## Diamond based photonic crystal microcavities

Optics Express, Vol. 14, Issue 8, pp. 3556-3562 (2006)

http://dx.doi.org/10.1364/OE.14.003556

Acrobat PDF (266 KB)

### Abstract

Diamond based technologies offer a material platform for the implementation of qubits for quantum computing. The photonic crystal architecture provides the route for a scalable and controllable implementation of high quality factor (Q) nanocavities, operating in the strong coupling regime for cavity quantum electrodynamics. Here we compute the photonic band structures and quality factors of microcavities in photonic crystal slabs in diamond, and compare the results with those of the more commonly-used silicon platform. We find that, in spite of the lower index contrast, diamond based photonic crystal microcavities can exhibit quality factors of Q=3.0×10^{4}, sufficient for proof of principle demonstrations in the quantum regime.

© 2006 Optical Society of America

## 1. Introduction

1. B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science **300**, 1537 (2003). [CrossRef] [PubMed]

2. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

2. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

*(Q)*of these cavities are modest, from a few hundred to a few thousand, depending on the structure. However, the microcavity

*Q*can be optimized by modifying the geometry of the lattice surrounding the cavity [2–8

2. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

*et al*constructed a cavity of three missing air holes in a row of a silicon slab with hexagonal lattice structure [2

**425**, 944–947 (2003). [CrossRef] [PubMed]

*Q*=4.5×10

^{4}, ten times higher than the quality factor of the un-optimized structure. Furthermore shifting three holes in the row next to the cavity they achieved a quality factor of

*Q*=1×10

^{5}[6

6. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**, 1202–1214 (2005). [CrossRef] [PubMed]

*Q*PCS based cavities have many applications, e.g. ultra-small filters in Ref. [1

1. B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science **300**, 1537 (2003). [CrossRef] [PubMed]

3. H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, and J-S. Kim, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. **79**, 3032–3034 (2001). [CrossRef]

8. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E **65**, 016608 (2001). [CrossRef]

16. M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, “Solid-state quantum computing using spectral holes,” Phys. Rev. A **66**, 032301 (2002). [CrossRef]

*Q*-factor.

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. **73**, 013818 (2006). [CrossRef]

*point cavity*designs. In this paper we only present designs that yield the highest

*Q*s within a modal volume of order one cubic wavelength. We compare the

*Q*that can be obtained to that in silicon that has a refractive index n=3.4 that typical for semiconductors.

## 2. PC geometry and method

*n*=

*2.4*, than semiconductors, leading to weaker TIR guidance in the vertical direction and a cavity mode that is less well confined in the vertical direction. .

*R*, with

*a*the lattice constant. In Fig. 1(b), modifications to the holes surrounding the cavity are illustrated. Neighbouring holes above and below the cavity have a reduced radius

*r*. The air holes in the

*x*-direction are shifted outwards by a distance

*d*. This type of the cavity modification, but in a silicon PCS, was first described by Zhang

*et al*[7

7. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

*et al*[6

6. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**, 1202–1214 (2005). [CrossRef] [PubMed]

23. M. Qiu, “Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately,” Microw. … Opt. Techn. Lett. **45**, 381–385 (2005). [CrossRef]

*UdV*/max(

*U*) where

*U*=

*ε*|

*E*|

^{2}/2 is the electric energy density [7

7. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

**425**, 944–947 (2003). [CrossRef] [PubMed]

7. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

## 3. Results

### 3.1 PBG calculations

*Q*, can be separated into the in-plane value,

*Q*

_{∥}, and out-of-plane value,

*Q*

_{⊥}.

*Q*

_{∥}can, in principle, be made arbitrarily high by increasing the number of periods. As the vertical confinement is ruled by TIR, the out-of-plane factor is crucial in designing high quality factor cavities [4

4. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express **11**, 579–593 (2003). [CrossRef] [PubMed]

*E*

_{x}. These modes have different bands and band gaps and the vertical radiation losses can be significantly reduced by choosing modes of a specific parity [5

5. O. Painter and K. Srinivasan, “Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis,” Phys. Rev. B **68**, 035110 (2003). [CrossRef]

*Q*

_{⊥}, and wide gap tend to have high

*Q*

_{∥}[4

4. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express **11**, 579–593 (2003). [CrossRef] [PubMed]

24. R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, “Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab,” JOSA B **17**, 629–633 (2000). [CrossRef]

*h*=

*0.7a*and for diamond slab it is

*h*=

*0.91a*. The diamond slab has the same optical thickness as the silicon slab. It will be shown in Section 3.2 that

*h*=

*0.91a*is the optimal thickness for a PCS with holes radius

*R*=

*0.29a*. The hole radius is varied between

*R*=

*0.2a*and

*R*=

*0.4a*. Figure 2 includes results for both silicon (grey) and diamond (black) PCSs. Note that the widths of the first gaps for the two materials are comparable for any size of the air holes. However, the gap centre of diamond occurs for higher frequencies than for silicon due to the smaller refractive index. Consequently, the in-plane quality factors

*Q*

_{∥}for the two materials are similar, whereas the out-of-plane quality factor

*Q*

_{⊥}, which relies on TIR, for diamond is expected to be smaller. For both materials, reducing the hole size lowers the central frequency of the gap and therefore the vertical confinement improves, increasing

*Q*

_{⊥}. But at the same time, the gaps become narrower and therefore

*Q*

_{∥}decreases and the modal volume increases. Therefore this in-plane mode delocalization contributes to higher

*Q*

_{⊥}, as pointed out in Ref. [25

25. H-Y Ryu, M. Notomi, and Y-H Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett **83**, 4294–4296 (2003). [CrossRef]

*Q*

_{∥}, but the modal volume is essentially not affected by this. In choosing the hole size we therefore have a trade off: small (large) holes lead to large (small)

*Q*, but also to large (small) modal volumes. Results presented in Fig. 2 indicate that there is a broad optimum for the holes size. Our choice of

*R*=

*0.29a*follows the most common choice.

### 3.2 Quality factors and modal volume

*et al*in Ref. [7

**12**, 3988–3995 (2004). [CrossRef] [PubMed]

*R*=

*0.29a*, and the thickness of the slab is

*h*=

*0.7a*for silicon and a thicker slab,

*h*=

*0.91a*, is used for diamond. The cavity is surrounded by 11 periods in all directions. Increasing the number of periods to 14 changes the quality factor by less than 2%. This means that the total

*Q*approaches

*Q*

_{⊥}. Numerical parameters for the calculations are presented in Table 1. At first we use a computational window of including the entire structure to find the resonances and their symmetries. Then the computational window is reduced eight times using the field symmetry properties. Satisfactory convergence is obtained by using 20 points per period when calculating quality factors up to a few thousand, and 32 points per period for higher quality factors. The perfectly-matched layer width and the height of the computational window are chosen to be quite large as they strongly affect convergence (see Table 1).

*Q*

_{dia}=

*144*is smaller than the silicon value

*Q*

_{Si}=

*400*. However, both values are too small to satisfy QIT requirements [19

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. **73**, 013818 (2006). [CrossRef]

*d*=

*0.15a*to

*d*=

*0.25a*. The optimum appears at

*d*=

*0.21a*, as for silicon [6

6. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**, 1202–1214 (2005). [CrossRef] [PubMed]

**12**, 3988–3995 (2004). [CrossRef] [PubMed]

*x*-direction is lifting the modes’ degeneracy [5

5. O. Painter and K. Srinivasan, “Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis,” Phys. Rev. B **68**, 035110 (2003). [CrossRef]

*ω*͂ =

*ωa*/2

*πC*=a/

*λ*=0.400. The calculated quality factor for that mode is

*Q*=

*2600*. As this mode has a higher quality factor than the modes in the gap center, we further concentrate on it. We next reduce the holes radius above and below the cavity. Figure 3(a) shows the calculated quality factors as a function of the hole radii r that are varied between

*r*≡

*R*=

*0.29a*, when the holes are not modified, and

*r*=

*0.19a*. The displacement is fixed at

*d*=

*0.21a*. The maximum,

*Q*=

*3.0*×10

^{4}, appears at

*r*=

*0.22a*. With this second modification, the modes from the band edge are pulled down towards the middle of the upper half of the band gap [7

**12**, 3988–3995 (2004). [CrossRef] [PubMed]

*ω*͂=0.389, has the same relative position within the band gap as in the silicon slab, but the diamond band gap is higher and therefore the quality factor is smaller.

*E*

_{x}, at the centre of the diamond slab is shown in Fig. 3(b). It is symmetric in the

*x*- and

*y*-directions and anti-symmetric in the

*z*-direction as for the corresponding mode of the silicon slab.

*et al*reported a quality factor

*Q*=

*5.85*×

*10*

^{5}within a small modal volume using a silicon based PCS [7

**12**, 3988–3995 (2004). [CrossRef] [PubMed]

*Q*=

*3.0*×

*10*

^{4}within a modal volume of

*V*=

*1.02*

*(λ/*

*n*)^{3}. The modal volume of the diamond slab is less than twice the modal volume of the equivalent mode in a silicon slab.

*h*=

*0.7*a to

*h*=

*1.1a*we find that the optimum does appear at

*h*=

*0.91a*for the given holes radius.

**425**, 944–947 (2003). [CrossRef] [PubMed]

**13**, 1202–1214 (2005). [CrossRef] [PubMed]

*et al*in Ref [6

**13**, 1202–1214 (2005). [CrossRef] [PubMed]

*et al*, we obtained

*Q*=

*6000*, compared to

*Q*=

*1*×

*10*

^{5}for silicon [6

**13**, 1202–1214 (2005). [CrossRef] [PubMed]

*d*

_{1}=

*0.21a*, the next one by

*d*

_{2}=

*0.025a*and the last one by

*d*

_{3}=

*0.2a*.

## 4. Conclusion

*Q*-values and modal volume that can be achieved in PCS made from silicon and from diamond, which has a somewhat lower refractive index. Using a number of different approaches to optimize these parameters, all originally developed for silicon PCS, we find that small modal volume microcavities can be designed in diamond PCS, with

*Q*=

*3.0*×

*10*

^{4}, smaller than in silicon. We believe that these results hold promise for future use of photonic crystal for quantum information purposes. Though this value is too small for full QIT experiments [19

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. **73**, 013818 (2006). [CrossRef]

*Q*may possibly be achieved using PCS heterostructures [26

26. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. **4**, 207–210 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science |

2. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

3. | H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, and J-S. Kim, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. |

4. | K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express |

5. | O. Painter and K. Srinivasan, “Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis,” Phys. Rev. B |

6. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express |

7. | Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express |

8. | J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E |

9. | K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high-Q photonic crystal microcavity,” Phys. Rev. B |

10. | H. Mabuchi and A. C. Doherty, “Cavity Quantum Electrodynamics: Coherence in Context,” Science |

11. | J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, “Quantum dot photonic-crystal-slab nanocavities: Quality factors and lasing,” Phys. Rev. B |

12. | S. Y. Kilin, “Entangled states and nanoojects in quantum optics,” Opt. and Spectr. |

13. | E. van Oort, N.B. Manson, and M. Glasbeek, “Optically detected spin coherence of the diamond N-V centre in its triplet ground state,” J. Phys. C |

14. | F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett |

15. | F. Jelezko, T. Gaebel, I. Popa, M. Domham, A. Gruber, and J. Wrachtrup, , “Observation of oherent oscillation of a single nuclar spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. |

16. | M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, “Solid-state quantum computing using spectral holes,” Phys. Rev. A |

17. | Y. L. Lim, A. Beige, and C. Kwek, “Repeat-until-success linear optics distributed quantum computing,” Phys. Rev. Lett |

18. | S. D. Barrett and P. Kok, “Efficient high-fidelity quantum computation using matter qubits and linear optics,” Phys. Rev. A |

19. | A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. |

20. | P. Barclay, O. Painter, and K. Srinivasan, “Nonlinear responseof silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” |

21. | J. Salzman, S. Prawer, and D. Jamieson, |

22. | D. F. Edwards and H. R. Philipp, |

23. | M. Qiu, “Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately,” Microw. … Opt. Techn. Lett. |

24. | R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, “Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab,” JOSA B |

25. | H-Y Ryu, M. Notomi, and Y-H Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett |

26. | B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 21, 2006

Revised Manuscript: March 17, 2006

Manuscript Accepted: April 12, 2006

Published: April 17, 2006

**Citation**

S. Tomljenovic-Hanic, M. J. Steel, C. Martijn de Sterke, and J. Salzman, "Diamond based photonic crystal microcavities," Opt. Express **14**, 3556-3562 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3556

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### References

- B. S. Song, and S. Noda, T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300,1537 (2003). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003). [CrossRef] [PubMed]
- H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, J-S. Kim, "Nondegenerate monopole-mode two-dimensional photonic band gap laser," Appl. Phys. Lett. 79, 3032-3034 (2001). [CrossRef]
- K. Srinivasan, and O. Painter, "Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003). [CrossRef] [PubMed]
- O. Painter and K. Srinivasan, "Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003). [CrossRef]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005). [CrossRef] [PubMed]
- Z. Zhang, and M. Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12, 3988-3995 (2004). [CrossRef] [PubMed]
- J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608 (2001). [CrossRef]
- K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, "Optical-fiber-based measurement of an ultrasmall volume high-Q photonic crystal microcavity," Phys. Rev. B 70, 081306 (2004). [CrossRef]
- H. Mabuchi, and A. C. Doherty, "Cavity Quantum Electrodynamics: Coherence in Context," Science 298, 1372 -1377 (2002). [CrossRef] [PubMed]
- J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, "Quantum dot photonic-crystal-slab nanocavities: Quality factors and lasing," Phys. Rev. B 72, 193303 (2005). [CrossRef]
- S. Y. Kilin, "Entangled states and nanoojects in quantum optics," Opt. and Spectr. 94, 709-710 (2003).
- E. van Oort, N.B. Manson and M. Glasbeek, "Optically detected spin coherence of the diamond N-V centre in its triplet ground state," J. Phys. C 21, 4385-4391 (1988). [CrossRef]
- F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillations in a single electron spin," Phys. Rev. Lett 92,076401 (2004). [CrossRef] [PubMed]
- F. Jelezko, T. Gaebel, I. Popa, M. Domham, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillation of a single nuclar spin and realization of a two-qubit conditional quantum gate," Phys. Rev. Lett. 93, 130501 (2004). [CrossRef] [PubMed]
- M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, "Solid-state quantum computing using spectral holes," Phys. Rev. A 66,032301 (2002). [CrossRef]
- Y. L. Lim, A. Beige, and C. Kwek, "Repeat-until-success linear optics distributed quantum computing," Phys. Rev. Lett 95, 030505 (2005). [CrossRef] [PubMed]
- S. D. Barrett, and P. Kok, "Efficient high-fidelity quantum computation using matter qubits and linear optics," Phys. Rev. A 71, 060310 (2005). [CrossRef]
- A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, "Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms," Phys. Rev. A. 73, 013818 (2006). [CrossRef]
- P. Barclay, O. Painter and K. Srinivasan, "Nonlinear responseof silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005).
- J. Salzman, S. Prawer, and D. Jamieson, Photonic crystal devices and systems in diamond, Provisional Patent, CCID 131000480.
- D. F. Edwards, and H. R. Philipp, Handbook of optical constants of solids, (Academic Press, 1985).
- M. Qiu, "Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately," Microw. & Opt. Techn. Lett. 45, 381-385 (2005). [CrossRef]
- R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, "Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab," JOSA B 17, 629-633 (2000). [CrossRef]
- H-Y Ryu, M. Notomi, and Y-H Lee, "High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities," Appl. Phys. Lett 83, 4294-4296 (2003). [CrossRef]
- B. S. Song, S. Noda, T. Asano and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Mater. 4, 207-210 (2005). [CrossRef]

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