Near-Field Radiative Heat Transfer between Black Phosphorus

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Near-field radiative heat transfer between black phosphorus sheets via anisotropic surface plasmon polaritons yong zhang, Hong-Liang Yi, and He-Ping Tan ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00776 • Publication Date (Web): 26 Jul 2018 Downloaded from http://pubs.acs.org on July 26, 2018

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Near-field radiative heat transfer between black phosphorus sheets via anisotropic surface plasmon polaritons Yong Zhang,1,2 Hong-Liang Yi,1,2,* He-Ping Tan1,2 1 2

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, P. R. China

Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology, Harbin 150001, P. R. China

*SSupporting Information ABSTRACT: Black phosphorus (BP), a novel natural two-dimensional layered material with intrinsic in-plane anisotropy, has been attracting significant research attention due to its outstanding electronic and optical properties and tunable bandgaps. Here, an enhancement of near-field radiative heat transfer (NFRHT) arising from a coupling of anisotropic surface plasmon polaritons (SPPs) between two layered BP sheets is demonstrated. The coupling of SPPs along armchair and zigzag directions dominate the NFRHT at near-infrared and mid-infrared frequencies, respectively. The dependence of NFRHT on the number of layers as well as the electron density of BP is then analyzed. It is found that at a small gap size the NFRHT between BP sheets with more number of layers and a higher electron density is lower. While this trend is reversed at a large gap size. Finally, the possibility of using BP to modulate the NFRHT by the mechanical rotation is explored. It is shown that the rotated system exhibits a non-monotonic dependency of its heat transfer coefficient on the rotation angle, which has never been noted in the noncontact heat exchanges at nanoscale before. This work opens the possibility to apply BP-based materials for active thermal management at the nanoscale. KEYWORDS:

thermal

management,

in-plane

anisotropy,

electron

density,

modulation

contrast,

two-dimensional material Radiative heat transfer between two bodies can be significantly enhanced in the near field, due to the tunneling effect of evanescent modes, especially when surface modes are excited, such as surface plasmon polaritons (SPPs) or surface phonon polaritons (SPhPs).1−7 Near-field radiative heat transfer (NFRHT) holds promise for next-generation energy conversion technologies, including thermophotovoltaics,8 thermal rectification,9 and noncontact refrigeration.10 To obtain large heat transfer rates that can benefit these applications, many efforts been devoted to exploring new materials or structures. Two-dimensional (2D) materials,11 such as graphene,12 hexagonal boron nitride (hBN),13 transition metal dichalcogenides (TMDs),14 and black phosphorus (BP),15 have been attracting enormous interest due to their exotic electronic and optical properties such as ultrahigh charge carrier mobility,16 anomalous quantum Hall

ACS Paragon Plus Environment

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17

effect,

18

and strong light-matter interaction.

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Such exciting properties are distinctively different from their bulk

parental materials, opening new opportunities for nanoscale electronics and photonics.19–22 For a particular set of 2D materials including graphene, light-matter interactions are very intense due to the excitation of surface plasmons.23 In contrast to conventional surface plasmons in noble metals, plasmons in 2D materials can provide unprecedented levels of light confinement and exhibit tunability by extrinsic doping.24 Recently, it was demonstrated that the coupling of surface plasmons between two graphene sheets can enhance the photon tunneling.25 NFRHT between suspended graphene sheets has been analyzed intensively,26,27 as well as that in configurations where graphene is deposited either on dielectric substrates,28 or on metamaterials.29 In addition to graphene, the NFRHT ability of single-layer MoS2 (a member of the TMDs family),30 as well as graphene/hBN heterostructures,31,32 has also been examined. However, to our knowledge, the study of NFRHT between BP has not been conducted yet. BP, an allotrope of phosphorus,33,34 has attracted enormous interests in recent years due to its preeminent electronic properties.15 Unlike the planar lattice of graphite, the crystal structure of BP exhibits a repeating puckered honeycomb structure along the armchair direction in each layer,35 hence an in-plane anisotropic optical and electrical natures. In contrast with graphene, due to the high electronic mobility, BP is a candidate for device applications such as field effect transistors (FETs), batteries, sensors, and thermoelectric applications.36,37 Thermal management plays an important role in the above mentioned applications. During the operation of the equipment, the increasing localized Joule heating can reduce device performance and reliability. Lattice thermal conduction is the common way for heat transfer. For materials such as graphene,38 due to the high in-plane thermal conductivity, waste heat can be dissipated efficiently. However, the theoretical studies reveal that the thermal conductivities of monolayer BP along zigzag and armchair are 15.33 and 4.59 Wm-1K-1, respectively, at 300K,39 which is three orders of magnitude lower than that of graphene. Thus, the problem in thermal management for the BP-based devices is emerged. Due to the several orders of enhanced heat transfer at the nanoscale, NFRHT can be a promised way in thermal management. Moreover, BP is a novel anisotropic plasmonic material.40−44 In contrast to the graphene, due to the intrinsic in- plane anisotropic structure, the plasmon dispersion supported by BP depends on the propagation direction,40 enabling the development of novel polarization dependent optoelectronic devices.45 Therefore, it is imperative to understand how the in-plane anisotropic plasmons enhance noncontact heat exchanges in BP at nanoscale. Moreover, previous results show that the optical conductivity of layered BP vary sensitively with thickness, and electron density.41 In addition, due to the intrinsic in-plane anisotropic structure of BP, as we change the relative orientation of the two aligned BP by external mechanical force, the symmetry of the system is broken. One can thus expect that thickness, electron density of layered BP as well as the mechanical rotation can affect the tunneling probability, hence not


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only the magnitude but also the spectral characteristic of thermal radiation, offering potential routes toward passive or active control of NFRHT. In this paper, we first present our physical system, and recall the definition of heat flux exchanged in the near field for anisotropic materials. After that, we investigate the NFRHT between two suspended monolayer BP sheets. We examine effects of the number of layers, electron density of BP and the mechanical rotation on the NFRHT, hoping to offer guidance to manipulate NFRHT. To get insight into the physical origin of the enhanced heat transfer and the tunable ability, we present a detailed analysis on the heat transfer coefficient, and energy transmission coefficient as well as the plasmon dispersion relation of BP. We finally summarize our results at the end.

RESULTS AND DISCUSSIONS

Figure 1. (a) Schematic of near-field radiative heat transfer between two BP sheets. (b) Side view and (c) top view of the crystalline structure of BP. (d) top view of the BP at a rotation angle of ϕ with respect to x-axis. The x and y axes stand for the armchair and zigzag crystalline directions, respectively, along the puckered atomic plane. The temperature of the top BP is higher than that of the bottom one, i.e., T1>T2. Let us consider a system composed of two N-layer BP sheets brought into close proximity with a vacuum gap size of d as sketched in Figure 1a. We show the side and top view, respectively, in Figure 1b and 1c. The x and y axes stand for the ‘armchair’ (AC) and ‘zigzag’ (ZZ) crystalline directions of the bottom BP, respectively, along the puckered atomic plane. The top BP can be rotated with respect to x-axis by an angle of ϕ from 0° to 90° as shown in Figure 1d. Our problem is to compute the NFRHT between two anisotropic parallel plates. This generic problem has been addressed by Biehs et al.46 The net power per unit of area exchanged between the two parallel plates at temperatures of T1 and T2 is given by the following Landauer-like expression:46,47


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q ( ω , T1 , T2 ) =

1 8π

3

∫

∞

0

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[Θ(ω , T1 ) − Θ(ω , T2 )] d ω ∫

∞

∫

∞

−∞ −∞

ξ (ω , k x , k y ) dk x dk y

(1)

where Θ(ω , T ) = hω [exp(hω kBT ) − 1] is the mean energy of a Planck oscillator at angular frequency ω and temperature T. ξ (ω, kx , k y ) is the energy transmission coefficient that describes the probability of two thermally excited photons (related to conductivity and permittivity tensor of BP, see SI).4,48 In this work, we focus on the analysis of the radiative linear heat conductance per unit of area, h (in units of Wm−2K−1), which is referred to as the heat transfer coefficient (HTC). This coefficient is given by

h (T , d ) = limT1 ,T2 →T

q ( ω , T1 , T2 ) T1 − T2

=

1 8π

3

∫

∞

0

∂Θ ( ω , T ) ∂T

dω ∫

∞

∫ ξ (ω, k , k )dk dk ∞

−∞ −∞

x

y

x

y

(2)

where T is the absolute temperature that we assume equal to 300 K (room temperature) throughout this work. Additionally, we define the spectral heat flux as the heat transfer coefficient per unit of frequency or photon energy.

Figure 2. (a) Heat transfer coefficient h as a function of the gap size d for the two suspended monolayer BP sheets. (b) Spectral HTC as a function of the photon energy for different gap sizes. The electron density is n = 5 × 1012 cm−2. We first consider the system with two suspended monolayer BP sheets, set the rotation angle ϕ to 0, and fix the electron density as n = 5 × 1012 cm−2. These parameters are used throughout the paper unless otherwise mentioned. We plot the heat transfer coefficient h as a function of the gap size d in Figure 2a. Here, the HTC is 3 normalized to the far-field limit of radiatively coupled blackbodies (with unity view factor) hbb ( T ) = 4σ SBT ,

where σSB is the Stefan-Boltzmann constant. We find that the radiative heat transfer rate of the monolayer BP system is significantly enhanced at the near-field region, which is many orders of magnitude higher than the


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blackbody limit. As the two BP bodies come closer to each other, the near-field enhancement gets more prominent. Figure 2b shows the spectral HTC with various gap sizes. According to the Wien’s displacement law, i.e., λmaxT = 2.898 × 10-3 m K, the value of the Wien’s frequency at 300K is 0.1284 eV. We can see that, the maximum of the spectral HTC at a small gap size (20 nm < d < 40 nm) is close to 0.1284 eV, which implies a thermal occupation at room temperature. With the increasing of gap size, not surprisingly, due to decreasing contribution of evanescent waves the spectral HTC decreases. Moreover, we observe that the maximum of the spectral HTC is redshifted from 0.12 eV for d = 20 nm up to around 0.056 eV for d = 100 nm. In this case, the conductivity along AC direction is much larger than that along ZZ direction (Figure S1). Note that this anisotropy is attributed to the different effective mass along different crystalline directions, which can be quantified via the Drude weight ratio DAC/DZZ.45 In the Drude regime, i.e., when the intra-band contributions dominate and inter-band transitions are negligible, the ratio σAC/σZZ is weakly dependent on frequency.49 Our calculations indicate that this ratio is about 5.19 (Figure S1). Moreover, we see that Im[σxx] > 0 and Im[σyy] > 0. One can thus expect an anisotropic elliptic surface plasmons supported by BP in the Drude regime.49 The physical mechanism of this enhanced heat transfer can be understood with an analysis of the energy transmission coefficient. In particular, we show ξ (ω , k x , k y ) with d = 20 nm at three different photon energies ω = 0.033, 0.066 and 0.132 eV in

(k

x

, k y ) plane, respectively, in Figure 3a−c. The wave vector is normalized

by the wave vector in vacuum κ0 ( ω c ). These contours clearly illustrate the coupling of anisotropy SPPs in the two monolayer BP system. We can see a bright region with high value of energy transmission coefficient in the contour, which can be understood as follows. Since the monolayer BP supports extremely confined anisotropic SPPs owing to its inductive nature, as we place the two BP sheets in close proximity to each other, due to the tunneling of the evanescent waves and hybridization mechanism of SPPs, the two branches, i.e., the symmetric and anti-symmetric branches, show up. Due to the anisotropic characteristic of BP, the isofrequency curves are elliptical. Thus, the two branches along ZZ direction would interact with each other, eventually forming a scenario as shown in Figure 3a−c. As noted above, the degree of anisotropy Im[σxx]/Im[σyy] is 5.19 in this scenario, which results in a significant canalization along the x-axis (AC direction) as shown in Figure 3a−c.49 Moreover, we notice that with the increasing of frequency the energy transmission coefficient reveals higher value along AC direction but lower value even equals to zero along ZZ direction as depicted in Figure 3c, which means that the coupling of SPPs along AC direction dominates the NFRHT at a higher frequency. We also show

ξ (ω , k x , k y ) along x-axis [ ξ (ω, kx ,0) ] and y-axis [ ξ (ω , 0, k y ) ], respectively, in Figure 3d and 3e. We see that the energy transmission coefficient distribution in the frequency-wave-vector plane is similar to that observed in


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26,31

the two graphene sheets system.

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However, unlike the isotropic SPPs of the graphene case, due to the strong

in-plane anisotropy of BP, the plots along AC and ZZ directions reveal great differences. We find that the transmission maxima along AC direction occupies in a broad spectral bandwidth while a low wave vector region. While those along ZZ direction locates in the region with a narrow spectral bandwidth but extends to a high wave vector. As the gap size gets larger, we find that the transmission maxima redshits and manifestly moves toward a low wave vector in both directions, not shown here, thus decreasing drastically the radiative transfer rate, which is consistent with our observation in the spectral HTC as depicted in Figure 2b.

Figure 3. Energy transmission coefficient at photon energies of (a) 0.033, (b) 0.066 and (c) 0.132 eV, and along (d) x-axis and (e) y-axis. The green curves correspond to the dispersion relations of the monolayer BP determined by eq 5 in SI. The blue and magenta lines in (d) and (e) correspond to the symmetric and anti-symmetric SPPs dispersion relations determined by eq 6 in SI. The gap size is d = 20 nm. To confirm that anisotropic SPPs of BP are indeed responsible for the NFRHT in this structure, we also show the plasmon dispersion relations in Figure 3 denoted by the green curves, which is obtained by calculating the dispersion formulation for anisotropic material (eqs 5 and 6 in SI). It is shown that the plasmon disperses differently due to their mass anisotropy. We can see from Figure 3a−c that the closed elliptic curve is enlarged proportionately to a larger range of wave vector with the increasing of frequency. This effect results from the decreasing in the BP conductivity (Figure S1 in SI). All these green curves nicely locate between the two bright


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branches, which unambiguously demonstrates that anisotropic SPPs dominate the NFRHT in our system. In addition, the blue and magenta lines in Figure 3d and 3e, respectively, correspond to the symmetric and anti-symmetric SPPs dispersion relations (see SI), which nicely predict the maximum of the energy transmission coefficient. Moreover, the green curves show that the plasmon along the ZZ direction is damped at mid-infrared frequencies, while the plasmon along AC persists up to the near infrared.41 Unlike the isotropic SPPs in graphene, the SPPs supported by the BP sheets are delocalized in frequency. Note that one important aspect of the electronic structure of BP is the dependence of the energy gap on the number of layers.43 Recently, BP was reintroduced in their multilayer thin film form, obtained from the simple mechanical exfoliation.15,33,34 In addition to the number of layers, electron density of BP can also be changed through electron doping, which can be induced either electrically or chemically by introducing donor or acceptor impurity atoms during the synthesis.50 Previous results show that the optical conductivity of multilayer BP varies sensitively with thickness, doping, and light polarization.41 Since many potential applications require an active tuning of the NFRHT, from a practical viewpoint, it is important to consider the effect of the electron density n, as well as the number of layers N, on the NFRHT. In our analysis, we restrict the electron density in the range of n ∈ [5, 50] × 1012 cm-2 which is reasonable in the Drude model,44 and the number of layers no more than 5 due to its experimental feasibility and reproducibility as well as the reasonability of the permittivity model.

(a)

(b)

104 d = 10 nm

10

30

N=1 N=2 N=3

d = 600 nm

10

h/hbb

h/hbb

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3

50

1000

80

102 10 20 30 40 Electron density, n ( ×1012cm−2 )

50

1

10 20 30 40 Electron density, n ( ×1012cm−2 )

50

Figure 4. Normalized HTC at room-temperature for the two aligned BP system with different number of layers and different electron densities at a gap size of (a) d = 10, 30, 50 and 80 nm or (b) d = 600 and 1000 nm. Figure 4 shows the normalized HTC at room-temperature for the two aligned BP system with different n and N at a gap size of d. We first notice that the NFRHT between the multilayer BP systems under all these parameters is several orders of magnitude higher than the blackbody limit, especially when the gap size is small. With the increasing of n and N, one can observe from Figure 4a that the HTC decreases. However, as the gap size gets larger approaching the far field, this trend is reversed as indicated in Figure 4b.


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Figure 5. (a) Normalized HTC at room-temperature as a function of the gap size at different number of layers and electron densities. Spectral HTC for a gap size of (b) d = 20 nm and (c) d = 600 nm. Different line colors indicate the number of layers N = 1 (black), 3 (red), and 5 (olive). The inset in (a) shows the detail at the gap size range of 600~900 nm. To get insight into the role of the number of layers and electron density, we show the normalized HTC at room-temperature as a function of the gap size at different N chosen from a set of 1, 3 and 5, and n taken from a set of 5 × 1012, 10 × 1012 and 15 × 1012 cm−2 in Figure 5a, and also the spectral HTC for a gap d = 20 nm and 600 nm, respectively, in Figure 5b and 5c with n = 5 × 1012 cm-2 and 10 × 1012 cm-2. As one can see in Figure 5a, with the increasing of gap size, the HTC decreases. For the BP with more number of layers and higher electron density, the HTC is higher at a small gap size but lower at a large gap size. Note that we calculate the HTC at room temperature in this work. Thus one can expect a larger thermal occupation when a higher maximum of the spectral HTC locates at a lower frequency. As shown in Figure 5b with d = 20 nm, although the spectral bandwidth is broadened, the maximum of the spectral HTC is blue shifted and decreases with the increasing of N, thereby reducing the HTC. As the electron density increases from 5 × 1012 to 10 × 1012 cm−2, the spectral bandwidth gets broader, while the maximum of spectral HTC decreases significantly, therefore HTC is reduced. For the case with a larger gap size as shown in Figure 5c, we see that the spectral HTC significantly decreases


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and is redshifted to a lower frequency. As N increases, we find that although the spectral bandwidth is broadened and the maximum of spectral HTC is blue shifted, the value of the maximum increases, hence increasing the HTC. Moreover, we observe that with the increasing of electron density, the spectral HTC is blue shifted, but increases and occupies a broader spectral bandwidth, thereby an increases in HTC is noticed in Figure 5a.

0.15

12

n = 5 × 10 cm

along ZZ

0.05

0.00 0

N=1 N=3 N=6 N=9

along AC

0.10

100

200

n (cm−2)

(b)

−2

300

kx/κ0(AC), ky/κ0(ZZ)

400

0.15

Energy (eV)

(a)

Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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along AC

5.0 × 1012 10 × 1012 15 × 1012

0.10

0.05

0.00 0

N=1 along ZZ

100 200 300 kx/κ0 (AC), ky/κ0(ZZ)

400

Figure 6. The plasmon dispersion for the BP at (a) different number of layers with a fixed n (5 × 1012 cm−2) and (b) different electron densities with N = 1. Here, 1, 3, 6, and 9 layers are considered, and the electron densities are taken as 5 × 1012, 10 × 1012 and 15 × 1012 cm−2, respectively. The above results can be further understood with an analysis of the dispersion relations of the SPPs supported by BP. By using eq 8 in SI, we plot the plasmon dispersion lines in Figure 6. Notice that during the calculation procedure we only use the conductivity of BP, we can thus ignore the effectiveness of eq 5 in SI while considering the case with more number of layers, i.e. N = 6 and 9. We see that the plasmon along AC and ZZ directions disperses differently due to their mass anisotropy, where the smaller mass along AC leads to a broader spectral bandwidth. Shown in Figure 6a is how the plasmon dispersion changes with the increasing of N. Here, we fixed n to be 5 × 1012 cm−2. For a larger N, since more sub-bands contribute to the optical absorption,44 the Drude weight ratio DAC/DZZ increases, hence increasing the asymmetry in conductivity. For the AC direction, this leads to a broader spectral bandwidth but a decrease in wave vector, hence reducing the spectral HTC at a fixed frequency with the increasing of number of layers. While for the ZZ direction, the plasmon dispersion moves toward a higher wave vector but occupies a narrower spectral bandwidth with the increasing of N. The plasmon dispersion at different n is shown in Figure 6b. With the increasing of n, we find both the plasmon lines along AC and ZZ directions manifestly move toward a lower wave vector hence reducing the NFRHT, but occupy a broader spectral bandwidth. The above analysis based on the plasmon dispersion is consistent with our observation about the spectral HTC at a small gap size [Figure 5]. The reversed trend in the HTC at a large gap size as depicted in Figure 4b and Figure 5a can be understood


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as follows. Note that the enhancement of the NFRHT is attributed to the tunneling of evanescent wave through the narrow gap. Notice that ξ (ω, k x , k y ) predicts an exponential decay of the evanescent wave [eq 7 in SI] which means that with the increasing of gap size the maximum of the energy transmission coefficient manifestly moves to the region with a lower wave vector. In Figure 6, we observe that compared to the ZZ direction, the plasmon dispersion for AC direction locates in the region with a lower wave vector and a broader spectral bandwidth. Thus we can stress that, at a large gap size, the coupling of the SPPs along AC direction dominates the NFRHT, while those along the ZZ direction are negligible. We can further observe that with the increasing of N and n, the plasmon dispersion line gets more close to the frequency-axis, which implies that at a fixed wave vector, the corresponding plasmon frequency is blue shifted, hence a broader spectral bandwidth and a bigger maximum of spectral HTC at a higher frequency as depicted in Figure 5c, as well as a higher HTC [Figure 5a]. Moreover, as d increases, one can expect that the differences between different number of layers and electron density get more prominent, which is consistent with our observation in Figure 5a. As we rotate the top BP with respect to x-axis at an angle of ϕ by an external mechanical force as depicted in Figure 1, due to the inherent in-plane anisotropy of BP, we break the symmetry of the system. One can thus expect that the NFRHT between the two BP would be significantly affected by manipulating the rotation angle. In the following work, based on the mechanical rotation, we explore the possibility of using BP, i.e., a novel natural two-dimensional layered material with intrinsic in-plane anisotropy, to modulate the NFRHT.

Figure 7. Normalized HTC for the two-layer BP with different electron densities by adjusting the rotation angle

ϕ from 0° to 90° at a gap size of (a) d = 20 nm, and (b) d = 600 nm. The green dot line in (a) represents the position for the maximum of HTC at a fixed electron density. In Figure 7, we show the HTC for the two-layer BP with different electron densities by adjusting the rotation angle ϕ from 0° to 90°. Note that the HTC for a fixed electron density is normalized by that of ϕ = 0°. We mark the corresponding rotation angle for the maximum of HTC denoted by the green dot line in Figure 7a for a fixed electron density. At a small gap size of d = 20 nm as shown in Figure 7a, interestingly, we find that


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the HTC for a specific n exhibits a non-monotonic dependency versus the rotation angle, especially for a large electron density. The maximum of HTC is observed at ϕmax > 0°, which implies that the asymmetrical structure of the system does not always reduce the NFRHT. In other words, the mechanical rotation provide us a passive way to further enhance the NFRHT. For instance, a 1.2-fold of HTC of that at ϕ = 0° can be achieved at ϕ max = 52° for BP with n = 50 × 1012 cm−2 as indicated in Figure 7a. Note that a monotonic decreasing of HTC on ϕ has been previously observed numerically,46,48 while such a non-monotonic dependency has never been noted in the noncontact heat exchanges at nanoscale before. One can further see that, with increasing electron density the rotation angle for the maximum of HTC moves toward a higher value. In spite of the non-monotonic trend, the lowest HTC is found at ϕ min = 90° for all the electron densities. Shown in Figure 7b is how HTC changes with n and ϕ at d = 600 nm. We can observe that at a large gap size the HTC decreases monotonically with respect to ϕ. (b)

0

400

13

2.5 × 10

30

ϕ (°)

60

200 90

−2

ϕ = 0°

3

30°

2

45° 1

60°

5 × 1012 cm−2 N=2

90° 0 0.0

0.32 0.24 0.16

0 30 45 60 90

0.08

25 × 1012 cm−2 N=2

−1

N=1 N = 2 800 N=3 600

−1

10

5 × 1012 cm−2

(c) 4

5

3

1000

Spectral HTC ( 10 W eV K m )

104

Spectral HTC ( 105 W eV−1 K−1 m−2)

(a)

h/hbb

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.1 0.2 Energy (eV)

0.3

0.00 0.0

ϕ (°)

0.1

0.2 0.3 Energy (eV)

0.4

Figure 8. (a) Normalized HTC at room-temperature as a function of the rotation angle at d = 20 nm for different electron densities and number of layers (as indicated in the figure). Spectral HTC for different rotation angles at d = 20 nm with an electron density of (b) 5 × 1012 cm−2 and (c) 25 × 1012 cm−2 for N = 2. To get insight into the role of the rotation angle on the NFRHT of the BP system we show the normalized HTC at room-temperature as a function of the rotation angle at d = 20 nm for n = 5 × 1012 and 25 × 1012 cm−2 with the number of layers selected as N = 1, 2 and 3 in Figure 8a, and also the spectral HTC at different rotation angles chosen from a set of 0°, 30°, 45°, 60° and 90° at d = 20 nm in Figure 8b and 8c, respectively, for n = 5×1012 and 25 × 1012 cm−2 with N = 2. As shown in Figure 8a, we find that, although with the increasing of N the HTC decreases, the HTC with different N exhibits a similar trend with respect to the rotation angle. Moreover, we can observe that all the curves exhibit a non-monotonic dependency versus the rotation angle. For a small electron density with n = 5 × 1012 cm−2, we find that the ascending trend is weak and the maximum of HTC is at a small angle of ϕ max = 10°, while the HTC decreases drastically with a further increasing of ϕ. However, for a higher electron density, the ascending trend is much more significant and the rotation angle corresponding to the maximum of HTC is larger. In the plot of spectral HTC [Figure 8b and 8c], we observe that, for n = 5 × 1012


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cm−2, the maximum of the spectral HTC decreases drastically and is redshifted upon increasing the rotation angle, hence reducing the HTC as depicted in Figure 8a. For a higher electron density of n = 25 × 1012 cm−2, the spectral bandwidth of the spectral HTC gets broader but the maximum decreases. Moreover, with the increasing of rotation angle, the maximum of spectral HTC firstly increases and then descends, which is in agreement with the non-monotonic trend of the HTC in Figure 8a.

Figure 9. Energy transmission coefficients at a photon energy of 0.12 eV for n = 5 × 1012 cm−2 and 0.1 eV for n = 25 × 1012 cm−2 with a rotation angle of (a) 0°, (b) 30°, (c) 45° and (d) 90°. The green and white curves correspond to the plasmon dispersions of the bottom and the top BP at a rotation angle of ϕ, respectively, determined by eq 8 in SI. The gap size is d = 20 nm. The origin of the dependency of the NFRHT on the rotation angle can be understood with a concrete analysis of the energy transmission coefficient and the dispersion relations of the SPPs supported by BP. In particular, we show ξ (ω, k x , k y ) with a rotation angle of 0°, 30°, 45° and 90°, respectively, in Figure 9a−d. A photon energy of 0.12 eV and n = 5 × 1012 cm−2 is considered for the above figures, and a photon energy of 0.1 eV and n = 25 × 1012 cm−2 for the below figures. Meanwhile, we plot the plasmon dispersion relations in Figure 9 where the green and white curves represent the plasmon dispersions of the bottom and the top BP at a rotation angle of ϕ, respectively. The gap size is selected as d = 20 nm and two-layer BP is considered. Through observation on Figure 9, it is demonstrated that by manipulating the rotation angle, the energy transmission


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between the two BP sheets is influenced significantly. Let us firstly focus on the analysis of the energy transmission coefficient. As mentioned above, due to the different plasmon dispersion relations of the BP with these two electron densities, the bright branches locate in the region with a high wave vector for 5 × 1012 cm−2, while a low wave vector for 25 × 1012 cm−2. Moreover, we can see that, for n = 5 × 1012 cm−2 with the increasing of rotation angle, the value of the energy transmission coefficient decreases significantly, and eventually it is negligible at ϕ = 90° as indicated in Figure 9d, leading to a remarkable decrease in HTC as indicated in Figure 8a. In addition, we notice that the decrease in the energy transmission coefficient is dominated along x-axis (ky = 0). While for a large electron density of n = 25 × 1012 cm−2, the value of the energy transmission coefficients with all these four rotation angles maintains at a high level. By increasing rotation angle, although the inner bright branch shrinks to a small circle gradually, the outer bright branch along x-axis not only has a maximum value of 1.0 but also extends to a larger wave vector region, particularly when ϕ varies from 0° to 45°, hence increasing the HTC. The above analysis is consistent with our observation above about the HTC. Now let us analyze the plasmon dispersion relations. One can see that with the increasing of rotation angle, the white plasmon dispersion curve along y-axis shrinks gradually, while enlarges along x-axis, eventually forms a graphic as shown in Figure 9d at ϕ = 90° which can be also obtained by rotating the green curve 90 degrees around the point of (kx, ky) = (0, 0). In addition, we can observe that the white graphic at ϕ = 45° is actually a circle, which means that the anisotropic plasmon dispersion of the top BP at ϕ = 45° is isotropic when we project it on the x-y plane. Notice that all these dispersion relations nicely locate between or coincide with the two bright branches, which contributes to a further confirmation and explaination of the NFRHT results.

Figure 10. Energy transmission coefficient distributions along x-axis at a rotation angle of 0°, 30°, 45°, 60° and 90° for BP with n = 25 × 1012 cm−2. To further give an explanation on the increasing trend of the HTC with the rotation angle, we present the energy transmission coefficient distributions for BP with n = 2.5 × 1013 cm−2 along x-axis in Figure 10 at a rotation angle of 0°, 30°, 45°, 60° and 90°. We can observe that, due to the mismatch of the plasmon dispersion


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relations between the bottom and the rotated top BP, the two branches do not merge together but move away from each other, especially for a large rotation angle. With the increasing of ϕ the symmetrical branch along x-axis moves toward a high wave vector, hence resulting in an increase in the HTC. Meanwhile, as we further increase ϕ, the two plasmon dispersions along x-axis get much more mismatched, therefore reducing the energy transmission coefficient of the symmetric branch as depicted in Figure 10. This reduction offsets the contributions from the increasing of wave vector, leading to the overall reduction of the HTC at a large ϕ. With the increasing of electron density, due to a lower wave vector, the HTC changes more sensitively with respect to

ϕ, leading to an apparent enhancement. Meanwhile, the maximum of HTC is observed at a larger ϕ.

Figure 11. (a) Modulation contrast η and (b) the rotation angle ϕmax corresponding to the maximum of HTC for the BP system with the electron densities in the range of [5, 50] × 1012 cm−2 at a gap size of d ∈ [10, 1000] nm. Based on the above analysis, we believe that this dependency of the HTC on the rotation angle for different electron densities may provide us a way to control the NFRHT. To characterize how much we can tune the NFRHT by externally acting on the electron density and the rotation angle, we define the modulation contrast η as the ratio between the maximum and minimum values of HTC, i.e., η = h(ϕ max)/h(ϕ min). We show this ratio for the BP system with the electron densities in the range of [5, 50] × 1012 cm−2 at a gap size of d ∈ [10, 1000] nm in Figure 11a. We see that in most cases, a large modulation contrast higher than 10 can be achieved, especially at a relatively large gap size. These high tunabilities enable substantial modulation of NFRHT. The BP with n = 5 × 1012 cm−2 exhibits a high modulation ability from near-field to far-field. While with the increasing of electron density, a better modulation ability is realized at a larger gap size as indicated in Figure 11a. We also find the rotation angle ϕmax corresponding to the maximum of HTC as shown in Figure 11b. With a big n and a small gap size, we obtain the maximum of HTC at a large value of ϕmax. Moreover, with the increasing of gap size, the ϕmax decreases for all the electron densities, and eventually equals to zero at far-field.

CONCLUSIONS


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We have demonstrated that the radiative heat transfer in the multilayer BP system at room temperature is significantly enhanced at the near-field region. By analyzing the energy transmission coefficient as well as the plasmon dispersion relation of BP, it has been shown that this enhancement of heat transfer results from a coupling of anisotropic SPPs supported by BP. The coupling of SPPs along AC direction dominates the NFRHT at near-infrared frequencies while that along ZZ direction is limited at mid-infrared frequencies. Further, we have investigated effects of the number of layers as well as the electron density of BP on the NFRHT. We find that at a small gap size the NFRHT is weaker between BP with more number of layers and a higher electron density. While this trend is reversed at a large gap size. Finally, we have explored the possibility of using BP to modulate the NFRHT by the mechanical rotation. The rotated system can exhibit a non-monotonic dependency in its heat transfer coefficient versus the rotation angle, which has never been noted in the noncontact heat exchanges at nanoscale before. The underlying mechanism is mainly attributed to the higher wave vector of the symmetrical branch of SPPs along x-axis with a bigger rotation angle. We have also shown that the rotated system can support a large modulation contrast higher than 10 at a relatively large gap size. Moreover, with the increasing of electron density, a better modulation ability is achieved at a larger gap size. Our results not only firstly give insight into the NFRHT between the BP sheets, i.e., a novel natural two-dimensional layered material with intrinsic in-plane anisotropy, but also pave the way to apply BP-based materials for active thermal management on the nanoscale, where one can, by means of mechanical exfoliation, electron doping or external mechanical rotation, tune the resonant coupling between the hot and the cold side.

METHODS Calculations in the present paper were performed by numerical evaluation of eqs 1−2. The reflection matrix

of the energy transmission coefficient ξ (ω, kx , k y ) are determined with the method presented in Ref. 47 (SI). The Drude model is used to approximate the conductivity of multilayer BP (see SI for details).44 For numerical calculations, BP is modeled as a slab with thickness t with an effective permittivity tensor (see SI for details).46 The plasmon dispersion relation is numerically solved with conventional root-finding algorithms in the complex plane.49

ASSOCIATED CONTENT

*S Supporting Information The Drude model for the conductivity of multilayer BP; conductivity and permittivity tensors for the thin film; expression for energy transmission coefficient of the anisotropic thin film; plasmon dispersion formulation for BP (PDF). This material is available free of charge via the Internet at http://pubs.acs.org

AUTHOR INFORMATION


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Corresponding Author *E-mail: [email protected]. ORCID Hong-Liang Yi: 0000-0002-5244-7117 Yong Zhang: 0000-0001-8835-7749 He-Ping Tan: 0000-0002-3461-0785 Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 51706053), as well

as the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 201842), and by the China Postdoctoral Science Foundation (Grant No. 2017M610208).

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For Table of Contents Use Only Near-field radiative heat transfer between black phosphorus sheets via anisotropic surface plasmon polaritons Yong Zhang, Hong-Liang Yi, He-Ping Tan

In this graphic, the schematic of near-field radiative heat transfer between two BP sheets is shown. In addition, the energy transmission coefficient in the wave-vector plane is presented, which shows clearly a coupling of anisotropic surface plasmon polaritons in this system. Thus, this graphic is in consistence with the title, and could provide the readers an intuition feeling of our work.


Near-Field Radiative Heat Transfer between Black Phosphorus

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