improving the efficiency of thermophotovoltaic devices ...

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Current Trends in Energy and Sustainability 2015 Edition Invited Editors:

Roberto Gómez-Calvet José M. Martínez-Duart

Symposium on Energy and Sustainability. XXXV Biennial. Spanish Royal Physical Society Gijón (Spain), July 13-17. 2015

Book title: Current Trends in Energy and Sustainability. 2015 Edition Invited Editors: Roberto Gómez-Calvet , José M. Martínez-Duart. ([email protected] - [email protected])

Copyright © 2015, Real Sociedad Española de Física ISBN: 978-84-608-5438-8 Depósito Legal: M-4058-2016 Dep. Legal: ALL RIGHTS RESERVED. This book contains material protected under International and Federal Copyright Laws and Treaties. Any unauthorized reprint or use of this material is prohibited. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author / publisher.

Current Trends in Energy and Sustainability. 2015 Edition

Current Trends in Energy and Sustainability 2015 Edition

Symposium on Energy and Sustainability. XXXV Biennial. Spanish Royal Physical Society Gijón (Spain), July 13-17. 2015

CONTENTS PREFACE

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PART I: PLENARY AND INVITED CONFERENCES Plenary Symposium Conference: Energy, Sustainability and Innovation. Cayetano López. CIEMAT

11

The three energy revolutions in Spain. Andrés Seco. RED ELECTRICA ESPAÑOLA

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Energy Group Activities of the Energy Group of the European Physical Society. Jef Ongena. Chairman of the EPS Energy Group

41

PART II: CONTRIBUTION CHAPTERS I

Economic growth, renewable energy and CO2 emissions: the Kaya identity and the environmental Kuznets curve. García-Ramos, JoséEnrique, Golpe, Antonio A., Mena-Nieto, Ángel and Robalino-López, Andrés

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II

Methods for capacitive energy production from salinity differences. Jiménez María L., Fernández María M., Ahualli Silvia, Iglesias Guillermo, Delgado Ángel V.

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III

Improving the efficiency of thermophotovoltaic devices: golden ratio based design. Macía, Enrique.

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IV

Liquid metal research for a fusion reactor: overview and CIEMAT activities. Martín-Rojo, A. B, Oyarzabal E., and Tabarés F.L.

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V

Magnetoresistive-based smart wattmeters network for active power measurement. Jaime Sánchez, Diego Ramírez, Sergio Ravelo, Susana Cardoso, Paulo P. Freitas

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VI

Integration of high-penetration intermitent renewables into the grid: technical and economic considerations. J. Martínez-Duart, S. SerranoCalle, J. Hernández-Moro, R. Gómez-Calvet.

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VII

Band alignment between CuGaS2 chalcopyrite and phovoltaic interesting heterointerfaces. Castellanos Águila, J. E., Palacios, P., Conesa, J. C., Arriaga, J., and Wahnón, P.

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VIII

Electronic structure of photovoltaic perovskites: the case of CH3NH3PBI3. Menéndez-Proupin, Eduardo, Palacios, Pablo, Wahnón, Perla, Conesa, José Carlos, Montero-Alejo, Ana Lilian, Beltrán Ríos, Carlos Leonardo

157

Progress and R&D challenges in solar thermal electricity. GonzálezAguilar, José, Romero, Manuel

171

Design and evaluation of a self-consumption photovoltaic installation. Ayala-Gilardón, Alejandro, Mora-López, Llanos, Sidrach-de-Cardona, Mariano

181

IX

X

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XI

Intrinsic energy performance of buildings by parameter identification of models with physical meaning. Naveros Mesa, Ibán, Ruíz Padillo, Diego Pablo, Ordoñez García, Javier, Ghiaus, Christian

193

XII

Global, ultraviolet solar irradiation and meteorological variable trends in Spain. Bilbao Santos, Julia, Román Diez, Roberto, De Miguel Castrillo, Argimiro

205

XIII

HTF system dynamic model in parabolic trough solar power plants. Álvarez Barcia, Lourdes, Peón Menéndez, Rogelio, Martínez Esteban, Juan Ángel, de Cos Juez, Francisco Javier, José Prieto, Miguel Ángel, Nevado Reviriego, Antonio

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XIV

Overview of SO2 and NOx emissions in Europe and Spain. Gómez-Calvet, R. and Dominguis Forquet, A.

237

XV

Characterization of smart windows based on polymer-dispersed liquid crystals and application of ITO low-emissivity coatings. Guillén, Cecilia, Trigo, Juan Francisco, Herrero, José.

243

XVI

Energetic, economic and environmental viability of a co-digestion plant of olive mill wastwater and maize for self-consumption in irrigation needs. Cuadros Blázquez, Francisco, Moreno Cordero, Laura, González González, Almudena, Cuadros Salcedo, Francisco

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XVII

Simulación termodinámica de una planta termosolar híbrida tipo Brayton. Santos Sánchez, María Jesús, Merchán Corral, Rosa Pilar, Medina Domínguez, Alejandro, Calvo Hernández, Antonio.

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XVIII

Modelos analíticos para la predicción de las prestaciones de turbinas eólicas de eje vertical y palas rectas. Meana-Fernández, Andrés, SolísGallego, Irene, Fernández Oro, Jesús M., Argüelles Díaz, Katia M., VelardeSuárez, Sandra.

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PREFACE The volume we are now introducing, “Current Trends in Energy and Sustainability. 2015 Edition” rises as a compilation of the contributions presented at the Biennial Conferences of the Spanish Physical Royal Society (RSEF) within the Energy and Sustainability Symposium organized by the Specialized Energy Group in Gijón (Spain), July 13-15, 2015. For the elaboration of the book, the Editors invited some of the participants to submit an expanded contribution of their presentation at the Symposium. The volume is divided into two parts, the first comprising the Invited Conferences, and the second one the regular contributions. In the first part, and in order to limit the length of this volume, the Editors asked the invited speakers to only submit the view graphs with the understanding that they should be selfexplanatory. On the other hand, for the regular contributions, the authors were asked to submit them in the form of individual chapters. As expected from the broad title of the Symposium, there is a large variety of topics presented, but all can be framed under the area of energy and sustainability. Accordingly the topics of this volume deal with renewable energies (especially solar), CO2 emissions, building conditioning, C-free technologies and fusion, energy generation and distribution (including smart grids and meters), bioenergy, etc. In addition to the contributors to this volume, the Editors would like to acknowledge other persons whom without their assistance this publication would have not been possible. Among them Prof. José Adolfo Azcárraga, President of the RSEF, Prof. Pedro Gorria, President of XXXV Biennial of the RSEF, Prof Joaquín Marro, General Editor RSEF, Itziar Serrano of the Editorial Department RSEF, Concepción Zocar, Manager RSEF, Marina Casanova, Secretary of the Symposium. Finally, we are especially indebted to Dr. Silvia Serrano who acted as general coordinator of the Symposium Committee.

THE EDITORS: Roberto Gómez-Calvet José Manuel Martínez-Duart Valencia, December 2015

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Improving the efficiency of thermophotovoltaic devices: golden ratio based designs

IMPROVING THE EFFICIENCY OF THERMOPHOTOVOLTAIC DEVICES: GOLDEN RATIO BASED DESIGNS Maciá, Enrique Departamento de Física de Materiales, Facultad CC. Físicas, Universidad Complutense Madrid Avda. Complutense s/n, E-28040. e-mail: [email protected]

Abstract. Photonic aperiodic structures based on the golden ratio can lead to more efficient photon management due to increased degrees of freedom in their design as compared to their periodic counterparts. In this work we will describe the thermal emission control capabilities related to the systematic use of Fibonacci multilayers and Penrose tilings in order to simultaneously improve spectral shapening of selective emitters and enhancing photon recycling in second generation thermophotovoltaic devices. Keywords: Thermophotovoltaic devices, Photonic quasicrystals, Golden ratio 1. INTRODUCTION Thermophotovoltaic (TPV) devices transform heat from high-temperature sources like furnaces or the sun into electricity via a two-step process. Firstly, input heat generates thermal radiation at the surface of a selective emitter. Then, this thermal radiation is absorbed by a photovoltaic (PV) cell that converts the high enough energy photons into electron-hole pairs. These charge carriers are finally conducted to the metallic leads to produce an electrical current (Bauer 2011, Chubb 2007). Therefore, the bandgap width of the PV cell semiconducting material introduces a reference energy scale (usually within the 0.2-1.4 eV interval) in TPV devices design. This scale determines the energy range of suitable incoming thermal photons from the selective emitter. Unfortunately, at working temperatures of interest (about 1000 K), most materials emit the vast majority of thermal photons with energies below the electronic bandgaps of typical PV cells. These photons are thereby absorbed as waste heat, which substantially reduces the efficiency of first generation TPV devices to figures close to just 1%. Two main strategies have been addressed in order to circumvent this shortcoming in the design of second generation TPV devices. On the one hand, the recourse to photon recycling via reflection of low-energy photons with a thin multilayered mirror placed on top of the PV material has been proved to significantly reduce radiative overheating. On the other hand, the concept of spectral shapening, aimed at directly suppressing emission of undesiderable photons with energies below the PV cell bandgap coming from the selective emitter, as well as enhancing emission of desiderable photons with energies above the PV cell bandgap, has also been demonstrated. A convenient way of getting this thermal emission control consists in coating the original emitter material with a properly designed multilayered thin film, in order to exploit the beneficial optical properties contributed by each component in a synergetic way. Within this approach the resulting thermal radiation spectrum of the composite structure can be substantially modified as compared to that corresponding to the original emitter alone, due to resonance effects within the multilayered coat (Cornelius 1999). Alternatively, the photon density of states and thus the thermal radiation spectrum can be strongly modified by placing a two-dimensional photonic crystal atop the original emitter in order to maximize the fraction

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of photons emitted within the useful energy range (Zhou 2015). Thus, by properly combining both phonon recycling (e.g., by using rugate IR filters on top of the PV material surface) and spectral shapening (e.g., by using drilled-hole based 2D photonic crystals) strategies, significantly higher efficiencies have been reported for micro-TPV reactors (26 %) and solar thermal TPV devices (45%), respectively (Bermel 2010). In doing so, one must simultaneously optimize the structural parameters of two separate optical structures, namely, the cylindrical hole depth, radius, and lattice parameter of the 2D photonic crystal atop the emitter along with the refraction index values and widths of the layered infrared mirror on the PV cell. This task typically pose a so-called non-convex optimization problem, in which many local optima can exist, usually requiring carefully designed global optimization algorithms (Bermel 2010). In this work we will follow a different approach based on a straightforward application of the so-called harmony principle (Stakhov 2009) in the design of optimized functional structures. This principle is inspired by the profusion of highly efficient designs found in nature under the driving force of genetic evolution, the overall arrangement of leaves and other botanical elements (phyllotaxis) in many plants being a very representative example (Zeng 2009). 2. APERIODIC ORDER BASED DESIGNS 2.1.

Aperiodic structures

In the previously described studies the required optical structures (i.e., photonic crystals and multilayered films) were implemented by means of chirped Bragg reflectors or sinusoidally rugated filters, which are both based on the periodic order notion. Nevertheless, during the last decade it has been progressively realized that ordered structures can be suitably expanded to embrace not only periodic arrangements but aperiodic ones as well. Accordingly, the very notion of aperiodic order, that is, order without periodicity, has been explored to properly describe a growing number of physical systems, including a broad collection of different optical structures (Dal Negro 2014, 2012; Maciá 2014, 2012, 2009, 2006). In this way, wellknown materials still remain in use, along with the previously gained technological expertise, as the attention is entirely focused on the (relatively low-cost) aspects related to device architecture optimization. To date two main classes of aperiodic structures have been extensively studied for optical applications, namely, structures based on the so-called quasiperiodic order (QPO), which is a diffracting-like long-range order generalizing the usual periodic one, on the one hand, and fractal structures based on the scale-invariance (self-similarity) property, on the other hand (Dal Negro 2014, Maciá 2014, 2012). As compared to the relatively complex chirped and rugated aforementioned designs, in this work we will focus on simpler aperiodic multilayers consisting of a number of layers stacked according to a certain deterministic rule, for instance, one given by a binary substitution sequence. A simple example of such a nanostructured material is a two-component Fibonacci multilayer, where layers of two different materials, say A and B, are arranged according to the Fibonacci substitution rule A→AB and B→A, whose successive application generates the sequence of layers A→AB→ABA→ABAAB→ABAABABA→ABAABABAABAAB→... and so on. The number of layers in a sequence of order n is given by a Fibonacci number Fn = {1,1,2,3,5,8,13,21,...}, where each number in the sequence is just the sum of the preceding two, according to the recurrence relation Fn = Fn-1 + Fn-2, n ≥ 2, with F0 = F1 ≡1. Fibonacci multilayers display both long-range QPO and self-similarity and show up a characteristic highly fragmented frequency spectrum, thereby providing more full transmission peaks

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(alternatively, absorption bands) than periodic ones in a given frequency range for a given system length. This feature stems from the presence of more resonant frequencies resulting from multiple interference effects throughout the structure. Indeed, the existence of more frequencies for operation becomes particularly useful when looking for selective enhancement (suppression) of thermal emission for some selected frequencies (Maciá 2015). 2.2.

The role of golden section in quasiperiodic structures

Nevertheless, many properties typically related to full-fledged long-range QPO require a threshold size to properly manifest themselves. In fact, only in the so-called asymptotic limit N→∞ (where N is the number of layers) it can be properly said that the highly fragmented transmission spectrum of a Fibonacci multilayer consists of a self-similar Cantor set with zero Lebesgue measure. Now, actual devices are always finite in length, whereas the mathematically derived results deal with ideal, arbitrarily long systems. Thus, in practical applications one must reach a balance between the unavoidable presence of energy losses and dispersion effects (demanding relatively small systems) and beneficial aspects stemming from quasiperiodicity related effects (requiring large enough systems). Fortunately, in order to get some of the characteristic features usually related to the presence of long-range QPO in a given structure it is not strictly necessary to attain the asymptotic limit. It suffices to introduce suitable values for certain geometrical parameters in the structure design. This property was first grasped from the study of quasicrystalline alloys (Elser 1985), and subsequently confirmed in artificially grown Fibonacci semiconductor superlattices, where a series of AlAs and GaAs bilayers with carefully choose thickness values were sequentially arranged according to the Fibonacci sequence (Todd 1986). If one connects all the vertices of a regular pentagon by diagonals one obtains the so-called Pythagorean pentagram (Fig.1, lower right corner). At their intersecting points the diagonals form a smaller pentagon at the center, and the diagonals of this pentagon will form a new pentagram enclosing a yet smaller pentagon. This progression can be continued ad infinitum, creating smaller and smaller pentagons and pentagrams in an endless succession exhibiting a self-similar nesting characteristic of fractal geometry.

Figure 1. Electron diffraction pattern corresponding to an AlNiCo decagonal quasicrystal. A 10-fold symmetry axis around the origin can be clearly appreciated. A Pythagorean pentagram is shown on the

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lower right corner and a series of three consecutive nested pentagons exhibiting a self-similar structure is shown on the lower left corner. On the upper right corner the radial arrangement of a series of diffraction peaks spaced according to a geometric series based on the golden mean (see Table 1) is drawn.

Surprisingly enough this geometrical construction can be easily recognized in the electron diffraction pattern of quasicrystals (Fig.1, lower left corner), which are characterized by nested series of pentagonal arrangements of Bragg peaks. In fact, the self-similar property leads to the appearance of a hierarchy of intensities extending several orders in the diffraction pattern. In the ideal case of a perfect quasicrystal structure these peaks arrangements will densely fill reciprocal space, though in actual samples most of them become extremely weak, making it possible to distinguish individual spots. By closely inspecting the diffraction pattern shown in Fig.1 one can disclose the characteristic inflation symmetry of quasicrystals. To this end, we measure the distances of successive main diffraction spots along the radial direction outwards from the centre, dn, and calculate the ratios dn/d1. The obtained values are listed in the second column of Table 1 and they clearly follow a non-periodic series. Quite remarkably, however, we can appreciate a close correspondence between the listed dn/d1 values and successive powers (n>1) of the so-called golden mean τ (rounded to the second decimal place), as it is seen in the third column of Table 1. n dn /d1 τn-1 (Fn, Fn-1) 1 1.00 1.00 2 1.42 1.62 (1,0) 3 2.63 2.62 (1,1) 4 4.25 4.24 (2,1) 5 6.87 6.85 (3,2) Tabla 1. The main spots in the diffraction pattern shown in Fig.1 along a radial direction can be arranged according to a power series related to the golden mean (more details in the text) .

The golden mean has been largely known from the ancient times, since it is of frequent occurrence in pentagon and decagon based polygons and polyhedra. This celebrated ratio can be derived in several ways. In fact, the golden mean can be obtained from the Pythagorean pentagram shown in Fig.1, where τ is given by the ratio between the diagonal and the side of the original pentagon. In general, a segment is said to be divided in the golden mean if the ratio of the whole segment to the larger part is equal to the ratio of the larger to the smaller one. If we take the smaller segment as unit and label the larger part as the unknown x, this geometrical definition can be expressed as (x+1)/x = x/1, which leads to the algebraic equation x² = x + 1, whose positive solution defines the irrational number τ = (√5+1)/2 = 1.6180339887.... Accordingly, we get the following expressions relating the golden mean, its square and its reciprocal, namely, τ² = τ + 1, and τ-1 = τ - 1. By successively multiplying the basic relation τ² = τ + 1 by τ we get, τ³ = τ ²+ τ = 2τ + 1, τ4 = 2τ² + τ = 3τ + 2, τ5 = 3τ² + 2τ = 5τ + 3, …, τⁿ = τFn-2 + Fn-3,

(1)

so that any power of τ can be expressed as a linear combination of two successive Fibonacci numbers in the base {1,τ}. Making use of Eq.(1) along with the data listed in Table 1 one can properly classify the different diffraction peaks along the radial directions of the diffraction pattern shown in Fig.1 in terms of a pair of Fibonacci numbers, as it is shown in the fourth column of Table 1. Thus, in quasicrystals the translation symmetry characteristic of periodic

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crystals is replaced by an inflation/deflation symmetry measured in terms of a characteristic scale factor given by the golden mean. A useful mathematical relationship between the Fibonacci series and the golden mean, is given by the asymptotic limit lim n 

Fn 1  Fn

(2)

which succinctly accounts for the property that the ratio of two successive larger and larger Fibonacci numbers comes closer and closer to the golden mean. This relationship underlies the emergence of characteristic pre-fractal features in relatively short structures scaled according to the golden ratio geometry as measured in terms of Fibonacci numbers arithmetics. Making use of Eq.(2) the relationships lim n 

Fn 1  2 , Fn 1

lim n 

Fn 1 3 , … Fn  2

(3)

relating successive powers of the golden mean with the limits of quotients respectively involving next and next-next neighbouring terms of the Fibonacci sequence, are readily obtained. 3. GOLDEN RATIO BASED INFRARED MIRRORS Quite remarkably, although it passed unnoticed by the authors of the original research, the fingerprints of a τn sequence can be found in a recently reported optimized structure for a daytime radiative passive cooler. A passive cooler is a device which decreases its temperature below that of the ambient air without any electricity (or any other form of energy) input (Rephaeli 2013). Making use of a suitable device a cooling of about 5º C below ambient air temperature, amounting to 40 Wm-2 cooling power, has been reported under direct sunlight (Raman 2014). These promising figures were obtained by properly combining material properties and interference effects in an integrated structure that collectively achieves high solar reflectance (97% at normal incidence) and strong thermal emission in a selected infrared frequency window. To this end, a thin film multilayer composed of seven alternating layers of hafnium dioxide (high refraction index) and silicon dioxide (low refraction index) of varying thicknesses were deposited on top of a 200 nm silver substrate. The resulting coating was in turn deposited on top of a 750 μm thick, 200 mm diameter silicon wafer. The top three layers are one order of magnitude thicker (within the range 200-700 nm) and are primarily responsible for thermal radiation from the cooler device. The bottom four layers in the coating film have thicknesses within the range 10-70 nm, and assist in optimizing solar reflection in a manner akin to that used in periodic one-dimensional photonic crystals. Thus, it is the presence of two different kinds of layer sizes which ultimately accomplishes the two functional tasks required for the coating, namely, high solar reflectance (due to the shorter layers stack) and high infra-red emissivity within the 8-13 μm window (due to the longer layers stack). Both the total number and the proper thickness values of the alternating low/high refraction index layers was determined by using a numerical optimization method (Raman 2014).

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dn (nm) 13 73 34 54

dn /d1 1.00 5.615 2.615 4.154

τn 1.00 τ2 + 3 τ2 = 2.618 τ3 = 4.236

Tabla 2. The first column list the optimized thickneses of the layers composing the solar reflector component on the radiative cooler arranged from top to bottom. Note that three of them correspond to Fibona cci numbers. Accordingly, their respective ratios (second column) approximate to successive powers of the golden mean, as it is indicated in the last column (more details in the text) .

In Table 2 we list the optimized thicknesses of the four layers comprising the solar reflector component. Albeit these layers are deposited according to the periodic sequence ABAB (where A stands for HfO2 and B for SiO2), we note that their thicknesses take on different values. Quite interestingly, we realize that three dn values are given by Fibonacci numbers, so that their respective ratios dn/d1 reasonably coincide with different golden ratio powers (allowing for the experimental ±1 nm tolerance), as it is prescribed by Eq.(3). Thus, the thicknesses distribution obtained from the optimization algorithm approximately satisfies the harmony principle in a natural way. In fact, from a fundamental point of view the presence of successive powers of the golden ratio among the solutions of a numerical optimization algorithm can be traced back to the very notion of mathematical harmony. When applied to design purposes this harmony refers to combinations of parts to form an orderly whole. Accordingly, mathematical harmony describes the proportionality of the parts between themselves and with the whole (Stakhov 2009). In particular, the harmony principle can be readily applied in order to get natural generalizations of the golden ratio obtained from the division of a line at different points defining a series of segments, say xj , such that all of them are related to each other according to a concatenated set of golden ratios. For instance, in the case of a line which is divided into four of such segments we have x1  x2  x3  1 x1  x2  x3  , x1  x2  x3 x1  x2

x1  x2  x3 x1  x2  , x1  x2 x1

x1  x2 x1  , x1 1

(4)

where, we assume that the ending segment in the series is taken as the unity (i.e., x1 ≡ 1). The above set of nested ratios can be reduced to the algebraic equation x 4 = x3 + 1, where for the sake of simplicity, we denote x ≡ x1 . It is readily shown by induction that in the case of a original line being divided into p segments following a nested golden ratio-like relationships we obtain the general algebraic equation xp+1 = xp + 1, whose positive real roots reduce to the well-known values x = ½ (the line is divided in just to equal halves) and x = τ (the line is divided in the classical golden section) for p = 0 and p = 1, respectively. For higher p values one gets the so-called generalized golden p-ratios (Stakhov 2009), whose values are given by τ2 = 1.465…, τ3 =1.380…, and τ4 = 1.324…, for p = 2, 3 and 4, respectively. As we see, one gets progressively smaller golden p-ratios as the p value is increased, so that τp ≤ τ, for p > 1. In Table 3 we list the solutions to the relationship given by Eq.(4). For the sake of comparison, in the third column we list the corresponding layer thickness values for a multilayer structure designed according to this general golden section mean geometry, where the thinner layer width is d2 = 13nm.

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n 1 2 3 4

xn τ3 =1.380 τ3 (τ3 -1) = 0.524 τ23 (τ3 -1) = 0.724 1

dn (nm) 34.5 13.0 18.1 25.0

Tabla 3. The second column list the values of the segments satisfying the generalized golden ratio series prescribed by Eq.(4). In the third column we list the layer thicknesses distribution for a possible infrared mirror component of a TPV device (more details in the text) .

It would be interesting to experimentally explore if thicknesses distributions based on these generalized golden mean τp values may outperform the purported reflectance figures for the infrared mirrors considered in (Bermel 2010). In particular, one may wonder if, rather than the reported 97% reflectance, a full solar reflectance could be obtained by further exploiting this design approach. 4. GOLDEN RATIO BASED SELECTIVE EMITTERS 4.1.

Thin film golden multilayers

An ingenious way to control the thermal emissivity of a given material at certain frequencies was proposed on the basis of coating the bulk substrate with a thin film made of alternating layers of different materials. In so doing, one aims to exploit quantum confinement effects in order to modify the emission processes through the multilayered structure, thereby properly engineering the thermal radiation characteristics of the overall system. In this way, the power spectrum of the multilayered film located in front of an emitting hot surface is given by the Equation (5) P(ω,T) = E(ω) PB(ω,T)

(5)

where PB (  

 3 2 2c 2

1 e

 k BT

1

(6)

is the Planck's law and E(ω) is the ratio of the optical power emitted at frequency ω into a spherical angle element dΩ by a unit surface area of the thin film, to the power emitted by a blackbody with the same area at the same temperature (Cornelius 1999). An illustrative result is shown in Fig.3., where the performances of a periodic and a Fibonacci multilayer are respectively compared. As we see the periodic film (Fig.3a) significantly blocks heat radiation emitted by the emitter at the frequencies corresponding to the photonic crystal bandgap (ω/ω0=1) as expected, but we also observe that the substrate's emission is enhanced from the gray-body level all the way up to the perfect blackbody rate at a number of frequencies corresponding to the pass-band transmission resonances of the multilayered film. This occurs because the thin film acts as an antireflective coating at these resonances. In this way, all the incident radiation tunnels through the multilayer structure into the emitter for these selected frequencies, so that the substrate effectively behaves as a perfectly absorbing blackbody in that case (Cornelius 1999, De Medeiros 2007). A similar enhancement of the substrate's thermal emittance at certain resonance frequencies, accompanied by the

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corresponding inhibition at the stop-bands, is observed in the quasiperiodically arranged thin film coating as well (Fig.3b), but in the case of the Fibonacci coating one finds a strong emittance within the spectral range corresponding to the midgap of the periodic multilayer.

Figure 3. Thermal radiation spectra as a function of the reduced frequency under normal incidence conditions for multilayered stacks (thin film) made of alternating layers of refractive indices nA = 1.45 (SiO2) and nB = 1.0+0.01i, coating a thick absorbing substrate (selective emitter) of refraction index nC = 3+0.03i. The entire structure is embedded in air (n = 1). The thin film coat layers are respectively arranged according to the following sequences: (a) periodic, and (b) Fibonacci (N = 377). The perfect blackbody thermal spectrum is given by the dotted curve, whereas the dashed curve gives the thermal spectrum of the substrate. The temperature is chosen so that the blackbody (Wien) peak is aligned with the midgap frequency ω0 = 2πc/λ0 (λ0 = 700 nm). All the curves are properly normalized by this peak power. (Adapted from (de Medeiros 2007). Courtesy of Eudenilson L. Albuquerque).

Furthermore, a characteristic trifurcation splitting (Dal Negro 2014, 2012; Maciá 2014, 2012, 2009, 2006) can be clearly appreciated around a number of frequencies close to ω/ω0 = 0.6 and ω/ω0 = 1.4. This triplet structure along with the significantly enhanced emission tail covering the overall spectral range up to ω/ω0 = 2.5, clearly enhance the global amount of higher energy photons coming from the selective emitter covered with a Fibonacci multilayer as compared to those coming from the periodic one. 4.2 Two-dimensional Penrose tiles Two-dimensional aperiodic tilings are collections of polygons capable of covering a plane with neither gaps nor overlaps in such a way that the resulting overall pattern lacks any translational symmetry. The simpler aperiodic tiling was discovered by Penrose in 1974 (Fig.4), and consists of just two different tiles: a skinny (acute angle π/5, dark gray) and a fat (acute angle 2π/5, light gray) rhombi with equal edge length. Since cos(π/5) = τ/2 and cos(2π/5) = τ-1/2, we realize that the golden ratio plays a significant role in the construction of the Penrose tiling. For instance the frequency of appearance of skinny and fat rhombi in the infinite tiling is 1:τ and their areas ratio is τ. From the solid state physicist's viewpoint, the most relevant feature of Penrose construction is not the fact that it lacks translational symmetry, but that it possesses long-range QPO, so that it is able to produce sharp diffraction peaks. Thus, the notion of repetitiveness, typical of

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periodic arrangements, should be replaced by that of local isomorphism, which expresses the occurrence of any bounded region of the tiling infinitely often across the whole tiling, irrespective of its size. In the particular case of Penrose tilings Conway's theorem states that given any local pattern having a certain characteristic length, L, several identical patterns can be found within a distance of τ3L/2 ≈ 2.12L.

Figure 4. Penrose tiling illustrating the local isomorphism characteristic of Conway's theorem. The entire tile is composed of two basic tiles.

One of the main differences between periodic and QP photonic crystals is that the latter lack translational symmetry, so that the notion of Brillouin zone becomes ill-defined in this case. To overcome this shortcoming two approaches have been usually considered. The first one relies on the notion of approximant unit-cell structure. The second one directly considers a finite structure. From the modeling point of view the second approach seems more appropriate to study actual physical realizations, which will always be finite, but it is constrained to the consideration of quite small systems due to computation limitations. The first approach is more convenient from both a fundamental point of view (since it allows for a systematic study about the QPO threshold) and a practical viewpoint (since the related optical band structures can be derived in a standard way). Nonetheless, the artificial periodicity introduced in the supercell approximation may lead to some spurious results which must be carefully distilled from the intrinsic QP features. In Fig.5 the optical band diagrams of a periodic square lattice and a Penrose approximant photonic crystals are compared. The square lattice exhibits a broad main photonic bandgap centered at about 0.33 normalized frequency (ωa/2πc, where a is the lattice constant), whereas Penrose lattice displays two narrower main photonic bandgaps with midgap normalized frequencies at 0.32 and 0.55 and relative widths of about 30% and 15%, respectively. The presence of more photonic bandgaps stems from the fact that the first Brillouin zone has more symmetries in quasiperiodic lattices as compared to conventional photonic crystals based on periodic square or hexagonal lattices, hence favouring the possible appearance of a complete gap in these systems. In way similar to that observed in the case of one-dimensional multilayers, two-dimensional lattices designed according to the golden mean geometry, give rise to a higher number of (narrower) bandgaps, leading to an overall shift of the allowed photon frequencies towards higher energies. As we have previously discussed,

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this feature is a very convenient one in order to improve the efficiency of TPV devices by means of a proper spectral shaping of the selective emitter contribution.

Figure 5. Optical band diagrams obtained for TM polarization for (a) a square lattice, and (b) a Penrose approximant lattice containing N = 76 dielectric cylinders placed perpendicularly to the substrate plane in an air background. (Adapted from Wang 2007).

5. CONCLUSIONS So far most studies on second generation TPV cells designs have considered photon recycling by reflecting back low energy photons to the thermal emitter by either a quarter-wave stack or a rugate filter. Rugate filter has significantly better performance compared to quarter-wave stack, though it requires a relative large amount of materials with different refractive indices. Therefore, the search for aperiodic multilayer structures, consisting of just two different materials, able to demonstrate a performance similar to that of rugate filters (namely, steep cut-off, no high order reflections and a broad stop-band) would be very appealing. (Zhou, private communication). It would also be interesting to experimentally explore if thicknesses distributions based on generalized golden mean τp values may outperform the purported reflectance figures for infrared mirrors. Acknowledgements I warmly thank Prof. Luca Dal Negro and Dr. Zhiguang Zhou for useful comments and for sharing different materials of common interest. I thank Eudenilson L. Albuquerque for kindly providing one of the pictures which illustrate the text. I am grateful to Victoria Hernández for a critical reading of the manuscript.

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