Universal features of coherent photonic thermal conductance in multilayer photonic band gap structures

We show that at the high-temperature limit, the coherent photonic thermal conductance of a multilayer photonic crystal can be significantly below the corresponding thermal conductance of vacuum. Moreover, the thermal conductance at this limit is independent of the thicknesses of the layers but dependent on the refractive indices of the layers only. Such universal features are directly related to the ergodic nature of the distribution of photonic bands in the frequency space.

Figure 1

Normalized three-dimensional thermal conductance $G_{3\text{D}}\left(T\right)/G_{\text{vac}}\left(T\right)$ versus normalized temperature $T/T_0$, for silicon-vacuum multilayer photonic crystals. Here $T_0\equiv hc/\left(k_{B}a\right)$, where $a$ is the lattice constant. The refractive index of silicon is $n_{\text{Si}}\equiv \sqrt{11.7}=3.42$. The black solid curves correspond to structures with silicon thicknesses $d_1=0.1a,0.2a,\dots ,0.9a$, respectively. The corresponding vacuum thickness for each $d_1$ is $d_2=a-d_1$. For all structures, ${\text{lim}}_{T\to \infty }{G}_{3\text{D}}\left(T\right)/G_{\text{vac}}\left(T\right)$ all converge to 0.548. The vacuum level $G_{3\text{D}}\left(T\right)/G_{\text{vac}}\left(T\right)=1$ is denoted by the dashed line.

Figure 2

(Color online) Normalized single-channel thermal conductance $G\left(T\right)/G_0\left(T\right)$ versus normalized temperature $T/T_0$ for silicon-vacuum multilayer photonic crystals with thermal photons of a single polarization at normal incidence $\left(\kappa =0\right)$. Here $T_0\equiv hc/\left(k_{B}a\right)$, where $a$ is the lattice constant. The index of silicon is $n_{\text{Si}}\equiv \sqrt{11.7}=3.42$. $d_1$ denotes the silicon thickness and the vacuum thickness is $d_2=a-d_1$, as shown in the inset. The black solid lines represent the cases with $d_1=0.1a,0.2a,\dots ,0.9a$, respectively. For each case, the characteristic ratio $\left(n_{\text{Si}}{d}_1-d_2\right)/\left(n_{\text{Si}}{d}_1+d_2\right)$ is irrational and ${\text{lim}}_{T\to \infty }G\left(T\right)/G_0\left(T\right)=0.540$. A special case $d_1=d_{\ast }\equiv \left[2/\left(n_{\text{Si}}+2\right)\right]a=0.369a$ is represented by the red dashed line. In this case, the characteristic ratio is a rational value of 1/3, which results in ${\text{lim}}_{T\to \infty }G\left(T\right)/G_0\left(T\right)=0.582$. The case of the homogeneous single channel $G\left(T\right)/G_0\left(T\right)=1$ is denoted by the black dotted line.

Figure 3

Plot of $f\left(\omega ,T\right)={\left[\hslash \omega /\left(k_{B}T\right)\right]}^2{e}^{\hslash \omega /\left(k_{B}T\right)}/{\left[e^{\hslash \omega /\left(k_{B}T\right)}-1\right]}^2$ versus $\omega$ for $T=0.1T_0$ (solid line), $T=T_0$ (dashed-dotted line), and $T=20T_0$ (dashed line). $\omega$ is in unit of $\left(2\pi c/a\right)$. $T_0\equiv hc/\left(k_{B}a\right)$, where $a$ is the lattice constant of the multilayer photonic crystal. The shaded regions denote the frequency ranges of the photonic bands of the crystal and the unshaded regions are the photonic band gaps. The case shown here is $d_1=d_2=0.5a$ and $n=n_{\text{Si}}\equiv \sqrt{11.7}$.

Figure 4

(Color online) (a) Trajectory of $z\left(\omega \right)=r_a\text{exp}\left(i{\tau }_a\omega \right)-r_b\text{exp}\left(i{\tau }_b\omega \right)$ for incommensurate ${\tau }_a$ and ${\tau }_b$. $z\left(\omega \right)$ is at $z=1$ when $\omega =0$ (small red ring) and is plotted as discrete points with frequency spacing $\Delta \omega =0.01\left(2\pi c/a\right)$. These points provide a dense coverage of the region $1\le \left|z\right|\le \xi$ and yield to a distribution pattern. The trajectory is generated using $n=n_{\text{Si}}\equiv \sqrt{11.7}$ and $d_1=d_2=0.5a$. The distribution pattern is the same for other values of $d_{1,2}$ of the incommensurate case. $z\left(\omega \right)$ is restricted within the region bounded between $\text{Re}\left(z\right)=-1$ (green solid line) and $\text{Re}\left(z\right)=1$ (brown solid line) when $\omega$ is in the photonic bands. (b) Graphical representation of $z\left(\omega \right)$. The large circle (golden solid line) with radius $r_a$ is centered at $z=0$. The small circle (orange solid line), with radius $r_b$, is centered at $O$ where $z=r_a\text{exp}\left(i{\varphi }_a\right)$ and $r_a-r_b=1$. In all cases, $z\left(\omega \right)$ always starts at $z=1$ when $\omega =0$ and is bounded within $1\le \left|z\right|\le \xi$ for all $\omega$. These boundaries are shown by gray dashed lines. When $\omega$ is in photonic bands, $z\left(\omega \right)$ is within the region between $\text{Re}\left(z\right)=-1$ (green solid line) and $\text{Re}\left(z\right)=1$ (brown solid line). The points $Q$, where $z\left(\omega ={\varphi }_a/{\tau }_a\right)$, and $S$, where $z\left(\omega ={\varphi }_a/{\tau }_a+2\pi /{\tau }_a\right)$, are on the same small circle. Here ${\varphi }_a=0.3\pi$. (c) Trajectory of $z\left(\omega \right)$ for commensurate ${\tau }_a$ and ${\tau }_b$. $z\left(\omega \right)$ is plotted as discrete points with frequency spacing $\Delta \omega =0.002\left(2\pi c/a\right)$ for $0\le \omega \le 2\pi N_a/{\tau }_a$. $z\left(\omega \right)$ intersects $\text{Re}\left(z\right)=-1$ at the points $X_{i=0,1,2}^{\{-\}}$ and $\text{Re}\left(z\right)=1$ at $X_{i=0,1,2}^{\{+\}}$, as marked by the small red rings. The case shown here is $n=n_{\text{Si}}\equiv \sqrt{11.7}$ and $d_1=d_{\ast }=\left[2/\left(n_{\text{Si}}+2\right)\right]a$ so that ${\tau }_b/{\tau }_a=N_b/N_a=1/3$. Other parameters are the same as in (a) and (b).

Figure 5

(Color online) Normalized single-channel thermal conductance $G\left(n,T\right)/G_0\left(T\right)$ at the high-temperature limit versus the index $n$ for dielectric-vacuum multilayer photonic crystals with various dielectric index and layer thicknesses. Here $T=T_{\text{h}}\equiv 20T_0$. The index of the dielectric layer $n$ is varied continuously from $0<n\le 20$. ${\text{lim}}_{T\to \infty }G\left(T,n\right)/G_0\left(T\right)$ from the left-hand side of Eq. (7) is denoted by the blue solid line while $\eta \left(n\right)$ from the right-hand side of Eq. (7) is denoted by the red dashed line. For each value of $n$, the size of the dielectric layer is set by a random generator, with equal probability distribution in the range of $0.1a\le d_1\le 0.9a$, as shown in the inset. The vacuum layer is set to be $d_2=a-d_1$ accordingly. The two most prominent peaks, corresponding to the cases of $N_b/N_a=0$ and $N_b/N_a=1/3$, are marked as brown dots.

Figure 6

(Color online) (a) Normalized single-channel thermal conductance $G\left(d_1,T\right)/G_0\left(T\right)$ at the high-temperature limit for various layer thicknesses (blue solid line), with silicon as the dielectric layers $\left(n_{\text{Si}}\equiv \sqrt{11.7}=3.42\right)$. Here $T=T_{\text{h}}\equiv 20T_0$. The size of dielectric layer is varied from $0\le d_1\le a$ and the vacuum layer is set at $d_2=a-d_1$ accordingly. Though not shown, $G\left(d_1,T_{\text{h}}\right)/G_0\left(T_{\text{h}}\right)$ approaches 1 at $d_1=0$ or $d_1=a$. The black dots are the theoretical values of ${\eta }^{\left[C\right]}\left(N_a,N_b\right)$ marked at $d_1=a\left(N_a+N_b\right)/\left[n_{\text{Si}}\left(N_a-N_b\right)+\left(N_a+N_b\right)\right]$, for any coprime integers $N_a$ and $N_b$ satisfying $N_a>0$ and $-N_a\le N_b\le N_a$. The values of ${\eta }^{\left[C\right]}\left(N_a,N_b\right)$ in the range $0\le \left|N_b\right|\le N_a\le 20$ are calculated. $N_b/N_a$ of the most pronounced peaks (0, 1/3, and $-1/3$) are labeled on the graph. (b) Same as (a), with $d_1=d^{\ast }\equiv \left[2/\left(n_{\text{Si}}+2\right)\right]a$. Shown here are the cases of $T=20T_0$ (blue solid line), $T=2T_0$ (gray dotted line), and $T=T_0$ (gray dashed line). The black dot is the theoretical value of ${\eta }^{\left[C\right]}\left(N_a=3,N_b=1\right)$.

Figure 7

(Color online) Normalized three-dimensional thermal conductance $G_{3\text{D}}\left(n,T\right)/G_{\text{vac}}\left(T\right)$ at the high-temperature limit versus $n$ for various dielectric index and layer thicknesses. $T=T_{\text{H}}\equiv 25T_0$. The index of the dielectric layer $n$ is varied continuously from $0<n\le 20$. ${\text{lim}}_{T\to \infty }{G}_{3\text{D}}\left(T,n\right)/G_{\text{vac}}\left(T\right)$ from the left-hand side of Eq. (20) is denoted by the blue solid line while ${\eta }_{3\text{D}}\left(n\right)$ from the right-hand side of Eq. (20) is denoted by the red dashed line. For each value of $n$, the size of the dielectric layer is set by a random generator, with equal probability in the range $0.1a\le d_1\le 0.9a$, as shown in the inset. The vacuum layer is set to be $d_2=a-d_1$ accordingly. At all data points, $G_{3\text{D}}\left(n,T_{\text{H}}\right)/G_{\text{vac}}\left(T_{\text{H}}\right)$ collapses onto the curve ${\eta }_{3\text{D}}\left(n\right)$ regardless of $d_1$.

Figure 8

(Color online) (a) Trajectory of $\alpha \left(\omega \right)=r_a\text{cos}\left[{\varphi }_a\left(\omega \right)\right]-r_b\text{cos}\left[{\varphi }_b\left(\omega \right)\right]$ for incommensurate ${\tau }_a$ and ${\tau }_b$. $\alpha \left(\omega \right)$ is at $\left[{\varphi }_a,{\varphi }_b\right]=\left[0,0\right]$ when $\omega =0$ and is plotted as discrete points with frequency spacing $\Delta \omega =0.01\left(2\pi c/a\right)$. These points distribute uniformly on the plane. The case shown here is $n=n_{\text{Si}}\equiv \sqrt{11.7}$ and $d_1=d_2=0.5a$ [same as Fig. 4a]. The area bounded by ${\varphi }_b^{\{-\}}\left({\varphi }_a\right)=\text{Re}\left\{{\text{cos}}^{-1}\left[\left(r_a\text{cos}{\varphi }_a+1\right)/r_b\right]\right\}$ (green solid curve) and ${\varphi }_b^{\{+\}}\left({\varphi }_a\right)=\text{Re}\left\{{\text{cos}}^{-1}\left[\left(r_a\text{cos}{\varphi }_a-1\right)/r_b\right]\right\}$ (brown solid curve) is directly proportional to ${\eta }^{\left[I\right]}$. These two curves correspond to the lines at $\text{Re}\left(z\right)=-1$ and $\text{Re}\left(z\right)=1$ in Fig. 4, respectively. For comparison, the solid orange line at ${\varphi }_a=0.3\pi$ is also drawn, which corresponds to the small circle (orange) in Figs. 4b, 4c. (b) Same as (a), except that ${\tau }_a$ and ${\tau }_b$ are commensurate. The case shown here is when $n=n_{\text{Si}}\equiv \sqrt{11.7}$ and $d_1=\left[2/\left(n_{\text{Si}}+2\right)\right]a$ such that ${\tau }_b/{\tau }_a=N_b/N_a=1/3$ [same as Fig. 4c]. $\alpha \left(\omega \right)$ is plotted as discrete points with frequency spacing $\Delta \omega =0.002\left(2\pi c/a\right)$ for $0\le \omega \le \pi N_a/{\tau }_a$. $\alpha \left(\omega \right)$ intersects ${\varphi }_b^{\{-\}}\left({\varphi }_a\right)$ at $U_{j=0,1,2}^{\{-\}}$ and ${\varphi }_b^{\{+\}}\left({\varphi }_a\right)$ at $U_{j=0,1,2}^{\{+\}}$, as marked by the small red rings. The total length of line segments bounded by ${\varphi }_b^{\{-\}}\left({\varphi }_a\right)$ and ${\varphi }_b^{\{+\}}\left({\varphi }_a\right)$ is directly proportional to ${\eta }^{\left[C\right]}\left(N_a,N_b\right)$.