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Abstract
We propose a scheme to generate ultra-strong four-wave mixing (FWM) signal based on a suspended monolayer graphene nanoribbon nanomechanical resonator (NR) coupled to an Au nanoparticle (NP). It is shown that, the FWM spectrum can switch among two-peaked, three-peaked, four-peaked or five-peaked via the modulation of exciton-phonon and exciton-plasmon couplings. This is mainly attributed to the vibrational properties of NR related to the exciton-phonon coupling, and the energy-level splitting of the localized exciton correlated to three classes of resonances consisting of three-photon resonance, Rayleigh Resonance, and AC-Stark atomic resonance. Especially, in a dual-strong coupling regime, the gains for these peaks can be as high as nine orders of magnitude (∼ 109) around the lower bistable threshold due to a combined effect of two couplings. Our findings not only offer an efficient way to measure the vibrational frequency of NR and the exciton-phonon coupling strength but also provide a possibility to fabricate high-performance optoelectronic nanodevices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Four-wave mixing (FWM) effect in nanosystems has created a new momentum because of its potential applications in photonics and optoelectronics [1–6]. Many studies on this topic have been performed, for example, in a coupled semiconductor quantum dot (SQD)-metal nanoparticle (MNP) system, the FWM response in this system can be enhanced greatly arising from the exciton-plasmon interaction [7–10]. This is quite different from the FWM enhancement in the gold NPs nanosystem [11,12]. In a typical work by Paspalakis et al., the evolution of the FWM spectrum has been illustrated. Specifically, the FWM spectrum can be freely switched between single-peaked and three-peaked form by only adjusting the spacing between two nanodimers (i.e. exciton-plasmon interaction) [7]. Also, it is realizable to boost the FWM signal by regulating the polarized direction of the pump field and structural parameters of this system [8,9]. Our previous work has also demonstrated the possibility of achieving highly-efficient four-wave optical parametric amplification in the SQD-MNP hybrid system [10].
Graphene nanoribbons possess many intriguing features such as low mass density, small size, high stiffness, and high quality factor [13–16], which can serve as promising candidates for fabricating nanomechanical systems. To date, there have been some interesting studies around optical properties in graphene nanoribbon-based nanomechanical resonators (NRs) [16–19]. For example, in a suspended graphene nanoribbon NR system, due to the exciton-phonon coupling, it is achievable to measure precisely the mass of small molecules by employing the linear absorption spectrum of the system [20]. This system has also been explored as a mean to build a highly-flexible optical bistable switch [21]. When a MNP approaches to a suspended monolayer Z-shaped graphene nanoribbon NR, a new coupling between the exciton and plasmons appears, inducing the modification of optical bistable properties of the NR system. The results show that it is feasible to control the bistable switch via a single channel or dual channels by only adjusting the intensity or frequency of the pump field [22]. However, to the best of our knowledge, a systematic investigation of evolution of the FWM signal in the MNP-monolayer graphene nanoribbon NR system has remained unaddressed so far and has been the focus of study in this paper.
The objective of this paper is to show how one can control the evolution of the FWM signal via the modulation of exciton-phonon and exciton-plasmon couplings in the Au NP-monolayer graphene nanoribbon NR hybrid system. For this we analyze the FWM response when no, one, or two couplings start to work, and find that the profile can switch among two, three, four and five peaks. Especially in a dual-strong coupling regime, we show that the heights and positions of peaks in the FWM spectra depend strongly on the pumping intensity and two coupling strengths. Due to a combined effect of two couplings, there is a great peak gain with an unexpected value of ∼ 109.
2. Theoretical model and method
A schematic of the setup is shown in Fig. 1(a). The setup consists of a doubly clamped suspended Z-shaped monolayer graphene nanoribbon NR and a spheroidal Au NP, which suffers simultaneously from a strong pump beam Epu with frequency ωpu and a weak probe beam Epr with frequency ωpr [23]. The Au NP with a radius of a is positioned above the NR at a center-to-center distance d. As for the Z-shaped junction device, its electronic states can be confined completely due to the topological structure of the junction. In addition, the spatial confinement and the number of discrete levels can be adjusted by changing the length of the junction [24]. When a Z-shaped monolayer graphene nanoribbon is clamped doubly and suspended, its center of mass of the exciton is localized via the spatial modulation due to a static inhomogeneous electric field [18]. For this reason, a quantum confinement effect appears. Therefore, the localized exciton can be modeled as a two-level system consisting of the ground state |0 > and the excited state |1>, which can be characterized by three pseudospin operators σ10, σ01, and σz. These three operators satisfy the commutation relation [σ10, σ01] = 2σz, [σ10, σz] = −σ10 and [σ01, σz] = σ01.
Fig. 1. (a) Schematic illustration of a doubly clamped suspended monolayer Z-shaped graphene nanoribbon NR coupled to an Au NP. (b) Energy-level diagram of a localized exciton while dressing phonons in the graphene nanoribbon NR and surface plasmons in the Au NP. Each energy-level of exciton is split into a doublet with separation of Ω. Ω is correlated to the pumping intensity, the exciton-phonon coupling strength, and exciton-plasmon coupling strength. TP refers to the three-photon resonance, RL denotes the Rayleigh resonance, and AC represents the AC-Stark-shifted atomic resonance [27].
The basic excitations in the MNP are named as surface plasmons, which possess a continuous energy spectrum. Especially, the radius of the MNP considered is much larger than 5 nm, so the classical treatment can be applied reasonably [25]. Here, we adopt a resonator-bath representation with the phonon mode. Its eigenmode can be described by a quantum harmonic oscillator [20]. The NR considered is assumed to possess a sufficiently high quality factor Q, whose lowest-energy resonance corresponds to the fundamental flexural mode with frequency ωn. In the NP-NR hybrid nanosystem, there is mainly two coupling types: exciton-phonon coupling and exciton-plasmon coupling [26]. The physical situation is displayed in Fig. 1(b).
Under the rotating frame at the frequency ωpu of the pump field, the Hamiltonian of the system coupled with two optical fields reads as follows [28]
According to the Heisenberg equations of motion dQ/dt = −i[Q, H]/h, the quantum Langevin equations can be expressed as [31]:
To solve Eqs. (2–4), we make the ansatz as follows: Θ = Θ0 + Θ1e−iδt + Θ−1eiδt, w = w0 + w1e−iδt + w−1eiδt and Π = Π0 + Π1e−iδt + Π−1eiδt, where Θ0 >> Θ1, Θ−1, w0 >> w1, w−1, and Π0 >> Π1, Π−1. δ = ωpr ‒ ωpu corresponds to the probe-pump detuning. Inserting these expressions into the Langevin Eqs. (2–4), we can derive the FWM signal as follows
The population inversion w0 of the exciton is given by the following cubic equation
3. Results and discussions
We consider a coupled Au NP-graphene nanoribbon NR system in the air (ɛB = 1). The Z-shaped graphene NR of dimensions are (l1, d) = (14.1, 0.7) nm which consists of 424 carbon atoms [32]. Other parameters are taken as r0 = 7 nm, ɛ∞ = 9.5, ћωp = 8.95 eV, ћγp = 0.149 eV [33], ɛs = 6, μ = 40 D, Γ1 = 2Γ2 = 2 GHz [34], ωn = 7.477 GHz [20], Q = 9000, and γn = ωn /Q [35].
One of our objectives in this paper is to investigate the evolution of the FWM signal in this NP-NR hybrid nanosystem. We implement this via exploring the variation of the FWM signal in four different cases: (1) two couplings are both absent (d→∝, g = 0); (2) only exciton-phonon coupling starts to work (d→∝, g = 3 GHz > Γ2, γn); (3) two couplings are both present (d = 17 nm, g = 3 GHz); (4) only exciton-plasmon coupling exists (d = 17 nm, g = 0 GHz). Before we get any further, it is necessary to explain a point: for g = 0, this suggests that the exciton-phonon coupling disappears. In other word, the real nanosystem evolves from a NP-NR hybrid system to a exciton-plasmon system. Only exciton-plasmon coupling provides a continuous effect on the system [36–38]. As shown in Fig. 2(a), for the case (1) the FWM spectrum exhibits a three-peaked structure (i.e. L, M0, R peaks), which are assigned to the TP resonance, stimulated RL resonance, and AC-Stark atomic resonance, respectively. Considering this, as g varies from zero to one (i.e. g = 0 → 3 GHz), the FWM spectrum changes from a three-peaked to five-peaked form. Two new peaks located at the position of ±ωn appear. These results are plotted in Fig. 2(b). When two couplings are both strong and start to work, the spectrum depicted in Fig. 2(c) converts to a four-peaked form, and the M0 peak is smeared out along with the invalidation of the RL resonance on the FWM effect. Unexpectedly, the heights of peaks in the FWM spectra are greatly suppressed (10−1 → 10−7). This suggests that the proposed system can behave as an ultra-sensitive FWM signal modulator. In Fig. 2(d), despite strong exciton-plasmon coupling, M1 and M2 peaks vanish simultaneously accompanied by the disappearance of the exciton-phonon coupling (g = 0). Under this condition, the Au NP-graphene NR hybrid system can be switched to a strong exciton-plasmon coupling system [39–42]. This provides a new possibility for detecting the vibrational frequency of graphene nanoribbon NR.
Fig. 2. Evolution of the FWM signal for the coupled Au NP-graphene nanoribbon NR hybrid system. We consider four different situations: (a) d = 100 nm and g = 0 GHz, (b) d = 100 nm and g = 3 GHz, (c) d = 17 nm and g = 3 GHz, (d) d = 17 nm and g = 0 GHz. The parameter used is taken as Ipu = 200 GHz2. L, M0, and R peaks are respectively ascribed to the three-photon resonance, Rayleigh resonance, and AC-Stark atomic resonance. M1 and M2 peaks located at the vibrational frequency of NR ±ωn result from the exciton-phonon coupling.
To further understand the contributions of the pumping intensity to the FWM signal, in Fig. 3 we mainly explore the FWM response when the system stays in the dual-strong coupling regime. As shown in Fig. 3(a), when the pumping intensity is relatively weak (Ipu = 0.1 GHz2), an almost invisible four-peaked spectrum arises. To clarify this, in Fig. 3(b) we carry out a study of the variation of the FWM signal with Ipu. The results exhibit that, with the increase of Ipu the heights of L and R peaks are both increased. As Ipu increases sharply to 100 GHz2, the heights for these two peaks both rise abruptly. However, their positions keep almost changeless. We also note, here, that M1 and M2 peaks also show a similar increase trend (see Fig. 3(c)). As expected, the L and R peaks (M1 and M2) are symmetric about the axis of δ = 0. In order to reveal clearly the amplification behavior of the FWM signal, we define two gain factors as follows: GI(L) = |FWM(Ipu)|/|FWM(IpuR)|L and GI(M1) = |FWM(Ipu)|/|FWM(IpuR)|M1, wherein IpuR denotes a referenced pumping intensity. Herein, IpuR is 0.1 GHz2. As seen in Fig. 3(d), GI(L) ≈ 2000 for Ipu = 200 GHz2. As a coincidence, GI(M1) ≈ GI(L) for a same given Ipu. This suggests that a strong pumping intensity can promote the amplifying extent of the FWM signal.
Fig. 3. (a) Variation of the FWM signal as a function of the probe-pump detuning δ for Ipu = 0.1 GHz2 and Ipu = 100 GHz2. (b) Dependence of the heights and positions of L, R (b) and M1, M2 (c) peaks on Ipu. (d) Gains for L and M1 peaks versus Ipu. Other parameters used are d = 17 nm and g = 3 GHz.
To better visualize the influence of the exciton-phonon coupling on the FWM signal, in Fig. 4(a) we explore the FWM response for g = 0 GHz and g = 1 GHz. Note that two very sharp peaks appear only when the exciton-phonon coupling exists (g > 0 GHz), which provides a feasible way for measuring the coupling strength between the exciton and phonons. As shown in Fig. 4(b), for d = 17 nm, the heights of L and R peaks increase monotonously as g increases. As a comparison, the scenario for their peak positions becomes quite different from that in Fig. 3(b). Herein, these two peaks move gradually to the axis of δ = 0 with increasing g. This suggests that the exciton-phonon coupling will affect significantly the energy-level splitting of the exciton (i.e. Ω), leading to the shifts of L and R peak positions. In other words, the roles of TP and AC resonances related to Ω in modulating the FWM signal will be changed via the modulation of g. As seen in Fig. 4(c), as g increases from 1 GHz to 7 GHz, there is a nearly 51.4 fold increase in the height of M1 peak. In order to easily illustrate this behavior, we rewrite the gain factors for L and M1 peaks with Gg(L) = |FWM(g)|/|FWM(gR)|L and Gg(M1) = |FWM(g)|/|FWM(gR)|M1, wherein gR = 0.1 GHz. As g increases to 10 GHz, Gg(L) and Gg(M1) are equal to 14.5, 10159.7, respectively. It is obvious that these two peaks located at ±ωn are strongly correlated to the exciton-phonon coupling strength g. However, g depends strongly on the length of graphene nanoribbon along the NR axis and its mass density. Therefore, the FWM signal can be modulated by only changing the structural parameters of the graphene nanoribbon NR.
Fig. 4. (a) Variation of the FWM signal as a function of the probe-pump detuning δ for g = 0 GHz and g = 1 GHz. Dependence of heights and positions of L, R peaks (b) and M1, M2 peaks (c) on the exciton-phonon coupling strength g. (d) Gains for L and M1 peaks versus g. For all plots, d = 17 nm and Ipu = 200 GHz2 are used.
Inspection of the results presented in Fig. 5 indicates that the FWM signal relies strongly on the exciton-plasmon coupling (i.e. d). In Figs. 5(a) and 5(b), for d = 10 nm, the FWM spectrum exhibits a four-peaked structure, while when d increases gradually to 18 nm, a new peak (M0) located at δ = 0 appears, and the corresponding spectrum evolves into a five-peaked structure. This result can be easily explained by taking into account the fact that the RL resonance comes into play. For dc0 = 26.75 nm ≤ d ≤ dc1 = 30.01 nm, a bistable effect occurs. dc0 and dc1 denote the lower and upper bistable thresholds, respectively. To provide more insight into the underlying nature of these peaks in Figs. 5(a) and 5(b), we explore the impact of d on the gains and positions for L and M1 peaks. Similarly, the gain factors can be rewritten as Gd (L) =|FWM(d)|/|FWM(dR)|L and Gd(M1) = |FWM(d)|/|FWM(dR)|M1 with dR = 10 nm. As shown in Fig. 5(c), as d increases from 10 nm to dc0, Gd(L) increases abruptly from 1 to 1.07 × 107, and the position for L peak (i.e. Ω) shifts from −12.06 THz to −0.03 THz. However, as d ≥ dc1, Gd(L) and Ω both keep almost unchanged. This indicates that, when the exciton-plasmon coupling becomes weak (i.e. d ↑), the TP resonance associated with Ω is switched from a dominate role to an insignificant one in modulating the FWM signal. The situation for M1 peak becomes a litter different. In Fig. 5(d), Gd(M1) can reach an ultra-large value of 4.44 × 109 at d = dc0. Its peak position only displays a slight fluctuation around the bistable region and keeps almost unchanged in the other region. The physical origin behind these results can be related to the fact that, the larger d, the weaker the exciton-plasmon coupling. For d ≤ dc0, the exciton-phonon and exciton-plasmon couplings provide a combined effect on modulating the FWM signal. While for d > dc1, the exciton-plasmon coupling starts to smear out, only the exciton-phonon coupling continues to make an impact on the FWM signal.
Fig. 5. (a, b) Variation of the FWM signal as a function of the probe-pump detuning δ for different d. Gains and positions for L (c) and M1 (d) peaks versus d. For all four panels, g = 3 GHz and Ipu = 200 GHz2.
4. Conclusions
We conducted a theoretical study of the evolution of the FWM signal in the Au NP-monolayer graphene nanoribbon NR hybrid system. We showed that, when the exciton-phonon and exciton-plasmon couplings are both absent, the FWM spectrum exhibits a three-peaked structure. When one or two couplings start to work, such a three-peaked spectrum can switch to two-peaked, four-peaked or five-peaked. This can be ascribed to the vibrational properties of NR and the energy-level splitting of the localized exciton. Especially, in a dual-strong coupling regime, there is a great peak gain with an exciting value of ∼ 109 around the lower bistable threshold due to a combined effect of two couplings. The results obtained in this paper both offers an efficient way to measure the vibrational frequency of NR and opens a new possibility to develop high-performance optical signal modulators.
Funding
National Natural Science Foundation of China (11404410, 12175315); Natural Science Foundation of Hunan Province (2020JJ4935, 2019JJ40533); Scientific Research Fund of Hunan Provincial Education Department (20B602);Undergraduate Innovation and Entrepreneurship Training Program of Hunan Province (2494).
Acknowledgments
The authors acknowledge fruitful discussions with Hong-Hua Zhong.
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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