Photonic Thermal Conductance in Multi-layer Photonic Crystals
- PDF / 514,854 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 92 Downloads / 296 Views
1162-J01-06
Photonic Thermal Conductance in Multi-layer Photonic Crystals Wah Tung Lau1,2, Jung-Tsung Shen1, and Shanhui Fan1,2 Edward L. Ginzton Laboratory, Stanford University, California 94305, USA Department of Electrical Engineering, Stanford University, California 94305, USA ABSTRACT Photonic thermal conductance of a multi-layer photonic crystal, when normalized with the corresponding thermal conductance of vacuum at each temperature, can be significantly below unity. The minimum of this normalized thermal conductance occurs at the high-temperature limit, and the conductance value at this limit is independent of the layer thicknesses. We give an analytic theory to explain such independence, and show that it is related to the ergodic nature of the distribution of photonic bands in frequency space. INTRODUCTION Vacuum is commonly believed to be of very low thermal conductance. It would be of great practical usage if there is a medium that is thermally less conducting than vacuum, for instance, in the realization of high-temperature thermal barrier coatings [1]. Since thermal energy is totally carried by photons in vacuum, in order to suppress below vacuum conductance, a natural idea is to use structures with photonic band gaps. The simplest photonic band gap structure is a multilayer photonic crystal, consisted of alternating dielectric and vacuum layers. In particular, intrinsic silicon, with ns = 11.7 = 3.4205... , is almost lossless for thermal spectrum at room temperature [2]. Coherent thermal transport, and the emergence of photonic band gaps, is thus possible in silicon-vacuum multi-layer photonic crystals. In this paper, we will see how the photonic band gaps effect can suppress, and influence the properties of thermal conductance in such multi-layer structures. THEORY
Photonic thermal conductance of the multi-layer photonic crystal is given by [3]: 2 hω /( k BT ) nω / c k dk ∞ d ω [ hω / ( k T )] e // // B G (T ) = ∫ Θ(ω , k/ / , σ ) , (1) 0 2π ∫0 2π (ehω /( kBT ) − 1) 2 where h is the reduced Planck constant, k B is the Boltzmann constant, T is the operating temperature. ω is photon frequency. k / / is the wavenumber parallel to the layers, and σ is either the TE, or TM polarizations. The function Θ(ω , k/ / , σ ) is 1 when the bloch wavenumber K is real, and 0 if K is imaginary. K is related to the thickness and index of the layers as: Pγ 1 γ α ≡ cos( Ka) = cos(γ s d s ) cos(γ v d v ) − ( s + v )sin(γ s d s ) sin(γ v d v ) , (2) 2 Pγ v γ s where γ s ,v = (ns ,vω / c) 2 − k /2/ , ns ,v are indices, while d s ,v are thicknesses of the dielectric and vacuum layers. P = 1 for TE, and P = (ns / nv ) 2 for TM modes. Note that | α |≤ 1 when K is real, which occurs at the photonic bands; | α |> 1 when K is imaginary, which is at the band gaps.
For vacuum, the conductance is Gvac (T ) = π 2 k B4T 3 / (15h3c 2 ) . Thus a natural quantity to examine is the normalized thermal conductance G (T ) / Gvac (T ) , as shown in figure 1. Here, we can identify an intrinsic temperature scale of the structure T0 = hc / (
Data Loading...
