measure for the hidden order in amorphous materials has drawn significant attention (18, 22, 23). HDPMs are disordered, but uniform without specific defects, and the DOS can be strictly zero. A key parameter controlling structural correlations of HDPM, and thus the band-gap width, is the degree of stealthiness

At the same time, experimental observations of Anderson localization of light have been questioned repeatedly, particularly in 3D systems (25, 26). The current state of the debate is that Anderson localization of light is absent in 3D fully random media (27) and thus can only be reached in the presence of correlations (26). However, correlated disordered media have also been shown to possess full band gaps for materials with similar optical contrast previously associated to Anderson localization (28, 29), both in two and three dimensions (12, 21). Currently, there exists no picture that may allow one to predict or classify the different optical transport regimes in disordered photonic materials with structural correlations. The main goal of our work is to address these fundamental questions and to propose a correlation–frequency (

## Results and Discussion

### Transport Phase Diagram.

A relatively simple example of a correlated disordered medium is a collection of infinite parallel cylinders of high-refractive index that are distributed according to a 2D stealthy hyperuniform (SHU) point pattern (Fig. 1, *Insets*) (18, 22, 23). The area of the pattern

Based on the NDOS and the results for 2D disordered crystals (30, 31), we first make a hypothesis about the transport phase diagram of 2D HDPM in Fig. 1. For strongly correlated, nearly crystalline materials (high

### Numerical Simulations.

Although the calculation of the standard transport mean free path can be useful to describe transport processes in disordered correlated media in frequency ranges where the system is almost transparent or transport is diffusive (9, 19), it loses its meaning in a region where the DOS is zero. Identifying the different transport regimes requires the numerical solution of the full multiple scattering problem in a wide spectral range. We calculate the decay of the intensity of the wave fields along the propagation direction and the full statistics of wave transport. First, we generate a statistical ensemble of 1,000 patterns of *A* shows a map of

To obtain the characteristic decay length *GSM Method* and Figs. S2 and S3). Periodic boundary conditions (Fig. 3, *Inset*) define a set of transversal propagation channels. The optical analog *B* shows a map of *Transport Characteristics*, Table S1, and Figs. S6 and S7).

### Transport Fluctuations and Conductance Distribution.

So far, we identified regions with exponential decay of the conductance, but did not reveal the physical mechanism responsible for this rapid decay. To this end, we analyze the optical transport statistics (34). Fig. 3 shows a color map of fluctuations in the logarithmic conductance

To discriminate between tunneling-like transport and classical Anderson localization (SPS), we consider the statistical distribution

## Isotropy of the SHU Seed Patterns

Eventually, for high enough values of

We find that nearly all considered samples are isotropic for

## GSM Method

We describe the GSM method used in the transport calculations (32). Throughout this work, TM polarization has been used. In this case, the electric fields are parallel to the axis of the cylindrical scattering units (axis along

All structures considered in this work have been generated in a square cell of size

In any given slice perpendicular to

The transverse modes **S10** in algebraic form,**S12** that is now suitable for numerical evaluation. Considering the expressions Eqs. **S13** and **S14**, the matrix

Once we have a complete description of the fields in the

To obtain the scattering matrix of a set of two adjacent slices (

In the calculation of the total scattering matrix of the system, we calculate the scattering matrix of each slice and then accumulate the total scattering by recursively applying the formulas **[S17]**. We set

One of the advantages of this method is that the accumulated scattering matrix at intermediate steps provides transport information for all accumulated lengths of the system (i.e., all of the way from *A* shows the average over 1,000 samples of *B* sketches the procedure followed in the GSM method. The scattering matrix of each system is calculated for lengths

## Transport Characteristics

We perform identical statistical analysis on the *A*–*C*). In Fig. S8 *D*–*F*, the corresponding color maps of

## Conclusions

In conclusion, our numerical results fully support the proposed transport phase diagram in Fig. 1 for 2D photonic structures derived from SHU point patterns. Based on the importance of positional correlations for the appearance of PBGs (13), we expect that other features of the transport phase diagram can also be transferred to disordered point patterns that possess different types of correlated disorder, as long as the patterns become uniform in the large-scale limit—in other words, they are, for example, hyperuniform, nearly hyperuniform (13, 21), or local self-uniform (40). In this case,

It will be interesting to establish a similar transport phase diagram in three dimensions. In this case, a sharp phase boundary, known as the mobility edge and set by the Ioffe–Regel criterion

An important direction for future work is the implications of the phase diagram for electronic transport. It has been argued that SHU plays a role in the formation of electronic band gaps, for example, in amorphous silicon (42). However, the influence of structural correlations and hyperuniformity on electron transport and localization in two or higher dimensions is far from being understood (43).

## Methods

### Generation of Stealthy Hyperuniform Point Pattern.

We use a simulated annealing relaxation scheme to generate disordered SHU patterns with

### Band Structure Calculation.

We calculate the NDOS using the supercell method (1) implemented in the open-source code MIT Photonic Bands (44). The supercell is repeated periodically, and the band structure is calculated by following the path

### GSM Method.

We use the improved GSM method. A summary of the method is provided in ref. 32. In the GSM, the system is discretized in slices in the propagation direction. For each slice, the wave equation is solved and the scattering matrix is calculated. By sequentially combining the corresponding scattering matrices, the total scattering matrix of the system up to a length

### DMPK Equation.

The DMPK equation is a Fokker–Planck equation describing the evolution of coherent wave transport statistics as a function of the ratio of the system length to the transport mean free path. It was shown to describe quantitatively the transport properties of disordered Q1D systems. There is evidence that the DMPK distributions retain the main properties of the conductance distributions in the metallic, critical, and localized regime also in higher dimensions (15, 38). The DMPK distributions were obtained as described in ref. 45 (*DMPK Equation* and Figs. S4 and S5).

### Similarity Analysis.

The similarity between the conductance distributions of correlated systems and the ideal DMPK distribution is characterized by a similarity function. We first create numerically a finite sample of DMPK conductance histograms with a certain bin size. We quantify differences between the ideal DMPK sample and the results obtained by the GSM by calculating a squared distance distribution function *Similarity Function S(𝝂,𝝌)*.

## Relations Between Degree of Stealthiness, Wave Numbers, and Frequencies

Stealthy hyperuniform structures are defined through their structure factor (18). There is a critical wave number

The parameter controlling the structural properties of the samples is the degree of stealthiness

We consider a length unit

If the maximum scattering wavenumber in the effective medium

## Generation of Point Patterns and Band Structure Calculations

We use a simulated annealing relaxation scheme to generate disordered SHU patterns with

NDOSs are calculated by using the supercell method (1) implemented in the open source code MIT Photonic Bands (44). The supercell is repeated periodically and the band structure is calculated by following the path

## DMPK Equation

The DMPK equation (35, 36) is a Fokker–Planck equation describing the evolution of coherent wave transport statistics as a function of the ratio of the system length to the transport mean free path. It is a suitable description for disordered Q1D systems.

We consider a Q1D system of length **S15**. By using the singular value decomposition of each of the scattering matrix entries, the scattering matrix can be expressed in terms of the polar decomposition

All statistical scattering magnitudes can be derived from the statistical distribution of the scattering matrix

We consider that the transport eigenvalues (radial variables) and the unitary matrices (angular variables) are statistically independent. We further assume that the unitary matrices

We further assume that the scattering matrix

The last assumption introduces a scaling length

With all these assumptions, applying **[S16]** to compute the scattering matrix **S20**. An approximate solution for the JPD of **[S21]** gives excellent results compared with microscopic statistical simulations. In particular, the conductance distribution that can be obtained from MC simulation of this JPD function is indistinguishable from the ones obtained from full wave simulations or by using more sophisticated statistical transport theories (37, 45) in the diffusive and localized regimes.

The DMPK conductance distribution at a fixed average **[S21]**.

Fig. S4 illustrates the results obtained after MC sampling of the JPD function **[S21]**. As can be seen, for

## Similarity Function S(𝝂,𝝌)

The quantitative evaluation of the similarity between conductance distributions of the correlated systems and those of DMPK are limited by finite sampling artifacts of the DMPK conductance histograms. We quantify the differences by calculating a squared distance distribution function

We further assume that any histogram showing the same *D*–*F*.

We furthermore assume that any particular conductance histogram should show the same

To assess the similarity between

We regard the overlapping integral (gray shaded area in Fig. S5 *D*–*F* as the probability of a given GSM method conductance histogram to belong to the DMPK ensemble. In fact, this procedure gives a good quantitative measure of the similarity that can be guessed by mere inspection. The red histograms in Fig. S5 *D*–*F* are the GSM-DMPK *D*–*F*, *Right*. In Fig. S5*D*, the similarity is S*A*) looks highly similar to the converged DMPK one (black distribution in Fig. S5 *A*–*C*). This last distribution is built out of *B*, we have a less similar GSM conductance distribution, although still resembling the main features of the DMPK one. In this case, S*F*, we have a null overlap between GSM and DMPK

This procedure for assessing the similarity of a particular GSM conductance distribution to the DMPK model is particularly sensitive to subtle variations in the parameters controlling the sampling and histogramming procedures. To compare both GSM and DMPK ensembles, sampling and histogramming have to be performed following the same procedure. For instance, comparing histograms with the same bin size but different number of samples gives rise to different results, even if the shape of the distribution is similar.

In all cases, the DMPK model results have been calculated by using an MC method using

## DOS

We have calculated the DOS as a function of the stealthiness parameter for systems of four different sizes ranging from *k*-points, the number of disorder realizations, and the number of rods in each pattern. For a given value of

### Local DOS.

Finite size effects are expected in related quantities like the local DOS (LDOS), *A*, we plot the *A*, *Inset* (

## Acknowledgments

This work was supported by the Swiss National Science Foundation through the National Center of Competence in Research Bio-Inspired Materials and through Projects 149867 and 169074 (to L.S.F.-P. and F.S.); Spanish Ministerio de Economía y Competitividad and European Regional Development Fund Project FIS2015-69295-C3-3-P and the Basque Departamento de Educación Project PI-2016-1-0041 (to J.J.S.). M.E. was supported by Deutsche Forschungsgemeinschaft through the Cluster of Excellence Engineering of Advanced Materials Grant EXC 315/2, the Central Institute for Scientific Computing, and the Interdisciplinary Center for Functional Particle Systems at Friedrich-Alexander University Erlangen-Nürnberg.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: frank.scheffold{at}unifr.ch.

Author contributions: L.S.F.-P. and F.S. designed research; L.S.F.-P. and M.E. performed numerical calculations; L.S.F.-P., J.J.S., and F.S. analyzed data; and L.S.F.-P., M.E., J.J.S., and F.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.T. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1705130114/-/DCSupplemental.

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