1. INTRODUCTION

To modulate light amplitude or phase and maintain the state afterwards, an on-chip optical memristor requires the real or imaginary refractive index of the active material to be modified in a nonvolatile state [1,2]. Moreover, from the viewpoint of future large-scale chip integration, its process flow should be simple and compatible with the standardized (or front-end/back-end) CMOS process [3,4], along with high scalability and reliability. For instance, phase-change materials, featuring substantial changes of real and imaginary refractive indices in the transition between amorphous and crystalline states, are a promising material platform for on-chip optical memristors, which can be physically deposited on large-scale photonic chips directly and compatible with the back-end CMOS process [5–11]. However, in the operation of phase-change transition, the local ramping up to several hundred degrees Celsius undoubtedly results in issues of thermal crosstalk, material fatigue, and equilibration period [2,10]. Mechanisms of resistive filamentations have been maturely employed in electrical memristors, which also support nonvolatile states of the modified dielectric function in the optical regime to implement optical memristors [12–14]. The challenge in this platform is the ineffective light-coupling with the nanogap of filamentary resistive memory. It could only be solved by employing plasmonic modes in the nanogap, which unfortunately involves large propagation loss and photonic-plasmonic conversion loss [1,15]. Polarization domain reconfigurations in ferroelectric optical materials, such as BaTiO3, have nonvolatile duration time in hundreds of nanoseconds, providing another strong and pivotal competitor of optical memristors [16]. Note that the trigger of polarization domain switching is determined by the properties of BaTiO3, including the coercive field, material defects, and crystallographic orientations, which complicates the memristor operation [17,18]. In addition, the wafer-scale integration of ferroelectric materials in photonic circuits still remains difficult [1,19]. Based on other technologies, such as MEMS [20] and magneto-optic effects [21,22], on-chip optical memories have been proposed, which, however, require complex fabrication processes and cannot perform multilevel analogue operations [16].

Here, we propose a new on-chip optical memristor based on graphene. As a semimetal hosting massless fermions, graphene represents one of the highest room-temperature mobilities of any material [23] and extremely strong on-resonance interaction with light across a wide range from ultraviolet to terahertz [24,25], which is well recognized as an excellent active material for chip-integrated optoelectronics [26–30]. With an external electric field, graphene’s carrier density could be readily boosted from 1012 to 4×1013cm−2, resulting in deep modifications of real (imaginary) refractive index from ∼3.0(∼2.8) to near zero (∼0.1) in the telecommunication band [31,32]. Leveraging this attribute, if the carrier doping state of graphene could be maintained, a promising optical memristor is expected, especially, with the assistance of chip integration to enhance light-graphene interaction. To implement it, we electrically gate the graphene integrated on a silicon nitride (SiN) waveguide with a ferroelectric thin film of poly (vinylidene fluoride trifluoroethylene) [P(VDF-TrFE)], as schematically demonstrated in Fig. 1. After the removal of the initially applied electrical voltage pulse over the ferroelectrics layer, there is a large remanent polarization at the interface between the ferroelectric film and graphene, which statically dopes graphene with a maintained certain carrier density and nonvolatile modified refractive index. As a result, the modulation states of transmission loss and phase delay of the hybrid graphene-SiN waveguide could be held, giving rise to the operation of optical amplitude and phase memristors. From the fabricated P(VDF-TrFE)-gated graphene memristor, a high nonvolatile modulation depth of ∼32.5dB is realized with a low programming energy of µ∼1.86pJ/µm2. Thanks to the multilevel polarization and stability of the ferroelectric film [33–35], the semicontinuous stored states are illustrated as well, with remarkable retention time (over 10 years) and cyclability (fluctuation ratio <0.9%). By further integrating the graphene-based optical memristor with cascaded microring resonators, the combined functions of optical parallel computing and in-memory computing are demonstrated by the implementations of analogue matrix-vector multiplications and digital logic gate operations. With the recently developed back-end CMOS foundry of graphene optoelectronics [36], the proposed graphene-based on-chip optical memristor has great potential in future large-scale manufacturing, and could be integrated with the state-of-the-art passive and active devices in silicon photonics to carry out high bandwidth neuromorphic computing [37], parallel convolutional processing [38], artificial intelligence [39], etc.

Fig. 1. Geometry and operation principle of the graphene-based optical memristor. (a) Schematic concept configuration. Light guided in the waveguide interacts with graphene via its evanescent field. Electrical pulses of the gate voltage (??) are applied to graphene via ferroelectric P(VDF-TrFE) to nonvolatile change graphene’s refractive index (?) and absorbance (?) to modulate the phase and amplitude of the optical output. (b) Operation mechanism of the graphene-based optical memristor. Nonvolatile variations of graphene’s ? and ? and waveguide transmission occur in respect to pulsed ??. ?up and ?down represent polarization-up and -down states in ferroelectrics after the SET and RESET operations, holding the carrier doping states in graphene for nonvolatile State 1 and State 2. Band structures of graphene with different doping levels are shown on the right side of the inset. (c) Schematic of the looped hysteretic responses of graphene’s ? and ? and resulting optical transmission states with ?? in the optical memristor.

2. RESULTS AND DISCUSSION

A. Device Geometry and Operation Principle

The concept geometry of the graphene-based on-chip optical memristor is shown schematically in Fig. 1(a). Monolayer graphene is integrated on a SiN waveguide (or a microring resonator composed by a waveguide as demonstrated later), which is then coated with a ferroelectric thin film of P(VDF-TrFE). The graphene layer is electrically contacted by drain and source electrodes to measure the carrier doping characteristics. A metal electrode is deposited on top of the P(VDF-TrFE) layer to apply a vertical gate voltage (??) over the graphene channel to control the memristor states. The light guiding in the SiN waveguide interacts with graphene via its evanescent field, which governs the dependence of the transmission loss (phase delay) of the hybrid graphene-SiN waveguide on the absorbance (refractive index) of graphene.

Figure 1(b) illustrates the operation principle of the proposed memristor. By applying a pulse of negative ?? (SET operation), the P(VDF-TrFE) is polarized up and maintains the state, as governed by the ferroelectric attribute. The remanent charges at the bottom surface of P(VDF-TrFE) dope graphene effectively without the requirement of external electrical voltage, which lifts its Fermi energy (??) away from the Dirac point. When ?? exceeds the half of photon energy (??), graphene’s absorbance (?) is suppressed due to the Pauli blocking [40], which increases the transmission of the hybrid graphene-SiN waveguide in a nonvolatile manner (State 1: HIGH Transmission). In addition, as described by the Kubo formulation, the refractive index (?) of graphene is modulated as well by the carrier doping, resulting in a varied phase delay of the waveguide mode [32,41,42]. By composing the graphene-SiN waveguide into a microring resonator, for instance, the resonant wavelength could be nonvolatile shifted for amplitude modulation. Then, if a positive ?? pulse is actuated (RESET operation), the polarization direction of the ferroelectric P(VDF-TrFE) is reversed to the polarization-down state. The heavily doping state on ?-type graphene is changed into almost intrinsic. The strong optical absorption in graphene is recovered, as well as the high refractive index, which resets the states of optical transmission loss and phase delay of the hybrid graphene-SiN waveguide (State 2: LOW Transmission). In a cycle of the above process, owing to the involvement of ferroelectrics to the graphene-SiN waveguide platform, both the optical properties (? and ?) of graphene and the waveguide transmission exhibit looped hysteresis responses to ??, as indicated in Fig. 1(c). Nonvolatile operations between two (State 1 and State 2) or more photonic states are accessible, providing two degrees (phase and amplitude) of freedom for advancements in photonics and optoelectronics (e.g., high-dimensional light manipulation).

B. On-Chip Graphene Amplitude Memristor Demonstration

Figure 2(a) displays the optical microscope image of a fabricated device with an µ∼80µm long graphene covered on a SiN strip waveguide, which has a width (height) of ∼1080(∼300)nm. A ∼300nm thick P(VDF-TrFE) layer is spin-coated on the whole chip, and a top electrode is subsequently deposited on it to overlap with the graphene-waveguide region. The manufacturing details and structural characterization are shown in Supplement 1 Figs. S1–S4. To confirm its ability as an optical memristor, we couple a laser at the wavelength of ∼1550nm into the graphene-SiN waveguide and monitor the modulation behavior of its transmission by cyclically scanning the gate voltage (??). The initial state of the device before applying ?? is shown in Supplement 1 Fig. S5. To evaluate the remanent polarization mechanism, we also measure the ??-dependent drain-source currents (DS?DS) of the graphene channel. The results are shown in Fig. 2(b).

Fig. 2. On-chip graphene amplitude memristor demonstration. (a) Optical microscope image of a fabricated device. The inset shows the full-scale view of the device. (b) Hysteresis of the variations in the measured waveguide transmission and source-drain current (DS?DS) of the graphene channel for ?? cyclically scanning in a range of −20 to 20 V at a specific wavelength of 1550 nm. Black arrow indicates the ?? sweep direction. (c) Demonstration of optical memristor operation at sub-millisecond level. The HIGH (LOW) transmission state is converted by ?? pulses of −20V (20 V) and maintained afterwards. (d) Cyclability of the SET/RESET operation for 400 switching events. The shaded area indicates fluctuations of 0.04 dB for cycling operations. (e) Transmission retention time for two states. Retention time is measured to over 24 h, and the dashed line marks extrapolated transmissions for over 10 years without crossing each other. (f) Broad-spectrum response of graphene-based optical memristors. Twelve switching transitions were performed randomly within 24 h. The shaded area indicates the standard deviation and the solid line indicates the average. (g) Semicontinuous nonvolatile operations of optical memristors. The device is returned to the base state by a RESET pulse of ??=20V. The intermediate state is determined by the amplitude of the SET pulse. The inset zooms into a complete operation of the intermediate state.

Both DS?DS and the optical transmission exhibit clear and distinct hysteresis. First, accounting for the SET operation implied in Fig. 1(b), when a SET voltage of ??=−20V is applied, the P(VDF-TrFE) is polarized up with negative charges bounded over its bottom surface to contact with graphene. The previously stored state of the graphene-based optical memristor is removed. Thus, graphene is heavily hole-doped and exhibits high conductivity with a large DS?DS and the graphene-SiN waveguide is set to the State 1 with HIGH transmission. When the applied ?? is scanned from −20 to 0 V, the electric field of ?? actuated on the graphene layer is eliminated gradually. However, DS?DS only exhibits a slight decrease, and the value at ??=0V is much larger than that of the intrinsic graphene. Based on the ferroelectric feature of P(VDF-TrFE), the bounded negative charges are retained even though the ?? is returned to 0 V, which holds the hole-doping state over graphene and thus makes it transparent. Consequently, the HIGH transmission state of the on-chip graphene memristor is constant for ?? changed from −20 to 0 V.

By applying ?? along the positive direction, the external electric field is reversed with respect to the internal electric field of the remanent polarization in P(VDF-TrFE). As long as ?? does not exceed the positive coercive voltage, the hole-doping in graphene and waveguide transmission remains at HIGH levels, governed by the residual poling field at the ferroelectrics interface. Around ??=11V, which is the coercive voltage of the 300 nm thick P(VDF-TrFE), the direction of ferroelectric polarization changes, which dopes graphene with electrons instead of holes. The decreased DS?DS indicates the strong hole-doping in graphene is modified into a light electron-doping, which lowers graphene’s ?? and makes it opaque. The waveguide transmission reduces to the LOW state sharply, and holds the state for ?? increased to 20 V. We notice that since graphene is slightly hole-doped by the SiN substrate and the wet transfer process, the polarization field of ferroelectrics will not dope graphene with strong electron concentration and thus not induce another HIGH transmission state of the waveguide. As ?? is gradually reduced from 20 to 0 V, similar to the aforementioned forward sweep process, the polarization charges of the ferroelectric layer keep graphene in the weak electron-doping state. The waveguide transmission is located in the LOW state. This process corresponds to the RESET operation indicated in Fig. 1(b).

With ?? applied from 0 to −20V, DS?DS increases gradually due to the strengthened carrier doping in graphene by the electric field of ?? via the P(VDF-TrFE) layer. For the transmission of the graphene-SiN waveguide, in the ?? range from 0 to −5V, it maintains a LOW state governed by graphene absorption. Further decreasing ?? from −5V, the transmission increases sharply to a HIGH state. The varied transmission is caused by the prohibition of graphene absorption since the high carrier density lifts the ?? larger than the half of photon energy. With the carrier doping in even higher levels, the HIGH transmission state retains, for instance, in the range of −20V<??<−12V. It will maintain until a positive ?? larger than the coercive value performs a RESET operation.

The above hysteresis curve over the broad ?? range indicates the P(VDF-TrFE)-gated graphene-SiN waveguide could modulate the transmission to a state and hold it afterwards, which is the characteristic of optical memristors. To further demonstrate its performance, we apply pulsed voltages ?? to SET and RESET the graphene doping levels and carry out the optical memristive transmission according to the illustration in Fig. 1. As shown in Fig. 2(c), the HIGH (LOW) transmission state is converted by ?? pulses of −20V (20 V) and maintained afterwards. Starting from a LOW transmission state, the transmission is raised to a HIGH transmission state immediately with the application of a ??=−20V. After a pulse width of 500 µs, ?? is removed (??=0V), and the HIGH transmission level remains unchanged due to the ferroelectric polarization. Similarly, a ?? of 20 V switches the transmission to a LOW state with the memory effect. The results demonstrate unequivocal mixed mode operation of binary data storage in the graphene-based optical memristor, with good reversibility and high extinction ratio.

Based on stable ferroelectric flipping of the well-developed P(VDF-TrFE) and excellent optoelectronic properties of graphene, the fabricated optical memristor has high cycling endurance and long retention time. We evaluate them by repeatedly switching the device between two states using the alternating 20 and −20V voltage pulses, as shown in Fig. 2(d). An increase in switching contrast is observed for the first 10 cycles, which is attributable to the wake-up effect of the ferroelectrics [43]. The repeatability of the device is so excellent that only a fluctuation of less than ∼0.04dB is seen in the whole testing period of up to ∼400cycles. The cyclic fluctuation ratio (i.e., the ratio of fluctuation to modulation depth) is less than 0.9%, highlighting the minimum degradation during the endurance test. Figure 2(e) depicts the measured retention time for each state, showing the retentions of two states over 105s (∼24h). The extrapolated transmission for all states does not intersect over a span of ∼10 years, which confirms the ferroelectrically driven graphene-based optical memristor has indisputable nonvolatility and high reliability. Such a linear extrapolation, commonly used for ferroelectric devices, provides an ideal prediction. Practical applications face more complex conditions such as degradation of polymers and graphene, as well as environmental factors such as temperature and pressure, which cannot be predicted in the laboratory and can affect retention time and device lifetime.

Optical memristors have the advantage of expanding the dimension of signal in optical wavelength over the electrical memristors, allowing parallel processing via wavelength division multiplexing (WDM) [1,38]. As elucidated in Fig. 2(f), it exhibits the nonvolatile reconfigurable switching of the waveguide spectrum between HIGH and LOW transmission states for 12 random cycles in 24 h in a range from ∼1500 to 1600 nm. The shaded regions of the spectra indicate the standard deviation for the 12 switching cycles, clearly revealing excellent cycle-to-cycle reproducibility. A uniform modulation depth of 4 dB can be observed across the entire telecommunication C-band. Here, the testing wavelength range is limited by the bandwidth of the grating couplers at the two ends of the waveguide. Considering graphene’s flat optical response over the spectral range from visible to mid-infrared [25], the memristive transmissions are expected to be further broadened.

For an ideal optical memristor, storing and addressing intermediate states in a deterministic and semicontinuous manner is critical [1]. P(VDF-TrFE) films allow partial polarization switching to support multilevel doping on graphene [34,35], resulting in semicontinuous nonvolatile modulation. Though the above demonstrated memristive transmissions are presented with binary storage, it also has the capability to program multiple intermediate nonvolatile states by setting the electrical pulse ?? step wisely. To demonstrate that, from the LOW transmission state, we send SET ?? pulses with varying amplitudes separated by a constant RESET pulse (??=20V) in order to return the device to the initial state. By continuously increasing and decreasing ?? pulses from −11 to −14V at different amplitudes, the transmissions with stable stored semicontinuous levels are obtained, as shown in Fig. 2(g). The use of a higher operating voltage compared to the threshold (−5V) of Fig. 2(b) is to minimize the pulse width by enhancing the action of the pulse. Since the graphene in the intermediate state has not reached full switching of modulation, the applied SET ?? will have an electrostatic effect on graphene in addition to the ferroelectric charge effect, which results in spikes in the transmission during the SET process [see inset of Fig. 2(g)]. This semicontinuous level of optical memristors can be matched to the requirements of neuromorphic devices [44,45]. Further tests involving continuous pulse stimulation for neuromorphic applications are detailed in the Supplement 1 Fig. S6. These tests verified the flexible manipulation of intermediate semicontinuous states and effectively expanded the number of bits operable in multi-value operations.

C. On-Chip Graphene Phase Memristor Demonstration

The above results verify the successful operation of the proposed graphene-based optical amplitude memristor by ferroelectrically gating graphene on a waveguide, i.e., modulating waveguide transmission in a nonvolatile and semicontinuous manner. It is achieved by exploiting the modification of graphene absorbance (?) with the nonvolatile carrier doping. Additionally, the strong carrier doping also gives rise to a considerable variation of graphene’s refractive index (?) [27,30], which provides another degree of freedom for realizing multi-function optical memristors via the varied phase delay of the graphene-SiN waveguide mode. To leverage it, we further integrate a graphene-based optical memristor on a microring resonator (MRR), which is sensitive to the phase delay and could greatly enhance graphene-light interaction for improving the modulation efficiency [27]. As shown in Fig. 3(a), the MRR is designed as a runway shape of the strip waveguide with a radius of µ∼50µm and a coupling length of µ∼30µm with the bus-waveguide. An 18 µm long graphene flake is integrated on the straight waveguide side of the MRR runway to form a graphene-MRR structure.

Fig. 3. On-chip graphene phase memristor demonstration. (a) Top-view optical microscope image of a fabricated graphene-memristor with an MRR. The inset shows an image of the device before the preparation of the gate electrode. (b) Transmission spectra of a graphene-MRR optical memristor in various levels. Three intermediate states are excited by −14,−13, and −12V pulses. Zoomed transmission spectra around the resonant wavelengths of 1531 nm (c) and 1561 nm (d). Hysteresis response of ? factors (e) and shift of resonant wavelength (f) with the cyclically scanned ?? at three specific resonances. (g), (h) Hysteresis response of transmission as a function of ?? at two resonant wavelengths. All curved arrows indicate the direction of the curve’s hysteresis.

The transmission spectrum ?(?) of the bus-waveguide coupled with the MRR is determined by ?(?)=?2+?2−2??⋅cos?(2???0?)1+?2?2−2??⋅cos?(2???0?), where ?0,?,?, and ? are the resonant wavelength, waveguide-MRR coupling coefficient, intra-MRR optical loss, and positive integers denoting resonant modes at different ?0. The modulation on ?(?) can be realized by changing ?0 and ?, which are determined by graphene’s refractive index (?) and absorbance (?) on the graphene-MRR structure. If the intra-MRR optical loss is increased gradually by the varied graphene absorbance, the coupling condition of the waveguide-MRR transitions from over-coupling (?<?), critical coupling (?=?), and finally to under-coupling (?>?) [46]. The transmission [?(?)] modulation can be obtained based on the classical Zeno effect [47,48]. As a result, the modulation of the graphene-MRR memristor by the ferroelectric P(VDF-TrFE) layer is determined by the combined effect of varied refractive index and absorbance (Supplement 1 Fig. S7), promising a remarkable nonvolatile modulation efficiency.

Figure 3(b) displays the transmission spectra over the wavelength range from ∼1520 to 1580 nm after the applications of specific ?? pulses. Shift of resonant wavelengths, variation of linewidths [quality (?) factor], and modulation of extinction ratio are observed from all of the whispering-gallery-type resonant modes. Because the waveguide-MRR coupling coefficients ? are varied for different resonant modes, their modulation trends vary. To analyze it, transmission spectra around the resonant wavelengths of 1531 and 1561 nm are zoomed in Figs. 3(c) and 3(d), respectively. Different transmission spectra, including three intermediate states, are observed after five different ?? pulses, demonstrating that graphene-MRR also exhibits semicontinuous nonvolatile modulation. As the absorbance (?) and refractive index (?) of graphene are increased gradually via ?? pulses (curves turn bluer), the tendency of the decrement of ? factor and red shift of ?0 is the same for the modes near 1531 and 1561 nm, but the modifications of the extinction ratios are different. It could be explained by the variations from critical-coupling (over-coupling) to under-coupling (critical-coupling) for the modes near 1531 nm (1561 nm), which is attributed to the combined effect of wavelength-dependent ? and ??-dependent ?.

To further characterize the memristive behavior of the graphene-MRR memristor gated by P(VDF-TrFE), we monitor the transmission spectra with the cyclically scanned ?? (Supplement 1 Fig. S8). The carrier doping in graphene by the ferroelectric field changes its refractive index and absorbance, determining the ? variation (? factor) and ?0 shift (Δ?0) of the resonant modes in a nonvolatile manner. Figures 3(e) and 3(f) plot ? factors and ?0 of resonant modes around the wavelengths of ∼1531, 1551, and 1561 nm versus the cycled ?? between −20 and 20 V. The ? factors are maximal at an initial ?? of −20V, relying on the high doping state in graphene, which maintains the high values when ?? is scanned back from −20 to 10 V. It is determined by the remnant surface polarization of the ferroelectric P(VDF-TrFE). With ?? changed from 10 to 20 V, the ? factors decrease as a result of enhanced intra-MRR loss by the recovered graphene absorption, which remains for ?? returned from 20 to 0 V. The ? factors begin to increase again as ?? is changed from 0 to −20V. The clockwise hysteresis of the ? factors in the graphene-MRR memristor has a similar trend as that of the transmission in graphene-waveguide [shown in Fig. 2(b)] because both are determined by the nonvolatile modification of graphene absorbance. In Fig. 3(f), the hysteresis curves of Δ?0 versus ?? are plotted. Heavily doped graphene causes a decrease in resonant wavelength (?0) compared to intrinsic graphene. Δ?0 are minimal at an initial ?? of −20V due to the heavily doping state in graphene, which mostly remains low for ?? scanned back from −20 to 10 V. With ?? changed from 10 to 20 V, Δ?0 increase sharply due to the weakened graphene doping. For ?? returned from 20 to 0 V, Δ?0 retain large values by the ferroelectric polarization. By applying ?? from 0 to −20V, Δ?0 increase to peak values first and decrease sharply to the initial minimum. These hysteresis loops can be explained by the Kubo formula [32,41], in which the refractive index of graphene increases to a peak and then decreases sharply as it goes from the intrinsic state to the heavily doping state.

The combination of the nonvolatile controlling in Δ?0 and the ? factor of resonant modes in the graphene-MRR memristor promises improved memristive modulation compared to that in the graphene-waveguide memristor (Fig. 2). In Figs. 3(g) and 3(h), we illustrate the transmission modulation at ∼1530.89 and 1560.73 nm, corresponding to the two resonant modes, respectively. Both of them show opposite hysteresis curves with respect to the cyclically scanned ??. A nonvolatile modulation with a high depth of ∼32.5dB featuring an anticlockwise hysteresis is realized at 1530.89 nm. Differently, at 1560.73 nm, the modulation hysteresis transits clockwise due to different coupling states at the two resonant wavelengths, which are critical-coupling to under-coupling at the former and over-coupling to critical coupling at the latter. It means both positive and negative ?? pulses can excite memristive transmissions with the highest or lowest values at specific wavelengths through the rational design, providing flexibility for multifunctional in-memory electro-optic modulations. In addition, the programming energy for each switching bit, as another important feature of the optical memristor, is evaluated as µ∼1.86pJ/µm2 by the product of the measured leaky current and ??, which is verified by calculating the device capacitance and doped charge volume (Supplement 1 Fig. S12). By employing thinner ferroelectric film, the energy consumption in programming could be further reduced.

D. On-Chip In-Memory Matrix-Vector Multiplication with Graphene Optical Memristors

The optical memristor based on the graphene-MRR memristor is encoded by its resonant wavelength. It promises the construction of integrated optical memristors with WDM capabilities, which is expected to supplement or even surpass the performance of electrical memristors [1,6]. As shown in Fig. 4(a), two graphene-MRR memristor cells are evanescently coupled with the same bus-waveguide. Relying on MRR’s wavelength-filtering function, the two memristor cells could be addressed individually by choosing input lasers at specific wavelengths. The two MRRs are marked as MRRa and MRRb. With the electrical inputs of ?? applied on graphene-MRRa and graphene-MRRb independently, memristive transmission from the bus-waveguide could be operated individually for the light on-resonant with each memristor. Figure 4(b) displays the transmission spectra from the bus-waveguide with sequential variations of ?? on MRRa and MRRb. The blue and red resonant dips correspond to the resonant modes in MRRa and MRRb, respectively, when memristive transmissions are SET and RESET successively. Combining with the possible multi-bit memory levels in each memristor cell, this framework, which integrates a single access waveguide with multiple graphene-MRRs, facilitates the development of cascaded memristors capable of multi-bit and multi-wavelength operations. This architecture is advantageous for advanced signal processing applications such as in-memory computing, parallel computing, and logical gate operations.

Fig. 4. Photonic architecture for in-memory matrix-vector multiplication (MVM) based on graphene optical memristor. (a) Optical micrograph of the device systems fabricated to demonstrate computational architectures. Two graphene-MRR memristor cells are evanescently coupled with the same bus-waveguide. By applying input voltage pulses (Input A and Input B) independently to the gates of the two MRRs, the transmittance values of their resonant modes are pre-SET and stored as Ta and Tb. Cooperating with the intensity of the input optical signals (Ia and Ib) at the resonant wavelengths of the two MRRs, individual multiplication is accomplished. A photodetector at the output port of bus-waveguide as an adder of optical signal to obtain the MVM calculation (Ia×Ta+Ib×Tb). (b) Demonstration of a WDM operations of the fabricated device systems. The blue and red resonant dips correspond to the modes in two MRRs. Each cell can be operated individually while another cell is not affected. (c) Demonstration of MVM. The figure is divided into four sections, corresponding to Ta and Tb being nonvolatile reprogrammed to four sets of values. A series of input optical signals is reused in each section to verify the accuracy of the MVM calculations. The measured values are compared with numerically calculated exact values.

As examples, we first demonstrate an arithmetic matrix-vector multiplication (MVM) with a nonvolatile matrix, which is a pivotal operation mechanism for image processing, machine learning, and data analytics [38,49]. The two cascaded memristor cells in Fig. 4(a) are employed. By applying gate voltages on them independently, the transmittance values of their resonant modes are pre-SET and stored. We mark them as Ta and Tb and represent them as a 1×2 matrix [Ta Tb]. Note that their values can be switched to intermediate levels and stored, as aforementioned in Figs. 3(b)–3(d). Two optical signals at the wavelengths on-resonant with MRRa and MRRb, respectively, are coupled into the bus-waveguide together, with different light intensities of Ia and Ib, constituting a 2×1 vector [IaIb]? for the MVM. After propagating through the two graphene-MRR memristor cells, the optical signals are weighted by the transmittance functions and, therefore, carry and transmit the information of each individual multiplication (i.e., Ia×Ta and Ib×Tb). With a photodetector at the output port of the bus-waveguide as an adder, the product photodetector signals, which are carried by two different wavelengths, can be combined to perform MVM (i.e., Ia×Ta+Ib×Tb).

Figure 4(c) demonstrates the MVM operation with four different in-memory sets of the matrix [Ta Tb]. The input vector is chosen with the optical signals at the resonant wavelengths of ∼1579.88 and 1581.12 nm for the two MRRs, respectively. After the transmittance values of the two graphene-MRR memristors [Ta Tb] are SET, the matrix elements are repeatedly used to make the MVM products with varied sets of the vector elements (Ia and Ib). As shown in the leftmost image of Fig. 4(c), by SET and storing the Ta=1 and Tb=0.85, a series of MVM products is obtained when Ia and Ib are varied. Next, the matrix is nonvolatile reprogrammed as Ta=0.6 and Tb=0.85, and the MVM products of the varied set of Ia and Ib can be carried out quickly without the holding energy budget, as shown in the middle image of Fig. 4(c). Similar MVM operations are also carried out with other matrixes of [Ta Tb]. In Fig. 4(c), the expected MVM products are also calculated from the transmittance values of the graphene-MRRs and the input light intensities, which are consistent with the measured outputs from the bus-waveguide. For matrix operations involving negative elements in practical applications, such as image recognition, the intermediate state of a single device’s optical transmission is set to “0”, with states above it representing positive numbers and those below representing negative numbers. This method has been proven feasible [38]. We note that this in-memory MVM operation could be generalized to a multiplication between an ?×? matrix and ?×1 vector using ? bus-waveguides independently coupled with ? graphene-MRR memristor cells. Larger-scale matrix operations imply the integration of more optical devices and corresponding control circuits, posing challenges in integration and manufacturing. However, in-memory computing can save on additional memory and data transfer costs, mitigating the challenges of large-scale integration.

E. On-Chip In-Memory Logic Gates with Graphene Optical Memristor

In addition to the MVM, a type of analogue computation [38], the device with cascaded graphene-MRR memristors is also capable of performing in-memory digital logic gate operations [50]. Different from the MVM addressed by individual resonant wavelengths of the two graphene-MRRs, the in-memory logic gate requires the correlation of the two graphene-MRRs at a common resonant wavelength. As shown in Fig. 5(a), thanks to the different free spectral ranges (FSR) of MRRa and MRRb, there could be a common resonance at a specific wavelength, e.g., ∼1586.77nm (the green shaded region). Other separately individual resonant wavelengths of MRRa and MRRb are also displayed, which are marked by blue and red shaded regions, respectively. To assist the discussion, the gate voltages applied on the graphene-MRRa and graphene-MRRb are independently denoted as Ga?Ga and Gb?Gb, respectively. Around the co-resonant wavelength of ∼1586.77nm, the transmission is the combined effect of the two graphene-MRRs. It exhibits HIGH transmission only if MRRa and MRRb simultaneously possess HIGH transmission. Therefore, after the application of Ga?Ga=20V and Gb?Gb=20V, a higher transmission is obtained compared to the other three cases, as indicated in Fig. 5(a).

To illustrate the memristive logic gate operation of the cascaded graphene-MRRs, we measure the transmission at 1586.77 nm by applying pulsed voltages of −20 or 20 V on Ga?Ga and cyclically scanning Gb?Gb, as shown in Fig. 5(b). Hysteresis curves at different levels are achieved, demonstrating that both Ga?Ga and Gb?Gb have modulating and storing effects at the common resonant wavelength. If the HIGH transmission obtained after ???=20V and ???=20V is defined as “1”, and the LOW transmission obtained from other gate pulse sets is the state of “0”, a logic AND gate could be implemented, as shown in Fig. 5(c). With (GaGb?Ga,?Gb) as the logic inputs (?,?), and the transmission (?) of the bus-waveguide as the output, the device remains in a LOW transmission state (logic output ? as “0”) for logic input (?,?) states of (0, 0), (0, 1), (1, 0) and turns to a HIGH transmission state (logic output ? as “1”) after logic input (1, 1), corresponding to the “AND” logic function. Note that the logic values ?/? are not required to be maintained by GaGb?Ga/?Gb. They can be excited by pulses and latched, which means that GaGb?Ga/?Gb are not necessary to arrive simultaneously. In Fig. 5(c), the Ga?Ga pulse of −20V is applied to set ?=0. The latched logic ?=0 is employed for the logic gate, with the logic ? subsequently set and latched by the Gb?Gb pulse, and thus, the corresponding result of the logic gate can be memorized after both ?? are removed. Then, the logic ?=0 is switched to ?=1 by a pulse of Ga?Ga=20V and latched again. Notably, it is demonstrated in Fig. 3 that the graphene-MRR memristor cells can also be modulated to a HIGH transmission state with ?? as −20V, allowing to obtain in-memory logic gates for other functions, such as “?⋅?¯”, “?¯⋅?”, “?¯⋅?¯” (i.e. NOR gate), as demonstrated in the Supplement 1 Fig. S14. Such an in-memory logic gate with electrical-input and optical-output has shown the advantages of reducing the common modulation effect of two optical memristors, resulting in savings of circuit complexity and static power dissipation.

Fig. 5. An architecture of in-memory logic gate operations based on two cascaded graphene-MRRs. (a) Transmission spectrum near the common resonant wavelength (1586.77 nm) of two graphene-MRRs. Two MRRs at their respective resonant wavelengths (the blue and red shaded areas) can be nonvolatile modulated by their respective VGs. They share a common resonant wavelength (the green shaded area) due to the different free spectral regions (FSR). The cascaded device system exhibits HIGH transmission at the common resonant wavelength only if both MRRs exhibit HIGH transmission (after Ga?Ga=20V & Gb?Gb=20V). (b) After setting the MRRa to two states by applying pulsed voltages of 20 (orange curve) or −20V (blue curve) on Ga?Ga, the hysteresis transmission response of the bus-waveguide with Gb?Gb at the common resonant wavelength of 1586.77 nm. (c) Optical output transmission states showing AND logic operations, with (GaGb?Ga,?Gb) as the logic inputs (?,?), and the bus-waveguide transmission (?) as the output.

Table 1. Comparison of Integrated Optical Memristors Technologies^a

3. DISCUSSION AND OUTLOOK

Leveraging graphene’s remarkable dependence of refractive index on the carrier doping levels and strong interfacial polarization field of ferroelectric P(VDF-TrFE), the proposed graphene-based optical memristor integrated on SiN waveguide presents high modulation depth exceeding 32.5 dB in a good deterministically semicontinuous nonvolatile manner. The good retention time (>10years) and cyclability (fluctuation ratio <0.9%) of the memristive modulations are demonstrated as well, relying on graphene’s stability and well-developed P(VDF-TrFE) material. With the assistance of multiple wavelength channels, optical parallel computing with in-memory attributes could also be implemented using cascaded memristor cells for functions of analogue MVM and digital logic gates. In Table 1, we compare various optical memristor technologies integrated on-chip, showing the advantages of this scheme over other platforms. Furthermore, the performance advantages of this scheme are demonstrated in Supplement 1 Fig. S13. With the CMOS compatibility of silicon photonics, graphene, and ferroelectric thin films, the proposed graphene-based optical memristor promises great potential to develop advanced optoelectronic integrated circuits for high bandwidth neuromorphic computing, parallel convolutional processing, artificial intelligence, etc.

Programming speed is another important figure of merit of an optical memristor. In the demonstrated graphene-based optical memristor, the speed is limited by the switching time of the ferroelectric polarization in P(VDF-TrFE), considering the ultrafast carrier mobility in graphene. With the programming voltage of 20 V, a speed around 1 kHz is obtained [Fig. 2(c)], which could be improved to 100 kHz for a higher gate voltage (Supplement 1 Fig. S11). In the next device development step, the polymer P(VDF-TrFE) film could be replaced by inorganic ferroelectric thin films to nonvolatile gate graphene, such as HfZrO2 and AlScN, which can enhance ferroelectric switching speed to Gigahertz [52]. In addition, by controlling thinner ferroelectric films, the operation voltages for switching the ferroelectric polarization could be reduced to sub-5 V, promising future optoelectronic packaging with CMOS electrical circuits. Considering the programming energy consumption of the proposed memristor is determined by the charging and discharging of the graphene layer, the thinner ferroelectrics would also reduce the operation energy dissipation considerably.

To the best of our knowledge, our results, as a proof of concept, demonstrate the first extension of the hysteresis response in ferroelectrics to photonic integrated circuits experimentally. Remarkably, our proposed nonvolatile optical memristor assigns the memory and the electro-optic modulation to the ferroelectric film and the active layer, respectively. It provides more design flexibility in that the combination of ferroelectric materials and active layers is diverse and optional. By replacing the active layer of graphene with other surface carrier-governed electro-optic materials, such as transparent conducting oxide [53], on-chip optical memristors are expected as well. With the CMOS compatibility of silicon photonics, graphene, and ferroelectric P(VDF-TrFE) or HfZrO2 thin films, the proposed graphene-based optical memristor promises great potential to enhance optical signal processing with the nonvolatile reconfigurable photonic circuits, in-memory computing, brain-inspired architectures, etc.