Near-field heat transfer between a nanoparticle and a rough surface

In this work we focus on the surface roughness correction to the near-field radiative heat transfer between a nanoparticle and a material with a rough surface utilizing a direct perturbation theory up to second order in the surface profile. We discuss the different distance regimes for the local density of states above the rough material and the heat flux analytically and numerically. We show that the heat transfer rate is larger than that corresponding to a flat surface at short distances. At larger distances it can become smaller due to surface polariton scattering by the rough surface. For distances much smaller than the correlation length of the surface profile, we show that the results converge to a proximity approximation, whereas in the opposite limit the rough surface can be replaced by an equivalent surface layer.

Figure 1

Sketch of the configuration considered here.

Figure 2

Simple sketch of the scattering processes (a) due to $M_{\text{s}/\text{p},0}$ and (b) due to $M_{\text{s}/\text{p},1}$. After averaging the translational symmetry is restored so that $\mathbf{\kappa }$ is the same before and after the scattering with the rough surface. Mathematically, this property follows directly from the statistical properties of the Gaussian surface roughness, i.e., ${⟨{\tilde{S}}^{\left(2\right)}\left(\mathbf{\kappa }-{\mathbf{\kappa }}^{\prime }\right)⟩}_{\text{p}}\propto \delta \left(\mathbf{\kappa }-{\mathbf{\kappa }}^{\prime }\right)$ and ${⟨{\tilde{S}}^{\left(1\right)}\left(\mathbf{\kappa }-{\mathbf{\kappa }}^{\prime }\right){\tilde{S}}^{\left(1\right)}\left({\mathbf{\kappa }}^{\prime }-{\mathbf{\kappa }}^{″}\right)⟩}_{\text{p}}\propto \delta \left(\mathbf{\kappa }-{\mathbf{\kappa }}^{″}\right)$.

Figure 3

(Color online) Plot of the modulus of $\Delta P$ for evanescent modes over the distance for SiC setting $T_{\text{B}}=0\text{K}$ and $T_{\text{P}}=300\text{K}$ for a rough surface with $\delta =5\text{nm}$ and $a=200\text{nm}$. The red part of the curve (for $d<2.8\mu m$) indicates positive and the blue one (for $d>2.8\mu m$) negative values. The PA and the LDA are included for comparison.

Figure 4

(Color online) As Fig. 3 but for evanescent and propagating modes.

Figure 5

(Color online) As Fig. 3 using $a=100$, 200, and 500 nm.

Figure 6

(a) Sketch of the physical situation encountered in the large distance regime for the heat transfer and (b) the replacement of the rough surface by an equivalent layer for surface roughness to determine the LDOS in that regime.

Figure 7

(Color online) Plot of $\Delta R_{\text{s}/\text{p}}$ as defined in Eq. (32) for (a) $s$- and (b) $p$-polarized modes using $\delta =5\text{nm}$ and $a=200\text{nm}$.

Figure 8

(Color online) Illustration of the excitation of surface polaritons by $s$-polarized light in the $\mathbf{\kappa }$ plane (dashed $\text{line}=\text{dispersion}$ relation of the surface phonon polariton, gray $\text{circle}=\text{propagating}$ waves with $\kappa <k_0$): an $s$-polarized wave with ${\mathbf{\kappa }}_i=\left(k_i,0\right)$ and an electric field in $y$ direction. Due to a first scattering process labeled as 1 with the rough surface the surface power spectrum provides the necessary extra momentum $\left({\mathbf{\kappa }}_i-{\mathbf{\kappa }}_{\text{sp}}\right)$ to match the phase with the surface polariton with wave vector ${\mathbf{\kappa }}_{\text{sp}}$. The incoming $s$-polarized wave has an electric field component in the direction of ${\mathbf{\kappa }}_{\text{sp}}$ so that it can excite the surface polariton. Due to the second scattering process 2 the surface power spectrum again provides the necessary extra momentum $-\left({\mathbf{\kappa }}_i-{\mathbf{\kappa }}_{\text{sp}}\right)$ resulting in a scattered $s$-polarized wave with $\mathbf{\kappa }={\mathbf{\kappa }}_i$.

Figure 9

(Color online) Plot of $\text{Im}\left(r_{\text{s}/\text{p}}\right)$ as defined in Eq. (33) for (a) $s$- and (b) $p$-polarized modes using $\delta =5\text{nm}$ and $a=200\text{nm}$.

Figure 10

Plot of $\text{Im}\left(r_{\text{p}}\right)$ and $\text{Im}\left({⟨r_{\text{p}}^{\left(0\right)-\left(2\right)}⟩}_{\text{p}}\right)$ for $\kappa a=1$ using $\delta =5\text{nm}$ and $a=200\text{nm}$.

Figure 11

(Color online) Plot of $D^{\text{E}}$ and $\Delta D^{\text{E}}$ as defined in Eqs. (10, 34) for propagating and evanescent modes (dashed line) and for propagating modes only (dotted line) using $\delta =5\text{nm}$ and $a=200\text{nm}$ and distance $z=5\mu \text{m}$.

Figure 12

(Color online) Plot of $D^{\text{E}}$ and $\Delta D^{\text{E}}$ as defined in Eq. (34) for $z=500\text{nm}$ using the same roughness parameters as in Fig. 11.

Figure 13

(Color online) Plot of $\Delta D^{\text{E}}$ over the distance for SiC using the frequency $\omega ={10}^{14}{\text{s}}^{-1}$ for a rough surface with $\delta =5\text{nm}$ for correlation length $a=100$, 200, and 500 nm. The thin solid line represents the PA result, whereas the thin dashed-dotted line is the LDA.

Figure 14

(Color online) As Fig. 13 but for $\omega =1.787\times {10}^{14}{\text{s}}^{-1}$. The red part of the curves indicates positive values and the blue one negative values.

Figure 15

(Color online) As Fig. 13 but with $\omega =1.8\times {10}^{14}{\text{s}}^{-1}$.