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We present two types of designs for plasmonic switches based on hybridization between single interface surface plasmon polaritons and modes of a thin film of transition metal oxide material, vanadium dioxide (VO2). The design includes integrated, localized heaters that activate the VO2 transition. The device operation is investigated and optimized by electromagnetic, electrical, and thermal simulations. The large change in the VO2 refractive index in the infrared wavelength range enables highly compact and efficient plasmonic switches. The proposed designs achieve extinction ratios of 23–32 dB using only a 5 μm active region, a switching voltage of about 60 mV, and a switching power of about 9 mW.
© 2012 Optical Society of America
1. Introduction
Surface plasmon polaritons (SPPs) are coupled electromagnetic density oscillations that propagate at the interface between a metal and a dielectric. SPPs concentrate optical energy to subwavelength scales, which can enable optoelectronic devices that are more compact than high-index contrast all-dielectric implementations [1–3]. However, active SPP-optoelectronic (plasmonic) devices remain a significant challenge due to the small active volumes. For example, plasmonic switches and modulators inevitably suffer from trade-offs between their size, extinction ratio, and insertion loss [4–7]. This trade-off is fundamentally due to the small electrically induced changes in the real and/or imaginary parts of the refractive index (of the order 10−4 – 10−2) under modest applied voltages in most common optical materials, such as electrooptic crystals, polymers, silicon, and III–V semiconductors. To counteract this trade-off, an efficient and compact plasmonic switch should be made with a material that can exhibit large changes in its optical properties subjected to an electrical control signal.
A promising material for plasmonic switches at telecommunication wavelengths is vanadium dioxide (VO2), which has garnered significant scientific attention recently for its electromagnetic and electronic properties [8,9]. VO2 is a transition metal oxide that undergoes a phase transition from an insulating dielectric state to a conductive metallic state at a critical temperature Tc ∼ 341 K. Moreover, applied electric fields [9–11] and electromagnetic radiation [8, 12] may also initiate the VO2 phase transition. The transition from the dielectric to the metallic state can be as fast as hundreds of femtoseconds while the transition from the metallic back to dielectric state is slower, with transition times ranging from nanosecond to millisecond timescales [13].
Accompanying the dramatic electrical resistivity change in VO2 is a large refractive index change of O(1) in the infrared wavelengths. At λ = 1.55 μm, VO2 has a refractive index of nd ≈ 3.24 + 0.30i in its dielectric state below Tc, and nm ≈ 2.03 + 2.64i in its metallic state above Tc [14]. However, despite the large change in the refractive index, using VO2 directly as a waveguide material is prohibitive for two reasons. First, the imaginary component of nd is too large for VO2 to be the core of a dielectric waveguide; and second, the imaginary part of nm is too small for VO2 to be the metal of a SPP waveguide without incurring substantial propagation losses. However, these limitations can be overcome by combining VO2 with low-loss metal films to take advantage of the strong electromagnetic confinement of SPPs and the large index change of VO2 to realize highly compact plasmonic switches with high extinction ratios at low switching powers. Hybrid geometries have been the basis of many recent plasmonic-based devices [15–18] since they benefit from or enhance the compact nature of SPPs while mitigating the strong absorption of the metal. Several VO2-based optical switches have also been studied recently in silicon waveguides [19, 20] and in SPP [21] geometries to investigate the feasibility of the material for modulation and switching.
In this article, we present designs for electrically controlled plasmonic switches that utilize mode hybridization and the thermally induced phase transition of VO2 to achieve switching with high extinction ratios in devices that are only a few wavelengths long. This study includes comprehensive thermal modeling to evaluate the efficiency of the switch and a comparison with other switching approaches. Our proposed geometries integrate a low-power heater with a hybrid VO2-SPP waveguide that, depending on the specific device geometry, operates in one of two regimes: “strongly-hybridized” or “weakly-hybridized.” The strongly- and weakly-hybridized designs differ mainly in their overlap with the VO2 region. Both designs can exhibit very high extinction ratios and moderate insertion losses at short device lengths ≲ 5 μm for an operation near λ = 1.55 μm. With a total active length of 5 μm, the devices have extinction ratios exceeding 30 dB with switching voltages of 60 mV and powers of only ∼ 9 mW. The results illustrate the potential for VO2 to be used in highly efficient plasmonic switches.
2. General device description
2.1. Device operation
Figure 1(a) shows the general proposed device geometry, which, for the simplicity of experimental demonstration, includes grating couplers to couple light to and from free-space. In an integrated circuit, the switch would instead be coupled to input and output waveguides. The device is composed of a single thin metal layer (silver, Ag) that is separated from the VO2 film by a silica (SiO2) buffer layer. The VO2 is deposited on an optically thick SiO2 layer. In the on-state of the switch, an input optical beam excites a propagating mode of the structure with the VO2 in its dielectric state and the light propagates along the waveguide with low loss. In the off-state, a current passes through the heater in the direction perpendicular to the light propagation. The resultant resistive heating induces the VO2 phase transition, which modifies the waveguide modes and increases their propagation losses. The optical transmission is consequently reduced and the output is switched off. Once the current is turned off, the VO2 dissipates the heat through thermal conduction and returns to its dielectric state, thereby reducing the propagating losses of the modes and restoring the transmission properties of the device to its original on-state.
Fig. 1 (a) Perspective of the integrated device including gratings for coupling light in and out. (b) Cross-section of the plasmonic waveguide.
2.2. Device geometry
The designed SPP waveguide is composed of a thin Ag strip with width wwg = 8 μm, which is large enough to ensure that the scattering losses at the intersection of the heater and the SPP waveguide are negligible (< 0.5 dB). The thickness of the Ag is tAg = 300 nm, which optimizes the grating input/output coupling efficiency and ensures that no light leaks out of the top surface of the metal. In an integrated switch, however, tAg ≳ 200 nm is sufficient to prevent optical leakage out of the top surface.
The heater width is wht = 5 μm and is chosen to provide a large enough resistance to ohmically heat the active VO2 region (i.e. the VO2 directly under the waveguide which has significant overlap with the plasmonic mode) beyond Tc while limiting the current density in the Ag to ≲ 107 A/cm2, below the damage threshold of thin Ag wires. The main advantages of this geometry are that it is simple to fabricate and power-efficient, since the Ag strip serves as both the SPP waveguide and integrated heater.
Figure 1(b) shows the cross-section of the SPP waveguide. An SiO2 buffer layer is inserted between the Ag layer and the VO2 film. The thicknesses of the SiO2 buffer layer and the VO2 films are respectively tSiO2 and tVO2. These two parameters play a key role in defining the electromagnetic modes of the structure and are varied in the rest of the discussion to identify different regimes of operation.
For the results to follow, the electromagnetic simulations were done using mode-solver and harmonic propagation techniques (2D, steady-state) of a commercial finite element method solver (COMSOL Multiphysics). Although 2D mode profiles were analyzed, the results show 1D cross-sections taken along the center (x = 0, Fig. 1(b)) of the device to more clearly illustrate the mode hybridization. Due to the waveguide width, these results closely resemble 1D calculations. Grating calculations were done with 2D harmonic propagation techniques, and the gratings were excited with a plane wave. The index of VO2 is taken from [14], the index of Ag from [22], and the index of SiO2 was set to 1.45.
3. Optical performance
3.1. Overview
The geometry of Fig. 1 supports two regimes of operation which we term “strongly-hybridized” and “weakly-hybridized.” In the strongly-hybridized case, the excited modes are hybrids of a VO2 waveguide mode and a single-interface SPP mode regardless of the state of the VO2. This regime occurs in thinner VO2 and SiO2 layers. In the weakly-hybridized case, the excited mode switches from being strongly hybridized when the VO2 is in its metallic state, to weakly hybridized when the VO2 is in its dielectric state. This regime occurs in thicker VO2 and SiO2 layers. The weakly-hybridized mode does not have significant overlap with the absorbing VO2, so it incurs less loss than the strongly-hybridized modes.
To illustrate the two regimes, Fig. 2(a)–2(b) show respectively the real, neff, and imaginary, keff, parts of the effective indices of the modes at λ = 1.55 μm as a function of tSiO2 for tVO2 = 120 nm and tVO2 = 260 nm. In each case, we identify and label four different modes: D1 and D2, when the VO2 is in its dielectric state; and M1 and M2, when the VO2 is in its metallic state. The shaded regions in Fig. 2(a)–2(b) show respectively the strongly and weakly hybridized cases that are discussed in more detail in Sections 3.2–3.3. To better understand the hybridization of these modes and their roles in efficient switching, Fig. 2(c) shows the constituent, non-hybridized modes and their effective indices corresponding to tSiO2 → ∞. The top shows the SPP mode at the Ag-SiO2 interface; the middle shows the mode of the VO2 waveguide in its dielectric state with a SiO2 cladding; and the bottom shows the even long-range SPP (LRSPP) supported by the VO2 in its metallic state with a SiO2 cladding.
Fig. 2 Real (top, neff) and imaginary (bottom, keff) parts of the effective index for (a) tVO2 = 120 nm and (b) tVO2 = 260 nm as a function of tSiO2 at λ = 1.55 μm. The shaded areas in (a) and (b) are studied in more detail in Section 3.2, which focuses on the strongly-hybridized design, and Section 3.3, which focuses on the weakly-hybridized design, respectively. (c) Individual modes which make up the hybrid modes of the VO2 modulator. As tSiO2 → ∞, D1 and M1 reduce to the Ag SPP mode, D2 reduces to the VO2 dielectric mode, and M2 reduces to the VO2 LRSPP mode.
For tSiO2 ≳ 2 μm, all modes are weakly hybridized and closely resemble their respective constituent modes. As illustrated by Fig. 2(a)–2(b), the effective indices of the modes become nearly constant as tSiO2 is increased. For these values of tSiO2, D1 and M1 correspond to the Ag SPP mode, D2 to the VO2 dielectric mode, and M2 to the VO2 LRSPP mode.
When tSiO2 is reduced, the spatial overlap between the modes increases and the modes hybridize. The hybridization can be quantified by the change in the effective indices of the modes as a function of tSiO2. As shown in Fig. 2(a)–2(b), as tSiO2 decreases, the D2, M1, and M2 modes become strongly hybridized while D1 (Ag SPP mode) remains largely unchanged and only interacts weakly with the VO2 layer. Thus, the D1 mode has a much lower propagating loss than the other modes, as can be seen in the plots of keff. However, for the thin VO2 variation (tVO2 = 120 nm), the D1 mode is cutoff for tSiO2 < 1.42 μm, so the D2 mode must be used.
For efficient switching, we require a large difference in keff between the D1,2 and M1,2 modes. The efficiency also depends on the overlap between the mode excited by the gratings and the mode in the waveguide region. At the same time, the insertion loss should be kept low. Thus, we define a figure of merit (FOM):
where is the extinction ratio per propagation length in dB, ILprop = − 10log10 (|Tmax|) is the insertion loss per propagation length in dB, Tmax is the transmittance per length when the VO2 is in its dielectric state, and Tmin is the transmittance per length when the VO2 is in its metallic state. The FOM represents a length-normalized trade-off between the extinction ratio and the insertion loss of the switch. Therefore, a high FOM is preferred. By varying (1) the separation between the VO2-Ag layers (tSiO2), and (2) the thickness of the VO2 layer (tVO2), the FOM can be optimized. Two distinct operation regimes emerge: the strongly-hybridized regime as shaded in Fig. 2(a) and the weakly-hybridized regime as shaded in Fig. 2(b).3.2. Strongly-hybridized design
Switching in the strongly-hybridized design is dominated by the change in the imaginary part of the VO2 refractive index. Here, the D1 mode is cutoff, and only the the higher loss D2 mode is supported when the VO2 is in its dielectric state. Figures 3(a)–3(b) show the electric and magnetic field intensities, respectively, of these modes normalized to the Poynting vector when the VO2 is in its dielectric and metallic states for tVO2 = 120 nm and tSiO2 = 160 nm at λ = 1.55 μm. neff and keff are also shown in Fig. 3(c)–3(d) at λ = 1.55 μm as a function of tSiO2 (magnified views of the shaded region in Fig. 2(a)).
Fig. 3 (a) Electric and (b) magnetic field intensity for the optimized strongly-hybridized device (tVO2 = 120 nm and tSiO2 = 160 nm). (c) neff and (d) keff for a range of tSiO2 for the strongly-hybridized device with tVO2 = 120 nm. Spectral dependence of (e) neff and (f) keff for the optimized device.
We perform a series of simulations, and find that the strongly-hybridized FOM reaches an optimal value of 6.8 for tVO2 = 120 nm, and tSiO2 = 160 nm. Figures 3(e)–3(f) show the spectral characteristics of the optimal design, which indicate the device supports broadband operation between 1.4 μm and 1.6 μm. At wavelengths below ∼ 1.31 μm, the real part of the VO2 refractive index in the metallic state is less than the index of SiO2, and the VO2 LRSPP mode becomes cut-off. Therefore, an efficient plasmonic switch operating on mode cut-off should be feasible for a wavelength range of several hundred nanometers up to ∼ 1.31 μm; however, this design is beyond the scope of the present work.
For this optimal device, ILprop = 0.9 dB/μm and ERprop = 6.1 dB/μm. However, there are two additional sources of loss. First, when integrated with the grating couplers, the gratings incur an additional insertion loss of 8.9 dB (5.2 dB at the input and 3.7 dB at the output). The grating couplers have 10 periods with a periodicity of 960 nm and a 50% duty cycle. Second, a mode mismatch exists when the active VO2 region is in its metallic state. Since the heating is highly localized (details in Section 4), the temperature of the VO2 at the gratings is below Tc. When the heater is activated, the input grating still excites the D2 mode, which, in the optimized design, then primarily excites the M1 mode. The M1 mode then propagates toward the output gratings where it is coupled back to the D2 mode and eventually to free space. The Poynting vector overlap of the D2 and M1 modes is 0.82 [23], which leads to a 1.7 dB loss when the VO2 is locally switched to the metallic state. Taken together, the optimal strongly-hybridized device with a 5 μm long modulation region (the shortest allowed as limited by the width of the integrated heater) including grating couplers has an insertion loss, IL, of 13.4 dB, and an extinction ratio, ER, of 32.1 dB. Additional 2D propagation simulations of the full device confirm these calculations, and show that light scattered from the input grating and, in the off-state, the mode-mismatch contribute negligibly to the output power, as most of the scattered light is directed into the substrate, and the remainder is not optimally matched to the output grating.
3.3. Weakly-hybridized design
As tVO2 increases beyond about 250 nm, the D1 mode becomes supported at all values of tSiO2. Figures 4(a)–4(b) show the Poynting vector normalized electric and magnetic field intensities respectively, of the D1, M1, and M2 modes for tVO2 = 260 nm and tSiO2 = 780 nm. The propagation loss of D1 is an order of magnitude less than that of D2, so D2 is not shown. neff and keff of the modes are shown as a function of the SiO2 layer thickness in Fig. 4(c)–4(d) (magnified views of the shaded region in Fig. 2(b)). As shown in Fig. 4(a)–4(b), and in contrast to D2 in the strongly-hybridized case, the field profile of D1 is significantly different from M1 and M2, due to its reduced coupling with the VO2 layer. Hence, when VO2 is switched between its dielectric and its metallic state, a different type of mode propagates. The D1 mode is weakly hybridized, whereas the M1 and M2 modes are strongly hybridized.
Fig. 4 (a) Electric and (b) magnetic field intensity for the optimized weakly-hybridized device (tVO2 = 260 nm and tSiO2 = 780 nm). (c) neff and (d) keff for a range of tSiO2 for the weakly-hybridized device. Spectral dependence of (e) neff and (f) keff for the optimized device.
Similar to Section 3.2, we find tVO2 and tSiO2 that optimize the FOM of Eq. (1) at λ = 1.55 μm. The FOM reaches an optimal value of 12.0 at tSiO2 = 780 nm and tVO2 = 260 nm. The spectral dependences of neff and keff are shown in Fig. 4(e)–4(f). Since the effective index does not deviate greatly over the wavelength range from 1.4 μm to 1.6 μm, the weakly-hybridized design is also capable of broadband performance.
In the optimal weakly-hybridized switch, ILprop = 0.3 dB/μm and ERprop = 3.6 dB/μm. The grating couplers add a loss of 15.5 dB (8.3 dB at the input, and 7.2 dB at the output). Each grating coupler has 10 periods of 1064 nm with a 50% duty cycle. Additionally, the overlap between the D1 and M1 modes is 0.58, and dominates over the overlap between the D1 and M2 modes. The mode mismatch leads to a 4.7 dB loss when the VO2 is in its metallic state. For this optimal geometry with a 5 μm long modulation region, IL = 17.0 dB and ER = 22.7 dB.
4. Integrated heater
A significant advantage of plasmonics is that an electrical signal can be sent along the same metal that supports the SPP. In our design, as shown in Fig. 1(a), an electrical wire runs perpendicular to the plasmonic waveguide and narrows at the modulation region. This functions as a localized heater which ohmically heats a narrow region around the thin Ag wire and has a minimal affect on the gratings. The length of the resistive heater is optimized to keep the heating volume to a minimum to reduce the consumed power, while ensuring that the electrical contacts are well separated from the plasmonic waveguide.
Figure 5(a) shows the relevant simulation space for the electrothermal simulations which were conducted in 3D under steady-state conditions. The entire simulation domain is 1200 μm × 1200 μm × ∼ 600 μm. The large volume accommodates the electrical contacts, which are tapered down to 5 μm at the device region. The boundaries of the simulation domain are sufficiently far away so that the results are unchanged when the domain size is increased further. A voltage is applied to a 50 μm × 50 μm electrical contact region on one contact, while a similar 50 μm × 50 μm region on the other contact is grounded. These contact regions marked in Fig. 5(a) represent the connection by electrical contact probe tips or wire bonds. All other external electrical boundaries are set to be electrically insulating. The SiO2 substrate is ∼ 600 μm thick, and the lower boundary is set to a constant temperature, T = 293 K, acting as a heat sink. All other external thermal boundaries are set to a heat flux representing air. The grating couplers are omitted in the thermal simulations for simplicity and to improve the convergence in the simulations.
Fig. 5 (a) Simulation space for the thermal calculations. (b) Powers and voltages required to switch the device from its dielectric to its metallic state (top), and the corresponding ER/IL ratio (bottom). (c) Perspective and (d) cross-section of the temperature distribution required for TVO2,avg = 355 K for the device with tVO2 = 260 nm, and tSiO2 = 780 nm. The temperature profiles for tVO2 = 120 nm and tSiO2 = 160 nm are similar.
To be as consistent as possible with the data from [14], we compute currents and powers at a fixed voltage to heat the active VO2 region to an average temperature of TVO2,avg = 355 K, the temperature at which the optical parameters of the metallic VO2 state were measured. This is roughly 14 K above the transition temperature. The thermal conductivity for VO2 is taken from [24], and the thermal conductivity for Ag is set to 429 W/(m·K). The electrical conductivity of Ag is represented with a linear model, with resistivity ρ = 1.59 × 108 Ω·m and temperature coefficient α = 0.0038 K−1. The electrical and thermal conductivities of SiO2 are taken from the built in parameters in COMSOL. The electrical conductivity of VO2 is neglected, as it is separated from the current-passing Ag wire by the electrically thick SiO2 layer.
Figure 5(b) (top) shows the required powers and voltages to switch the VO2 from its dielectric to its metallic state. The required power increases with device length, since a greater volume of VO2 must be heated. The strongly-hybridized device requires less power since it has a smaller separation between the Ag and VO2 layers and has a thinner VO2 film to heat. Generally, the devices require about 60 mV and 10 mW for switching. Figure 5(b) (bottom) shows the ratio ER/IL (different than the FOM = ERprop/ILprop, which is a length-normalized ratio) for the devices. As the power consumption and applied voltages of the devices increase, this ratio also increases.
The temperature distribution of the optimized weakly-hybridized device with a waveguide length of 15 μm is shown in Fig. 5(c)–5(d). This device requires a power of 11.2 mW, and a voltage of 62.7 mV for switching. Figure 5(c) illustrates a perspective view of the heater and the localized nature of the heating. Figure 5(d) is a cross-section, showing that the temperature is nearly uniform across the active VO2 region. The temperature drops below the transition temperature Tc = 341 K within roughly one grating period. As a result, the gratings are relatively unaffected by the ohmic heating and the dielectric modes (D1 for weakly-hybridized, and D2 for strongly-hybridized) are mainly excited at the grating couplers, as we assumed earlier. The temperature profile of the optimized strongly-hybridized device is similar (not included here).
5. Discussion
To evaluate the functionality of our proposed VO2 switch, we compare the strongly- and weakly-hybridized devices presented here with other devices in literature. The summary of the comparison is presented in Table 1. We present the data for our designs including and excluding the grating couplers. The performance without gratings characterizes the behavior of the modulation region on its own to indicate how an integrated version of the device may perform.
We observe that the strongly-hybridized device performs better than the weakly-hybridized device in most aspects when the gratings are included, as it possesses a superior ERprop, ER, IL, and FOM. It is also shorter, and requires smaller switching powers and voltages. The weakly-hybridized device has a smaller ILprop. Excluding the gratings, the weakly-hybridized device has a superior IL and FOM for the same device size. This difference is due to the increased coupling efficiency of the grating for the strongly-hybridized device, where the plasmonic mode is more confined to the Ag surface, and hence, better matched to the gratings. For an isolated device excited by free-space optics, the strongly-hybridized device is preferable. However, if the device is integrated, and if the coupling efficiency is equal for both devices, the weakly-hybridized device is more attractive. The ILprop of the weakly-hybridized case is nearly three times less than that of the strongly-hybridized case. Even though the ER is less than for a strongly-hybridized section, a 5 μm long weakly-hybridized section can achieve an ER > 20 dB, which is sufficient for many applications.
Table 1. Comparison of devices presented in this work with recent SPP switches and modulators. Where the numbers are absent, data is not available.
Also included in Table 1 are several recent plasmonic switches and modulators. The metal-dielectric-metal (MDM) geometry in [21] is most comparable to the work presented here, as it utilizes a thin VO2 layer in the dielectric region. The maximum quoted ER is > 20 dB and coincides with an ILprop of ∼ 3.1 dB/μm. However, the length required to reach this ER is not specified, and neither are the methods for exciting modes and VO2 switching. Hence, the comparison is incomplete. A second design utilizing a thermally-induced phase change in gallium (Ga) on silicon-on-insulator (SOI) is presented in [25]. When Ga is switched from its metallic to its α-state, the optical mode is extinguished. This device is highly compact (only 400 nm long), but has a value of ER/IL barely above 1, even though input/output coupling should be more efficient than a grating coupler. Also, the electrical characteristics and integrated heating are not specified in [25].
Some designs based on modulating the transmission through thin metallic slits have been proposed [6,26,27]. Reference [26] is given as an example in Table 1. These designs are highly compact, but rely on 3-dimensional geometries and are not well-suited for on-chip integration. The designs suffer from low ERs (∼ 3 dB) [6], or require high applied voltages (∼ 45 V) [26]. The design in [27] is not fully described, but may have an IL comparable to or exceeding the ER.
Experimental [4] and theoretical [7,28] examples utilizing electrooptic effects have also been investigated. Reference [28] is given in Table 1. These designs can be much more compact than their dielectric mode counterparts. However, electrooptic effects produce a much smaller change in the refractive index/loss compared to the VO2 phase change. As a result, higher voltages are required to reach modest ERs.
Finally, several other approaches to plasmonic switches have shown various degrees of success. Reference [5] proposes a design in which an indent of a MDM waveguide is filled with an active material. The device promises excellent ERs and moderately low ILs. However, it relies on optical pumping and precise fabrication. [29] proposes an interferometric device switched by a thermooptic effect, which must be relatively large (tens of micrometers long) to achieve ERs ∼ 1 dB. Reference [30] proposes to use liquid crystals as the active medium for modulation. However, the modulation requires a high applied voltage of ∼ 25 V.
6. Conclusion
In summary, we have presented two designs for a hybrid VO2-plasmonic switch. These are the first VO2-based plasmonic device designs to include integrated heaters. Both designs rely on the large refractive index difference between the dielectric and metallic states of VO2 to drastically enhance the optical absorption. The strongly-hybridized device switches between two strongly hybridized modes. The design is highly efficient, reaching an extinction ratio of 32.1 dB and an insertion loss of 13.4 dB within an active region of 5 μm. The weakly-hybridized device switches between a strongly and weakly hybridized mode. It has an extinction ratio of 22.7 dB and an insertion loss of 17.0 dB in an active region of 5 μm. In both cases, the switching voltage is about 60 mV and the power required is about 9 mW. The large index change of VO2 presents significant advantages in power efficiency, optical loss, and extinction ratio compared to many existing plasmonic modulators. The device operation speed is limited to tens of kilohertz by the thermal transport and VO2 phase transition times. Higher modulation speeds can be achieved by changing the mechanism of phase transition from a thermal process to a field effect process [9]. Experimental verification of these designs will be reported in the near future.
Acknowledgments
The authors are grateful for the financial support of the Natural Sciences and Engineering Research Council of Canada.
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