Optical Properties of Non-Periodic Dielectric Systems Made of Nanostructured Porous Silicon

Optical Properties of Non-periodic Dielectric Systems Made of Nanostructured Porous Silicon

B. Alvarado Tenorio1, J. Escorcia-Garcia1, Miguel Eduardo Mora-Ramos2,

and V. Agarwal1,a

1Centro de Investigacion en Ingenieria y Ciencias Aplicadas Universidad Autonoma del Estado de Morelos

Av. Universidad 1001, Col. Chamilpa, CP 62209, Cuernavaca, Morelos, Mexico 2Facultad de Ciencias, Universidad Autonoma del Estado de Morelos

Av. Universidad 1001, Col. Chamilpa, CP 62209, Cuernavaca, Morelos, Mexico [email protected] (corresponding author)

Submitted: July 11

th, 2007, revised: December 20

th, 2007, accepted: January 12

th, 2008)

Keywords: porous silicon, Multilayers, Optical properties, non-periodic nanostructures, heterostructures

Abstract. To investigate the optical properties in non-periodic dielectric systems, we study here the

reflection of light from nanostructured porous-silicon-based period doubling heterostructures. The

multilayered systems are fabricated in such a way that the optical thickness of each layer is one

quarter of 650nm. The results for the optical reflectance are presented and compared with that of

Fibonacci, Thue-Morse, and random structures fabricated under the same conditions. Numerical

simulation for the reflectance along the lines of the transfer matrix approach is performed. In

addition, optical reflection from Gaussian porous silicon multilayers is also briefly discussed. We

find that porous silicon Period Doubling dielectric multilayers could demonstrate the optical

properties similar to the classical periodic Febry-Perot interference filters with one or multiple

resonant peaks, but with an advantage of having total optical thickness much lesser than that of the

periodic structures.

Introduction

Formation of a periodic dielectric structure is done by repeating a finite arrangement several times

which leads to the formation of a photonic crystal. On the other hand, a random structure follows no

particular rule and cannot be reduced to a single cell that is repeated several times. Another

category of quasiperiodic or quasiregular (non-periodic) structures are made up of two or more

incomplete periods joined together to form an overall system that is neither fully periodic nor

random but somewhere in between [1]. Such complex dielectrics are often considered as

quasicrystals, and can be more easily obtained in a one-dimensional multilayered configuration.

Several resonant peaks appear on breaking the periodicity of the structure, thus creating propagation

modes within the structure. This property opens the doors for many useful applications. Typical

binary systems of this nature are the Fibonacci (FN), Thue-Morse (TM), and Period Doubling (PD).

All of them exhibit a singular continuous spectrum of the elementary excitations, which is the

consequence of a multi-fractal character. A simple way of visualizing the generation of these

quasiregular sequences in one dimension is through the definition of the so-called substitutional

rules [2-5]. In particular, the Period Doubling sequence is a quasiperiodic system generated by

iterated applications of the substitutional rule of the type (A → AB, B → AA). On the other hand,

Fibonacci (FN) structure is also a quasiperiodic system whose j-th generation, Fj, can be constructed

by defining F1 = A, F2 = BA, together with the addition rule, Fj = Fj−2 + Fj−1. The Thue-Morse (TM)

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sequence is generated by iterated applications of the substitutional rule of the type (A →AB, B →

BA). The TM structure has a singular continuous Fourier spectrum without the presence of Bragg

peaks. It is not quasiperiodic, and can be considered as a critical aperiodic system [6].

On the other hand nanostructured PS has been a subject of extensive study for the last 17

years [7-12]. There has been a lot of interest in the photonic applications of porous silicon (PS).

The high optical sensitivity of PS is due to the possibility of producing the layers with the large

refractive index contrast which is made possible by producing a variation in the porosity of the

structure. This quality of porous silicon makes it a promising material for photonic devices like

Bragg filters and optical microcavities. Multilayered structures with high reflectivity [8], efficient

visible photoluminescence [7], compatibility with standard silicon processes for integrated

optoelectronics [9] and photonic applications [8,10,12], have been major attractions of this field.

The porous silicon can be fabricated, with an easy and cheap process i.e. using electrochemical

etching of crystalline silicon (c-Si) with HF. Moreover, this method gives us the ability of having a

wide refractive index contrast within the same material without having any the problem of inter-

diffusion between the dielectric layers. The advantage of making 1D structures with porous silicon

is that the optical properties can be measured accurately and the samples can be designed and

realized in any desired configuration. Since we can control both, the layer thickness and refractive

index, in principle, from completely ordered to completely disordered structure can be studied

experimentally.

Some previous works on FN and TM dielectric heterostructures based on porous silicon

have been put forward [13-18]. In the case of the PD sequence, only very scarce reports are found

regarding the analysis of the optical transmission of dielectric multilayers [16,19]. Within a

different framework, it is possible to mention the study, regarding the energy spectrum of Period

Doubling quasiperiodic structures, made by Luck [20] and A. Chakrabarti et al. [21], while their

dynamic structure factor has been calculated by A. Ghosh et al [22]. In addition, the propagation

and localization of electromagnetic waves in Period Doubling serial loop structures was recently

presented [23].

Apart from that the idea of Gaussian layered structures has several antecedents in the field of

electronics [24,25]. In the case of dielectric multilayer systems it introduces a novel approach in the

search for application in optics, and optoelectronics. For example, the work proposed by Arriaga et

al. [26] demonstrates the possibility of the formation of omnidirectional photonic band gap with the

PS-based one-dimensional photonic crystals with Gaussian refractive index profile.

In general, the non-regularity associated to the structure of these systems provides a special

kind of dielectric contrast that throughout reflects in the propagation of the light. Therefore, the

random character of the multilayers distribution should also lead to interesting features in its optical

properties. In this subject, disordered dielectric layered systems have been studied in the past by

randomly choosing the thickness of the layers and keeping the optical length of each ingredient

layer approximately equal to one quarter of a given wavelength [27,28]. Random Bragg reflectors

have been studied by Pavesi group [29] by keeping the thickness and the porosity of the low/high

porosity layer constant. The thicknesses of the high porosity layers were generated randomly in

such a way that their histograms form a Gaussian distribution.

So far there have been no previous reports on PS-based PD dielectric structures. Therefore,

the main motivation behind the present research is to present experimental measurements of the

properties of light propagation in this kind of structures and to compare their output with those

arising from some other non-regular systems fabricated from PS multilayers that follow the building

patterns of FN, TM, and Random Binary (RM) systems. They are all fabricated under the same

conditions. This means that the optical thickness of each ingredient layer is one quarter of 610 nm.

The comparison with the periodic structures is presented as well. In addition, some results regarding

light propagation in PS multilayers with Gaussian thickness profiles are also presented.

70 Journal of Nano Research Vol. 5



Experimental Details

For fabricating nanostructured PS-based multilayers, we have used boron doped, p++

type

crystalline silicon with resistivity 0.001−0.005 ohm-cm, (100) oriented substrates. For

electrochemical anodization process at room temperature, the electrolyte with volumetric ratio of

3:7 of HF (48 wt%) and ethanol (98 wt%) respectively was taken. The current density was

controlled by a computer interfaced electronic circuit. The 1D - PD quasiperiodic photonic

structures can be generated by stacking together the layers of different dielectric materials namely A

and B. The refractive index corresponding to each porosity was calculated by interferometric

method for a monolayer of approximately two microns at 1500nm of wavlength. The two current

densities used for making all the quasiregular heterostructures considered were (a) 5 mA/cm2,

refractive index nA = 2.0 (anodization time: 22 sec), and (b) 80 mA/cm2 (nB = 1.2, anodization

time: 4.6 sec). The thicknesses of the layers (as measured using Scanning Electron Microscope

(SEM)) were 80 nm for the layer A (5 mA/cm2) and 140 nm for B (80 mA/cm

2). The thicknesses of

Figure 1. Cross-sectional SEM image of a typical

(a) Fifth generation period doubling structure (abaaabababaaabaaabaaabababaaabab).

(b) Sixth generation Fibonacci structure (abaababaabaababaababa).

(c) Fifth generation Thue-Morse structure (abbabaabbaababbabaababbaabbabaab).

(d) periodic structure (abababababababababab).

(a)

(a) (b)

(d) (c)

Journal of Nano Research Vol. 5 71



the layers are chosen in such a way that the optical thickness of each layer is approximately one

quarter of 610 nm. For the characterization and analysis of the optical properties of period doubling

dielectric heterostructures, absolute reflectance spectra was measured using specular reflectance

accessory (URA) of a Perkin Elmer UV-Vis-NIR spectrophotometer (Lambda 950) at 80 angle of

incidence.

Results and Discussion

Here we present some SEM images and the results of the optical measurements of the

different non-periodic PS-based multilayer structures considered in this work. Some numerical

simulations are included in order to compare with the experimental output. The calculation of

reflectance properties in these systems is based on the transfer matrix model [5,30,31].

Fig. 1 shows a typical scanning electron micrograph of fifth generation of PD sequence

(Figure 1a), sixth generation of FN (Figure 1b), fifth generation of TM series (Figure 1c) and a

periodic structure (Figure 1d). White and black layers represent the low and high porosity layers

respectively.

Fig. 2 shows the calculated and experimentally measured reflectance of PSPD

heterostructures fabricated in such a way that the optical thickness of each layer is one quarter of

610 nm. The orders of the PD generations in the multilayered structures are 5 (32 blocks), 6 (64

blocks) and 7 (128 blocks), respectively.

Figure 2. Theoretical and experimental reflectivity spectra of 5th, 6

th and 7

th generation of Period Doubling.




Taking into account the strong light absorption of the PS in the short wavelength regions, we

were only interested in the reflectance signal above 500 nm. It can be seen that the fabricated

samples show rather wide regions of high reflectivity, exhibiting a good accordance with the

theoretically predicted signal. As expected, higher generations lead more directly to the appearance

of pseudo-photonic band gap (PBG) regions because of the increasing dielectric contrast in the

system.

It can be observed in the fifth generation of PD structures (PD-32) that the main pseudo

PBG in the visible region appears in the wavelength range of 570 nm to 700 nm, with a fairly strong

photon confinement at 650 nm. The strongest confinement (resonant peak) is observed in the near

infra-red region at 854 nm, which leads to the observation of microcavity configuration (from 760

to 930 nm). In the sixth generation of period doubling structure (PD-64), the position of the main

pseudo PBG remains almost unchanged, but an additional photonic confinement can be observed at

620nm. The PBGs appearing in the NIR range are found to move slightly towards the visible

region, with an additional confinement at approximately 864nm.

Observing the wavelength range from 754nm to 960nm of the PD-32 and 720 to 940nm of

the PD-64, one can deduce that the structures are capable of producing resonant peaks in a

microcavity configuration. Analyzing in terms of optical thickness, PD-32 sequence has an

advantage, over the classical periodic Febry-Perot interferometric structure, fabricated for 848 nm

as a resonant frequency, of having lesser optical thickness of the structure (4880 nm) than the

periodic structure (5088 nm for a complete wavelength cavity structure at 848nm, added to another

few thousand nanometers more for creating any confinement mode in the visible region).

Figure 3. Theoretical and experimental reflectivity spectra of 6th

and 7th

generation of Fibonacci.




The seventh generation of the period doubling structure (PD-128) also shows a highly

fragmented spectrum. The main pseudo band gap appears in the visible range from 552 nm to 634

nm. Apart from that strong light confinement is observed at 640, 670 and 698 nm. The percentage

reflection of peaks appearing in the visible region (apart from the main PBG) is observed to be in

the range of 70-85%. Another band gap appears in the visible region extending up to the NIR region

(750-974 nm). Within this band gap few resonant modes appear at 814, 842, 876 and 890 nm of

wavelength.

In these PD heterostructures one observes that the experimental reflectance signals in the

PBG regions fall short to achieve the unity. This can be explained by assuming the presence of

optical losses in the system. In general, the losses detected can be attributed to different effects. In

our case, knowing that light absorption will occur for high frequencies, we are inclined to consider

the leading scattering mechanism as to be due to interface roughness. However, there is always

some kind light of light scattering due to the microscopic structure disorder.

For the sake of comparison, Fig.3 presents the reflectance of PS-based FN multilayer

heterostructures of 6th and 7th generation, respectively. Contrary to the PD structures, sixth

generation of Fibonacci structure (FN-21), having in total 21 layers, presents only one pseudo band

gap from 724 to 940 nm, with no resonant peaks. Similarly the seventh 5 generation of FN (FN-34)

shows a pseudo band gap from 688-910 nm with a resonant peak at 876 nm.

Besides, Fig. 4 shows the reflectance for the 5th (TM-32) and 6th (TM-64) PS based Thue-

Morse generations respectively. In these structures one can see the presence of wider photonic band

gaps, in the near infrared region (TM-64) 824-1106 nm, with two modes at 864 and 1004nm.

Another strong mode appears in the range of 700-797nm at 746nm. Furthermore, the agreement

between theory and experiment is even better in the FN and TM series than in PD series. However,

the damping effects associated to the light absorption (extending over a larger range towards the red

part of the spectrum) are more apparent for lower wavelengths in the FN structures than in PD

structures.

Figure 4. Theoretical and experimental reflectivity spectra of 5th and 6

th generation of Thue-Morse.




From the analysis of SEM micrographs of Fig 1, it is seen that interface quality is better for

the TM sample as compared to the interfaces present in PD sample. This helps to support the

interface scattering argument given above, regarding to the losses detected in the PD signals of

Fig.2.

Similarly some random structures, made under the same conditions were also found to

demonstrate interesting optical properties. Fig.5 (R-i10) shows a sequence generated from partially

(10 layers from the middle part of the sequence) randomizing periodic 20 layered ‘ab’ system. The

sequence is as follows: ‘ababababaabbbaababab’. The constituent a and b blocks are fabricated

bearing the same optical thickness than in the quasi regular systems discussed above. The structure

shows two sharp resonances at 708 and 790 nm, which is similar to double microcavity

configuration. In this sense, it is more like the behavior of the PD structures studied here. That is,

multiple resonance peaks appear even for a small number of active dielectric layers involved, which

implies that the similar properties can be achieved with much lesser optical thickness than the

periodic structure. Second part of Fig.5 (R-i20) shows the reflectance spectrum of a partially

randomized 40 layered sequence ‘bbabbbabaaabaabababab-babbbabaaabaabababa’ fabricated

with the same layers as the other non-regular structure. It demonstrates the formation of three major

confinement modes at 540, 625 and 700 nm respectively.

Figure 5. Theoretical and experimental reflectivity spectra of two partially randomized sequences

(a) 20 layered structure: bbabbbabaaabaabababa and (b) 40 layered structure: bbabbbabaaabaabababa-

bbabbbabaaabaabababa.

Some other configurations, which were made for the sake of comparison, consisted of a

periodic arrangement of 20 layers with a Gaussian distribution in thicknesses. First row of the Fig.6

(G-20) shows the reflectance spectra of the structure with a periodic variation in refractive index

and a Gaussian variation in thicknesses of the low porosity layer. The thickness of the low porosity




layer was varied from 175 nm to 290 nm and the thickness of the high porosity layer was kept

constant at 110 nm. One can observe (G-20) that there are no confinement modes in visible region.

Finally, Fig.6 (G-40) presents the experimental and theoretical results of light transport in a

structure with a Gaussian profile in the thickness of the low porosity layer (similar to the one

presented in G-20), but the sequence is repeated twice i.e. now the structure has total 40 layers. The

values of the refraction indices na and nb are kept equal to the corresponding ones in the other non-

regular heterostructures investigated here. One can observe the appearance of three resonance

peaks, at 904, 1042 and 1132 nm within the photonic bandgap of 424 nm in the infrared region.

Figure 6. Theoretical and experimental reflectivity spectra of (a) G-20, 20 layered periodic sequence having

a Gaussian distribution in the thickness of the low porosity layer (b) G-40, 40 layers in a periodic sequence

having a Gaussian distribution in the thickness of the low porosity layer for each 20 layers i.e. G-20 made

twice.

Conclusions

In an attempt to create resonant peaks, period doubling structures seem to be appropriate in the

visible wavelength range of 600-700nm and in the NIR range of 800-900nm. On the contrary FN

structures did not present any sharp resonant peaks. TM structures also present resonant peaks in the

visible (only for the higher generations) and near infrared region. These proposed structures can be




useful for the applications in optical modulation and biosensors. Period doubling sequences (in

particular PD-5 sequence) can be used as a reflector in a microcavity configuration, with an

advantage of having the total optical thickness of the structure much lesser than the classical

periodic Fabry-Perot interferometric structure and can be more sensitive than the periodic structures

for biosensing due to the lower number of interfaces.

The above discussed results allow us to conclude that PD PS-based dielectric aperiodic

structures depart more from ”periodicity” than FN and TM structures do. Then it appears that with

an easily controllable structure like PD, one can get results analogous to the random systems,

viewing the occurrence of multiple and sharp resonant peaks. In general, non-periodic multilayers

reveal to be a promising ingredient to fabricate porous silicon microcavities with multiple

resonances.

Acknowledgements

We acknowledge the SEM facility provided by Dr. M Terrones (IPICyT, SLP) and the technical

help provided by Daniel Ramirez Gonzalez (IPICyT, SLP). VA and M.E.M.R acknowledge the

financial support from CONACyT, 57631 and 52304 respectively.

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