Computational analysis of cooling dynamics in photonic-crystal-based thermal switches
- Principia Dardano^{1},
- Massimo Borrelli^{2},
- Marilena Musto^{2},
- Giuseppe Rotondo^{2} and
- Mario Iodice^{1}Email author
https://doi.org/10.1186/s41476-016-0001-0
© The Author(s) 2016
Received: 9 November 2015
Accepted: 18 April 2016
Published: 23 June 2016
Abstract
The paper presents optical and thermal simulations of two kinds of thermally controlled silicon Photonic Crystal devices: an air hole in silicon slab that switches between two refractive behaviors, and a T-shaped circuit made by silicon rods in air that has an on-off behavior. Both effects have been obtained by increasing the device temperature, then exploiting the thermo-optic effect. Theoretical models of thermal dynamics, in particular for natural convective heat exchange on micrometric and sub-micrometric scale, have been validated by means of numerical simulations.
Keywords
Photonic crystals Thermo-optic effect Scaling of thermal parameters Photonic switchesBackground
The control of light propagation in a Photonic Crystal (PhC) slab can occur through the modulation of several parameters [1]. Indeed, both structural parameters (as refractive index of constituent materials, array geometry, etc.) and light parameters (as incident angle, polarization status and wavelength) influence the light propagation inside the PhCs [1–3]. Fixed the light parameters, the refractive index varies with the temperature, according to the thermo-optic effect [4, 5]. Therefore, it is possible to control light propagation in a PhC by means of temperature variations. This paper presents theoretical results about two different thermally modulated PhC devices: a PhC slab based on air holes in silicon that exploits the negative refraction behavior [6, 7] and a T-shaped PhC circuit based on silicon rods in air that exploits the band gap properties [8]. Both devices have been simulated thermally and optically using commercial software, COMSOL multiphysics® and RSoft™, respectively. In particular, a rigorous numerical analysis of the cooling phase of these structures has been performed in order to determine the dynamic and the thermal time constant at micrometric and sub-micrometric characteristic length. In [9], the trend of the natural convective heat exchange coefficient h _{ c } has been obtained analytically from the relations between Nusselt (Nu), Reynolds (Re) and Prandtl (Pr) numbers at characteristic length in the range of (0.01÷1000) mm. In this range, the h _{ c } coefficient is inversely proportional to the characteristic lengths below 10 mm and almost constant for higher dimensions. In [10], an experimental validation at the same characteristic length has been reported. However, the interesting range of PhC-based devices was almost unexplored. In this paper, the cooling dynamics of sub-micrometric devices has been investigated using a thermal-fluid-dynamic approach. Results emphasize that the dominant contribution is due to the conductive mechanism, in contrast to the statements of J. Peirs et al. in [9]. Moreover, using thermo-optical effect and optical Finite Difference Time Domain (FDTD) simulations, the optical efficiency of two analyzed devices and the value of the switching bandwidth of both geometries have been calculated.
Methods
Changing temperature is a simple way to obtain a refractive index change and then a photonic modulation of silicon based devices. However, for macroscopic devices, the relatively slow heat diffusion in solids strongly limits their performances. In this frame, PhC devices are promising to overcome the problem, because they are constituted by an array of elements of micrometric or sub-micrometric dimensions. Before to simulate light propagation and calculate efficiency of two specific devices, two preliminary analyses have to be done. The first is needed to establish if and which thermal exchange mechanism is predominant in a generic pillar-based device. The second has been done to simulate both thermal and optical behavior for specific PhC structures.
Thermal analysis
- A
Conjugate heat transfer (conduction in solids, modeling the convective flow of air surrounding the pillar to describe the thermal dissipation);
- B
Conduction in solids and in fluid (neglecting convection mechanism, considering air as a stationary fluid);
- C
Conduction in solids and convection in fluid (using heat transfer coefficients h _{ c } at surface to describe the thermal dissipation in fluid);
- D
Conduction in solids and radiation (neglecting convection in fluid).
The first approach solves the total energy balance and the flow equations of the outside cooling air. In this case, a Computational Fluid Dynamics (CFD) analysis has been performed, producing detailed results for the flow field around the pillar, the temperature distribution and the cooling power. However, this approach is more complex and requires more computational resources than the others, but it can be considered as a rigorous model for the actual heat dissipation mechanism. The second approach assumes that heat flux from solid to air is not enough large to induce fluid density changes and consequently laminar convection cells initiation; in this frame, air is considered as a still medium where heat is dissipated only by conduction. The third approach describes the outside heat flux by means of a heat transfer coefficient function, that is automatically calculated by the software as function of surface area, geometry and orientation. This assumption is generally valid for macroscopic objects and has been numerically verified in this work for micrometric and sub-micrometric structures. The fourth approach is carried out with the aim to evaluate the relative influence of the radiative heat exchange, compared to the other mechanisms.
dimensions of PhC elementary cell
Object | Radius [μm] | Heighy [μm] | Material |
---|---|---|---|
Pillar | h _{ p }/5 | h _{ p } | Silicon |
Buffer layer | h _{ p }/2 | h _{ p }/5 | Silicon dioxide |
Substrate | h _{ p }/2 | h _{ p }/2 | Silicon |
Air volume | 20 · h _{ p } | 20 · h _{ p } | Air |
Thermal properties of materials at 20 °C
Density ρ [kg m^{-3}] | Specific heat c _{ p } [J g^{-1} K^{-1}] | Thermal conductivity k [W m^{-1} K^{-1}] | |
---|---|---|---|
Air | 1.205 | 1.005 | 0.0257 |
Silicon | 2329 | 0.712 | 149 |
Silicon dioxide | 2200 | 0.73 | 1.4 |
Mathematical model and boundary conditions
Boundary conditions at pillar/air interface
Case B | Case C | Case D | |
---|---|---|---|
r = h _{ p } /5 | \( -\left({k}_p\frac{\partial T}{\partial r}\right)=\left({k}_a\frac{\partial T}{\partial r}\right) \) | \( - \left(k\frac{\partial T}{\partial r}\right)=h\left({T}_p-{T}_a\right) \) | \( -\left({k}_p\frac{\partial T}{\partial r}\right)=\sigma \varepsilon \left({T}_p^4-{T}_a^4\right) \) |
0 ≤ φ ≤ 2π | |||
0 ≤ z ≤ h _{ p } |
FEA analysis results
First of all, Fig. 3 shows that the time required for pillar cooling, when only radiative heat exchange mechanism is considered (case D), is about one hundred times longer than other three cases. Hence, for the considered geometry and temperature, the radiative contribution to the cooling dynamics can be neglected. Moreover, the A and B curves overlap, at least when the pillar temperature is below 380 K; this implies that modeling air as a stationary medium and heat dissipation only by means of conduction is a good approximation of the real behavior. Instead, the C curve shows an evident delay compared to the A curve. This delay confirms the hypothesis that a laminar convective heat exchange doesn’t occur for the considered geometry and temperature. The well-known theory of convection, which well describes the heat exchange from hot macroscopic objects in air, underestimates the real cooling rate in the micron and sub-micron dimensional range. In fact, for sub-micron-scaled devices and relatively small thermal gradient, recent studies report that conduction is the dominant mechanism of heat transfer [12], even with surrounding air considered as static fluid. The presented fully coupled thermo-fluid-dynamic FEA of a single silicon rod immersed in air (case A) confirms the preliminary results presented in [12]: for sub-micron range, air can be considered as a static medium where the heat is dissipated principally by conduction. Moreover, the simple scaling of classical convective coefficient produces incorrect results, underestimating the efficiency of the heat exchange. From these considerations it can be concluded that the model B is an enough accurate approximation of the real heat exchange mechanism which is rigorously described by the fully coupled thermo-fluidic model A. Therefore, in order to reduce the computational time without significantly compromising the accuracy of the simulation, for the thermal analysis performed on such sub-micrometric structures, we will adopt the model B, neglecting natural convective and radiative heat exchange mechanisms.
Band structure analysis
Results and Discussions
In this case, using the condition of predominant conduction in the heat transfer, simulations show (Fig. 7c) that the second structure has smaller thermal transient time than the first (about 2 μs versus 10 μs), in despite of a greater required temperature increase.
Conclusions
PhC devices can be controlled by changing their temperature: it is not particularly difficult to control it in an extremely small region (few tens of μm^{2}). The optical efficiency of two analyzed devices is up to 80 % and the value of the bandwidth of both devices is in the range of hundreds of kHz.
Declarations
Acknowledgements
The research activity is supported by Italian Ministry of Education, University and Research (MIUR) through the AQUASYSTEM project, in the framework of the National Operational Program – PON.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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