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Abstract
We theoretically propose a dual-channel bistable switch based on a monolayer Z-shaped graphene nanoribbon nanoresonator (NR) coupled to a metal nanoparticle (MNP). We show that the bistable nonlinear absorption response can be realized due to a competition and combination of the exciton-plasmon and exciton-phonon interactions. We map out two-dimensional and three-dimensional bistability phase diagrams, which reveal clearly the dynamical evolution of the roles played by these two interactions in managing optical bistability (OB) at all stages. Specifically, the bistable switch proposed can be controlled via a single channel or dual channels by only adjusting the intensity or frequency of the pump field. In/outside these channels, the switch will be turned on/off. The results obtained here not only can be employed to measure precisely the distance between the MNP and the NR but also provide promising applications in optical switching and optical storage.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The study of exciton-plasmon interaction has been a subject of considerable attention in the last decades [1–5]. In exciton-plasmon hybrid systems, many interesting nonlinear optical phenomena have been studied including nonlinear Fano effect [6,7], electromagnetically induced transparency [8], four-wave mixing [9,10], saturable absorption [11], second-harmonic generation [12], Kerr nonlinearity [13,14], ultrafast plasmonic field oscillations [15], Berry phase control [16], and so on. Especially, strong exciton-plasmon interaction can be used to realize and modulate optical bistability (OB). In a strongly coupled semiconductor quantum dot (SQD)-metal nanoparticle (MNP) hybrid system, Artuso and Bryant found that, the self-interaction of the SQD can result in the appearance of OB when the coupling between the SQD and the MNP is strong enough [17]. Malyshev et al. showed that OB in such a system can be revealed by measuring the optical hysteresis of Rayleigh scattering [18]. The bistable nonlinear absorption and refraction responses were also reported by Li et al. [19]. Nugroho et al. calculated two-dimensional bistability phase diagrams within the system’s parameter space and calculated the switching times from the lower stable branch to the upper one (and vice versa) [20]. Then they demonstrated that a molecular dimmer-MNP hybrid system leads to OB and hysteresis for the one-exciton transition [21]. Carreño et al. studied the nonlinear optical rectification and bistable effects and explored the relationship between these effects and the excitation conditions and structural parameters [22].
Graphene nanoribbon, as an attractive two-dimensional semiconductor material, possesses many outstanding properties such as low weight, high stiffness, large Kerr nonlinear coefficients and high quality factor [23–26]. Owning to these excellent properties, on the one hand, graphene nanoribbon can be expected to behave as a nanoresonator (NR) [27–29]. In a doubly clamped suspended graphene nanoribbon, a localized exciton will be formed due to the spatial modulation of the Stark-shift induced by a static inhomogeneous electric field. Therefore, there exist two kinds of excitations consisting of exciton and phonons in such a NR [30]. The exciton-phonon interaction plays an important role in modulating optical properties of the graphene nanoribbon NR. This point is different from the situation in the microresonators [31]. On the other hand, graphene nanoribbons can be used as a promising candidate for optical bistable devices. Peres et al. reported a bistable behavior in monolayer graphene in the terahertz range [32]. Xiang et al. investigated theoretically the OB of reflection at the interface between graphene and Kerr-type nonlinear substrates, and demonstrated that the bistable thresholds can be lowered markedly by increasing the Fermi energy in the THz range [33]. Then, some other schemes were proposed to manipulate OB such as a Fabry-Perot cavity with graphene in the THz frequency [34], a modified Kretschmann-Raether configuration [35], the dielectric/nonlinear graphene/dielectric heterostructures [36] and a nonlinear metasurface made of a few-layer graphene and a metal grating [37]. Specifically, OB of graphene SPs at mid-IR wavelength can also be observed when the graphene nanoribbon is covered on a dielectric grating [38]. Zhang et al. proposed a scheme for realizing OB and OM in a monolayer graphene system [39]. Gao et al. revealed that, when the dielectric spherical inclusions are coated by graphene layers, the hybrid system exhibits effective third-order nonlinear response and OB at terahertz frequencies [40]. Optical tristability in this system was also predicted [41]. Very recently, we proposed a highly-flexible bistable switch based on a suspended monolayer Z-shaped graphene nanoribbon NR [42].
Inspirited by the work of [42], we further theoretically design a dual-channel bistable switch based on a monolayer graphene nanoribbon NR coupled to a MNP. There exist simultaneously exciton-plasmon and exciton-phonon interactions in the graphene nanoribbon NR-MNP coupled system. This point is obviously different from other coupled dimers such as SQD-MNP and SQD-DNA hybrid systems with only a single interaction [17–22,43]. We map out two-dimensional and three-dimensional bistability phase diagrams in the system’s parameter subspace, which provides a clear picture about the dynamical evolution of the roles played by these two interactions in controlling OB at all stages. Detailed analysis also shows that the bistable switch proposed can be controlled via a single channel or dual channels. In these channels, the switch is turned on; outside these channels, the switch is turned off. Our findings will open a new horizon for measuring precisely the distance between two systems at the nanoscale.
2. Model and formalism
We consider a system consisting of a monolayer Z-shaped graphene nanoribbon NR coupled to an Au NP. This hybrid system is subjected to a strong pump laser Epu with frequency ωpu and a weak probe laser Epr with frequency ωpr, as shown in Fig. 1(a). The Au NP is attached to a sharp glass fiber tip and placed above the graphene NR. The distance between the NP and the NR can be adjusted by changing the position of the glass fiber tip via an atomic force microscope (AFM) [44,45]. The length of the glass fiber tip is far larger than the radius of Au NP. It would be reasonable to neglect the influence of this tip on the total system and treat it only as an auxiliary. It is reported that, the center of mass of the exciton in the graphene nanoribbon is localized arising from the spatial modulation of the Stark-shift induced by a static inhomogeneous electric field [46]. Due to the inhomogeneity of the applied field, the quantum confinement effect appears. Therefore, the graphene nanoribbon can behave as a quantum dot when its electrons are confined to a small region in this nanoribbon [46]. A small-scale localized exciton consisting of a ground state |0 > and the first excited state |1 > will be formed. Such an exciton can be characterized by three pseudospin operators σ01, σ10 and σz. In this NR, its fundamental flexural mode with the frequency ωn corresponds to its lowest-energy resonance. Here, the NR is assumed to possess a sufficiently high quality factor Q. In the hybrid system, there is a localized exciton, surface plasmons in the Au NP, and phonons in the graphene NR. Figure 1(b) displays the physical situation when the exciton interacts simultaneously with plasmons and phonons.
Fig. 1. (a) Sketch of a proposed dual-channel bistable switch based on a suspended monolayer Z-shaped graphene nanoribbon NR coupled to an Au NP in the simultaneous presence of a strong pump beam and a weak probe beam [47]. (b) The energy level scheme of a localized exciton coupled to surface plasmons in the Au NP and phonons in the graphene nanoribbon NR.
In a frame rotating at the pump field frequency ωpu, the total Hamiltonian of the system can be written as [48]
Making use of the Heisenberg equations of motion and bringing the noise and damping terms into the system, we arrive at the quantum Langevin equations as follows [51]:
The steady-state solution is denoted as Λ0, w0 and ϑ0, which can be derived from Eqs. (2)–(4) by setting all the time derivatives equal to zero. Therefore, we can rewrite the solutions around the steady state as follows: Λ = Λ0 + Λ1e−iδt + Λ−1eiδt, w = w0 + w1e−iδt + w−1eiδt and ϑ = ϑ0 + ϑ1e−iδt + ϑ−1eiδt. Here Λ0 >> Λ1, Λ−1, w0 >> w1, w−1, and ϑ0 >> ϑ1, ϑ−1. Upon working to the lowest order in Epr but to all orders in Epu, we can obtain the effective three-order nonlinear susceptibility given by
The exciton-population inversion w0 is determined by the following equation:
3. Results and discussions
We perform numerical calculations for a coupled Au NP-graphene nanoribbon NR system in the air environment (ɛB = 1). For the Au NP, we take r0 = 7 nm, ɛ∞ = 9.5, ћωp = 8.95 eV, ћγp = 0.149 eV [54]. For the localized exciton in graphene nanoribbon, ɛs = 6, μ = 40 D, Γ1 = 2 GHz and Γ2 = 1 GHz [30]. For the graphene NR, ωn = 7.477 GHz [55], Q = 9000, and γn = ωn /Q [56].
We start our investigation considering the impact of exciton-plasmon interaction and exciton-phonon interaction on the exciton population dynamics. But, before that, it is necessary to define three different exciton-phonon coupling regimes: (1) weak coupling regime (g < Γ2, γn); (2) intermediate coupling regime (g ∼ Γ2, γn); (3) strong coupling regime (g > Γ2, γn). In fact, in our system Γ2 >> γn. Figure 2(a) shows the variation of w0 with the pumping intensity Ipu for d = 20 nm and different g. The results show that in a strong exciton-plasmon coupling case (d = 20 nm) w0 exhibits a serial of standard S-shaped bistable curves by varying g from 0 GHz to 50 GHz. This indicates that an ultrastrong exciton-plasmon interaction plays a more important role than the exciton-phonon interaction in the generation of OB. As a comparison, we discuss two other cases of d = 30 nm and d = 40 nm so as to reveal completely the influence of the exciton-plasmon interaction on OB. The corresponding results are illustrated in Figs. 2(b) and 2(c). From Figs. 2(a)–2(c), we note that, in the weak and intermediate coupling regimes (g = 0 GHz and g = 1 GHz), the bistable effect will disappear gradually and the width of the bistable region becomes narrower and narrower with d increasing. However, the scenario for an ultrastrong exciton-phonon coupling regime (eg. g = 50 GHz) is rather different. The width of its bistable region becomes larger and larger as d increases. Moreover, an optical hysteresis loop of w0 with Ipu is plotted in the inset of Fig. 2(c). As Ipu increases, the system firstly follows the lower (stable) branch and then jumps to the upper (stable) branch at Ipu = 95.85 GHz2. With sweeping Ipu back, the system retains on the upper branch and then makes a transition to the lower branch at Ipu = 2.35 GHz2. A hysteresis loop has been completed. The intermediate branch is unstable.
Fig. 2. Variation of the exciton-population inversion w0 as a function of the pumping intensity Ipu for different exciton-phonon coupling strengths in three cases: (a) d = 20 nm, (b) d = 30 nm, and (c) d = 40 nm. (d) Variation of the nonlinear absorption Imχeff(3) as a function of the pumping intensity Ipu for d = 20 nm, 30 nm, and 40 nm. The other parameters are g = 10 GHz and Δpu = 0.
Figure 2(d) shows how the nonlinear absorption Imχeff(3) changes with Ipu when d = 20 nm, 30 nm, and 40 nm in a strong exciton-phonon coupling regime (i.e. g = 10 GHz >> Γ2, γn). For d = 20 nm, the nonlinear absorption spectrum depicted in Fig. 2(d) exhibits an “unfamiliar” bistable region. This region is almost overlapped. This suggests that an ultrastrong exciton-plasmon interaction plays a dominate role in producing OB. When d increases to 30 nm, however, the situation becomes completely different. Unexpectedly, the bistable region disappears entirely. The physical cause is mainly ascribed to a competition and cooperation between these two interactions. The exciton-plasmon interaction may counteract the exciton-phonon interaction and makes the total interaction weaker, suppressing greatly the appearance of OB. When d further increases to 40 nm, the bistable region reappears. The corresponding bistable nonlinear absorption response curve traces out an interesting path ➀→➁→➂→➃→➄→➅ by screening the value of Ipu. This indicates that the strong exciton-phonon interaction takes the place of the exciton-plasmon interaction and plays a leading role in controlling OB. The physics behind Fig. 2(d) can be understood as follows: When the spacing between the Au NP and graphene nanoribbon NR d is small (d = 20 nm), the exciton-plasmon interaction is relatively strong. When a strong exciton-plasmon interaction comes across a strong exciton-phonon interaction, the total interaction is weak due to a counteraction between them. In this case, the exciton-plasmon interaction plays a dominate role in the generation of OB. While when the spacing d increases to 30 nm, the exciton-plasmon interaction would be weakened, inducing that the total interaction becomes weaker. This may be the main cause of disappearance of OB. When d further increases to 40 nm, the exciton-plasmon interaction becomes relatively weak. However, the total interaction becomes strong. Under this case, the strong exciton-phonon interaction will take the place of the exciton-plasmon interaction and play a leading role in the appearance of OB. Therefore, OB reappears.
To obtain a clear and integrated picture of the dynamical evolution of the roles played by the exciton-plasmon interaction and exciton-phonon interaction in managing OB, we plot three-dimensional bistability phase diagrams of nonlinear absorption in the system’s parameter subspace (Iput; d; different g; Δpu = 0). As shown in Fig. 3(a), as the exciton-phonon interaction is absent (g = 0), a bistable region occurs and its critical point locates at (dc = 35.15 nm, Iput = 27.62 GHz2). For smaller d, the bistable thresholds both become quite large (i.e. Iput2 > Iput1 > 105 GHz2). This shows that a strong exciton-plasmon interaction is beneficial to generate OB. As g increases to 6.24 GHz, the scenario becomes rather different. A new added bistable region will appear within a window of 120.3 nm ≤ d ≤ 150 nm. In fact, the exciton-plasmon interaction in this window is rather weak as d is large enough. This indicates that such a new bistable region can be mainly ascribed to the contribution of the strong exciton-phonon interaction. As g further increases, the area of this new bistable region becomes larger and larger, suggesting that the exciton-phonon interaction becomes dominant in controlling OB compared to the exciton-plasmon interaction. Corresponding two-dimensional bistability phase diagrams are plotted in Fig. 3(b). For g < 6.24 GHz, there only exists a bistable region, while when g ≥ 6.24 GHz, another new bistable region appears in a rather weak exciton-plasmon coupling regime (i.e. d > 100 nm). Under this condition, the exciton-phonon interaction becomes decisive in terms of the appearance of OB. Unexpectedly, when g = 10 GHz and 28.41 nm < d < 32.66 nm, the bistable region disappears entirely. This suggests that, although these two interactions are both strong, the total interaction is still very weak due to a counteraction between them. Therefore, OB is significantly suppressed.
Fig. 3. (a) Three-dimensional bistability phase diagrams of the Au NP-graphene NR hybrid system in the system’s parameter subspace (Iput; d; different g; Δpu = 0). Iput1, Iput2, Iput3 and Iput4 represent the bistable thresholds of two regions, respectively. (b) Corresponding two-dimensional bistability phase diagrams. The colored area represents the region where the bistability exists. In/outside these areas, OB is switched on/off.
As a complement of Fig. 3, next we plot three-dimensional bistability phase diagrams in the system’s parameter subspace (Iput; g; different d; Δpu = 0). As seen in Figs. 4(a)–4(b), for d = 15 nm, there is a continuous bistable region ranging from g = 0 GHz to g ≥ 100 GHz, however, as d increases to 20 nm, this bistable region will be divided into two separated parts: Left bistable region and right bistable region. With d further increasing, the left bistable region moves to a direction with smaller g and its area becomes smaller and smaller. This region will shrink to a point at d = 35.15 nm, and then vanishes completely with further increase in d. As we expected, the right bistable region becomes larger and larger as d increases. It is worth mentioning that, as d ≥ 35.15 nm, the right bistable region nearly keeps unchanged. Under this condition, in comparison to the exciton-plasmon interaction, the exciton-phonon interaction plays a more important role than the exciton-plasmon interaction in producing OB. In a word, the dynamical evolution of the left and right bistable regions is accompanied by the change of the roles of two interactions.
Fig. 4. (a) Three-dimensional bistability phase diagrams of the Au NP-graphene NR hybrid system in the system’s parameter subspace (Iput; g; different d; Δpu = 0). (b) Corresponding two-dimensional bistability phase diagrams. The points P1 and P2 denote the critical locations that the bistability just appears. (c) Critical bistability conditions for a given Au NP-NR distance. (d) Dynamical evolution of the control channel of optical bistable switch for different g. The other parameter is Δpu = 0.
Having the results in Figs. 4(a)–4(b), we further extract information about the critical conditions for OB. The results are illustrated in Fig. 4(c). It is necessary to mention that, two critical points for bistability are labeled as P1 (d, gc1, Ipuc1) and P2 (d, gc2, Ipuc2). Ipuc1 and Ipuc2 denote the critical pumping intensities for bistability. By doing so, we can clarify and identify the critical conditions for bistability. When OB just arises, Ipuc1 and Ipuc2 both increase with d increasing. However, gc1 and gc2 both decrease. This indicates that, although these two interactions are both strong and weak, the total interaction is still very weak due to a counteraction between them. Therefore, OB is greatly suppressed. In addition, we also note that, when the exciton-plasmon interaction is weak (i.e. d is large), a large pumping intensity is beneficial to generate OB.
Figure 4(d) shows the dynamical evolution of the control channel of optical bistable switch for different g. For g = 10 GHz, OB will be triggered when Ipu is closed to 10 GHz2. As Ipu further increases, the switch exhibits two controllable channels accompanied by the appearance of two separated bistable regions. Taking Ipu = 70 GHz2 for example, when 27.06 ≤ d ≤ 27.58 or 35.87 nm ≤ d ≤ 40.36 nm, the bistable switch is turned on; while when d is outside these intervals, the bistable switch is turned off. Obviously, the system proposed can act as a dual-channel switch. Additionally, when Ipu ≥ 87.5 GHz2, the threshold of the upper bistable channel dput becomes infinitely large. This indicates a combination of a strong exciton-phonon interaction and a weak exciton-plasmon interaction is beneficial for broadening the controllable channel of the switch. For g = 5 GHz, the system can behave as a single-channel bistable switch. Its channel becomes wider and wider with Ipu increasing. It is easy to realize the switching between a single channel and dual channels by only adjusting the coupling strength between the exciton and phonons. These presented results with significant impact, bring highly feasibility and flexibility to develop the bistable devices with high-performance.
Inspection of Eq. (6) shows that OB depends strongly on the structural parameters (eg. g, d) and excitation conditions (eg. Ipu, Δpu). Considering this, we plot two-dimensional bistability phase diagrams in the system’s parameter subspace (Δpu; gt; different Ipu; d = 30 nm). The results in Fig. 5(a) show that, for Ipu = 10 GHz2 there exist upper and lower bistable regions. This is rather unexpected, since such a pumping intensity is relatively weak. As Ipu increases, the upper bistable region becomes larger and larger, however, the situation for the lower bistable region becomes completely different. The area of the lower bistable region becomes smaller and smaller as Ipu further increases. Until Ipu = 248.02 GHz2, this region is degraded to a point with gt ≈ 0 GHz and Δpu ≈ −1.48 GHz, while when Ipu > 248.02 GHz2, it is smeared out. These results in Fig. 5(a) demonstrate that, for a given exciton-plasmon coupling strength (i.e. d), a large pumping intensity Ipu will suppress significantly the appearance of the lower bistable region and broaden the upper bistable region to some extent. Obviously, the generation of OB can be attributed to a combined effect of the excitation conditions and two interaction mechanisms. Figure 5(b) shows the dynamical evolution of the control channel of optical bistable switch for different Δpu. For a given Ipu ( = 100 GHz2), the bistable switch undergoes a single-dual-dual-single-no channel switching process, corresponding to Δpu = −2, −0.55, 0, 0.55, 2 GHz, respectively. Also, for a given Δpu (= −0.55), OB just appears when Ipu = 10 GHz2, while when Ipu varies from 50 GHz2 to 100 GHz2, the switch will be switched from a single-channel to dual-channel. Interestingly, the switch will be converted from dual-channel to single-channel when Ipu further increases from 150 GHz2 to 200 GHz2. It is evident that the bistable switch can be controlled via a single channel or dual channels by only adjusting the intensity or frequency of the pump field.
Fig. 5. (a) Two-dimensional bistability phase diagrams (gt; Δpu; different Iput; d = 30 nm). (b) Dynamical evolution of the control channel of optical bistable switch for different Δpu. The other parameter is d = 30 nm.
Experimentally, Au NP-monolayer graphene nanoribbon NR hybrid systems can be prepared using the recent development. Regarding this, some related research has already been conducted. For example, in [27], Chen et al. proposed a scheme for realizing the fabrication and electrical readout of monolayer graphene NRs. Such graphene nanoribbon NRs can be decorated with Au@SiO2 core-shell NPs [57]. At the device level, the results presented in this paper can be utilized in two aspects: (1) The ratio of nonlinear absorption in the two stable states is as high as about 102, indicating that our model can serve as an optical memory cell: the lower and higher absorption can denote a logical 0 and 1, respectively. These two states of the memory cell can be switched by sweeping the pumping intensity. (2) Our system proposed can be used as a high-performance bistable switch. Such a switch can be controlled via a single channel or dual channels. In these channels, this switch is turned on; outside these channels, this switch is turned off. In fact, the switching of the control channel can be realized easily by only adjusting the intensity or frequency of the pump field. Also, the bistable switch can be controlled by changing some parameters, such as the spacing between the Au NP and the graphene NR and graphene-based parameters. We note that, the coupling strength between the exciton and the phonon modes is denoted as $g \propto (1 - \Lambda )\frac{{e\rho _G^{1/4}E{\varepsilon _ \bot }}}{{E_G^{3/4}({{q_0}L} )}}\sqrt {\frac{L}{{\pi \hbar }}}$. For example, g is proportional to L−1/2. In other word, g can be manipulated by only changing the length of the graphene nanoribbon along the NR axis L.
4. Conclusions
We theoretically proposed a scheme of a dual-channel bistable switch based on a suspended monolayer Z-shaped graphene nanoribbon NR coupled to an Au NP. The results showed that, the bistable nonlinear absorption response of the system appears due to a competition and cooperation between the exciton-plasmon interaction and exciton-phonon interaction. We calculated two-dimensional and three-dimensional bistability phase diagrams, which reveal fully the dynamical evolution of the roles played by these two interactions in managing OB. Furthermore, our model can be used as a high-performance bistable switch with controllable channels. This switch can be controlled through a single channel or dual channels. In/outside these channels, the switch can be switched on/off. Finally, we hope that our proposed device can be realized by current experiments in the near future.
Funding
National Natural Science Foundation of China (11404410, 11805283, 51701243, 11174372); Foundation of Talent Introduction of Central South University of Forestry and Technology (104-0260); Undergraduate Innovation and Entrepreneurship Training Program of Central South University of Forestry and Technology (72).
Disclosures
The authors declare no conflicts of interest.
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