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Photocurrent in plasmonic nanofibers

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Abstract

We developed a theory of photoresponse and photocurrent in photonic nanofibers. Photonic nanofiber is a compound system doped with an ensemble of quantum dots and metallic nanoparticles, where they interact with each other via the dipole–dipole interaction. The bound states of the confined probe photons in the nanofiber hybrid are calculated using the transfer matrix method based on Maxwell’s equations. It is found that the density of states of photons in the nanofiber depends on the dipole–dipole interaction coupling. The nonradiative decay rate due to dipole–dipole interaction rates is calculated using the quantum mechanical perturbation theory. An analytical expression of the photoresponse coefficient and the photocurrent is calculated using the density matrix method. We predicted that the quenching in photocurrent is due to the dipole–dipole interaction. Furthermore, we have shown that the photoluminescence quenching increases as the strength of the dipole–dipole coupling increases. We also compared our theory with the experimental results of the photocurrent in a nanofiber doped with Al metallic nanoparticle nanodisks and Ge/Si quantum dots. A good agreement between theory and experiment is found. Our analytical expressions can be used by experimentalists to perform new types of experiments and for inventing new types of nanosensors and nanoswitches.

© 2021 Optical Society of America

1. INTRODUCTION

Research into the enhancement of photocurrents due to plasmon interaction with metallic nanoparticles (MNPs) has recently been of great interest, especially concerning electrospun nanofibers [124]. The study of plasmonic nanofibers shows great potential for photonic sensing, optical switching, and solar cell applications, and furthermore, the development of new manufacturing techniques such as electrospinning allows for the creation of complex nanohybrid systems relatively easily and cheaply. These nanofiber systems are often biocompatible and have potential applications in biomedical imaging. Additionally, there has also been an increase in the study of the photoresponse in confined systems such as core–shell nanowires, sometimes in conjunction with quantum dots (QDs), to increase photoemission or detection [811].

Recently, Ding et al. [12] studied ZnO-CdS core–shell nanorods and found that the photocurrent was increased on the order of ${{10}^2}$ compared to bare ZnO nanorods. Guan et al. [13] studied CdSe-CdS core–shell QDs films layered on top of an Au MNP lattice and found that it produced directional lasing whose polarization was dependent upon the thickness of the film. In addition, Liu et al. [14] examined the interaction between ZnO and graphene QDs, when exposed to ultraviolet radiation. They found that the inclusion of the QDs resulted in ultrahigh conductivity and high photoresponse current. Wang et al. [15] created a polymer nanofiber embedded with gold nanorods and found a photon-to-plasmon conversion rate of up to 70% efficiency. Furthermore, Kumari et al. [16] fabricated solar cells created with Ag nanowires and ${{\rm TiO}_2}$ nanofibers and found an efficiency increase of 45% due to plasmon-enhanced photocurrent. Dissanayake et al. [17] studied PbS:Hg QDs and Au MNPs doped solar cells with a ${{\rm TiO}_2}$ nanostructure and found both efficiency and photocurrent density increased by around 15% and 38%, respectively. Chen et al. [18] created poly(vinyl pyrrolidone) electrospun nanofibers embedded with Ag nanoparticles, resulting in improved power conversion efficiency stemming from increased photocurrent caused by surface plasmons.

Yakimov et al. [19] created a planar QD detector composed of Ge/Si heterostructures coupled to a periodic Al lattice of disks with diameters varying between 150 nm and 225 nm. A significant increase in sensitivity is found due to plasmon resonances near the Si–Al interface. Also, Yakimov et al. [20] studied the interaction of Ge QDs and both an array of Au nanodisks with a diameter of 0.7 µm and an Au sheet with nanoholes of the same diameter. They found a 3.7-factor improvement in the plasmonic response for the array and an 11 times increase for the sheet. Jung et al. [21] studied ${{\rm TiO}_2}$ nanotubes embedded with Ag nanoparticles and dye molecules. They found an enhanced photocurrent and showed that plasmonic effects could be controlled by modifying the thickness of a dielectric spacer between the nanoparticles and the dye. Sugimoto et al. [22] fabricated a biocompatible polymer nanofiber via electrospinning doped with Si nanocrystals and Ag MNPs and demonstrated an increase in photoluminescence (PL) of 2.2 times. Sobti et al. [23] examined ${{\rm Fe}_2}{{\rm O}_3}$ nanoparticles deposited on ${{\rm TiO}_2}$ nanofibers and showed that this both moved the absorption spectrum into the visible wavelengths of light as well as enhanced photocurrent. Finally, Sebo et al. [24] studied solar cells doped with Ag MNPs encased in thin shells of ${{\rm TiO}_2}$, whose plasmonic effects generated an 18% rise in photocurrent and a roughly 6% rise in efficiency.

In this paper, we have developed a theory of the photocurrent for photonic nanofibers (PNFs) in the presence of surface plasmon polaritons (SPPs) coupling and dipole–dipole interaction (DDI) coupling. Recently, SPPs in MNPs have been widely studied in plasmonic literature [2532]. In Section 1, we have already surveyed the literature for the photocurrent in nanofiber hybrids. A theory of the photocurrent in a nanofiber doped with an ensemble of quantum emitters (QEs) such as QDs and an ensemble of MNPs has been proposed in Section 2. In Section 3, a theory of the photocurrent quantum yield and nonradiative decay rates due to the exciton–DDI coupling is developed using the time-dependent quantum mechanical perturbation method. In Section 4, an analytical expression of the photoresponse coefficient and photocurrent is derived using the density matrix method. In Section 5, we compared our theory with the experimental results of [19]. Finally, a summary of the findings of our paper is presented in Section 6.

2. PHOTOCURRENT IN METALLIC NANOFIBERS

In this section, we develop a theory of photocurrent in the plasmonic nanofibers. Recently, some experimental work is done on photocurrent spectral response in metallic nanofibers doped with MNPs and QDs [19,20]. For example, Yakimov et al. [19] measured the photoresponse spectrum, photocurrent in nanowaveguide containing an ensemble of Ge/Si QDs and MNPs fabricated from an Al disk array. The ensemble of Ge/Si QDs is located in the core of the nanofiber. The shell is fabricated from Al MNP discs and Si layers. They showed that electrons and holes are created in Ge/Si QDs which are responsible for the photocurrent.

We consider a nanofiber made of a core and a shell (cladding). The core of the nanofiber is doped with an ensemble of QDs and MNPs. They are densely populated, interacting with each other via DDI. The cross-sectional area of the core is denoted as $A$ and the length of the nanofiber is denoted as $L$. The nanofiber lies along the ${z}$ direction. We consider that ${\epsilon _c}{\rm \;and\;}{\epsilon _s}$ are the dielectric constant of the core and the shell, respectively. A schematic diagram of the nanofiber hybrid is shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the nanofiber hybrid. The nanofiber is composed of a core and a shell (cladding), both of which are made of dielectric materials. The core of the nanofiber is doped with an ensemble of QDs and MNPs.

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It is well known that photocurrent is generated by the absorption of electromagnetic fields (i.e., laser light) in electronic materials such as quantum dots. We consider that an external probe electromagnetic field with amplitude ${E_P}$, energy/frequency ${\varepsilon _k}/{\omega _k}$, and wave vector $k$ is applied to study the photocurrent. The probe field wave propagates along the length direction (i.e., ${z}$ direction) of the PNF. In the presence of the external probe field, the MNP produced the induced SPP field ${E_{\rm{SPP}}}$ [3236]. The ensemble of MNPs produces the DDI electric field denoted as $E_{\rm{DDI}}^{\rm{MNP}}$. Similarly, the ensemble of QDs produces the DDI electric field denoted as $E_{\rm{DDI}}^{\rm{QD}}$ [3436].

When an electromagnetic field falls on QDs, electrons and holes are created. The photocurrent is also produced due to the motion of free electrons and holes. We consider that a QD has four energy levels, which are denoted as $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$. These types of QDs are studied in the plasmonic literature [32,33]. The energy difference between levels $|1\rangle$ and $|2\rangle$ is expressed as ${\varepsilon _{12}}$, the energy difference between levels $|1\rangle$ and $|3\rangle$ is expressed as ${\varepsilon _{13}}$, and the energy difference between levels $|1\rangle$ and $|4\rangle$ is expressed as ${\varepsilon _{14}}$. We consider that the probe field ${E_P}$ and the SPP field ${E_{\rm{SPP}}}$ are acting between transition $| {1\rangle \leftrightarrow} |2\rangle$, the DDI-MNP field between transition $| {2\rangle \leftrightarrow} |3\rangle$, and the DDI-QD field between transition $| {3\rangle \leftrightarrow} |4\rangle$. A schematic diagram of the QD is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Schematic diagram of a four-level QD. The energy levels are denoted as $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$. The probe field and the SPP field are acting in transition $| {1\rangle \leftrightarrow} |2\rangle$. The DDI-MNP and the DDI-QD fields are acting in transitions $| 2\rangle \leftrightarrow | 3\rangle$ and $| {3\rangle \leftrightarrow} |4\rangle$, respectively.

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In the present case, the probe and SPP fields are acting between $| {1\rangle \leftrightarrow} |2\rangle$. When photons from the probe field and polaritons from the SPP field are absorbed, electrons are excited from the ground state $|1\rangle$ to the excited state $|2\rangle$. Charge carriers (electrons and holes) are generated in the ensemble of QDs and these charge carriers are responsible for the photocurrent in the PNF. Let us consider that the number of electrons and holes emitted by light is denoted as ${{\boldsymbol n}_{\boldsymbol e}}$ and ${{\boldsymbol n}_{\boldsymbol h}}$, respectively. Therefore, the photocurrent density (${{\boldsymbol j}_{{\boldsymbol pc}}}$) due to electrons and holes flowing in the nanofiber is found as

$${j_{\textit{pc}}} = {n_e}ev_d^e + {n_h}ev_d^h,$$
where $e$ is the charge for the electron and hole. Here $v_d^e$ and $v_d^h$ are the drift speed of electrons and holes, respectively, and their expression can be written as follows:
$$\begin{split}v_d^e & = \frac{{e{\tau _e}}}{{{m_e}}}\left({{E_P} + {E_{\rm{SPP}}}} \right),\\v_d^h & = \frac{{e{\tau _h}}}{{{m_h}}}\left({{E_P} + {E_{\rm{SPP}}}} \right),\end{split}$$
where ${\tau _e}$ and ${\tau _h}$ are the decay rates for electrons and holes, respectively. Here ${m_e}$ and ${m_h}$ are the mass of an electron and a hole, respectively. Inserting Eq. (2) into Eq. (1), we get
$${j_{\textit{pc}}} = {\sigma _e}\left({{E_P} + {E_{\rm{SPP}}}} \right) + {\sigma _h}\left({{E_P} + {E_{\rm{SPP}}}} \right),$$
where ${\sigma _e}$ and ${\sigma _h}$ are the photoconductivity of electrons and holes, respectively, and their expressions are found as
$${\sigma _e} = \frac{{{n_e}{e^2}{\tau _e}}}{{{m_e}}}\quad {\sigma _h} = \frac{{{n_h}{e^2}{\tau _h}}}{{{m_h}}}.$$

We know that the number of electrons and holes are the same then we have ${n_e} = {n_h}$. We consider that the decay rates for the electrons and holes are the same (i.e., ${\tau _e} = {\tau _h}$). Inserting Eq. (4) into Eq. (3), we get

$$\begin{split}{j_{\textit{pc}}} & = \frac{{{n_e}{e^2}{\tau _e}}}{{{m^*}}}\left({{E_P} + {E_{\rm{SPP}}}} \right), \\\frac{1}{{{m^*}}} & = \frac{1}{{{m_e}}} + \frac{1}{{{m_h}}},\end{split}$$
where ${m^*}$ is the effective mass of the charge carrier.

To calculate the photocurrent in Eq. (5), we need to calculate the electron density ${n_e}$. Two factors control the electron density of the emitted electrons in the ensemble of QDs. The first is the continuous creation of free electron and hole pairs due to the probe field ${E_P}$ and SPP field ${E_{\rm{SPP}}}$. The second is the continuous annihilation due to the recombination of electrons and holes. The first process increases the electron concentration, and the second process reduces the electron concentration. The second process depends on the recombination time of the electron–hole annihilation. The variation of concentration with time is governed by the following equation:

$$\frac{{d{n_e}}}{{dt}} = {g_e} - \frac{{{n_e}}}{{{\tau _r}}},$$
where ${\tau _r}$ is called the recombination time and ${g_e}$ is the generation rate related to the power absorption due to the probe and SPP fields. In Eq. (6), the first term is the generation rate, and the second term is the recombination rate. In the steady state, we have ($d{n_e}/dt$) equals to zero, thus Eq. (6) reduces to
$${n_e} = {g_e}{\tau _r}.$$

The generation rate ${g_e}$ is calculated as follows. We denote the power absorbed by the QDs due to the probe field as ${P_P}$. Similarly, the power absorbed by the QDs due to the SPP field is denoted as ${P_{\rm{SPP}}}$. The probability of the electron excitation achieved by absorbing probe photons and SPP polaritons is denoted as ${\eta _{\rm{QE}}}$, which is also known as the quantum yield. Then the carrier generation rate can be written as

$${g_e} = {\eta _{\rm{QE}}}\left({\frac{{{P_P}({\omega _P})}}{{\hbar {\omega _P}}} + \frac{{{P_{\rm{SPP}}}({\omega _P})}}{{\hbar {\omega _{\rm{SP}}}}}} \right),$$
where ${\omega _{\rm{SP}}}$ is the resonance frequency of SPP polaritons. It is known that the SPP field reaches its maximum at the SPP resonance frequency ${\omega _{\rm{SP}}}$. We can calculate the concentration of electrons by inserting Eq. (8) into Eq. (7) as
$${n_e} = {\eta _{\rm{QE}}}{\tau _r}\left({\frac{{{P_P}({\omega _P}) + {P_{\rm{SPP}}}({\omega _P})}}{{\hbar {\omega _P}}}} \right).$$

Finally, we can obtain the following expression of the photocurrent by substituting Eq. (9) into Eq. (5) as

$${j_{\textit{pc}}} = \left[\begin{array}{l}\left({\frac{{{e^2}{\tau _e}{\tau _r}}}{{{m^*}}}} \right){\eta _{\rm{QE}}}\left({{E_P} + {E_{\rm{SPP}}}} \right) \times \\\left({\frac{{{P_P}({\omega _P}) + {P_{\rm{SPP}}}({\omega _P})}}{{\hbar {\omega _P}}}} \right)\end{array} \right].$$

Note that the photocurrent density depends on the quantum yield.

Let us find the expression of the quantum yield (${\eta _{\rm{QE}}}$) appearing in Eq. (10). Note from Fig. 2 that the decay of excited electrons has two channels: radiative decay (${\gamma _r}$) and nonradiative decay $({{\gamma _{\textit{nr}}}})$. The radiative decay is caused by the spontaneous emission of the electron transition from $|2\rangle$ to $|1\rangle$, while the nonradiative decay rate is a result of the exciton electron decay from $|2\rangle$ to $|3\rangle$ due to the electron scattering with DDI-MNP field interaction. Similarly, the electron decay from $|3\rangle$ to $|4\rangle$ is also responsible for the nonradiative decay rate due to the electron–DDI-QD interaction. The expression of the quantum yield can be written as

$${\eta _{\rm{QE}}}= \frac{{{\gamma _r}}}{{{\gamma _r} + {\gamma _{\textit{nr}}}}}.$$

Note that when the nonradiative decay rate is absent (${\gamma _{\textit{nr}}} = 0$), the quantum yield is real to one (i.e., ${\eta _{\rm{QE}}} = 1$). The expression of the decay rates will be calculated in Section 3.

Next, we use the density matrix method to calculate the expression of the power absorption ${P_P}$ and ${P_{\rm{SPP}}}$ appearing in Eq. (10). Using the density matrix method [3741], we can find the following expressions of power absorption due to the probe and SPP fields as

$${P_P} + {P_{\rm{SPP}}} = \frac{{{\mu _{12}}{\omega _p}}}{2}{\mathop{\rm Im}\nolimits} ({\rho _{12}}){\left({{E_P} + {E_{\rm{SPP}}}} \right)^*},$$
where ${\rho _{12}}$ and ${\mu _{12}}$ are the matrix elements of the density matrix operator and the dipole moment of the QD for transition $| {1\rangle \leftrightarrow} |2\rangle$, respectively.

Inserting Eq. (12) into Eq. (11), we get the expression of the photocurrent as

$${j_{\textit{pc}}} = \left({\frac{{{\mu _{12}}{e^2}{\tau _e}{\tau _r}}}{{2\hbar {m^*}}}} \right){\eta _{\rm{QE}}}{\mathop{\rm Im}\nolimits} ({\rho _{12}}){\left| {{E_P} + {E_{\rm{SPP}}}} \right|^2}.$$

The above expression can be rewritten as

$${j_{\textit{pc}}} = {j_o.P_{\rm{PRC}}}\quad{j_o} = \left({\frac{{{\mu _{12}}{e^2}{\tau _e}{\tau _r}}}{{2{m^*}}}} \right)E_P^2,$$
where ${P_{\rm{PRC}}}$ is called the photoresponse coefficient (PRC):
$${P_{\rm{PRC}}} = {\eta _{\rm{QE}}}{\mathop{\rm Im}\nolimits} ({\rho _{12}}){\left({1 + {E_{\rm{SPP}}}/{E_P}} \right)^2}.$$

The density matrix element ${\rho _{12}}$ will be calculated in Section 4.

It is worth pointing out in Eq. (13) that there is an enhancement in the photocurrent due to the presence of the SPP field ${E_{\rm{SPP}}}$. This is one of the new findings of the paper.

3. QUANTUM YIELD

In this section, we calculate the expressions of the radiative and nonradiative decay rates appearing in the quantum yield appearing in Eq. (11). First, let us calculate the SPP and DDI fields acting on the QE. The expression of the SPP field is calculated with the help of [37,42], and the DDI fields are calculated using the method of [3436] and they are found as

$$\begin{split}{E_{\rm{SPP}}} & = \Pi _{\rm{SPP}}E_P, \\\Pi _{\rm{SPP}} & = \frac{{R_{\rm{MNP}}^3{g_l}}}{{r_s^3}}\varsigma _{\rm{MNP}},\\ \varsigma _{\rm{MNP}}& = \frac{{{\in _m} - {\in _b}}}{{{\in _m} + 2{\in _b}}},\end{split}$$
$$\begin{split}E_{\rm{DDI}}^{\rm{MNP}} & = \Phi _{\rm{DDI}}^{\rm{MNP}}{E_P},\\\Phi _{\rm{DDI}}^{\rm{MNP}} & = \frac{{{\lambda _{\rm{DDI}}}{g_l}}}{3}\varsigma _{\rm{MNP}},\end{split}$$
$$\begin{split}E_{\rm{DDI}}^{\rm{QE}} & = \Phi _{\rm{DDI}}^{\rm{QE}}E_P,\\\Phi _{\rm{DDI}}^{\rm{QE}} & = \frac{{{\lambda _{\rm{DDI}}}{g_l}}}{3}\varsigma _{\rm{QE}},\\\varsigma _{\rm{QE}} & = \frac{{{\in _q} - {\in _b}}}{{{\in _q} + 2{\in _b}}}.\end{split}$$

Here ${\epsilon _m}$, ${\epsilon _b},$ and ${\epsilon _q}$ are the dielectric constant of the MNP, the core dielectric material, and the QE, respectively. The constant ${r_s}$ is the distance between the center of the QE and the surface of the MNP. Here ${\lambda _{\rm{DDI}}}$ is the DDI constant. The parameter $\Pi _{\rm{SPP}}$ is called the SPP coupling constant, ${\Phi}_{\rm{DDI}}^{\rm{MNP}}$ is called the DDI coupling constant for MNPs (DDI-MNP) and ${\Phi}_{\rm{DDI}}^{\rm{QE}}$ is called the DDI coupling constant for QEs (DDI-QE).

Solving Maxwell’s equations in the quasi-static approximation [37,42], one can derive the polarization of a QE (${P_{\rm{QE}}})$ and an MNP (${P_{\rm{MNP}}})$. The following expressions for ${P_{\rm{QE}}}$ and ${P_{\rm{MNP}}}$ are derived:

$$\begin{split} P_{\rm{QE}} = {\in _0}{\in _c}{g_l}\varsigma _{\rm{QE}} \left(\begin{array}{l}E_P + E_{\rm{SPP}}, \\ + E_{\rm{DDI}}^{\rm{MNP}} + E_{\rm{DDI}}^{\rm{QE}}\end{array} \right),\\P_{\rm{MNP}} = {\in _0}{\in _c}{g_l}\varsigma _{\rm{MNP}} \left(\begin{array}{l}E_P + \\E_{\rm{DDI}}^{\rm{QE}} + E_{\rm{DDI}}^{\rm{MNP}}\end{array} \right),\end{split}$$
where ${g_l}$ appearing in Eqs. (15) and (16) is called the polarization parameter and it has a value either ${g_l} = 2$ or ${g_l} = - 1$ for ${P_i}\parallel {E_P}$ and ${P_i} \bot {E_P}$ (${\rm i} = {\rm QE}$, MNP), respectively. Here ${\in _0}$ is the dielectric constant of the vacuum.

Inserting Eq. (14) into Eq. (15), we get the expression for polarization of QEs and MNPs as follows:

$$\begin{split} P_{\rm{QE}} & = {\in _0}{\in _c}\alpha _{\rm{QE}} E_P,\\\alpha _{\rm{QE}} & = {g_l}\varsigma _{\rm{QE}} \left(\begin{array}{l}1 + \Pi _{\rm{SPP}} + \\\Phi _{\rm{DDI}}^{\rm{MNP}} + \Phi _{\rm{DDI}}^{\rm{QE}}\end{array} \right),\\P_{\rm{MNP}}& = {\in _0}{\in _c}\alpha _{\rm{MNP}} E_P, \\ \alpha _{\rm{MNP}} & = {g_l}\varsigma _{\rm{MNP}} \left(\begin{array}{l}1 + \Pi _{\rm{QE}} + \\\Phi _{\rm{DDI}}^{\rm{QE}} + + \Phi _{\rm{DDI}}^{\rm{MNP}}\end{array} \right),\end{split}$$
where ${\alpha _{\rm{QD}}}$ and ${\alpha _{\rm{MNP}}}$ are the polarizability of QE and MNP, respectively.

Now, we calculate the photonic bound states present in the PNF by using Maxwell’s equation for the core and cladding. Note that the polarization due to the QEs and MNPs (${P_{\rm{QE}}}$, ${P_{\rm{MNP}}}$) are present in the core along with the probe electric field. With help of Eq. (16), the Maxwell equations for the core and the cladding are written as

$$\begin{split} {\nabla ^2}{E_c} & = - \frac{{\omega _{{^k}}^2}}{{{c^2}}}\left(\begin{array}{l}{\in _c} + \\{\in _c}\alpha _{\rm{QE}} + {\in _c}\alpha _{\rm{MNP}}\end{array} \right){E_c},\\{\nabla ^2}{E_s} & = - \left({\frac{{\omega _{{^k}}^2}}{{{c^2}}}} \right){\in _s}{E_s},\end{split}$$
where ${E_c}$ and ${E_s}$ are the electric fields inside the core and the cladding, respectively. Here $c = {({{\epsilon _0}{\mu _0}})^{- 1/2}}$ is the speed of light in the vacuum.

We know that the electric field propagates through the core and decays exponentially in the shell. Matching the boundary conditions at the interface between the core and the shell and using the transfer matrix method based on Maxwell’s equations [4345], we obtain the following dispersion relation of photons in the nanofiber as

$$\begin{split}k_z & = {\left[{F_c^2({\varepsilon _{nm,{k_z}}}) - F_c^2({\varepsilon _{\textit{nm}}})} \right]^{1/2}},\\F_c ({\varepsilon _k}) & = \frac{{{\varepsilon _k}\sqrt {\left({{\in _c} + {\in _c}\alpha _{\rm{QE}} + {\in _c}\alpha _{\rm{MNP}} } \right)}}}{{\hbar c}},\end{split}$$
where ${\varepsilon _{\textit{nm}}}$ is bound state energy perpendicular to the photon propagating vector ${k_z}$ (i.e., ${z}$ direction). Therefore, photons propagating inside the PNF have wave vector ${k_{nm,z}}=({k_n},\;{k_m},{k_z})$ and energy ${\varepsilon _{nm,{k_z}}}$. Note that ${\varepsilon _{nm,{k_z}}} = {\varepsilon _{\textit{nm}}}$ when ${k_z} = 0$.

To calculate the decay rates, we write the interaction Hamiltonian of propagating photons inside the PNF. The decay interaction Hamiltonian can be written in the dipole and rotating wave approximation as

$$\begin{split} {H_r} & = \left(\begin{array}{l}\sum\limits_{nm,{k_z}} V_Pp_{nm,{k_z}}\sigma _{21}^{{\dagger}} + \\\sum\limits_{nm,{k_z}} V_{\rm{SPP}}^{\rm{MNP}}p_{nm,{k_z}}\sigma _{21}^{{\dagger}} + hc\end{array} \right),\\[-2pt]{H_{\textit{nr}}} & = \left(\begin{array}{l}\sum\limits_{nm,{k_z}} V_{\rm{DDI}}^{\rm{MNP}}p_{nm,{k_z}}\sigma _{23}^{{\dagger}}\\ + \sum\limits_{nm,{k_z}} V_{\rm{DDI}}^{\rm{QE}}p_{nm,{k_z}}\sigma _{34}^{{\dagger}} + hc\end{array} \right),\end{split}$$
where the operator $p_{nm,{k_z}}$ is the photon annihilation operator for energy $\varepsilon _{nm,{k_z}}$ and the operator $\sigma _{\textit{ij}}^{{\dagger}}$ (${ij} = {21}$, 23, 34) is the exciton creation operator. Here $hc$ stands for the Hermitian conjugate.

In Eq. (19), coupling terms ${V_P}$ and ${V_{\rm{SPP}}}$ are the exciton–probe field and exciton–SPP field interactions, respectively, and they are responsible for radiative decay emission. Coupling terms ${\rm V}_{\rm{DDI}}^{\rm{MNP}}$ and ${\rm V}_{\rm{DDI}}^{\rm{QE}}$ are the exciton–MNP-DDI field and exciton–QE-DDI field interactions, respectively, and they are responsible for the nonradiative emission. Their expressions are found as

$$\begin{split}{V_P} & = i{\left({\frac{{\mu _{21}^2{\varepsilon _{nm,{k_z}}}}}{{2{\in _0}{\in _b}{V_{\rm{QE}}}}}} \right)^{1/2}},\\[-2pt]V_{\rm{SPP}} & = V_P\Pi _{\rm{SPP}}, \\[-2pt] V_{\rm{DDI}}^{\rm{MNP}} & = i{\left({\frac{{\mu _{23}^2{\varepsilon _{nm,{k_z}}}}}{{2{\in _0}{\in _b}{V_{\rm{QE}}}}}} \right)^{1/2}}\Phi _{\rm{DDI}}^{\rm{MNP}}, \\[-2pt] V_{\rm{DDI}}^{\rm{QE}} & = i{\left({\frac{{\mu _{34}^2{\varepsilon _{nm,{k_z}}}}}{{2{\in _0}{\in _b}{V_{\rm{QE}}}}}} \right)^{1/2}}\Phi _{\rm{DDI}}^{\rm{QE}},\end{split}$$
where ${V_{\rm{QE}}}$ is the volume of the QE. Here ${\mu _{21}},\;{\mu _{23}}$, and ${\mu _{34}}$ are the dipole moments of the QE for transitions $| {2\rangle \leftrightarrow} |1\rangle$, $| {2\rangle \leftrightarrow} |3\rangle$, and $| {3\rangle \leftrightarrow} |4\rangle$, respectively.

We use the Golden rule method of the quantum mechanical perturbation theory to calculate the radiative and nonradiative decay rates. The result is written as follows:

$${\Gamma _{\textit{ij}}} = 2\pi \sum\limits_{\textit{nm}} \left[\begin{array}{l}\int {{\rm d}{\varepsilon _{nm,{k_z}}}} {D_{\textit{nm}}}\left({{\varepsilon _{nm,{k_z}}}} \right)\\ \times {\left| {V_{{\mathop{\rm int}}}} \right|^2}\delta \left({{\varepsilon _{nm,{k_z}}} - {\varepsilon _{\textit{ij}}}} \right)\end{array} \right],$$
where ${\varepsilon _{nm,{k_z}}}$ is the quantized energy of photons propagating in the PNF. Here ${V_{\textit{int}}}$ is the coupling constant for the radiative and nonradiative decays and given in Eq. (20). Here ${D_{\textit{nm}}}$ is the density of states (DOS) of the propagating photons inside the PNF.

Let us calculate the expression of the DOS for photons with energy ${\varepsilon _{nm,{k_z}}}$ propagating along the ${ z}$ direction. With help of Eq. (18), we have calculated the following expression of the DOS as

$$\begin{split}{D_{\textit{nm}}} &= {D_0}\left({\frac{{{G_c}{F_c}\sqrt {\left({1 + \alpha _{\rm{QE}} + \alpha _{\rm{MNP}}} \right)}}}{{{{\left[{F_c^2({\varepsilon _{nm,{k_z}}}) - F_c^2({\varepsilon _{\textit{nm}}})} \right]}^{1/2}}}}} \right), \\[-2pt] {D_0} &= \frac{{{V_{\textit{PNF}}}}}{{{\pi ^2}}}\frac{{\varepsilon _{nm,{k_z}}^2}}{{{\hbar ^3}{c^3}}},\\[-2pt]G_c &= \frac{{\pi {\hbar ^2}{c^2}\sqrt {{\in _c}}}}{{A\varepsilon _{nm,{k_z}}^2}}.\end{split}$$

In the above expression, ${D_0}$ is the density of the photons in the vacuum, and ${V_{\rm{PNF}}}$ is the volume of the PNF. The DOS is dependent on the SPP and DDI coupling and possesses an extremely large value when ${\varepsilon _{nm,{k_z}}}$ is equal to ${\varepsilon _{\textit{nm}}}$.

With the help of Eqs. (20)–(22) and after extensive mathematical manipulations, the radiative and nonradiative decay rates for excitons are found as

$${\gamma _r} = \Gamma _P\left({1 + {{\left| {{\Pi _{\rm{SPP}}}} \right|}^2}} \right)\begin{array}{*{20}{c}},&\end{array}\\\Gamma _P = {\gamma _0}\sum\limits_{\textit{nm}} \left\{{\frac{{{G_c}\left({{\varepsilon _{21}}} \right){F_c}\left({{\varepsilon _{21}}} \right)}}{{{{\left[{F_c^2({\varepsilon _{21}}) - F_c^2({\varepsilon _{\textit{nm}}})} \right]}^{1/2}}}}} \right\},$$
$${\gamma _{\textit{nr}}} = \gamma _{\textit{nr}}^{23} + \gamma _{\textit{nr}}^{34}\begin{array}{*{20}{c}},&\end{array}\\\gamma _{\textit{nr}}^{23} = \Gamma _0{\left| {\Phi _{\rm{DDI}}^{\rm{MNP}}} \right|^2},\\\gamma _{\textit{nr}}^{34} = \Gamma _0{\left| {\Phi _{\rm{DDI}}^{\rm{QE}}} \right|^2},$$
$$\Gamma _0 = {\gamma _0}\sum\limits_{\textit{nm}} \left\{{\frac{{{G_c}\left({{\varepsilon _{23}}} \right){F_c}\left({{\varepsilon _{23}}} \right)}}{{{{\left[{F_c^2({\varepsilon _{23}}) - F_c^2({\varepsilon _{\textit{nm}}})} \right]}^{1/2}}}}} \right\}.$$
Here ${\gamma _0}$ is the radiative decay rate when the QD is in the vacuum. The quantum yield depends on the radiative decay rate due to exciton–SPP interaction and the nonradiative decay rate due to exciton–DDI-MNPs interaction and exciton–DDI-QDs interaction. It also depends on the SPP and DDI couplings.

4. DENSITY MATRIX METHOD AND PHOTOCURRENT

Now, we evaluate expressions of the density matrix element $\rho _{21}$ appearing in the expression of photocurrent [see Eq. (12)]. To evaluate this density matrix, we need to calculate the interaction Hamiltonian between excitons in the QE and the ensemble of MNPs in the PNF. The exciton is interacting with the probe field, the SPP field, the MNP-DDI field, and the QD-MNP field. These electric fields induce the dipole in the QE and this dipole, in turn, interacts with these fields. With the help of Eq. (13) and using the dipole and rotating wave approximation, the interaction Hamiltonian is derived as follows:

$$H_0 = \left[\begin{array}{l}\hbar {\Omega _P}{\sigma _{21}} + \hbar {\Omega _P}\left({{\Pi _{\rm{SPP}}}} \right){\sigma _{21}}\\ + \hbar {\Omega _p}\Phi _{\rm{DDI}}^{\rm{MNP}}{\sigma _{23}}\\ + \hbar {\Omega _P}\left({\Phi _{\rm{DDI}}^{\rm{QE}}} \right){\sigma _{34}} + hc\end{array} \right],$$
where ${\Omega _P} = {\mu _{21}}{E_P}$ is the Rabi frequency.

Using the density matrix method of [38] and Eq. (20), equations of motion for density matrix elements are found after extensive mathematical manipulations as follows:

$$\begin{split}\frac{{d{\rho _{11}}}}{{dt}} &= 2\gamma _{\textit{nr}}^{23}{\rho _{22}} - i\Omega _P\left({1 + {\Pi _{\rm{SPP}}}} \right)\left({{\rho _{12}} - {\rho _{21}}} \right)*\\\frac{{d{\rho _{22}}}}{{dt}}& = \left[\begin{array}{l} - 2\gamma _r{\rho _{22}} + i\Omega _P\left({1 + {\Pi _{\rm{SPP}}}} \right)\left({{\rho _{12}} - {\rho _{21}}} \right)\\ + i\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\left({{\rho _{32}} - {\rho _{23}}} \right)\end{array} \right],\end{split}$$
$$\begin{split}\frac{{d{\rho _{33}}}}{{dt}} &= \left[\begin{array}{l}2{\gamma _{31}}{\rho _{22}} + i\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\left({{\rho _{23}} - {\rho _{32}}} \right)\\ + i\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}}\left({{\rho _{43}} - {\rho _{34}}} \right)*\end{array} \right],\\\frac{{d{\rho _{12}}}}{{dt}} &= \left[\begin{array}{l} - {d_{21}}{\rho _{12}} - i\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}{\rho _{13}}\\ + i({\rho _{22}} - {\rho _{11}}){\Omega _p}\left({1 + {\Pi _{\rm{SPP}}}} \right)\end{array} \right],\end{split}$$
$$\begin{split}\frac{{d{\rho _{13}}}}{{dt}} &= \left[\begin{array}{l} - {d_{13}}{\rho _{13}}\, - i{\rho _{12}}\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\\ + i{\rho _{23}}\Omega _p\left({1 + {\Pi _{\rm{SPP}}}} \right) - i{\rho _{14}}\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}}\end{array} \right],\\[-4pt]\frac{{d{\rho _{23}}}}{{dt}} &= \left[\begin{array}{l} - {d_{23}}{\rho _{23}}\, + i({\rho _{33}} - {\rho _{22}})\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\\ + i{\rho _{13}}\Omega _p\left({1 + {\Pi _{\rm{SPP}}}} \right) - i{\rho _{24}}\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}}\end{array} \right],\end{split}$$
$$\begin{split}\frac{{d{\rho _{14}}}}{{dt}} &= \left[\begin{array}{l} - {d_{14}}{\rho _{14}}\, - i{\rho _{13}}\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}}\\ + i{\rho _{24}}\Omega _p\left({1 + {\Pi _{\rm{SPP}}}} \right)\end{array} \right],\\[-4pt]\frac{{d{\rho _{24}}}}{{dt}} &= \left[\begin{array}{l} - {d_{24}}{\rho _{24}}\, + i{\rho _{34}}\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\\ - i{\rho _{23}}\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}} + i{\rho _{13}}\Omega _p\left({1 + {\Pi _{\rm{SPP}}}} \right)\end{array} \right],\\[-4pt]\frac{{d{\rho _{34}}}}{{dt}}& = \left[\begin{array}{l} - {d_{34}}{\rho _{34}} + i{\rho _{24}}\Omega _P\Phi _{\rm{DDI}}^{\rm{QE}}\\ + i\Omega _P\Phi _{\rm{DDI}}^{\rm{MNP}}\left({{\rho _{44}} - {\rho _{33}}} \right)\end{array} \right].\end{split}$$

Parameters appearing in Eq. (26) are found as

$$\begin{split}{d_{21}} &= {\delta _r} + i{\gamma _r},{d_{23}} = {\delta _{23}} + i\gamma _{\textit{nr}}^{23},\\[-4pt]{d_{43}} &= {\delta _{43}} + i\gamma _{\textit{nr}}^{34},\\[-4pt]{d_{41}}& = {\delta _{41}} + i(\gamma _r + \gamma _{\textit{nr}}^{34})/2,\\[-4pt]{d_{42}} &= {\delta _{42}} + i({\gamma _r} + \gamma _{\textit{nr}}^{23})/2,\\[-4pt]{d_{31}} &= {\delta _{31}} + i({\gamma _r} + \gamma _{\textit{nr}}^{34})/2,\end{split}$$
$$\begin{split}{\delta _{21}} &= {\omega _p} - {\omega _{21}},{\delta _{23}} = {\omega _d} - {\omega _{23}},\\[-4pt]{\delta _{34}} &= {\omega _{\textit{sp}}} - {\omega _{34}},{\delta _{31}} = 0,\\[-4pt]{\delta _{42}} &= {\delta _{23}} + {\delta _{34}},\\[-4pt]{\delta _{14}} &= {\delta _{21}} - {\delta _{24}},{\delta _{13}} = {\delta _{21}} - {\delta _{23}}.\end{split}$$
Here ${\delta _{\textit{ij}}}$ is called the field detuning.

We solve the density matrix equations given in Eq. (26) in the steady state and consider that the electron population of the ground state is larger than the excited states. After extensive mathematical manipulations, we find the following analytical expression of the density matrix element ${\rho _{12}}$ as

$$\rho _{12} = - \frac{{i\left({{\Omega _P} + {\Omega _P}\Pi _{\rm{SPP}}} \right)\left({{d_{13}}{d_{14}} - {{({\Omega _P}\Phi _{\rm{DDI}}^{\rm{MNP}})}^2}} \right)}}{{{d_{21}}\left({{d_{13}}{d_{14}} - {{({\Omega _P}\Phi _{\rm{DDI}}^{\rm{MNP}})}^2}} \right) + {d_{14}}{{({\Omega _P}\Phi _{\rm{DDI}}^{\rm{QD}})}^2}}}.$$

Inserting Eq. (28) into Eq. (13), we get the analytical expression of the photoresponse coefficient as

$$\begin{split}{P_{\rm{PRC}}}& = {\eta _{\rm{QE}}}{\left({1 + \Pi _{\rm{SPP}}} \right)^2}{\rm Im}\left({\frac{{{R_{\textit{num}}}}}{{{R_{\textit{den}}}}}} \right),\\[-4pt]{R_{\rm{num}}} &= \left[\begin{array}{l}i{\Omega _P}\left({1 + \Pi _{\rm{SPP}}} \right) \times \\\left({{d_{13}}{d_{14}} - \Omega _P^2{{(\Phi _{\rm{DDI}}^{\rm{MNP}})}^2}} \right)\end{array} \right],\\[-4pt]{R_{\rm{den}}}& = \left[\begin{array}{l}{d_{21}}\left({{d_{13}}{d_{14}} - \Omega _P^2{{(\Phi _{\rm{DDI}}^{\rm{MNP}})}^2}} \right)\\ + {d_{14}}\Omega _P^2{(\Phi _{\rm{DDI}}^{\rm{QE}})^2}\end{array} \right].\end{split}$$

We found that the analytical expression of the photocurrent depends on the SPP field, the DDI-MNP field, and the DDI-QD field.

Note that the photocurrents depend on the quantum yield. In the presence of the nonradiative decay rates due to the DDI couplings, the value of the quantum yield is less than 1 (${\eta _{\rm{QE}}} \lt 1)$; meanwhile, when the DDI coupling is absent then we have ${\eta _{\rm{QE}}} = 1$. This means that the DDI coupling is responsible for photocurrent quenching. We also established that the SPP coupling is mainly responsible for photocurrent enhancement. Therefore, there is a competition between the DDI nonradiative decay and the SPP coupling, which decides whether there will be enhancement or quenching in photocurrent in the plasmonic nanofibers.

5. RESULTS AND DISCUSSION

In this section, we compare our theory with the experimental data from [19] for photoresponse. Yakimov et al. [19] measured the photoresponse spectrum, photocurrent, and extinction cross section in a nanowaveguide containing an ensemble of Ge/Si QDs and MNPs fabricated from an Al disk array. The ensemble of Ge/Si QDs is located in the core of the nanofiber. The shell is fabricated from Al-MNP discs and Si layers. The Si layer lies in between the Ge/Si QDs and Al-MNP discs. A schematic diagram of the nanowaveguide is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Schematic diagram of the nanowaveguide coating with Al-MNPs and Ge-Si QDs.

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Yakimov et al. [19] observed a significant enhancement in the photoresponse spectrum and the extinction cross section of Ge/Si QDs due to the presence of Al-MNP discs and radiative coupling to the nanowaveguide. The linewidth and the exciton energy for the Ge/Si QDs is found as 1055 nm and ${\Gamma _p} =$ 0.07 ${\Omega _p}$. All the energies/frequencies in the numerical calculations are measured with respect to the Rabi frequency ${\Omega _p}$. The SPP resonance wavelength (frequency) for disks of 150 nm, 200 nm, and 225 nm diameter are found as 1140 nm, 1160 nm, and 1165 nm, respectively.

The theoretical and experiential values of the PRC spectrum are plotted in Fig. 4 as a function of wavelength. The open circles correspond to the Ge-Si quantum dots alone. The diamonds, squares, and asterisks correspond to the doped Al-MNPs with disk diameters 150 nm, 200 nm, and 225 nm, respectively. The solid, dotted, dashed, and dashed–dotted are theoretical curves. The physical parameters are for the sample. The SPP coupling parameters (${\Pi _{\rm{SPP}}}$) for the sample 150 nm, 200 nm, and 225 nm are found as 1.10, 1.20, and 1.85, respectively. The linewidth of these samples has increased about 12%–40% due to the presence of the SPP decay rate. The DDI coupling constant is taken the same for all samples since the concentration of the Al-MNPs and QDs does not change. The quantum yield is taken as unity since the quenching does not occur in these experiments. Note that a good agreement between our theory and experiment is found. Our theory predicts that there is an enhancement in photoresponse spectrum extinction cross sections in the presence of Al-MNPs. We have also found that as the size of the MNP increases the enhancement also increases.

 figure: Fig. 4.

Fig. 4. Theoretical and experiential values for the photoresponse spectrum (A.U.) are plotted as a function of wavelength (nm). The open circles correspond to the Ge-Si quantum dots alone. The diamonds, squares, and asterisks correspond to the doped Al-MNPs with disk diameters 150 nm, 200 nm, and 225 nm, respectively. The solid, dotted, dashed, and dashed–dotted lines are theoretical curves.

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A physical explanation of the mechanism of enhancement is given as follows. In the absence of the Al-MNPs, the SPP and DDI couplings are absent as well. Therefore, the photoresponse spectrum extinction cross section is due to the external field. On the other hand, when the Al-MNP is doped, the SPP field appears in the system, which also falls on the QDs. Therefore, the SPP field interacts with the exciton of the QDs. The intensity of this coupling is denoted by the SPP coupling parameter ${{\Pi}_{\rm{SPP}}}$. According to the aforementioned theory, the SPP coupling parameter appears in the numerator. Therefore, in the presence of Al-MNPs, the two fields are falling on the QDs, and that is why there is an enhancement in the photoresponse spectrum extinction cross section.

We also found that as the radius of the Al-MNP increases, the photoresponse spectrum and the extinction cross section also experience an increment. The SPP field and the SPP coupling parameter ${{\Pi}_{\rm{SPP}}}$ are proportional to ${\rm R}_{\rm{MNP}}^3$. Hence, as the radius of the Al-MNP increases, so does the SPP coupling. This in turn enhances the photoresponse spectrum extinction cross section. Note that the concentration of Al-MNPs is small and invariant in the experiment. Hence the DDI and the quenching term do not have a significant effect on the experimental results.

We would like to investigate the effect of SPP interaction on the PRC enhancement of a general PNF. Here the PNF is made of a nanofiber containing an ensemble of QDs and MNPs. We have plotted a three-dimensional figure of the normalized power term (${W_{\rm{QE}}}/{W_0}$) as a function of the probe detuning and the SPP coupling. The results are plotted in Fig. 5. We have considered that the DDI-MNP and DDI-QD fields are in resonance with the exciton transitions $| {2\rangle \leftrightarrow} |3\rangle$ and $| {4\rangle \leftrightarrow} |3\rangle$, respectively. The physical parameters such as decay rate and coupling constant used in the calculations are measured with respect to the exciton decay rate $\Gamma_{P}$. For example, the probe detuning ($\delta_{21}$) is normalized as ${\delta _{21}}/\;{\Gamma _P}$, which is a unitless quantity. We have considered $\Phi _{\rm{DDI}}^{\rm{MNP}} = \Phi _{\rm{DDI}}^{\textit{QD}} = 0.2,$ $\;{\Pi _{\rm{SPP}}} = {\Omega _P}/2$, ${{\Gamma}_{\rm{SPP}}}/{{\Gamma}_P}= {\Gamma}_{\rm{DDI}}^{\rm{MNP}}/{{\Gamma}_P} = {\Gamma}_{\rm{DDI}}^{\textit{QD}}/{{\Gamma}_P} = 0.1$. As shown in Fig. 5, there is an enhancement in the power term due to the exciton–SPP coupling, which in turn enhances the PL spectrum.

 figure: Fig. 5.

Fig. 5. The power term (${{\rm W}_{QD/}}{{\rm W}_0}$) (A.U.) is plotted as a function of the normalized probe detuning ${\delta _{21}}/\;{\Gamma _P}$ (unitless) and the normalized SPP coupling ${\Pi _{\rm{SPP}}}$ (unitless). One can see from the figure that the PL spectrum is enhanced.

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Next, we study the effect of DDI interaction on the nonradiative decay rates and the quantum yield of a general PNF. We have plotted a three-dimensional figure for the PRC. The results are shown in Fig. 6 where the PRC is plotted as a function of the normalized probe detuning and the DDI coupling. The physical parameters used are the same as that of Fig. 5. Here we have considered $\Phi _{\rm{DDI}}^{\textit{QD}} = \Phi _{\rm{DDI}}^{\rm{MNP}}$ in the calculation of DDI-QD and DDI-MNP decay rates. One can see from Fig. 6 that as the DDI decay rate increases the PRC decreases. The photocurrent also decreases with the increase of the radiative DDI decay rate since it is directly related to the PRC. The decrease of the PRC and the photocurrent is called quenching. In other words, our theory predicts that radiative DDI decay rate can quench the photocurrent in the MNFs; meanwhile, the quantum yield is responsible for the quenching phenomenon.

 figure: Fig. 6.

Fig. 6. The PL spectrum (A.U.) is plotted as a function of the normalized probe detuning ${\delta _{21}}/\;{\Gamma _P}$ (unitless) and the normalized DDI-QD coupling $\Phi _{\rm{DDI}}^{\textit{QD}}$ (unitless). The PL spectrum splits into two quenched states at higher values of the DDI-QD coupling.

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Finally, we study the effect of DDI coupling on the power term ($W_{QD}/W_{0}$) and the PL quenching. We consider that $\Phi _{\rm{DDI}}^{\rm{MNP}} = \Phi _{\rm{DDI}}^{\textit{QD}}$=${\Phi _{\rm{DDI}}}$. Note that the DDI coupling ${\Phi _{\rm{DDI}}}$ depends on the concertation of MNPs. As the concentration of MNPs increases, so does the DDI coupling. In Fig. 7, we have plotted the power term (${{\rm W}_{QD/}}{{\rm W}_0}$) as a function of the DDI coupling for different values of the magnitude of the probe field detunings ($|{\delta _{21}}|/{\Gamma _P}$). The solid line, dashed line, and dashed-dotted line correspond to $|{\delta _{21}}|/{\Gamma _{P}} = {0}$, 0.5, and 1.0. To show the effect of the DDI coupling more clearly, we have also plotted a three-dimensional figure for the power term as a function of the magnitude of probe detuning and DDI coupling ($\Phi _{\rm{DDI}}^{\rm{QD}}$) in Fig. 8. It is interesting to note that a peak appears in the PL spectrum along the DDI coupling axis.

 figure: Fig. 7.

Fig. 7. The power term (${{\rm W}_{\rm QD/}}{{\rm W}_0}$) is plotted as a function of the DDI coupling (${\Phi _{\rm{DDI}}})$ for different values of the magnitude of the probe field detunings ($|{\delta _{21}}|/{\Gamma _P}$). The solid line, dashed line, and dashed–dotted line correspond to $|{\delta _{21}}|/{\Gamma _{P}} = {0}$, $|{\delta _{21}}|/\;{\Gamma _{P\:}} = {0.5}$, and $|{\delta _{21}}|/{\Gamma _{P}} = {1.0}$, respectively.

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 figure: Fig. 8.

Fig. 8. The power term (${{\rm W}_{\rm QD/}}{{\rm W}_0}$) is plotted as a function of the DDI coupling (${\Phi _{\rm{DDI}}})$ and the magnitude of the probe field detunings ($|{\delta _{21}}|/{\Gamma _P}$).

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The physics of Figs. 7 and 8 can be understood as follows. When the probe field energy is in resonance with the QD exciton energy (i.e., zero detuning, $|{\delta _{21}}|/{\Gamma _P} = 0$), the power term ${{\rm W}_{\rm QD/}}{{\rm W}_0}$ decreases as the strength of the DDI coupling increases (see the solid line), which causes a rise in the PL quenching. The power term has a maximum at zero detuning in the absence of the DDI coupling (i.e., ${\Phi _{\rm{DDI}}} = 0$). The following decrease can be attributed to the presence of the DDI terms ($\Phi _{\rm{DDI}}^{\rm{MNP}},\Phi _{\rm{DDI}}^{\rm{QD}}$) in the denominator of the power term. On the other hand, when the probe field energy is not in resonance (i.e., $|\delta_{21}|/\Gamma_{P}\ne 0$), the power term (or PL intensity) first increases and reaches the maximum, and thereafter decreases as the strength of the DDI coupling increases.

We found that the maximum value of the power spectrum in Figs. 7 and 8 shifts from the zero detuning ($|\delta_{21}|/\Gamma_{P}=0$) to the nonzero detuning ($| \delta_{21}|/\Gamma_{P}\ne 0$). Note that the maximum at the zero detuning is located where the DDI coupling is zero (${\Phi _{\rm{DDI}}} = 0$), while the maximum at the nonzero detuning is located in the presence of the DDI coupling (${\Phi _{\rm{DDI}}} \ne 0$), which means that the cause of the shift of the maximum is the DDI coupling. The maximum at zero detuning is due to the transition $| {2\rangle \leftrightarrow} |1\rangle$. In the presence of strong DDI coupling, level $|2\rangle$ splits into two dressed states $|{2_ -}\rangle$ and $|{2_ +}\rangle$. Therefore, the maxima at nonzero detuning are due to the transition $| {{2_ \pm}\rangle \leftrightarrow} |1\rangle$.

Note also in Figs. 7 and 8 that as the strength of the DDI coupling increases the distance between the zero-detuning maxima and the nonzero maxima increases. For example, see the dashed and dashed-dotted curves in Fig. 7. This is because the splitting of the dressed states increases as the DDI coupling increases. In other words, the energy difference between levels $|2\rangle$ and $|{2_ \pm}\rangle$ increases with the strength of the DDI coupling. The shift of the zero-detuning maxima (OFF) and the nonzero maxima (ON) due to the DDI coupling can be used to make nanoswitches from present PNFs.

Finally, we comment on the cylindrical geometry and the planar geometry shown in Fig. 1 and Fig. 3, respectively. The main aim of the paper is to develop a theory of photocurrent for PNFs doped with MNPs and QDs and compare the theory with experiments having planer geometry. Generally, nanofibers have cylindrical geometry, which is why we have plotted Fig. 1 as a schematic diagram for PNFs. However, in comparing our theory with experiments, we have calculated the bound states for experimental planar geometry not for the cylindrical geometry. For example, see Eq. (18), where propagating vector ${ k}\;({{ k}_x},\;{{ k}_y},\;{{ k}_z})$ is written in the ${x},\;{ y},\;{ z}$ coordinates instead of the cylindrical coordinates. That is why ${ k}\;({{ k}_x},\;{{ k}_y},\;{{ k}_z})$ is quantized to ${k}\;({{k}_n},\;{{ k}_m},\;{{ k}_z})$ because of the planar geometry. In other words, we have used the planer geometry to compare our theory with experiments. The distance ${{ r}_s}$ given in Eq. (14) is taken as the thickness of the Si Cap as shown Fig. 3. The SPP coupling parameters taken as 1.10, 1.20, and 1.85 for the Al-MNP disks of different radii are determined by a fitting to the experimental curves in Fig. 4.

6. SUMMARY

The plasmonic properties of photonic nanofibers doped with an ensemble of quantum dots and metallic nanoparticles are investigated in this paper. We have shown that the number of bound states in the nanofiber hybrid depends on the dielectric constant of the ensemble of metallic nanoparticles and quantum dots. The photoluminescence and photoresponse spectrum are calculated using the density matrix method in the presence of radiative and nonradiative scattering processes. The radiative and nonradiative decay rates are calculated using quantum mechanical perturbation theory due to the scattering of excitons with surface plasmon polaritons and dipole–dipole interactions. We predicted that the exciton–DDI coupling is responsible for the quenching of photocurrent. On the other hand, the exciton–SPP coupling is responsible for the enhancement of the PL spectrum. We found that when the probe field energy is in resonance with that of the QD exciton, the photocurrent quenching increases as the strength of the DDI coupling increases. Meanwhile, when the probe field energy is not in resonance, the PRC spectrum first experiences growth and reaches the maximum value. After that, the photocurrent quenching increases as the strength of the DDI coupling increases. We also compared our theory with experimental results of the photocurrent of a nanofiber doped with Al-MNP nanodisks and Ge/Si QDs. A good agreement between theory and experiment is found.

Funding

Natural Sciences and Engineering Research Council of Canada (RGOIN-2018-05646).

Acknowledgment

The authors are thankful to Mr. Grant Brassem for helping us to rewrite the introduction section and correcting the typing and English errors. The authors are also thankful to Ms. Ningyan Fang for retyping the references in the present format and correcting typing errors. Finally, one of the authors (MRS) is grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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19. A. I. Yakimov, V. V. Kirienko, A. A. Bloshkin, A. V. Dvurechenskii, and D. E. Utki, “Near-infrared photoresponse in Ge/Si quantum dots enhanced by localized surface plasmons supported by aluminum nanodisks,” J. Appl. Phys. 128, 143101 (2020). [CrossRef]  

20. A. I. Yakimov, V. V. Kirienko, and A. V. Dvurechenskii, “Comparison of two-dimensional arrays of gold disks and holes for plasmonic enhancement of Ge/Si quantum dot mid-infrared photodetectors,” Opt. Mater. Express 8, 3479–3484 (2018). [CrossRef]  

21. M.-H. Jung, Y. J. Yun, M.-J. Chu, and M. G. Kang, “Fabrication of Ag nanoparticles embedded in TiO2 nanotubes: using electrospun nanofibers for controlling plasmonic effects,” Chem. Eur. J. 19, 8543–8549 (2013). [CrossRef]  

22. H. Sugimoto, R. Zhang, B. M. Reinhard, M. Fujii, G. Perotto, B. Marelli, F. G. Omenetto, and L. NegroDal, “Enhanced photoluminescence of Si nanocrystals-doped cellulose nanofibers by plasmonic light scattering,” Appl. Phys. Lett. 107, 41111 (2015). [CrossRef]  

23. N. Sobti, A. Bensouici, F. Coloma, C. Untiedt, and S. Achour, “Structural and photoelectrochemical properties of porous TiO2 nanofibers decorated with Fe2O3 by sol-flame,” J. Nanopart. Res. 16, 2577 (2014). [CrossRef]  

24. B. Sebo, N. Huang, Y. Liu, Q. Tai, L. Liang, H. Hu, S. Xu, and X.-Z. Zhao, “Dye-sensitized solar cells enhanced by optical absorption, mediated by TiO2 nanofibers and plasmonics Ag nanoparticles,” Electrochim. Acta 112, 458–464 (2013). [CrossRef]  

25. M. R. Singh, G. Brassem, and P. Persaud, “Spontaneous emission in nanofibers doped with ensemble of quantum dots and quantum emitters,” Phys. Rev. A 102, 063704 (2020). [CrossRef]  

26. 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. B 82, 195419 (2010). [CrossRef]  

27. B. D. Fainberg, N. N. Rosanov, and N. A. Veretenov, “Light-induced ‘plasmonic’ properties of organic materials: surface polaritons and switching waves in bistable organic thin films,” Appl. Phys. Lett. 110, 203301 (2017). [CrossRef]  

28. Y. Suganuma, P. E. Trudeau, B. Leathem, B. Shieh, and A. Dhirani, “Discrete electron forces in a nanoparticle-tunnel junction system,” J. Chem. Phys. 118, 9769–9772 (2003). [CrossRef]  

29. A. F. Terzis, S. G. Kosionis, J. Boviatsis, and E. Paspalakis, “Nonlinear optical susceptibilities of semiconductor quantum dot–metal nanoparticle hybrids,” J. Mod. Opt. 63, 451–461 (2016). [CrossRef]  

30. M. R. Singh, M. SekharChandra, S. Balakrishnan, and S. Masood, “Medical applications of hybrids made from quantum emitter and metallic nanoshell,” J. Appl. Phys. 122, 034306 (2017). [CrossRef]  

31. M. R. Singh, J. D. Cox, and M. Brzozowski, “Photoluminescence and spontaneous emission enhancement in metamaterial nanostructures,” J. Phys. D 47, 085101 (2014). [CrossRef]  

32. M. R. Singh and S. Yastrebov, “Switching and sensing using Kerr nonlinearity in quantum dots doped in metallic nanoshells,” J. Phys. Chem. C124, 12065–12074 (2020). [CrossRef]  

33. M. R. Singh, “Theory of all-optical switching based on Kerr nonlinearity in metallic nanohybrids,” Phys. Rev. A 102, 013708 (2020). [CrossRef]  

34. M. R. Singh and P. D. Persaud, “Dipole–Dipole interaction in two-photon spectroscopy of metallic nanohybrids,” J. Phys. Chem. 124, 6311–6320 (2020). [CrossRef]  

35. J. Guo, K. Black, J. Hu, and M. Singh, “Study of plasmonics in hybrids made from a quantum emitter and double metallic nanoshell dimer,” J. Phys. Condens. Matter. 30, 185301 (2018). [CrossRef]  

36. M. R. Singh, “The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials,” J. Mod. Opt. 54, 1739–1757 (2007). [CrossRef]  

37. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University, 2006), Section 8.3.3 on page 266.

38. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

39. M. Singh, Electronic, Photonic, Polaritonic and Plasmonic Materials (Wiley Custom, 2014).

40. E. Hamura, Y. Kawabe, and A. Yamanaka, Quantum Nonlinear Optics (Springer, 2007), Chap. 6.

41. R. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Chap. 2.

42. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications (Cambridge University, 2010).

43. K. Okamoto, Fundamentals of Optical Waveguides (Elsevier, 2006), Chap. 2.

44. M. J. Adam, Introduction to Optical Wave Guide (Wiley, 1981).

45. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2007).

References

  • View by:

  1. J. Ding, Y. Sui, W. Fu, H. Yang, B. Zhao, and M. Li, “ZnO Nanorod array/CuAlO2 nanofiber heterojunction on Ni substrate: synthesis and photoelectrochemical properties,” Nanotechnology 22, 295706 (2011).
    [Crossref]
  2. D. Choi, Y. Jang, J. H. Lee, G. H. Jeong, D. Whang, S. W. Hwang, K. S. Cho, and S. W. Kim, “Diameter-controlled and surface-modified Sb2Se3 nanowires and their photodetector performance,” Sci. Rep. 4, 6714 (2014).
    [Crossref]
  3. F. O. Ochanda and M. R. Barnett, “Synthesis and characterization of silver nanoparticles and Titanium oxide nanofibers: toward multifibrous nanocomposites,” J. Am. Ceram. Soc. 93, 2637–2643 (2010).
    [Crossref]
  4. Z. Haiguang, L. Guiju, Y. Shujie, F. V. A. Camargo, Z.-R. Margherita, W. Xiaohan, S. Changchun, L. Bing, Z. Yuanming, H. Guangting, V. Alberto, and G. Xiao, “Gram-scale synthesis of carbon quantum dots with a large Stokes shift for the fabrication of eco-friendly and high-efficiency luminescent solar concentrators,” Energy Environ. Sci. 14, 396–407 (2021).
    [Crossref]
  5. Z. Haiguang, S. Ruijia, W. Zhaofen, F. Kaifang, H. Xun, and Z. Yuhai, “Zero-dimensional perovskite nanocrystals for efficient luminescent solar concentrators,” Adv. Funct. Mater. 29, 1902262 (2019).
    [Crossref]
  6. A. Baranowska-Korczyc, K. Sobczak, P. Dłuzewski, A. Reszka, B. J. Kowalski, Ł. Kłopotowski, D. Elbaum, and K. Fronc, “Facile synthesis of core/shell ZnO/ZnS nanofibers by electrospinning and gas-phase sulfidation for biosensor applications,” Phys. Chem. Chem. Phys. 17, 24029–24037 (2015).
    [Crossref]
  7. H. Liu, N. Gao, M. Liao, and X. Fang, “Hexagonal-like Nb2O5 nanoplates-based photodetectors and photocatalyst with high performances,” Sci. Rep. 5, 7716 (2015).
    [Crossref]
  8. J. A. Alexander-Webber, C. K. Groschner, A. A. Sagade, G. Tainter, M. F. ZalbaGonzalez-, R. PietroDi, J. Wong-Leung, H. H. Tan, C. Jagadish, S. Hofmann, and H. J. Joyce, “Engineering the photoresponse of InAs nanowires,” ACS Appl. Mater. Interfaces 9, 43993–44000 (2017).
    [Crossref]
  9. J. Treu, T. Stettner, M. Watzinger, S. Morkötter, M. Döblinger, S. Matich, K. Saller, M. Bichler, G. Abstreiter, J. J. Finley, J. Stangl, and G. Koblmüller, Lattice-matched InGaAs-InAlAs core-shell nanowires with improved luminescence and photoresponse properties,” Nano Lett. 15, 3533–3540 (2015).
    [Crossref]
  10. K. Erotokritou, R. M. Heath, G. G. Taylor, C. Tian, A. Banerjee, A. Casaburi, C. M. Natarajan, S. Miki, H. Terai, and R. H. Hadfield, “Nano-optical photoresponse mapping of superconducting nanowires with enhanced near infrared absorption,” Supercond. Sci. Technol. 31, 125012 (2018).
    [Crossref]
  11. F. Fang, X. J. Wang, X. Fang, T. Li, Z. Wei, J. Li, and X. Chu, “Effects of shell characteristics on the UV photoresponse behaviors of photoconductivity based on ZnO/ZnMgO core-shell nanowire arrays,” Mater. Res. Express. 2, 076201 (2015).
    [Crossref]
  12. M. Ding, N. Yao, C. Wang, J. Huang, M. Shao, S. Zhang, P. Li, X. Deng, and X. Xu, “ZnO@CdS core-shell heterostructures: fabrication, enhanced photocatalytic, and photoelectrochemical performance,” Nanoscale Res. Lett. 11, 205 (2016).
    [Crossref]
  13. J. Guan, L. K. Sagar, R. Li, D. Wang, G. Bappi, W. Wang, N. Watkins, M. R. Bourgeois, L. Levina, F. Fan, S. Hoogland, O. Voznyy, J. M. PinaDe, R. D. Schaller, G. C. Schatz, E. H. Sargent, and T. W. Odom, “Quantum dot-plasmon lasing with controlled polarization patterns,” ACS Nano 14, 3426–3433 (2020).
    [Crossref]
  14. D. Liu, H. J. Li, J. Gao, S. Zhao, Y. Zhu, P. Wang, D. Wang, A. Chen, X. Wang, and J. Yang, “High-performance ultraviolet photodetector based on graphene quantum dots decorated ZnO nanorods/GaN film isotype heterojunctions,” Nanoscale Res. Lett. 13, 261 (2018).
    [Crossref]
  15. P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
    [Crossref]
  16. M. G. C. M. Kumari, C. S. Perera, B. S. Dassanayake, M. A. K. L. Dissanayake, and G. K. R. Senadeera, “Highly efficient plasmonic dye-sensitized solar cells with silver nanowires and TiO2 nanofibres incorporated multi-layered photoanode,” Electrochim. Acta 298, 330–338 (2019).
    [Crossref]
  17. M. A. K. L. Dissanayake, T. Jaseetharan, G. K. R. Senadeera, J. M. K. W. Kumari, C. A. Thotawatthage, B. E. Mellander, I. Albinson, and M. Furlani, “Highly efficient, PbS:Hg quantum dot–sensitized, plasmonic solar cells with TiO2 triple-layer photoanode,” J. Solid State Electrochem. 23, 1787–1794 (2019).
    [Crossref]
  18. J. Y. Chen, H. C. Wu, Y. C. Chiu, and W. C. Chen, “Plasmon-enhanced polymer photovoltaic device performance using different patterned Ag/PVP electrospun nanofibers,” Adv. Energy Mater. 4, 1301665 (2014).
    [Crossref]
  19. A. I. Yakimov, V. V. Kirienko, A. A. Bloshkin, A. V. Dvurechenskii, and D. E. Utki, “Near-infrared photoresponse in Ge/Si quantum dots enhanced by localized surface plasmons supported by aluminum nanodisks,” J. Appl. Phys. 128, 143101 (2020).
    [Crossref]
  20. A. I. Yakimov, V. V. Kirienko, and A. V. Dvurechenskii, “Comparison of two-dimensional arrays of gold disks and holes for plasmonic enhancement of Ge/Si quantum dot mid-infrared photodetectors,” Opt. Mater. Express 8, 3479–3484 (2018).
    [Crossref]
  21. M.-H. Jung, Y. J. Yun, M.-J. Chu, and M. G. Kang, “Fabrication of Ag nanoparticles embedded in TiO2 nanotubes: using electrospun nanofibers for controlling plasmonic effects,” Chem. Eur. J. 19, 8543–8549 (2013).
    [Crossref]
  22. H. Sugimoto, R. Zhang, B. M. Reinhard, M. Fujii, G. Perotto, B. Marelli, F. G. Omenetto, and L. NegroDal, “Enhanced photoluminescence of Si nanocrystals-doped cellulose nanofibers by plasmonic light scattering,” Appl. Phys. Lett. 107, 41111 (2015).
    [Crossref]
  23. N. Sobti, A. Bensouici, F. Coloma, C. Untiedt, and S. Achour, “Structural and photoelectrochemical properties of porous TiO2 nanofibers decorated with Fe2O3 by sol-flame,” J. Nanopart. Res. 16, 2577 (2014).
    [Crossref]
  24. B. Sebo, N. Huang, Y. Liu, Q. Tai, L. Liang, H. Hu, S. Xu, and X.-Z. Zhao, “Dye-sensitized solar cells enhanced by optical absorption, mediated by TiO2 nanofibers and plasmonics Ag nanoparticles,” Electrochim. Acta 112, 458–464 (2013).
    [Crossref]
  25. M. R. Singh, G. Brassem, and P. Persaud, “Spontaneous emission in nanofibers doped with ensemble of quantum dots and quantum emitters,” Phys. Rev. A 102, 063704 (2020).
    [Crossref]
  26. 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. B 82, 195419 (2010).
    [Crossref]
  27. B. D. Fainberg, N. N. Rosanov, and N. A. Veretenov, “Light-induced ‘plasmonic’ properties of organic materials: surface polaritons and switching waves in bistable organic thin films,” Appl. Phys. Lett. 110, 203301 (2017).
    [Crossref]
  28. Y. Suganuma, P. E. Trudeau, B. Leathem, B. Shieh, and A. Dhirani, “Discrete electron forces in a nanoparticle-tunnel junction system,” J. Chem. Phys. 118, 9769–9772 (2003).
    [Crossref]
  29. A. F. Terzis, S. G. Kosionis, J. Boviatsis, and E. Paspalakis, “Nonlinear optical susceptibilities of semiconductor quantum dot–metal nanoparticle hybrids,” J. Mod. Opt. 63, 451–461 (2016).
    [Crossref]
  30. M. R. Singh, M. SekharChandra, S. Balakrishnan, and S. Masood, “Medical applications of hybrids made from quantum emitter and metallic nanoshell,” J. Appl. Phys. 122, 034306 (2017).
    [Crossref]
  31. M. R. Singh, J. D. Cox, and M. Brzozowski, “Photoluminescence and spontaneous emission enhancement in metamaterial nanostructures,” J. Phys. D 47, 085101 (2014).
    [Crossref]
  32. M. R. Singh and S. Yastrebov, “Switching and sensing using Kerr nonlinearity in quantum dots doped in metallic nanoshells,” J. Phys. Chem. C124, 12065–12074 (2020).
    [Crossref]
  33. M. R. Singh, “Theory of all-optical switching based on Kerr nonlinearity in metallic nanohybrids,” Phys. Rev. A 102, 013708 (2020).
    [Crossref]
  34. M. R. Singh and P. D. Persaud, “Dipole–Dipole interaction in two-photon spectroscopy of metallic nanohybrids,” J. Phys. Chem. 124, 6311–6320 (2020).
    [Crossref]
  35. J. Guo, K. Black, J. Hu, and M. Singh, “Study of plasmonics in hybrids made from a quantum emitter and double metallic nanoshell dimer,” J. Phys. Condens. Matter. 30, 185301 (2018).
    [Crossref]
  36. M. R. Singh, “The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials,” J. Mod. Opt. 54, 1739–1757 (2007).
    [Crossref]
  37. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University, 2006), Section 8.3.3 on page 266.
  38. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  39. M. Singh, Electronic, Photonic, Polaritonic and Plasmonic Materials (Wiley Custom, 2014).
  40. E. Hamura, Y. Kawabe, and A. Yamanaka, Quantum Nonlinear Optics (Springer, 2007), Chap. 6.
  41. R. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Chap. 2.
  42. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications (Cambridge University, 2010).
  43. K. Okamoto, Fundamentals of Optical Waveguides (Elsevier, 2006), Chap. 2.
  44. M. J. Adam, Introduction to Optical Wave Guide (Wiley, 1981).
  45. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2007).

2021 (1)

Z. Haiguang, L. Guiju, Y. Shujie, F. V. A. Camargo, Z.-R. Margherita, W. Xiaohan, S. Changchun, L. Bing, Z. Yuanming, H. Guangting, V. Alberto, and G. Xiao, “Gram-scale synthesis of carbon quantum dots with a large Stokes shift for the fabrication of eco-friendly and high-efficiency luminescent solar concentrators,” Energy Environ. Sci. 14, 396–407 (2021).
[Crossref]

2020 (6)

J. Guan, L. K. Sagar, R. Li, D. Wang, G. Bappi, W. Wang, N. Watkins, M. R. Bourgeois, L. Levina, F. Fan, S. Hoogland, O. Voznyy, J. M. PinaDe, R. D. Schaller, G. C. Schatz, E. H. Sargent, and T. W. Odom, “Quantum dot-plasmon lasing with controlled polarization patterns,” ACS Nano 14, 3426–3433 (2020).
[Crossref]

A. I. Yakimov, V. V. Kirienko, A. A. Bloshkin, A. V. Dvurechenskii, and D. E. Utki, “Near-infrared photoresponse in Ge/Si quantum dots enhanced by localized surface plasmons supported by aluminum nanodisks,” J. Appl. Phys. 128, 143101 (2020).
[Crossref]

M. R. Singh, G. Brassem, and P. Persaud, “Spontaneous emission in nanofibers doped with ensemble of quantum dots and quantum emitters,” Phys. Rev. A 102, 063704 (2020).
[Crossref]

M. R. Singh and S. Yastrebov, “Switching and sensing using Kerr nonlinearity in quantum dots doped in metallic nanoshells,” J. Phys. Chem. C124, 12065–12074 (2020).
[Crossref]

M. R. Singh, “Theory of all-optical switching based on Kerr nonlinearity in metallic nanohybrids,” Phys. Rev. A 102, 013708 (2020).
[Crossref]

M. R. Singh and P. D. Persaud, “Dipole–Dipole interaction in two-photon spectroscopy of metallic nanohybrids,” J. Phys. Chem. 124, 6311–6320 (2020).
[Crossref]

2019 (3)

M. G. C. M. Kumari, C. S. Perera, B. S. Dassanayake, M. A. K. L. Dissanayake, and G. K. R. Senadeera, “Highly efficient plasmonic dye-sensitized solar cells with silver nanowires and TiO2 nanofibres incorporated multi-layered photoanode,” Electrochim. Acta 298, 330–338 (2019).
[Crossref]

M. A. K. L. Dissanayake, T. Jaseetharan, G. K. R. Senadeera, J. M. K. W. Kumari, C. A. Thotawatthage, B. E. Mellander, I. Albinson, and M. Furlani, “Highly efficient, PbS:Hg quantum dot–sensitized, plasmonic solar cells with TiO2 triple-layer photoanode,” J. Solid State Electrochem. 23, 1787–1794 (2019).
[Crossref]

Z. Haiguang, S. Ruijia, W. Zhaofen, F. Kaifang, H. Xun, and Z. Yuhai, “Zero-dimensional perovskite nanocrystals for efficient luminescent solar concentrators,” Adv. Funct. Mater. 29, 1902262 (2019).
[Crossref]

2018 (4)

D. Liu, H. J. Li, J. Gao, S. Zhao, Y. Zhu, P. Wang, D. Wang, A. Chen, X. Wang, and J. Yang, “High-performance ultraviolet photodetector based on graphene quantum dots decorated ZnO nanorods/GaN film isotype heterojunctions,” Nanoscale Res. Lett. 13, 261 (2018).
[Crossref]

K. Erotokritou, R. M. Heath, G. G. Taylor, C. Tian, A. Banerjee, A. Casaburi, C. M. Natarajan, S. Miki, H. Terai, and R. H. Hadfield, “Nano-optical photoresponse mapping of superconducting nanowires with enhanced near infrared absorption,” Supercond. Sci. Technol. 31, 125012 (2018).
[Crossref]

J. Guo, K. Black, J. Hu, and M. Singh, “Study of plasmonics in hybrids made from a quantum emitter and double metallic nanoshell dimer,” J. Phys. Condens. Matter. 30, 185301 (2018).
[Crossref]

A. I. Yakimov, V. V. Kirienko, and A. V. Dvurechenskii, “Comparison of two-dimensional arrays of gold disks and holes for plasmonic enhancement of Ge/Si quantum dot mid-infrared photodetectors,” Opt. Mater. Express 8, 3479–3484 (2018).
[Crossref]

2017 (3)

B. D. Fainberg, N. N. Rosanov, and N. A. Veretenov, “Light-induced ‘plasmonic’ properties of organic materials: surface polaritons and switching waves in bistable organic thin films,” Appl. Phys. Lett. 110, 203301 (2017).
[Crossref]

M. R. Singh, M. SekharChandra, S. Balakrishnan, and S. Masood, “Medical applications of hybrids made from quantum emitter and metallic nanoshell,” J. Appl. Phys. 122, 034306 (2017).
[Crossref]

J. A. Alexander-Webber, C. K. Groschner, A. A. Sagade, G. Tainter, M. F. ZalbaGonzalez-, R. PietroDi, J. Wong-Leung, H. H. Tan, C. Jagadish, S. Hofmann, and H. J. Joyce, “Engineering the photoresponse of InAs nanowires,” ACS Appl. Mater. Interfaces 9, 43993–44000 (2017).
[Crossref]

2016 (2)

M. Ding, N. Yao, C. Wang, J. Huang, M. Shao, S. Zhang, P. Li, X. Deng, and X. Xu, “ZnO@CdS core-shell heterostructures: fabrication, enhanced photocatalytic, and photoelectrochemical performance,” Nanoscale Res. Lett. 11, 205 (2016).
[Crossref]

A. F. Terzis, S. G. Kosionis, J. Boviatsis, and E. Paspalakis, “Nonlinear optical susceptibilities of semiconductor quantum dot–metal nanoparticle hybrids,” J. Mod. Opt. 63, 451–461 (2016).
[Crossref]

2015 (5)

H. Sugimoto, R. Zhang, B. M. Reinhard, M. Fujii, G. Perotto, B. Marelli, F. G. Omenetto, and L. NegroDal, “Enhanced photoluminescence of Si nanocrystals-doped cellulose nanofibers by plasmonic light scattering,” Appl. Phys. Lett. 107, 41111 (2015).
[Crossref]

F. Fang, X. J. Wang, X. Fang, T. Li, Z. Wei, J. Li, and X. Chu, “Effects of shell characteristics on the UV photoresponse behaviors of photoconductivity based on ZnO/ZnMgO core-shell nanowire arrays,” Mater. Res. Express. 2, 076201 (2015).
[Crossref]

J. Treu, T. Stettner, M. Watzinger, S. Morkötter, M. Döblinger, S. Matich, K. Saller, M. Bichler, G. Abstreiter, J. J. Finley, J. Stangl, and G. Koblmüller, Lattice-matched InGaAs-InAlAs core-shell nanowires with improved luminescence and photoresponse properties,” Nano Lett. 15, 3533–3540 (2015).
[Crossref]

A. Baranowska-Korczyc, K. Sobczak, P. Dłuzewski, A. Reszka, B. J. Kowalski, Ł. Kłopotowski, D. Elbaum, and K. Fronc, “Facile synthesis of core/shell ZnO/ZnS nanofibers by electrospinning and gas-phase sulfidation for biosensor applications,” Phys. Chem. Chem. Phys. 17, 24029–24037 (2015).
[Crossref]

H. Liu, N. Gao, M. Liao, and X. Fang, “Hexagonal-like Nb2O5 nanoplates-based photodetectors and photocatalyst with high performances,” Sci. Rep. 5, 7716 (2015).
[Crossref]

2014 (4)

D. Choi, Y. Jang, J. H. Lee, G. H. Jeong, D. Whang, S. W. Hwang, K. S. Cho, and S. W. Kim, “Diameter-controlled and surface-modified Sb2Se3 nanowires and their photodetector performance,” Sci. Rep. 4, 6714 (2014).
[Crossref]

J. Y. Chen, H. C. Wu, Y. C. Chiu, and W. C. Chen, “Plasmon-enhanced polymer photovoltaic device performance using different patterned Ag/PVP electrospun nanofibers,” Adv. Energy Mater. 4, 1301665 (2014).
[Crossref]

N. Sobti, A. Bensouici, F. Coloma, C. Untiedt, and S. Achour, “Structural and photoelectrochemical properties of porous TiO2 nanofibers decorated with Fe2O3 by sol-flame,” J. Nanopart. Res. 16, 2577 (2014).
[Crossref]

M. R. Singh, J. D. Cox, and M. Brzozowski, “Photoluminescence and spontaneous emission enhancement in metamaterial nanostructures,” J. Phys. D 47, 085101 (2014).
[Crossref]

2013 (2)

B. Sebo, N. Huang, Y. Liu, Q. Tai, L. Liang, H. Hu, S. Xu, and X.-Z. Zhao, “Dye-sensitized solar cells enhanced by optical absorption, mediated by TiO2 nanofibers and plasmonics Ag nanoparticles,” Electrochim. Acta 112, 458–464 (2013).
[Crossref]

M.-H. Jung, Y. J. Yun, M.-J. Chu, and M. G. Kang, “Fabrication of Ag nanoparticles embedded in TiO2 nanotubes: using electrospun nanofibers for controlling plasmonic effects,” Chem. Eur. J. 19, 8543–8549 (2013).
[Crossref]

2012 (1)

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref]

2011 (1)

J. Ding, Y. Sui, W. Fu, H. Yang, B. Zhao, and M. Li, “ZnO Nanorod array/CuAlO2 nanofiber heterojunction on Ni substrate: synthesis and photoelectrochemical properties,” Nanotechnology 22, 295706 (2011).
[Crossref]

2010 (2)

F. O. Ochanda and M. R. Barnett, “Synthesis and characterization of silver nanoparticles and Titanium oxide nanofibers: toward multifibrous nanocomposites,” J. Am. Ceram. Soc. 93, 2637–2643 (2010).
[Crossref]

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. B 82, 195419 (2010).
[Crossref]

2007 (1)

M. R. Singh, “The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials,” J. Mod. Opt. 54, 1739–1757 (2007).
[Crossref]

2003 (1)

Y. Suganuma, P. E. Trudeau, B. Leathem, B. Shieh, and A. Dhirani, “Discrete electron forces in a nanoparticle-tunnel junction system,” J. Chem. Phys. 118, 9769–9772 (2003).
[Crossref]

Abstreiter, G.

J. Treu, T. Stettner, M. Watzinger, S. Morkötter, M. Döblinger, S. Matich, K. Saller, M. Bichler, G. Abstreiter, J. J. Finley, J. Stangl, and G. Koblmüller, Lattice-matched InGaAs-InAlAs core-shell nanowires with improved luminescence and photoresponse properties,” Nano Lett. 15, 3533–3540 (2015).
[Crossref]

Achour, S.

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ACS Appl. Mater. Interfaces (1)

J. A. Alexander-Webber, C. K. Groschner, A. A. Sagade, G. Tainter, M. F. ZalbaGonzalez-, R. PietroDi, J. Wong-Leung, H. H. Tan, C. Jagadish, S. Hofmann, and H. J. Joyce, “Engineering the photoresponse of InAs nanowires,” ACS Appl. Mater. Interfaces 9, 43993–44000 (2017).
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ACS Nano (1)

J. Guan, L. K. Sagar, R. Li, D. Wang, G. Bappi, W. Wang, N. Watkins, M. R. Bourgeois, L. Levina, F. Fan, S. Hoogland, O. Voznyy, J. M. PinaDe, R. D. Schaller, G. C. Schatz, E. H. Sargent, and T. W. Odom, “Quantum dot-plasmon lasing with controlled polarization patterns,” ACS Nano 14, 3426–3433 (2020).
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J. Y. Chen, H. C. Wu, Y. C. Chiu, and W. C. Chen, “Plasmon-enhanced polymer photovoltaic device performance using different patterned Ag/PVP electrospun nanofibers,” Adv. Energy Mater. 4, 1301665 (2014).
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Adv. Funct. Mater. (1)

Z. Haiguang, S. Ruijia, W. Zhaofen, F. Kaifang, H. Xun, and Z. Yuhai, “Zero-dimensional perovskite nanocrystals for efficient luminescent solar concentrators,” Adv. Funct. Mater. 29, 1902262 (2019).
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H. Sugimoto, R. Zhang, B. M. Reinhard, M. Fujii, G. Perotto, B. Marelli, F. G. Omenetto, and L. NegroDal, “Enhanced photoluminescence of Si nanocrystals-doped cellulose nanofibers by plasmonic light scattering,” Appl. Phys. Lett. 107, 41111 (2015).
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M.-H. Jung, Y. J. Yun, M.-J. Chu, and M. G. Kang, “Fabrication of Ag nanoparticles embedded in TiO2 nanotubes: using electrospun nanofibers for controlling plasmonic effects,” Chem. Eur. J. 19, 8543–8549 (2013).
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B. Sebo, N. Huang, Y. Liu, Q. Tai, L. Liang, H. Hu, S. Xu, and X.-Z. Zhao, “Dye-sensitized solar cells enhanced by optical absorption, mediated by TiO2 nanofibers and plasmonics Ag nanoparticles,” Electrochim. Acta 112, 458–464 (2013).
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M. R. Singh and S. Yastrebov, “Switching and sensing using Kerr nonlinearity in quantum dots doped in metallic nanoshells,” J. Phys. Chem. C124, 12065–12074 (2020).
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F. Fang, X. J. Wang, X. Fang, T. Li, Z. Wei, J. Li, and X. Chu, “Effects of shell characteristics on the UV photoresponse behaviors of photoconductivity based on ZnO/ZnMgO core-shell nanowire arrays,” Mater. Res. Express. 2, 076201 (2015).
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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of the nanofiber hybrid. The nanofiber is composed of a core and a shell (cladding), both of which are made of dielectric materials. The core of the nanofiber is doped with an ensemble of QDs and MNPs.
Fig. 2.
Fig. 2. Schematic diagram of a four-level QD. The energy levels are denoted as $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$. The probe field and the SPP field are acting in transition $| {1\rangle \leftrightarrow} |2\rangle$. The DDI-MNP and the DDI-QD fields are acting in transitions $| 2\rangle \leftrightarrow | 3\rangle$ and $| {3\rangle \leftrightarrow} |4\rangle$, respectively.
Fig. 3.
Fig. 3. Schematic diagram of the nanowaveguide coating with Al-MNPs and Ge-Si QDs.
Fig. 4.
Fig. 4. Theoretical and experiential values for the photoresponse spectrum (A.U.) are plotted as a function of wavelength (nm). The open circles correspond to the Ge-Si quantum dots alone. The diamonds, squares, and asterisks correspond to the doped Al-MNPs with disk diameters 150 nm, 200 nm, and 225 nm, respectively. The solid, dotted, dashed, and dashed–dotted lines are theoretical curves.
Fig. 5.
Fig. 5. The power term (${{\rm W}_{QD/}}{{\rm W}_0}$) (A.U.) is plotted as a function of the normalized probe detuning ${\delta _{21}}/\;{\Gamma _P}$ (unitless) and the normalized SPP coupling ${\Pi _{\rm{SPP}}}$ (unitless). One can see from the figure that the PL spectrum is enhanced.
Fig. 6.
Fig. 6. The PL spectrum (A.U.) is plotted as a function of the normalized probe detuning ${\delta _{21}}/\;{\Gamma _P}$ (unitless) and the normalized DDI-QD coupling $\Phi _{\rm{DDI}}^{\textit{QD}}$ (unitless). The PL spectrum splits into two quenched states at higher values of the DDI-QD coupling.
Fig. 7.
Fig. 7. The power term (${{\rm W}_{\rm QD/}}{{\rm W}_0}$) is plotted as a function of the DDI coupling (${\Phi _{\rm{DDI}}})$ for different values of the magnitude of the probe field detunings ($|{\delta _{21}}|/{\Gamma _P}$). The solid line, dashed line, and dashed–dotted line correspond to $|{\delta _{21}}|/{\Gamma _{P}} = {0}$, $|{\delta _{21}}|/\;{\Gamma _{P\:}} = {0.5}$, and $|{\delta _{21}}|/{\Gamma _{P}} = {1.0}$, respectively.
Fig. 8.
Fig. 8. The power term (${{\rm W}_{\rm QD/}}{{\rm W}_0}$) is plotted as a function of the DDI coupling (${\Phi _{\rm{DDI}}})$ and the magnitude of the probe field detunings ($|{\delta _{21}}|/{\Gamma _P}$).

Equations (38)

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j pc = n e e v d e + n h e v d h ,
v d e = e τ e m e ( E P + E S P P ) , v d h = e τ h m h ( E P + E S P P ) ,
j pc = σ e ( E P + E S P P ) + σ h ( E P + E S P P ) ,
σ e = n e e 2 τ e m e σ h = n h e 2 τ h m h .
j pc = n e e 2 τ e m ( E P + E S P P ) , 1 m = 1 m e + 1 m h ,
d n e d t = g e n e τ r ,
n e = g e τ r .
g e = η Q E ( P P ( ω P ) ω P + P S P P ( ω P ) ω S P ) ,
n e = η Q E τ r ( P P ( ω P ) + P S P P ( ω P ) ω P ) .
j pc = [ ( e 2 τ e τ r m ) η Q E ( E P + E S P P ) × ( P P ( ω P ) + P S P P ( ω P ) ω P ) ] .
η Q E = γ r γ r + γ nr .
P P + P S P P = μ 12 ω p 2 Im ( ρ 12 ) ( E P + E S P P ) ,
j pc = ( μ 12 e 2 τ e τ r 2 m ) η Q E Im ( ρ 12 ) | E P + E S P P | 2 .
j pc = j o . P P R C j o = ( μ 12 e 2 τ e τ r 2 m ) E P 2 ,
P P R C = η Q E Im ( ρ 12 ) ( 1 + E S P P / E P ) 2 .
E S P P = Π S P P E P , Π S P P = R M N P 3 g l r s 3 ς M N P , ς M N P = m b m + 2 b ,
E D D I M N P = Φ D D I M N P E P , Φ D D I M N P = λ D D I g l 3 ς M N P ,
E D D I Q E = Φ D D I Q E E P , Φ D D I Q E = λ D D I g l 3 ς Q E , ς Q E = q b q + 2 b .
P Q E = 0 c g l ς Q E ( E P + E S P P , + E D D I M N P + E D D I Q E ) , P M N P = 0 c g l ς M N P ( E P + E D D I Q E + E D D I M N P ) ,
P Q E = 0 c α Q E E P , α Q E = g l ς Q E ( 1 + Π S P P + Φ D D I M N P + Φ D D I Q E ) , P M N P = 0 c α M N P E P , α M N P = g l ς M N P ( 1 + Π Q E + Φ D D I Q E + + Φ D D I M N P ) ,
2 E c = ω k 2 c 2 ( c + c α Q E + c α M N P ) E c , 2 E s = ( ω k 2 c 2 ) s E s ,
k z = [ F c 2 ( ε n m , k z ) F c 2 ( ε nm ) ] 1 / 2 , F c ( ε k ) = ε k ( c + c α Q E + c α M N P ) c ,
H r = ( n m , k z V P p n m , k z σ 21 + n m , k z V S P P M N P p n m , k z σ 21 + h c ) , H nr = ( n m , k z V D D I M N P p n m , k z σ 23 + n m , k z V D D I Q E p n m , k z σ 34 + h c ) ,
V P = i ( μ 21 2 ε n m , k z 2 0 b V Q E ) 1 / 2 , V S P P = V P Π S P P , V D D I M N P = i ( μ 23 2 ε n m , k z 2 0 b V Q E ) 1 / 2 Φ D D I M N P , V D D I Q E = i ( μ 34 2 ε n m , k z 2 0 b V Q E ) 1 / 2 Φ D D I Q E ,
Γ ij = 2 π nm [ d ε n m , k z D nm ( ε n m , k z ) × | V int | 2 δ ( ε n m , k z ε ij ) ] ,
D nm = D 0 ( G c F c ( 1 + α Q E + α M N P ) [ F c 2 ( ε n m , k z ) F c 2 ( ε nm ) ] 1 / 2 ) , D 0 = V PNF π 2 ε n m , k z 2 3 c 3 , G c = π 2 c 2 c A ε n m , k z 2 .
γ r = Γ P ( 1 + | Π S P P | 2 ) , Γ P = γ 0 nm { G c ( ε 21 ) F c ( ε 21 ) [ F c 2 ( ε 21 ) F c 2 ( ε nm ) ] 1 / 2 } ,
γ nr = γ nr 23 + γ nr 34 , γ nr 23 = Γ 0 | Φ D D I M N P | 2 , γ nr 34 = Γ 0 | Φ D D I Q E | 2 ,
Γ 0 = γ 0 nm { G c ( ε 23 ) F c ( ε 23 ) [ F c 2 ( ε 23 ) F c 2 ( ε nm ) ] 1 / 2 } .
H 0 = [ Ω P σ 21 + Ω P ( Π S P P ) σ 21 + Ω p Φ D D I M N P σ 23 + Ω P ( Φ D D I Q E ) σ 34 + h c ] ,
d ρ 11 d t = 2 γ nr 23 ρ 22 i Ω P ( 1 + Π S P P ) ( ρ 12 ρ 21 ) d ρ 22 d t = [ 2 γ r ρ 22 + i Ω P ( 1 + Π S P P ) ( ρ 12 ρ 21 ) + i Ω P Φ D D I Q E ( ρ 32 ρ 23 ) ] ,
d ρ 33 d t = [ 2 γ 31 ρ 22 + i Ω P Φ D D I Q E ( ρ 23 ρ 32 ) + i Ω P Φ D D I M N P ( ρ 43 ρ 34 ) ] , d ρ 12 d t = [ d 21 ρ 12 i Ω P Φ D D I Q E ρ 13 + i ( ρ 22 ρ 11 ) Ω p ( 1 + Π S P P ) ] ,
d ρ 13 d t = [ d 13 ρ 13 i ρ 12 Ω P Φ D D I Q E + i ρ 23 Ω p ( 1 + Π S P P ) i ρ 14 Ω P Φ D D I M N P ] , d ρ 23 d t = [ d 23 ρ 23 + i ( ρ 33 ρ 22 ) Ω P Φ D D I Q E + i ρ 13 Ω p ( 1 + Π S P P ) i ρ 24 Ω P Φ D D I M N P ] ,
d ρ 14 d t = [ d 14 ρ 14 i ρ 13 Ω P Φ D D I M N P + i ρ 24 Ω p ( 1 + Π S P P ) ] , d ρ 24 d t = [ d 24 ρ 24 + i ρ 34 Ω P Φ D D I Q E i ρ 23 Ω P Φ D D I M N P + i ρ 13 Ω p ( 1 + Π S P P ) ] , d ρ 34 d t = [ d 34 ρ 34 + i ρ 24 Ω P Φ D D I Q E + i Ω P Φ D D I M N P ( ρ 44 ρ 33 ) ] .
d 21 = δ r + i γ r , d 23 = δ 23 + i γ nr 23 , d 43 = δ 43 + i γ nr 34 , d 41 = δ 41 + i ( γ r + γ nr 34 ) / 2 , d 42 = δ 42 + i ( γ r + γ nr 23 ) / 2 , d 31 = δ 31 + i ( γ r + γ nr 34 ) / 2 ,
δ 21 = ω p ω 21 , δ 23 = ω d ω 23 , δ 34 = ω sp ω 34 , δ 31 = 0 , δ 42 = δ 23 + δ 34 , δ 14 = δ 21 δ 24 , δ 13 = δ 21 δ 23 .
ρ 12 = i ( Ω P + Ω P Π S P P ) ( d 13 d 14 ( Ω P Φ D D I M N P ) 2 ) d 21 ( d 13 d 14 ( Ω P Φ D D I M N P ) 2 ) + d 14 ( Ω P Φ D D I Q D ) 2 .
P P R C = η Q E ( 1 + Π S P P ) 2 I m ( R num R den ) , R n u m = [ i Ω P ( 1 + Π S P P ) × ( d 13 d 14 Ω P 2 ( Φ D D I M N P ) 2 ) ] , R d e n = [ d 21 ( d 13 d 14 Ω P 2 ( Φ D D I M N P ) 2 ) + d 14 Ω P 2 ( Φ D D I Q E ) 2 ] .

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