## Optical properties of excitons in metal-insulator-semiconductor nanowires |

Optics Express, Vol. 21, Issue 21, pp. 25607-25618 (2013)

http://dx.doi.org/10.1364/OE.21.025607

Acrobat PDF (1030 KB)

### Abstract

The theoretical model for the metal-insulator-semiconductor nanowires is established and the optical properties are investigated. The linear absorption of the hybrid excitons, formed due to the exciton-plasmon interaction, shows obvious red shift on the magnitude of several meVs. The mechanism of the red shift is found to be the joint action of the increased excitonic binding energy attributed to the indirect Coulomb interaction and the decreased effective bandgap caused by the additional self-energy potential. The conclusion is also supported by the evolution of the absorption spectra with the adjustable structural parameters.

© 2013 OSA

## 1. Introduction

1. J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: a tunable superemitter,” Phys. Rev. Lett. **95**, 017402 (2005). [CrossRef] [PubMed]

3. A. O. Govorov, J. Lee, and N. A. Kotov, “Theory of plasmon-enhanced Forster energy transfer in optically excited semiconductor and metal nanoparticles,” Phys. Rev. B **76**, 125308 (2007). [CrossRef]

4. W. Zhang, A. O. Govorov, and G. W. Bryant, “Semiconductor-metal nanoparticle molecules: hybrid excitons and the nonlinear fano effect,” Phys. Rev. Lett. **97**, 146804 (2006). [CrossRef] [PubMed]

5. W. Zhang and A. O. Govorov, “Quantum theory of the nonlinear Fano effect in hybrid metal-semiconductor nanostructures: the case of strong nonlinearity,” Phys. Rev. B **84**, 081405 (2011). [CrossRef]

6. H. T. Dung, L. Knoll, and D.-G. Welsch, “Spontaneous decay in the presence of dispersing and absorbing bodies: general theory and application to a spherical cavity,” Phys. Rev. A **62**, 053804 (2000). [CrossRef]

9. Y. He, C. Jiang, B. Chen, J.-J. Li, and K.-D. Zhu, “Optical determination of vacuum Rabi splitting in a semiconductor quantum dot induced by a metal nanoparticle,” Opt. Lett. **37**, 2943–2945 (2012). [CrossRef] [PubMed]

10. K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. **89**, 117401 (2002). [CrossRef] [PubMed]

13. J.-Y. Yan, W. Zhang, S. Duan, and X. G. Zhao, “Plasmon-enhanced midinfrared generation from difference frequency in semiconductor quantum dots,” J. Appl. Phys. **103**, 104314 (2008). [CrossRef]

14. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface plasmon enhanced light-emitters based on InGaN quantum wells,” Nat. Mater. **3**, 601–605 (2004). [CrossRef] [PubMed]

19. Y. Fedutik, V. V. Temnov, O. Schöps, U. Woggon, and M. V. Artemyev, “Exciton-plasmon-photon conversion in plasmonic nanostructures,” Phys. Rev. Lett. **99**, 136802 (2007) [CrossRef] [PubMed]

20. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and A. Atwater, “Plasmonics - a route to nanoscale optical devices,” Adv. Mater. **13**, 1501–1505 (2001). [CrossRef]

23. P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Photon. **1**, 484–588 (2009). [CrossRef]

4. W. Zhang, A. O. Govorov, and G. W. Bryant, “Semiconductor-metal nanoparticle molecules: hybrid excitons and the nonlinear fano effect,” Phys. Rev. Lett. **97**, 146804 (2006). [CrossRef] [PubMed]

12. J.-Y. Yan, W. Zhang, S. Duan, X. G. Zhao, and A. O. Govorov, “Optical properties of coupled metal-semiconductor and metal-molecule nanocrystal complexes: role of multipole effects,” Phys. Rev. B **77**, 165301 (2008). [CrossRef]

13. J.-Y. Yan, W. Zhang, S. Duan, and X. G. Zhao, “Plasmon-enhanced midinfrared generation from difference frequency in semiconductor quantum dots,” J. Appl. Phys. **103**, 104314 (2008). [CrossRef]

24. R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum-dot/metal nanoparticle systems,” Phys. Rev. B **83**, 235406 (2011). [CrossRef]

35. D. E. Gómez, A. Roberts, T. J. Davis, and K. C. Vernon, “Surface plasmon hybridization and exciton coupling,” Phys. Rev. B **86**, 035411 (2012). [CrossRef]

19. Y. Fedutik, V. V. Temnov, O. Schöps, U. Woggon, and M. V. Artemyev, “Exciton-plasmon-photon conversion in plasmonic nanostructures,” Phys. Rev. Lett. **99**, 136802 (2007) [CrossRef] [PubMed]

36. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) **450**, 402–406 (2007). [CrossRef]

39. I. D. Rukhlenko, D. Handapangoda, M. Premaratne, A. V. Fedorov, A. V. Baranov, and C. Jagadish, “Spontaneous emission of guided polaritons by quantum dot coupled to metallic nanowire: beyond the dipole approximation,” Opt. Express **17**, 17570–17581 (2009). [CrossRef] [PubMed]

40. N. T. Fofang, N. K. Grady, Z. Fan, A. O. Govorov, and N. J. Halas, “Plexciton dynamics: exciton plasmon coupling in a J-aggregate Au nanoshell complex provides a mechanism for nonlinearity,” Nano Lett. **11**, 1556–1560 (2011). [CrossRef] [PubMed]

41. J. M. Slocik, F. Tam, N. J. Halas, and R. R. Naik, “Peptide-assembled optically responsive nanoparticle complexes,” Nano Lett. **7**, 1054–1058 (2007). [CrossRef] [PubMed]

12. J.-Y. Yan, W. Zhang, S. Duan, X. G. Zhao, and A. O. Govorov, “Optical properties of coupled metal-semiconductor and metal-molecule nanocrystal complexes: role of multipole effects,” Phys. Rev. B **77**, 165301 (2008). [CrossRef]

42. J. Lee, P. Hernandez, J. Lee, A. O. Govorov, and N. A. Kotov, “Exciton-plasmon interactions in molecular spring assemblies of nanowires and wavelength-based protein detection,” Nat. Mater. **6**, 291–295 (2007). [CrossRef] [PubMed]

44. J.-Y. Yan, “Strong exciton-plasmon interaction in semiconductor-insulator-metal nanowires,” Phys. Rev. B **86**, 075438 (2012). [CrossRef]

45. V. I. Sugakov and G. V. Vertsimakha, “Localized exciton states with giant oscillator strength in quantum well in vicinity of metallic nanoparticle,” Phys. Rev. B **81**, 235308 (2010). [CrossRef]

47. P. Vasa, R. Pomraenke, S. Schwieger, Yu. I. Mazur, Vas. Kunets, P. Srinivasan, E. Johnson, J. E. Kihm, D. S. Kim, E. Runge, G. Salamo, and C. Lienau, “Coherent exciton-surface-plasmon-polariton interaction in hybrid metal-semiconductor nanostructures,” Phys. Rev. Lett. **101**, 116801 (2008). [CrossRef] [PubMed]

## 2. Model and solution

*z*direction, covered in order with a coaxial cylindrical IL and an SL, as shown in Fig. 1(a). The outer space can be the air or other dielectric materials. The radius of the MNW is denoted as

*R*

_{1}, and the outer radii of the IL and the SL are

*R*

_{2}and

*R*

_{3}, respectively. Figure 1(b) plots the dielectric distribution in the multi-layer structure, where

*ε*,

_{s}*ε*are those of the SL and the IL respectively, and

_{i}*ε*(Ω) is that of the MNW, depending on the frequency of plasmons, or that of the detecting optical field Ω if the plasmons could totally follow the oscillation of the external field. The MNW is made of Au or Ag, or other metamaterial, whose dielectric function

_{m}*ε*(Ω) is negative when the photonic energy of the excitation locates near the bandgap of a typical semiconductor. The semiconductor cylinder (SC) forms a quantum well, in which the excited excitons are confined in the radial direction. In the absence of the MNW, the system is actually a hollowed semiconductor cylinder (SC), which is expected to gradually demonstrate the optical properties of a typical planar quantum well of width

_{m}*R*

_{3}−

*R*

_{2}with

*R*

_{2}increasing.

*z*direction,

*E*(

*t*) =

*E*

_{0}

*e*

^{−iΩt}, both excitons in the SC and plasmons in the MNW are excited, and consequently the EPI begins to play through the dipole-dipole coupling between them, expressed in Fig. 1(c). Beyond the dipole approximation, these excitons could be treated as the electron-hole pairs confined in the SC, which satisfy the equation of motion with the EPI modified potentials. As we want to find the linear absorption of the system, the quasistatic approximation is applicable. It means both the excitons and the plasmons oscillate with the frequency of Ω. Therefore the polarization of the system

*P*(

*t*) can be written as

*P*(

*t*) =

*P̃*

_{0}

*e*

^{−iΩt}where

*P̃*

_{0}is time independent. [12

12. J.-Y. Yan, W. Zhang, S. Duan, X. G. Zhao, and A. O. Govorov, “Optical properties of coupled metal-semiconductor and metal-molecule nanocrystal complexes: role of multipole effects,” Phys. Rev. B **77**, 165301 (2008). [CrossRef]

**77**, 165301 (2008). [CrossRef]

49. E. A. Muljarov, E. A. Zhukov, V. S. Dneprovskii, and Y. Masumoto, “Dielectrically enhanced excitons in semiconductor-insulator quantum wires: theory and experiment,” Phys. Rev. B **62**, 7420–7432 (2000). [CrossRef]

*ε*=

_{i}*ε*, and the outer space is assumed to be the same insulator material in order to further simplify the theoretical solution.

_{s}_{2}, which has an energy gap about 9.0 eV. So the tunnelling probability of the charges into the IL is small enough to be neglected. So it is reasonable to approximate the bandgap of the IL as infinity and thus totally neglect the tunneling effect. Of course a more powerful theory to consider the tunneling effect is expected, but beyond our consideration in the paper. As the emphasis is mainly put on the EPI, the size dependence effect of the electron-electron interaction which becomes important for the structures of less than 5nm [50

50. L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. **80**, 4403–4407 (1984). [CrossRef]

48. Y. Fedutik, V. Temnov, U. Woggon, E. Ustinovich, and M. Artemyev, “Exciton-plasmon interaction in a composite metal-insulator-semiconductor nanowire system,” J. Am. Chem. Soc. **129**, 14939–14945 (2007). [CrossRef] [PubMed]

*q′*inside the SC can be expressed analytically. It is found that the indirected Coulomb potential

*q*and the extra self-energy potential

*q′*in the SC, whose detailed derivations are given in Appendix. The new produced potentials are just the feedback of the field’s change in the MNW induced by the charge.

*r**,*

_{e}

*r**,*

_{h}*t*), where

*r**(*

_{i}*i*=

*e*,

*h*) represents (

*ρ*,

_{i}*θ*,

_{i}*z*) and

_{i}*R*

_{2}≤

*ρ*≤

_{i}*R*

_{3}. Based on the symmetry of the system, the wavefunction is assumed to be [49

49. E. A. Muljarov, E. A. Zhukov, V. S. Dneprovskii, and Y. Masumoto, “Dielectrically enhanced excitons in semiconductor-insulator quantum wires: theory and experiment,” Phys. Rev. B **62**, 7420–7432 (2000). [CrossRef]

*z*is the relative coordinate of the electron-hole pair along the axis. Under this assumption, the Coulomb interaction

*C*(

*ρ*,

_{e}*ρ*) reads

_{h}*d*is the interband dipole matrix element and

_{cv}*g*

_{2}is the damping factor of the excitons. The Hamiltonian is where and

*μ*is the reduced mass of the electron-hole pair

*z*direction and

**plane. The center-of-mass motion of the electron-hole pair is neglected.**

*ρ*51. S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B **54**, 11592–11601 (1996). [CrossRef]

53. J.-Y. Yan, R. B. Liu, and B. F. Zhu, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. **11**, 083004 (2009). [CrossRef]

*P*(Ω) is the Fourier transformation of the polarization

## 3. Results and discussions

*U*refers to

^{S}*R*

_{1},

*R*

_{2}and

*R*

_{3}are 10 nm, 15nm, and 20 nm, respectively. The effective mass of the electron is

*m*

_{0}is the mass of free electron. The heavy-hole exciton are considered here. The effective masses parallel and perpendicular to the

*z*direction of the heavy holes are expressed as

*γ*

_{1}= 6.85 and

*γ*

_{2}= 2.1 for GaAs material. The bandgap of the semiconductor

*E*is 1.6 eV. The dielectric constant

_{g}*ε*is set as 10

_{s}*ε*

_{0}(

*ε*

_{0}is the vacuum dielectric constant). The MNW is made of Au, whose dielectric constant

*ε*is about −22

_{m}*ε*

_{0}for the photonic energy near the bandedge excitation. In this paper the dielectric function of the metal is taken from the Ref. [54

54. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*g*

_{2}is set as 1 meV for a clear spectral resolution. The dipole matrix element

*d*is 0.7 nm/

_{cv}*e*.

*U*separately. Compared the case of

^{S}*U*also causes a red-shift in the absorption. For a better understanding of this, the different confinement potentials for both the unaffected exciton and the hybrid one in the corresponding structures are plotted in Fig. 3. The unaffected exciton in Fig. 3(a) is a two-dimension one confined in the SC sandwiched by the ILs, while the hybrid exciton feels extra potential

^{S}*U*produced by the MNW. The real potentials

^{S}*U*are drawn in Fig. 3(b) with red lines with the same parameters. They lower the energy of the hybrid exciton and therefore narrow the effective bandgap of the semiconductor quantum well to some degrees. More closer to the MNW the SC is, more significant the effect is. The decreasing of the bandgap is well-understood because the attraction between the carriers and their self-images with opposite charges weakens the confinement potential. It results in the red-shift of the lowest excitonic level in linear absorption. In all, the excitonic absorption experiences red-shift under the cooperation of the indirect Coulomb potential and the self-image potential. The magnitude of the red-shift is about several meVs, which is far bigger than that in the SQD/MNP complexity. [12

^{S}**77**, 165301 (2008). [CrossRef]

*R*

_{2}as 25nm, 18nm, 16nm, 14nm, and 12nm in order, while keeping the radius of the MNW

*R*

_{1}as 10nm and the width of the SC

*R*

_{3}−

*R*

_{2}as 5nm. The linear absorptions of both the hybrid exciton (solid lines) and the unaffected exciton (dashed lines) are shown in Fig. 4 for each case. Other parameters are the same as used in Fig. 2. The spectra are normalized to their maximum values, respectively. It is seen that the absorption of the unaffected exciton is almost unchanged in the process. This is because the width of the SC

*R*

_{3}−

*R*

_{2}is relatively small when compared with the length 2

*πR*

_{2}so that the unaffected exciton can be totally considered as a two-dimension one confined in a rectangular quantum well with the same width

*R*

_{3}−

*R*

_{2}in each case. It also implies that the change of the hybrid excitonic absorptions in these structures can be totally attributed to the EPI. For the hybrid exciton, the red-shift of the absorption gets bigger when the width of the IL decreases, so does the EPI strength. The shift can reach several meVs for a set of typical parameters, which is expected to be easily observed in practical measurements. These optical properties are the basis for the possible optical detectors in future.

*R*

_{2}−

*R*

_{1}and

*R*

_{3}−

*R*

_{2}are 5nm, and

*R*

_{1}is in turn 6nm, 10nm, 16nm, 25nm and 40nm from top to bottom. The linear absorption of the unaffected exciton is also given for comparison in each case. The red-shift in the linear absorption gets bigger with the radius of the MNW increasing. It tells us that the EPI becomes stronger for a system with a larger MNW. However, the acceleration of the shift slows down when

*R*

_{1}increases. It is because that when the radius of the MNW is big enough, the system can be considered as a rectangular quantum well in the vicinity of a metal slab, which is obviously independent on the width of the metal as the EPI works through the self-images of the excitons locating near the surface of the MNW.

*R*

_{1},

*R*

_{2}and

*R*

_{3}while optimizing as far as possible the propagation of the involved plasmons. [55

55. D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain-assisted waveguiding in metal-dielectric nanowires,” Opt. Lett. **35**, 4190–4192 (2010). [CrossRef] [PubMed]

56. D. Handapangoda, M. Premaratne, I. D. Rukhlenko, and C. Jagadish, “Optimal design of composite nanowires for extended reach of surface plasmon-polaritons,” Opt. Express **19**, 16058–16074 (2011). [CrossRef] [PubMed]

## 4. Conclusion

## Appendix

*q′*locates at

**= (**

*r′**ρ′*,

*θ′*,

*z′*) inside the SC, the generated electric potential Φ at

**= (**

*r**ρ*,

*θ*,

*z*) should satisfy the Poisson equation with the following boundary conditions: and where Φ

*and Φ*

_{i}*refer, respectively, to the potentials Φ(*

_{o}*ρ*<

*R*

_{1},

*θ*,

*z*) and Φ(

*ρ*>

*R*

_{1},

*θ*,

*z*). One particular solution to Eq. (11) is the Coulomb potential of itself

*z*, the potential has the form of Φ

*=*

_{i}*ϕ*(

_{i}*ρ*,

*ρ′*,

*θ*−

*θ′*,

*z*−

*z′*) inside the MNW and Φ

*=*

_{o}*ϕ*(

_{c}*ρ*,

*ρ′*,

*θ*−

*θ′*,

*z*−

*z′*)+

*ϕ*(

_{o}*ρ*,

*ρ′*,

*θ*−

*θ′*,

*z*−

*z′*) in the dielectric environment of

*ε*, where

_{s}*ϕ*

_{i}_{(}

_{o}_{)}can be expanded as where constants

*A*(

*B*)

_{i}_{(}

_{o}_{)}are determined by the boundary conditions Eqs. (12) and (13). Here

*I*and

_{n}*K*are modified Bessel functions of, respectively, the first and second kind of order

_{n}*n*, and

*I′*and

_{n}*K′*are their derivatives correspondingly.

_{n}*ϕ*can also be expanded as where

_{c}*ρ*

_{<}_{(}

_{>}_{)}refers to the larger (smaller) radial coordinate between

*ρ*and

*ρ′*. Finally we can find that the potential inside the SC created by the charge

*q′*has the following constants: and

*A*is zero. The function

_{o}*X*(

_{n}*x*) is defined here as

*ϕ*is the Coulomb potential of the point charge

_{c}*q′*,

*ϕ*is the potential produced by the MNW through the image of charge

_{i}*q′*. For a charge

*q*locating at

**, it would have the potential energy**

*r**q′*,

*q′*also feels its own field. As the existence of the MNW modifies the otherwise exclusive Coulomb potential, the additional self-energy of the charge at current position

**equals This is usually called the self-image potential, which can be explicitly expressed as**

*r′*## Acknowledgments

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40. | N. T. Fofang, N. K. Grady, Z. Fan, A. O. Govorov, and N. J. Halas, “Plexciton dynamics: exciton plasmon coupling in a J-aggregate Au nanoshell complex provides a mechanism for nonlinearity,” Nano Lett. |

41. | J. M. Slocik, F. Tam, N. J. Halas, and R. R. Naik, “Peptide-assembled optically responsive nanoparticle complexes,” Nano Lett. |

42. | J. Lee, P. Hernandez, J. Lee, A. O. Govorov, and N. A. Kotov, “Exciton-plasmon interactions in molecular spring assemblies of nanowires and wavelength-based protein detection,” Nat. Mater. |

43. | Q. Zhang, X.-Y. Shan, X. Feng, C.-X. Wang, Q.-Q. Wang, J.-F. Jia, and Q.-K. Xue, “Modulating resonance modes and Q value of a CdS nanowire cavity by single Ag nanoparticles,” Nano Lett. |

44. | J.-Y. Yan, “Strong exciton-plasmon interaction in semiconductor-insulator-metal nanowires,” Phys. Rev. B |

45. | V. I. Sugakov and G. V. Vertsimakha, “Localized exciton states with giant oscillator strength in quantum well in vicinity of metallic nanoparticle,” Phys. Rev. B |

46. | H.-C. Wang, X.-Y. Yu, Y.-L. Chueh, T. Malinauskas, K. Jarasiunas, and S.-W. Feng, “Suppression of surface recombination in surface plasmon coupling with an InGaN/GaN multiple quantum well sample,” Opt. Express |

47. | P. Vasa, R. Pomraenke, S. Schwieger, Yu. I. Mazur, Vas. Kunets, P. Srinivasan, E. Johnson, J. E. Kihm, D. S. Kim, E. Runge, G. Salamo, and C. Lienau, “Coherent exciton-surface-plasmon-polariton interaction in hybrid metal-semiconductor nanostructures,” Phys. Rev. Lett. |

48. | Y. Fedutik, V. Temnov, U. Woggon, E. Ustinovich, and M. Artemyev, “Exciton-plasmon interaction in a composite metal-insulator-semiconductor nanowire system,” J. Am. Chem. Soc. |

49. | E. A. Muljarov, E. A. Zhukov, V. S. Dneprovskii, and Y. Masumoto, “Dielectrically enhanced excitons in semiconductor-insulator quantum wires: theory and experiment,” Phys. Rev. B |

50. | L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. |

51. | S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B |

52. | J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B |

53. | J.-Y. Yan, R. B. Liu, and B. F. Zhu, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. |

54. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

55. | D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain-assisted waveguiding in metal-dielectric nanowires,” Opt. Lett. |

56. | D. Handapangoda, M. Premaratne, I. D. Rukhlenko, and C. Jagadish, “Optimal design of composite nanowires for extended reach of surface plasmon-polaritons,” Opt. Express |

**OCIS Codes**

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

**ToC Category:**

Plasmonics

**History**

Original Manuscript: September 12, 2013

Revised Manuscript: October 10, 2013

Manuscript Accepted: October 10, 2013

Published: October 18, 2013

**Citation**

Jie-Yun Yan, "Optical properties of excitons in metal-insulator-semiconductor nanowires," Opt. Express **21**, 25607-25618 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25607

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