Effect of orientation on the directional and hemispherical emissivity of hyperbolic metamaterials

International Journal of Heat and Mass Transfer 135 (2019) 1207–1217

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of orientation on the directional and hemispherical emissivity of hyperbolic metamaterials Xiaohu Wu a,b, Ceji Fu a,⇑, Zhuomin M. Zhang b,⇑ a b

LTCS and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

a r t i c l e

i n f o

Article history: Received 21 November 2018 Received in revised form 31 January 2019 Accepted 20 February 2019

Keywords: Hemispherical emissivity Metamaterials Thermal radiation Uniaxial crystal

a b s t r a c t We investigate the spectral directional and hemispherical emissivity of hyperbolic metamaterials with arbitrary orientation of the optic axis. The 4
4 matrix method is combined with coordinate rotational transforms to circumvent projection operations. The two hyperbolic bands of hexagonal boron nitride (hBN) are examined in detail to elucidate the influence of the orientation of the optic axis on the emissivities for both transverse magnetic (TM) and transverse electric (TE) waves. The results show that the orientation of the optic axis can greatly affect the hemispherical emissivity in the two hyperbolic bands of hBN. The directional emissivity varies periodically with the azimuthal angle when TM and TE waves are coupled, resulting in conversion of polarization in the medium. For both TM and TE waves, the matching of impedance and admittance at the surface of the material is better for smaller tilting angle in the type I hyperbolic band, but the opposite is true in type II hyperbolic band where the matching becomes better for larger tilting angles. As the tilting angle increases, the hemispherical emissivity decreases in the type I hyperbolic band but increases in the type II hyperbolic band. Our conclusions can be extended to other hyperbolic materials. Therefore, this work may provide valuable guidance on characterization, measurement and tailoring of the directional and hemispherical emissivity for hBN and other hyperbolic materials with arbitrary orientation and application of such materials for manipulation of radiative heat transfer. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The emissivity is a vital parameter for radiative heat transfer analysis in numerous practical applications such as space cooling, materials processing, and energy harvesting [1–5]. Both theoretical calculation methods [6–11] and experimental measurement methods [12–15] have been extensively pursued to investigate the emissivity mainly for isotropic materials. For example, Zhou et al. [5] modeled the effective emissivity of silicon wafers considering multiple reflections in the rapid thermal processing chamber. Wen and Mudawar [8] investigated the effects of surface roughness on the emissivity of aluminum alloys and compared different models with experiments. King et al. [6] developed a computational algorithm based on geometric optics model for calculating the total hemispherical emissivity from the directional spectral models. Wang et al. [11] have shown the equivalence between the direct method and indirect method for layered structures made ⇑ Corresponding authors. E-mail addresses: [email protected] (C. Fu), [email protected] (Z.M. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.066 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

of isotropic materials, where the latter based on the reflectivity and Kirchhoff’s law [2] is much simpler than the former based on the fluctuational electrodynamics [9]. Zhang and Wang [4] provided an extensive review of the modeling methods and measurement techniques for the spectral and directional radiative properties of micro/nanostructured surfaces. Watanabe et al. [15] described the wide variety of experimental instruments and methods used for spectral emissivity measurements. Jones et al. [16] compiled the high-temperature emissivity data with an extensive review of the relative literature. However, most of those works dealt only with isotropic materials. With the development of optically anisotropic metamaterials [17–22] and 2-D materials, such as hexagonal boron nitride (hBN) [23,24] and black phosphorus (BP) [25,26], more and more attention has been paid to the exotic properties and promising applications of such anisotropic materials, especially for optoelectronics and energy harvesting. Thus, it is critical to determine the emissivity of anisotropic materials. Unlike isotropic materials, it is challenging to calculate the emissivity of anisotropic materials because the transverse electric (TE) and the transverse magnetic (TM) waves can couple in the materials, resulting in

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Nomenclature A E H j k0 kx Kx p q R r S s Ty ; Tz U w W x, y, z Y Z

aTM;k

coefficient matrix electric field magnetic field pffiffiffiffiffiffiffi 1 wavevector in vacuum, m1 wavevector component along the x-axis, m1 normalized wavevector component along the x-axis transverse magnetic wave eigenvalue reflection matrix reflection coefficient electric field component transverse electric wave rotational transformation matrix magnetic field component eigenvector element eigenvector space coordinate, m normalized admittance normalized impedance spectral directional absorptivity of transverse magnetic wave

cross-polarization effects. For a uniaxial crystal, its emissivity depends strongly on the orientation of its optic axis, making the calculation of the emissivity very complicated. Nevertheless, in the case of a semi-infinite uniaxial crystal, explicit expressions of emissivity have been obtained when the optic axis is either perpendicular or parallel to the material surface [27]. Besides, Autio and Scala [28] experimentally demonstrated that the emissivity of a uniaxial crystal is different when its optic axis is perpendicular or parallel to the surface. However, very few studies have discussed the emissivity of hBN and other hyperbolic materials when the optic axis is tilted with respective to the surface. The 4
4 matrix method is commonly used for calculating the optical properties of anisotropic materials with several different formulations [29–32]. However, these formulations are quite complex and tedious for use to calculate the hemispherical emissivity of an anisotropic material. For example, the applied boundary conditions need to be projected onto the coordinate axes if the plane of incidence is not parallel or perpendicular to the coordinate axes, since the incident TE and TM waves are always defined based on the plane of incidence. Besides, the TE (TM) wave will have two electric (magnetic) field components along the coordinate axes if the plane of incidence is neither parallel nor perpendicular to the coordinate axes. On the other hand, the calculated electromagnetic field components along the coordinate axes should be projected back onto the plane of incidence if one wants to define the reflected TE and TM waves. These two projections make the calculations and interpretations rather complicated. In this paper, we numerically investigate the spectral directional and hemispherical emissivity of hBN assumed in the form of bulk, with emphasis on the effect of its optic axis orientation. Since hBN is a natural hyperbolic material with two hyperbolic bands of different types, the results obtained here are expected to provide some general guidelines on understanding other hyperbolic materials. A modified 4
4 matrix method is employed to calculate the spectral directional emissivity of the material for each polarization, in which a rotational transformation technique is implanted to avoid the projection operations when solving for

aTE;k

spectral directional reflectivity of transverse electric wave tilting angle, degree relative permittivity matrix relative dielectric function permittivity of vacuum spectral directional emissivity of transverse magnetic wave spectral directional emissivity of transverse electric wave spectral hemispherical emissivity angle of incidence, degree wavelength, m permeability of vacuum spectral directional reflectivity of transverse magnetic wave spectral directional reflectivity of transverse electric wave azimuthal angle, degree angular frequency, rad s1 parallel component vertical component

b

e e e0 eTM;k eTE;k ek

h k

l0 qTM;k qTE;k u x jj ?

the reflection coefficient matrix. The hemispherical emissivity is calculated by integration of the directional emissivity. The effect of the optic axis orientation on the hemispherical emissivity of hBN is illustrated in its two hyperbolic bands. The spectral directional emissivity for TM and TE waves is also calculated together with the normalized impedance and admittance to elucidate the effect of the optic axis orientation on the emissivity in the hyperbolic bands of hBN. 2. Methodology The emissivity of a semi-infinite medium can be obtained from the reflectivity based on Kirchhoff’s law [1–3]. For an isotropic medium, the reflectivity for each polarization can be obtained based on Fresnel’s coefficients. For an anisotropic medium, however, cross-polarization can occur. Therefore, one needs to obtain the reflection coefficient matrix defined as follows:




R¼

r pp

r ps

r sp

r ss



ð1Þ

Here, the first and the second subscripts of each reflection coefficient describe the polarization status of the incident and the reflected waves, respectively. Although there have been numerous studies on the optical properties of anisotropic materials, conventional approaches are usually cumbersome and complicated due to anisotropy [29,30]. Here, we present a concise procedure for calculating the spectral hemispherical emissivity of a semi-infinite uniaxial crystal. The permittivity tensor of the   uniaxial medium can be expressed as e ¼ diag e? ; e? ; ejj when its optic axis is along the z-axis of the coordinate system. When the optic axis is tilted off the z-axis by an angle b in the x-z plane, as shown in Fig. 1(a), the permittivity tensor in the xyz coordinate system can be expressed as

0

e?

0

0

1

0

cosb 0 sinb

B C 1 e ¼ Ty B 1 @ 0 e? 0 ATy ; Ty ¼ @ 0 0 0 ejj sinb 0

0 cosb

1 C A

ð2Þ

X. Wu et al. / International Journal of Heat and Mass Transfer 135 (2019) 1207–1217

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only two components with respect to the x0 y0 z0 coordinate system. Secondly, if the calculations are performed in the xyz coordinate system, all the electromagnetic field vectors need to be projected onto the coordinate axes when dealing with the boundary conditions. This process can be avoided if the calculations are performed in the x0 y0 z0 coordinate system. Similar technique has been used by Rosa et al. [33] in their analysis of the Casimir interactions for anisotropic magnetodielectric metamaterials, although their analysis is only for the case when the optic axis is either parallel or perpendicular to the surface of the medium. Here, we consider the general case where the optic axis can be arbitrarily oriented with respect to the surface. Consider a TM wave incident on the surface of a uniaxial medium at an angle of incidence h, the electromagnetic fields in the medium can be expressed in the x0 y0 z0 coordinate system in the following form [34], where and hereinafter the symbol 0 is omitted to keep the writing succinct and for brevity:

  H ¼ UðzÞexpðjxt  jkx xÞ; where U ¼ U x ; U y ; U z and E ¼ j



l0 =e0

1=2

  SðzÞexpðjxt  jkx xÞ; where S ¼ Sx ; Sy ; Sz

ð5Þ ð6Þ

Fig. 1. Schematic of the coordinate systems used for calculating the reflection coefficients. Here, the upper half is air (or vacuum) and the lower half is a uniaxial crystal whose optic axis is tilted in the x-z plane by an angle b with respect to the zaxis. (a) The plane of incidence is the x-z plane with an incidence (polar) angle h; (b) The plane of incidence is rotated around z-axis by an azimuthal angle /. In the x0 y0 z0 coordinate system, the plane of incidence in the x0 -z0 plane, where the z0 -axis and zaxis are the same.

When b is equal to 0° or 90°, the optic axis of the crystal is perpendicular or parallel to the surface of the material, respectively. These two cases have been considered previously [27]. Here, we consider the general case whenb is between 0° and 90°. In order to calculate the hemispherical emissivity of the uniaxial medium, it is necessary to calculate the emissivity for different combinations of polar angle h and azimuthal angle /. When the azimuthal angle / is 0°, the plane of incidence is the x-z plane, as shown in Fig. 1(a). When / is not equal to 0°, the plane of incidence is rotated off the x-z plane, as shown in Fig. 1(b). It is cumbersome to calculate the emissivity in the xyz coordinate system when / is not equal to 0°. Here, we choose the x0 y0 z0 coordinate system to evaluate the reflection coefficients and use them to obtain the directional emissivity. The permittivity tensor in the x0 y0 z0 coordinate system can be expressed as

0

1

2 0

1

3

e? 0 0 exx exy exz C 6 B C 1 7 1 e¼B @ eyx eyy eyz A ¼ Tz 4Ty @ 0 e? 0 ATy 5Tz 0 0 ejj ezx ezy ezz

ð3Þ

where Tz is the coordinate rotational transformation matrix, which is given by

0

cos/

sin/

0

0

0

1

1

B C Tz ¼ @ sin/ cos/ 0 A

ð4Þ

It should be noted that when / is 0°, Tz is a unit matrix, and the plane of incidence is the x-z plane. So we can always calculate the emissivity in the x0 y0 z0 coordinate system no matter whether the azimuthal angle is 0° or not. While the two approaches are equivalent, carrying out the calculations in the x0 y0 z0 coordinate system has two advantages. Firstly, in the xyz coordinate system, the wavevector has three components. However, the wavevector has

Fig. 2. Calculated emissivity spectra of hBN for different titling angles: (a) hemispherical emissivity; (b) normal emissivity, i.e., h ¼ 0 .

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Fig. 3. Contour plots for the spectral directional emissivity of hBN with respect to the azimuthal angle / and tilting angle b at wavenumber of 1400 cm1: (a) TM wave, h ¼ 0 ; (b) TE wave, h ¼ 0 ; (c) TM wave, h ¼ 60 ; (d) TE wave, h ¼ 60 .

Note that e0 and l0 are the absolute permittivity and permeability of vacuum, respectively, xis the angular frequency and kx is the wavevector component along the x-axis given by pffiffiffiffiffiffiffiffiffiffi kx ¼ k0 sinh, k0 ¼ x e0 l0 . By substituting Eqs. (5) and (6) into the Maxwell equations and setting K x ¼ kx =k0 , we obtain the following differential equations:

0

Sx

1

0

Sx

1

C BS C dB B Sy C B yC B C ¼ k0 AB C A @ @ Ux A dz U x Uy

ð7Þ

Uy

jK x ezx =ezz

6 6 0 A¼6 6 e e =e  e 4 yz zx zz yx exx  exz ezx =ezz

ð11Þ

and U y ðzÞ ¼ w41 c1 expðk0 q1 zÞ þ w42 c2 expðk0 q2 zÞ

ð12Þ

where wim is the corresponding element of the eigenvector matrix W of matrix A in Eq. (8), q1 and q2 are the two eigenvalues of matrix A with negative real parts, c1 and c2 are unknowns. By applying the boundary conditions, i.e., the tangential components of the magnetic and electric field vectors should be continuous at the vacuum/hBN interface, it can be shown that

0

where the coefficient matrix is

2

U x ðzÞ ¼ w31 c1 expðk0 q1 zÞ þ w32 c2 expðk0 q2 zÞ

jK x ezy =ezz 0

0 K 2x =ezz  1 1

0

eyz ezy =ezz þ  eyy 0 jK x eyz =ezz exy  exz ezy =ezz 0 jK x exz =ezz K 2x

jcosh 0 w11 B 0 j w21 B B @ 0 cosh w31

3 7 7 7 7 5

ð8Þ

The electromagnetic fields in the medium can be described by the eigenvalues and eigenvectors of the coefficient matrix A. The transmitted fields into the semi-infinite uniaxial medium can be expressed as

Sx ðzÞ ¼ w11 c1 expðk0 q1 zÞ þ w12 c2 expðk0 q2 zÞ

ð9Þ

Sy ðzÞ ¼ w21 c1 expðk0 q1 zÞ þ w22 c2 expðk0 q2 zÞ

ð10Þ

1

0

w41

1 10 1 0 r pp jcosh w12 C C B B w22 C CB r ps C B 0 C C CB C¼B w32 A@ c1 A @ 0 A w42

c2

ð13Þ

1

From the above equation, the reflection coefficients along with c1 and c2 can be obtained. Likewise, when the incident wave is a TE wave, the electromagnetic fields in the medium can be written as follows:

  E ¼ SðzÞexpðjxt  jkx xÞ; where S ¼ Sx ; Sy ; Sz

ð14Þ

 1=2 UðzÞexpðjxt  jkx xÞ; where U and H ¼ j e0 =l0   ¼ Ux ; Uy ; Uz

ð15Þ

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According to Kirchhoff’s law, the directional emissivity for each polarization reads [1,2]

eTM;k ðh; /Þ ¼ aTM;k ðh; /Þ; eTE;k ðh; /Þ ¼ aTE;k ðh; /Þ

ð19Þ

Finally, the hemispherical emissivity can be calculated from the spectral directional emissivity as [1,2]

ek ¼

1 2p

Z 2p Z p=2

eTM;k ðh; /Þ þ eTE;k ðh; /Þ 2

0

0

sinð2hÞdhd/

ð20Þ

All the above-mentioned radiative properties are spectral quantities. If the total emissivity is needed, one can perform a weighted integration over Planck’s blackbody distribution function evaluated at a given surface temperature. This algorithm has been validated by comparing Fresnel’s coefficients obtained with it with those obtained with the traditional 4
4 matrix method [31], and by comparing the calculated spectral emissivity of a uniaxial medium when the optic axis is either perpendicular or parallel to the surface with published results [27]. It should be pointed out that the spectral hemispherical emissivity is the same as long as the tilting angle is the same, no matter whether the optic axis is tilted in the xz plane or not. Although other methods have also been developed for calculation of Fresnel’s coefficients of uniaxial materials with arbitrarily oriented optic axis [35,36], it is very convenient to calculate the spectral hemispherical emissivity of such materials with the algorithm presented here. 3. Numerical results and discussion When the optic axis of hBN is along the z-axis, its (relative) permittivity tensor can be expressed as

0

e?

0

0

1

C e¼B @ 0 e? 0 A 0 0 ejj

ð21Þ

where the subscript ? or k indicates the component perpendicular or parallel to the optic axis. The components e? and ejj are given as a function of frequency as [24]

Fig. 4. The emissivity averaged over TM and TE waves as a function of / and b at the wavenumber of 1400 cm1: (a) h ¼ 0 ; (b) h ¼ 60 .

By substituting Eqs. (14) and (15) into the Maxwell equations, the same differential equations as those in Eq. (7) can be obtained with exactly the same coefficient matrix shown in Eq. (8). One can express the fields in the uniaxial medium in the same forms as in Eqs. (9)–(12) and apply the boundary conditions. Similar to the case with TM wave incidence, the following equation can be obtained and solved for rss ,rsp , c1 andc2 :

0 B B B @

cosh

0

w11

0

1

w21

0 j

jcosh w31 0

w41

w12

10

r sp

0

1

0

1

C C B B w22 C CB r ss C B 1 C C CB C¼B w32 A@ c1 A @ jcosh A w42

c2

ð16Þ

0

Once the reflection coefficients are obtained, the spectral directional-hemispherical reflectivity for TM and TE waves are given respectively as [27]













qTM;k ðh; /Þ ¼ rpp 2 þ rps 2 ; qTE;k ðh; /Þ ¼ rsp 2 þ jrss j2

ð17Þ

The directional absorptivity is obtained by applying the energy balance at the interface for each polarization, viz.

aTM;k ðh; /Þ ¼ 1  qTM;k ðh; /Þ; aTE;k ðh; /Þ ¼ 1  qTE;k ðh; /Þ

ð18Þ

em ðxÞ ¼ e1;m 1 þ

x2LO;m  x2TO;m 2 xTO;m  x2 þ jxCm

! ð22Þ

where m ¼? or k, xTO;? ¼ 1370 cm1 , xTO;jj ¼ 780 cm1 , 1 1 xLO;? ¼ 1610 cm , xLO;jj ¼ 830 cm , e1;? ¼ 4:87, e1;jj ¼ 2:95, C? ¼ 5 cm1 , and Cjj ¼ 4 cm1 . Note that for convenience, the unit of angular frequency x is taken as wavenumber (1 cm1 ¼ 1:884
1011 rad=s), as often done in the literature [1]. In the following discussions, the term frequency and wavenumber are used interchangeably. The components e? and ejj are related respectively to the in-plane and out-of-plane optical phonon modes in hBN, resulting in two mid-infrared Reststrahlen bands: one is between xTO;? and xLO;? for e? and the other is between xTO;jj and xLO;jj for ejj . One can find from Eq. (22) that these two Reststrahlen bands are the two hyperbolic bands of hBN, since e? and ejj possess opposite signs in these two bands, which results in the dispersion relation for electromagnetic wave propagation in the medium exhibiting the hyperbolicity of different type. The hyperbolic band with ejj < 0 is type I while that with e? < 0 is type II. These two hyperbolic bands have been discussed in detail by Zhao and Zhang [37] and the dielectric functions of hBN have been plotted as Fig. 3 in Ref. [37]. The calculated spectral hemispherical emissivity of the hBN varying with wavenumber for different tilting angles is shown in Fig. 2(a). It can be seen that the emissivity drops abruptly and is also affected significantly by the tilting angle in the two hyperbolic

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 2  2  2 Fig. 5. Reflectivity components for a plane wave at normal incidence and at 1400 cm1: (a) r pp  ; (b) r ps  ; (c) jr ss j2 ; (d) r sp  .

bands. The normal emissivity of the hBN medium is shown in Fig. 2 (b) for comparison. It can be seen that the normal and hemispherical emissivities have very similar trends, except at 830 cm1 when the tilting angle is 0° and at 1610 cm1 when the tilting angle is 90°. Similar feature to the drop of emissivity for hBN in the two hyperbolic bands has been demonstrated in the experimental study of pyrolytic boron nitride (pBN) at elevated temperatures [38]. However, since their study dealt with a randomly oriented polycrystalline material, the dielectric function of pBN was modeled based on the effective medium approaches using a weighted average of the two polarizations. Therefore, all the anisotropic features were obscure and only the average effect was observed in the experiment. It is impossible to give quantitative comparison. Our numerical model allows the calculation of the detailed contributions from different hBN orientation as well as emission angles. In the following, we pick one wavenumber in each of the hyperbolic band to illustrate the effect of orientation and polarization on the emissivity. Taking x ¼1400 cm1 as an example, which is within the type II hyperbolic band, we calculate the spectral directional emissivity as a function of the tilting angle b and the azimuthal angle /. The results for TM and TE waves at h ¼ 0 are shown in Fig. 3(a) and 3(b), and those at h ¼ 60 are in Fig. 3(c) and 3(d), respectively. It can be seen that when h ¼ 0 , the emissivity patterns for TM and TE waves are the same except that there is a shift in the azimuthal angle of D/ = 90 between the two patterns. As shown in Fig. 3(a) and 3(b), the emissivity for TM wave is high only in the regions

when / is around 0° and 180° and b > 20°, while the emissivity for TE wave is high only in the regions when / is around 90° and 270° and b > 20°. When h ¼ 60 , the regions of high emissivity for TM wave becomes plumper but the regions of high emissivity for TE wave shrinks compared to the cases at h ¼ 0 . This suggests that the emissivity increases for TM wave but decreases for TE wave as h increases from 0° to 60°. The spectral directional emissivity averaged over TM and TE waves is shown in Fig. 4(a) and (4b) for h ¼ 0 and h ¼ 60 , respectively. It can be seen that the averaged emissivity is independent of the azimuth angle when h = 0°. However, when h = 60°, the averaged emissivity becomes a periodic function of the azimuthal angle / with a period of 180°. It should be pointed out that occurrence of polarization conversion between TM and TE waves gives rise to a small difference of the emissivity pattern in the azimuthal angle region [0°, 180°) from that in [180°, 360°). Consequently, the emissivity for a TM or TE wave is in fact a periodic function of / with a period of 360°. But the difference is so small that the emissivity patterns shown in Fig. 3 exhibit a period of 180°. Similar results have been found for the averaged reflectivity of an anisotropic metamaterial [17]. In order to weight the impact of polarization conversion on the  2 reflectivity and emissivity of the medium, the values of rpp  and  2 r ps  varying with the angles / and b for normal incidence of a  2 TM wave, and those of jrss j2 and rsp  for normal incidence of a TE wave are plotted in Fig. 5(a–d), respectively. The reflectivity

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Fig. 6. The spectral directional emissivity of hBN at of the medium varying with at the wavenumber of 800 cm1 as a function of / and b: (a) TM wave, h ¼ 0 ; (b) TE wave, h ¼ 0 ; (c) TM wave, h ¼ 60 ; (d) TE wave, h ¼ 60 .

components resulting from polarization conversion can be clearly observed in Fig. 5(b) and 5(d) for / around 45°, 135°, 225° and  2  2 315°. In addition, the values of r ps  and r sp  increase with the tilting angle and their maxima appear at b ¼ 90 . The occurrence of the maximum polarization conversion can also impact nearfield radiative heat transfer as demonstrated by Liu et al. [18]. The directional emissivity of the medium at 800 cm1, which is within the type I hyperbolic band, varying with the azimuthal angle and the tilting angle is shown in Fig. 6(a) and 6(b) for normal incidence of TM and TE waves, respectively. The corresponding results for incidence at h ¼ 60 are plotted in Fig. 6(c) and 6(d). Interestingly, the emissivity patterns are found to be complementary to those at wavenumber 1400 cm1, i.e., the regions of high emissivity in Fig. 6 are almost those exhibiting low emissivity in Fig. 3. As shown in Fig. 6(a), the emissivity for TM wave is small only in the regions when / is around 0° and 180° and b > 40°. But Fig. 6(b) reveals that the emissivity for TE wave is small only in the regions when / is around 90° and 270° and b > 40°. This is anticipated since the lattice vibration for hBN is parallel to the optic axis in the type I hyperbolic band, while it is perpendicular to the optic axis in the type II hyperbolic band. To gain a deeper understanding of these results, we analyze the impedance of the medium for incidence of a TM wave and the admittance of the medium for incidence of a TE wave on it. For convenience, let us focus on the four azimuthal angles: / ¼ 0 , 90°, 180° and 270°. When / ¼ 0° or 180°, the optic axis is in the plane of incidence and there is no coupling between TM and TE waves in this case regardless of the angle of incidence. The

normalized impedance is defined as the ratio of the impedance of hBN over that of vacuum. For TM wave incidence, the normalized impedance for / ¼ 0° or 180° is expressed as [39,40]

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ejj cos2 b þ e? cos2 b  sin2 h Z¼ e? ejj cos2 h

ð23Þ

Similar definition applies for the admittance of the medium for TE wave incidence. Therefore, the normalized admittance for / ¼ 0 or 180° is written as [40]

Y¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e?  sin2 h cosh

ð24Þ

The reflection coefficients r pp and r ss can be expressed respectively in terms of the normalized impedance and the normalized admittance as follows [40].

rpp ¼

1Z 1Y and rss ¼ 1þZ 1þY

ð25Þ

The normalized impedance of the hBN medium for / ¼ 0 or 180° is calculated using Eq. (23) for a TM wave incidence at 1400 cm1. The real and imaginary parts of the normalized impedance versus the tilting angle b at h = 0 and 60 are shown in Fig. 7(a). It can be seen that when h ¼ 0 , the imaginary part of the normalized impedance is very small when b > 20°, while the real part increases monotonically with b and exceeds 0.5 when b > 60 . This implies that matching of impedance between vacuum and hBN is getting better and better as b increases, resulting in a

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Fig. 7. The normalized (a) impedance for TM wave incidence and (b) admittance for TE wave incidence at 1400 cm1, and the normalized; (c) impedance for TM wave incidence and (d) admittance for TE wave incidence at 800 cm1.

reduction of the reflectivity and enhancement of the absorptivity, according to Eq. (25). When h ¼ 60 , Im(Z) is also very small when b > 20°, but Re(Z) is closer to unity than the case for normal incidence. Therefore, the emissivity is larger than the case for h ¼ 0 , as revealed in Fig. 3. Hence, applying the concept of impedance helps to elucidate the trend of the emissivity varying with the angle of incidence. When the incidence is a TE wave, the real and imaginary parts of the normalized admittance are plotted in Fig. 7(b) for h = 0 and 60 , respectively, when / ¼ 0 or 180°. The admittance is independent of b in this case. While the real part is small, the magnitude of the imaginary part is much larger than unity, especially for the case when h ¼ 60 . This is due to the fact that the real part of e? is negative at 1400 cm1. A large admittance will result in a large reflection from the surface and thus the emissivity is small, as already seen from Fig. 3(b) and 3(d) at / = 0 or 180 . When the azimuthal angle / is 90° or 270°, TE and TM waves inside the medium are decoupled only when the angle of incidence is 0°. In this case, the normalized impedance for a TM wave incidence is

1 Z ¼ pffiffiffiffiffi

ð26Þ

e?

and the normalized admittance of the medium for a TE wave incidence is

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Y¼

ejj

e? ejj 2 þ e? sin b

cos2 b

ð27Þ

Note that the impedance and the admittance in Eqs. (26) and (27) are respectively the reciprocals of the admittance in Eq. (24) and the impedance in Eq. (23) when h ¼ 0 . Therefore, it can be inferred from Eq. (25) that the emissivity for TM wave at / = 90 or 270 is the same as the emissivity for TE wave at / = 0 or 180 and vice versa, as seen in Fig. 5(a) and 5(c). Impedance and admittance cannot be defined when TM and TE waves inside the medium are coupled to each other. Therefore, they fail to be applicable for interpreting the reflectivity and emissivity when coupling and polarization conversion occur between TM and TE waves. Similar discussion can be made for the case at wavenumber equal to 800 cm1. The real and imaginary parts of the normalized impedance varying with the tilting angle b for a TM wave at h = 0 and 60 are shown in Fig. 7(c), when the azimuthal angle is 0° or 180°. It can be seen that for either value of the angle of incidence, the real part of the normalized impedance is far away from unity when b > 40°. On the other hand, the imaginary part increases with b. Hence, the reflectivity is large and the emissivity is small, which agrees with the results shown in Fig. 6(a) and 6(c). The corresponding results of the admittance for incidence of a TE wave are shown in Fig. 7(d), where the flat lines indicate that the admittance is independent of b. However, in contrast to the case at 1400 cm1, the imaginary part here is close to zero while the real part is larger than unity. This is because e? has a positive real part at 800 cm1. From these values of the admittance and according to Eq. (25), the emissivity at 800 cm1 is not so small, especially for the case when h ¼ 0 . This explains the calculated results shown in Fig. 6(b) and 6 (d). When the azimuthal angle / is 90° or 270°, and the angle of

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Fig. 8. The spectral directional emissivity at 1400 cm1: (a) TM wave, b = 0°; (b) TE wave, b = 0°; (c) TM wave, b = 45°; (d) TE wave, b = 45°; (e) TM wave, b = 90°; (f) TE wave, b = 90°.

incidence is 0°, the emissivity for TM wave is the same as that for TE wave at / = 0° or 180° and vice versa. The same reason could be used to explain the result at wavenumber 1400 cm1. We now turn our attention to the directional emissivity of the medium when the tilting angle b is fixed. Fig. 8(a), 8(c) and 8(e) show the emissivity as a function of the angles / and h for TM wave at the wavenumber of 1400 cm1, when b is 0°, 45°, and 90°, respectively. It can be seen from Fig. 8(a) that the emissivity is independent of /. This is because the optic axis is along the

z-axis when b is 0°; thus, there is no coupling effect between TE and TM waves. However, the emissivity first increases with h, reaching a maximum (though small) at an angle greater than 80°, then drops rapidly to zero when h = 90°. In contrast, due to existence of polarization conversion, the emissivity depends on both / and h when b is greater than 0°, as shown in Fig. 8(c) and 8(e). In addition, the emissivity is high around / = 0° and 180°. It first increases with h and then drops rapidly when h exceeds about 80°. The corresponding results for TE wave at wavenumber

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Fig. 9. The directional emissivity at 800 cm1: (a) TM wave, b = 0°; (b) TE wave, b = 0°; (c) TM wave, b = 45°; (d) TE wave, b = 45°; (e) TM wave, b = 90°; (f) TE wave, b = 90°.

1400 cm1 are shown in Fig. 8(b), 8(d) and 8(f), respectively. In this case, the dependence of the emissivity on / is similar to that for TM wave, except that the regions for high values of the emissivity are around / = 90° and 270°, as those shown in Fig. 3(b) and 3(d). Furthermore, the emissivity decreases as h increases, which is different from the case for TM wave. The corresponding cases at wavenumber 800 cm1 are shown in Fig. 9(a–f) for comparison. It can be seen that the distribution patterns are quite similar to those in Fig. 8(a–f). However, when b = 0°, the emissivities for both TM and TE waves are much higher

than that at wavenumber 1400 cm1. In addition, for the cases of b = 45° and 90°, high emissivity values for TM wave are found around / = 90° and 270°, while those for TE wave are found around / = 0° and 180°, in contrast to those at wavenumber 1400 cm1. The above analysis can be applied to the spectral directional emissivity of hBN in other wavenumber regions. However, the influence of the optic axis orientation in other wavenumber regions is not as significant as in the hyperbolic bands, since the real parts of e? and ejj have the same sign and the imaginary parts are relatively small.

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4. Conclusions The spectral directional and hemispherical emissivities of hBN with arbitrary orientation of the optic axis have been extensively investigated, focusing on the hyperbolic bands, using the 4
4 matrix method in combination with coordinate rotational transforms to circumvent projection operations. It is found that the orientation of the optic axis can greatly affect the emissivity in the two hyperbolic bands of hBN. The spectral directional emissivity for TM and TE waves varies periodically with / due to conversion of polarization when the optic axis is tilted off the z-axis. The switching of the positive and negative signs of ejj and e? between the two hyperbolic bands gives rise to opposite trends in matching of impedance and admittance at the surface of the material. As the tilting angle increases, the matching becomes worse for type I but better for type II hyperbolic region. Subsequently, as the tilting angle increases, the spectral hemispherical emissivity decreases in type I hyperbolic band but increases in type II hyperbolic band. The conclusions drawn from the present study using hBN should apply for the spectral directional and hemispherical emissivity of other hyperbolic materials. Therefore, the results obtained in this work may provide valuable guidance on characterization, measurement and tailoring of the directional and hemispherical emissivity for hBN and other hyperbolic materials with arbitrary orientation and application of such materials for manipulation of radiative heat transfer. Conflicts of interest The authors declare no conflicts of interest. Acknowledgements X. Wu was supported by the China Scholarship Council (No. 201706010271); C. Fu was supported by the National Natural Science Foundation of China (No. 51576004); and Z.M. Zhang was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, United States (DE-SC0018369). References [1] Z.M. Zhang, Nano/Microscale Heat Transfer, McGraw-Hill, New York, 2007. [2] J.R. Howell, M.P. Menguc, R. Siegel, Thermal Radiation Heat Transfer, sixth ed., CRC Press, Boca Raton, FL, 2016. [3] M.F. Modest, Radiative Heat Transfer, third ed., Academic Press/Imprint of Elsevier, Amsterdam, Netherlands, 2013. [4] Z.M. Zhang, L.P. Wang, Measurements and modeling of the spectral and directional radiative properties of micro/nanostructured materials, Int. J. Thermophys. 34 (12) (2013) 2209–2242. [5] Y.H. Zhou, Y.J. Shen, Z.M. Zhang, B.K. Tsai, D.P. DeWitt, A Monte Carlo model for predicting the effective emissivity of the silicon wafer in rapid thermal processing furnaces, Int. J. Heat Mass Transf. 45 (9) (2002) 1945–1949. [6] J.L. King, H. Jo, S.K. Loyalka, R.V. Tompson, K. Sridhatan, Computation of total hemispherical emissivity from directional spectral models, Int. J. Heat Mass Transf. 109 (2017) 894–906. [7] M. Rosenberg, R.D. Smirnov, A.Y. Pigarov, On thermal radiation from fusion related metals, Fusion Eng. Des. 84 (1) (2009) 38–42. [8] C.D. Wen, I. Mudawar, Modeling the effects of surface roughness on the emissivity of aluminum alloys, Int. J. Heat Mass Transf. 49 (23) (2006) 4279– 4289. [9] Y. Guo, Z. Jacob, Fluctuational electrodynamics of hyperbolic metamaterials, J. Appl. Phys. 115 (2014) 234306. [10] L. Gu, J.E. Livenere, G. Zhu, T.U. Tumkur, H. Hu, C.L. Cortes, Z. Jacob, S.M. Prokes, M.A. Noginov, Angular distribution of emission from hyperbolic metamaterials, Sci. Rep. 4 (2014) 7327. [11] L.P. Wang, S. Basu, Z.M. Zhang, Direct and indirect methods for calculating thermal emission from layered structures with nonuniform temperatures, J. Heat Transf. 133 (7) (2011) 072701.

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