## Thermo-optic characteristics and switching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform |

Optics Express, Vol. 20, Issue 4, pp. 4225-4231 (2012)

http://dx.doi.org/10.1364/OE.20.004225

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### Abstract

Employing a semi-analytic approach, we study the influence of key structural and optical parameters on the thermo-optic characteristics of photonic crystal waveguide (PCW) structures on a silicon-on-insulator (SOI) platform. The power consumption and spatial temperature profile of such structures are given as explicit functions of various structural, thermal and optical parameters, offering physical insight not available in finite-element simulations. Agreement with finite-element simulations and experiments is demonstrated. Thermal enhancement of the air-bridge structure is analyzed. The practical limit of thermo-optic switching power in slow light PCWs is discussed, and the scaling with key parameters is analyzed. Optical switching with sub-milliwatt power is shown viable.

© 2012 OSA

## 1. Introduction

1. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1678–1687 (2006). [CrossRef]

2. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. **24**(12), 4600–4615 (2006). [CrossRef]

3. G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys. **93**(9), 4955–4978 (2003). [CrossRef]

4. M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. **3**(4), 211–219 (2004). [CrossRef] [PubMed]

5. E. A. Camargo, H. M. H. Chong, and R. M. De La Rue, “2D Photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure,” Opt. Express **12**(4), 588–592 (2004). [CrossRef] [PubMed]

10. Y. Cui, K. Liu, D. L. MacFarlane, and J.-B. Lee, “Thermo-optically tunable silicon photonic crystal light modulator,” Opt. Lett. **35**(21), 3613–3615 (2010). [CrossRef] [PubMed]

11. L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. **90**(7), 071105 (2007). [CrossRef]

13. J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. **92**(10), 103114 (2008). [CrossRef]

*κ*for a silicon photonic crystal slab is determined through the lateral thermal spreading length. Physical properties such as the spatial temperature profile and the power consumption required to induce a

_{eff}*π*phase shift can be described semi-analytically based on a quasi-1D model with numerically determined

*κ*. The results agree well with 3D simulations based on the finite element method (FEM). The theoretical results also explain the low switching power observed in an air-bridge structure [6

_{eff}6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**(7064), 65–69 (2005). [CrossRef] [PubMed]

## 2. Analysis of SOI photonic crystal thermo-optic structures

*W*and length

*L*is assumed to be embedded in the top silicon layer. Such a heat source can be formed by a lightly doped (e.g. ~10

^{14}cm

^{−3}) Si strip surrounded by a relatively highly doped (e.g. ~10

^{17}cm

^{−3}) silicon on both sides [6

6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**(7064), 65–69 (2005). [CrossRef] [PubMed]

*κ*. This is valid because the temperature varies spatially on a scale much larger than the typical photonic crystal lattice constant

_{eff}*a*. To determine

*κ*, the heat transfer process is simulated using the finite element method for one period of the PCW structure, as shown in the inset of Fig. 2 . The thicknesses of the top Si layer and buried oxide layer are

_{eff}*t*= 250 nm and

_{Si}*t*= 2 μm respectively. The hexagonal lattice has a lattice constant

_{ox}*a*= 400nm. The simulations indicate that the vertical temperature variation in the top Si layer and the in-plane temperature variation in each unit cell are small. The temperature of the top Si layer varies significantly only along the

*x*axis, as plotted in Fig. 2. Outside the heater (centered at

*x*= 0), it closely follows an exponential formwhere

*X*(

_{spr}*r*) is the thermal spreading length. For an unpatterned SOI structure, it is given by [14

14. V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express **13**(9), 3429–3437 (2005). [CrossRef] [PubMed]

15. L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. **41**(1), 69–75 (1994). [CrossRef]

*κ*and

_{Si}*κ*are the thermal conductivities of silicon and SiO

_{ox}_{2}respectively (values from Ref [14

14. V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express **13**(9), 3429–3437 (2005). [CrossRef] [PubMed]

*X*(

_{spr}*r*) depends on the hole radius

*r*and it can be obtained from an exponential fit of the lateral temperature profile in the slab. The effective thermal conductivity of a Si photonic crystal slab can then be calculated from

*κ*/

_{eff}*κ*and

_{Si}*X*determined from the plots are given in Table 1 . To further verify the results, homogenized slab structures with the tabulated

_{spr}*κ*

_{eff}(

*r*) are simulated, with all other parameters unchanged. The lateral temperature profile in the homogenized slab is generally in good agreement (within 6%) with that of the original photonic crystal slab, as shown in Fig. 2.

14. V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express **13**(9), 3429–3437 (2005). [CrossRef] [PubMed]

15. L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. **41**(1), 69–75 (1994). [CrossRef]

*A*[

_{eff}= L*W + 2X*]. For the photonic crystal structure in Fig. 1(a), this model yieldswhere

_{spr}*Q*is the heat transfer rate (equal to the heating power in steady state) and Δ

*T*the temperature difference between the top and bottom of the oxide at

_{ox}*x*= 0. To verify Eq. (4), 3D FEM steady-state simulations are performed for an SOI chip having a homogenized top layer with

*κ*(Fig. 3 inset). The absence of small holes significantly mitigates the difficulty in mesh generation for multi-scale structures, and reduces the simulation time significantly.

_{eff}7. M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express **13**(18), 7174–7188 (2005). [CrossRef] [PubMed]

17. M. Iodice, G. Mazzi, and L. Sirleto, “Thermo-optical static and dynamic analysis of a digital optical switch based on amorphous silicon waveguide,” Opt. Express **14**(12), 5266–5278 (2006). [CrossRef] [PubMed]

*T*per unit heating power

_{ox}*Q*and the results based on Eq. (4) agree well (within 3%), as shown in Fig. 3 for various lengths of the heat source.

## 3. Thermo-optic characteristics and switching power for SOI and air-bridge structures

4. M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. **3**(4), 211–219 (2004). [CrossRef] [PubMed]

*n*is the group index of the mode,

_{g}*λ*the wavelength, and

*σ*the fraction of the mode energy stored in the region where the refractive index change

*∆n =*(

*dn/dT*)

*∆T*occurs. By virtue of Eqs. (4) and (5), the power required to induce a phase shift of

*π*for a structure in Fig. 1(a) is given by

*Q*actually scales as

_{π}*W*<<

*X*. Figure 4 shows the results for

_{spr}*σ =*0.9,

*λ =*1.55μm and

*dn/dT*= 1.86 × 10

^{−4}K

^{−1}with different values of oxide layer thickness. For

*n*= 60,

_{g}*r*/

*a*= 0.25 and

*t*= 2μm,

_{ox}*Q*is less than 2.5mW.

_{π}*W*is the membrane width,

_{membrane}*X*is given by Eq. (2), (Δ

_{Si}*T*)

*is the membrane temperature rise evaluated at the PCW core and (*

_{membrane}*∆T*)

*at the membrane edge. Eliminating (*

_{edge}*∆T*)

*, we find*

_{edge}*Q*, the membrane structure may enhance the temperature rise by a factor

*Q*of the membrane structure is reduced by this factor. The enhancement factors obtained from Eq. (9) agree very well (within 6%) with the simulation results, as shown in Fig. 4 inset. Based on Fig. 4, the attainable power consumption for a Si air-bridge PCW thermo-optic Mach-Zehnder switch is estimated between 1~2mW for

_{π}*n*~60 and

_{g}*t*= 2μm, which agrees well with the experimental result [6

_{ox}6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**(7064), 65–69 (2005). [CrossRef] [PubMed]

## 4. Discussions

**438**(7064), 65–69 (2005). [CrossRef] [PubMed]

8. L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett. **19**(5), 342–344 (2007). [CrossRef]

*x*= 6μm from the core has an efficiency reduction by exp(−6μm/

*X*)≈0.3~0.4 for

_{spr}*r*/

*a*= 0.25~0.35. The buried oxide thickness is another crucial factor. Generally, a thicker oxide is preferred for lower power consumption according to Eqs. (6) and (8). However, the thermal time constant of an SOI chip increases with the oxide thickness. Therefore, some trade-off must be made in realistic device design to balance power consumption and speed. For the membrane structure, the enhancement factor in Eq. (9) is found to weaken the scaling of

*Q*with

_{π}*t*due to

_{ox}*X*~

_{Si}*Q*scales slower than

_{π}*W*. Ultimately, the reduction of

_{membrane}*Q*based on the slow light effect is limited by optical loss, which increases with

_{π}*n*. The optical loss of a PCW can be attributed to a number of factors, such as random variation of hole positions due to fabrication tolerances, sidewall roughness, and the input/output coupling. The random variation of the hole positions in fabricated PCWs can be controlled to be within a small range (<1nm) with high-end e-beam lithography tools [18

_{g}18. E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B **72**(11), 115102 (2005). [CrossRef]

*n*. The estimated PCW length for 3dB propagation loss is plotted against

_{g}*n*in Fig. 4 based on theoretical calculations with experimentally achievable rms roughness

_{g}*σ*= 3nm and correlation length

*l*= 40nm [19

_{c}19. W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B **82**(23), 235306 (2010). [CrossRef]

**438**(7064), 65–69 (2005). [CrossRef] [PubMed]

20. C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. **97**(18), 183302 (2010). [CrossRef]

**438**(7064), 65–69 (2005). [CrossRef] [PubMed]

*n*~110 for

_{g}*L*= 50μm and 250μm and shows weak dependence on the PCW lengths. This indicates that most of the observed loss is due to input/output coupling [6

**438**(7064), 65–69 (2005). [CrossRef] [PubMed]

20. C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. **97**(18), 183302 (2010). [CrossRef]

19. W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B **82**(23), 235306 (2010). [CrossRef]

*n*

_{g}^{2}) will be a primary limiting factor. Hence the roughness-induced loss (including backscattering and out-of-plane scattering loss) is considered in Fig. 4 to explore the limit of

*Q*in connection with

_{π}*n*. Considering all the factors discussed above, a practical lower limit of

_{g}*Q*is estimated on the order of 0.5mW for a reasonable

_{π}*t*~5μm,

_{ox}*L*~10μm, and

*n*~110. Our calculation also shows that for

_{g}*n*~60,

_{g}*Q*already enters the sub-milliwatt regime for the

_{π}*t*~5μm case.

_{ox}*Q*is insensitive to the choice of the heater width

_{π}*W*as long as

*W*<<2

*X*(~12μm). Also,

_{spr}*Q*varies only ~20% for the typical radius range of

_{π}*r*/

*a*= 0.25~0.35. Note that typical silicon photonic crystal waveguides used for the 1550nm communications window have

*a*= 380nm to 440nm. As the lattice constant is much smaller than the scale of temperature variation (

*a*<<

*X*), this approach works well for this range of

_{spr}*a*. For a given lattice structure, when

*a*and

*r*vary simultaneously while maintaining a fixed ratio of

*r/a*,

*X*is essentially invariant. Note that the power

_{spr}*Q*given above is for switching and steadily holding a state. This is pertinent for most optical switching applications that require holding a switching state steadily over an extended period. The thermal time constant of an SOI structure is

_{π}*t*= 1~2μm), where

_{ox}*ρ*is the density and

_{ox}*c*the specific heat capacity of SiO

_{ox}_{2}. Our simulations confirm that

*τ*is relatively insensitive to the details of a photonic crystal structure. Although the heating transient can be shortened [7

7. M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express **13**(18), 7174–7188 (2005). [CrossRef] [PubMed]

**13**(9), 3429–3437 (2005). [CrossRef] [PubMed]

*Q*and

_{π}*τ*given above. The effect of the temperature drop in the substrate is less than 10% for all cases we simulated. Note that

*κ*used in this work is obtained based on the structured “porosity” of materials within the framework of classical heat transfer theory, neglecting quantum mechanical effects such as phonon scattering in a periodic structure [21

_{eff}21. C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances **1**(4), 041403 (2011). [CrossRef]

*κ*values from quantum mechanical calculations will be used.

_{eff}## 5. Summary

*Q*and spatial temperature profile are given as explicit functions of structural, thermal, and optical parameters. The results agree well with FEM simulations and also explain the low switching power in air-bridge structures. The scaling of

_{π}*Q*with key physical parameters is analyzed. The practical limit of

_{π}*Q*is estimated on the sub-milliwatt level considering all key factors.

_{π}## Acknowledgments

## References and links

1. | R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. |

2. | B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. |

3. | G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys. |

4. | M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

5. | E. A. Camargo, H. M. H. Chong, and R. M. De La Rue, “2D Photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure,” Opt. Express |

6. | Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

7. | M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express |

8. | L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett. |

9. | D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. |

10. | Y. Cui, K. Liu, D. L. MacFarlane, and J.-B. Lee, “Thermo-optically tunable silicon photonic crystal light modulator,” Opt. Lett. |

11. | L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. |

12. | L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron. |

13. | J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. |

14. | V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express |

15. | L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. |

16. | Y. Jaluria, |

17. | M. Iodice, G. Mazzi, and L. Sirleto, “Thermo-optical static and dynamic analysis of a digital optical switch based on amorphous silicon waveguide,” Opt. Express |

18. | E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B |

19. | W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B |

20. | C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. |

21. | C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances |

**OCIS Codes**

(130.4815) Integrated optics : Optical switching devices

(130.5296) Integrated optics : Photonic crystal waveguides

(230.5298) Optical devices : Photonic crystals

(130.4110) Integrated optics : Modulators

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 15, 2011

Revised Manuscript: January 26, 2012

Manuscript Accepted: January 27, 2012

Published: February 6, 2012

**Citation**

Manjit Chahal, George K. Celler, Yogesh Jaluria, and Wei Jiang, "Thermo-optic characteristics and switching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform," Opt. Express **20**, 4225-4231 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4225

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### References

- R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006). [CrossRef]
- B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol.24(12), 4600–4615 (2006). [CrossRef]
- G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys.93(9), 4955–4978 (2003). [CrossRef]
- M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3(4), 211–219 (2004). [CrossRef] [PubMed]
- E. A. Camargo, H. M. H. Chong, and R. M. De La Rue, “2D Photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure,” Opt. Express12(4), 588–592 (2004). [CrossRef] [PubMed]
- Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005). [CrossRef] [PubMed]
- M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express13(18), 7174–7188 (2005). [CrossRef] [PubMed]
- L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007). [CrossRef]
- D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett.33(2), 147–149 (2008). [CrossRef] [PubMed]
- Y. Cui, K. Liu, D. L. MacFarlane, and J.-B. Lee, “Thermo-optically tunable silicon photonic crystal light modulator,” Opt. Lett.35(21), 3613–3615 (2010). [CrossRef] [PubMed]
- L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007). [CrossRef]
- L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008). [CrossRef]
- J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett.92(10), 103114 (2008). [CrossRef]
- V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express13(9), 3429–3437 (2005). [CrossRef] [PubMed]
- L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev.41(1), 69–75 (1994). [CrossRef]
- Y. Jaluria, Natural Convection Heat and Mass Transfer (Pergamon Press, Oxford, UK, 1980).
- M. Iodice, G. Mazzi, and L. Sirleto, “Thermo-optical static and dynamic analysis of a digital optical switch based on amorphous silicon waveguide,” Opt. Express14(12), 5266–5278 (2006). [CrossRef] [PubMed]
- E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B72(11), 115102 (2005). [CrossRef]
- W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B82(23), 235306 (2010). [CrossRef]
- C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett.97(18), 183302 (2010). [CrossRef]
- C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances1(4), 041403 (2011). [CrossRef]

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