OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 13 — Jun. 18, 2012
  • pp: 13847–13856
« Show journal navigation

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

Yiling Yu and Linyou Cao  »View Author Affiliations


Optics Express, Vol. 20, Issue 13, pp. 13847-13856 (2012)
http://dx.doi.org/10.1364/OE.20.013847


View Full Text Article

Acrobat PDF (1728 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Light-matter interactions at sub-wavelength structures exhibit significant resonances. [1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

, 2

2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).

] One interesting application of the resonance is to enhance light absorption. Recent works have demonstrated strong enhancements in the absorption by leveraging on the resonant light-matter interaction at nanostructures [3

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

, 4

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

].This absorption enhancement is critical for the development of many high-performance absorption-based photonic devices, such as photo detectors [5

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]

], electro-optic modulators [6

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

], and solar cells [3

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

, 7

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]

]. In order to explore the full potential of nanostructures for absorption enhancement, it is very important to obtain better physical insights into the resonant light absorption.

Here we demonstrate an intuitive, simple theoretical model to analyze the resonant light absorption of semiconductor nanostructures, including two-dimensional (2D) planar films, one-dimensional (1D) nanowires, and zero-dimensional (0D) nanoparticles. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure, hence termed as coupled leaky mode theory (CLMT). In stark contrast to conventional methods (i.e., Mie theory [1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

], finite difference time domain (FDTD) [8

8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

]) that request specific physical features of the nanostructure for calculating the light absorption, this CLMT model provides an unprecedented capability to evaluate light absorption using eigenvalues of the leaky modes. Significantly, the eigenvalue only shows very mild dependence on the physical features of nanostructures. As a result, we can generally apply one set of eigenvalues calculated using a real and constant refractive index to calculations for the absorption of nanostructures with different sizes, different semiconductor materials, and wavelength-dependent complex refractive index. This CLMT model presents a general, simple, yet reasonably accurate method to analyze the light absorption of semiconductor nanostructures. It 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.

2. General existence of leaky mode resonances in semiconductor nanostructures

Solving Eqs. (1)-(6) gives complex values for a normalized parameter nkr (nkr = Nreal-Nimagi). These complex values are eigenvalues of the leaky modes. Table 1

Table 1. Eigenvalue of leaky modes in nanostructures

table-icon
View This Table
lists the solution for typical leaky modes calculated using a constant refractive index of 4, i.e. n = 4. The real part of the eigenvalueNreal indicates the condition for leaky mode resonances (LMRs). For instance, we can expect to observe TM11 leaky mode resonance in 1D wires when the condition of nkr = 2.30 is satisfied. The imaginary part Nimag 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.

The leaky mode of 2D planar films can be labeled with a mode number of m, as TEMm. 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 TMml or TEml [1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

, 2

2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).

].

The eigenvalue of leaky modes shows interesting dependence on the subscript numbers of m and l as well as the refractive index n. For all the 2D, 1D and 0D structures, the real part of the eigenvalue (Nreal) is always linearly dependent on the mode number m (upper panels in Fig. 1
Fig. 1 Eigenvalues of leaky modes in 2D (left), 1D (middle), and 0D (right) structures. (a-c) the real part of the eigenvalue, Nreal, is plotted as a function of the mode number m with different refractive indexes n = 2 (red curve), 3(black curve), and 4(blue curve). In the result for the 2D film, the calculations for different refractive index perfectly overlap each other. (d-f) the imaginary part of the eigenvalue, Nimag, is plotted as a function of the mode number m with different refractive index n = 2 (red curve), 3(black curve), and 4(blue curve). For 1D and 0D structures, Nimag is plotted in log scale for visual convenience, and results for only one polarization (TM for 1D, and TE for 0D) are given.
) and the order number l (not shown). Additionally, for a given leaky mode, the real part Nrealis essentially independent of the refractive index of the material (upper panels of Fig. 1). For instance, the Nreal of TEMm leaky modes in 2D structures is always equal to an integer number m of π, Nreal = mπ, 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 (Nimag) of the eigenvalue shows substantial dependence on both the subscript numbers and the refractive index. Interestingly, Nimag 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, Nimag 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

The framework of LMRs indicates that the light absorption of nanostructures is governed by the resonance of incident light with leaky modes of the nanostructure. As a further step, we develop a model of coupled leaky mode theory (CLMT) to quantitatively evaluate the light absorption from the perspective of LMRs. In contrast to typical rigorous methods (i.e., Mie theory, FDTD) for analyzing the light absorption that requests pecitic physical features of nanostructures, this CLMT model provides a capability of analyzing the light absorption only using eigenvalues of the leaky modes.

Substituting the expressions of |Wi| into Eq. (11), we can find out the light absorption cross section Cabs using Cabs = Pabs/I0. Subsequently, we can derive the absorption efficiency Qabs as Qabs = Cabs/G.G is the geometrical cross section of nanostructures, which is unity, 2r and πr2 for 2D, 1D and 0D structures, respectively. Expressions for the absorption efficiency Qabs can be written as
2DfilmQabs=2Nimag/Nreal.nimag/nreal(nrealkr/Nreal1)2+(Nimag/Nreal+nimag/nreal)2
(15)
1DwireQabs=2krNimag/Nreal.nimag/nreal(nrealkr/Nreal1)2+(Nimag/Nreal+nimag/nreal)2
(16)
0DparticleQabs=(2m+1)2(kr)2Nimag/Nrealnimag/nreal(nrealkr/Nreal1)2+(Nimag/Nreal+nimag/nreal)2
(17)
Equations (15)-(17) indicate that, for a given material (fixed nreal and nimag), the absorption of nanostructures for a given frequency ω is dictated by the eigenvalue of leaky modes, Nreal and Nimag. 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 Qabs,ml of each leaky mode to get the total absorption efficiency QabsT,
QabsT=mlQabs,ml
(18)
To minimize the interference between different leaky modes, the absorption efficiency for a single mode Qabs,mlneeds be corrected from the value calculated using Eqs. (15-17) by Qabs,ml=Qabs/[1+2(nrealkrNreal)2]. This essentially limits every leaky mode can only interact with incident wavelengths at the proximity of each resonant wavelength.

Equation (18) can nicely reproduce the absorption efficiency calculated using conventional rigorous methods, such as Lorentz-Mie theory for 1D and 0D structures [1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

]. Figure 3
Fig. 3 Calculated absorption spectra of 2D (left), 1D (middle), and 0D (right) silicon nanostructures under normal illumination of a plane wave using conventional analytical methods and the CLMT model. The analytical method for the 2D structure is transfer matrix, and Mie theory for the 1D and the 0D structures. For the 1D and 0D structures, calculations for only one polarization (TM for 1D, and TE for 0D) are given. The thickness of the 2D film is 100 nm, and the radii of the 1D wire and the 0D particle both are 100 nm.
shows calculated absorption spectra of 2D (left panel), 1D (middle panel) and 0D (right panel) of silicon nanostructures using the conventional analytical methods (blue curve) and the CLMT model (red curve). Again, without losing generality, the thickness of the 2D thin film is set 100 nm, and the radii of the 1D wire and the 0D particle both are 100 nm. For simplicity, we only calculate the absorption for normal incidence of plane waves with linear polarization (TM polarization for 1D wire, and TE polarization for 0D particle). In the calculations using the CLMT model, we use the eigenvalues of leaky modes listed in Table 1, which are substituted into Eq. (18) along with the intrinsic refractive index of silicon. We can find from Fig. 3 that the CLMT calculations for all the structures are reasonably consistent with the rigorous calculations.

Notably, while the refractive index of semiconductor materials typically shows substantial wavelength dependence, the CLMT model can reproduce the rigorous solution using the eigenvalue of leaky modes that are calculated with a constant refractive index. For instance, the CLMT calculations shown in Fig. 3 use the eigenvalues of leaky modes (Nreal and Nimag) 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

13. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

].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 (nimag, nreal are fixed) is essentially dictated by Nimag and Nreal. We have demonstrated in Fig. 2 that Nreal of a given leaky mode is approximately independent of the refractive index of materials. This leaves Nimag 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 Nimag of these leaky modes show moderate variation for different refractive indexes. For instance, Nimag of the TM11 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 Nimag can causes only minor change in the overall absorption efficiency.

The CLMT model can generally apply to 2D, 1D, and 0D nanostructures of all kinds of semiconductor materials. Most semiconductor materials have a refractive index in the range of 3~5. We find that the eigenvalue of leaky modes calculated with a constant refractive index of 4 can reasonably reproduce the spectra absorption of a wide range of semiconductor nanostructures. Figure 4
Fig. 4 Calculated absorptions for 100-nm-radius nanowires of different materials under normal illumination of a TM-polarized plane wave with Mie theory (blue) and the CLMT model we propose (red). The materials of the nanowire are indicated in the corresponding panel. Most of the CLMT calculations use the eigenvalue of leaky modes for a constant refractive index of 4. For a-Si and CuInGaSe, the CLMT calculations with n = 3 are also given as black dashed curves.
show the calculated absorption spectral of 1D nanostructures of a variety of materials, including amorphous silicon (a-Si), gallium arsenide (GaAs), germanium (Ge), and copper indium gallium selenide (CuInGaSe), using rigorous analytical methods (blue curve) and the CLMT model (red curve). The CLMT calculations use the eigenvalues of leaky modes for a constant refractive index of 4 (listed in Table 1). To illustrate the robustness of the CLMT model, the CLMT calculations using the eigenvalues of leaky modes for a constant refractive index of 3 are also given for a-Si and CuInGaSe (black dashed line). We can find that the CLMT calculations generally show reasonable consistence with the results calculated from rigorous analytical methods. For a-Si, both CLMT calculations with n = 4 (red solid curve) and n = 3 (black dashed line) are reasonable approximations for the Mie calculation. However, for CuInGaSe, the CLMT calculation with n = 3 shows a better approximation for the Mie calculation than the one with n = 4. This is because the real part of the refractive index of CuInGaSe is close to 3 across the whole spectrum.

4. Conclusion

We demonstrate anew theoretical model, coupled leaky mode theory (CLMT), to analyze the light absorption in 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. The CLMT model provides a capability of evaluating the light absorption of nanostructures using the eigenvalue of leaky modes, instead of specific physical features of the nanostructure as conventional analytical methods. Significantly, the eigenvalue only shows mild dependence on the physical features of nanostructures. As a result, we can generally apply one set of eigenvalues calculated using a real and constant refractive index to calculations for the absorption of nanostructures with different sizes, different semiconductor materials, and wavelength-dependent complex refractive index. This CLMT model provides a general, simple and reasonably accurate approach for the analysis of light absorption in nanostructures as an alternative to existing methods that are typically computation intensive.

More importantly, the CLMT provides new physical insights into the light absorption that cannot be obtained from existing analytical methods. It reveals that the light absorption of nanostructures is determined by the coupling between incident light and leaky modes of the structures. This insight opens a new door for the development of high-performance absorption-based photonic devices. For instance, we can see from Eqs. (15)-(17) that, upon resonances (nrealkr/Nreal-1 = 0), the absorption efficiency is dictated by (Nimag/Nreal).(nimag/nreal)/(Nimag/Nreal + nimag/nreal)2. This absorption can be maximized by tuning the radiative quality factor qrad = Nreal/(2*Nimag)of leaky modes equal to the absorption quality factor qabs = nreal/(2*nimag). 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/(2kr), 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 qrad comparable to the intrisinc absorption quality factor qabs of the materials at this wavelength.

Acknowledgments

This work has been supported by start-up fund from North Carolina State University. L.C. acknowledges the Ralph E. Powe Junior Faculty Enhancement Award.

References and links

1.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2.

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).

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]

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]

8.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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. 8(8), 643–647 (2009). [CrossRef] [PubMed]

10.

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]

11.

A. W. Snyder, Optical Waveguide Theory (Springer, Berlin, 1983).

12.

U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice Hall, 2000).

13.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

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 75(5), 053801 (2007). [CrossRef]

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 114(16), 7324–7329 (2010). [CrossRef]

16.

H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).
  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. Photonics5(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. Photonics4(8), 518–526 (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]
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  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.8(8), 643–647 (2009). [CrossRef] [PubMed]
  10. 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]
  11. A. W. Snyder, Optical Waveguide Theory (Springer, Berlin, 1983).
  12. U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice Hall, 2000).
  13. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
  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. A75(5), 053801 (2007). [CrossRef]
  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. C114(16), 7324–7329 (2010). [CrossRef]
  16. H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited