Finite-difference time-domain analysis of the tunability of Anderson localization of light in self-organized GaN nanowire arrays Free

The design and implementation of photonic devices have conventionally relied on ordered materials and nanostructures, where imperfections are generally considered undesirable. Disordered photonics, the area of photonic research which investigates the complex behavior of light in random or disordered media, offers a paradigm shift in how photonic components can be realized. Though this branch of photonics emerged primarily out of scientific curiosity, research in this area over the past decade has resulted in theoretical propositions, as well as experimental demonstrations of diversified applications based on the complex interplay of multiply scattered light waves in random medium.^1,2 The rich physics inherent in this field of research has already been exploited to experimentally realize random lasers,^3–6 resonators,⁷ optical fibers,^8–10 light harvesters for solar cells,^11,12 and non-invasive imaging systems.^13–15 A phenomenon intricately related to the study and application of disorder photonics is Anderson localization of light in a random medium.^16,17 This effect was in fact originally proposed to explain the behavior of electrons in a disordered solid-state system, where interference of multiply scattered electron waves can result in a complete halt of electron transport.¹⁸ The underlying wave-mechanics dictates that similar effects should be observable in a disordered photonic medium as well, where multiple scattering of light waves should create a spatially localized mode.^19–21 In fact, photonic systems in themselves offer a more conducive environment for observing Anderson localization than does an electronic system as the prerequisites of non-interacting particles and the absence of temporal fluctuations of scattering potential greatly diminishes the possibility of interference mediated localization effects during electron transport, whereas these conditions are readily attained in optics because of the non-interacting Bosonic nature of photons and natural preservation of coherence in the photonic system.¹⁶ 

The optical analogy of Anderson localization has already been studied in a number of materials and nanostructures employing different theoretical and experimental techniques.^19–23 The observation of this phenomenon is governed by the Ioffe-Regel criterion,²⁴ which states that the mean free path of scattered photons must be approximately in the same order of the wavelength of light for localization to occur. Fulfillment of this criterion requires considerably high refractive index contrasts for a three-dimensional (3D) system, thereby making it unfavorable for observing Anderson localization.^25–27 Of particular interest has been the Anderson localization in one-dimensional (1D) and two-dimensional (2D) disordered media, where the confinement of the optical field is more easily achieved than in 3D systems. In fact, coherent waves are found to be readily localized in unbounded 1D and 2D disordered media, whereas in bounded media, confinement can still be observed if the sample size is made significantly larger than the localization radius so that boundary effects are minimized. Based on these principles, Anderson localization of light has been studied in a number of 2D disordered systems, which include GaN and ZnO nanocolumns,^28,29 random fractal array of silicon nanowires,³⁰ core-shell TiO₂ nanowires,³¹ silicon-nitride photonic crystal waveguides,³² and AlGaN nanowires arrays.³³ Different experimental techniques, such as molecular beam epitaxy (MBE),^28,33 metalorganic vapor phase epitaxy,²⁹ metal-assisted etching,³⁰ and atomic layer deposition,³¹ have been employed to grow or fabricate these nanostructures. Among these approaches, molecular beam epitaxial growth is an exceptionally powerful technique for realizing disordered photonic systems owing to its ability to grow nanostructures in a self-organized manner. In particular, a self-organized nanowire-array grown by molecular beam epitaxy offers in itself an ideal 2D random medium for Anderson localization, thereby eliminating the need for any lithography or patterning.

In the present study, Anderson localization of light is investigated in self-assembled GaN nanowires grown by molecular beam epitaxy. Experimentally obtained dimensionalities of these nanowires have been employed to study light trapping in their disordered arrays using the finite-difference time domain (FDTD) technique. Besides studying the case of clearly identifiable discrete nanowires, this study also considers the case of coalesced structures, which are very likely to appear during self-organized growth. The study indicates that in spite of the inherent randomness of the medium, significant control over the localization wavelength and photon lifetime can be obtained by tuning dimensional features of nanowires, which can be effectively controlled during epitaxial growth. Based on the results of FDTD analysis, an empirical model has been presented which shows that within some degree of variability, it is possible to predict both the wavelength and strength of localization in randomly oriented nanowires. The derived relation also points toward the presence of Bragg-diffraction related interference effects, which results in a systematic variation of localization characteristics as nanowire dimensions are varied.

In what follows, the experimental and theoretical techniques employed and also the structures analyzed in this work are discussed in Sec. I. In this section, MBE growth of the self-organized nanowire array with respect to the generation of the random pattern is discussed. The FDTD-based analysis techniques applied in this work are also discussed in this section. In Sec. III, the results of numerical analysis are discussed in detail. This section describes how the dimensional features of the nanowire arrays influence the light trapping phenomenon and how these localization characteristics can be controlled. Finally, in Sec. IV, conclusion of the work is drawn.

The molecular beam epitaxial growth technique of self-organized nanowires has already been extensively utilized to experimentally realize monolithic III-Nitride nanowire arrays on silicon for applications ranging from deep-ultraviolet to near-infrared regime of the electromagnetic (EM) spectrum.^34–36 In the present study, self-organized GaN-nanowire arrays were grown by plasma assisted molecular beam epitaxy on the (001) Si substrate, under nitrogen-rich condition in the absence of any foreign- or Ga self-catalyst. The growth of these nanowires is initiated by the formation of a thin amorphous Si_xN_y layer which facilitates the incorporation of Ga atoms on the growth plane. It is important to note that these nanowires nucleate on the growth substrate spontaneously in a random manner. This is confirmed by scanning electron microscopy (SEM) imaging, which shows that the as-grown nanowires do not possess any preferential nucleation site and hence they naturally form a random pattern. Under optimized growth conditions, the growth of these nanowires is governed by the higher sticking coefficient along the polar c-plane than on the non-polar m-plane. Though the nanowire assembly is expected to grow vertically with a uniform height, non-uniformity of the initial Si_xN_y layer may lead to their tilted growth and consequent coalescing. Coalescence can also arise from high density or large diameters of the nanowires.³⁷ During MBE growth, density and diameter of GaN nanowires are controlled by varying Ga- and nitrogen fluxes and also the temperature of the growth substrate. By tuning these growth parameters, diameters of self-assembled nanowires have been reportedly varied from 25 to 200 nm.³⁸ In this work, the nanowires were grown at a substrate temperature of 800 °C with a Ga-flux of 1.65 × 10⁻⁷ Torr. A top view scanning electron microscopy (SEM) image of the as-grown nanowires is shown in Fig. 1(a).

To study localization of light in the self-organized nanowire structure, the SEM image shown in Fig. 1(a) is transferred onto the FDTD analysis domain as shown in Fig. 1(b). For simplicity of pattern transfer, the nanowires are approximated here as circles. As can been seen from the transferred pattern, a significant number of nanowires are coalesced because of the considered growth conditions. Next, to explain the effect of coalescing and also to investigate the possibility of tuning the localization wavelength, random patterns of GaN nanowires having diameters ranging from 60 to 80 nm and fill factors in between 30% and 50% are generated using a uniform random distribution. These values of diameter and fill factor have been chosen in accordance with experimentally measured dimensions of MBE grown self-assembled GaN nanowires.^33–36 Considering growth-axis to be oriented along the z-direction, a 3D schematic diagram of such a pattern is shown in Fig. 1(c). These nanowires form a 2D random distribution in the x-y plane as shown in Fig. 1(d). It is known that for strong localization in a finite-size disordered photonic system, the localization length should be much smaller than that of the system size.³⁹ The system area considered here is 3 μm × 3 μm, which is expected to be significantly larger than the localization area for the considered dimensions of the nanowires.

The temporal and spatial behavior of light in the self-organized GaN nanowire array is studied using open-source software package MEEP available for electromagnetic simulation using the FDTD technique.⁴⁰ Considering z-axis as the out-of-plane direction, a Gaussian pulse-source is considered at the center of the disordered system such that there is a concentric diffusion of light along the x-y plane. Perfectly matched layers are placed surrounding the computational region to model an open system, which ensures that non-localized EM wave will leave the computational region quickly (in the timescale of few femtoseconds), whereas photons trapped by multiple scattering will stay in the system longer. These confined photons correspond to Anderson localization related resonant modes of the disordered medium. Two different approaches have been adopted to estimate these modes. In the first approach, the transmission spectrum is calculated by placing the Gaussian source at the center of the nanowires assembly. In this case, the flux emanating from the system is recorded till at least 900 time periods (≈1 ps) after the source has turned off. By normalizing it with respect to the input, the transmitted power spectrum is obtained. The second approach of identifying the resonant modes is based on the harmonic inversion of EM wave in time domain. In this technique, the exact frequencies and lifetimes of the modes are obtained by spectral analysis of electric and magnetic fields after the source is turned off. This spectral analysis is done by taking the values of the finite-length fields as a function of time and decomposing it by the filter diagonalization method,⁴¹ which determines the amplitudes, frequencies, phases, and decay constants of the sinusoids corresponding to the resonant modes.

To compare and relate these two approaches, FDTD analysis is performed for the distribution shown in Fig. 1(d), where the nanowires have an average diameter of 70 nm and a fill factor of 40%. Figure 2(a) shows the input Gaussian flux and corresponding transmitted flux obtained from FDTD analysis. The calculated transmission spectrum is shown in Fig. 2(b), which indicates that a photonic bandgap is formed between 305 nm and 335 nm. The sharp peaks observed in the middle of this bandgap correspond to the resonant modes of the system. To obtain wavelength and quality factor of the resonant modes, harmonic inversion and the filter diagonalization method is applied. Results of this analysis show that a resonant mode at λ_localized ≈ 326.6 nm is obtained with a quality factor of 7 × 10⁵ [Fig. 2(b)], which corresponds to a cavity photon lifetime of 121.3 ps. It is to be noted that this resonant mode lies within the transmission gap shown in Fig. 2(b). To understand temporal behavior of the localized mode, time evolution of the electric field component of the EM wave in the disordered medium is also analyzed (animation available in Videos 1 and 2 in the supplementary material). A narrowband Gaussian pulse source, centered around the resonant-mode wavelength of 326.6 nm, is placed in the center of the random array and the electric field amplitude in the vicinity of source is recorded. For comparison, similar analysis is performed for a non-resonant mode centered at 312 nm. As shown in Fig. 2(c), electric field corresponding to the non-localized mode leaks out of the system within few femtoseconds, whereas for the localized mode, the electric field component takes more than hundreds of femtoseconds to leak out. From the slope of the decaying part of the localized mode's electric field, the cavity quality factor and photon lifetime are estimated to be 9 × 10⁵ and 156 ps, respectively. These values are in accordance with the results obtained from harmonic inversion and the filter diagonalization method discussed previously.

To study localization of light in the experimentally grown self-organized nanowire array, FDTD analysis is performed on the transferred pattern shown in Fig. 1(b) for transverse magnetic (TM) electromagnetic waves spanning over 250 nm to 500 nm. The resultant distribution of the electric field more than 900 cycles after the source is turned off is shown in Fig. 3(a). As can be seen, multiple scattering of light leads to interference of counter-propagating waves and consequent spatial localization of the electric field in the random array. Though explicit propagation of light along the z-direction is not considered here, the x-y plane is in fact the transverse plane comprising of the disordered medium, where light is effectively diffused and localized upon random scattering. The quality factor corresponding to this localized mode is calculated using harmonic inversion and the filter diagonalization method and the results are shown in Fig. 3(b). As can be seen, a resonant mode is obtained at 428.1 nm with a quality factor (Q) of 3610.8. Though this value of Q is at least twice of the value obtained for distributed Bragg reflector (DBR)-based resonant cavities of GaN edge-emitting lasers,^34,35,38 higher Q-factors are expected for Anderson localized resonant modes.³³ The obtained diminished Q-factor can be attributed to the relatively small size of the nanowire array in comparison with the localization wavelength, which altogether results in insufficient number of multiple scattering events. The diameter and fill factor of the random array are also expected to have a significant influence herein. To investigate the effect of diameter and fill factor variation, a series of 2D random arrays of GaN nanowires have been considered for FDTD analysis. Based on experimentally obtained dimensions of self-organized nanowires, nanowire arrays having average diameters ranging from 60 to 80 nm and fill factors varying between 30 and 50% are considered. In order to estimate the localized modes in these arrays, the transmittance is calculated using the previously discussed approach in Sec. II. In Figs. 4(a)–4(c), false color plots of the transmittance are shown as a function of wavelength and fill factor for the considered nanowire arrays. As can be observed, low values of transmittance are obtained over wavelength ranges of 250-315 nm, 300-360 nm, and 350-420 nm for respective average diameters of the nanowires. These low-transmittance values or transmission gaps correspond to the phenomenon of light trapping and the consequent formation of resonant modes in the medium. It is noteworthy that in spite of the randomness of the system, the transmission gaps appearing in Figs. 4(a)–4(c) shift toward a higher wavelength as the nanowire diameter is increased.

To further investigate the effect of diameter variation, the wavelength and quality factor corresponding to the longest lived modes in nanowire arrays having different dimensions and random distributions are calculated using the harmonic inversion and the filter diagonalization method. In order to eliminate artifact of the random orientation of the arrays, for each pair of diameter and fill factor, configurational averaging has been performed for eight random realizations of nanowire arrays. Figure 5(a) shows the change of peak wavelength of the highest quality factor mode for a given pair of diameter and fill factor, whereas the effect of fill factor variation on localization strength is shown in Fig. 5(b). Here, data points corresponding to the resonant wavelength and the Q-factor represent average values, whereas the associated error bars indicate the range of variation for each pair of diameter and fill factor. As can be observed, the localization wavelength shifts toward a higher wavelength as the nanowire diameter is increased, while keeping the fill factor constant. These results are in accordance with the observations of the false color plots shown in Figs. 4(a)–4(c), which show red-shift of the transmission gap with an increasing nanowire diameter. According to Fig. 5(b), the average quality factor obtained for these modes is approximately 10⁵, which is consistent with the values reported in Ref. 33. Such high values of Q suggest that the corresponding modes will be strongly localized in the random array. This is further illustrated in Fig. 6, which shows spatial distribution of the electric field in nanowire arrays having diameters and fill factors considered in Fig. 5(a). As can be seen, the electric field component of optical wave remains spatially localized in the random arrays for the diameter and fill factor combinations considered in Fig. 5. It is noteworthy that the Q-factors obtained for these arrays are one to two orders of magnitude higher than the value obtained for the coalesced case of self-organized nanowires shown in Figs. 3(a) and 3(b). This can be attributed to the inhomogeneity in the shape of the scatterers, which arises from random coalescing of adjacent nanowires during self-organized growth. Therefore, coalescing is expected to have significant influence on light localization characteristics in self-assembled nanowires.

According to Fig. 5(b), quality factors of the localized modes apparently randomly fluctuate between 3 × 10³ and 10⁶ as the fill factor is varied while keeping the diameter constant. However, it is important to note that in deriving these results, different random distributions have been considered for each pair of diameter and fill factor. In order to examine the effect of orientation of random scatterers, localized modes have been calculated for two different random configurations of the nanowires having identical diameter and fill factor. These arrays, denoted as random configuration 1 and 2, respectively, are shown in Figs. 7(a) and 7(b) along with electric field distributions of the localized modes. It is obvious that in spite of identical diameter and fill factor, the shape and spatial location of the optical field distributions in these arrays are significantly different because of the difference in the orientation of the scatterers.^42,43 Consequently, a significant difference in both peak wavelength [Fig. 7(c)] and quality factor [Fig. 7(d)] of the resonant modes are observed for these two un-correlated random distributions as fill factors and diameters are varied. To further study the effect of nanowire distribution, correlated random arrays are next considered such that the change in random arrangement of the nanowires do not mask the effect of changing fill factor. To create such distributions, a nanowire array having a fill factor of 5% and a fixed diameter of 70 nm is first generated. Next, the fill factor (Φ) is increased and new patterns are generated by randomly placing nanowires in the existing gaps while keeping the locations of previously placed nanowires unchanged. Thus correlated random patterns having fill factors ranging from 5% to 55% are created and localized modes are estimated for each pattern. Figures 8(a)–8(c) show the electric field distributions of localized modes in three such correlated arrays. The location and shape of the localized field are dependent on the local randomness of these arrays and hence cannot be generalized. However, it is obvious that as the fill factor is increased keeping previous locations of the scatterers intact, the presence of new scatterers in previously unoccupied locations results in enhanced multiple scattering and consequent trapping of the optical field with stronger confinement. Consequently, a rather monotonic increase of the Q-factor with respect to Φ is observed [Fig. 8(d)], which is quite contrary to the random fluctuations observed in Fig. 5(b). This result is a direct consequence of the increase in the number of recurring scattering events resulting from larger number of random scatterers in the medium. For Φ < 10%, the quality factor is ∼100, which practically indicates no localization at all because of the lack of sufficient number of random scatterers. Figure 8(d) also elucidates the variation of localization wavelength as a function of fill factor in correlated random arrays. As can be seen, the localization wavelength though fluctuates over a wide range for lower values of Φ, it becomes confined within the wavelength range of 330-350 nm for Φ > 30%. Hence Φ should be around 30%-50% for the optimum condition of strong localization in self-assembled GaN nanowires. It is promising that this desired range of the fill factor is very much within the gamut of experimentally reported fill factors of MBE grown self-assembled nanowire arrays.^34–38 

The results discussed so far in this study have been based on Anderson localization of the TM polarized wave. To study the effect of polarization, similar analysis has been performed for the Gaussian pulse source of the TE polarized wave. Figure 9(a) shows the false color plot of transmittance for varying fill factors and diameters of the nanowire array with a TE polarized light source placed in the center of the disordered medium. Contrary to the case of TM polarization, no clear transmission gap is observed at any fill factor or diameter, which indicates that the TE polarized wave is only weakly localized in the system. To verify this observation, wavelengths and Q-factors of resonant modes corresponding to TE polarization are calculated using harmonic inversion and the filter diagonalization method, as shown in Figs. 9(b) and 9(c). For all values of diameter and fill factor of the nanowires, the quality factor remains in the order of 10² or lower, confirming weak localization of light wave in the medium. These results are consistent with the findings of previous studies,⁴³ which showed that for the TE polarized wave, no transmission gap is expected for any value of the fill factor or diameter when the refractive index of the scatterers is less than 3.87. Since the refractive index of GaN is 2.6 over the considered wavelength range of interest,⁴⁴ strong localization has not been observed for the TE polarized light source.

Localization characteristics obtained for the TM polarized wave in this study are further analyzed employing the theoretical formulation reported by Arya et al.³⁹ In this approach, the scattering length of the light wave in the medium is calculated using the scattering t-matrix, within the framework of the multiple scattering theory. According to this analysis, strong localization is governed by the condition $l/lc≤1$⁠,where l and l_c denote the scattering length (also known as the diffusion length) and critical scattering length, respectively. These lengths can be calculated using the relations $l=c/(2γ)$ and $lc=3/π(c/ω)$⁠, where c is the velocity of EM wave in the background medium of the dielectric particles, ω is the angular frequency of EM wave, and the parameter γ is given as^39,45

In this expression, the average diameter of the circular scatterers is denoted by d, whereas Φ is the filling factor and ɛ_r is the relative dielectric constant of the scattering material. Based on these relations, the values of l/l_c are calculated and plotted in Fig. 10(a) as a function of wavelength for different diameters and fill factors. It is obvious that for the systems considered in this study, the condition of strong localization $(l/lc≤1)$ is fulfilled for a narrow wavelength range of the EM spectra. Also, the minima of l/l_c shifts toward higher wavelength with the increase of scatterer diameter d and the values of l/l_c decreases with the increase of Φ, while the fill factor and the diameter are, respectively, kept constant. These trends confirm our observation that localized modes move toward a higher wavelength with the increase of nanowire diameters and for a particular random distribution, a relatively higher value of Φ is favored to attain strong localization.

Based on the results of numerical analysis of this work, an empirical relation is formulated to approximate the localization wavelength in a random array of nanowires. To formulate this relation, resonant modes in different random arrays of nanowires having diameters ranging between 60 and 80 nm and fill factor spanning over 30% to 50% are first numerically calculated using the FDTD technique. The obtained localization wavelengths are next analyzed with the following relation such that the root mean square error is minimized:

Here, λ_localized and d are in units of nanometer. Figure 10(b) shows FDTD analysis based numerically calculated values along with the empirical fitting obtained from Eq. (2). It is observed that the localization wavelength is only weakly dependent on the considered range of fill factors. However, it is noteworthy that the nanowire diameter and the localization wavelength are related by an approximate factor of 4, which is similar to the well-known t = λ/4 relation between individual layer thickness (t) and Bragg wavelength (λ) of DBRs. This indicates that in spite of the absence of periodicity, light trapping in the medium is related to interference effects resulting from Bragg-like diffractions. Similar observations were reported in Ref. 46, which showed that the frequency dependence of light trapping in the disordered medium is governed by collective Bragg-diffraction related interference effects, rather than by random scattering of individual scatterers. This explains the observed red-shift of the localization wavelength as the nanowire diameter is increased. To check the validity of this empirical relation for estimating localization wavelength in the long wavelength regime, a random nanowire array having d = 320 nm and Φ = 45% is considered [Fig. 11(a)]. Applying harmonic inversion and filter diagonalization for this configuration, a resonant mode is obtained at 1373.7 nm [Fig. 11(b)], which is 1% within the value calculated using Eq. (2). Attainment of light trapping for this combination of the diameter and the fill factor is further confirmed by the observation of strongly localized electric field distribution in this array [Fig. 11(a)]. The model presented here is also verified against previously reported experimental observation of the Anderson localized mode in self-organized GaN-based nanowires, which had a fill factor of 30% and diameters ranging from 70 to 75 nm.³³ The experimentally measured resonant mode for this array was 334.1 nm, which is well within the range calculated from the derived empirical relation. Though the model formulated here is for an array of clearly identifiable discrete nanowires, it is applicable to the case of coalesced nanowires as well. This is confirmed using the transferred pattern shown earlier in Fig. 1(b), for which a fill factor of 48.5% is calculated and a localization wavelength of 428.1 nm is obtained by FDTD analysis [Fig. 3(b)]. Using these values in Eq. (2), we obtain an average diameter of 93.97 nm, which is well within the diameter range of coalesced structures measured by SEM imaging. Therefore, a formulation like this allows light localization to be treated as an inverse problem, in which the localization characteristics of a random array of scatteres can be employed to estimate its dimensional attributes. A more rigorous and accurate relationship between localization characteristics and dimensional features of the nanowires can be formulated by considering shapes of individual scatterers, a study which is beyond the scope of the present work. Moreover, though we present here the tunability of Anderson localization characteristics in random nanowire arrays, similar tunability should be achievable in their inverted structures, such as random air-holes in semiconductors⁴⁷ or dielectrics, as well. This opens the possibility of designing and realizing disorder based novel photonic structures and integrated systems, where strong-localization characteristics can be tuned as per the requirements of cavity quantum electrodynamics and random lasing related applications.

Anderson localization characteristics in random arrays of self-assembled nanowires have been studied with respect to the variation of dimensional features of the nanowire array. Experimentally, measured dimensions of self-assembled GaN nanowires have been used to define disordered patterns, where the propagation and localization of light are analyzed using the finite difference time domain technique. The results of this work indicate that because of interference resulting from Bragg-like diffraction in the medium, the localization wavelength gradually shifts toward a higher wavelength as the nanowire diameter is increased. Different localization characteristics are observed in the case of un-correlated random arrays having identical diameter and fill factor. However, in the case of correlated random arrays, the localization strength increases with the fill factor because of recurring scattering events caused by additional nanowires. Based on numerical calculations, an empirical model has been developed to estimate the localization wavelength over the range of the ultra-violet to near-infra-red regime of the optical spectrum and the model has been employed to characterize light localization in experimentally grown self-assembled GaN nanowires. The theoretical framework presented here offers the possibility of treating Anderson localization in a random medium as an inverse problem, where light trapping characteristics can be utilized to predict dimensional features of the scatterers. It is also envisaged that the tunability of high-Q resonant modes in the absence of mirrors can be effectively utilized for designing and experimentally realizing disorder-based photonic devices which can operate from the ultra-violet to near-infrared regime of the EM spectra.

Supplementary material includes two video files displaying the temporal behavior of the normalized electric field amplitude within the sample area for localized and leaky modes in a random GaN-nanowire array.

D.J.P., A.A.M., and M.Z.B. thankfully acknowledge the support and facilities obtained from the Department of Electrical and Electronics Engineering, BUET during the course of this research work. A.H. and P.B. acknowledge the support received from the National Science Foundation (NSF) under the MRSEC program (Grant No. DMR-1120923).