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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25607–25618
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Optical properties of excitons in metal-insulator-semiconductor nanowires

Jie-Yun Yan  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 25607-25618 (2013)
http://dx.doi.org/10.1364/OE.21.025607


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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

The optical properties of excitons in various semiconductor nanostructures have been intensively investigated as the fundamentals of most optoelectronic applications. When they are combined with the metallic nanostructures, hybrid excitons are formed in these superstructures due to the exciton-plasmon interactions (EPIs) inside. Both theoretical and experimental researches show that the hybrid excitons exhibit novel optical properties, such as but definitely not limited to Förster energy transfer, [1

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

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]

] nonlinear Fano effect, [4

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

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]

] vacuum Rabi oscillations, [6

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

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]

] local field enhancement, [10

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

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]

] enhancement of luminescence, [14

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

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]

] etc.. The prominent optical properties can be further controlled by changing the materials, sizes, shapes, and other parameters, which makes the EPI a hot topic in the field of nanoscience and plasmonics. [20

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

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

] These diverse superstructures could be classified according to the dimensionality of the excitons, among which those involving the zero-dimension excitons, usually residing in the semiconductor quantum dots (SQDs), have drawn lots of attention, such as the SQD/metal nanoparticle (MNP) system, [4

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

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

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

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

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]

] SQD/metal nanowire (MNW) system, [19

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

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

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]

] SQD/metal core-shell structure, [40

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

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]

] etc.. This kind of exciton, usually treated as a dipole under the external field, becomes the hybrid exciton whose optical properties are largely modified by the EPI. For example, the hybrid exciton in the SQD/MNP complexity has shortened lifetime and shifted energy level. [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]

] By contrast, the researches on the EPI involving high dimensional excitons, such as excitons in the semiconductor quantum wires [42

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

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

] or quantum wells [45

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

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]

] coupled to the metamaterials are relatively limited, especially the theoretical works. On the other hand, to develop the new optoelectronic devices based on the plasmonics necessitates the theoretical studies on how the EPI modifies the optical properties of the excitons.

In this paper, the EPI modified optical properties of the two-dimension excitons are investigated in the composite metal-insulator-semiconductor nanowire, which is made of an MNW covered in turn with an insulator layer (IL) and a semiconductor layer (SL). Both the radius of the MNW and the widths of the IL and the SL are at the scale of several or tens of nanometers. The excitons, two-dimension ones strongly confined in the cylindrical SL, become the hybrid excitons due to the EPI. The structure is worthy of attention because the EPI with the interaction distance of several nanometers allows for an effective transformation of the light absorbed by the SL into the plasmons in the MNW, and vise versa. Moreover, the one-dimension structure provides waveguide to the propagating plasmons, which can efficiently turn back to photons at its tips. All these characteristics make the cylindrical multi-layer nanostructures perspective in the optoelectronic applications.

As a fact, the same structure has been synthesized in experiments by the wet-chemically grown silver wire surrounded by width-controllable SiO2 shell and CdSe nanocrystals. [48

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]

] However, the theoretical works to find out how and to what extent the excitonic optical properties are changed are lack. Now a theoretical model is established to study the role of the EPI in this system. Compared with the zero-dimension exciton systems, the key is to give a proper description of the two-dimension exciton rather than a simple dipole approximation. Here the author manages to get the optical properties of the excitons by obtaining the excitonic wave-function through solving the excitonic equation of motion in electron-hole-pair representation in real space with the exact potentials presented analytically. Due to the EPI, the calculated linear absorption shows red-shift with several meVs for typical parameters, far larger than that in the SQD/MNP system (on the scale of 0.1 meV [4

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

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]

]). The underlying mechanism is analyzed to be the joint action of the indirect Coulomb interaction and the self-image potential based on the numerical results.

The paper is organized as follows: In Sec. 2, the theoretical model of the system is established and the method to get the optical properties is introduced. The underlying mechanism of the modification of the excitonic absorption is discussed in Sec. 3. Finally a brief conclusion is presented in Sec. 4. The detailed analytical solutions of these potentials felt by the hybrid excitons are given in Appendix.

2. Model and solution

The superstructure is composed of an MNW, infinite in 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 R1, and the outer radii of the IL and the SL are R2 and R3, respectively. Figure 1(b) plots the dielectric distribution in the multi-layer structure, where εs, εi are those of the SL and the IL respectively, and εm(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 εm(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 R3R2 with R2 increasing.

Fig. 1 (a) The structure of the metal-insulator-semiconductor nanowires, infinite in the z direction. The sizes of different materials in the ρ direction are labelled as Ri (i = 1, 2, 3). (b) The dielectric distribution of the composite nanowires in the ρ direction. (c) The illustration of the interaction between the excitons (the electron and hole pair in the SC) and the plasmons (oscillating field in the MNW).

When the structure is radiated with the excitation polarized in z direction, E(t) = E0eiΩt, 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) = 0eiΩt where 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]

] Therefore the EPI takes effects on the excitons mainly through the modification of the strengh of the field in the SC. The EPI includes the following three aspects: (A) The plasmons enhance the electric field felt by the excitons; This field enhancement effect would not influence the characteristics of the excitonic optical properties, such as the peak position and the lineshape of the absorption. (B) The field in the MNW is modified by the excitons. The modification would only influence the absorption rate of the MNW. (C) The exciton-induced modification of the electric field in the MNW as mentioned in (B) will in turn react on the excitons in the SC. The feedback of the electric field felt by the excitons is the most important part of the EPI. Due to the EPI, the excitons become the hybrid excitons, whose optical properties are thus changed. The effect of the EPI can be studied by observing the linear absorption of the hybrid excitons, which is easily measured in experiments. Because of the nanoscale of the interaction distance, determined by the width of the IL here, the EPI is very strong. Although the MNW also makes a contribution to the absorption of the whole system, it only provides a relatively smooth background when compared with the obvious absorption peak of the excitons in this kind of system and therefore can be neglected, especially when the excitonic energy is far from the resonance of plasmons in metamaterials [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]

]. The plasmon resonance frequency of Au is about 2.5 eV, which is far large than the bandgap of a typical semiconductor material, such as GaAs (with the bandgap of about 1.6 eV) here. Moreover, the SC is the outer layer, so that one can separate its signal from that of the MNW in practical measurements by some ways.

Now that the EPI takes effect mainly through the self-modification of field felt by the excitons, the potentials of the excited electrons and holes constituting the excitons are correspondingly modified by the MNW. Due to the dielectric contrast, these electrons and holes inside the SC will induce their images (usually called self-images) in the MNW, which bridge the interaction between the excitons and the plasmons. To highlight the role of EPI for the existence of the MNW and eliminate the possible dielectric confinement effect [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]

] induced by the IL and outer air, the dielectric constant of the IL is set equal to that of the SC in our model, i.e., εi = εs, and the outer space is assumed to be the same insulator material in order to further simplify the theoretical solution.

The main role of the ILs is to provide the confinements to the electrons and holes in the SC. Compared with the bandgap of a typical semiconductor material, such as GaAs with the bandgap of about 1.6 eV, that of the IL is usually above 4.0 eV. For the same superstructure synthesized in experiments, the IL is made of SiO2, 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]

] is also neglected. The model is also supported by the fact that the effective EPI is experimentally testified in the nanowires. [48

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]

]

Under these conditions, the potential of a charge q′ inside the SC can be expressed analytically. It is found that the indirected Coulomb potential VcS=qϕi with another charge q and the extra self-energy potential UqS(ρ) are added to the charge 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.

The linear absorption is totally determined by the evolution of the electron-hole pair wave-function Ψ(re, rh, t), where ri (i = e, h) represents (ρi, θi, zi) and R2ρiR3. 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]

]
Ψ(ρe,ρh,z,t)=ψe(ρe,t)ψh(ρh,t)ψz(z,t),
(1)
where z is the relative coordinate of the electron-hole pair along the axis. Under this assumption, the Coulomb interaction Vc0 and indirect Coulomb interaction VcS can be written as
Vc0(ρe,ρh,z)=14πεse2(ρeρh)2+z2;
(2)
VcS(ρe,ρh,z)=e24πεs0+2(εsεm)πC(ρe,ρh)coskzdk,
(3)
respectively, where the factor C(ρe, ρh) reads
C(ρe,ρh)I0(kR1)I0(kR1)K0(kρe)K0(kρh)X0(kR1).
(4)

With the effective-mass theory, the wavefunction obeys an inhomogeneous Schrödinger equation in the real-space representation:
i(t+g2)ψ(ρe,ρh,z,t)=Hψ(ρe,ρh,z,t)dcvE(t)δ(rerh),
(5)
with the initial condition
Ψ(ρe,ρh,z,)=0,
(6)
where dcv is the interband dipole matrix element and g2 is the damping factor of the excitons. The Hamiltonian is
H=H0+Vc0(ρe,ρh,z)+VcS(ρe,ρh,z)+UeS(ρe)+UhS(ρh),
(7)
where
H0=h¯22meρe2h¯22mhρh2h¯22μz2,
(8)
and μ is the reduced mass of the electron-hole pair μ=(1/me||+1/mh||)1, and me(h)|| is the electron (hole) effective mass in the z direction and me(h) in the ρ plane. The center-of-mass motion of the electron-hole pair is neglected. Ue(h)S is the self-image potential of the electron (hole). The wavefunction can be obtained by solving the equation with the finite-difference time-domain method in real space. [51

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

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]

] The optical absorption spectrum is determined by
χ(Ω)[P(Ω)/E0],
(9)
where P(Ω) is the Fourier transformation of the polarization
P(t)=1V0+dcv*Ψ(ρ,ρ,z=0,t)2πρdρ.
(10)

3. Results and discussions

In the absence of the MNW the excitons are only confined by two ILs and demonstrate the properties of two-dimension ones, so called as the unaffected excitons hereafter. As the existence of the MNW brings about the potentials Ue(h)S and VcS, so the task is to find out how the EPI works through these potentials in the linear absorption of the hybrid exciton. The linear absorption spectra by adding four different potentials ( Vc0, Vc0+US+VcS where US refers to UeS+UhS, Vc0+US, and Vc0+VcS) respectively to the Hamiltonian are calculated in Fig. 2. A typical set of parameters is used here: the radii R1, R2 and R3 are 10 nm, 15nm, and 20 nm, respectively. The effective mass of the electron is me||,=0.067m0, where m0 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 mh||=(γ1+γ2)1m0 and mh=(γ12γ2)1m0, respectively, with the Luttinger parameters γ1 = 6.85 and γ2 = 2.1 for GaAs material. The bandgap of the semiconductor Eg is 1.6 eV. The dielectric constant εs is set as 10ε0 (ε0 is the vacuum dielectric constant). The MNW is made of Au, whose dielectric constant εm is about −22ε0 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]

]. The dephasing rate g2 is set as 1 meV for a clear spectral resolution. The dipole matrix element dcv is 0.7 nm/e.

Fig. 2 The excitonic linear absorptions with four different Hamiltonians, whose detailed expressions are listed on it. Two diagrams of the structures corresponding to the unaffected exciton and the hybrid one are drawn on top of their spectra, respectively. Ω is the excitation energy and Eg is the bandgap. Other parameters are stated in the context.

The case with the potential Vc0 only is just the linear absorption of the unaffected exciton in the SC embedded in the same dielectric background. Due to the confinement of the ILs, the photonic energy at the peak of absorption is about 50 meV above the bandgap of the semiconductor. While the case of Vc0+US+VcS gives the absorption of the hybrid exciton, with a clear red-shift of the peak. The mechanism of the red-shift can be find out by checking the roles of the potentials VcS and US separately. Compared the case of Vc0+VcS with that of Vc0, it is concluded that the indirect Coulomb interaction VcS enhances the Coulomb interaction, leading to the increased excitonic binding energy. The conclusion can also be drawn from the analysis of Eqs. (3) and (4), where the characteristic of the negative dielectric constant of the MNW makes the indirect Coulomb interaction an attractive one. The role of Ue,hS can be found by comparing the absorption in the case of Vc0+US with the case of Vc0, as shown in Fig. 2. The potential US 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 US produced by the MNW. The real potentials US 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

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]

]

Fig. 3 (a) Schematics of the energy levels and confinement potentials of the unaffected excitons in the absence of the MNW (a) and the hybrid excitons in the composite MNW-IL-SC-IL nanowire (b). The real self-energy potentials Ue,hS for the hybrid excitons are plotted in (b). The red arrows represent the excitations at the absorption peak for both cases. Parameters are same as that in Fig. 2.

The interaction distance is an important factor for the EPI, which is determined by the width of the IL sandwiched between the MNW and the SC in the system. Intuitively, when the interaction distance increases, the strength of the EPI will diminish and finally disappear, so will the shift of linear absorption. On the other hand, one wants to know how large the shift is expected to be when the interaction distance decreases. For this purpose, the linear absorption spectra are calculated by decreasing the width of the IL, for example with the values of R2 as 25nm, 18nm, 16nm, 14nm, and 12nm in order, while keeping the radius of the MNW R1 as 10nm and the width of the SC R3R2 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 R3R2 is relatively small when compared with the length 2πR2 so that the unaffected exciton can be totally considered as a two-dimension one confined in a rectangular quantum well with the same width R3R2 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.

Fig. 4 The absorption spectra (solid lines) of the hybrid excitons with the width of the IL (R2R1) changing. The radius of the MNW (R1) is 10nm and the width of the SC (R3R2) is fixed to 5nm. The outer radius of the IL (R2) is set to 25nm, 18nm, 16nm, 14nm, and 12nm, respectively, from top to bottom. The change of these structures can be seen from their diagrams given aside. The absorption spectra of the unaffected excitons are also plotted with dashed lines for comparison in each case.

The size of the MNW determines the feature of the plasmons and thus the strength of the EPI. Its influence can be analyzed by the dependence of the potentials on the radius of the MNW. A more direct way to show how the EPI changes with the radius of the MNW is to observe the linear absorption. The absorption spectra, shown in Fig. 5, are calculated with the following parameters: both R2R1 and R3R2 are 5nm, and R1 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 R1 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.

Fig. 5 The absorption spectra of the hybrid excitons (solid lines) with the radius of the MNW changing. The radius of the MNW changes as 6nm, 10nm, 16nm, 25nm and 40nm from top to bottom. Both the width of the IL R2R1 and that of the SC R3R2 are set to 5nm. The absorption spectra of the unaffected excitons (dashed lines) in the same structures are also calculated for reference.

From above analysis, the maximization the excitonic absorption shift can be achieved by both decreasing the width of the middle IL and increasing the radius of the MNW within the model-permitted range. Therefore in practical nano-photonic applications, the absorption peak can be designed by choosing appropriate semiconductor material and adjusting the parameters R1, R2 and R3 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

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

The optical properties of the excitons in the composite metal-insulator-semiconductor nanowires are studied with the established theoretical model. Due to the electron-plasmon interaction, the excitons additionally feel the indirect Coulomb interaction and the extra self-energy potential and thus become the hybrid excitons with the modified optical properties. The linear absorption is calculated by numerically solving the excitonic equation of motion in the electron-hole-pair representation in real space with all the potentials presented analytically. The results show that the linear absorption shows red-shift due to the electron-plasmon interaction, which is found to be the joint action of the increased excitonic binding energy attributed to the indirect Coulomb interaction and the decreased effective band-gap induced by the self-energy potential. The red-shift can reach the scale of 10 meV, which is strong enough when compared with that in the SQD/MNP system. The adjustable optical properties as well as the structural virtue of the nanowires make the structure a prospective candidate in the optical applications.

Appendix

The Appendix gives the derivation of the potentials felt by the excitons in the SC. In the structure, suppose a carrier with the charge 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
ΔΦ=qεsδ(rr),
(11)
with the following boundary conditions:
Φi|R1=Φo|R1,
(12)
and
εmΦiρ|ρ=R1=εsΦoρ|ρ=R1,
(13)
where Φi and Φo refer, respectively, to the potentials Φ(ρ < R1, θ, z) and Φ(ρ > R1, θ, z). One particular solution to Eq. (11) is the Coulomb potential of itself ϕcq4πεs1|rr|. According to the cylindrical symmetry of the system and the homogeneity along z, the potential has the form of Φi = ϕi(ρ, ρ′, θθ′, zz′) inside the MNW and Φo = ϕc(ρ, ρ′, θθ′, zz′)+ϕo(ρ, ρ′, θθ′, zz′) in the dielectric environment of εs, where ϕi(o) can be expanded as
ϕi(o)=2πn=+0+dkein(θθ)(Ai(o)In(kρ)+Bi(o)Kn(kρ))cosk(zz),
(14)
where constants A(B)i(o) are determined by the boundary conditions Eqs. (12) and (13). Here In and Kn are modified Bessel functions of, respectively, the first and second kind of order n, and I′n and K′n are their derivatives correspondingly.

The Coulomb potential ϕc can also be expanded as
ϕc=q4πεs2πn=+0+ein(θθ)In(kρ<)Kn(kρ>)cosk(zz)dk,
(15)
where 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:
Bo=q(εsεm)4πεsIn(kR1)In(kR1)Kn(kρ)Xn(kR1),
(16)
and Ao is zero. The function Xn(x) is defined here as
Xn(x)εmIn(x)Kn(x)εsIn(x)Kn(x).
(17)

Whereas ϕc is the Coulomb potential of the point charge q′, ϕi is the potential produced by the MNW through the image of charge q′. For a charge q locating at r, it would have the potential energy VcqΦi=Vc0+VcS, of which the first term is just the direct Coulomb interaction Vc0=qϕc and the second one is the so called indirect Coulomb interaction bridged by the self-image of q′, VcS=qϕi.

Besides the interaction with another charged particle via the Coulomb interaction, the charge 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 r′ equals
UqS(ρ)=limrr0qϕidq=q2ϕi(ρ,ρ,0,0).
(18)
This is usually called the self-image potential, which can be explicitly expressed as
UqS(ρ)=q24πεsn=+0+εsεmπIn(kR1)In(kR1)Kn2(kρ)Xn(kR1)dk.
(19)

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11004015), the Doctoral Fund of the Ministry of Education of China (Grant No. 20100005120017), and the National Basic Research Program of China (Grant No. 2010CB923200).

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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]

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56.

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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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References

  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]
  2. S. M. Sadeghi and R. G. West, “Coherent control of Forster energy transfer in nanoparticle molecules: energy nanogates and plasmonic heat pulses,” J. Phys.: Condens. Matter23, 425302 (2011). [CrossRef]
  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. B76, 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. B84, 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. A62, 053804 (2000). [CrossRef]
  7. G. S. Agarwal and S. V. ONeil, “Effect of hydrodynamic dispersion of the metal on surface plasmons and surface-enhanced phenomena in spherical geometries,” Phys. Rev. B28, 487–493 (1983). [CrossRef]
  8. V. V. Klimov, M. Ducloy, and V. S. Letokhov, “Strong interaction between a two-level atom and the whispering-gallery modes of a dielectric microsphere: quantum-mechanical consideration,” Phys. Rev. A59, 2996–3014 (1999). [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]
  11. H. Mertens, J. S. Biteen, H. A. Atwater, and A. Polman, “Polarization-selective plasmon-enhanced silicon quantum-dot luminescence,” Nano Lett.6, 2622–2625 (2005). [CrossRef]
  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. B77, 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]
  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]
  15. J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of Si quantum dots: simulation and experiment,” J. Phys. Chem. C111, 13372–13377 (2007). [CrossRef]
  16. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B60, 11564–11567 (1999). [CrossRef]
  17. A. Neogi, C.-W. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B66, 153305 (2002). [CrossRef]
  18. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett.96, 113002 (2006). [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]
  21. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010). [CrossRef]
  22. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express19, 22029–22106 (2011). [CrossRef] [PubMed]
  23. P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Photon.1, 484–588 (2009). [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. B83, 235406 (2011). [CrossRef]
  25. A. Ridolfo, O. Di Stefano, N. Fina, R. Saija, and S. Savasta, “Quantum plasmonics with quantum dot-metal nanoparticle molecules: influence of the Fano effect on photon statistics,” Phys. Rev. Lett.105, 263601 (2010). [CrossRef]
  26. M. R. Singh, D. G. Schindel, and A. Hatef, “Dipole-dipole interaction in a quantum dot and metallic nanorod hybrid system,” Appl. Phys. Lett.99,181106 (2011). [CrossRef]
  27. A. O. Govorov, “Semiconductor-metal nanoparticle molecules in a magnetic field: spin-plasmon and exciton-plasmon interactions,” Phys. Rev. B82, 155322 (2010). [CrossRef]
  28. R. D. Artuso and G. W. Bryant, “Strongly coupled quantum dot-metal nanoparticle systems: exciton-induced transparency, discontinuous response, and suppression as driven quantum oscillator effects,” Phys. Rev. B82,195419 (2010). [CrossRef]
  29. A. Manjavacas, F. J. García de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011). [CrossRef] [PubMed]
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