Near-field radiative transfer between two semi-infinite media

Figure 1. Schematic of near-field radiative heat transfer between two closely spaced semi-infinite plates, at temperatures T1 and T2, separated by a vacuum gap d.

Consider the structure shown in Fig. 1, where both the emitter and receiver are n-doped silicon. The emitter and receive are assumed to be at 400 and 300 K, respectively. The dielectric function model of doped Si may be modeled as a combination of Drude term and other contributions [1] and the details are described in [2,3]. The spectral energy transfer per unit area $q _{\omega}^{''}$ can be obtained as [4]

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

(1)

so that   $q''_{net} =\int _{0}^{\infty }q''_{\omega } \, {\rm d} \omega$   . Note that the integration of s(ω,β) over ω gives a weighted function to modify the Planck blackbody distribution function. Expression of s(ω,β) is different for propagating (β < ω/c) and evanescent (β > ω/c) waves,

$s_{\rm prop} \left(\omega ,\beta \right)=\frac{\beta (1-\rho _{{\rm 01}}^{{\rm s}} )(1-\rho _{{\rm 02}}^{{\rm s}} )}{4\left|1-r_{{\rm 01}}^{{\rm s}} r_{{\rm 02}}^{{\rm s}} {\rm e}^{{\rm i2}\gamma _{{\rm 0}} d} \right|^{2} } +\frac{\beta (1-\rho _{{\rm 01}}^{{\rm p}} )(1-\rho _{{\rm 02}}^{{\rm p}} )}{4\left|1-r_{{\rm 01}}^{{\rm p}} r_{{\rm 02}}^{{\rm p}} {\rm e}^{{\rm i2}\gamma _{{\rm 0}} d} \right|^{2} }$

(2a)

and,

$s_{\rm evan} \left(\omega ,\beta \right)=\frac{Im(r_{{\rm 01}}^{{\rm s}} )Im(r_{{\rm 02}}^{{\rm s}} )\beta {\rm e}^{-2{\rm Im(}\gamma _{{\rm 0}} )d} }{\left|1-r_{{\rm 01}}^{{\rm s}} r_{{\rm 02}}^{{\rm s}} {\rm e}^{-{\rm 2Im}\left(\gamma _{{\rm 0}} \right)d} \right|^{2} } +\frac{Im(r_{{\rm 01}}^{{\rm p}} )Im(r_{{\rm 02}}^{{\rm p}} )\beta {\rm e}^{-{\rm 2Im}\left(\gamma _{{\rm 0}} \right)d} }{\left|1-r_{{\rm 01}}^{{\rm p}} r_{{\rm 02}}^{{\rm p}} {\rm e}^{-{\rm 2Im}\left(\gamma _{{\rm 0}} \right)d} \right|^{2} }$

(2b)

In Eqs. (2a,b), the first term on the right-hand side refers to the contribution of s polarization or TE wave, while the second term refers to the contribution of p polarization or TM wave. Note that $r_{0j}^{{\rm s}} =(\gamma _{0} -\gamma _{j} )\mathord (\gamma _{0} +\gamma _{j} )$   and   $r_{0j}^{{\rm p}} =(\varepsilon _{j} \gamma _{0} -\gamma _{j} )\mathord/ (\varepsilon _{j} \gamma _{0} +\gamma _{j} )$   are the Fresnel reflection coefficients for s and p polarization, respectively, at the interface between vacuum and medium j (1 or 2). On the other hand, $\rho _{0j}^{} =\left|r_{0j} \right|^{2}$ is the far-field reflectivity at the interface between vacuum and medium j. When different doping levels are considered, the location of the peak in s(ω,β) shifts towards higher frequencies with increased doping level.

Notice that s(ω,β) is independent of temperature and contains all the information about the material properties as well as the geometry of the emitting media. The predicted radiative heat transfer between two doped Si plates is plotted in Fig. 2(a) as a function of the vacuum gap width [3]. Both plates are maintained at the same doping level, which is varied from 1018 to 1021 cm-3. The dotted line with circles is the radiative heat flux between two blackbodies. The net heat flux at d = 1 nm between 1019 or 1020 cm-3 doped Si plates can exceed that between two blackbodies by five orders of magnitude, because of photon tunneling and surface waves. Increase in the doping level of Si does not always enhance the energy transfer. In fact, the radiative heat transfer is the smallest for 1021 cm-3 doped Si plates as compared with other doping levels considered here. At d > 200 nm, doping concentrations between 1018 and 1019 cm-3 yield the largest radiative heat transfer, which is comparable to that between SiC and SiC. A detailed parametric study has been performed to determine the ideal Drude or Lorentz dielectric functions that yield the largest near-field enhancement [5]. Figure 2(b) illustrates the effect of doping concentration on nanoscale energy flux when the vacuum gap width is fixed at d = 1 nm. The doping level of medium 1 is represented as N1 while that for medium 2 is represented by N2. Generally speaking, surface waves are better coupled when the two media have similar dielectric functions. As a result there exist peaks when N1N2, at doping levels up to 1020 cm-3.

Figure 2. (a) Net energy flux between medium 1 (at 400 K) and medium 2 (at 300 K) versus the gap width for Si at different doping levels. The dash-dotted line refers to the net energy transfer between two blackbodies maintained at 400 and 300 K, respectively; and (b) effect of doping on the net energy transfer between two doped Si plates separated by 1 nm vacuum gap [3].

The enhancement of near-field radiation in metallic and polar materials can be well explained by surface polaritons. The coupling of SP(h)Ps allows a significant increase in the function given in Eq. (2b) for evanescent waves. Furthermore, for magnetic materials, the enhancement can occur for both s- and p-polarizations, resulting in multiple spectral peaks in near-field radiative transfer [5-7]. It should be noted that for intrinsic Si or dielectric materials without strong phonon absorption bands, the tunneling is limited and saturate at extremely small distances. Also, for good metals, the SPP excitation frequency is too high that does not result significant enhancement unless the distance is less than 1 nm [5,8]. To illustrate surface wave effect, Figure 3 displays the contour plots of s(ω,β) / 2π for SiC plates separated at 100 nm (Figure 3a) and for 1020 cm-3 n-doped Si plates separated at 10 nm for TM waves since the contribution of TE waves is negligibly small. Here β is normalized with respect to ω/c. The brightest color represents the peak value at ωm = 1.79×1014 rad/s, βm = 50ω/c for SiC and ωm = 2.67×1014 rad/s, βm = 62ω/c for doped Si. The contribution of propagating waves (β < ω/c) to the overall heat transfer is negligible. As mentioned earlier, the resonance energy transfer in the near field around ωm is due to SPhP for SiC and SPP for doped Si.

The calculated dispersion curves for surface polaritons between two SiC and doped Si plates are also plotted as dashed lines in Fig. 3(a) and 3(b), respectively. Due to the coupling of surface polaritons at vacuum-SiC and vacuum-doped Si interfaces, there exist two branches of dispersion curves for the p polarization as follows:

Symmetric mode:

${\frac{\gamma _{0} }{\varepsilon _{0} } +\coth \left(-\frac{{\rm i}\gamma _{0} d}{2} \right)\frac{\gamma _{1} }{\varepsilon _{1} }} =0$

(3a)

Asymmetric mode:

${\frac{\gamma _{0} }{\varepsilon _{0} } +\tanh \left(-\frac{{\rm i}\gamma _{0} d}{2} \right)\frac{\gamma _{1} }{\varepsilon _{1} }} =0$

(3b)

The lower-frequency branch corresponds to the symmetric mode, and the higher-frequency branch represents the asymmetric mode [9]. Note that for both doped Si and SiC, the dispersion relation becomes almost flat at ωm implying that at ωm surface polaritons can be excited in a wide range of β. At this frequency, evanescent waves in a broad range of β, whose values are much greater than ω/c, are coupled with surface polaritons and therefore, is responsible for the enhancement of thermal radiation through photon tunneling [10].

Figure 3. Contour plot of s(ω,β)/2π for (a) SiC and (b) doped Si for doping concentration of 1020 cm-3. Note that the parallel wavevector component is normalized to the frequency. The dashed curves represent the two branches of the surface-polariton dispersion [3,10].

References

[1] Fu, C. J., and Zhang, Z. M., 2006, "Nanoscale Radiation Heat Transfer for Silicon at Different Doping Levels," International Journal of Heat and Mass Transfer, 49, pp. 1703-1718.

[4] Basu, S., Zhang, Z. M., and Fu, C. J., 2009, "Review of Near-Field Thermal Radiation and Its Application to Energy Conversion," International Journal of Energy Research, 33, pp. 1203-1232.

[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.

[6] Joulain, K., Drevillon, J., Ben-Abdallah, and Greffet, J.-J., 2010, “Noncontact Heat Transfer between Two Metamaterials,” Physical Review B, 81, p. 165119.

[7] Zheng, Z., and Xuan, Y., “Theory of Near-Field Radiative Heat Transfer for Stratified Magnetic Media,” International Heat and Mass Transfer, 54, pp. 1101-1110.

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

[9] Park, K., Lee, B. J., Fu, C. J., and Zhang, Z. M., 2005, “Study of the Surface and Bulk Polaritons with a Negative Index Metamaterial,” Journal of the Optical Society of America B, 22, pp. 1016–1023.

[10] Lee, B. J., and Zhang, Z. M., 2008, “Lateral Shift in Near-Field Thermal Radiation with Surface Phonon Polaritons,” Nanoscale and Microscale Thermophysical Engineering, 12, pp. 238-250.