# Penetration depth in nanoscale thermal radiation

Traditionally, radiation penetration depth in a solid, also called skin depth or photon mean free path, is defined as δλ = λ / (4πκ), where κ is the extinction coefficient as discussed earlier. A film whose thickness is six times the skin depth can be treated as opaque in most applications. In the optical spectrum, δλ of noble metals is usually 10-20 nm. For an evanescent wave, such as that induced under the total internal reflectance setup when light is incident from an optically denser medium to a rarer medium, the skin depth may be defined according to the 1 / e attenuation of the field as δ = 1 / Im(kz) , where kz (purely imaginary) is the wavevector component perpendicular to the interface in the optically rarer medium. The electric and magnetic fields will decay exponentially and become negligible at a distance greater than about one wavelength. Hence, the skin depth is expected to be several tenths of a wavelength in a dielectric medium. However, in near-field radiation especially when SP(h)Ps are excited, an extremely small skin depth (on the order of the vacuum gap d) may exist even though the dominant wavelengths are in the infrared [1]. Furthermore, the skin depth is proportional to the separation distance. In essence, the skin depth in near-field thermal radiation is a function of the vacuum gap as well as material properties [2].

For very small gap, while ωm (see Fig. 1) remains constant as d decreases, the energy transfer is shifted towards larger β values, leading to greater near-field enhancement. For $\beta >>\sqrt{\varepsilon _{j} } \omega /c$ ,   $k_{zj} \approx i\beta$   or   $Im(k_{zj}^{} )\approx \beta$ . There exists an evanescent wave (in medium 3) whose amplitude decays according to e-β(z-d). Hence, the skin depth of the field becomes $\delta _{{\rm F}} \approx 1/\beta$ and the power penetration depth becomes $\delta _{{\rm P}} \approx 1/(2\beta )$ .

Using the multilayer Green’s function, the z-component of the Poynting vector, which is proportional to the heat flux, can be calculated both inside the emitter and the receiver. The spectral and total Poynting vector distributions are plotted in Fig. 2 for SiC. The ordinate is normalized to the Poynting vector inside the vacuum gap. The energy flux increases in the emitter towards the surface, as more and more energy is emitted, remains constant in the vacuum gap, and decreases in the receiver away from the surface. When the abscissa is z / d, the results are nearly the same for 1 nm < d < 100nm. Surprisingly, the distributions are “symmetric” in the emitter and the receiver. The 1 / e decay line is shown as the horizontal dashed line so that the penetration depth can be evaluated. Note that the calculated Poynting vector is integrated over all β values. As mentioned earlier, when SPP is excited, the energy transfer is pushed towards large β values; hence, the spectral penetration depth has a minimum near ωm.

As shown in Fig. 2, the penetration depth is approximately 0.19d at 10.54 μm, where SPP is excited at the vacuum-SiC interfaces. The actual minimum penetration depth is located at 10.47 μm, corresponding to the maximum of X(ω). The penetration depth increases towards longer or shorter wavelengths, and the overall penetration depth based on the total energy flux is 0.25d, which is about 30% greater than δP evaluated at ωm and βm. For a thin vacuum gap, the SPhP dispersion is shifted to large β values, resulting in a shorter penetration depth. Hence, a 10-nm coating of SiC can act as an optically thick medium when d = 10 nm as predicted in refs. [3,4]. When d < 1 nm, the penetration depth is less than a monolayer, implying that the SiC emitter is completely a 2D solid. It should be mentioned that δP cannot be arbitrarily small. When d is comparable to or less than the interatomic distance, the radiative transfer cannot be explained by the local electromagnetic theory. Obviously, in such case, one cannot use its bulk dielectric function and also cannot set βc as infinity. Note that with magnetic materials, surface waves can be excited for both TE and TM waves. Recently, the penetration depth in near-field radiation between metamaterials has also been examined [2].

Figure 2. The distributions of the spectral and total Poynting vector (z component) near the surfaces of the emitter and receiver, both made of SiC, normalized to that in the vacuum [1].

## References

[1] Basu, S., and Zhang, Z. M., 2009, “Ultrasmall Penetration Depth in Nanoscale Thermal Radiation,” Applied Physics Letters, 95, p. 133104.

[2] Basu, S., and Francoeur, M., 2011, “Penetration Depth in Near-Field Radiative Heat Transfer between Metamaterials,” Applied Physics Letters, 99, p. 143107.

[3] Francoeur, M., Menguc, M. P., and Vaillon, R., 2008, “Near-Field Radiative Heat Transfer Enhancement via Surface Phonon Polaritons Coupling in Thin Films,” Applied Physics Letters, 93, p. 043109.

[4] Fu, C. J., and Tan, W. C., 2009, “Near-Field Radiative Heat Transfer Between Two Plane Surfaces with One Having a Dielectric Coating,” Journal of Quantitative Spectroscopy and Radiative Transfer, 110, pp. 1027-1036.