Anisotropic radiative heat transfer between nanoparticles mediated by a twisted bilayer graphene grating

Twisted two-dimensional bilayers exhibit many intriguing physical phenomena through interlayer twisting and coupling. Manipulating the “twisted angle” between the two evanescently coupled layers enables the hybridization of polaritons, and the dispersion engineering of polaritons in these structures can be well controlled. In this paper, we study the near-field radiative heat transfer (NFRHT) between two nanoparticles in the presence of a bilayered hyperbolic metasurface, which is modeled as two arrays of graphene strips (GSs) in parallel. We prove that the topological transition of the surface state under different twisted angles [from open (hyperbolic) to closed (elliptical) contours] has significant effect on NFRHT between nanoparticles. When both the integral and interlayer twisted angles are adjusted to proper values, the bilayered GS can canalize the energy transmission channel of the nanoparticles, hence significantly amplifying the NFRHT. A modulation factor beyond five orders of magnitude is achieved. We also demonstrate that, when the nanoparticles are arranged in different directions along the twisted bilayered system, NFRHT between nanoparticles can be strongly enhanced or inhibited. By changing the chemical potential and filling factor of the twisted bilayered GS, NFRHT can be effectively modulated. This paper reveals a hybridization effect of polaritons on NFRHT between nanoparticles, giving a degree of freedom (twisted angle) for controlling the heat transfer at the nanoscale with potential for effective energy management.

Figure 1

Schematic of near-field heat transfer between two nanoparticles ($p_1$ and $p_2$) separated by an interparticle distance of $d_{\mathrm{P}}$ above the bilayered system. The twisted bilayered system is composed of two identical graphene strips (GSs; GS1 and GS2), where W and G are the strip width and air gap distance, respectively. GS1 and GS2 are separated by a thin dielectric spacer with a thickness of $d_{\mathrm{G}}$ and a dielectric constant of ${\varepsilon }_d$. $d_{\mathrm{Z}}$ is the particle-GS distance for the two nanoparticles. (b) Top view of the particle-GS structure without twist. (c) Top view of the twisted particle-GS system at an interlayer twisted angle ${\phi }_{\mathrm{L}}$ of with respect to the α axis.

Figure 2

Effective optical conductivity of graphene strip (GS) for different filling factors. Here, the unit of the surface conductivity is millisiemens (mS). The orange-shaded region denotes where $\mathrm{sgn}\left(\mathrm{Im}\left[{\sigma }_{xx}^{\mathrm{eff}}\right]\right)\ne \mathrm{sgn}\left(\mathrm{Im}\left[{\sigma }_{yy}^{\mathrm{eff}}\right]\right)$, where hyperbolic graphene plasmons are supported.

Figure 3

Twist-induced topological transition of hyperbolic surface plasmon polaritons (HSPPs). (a)–(e) The upper panel shows the formed twisted fringe patterns of the twisted bilayered system. The bottom panel shows the corresponding contours of the imaginary part of the reflection coefficient, viz., $\mathrm{Im}\left[r_{pp}\right]$, in the wave vector space at a frequency of ${\omega }_{\mathrm{res}}=1.756\times {10}^{14}\mathrm{rad}/\mathrm{s}$. The parameters are set as $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $W=10\mathrm{nm}$, $f=0.5$, and ${\varepsilon }_d=1$.

Figure 4

Total normalized heat transfer conductance (HTC) $h$ with respect to the interlayer twisted angle ${\phi }_{\mathrm{L}}$ for different filling factors $f$. Normalization factor is the total HTC $h_0$ between nanoparticles without surface. The gray and orange dash-dot lines correspond to results in the case of vacuum and bilayered unpatterned graphene sheets. The SiC nanoparticles with an interparticle distance of $d_{\mathrm{P}}=1\mu \mathrm{m}$ are placed at a particle-surface distance of $d_{\mathrm{Z}}=60\mathrm{nm}$ above the twisted bilayered graphene strip (GS). The chemical potential of the GS is fixed at 0.5 eV.

Figure 5

(a) The spectral normalized heat transfer conductance (HTC) $h\left({\omega }_{\mathrm{res}}\right)$ at ${\omega }_{\mathrm{res}}=1.756\times {10}^{14}\mathrm{rad}/\mathrm{s}$ with respect to the interlayer twisted angle ${\phi }_{\mathrm{L}}$ for different filling factors $f$, normalized by the spectral HTC $h{\left({\omega }_{\mathrm{res}}\right)}_0$ between nanoparticles without surface. The distributions of the spectral HTC vs the frequency and the interlayer twist for (b) the bilayered unpatterned graphene sheets and (c) the bilayered graphene strip (GS).

Figure 6

The dispersion relations of the twisted bilayered graphene strip (GS) for different filling factors in the wave vector space at the interlayer twisted angle of (a) and (b) ${\phi }_{\mathrm{L}}=0^{\circ }$ and (c) and (d) ${\phi }_{\mathrm{L}}={90}^{\circ }$ at ${\omega }_{\mathrm{res}}=1.756\times {10}^{14}\mathrm{rad}/\mathrm{s}$. The parameters are $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $W=10\mathrm{nm}$, $f=0.5$, and ${\varepsilon }_d=1$.

Figure 7

(a)–(e) Wave vector contours of the real part of the first component of the reflected Green's function (GF) $\mathrm{Re}\left[G_R\left(1,1\right)\right]$ for the case with unpatterned bilayered graphene sheets and bilayered graphene strips (GSs) at different interlayer twisted angles. Black dashed lines represent the polariton wave vectors, and red solid lines denote the group velocity directions. (f)–(j) Spatial contours of the electric field energy density $U_e$ at $d_{\mathrm{Z}}=60\mathrm{nm}$ corresponding to the cases in (a)–(e) at ${\omega }_{\mathrm{res}}=1.756\times {10}^{14}\mathrm{rad}/\mathrm{s}$. In the upper panel, the blue dashed lines correspond to the isofrequency curves. For the bottom panel, the temperatures of the left and right nanoparticles are kept at 300 and 0 K, respectively. The green dashed arrow indicates the direction of maximum energy density. The parameters for these plots are $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $f=0.5$, and ${\varepsilon }_d=1.0$.

Figure 8

(a) The spectral normalized heat transfer conductance (HTC) $h\left({\omega }_{\mathrm{res}}\right)$ with respect to the interparticle distance $d_{\mathrm{P}}$ for different interlayer twisted angle ${\phi }_{\mathrm{L}}$. (b) The spectral HTC $h\left({\omega }_{\mathrm{res}}\right)$ with respect to the interlayer twisted angle ${\phi }_{\mathrm{L}}$ for different particle-interface distance $d_{\mathrm{Z}}$. The normalization factor is the spectral HTC $h{\left({\omega }_{\mathrm{res}}\right)}_0$ between nanoparticles without surface. The parameters for these plots are $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $f=0.5$, $\omega ={\omega }_{\mathrm{res}}$, and ${\varepsilon }_d=1.0$.

Figure 9

(a) Definition of the propagation direction of the hybrid hyperbolic surface plasmon polaritons (HSPPs) excited at an angle of ${\phi }_{\mathrm{PL}}$. (b) Propagation length of the hybrid HSPPs along the surface at ${\omega }_{\mathrm{res}}$ calculated from $L=1/\mathrm{Im}\left(K\right)$ in different directions. (c) Decay length of the hybrid HSPPs along the $z$ direction at ${\omega }_{\mathrm{res}}$ calculated from $\delta =1/\mathrm{Im}\left(k_z\right)$ in different directions. The parameters for these plots are $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $f=0.5$, and ${\varepsilon }_d=1.0$.

Figure 10

Schematic of near-field radiative heat transfer (NFRHT) between two nanoparticles separated by an interparticle distance of $d_{\mathrm{P}}$ above the twisted bilayered system. ${\phi }_{\mathrm{B}}$ and ${\phi }_{\mathrm{L}}$ denote the integral twisted angle of the bilayered graphene strip (GS) relative to the two nanoparticles, and the interlayer twisted angle of GS2 relative to GS1, respectively. The twist of the GS is executed clockwise. The connecting line between nanoparticles $p_1-p_2$ is aligned with the $x$ axis.

Figure 11

The ratio between the spectral conductance $h\left({\omega }_{\mathrm{res}}\right)$ of the configuration in the presence of bilayered graphene strip (GS) and the one in the presence of bilayered graphene as a function of integral twisted angle ${\phi }_{\mathrm{B}}$ and interlayer twisted angle ${\phi }_{\mathrm{L}}$ at an interparticle distance of (a) 0.5 μm and (b) 1.0 μm. The green dashed line corresponds to the maximum spectral conductance at different ${\phi }_{\mathrm{B}}$ and ${\phi }_{\mathrm{L}}$. The parameters are set as $d_{\mathrm{Z}}=60\mathrm{nm}$, $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $f=0.5$, and ${\varepsilon }_d=1$.

Figure 12

The spatial contours of the electric field energy density $U_e$ of the bilayered graphene strip (GS) at ${\omega }_{\mathrm{res}}=1.756\times {10}^{14}\mathrm{rad}/\mathrm{s}$ under different interlayer twisted angles and integral twisted angles, viz., $\left({\phi }_{\mathrm{L}},{\phi }_{\mathrm{B}}\right)$. The parameters are set as $d_{\mathrm{Z}}=60\mathrm{nm}$, $d_{\mathrm{G}}=10\mathrm{nm}$, $\mu =0.5\mathrm{eV}$, $f=0.5$, and ${\varepsilon }_d=1$.

Figure 13

The ratio between the spectral conductance $h\left({\omega }_{\mathrm{res}}\right)$ of the configuration in the presence of bilayered graphene strip (GS) and the one in the presence of bilayered graphene as a function of the filling factor $f$ and the chemical potential μ of the bilayered GS at an interparticle distance of 1.0 μm. The parameters are set as ${\phi }_{\mathrm{L}}={\phi }_{\mathrm{B}}={60}^{\circ }$, $d_{\mathrm{Z}}=60\mathrm{nm}$, $d_{\mathrm{G}}=10\mathrm{nm}$, and ${\varepsilon }_d=1$.

Figure 14

Wave vector contours of the real part of the first component of the reflected Green's function (GF) $\mathrm{Re}\left[G_R\left(1,1\right)\right]$ at a chemical potential of (a) 0.2 eV, (b) 0.5 eV, (c) 0.7 eV, and (d) 1.0 eV. The blue dashed line corresponds to the dispersion relations of the hybrid hyperbolic surface plasmon polaritons (HSPPs). The parameters are set as $d_{\mathrm{G}}=10\mathrm{nm}$, $d_{\mathrm{Z}}=60\mathrm{nm}$, and $f=0.5$.