Introduction

For more than four decades, Moore’s law has been the driving force for the semiconductor industry. However, as CMOS circuits approach the scaling limitations due to fundamental physical constraints, this scaling trend is eventually going to end at some point in the future1,2. Photonic devices and circuits are a promising alternative due to their high speed and low propagation loss3. However, the diffraction limit of light proves to be an obstacle for realizing nanoscale photonic devices as the dimensions approach the wavelength of light in the material. Surface plasmon polaritons (SPP) - electromagnetic waves propagating at the interface between a metal and a dielectric - can circumvent this problem by localizing the electromagnetic energy in dimensions much smaller than the diffraction limit4,5. Recent advances in the field of plasmonics have witnessed the development of innovative waveguiding schemes6,7,8,9,10,11,12 and devices13,14,15,16,17,18,19,20. A complete set of fundamental logic gates (basic and universal) have been realized by exploiting phase-dependent interference of SPP waves18,19,20. However, these CMOS based digital logic gate designs cannot exploit an important feature of plasmonic logic and wave computing, namely, the ability to execute majority voting efficiently. Deviating from the traditional path of CMOS oriented chip design, new logic abstractions and synthesis techniques are being developed for the emerging nano-devices that are capable of reproducing the same CMOS logic circuitry with lower footprint and higher performance with only two building blocks – inverter and majority logic gate21. While recent works in the analogous field of spin wave based computing22 have seen the proposal for a spin wave majority logic gate23, a wave computing paradigm based on majority gates has not yet been implemented with plasmons.

In this work, we aim at highlighting the enhanced native functionality of plasmonics based logic - the majority gate - utilizing the phase of the SPP wave as the state variable. Using finite-difference-time domain (FDTD) simulations performed in commercially available software Lumerical Solutions24, we provide a comprehensive scheme for building a nanoscale cascadable plasmonic majority logic gate. We start by characterizing the waveguiding properties of the chosen slot waveguide followed by the design and simulation results of a single stage 3-input plasmonic majority gate. We demonstrate that besides Boolean logic, our proposed plasmonic majority gate is also capable of performing non-Boolean computing due to multi-level output denoting the strength of the majority. We investigate the cascadability of these gates by studying up to 3 stages of cascaded logic. This is sufficient to support interesting arithmetic primitives like adders and multipliers with limited bit-width. Larger arithmetic processor data-paths can then be composed from these primitives, however this lies outside the scope of this paper. We also propose a unique referencing scheme at the output that can directly translate the information encoded in the amplitude and phase of the output SPP wave into the intensity of the output electric field or the output power, thus circumventing the challenging problem of tera-Hertz phase-detection of the SPP waves. Due to high throughput, the proposed scheme can be of use in highly parallel real-time signal processing applications that are arithmetic-heavy with strict timing requirements. A representative example of this, namely a pattern recognition system, has been briefly discussed towards the end of the paper. Note that since the emphasis of this work is on building a nanoscale cascadable plasmonic majority logic, we do not focus on the mechanism or circuits for excitation and final detection of SPP which are the focus of separate works in the literature25,26,27,28,29,30,31,32,33,34,35,36,37.

Results

Characterization of Slot Plasmonic Waveguide

We start by first describing the basic building block of a plasmonic majority gate – the plasmonic waveguide, that acts as a conduit for transmission of information. We choose a slot (metal-insulator-metal MIM) waveguide configuration due to its high field confinement capability (in nanometers) that aids the designing of nanoscale logic devices12,20,38. Figure 1(a) shows the schematic of the slot plasmonic waveguide. Silver (Ag) on top of a silicon dioxide (SiO2) substrate is used for the metallic structure while the dielectric is assumed to have a refractive index of 1.5. The refractive index of the surrounding air is set to 1. We perform finite-difference time-domain (FDTD) simulations in Lumerical Solutions24 (see method section for simulation details). We choose the height (h) of the metal slot as 100 nm throughout our simulations while the width (w) of the waveguide is varied between 60, 120 and 180 nm as explained later.

Figure 1
figure 1

(a) Schematic of the metal-slot MIM plasmonic waveguide consisting of Ag on top of SiO2 substrate and a dielectric of 1.5 refractive index, acting as a channel for transmission of information. (b) Calculated dispersion relation of the slot waveguide for gap widths of 60, 120 and 180 nm. (c) Fundamental plasmonic mode in a 60 nm wide metal-slot waveguide at λ 0 = 1.55 μm. (d) Two dimensional figure of merit (FOM) graph as a function of the wavelength λ 0 illustrating for our exploration space a good trade-off between the propagation and confinement at the chosen excitation wavelength of λ 0 = 1.55 μm (e) Plot showing coupling length L p as a function of the waveguide pitch p for gap widths of 60, 120 and 180 nm, illustrating the trade-off between crosstalk and on-chip packing density.

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In a MIM geometry, the two identical SPP modes overlap with each other for small dielectric (insulator) layer width, resulting in a so-called gap SPP (G-SPP). The G-SPP mode displays an odd symmetry of the longitudinal electric field component E x and even symmetry of the transverse field component E y . The electromagnetic fields are predominantly confined within the slot with the evanescent decay \({e}^{-{k}_{y}y}\) inside the metal, where ky is the transverse component of the wave vector (y-direction). Along the direction of propagation, the electromagnetic fields vary harmonically as \({e}^{i({k}_{x}x-\omega t)}\), where kx is the longitudinal component of the wave vector (x-direction). The longitudinal wave vector k x can be defined in terms of the permittivities of the metal ∈ m and dielectric ∈ m and the free-space wavelength of light λ 0. The real part of the SPP wave vector can be further used to define the effective index of the plasmonic mode inside the waveguide as \({n}_{SPP}^{eff}=\frac{\lambda }{2\pi }Real({k}_{x}).\,\,\)Using appropriate boundary conditions for the normal and tangential electric field components and the aforementioned symmetry of the corresponding field component distributions, the G-SPP dispersion relation can be obtained as \(\tanh (\frac{{k}_{y}^{d}w}{2})=\,-\frac{{{\epsilon }}_{d}{k}_{y}^{m}}{{{\epsilon }}_{m}{k}_{y}^{d}}\),39,40 where, \({k}_{y}^{d,m}=\sqrt{{k}_{x}^{2}-{{\epsilon }}_{d,m}{k}_{0}^{2}}\) and \({k}_{0}=\frac{2\pi }{{\lambda }_{0}}\). Approximate analytical expressions for the G-SPP dispersion relation for sufficiently small or relatively large gap width have been obtained in refs14,40. Figure 1(b) shows the numerically calculated dispersion relation of the slot waveguide for three different widths (60, 120 and 180 nm).

Since the emphasis of this work is on building a nanoscale cascadable plasmonic majority logic, we do not refer to a particular technique of excitation (see supplementary section S1 for possible ways of exciting on-chip SPP waves) and use the standard mode source in our Lumerical simulations to inject a fundamental guided mode into the plasmonic waveguide (see supplementary section S2 for details about the excitation source). We choose an operating wavelength of λ 0 = 1.55 μm (corresponding to a frequency of 193 THz) for the mode source in Lumerical simulations. The corresponding wavelength dependent complex permittivity of Ag is −116.676 + 11.6522i41 while the permittivity of SiO2 used here is 2.08. Note that the wavelength of the propagating SPP mode (λ SPP ) is different from λ 0 depending on the width of the waveguide as indicated by the dispersion relation in Fig. 1(b). Figure 1(c) shows the fundamental plasmonic mode (electric field) distribution inside a 60 nm wide metal-slot waveguide at λ 0 = 1.55 μm suggesting deep subwavelength confinement. The maximum electric field intensity is located at the metal-dielectric interface around the two lower vertexes, consistent with the results of Pan et al.20.

The development of plasmonic logic is hindered due to the presence of dissipative losses. Dissipative losses arise due to ohmic losses encountered by surface plasmons propagating along the interface of metal and dielectric42. The propagation length of surface plasmons depends on the properties of the waveguide – material, geometry, and mode profile of the propagating surface plasmon polariton, and is calculated as the distance over which the propagating power decays up to 1/e: \({L}_{p}=\frac{1}{2{\rm{Im}}[{k}_{x}]}\). The evanescent decay length inside the metal \(\delta =\frac{1}{{\rm{Im}}[{k}_{y}^{m}]}\) is defined as the distance over which the field decays up to 1/e in the transverse direction43. Both the propagation length (L p ) and confinement (δ) show strong frequency dependence. A higher frequency provides better confinement. However, this would imply an increase in the damping of electrons and energy dissipation in the metal, causing a decrease of the propagation length. Thus, there exists a trade-off between L p and δ. To quantify this trade-off, next we look at the individual figures of merits (FOM). The figures of merit for propagation and confinement are given by \(FO{M}_{prop}=\frac{{L}_{p}}{{\lambda }_{spp}}\) and \(FO{M}_{conf}=\frac{{\lambda }_{0}}{\delta }\) respectively. Note that there are various other definitions of plasmonic mode confinement involving the spatial extent of the energy38,44. However, all of them are largely determined by the same exponentially decaying field outside the waveguide, so we believe that our definition of δ is sufficiently representative. Figure 1(d) shows the two dimensional figure of merit (FOM) graph45 for our exploration space as a function of the excitation wavelength λ 0 of the mode source in Lumerical simulations for the three different widths of the waveguide. The chosen wavelength of λ 0 = 1.55 μm illustrates a good trade-off between the propagation and the confinement. While the MIM waveguides feature lower propagation lengths compared to insulator-metal-insulator (IMI) waveguides, the higher confinement results in a tighter pitch (center-to-center distance between the adjacent waveguides) while maintaining a low crosstalk noise. This is critical for signal integrity. The crosstalk noise, defined as the coupling or overlap of modes between the adjacent waveguides resulting in a transfer of power from one waveguide to another, is measured in terms of the coupling length \({L}_{c}=\frac{{\lambda }_{0}}{\pi \Delta n}{si}{{n}}^{-1}(\sqrt{\frac{{P}_{V}}{{P}_{A}}})\) where P A and P V are the powers in the active and victim waveguides respectively and Δn is the index difference between the two coupled modes. For the worst case scenario, we calculate the length for 100% coupling of power as \({L}_{c}=\frac{{\lambda }_{0}}{2\Delta n}\). Figure 1(e) shows the coupling length L c as a function of the waveguide pitch p for three different widths of the waveguide. While a larger pitch minimizes crosstalk, it puts a limitation on the on-chip footprint area and the achievable on-chip packing density which is another important figure-of-merit for us. Note that while the FOM graph in Fig. 1(d) shifts towards the right for an increasing width of the waveguide illustrating an increase in the propagation length, the coupling length for a given pitch decreases with the increasing width of the waveguide as shown in Fig. 1(e). This suggests the requirement of higher pitch for wider waveguides for minimizing crosstalk noise, thus increasing the footprint area and reducing the on-chip packing density.

Realizing Majority Logic Using Surface-Plasmon-Polariton (SPP) Wave

The functionality of a majority logic gate with N (odd) number of inputs is to return a true output if and only if more than half of its inputs (N/2) are true. For a SPP wave propagating through a MIM waveguide, the transverse electric field component E y can be approximated as an exponentially decaying harmonic wave \({E}_{i}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+{\varphi }_{i})}\) where E 0 depends on the strength of the optical or electrical stimulus at the point of excitation and the frequency of the wave ω is related to the excitation frequency/wavelength. The phase of the wave ϕ may be specified by the source or by an additional phase-shifter introduced to achieve a specific phase-shift. We propose the utilization of the phase of the wave instead of the intensity as the state or computational variable. The information bit is encoded in the phase of the wave. Hence, a phase “ϕ” represents a logic “1” and a phase “ϕ + π” represents a logic “0”. The principle phenomena guiding wave-based computing is interference of waves. In general, consider a plasmonic logic gate with odd number of inputs, allowing the input waves to interfere in a “combiner” region and extracting the resultant wave the output end. In general, if m is the number of inputs with phase ϕ and n is the number of inputs with phase ϕ + π, the resultant output SPP wave can be approximated as \({E}_{out}=(m-n){E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) with the peak amplitude depending on the number of inputs with phases ϕ and ϕ + π as \({E}_{out}^{peak}=(m-n){E}_{0}\). A critical requirement of such wave-computing utilizing the phase as the state variable is to have odd number of inputs. An even number of inputs would result in a complete destructive interference (with no output signal) giving rise to an in-determinant state. For the case of 3-input majority logic, the three input SPP waves with the same frequency and amplitude and a certain phase shift relative to each other can be approximated by \({E}_{i}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+{\varphi }_{i})}\,s\), where i = A, B, C relates to the three input waves. The output SPP wave resulting from the interference is then, \({E}_{out}={E}_{A}+{E}_{B}+{E}_{C}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t)}[{e}^{i{\varphi }_{A}}+{e}^{i{\varphi }_{B}}+{e}^{i{\varphi }_{C}}]\). One of the following four possible scenarios arises:

  1. (i)

    \({\varphi }_{A}={\varphi }_{B}={\varphi }_{C}=\varphi \)

    For the case when all the inputs have the same phase ϕ (bit “1”), the resultant SPP wave \({E}_{out}=3{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has thrice the amplitude and phase ϕ resulting in a Boolean logic “1”.

  2. (ii)

    \({\varphi }_{A}={\varphi }_{B}=\varphi ,\,{\varphi }_{C}=\varphi +\pi \)

    When two of the inputs are in the same phase ϕ (bit “1”) and one is out of phase ϕ + π (bit “0”), the output SPP wave \(\,{E}_{out}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has the same amplitude as the inputs and phase ϕ resulting in a Boolean logic “1”.

  3. (iii)

    \({\varphi }_{A}={\varphi }_{B}=\varphi +\pi ,\,{\varphi }_{C}=\varphi \)

When two of the inputs are in the same phase ϕ + π (bit “0”) and one is out of phase ϕ (bit “1”), the output SPP wave \({E}_{out}=-{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has the same amplitude as the inputs and phase ϕ + π resulting in a Boolean logic “0”.

(i) \({\varphi }_{A}={\varphi }_{B}={\varphi }_{C}=\varphi +\pi \)

For the case when all the inputs have the same phase ϕ + π (bit “0”), the resultant SPP wave \({E}_{out}=\) \(-3{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has thrice the amplitude and phase ϕ + π resulting in a Boolean logic “0”.

Single Stage 3-input Plasmonic Majority Logic Gate

Next, we investigate a single stage 3-input plasmonic majority logic gate structure illustrated in Fig. 2(a,b). Coherent SPP waves are injected in three parallel input metal-slot waveguides using a standard mode source. The standard mode source in our simulations operate at a wavelength of λ 0 = 1.55 μm (corresponding to a frequency of 193 THz) and inject a fundamental guided mode into the three plasmonic waveguides. The phase of the mode source is specified to be either 0° (from here on referred to as ϕ) or 180° (ϕ + π) for simulating a logic 1 or 0, respectively. Note that here we rely on the phase of the SPP wave as the computational or state variable. We choose the input waveguide width (w in ) to be 60 nm and height (h) 100 nm. The effective refractive index of the injected SPP mode is calculated to be 1.79 + 0.0231i which results in a considerable range of usable propagation length of L P  = 5.31 μm (around 6 times the wavelength of the propagating SPP wave λ SPP and 5 times the length of the majority gate ~ 1 μm) and a sufficiently high degree of confinement of δ = 22 nm. The length of input and output waveguide regions (x1) are 200 nm long. The center-to-center distance (pitch p) between the adjacent input waveguides in the input region is chosen as 360 nm which corresponds to a coupling length L c  ~ 10 μm. Since the coupling length is L c is almost an order of magnitude higher than the length of the majority gate (~1 μm), this choice of pitch size provides a good trade-off between crosstalk and on-chip packing density. The combiner region consisting of two bends where the SPPs merge into an output waveguide as shown schematically in Fig. 2(b). To ensure smooth merging of the waveguides with minimum dissipative and backpropagation loss, we choose the merging angle to be 35° between the waveguides which corresponds to a 500 nm long combiner region (x2). The cross-sectional dimensions of the waveguide (60 nm × 100 nm), pitch (360 nm) and merging angle (35°) chosen in this work are comparable to that used by Pan, D. et al.20 for MIM waveguides resulting in a similar propagation and coupling length. However, while the authors of ref.20 relied on the intensity of the SPP wave for designing CMOS-oriented logic gates, here we utilize the phase of the SPP wave as the state variable to design majority logic gate.

Figure 2
figure 2

(a, b) Illustration of a single stage 3-input plasmonic majority logic gate. (c) Simulation result for a 3-input plasmonic majority gate for 23 input combinations in terms of the time-domain electric field component EY at the output, normalized to the total source electric field and integrated over the cross-section of the output waveguide. (d) Calculated peak values of the normalized integrated electric field component EY at the output for different combinations of the input phases. (e) Time-lapse simulation results in terms of the distribution of EY in the x-y plane showing the propagation and interference of the SPP waves.

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As the three input waveguides inject power into a single output waveguide, a considerable backflow can occur due to reflection from the merging point. To improve transmission, we increase the gap width of the output waveguide. Numerical simulations (see supplementary section S3 for results) show that the transmitted power through the 3-input junction increases with the increase in the gap width of the output waveguide. However, to avoid mode splitting and large increase in the required pitch for the next stage, we choose a moderately increased output gap width of 120 nm. To better elucidate the result, we resort to an approach of impedance matching put forward by Cai et al.46 which shows an increase in the transmission due to an increase in the impedance of the waveguide (see supplementary section S3 for further details and results). The SPPs propagating through the two side arms of the majority gate have to cover an extra distance introduced by the bends. Since we rely purely on a constructive (phase difference = 0) or destructive interference (phase difference = π) between the input SPP waves, we compensate the path difference δl between the middle and side arms and make all the three inputs equal in phase and strength by shifting the position of the mode source for the middle input waveguide to the left by a distance \(\delta l=(\frac{p}{\sin [{ta}{{n}}^{-1}(\frac{p}{{x}_{2}})]}-{x}_{2})\) as shown in Fig. 2(b).

Figure 2(c) shows the numerical simulation result for a 3-input plasmonic majority gate for all 23 possible input combinations in terms of the time-domain electric field component EY at the output, integrated over the cross-section of the output waveguide and normalized to the total source electric field and cross-sectional area of the output waveguide (see method section for details). The input logic combinations (1,1,1), (1,1,0), (1,0,1) and (0,1,1) which correspond to majority of the input being logic 1, i.e., SPP waves having a phase ϕ, result in logic 1 as the output, i.e., an output SPP wave having a phase ϕ. Similarly, the input combinations (0,0,0), (0,0,1), (0,1,0) and (1,0,0) result in an output SPP wave with phase ϕ + π depicting a logic 0 as the output. In addition to generating Boolean outputs, the proposed majority logic gate also has the capability to distinguish between a strong and a weak majority. As highlighted by the simulation results in Fig. 2(c), an input logic combination of (1,1,1) corresponding to all the SPP waves having a phase ϕ, gives rise to maximum constructive interference resulting in a high amplitude output with the phase ϕ, i.e., a strong 1. On the other hand, for an input logic combination of (1,1,0), two of the SPP waves having phase ϕ and ϕ + π undergo destructive interference resulting in a low amplitude output with the phase ϕ, i.e., a weak logic 1. Figure 2(d) shows the peak values of the integrated electric field component EY at the output, normalized to the source electric field, for different combinations of the input phases. As can be seen, the output of a single stage 3-input majority gate has four different levels denoting a combination of Boolean output 1 and 0 and the strength of the majority. Also, the normalized peak output EY for strong majority (all inputs in phase) is 3 times stronger than that of the weak majority (two in phase, one out of phase) as explained earlier. To further study the propagation and interference of SPP waves in the majority logic structure, we plot the time-lapse simulation results in terms of the distribution of the electric field component EY in the x-y plane at different snapshots in time for 4 different input combinations (1,1,1), (1,0,1), (1,0,0) and (0,0,0) as seen in Fig. 2(e) (see supplementary section S4 for further simulation results).

Note that the majority gate also has the capability to perform “AND” and “OR” operation if one of the inputs is used as a control input. Additionally, due to the wave nature of this computation scheme, an inverter (INV) operation can be simply implemented using a waveguide of length equal to half of the SPP wavelength. This set of logic primitives allows to effectively map most practical arithmetic functions, even when they cannot be directly matched well to just a cascaded majority logic structure. The multi-level output of the majority gate depicts a combination of Boolean output 1 and 0 and the strength of the majority. In order to utilize such a majority gate for Boolean computation, one needs to renormalize the output before feeding it to the next stage. However, here we strive to utilize this multi-level output to our advantage for non-Boolean computing. Since each of the stages in a cascaded structure (see Fig. 3(a)) performs the dual functionality of Boolean output and strength of majority, the overall final result will display the Boolean logic output 1 or 0 and the overall strength the majority of all the inputs.

Figure 3
figure 3

(a) Illustration of a 2-stage cascaded plasmonic majority logic gate. (b) Simulation results for the 10 representative input phase combinations in terms of the time-domain electric field component EY at the output, normalized to the total source electric field and integrated over the cross-section of the output waveguide. (c) Calculated peak values of the normalized integrated electric field component EY at the output for different combinations of the input phases. (d) Time-lapse simulation results in terms of the distribution of EY in the x-y plane showing the propagation and interference of the SPP waves.

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2-Stage Cascaded Plasmonic Majority Logic Gate

Next, we analyze a 2-stage cascaded majority gate structure shown in Fig. 3(a). The first stage consists of three ‘3-input majority gates’, each of whose outputs are combined in a majority gate fashion to form the second stage (see supplementary section S5 for the dimensional scaling implemented at each stage). In our numerical simulations, all nine inputs are excited with the same stimulus in terms of source amplitude. The phase of the injected SPP waves are chosen to be either ϕ or ϕ + π representing logic 1 and 0 respectively. The final output is monitored at the end of the second stage as indicated in Fig. 3(a). As mentioned earlier, the propagating SPP waves can be considered as exponentially decaying harmonic waves with the same frequency and amplitude and a certain phase shift relative to each other and can be approximated as \({E}_{i}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+{\varphi }_{i})}\,{\rm{s}}\), where i = 1, …, 9 relates to the nine input waves for each of the distinct arms. The resultant output SPP wave due to wave interference can be written as \({E}_{out}={E}_{1}+{E}_{2}+\cdots +{E}_{8}+{E}_{9}={E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t)}[{e}^{i{\varphi }_{1}}+{e}^{i{\varphi }_{2}}+\cdots +{e}^{i{\varphi }_{8}}+{e}^{i{\varphi }_{9}}]\). There are 29 possible input combinations for such a 2-stage cascaded majority logic gate, however here we only concentrate on 10 representative cases as highlighted in Fig. 3(b) and (c) that capture all the possible combinations of input phases required for studying the cascaded majority gate structure.

Figure 3(b) shows the numerical simulation results for the 10 representative input combinations in terms of the time-domain normalized integrated electric field component EY at the output. As explained earlier, the resultant output SPP wave can be approximated as \({E}_{out}=(m-n){E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\), m being the number of inputs with phase ϕ and n is the number of inputs with phase ϕ + π. For the case when all the inputs are logic “1”, i.e., having the same phase ϕ, the resultant SPP wave \({E}_{out}=9{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has nine time the amplitude and phase ϕ resulting in a Boolean logic “1”. In the extreme opposite case when all the inputs are logic “0”, i.e., having the same phase ϕ + π, the resultant SPP wave \({E}_{out}=-9{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) has nine time the amplitude but phase ϕ + π resulting in a Boolean logic “0”. The peak amplitude of the output electric field depends on the number of inputs with phases ϕ (m) and ϕ + π (n) and varies as \({E}_{out}^{peak}=(m-n){E}_{0}\) as seen in Fig. 3(b,c). Overall, we obtain 10 different levels of output electric field component EY which directly corresponds to the strength of the majority of the input. The strongest majorities where all the 9 inputs have the same phase ϕ or ϕ + π produce the highest magnitude of output amplitude with opposite sign as indicated in Fig. 3(c). Likewise, the weakest majorities where only 5 inputs have phase ϕ while the remaining 4 have phase ϕ + π or vice-versa produce the lowest magnitude of output amplitude. The time-lapse simulation results in terms of the distribution of the electric field component EY in the x-y plane for 4 representative input combinations (111,111,111), (111,110,110), (000,001,001) and (000,000,000) are shown in Fig. 3(d) depicting the propagation and interference of the SPP waves (see supplementary section S6 for further simulation results).

Referencing Technique for Detection

As mentioned earlier, the binary data is encoded in the phase of the excited SPP wave (phase ϕ representing logic 1 and phase (ϕ + π) representing logic 0) which we use as the state variable for computing. Hence, after wave interference, the relevant parameter to extract is the phase of the output SPP wave which gives the Boolean output of 1 or 0. However, it may be challenging to devise a precise tera-Hertz phase-detection scheme for our plasmonic logic gate operating at 193 THz (see dispersion plot in Fig. 1(b)). In addition, our proposed plasmonic majority gate also displays a non-Boolean characteristic in terms of indicating the strength of the majority which further adds to the expressive power of the plasmonic majority gate. As such, in this work, we propose a novel approach to extract both the amplitude and phase information from the output by using a “referencing” technique. An illustration for using the referencing technique at the end of the 2-stage cascaded structure is shown in Fig. 4(a). In addition to the 9 input SPP waves, we also inject a reference signal E ref that merges at the output of the second stage. We choose the gap width of the reference waveguide to be 3  w. We adjust the amplitude and the phase of the injected reference signal \({E}_{ref}\approx 9{E}_{0}{e}^{-{k}_{y}y}{e}^{i({k}_{x}x-\omega t+\varphi )}\) to match the output of the cascaded gate for the case of the strongest majority logic 1 (the case of all inputs 1 in Fig. 3(b)). This can be done by using higher excitation power for the reference source and accounting for the total path delay for the reference signal. Figure 4(b) shows the numerical simulation result for the 2-stage cascaded majority with reference for the 10 representative input combinations in terms of the time-domain normalized integrated electric field component EY at the output. Since the reference signal has been adjusted to have a phase ϕ, all the input combinations having majority of the input as logic 1, i.e., SPP waves having a phase ϕ, result in a constructive interference while all majority 0 cases having phase ϕ + π result in destructive interference. However, since we set the peak amplitude of the reference signal to be \({E}_{ref}^{peak}\ge 9{E}_{0}\), we get a Boolean output of logic 1 for all the 10 cases (output SPP waves having phase ϕ), but with varying levels of output amplitude of the electric field component EY as indicated in Fig. 4(b) and (c). In contrast to Fig. 3(c) and (d), in this case the strongest majority with 9 inputs having phase ϕ produces the highest magnitude of output electric field EY and maximum transmission of output power while the strongest majority with 9 inputs having phase ϕ + π produces the lowest magnitude of output EY and minimum power transmission as indicated in Fig. 4(c) and (d). Thus, the referencing technique directly translates the information encoded in the amplitude and phase of the output SPP wave into the intensity of the output electric field or the output power as illustrated in Fig. 4(c) and (d). We can further define the peak amplitude of the output electric field or the output power transmitted for the case when only the reference signal is present as the threshold level. As such, anything above the defined threshold can be considered as a logic 1 while anything below gives a logic 0. Figure 4(e) shows the time-lapse simulation results in terms of the distribution of the electric field component EY in the x-y plane for 4 representative input combinations (111,111,111), (111,110,110), (000,001,001) and (000,000,000), depicting the propagation and interference of the SPP waves along with the reference signal (see supplementary section S7 for further simulation results).

Figure 4
figure 4

(a) Illustration of a 2-stage cascaded plasmonic majority logic gate with the reference signal. (b) Simulation results for the 10 representative input phase combinations in terms of the time-domain electric field component EY at the output, normalized to the total source electric field and integrated over the cross-section of the output waveguide. (c) Calculated peak values of the normalized integrated electric field component EY at the output for different combinations of the input phases. The peak amplitude of the output electric field in (c) for the case when only the reference signal is present is defined as the threshold level. (d) Time-lapse simulation results in terms of the distribution of EY in the x-y plane showing the propagation and interference of the SPP waves.

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Discussion

Number of Cascadable Stages

While it is highly desired to have a multi-staged cascaded plasmonic logic without intermediary signal conversion between plasmon and charge domain, the propagation loss of SPP puts a limitation on the number of feasible cascaded stages. As shown in Fig. 5(a) (and Fig. 4(b) in supplementary information), the size of majority logic gate increases with the number of stages from an estimated value of 0.636 μm 2 for the first stage to 4.66 μm 2 and 38.24 μm 2 for the second and third stage, respectively. The increase in the path-length travelled by the SPP compared to the propagation length LP increases the transmission loss from around 30% in the first stage to more than 50% in the third stage. Hence, it is inefficient to go beyond the third stage without using either amplifiers to boost the signal amplitude or convert plasmonic signal to voltage signal at the end of the third stage. An additional constraint comes from distinction or separation between output levels after referencing. The number of output levels (amplitude of electric field EY or transmitted power) increases with the number of stages and input, from 4 in 1st stage to 10 in the 2nd and so on. Figure 5(b) shows the range of amplitude of the output electric field EY for logic 1 and 0 obtained at the end of each stage. Note that the range of output for both logic 1 and 0 decreases due the propagation loss from one stage to the next. Hence, even though the referencing technique will translate the information encoded in the amplitude and phase of the output SPP wave into electric field intensity, it will be difficult to separate or distinguish between the output levels for logic 1 and 0 as they get closer to the threshold level (case of weakest majority). We further investigate the possibility of separation of states above and below the threshold level (corresponding to logic 1 and 0) by plotting the resolution as a function of the number of stages shown in Fig. 5(c). We define the resolution as the difference between the minimum value of peak output EY for logic 1 and the maximum value of peak output EY for logic 0 (case of weakest majority outputs), \(resolution={\rm{\Delta }}{E}_{Y,out}={E}_{Y,min}^{logic\,1}-{E}_{Y,max}^{logic\,0}\,\).

Figure 5
figure 5

(a) Illustration of a multi-stage cascaded plasmonic majority logic. (b) Plot showing the range of amplitude of the output electric field EY for logic 1 and 0 obtained at the end of each stage. The range of output decreases due to propagation loss at each stage. (c) Resolution, defined as the difference between the minimum value of peak output EY for logic 1 and the maximum value of peak output EY for logic 0 (case of weakest majority outputs), as a function of the number of stages.

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Comparison with other proposals for plasmonic logic and device

Our work on majority logic gate design focuses on metal-insulator-metal (MIM) wave-guiding configuration for high field confinement capability (in nanometers) that aids the designing of nanoscale logic devices. Recently, a work on plasmonic majority gate using long-range dielectric-loaded surface plasmon polariton (LR-DLSPP) waveguide has also been proposed47. While the latter supports long range propagation of plasmon polaritons (LP ~ 3 mm), it has a rather poor confinement (lateral mode width, δ = ~1.6 μm) when compared to the high degree of confinement (δ = 22 nm) achieved in our proposal. With a waveguide length and width of 14 μm and 5.5 μm, respectively, in addition to a pitch of 0.5 μm, the single-stage 3-input majority gate proposed in47 requires an area of 77 μm2. On the contrary, our single-stage MIM-based 3-input majority gate has a highly improved overall area of only 0.636 μm2. The cross-sectional dimensions of the waveguide (60 nm × 100 nm) chosen in this work are comparable to that used by Pan et al.20 resulting in a similar coupling length of around 12 μm for a chosen pitch of 300 nm. Similar investigations using MIM geometry include a 200 nm × 100 nm air groove based cascaded XOR gate resulting in an all-optical logic parity checker where an overall minimum feature size of 15 μm for the logic device was achieved48. A 320 nm × 300 nm dielectric crossed waveguide structure enabling all-optical NOT, AND, OR, and XOR gate with lateral area of 200 μm2  49 and a half-adder structure of area 280 μm2 was proposed. The all-optical realization of XNOR, XOR, NOT, and OR logic gates by Fu. et al. used 100 × 100 nm slot waveguide50. However, the latter used a pitch of 2 μm resulting in the lateral dimension of the logic gate to be close to 5 μm. The experimental demonstration of OR/NOT/NOR using cascaded plasmonic logic using Ag nanowires (NWs)18,19 reports using 200–350 nm diameters NWs with a few to tens of micron length. Recently, the surface plasmon two-mode interference (SPTMI) coupler featuring a silicon core (width 0.22–0.48 μm), a silver upper and lower cladding, and GaAsInP left and right cladding has been proposed as an alternative for low power consumption that realizes the complete set of logic operations. However, the device dimensions still remain comparatively large: 1 × 2 SPTMI requiring coupling length of 93.1 μm results in a total length of 143.06 um51, 2 × 2 SPTMI single stage has a coupling length 38.4 μm and total length of 88.34 μm52, and a two-stage cascaded structure has a coupling length of 92.35 μm resulting in total length of device equal to 284.58 μm53.

Application Illustration

Due to high throughput, the proposed cascaded plasmonic majority gate can be of great use in highly parallel real-time signal and data processing applications. One good illustration of this usage is present in the non-boolean decision making process of a pattern recognition system. The non-boolean decision making process involves counting the number of matches and mismatches and determining the degree of match or mismatch between the input and the reference pattern. We believe the proposed cascadable plasmonic majority logic gate can find a direct utility here. The patterns can be considered as binary valued matrices with black and white pixels represented as logic “1” and “0”, respectively. Using the majority voting capability of the gate along with the referencing technique, it would be possible to count the number of match or mismatch at each stage with the final output portraying an overall match or mismatch between the input and the reference pattern and the degree of match or mismatch found.

In conclusion, using numerical FDTD simulations, we have demonstrated the possibility of building a nanoscale cascadable plasmonic majority gate. We utilize the phase of the SPP wave, instead of the intensity, as the state or computational variable that allows us to exploit the majority voting capability of the device inaccessible to amplitude-based wave computing. We choose the MIM geometry that allows nanometer scale plasmonic mode confinement, sub-micrometer pitch and propagation lengths over several micrometers. The dimensions and pitch of the waveguide chosen are comparable to that used by Pan, D. et al.20 for MIM waveguides. However, while the authors of ref.20 relied on the intensity of the SPP wave for designing CMOS-oriented logic gates, here we utilize the phase of the SPP wave as the state variable to design majority logic gate. Comparison with other previous works of plasmonic logic47,48,49,50,51,52,53 reveals that our single-stage MIM-based 3-input majority gate has a highly improved overall area of only 0.636 μm2. Performing 3-D FDTD simulations, we illustrate the majority logic functionality of the proposed gate and its cascadability up to 3 stages. In addition to boolean logic, the proposed plasmonics majority gate is also capable of performing non-boolean computing due to multiple output levels denoting the strength of the majority. We also propose a novel referencing scheme at the output that can translate the information encoded in the phase of the output SPP wave into the intensity of the output electric field. The extremely high throughput of plasmonic based logic finds direct usage in high throughput low latency signal processing applications which are arithmetic-heavy with strict timing requirements, like a pattern recognition system.

Methods

Numerical simulation

We perform full-wave 3-D simulations using the commercially available software Lumerical Solution24. The finite difference time domain (FDTD) method is used to solve the fully vectorial Maxwell equation on a discrete spatial (Yee cell) and temporal grid. We use the conformal mesh algorithm with a specified mesh size of 5 nm for regular straight waveguides and a reduced mesh of 2 nm or 2.5 nm for bends. Absorbing boundary conditions based on perfectly matched layers (PML) are used to minimize reflections. We use the mode source to inject the fundamental guided mode into the plasmonic waveguide. The standard mode source injects a broadband Gaussian pulse signal, but the results and the conclusion of the paper remain same when using a narrower-band mode source (see supplementary section S2 and S8 for further details). The phase of the source is specified as either 0 or 180° for simulating either a logic 1 or 0. The refractive index of Ag in our simulations is obtained from Palik’s Handbook of Optical Constants41. We use the electromagnetic field and power monitor (2D x-normal) in Lumerical to record the normalized transmission coefficients at the output. The time-domain electric field results are obtained using the time-domain monitor in Lumerical (2D x-normal for output EY and 2D z-normal for time-lapse results). The output electric field component EY is calculated as a spatial average of the y-component of the electric field distribution, obtained from the time-domain monitor, over the cross-section of the output waveguide.