## 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, [1–3] nonlinear Fano effect, [4, 5] vacuum Rabi oscillations, [6–9] local field enhancement, [10–13] enhancement of luminescence, [14–19] 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. [20–23] 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, [4,12,13,24–35] SQD/metal nanowire (MNW) system, [19,36–39] SQD/metal core-shell structure, [40, 41] 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] By contrast, the researches on the EPI involving high dimensional excitons, such as excitons in the semiconductor quantum wires [42–44] or quantum wells [45–47] 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 SiO_{2} shell and CdSe nanocrystals. [48] 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 [4,12]). 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 *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.

When the structure is radiated with the excitation polarized in *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] 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]. 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] 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., *ε _{i}* =

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

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

Under these conditions, the potential of a charge *q′* inside the SC can be expressed analytically. It is found that the indirected Coulomb potential
${V}_{c}^{S}=q{\varphi}_{i}$ with another charge *q* and the extra self-energy potential
${U}_{{q}^{\prime}}^{S}({\rho}^{\prime})$ 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(*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]

*z*is the relative coordinate of the electron-hole pair along the axis. Under this assumption, the Coulomb interaction ${V}_{c}^{0}$ and indirect Coulomb interaction ${V}_{c}^{S}$ can be written as

*C*(

*ρ*,

_{e}*ρ*) reads

_{h}With the effective-mass theory, the wavefunction obeys an inhomogeneous Schrödinger equation in the real-space representation:

*d*is the interband dipole matrix element and

_{cv}*g*

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

*μ*is the reduced mass of the electron-hole pair $\mu ={\left(1/{m}_{e}^{\left|\right|}+1/{m}_{h}^{\left|\right|}\right)}^{-1}$, and ${m}_{e(h)}^{\left|\right|}$ is the electron (hole) effective mass in the

*z*direction and ${m}_{e(h)}^{\perp}$ in the

**plane. The center-of-mass motion of the electron-hole pair is neglected. ${U}_{e(h)}^{S}$ 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. [51–53] The optical absorption spectrum is determined by where**

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

## 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
${U}_{e(h)}^{S}$ and
${V}_{c}^{S}$, 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 (
${V}_{c}^{0}$,
${V}_{c}^{0}+{U}^{S}+{V}_{c}^{S}$ where *U ^{S}* refers to
${U}_{e}^{S}+{U}_{h}^{S}$,
${V}_{c}^{0}+{U}^{S}$, and
${V}_{c}^{0}+{V}_{c}^{S}$) respectively to the Hamiltonian are calculated in Fig. 2. A typical set of parameters is used here: the radii

*R*

_{1},

*R*

_{2}and

*R*

_{3}are 10 nm, 15nm, and 20 nm, respectively. The effective mass of the electron is ${m}_{e}^{\left|\right|,\perp}=0.067{m}_{0}$, where

*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 ${m}_{h}^{\left|\right|}={({\gamma}_{1}+{\gamma}_{2})}^{-1}{m}_{0}$ and ${m}_{h}^{\perp}={({\gamma}_{1}-2{\gamma}_{2})}^{-1}{m}_{0}$, respectively, with the Luttinger parameters

*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]. The dephasing rate

*g*

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

*d*is 0.7 nm/

_{cv}*e*.

The case with the potential
${V}_{c}^{0}$ 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
${V}_{c}^{0}+{U}^{S}+{V}_{c}^{S}$ 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
${V}_{c}^{S}$ and *U ^{S}* separately. Compared the case of
${V}_{c}^{0}+{V}_{c}^{S}$ with that of
${V}_{c}^{0}$, it is concluded that the indirect Coulomb interaction
${V}_{c}^{S}$ 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
${U}_{e,h}^{S}$ can be found by comparing the absorption in the case of
${V}_{c}^{0}+{U}^{S}$ with the case of
${V}_{c}^{0}$, as shown in Fig. 2. The potential

*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}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 *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.

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 *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.

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 *R*_{1}, *R*_{2} and *R*_{3} while optimizing as far as possible the propagation of the involved plasmons. [55, 56]

## 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

*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 ${\varphi}_{c}\equiv \frac{q}{4\pi {\epsilon}_{s}}\frac{1}{\left|\mathit{r}-{\mathit{r}}^{\prime}\right|}$. According to the cylindrical symmetry of the system and the homogeneity along

*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

*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}The Coulomb potential *ϕ _{c}* can also be expanded as

*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:

*A*is zero. The function

_{o}*X*(

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

Whereas *ϕ _{c}* is the Coulomb potential of the point charge

*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 ${V}_{c}\equiv q{\mathrm{\Phi}}_{i}={V}_{c}^{0}+{V}_{c}^{S}$, of which the first term is just the direct Coulomb interaction ${V}_{c}^{0}=q{\varphi}_{c}$ and the second one is the so called indirect Coulomb interaction bridged by the self-image of**

*r**q′*, ${V}_{c}^{S}=q{\varphi}_{i}$.

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

## 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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