Terahertz band communication using plasma wave propagation in multilayer graphene heterostructures

Abstract

Graphene‐based heterostructures provide a viable platform to implement optoelectronic devices that can operate in the terahertz (THz) band. In this study, the authors focus on multilayer (ML) graphene as the building block to implement high‐frequency and low‐energy plasmonic interconnects for on‐chip signalling in next‐generation systems. Two specific plasmonic interconnect geometries are analysed: single waveguide (SWG) and parallel‐plate waveguide (PPWG). While SWG interconnects support propagating surface plasmons that are polarised in the transverse magnetic direction, in PPWG interconnects, nearly dispersion‐less quasi‐transverse electromagnetic modes are supported. The dispersion characteristics are derived by solving Maxwell's equations in the device setup in which ML graphene presents an impedance boundary condition. The effects of number of layers, electrostatic screening, and Fermi level are included in the model of intra‐band dynamical surface conductivity of ML graphene. The authors also develop analytical models of energy‐per‐bit and bandwidth density for both SWG and PPWG interconnects. The energy dissipation includes the effect of plasmon generation, detection, and modulation circuitry within a thermal‐ and shot‐noise‐limited transmission of information. They quantify optimal interconnect length scales for which plasmonic interconnects provide lower energy and higher bandwidth when compared against their electrical (copper/low‐κ) counterparts at the 2020 ITRS technology node.

1 Introduction

The remarkable growth of silicon semiconductor industry over the last several decades is driven by Moore's law and the dimensional scaling of on-chip components, which yields consistent improvement in the binary information throughput of microprocessor chips. However, quantum mechanical leakage and energy concerns will be prohibitive to further miniaturisation of devices, thereby slowing down the momentum of Moore's law [1–3]. While transistor scaling improves their performance, the performance of interconnects degrades with scaling [4]. As such, interconnects impose limits on the overall throughput and efficiency of microprocessors. Several potential solutions including the use of ballistic low-dimensional conductors (carbon nanotubes and graphene), less aggressive wire width and thickness scaling, and low-swing signalling have been proposed to overcome on-chip interconnect challenges [5–7]. At the same time, system-level performance is limited by the bandwidth of off-chip interconnects. To address system-level interconnect challenges, three-dimensional (3D) integration and airgap interconnects are being investigated [8, 9]. As the frequency of operation increases, electrical interconnects also suffer from signal distortion resulting from interconnect dispersion, which sets an upper bound on the maximum achievable signal bandwidth [10]. Going forward, terahertz (THz) band communication is envisioned as a key wireless technology to handle bandwidth-greedy applications and delay-critical communication needs for next-generation systems [11, 12].

In this paper, we propose to exploit plasma waves in multilayer (ML) graphene heterostructures for on-chip signalling purposes in the THz band. Graphene, which is an atomically thin sheet of carbon atoms arranged in a honeycomb lattice, displays strong light–matter interaction over a broad frequency range [13–16]. Graphene can be easily combined with other 2D materials such as transition metal dichalcogenides, phosphorene, and hexagonal boron nitride, to develop a fully integrated solution for the generation, detection, modulation, and propagation of plasma waves [14, 17]. At the lower end of the THz frequency spectrum (<10 THz), a sheet of graphene sandwiched between two dielectrics supports the propagation of plasma waves that are polarised in the transverse magnetic (TM) direction [18]. However, a graphene-based parallel-plate waveguide (PPWG) structure is able to support quasi-transverse electromagnetic (TEM) waves in the THz band [19, 20]. These plasma waves can be guided along the waveguide for on-chip local and global communication. The propagation characteristics of graphene plasmons can be configured through electrical or chemical doping. This provides opportunities for reconfigurable plasmonics – something that is not feasible with conventional metal plasmonics [21–24]. For electrostatically tuning the characteristics of plasma waves, a gate voltage is applied through a metal–oxide–semiconductor setup in PPWG geometry.

To quantify the performance of plasmonic interconnects, we analytically model their energy-per-bit and bandwidth density. The energy dissipation includes the energy required for generation, detection, and modulation of the plasmonic signal. We focus on the thermal- and shot-noise limited transmission of plasmons. To derive the bandwidth density, we consider the propagation of a narrowband Gaussian signal centred at THz frequency for both single waveguide (SWG) and PPWG interconnects. The distortion in the pulse width and amplitude reduction due to graphene ohmic losses is obtained by the convolution of the channel impulse response and the input envelope signal. At THz frequencies, only the intra-band dynamical conductivity of ML graphene is considered, while inter-band contribution can be neglected. The conductivity model of graphene includes the effect of carrier scatterings due to acoustic phonons and charged impurities. The effect of finite electrostatic screening between the multiple layers in ML graphene stack is also included. Lossy propagation resulting from energy-dependent carrier scatterings in graphene is studied for both TM and TEM modes as a function of operating frequency and material parameters of ML graphene stack. Owing to their nearly dispersion-less propagation, PPWG plasmonic interconnects offer significantly higher bandwidth density as compared to their electrical counterparts at the 2020 ITRS technology node [25]. Therefore, PPWG interconnects are more suitable for global level or even off-chip signalling purposes. On the other hand, TM plasmons propagating in SWG interconnects suffer much less degradation due to carrier scatterings. As such, they can be used for energy-efficient signalling at the local and semi-global length scales on the chip.

2 Guided modes in plasma wave interconnects

The device setup for the propagation of TM-polarised plasma waves consists of an ungated graphene sheet within a dielectric referred to as SWG geometry. To support quasi-TEM modes, we focus on a gated graphene-based PPWG geometry. The two waveguide structures are shown in Fig. 1. In the case of SWG, the electric field has components in both z- and x-directions. However, in the case of PPWG, the electric field is predominantly in the z-direction that is perpendicular to the graphene sheet. In both setups, x-direction is defined along the length of the graphene sheet and corresponds to the direction of propagation of plasma waves. In general, the dielectric and the substrate have different dielectric permittivities. However, in this paper we consider that ungated graphene is embedded in a linear, homogeneous dielectric media, which simplifies analytical calculations. In both device setups, graphene is modelled as a stack of multiple layers with a linear energy-dispersion relationship. In ML graphene grown epitaxially on silicon carbide or via chemical vapour deposition, it is shown that layers in graphene are decoupled. As such, Bernal stacking no longer exists and the layers are rotated relative to each other. Essentially, the decoupling of layers preserves the linear energy dispersion in ML graphene [26, 27].

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To obtain the plasma wave dispersion relationship for the device setups in Fig. 1, we solve Maxwell's equations and apply impedance boundary condition on graphene–dielectric interface. This is mathematically expressed as $\hat{n}\times ({\mathit{H}}_1-{\mathit{H}}_2)={\sigma }_g(\omega ){\mathit{E}}_{\parallel }$. Here, ${\sigma }_g(\omega )$ is the complex dynamical conductivity of ML graphene, ${\mathit{E}}_{\parallel }$ is the tangential electric field at the graphene interface, and ${\mathit{H}}_i$ (i = 1, 2) is the magnetic field on either side of the graphene sheet. The TM and quasi-TEM dispersion relationships, under long-wavelength approximation, are given as [28]1a$\frac{k_{\text{TM}}(\omega )}{{\mathit{k}}_d}=\sqrt{1-{ϵ}_{ox}{\left(\frac{2}{{\sigma }_g(\omega ){\eta }_0}\right)}^2},$1b$\begin{array}{rl}& \frac{k_{\text{TEM}}(\omega )}{{\mathit{k}}_d}\\ & =\sqrt{1+\frac{1}{{\eta }_0^2{\sigma }_m(\omega ){\sigma }_g(\omega )}\left[{ϵ}_{ox}-j\frac{{\eta }_0\left({\sigma }_m(\omega )+{\sigma }_g(\omega )\right)}{k_{0}d}\right]}.\end{array}$ In the above set of equations, $k(\omega )=\beta (\omega )+j\alpha (\omega )$ is the complex propagation constant of the wave at frequency $\omega$, ${\mathit{k}}_d=(\omega /c)\sqrt{{ϵ}_{ox}}$ is the wave vector in the dielectric media, ${\eta }_0$ is the impedance of free space, d is the thickness of the dielectric, ${ϵ}_{ox}$ is the relative permittivity of the dielectric, and ${\sigma }_m(\omega )$ is the conductivity of metal plate. We consider that the metal plate is a perfect electric conductor (PEC) at THz frequencies; therefore, ${\sigma }_m(\omega )\to \mathrm{\infty }$. Under the PEC boundary condition, (1b) can be simplified to2$k_{\text{TEM}}(\omega )=k_0\sqrt{{ϵ}_{ox}\left(1-\frac{j}{{\eta }_0{k}_{0}d{\sigma }_g(\omega )}\right)}.$ The quasi-TEM mode can be considered as a perturbation of the pure TEM mode supported by a perfectly conducting PPWG geometry. For bound TM modes, the magnetic field changes sign across the graphene sheet, when the surrounding dielectric medium is lossless. In quasi-TEM modes, the electric field profile has an even symmetry. Furthermore, if long-wavelength approximation is used then the quasi-TEM mode characteristics only depend on the inner dielectric media (gate dielectric in Fig. 1), which means that the mode is confined strongly between the metal plate and the graphene sheet. In this paper, we will specifically focus on the quasi-TEM modes for which the inter-plate dielectric thickness is small such that the condition ${\beta }_{\text{TEM}}(\omega )d\ll 1$ is satisfied for all frequencies of operation. The advantage of the gated graphene structure is that it allows modifying the Fermi level in graphene post-fabrication by changing the gate voltage, $V_{\mathrm{G}}$ (discussed in Section 2.3). Therefore, the gated graphene setup enables reconfigurable graphene optoelectronic devices.

2.1 ML graphene conductivity

To study the mode characteristics of plasma waves in graphene, it is important to model the dynamical conductivity of the graphene sheet. In this work, we ignore non-local effects, related to spatial dispersion, in graphene to model the conductivity. Instead we focus on the random-phase approximation in long-wavelength limit for which $\omega \gtrsim \beta (\omega )v_{\mathrm{F}}$. Here, $v_{\mathrm{F}}$ is the Fermi velocity of carriers in graphene. If we define the effective plasmon mode index as $n=\beta (\omega )/k_0$, to ensure the validity of local conductivity model, $n\lesssim 300$. This condition is typically satisfied for both TM and quasi-TEM modes in graphene at THz frequencies. Mathematically, the frequency-dependent intra-band conductivity of monolayer graphene is given as [29]3${\sigma }_{\text{mono}}(\omega )=j\frac{e^2\omega }{\pi \hslash }{\int }_0^{\mathrm{\infty }}\mathrm{d}E\frac{|E|}{{\omega }^2}\frac{\mathrm{d}{f}_0(E)}{\mathrm{d}E},$ where e is the elementary charge, $\hslash$ is the reduced Planck's constant, $f_0(E)=(1+\exp ((E-E_f)/k_{\mathrm{B}}T){\hspace{0.17em})}^{-1}$ is the Fermi function, and $E_f$ is the Fermi level in graphene. Carrying out the integration4${\sigma }_{\text{mono}}(\omega )=-j\frac{2e^2}{\pi {\hslash }^2}\frac{k_{\mathrm{B}}T}{\omega -j\mathrm{\Gamma }}\ln \left[2\cosh \left(\frac{|E_f|}{k_{\mathrm{B}}T}\right)\right],$ where $k_{\mathrm{B}}T$ is thermal energy, $\mathrm{\Gamma }$ (${\tau }^{-1}$) is the carrier relaxation rate in graphene. Typically, $\mathrm{\Gamma }$ depends on temperature, sample quality (defects and charged impurities), and carrier concentration or $E_f$. Details are presented in Section 2.2. In the case of ML graphene with finite electrostatic screening between the multiple layers, the Fermi level of each layer is uniquely defined. The conductivity of ML graphene is, therefore, obtained by summing up the conductivity of each layer according to5${\sigma }_g(\omega )=\sum _n^{N_{\text{layer}}}{\sigma }_{\text{mono},n}(\omega ).$ Substituting (4) in (5), ${\sigma }_g(\omega )$ is simplified as6a${\sigma }_g(\omega )=\sum _n^{N_{\text{layer}}}{\chi }_n\frac{{\mathrm{\Gamma }}_n-j\omega }{{\mathrm{\Gamma }}_n^2+{\omega }^2},$6b${\chi }_n=\frac{2e^2{k}_{\mathrm{B}}T}{\pi {\hslash }^2}\left[\frac{|E_{fn}|}{2k_{\mathrm{B}}T}+\ln \left(1+{\mathrm{e}}^{-|E_{fn}|/k_{\mathrm{B}}T}\right)\right],$ where $E_{fn}$ and ${\mathrm{\Gamma }}_n$ are the Fermi level and carrier relaxation rate in the nth graphene layer. The Fermi level $E_{fn}$ is given as7$E_{fn}=E_f\exp \left(-\frac{d_0}{{\lambda }_s}(2n-1)\right),$ where $d_0=0.35\text{nm}$ is the spacing between the layers in ML graphene, ${\lambda }_s$ is the electrostatic screening length, and $E_f$ is the Fermi level in the lowest layer set by electrostatic (chemical) doping in gated (ungated) graphene structure. In [30], ${\lambda }_s$ is experimentally found to be 0.5–0.7 nm. The consequence of electrostatic screening in ML graphene is that the effective number of layers available for conduction is less than the total number of layers physically present in the stack.

2.2 Carrier relaxation rate

We consider elastic momentum relaxation processes, including acoustic phonons and charged impurities, in graphene to model the carrier relaxation rate. In graphene, scattering due to longitudinal acoustic phonon modes dominates as coupling of electron–phonon states for other phonon modes is too weak or the energy scales of the optical phonon modes are too high for the frequency range of interest. Carrier scattering time due to acoustic phonons is given as8${\tau }_k=\frac{4{\hslash }^2\rho v_s^2{v}_f}{D_{\mathrm{A}}^2{k}_{\mathrm{B}}T}\frac{1}{k},$ where $v_s$ is the velocity of acoustic phonons, $v_f={10}^6\mathrm{m}/\mathrm{s}$ is the Fermi velocity of carriers in graphene, $D_{\mathrm{A}}$ is the electron acoustic deformation potential, estimated to be of the order of 3t, where t is the nearest neighbour hopping integral, $\rho$ is the material density of graphene, k is the wave vector of electrons. Equation (8) can be re-written in terms of the Fermi level $E_f$ in the graphene sheet according to9${\tau }_{\text{ac}}=\frac{4{\hslash }^3}{D_{\mathrm{A}}^2}\frac{\rho {(v_s{v}_f)}^2}{k_{\mathrm{B}}T}\frac{1}{|E_f|}.$ In the case of graphene $D_{\mathrm{A}}\simeq 9\text{eV}$, $v_s=7.33\times {10}^3\mathrm{m}/\mathrm{s}$, and $\rho =7\times {10}^{-7}\text{kg}/{\mathrm{m}}^2$ [31].

The scattering time due to long-range Coulomb potential of charged impurities for large doping ($|E_f|\gg k_{\mathrm{B}}T$) is given as [32]10a${\tau }_{\text{imp}}={\tau }_{k_f}=\frac{{\hslash }^2{v}_f{\mathit{k}}_{\mathrm{F}}}{u_0^2},$10b$u_0=\frac{\sqrt{N_{\text{imp}}}Z{e}^2}{4{ϵ}_0ϵ(1+\gamma )},$10c$\gamma =\frac{\rho (E_f)e^2}{2{ϵ}_0ϵ{\mathit{k}}_{\mathrm{F}}},$ where ${\mathit{k}}_{\mathrm{F}}$ is the Fermi wave vector, $ϵ={ϵ}_{ox}$ is the relative dielectric permittivity of media surrounding graphene, Ze is the net charge of the impurity atom, and $N_{\text{imp}}$ denotes the concentration of charged impurities in the sample.

The net relaxation rate is computed using the Mattheissen's sum rule. That is, $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\text{ac}}+{\mathrm{\Gamma }}_{\text{imp}}={\tau }_{\text{ac}}^{-1}+{\tau }_{\text{CI}}^{-1}$. As shown in Fig. 2, carrier relaxation rate due to charged impurities decreases as the carrier concentration increases. This is because carriers screen electric field lines, which increases the scattering time. On the other hand, carrier relaxation rate due to acoustic phonons increases with an increase in Fermi level. For typical values of Fermi level ($E_f\simeq (0.2-0.6)\text{eV}$) and charged impurity concentration ($N_{\text{imp}}\simeq ({10}^{14}-{10}^{15}{\mathrm{m}}^{-2}$), the net carrier relaxation rate is around 1–3 THz. Due to electrostatic screening between the layers in ML graphene, the carrier relaxation rate for each layer will be different and will be evaluated accordingly. For all analyses in this paper, $N_{\text{imp}}={10}^{15}{\mathrm{m}}^{-2}$ and $ϵ=3.9$, unless otherwise noted.

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Using the model of carrier relaxation rate, we obtain the dynamical conductivity of graphene as shown in Fig. 3 for various values of $E_f$ and ${\lambda }_s$ and $N_{\text{layer}}=5$. An increase in ${\lambda }_s$ increases the conductivity as more layers tend to contribute to current conduction. This fact is also evidenced by the increase in the number of effective layers. For lower value of ${\lambda }_s$, $N_{\text{eff}}/N_{\text{layer}}\ll 1$, but this ratio saturates to unity for ${\lambda }_s\gg d_0$, where $d_0=0.35\text{nm}$ in ML graphene. Both real and imaginary parts of graphene conductivity increase with an increase in $E_f$ at all values of ${\lambda }_s$. While ${\mathrm{\Gamma }}_n$ has a non-monotonic dependence on $E_{fn}$, the factor ${\chi }_n$ in (6b) increases with an increase in Fermi level. This leads to a monotonic increase in ${\sigma }_g(\omega )$ with $E_f$.

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2.3 Electrostatic doping

In the case of PPWG geometry, electrostatic gating can be used to adjust the Fermi level $E_f$ defined in (7). In the absence of interface and substrate traps, the Fermi level in the graphene sheet as a function of gate voltage ($V_{\mathrm{G}}$) is given as [24]11$\frac{E_f}{e}=\frac{(V_g-V_{\text{min}})C_{\text{ox}}}{C_{\text{ox}}+C_Q},$ where $V_{min}$ is the Dirac point voltage, $C_{\text{ox}}={ϵ}_{\text{ox}}/d$ is the oxide capacitance, $C_Q=q^2\partial n_s/\partial E_f$ is the quantum capacitance of graphene, and $n_s$ is the carrier concentration in the graphene sheet. The total carrier concentration, $n_s=|n_{\text{elec}}-n_{\text{hole}}|$, where $n_{\text{elec}}$ and $n_{\text{hole}}$ are the electron and hole concentrations, respectively. The carrier concentration can be obtained using the Fermi-Dirac integral using an appropriate density of states that incorporates broadening due to electron–hole puddles near the Dirac point [33]. The variation of the Fermi level and total capacitance with gate voltage is shown in Fig. 4. The effect of quantum capacitance is most apparent for gate voltages close to the Dirac point. Quantum capacitance acts to reduce the total gate capacitance and is more critical to consider for thin dielectrics. For all analyses in this paper, we choose oxide thickness d = 5 nm. For such a thin dielectric, $V_g$ must be limited to <5 V to prevent dielectric breakdown.

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The complex conductivity of ML graphene as a function of gate voltage evaluated at a frequency of 1 THz is shown in Fig. 5. There exists a minimum conductivity around the Dirac point due to the formation of electron–hole puddles. The imaginary part of conductivity increases more rapidly than the real part as gate voltage increases. The real part of conductivity is related to in-phase current which produces Joule heating, while the imaginary part represents the out-of-phase current or the inductive response of electrons. It can be seen from Fig. 5 that inductive effects become more important at higher end of the THz frequencies. In fact, in the low-loss scenario ($\mathrm{\Im }({\sigma }_g(\omega ))\gg \mathrm{\Re }({\sigma }_g(\omega ))$), per-unit-length resistance ($R_k$) and kinetic inductance ($L_k$) of the graphene sheet supporting plasmons can be defined as12a$R_k\simeq \frac{2}{W}\frac{{\sigma }_r(\omega )}{{\sigma }_i{(\omega )}^2},$12b$L_k\simeq \frac{1}{\omega W{\sigma }_i(\omega )},$ where W is the width of graphene sheet, ${\sigma }_r(\omega )$ (${\sigma }_i(\omega )$) is the real (imaginary) part of graphene conductivity. The above set of equations can be used to study the propagation of TM plasmons in graphene using the concept of equivalent circuit models as discussed in prior work [34, 35]. The ratio $L_k/R_k\propto {\sigma }_i(\omega )/{\sigma }_r(\omega )$ increases at higher frequencies as observed in Fig. 5.

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3 Performance modelling

Using the conductivity model derived in prior sections, we obtain the characteristics of propagating plasmon modes in ML graphene using (1b). The advantage of graphene plasmons is that they have smaller mode areas, i.e. higher value of $\beta (\omega )/k_0$ compared to noble metals. In Fig. 6, we plot the wave vector of both TM and TEM plasmons in graphene as a function of frequency. Material parameters chosen for this analysis are indicated in the figure caption. The real part of the wave vector is related to the plasmon wavelength, group velocity, and wave-front velocity, while its imaginary part determines the propagation length of plasmons owing to finite ohmic losses in graphene. Propagation length is defined as the length scale at which the plasmon field amplitude falls to 1/e of its initial value.

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The most obvious difference in the propagation characteristics of TM and TEM plasmons is related to their frequency dependence. While the real part of the wave vector of TM plasmons displays ${\omega }^2$ behaviour, TEM plasmons have a nearly dispersion-less propagation. That is, $\beta (\omega )$ increases linearly with frequency for TEM modes. Unfortunately, at the lower end of the THz spectrum, TEM plasmons display high imaginary part of the wave vector. This means that TEM plasmon modes are extremely lossy and may not be suitable for information propagation for low THz frequencies.

To compare the performance of TM and TEM plasmon modes, we consider two metrics: propagation length ($L_{\text{prop}}$) and localisation length (LL). Mathematically13a$L_{\text{prop}}=\frac{1}{2|\alpha (\omega )|},$13b$\text{LL}=\frac{1}{|\mathrm{\Im }(k_z)|},$ where $\alpha (\omega )$ is the imaginary part of the wave vector defined in (1) for both SWG and PPWG geometries. The wave vector $k_z$ is obtained in the direction perpendicular to the propagation of plasmon modes. That is14$k_z=\sqrt{{ϵ}_{ox}{k}_0^2-k{(\omega )}^2}.$ While a large $L_{\text{prop}}$ is desired so that the plasmonic signal can be communicated over long distances on the waveguide without loss, it is preferred to have a smaller localisation length so that the plasmon modes remain tightly bound to the graphene sheet and their energy does not leak into the neighbouring dielectric. As shown in Fig. 7, SWG interconnects have superior propagation length compared to PPWG interconnects. Unfortunately, mode confinement is weaker in the case of SWG interconnects. The parallel-plate configuration squeezes light into the region between the plates and allows for much tighter confinement. As expected, there is a trade-off between localisation and propagation. That is, modes that are less leaky also suffer more ohmic losses as they propagate. The electrostatic screening has a significant impact on both LL and $L_{\text{prop}}$ at all values of $E_f$. As ${\lambda }_s$ increases, LL and $L_{\text{prop}}$ improve for both TM and TEM modes. Therefore, it is important to have graphene heterostructures in which the layers are well coupled to each other. An increase in Fermi level decreases ohmic losses in the system, thereby improving the metrics LL and $L_{\text{prop}}$.

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We also define a figure-of-merit (FOM) as the ratio $(L_{\text{prop}}/{\lambda }_{sp})$, which indicates the number of wavelengths (${\lambda }_{sp}$) that the plasma wave could travel on the waveguide before losing most of its energy. The FOM incorporates the impact of Fermi level and other material parameters on the performance of the waveguide (Fig. 8). The FOM improves for TEM waves as a function of frequency. This means that TEM plasmons are more advantageous for information propagation at higher frequencies. On the other hand, the FOM corresponding to TM plasmons exhibits a non-monotonic dependence on frequency. In general, TM modes are preferred at the lower end of the THz spectrum for frequencies on the order of a few 100s of GHz. Beyond a frequency of 1–2 THz, the FOM of TEM modes exceeds that of TM modes. If the FOM is less than unity, it indicates that the plasmon modes are too lossy and unsuitable for waveguiding application. For the chosen material parameters, TM modes always have a FOM higher than unity. However, TEM modes display huge ohmic losses for frequency <1 THz. We also note the distinct behaviour of TM and TEM modes on the Fermi level in the graphene sheet. The FOM of TEM modes degrade with an increase in Fermi level at all frequencies, while the FOM of TM modes improve with an increase in Fermi level for frequencies <7 THz. For higher frequencies, the FOM of TM modes has the same dependence on Fermi level as that of TEM modes. TEM modes tend to be more confined for lower Fermi level and their confinement degrades rapidly with an increase in Fermi level.

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3.1 Energy-per-bit

The propagation length plays an important role in determining the energy dissipation of plasmonic interconnects as it limits the maximum distance between transceivers communicating using plasma waves. The energy dissipation of the plasmonic interconnect is given as [36, 37]15$E_{\text{pl}}=E_{\text{source},\text{detect}}+E_{\text{mod}},$ where $E_{\text{source},\text{detect}}$ is the energy consumed in the excitation and detection circuitry for plasmons, and $E_{mod}$ is the energy consumed in the modulator for converting electrical information into plasmonic information. Photodetectors based on graphene and other 2D materials operating in the THz frequency range are being actively investigated [38]. Due to the high carrier mobility in graphene, the conversion of photons into electrical current or voltage is an ultrafast and ultra-sensitive process. Different mechanisms including photovoltaic effect, photo-thermoelectric effect, bolometric effect, and plasma-wave assisted mechanisms, have been achieved in graphene [39–41]. Researchers have recently demonstrated that Dyakonov–Shur (DS) instability in a high electron mobility transistor (HEMT) with a graphene gate can be used as an efficient plasma wave detector in the THz range [42, 43]. Based on the DS mechanism, a DC voltage is generated across the source–drain contacts in the HEMT due to incident THz radiation. In the resonant detection regime, the DC source–drain voltage exhibits peaks at odd multiples of the lowest plasma wave frequency. Graphene-based electro-optic modulators from the THz to visible light regimes have been designed using the gate-tunable electro-absorption effect in graphene [44]. Light modulation using graphene has the advantage of faster speed and broadband operation when compared to conventional semiconductor-based modulators. However, one of the challenges of graphene electro-optic modulator is the low absorption in its monolayer form, which reduces the modulation depth – a critical parameter that determines the net energy consumption of the plasmonic interconnect. Theoretical calculations performed in [45] predict >100 GHz operation and sub 1 fJ/bit energy dissipation of electro-absorption graphene modulator operating at 1550 nm. A THz graphene modulator monolithically integrated with a quantum cascade laser with a modulation depth in excess of 95% was demonstrated in [46]. While the modulation speed of the design is limited to 100 MHz, it is expected to improve with optimised laser cavity design and modulator architectures.

A lower bound on the energy dissipation of plasmonic interconnects can be obtained by considering the transmission of plasmons to be thermal- and shot-noise limited. That is, the bit-error rate (BER) requirement supersedes the requirement for the received optical power and extinction ratio. In this case, the energy dissipation $E_{\text{source},\text{detect}}$ for a plasmonic interconnect of length L is given as [47]16$E_{\text{source},\text{detect}}>\frac{\hslash \omega }{{\eta }_{\mathrm{L}}{\eta }_{\mathrm{C}}{\eta }_{\mathrm{M}}}\left(n_{min,\text{avg}}\right)\exp \left(\frac{L}{L_{\text{prop}}}\right),$ where ${\eta }_{\mathrm{L}}$ is the laser quantum efficiency, ${\eta }_{\mathrm{C}}$ is the laser to waveguide coupling efficiency, ${\eta }_{\mathrm{M}}$ is the modulator insertion loss, and $n_{min,\text{avg}}$ is the mean number of plasmons needed to encode one bit of information. Considering thermal- and shot-noise limits, the fluctuation in the number of plasmons detected is given as17$⟨\mathrm{\Delta }{n}^2⟩=⟨n⟩+kT/(q^2/2C_d),$ where $⟨n⟩$ is the mean number of detected plasmons due to the Poisson-distributed shot noise, kT is the thermal energy, $C_d$ is the capacitance of the photodetector. The second term in the above equation is due to thermal noise which follows a Gaussian distribution. For BER $P_{\text{err}}<\exp (-b)$, the mean number of plasmons, $n_{min}$ required in ‘on’ pulse is given as [36, 47]18$n_{min}\simeq \frac{2b}{{\eta }_d{M}^2}\left(2-M+2\sqrt{1-M+\frac{M^2}{2b}{\sigma }_{\text{th}}^2}\right),$ where M is the modulation depth, ${\eta }_d$ is the quantum efficiency of the photodetector, ${\sigma }_{\text{th}}^2$ is the thermal noise [second term in (17)]. The mean number of plasmons in ‘off’ pulse is equal to $(1-M)n_{min}$. The average number of plasmons in (16) is, therefore, $n_{min,\text{avg}}=n_{min}(1-0.5M)$. The BER is given as [5]19$P_{\text{err}}<\frac{R}{NfT},$ where N is the number of nodes in the interconnection network, f is the operation frequency, R is the target failure rate, and T is the lifetime of operation. Using $N={10}^4$, f = 5 GHz, $R={10}^{-6}$, T = 10 years, we obtain $P_{\text{err}}=6.3\times {10}^{-30}$. Using a value of $P_{\text{err}}={10}^{-30}$, the parameter b to be used in (18) is equal to 70.

The product $n_{min,\text{avg}}{\eta }_d$ as a function of the modulation depth for various values of the detector capacitance is shown in Fig. 9. As modulation depth increases, the mean number of plasmons needed for detection reduces. In the high modulation depth regime, the detector capacitance significantly impacts the value of mean number of plasmons. The specific choice of $P_{\text{err}}$ does not significantly impact the value of $n_{min,\text{avg}}{\eta }_d$ as shown in the inset plot of Fig. 9.

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A lower bound on the energy dissipation of the modulator is given as [37, 48, 49]20$E_{mod}=Q_m{V}_m=C_m{V}_m^2,$ where $Q_m$ is the charge through the modulator, $C_m$ is the modulator capacitance, and $V_m$ is the drive voltage of the modulator. It is expected that as the 2D optoelectronic technology matures, $E_{mod}$ will scale down to values lower than 10 fJ/bit. Further, if the modulator is integrated with the off-chip quantum cascade laser, it may be prudent to omit its energy dissipation in the calculation of on-chip energy dissipation as done previously.

The energy dissipation of SWG and PPWG interconnects of length 10 μm is shown in Fig. 10 as a function of frequency of operation. At the lower end of the THz spectrum, PPWG interconnects consume approximately three orders of magnitude higher energy compared to SWG interconnects. At shorter interconnect length scales, the overall energy consumption of SWG plasmonic interconnects tends to be dominated by the modulator energy dissipation; therefore, for a fixed value of $E_{mod}=5\text{fJ}/\text{bit}$, the energy dissipation of SWG interconnects becomes nearly independent of frequency. The excessive energy dissipation of PPWG interconnects is a result of the short propagation length, which could limit the use of PPWG interconnects for energy-efficient signalling.

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3.2 Bandwidth density

The bandwidth of a single channel in nanophotonic interconnects is typically limited by the operation speed of the transceiver circuits at the ends of the interconnect. However, in this paper, we will obtain an upper bound on the bandwidth of plasmonic interconnects by considering only the interconnect-dominated limits. To evaluate the bandwidth of a plasmonic interconnect, we consider the propagation of a narrow-band Gaussian signal centred at the THz frequency through SWG and PPWG geometries. As the Gaussian pulse propagates, it suffers from attenuation due to ohmic losses in graphene and the pulse shape distorts due to waveguide dispersion. The amplitude of the Gaussian signal, $|F(z,t)|$, at any location z on the waveguide measured at time t can be obtained by the convolution of the channel impulse response and the input signal. Details can be found in prior works [50, 51]. Mathematically, $|F(z,t)|$ is given as21$\begin{array}{rl}|F(z,t)|=& {\left[\frac{{\tau }_0^4}{{({\tau }_0^2-{\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}z)}^2+({\beta }_0^{\mathrm{\prime }\mathrm{\prime }}z{\hspace{0.17em})}^2}\right]}^{1/4}\\ & \times \exp \left(\frac{-{\alpha }_0^{\mathrm{\prime }2}{z}^2}{2({\tau }_0^2-{\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}z)}\right)\exp \left(-\frac{{(t-t_g)}^2}{2{\tau }^2}\right).\end{array}$ In the above equation, ${\tau }_0$ is the pulse width of the input signal, ${\beta }_0^{\mathrm{\prime }}$ (${\beta }_0^{\mathrm{\prime }\mathrm{\prime }}$) and ${\alpha }_0^{\mathrm{\prime }}$ (${\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}$) are the first (second) derivatives of the real and imaginary part of the wave vector evaluated at $\omega ={\omega }_0$, $t_g$ is the group delay, and $\tau$ is the pulse width of the distorted Gaussian signal. The group delay consists of two components: one that scales linearly with interconnect length and another that scales quadratically with interconnect length. The quadratic component is a direct result of ohmic losses in the medium. As such22a$t_g={\beta }_0^{\mathrm{\prime }}z-\frac{{{\alpha }^{\mathrm{\prime }}}_0{{\beta }^{\mathrm{\prime }\mathrm{\prime }}}_0{z}^2}{{\tau }_0^2-{{\alpha }^{\mathrm{\prime }\mathrm{\prime }}}_{0}z},$22b$\tau =\sqrt{{\tau }_0^2-{{\alpha }^{\mathrm{\prime }\mathrm{\prime }}}_{0}z+\frac{{({{\beta }^{\mathrm{\prime }\mathrm{\prime }}}_{0}z)}^2}{{\tau }_0^2-{{\alpha }^{\mathrm{\prime }\mathrm{\prime }}}_{0}z}}.$ The time evolution of the Guassian envelope for both SWG and PPWG interconnects is shown in Fig. 11 at various locations on the waveguide. While the Gaussian pulse in SWG interconnects broadens out as it propagates, in PPWG interconnects, the pulse may even undergo compression. This is because of the positive value of ${\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}$ in PPWG interconnects. For (22b) to be valid, the condition $({\tau }_0^2-{\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}z)>0$ must be satisfied. For SWG interconnects, this condition is always true due to the negative value of ${\alpha }_0^{\mathrm{\prime }\mathrm{\prime }}$. For PPWG interconnects, this condition fails only for interconnects longer than several millimetres, which is beyond the scope of this work.

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Due to pulse shape distortion, successive pulses launched into the waveguide may overlap if inter-pulse timing is not enough. The maximum achievable signal bandwidth or bitrate, $F_b$, for plasmonic interconnects is given as23$F_b\le \frac{1}{\mathrm{\Delta }{t}_g}=\left(\frac{\mathrm{d}\omega }{\mathrm{d}{t}_g}\right)\frac{1}{\mathrm{\Delta }\omega }.$ The bandwidth density of an interconnect is defined as the bitrate of an interconnect normalised to the wiring pitch. Essentially, the bandwidth density represents the number of bits that can be transferred across a unit length bisectional line on the chip. Mathematically, the bandwidth density is given as24${\mathrm{\Phi }}_D=\frac{F_b}{P},$ where $F_b$ can be found from (23), and P is the wiring pitch. A typical cross-section of plasmonic interconnects is shown in Fig. 12. The pitch in this case is given as P = (W + S), where W is the width and S is the spacing between the wires. For analysis in this paper, we choose, $W=5\beta (\omega {\hspace{0.17em})}^{-1}$, and S is taken as twice of the localisation length. Therefore25$P=5\beta (\omega {\hspace{0.17em})}^{-1}+\frac{2}{|\mathrm{\Im }({\mathit{k}}_z(\omega ))|},$ where ${\mathit{k}}_z(\omega )$ is the plasmon wave vector in the surrounding dielectric media as discussed in (12). This choice of pitch ensures minimum leakage and coupling between neighbouring plasmonic wires. Since $\beta (\omega )$ and ${\mathit{k}}_z(\omega )$ vary with the frequency of operation, the design of plasmonic interconnect structures will depend on the frequency of operation.

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Fig. 13 shows the bandwidth density versus frequency of both SWG and PPWG plasmonic interconnects. While the bandwidth density of SWG interconnects supporting TM plasmon modes is nearly independent of frequency, for PPWG interconnects supporting TEM modes bandwidth density increases with an increase in frequency. Due to the dispersion-less propagation of TEM modes, the bandwidth density of PPWG interconnects is significantly higher than that of SWG interconnects. At a frequency of 2 THz, the bandwidth density of PPWG interconnects is ${10}^4\text{Gbit}/\mathrm{s}/\mu \mathrm{m}$ for a channel length of 100 μm. However, for the same parameters, the bandwidth density of SWG interconnects is 117 Gbit/s/μm. As such, PPWG interconnects are more appropriate for global or off-chip signalling purposes, where higher bandwidth density is desirable.

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4 Comparison with electrical interconnects

The performance of plasmonic and electrical interconnects is compared with respect to energy-per-bit and bandwidth density. We also quantify interconnect length scales for which plasmonic transmission is more advantageous compared to electrical transmission on the chip.

4.1 Energy dissipation

The energy-per-bit of plasmonic interconnects is discussed in Section 3.1. For electrical interconnects, the energy dissipation is determined by the charging and discharging of interconnect and transistor capacitances. As such, the energy-per-bit of electrical interconnects can be stated as26$E_{\text{elec}}=(C_{\text{tx},\text{rx}}+c_{w}L)V_{DD}^2,$ where $C_{\text{tx},\text{rx}}$ is the total capacitance of the transceiver circuits, $c_w$ is the wire capacitance per-unit-length, $V_{DD}$ is the supply voltage. In writing the above equation, we assume that the wires and all transceiver units have a full signal swing ($0\to V_{DD}$) at their output nodes, and static power dissipation is neglected. The value of electrical interconnect capacitance depends on its dimensions. In this analysis, $c_w=1.8\text{pF}/\text{cm}$ for global wires and the capacitance of a minimum-sized n-type transistor at the 2020 ITRS technology node is 6.193 aF. Considering that the transmitter and receiver circuits are sized as $5\times$ minimum-sized inverter, $C_{\text{tx},\text{rx}}\simeq 62\text{aF}$. For interconnect lengths $L>C_{\text{tx},\text{rx}}/c_w=0.7\mu \mathrm{m}$, the total energy is dominated by the interconnect energy.

In Fig. 14, the energy-per-bit of electrical and plasmonic interconnects is shown as a function of interconnect length. For plasmonic interconnects, the frequency of operation is selected as 1 THz to achieve a reasonable trade-off between energy-per-bit and bandwidth density. While the energy dissipation of electrical interconnects increases linearly with interconnect length, the energy dissipation of plasmonic interconnects grows exponentially with interconnect length. If the energy dissipation associated with modulation is not included, SWG interconnects are more energy efficient compared to electrical interconnects up to a few millimetre length scale. For an energy dissipation of 5 fJ/bit of the modulator, SWG interconnects have lower energy than electrical interconnects only up to 100 μm as indicated by point ‘a’ in the figure. PPWG plasmonic interconnects have a lower propagation length; therefore, their energy dissipation increases much more rapidly with interconnect length. As such, PPWG plasmonic interconnects have lower energy dissipation compared to electrical interconnects only up to 20 μm when $E_{mod}=0$ as shown by point ‘b’ in the figure. In the presence of modulator energy dissipation, PPWG plasmonic interconnects consume more energy than their electrical counterparts for lengths >1 μm. For energy-efficient communication using plasmonic interconnects, it is better to exploit TM plasmons in SWG interconnect geometries owing to the larger propagation length. Further, for competitive advantage against electrical interconnects, it is important to bring the energy dissipation of modulators to below 1 fJ/bit.

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4.2 Bandwidth density

For digital electrical interconnects in which signals are sent as ‘on’ and ‘off’, the bitrate capacity depends on the rise time of the signal. The rise time of the electrical interconnects depends on whether the interconnect is operating in the RC or LC regime. In general, the bitrate of electrical interconnects with a cross-sectional area of $A_{\text{cross}}$ and length L is given as27a$f<f_{\text{cutoff}}(\text{RC}-\text{region}):B_0\cong \frac{1}{8r_w{c}_w}\frac{A_{\text{cross}}}{L^2},$27b$f>f_{\text{cutoff}}(\text{LC}-\text{region}):B_0\cong \frac{1}{k}\left(\frac{Z_0^2}{{\mu }_0{\rho }_{e}ff}\right)\frac{A_{\text{cross}}}{L^2},$ where $r_w$ and $c_w$ are the per-unit-length resistance and capacitance values of the interconnect, ${\rho }_{\text{eff}}$ is the effective resistivity of the interconnect, ${\mu }_0$ is the magnetic permeability of the medium, $Z_0$ is the lossless characteristic impedance of the line and is given as $Z_0=\sqrt{{\epsilon }_d}/(c_0{c}_w)$, where $c_0$ is the speed of light in vacuum. The cutoff frequency, $f_{\text{cutoff}}=4{\rho }_{\text{eff}}/(A_{\text{cross}}{\mu }_0)$. The factor k depends on the desired eye opening and is usually chosen between 50 and 100. The effective resistivity is computed by taking into account both grain boundary and sidewall scatterings in scaled interconnects [52]. The cut-off frequency is $\simeq 6\text{GHz}$. The wiring pitch is taken to be twice the width of the interconnect, which is selected as 100 nm for this analysis. As shown in Fig. 15, the bandwidth density of RC electrical interconnects exceeds that of LC electrical interconnects. As expected, the bandwidth density of PPWG plasmonic interconnects is higher than that of SWG plasmonic and electrical interconnects for all length scales considered in this analysis. On the other hand, SWG plasmonic interconnects offer superior bandwidth density compared to their electrical counterparts for length scales greater than $\simeq 100\mu \mathrm{m}$.

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4.3 Partition length

We obtain a partition length, $L_{p,\text{energy}}$ corresponding to energy dissipation of plasmonic and electrical interconnects. $L_{\text{part},\text{energy}}$ is the interconnect length beyond which plasmonic interconnects consume more energy than their electrical counterparts. Therefore, a higher value of $L_{p,\text{energy}}$ is preferred for energy-efficient plasmonic signalling. As shown in Table 1, modulator energy dissipation strongly affects the value of $L_{p,\text{energy}}$ for both TEM and TM modes. In general, for a fixed value of $E_{mod}$, $L_{p,\text{energy}}$ for TM modes is higher than for TEM modes. When $E_{mod}$ dominates the total energy dissipation in the plasmonic domain, it is found that TEM modes will always consume more energy than their electrical counterparts even for the shortest length scale. This is indicated using ‘ $\times \times$’ in the table for TEM modes when $E_{mod}=5\text{fJ}/\text{bit}$. Interestingly, TM modes becomes more energy efficient at longer length scales when $E_{mod}=5\text{fJ}/\text{bit}$. This is contrary to the behaviour observed when $E_{\text{source},\text{detect}}$ dominates the overall energy dissipation. Our results indicate that SWG plasmonic interconnects are more suited for energy-efficient signalling purposes for intermediate and semi-global length scales on the chip.

Table 1 Partition length, $L_{p,\text{energy}}$, corresponding to energy dissipation

	TEM modes	TM modes
$E_{\text{mod}}$	$E_f=0.2\text{eV}$	$E_f=0.4\text{eV}$	$E_f=0.2\text{eV}$	$E_f=0.4\text{eV}$
0	13.8 μm	16.5 μm	1.63 mm	5.4 mm
1 fJ/bit	9.4 μm	11.4 μm	19.2 μm
5 fJ/bit	$\times \times$	$\times \times$	>100 μm

We also quantify an interconnect length scale, $L_{p,\text{BWD}}$, beyond which the bandwidth density of plasmonic interconnects exceeds that of electrical interconnects. A lower value of $L_{p,\text{BWD}}$ indicates superior performance of plasmonic interconnects. Results are given in Table 2. The wiring pitch of electrical interconnects operating in the RC regime is chosen between the minimum and maximum allowed values for global interconnects per the 2020 ITRS technology roadmap. As the results show, TEM modes offer very high bandwidth density and are more advantageous than electrical interconnects for length scales as short as a few micrometres. By up-sizing the wiring pitch of electrical interconnects, their bandwidth density improves, which means $L_{p,\text{BWD}}$ values would increase. The dependence of $L_{p,\text{BWD}}$ on $E_f$ is different for TM and TEM modes. While an increase in $E_f$ reduces the value of $L_{p,\text{BWD}}$ for TM modes, the corresponding value for TEM modes increases. This is due to the different dispersive features of TM and TEM modes as discussed in Section 3.

Table 2 Partition length, $L_{p,\text{BWD}}$, corresponding to bandwidth density

	TEM modes	TM modes
Pitch	$E_f=0.2\text{eV}$	$E_f=0.4\text{eV}$	$E_f=0.2\text{eV}$	$E_f=0.4\text{eV}$
32 nm	0.21 μm	0.31 μm	13.7 μm	6.0 μm
200 nm	4.8 μm	7.1 μm	308.0 μm	135.2 μm
2000 nm	80.9 μm	118.6 μm	5.2 mm	2.27 mm

5 Conclusion

In this paper, graphene-based plasmonic interconnects are investigated for low-energy and high-bandwidth on-chip signalling purposes for next-generation systems. Two geometries of the plasmonic interconnects, namely SWG and PPWG, are investigated. SWG interconnects support the propagation of TM modes, while PPWG interconnects support nearly dispersion-less TEM modes. The energy dissipation and bandwidth density of both modes is analytically modelled as a function of material (Fermi level, scattering rate, electrostatic screening) and geometrical parameters of the waveguides. It is shown that due to their parallel-plate structure, PPWG interconnects confine plasmons better yielding a superior localisation length. The improvement in localisation reduces the propagation length significantly for PPWG interconnects. The energy dissipation of plasmonic interconnects is modelled by incorporating the energy consumed by source, detection, and modulation circuitry within the regime of noise-limited plasmon transmission. We show that when the source/detect circuit is the dominant component of energy dissipation, SWG interconnects have lower energy dissipation compared to their electrical counterparts up to length scales of few millimetres. This energy advantage drops rapidly with growing modulator energy dissipation. On the other hand, PPWG interconnects have limited propagation length. Therefore, their energy advantage ceases beyond length scales of $\simeq 10\mu \mathrm{m}$. To model bandwidth density, we focus on the propagation of a narrowband Gaussian signal centred at THz frequency through both waveguide geometries. Our results clearly show that PPWG interconnects exceed the bandwidth density of RC-limited electrical interconnects for interconnects as short as few micrometres at the 2020 ITRS technology node. Hence, PPWG interconnects are more suitable for global chip-level signalling, while SWG interconnects can achieve low-energy operation at the local and semi-global length scales.

References

Cavin, R.K., Lugli, P., Zhirnov, V.V.: ‘Science and engineering beyond moore's law’, Proc. IEEE, 2012, 100, pp. 1720–1749 (Special Centennial Issue)

Cumming, D.R.S., Furber, S.B., Paul, D.J.: ‘Beyond moore's law’, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., 2014, 372, (2012), p. 20130376

Markov, I.L.: ‘Limits on fundamental limits to computation’, Nature, 2014, 512, (7513), pp. 147–154

Meindl, J.D., Chen, Q., Davis, J.A.: ‘Limits on silicon nanoelectronics for terascale integration’, Science, 2001, 293, (5537), pp. 2044–2049

Chen, J.: ‘Self‐calibrating on‐chip interconnects’. PhD thesis, Stanford University, 2012

Ceyhan, A., Naeemi, A.: ‘Cu interconnect limitations and opportunities for swnt interconnects at the end of the roadmap’, IEEE Trans. Electron Devices, 2013, 60, (1), pp. 374–382

Prasad, D., Ceyhan, A., Pan, C. et al.: ‘Adapting interconnect technology to multigate transistors for optimum performance’, IEEE Trans. Electron Devices, 2015, 62, (12), pp. 3938–3944

Kumar, V., Bashirullah, R., Naeemi, A.: ‘Modeling, optimization and benchmarking of chip‐to‐chip electrical interconnects with low loss air‐clad dielectrics’. IEEE 61st Electronic Components and Technology Conf. (ECTC), 2011, 2011, pp. 2084–2090

Kumar, V.: ‘Modeling and optimization approaches for benchmarking emerging on‐chip and off‐chip interconnect technologies’. PhD thesis, Georgia Institute of Technology, 2014

Miller, D.A.B., Ozaktas, H.M.: ‘Limit to the bit‐rate capacity of electrical interconnects from the aspect ratio of the system architecture’, J. Parallel Distrib. Comput., 1997, 41, (1), pp. 42–52

Jornet, J.M., Akyildiz, I.F.: ‘Fundamentals of electromagnetic nanonetworks in the terahertz band’, Found. Trends Netw., 2013, 7, (2‐3), pp. 77–233, ISSN 1554‐057X, doi: 10.1561/1300000045, available at [DOI: https://dx.doi.org/10.1561/1300000045] , 2004

Rakheja, S.: ‘On the Gaussian pulse propagation through multilayer graphene plasmonic waveguides – impact of electrostatic screening and frequency dispersion on group velocity and pulse distortion’, IEEE Trans. Nanotechnol., 2016, 15, (6), pp. 936–946

Ceyhan, A., Naeemi, A.: ‘Overview of the interconnect problem’, in Todri‐Sanial, Aida, Dijon, Jean, Maffuci, Antonio (Ed.): ‘Carbon nanotubes for interconnects’ (Springer, Switzerland, 2017), pp. 3–36