Enhanced Near-Field Thermal Radiation Based on Multilayer

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Enhanced Near-Field Thermal Radiation Based on Multilayer Graphene-hBN Heterostructures Kezhang Shi, Fanglin Bao, and Sailing He ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00037 • Publication Date (Web): 22 Mar 2017 Downloaded from http://pubs.acs.org on March 30, 2017

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Enhanced Near-Field Thermal Radiation Based on Multilayer Graphene-hBN Heterostructures Kezhang Shi,1 Fanglin Bao,2,* and Sailing He1,2,3,* Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, JORCEP, Zhejiang University, Hangzhou 310058, China 2 Centre for Optical and Electromagnetic Research, ZJU-SCNU Joint Center of Photonics, South China Academy of Advanced Optoelectronics, South China Normal University, Guangzhou 510006, China 3 Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, Stockholm S-100 44, Sweden 1

ABSTRACT: Graphene-covered hexagonal boron nitride (hBN) can exceed black body thermal radiation in near-field due to the coupling of surface plasmon polaritons (SPPs) and hyperbolic phonon polaritons (HPPs). As previous research found that the thickness of hBN in a graphene-hBN cell can be very thin while still presenting strong radiation enhancement, multilayer graphene-hBN heterostructures are proposed in this paper to further enhance the near-field thermal radiation. We found that a heterostructure consisting of five or more graphene-hBN cells performs better than all existing graphene-hBN configurations, and the infinite-cell limit exhibits 1.87- and 2.94-fold larger heat flux at 10 nm separation than sandwich and mono-cell structures do, respectively, due to the continuously and perfectly coupled modes. The heat flux is found to be 4 orders larger of magnitude larger than that of the blackbody. The effective tunability of the thermal radiation of the multi-cell structure is also observed by adjusting the chemical potentials of graphene with an optimized thickness of 20 nm on each hBN, which is instructive for both experimental design and fabrication of thermal radiation devices. KEYWORDS: near-field, thermal radiation, graphene-hBN heterostructure Near-field thermal radiation can exceed the black body limit, which is governed by the well-known Stefan-Boltzmann Law, by several orders of magnitude1-3, due to the tunneling effect of evanescent modes, especially when surface modes or hyperbolic modes are excited4. The enhancement and further manipulation of near-field thermal radiation has promising wide range applications, from thermal signature control5 and infrared nano-imaging6 to near-field thermophotovoltaics2,7, for which a number of hyperbolic metamaterials, such as multilayer structures, nanowires, nanoholes, and 2D-gratings composed of D-Si or SiC, have been investigated both theoretically and experimentally8-10. Recently, hexagonal boron nitride (hBN) has been proposed as a natural hyperbolic material to enhance near-field thermal radiation. The permittivity tensor of hBN has components of the same value in the basal plane and of an opposite sign in the normal plane. Hyperbolic phonon polaritons (HPPs) supported by hBN open more paths for the frustrated total internal reflection modes to heat transfer. However, the tunability of HPPs excited by resonance between photons and hBN remained a challenge due to its inherent crystal lattice11, leading to inefficient tunability of near-field thermal radiation. When hBN is covered with a single sheet of graphene, HPPs coupled with surface plasmon polaritons (SPPs) of graphene have shown a significant impact on

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enhancing the near-field thermal radiation, which can be optimized by adjusting the chemical potential of the graphene. Worth noting is that if the thickness of the hBN is decreased, though fewer HPPs are supported6,12-13, SPP-HPP coupling still results in a considerable enhancement of near-field thermal radiation. In view of this, as well as the fact that a sheet of graphene in the proximity may produce strong near-field interaction14, in this paper we study multilayer graphene-hBN heterostructures for further near-field thermal radiation enhancement. Explicitly, 4 types of heterostructures15 are proposed to compare with each other: the simple graphene-hBN cell (mono-cell), the graphene-hBN-graphene sandwich configuration (sandwich), the graphene-hBN-graphene-hBN configuration (double-cell), and the infinite-cell configuration consisting of infinite number of graphene and hBN layers. Multi-cell heterostructures provide more tunable parameters than mono-cells, such as the thickness of hBN layers, the ratio of thicknesses, the differences in chemical potentials16 of graphene sheets, etc. In our work, (i) the mechanism of near-field thermal radiation enhancement from bulk hBN to those of the four kinds of heterostructures is properly discussed, by analyzing photon tunneling probability. (ii) The fact that the infinite-cell shows the best performance in enhancing near-field radiation than others is demonstrated, and the effective tunability of the thermal radiation of the infinite-cell is observed by varying the chemical potentials of the graphene sheets. (iii) The optimized thermal radiation is calculated by choosing the proper thickness of hBN for four kinds of graphene-hBN heterostructures. This work may have guiding significance both for experimental design and choice of near-field thermal radiation devices. Theory As a natural hyperbolic material, hBN exists in two hyperbolic forms (Type I,

ε ⊥ hBN >0,

ε ||hBN k0  2 2 ik z 0 d 2 − r e 1 j 

(4)

Note that k z 0 = ( k0 2 − β 2 )1/2 , d represents the distance between the two bodies, and rj is the Fresnel reflection coefficient, which has distinct expressions according to various kinds of structures in our paper. When β > k0 ,

ξ j (ω , β ) is also called photon tunneling probability.

Initially, for an anisotropic medium like hBN, the Fresnel reflection coefficients rj can be simply derived as3,29

rs =

k z(0 s ) − k z(1s ) k z(0 s ) + k z(1s )

k (0 p )ε (1) − k (1 p )ε (0) rp = z(0 p ) ⊥(1) z(1 p ) ⊥(0) kz ε ⊥ + kz ε ⊥

(5)

For a mono-cell with a sheet of graphene (about 0.34 nm) layered on hBN, the expression is modified slightly, and rj becomes7

rs =

k z(0 s ) − k z(1s ) − σµ0ω k z(0 s ) + k z(1s ) + σµ0ω

σ k z(0 p ) k z(1 p ) ε 0ω rp = (0 p ) (1 p ) σ k kz k z(0 p )ε ⊥(1) + k z(1 p )ε ⊥(0) + z ε 0ω k z(0 p )ε ⊥(1) − k z(1 p )ε ⊥(0) +

( np )

where k z

= (ε ⊥( n ) k02 −

(6)

ε ⊥( n ) 2 1/2 ( ns ) β ) , k z = (ε ⊥( n ) k02 − β 2 )1/2 , n = 0,1, 2 is the number of the ε ||( n )


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layers, ε ⊥

(n)

and

tensor (for air,

ε ||( n ) are the perpendicular and parallel components of the relative dielectric

ε ⊥(0) = ε ||(0) = 1 ), and σ is the frequency-dependent conductivity of graphene. To

derive the above reflection coefficients, we have imposed the continuity of component of the electric field parallel to the graphene sheet and connected the discontinuity of the magnetic field to the surface current on the sheet (which is proportional to the electric field on the sheet) through the conductivity σ . This approach has been justified by a number of experiments. Dai et al.11 demonstrated a tunable hyperbolic metamaterial using graphene and hBN, and they have found good agreement between experimental data and the calculations based on the above approach. Fei et al.30-31 investigated infrared nanoscopy of graphene plasmons and have also demonstrated the validity of the above approach. In the present work, when considering the sandwich or double-cell structure, rj is derived as (1 s )

rs =

rs 01 + rs12 (1 + rs 01 + rs10 )e2ikz (1 s )

1 − rs10 rs12e 2ikz

h1

h1 (1 p )

rp =

rp 01 + rp12 (1 − rp 01 − rp10 )e2ikz (1 p )

1 − rp10 rp12 e2ikz

where h1 is the thickness of the hBN, and

(7)

h1

h1

r( s / p ),n1 ,n2

represents the Fresnel reflection

coefficient from layer n1 to layer n2 of s- or p-polarization mode which can be attained from equations (5) and (6) respectively. Particularly, the

r( s / p ),n1 ,n2

of the infinite-cell configuration derived in our work are given as (1 s )

rs =

rs 01 + rs1Ⅱ(1 + rs 01 + rs10 )e 2ikz (1 s )

1 − rs10 rs1Ⅱe2ik z

h1

h1 (1 p )

rp =

rp 01 + rp1Ⅱ(1 − rp 01 − rp10 )e 2ikz (1 p )

1 − rp10 rp1Ⅱe2ikz

(8)

h1

h1

where (2s)

rs1Ⅱ = [rs12 + rs1Ⅱ(1 + rs 21 + rs12 )e2 ik z rp1Ⅱ = [rp12 + rp1Ⅱ(1 − rp 21 − rp12 )e

h2

2 ik z( 2 p ) h2

(2s)

] / (1 − rs 21rs1Ⅱe2 ik z ] / (1 − rp 21rp1Ⅱe

h2

)

2 ik z( 2 p ) h2

(9)

)

and h2 = h1 is set in our calculation.

Results and discussion Above, we have described the calculation of the near-field heat flux between two anisotropic bodies. Figures 3a-c show the heat flux as a function of gap distance for the mono-cell, sandwich, double-cell, and infinite-cell configurations with different chemical potentials at µ = 0.1 eV,


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0.37 eV and 0.6 eV, respectively, compared against bulk hBN and a blackbody (BB). Due to the excitation of HPPs, heat flux between hBN (thickness h1 = 20 nm) is remarkably enhanced compared to that between blackbodies. When a graphene monolayer is added to hBN, SPPs in graphene play an important role in the mid-infrared region and make crucial contributions to near-field radiation enhancement32. We see that the infinite-cell configuration shows greater heat radiation than other configurations at small gap distances (below 50 nm) at different µ , especially at d = 10 nm. Its heat flux reaches 5.43×104 W/m2, which is 4 orders larger in magnitude than that of BB and is 1.87- and 2.94-fold higher than those of the sandwich and mono-cell structures, respectively, at d = 10 nm and µ = 0.6 eV. As shown in inset A of Figure 3a, the magnitude of thermal radiation at d = 10 nm is, from highest to lowest: infinite-cell > sandwich > double-cell > mono-cell. This phenomenon can be confirmed by the photon tunneling probability ξ p at ω = 1×1014 rad/s in Figure 3d. The sandwich structure has three peaks (blue line) of high

ξ p versus the transverse wave vector β , while the mono-cell has

only two peaks (cyan line), leading to smaller photon tunneling probability regions. Surprisingly, although the double-cell structure possesses one hBN layer more than the sandwich structure, its photon tunneling probability displays only two high peaks and one side low peak. It seems that the second hBN layer decreases the contribution to the thermal radiation instead of enhancing it. However, when more graphene-hBN cells are built, up to the infinite-cell configuration, the photon tunneling probability curve becomes broadband, and its value is even higher than 0.95, with quite a large range, from β = 27 k0 to 137 k0 . Obviously, the infinite-cell shows near-perfect photon tunneling probability due to the ideal coupling of SPPs and HPPs of infinite graphene and hBN layers. In addition, different heat flux curves are observed by varying µ , which shows the tunability of the thermal radiation of graphene-hBN heterostructures.


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Figure 3. (a)-(c) Heat flux versus gap distance from 10 nm to 1000 nm with chemical potential

µ

= 0.1 eV, 0.37 eV, and 0.6 eV, respectively. Curves from top to bottom: infinite-cell (red curve), sandwich (blue curve), double-cell (purple curve), mono-cell (cyan curve), bulk hBN (green curve), and blackbody (black curve). Note that for all configurations, the thickness of each hBN layer is h1 = 20 nm. Inset A shows the magnified schematic of (a) near the gap distance d = 10 nm. Units of the inset A are the same as in the main figure. (d) Comparison of photon tunneling probability ξ p of mono-cell, sandwich, double-cell, and infinite-cell configurations at ω = 1×1014 rad/s. In order to illustrate the enhancement more intuitively, the contour maps of the photon tunneling probabilities of the four kinds of heterostructures with d = 10 nm, h1 = 20 nm, and

µ = 0.1 eV are shown in Figure 4. Results based on fitted hBN parameters from both first-principles calculations and experimental data are provided for comparison. We see that the main contribution to the near-field thermal radiation comes from the SPP modes of the graphene layers below the Type I band. There, the photon tunneling probability ξ p of the mono-cell is large and close to unity, even at a large transverse wave vector β = 250 k0 , due to the coupling of SPPs and HPPs. When another graphene monolayer is added and the mono-cell becomes a sandwich configuration, coupling of the two graphene sheets and hBN results in SPP-HPP modes branch G with near-unity photon tunneling probability from β = 150 k0 to 200 k0 , and thus more contributions to the thermal radiation are observed. However, in the double-cell structure, though extra hBN makes the SPP-HPP coupling stronger, thermal radiation over mode branch G is unexpectedly suppressed. Extra hBN in a double-cell structure makes no contribution to thermal radiation enhancement. This also accounts for the photon tunneling probability curve of the double-cell in Figure 3d possesses only two high peaks and one side low peak (while the sandwich case has three high peaks). When more graphene-hBN cells are added to build an infinite-cell configuration, as is shown in Figure 4d, more HPP modes are supported because of the infinite number of hBN layers, the photon tunneling probability ξ p within the hyperbolic bands becomes continuous. In addition, the SPP modes of all these graphene sheets make up a broad region of

ξ p with a value close to unity, which is the main contributor to the enhancement. Therefore, the SPP modes from infinite graphene sheets with

µ = 0.1 eV can compensate for the suppression

from the extra hBN, and in total the infinite-cell configuration shows the best performance to enhance near-field thermal radiation. Note that the above conclusions are true for hBN parameters that are derived from either first-principles calculations or experimental data. The stability of our calculation results to small variations in hBN parameters (comparing Figures 4(e)-(h) to (a)-(d), respectively) means that our conclusions are robust and can give good guidance to experimental designs.


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Figure 4. Contour map of photon tunneling probability

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ξ p ( β > k0 ) of the mono-cell, sandwich,

double-cell, and infinite-cell configurations at µ = 0.1 eV, d = 10 nm, and h1 = 20 nm. Capital G represents the extra SPP-HPP mode branch when a second graphene sheet is present. Regions of hyperbolic band Type I and II are marked with white dashed lines in (d). The heat fluxes are (4.59×104 W/m2, 5.01×104 W/m2, 4.79×104 W/m2, 5.43×104 W/m2) for (a)-(d), and (4.47×104 W/m2, 4.53×104 W/m2, 4.46×104 W/m2, 5.05×104 W/m2) for (e)-(h) respectively. Furthermore, in order to see clearly (with thinner band structures) the coupling process of SPPs and HPPs from mono-cell to infinite-cell configurations, the contour maps of


ξ p at higher

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chemical potential

µ = 0.6 eV (the other parameters are the same, e.g. d = 10 nm, and h1 =

20 nm), which has thinner band structures for the relevant modes, are illustrated in Figure 5. As shown in Figure 5a, bulk hBN of 20 nm thickness has finite HPP branches in two hyperbolic bands (Type I at low frequencies and Type II at high frequencies). In Figures 5a-f, the first five HPP branches and all SPP branches of one to five graphene sheets have been marked out. When a graphene sheet is added to decorate hBN, the SPP modes supported by graphene cover the shortages of bulk hBN outside the hyperbolic regions and couple HPP modes in H1 and H2 as SH1, SH2 respectively, leading to an improvement of the photon tunneling probability over a wide frequency range. Furthermore, when another graphene sheet joins the mono-cell and creates a sandwich configuration, HPP modes in branches H3 and H4 are also coupled with SPP modes as SH3 and SH4 respectively. A similar phenomenon occurring near the Type I hyperbolic band can be observed as well. Moreover, we see that the

ξ p contour map of the double-cell is almost the

same as that of the sandwich configuration, even over the G2 branches, which are quite different from Fig. 4 when µ =0.1 eV. This means that as µ rises up to 0.6 eV, the second hBN layer does not suppress the coupled modes in G2 in the low frequency band while strongly enhance the coupling of SPPs and HPPs in the higher frequency region above the Type I hyperbolic band. As is shown in Figures 5e-h, the three-cell structure supports coupled modes from SH1 to SH6 within the Type I and Type II regions and in G1 to G3 within the low frequency band. The five-cell structure supports even more discrete mode branches, and in the infinite-cell limit, the high

ξ p region

becomes continuous and broadband. From the inset of Figure 5g, we can see that hBN spacers would extend strong photon tunnelling to higher

β regions due to its hyperbolicity though

suppressing photon tunnelling at very narrow bands around ω = 1.47×1014 rad/s and 2.58×1014 rad/s due to its absorption. The trade-off of the above two effects usually makes the infinite-cell configuration better than the corresponding ones with hBN spacers replaced with some non-resonance materials (particularly when a non-vacuum non-resonance material is used for spacers to mechanically support the structure in an experiment). When the spacer is chosen to be other absorptive materials without hyperbolicity, for example, silicon, the photon tunnelling probability becomes much worse than that of vacuum spacers. At µ = 0.6 eV, the heat flux is 1.13×104 W/m2 for our infinite-cell, 6.69×103 W/m2 for infinite-graphene and 2.60×103 W/m2 for infinite-graphene-silicon configuration. The near-unity value of

ξ p due to the strong coupling of

SPPs and HPPs covers such a wide region that it produces the highest enhancement of thermal radiation reported to date. From a more practical point of view, we find that a 100-cell structure already gives a convergent result with the infinite-cell configuration.


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Figure 5. Contour map of photon tunneling probability

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ξ p ( β > k0 ) of (a) bulk-hBN, (b)

mono-cell, (c) sandwich, (d) double-cell, (e) three-cell, (f) five-cell, (g) infinite-cell, and (h) 100-cell configuration versus transverse wave vector β at d = 10 nm, µ = 0.6 eV, and h1 = 20 nm. Hi is the ith HPP branch of hBN in band Type II, while Si is the ith SPP branch supported by suspended graphene sheets. SHi are the coupled modes of SPPs and HPPs near the hyperbolic bands, while Gi are ith coupled modes within the low frequency Type I band. Insets in Figures 5b-g show the

ξ p distribution of suspended graphene sheets with separation of 20 nm (thickness

of the vacuum). Units of the insets are the same as the main schematics.


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We investigate the optimized thickness h1 of the hBN at a gap distance d = 10 nm with

µ = 0.1 eV. As is shown in Figure 6, when h1 is 3.8 nm (note that at this ultrathin thickness, hBN can still support HPP modes, which has been demonstrated experimentally12), which is also the separation of each graphene sheet, the heat flux of a mono-cell is 4.15×104 W/m2, much greater than that of other configurations. It seems that if the graphene sheets are too close to each other, they might interact with each other and act as graphite, which does not support SPP modes, and thus multi-cells are worse than a mono-cell for enhancing thermal radiation33. When h1 increases, multi-cell heterostructures reach an optimal value and finally converge to certain values (varying by only in a few tens) after h1 = 200 nm. As the number of graphene-hBN cells increases, the heat flux of the five-cell structure begins to outperform the sandwich configuration, and the infinite-cell limit performs the best, with the highest thermal radiation of all configurations. In addition, we have found that the optimized peak of h1 shifts from 30 nm to 20 nm, corresponding to the double-cell and infinite-cell configurations (see the inset in Fig 6). This reflects a dynamic trade-off process between the contributions of SPPs and HPPs to near-field thermal radiation, noting that thicker hBN supports more HPPs but means weaker SPP coupling, and thinner hBN results in stronger SPP coupling, but fewer HPPs. In addition, we have calculated the heat flux when µ is different for different graphene sheets for the double-cell and sandwich structures at d = 10 nm; the change is not distinct when the chemical potential

µ of the first

graphene sheet is fixed (not shown here), i.e., the heat flux is not so sensitive to µ values for other graphene sheets.

Figure 6. Heat flux curves (from top to bottom) of infinite-cell, five-cell, sandwich, double-cell, and mono-cell configurations versus the thickness of h1 at µ = 0.1 eV and d = 10 nm. Inset shows the magnified schematic of optimized situation of all kinds of configurations. Unit of the


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inset is the same as the main schematic. Conclusion In conclusion, we have demonstrated that multilayer graphene-hBN heterostructures consisting of five or more graphene-hBN cells perform better in enhancing near-field thermal radiation than all other existing graphene-hBN configurations, such as the mono-cell or sandwich structures. Four kinds of graphene-hBN heterostructures have been explicitly investigated and compared by analyzing the photon tunneling probability. The influence of the chemical potential

µ and the thickness of hBN h1 have also been studied. Our results show that the infinite-cell limit, which can be approximated by a 100-cell configuration, performs 1.87- and 2.94-fold better in thermal radiation due to the continuously and perfectly coupled SPPs and HPPs than the sandwich and mono-cell structures, respectively, at d = 10 nm and

µ = 0.6 eV. The heat flux

is 4 orders larger in magnitude than that of a blackbody. In addition, the optimized thermal radiation is calculated by designating an optimized thickness of 20 nm of hBN for our proposed multilayer graphene-hBN heterostructures, which may have guiding significance both for experimental designs and choices of near-field thermal radiation devices. Author information Corresponding authors *E-mail: [email protected] *E-mail: [email protected] Notes The authors declare no competing financial interest. Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Nos. 11621101 and 91233208). References [1] Kralik, T.; Hanzelka, P.; Zobac, M.; Musilova, V.; Fort, T.; Horak, M. Strong Near-Field Enhancement of Radiative Heat Transfer between Metallic Surfaces. Phys. Rev. Lett. 2012, 109, 224302. [2] Biehs, S. A.; Tschikin, M.; Messina, R.; Ben-Abdallah, P. Super-Planckian Near-Field Thermal Emission with Phonon-Polaritonic Hyperbolic Metamaterials. Appl. Phys. Lett. 2013, 102, 131106. [3] Biehs, S. A.; Tschikin, M.; Ben-Abdallah, P. Hyperbolic Metamaterials as an Analog of a Blackbody in the Near Field. Phys. Rev. Lett. 2012, 109, 104301. [4] Liu, X. L.; Zhang, R. Z.; Zhang, Z. M. Near-Perfect Photon Tunneling by Hybridizing Graphene Plasmons and Hyperbolic Modes. ACS Photonics 2014, 1, 785-789. [5] Ben-Abdallah, P.; Biehs, S. A. Near-Field Thermal Transistor. Phys. Rev. Lett. 2014, 112, 044301. [6] Dai, S.; Ma, Q.; Andersen, T.; Mcleod, A. S.; Fei, Z.; Liu, M. K.; Wagner, M.; Watanabe, K.;


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Enhanced Near-Field Thermal Radiation Based on Multilayer Graphene-hBN Heterostructures Kezhang Shi,1 Fanglin Bao,2,* and Sailing He1,2,3,* 1 Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, JORCEP, Zhejiang University, Hangzhou 310058, China 2 Centre for Optical and Electromagnetic Research, ZJU-SCNU Joint Center of Photonics, South China Academy of Advanced Optoelectronics, South China Normal University, Guangzhou 510006, China 3 Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, Stockholm S-100 44, Sweden


Enhanced Near-Field Thermal Radiation Based on Multilayer

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