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Abstract
Graphene has emerged as an ultrafast photonic material for on-chip all-optical switching applications. However, its atomic thickness limits its interaction with guided optical modes, resulting in a high switching energy per bit. Herein, we propose a novel technique to electrically control the switching energy of an all-optical graphene switch on a silicon nitride waveguide. Using this technique, we theoretically demonstrate a 120 µm long all-optical graphene switch with an 8.9 dB extinction ratio, 2.4 dB insertion loss, a switching time of <100 fs, a fall time of <5 ps, and a 235 fJ switching energy at 2.5 V bias, where the applied voltage reduces the switching energy by $\sim 16\times$. This technique paves the way for the emergence of ultra-efficient all-optical graphene switches that will meet the energy demands of next-generation photonic computing systems, and it is a promising alternative to lossy plasmon-enhanced devices.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Global internet traffic has grown by 12$\times$ since 2010 [1]. Meanwhile, electricity use by the information and communications technology industry remained nearly flat owing to the simultaneous enhancement of the energy efficiency of data centers [2]. Nonetheless, the rise of computationally demanding applications like artifical intelligence and cryptocurrencies is likely to increase data-center electricity use by 15$\times$ by 2030 [3], and this in turn necessitates the development of ultra-efficient data-center architectures and components. Modern data centers make use of silicon photonic interconnects to communicate between servers, racks, and digital logic chips, where the transmission loss can be as low as 0.045 dB/m using silicon nitride (SiN) waveguides [4]. On-chip processing of data is performed by an ultrafast optical switch, which actively controls the flow of light in a photonic link. Electro-optic switches are commonly employed in these links [5–10], but they are usually enclosed within bulky transceiver systems that perform the necessary optical-electric-optical conversions, which leads to additional radio-frequency time delays in the signal processing [11]. In recent years, attention has been drawn to all-optical switching technologies [12–17]. These switches can operate with sub-picosecond switching times, but consume several picojoules of energy per bit [18–20]. Energy-efficient switches with a sub-pJ switching energy have been demonstrated in Refs. [21–23], but were limited by relatively high switching times (>1 ps). This energy-speed trade-off was recently broken by the demonstration of an ultrafast (260 fs) and energy-efficient (35 fJ) all-optical switching device that is based on graphene-loaded deep-subwavelength plasmonic waveguides [24]. Other plasmon-enhanced all-optical graphene switches have been reported in Refs. [25,26]. In these devices, the presence of plasmonic nanostructures boosts the interaction of graphene with the guided optical mode, which effectively reduces the switching energy per bit. However, much of the propagating light energy in these devices is lost due to ohmic and coupling losses, which might hinder their adoption in practical photonic links. Increasing the interaction between graphene and the guided optical mode is possible by elongating the switch waveguide. Nevertheless, the switching energy is also proportional to the graphene sheet area [24,26], so a larger switching energy would be required to operate a larger graphene switch. Because of its ultrafast electron heating times (<150 fs) [27–29], graphene is an excellent candidate material for ultrafast all-optical switching, yet there needs to be alternative methods to build energy-efficient graphene switches. In this work, we propose a novel technique to significantly reduce the switching energy of an on-chip all-optical graphene switch on a SiN waveguide. The reduction in the switching energy is achieved by applying a bias voltage. Using this technique, we theoretically demonstrate the reduction of the switching energy by $\sim 16\times$ by applying a bias voltage of a few volts, depending on the device configuration.
In the next section, the device structure and its constituent materials are presented. Following that, the all-optical switching mechanism is introduced and explained, and the device efficiency is characterized by calculating its switching energy and extinction ratio. Then, the impact of applying a bias voltage on the graphene switch is explored by investigating the resulting switching energy. To efficiently harness this effect, an alternative, more effective configuration is proposed. Subsequently, some practical considerations that might limit the effectiveness of this approach at certain bias voltages are discussed. In addition, the influence of the applied voltage on the switching performance is investigated. Finally, the report is concluded by pointing out the major findings of this study.
2. Device structure
The structure of the on-chip switch is illustrated in Fig. 1. A 600$\times$1100 nm SiN wire waveguide on top of a 1.5$\,\mathrm{\mu}$m buried oxide (BOX) layer, guides the incoming light to the graphene switch section. Silicon dioxide (SiO$_2$) is placed on the sides of the SiN wire to facilitate the placement of graphene on top of the waveguide. The waveguide supports a quasi transverse-magnetic (TM) mode, which is chosen for its high absorption by graphene (0.032 dB/µm), in comparison to the quasi transverse-electric (TE) mode (0.028 dB/µm) at $\lambda = 1550$ nm. The electric field distribution of the TM mode with graphene is presented in the inset of Fig. 1. The guided mode couples efficiently to the switch section, with a computed coupling efficiency of 99.99%. To achieve a high switching efficiency and to demonstrate the effectiveness of our method, we set the device length ($L$) to 120 µm. Discussion of the switching efficiency is presented in the next section. An electrical voltage ($V_{G}$) is applied through a Si gate to electrically control the switching energy. To maximize the impact of $V_{G}$, the BOX thickness ($d$) is set to a minimum of 1.5 µm to prevent significant substrate leakage [30]. In the next section, the operation principle and the switching energy of the device are investigated.
Fig. 1. All-optical graphene switch on a silicon nitride waveguide. The inset shows the electric field profile of the propagating quasi-TM mode in the waveguide-integrated switch. The dashed white line represents the graphene sheet plane. $\lambda =$ 1550 nm. Si: silicon, SiO$_{2}$: silicon dioxide, BOX: buried oxide, Gr: graphene, Au: gold, $V_{G}$: gate voltage, $L$: device length, and $d$: BOX thickness.
3. Operating principle
The absorption of graphene can be tuned based on the principle of Pauli-blocking [31–33], which occurs when photoexited electrons fill the conduction band states of graphene following a sufficiently intense pump excitation, thereby blocking the interband transition of other electrons. The chemical potential of graphene ($\mu$) is given by [34]:
where $v_{F}$ is the Fermi velocity, and $n_0$ is the carrier density of graphene. At telecom wavelengths, the optical absorption of graphene is dominated by interband absorption [28], which becomes saturated when $\mu$ reaches a value that is given by $\mu ^\prime = \hbar \omega _{\textrm {pump}}/2$ [24]. As a consequence, incoming pump photons cannot induce an interband transition because $\hbar \omega _{\textrm {pump}} \leq 2|\mu ^\prime |$ [34], and are thus transmitted through graphene. Similarly, a probe photon with an energy $\hbar \omega _{\textrm {probe}} \leq \hbar \omega _{\textrm {pump}}$ is also transmitted because $\hbar \omega _{\textrm {probe}} \leq 2|\mu ^\prime |$. By utilizing this phenomenon, all-optical switching is realized: the probe signal is transmitted when the pump signal is HIGH, or absorbed when the pump signal is LOW [26].The increase in carrier density ($\Delta n$) that is needed to saturate the absorption of graphene is given by [26]:
For a graphene sheet with an area $A=WL$, the number of carriers that are needed to reach $\mu ^\prime$ is $m = \Delta n W L$. Because each absorbed photon generates an electron-hole pair, the energy that is required to saturate the absorption of graphene ($U_{\textrm {sw}}$) can be expressed as [24]:
where $U_{\textrm {sw}}$ is the switching energy, and $\hbar \omega$ is the pump photon energy. We consider the portion of graphene that interacts with the optical mode. For the switch structure of Fig. 1, the interaction area is the waveguide width ($W$) times the modulator length ($L$). More electrons are required to fill the conduction band states up to $\mu ^\prime$ when $\mu$ is low, resulting in higher switching energies. While the maximum value of $U_{\textrm {sw}}$ is $\sim 2\,$pJ, the effective switching energy ($U_{\textrm {eff}}$) of this device theoretically exceeds 3.5 pJ (see Fig. 2). To calculate $U_{\textrm {eff}}$, we consider the actual percentage of light that is absorbed by graphene and the non-saturable component of the graphene absorption. First, the percentage of light that is absorbed by graphene ($A_{G})$ is calculated by where $\alpha = 0.032\,$dB/µm in our configuration. Next, $U_{\textrm {eff}}$ is given by the following relation [26]: where $A_{\textrm {ns}}$ is the non-saturable fraction of $A_G$ [35–37], and is taken as 5% [26,36]. Our calculated switching energies are comparable to the saturation thresholds that were experimentally reported in Ref. [38]. Before concluding this section, we quantify the switching efficiency of the device by its extinction ratio (ER), which is given by [24]: where $T_{\textrm {on}}$ and $T_{\textrm {off}}$ represent the transmitted power of the probe signal when the pump signal is turned on and off, respectively. The pump signal wavelength is 1550 nm, while the probe signal wavelength is taken as 1560 nm. The propagation loss at $\lambda _{\textrm {probe}} = 1560\,$nm is also equal to 0.032 dB/µm in our configuration. Because the pump and probe signal wavelengths are very close to one another, both signals experience a similar optical loss. The performance of the device at other wavelengths can be quantified using the methods that we presented in our previous work [26]. The absorption of graphene is maximized when the pump signal is turned off, and thus $T_{\textrm {off}}$ is expressed as:The maximum ER is obtained when a pump signal is applied with an energy $U \geq U_{\textrm {eff}}$, because the optical absorption of graphene saturates and $T_{\textrm {on}}$ is maximized:
where $A_{\textrm {ns}}$ has been included to account for the non-saturable fraction of the graphene absorption. For $L=120\,$µm, the maximum ER is 3.8 dB.Fig. 2. Switching energy ($U_{\text{sw}}$) and effective switching energy ($U_{\text{eff}}$) as a function of chemical potential ($\mu$) in (a) and as a function of the applied bias voltage ($V_G - V_D$) in (b). $\lambda_{\text{pump}} = 1550\,$nm.
4. Electrical control
It is observed in Fig. 2 that the switching energy is determined by the chemical potential of graphene. Consequently, it is possible to tune the switching energy of a graphene switch by manipulating the chemical potential. Applying a back-gate voltage $V_{G}$, as was shown in Fig. 1, is one practical way to accomplish that. When graphene is placed on a back-gated dielectric substrate, a parallel plate capacitor is formed. In this case, the carrier density of graphene is controlled by the back-gate voltage according to the following relation [34]:
Using this relation, the switching energy in Fig. 2(b) is plotted as a function of $V_{G} - V_{D}$, where $V_{D}$ is the gate voltage for the chemical potential to be tuned at the Dirac point [34]. As expected, the switching energy depends on the back-gate voltage, such that a higher $V_G$ decreases the energy that is necessary to switch on the device. When $V_{G}-V_{D}=100\,$V, the switching energy decreases by a factor of $\sim 4 \times$. Hence, the switching energy of an all-optical graphene switch can be electrically tuned by applying a bias voltage. Despite the effectiveness of this method, the device that is shown in Fig. 1 demands relatively high voltages to produce efficient switching energies, e.g. $V_{G}-V_{D} = 100\,$V for $U_{\textrm {eff}}\approx 799\,$fJ. Considering that this device might be integrated into optoelectronic systems with advanced CMOS technology nodes, the voltage values that are presented in Fig. 2(b) might not be practical. As such, there needs to be an alternative configuration where voltages as low as a few volts are applied to achieve energy-efficient switching.
Fig. 3. (a) Front view of the all-optical graphene switch. The structure is divided into three regions; regions 1 & 3 are identical. (b) Equivalent circuit model for the same structure.
5. Top-gate configuration
The previous section discussed the role of electrical biasing in tuning the chemical potential of graphene, in a parallel plate capacitor configuration. According to Eq. (12), the gate voltage is inversely proportional to the capacitance per unit area. Taking that into account, we propose a top-gate configuration with a relatively high $C$ to reduce the amount of $V_{G}$ that would be necessary to reach high carrier densities or chemical potentials. The capacitance can be increased by reducing the separation between the electrodes or by incorporating an insulating material with a high dielectric constant. Figure 4 presents a front-view of the top-gate configuration, which is inspired by the device that is reported in Ref. [42]. In this configuration, the graphene sheet forms a parallel plate capacitor with a 40 nm thick layer of indium tin oxide (ITO), which is chosen for its high optical transmittance and high electrical conductivity [43]. Titanium dioxide (titania) is an insulating material with a very high dielectric constant ($\epsilon _{\textrm {Ti}} \approx 170$) along the c-axis [44,45], and is therefore employed in this configuration as a high-$\epsilon$ dielectric. A 60 nm Au/Cr contact supplies the gate voltage to the ITO electrode. The device performance is studied at multiple titania thicknesses: 350, 450, 550 and 650 nm. The propagating modes in this structure are presented in Fig. 12 in the Appendix. To determine which mode is optimal, we first find out the propagation loss that is related to graphene ($\alpha _G$), that is [26,46–48]:
where $\alpha _{\textrm {tot}}$ is the total propagation loss in the switch waveguide due to graphene and ITO, and $\alpha _{\textrm {ITO}}$ is the propagation loss of the switch waveguide after removing graphene or when graphene’s absorption is totally saturated, as shown in Figs. 12 & 13. Using this information, we plot the absorption efficiency of graphene ($\alpha _G / \alpha _{\textrm {tot}}$) as a function of titania thickness ($t_{\textrm {Ti}}$) for both propagation modes in Fig. 5. Clearly, the TE-mode yields a higher absorption efficiency at all titania thicknesses. The lossy ITO electrode is placed further away from the SiN waveguide for a thicker titania layer. Hence, in this case, the waveguide mode experiences a lower ohmic loss due to ITO and more absorption by graphene. On the other hand, for a thinner titania layer, the ITO electrode is placed closer to the SiN waveguide. As a result, it induces a higher ohmic loss to the propagating mode, which reduces the absorption efficiency of graphene. The coupling efficiency ($\kappa$) of the switch waveguide with the original SiN waveguide is also computed and shown in Fig. 5(b), where $\kappa$ is higher at small titania thicknesses. In Fig. 12 in the Appendix, it is observed that a portion of the waveguide mode propagates in the titania layer that is placed above SiN; this observation is more visible for the case of the TM-mode. A higher portion of the waveguide mode propagates in the titania layer at thicker $t_{\textrm {Ti}}$, which results in a lower coupling efficiency with the original SiN waveguide mode. The TE-mode yields a higher absorption and coupling efficiencies at all titania thicknesses, and is therefore chosen as the optimal mode. The Au/Cr contact is placed 1 µm away from the waveguide edge to ensure that it does not introduce ohmic losses, as explained in the Appendix. Other details of the modeling parameters are also provided in the Appendix.Fig. 4. Top-gate configuration of the switch waveguide. Voltage is applied through a Au/Cr top gate that is connected to an indium tin oxide (ITO) electrode. Titania is used as the dielectric material between ITO and graphene.
Fig. 5. (a) Absorption efficiency of graphene ($\alpha_G / \alpha_{\text{tot}}$) and (b) coupling efficiency ($\kappa$) as a function of titania thickness ($t_{\text{Ti}}$) for the TE and TM modes. $\lambda = 1550\,$nm.
In this configuration, the ITO electrode contributes to the absorption of the waveguide mode and introduces some coupling losses, resulting in a notable insertion loss ($IL$). Therefore, Eqs. (7) & (8) are modified to include these losses [26]:
By incorporating Eqs. (14)–18 into Eq. (6), the maximum $ER$ is calculated. In addition, the $IL$ is calculated using the following relation [49]:
Figure 6 shows the calculated $ER$ and $IL$ at multiple titania thicknesses. It is observed that the $ER$ increases with the titania thickness. The device switching mechanism is based on the saturable absorption of graphene. As previously explained, the absorption efficiency of graphene ($\alpha _{G}/\alpha _{\textrm {tot}}$) increases at thicker $t_{\textrm {Ti}}$, and the absorption of graphene is saturated after applying a pump signal with an energy $U>U_{\textrm {eff}}$. Consequently, at thicker $t_{\textrm {Ti}}$, a higher contrast exists between the ON and OFF states, because of the high $\alpha _{G}/\alpha _{\textrm {tot}}$ ratio, which in turn leads to a higher $ER$. The $IL$ slightly increases at thicker $t_{\textrm {Ti}}$, following the trend of the decreasing coupling efficiency, as shown in Fig. 5(b). For a $350\,$nm thick titania, the $ER$ is 8.9 dB, and the $IL$ is 2.4 dB. Figure 7 plots the $ER/IL$ ratio as a function of the titania thickness, which is maximized at thicker $t_{\textrm {Ti}}$ in the studied range. Nevertheless, the capacitance per unit area ($C$) is higher at smaller $t_{\textrm {Ti}}$, as shown in Fig. 7(b). As was previously explained, the gate voltage ($V_{G}$) is inversely proportional to $C$. Consequently, a smaller amount of voltage is required at smaller $t_{\textrm {Ti}}$ to increase the chemical potential. To check this out, the effective switching energy is calculated for this configuration using the following relation [26]:
Fig. 6. Extinction ratio ($ER$) and insertion loss ($IL$) as a function of the titania layer thickness ($t_{\textrm {Ti}}$). $\lambda _{\textrm {probe}} = 1560\,$nm.
Fig. 7. (a) $ER/IL$ ratio for $\lambda_{\text{probe}}=1560\,$nm and (b) capacitance per unit area ($C$) as a function of titania thickness ($t_{\text{Ti}}$).
The resulting $U_{\textrm {eff}}$ at multiple titania thicknesses is shown in Fig. 8. It is observed that $U_{\textrm {eff}}$ significantly drops by applying a few volts, especially at smaller $t_{\textrm {Ti}}$. For instance, for a 350 nm thick titania layer, $U_{\textrm {eff}}$ drops from the initial $\sim 3.67\,$pJ to $235\,$fJ at $V_G - V_D = 2.45\,$V, thereby reducing $U_{\textrm {eff}}$ by $\sim 16\times$. Therefore, using this configuration, it is possible to effectively control the switching energy of an all-optical graphene switch by applying CMOS-compatible voltages of a few volts. It is noted that the capacitor functions as an open circuit when the bias voltage is electrostatic. Consequently, the consumed electrical power is zero based on $P = IV$. The effectiveness of this control mechanism is paid for by the insertion loss that is introduced by the ITO electrode. Depending on the application scope and the range of available bias voltages, the titania thickness can be adjusted to meet a practically satisfactory output. Other practical considerations are presented in the next section.
Fig. 8. Effective switching energy ($U_{\textrm {eff}}$) as a function of the bias voltage ($V_G - V_D$) at multiple titania thicknesses. $\lambda _{\textrm {pump}} = 1550\,$nm.
6. Practical considerations
In the previous section, it was shown that the switching energy can be controlled by applying a bias voltage of a few volts. From Fig. 8, it is also observed that $U_{\textrm {eff}}$ can be $<5\,$fJ at certain values of $V_G - V_D$. However, in practice, this technique is not omnipotent, because the optical absorption of graphene is a function of the chemical potential, scattering rate and temperature. To demonstrate that, we first compute the surface optical conductivity of graphene ($\tilde {\sigma }$), which is given by [50]:
The normalized $\alpha$ is plotted in Fig. 9(b) for $\lambda = 1550\,$nm and $0.2 \leq \mu \leq 0.6$ at multiple operating temperatures: 10 K, 100 K, 200 K and 300 K, while assuming a fixed scattering time of 100 fs. As previously explained, graphene absorbs photons with an energy $\hbar \omega > 2|\mu |$, and thus the cutoff energy for absorbing a photon is $|\mu | = \hbar \omega /2$, which is $\sim 0.4\,$eV in our case. However, in practical operating conditions where the scattering rate and the temperature are not zero, a relatively smooth cutoff curve is obtained (see Fig. 9). Because of that, $\alpha$ can be notably reduced at a $\mu$ value that is less than $\hbar \omega /2$. For the case of $T=300\,$K, we set $\mu \approx 0.3\,$eV as the maximum chemical potential at which the graphene switch practically operates, where the corresponding normalized $\alpha > 0.95$. This is because at chemical potentials beyond $0.3\,$eV, the noteworthy drop in $\alpha$ results in a poor extinction ratio, according to Eqs. (6), (14) and (15). Using Eq. (12), the normalized $\alpha$ is plotted as a function of $V_{G} - V_D$ for a $350\,$nm thick titania layer in Fig. 10(a). For $t_{\textrm {Ti}} = 350\,$nm, the $V_G - V_D$ value that corresponds to $\mu = 0.3\,$eV is 2.45 V (see Fig. 10(b)), which yields an effective switching energy of $235\,$fJ and a $\sim 16\times$ reduction factor based on the data presented in Fig. 8. For larger bias voltages, $U_{\textrm {eff}}$ is lower, but so is the $ER$ of the switch. Therefore, the absorption mechanism in graphene imposes a fundamental trade-off between $U_{\textrm {eff}}$ and $ER$ at high bias voltages. This trade-off is reduced for a sharper cutoff curve, which can be done by either reducing $T$ or $\Gamma$. In practice, tuning $\Gamma$ might not be feasible, but it is possible to sharpen the cutoff curve by operating at low temperatures. For instance, for $t_{\textrm {Ti}} = 350\,$nm, the normalized $\alpha > 0.95$ at $V_{G} - V_D = 3.6\,$V when $T = 10\,$K (see Fig. 10(b)), which yields $U_{\textrm {eff}} = 32.2\,$fJ, thereby substantially reducing $U_{\textrm {eff}}$ by $\sim 114\times$ from the initial $\sim 3.67\,$pJ. Therefore, with this technique, subpicojoule-efficient all-optical switching is obtainable by applying CMOS-compatible voltages of a few volts. Furthermore, the ultra-efficient operation that is achievable at low temperatures might potentially be of interest in applications that necessitate cryogenic temperatures, e.g. integrated quantum photonics. In the next section, the influence of electrical gating on the switching performance of the device is investigated.
Fig. 9. (a) Surface optical conductivity of graphene ($\tilde{\sigma}$) as a function of chemical potential ($\mathrm{\mu}$) for $T = 300\,$K, $\lambda = 1550\,$nm and a fixed scattering time of $100\,$fs. The y-axis is divided by $\sigma_0 = e^2 / 4\hbar \approx 60 \, \mathrm{\mu}$S. (b) Normalized absorption coefficient of graphene ($\alpha$) as a function of chemical potential ($\mathrm{\mu}$) at multiple temperatures: 10 K, 100 K, 200 K and 300 K for a fixed scattering time of $100$ fs and $\lambda = 1550$ nm.
Fig. 10. (a) Normalized absorption coefficient of graphene ($\alpha$) as a function of the bias voltage ($V_G - V_D$) for a $350\,$nm thick titania layer at multiple temperatures: 10 K, 100 K, 200 K and 300 K for a fixed scattering time of $100\,$fs and $\lambda = 1550\,$nm. (b) Bias voltage ($V_G - V_D$) as a function of the chemical potential ($\mu$) for multiple titania thicknesses: 650 nm, 550 nm, 450 nm and 350 nm.
7. Switching performance
The switching performance can be characterized by the rise and fall times of the all-optical switch, which are fundamentally limited by the electron heating and cooling mechanisms in graphene [26]. Graphene has a unique conical dispersion, where its density of states vanishes at the Dirac point. Consequently, electrons in the vicinity of the Dirac point have a relatively low heat capacity. Following a pump excitation, these electrons immediately scatter with one another, creating an ephemeral Fermi-Dirac distribution of hot thermalized electrons within a few tens to 150 fs [27–29]. Thus, an electron becomes "hot" by absorbing the energy of an incident photon and then heats up other electrons by electron-electron scattering events. These hot electrons later cool down in a few picoseconds by emitting optical and acoustic phonons, coupling with surface optical phonons, and most importantly through disorder-assisted scattering which dominates at room temperature [53–56]. In order to calculate the switching time of the device, the electrical conductivity of graphene is first calculated using the following relation [57–59]:
Next the electron-electron scattering time is calculated using the Boltzmann transport theory [61]:
In addition, the electron cooling time ($\tau _{\textrm {cool}}$), which is the inverse of the the electron cooling rate ($\gamma _{\textrm {cool}}$), is calculated using the following relations [28,54,57]:
Fig. 11. Electron-electron scattering time and electron cooling time as a function of chemical potential ($\mu$) in (a), and as a function of the applied bias voltage ($V_G - V_D$) for a 350 nm thick titania layer in (b).
To understand the influence of electrical gating on the switching performance, we plot $\tau _{\textrm {scat}}$ and $\tau _{\textrm {cool}}$ as functions of the gate voltage. This is simply done by computing Eqs. (29) & (32) with $V_G$ instead of $\mu$. The relation between $\mu$ and $V_G$ was derived in Eq. (12). Figure 11(b) shows the resulting plot for a 350 nm thick titania layer. It is observed that $\tau _{\textrm {scat}}$ and $\tau _{\textrm {cool}}$ vary with $V_G - V_D$, yet this variation is practically not significant, because the switch still preserves its ultrafast electron heating and cooling dynamics. The switch is not operated under bias at $V_G - V_D = 0$, because that is where the switching energy is maximized (see Figs. 2(b) & 8), and as such that region is not considered. Hence, it is concluded that the electrical control of the switching energy of an all-optical graphene switch does not compromise its switching performance.
For electro-optic switches, a smaller dielectric layer thickness results in a higher capacitance, and a higher resistance-capacitance (RC) product reduces the device speed [64]. On the other hand, for all-optical graphene switches, the device speed is determined by the electron heating and cooling dynamics in graphene, as was previously explained. In the top-gate configuration shown in Fig. 4, graphene is placed on top of the oxide substrate, which provides an efficient cooling pathway for photoexcited hot carriers through disorder-assisted scattering [28,53,65]. The presence of the oxide substrate ensures that the switching mechanism of the device is ultrafast at all titania thicknesses. Therefore, it is also concluded that varying the titania layer thickness does not compromise the switching performance of the device.
8. Conclusion
In conclusion, we propose a novel technique to electrically control the switching energy of a chip-integrated all-optical graphene switch on a silicon nitride waveguide. The electrical control is achieved by applying a bias voltage in a back-gate or a top-gate configuration. The back-gate configuration is less effective than the top-gate configuration, because it is limited by the large thickness of the buried oxide (BOX) layer. Using the top-gate configuration, we theoretically demonstrated a 120 µm long all-optical graphene switch with an 8.9 dB extinction ratio, a 2.4 dB insertion loss, a switching time of < 100 fs, a fall time of < 5 ps, and a 235 fJ switching energy at a 2.5 V bias, hence achieving a $\sim$16$\times$ reduction in the switching energy. Furthermore, by operating the switch at a low temperature of 10 K and 3.6 V bias, we theoretically demonstrated a $\sim 114\,\times$ reduction factor, where the resulting switching energy is merely 32.2 fJ. Moreover, it was shown that at high bias voltages, further reduction in the switching energy is accompanied by a reduction in the extinction ratio because the absorption coefficient of graphene depends on the chemical potential, scattering rate and temperature. In addition, it was numerically demonstrated that the electrical control of the switching energy of the device does not compromise its switching performance. Further improvement of the switching efficiency is attainable by incorporating a dielectric material with a higher $\epsilon$ value and enhancing graphene’s interaction with the optical mode by exploring alternative waveguide structures, e.g. slot waveguides. The effectiveness of this technique in significantly reducing the switching energy at a few volts bias makes it a highly promising alternative to lossy plasmon-enhanced graphene switches. Therefore, this technique is likely to find applications in next-generation photonic computing systems where ultrafast and energy-efficient switches will be sought after.
Appendix
Figure 12 shows the propagating quasi-TE and quasi-TM modes in the switch waveguide for a 350 nm thick titania layer at $\lambda = 1550\,$nm. Figure 13 shows the same modes after removing graphene or when graphene’s absorption is totally saturated. Figure 14 shows the propagation loss of the TE-mode as a function of the 300 nm wide Au/Cr contact spacing from the waveguide edge, for a $350\,$nm thick titania layer. It is observed that the curve flattens at large spacing distances, indicating that the Au/Cr contact is no more contributing to the propagation loss. The contribution of the Au/Cr contact is practically negligible for a 500 nm spacing distance and above. However, fabrication variations might introduce some errors in the actual position of the contact; thus, a 1 µm spacing distance is preferably used in our configuration to avoid these issues. The Au/Cr contact is modeled as a pure Au contact, because Au exhibits a strong plasmonic response at near-infrared wavelengths [66]. The refractive index of Au at 1550/1560 nm is taken as $n_{\textrm {Au}} = 0.47 + j11$, based on the data that are experimentally reported for a 53 nm thick Au film in Ref. [67]. The refractive index of ITO at 1550/1560 nm is taken as $\: n_{\textrm {ITO}} = 1.68 + j0.11 \:$ [68].
Fig. 12. (a) Quasi transverse-electric (TE) and (b) quasi transverse-magnetic (TM) modes that propagate in the switch waveguide. The dashed white line represents the graphene sheet plane. $\alpha$ is the propagation loss that is experience by the mode. The titania layer is 350 nm thick. The white line represents the graphene sheet plane. $\lambda=1550\,$nm.
Fig. 13. Propagation loss of the quasi TE-mode in (a) and the quasi TM-mode in (b) after removing graphene. The titania layer is 350 nm thick. $\lambda=1550\,$nm.
Fig. 14. Propagation loss of the TE-mode as a function of the Au/Cr contact spacing from the waveguide edge, for a 350 nm thick titania layer. $\lambda = 1550\,$nm. The inset shows the propagating mode with the Au/Cr contact (red outline) placed 1 µm away from the waveguide edge.
The refractive index data of SiN are taken from Ref. [69] in Lumerical. Graphene is modeled in Lumerical as a 2D material using the model that is described in Ref. [70].
Funding
New York University Abu Dhabi.
Acknowledgments
Support from the NYUAD Center for Cyber Security research grant is gratefully acknowledged.
Disclosures
The authors declare no conflicts of interest.
Data availability
The authors confirm that the data supporting the findings of this study are available within the article.
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