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Radiative thermal switch via asymmetric black phosphorus gratings

Active control of heat transfer at the nanoscale has great potentials in thermal logic and energy conversion devices. In the present work, we theoretically propose a radiative thermal switch (RTS) composed of a pair of asymmetric black phosphorus (BP) gratings, with BP nanoribbons periodically patterned in different directions. The simply mechanical rotation between the gratings enables substantial modulation of near-field radiative heat transfer, especially when combined with the use of non-identical parameters, i.e., filling factors and electron densities of BP. Among all the cases including asymmetric BP gratings, symmetric BP gratings, and BP films, we find that the asymmetric BP gratings possess the most excellent switching performance. The optimized switching factors can be as high as 90% with the vacuum separation d=50 nm and higher than 70% even in the far-field regime. The high-performance switching is basically attributed to the rotatable-tunable matching degree between the surface characteristics of the two asymmetric gratings. Moreover, due to the twisting principle, the RTS can work at any temperature, which has great advantage over the phase change materials-based RTS. The proposed switching scheme has great significance for the applications in thermal management and thermal circuits.

Two-dimensional materials have received enormous amount of interest in recent years due to their unique optoelectronic, mechanical and thermal properties, which are not found in bulk forms [37]. Among these materials, black phosphours (BP), an allotrope of phosphorus, exhibits different repeating structures along the armchair and zigzag directions in each layer, hence, an in-plane anisotropy exist [38][39][40]. Inspired by the unique characteristics of BP, the NFRHT between two mono/multilayer BP sheets is investigated and an enhancement of heat transfer is found, which results from a coupling of anisotropic SPPs of BP [41,42]. Recently, by patterning BP into nanoribbon structures, the topological transition from quasi-ellipses to quasi-hyperbolas of BP SPPs has been observed between two BP gratings [43]. In addition, the topological transitions induced by doping or nanopatterning are found to lead to giant enhancement of radiative heat transfer. The results in the above work imply that we can find two types of anisotropies in BP gratings: (1) the intrinsic anisotropy resulted from different lattice structures in the armchair and zigzag directions, and (2) the anisotropic structures caused by nanopatterning.
These two types of anisotropies can couple with each other and therefore result in more complicated BP plasmons, which can be tuned by changing the electron density and structural parameters of the gratings.
In the present work, a near-field RTS is demonstrated, which is made of two asymmetric BP gratings separated by vacuum separation. It should be pointed out that the asymmetric gratings are made of BP nanoribbons periodically patterned in different directions. The switching can be realized just by simply mechanical rotation between the two gratings, i.e., the heat flux reaches the maximum and the minimum when the two gratings are orthogonal and parallel to each other. The mismatch between the two interfaces of the system is the key to realize the switching function, which benefits from the twofold anisotropy in BP gratings. The paper is structured as follows. In Section 2, we introduce the physical system by presenting the configuration of the RTS, the conductivity model of BP, and the definition of heat transfer between two gratings made of two-dimensional materials. Then in Section 3, the switching performance of the proposed RTS is demonstrated and then detailedly analyzed by the spectral radiative heat transfer coefficients, the energy transmission coefficients, the dispersion relations of surface characteristics and the reflection coefficients. The effects of the BP structures on the switching performance, including BP films, symmetric gratings, and asymmetric gratings, are compared. In addition, the switching performance is examined at different operating temperatures and with the substrate to identify the universality of the switching scheme. Fig. 1 The configuration of the proposed RTS is illustrated in Fig. 1(a), which is composed of a pair of asymmetric BP gratings. The bottom and upper BP gratings are both made of periodically patterned BP nanoribbons with periodicity and width L 1 (L 2 ) and L g,1 (L g,2 ), respectively. The two gratings are separated by a vacuum separation denoted as d, and kept at temperature T 1 and T 2 . It should be mentioned that the proposed RTS is at the ON and OFF modes when the two gratings are perpendicular and parallel to each other, respectively. It means that a simply mechanical rotation can switch the modes of the device. The detailed drawings of the BP nanoribbons in grating 1 and grating 2 are given in Fig. 1(b), which are respectively patterned along two principal lattice axes of BP, x (Armchair)-and y (Zigzag)-directions. In addition, in Fig. 1(c), the operating principle of the RTS is demonstrated by the top view of the device at the ON and OFF modes. The conductivity of BP we used for numerical simulations is taken from Ref. [38], in which the Drude model suffice to model the conductivity of BP for photon energies of up to 0.3 eV [39] ( )

PHYSICAL SYSTEM
where D j is the Drude weight along the j-axis with j∈(AC, ZZ). A damping constant of δ =10 meV is selected, which has been widely used in the studies of BP [37,41,42].
is the density of states effective mass. E i (eV)=0.58ς i +0.505 is the minimum energy of the i-th conduction subband. µ is the chemical potential and the total electron density is given by For gratings made of anisotropic two-dimensional materials, the conductivity should be modified with the effective conductivities along (σ∥) and across (σ⊥) the main axis of the gratings, which are given by [45] g m where f g =L g /L is the BP filling factor, and f c =1-f g is the filling factor of the free space between adjacent nanoribbons. Throughout this work, we focus on the case with the parameters selected as L 1 =L 2 =10 nm, which has been validated by the previous work in the NFRHT between BP gratings [43]. To ensure the accuracy of the calculations based on the effective medium theory, the vacuum gap distance d should be several times larger than the nanoribbon periodicity. As given by Ref. [43], for L=10 nm, the effective medium theory predicts the real heat flux well when d ≥ 50 nm. In the following results, the parameters are selected as L=10 nm with d ≥ 50 nm, and T=300 K unless otherwise noted. For x (Armchair)-patterning BP gratings, the conductivities obey the relations σ m =σ AC and σ n =σ ZZ , and for y (Zigzag)-patterning BP gratings, the relations are σ m =σ ZZ and σ n =σ AC [43]. 45] is an equivalent conductivity associated with the near-field coupling between adjacent nanoribbons, where ε eff denotes the relative permittivity of the dielectric medium surrounding BP. For two suspended BP sheets or gratings, ε eff is equal to 1.
In Fig. 2, the effective conductivities σ∥ and σ⊥ of x -patterning BP gratings are given with different parameters to reveal the surface characteristics. The results for the y -patterning BP gratings have the same trends with the following results, and therefore they are ignored in the following discussion. For f g =0.3-1, the real and imaginary parts of σ∥ are given in Figs. 2(a) and 2(b), and the the real and imaginary parts of σ⊥ are given in Figs. 2(c) and 2(d), respectively. As can be seen from the effective conductivities, the Real(σ∥) and Imag(σ∥) both increase with the filling factors of BP at all frequencies. When f g <1, the Real(σ⊥) exhibits a single peak, and then the peak fades away when f g reaches unity. In addition, the attenuation of the peak, together with the red shift is observed when f g is getting larger. As is known, for gratings made of two-dimensional materials like graphene [11,23,46], the closed-form SPPs converts into open hyperbolic SPPs, which enables the excitation of surface resonances at extremely high wavevectors. Recently, the topological transition from quasi-ellipses to quasi-hyperbolas of BP SPPs has been observed in NFRHT between two BP gratings [43]. The transition is mainly attributed to the positive-negative transformation observed in Imag(σ⊥). When Imag(σ⊥)<0 and Imag(σ ∥)>0, i.e., Imag(σ⊥)•Imag(σ∥)<0, the BP SPPs exhibit hyperbolic characteristics. When Imag(σ⊥)•Imag(σ∥)>0, the anisotropic BP SPPs are elliptic, which are similar to those of BP films. The transition between hyperbolic and elliptic modes are indicated by dashed lines and arrows in the results of Imag(σ⊥). For BP, the doping can be induced either electrically or chemically by introducing donor or acceptor impurity atoms during the synthesis [40]. To demonstrate the effects of electron density on the effective conductivity, in Figs. 2(e)-2(h), the results are given for electron densities of BP n=1×10 12 , 1×10 13 , 2×10 13 , 3×10 13 , 4×10 13 , 5×10 13 cm -2 with f g =0. 8. The evolutions of effective conductivities with electron densities are found to be similar with the variation tendency of filing factors. Differently, as n increases, the peak of Real(σ⊥) enhances and reveals a blue shift at the same time.
In addition, the dividing line between hyperbolic and elliptic modes gradually moves to higher frequencies. The above results imply that the effective conductivities of BP gratings can be easily tuned by changing the filling factors and the electron densities. It means that the anisotropic surface characteristics can be highly induced by selecting the proper parameters, which is the basic mechanism of the proposed RTS.
In the present work, the radiative heat transfer coefficient is utilized to evaluate the NFRHT between two twisted BP gratings as [47] ( ) where ∆Φ is the net radiative heat flux and 1 2 T T T ∆ = − is the temperature difference between the two gratings.

I R R T T D I R R T T D R R D R R D
where κ and φ are the surface-parallel wave vector and the azimuthal angle. κ 0 =ω/c is the wave vector in vacuum, is the normal component of the wave vector in vacuum.
( ) is the Fabry-Perot-like denominator matrix and R j (j=1, 2) is the reflection coefficient matrix for the j-th BP grating, with the form where the superscripts s and p represent the polarizations of transverse electric and transverse magnetic modes, respectively. The reflection coefficients are given by [48] φ j in Eq. (12) denotes the azimuthal angle of the main axis for grating j (j=1, 2). When the gratings are aligned, the relationship of azimuthal angles for the two gratings is φ 2 = -φ 1 . If a twisted angle θ exists between the main axes of the two gratings, then φ 2 =-φ 1 + θ [47]. In view of Eq. (12), integration over azimuthal angles is necessary for the energy transmission coefficients because Fresnel's coefficients and conductivity of BP gratings depend on φ.
Hence, the energy transmission coefficients considering the integration over azimuthal angles are given as ( ) The NFRHT between two twisted anisotropic bulk materials [50], gratings made of isotropic materials [51], and twisted gratings made of two-dimensional mater ials [23,46] have been investigated in the previous studies.
The results show that the NFRHT can be largely modulated by rotating the upper and bottom objects, which benefits from the anisotropic properties of the materials. To measure the performance of the RTS, the switching factor is defined as follows

RESULTS AND DISCUSSION
where h off and h on denotes the heat transfer coefficients of the RTS at the OFF and ON mode, respectively. As is demonstrated in Fig. 1, when the vacuum distance d is settled, the parameters that can influence the NFRHT are the BP filling factors (f g,1 /f g,2 ) and the electron densities (n 1 /n 2 ) of the two BP gratings. Due to the asymmetric structure, these four parameters have different influences on the NFRHT and therefore the switching factor has different trends with the parameters. To obtain the optimized performance of the RTS, the η should be evaluated at different combinations of (f g,1 , f g,2 , n 1 , n 2 ), which is a time-consuming work. To solve this problem, a stochastic particle swarm optimizer (SPSO) algorithm [52], which can guarantee the convergence of the global optimization solution, is adopted to optimize the performance of the RTS. Compared with standard particle swarm optimizer algorithm, SPSO eliminates the historical velocity term which makes the particle lose velocity memory to decrease the global searching ability. But it guarantees at every generation one particle stops evolution for locating at the best position. By utilizing the optimized method, we could explore the influences of the different ].
Conversely, the electron density n 2 of grating 2, which is y-patterning, is nearly at the lower boundary 1×10 12 cm -2 of the scope. Inspired by the phenomenon, in the subsequent optimizing process, the electron densities of BP grating 1 and grating 2 are set as 5×10 13 cm -2 and 1×10 12 cm -2 , respectively. Thus the four-parameter optimizing problem reduces to a two-parameter optimization, which is easy to obtain the global optimization solution.
In Fig. 3(a), the optimized switching performance of the RTS, is demonstrated by the switching factors at different vacuum separation d. The results show that η reaches nearly 90% at the vacuum separation d=50 nm. As d increases, the switching performance shows a monotonically decrease due to the attenuation effect of BP SPPs at the long distance. However, the η is still larger than 70% even at the large distance d=1 µm. The heat transfer coefficients at the ON and OFF modes are also demonstrated in Fig. 3(a) to reveal the trend of the NFRHT in the system. An interesting phenomenon can be observed that near d=300~400 nm, a rapid growth of the heat transfer coefficients emerges. It implies that the RTS can still maintain a relatively strong heat transport in the far-field, and thus the practicality of the RTS is guaranteed. The optimized filling factors of BP gratings for the gratings 1 and grating 2 are given in Fig. 3(b). The lower and upper boundaries of the parameters selected in the optimization are labeled with dashed lines in blue and red, respectively. As discussed above, the optimized electron densities of gratings 1 and 2 are respectively settled at n 1 =5×10 13 cm -2 and n 2 =1×10 12 cm -2 . The filling factors of x-patterning grating f g,1 varies around 0.8 at different d, while f g,2 has an entirely different trend with d.
In the near-field regime, f g,1 gradually increases, conversely, f g,2 decreases rapidly and stays at f g,2 =0.1, which is the lower boundary of the parameter range. Around d=300~400 nm, a sudden change of f g,2 emerges and it jumps to 0.64 at d=331 nm. Then as d increases, f g,1 and f g,2 have the same trend with d and they both rise. Thus, the sudden change of heat transfer coefficients observed in Fig. 3(a) can be attributed to the variety of the BP filling factors. To explore the NFRHT in the system, in Fig. 3(c), the ratio of spectral heat transfer coefficients at the ON mode (h on (ω)) to that of the OFF mode (h off (ω)) is demonstrated at different d. It shows that at d=50 nm, the ratio h on (ω)/h off (ω) can reach as high as 16. Around d=300~400 nm, the spectral ratio also reveals a sudden change of NFRHT, i.e., the single-peak transforms to double-peak. To specify the two types of performances of the RTS in detail, in Fig. 3(d), the spectral heat transfer coefficients at the ON mode and OFF mode are plotted for d=50 nm and 1000 nm, respectively. Moreover, the optimized parameters for the two cases are indicated in the figure. The results show that the NFRHT in the system reveals two different mechanisms in the near-and far-field regime. In spite of this, the optimized switching factors monotonically changes with the vacuum distance and stays above 70%. Additionally, the transition from single-peak to double-peak in spectral heat transfer coefficients can also contributed to the enhancement of thermal channel of the RTS.  As discussed in Fig. 2, the BP SPPs transform from elliptic to hyperbolic forms at low frequencies when BP is patterned into gratings. By tuning the electron density of BP or changing the filling factors of the gratings, the topological transitions of BP SPPs can be adjusted [43]. The optimized parameters at d=50 nm (f g,1 =0.76, f g,2 =0.23, n 1 =5×10 13 cm -2 and n 2 =1×10 12 cm -2 ) obtained in Fig. 3 show that, both the filling factors and the electron densities tend to be the opposite trend for the two gratings. It means that the surface characteristics of the x-patterning and Specifically, r pp plays a crucial role in the dispersion relations and thus the NFRHT through the channel of SPPs.
The results in the left column and right column represent the Im(r pp ) of x-patterning and y-patterning gratings, respectively. The Im(r pp ) is illustrated when the two gratings are perpendicular to each other. An obvious mismatch of reflection coefficients between the two gratings is observed in the results, which is also resulted from the great difference between the two gratings discussed above. For the x-patterning grating, due to the high electron density and large BP filling factors, we show that the Im(r pp ) is larger than those of y-patterning grating.
In addition, the hyperbolic r pp is much narrower than that of y-patterning grating and is restricted in narrow κ y space. As ω increases, the Im(r pp ) enhances and expands to higher κ y . For y-patterning grating, the spreading effect of Im(r pp ) is much more obvious than that of x-patterning grating. When the r pp of y-patterning grating rotates 90°, we can expect that the two reflection coefficients will have no overlap at all. In the previous studies between two gratings with identical parameters [23,46], the heat flux reaches the maximum and the minimum when the gratings are parallel and perpendicular to each other. When the two gratings are parallel, the plasmons are strongly excited due to the match between the two interfaces. However, the modes dramatically decay when the two gratings are perpendicular to each other, for the reason that the couple modes are broken by the mismatch.
As for the proposed RTS in the present work, the mechanism is totally contrary to that of the previous studies. The strong mismatch in the reflection coefficients between the two gratings makes it possible to realize broken coupling of BP SPPs between the two interfaces. For the reason that SPPs can only be excited near the region where the dispersion relations of the interfaces intersect, the mismatched surface characteristics break the coupling when the dispersion relations cannot meet at θ =0°, as demonstrated in Fig. 5(b).

Fig. 6
To reveal the rotation effect on the switching performance, in Fig. 6(a), the spectral heat transfer coefficients are plotted for different twisted angles θ between the two BP gratings at d=50 nm. The twisted angle θ between the two BP gratings is illustrated by the inset in Fig. 6(a). The spectral results show that as θ increases, the heat transfer enhances gradually and reaches the maximum at θ =90°. The enhancing trend of NFRHT with twisted angles, has never been observed in the previous studies considering twisting between two anisotropic materials [23,50,51,53,54]. In Fig. 6  To highlight the superiority of the proposed asymmetric structure, the switching factors for different cases are demonstrated in Fig. 7(a): the optimized asymmetric BP gratings, symmetric BP gratings, BP films at different electron densities. The symmetric gratings are composed of two identical gratings, which are both in x-patterning or y-patterning, respectively. It is important to note that for symmetric BP gratings, the heat transfer reaches the maximum and minimum when the two gratings are parallel and perpendicular to each other, respectively. For BP films, the maximum and minimum NFRHT are achieved when the two films have the same and orthogonal directions of lattice axes. The mechanism obeys the same relation with the previous studies about NFRHT between anisotropic materials [23,46]. Therefore, the switching factors for the two cases are defined as η=1-h(90°)/h (0). The SPSO method is utilized to optimize the switching performance of the symmetric gratings at different combinations of (f g,1 , f g,2 , n 1 , n 2 ). The optimized results show that when the electron densities of BP and the filling factors of the bottom and upper BP gratings are identical, the switching performance is strongest. In addition, the best performance is achieved with the maximum electron densities n 1 =n 2 =5×10 13 cm -2 . The optimized switching performance of symmetric BP gratings in x-patterning varies between 62.4% and 77.8%, while in y-patterning it varies between 66.7% and 74.6%. A cross has been observed for the two lines of x-patterning and y-patterning gratings near d=300 nm. We show that by means of asymmetric configurations, in the near-field regime, the switching factors can be increased by about 15% than the symmetric gratings and about 25% than the BP films. In the far-field regime d=1 µm, the enhancement of the switching performance caused by the asymmetric configuration is still strong. In Fig. 7(b), the optimized filling factors of symmetric BP gratings are given to demonstrate the different mechanisms in different patterning manners. The filling factors of x-patterning gratings, f g,x , varies in a small range 0.9~1. Nevertheless, the range is much larger for the filling factors of y-patterning gratings, which is nearly 0.5~0.9. The differences in optimized parameters and switching performance between x-patterning and y-patterning symmetric gratings, reveal the strong mismatch induced by different patterning manners. This can also contribute to the explanation of the excellent performance of the asymmetric gratings. In Fig. 7(a), the results of BP films are always worse than those of gratings, no matter in any grating structures. It is mainly attributed to the weak anisotropy of BP films, which can only be induced by the armchair and zigzag lattices in BP. As discussed above, the grating configuration endows the system with another manner to enhance the anisotropy in the in-plane directions. It is just the strong and twofold anisotropy that results in the high performance of RTS. The results corresponding to different n of BP show that, as n increases, the switching factors are enhanced greatly. The trend with electron density also contributes to the switching performance of asymmetric RTS obtained in Fig. 3, which has two extreme values n 1 =5×10 13 cm -2 and n 2 =1×10 12 cm -2 in the settled range of n.

Fig. 8
To examine the universality of the proposed RTS, the switching factors at different temperatures are demonstrated in Fig. 8(a) when the parameters of the RTS are same as those of Fig. 3. An interesting phenomenon has been observed that the η exhibits an obvious step with d. As the temperature of BP increases, the switching factors slightly decay in the near field regime, while in the far-field regime they are enhanced. The different trends of switching factors with T are mainly attributed to the different mechanisms observed in Fig. 3 in different ranges of d, i.e., the single-and double-peak characteristics. Although the switching factors vary with the operating temperatures, they still stay at relatively high values at different d. We show that the RTS can work at any temperature due to the twisting principle, which has great advantage over the phase change materials-based RTS.
In the actual devices, two-dimensional materials are always coated on the substrate. In Fig 8( [15] or silicon carbide [3,23], which can strongly interact with the coated films with surface phonon polaritons, will introduce totally different physical regimes in the NFRHT. Then the RTS, composed of BP grating-covered polar substrate, will have different switching trend with the parameters of BP gratings. To develop the switching performance of the BP grating-covered RTS will then convert into a totally different physical problem. In addition, we think that the complex factors, including operating temperatures, the thickness of the substrate, and the category of the substrate materials also have great influence on the switching performance of the RTS. Therefore, in the present work, we restrict our attention only on the BP gratings, which will have guidelines for developing BP grating-covered RTS with different substrates in the future research. The results obtained in the present work are not influenced by the other materials and thus the pure effects of BP can be understood well, which paves a way for the BP-based technology.

CONCLUSION
A radiative thermal switching scheme is theoretically proposed in the present work, which is based on simply mechanical rotation between two BP gratings. The RTS is composed of two asymmetric BP gratings, which are made of BP nanoribbons periodically patterned along different lattice axes. A simply mechanical rotation can switch the modes of the device, i.e., the NFRHT reaches the maximum and minimum when the two gratings are orthogonal and parallel, respectively. The enhancing trend of NFRHT with twisted angle, has never been observed in the previous studies considering rotation between two anisotropic materials. In addition, the switching function slightly relies on the precision of the mechanical rotation. By tuning the parameters of the two BP gratings, the switching performance of the proposed RTS is optimized, which can be as high as 90% with the vacuum separation d=50 nm and higher than 70% even in the far-field regime. The spectral radiative heat transfer coefficients, the energy transmission coefficients, the dispersion relations of surface characteristics and the reflection coefficients are analyzed in detail to take a deep insight into the NFRHT in the system. It is found that the mismatch between the two interfaces is the key to realize the switching function, which benefits from the twofold anisotropy in BP gratings. Among all the cases including asymmetric BP gratings, symmetric BP gratings, and BP films, we find that asymmetric BP gratings possess the most excellent switching ability. Moreover, due to the twisting principle of the RTS, the performance can be slightly influenced by the temperature of the operating temperature, which has great advantage over the phase change materials-based RTS.
Due to the limitations of the effective medium theory used in the present work, the switching performance cannot be precisely simulated below d=50 nm. As a result of the monotonicity of switching factors with d, we expect a better switching performance in the ultra near-field regime d<50 nm. In the future research, one can explore the switching performance of the proposed RTS in extremely small separation via accurate scattering theory based on the rigorous coupled-wave analysis method [43] or the full many-body radiative heat transfer theory [55]. Additionally, we expect the performance of the RTS can be improved when the substrate is made of polar materials, which can strongly interact with the coated gratings [15,23]. The factors mentioned above will provide the possibility to improve the switching performance and serviceability of the RTS, which can help to develop novel devices in thermal logic and energy conversion.

Conflict of Interest
The authors declare no competing financial interest.       gratings in x-patterning and y-patterning, BP films at different electron densities. For symmetric BP gratings and BP films, the switching factor is defined as η=1-h(90°)/h (0), for the reason that the heat transfer reaches the maximum and minimum at θ =0° and θ =90°, respectively. The trend is inverse to that of asymmetric BP gratings.
(b) The optimized filling factors of symmetric BP gratings in x-patterning and y-patterning, respectively.