1. Introduction

Interconnects – the metal wires that connect two data-processing units on a semiconductor chip – have long been heralded as the eventual bottleneck of an industry that is nearing the outer limits of Moore’s Law. While the industry is adopting multi-core and co-processor based architecture to keep Moore’s law alive, it is pointed out that a data transfer bottleneck is the main reason behind the comparatively slow adoption of co-processor-enabled databases [1,2]. This bottleneck has so far been primarily dealt with by two approaches: one, by scaling up the frequency limit of electrical interconnects (e.g. by replacing copper with higher conductive metal such as Ruthenium) [3–5], and two, by modulating high frequency carrier signal with multiplexed electrical signals to harness the bandwidth capacity of optical interconnect [6,7]. While both these approaches are facing tremendous challenges in terms of CMOS process integration, a novel approach to solve the interconnect bottleneck involves the act of harnessing a different kind of electromagnetic mode, known as the spoof surface plasmon polariton.

Researchers around the world, including the authors of this work, have demonstrated that spoof-surface-plasmon-polariton (SSPP), a unique mode of coupled electromagnetic resonance hosted by patterned copper metal, offers significantly smaller signal delay and parasitic effects compared to conventional RC delay-limited metal interconnects [8–14]. The experimental realization of flexible SSPP interconnects [15,16], along with reports of CMOS compatibility for SSPP transmission lines [17,18], positions SSPP interconnects as highly promising candidates to address the "last centimeter barrier" in future high-speed chip-to-chip communication. These achievements have been acknowledged in recent review articles on spoof plasmonics [19,20]. However, the success of any emerging interconnect technology for high-speed communication will be largely contingent upon the feasibility of the modulation technique employed and its integrability with existing technology [21–23]. Hence, building an SSPP-based electro-optic modulator is instrumental in realizing SSPP-interconnect technology at the chip-scale.

Different techniques of electro-optic modulation, such as quantum confined Stark effect (QCSE), Pockel effect, and Kerr effect, cost tens of femto-joule-per-bit energy consumption while providing few tens of gigahertz modulation [24]. The problem of slow modulation speed mainly arises from the fundamental nature of weak electron-light interaction owing to the many orders of wavelength-scale mismatch between electron and photon [25–27]. This interaction can be greatly enhanced in spoof surface plasmon polariton (SSPP) based interconnects, where the electromagnetic wave is confined into a volume beyond the diffraction limit [28–30]. As of now, there have been a few reports on signal modulation in SSPP-based communication channels. Notable examples include the proposal of dynamic manipulation of SSPP waves by controlling the dielectric permittivity of a $Ba_{0.6}Sr_{0.4}TiO_3$ film embedded between corrugated interconnects [31], and a multi-scheme digital modulation technique of SSPP wave using switchable PIN diodes [32] and varactors [33]. What has been missing all along is the theoretical underpinning of the SSPP modulation for a comprehensive guideline enabling optimal modulator design, predictions of the geometric dispersion-limited maximum possible modulation speed, and the quantification of the fundamental trade-off relation between modulation speed and energy-efficiency for an arbitrary design choice of SSPP structure. The current work stands out to serve all these critical purposes.

In addition to developing the foundational theory for performance analysis of the existing spoof-surface-plasmon-polariton modulator, this work introduces a novel method of surface wave modulation based on a variable resistor-induced change in the boundary condition of the resonant groove, where the length of the boundary (i.e., the groove width) can be much smaller than the SSPP wavelength. Instead of relying on the inherently slow change of the material’s refractive index by electron injection, we have capitalized on the unique property of high-sensitive boundary condition of the grooves in SSPP interconnects in order to design the principle of an SSPP modulator with high modulation speed. Guided by our theory, an appropriate design of the SSPP structure can achieve a modulation speed ranging from hundreds of Mbps to tens of Gbps, while maintaining a reasonably high energy efficiency. The numerical analysis of this work shows that the proposed technique can be applied to implement the digital modulation scheme of amplitude shift key (ASK). Considering practical aspects of a communication system, the ASK modulation scheme has been analyzed in terms of thermal-noise-limited bit-error-rate and associated eye-diagrams.

2. Theoretical framework

In this section, we present the theoretical formalism of the proposed modulation scheme. The formalism is followed by numerical analysis which qualitatively describes the validity of the scheme.

2.1 SSPP dispersion in the presence of modulator

A schematic illustration of the proposed structure for signal modulation of the SSPP interconnect is shown in Fig. 1(a). The structure consists of periodic rectangular grooves of length $h_0$ with variable resistors connected in between the teeth at position $|y|=h_m$, where $h_m<h_0$ (Fig. 1(b)). The variable resistor can be implemented by the gate-controlled channel resistance of a field-effect transistor (FET). The basic mechanism of the proposed SSPP modulator is as follows. The conductivity state of the electronic modulator would alter the cut-off frequency of the structure and thus would modulate the transmission of a carrier frequency. The cut-off frequency of a static SSPP structure of groove length $h_0$ is approximately $f_c\approx \frac {c}{4h_0}$ where $c$ is the free space velocity of electromagnetic wave (a more accurate estimation would be given in section 2.2). During OFF-state of the electronic modulator (i.e. when gate voltage is less than the threshold voltage of FET), it provides no conductive path across the teeth, and therefore does not impact the electromagnetic state of the system. On the contrary, during ON-state, it creates a very low impedance path between the teeth at the position $|y|=h_m$, thereby effectively reducing the optical length of the groove to $h_m$. This in-turn increases the cut-off frequency of the structure upto $\frac {c}{4h_m}$. Thus a carrier frequency chosen between $\frac {c}{4h_0}$ and $\frac {c}{4h_m}$ can alternate between high transmission and high back-scattering if the controllable resistor is switched between a low and a high impedance value. In a real system, the SSPP modulator is supposed to feed a static SSPP interconnect of groove length $h_r$. It can be shown that, in order to match the mode between the transmission state of the SSPP modulator and the static SSPP interconnect, the best position of placing the controllable resistor inside the SSPP modulator is $h_m=h_r$. Thus a natural design choice for the controllable resistor-implanted SSPP modulator would be having its groove-length $h_0$ longer than the groove-length $h_r$ of the static SSPP interconnect to which the modulating signal is fed.

 

Fig. 1. Schematic illustration of the SSPP interconnect with modulator (a) Three-dimensional schematic of the proposed SSPP interconnect with adjustable resistor across the teeth, envisioned to be implemented by a gate-controlled Field-Effect-Transistor.. (b) Two-dimensional (2D) view showing groove length (h), period (d), groove width (a) and modulating resistors ($Z_M$) located at position $h_m$ of the anomalous grooves; the dashed box represents the region that enables signal modulation. (c) Extended 2D view of the dashed region shown in (b), with a metal-bar placed adjacent to the interconnect at $y=g$

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In our subsequent analysis, the width of all grooves are considered as $a$, and the groove-periodicity as $d$ (Fig. 1(b)). It may be noted that SSPP interconnects are not limited by RC-delay as they can transmit signals without metallic ground. Hence in accordance with previous studies [8,34–36], no metallic ground is considered in this work. It has been shown that only a handful of grooves suffices to produce the transmission properties of a chain of periodic grooves in SSPP structure [36]. Therefore for the rest of the paper, unless otherwise mentioned, we designed the SSPP modulator consisting of two grooves, slightly elongated from the rest of the grooves in the static SSPP interconnect. The geometry of the modulating segment of the structure is shown in Fig. 1(c). To generalize our theoretical framework, the modulator is considered to have a metallic stripline situated next to the grooves at the same plane and at a distance of $y=g$. While the parameter $g$ gives us another knob for controlling the electromagnetic properties of the interconnect, it will be shown that the SSPP waveguide can propagate signals without the presence of the metallic bar.

a. Impedance without modulator: the insertion of a modulating resistor having impedance $Z_{M}$ will change the input impedance of the groove at position $y = -h_m$. For a TEM mode, the equivalent input impedance ($Z_{in}$) looking through the anomalous groove is given by:

Here $Z_0$ is the impedance of the groove looking at the position $y=-h_m$ in the absence of modulating resistance. $Z_0$ is defined as:

For a perfect metal, the tangential field component $E_{g,y}$ inside the groove vanishes at the metal interface at $x=\pm \frac {a}{2}$. Consequently, the first mode of nonzero field component $E_{g,y}$ would correspond to higher order modes having mode numbers $m\geq 1$, and frequencies higher than $\frac {\pi }{a}$. Considering narrow groove approximation [37], which suggests that the operating frequency for information transfer through the SSPP channel is lower than $\frac {\pi }{a}$, we can set $E_{g,y}=0$ in our frequency regime of interest.

The magnetic field component inside the grooves in the absence of modulating resistance is given by:

b. Impedance with modulator: for impedance $Z_M$ inserted at the position $y=-h_m$, Eqs. (3) and (4) can be re-written by decomposing the fields into upward (+y direction) and downward (-y direction) propagating components having amplitudes $G_{m+}$ and $G_{m-}$ respectively, as follows:

Assuming fundamental mode to be the dominant mode, we obtain the expression of the equivalent impedance to be:

Considering $\tilde {Z}_{i n}=\frac {Q Z_{i n}}{\omega \mu }$ to be the normalized input impedance, the terms $r$ and $\theta$ can be defined as follows:

According to these definitions, $r$ depends on the magnitude of $Z_M$, and $\theta$ depends on the location of inserting $Z_M$ inside the groove. Utilizing $r$ and $\theta$ in Eq. (8), it can be shown that:

c. Relating field components inside and outside the groove: outside the groove, i.e., in the region $0\leq y \leq g$, assuming no variation of the field components along $z$ direction (i.e., $k_z=0$), the electric field component can be written as follows

Taking into account the time dependence, the magnetic field component can be written as:

From the boundary condition that the tangential electric field at the metal wall vanishes, we obtain the following relation from Eq. (12), between upward and downward propagating electric field components outside the groove.

Again, from the continuity condition of tangential electric field $E_x$ at the SSPP-air interface ($y=0$), we get the following from Eqs. (6) and (12):

Multiplying Eq. (16) with $e^{-i\beta _{n'}x}$, and integrating both sides of the equation over the SSPP period $d$, we get the following:

To define coupling between m-th mode of the field inside the groove, and n-th mode of the field outside, we introduce the shorthand notation $\mathcal {K}_{mn}= \int _{-a/2}^{a/2}e^{-i\beta _n x}\cos (\frac {m\pi }{a}(x-\frac {a}{2}))dx$, where $n'$ has been replaced by $n$. Then Eq. (17) reduces to the following form:

Considering only the dominant fundamental modes, the relation can be simplified as:

Similarly, by utilizing the continuity condition of the tangential magnetic field at the SSPP-air interface at $y=0$, Eqs. (7) and (14) can be equated and integrated over the limit $-d/2\leq x \leq d/2$ to arrive at the following relation:

d. Dispersion relation with modulator: from Eqs. (20) and (21), we obtain the following relation between amplitudes of Fourier components of the field inside and outside the groove:

Using Eqs. (11) and (15), and applying the mathematical identity that $\cot (x)=i\frac {e^{ix}+e^{-ix}}{e^{ix}- e^{-ix}}$, we obtain the dispersion relation of an SSPP waveguide having a lump impedance $Z_M$:

Here the longitudinal propagation constant $\beta$ is explicitly shown to have a real ($\beta '$) and imaginary ($\beta ''$) component. As most works on SSPP interconnects focus on the passband, usually the imaginary part of $\beta$ is not considered. In contrast, here we focus on the imaginary component of $\beta$ and show that $\beta ''$ can be altered by changing the value of $Z_M$ such that the information transmitted through the SSPP channel is modulated.

Finally, if the metallic bar on the top of the interconnect is removed, $k_y$ for a non-radiating SSPP mode becomes imaginary such that $k_y=i\kappa$, where $\kappa$ is the attenuation constant along $+y$ axis away from the SSPP-air interface. Under this condition, the dispersion relation of the SSPP waveguide is obtained by evaluating Eq. (23) at the limit of $g\rightarrow \infty$:

If the grooves are terminated with a short- circuit (i.e., $Z_{in}=0$) at the point $y=-h_m$, then $r=-1$, and $\theta =-2Q_0 h_m$. Under this condition, the dispersion relation reduces to the following well-known form of a regular SSPP interconnect having groove width $a$ and groove periodicity $d$:

2.2 Prediction of dispersion-limited modulation speed

In the previous Section, we discussed how the dispersion relation of the SSPP waveguide is drastically altered by the introduction of a switching impedance $Z_{M}$. In this Section, we will show, how the SSPP signal modulation speed can be estimated from the modulated SSPP interconnects transmission characteristics. For this, we consider a carrier signal of unity amplitude with frequency $\omega _c$, which coincides with the upper edge ($\omega _u$) of the SSPP channel’s bandgap. This bandgap is centered around the resonant frequency ($\omega _r$) of the SSPP waveguide. The carrier signal is modulated by the mechanism of abrupt impedance change over a range of $Z_{M}=[Z_{off},Z_{on}]$, where $Z_{off}$ and $Z_{on}$ are open and short circuit impedance, respectively. The carrier signal can pass through the waveguide if $Z_M=Z_{on}$ and is blocked when $Z_M=Z_{off}$. If the modulation frequency is $\omega _{m}$, for a square-wave message signal $\Pi (t)$, the modulated signal $S_m(t)$ becomes:

Here $\tau =\frac {\pi }{\omega _m}$ and $S_{21,on}$ and $S_{21,off}$ are transmission coefficients of the waveguide for $Z_M=Z_{on}$ and $Z_M=Z_{off}$, respectively. Considering the first Fourier component of $\Pi (t)$, we obtain:

It can be seen that the modulated signal essentially becomes a monotonic wave of frequency $\omega _c$ if the Fourier component $c_1$ vanishes. Therefore, $c_1$ essentially corresponds to the modulation depth at the critical limit of modulation frequency. According to Eq. (28), $c_1$ would drastically drop once the modulation frequency $\omega _m$ becomes large enough so as to make $S_{21,off}=S_{21,on}$ at the frequency $\omega =\omega _c-\omega _m$. This limiting frequency corresponds to the lower edge of the SSPP bandgap, which we denote as $\omega _e$. Therefore, the theoretical maximum of modulation frequency in the SSPP waveguide will be approximately equal to the bandgap, $BG=\omega _u-\omega _e$ of the SSPP interconnect when $Z_M=Z_{off}$.

In order to estimate this bandgap, we set $Z_{off}\rightarrow \infty$ and obtain the following dispersion relation from the general expression of Eq. (23):

The bandgap around the resonant frequency $\omega _r = \frac {\pi c}{2h}$ has two regions, namely $\omega _e<\omega <\omega _r$ and $\omega >\omega _r>\omega _u$. The lower band-edge frequency $\omega _e$ is given by:

If $\delta _L$ is small, then $k_y^2(\omega _e) \approx (\frac {\pi }{2h}^2-\beta ^2(\omega _e)) = (\frac {\pi }{2h}^2-(\frac {\pi }{d})^2)$, and $\tan (Qh)\approx \frac {1}{\delta _L}$. Thus the lower detuning parameter becomes:

This relation is obtained within the approximation $\tan (Q_0 h)=\tan (\frac {\pi }{2}+\delta _u)\approx - \frac {1}{\delta _u}$. By replacing $\delta _L$ and $\delta _u$ of Eq. (31), with Eqs. (32) and (33), the bandgap — i.e. the maximum modulation frequency — can be expressed in terms of geometrical parameters of the SSPP interconnect. In subsequent sections, we shall validate this prediction by numerical analysis and finite element method based simulations.

2.3 Analysis of energy efficiency

In order to analyze the energy efficiency of the modulation scheme, we need to account for the various power loss mechanism in the system. Power loss can occur through the following pathways: Joule heating loss in the controllable resistor (i.e., channel of FET), carrier signal absorption loss in the patterned metal of SSPP waveguide, insertion loss due to modal mismatch between the TEM wave feeder and the SSPP modulator, radiation loss due to waveguide bending, and interference loss due to electromagnetic coupling with neighboring channel. The radiation and interference loss analysis is not within the scope of the current work, as we focus on analysis of an isolated SSPP modulator in its most basic form. The insertion loss has been reduced to have nominal effect by designing appropriate tapered adapter, as shown in previous works [35]. Thus the most salient form of power loss in our considered structure is the $I^2R$ loss in modulator’s resistance ($P_{loss,M}$) and the metallic absorption in SSPP waveguide ($P_{loss,wg}$). Then the energy efficiency $\eta$ can be calculated as:

In order to calculate the loss in SSPP wave propagation due to metallic absorption, let us assume we are applying a rectangular pulse of width $\tau$ in time-domain on the control terminal of the electronic modulator. Let us also assume that the electronic modulator’s resistance is very low ($Z_M<<Z_c$) during the ON-state of the pulse. Here $Z_c= \frac {\omega \mu }{Q}$ is the characteristic impedance of the medium. For a real metal, its resistive loss can be accounted for by its skin depth $l_s$. It is shown in [38], that the wavevector $Q$ inside the SSPP groove can be expressed as a complex quantity for a lossy metal as,

For the is low impedance condition of the electronic modulator during the on-state ( i.e., $Z_M<<Z_c$), we can assume $r\approx -1$. Then, by replacing the term $(Q_0)$ in Eq. (25) with the complex value of $Q$, we can solve for a complex value of $\beta$. Following [8], the attenuation constant, represented by $Im[\beta ]$ along the direction of SSPP signal propagation can be related to $Q$ as,

Hence, if the carrier signal is modulated by a rectangular pulse of message signal where the pulse-width of the message is $\tau$ in time domain, then the normalized power loss due to the propagation of double sideband modulated signal with carrier through lossy SSPP waveguide of length $L_{wg}$ would be,

2.4 Numerical analysis

It is well known that the cut-off frequency ($\omega _c$) of the fundamental mode of an SSPP waveguide is inversely proportional to its groove length. Quantitatively, $\omega _c=\frac {\pi c}{2\epsilon _r h}$, where $c$ is the free-space velocity of the electromagnetic wave and $\epsilon _r$ is the dielectric constant of the material in between the grooves. The longitudinal attenuation constant ($\beta ''$), which is the imaginary part of the longitudinal propagation constant $\beta$, is supposed to be high around $\omega _c$. This is confirmed by Fig. 2(a), which shows the change of attenuation constant for groove lengths of $5$ and $20$ mm. Here it has been assumed that the variable resistor connected between the grooves has an impedance of $Z_M=5$ $k\Omega$, which effectively serves as a high-impedance for the interconnect and therefore does not impact the field interior to the groove. As observed in Fig. 2(a), the stop band of SSPP waveguide appears at every odd integer multiple of the frequency $\frac {c}{4h}$. For $h=20$ mm, the first three stop bands occur at $3.75$ GHz, $3\times 3.75$ GHz, and $5\times 3.75$ GHz. The first stop band of the interconnect coincides with the cut-off frequency of 3.75 GHz corresponding to $h=20$ mm, implying that in principle, a carrier signal of $3.75$ GHz can be modulated if the length of the SSPP groove can be dynamically changed from $h=20$ mm. However, because the groove length cannot be physically changed in real-time, we aim to mimic the same effect by electrically changing the impedance $Z_M$.

 

Fig. 2. Frequency dependence of attenuation constant (a) Normalized attenuation constant ($\beta ''$) for two interconnects having different groove lengths ($h$) and a fixed modulating resistance $Z_M=5000$ $\Omega$. Normalized attenuation constant ($\beta ''$) of an interconnect having $h$=20 mm, (b) as a function of frequency for modulating resistances, $Z_M=5$ $\Omega$ and $Z_M=5000$ $\Omega$, and (c) as a function of $Z_M$, for three carrier frequencies within the stop band.

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To numerically calculate the effect of changing $Z_M$, the input impedance $Z_{in}$ is calculated based on Eq. (1), and the result has been utilized to evaluate $r$ and $\theta$ from Eqs. (9) and (10). Next, using Eq. (24), and based on the relation between wave-vector components of the SSPP mode ($\beta ^2-\kappa ^2= Q^2=(\frac {\omega }{c})^2$), we obtain the change of attenuation constant for two different values of modulating impedance $Z_M$ in an SSPP channel of $h=20$ mm. According to Fig. 2(b), when $Z_M=5k\Omega$, the attenuation constant is high around $\omega _c=3.75$ GHz and hence a stop-band exists around this frequency. However, when $Z_M$ is reduced to $5\Omega$, the cut-off frequency shifts to a higher frequency, effectively making the frequency region below $16$ GHz a pass-band. This suggests that for a fixed groove length, it is possible to attain the function of modulation by changing the impedance of the variable resistor. Hence for a given carrier frequency within the stop-band, the act of switching the modulating impedance $Z_M$ would encode the message signal into the amplitude of the carrier.

2.5 Trade-off between modulation speed and energy-efficiency

If we can drastically switch the value of $Z_M$ such that $Z_{M,on}<<Z_c$ and $Z_{M,off}>>Z_c\tan (\frac {\omega }{c}(h_0-h_m))$, then both the on-state and off-state of the SSPP modulator can be described with Eq. (25). However, for off-state, the variable $h_m$ in Eq. (25) has to be replaced by $h_0$. Then the additional band-gap $\Delta _{BG}$ induced by the switching of $Z_M$ from on-state to off-state would be:

As evident from Eqs. (37) and (38), the maximum possible modulation speed would be inversely propertional to groove length $h$, while the power loss increases with higher frequencies and therefore with shorter groove length. Thus modulation-speed and energy-efficiency of an SSPP modulator holds a direct trade-off relation. In Fig. 3(a), based on our theory, we show how the act of scaling of the groove-length $h_0$ modifies the modulation speed and how the it takes a toll on the energy-efficiency of the structure. While scaling the groove length $h_{0}$, we also scaled the other feature sizes of the SSPP waveguide, such as groove width $a$, groove period $d$, and modulator position $hm$ in a way so that the following geometric ratio always holds: $a/h_{0}=0.2, h_m/h_{0}=0.8, d/h_{0}=1$. For all cases, we considered the total length of the SSPP waveguide to remain as $L_{wg}=100 mm$. For calculation of this trade-off relation, the carrier frequency for modulation is chosen as half of the summation of the band-edge frequencies ($f_{e,ON},f_{e,OFF}$) of SSPP waveguide (i.e., the frequncy of surface bound mode at longitudinal phase of 180 degree) corresponding to OFF state and ON state of electronic modulator, respectively. For the double side band modulation scheme considered here, the switching frequency of the controllable resistor are taken to be $f_m=f_{e,ON}-f_c = f_c-f_{e,OFF}$. It can be seen from Fig. 3(a) that, a double-sideband modulation scheme can yield $\sim 10$ Gbps data speed with $\geq 80{\% }$ energy-efficiency. By increasing the ratio of $a/h_0$ from $0.2$ to $0.4$, we can increase the energy efficiency by $10{\% }$ for $1$ mm long groove length (corresponding to 75 GHz cut-off frequency) with nominal change in modulation speed. We may also increase energy-efficiency by $20{\% }$ by increasing the ratio $d/h_0$ from $1$ to $1.5$, albeit at the expense of reduced surface-confinement.

 

Fig. 3. Trade-off between modulation speed and energy efficiency in controllable-resistor based SSPP modulator and modulation of SSPP dispersion using different modulators. (a) The modulation speed in double-sideband (DSB) scheme increases with an inverse proportion to the scaling of groove length $h$, while the other geometric features of the SSPP waveguide are scaled accordingly. The energy efficiency is calculated for the corresponding SSPP structure having a length of 100mm. (b) Modulation of geometric dispersion relation in a SSPP waveguide with the switching of a variable resistor between $5\Omega$ and $5K\Omega$ (c) Modulation of dispersion relation in a SSPP waveguide with the switching of a varactor between 1 $pF$ and 6 $pF$.

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2.6 Extension of the theory for a broad class of SSPP modulators

Our theoretical framework offers a unified approach to understanding the properties of various SSPP modulators, encompassing both previously reported behaviors from existing literature and the novel behavior introduced by our proposed modulator. Thus our theoretical machineries provide a comprehensive explanation of modulator behavior. In order to prove our statement, we will show how our theory can be readily extended to produce the dispersion properties of a varactor diode controlled SSPP modulator, previously reported in [33]. The extension of the theory would require to modify the input impedance of the groove, represented by Eq. (5) as follows,

In Fig. 3(b) and 3(c), we plotted the dispersion diagram of an SSPP waveguide for different electrical impedance state of controllable resistors and varactors, respectively. For both the diagram, the sample SSPP waveguide has the following geometry: groove length $h = 10 mm$, groove width $a= 2 mm$, period $d = 10mm$. The modulating component (either transistor or varactor) is placed at the position of $h_{m} = 8 mm$ inside into the groove. The controllable resistor is connected as a bridge between two neighbor teeth of the grooves, while the varactor is connected by splitting each tooth of the groove at $h_{m}=8 mm$ position and connecting the split parts of the tooth with the varactor.

In case of the resistor-controlled modulation scheme, the switching of the electronic modulator from 5000 $\Omega$ (OFF-state) to 5 $\Omega$ (ON-state) increases the cut-off frequency from $6.75$ GHz to $\sim 8.625$ GHz, a $\sim 28{\% }$ change in transmission band. Sweeping the value of resistance in this range can be practically achieved by state-of-the-art transistors with submicron gate length and a hundred microns of gate width [39,40]. It can be readily seen from Eq. (38) that the controllable resistor-induced change in transmission band can be enhanced even further by placing the electronic modulator further away from the short-circuited terminal of the groove (i.e., by choosing smaller $h_m$ in the design), albeit at the price of reduced surface confinement of the signal. In case of varactor-controlled modulation scheme, our theory shows how the transmission band decreases with the increase of the value of its capacitance (Fig. 3(c)), which agrees with the findings of varactor-based SSPP modulation reported in [33]. For brevity, the key equations used in the analysis of the SSPP modulation scheme are shown in Table 1.

Table 1. Key equations for analyzing the SSPP modulation scheme

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3. Experimental considerations and signal modulation

3.1 Design considerations

Our vision for the SSPP modulator involves its integration into chip-scale communication systems, necessitating the selection of a modulating device with low on-resistance and minimal switching latency. Advancements in transistor scaling over the past years have yielded MOSFETs and HEMTs with exceptional switching speeds and reduced parasitic capacitances. Very recently, a record bandwidth of 400 GHz and a record low ON-state impedance of $500\Omega -\mu$m have been reported with GaN MOSHEMTs [40]. By capitalizing on the switching speed of such state-of-the-art transistors, it is anticipated that the choice of FET as the modulating component for SSPP waveguides will be highly advantageous. A schematic illustration of the envisioned scheme (shown in Fig. 1(a)) suggests that during experimental realization, the source and drain regions of the FET can posit right below the neighboring teeth of the groove to make good contact with the metallic part of the SSPP waveguide. One may choose the n-channel enhancement mode FET to capitalize on the relatively higher mobility of electrons compared to holes for designing a high-speed modulator. The following factors should play an important role in the design of the FET-based SSPP modulator:

3.2 Transmission characteristics

In order to match the modes between the feeder and the SSPP waveguide, we designed an adapter utilizing the structure of a vivaldi antenna [34,35]. The geometry of the converter is designed in a way so as to achieve approximately 50 $\Omega$ input impedance. The matching with the input impedance was confirmed by varying the impedance of a lump element at the output port and monitoring the minima of reflection coefficient at frequencies close to the spoof plasmon resonance. Figure 4(a) and 4(b) demonstrate the electric field distribution in logscale for the ON- and OFF-states of the modulator, respectively. The ON- and OFF-states of the modulator are mimicked by setting $Z_M<<Z_C$, and $Z_M>>Z_{C}$, respectively, where $Z_C$ is the characteristic impedance of the medium inside the groove for TEM mode propagation. It can be readily seen that the ON-state of the modulator corresponds to high transmission of surface wave, while the OFF-state corresponds to signal propagation up to the elongated groove (where the variable resistor is located), followed by dominant back-scattering.

 

Fig. 4. Field distribution in SSPP waveguide during ON- and OFF-state The center groove is elongated and incorporates a variable resistor (such as, a gate controlled FET) inside it. (a) High transmission coefficient is observed at carrier frequency when the modulator is switched ON (b) Suppressed signal propagation through SSPP waveguide during OFF-state of the modulator.

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Based on the formalism discussed in Section 2, three types of SSPP interconnects have been considered for finite element method-based numerical analysis. These structures have been labeled and shown as Type-I, II, and III in Fig. 5. Here Type-I is a reference static structure with two elongated grooves, Type-II is a dynamic structure with one elongated groove and a variable resistor inside it, and Type-III is an extension of Type II structure with multiple modulating grooves. All the structures, unless otherwise specified, have a regular groove length of 20 mm, elongated groove length of 24 mm, groove width of 2 mm, and periodicity of 20 mm. The objective is to show how the transmission bandwidth and roll-off of band-edge can be impacted by the in-situ variation of the resistance $Z_M$ of implanted electronic modulators and also by the design choice of total number of modulating cells.

 

Fig. 5. Design optimization of SSPP interconnects for signal modulation (a-c) Schematic illustration of the three types of SSPP interconnects, labeled as Type-I, Type-II and Type-III, depending on the presence and number of modulating resistors. (d) Comparison of $S_{21}$ parameters of Type-I and Type-III interconnects, considering identical values of modulating resistance $Z_{M1}$ and $Z_{M2}$. (e) Comparison of $S_{21}$ parameters of Type-I, Type-II and Type-III interconnects; both equal and non-equal values of $Z_{M1}$ and $Z_{M2}$ have been considered for the Type-III interconnect. (f) Phase response of $S_{21}$ parameters corresponding to the amplitude response shown in (e).

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The bandwidth ($BW_{sp}$) of the Type I interconnect is given by the following relation [37].

To obtain further insight into the prospect of amplitude modulation, transmission characteristics of Type-II interconnect are analyzed (Fig. 6). The −3 dB cutoff changes by $0.4$ GHz as $Z_M$ is changed from $5$ $\Omega$ to $5$ k$\Omega$. The shaded region of Fig. 6(a), extending from $\sim$2.4 GHz to 2.8 GHz, posits the region of interest for signal modulation. To understand how a continuous change of channel resistance would impact signal transmission, the $S_{21}$ parameter is plotted as a function of $Z_M$ in Fig. 6(b) for SSPP carrier frequencies of $2.45$, $2.75$, and $2.9$ GHz. While transmittance at $2.75$ GHz decreases by more than an order of magnitude as $Z_M$ increases from low to high resistance, at $2.45$ GHz the transmittance is relatively less sensitive to the change of $Z_M$. According to the prediction of our theory established in Section 2, for a choice of $2.75$ GHz carrier frequency, the dispersion limited maximum modulation speed would be (2.75-2.45) Gbps = 300 Mbps. In the next section, we will verify this prediction by time domain analysis.

 

Fig. 6. Transmission characteristics for amplitude modulation (a) Transmission characteristics of the Type-III SSPP interconnect for $Z_M=5 \Omega$ and $Z_M=5000 \Omega$ (inset shows corresponding phase response). (b) $S_{21}$ parameters obtained for various frequencies around the resistor-induced bandgap regime while $Z_M$ is varied up to $5000 \Omega$

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3.3 Modulation characteristics

In this Section, based on time-domain analysis, we demonstrate how the proposed design can be utilized to implement the amplitude shift keying (ASK) modulation. The corresponding SSPP structure with adapter at input and output port and the modulating impedance halfway through the structure is shown in Fig. 7(a). For a carrier signal having frequency $f_{c}$, the message signal shown in Fig. 7(b) is given as input to the control terminal of the electronic modulator. The ON and OFF states of the modulator correspond to modulating resistance $Z_M=5$ $\Omega$ and $Z_M=5000$ $\Omega$ respectively. Pseudo-random bit streams of OOK-NRZ encoded message signal ($m(t)$) have been considered in the time domain analysis. The carrier frequency is maintained at $2.76$ GHz and the modulating signal speed is varied from $25$ to $500$ Mbps.

 

Fig. 7. Amplitude modulation and time-domain analysis (a) 3D schematic of the SSPP communication system comprising the SSPP interconnect, input-output couplers or adapters, and modulating variable resistor. (b) The message signal ($m(t)$) and (c-e) corresponding output signal ($y(t)$), obtained with a SSPP carrier frequency of $f_c$=2.76 GHz and two-cell modulation. Here (c) and (d) show modulated ASK output for carrier power ($P_{c}$) of 1 mW and 0.01 mW respectively, at a bit-rate of BR=25 Mbps. In (e), the case of $P_c$= 0.01 mW and BR=300 Mbps is shown. (f) Extinction Ratio (ER), which is defined as the ratio of output power during ON and OFF states of the electronic modulator (denoted as $P_{ON}$ and $P_{OFF}$ respectively), has been plotted as a function of modulation speed for two cases: two-cell modulation with a carrier frequency of $2.76$ GHz, and five-cell modulation with a carrier frequency of $2.86$ GHz.

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The output signal ($y(t)$), obtained for two different power levels ($P_{c}$) of the carrier is shown in Fig. 7. For $P_{c}$=1 mW, the transmitted message signal of 25 Mbps speed remains nearly identical to the original message signal after demodulation at the receiving end (Fig. 7(c)). At such a speed, which is significantly lower than the electronic modulator-switching induced bandgap, the transmitted signal remains quite similar to the original signal even after a reduction of carrier power by 1000 times (Fig. 7(d)). However, when the bit-rate is ramped up to 300 Mbps, the distortion and attenuation of the transmitted signal becomes significant (Fig. 7(e)). To estimate the maximum bit-rate for the considered design parameters, we calculate the signal extinction ratio (ER) as the ratio of output amplitude level during the ON- and OFF-states of the modulator. The extinction ratio decreases at high bit-rates, with a drastic decline within the $200-300$ Mbps range. These characteristics can be correlated with the frequency response of the SSPP interconnect shown earlier in Fig. 6(a). According to the frequency response, the −3 dB cut-off frequency of the modulator during its OFF-state is 2.45 GHz. Therefore, for a choice of 2.76 GHz carrier signal, the lower band edge of the OFF-state of modulator is approximately 300 MHz away from the carrier frequency, limiting the modulation speed within 300 Mbps.

However, the precise value of modulation bandwidth and degree of modulation depth (i.e. the extinction ratio) for a given carrier frequency additionally depends on the number of modulated grooves. To demonstrate this, in Fig. 7(f), we have compared the extinction ratio (ER) of two different cases of the Type-III system discussed in Section 3.2. It is observed that at any given bit-rate, the ER increases as the number of modulating cells is increased. For an interconnect having 5 modulating cells – each comprising a variable resistor as modulator – the highest enhancement of ER is achieved around the modulation speed of $300$ Mbps. Nonetheless, it is important to note that a power penalty is associated with the incorporation of electronic modulators in additional groves of the interconnect. Each electronic modulator within the SSPP cell introduces its own $I^2R$ loss, thereby contributing to the system’s overall power consumption. Furthermore, achieving synchronization among multiple electronic modulators during high-speed modulation necessitates the design of suitable signal delay lines for each modulator. This would invariably increase the design complexity and footprint of the system.

To further evaluate the ASK modulation scheme, a standard communication system comprising a transmitter, a modulated SSPP interconnect, and a receiver is considered (Fig. 8(a)). For simplicity, the channel is considered distortion-free with an attenuation of 10 dB. The detector is modeled with shot noise spectral current density of $N=\sqrt {2} {\times } 10^{-11}\frac {A}{\sqrt {Hz}}$, and responsitivity of, $R$ = 1 A/W. The detector is followed by a low-pass filter (LPF), the cut-off frequency ($f_{LC}$) of which is approximately 80% of BR. The bit error rate (BER), calculated for different bit rates and carrier powers, is shown in Fig. 8(b). In accordance with previous studies [41,42], BER$\le 10^{-13}$ is taken as the threshold for error-free data transmission. For, BR=25 Mbps, the minimum carrier launch power required to maintain BER$\ge 10^{-13}$ is −8.46 dBm (0.14 mW). As the BR is increased to 200 Mbps, this power requirement increases to −1.63 dBm (0.7 mW). For signal frequencies at 300 Mbps or higher, BER for a given carrier power drastically increases beyond $>10^{-13}$ prohibiting the modulation at such speed. Thus we validate our theoretical prediction that the maximum modulation speed for an SSPP waveguide with groove length $20$ mm would be $\leq 300$ Mbps.

 

Fig. 8. SSPP-based communication system and its figure of merit (a) Schematic of an SSPP interconnect-based communication system comprising channel-loss of 10 dB, detector responsivity $R$ = 1 A/W, noise spectral current density $N=\sqrt {2} {\times } 10^{-11}\frac {A}{\sqrt {Hz}}$; cut-off frequency ($f_{LC}$) of the low-pass filter is 80% of bit rate. (b) Bit error rate (BER) – obtained for different bit rates of the message signal – plotted as a function of carrier power (the dashed line indicates a threshold of $10^{-13}$ BER) and (c) corresponding eye diagrams considering a fixed carrier power and frequency of 0 dBm and 2.76 GHz respectively.

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The eye diagram of Fig. 8(c) is obtained using a virtual test bench of a commercial software. The eye diagram shows the transmission characteristics for a carrier power of 0 dBm (1 mW) at different bit-rates of the message signal. The eye opening decreases by 26.8% as the BR increases from 25 Mbps to 300 Mbps. Also jitter – normalized by corresponding symbol period – increases from 4% to about 17.5% as the BR is increased. With 1 mW power of carrier signal and 300 Mbps bit-rate of the message signal, BER is $\sim 10^{-3}$. The BER at such high bit-rates can be reduced by increasing the carrier power, as has been demonstrated in Fig. 8(b). The modulation speed of SSPP interconnects can be extended from hundreds of Mbps to hundreds of Gbps by scaling down the groove length from millimeter to micrometer dimensions. However, at the sub-THz regime of data transfer rate, the low energy efficiency of the modulator due to ohmic loss in metal would become a bottleneck for the system, as explicated by our theoretical analysis.

4. Conclusion

This study presents a generalized theoretical framework for the analysis of critical performance metrics of signal modulation in spoof-surface plasmon polariton interconnects. The theoretical formalism of this work can be utilized to predict the dispersion-limited maximum achievable modulation speed and the corresponding energy efficiency of arbitrary modulators in an SSPP-based communication channel. The developed theory is agnostic to the choice of specific modulating components, such as transistors or varactors, making it versatile for analyzing various modulating devices. In conjunction, the study introduces a novel SSPP modulation technique based on the dynamic alteration of the effective optical length of the grooves through implanted adjustable resistors. The theoretical predictions of this work have been validated using numerical analysis and finite element method-based electromagnetic simulations. Time and frequency domain analyses have also been employed to estimate the figure of merits of amplitude shift keying (ASK) digital modulation in an SSPP channel. The analysis reveals a potential $\sim 28{\% }$ change in the SSPP channel’s transmission-band when implanted resistance is sweeped from $5$ k$\Omega$ to $5 \Omega$ – a range of resistance that can be practically achieved using gate-controlled, state-of-the-art submicron scale field-effect transistors. For a representative SSPP structure, up to $300$ Mbps speed is achieved with cut-off frequencies of around $2.75$ GHz. Numerical analysis shows that scaling down the geometric features of the structure allows $\sim 10$ Gbps modulation speed for point-to-point data transfer over $100$ mm distance while maintaining over $80{\% }$ energy efficiency. We envision that the developed theory would facilitate design and optimization of surface-plasmon based interconnect technologies. Furthermore, the proposed variable resistor-based modulation technique would leverage the advantages of CMOS-compatible process technology for future high-speed, high-volume communication networks at the chip scale.