Fundamental limit to the rectification of near-field heat flow: The potential of intrinsic semiconductor films

We derive the fundamental limit to near-field radiative thermal rectification mediated by an intrinsic semiconductor film within the framework of fluctuational electrodynamics. By leveraging the electromagnetic local density of states, we identify ${\varepsilon }_{\mathrm{H}}^{″}/{\varepsilon }_{\mathrm{L}}^{″}$ as an upper bound on the rectification magnitude, where ${\varepsilon }_{\mathrm{H}}^{″}$ and ${\varepsilon }_{\mathrm{L}}^{″}$ are the imaginary parts of the film permittivity at high and low temperatures, respectively. This bound is tight and can be approached regardless of whether the film is suspended or supported. For intrinsic silicon the limit can in principle exceed ${10}^9$. Our work highlights the possibility of controlling heat flow as effectively as electric current, and offers guidelines to potentially achieve this goal.

Figure 1

Conceptual design and LDOS analysis. (a) Schematic of the thermal diode. (b) LDOS contrast of $p$-polarized evanescent modes as a function of ${\varepsilon }_{\mathrm{H}}^{″}/{\varepsilon }_{\mathrm{L}}^{″}$ at $\omega =1\times {10}^{13}$ $\mathrm{rad}{\mathrm{s}}^{-1}$ for a hypothetical thin film. Here, ${\varepsilon }^{\prime }=10$ and ${\varepsilon }_{\mathrm{L}}^{″}=2\times {10}^{-4}$ (close to that of $i$-Si at 300 K) are fixed while ${\varepsilon }_{\mathrm{H}}^{″}$ is varied. Inset shows the case of an $i$-Si film at three representative frequencies incorporating all contributing modes, with $T_{\mathrm{L}}$ fixed at 300 K and $T_{\mathrm{H}}$ varied from 350 K to 1000 K.

Figure 2

Thermal diode pairing a suspended $i$-Si film with a polar dielectric film. (a) Contrast of ${\varepsilon }^{″}$ and $q$ as a function of $\omega$, and normalized LDOS above the polar dielectric films. (b) Contrast of ${\varepsilon }^{″}$ and $q$ at $\omega ={\omega }_{\mathrm{r}}$, and contrast of $Q$, as a function of $T_{\mathrm{H}}$. The $i$-Si film is 10 nm thick, and the polar dielectric film is 100 nm thick.

Figure 3

Thermal diode with the semiconductor film supported on a substrate. (a) LDOS contrast map at 10 nm above a 10-nm-thick semiconductor film on a bulk substrate, as a function of ${\varepsilon }_{\mathrm{s}}^{\prime }$ and ${\varepsilon }_{\mathrm{s}}^{″}$. (b) Contrast of $q$ when pairing a polar dielectric film (${\omega }_{\mathrm{r}}=5\times {10}^{13}$ $\mathrm{rad}{\mathrm{s}}^{-1}$) with an $i$-Si film on a thin substrate with fixed ${\varepsilon }_{\mathrm{s}}^{\prime }$ and varying ${\varepsilon }_{\mathrm{s}}^{″}$, as a function of $\omega$.

Figure 4

Thermal rectification potential of intrinsic silicon. (a) Contrast of ${\varepsilon }^{″}$ for $i$-Si as a function of $\omega$ and $T_{\mathrm{H}}$, with $T_{\mathrm{L}}$ fixed at 300 K. Inset shows the case when $T_{\mathrm{H}}=1000\mathrm{K}$. Short dashed lines mark the optimized rectification magnitudes when pairing the $i$-Si film with a film of some representative polar dielectrics such as potassium bromide, potassium chloride, barium fluoride, magnesium oxide, and cubic boron nitride [54]. The position and width of each dashed line mark the corresponding Reststrahlen band. (b) Rectification magnitudes when pairing the $i$-Si film with a film of hypothetical polar dielectric as a function of ${\omega }_{\mathrm{r}}$ and $T_{\mathrm{H}}$, with $T_{\mathrm{L}}$ fixed at 300 K. Inset shows the case when $T_{\mathrm{H}}=1000\mathrm{K}$. To achieve appropriate narrowband filters at different frequencies, for the polar dielectric we assume ${\varepsilon }_{\infty }=1$ and ${\omega }_{\mathrm{LO}}=1.01{\omega }_{\mathrm{TO}}$ with $\mathrm{\Gamma }={10}^{-3}{\omega }_{\mathrm{TO}}$ for ${\omega }_{\mathrm{r}}<{10}^{14}$ $\mathrm{rad}{\mathrm{s}}^{-1}$, $\mathrm{\Gamma }={10}^{-5}{\omega }_{\mathrm{TO}}$ for ${\omega }_{\mathrm{r}}$ from ${10}^{14}$ to ${10}^{15}$ $\mathrm{rad}{\mathrm{s}}^{-1}$, and $\mathrm{\Gamma }={10}^{-7}{\omega }_{\mathrm{TO}}$ for ${\omega }_{\mathrm{r}}>{10}^{15}$ $\mathrm{rad}{\mathrm{s}}^{-1}$. The gap size and film thicknesses are all 1 nm.