# Upper limit of near-field heat flux

For nonmagnetic materials, when β >> ω/c (evanescent waves), we have $\gamma _{1} \approx \gamma _{2} \approx \gamma _{0} \approx {\rm i}\beta$. As a result, for dielectrics, $r_{01}^{s}$ and $r_{02}^{s}$ are negligibly small, and the contribution of TE waves can be ignored. Furthermore, ${r_{01}^{{\rm p}} \approx ({\it \varepsilon }_{1} -1)/({\it \varepsilon }_{1} +1)}$ and ${r_{02}^{{\rm p}} \approx ({\it \varepsilon }_{2} -1)/({\it \varepsilon }_{2} +1)}$ are independent of β. Hence sevan(ω,β) can be simplified as,

$s_{{\rm evan}} (\omega ,\beta )\approx \frac{\beta Im(r_{01}^{{\rm p}} )Im(r_{02}^{{\rm p}} )e^{-2\beta d} }{\left|1-r_{01}^{{\rm p}} r_{02}^{{\rm p}} e^{-2\beta d} \right|^{2} }$

(1)

However, for metals, the contribution from TE waves is more significant when ${\omega / c} \gg \beta \gg \sqrt{\left|\varepsilon _{1} \right|} {\omega / c }$ whereas the contribution from TM waves is more important for ${\beta \gg \sqrt{\left|\varepsilon _{1} \right|} {\omega / c } \gg {\omega / c}}$ [1]. As a result for metals, heat transfer due to TM waves becomes dominant at very short distances.

Using the relation, $Im\left(\frac{\varepsilon -1}{\varepsilon +1} \right)=\frac{2\varepsilon ''}{\left|\varepsilon +1\right|^{2} }$, and assuming identical permittivity for both media, the spectral heat flux from 1 to 2 in the limit d → 0 is given by [2],

(2)

where ξ=βd ,   ξ0=dω/c and ε" is the imaginary part of the dielectric function. As observed from Eq. (2) the heat flux will be inversely proportional to d2 in the proximity limit. This means that the heat flux will diverge as d → 0 and its physical significance has been debated among researchers. It should be noted that the d-2 dependence is for contribution from the p-polarized electromagnetic waves only, since the contribution from the s-polarized waves will asymptotically reach a constant as d → 0. As the vacuum gap decreases, the energy transfer shifts to large values of the parallel wavevector component. A cutoff in the order of the lattice constant is imposed as the minimum spatial wavelength, which subsequently sets a maximum wavevector component parallel to the interfaces [3]. The imposed cutoff limits the number of modes for photon tunneling. Consequently, the radiative heat flux will experience a reduction as d → 0.

The total heat transfer between two media as given by Eq. (16) can be rewritten as

$q''_{{\rm net}} =\int _{ 0}^{ \infty }\, q''_{\omega } (\omega ){\rm d}\omega =\frac{1}{\pi ^{2} } \int _{ 0}^{ \infty }\, \left[\Theta \left(\omega ,T_{1} \right)-\Theta \left(\omega ,T_{2} \right)\right] X(\omega ){\rm d}\omega$

(3)

where ${X(\omega )}={\int _{0}^{\beta _{{\rm c}} }s(\omega ,\beta ){\rm d}\beta }$ and the upper limit of the integration in β is restricted to βc. Electrons in solids move in a periodic potential characterized by the Bloch wave, with a maximum wavevector of π/a at the edge of the first Brillouin zone. Here, a is the lattice constant, which is on the order of interatomic distance. This posts a limit on the smallest surface wavelength or cutoff wavevector parallel to the surface βc= π / a [3]. Take a typical value of a as 0.5 nm and note that there exists a maximum of X, i.e., $X_{\max } =\beta _{{\rm c}}^{2} /8$. There exists an upper limit of the near-field radiative heat flux given by [2,3],

$q''_{max } =X_{\max } \frac{k_{{\rm B}}^{2} }{6\hbar } (T_{1}^{2} -T_{2}^{2} )=\frac{k_{{\rm B}}^{2} \beta _{{\rm c}}^{2} }{48\hbar } (T_{1}^{2} -T_{2}^{2} )$

(4)

for nonmagnetic materials. Note that q''max is the ultimate maximum heat flux and is only achievable when d → 0. It is found [4] that metals with a large imaginary part in the infrared can help reach such a limit at extremely small distances. For distances that are meaningful, however, the situation is different. Basu and Zhang [2] considered a case in which both the emitter and receiver are assumed to have frequency-independent permittivity in order to identify the expression of the complex dielectric constant that will results in maximum heat flux. It should be noted that such a constant dielectric function cannot exist in reality because of the violation of Causality. When X is plotted against ε' and ε" in a 3D plot or a 2D contour for given d (say 10 nm), it was found that the maximum of X is corresponding to ε' = -1 for which surface waves exist. Figure 1 shows the calculated radiative heat flux between the two media ( T1= 300 K and T2= 0 K ) as a function of the vacuum gap for different values of ε' and ε" [2]. In most cases, ε' is fixed at -1. For the sake of comparison, the energy transfer between two SiC plates is also shown in the figure using a frequency-dependent dielectric function. At 300 K, the upper limit of near-field heat flux is 1.4×1011 W/m2, which is represented as the dashed horizontal line. The radiation flux between two blackbodies is 459 W/m2, which is several orders of magnitude smaller than near-field radiative transfer. The cutoff in β sets an upper limit on the maximum energy transfer between the two media. Hence, for each of the dielectric functions, there exists an optimal vacuum gap width (dm) for the maximum energy transfer. For ε = -1 + i0.1 , it can be seen from Fig. 1 that dm = 0.6 nm , which also maximizes X. The value of dm decreases with increasing ε" , implying that the reduction in the energy transfer begins to take place at smaller vacuum gaps. Furthermore, the d-2 dependence in the energy transfer exists only when d>dm. At d>2nm , increasing ε" results in a decrease of the heat flux. When ε = 0 + i10 , the radiative heat flux is generally much smaller than those with ε' = -1 but will keep increasing towards the upper limit as d unrealistically approaches zero. For the selected dielectric functions with ε' = -1 and ε" << 1 , the energy transfer can be orders of magnitude greater than that between SiC plates. This is due to the assumed frequency-independent dielectric function, which results in the excitement of surface waves at almost every frequency. While no such materials can exist, the results provide some hints of appropriate dielectric functions that will result in optimal heat flux at different vacuum gaps. By introducing the cutoff in β , even for SiC, the d-2 trend ceases to exist at d < 0.6 nm. Instead, the near-field radiative transfer reaches a plateau below d = 0.5 nm. Wang et al. [5] performed a design optimization of the parameters in the Drude model and Lorentz model that can result in the highest near-field radiative flux at given distance and temperatures.

Figure 1. Plot of radiative heat flux versus gap width for different dielectric functions [2]. The temperatures of the two media are set to be 300 K and 0 K. The dielectric function is assumed to be frequency independent, except that for SiC for which the Lorentz model is used.

## References

[1] Chapuis, P. O., Volz, S., Henkel, C., Joulain, K., and Greffet, J.-J., 2008, “Effects of Spatial Dispersion in Near-Field Radiative Heat Transfer between Two Parallel Metallic Surfaces,” Physical Review B, 77, p.035431.

[2] Basu, S., and Zhang, Z. M., 2009, “Maximum Energy Transfer in Near-Field Thermal Radiation at Nanometer Distances,” Journal of Applied Physics, 105, p. 093535.

[3] Volokitin, A. I., and Persson, B. N. J., 2004, “Resonant Photon Tunneling Enhancement of the Radiative Heat Transfer,” Physical Review B, 69, p. 045417.

[4] Pendry, J. B., 1999, “Radiative Exchange of Heat between Nanostructures,” Journal of Physics-Condensed Matter, 11, pp. 6621-6633.

[5] Wang, X. J., Basu, S., and Zhang, Z. M., 2009, “Parametric Optimization of Dielectric Functions for Maximizing Nanoscale Radiative Transfer,” Journal of Physics D: Applied Physics, 42, p. 245403.