## Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures |

Optics Express, Vol. 20, Issue 13, pp. 13847-13856 (2012)

http://dx.doi.org/10.1364/OE.20.013847

Acrobat PDF (1728 KB)

### Abstract

We present an intuitive, simple theoretical model, coupled leaky mode theory (CLMT), to analyze the light absorption of 2D, 1D, and 0D semiconductor nanostructures. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure. Unlike conventional methods such as Mie theory that requests specific physical features of nanostructures to evaluate the absorption, the CLMT model provides an unprecedented capability to analyze the absorption using eigen values of the leaky modes. Because the eigenvalue shows very mild dependence on the physical features of nanostructures, we can generally apply one set of eigenvalues calculated using a real, constant refractive index to calculations for the absorption of various nanostructures with different sizes, different materials, and wavelength-dependent complex refractive index. This CLMT model is general, simple, yet reasonably accurate, and offers new intuitive physical insights that the light absorption of nanostructures is governed by the coupling efficiency between incident light and leaky modes of the structure.

© 2012 OSA

## 1. Introduction

3. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. **9**(3), 205–213 (2010). [CrossRef]

4. L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics **5**(2), 83–90 (2011). [CrossRef]

5. L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. **10**(4), 1229–1233 (2010). [CrossRef] [PubMed]

6. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics **4**(8), 518–526 (2010). [CrossRef]

3. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. **9**(3), 205–213 (2010). [CrossRef]

7. L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. **10**(2), 439–445 (2010). [CrossRef] [PubMed]

## 2. General existence of leaky mode resonances in semiconductor nanostructures

*nkr*(

*nkr*=

*N*

_{real}-

*N*

_{imag}

*i*). These complex values are eigenvalues of the leaky modes. Table 1 lists the solution for typical leaky modes calculated using a constant refractive index of 4, i.e.

*n*= 4. The real part of the eigenvalue

*N*

_{real}indicates the condition for leaky mode resonances (LMRs). For instance, we can expect to observe TM

_{11}leaky mode resonance in 1D wires when the condition of

*nkr*= 2.30 is satisfied. The imaginary part

*N*

_{imag }refers to the radiative leakage of the electromagnetic energy stored in leaky modes. For materials without intrinsic absorption loss, this imaginary part indicates spectral width of the leaky mode resonance.

*m*, as TEM

_{m}. The mode number

*m*corresponds to the number of half wavelength in the transverse direction of the planar film. Leaky modes in 1D wires and 0D particle scan be characterized by an azimuthal mode number,

*m*, and a radial order number,

*l*. Physically, the azimuthal mode number

*m*indicates the number of effective wavelength around the circumference of the structure, while the radial order number

*l*describes the number of radial field maxima within the structure. As a result, the modes can be termed as TM

_{ml}or TE

_{ml}[1, 2].

*m*and

*l*as well as the refractive index

*n*. For all the 2D, 1D and 0D structures, the real part of the eigenvalue (

*N*

_{real}) is always linearly dependent on the mode number

*m*(upper panels in Fig. 1 ) and the order number

*l*(not shown). Additionally, for a given leaky mode, the real part

*N*

_{real}is essentially independent of the refractive index of the material (upper panels of Fig. 1). For instance, the

*N*

_{real }of TEM

_{m}leaky modes in 2D structures is always equal to an integer number

*m*of

*N*

_{real}=

*m*π ,

*m*= 1, 2, 3 ……regardless the refractive index (the upper left panel of Fig. 1). This means that, no matter whatever the refractive index is, given LMRs always happen at fixed values of

*nkr*. Changing the refractive index

*n*can cause the value of

*kr*for LMRs shift to keep the value of

*nkr*invariant. In contrast, the imaginary part (

*N*

_{imag}) of the eigenvalue shows substantial dependence on both the subscript numbers and the refractive index. Interestingly,

*N*

_{imag}is constant for all leaky modes in the 2D structure, and increases with the refractive index increasing (the lower left panel of Fig. 1). This indicates identical radiative leakage for all the leaky modes, and the leakage is lower for materials with lower refractive index. In the 1D and0D structures,

*N*

_{imag}exponentially decreases with the mode number

*m*and the refractive index

*n*increasing (the lower middle and right panels of Fig. 1). This suggests stronger optical confinement at higher order modes and larger refractive index.

## 3. Coupled leaky mode theory

*C*

_{abs }using

*C*

_{abs}=

*P*

_{abs}/

*I*

_{0}. Subsequently, we can derive the absorption efficiency

*Q*

_{abs}as

*Q*

_{abs}=

*C*

_{abs}/

*G*.

*G*is the geometrical cross section of nanostructures, which is unity, 2

*r*and π

*r*

^{2}for 2D, 1D and 0D structures, respectively. Expressions for the absorption efficiency

*Q*

_{abs }can be written as

*n*

_{real}and

*n*

_{imag}), the absorption of nanostructures for a given frequency

*ω*is dictated by the eigenvalue of leaky modes,

*N*

_{real}and

*N*

_{imag}. These equations calculate the absorption efficiency contributed by one single leaky mode. For nanostructures that typically involve multiple leaky modes, we need sum up the absorption efficiency

*Q*of each leaky mode to get the total absorption efficiency

_{abs,ml }*Q*needs be corrected from the value calculated using Eqs. (15-17) by

_{abs,ml}*N*

_{real}and

*N*

_{imag}) calculated with a constant refractive index of 4.As a reference, the refractive index of silicon materials varies in a range of 4.6-3.5 in this same spectral range [13].We can get similar CLMT calculations by assuming the refractive index as other constants, for example, 3.5, or 5 when calculating the eigenvalue. This robustness of the CLMT calculation over the refractive index can be understood from Eqs. (15)-(17). The absorption of nanostructures of given materials (

*n*

_{imag},

*n*

_{real}are fixed) is essentially dictated by

*N*

_{imag}and

*N*

_{real}. We have demonstrated in Fig. 2 that

*N*

_{real }of a given leaky mode is approximately independent of the refractive index of materials. This leaves

*N*

_{imag }as the only variables that could vary with the refractive index. The leaky modes involved in the light absorption of nanostructures are typically low order modes (

*m*< 3 and

*l*< 3), and the

*N*

_{imag }of these leaky modes show moderate variation for different refractive indexes. For instance,

*N*

_{imag}of the TM

_{11}leaky mode in 1D wires can be found equal to 0.163 and 0.113 for a refractive index of 4 and 5, respectively. Such moderate variation in

*N*

_{imag }can causes only minor change in the overall absorption efficiency.

## 4. Conclusion

*n*

_{real}

*kr*/

*N*

_{real}-1 = 0), the absorption efficiency is dictated by (

*N*

_{imag}/

*N*

_{real}).(

*n*

_{imag}/

*n*

_{real})/(

*N*

_{imag}/

*N*

_{real}+

*n*

_{imag}/

*n*

_{real})

^{2}. This absorption can be maximized by tuning the radiative quality factor

*q*

_{rad}=

*N*

_{real}/(2*

*N*

_{imag})of leaky modes equal to the absorption quality factor

*q*

_{abs}=

*n*

_{real}/(2*

*n*

_{imag}). Regardless the intrinsic absorption of materials for specific incidence, properly nanostructuring the materials in nanostructures can always maximize the absorption efficiency to the same level as ½, 1/(2

*kr*), and (2m + 1)/[2(kr)

^{2}] for 2D, 1D and 0D structures, respectively. Therefore, to design high-performance nanostructure photodetectors for a specific wavelength, we need tune the wavelength close to a leaky mode resonance with a radiative quality factor

*q*

_{rad }comparable to the intrisinc absorption quality factor

*q*

_{abs}of the materials at this wavelength.

## Acknowledgments

## References and links

1. | C. F. Bohren and D. R. Huffman, |

2. | P. W. Barber and R. K. Chang, eds., |

3. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. |

4. | L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics |

5. | L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. |

6. | G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics |

7. | L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. |

8. | A. Taflove and S. C. Hagness, |

9. | L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. |

10. | L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. |

11. | A. W. Snyder, |

12. | U. S. Inan and A. S. Inan, |

13. | E. D. Palik, |

14. | R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A |

15. | Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C |

16. | H. A. Haus, |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(290.4020) Scattering : Mie theory

(160.4236) Materials : Nanomaterials

**ToC Category:**

Scattering

**History**

Original Manuscript: May 1, 2012

Revised Manuscript: May 26, 2012

Manuscript Accepted: May 27, 2012

Published: June 7, 2012

**Citation**

Yiling Yu and Linyou Cao, "Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures," Opt. Express **20**, 13847-13856 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-13847

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### References

- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater.9(3), 205–213 (2010). [CrossRef]
- L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011). [CrossRef]
- L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett.10(4), 1229–1233 (2010). [CrossRef] [PubMed]
- G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics4(8), 518–526 (2010). [CrossRef]
- L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett.10(2), 439–445 (2010). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
- L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater.8(8), 643–647 (2009). [CrossRef] [PubMed]
- L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett.10(7), 2649–2654 (2010). [CrossRef] [PubMed]
- A. W. Snyder, Optical Waveguide Theory (Springer, Berlin, 1983).
- U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice Hall, 2000).
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
- R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A75(5), 053801 (2007). [CrossRef]
- Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C114(16), 7324–7329 (2010). [CrossRef]
- H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).

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