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Energy 154 (2018) 488e497

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Energy

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Numerical investigation of thermodynamic properties in 2D poroussilicon photonic crystals integrated in thermophotovoltaic energyconversion system

Kossi Aniya Amedome Min-Dianey a, Hao-Chun Zhang a, *,No�e Landry Privace M'Bouana b, Chengshuai Su c, Xinlin Xia a

a School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Chinab Institut Sup�erieur de Technologie, Universit�e de Bangui, BP 892, Bangui, Central African Republicc Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China

a r t i c l e i n f o

Article history:Received 19 December 2017Received in revised form24 April 2018Accepted 26 April 2018Available online 30 April 2018

Keywords:Porous siliconPhotonic crystalsThermal radiationThermodynamic propertiesPlane wave expansion

* Corresponding author.E-mail address: zhc5@vip.163.com (H.-C. Zhang).

https://doi.org/10.1016/j.energy.2018.04.1560360-5442/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t

The prediction of thermodynamic properties performance in porous silicon (pSi) photonic crystals (PhCs)was studied for enhancement of light conversion into different types of energy based thermophoto-voltaic energy conversion system. The unit-cell for 2D square and triangular lattices of circular air holeswere formulated and solved using plane wave expansion (PWE) method. This was used to investigate thethermodynamic properties and the effect of lattice dynamic on these properties at different porosities insilicon PhCs; which are regarded as isolated and non-interacting particles systems. This was achieved byconnecting density of states (DOS) and thermodynamic quantities as described in statistical physics. Itrevealed that irrespective of lattice type, increasing porosity ensued a decline in thermodynamicproperties. Novel insights and theoretical concepts that could be integrated in thermophotovoltaicsystem were revealed. Regarding the square lattice, the optimum value of these properties except thefree energy was obtained at R=a ¼ 0:30 for 20% and 50% porosities, and at R=a ¼ 0:25 for 80% porosity.Regardless of the porosity, the optimum value for the triangular lattice was found at R=a ¼ 0:25. Thesecharacteristics have attempted to provide contributions with regards to the selection of the appropriatedesign for PhC in order to achieve high efficiency conversion.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Recent strides in silicon technology has brought about poroussilicon (pSi); a prodigious material which has been introduced invarious scientific and engineering fields including photonics andenergy technologies [1e11]. Thermal radiation from photoniccrystal (PhC); often referred as complex electromagnetic structures,has recently attracted the attention of several researchers [12e18].Lin et al. indicated that the thermal radiation from metallic PhCsprobably exceeds that of a blackbody in free space [15]. The effect ofthermal oxidation and oxide etching on silicon PhCs of triangularlattice has been studied. Using plane wave expansion (PWE) tech-nique, Thitsa & Sacharia [19], modeled the latter and proposedadvantageous processes for tuning the photonic band gap and

defect frequency. Indeed, the knowledge on the propagation ofelectromagnetic fields in PhCs, analytical and numerical methodsare available. These methods allow the access to quantities deemedinaccessible by experimental measurements but which can benecessary for understanding the behavior of PhCs. Among thesedifferent methods, the PWE remains the most commonly used instudying the band structure of PhCs and has been employed bydifferent authors in the investigation of PhCs [20e24]. Currently,there are numerous proposed applications relating to pSi basedPhCs. This is largely due to their ability to control light propagationwithin them as reported in Ref. [5]. Luo et al. presented a classicalsimulation of equilibrium thermal emissivity from dispersive lossyPhCs [25]. They indicated the potential usefulness of PhCs in in-candescent lighting and thermal photovoltaic applications.Furthermore, a basis for manipulating the thermal emission andabsorption of radiation in complex photonic structures and thedesign of novel solar cell devices has been reported by Florescuet al. [26]. These authors revealed that controlling the thermal

Table 1Refractive index and dielectric constant versus porosity in Bruggeman model[41,42].

Porosity (%) Refractive index ðnSiÞ Dielectric constant ðεSiÞBulk 3.47 12.0410 3.23 10.4320 2.98 8.8830 2.72 7.4050 2.14 4.5870 1.56 2.4380 1.32 1.7490 1.12 1.25

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497 489

emission and absorption of radiation in a PhCs enables the reali-zation of high-efficiency solar cells. Florescu et al. in another study,analyzed the origin of thermal radiation enhancement and sup-pression inside PhCs as a prerequisite for the understanding thethermal radiation properties of finite PhCs [27]. Moreover, Gese-mann et al. presented measurements of the thermal emissionproperties of 2D and 3D silicon PhCs which depended on substrate

Table 2List of relevant research works and key findings.

Authors Type of research Type ofdevices

Key findings

Schilling & Scherer [8] Experimental 3D PhC Reflection measurementsfor an extended fabricate

Bicer et al. [40] Numerical Photovoltaiccell

The PV generator-photo cprocesses.

Shimizu et al. [34] Experimental Periodicmicrocavities

The effect of spectrally seblack painted sample.

Rinnerbauer et al [31] Numerical &Experimental

2D PhC The efficiency of selective

Mogami et al. [33] Experimental Opticalwaveguidedevices

Their Si photonics platfor

Florescu et al. [26] Numerical &Experimental

Solar cell Strong enhancement of thachieved by PhC.

Latella et al. [57] Numerical Energyconversiondevices

The maximum work fluxmore than the correspondother materials with a lowhigher.

Wang et al. [13] Numerical 2D & 3D PhCs In 2D, using van Hove sinfactor can exceed the conIn 3D, it is more difficult

Recio-Sanchez et al [42] Numerical 2D & 3D PhCs These structures are veryKordas et al. [61] Numerical &

ExperimentalpSi Layer Normal optical dispersion

Fu et al. [7] Numerical 2D PhC The operation of logic gatJohnson et al. [12] Numerical &

ExperimentalPhC Enhancement of Er3þ emi

factors seen for longer waFlores & Palma-Chilla [38] Numerical Dual & direct

systemsEntropies correlated beinexplicitly showed that the

Chan et al. [17] Numerical 2D PhC The ability to design therapplications.

Gesemann et al [28] Numerical &Experimental

2D & 3D PhCs It turned out that for thevisible. For the out-of-plasilicon oxide emission fro

Hu et al. [11] Numerical &Experimental

Siliconphotovoltaic

This study provides a basitransfer coefficient.

Florescu et al. [27] Numerical 2D PhC The spectral energy densithe total number of availaThe central quantity thatsurfaces and not the phot

Lourek & Tribeche [54] Numerical Blackbody The reexamination of theenergy with an increase othe effects of the deforme

Lin et al. [15] Experimental 3D PhC A 3D tungsten PhC sampleexperimentally shown toassociated with the propa

heating (resistively and passively) with an aluminum hotplate [28].A decrease in the averaged energy of 2D PhCs based on bulk and pSimaterials was revealed by Szabo et al. who applied finite differ-ential time domain (FDTD)method in calculating wave propagation[29]. Tong et al. indicated that both dispersion characteristic anal-ysis and numerical simulation of field patterns can verify theeffective phase indexes of 2D triangular PhCs with dielectric rods inair background [30].

Currently, applications of thermal light sources through PhCsare being explored, mainly in the field of thermophotovoltaic po-wer conversion. However, the demands of high-operating tem-peratures inevitably leads to nano and microscopic materialdegradation which are challenging [31e35]. However, the estimateof optimum thermodynamic properties of materials is theemphasis of solid-state knowledge and industrial research which isapplying in several modern technologies at high temperature aswell as low temperature configuration [36e40]. Other relevantworks and their findings are summarized in Table 2.

Based on existing knowledge, thermodynamics is a subject ofgreat generality which is applicable to systems of elaborate

along different directions indicate the onset of the 3D bandgap at low frequenciesd structure.urrent generation process has the highest exergy destruction rate among the sub-

lective property is appeared as increasing of the temperature comparing with the

emitters based on 2D PhCs in refractory metals.

m is very useful for optical multi-applications.

e conversion efficiency of solar cell devices without using concentrators was

that can be obtained from near-field radiation is almost two orders of magnitudeing for blackbody radiation if one considers sources of hexagonal boron nitride. Forer resonance frequency, e.g., silicon carbide, the maximum work flux is even

gularity in the DOS, the angle-integrated light trapping absorption enhancementventional limit over a substantial bandwidth.to use PhCs to overcome the conventional limit.promising candidates for the development of low-cost photonic devices.on the contrary to the anomalous dispersion obtained using the envelope method.

es does not require high power excitation.ssion intensity is observed for the 550-nm transition, with lower enhancementvelengths.g a basic tool to attach thermodynamically both categories of systems. It isDual has negative temperatures and positive pressures.

mal emission could well find uses in thermophotovoltaic systems and defense

in-plane 2D PhC and out-of-plane 3D PhC emission a photonic stop gap effect isne 2D PhC emission, no photonic bandgap effect is observable but instead strongm native oxide inside the pores of silicon are observable.c understanding of thermal performances, such as the temperature field and heat

ty, the spectral intensity, and the spectral hemispherical power are only limited byble photonic states and their propagation characteristics.determines these thermal radiation characteristics is the area of the isofrequencyonic density of states as it is generally assumed.thermodynamic properties of the blackbody radiation shows that it emits moref the value of |k| in comparison with the standard Planck radiation law. Moreover,d Kaniadakis statistics are shown to be more appreciable for high temperatures.is thermally excited and exhibits emission at a narrow band. The sharp emission isexceed the free-space Planck radiation. It is proposed that an enhanced DOSgating electromagnetic Bloch waves is responsible for the observed effect.

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497490

structurewith all manner of complex thermal properties. Using thisas a basis, we extended our scientific curiosity in investigating athigh temperature thermodynamic properties through 2D pSi PhCs.Accordingly, this work attempted to investigate thermodynamicquantities of silicon material based on square and triangular latticePhCs in different porosities consideration. This is aimed atexploring new concepts on porosity effect that could be taking intoaccount in thermophotovoltaic system for light conversion basedupon the interaction of electromagnetic radiationwith PhCs. It is toalso apply the effect of lattice dynamic on these quantities thoughpSi PhC in order to retain the reliable design for optimum conver-sion. These objectives have been achieved by using a PWE algo-rithm from wave equation based Fourier expansion method to thefirst kind of Bessel function. The algorithm was implemented inMatlab for the density of states (DOS) from which the thermody-namic quantities were computed. The results have been comparedwith existing published works in statistical physics.

2. Numerical methods

2.1. Effective refraction index versus porosity

Indeed, the dielectric properties of pSi are mainly governed bythe porosity. Moreover, several models that relate the porosity of apSi layer and its refractive index exist. One of such is the Vegard'slaw which considers pSi as a homogeneous mixture of silicon andair [5]. Furthermore, Maxwell-Garnett or the Bruggeman modelscan be used to describe the dielectric constant of a two-componentmaterials system [41,42]. The latter is used in this work and relatesporosity and effective refractive index through the followingexpressions:

ð1� pÞ εSi � εpSi

εSi þ 2εpSiþ p

εair � εpSi

εair þ 2εpSi¼ 0 (1)

ε ¼ n2 (2)

ð1� pÞn2Si � n2pSin2Si þ 2n2pSi

þ p1� n2pSi1þ 2n2pSi

¼ 0 (3)

p ¼ 1�24�1� n2pSi

��n2Si þ 2n2pSi

�

3n2pSi�1� n2Si

�35 (4)

where porosity p is the porosity, εSi is the dielectric constant ofsilicon, εpSi is the dielectric constant of pSi, εair is the dielectricconstant of air equal to 1, nSi is the refractive index of silicon, and

26666664

Fðm;nÞjm¼�2;n¼�2zðm;n; o; pÞjm¼�2;n¼�2;ðo¼�2…þ2;p¼�2…þ2Þ / /««

Fðm;nÞjm¼2;n¼2zðm;n; o; pÞjm¼�2;n¼�2;o¼2;p¼2

37777775

26666664

Am¼�2;n¼�2Am¼�2;n¼�1

««

Am¼2;n¼1Am¼2;n¼2

37777775¼ u2

26666664

Am¼�2;n¼�2Am¼�2;n¼�1

««

Am¼2;n¼1Am¼2;n¼2

37777775

(10)

npSi is the effective refractive index of pSi. Table 1 depicts therefractive index and dielectric constant as function of porosity inBruggeman model and Fig. 1 presents scanning electron micro-scope images of pSi morphology found in Ref. [43].

2.2. Conjugate PWE and Fourier expansion methods

In the electromagnetic theory, Maxwell's equations in free spacedescribed the propagation of electromagnetic waves and the so-lutions of these equations can be approximated by conjugatingPWE and Fourier expansion methods [44e48]. Thus, the governingequation for the electric andmagnetic component of the light waveis given by Eq. (5):

V2Apol � m0εð r!Þ v2Apol

vt2¼ 0 (5)

m0εð r!Þ ¼ 1

cð r!Þ2(6)

whereV2designed the Laplace operator, r!is the position, Apolthevector potential, pol stands for the polarization, m0is the perme-

ability, εð r!Þis the permittivity and cð r!Þ2is the phase velocity. In thePWE method the phase velocity can be expanded as Fourier series,

using the reciprocal lattice vector G!¼ mb

!1 þ n b

!2 withm;n2ℤ to

ensure invariance of the function to displacement:

cð r!Þ2 ¼XG!

FG!eið G

!$ r!Þ ¼

Xm;n

Fm;nei�m b!

1þn b!

2

�$ r! (7)

Fig. 2 illustrates the passage from the real space to the reciprocalspace in the square lattice and triangular lattice. The vector po-tential in the wave equation can also be written as a Fourier series:

Apol ¼Xk!

Ak!ei

�k!

$ r!�ut�

Apol ¼Xo;p

Ao;pei��

o b!

1þp b!

2þ k!�

r!�ut� (8)

Introducing Eqs. (7) and (8) into Eq. (5) and comparing the co-efficients, the central equation can be represented as:

Xm

XnFm;nzAo�m;p�n ¼ u2Ao;p (9)

where z ¼ ½ðb1;xðo�mÞ þ kxÞ2 þ ðb1;yð2ðp� nÞ � ðo�mÞÞ þ kyÞ2� inthe triangular lattice and z ¼ ½ðb1;xðo�mÞ þ kxÞ2 þðb1;yðp� nÞ þ kyÞ2�into the square lattice. Eq. (9) presents theeigenvalue problem that leads to linear system equation that can besolved numerically by using matrix formulation expressed in Eq.(10):

The matrix coefficients represent the Fourier expansion co-efficients wherem and n are truncated symmetrically about zero tofive terms (�2, �1, 0, 1, 2). Explicitly, the problem is simplified byneglecting all matrix elements with index pairs (o ¼ 0 and p ¼ 0 up

Fig. 1. Example of SEM images of porous silicon morphology view in different configurations as reported in Ref. [43] of different relevant works. Morphologies for (100) p-Si: (a)Plan view and (b) cross section. (c) Macropores on p-Si, view after cleavage, for samples prepared from p-Si (400U cm, (100)-oriented), 100 mA/cm2, 6min, 15% ethanolic HF.Macropores on (100) n-Si etched in ethanolic hydrofluoric solution with frontside illumination and with an anodization current J¼ 20mA/cm2 for t¼ 45min. (d) Cross-section and(e) plan view. (f) Macropore on p-Si (org) prepared from p-Si (100U cm, 20mA/cm2, 40min, and HF/ethylene glycol 50/50 by vol).

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497 491

to the first order). Subsequently, the expansion of c2ð r!Þ in a Fourierseries with the coefficients Fa;b is given in following steps:

Fa;b ¼ ð1=AcellÞZcell

cð r!Þ2e�i ab1þbb2� �

·rd r! (11)

Fig. 2. Illustration of the passage from the real space to the reciprocal space andBrillouin zone: (a) square lattice and (b) triangular lattice. The triangleGMK representsthe irreducible Brillouin zone.

Considering unit cell with an area A that is represented by thephase velocity, the following equation satisfies this assumption;

c2ð r!Þ ¼�c2M; r � Rc2; r<R

(12)

where R is the radius of the holes; c2M and c2 represent the phasevelocity in dielectric material and phase velocity in vacuumrespectively. By inserting Eq. (12) into Eq. (11) and satisfyinga ¼ b ¼ 0 this equation can be achieved;

F0;0 ¼ c2M.Acell

Zcell

d r!þ�c2 � c2M

�.Acell

Zcell

d r!

F0;0 ¼ c2M þ�c2 � c2M

�Ahole=Acell

(13)

Finally, making the assumption that as0 or bs0, the resultingequation will be;

Fa;b ¼�c2 � c2M

�.Acell

Zp

�p

ZR

0

drdfre�iGcocðfÞ

Fa;b ¼ 2p�c2 � c2M

�.Acell

RGJ1ðGRÞ

(14)

where J1 denotes the Bessel function of the first kind, Acell equal to

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497492

a2ffiffiffi3

p=2 and a2for the triangular and the square lattice

respectively.

2.3. Thermal radiation and density of states

The concepts of DOS used in this study are essentially found inRefs. [49,50]. These concepts are potential in understanding theunderlying physics of near-field thermal radiation. Similarly, it hasestablished essentially in the study of electromagnetic wavepropagation through periodic structures such as mode confine-ment in PhC. The DOS is fundamental in light trapping for solar cellsand in mode confinement in PhC. It is usually defined as:

NðuÞ ¼Xm

1ABZ

ZBZ

d�u� um

�k!��

d2k (15)

where the integral is taken over them� th band and ABZ is the areaof the Brillouin zone (BZ). Eq. (15) can be written as:

NðuÞ ¼Xm

1=ABZ

ZEFSm

dkduds (16)

where the integral is taken along the m� th equifrequency sur-faces (EFS) at frequencyu and v�1

g ¼ dk=du is the inverse groupvelocity. However, there are related problems in the existingmethods for computing the DOS of PhC. The definition in Eq. (15)suggests the typical method by which the DOS is computed; us-ing the full band structure and binning by frequency to approxi-mate the integral. The frequency binning method can be improvedif the group velocities are also available. The DOS can also be ob-tained from the local density of states (LDOS) by an integral over thereal space unit cell or from the angular DOS by an integral over theBZ [51]. These integration methods are computationally intensivesince they require repeated computation of Green's functions in theprocess of performing the integration [50,52]. The integrationtechniques of DOS was presented in Ref. [53].

2.4. Thermodynamic quantities as function of DOS in thermalradiation field

Expending the principles of energy quantization and statisticalthermodynamics approach [54,55], the energy density of a systemat a given frequency u is obtained as the product of the DOS and themean energy of a state at frequency u and temperature T [56]. Themean energy of a state qðu; TÞ also known as mean energy of aPlanck oscillator at u and T is given by;

qðu; TÞ ¼ Zu

expðZu=kBTÞ � 1(17)

Furthermore, Eqs. (18)e(21) are essentially found in Ref. [57]where the internal energy density (per unit volume) uðTÞ of ther-mal radiation at temperature Tis written as:

uðTÞ ¼Z∞

0

Zunðu; TÞNðuÞdu (18)

where nðu;TÞ ¼ ðeZu=ðkBTÞ � 1Þ�1is the mean occupation number ofphotons in a mode of frequency u and can be interpreted as thenumber of photons in the state characterized by the photon energyZu; NðuÞ is the DOS with frequencyu;Z and kB are the reduced

Planck's and Boltzmann's constant respectively. The function de-pends on the microscopic details of how the radiation is emitted bythe body. The specific heat at constant volume can be acquired fromEq. (18) as follows.

cV ¼ vuðTÞvT

cV ðTÞ ¼Z∞

0

kB

�Zu

kBTnðu; TÞ

�2e

ZukBTNðuÞdu

(19)

In deriving Eq. (19) it is assumed that the DOS does not dependon temperature. Thus, according to usual thermodynamic relations,the entropy density of the radiation is readily given by sðTÞ ¼R T0 cV ðT 0Þ=T 0dT 0. Siting x ¼ Zu=ðkBT 0Þ in the integration over the

temperature in the previous expression, taking into account Eq.(19) and introducing the following Eq. (20),

mðu; TÞ ¼Z∞

Zu=ðkBTÞ

xdx

4 sinh2ðx=2Þ

mðu; TÞ ¼ ½1þ nðu; TÞ�ln½1þ nðu; TÞ� � nðu; TÞln nðu; TÞ

(20)

One can achieve the entropy density sðTÞ and the Helmholtz freeenergy density FðTÞ respectively by the following form;

sðTÞ ¼Z∞

0

kBmðu; TÞNðuÞdu (21)

FðTÞ ¼ kBTZ∞

0

ln�1� e�

ZukBT

�NðuÞdu (22)

3. Results and discussions

The graph of DOS for both lattices at low, half and high porosity(10%, 50% and 90% respectively) are displayed in Fig. 3(a)e(c), forsquare lattice and Fig. 3(d)e(f), for the case of triangular lattice. It isworthy to mention that information on DOS in the case of vacuumand bulk pSi is presented in our previous study [58]. Furthermore,this present study shows a similar trend to that calculated by Sunand Stirner (shown in Fig. 3 of Ref. [20]) who investigated a trian-gular lattice of air holes etched into macroporous silicon. However,the absence of the eigen-states that corresponds to the low or zeroDOS is more found in the case of square lattice at low frequencyrange regardless of the degree of porosity by comparing to thetriangular lattice. Moreover, the number of eigen-states reduceswith increase in porosity. Consequently, the square lattice forecaststhe optimum performance for radiation in the PhCs as well for allthermodynamic properties that can be computed as integrals of theDOS in which the information of these quantities is contained.However, only the specific heat, the internal energy, the Helmholtzfree energy and entropy density are presented in this work astypical thermodynamic properties investigated in unit areathrough both lattices of pSi PhC.

In addition, the unit cell band structure profile of pSi PhCs wascomputed for 10%, 50% and 90% porosity considerations for bothsquare and triangular lattices as displayed in Fig. 9. This is aimed atproviding better understanding of the accuracy of our

Fig. 3. 2D DOS of unit cell of pSi PhCs: (Left) Square lattice of circular air holes withP¼ 10% (a), P¼ 50% (b), P¼ 90% (c). (Right) Triangular lattice of circular air holes withP¼ 10% (d), P¼ 50% (e), P¼ 90% (f). The holes radiusR ¼ 0:22 mm; the lattice con-stanta ¼ 0:5 mm.

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497 493

computational results. From the results, a gradual disappearance ofthe bands with respect to the increase in porosity was found. Thisphenomenon occurred regardless the lattice and thus agrees with

Fig. 4. Specific heat at constant volume versus temperature of unit cell of pSi PhCs indifferent porosities: (a) and (b) Correspond to specific heat in the square lattice. (c) and(d) Correspond to specific heat in the triangular lattice. The holes radiusR ¼0:22 mmand the lattice constanta ¼ 0:5 mm.

that seen in DOS results where a reduction in the number of eigen-states with increase in porosity was revealed.

In the case of square lattice at low temperature seen in Fig. 4(a)and (b), the specific heat is sufficiently low, regardless of the degreeof silicon porosity. Beyond temperatures of 1300 K appears a dif-ference in the conduct of the specific heat respect to the porosity.However, a growth in the specific heat with increase in tempera-ture was observed. This growth is much more pronounced whenthe porosity of the material decreases. For example, the graph ofthe specific heat corresponding to the porosity of 10% is underneaththe one corresponding to the porosity of 90%. Hence, the influenceof porosity on the specific heat is well related as well on the otherthermodynamics properties. This is justified by the fact that theseproperties are linked to each other by Maxwell's relations in ther-modynamic concepts. Also, in the case of triangular lattice asdepicted in Fig. 4(c) and (d), a slight instability in the trends ofspecific heat is observed at porosity less than 50%. In addition, theability of the PhCs to introduce or provide the energy necessary tochange the temperature of DT is required at low porosity. In otherwords, the lower the porosity the greater the specific heat. Thisnovel interpretation which introduces the porosity quantity forthermodynamic concepts in PhCs has been made possible here byapplying Bruggeman model that links the porosity to the refractiveindex.

Furthermore, Fig. 5(a) and (b) presented in the square lattice theresult of entropy density which evaluates the quantity that mea-sures the degree of disorder in the PhC and Fig. 5(c) and (d)depicted the trends of entropy density in the case of triangularlattice. Thus, an observed low entropy density is seen in both lat-tices at low temperature. While, the entropy density increase whenthe temperature becomes sufficiently great. This could be inter-preted by the high thermal agitation of the molecules in the crystalat high temperatures due to intense collision and expansion phe-nomena, which would conduct the crystal at important disorderstate. This is verified in fact that, in a crystal, the molecules are wellarranged one next to the other by forming the well-defined pat-terns. A crystal will therefore have entropy lower than gas wherethe molecules are agitated in all directions; in a disorderly manner.Furthermore, it is fundamental to note that the increase of entropy

Fig. 5. Entropy density versus temperature of unit cell of pSi PhCs in different po-rosities: (a) and (b) Correspond to entropy density in the square lattice. (c) and (d)Correspond to entropy density in the triangular lattice. The holes radiusR ¼0:22 mmand the lattice constanta ¼ 0:5 mm.

Fig. 6. Internal energy density versus temperature of unit cell of pSi PhCs in differentporosities: (a) and (b) Correspond to internal energy density in the square lattice. (c)and (d) Correspond to internal energy density in the triangular lattice. The holesradiusR ¼ 0:22 mmand the lattice constanta ¼ 0:5 mm.

Fig. 8. Thermodynamic properties as function of the ratio of the holes radius to thelattice constantR=ain 20%, 50% and 80% porosities for square lattice (left) and triangularlattice (right) pSi PhCs: (a) and (b) Correspond to specific heat. (c) and (d) Correspondto the internal energy density. (e) and (f) Correspond to the entropy density.

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497494

is a consequence of decreasing material porosity; implying that thepresence of pores in the material allow to model and control theentropy response in porous PhCs. These results are in accordance tothe second law of thermodynamic for which, in any process inwhich a thermally isolated system goes from one macrostate toanother, the entropy tends to increase or remain the same, that is,entropy never decreases [59]. In other terms, in any physical pro-cess, the total entropy of a system and its surroundings never de-creases; what is confirmed here.

The distribution of internal energy in the PhCs is projected inFig. 6(a) and (b) for the square lattice consideration. Fig. 6(c) and(d) highlights the distribution of internal energy in the case oftriangular lattice which is regarded as the energy content in thecrystal which is considered as objects collection, such as atoms invibratory state. It presents the similar characteristics with that ofspecific heat conducts depicted in Fig. 4 regardless of the porosityand the type of lattice. This provides some insights in theimprovement of PhCs based thermophotovoltaic energy conver-sion system. Since the internal energy is what is used by theconverter to transform an energy input taken as initial internalenergy of the system into work in energy conversion process.Thus, as for our results, the square lattice predicted optimum in-ternal energy at low porosity from which the useable work couldbe extracted for the energy conversion practice as well as forthermal emission control in solar thermophotovoltaic conversionefficiency.

Fig. 7. Helmholtz free energy density versus temperature of unit cell of pSi PhCs indifferent porosities: (a) Correspond to the square lattice and (b) correspond to thetriangular lattice. The holes radiusR ¼ 0:22 mmand the lattice constanta ¼ 0:5 mm.

Fig. 9. Band structure profile of unit cell of pSi PhCs in 10%, 50% and 90% porositiesfrom top to bottom respectively: Square lattice (left) and triangular lattice (right). Theholes radiusR ¼ 0:22 mmand the lattice constanta ¼ 0:5 mm.

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497 495

Considering the trends Figs. 4e6, they illustrate that the lowerthe porosity, the greater the thermodynamic properties in the caseof the square lattice. Simply, the largest value of thermodynamicquantities is obtained at the lowest porosity, which is estimated at10% for the pSi PhCs. In addition, the analysis of these quantities inthe case of triangular lattice not only presents the identical conductas is observed in the case of square lattice, but also reveals aparticular trend that focuses on porosity effect which was notobserved in the case of square lattice. Indeed, when the porositydecreases from 90% to 30%, these quantities rise until reachingoptimal value at 30% porosity and then become considerably low asporosity decreases to 10%, therefore, posing a challenge. Similarly,in the triangular lattice, the largest value of each of these param-eters is found at 30% of porosity. Below that percentage, the value ofparameters reduces drastically once the porosity tends towards10%. Curiously, it appears to note the presence of instability phe-nomenon of thermodynamic properties with respect to theporosity in a triangular lattice of pSi PhC (circular air holes built inpSi matrix in triangular arrangement).

In both lattices, the all thermodynamic parameters investigatedat low temperature are seemingly low while elevated at hightemperature except for the negative Helmholtz free energy shownin Fig. 7(a) and (b) which is seen to be substantially low. This droprepresents the maximum amount of useful work that can beextracted from the PhCs. The value of thermodynamic propertiesrevealed in the case of square lattice is ten times larger compared tothose observed in the case of triangular lattice. The instability ofthese properties is revealed in the case of the triangular lattice, andabsent in the case of square lattice, therefore, the lattice geometryremains as an influencing factor on the performance of thermo-dynamic properties in a PhCs. This could be justified by the dif-ference within the vectors components in the various latticeswhich are involved in the calculation of the DOS.

Furthermore, in order to investigate the influence of latticedynamic on the prediction of thermodynamic quantities, thenecessary boundary condition was applied to the ratio R=a of theholes radius R to the lattice constant a. The lattice constant ismaintained at 0:5 mmwhile the holes radius varies. In this case, R=adoes not exceed 0.5 in order to provide a physical meaning to themodeling of the PhC structure; i.e. a� 2R>0 according to thedefinition of the lattice constant and the fact that the holes has acylindrical geometry of radius R. Thus, R=a<0:5 is the satisfactorycondition for the ratio R=a. Consequently, this has been taken ac-count in the result depicted in Fig. 8. However, in the both latticesconsidered, the specific heat is higher at low porosity and decreaseswhen R=a tends to its maximum value which is limited to 0.5 (seeFig. 8(a) and (b)). It is worthy to also note that irrespective of type oflattice, a trend of increasing porosity ensuing a decrease in ther-modynamic properties i.e. 20%> 50% > 80% is observed. Withrespect to the square lattice shown in Fig. 8(a), the optimum valueof specific heat is obtained at R=a ¼ 0:30 for 20% and 50% porosities,whereas, at 80% porosity this value is found at R=a ¼ 0:25. In thecase of triangular lattice shown in Fig. 8(b), the specific heat re-mains the same for 20% and 50% porosities when 0:35<R=a<0:45.The optimum value of specific heat is obtained at R=a ¼ 0:25regardless of the porosity. Similar trends as those discussed for thespecific heat in both lattices are found in the case of internal energydensity as shown in Fig. 8(c) and (d), as well in the case of entropydensity as presented in Fig. 8(e) and (f). Besides, the entropy densityis relative to the energy density, and the change of the entropydensity with the ratioR=a is also relative to that of the energydensity. These similar trends found in both entropy density andenergy density implies that the evaluated entropy density isequivalent to the thermal entropy density. This fact is based onYang R [60] extensive discussion on the relationship of both

densities. Overall, the entropy density is low regardless of porosityand lattice configuration. This can be explained by the small vol-ume at micro-scale of the entire photonic crystal considered in thisanalysis. As observed in Fig. 8; which summarizes the overallthermodynamic properties highlighted by the influence of the ratioR=a, the behavior at different porosity revealed the optimum per-formance of thermodynamic properties based pSi PhCs. This can beachieved by considering low silicon porosity as well as low value ofR=a in square lattice for the PhC design. Since low porosity isrequired for high refractive index according to Bruggemanmodel, itis beneficial to optimize the thermodynamic quantities based PhCsby subjecting the design into square lattice at low ratio R=a of cir-cular air holds built in high refractive index materials.

Once the internal energy and entropy densities are determined,other thermodynamic potentials can be obtained via Legendretransformations as mentioned by Ref. [57]. As an example; theenergy flux _UðTÞ can be defined by _UðTÞ≡ R∞

0 Zunðu;TÞfðuÞdu; thisleads to an associated entropy flux _SðTÞwhich can also bewritten as_SðTÞ≡ R∞

0 kBmðu; TÞfðuÞdu. The function fðuÞ is called the spectralflux of modes. If only propagative modes are considered, thespectral flux of modes is related to the DOS through fpropðuÞ ¼cNðuÞ=4. The case of blackbody radiation which corresponds topropagative electromagnetic waves is obtained by consid-eringfðuÞ ¼ cNbbðuÞ=4, with the blackbody DOS NbbðuÞ ¼u2=ðp2c3Þ. Taking this DOS into account, the energy and entropyfluxes become _UbbðuÞ ¼ sT4 and _Sbb ¼ 4sT4=3 respectively, where;sdenotes Stefan's constant. Therefore, these thermodynamic po-tentials could be subject for further investigations in pSi PhCs byachieving a complete improvement in thermo-photovoltaic systemconversion using pSi PhCs.

Some numerical results are worth noting. Excluding the Helm-holtz free energy, increasing in porosity ensued a decrease inthermodynamic properties in both square and triangular latticespSi PhCs. There was slight instability in thermodynamic propertieswith respect to porosity in triangular lattice. In addition, the squarelattice predicted optimal thermodynamic properties at lowporosity and high temperature. This provides an interpretation ofthe thermodynamic properties in pSi based PhCs at known porosityinstead of refractive index. Therefore, one can predict the optimumdegree of material porosity in attaining high performance of ther-modynamic properties which can be used in the selecting of thePhCs to achieve the high efficiency conversion in enhancementsolar cell efficiency.

4. Conclusion

This paper predicted the performance of thermodynamicproperties in pSi based PhCs which was regarded as isolated andnon-interacting particle systems. The results of both latticesrevealed that a low temperature range corresponds to low ther-modynamic quantities in pSi PhCs whatever the degree of porosity.Whereas, the increasing of these properties except the Helmholtzfree energy is related by the more the temperature increase. Theoptimization of these quantities is directly predicted with respectto the degree of porosity which consists to sufficiently decrease theporosity to adapt the PhCs for more efficiency thermal radiationissues. However, the case of triangular lattice revealed a particulartrend within porosities of 30% and 10% which consisted of unstablestate of pSi PhC with respect to the thermodynamic properties athigh temperature. A noteworthy trend of; increasing porosityensuing a decrease in thermodynamic properties was observedirrespective of type of lattice. In the square lattice, the optimumvalue of specific heat is achieved at the ratio R=a ¼ 0:30 for 20% and50% porosities, whereas, at 80% porosity this value is satisfied at R=a ¼ 0:25. In the case of triangular lattice, the specific heat remains

K.A. Amedome Min-Dianey et al. / Energy 154 (2018) 488e497496

the same for 20% and 50% porosities when 0:35<R=a<0:45. Theoptimum value of specific heat is obtained at R=a ¼ 0:25 regardlessthe porosity. These predicted values are also satisfied in the case ofinternal energy density as well as in the case of entropy density inboth lattices studied. The rationale for the use of porosity in thiswork as a means of investigation is according to the Bruggemanwhose approximation model relates the porosity of material and itseffective refractive index. Furthermore, this study proposes that thedesign of such crystal under low porosity as well at low ratio R=aand square lattice considerations will serve fundamental elementsin enhancement of PhCs based thermophotovoltaic energy con-version system and can act as an active layer that could interposebetween emitter and mirror for optimum conversion. Future workscould investigate these aspects in evaluating the detailed process ofPhCs integrated in thermophotovoltaic energy conversion systemwhere the thermal performances as well as the role of thermody-namic properties will be highlighted.

Acknowledgments

This work is supported by the National Natural Science Foun-dation of China (NSFC) (Grant No. 51776050, 51536001).

Nomenclature

a lattice constanta!1; a

!2 primitive basis vectors in real space

A areaApol potential vector

b!

1; b!

2 reciprocal basis vectorsc2 phase velocity in vacuumc2M phase velocity in dielectric material

cð r!Þ2 phase velocityFa;b coefficients in Fourier series

G!

reciprocal lattice vectorZ reduced Planck's constantJ1 First kind of Bessel function

k!

wave vectorkB Boltzmann constantn refractive indexn2 dielectric constantp porositypol polarizationpSi porous siliconr! positionR hole radiusSi siliconT temperatureuðTÞ internal energy densitycV ðTÞ specific heatsðTÞ entropy densityf ðTÞ Helmholtz free energy densityV2 Laplace operator

Greek lettersa; b arbitrary coefficients for reciprocal lattice vectorε permittivitym0 permeabilityz arbitrary coefficient in the eigenvalue problemu Frequency

AbbreviationBZ Brillouin zone

DOS density of statesPhCs photonic crystalsPWE plane wave expansion

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