The Ultra-Low-k Dielectric Materials for Performance Improvement in Coupled Multilayer Graphene Nanoribbon Interconnects

Abstract

The ultra-low-k dielectric material replacing the conventional SiO₂ dielectric medium in coupled multilayer graphene nanoribbon (MLGNR) interconnects is presented. An equivalent distributed transmission line model of coupled MLGNR interconnects is established to derive the analytical expressions of crosstalk delay, transfer gain, and noise output for 7.5 nm technology node at global level, which take the in-phase and out-of-phase crosstalk into account. The results show that by replacing the SiO₂ dielectric mediums with the nanoglass, the maximum reduction of delay time and peak noise voltage are 25.202 ns and 0.102 V for an interconnect length of 3000 µm, respectively. It is demonstrated that the ultra-low-k dielectric materials can significantly reduce delay time and crosstalk noise and increase transfer gain compared with the conventional SiO₂ dielectric medium. Moreover, it is found that the coupled MLGNR interconnect under out-of-phase mode has a larger crosstalk delay and a lesser transfer gain than that under in-phase mode, and the peak noise voltage increases with the increase of the coupled MLGNR interconnect length. The results presented in this paper would be useful to aid in the enhancement of performance of on-chip interconnects and provide guidelines for signal characteristic analysis of MLGNR interconnects.

1. Introduction

As the feature size of very large-scale integrated (VLSI) circuits is scaled down to the nanometer order, various performance degradation and stability problems on the conventional Cu interconnects have emerged in recent years [1,2,3]. Graphene, as a promising candidate for replacing the common copper, has attracted the intensive interest of many researchers in terms of its excellent electrical, mechanical and thermal properties [4,5]. Compared with the copper material, high quality graphene has a long mean free path on the order of several micrometers, which can result in a lower resistivity and achieving the ballistic transport at shorter interconnects. The current density of graphene can reach 10⁹ A/cm² in comparison to its Cu counterpart, thereby, eliminating the electromigration and skin effect of Cu interconnects [6]. Graphene nanoribbon (GNR) is a narrow strip of graphene sheet, which can be classified into multilayer GNR (MLGNR) and single-layer GNR (SLGNR) depending on its stacked number of layers. SLGNR is not suitable for on-chip interconnects owing to its larger intrinsic resistance [7,8,9]. Based on the types of connection with other devices or interconnects, MLGNR can be further categorized into top contact MLGNR (TC-MLGNR) and side contact MLGNR (SC-MLGNR) [10,11,12]. TC-MLGNR has only the top most layer connected to surrounding contacts while all layers of SC-MLGNR are coupled with the other contacts, which results in the distributed scattering resistance of the former over the latter. Hence the SC-MLGNR is adopted as interconnect material in this paper.

With the interconnect width shrinking into the nanometer scale, the crosstalk delay, crosstalk noise, and transfer gain of global interconnects in VLSI circuits have become major performance concerns owing to the longer distance. To date, a series of studies about these issues on MLGNR interconnects have been done. In reference [13], Agrawal et al. presented the analytical model of crosstalk delay and crosstalk noise based on the FDTD method. In references [14,15,16], the propagation delay and transfer gain for a single-line of MLGNR interconnect were investigated. In reference [17], Sahoo et al. analyzed the characteristic of crosstalk noise of coupled MLGNR interconnects, considering the coupling capacitance. In reference [1], Zhao et al. analyzed and compared the performance difference of crosstalk noise and crosstalk delay between coupled MLGNR and Cu interconnects, considering the impact of coupling capacitance.

However, the works mentioned above only focus on how to establish the analytical model but do not propose the optimization method to reduce crosstalk delay and crosstalk noise and increase transfer gain of on-chip interconnects. Apparently, it is crucial to further enhance the MLGNR interconnects performance at the end of the roadmap. There are some reports that replacing the SiO₂ with the ultra-low-k dielectric materials (k ≤ 2) as the dielectric medium can reduce the delay of on-chip interconnects [18,19,20,21,22]. Hence, it is also necessary to investigate the effects of the ultra-low-k dielectric materials on the crosstalk noise and transfer gain of on-chip interconnects. So far, to the best of our knowledge, there is no research to propose the analytical model of coupled MLGNR interconnects, considering the impact of the ultra-low-k dielectric materials. Therefore, in this paper, an analytical model for crosstalk delay, noise output, and frequency response of coupled MLGNR interconnects with different dielectric materials is proposed.

The overview of development about the ultra-low-k dielectric constant materials is shown as follows. The relative dielectric constant of SiO₂ as the most common insulator material is usually equal to 3.9 [20]. The low dielectric constant (low-k) SiCOH (k = 3.0) is regarded as an effective substitute for the conventional SiO₂ and has been successfully integrated in some 130 nm and 90 nm VLSI circuits [21]. The emergence of the porosity fabrication technology by plasma-enhanced chemical vapor deposition (PECVD) can reduce the relative dielectric constant of insulator material [21]. By applying the fabrication technology, SiCOH is processed into porous p-SiCOH (k = 2.4) that has been applied at IBM’s P7 microprocessor [21]. With the continuous advancement of the porosity fabrication technology, the inorganic ultra-low-k dielectrics, such as nanoglass, have the relative dielectric constant of 1.3 [22]. In view of the current research progress of insulator material of VLSI circuits, the nanoglass is the lowest dielectric constant material. Thus, nanoglass was adopted as the ultra-low-k dielectric constant material for analysis of performance of on-chip interconnect compared with the conventional SiO₂ in the paper.

2. Interconnect Model

A typical geometry of two-line coupled MLGNR interconnects placed above the ground plane and surrounded by a dielectric medium is exhibited in Figure 1, where the common SiO₂ dielectric material is replaced by the ultra-low-k dielectric material. The W, S, T_gnr, T_ox, and ε_r are line width, line space, line height, thickness of insulator dielectric, and relative dielectric constant of the ultra-low-k dielectric material, respectively. The total number of layers for MLGNR depends on the line thickness T_gnr and can be expressed as N_layer = 1 + Integer[T_gnr/δ] [23]. Herein the operator Integer[.] means that only the integer part is considered and δ (=0.34 nm) is the interlayer spacing between successive GNR layers [24].

As shown in Figure 2, the equivalent circuit model for two-line coupled MLGNR interconnects is configured with the same effective driver resistance R_d, driver capacitance C_d, and load capacitance C_l for aggressor and victim lines. The coupled MLGNR interconnects are comprised of lumped and distributed parts.

For MLGNR coupled interconnects, R_lu and R_E represent the lumped resistance and per unit length (p.u.l.) equivalent distributed scattering resistance, respectively. They can be given by [7],

$$R_{lu}=\frac{R_{cm}+R_{qm}}{N_{layer}\times N_{ch}},$$

(1)

$$R_E=\{\begin{array}{cc}0& L_{gnr}<{\lambda }_{eff}\\ \frac{R_{qm}}{N_{layer}\times N_{ch}\times {\lambda }_{eff}}& L_{gnr}>{\lambda }_{eff}\end{array}.$$

(2)

Here, R_cm denotes the imperfect contact resistance and its value ranges from 1 KΩ to 20 KΩ [13]. R_qm is the monolayer quantum resistance and can be defined as R_qm = h/2e² (herein h is Plank’s constant and e is charge of electron). λ_eff denotes the effective mean free path (MFP). N_ch represents the total number of conducting channels in the monolayer GNR and can be approximated as below [25,26],

$$N_{ch}=a_0+a_{1}W+a_2{W}^2+a_3{E}_f+a_4{E}_{f}W+a_5{E}_f{}^2.$$

(3)

where a₀ to a₅ are the parameters for zigzag MLGNR (zz-MLGNR) at room temperature (300 K) with the Fermi energy E_F over 0 [25]. In the light of the total number of conducting channels of zigzag MLGNR over armchair MLGNR (ac-MLGNR), hence, only the zz-MLGNR is investigated in the paper. L_gnr represents the length of MLGNR interconnects. The effective mean free path (MFP) for the ith-subband can be expressed as,

$${\lambda }_{eff,i}={\left(\frac{1}{{\lambda }_d}+\frac{1}{{\lambda }_s}+\frac{1}{{\lambda }_{e,i}}\right)}^{-1}.$$

(4)

Herein, λ_d represents the mean free path due to the scattering effects by the static impurities and crystal defects (=1 µm) [1,10]. λ_s denotes the mean free path induced by electron–phonon scattering (=70 μm) [27]. λ_e,i is the mean free path contributed to the scattering of edge roughness, which is depended on the interconnect width and the backscattering probability P at the edges [28].

$${\lambda }_{e,i}=\frac{W}{P}\sqrt{{\left(\frac{2W{E}_f}{ih{v}_f}\right)}^2-1},$$

(5)

where v_f is the Fermi velocity of electrons in graphene (=8 × 10⁵ m/s) [12]. The value of P will change when the edge roughness situation is different. Especially, P = 0 and P = 1 demonstrate that the edge of MLGNR are fully specular and fully diffusive, respectively [1].

The distributed capacitance C_E of MLGNR interconnects comprises of the equivalent quantum capacitance C_eq and the electrostatic capacitance C_el. The p.u.l. equivalent quantum capacitance can be obtained by using a recursive scheme as [1,7,17],

$$C_{{}_{rec}}^1=C_q=\frac{4e^2{N}_{ch}}{h{v}_f},$$

(6)

$$C_{{}_{rec}}^i={\left(\frac{1}{C_{{}_{rec}}^{i-1}}+\frac{1}{C_m}\right)}^{-1}+C_q,$$

(7)

$$C_{eq}=C_{{}_{rec}}^{N_{layer}}.$$

(8)

wherein, C_q is the p.u.l. length quantum capacitance of monolayer GNR. C_m is the p.u.l. coupling capacitance between successive GNR layers and can be defined as C_m = ε₀W/δ (here ε₀ = 8.854 × 10⁻¹² is the vacuum dielectric constant). In order to investigate the impacts of ultra-low-k dielectric material materials, thus the relative dielectric constant ε_r can be applied to distinguish different dielectric mediums in this paper. The p.u.l. electrostatic capacitance C_el is determined by the interconnect dimension and relative dielectric constant ε_r of medium material, and can be derived as [29],

$$C_{el}={\epsilon }_r{\epsilon }_{0}M\left[\tanh \left(\frac{\pi W}{4T_{ox}}\right)\right],$$

(9)

where M [.] can be described as [30],

$$M[k]=\{\begin{array}{cc}\frac{2\pi }{\ln \left((2+2\times \sqrt[4]{1-k^2})/(1-\sqrt[4]{1-k^2})\right)}\hspace{1em}& 0\le k\le \frac{1}{\sqrt{2}}\\ \frac{2}{\pi }\ln \left(\frac{2+2\sqrt{k}}{1-\sqrt{k}}\right)& \frac{1}{\sqrt{2}}\le k\le 1\end{array}.$$

(10)

Therefore, the p.u.l. distributed capacitance C_E can be calculated as,

$$C_E={\left(\frac{1}{C_{eq}}+\frac{1}{C_{el}}\right)}^{-1}.$$

(11)

The distributed inductance L_E of MLGNR interconnects consists of the equivalent kinetic inductance L_eq and magnetic inductance L_ma, and their relationship can be described as in Equation (12). Similarly, the p.u.l. equivalent kinetic inductance L_eq also can be computed by using a recursive method as [1,17],

$$L_E=L_{eq}+L_{ma}=L_{eq}+\frac{{\mu }_0{T}_{ox}}{W},$$

(12)

$$L_{{}_{rec}}^1=L_k=h/4e^2{v}_f{N}_{ch},$$

(13)

$$L_{{}_{rec}}^i={\left(\frac{1}{L_{{}_{rec}}^{i-1}+L_m}+\frac{1}{L_k}\right)}^{-1},$$

(14)

$$L_{eq}=L_{{}_{rec}}^{N_{layer}}.$$

(15)

Herein, L_k represents the p.u.l. kinetic inductance of monolayer GNR. L_m denotes the p.u.l. coupling inductance between successive GNR layers and can be expressed as L_m = µ₀δ/W (here µ₀ = 8.854 × 10⁻¹² is the vacuum magnetic permeability).

As illustrated in Figure 2, the impacts of mutual inductance M_m and coupling capacitance C_c on the dynamic and functional crosstalk of the coupled MLGNR interconnects are taken into consideration. The analytical expressions of p.u.l. M_m and C_c are defined as follows [13],

$$M_m=\frac{{\mu }_0}{2\pi }\ln \left(\frac{2}{S+W}-1\right),$$

(16)

$$C_c=\frac{0.5}{1+{\left(\frac{S}{(T_{gnr}+T_{ox})}\right)}^2}{C}_{\left[BCP\right]}\left(\frac{T_{gnr}}{S/2},\frac{2T_{ox}}{S/2}\right)+\frac{0.87}{1+{\left(\frac{S/2}{(T_{gnr}+T_{ox})}\right)}^2}{C}_{\left[CP\right]}\left(\frac{W}{S}\right).$$

(17)

Herein C_[BCP](z, y) and C_[CP](z) are written as below,

$$C_{\left[BCP\right]}(z,y)=\frac{{\epsilon }_r{\epsilon }_0}{2}M\left[K_{\left[BCP\right]}(z,y)\right],$$

(18)

$$C_{\left[CP\right]}(z)=\frac{{\epsilon }_r{\epsilon }_0}{4}M\left[K_{\left[CP\right]}(z)\right],$$

(19)

wherein the function M [.] is shown in Equation (10), K_[BCP](z,y) and K_[CP](z) are given in reference [30].

3. Crosstalk Delay Model

Based on the single-line delay model [31], we derived a 50% crosstalk delay model of the coupled MLGNR interconnects considering the in-phase crosstalk and out-of-phase crosstalk as,

$$T_{delay}=(1.48\xi +e^{-2.9{\xi }^{1.35}})\sqrt{L_T{L}_{gnr}\left(C_T{L}_{gnr}+C_l\right)}.$$

(20)

Here, the total equivalent capacitance C_T = C_E + (1 − β)C_c, the total inductance L_T = L_E + βM_m and $\xi =\frac{1}{2}{\left(1+\frac{C_l}{C_T{L}_{gnr}}\right)}^{-0.5}\left[\begin{array}{l}(\frac{1}{2}{R}_E{L}_{gnr}+2R_{lu}+R_d)\sqrt{C_T/L_T}+\\ (R_E{L}_{gnr}+2R_{lu}+R_d)\sqrt{C_l{}^2/\left[L_T\times C_T\times L_{gnr}{}^2\right]}\end{array}\right]$.

The dynamic crosstalk consists of in-phase crosstalk and out-of-phase crosstalk schemes. For the switching factor β, β = 1 and β = −1 are introduced to distinguish the corresponding schemes, namely, the aggressor and victim lines switching in the same direction and opposite direction, respectively.

4. Crosstalk Noise Model

The ABCD parameter matrix for the MLGNR victim interconnect excluding the driver and load terminals under in-phase crosstalk and out-of-phase crosstalk schemes, respectively, can be expressed as,

$$\left[\begin{array}{cc}{A}_{in}& B_{in}\\ C_{in}& D_{in}\end{array}\right]=\left[\begin{array}{cc}\cosh ({\theta }_{in}{L}_{gnr})& Z_{in}\sinh ({\theta }_{in}{L}_{gnr})\\ \frac{\sinh ({\theta }_{in}{L}_{gnr})}{Z_{in}}& \cosh ({\theta }_{in}{L}_{gnr})\end{array}\right],$$

(21)

$$\left[\begin{array}{cc}{A}_{out}& B_{out}\\ C_{out}& D_{out}\end{array}\right]=\left[\begin{array}{cc}\cosh ({\theta }_{out}{L}_{gnr})& Z_{out}\sinh ({\theta }_{out}{L}_{gnr})\\ \frac{\sinh ({\theta }_{out}{L}_{gnr})}{Z_{out}}& \cosh ({\theta }_{out}{L}_{gnr})\end{array}\right],$$

(22)

where θ_in and θ_out are propagation constant of the MLGNR victim interconnect under in-phase crosstalk and out-of-phase crosstalk models, respectively. Similarly, Z_in and Z_out represent the corresponding characteristic impedance, respectively. They are given by,

$${\theta }_{(in,out)}=\sqrt{\left[R_E+s(L_E+\beta M_m)\right]\left[s(C_E+(1-\beta )C_c)\right]},$$

(23)

$$Z_{(in,out)}=\sqrt{\frac{R_E+s(L_E+\beta M_m)}{s(C_E+(1-\beta )C_c)}}.$$

(24)

Herein θ_(in,out) is composed of θ_in and θ_out cases, similarly, Z_in and Z_out are expressed as Z_(in,out). β = 1 is for the in-phase crosstalk scheme while β = −1 represents the out-of-phase crosstalk case.

Taking the effect of the driver terminals into account, the total ABCD parameter matrix of the MLGNR victim interconnect at different crosstalk models can be written by,

$$\begin{array}{l}\left[\begin{array}{cc}{A}_{T(in,out)}& B_{T(in,out)}\\ C_{T(in,out)}& D_{T(in,out)}\end{array}\right]=\left[\begin{array}{cc}1& R_d\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ s{C}_d& 1\end{array}\right]\left[\begin{array}{cc}1& \frac{R_{lu}}{2}\\ 0& 1\end{array}\right]\\ \left[\begin{array}{cc}\cosh ({\theta }_{(in,out)}{L}_{gnr})& Z_{(in,out)}\sinh ({\theta }_{(in,out)}{L}_{gnr})\\ \frac{\sinh ({\theta }_{(in,out)}{L}_{gnr})}{Z_{(in,out)}}& \cosh ({\theta }_{(in,out)}{L}_{gnr})\end{array}\right]\left[\begin{array}{cc}1& \frac{R_{lu}}{2}\\ 0& 1\end{array}\right]\end{array}$$

(25)

Being similar to the situation mentioned above (θ_(in,out) and Z_(in,out)), A_T(in,out) consists of A_T(in) and A_T(out) cases, which represents the parameters of the total ABCD matrix under in-phase crosstalk and out-of-phase crosstalk models. The meaning of parameters B_T(in,out), C_T(in,out), and D_T(in,out) are the same as the A_T(in,out) that contain two cases of in-phase crosstalk and out-of-phase crosstalk_. They can be solved by matrix computation as follows,

$$\begin{array}{l}{A}_{T(in,out)}=\frac{\left[\frac{1}{2}(s{C}_d{R}_d+1)R_{lu}+R_d\right]\sinh ({\theta }_{(in,out)}{L}_{gnr})}{Z_{(in,out)}}\\ +(s{C}_d{R}_d+1)\cosh ({\theta }_{(in,out)}{L}_{gnr})\end{array},$$

(25a)

$$\begin{array}{l}{B}_{T(in,out)}=\left[\frac{\left(\frac{1}{2}(s{C}_d{R}_d+1)R_{lu}+R_d\right)R_{lu}}{2Z_{(in,out)}}+(s{C}_d{R}_d+1)Z_{(in,out)}\right]\sinh ({\theta }_{(in,out)}{L}_{gnr})\\ +\left((s{C}_d{R}_d+1)R_{lu}+R_d\right)\cosh ({\theta }_{(in,out)}{L}_{gnr})\end{array},$$

(25b)

$$C_{T(in,out)}=\frac{\left(\frac{1}{2}s{C}_d{R}_{lu}+1\right)\sinh ({\theta }_{(in,out)}{L}_{gnr})}{Z_{(in,out)}}+s{C}_d\cosh ({\theta }_{(in,out)}{L}_{gnr}),$$

(25c)

$$\begin{array}{l}{D}_{T(in,out)}=\left[\frac{\left(\frac{1}{2}s{C}_d{R}_{lu}+1\right)R_{lu}}{2Z_{(in,out)}}+s{C}_d{Z}_{(in,out)}\right]\sinh ({\theta }_{(in,out)}{L}_{gnr})\\ +\left(s{C}_d{R}_{lu}+1\right)\cosh ({\theta }_{(in,out)}{L}_{gnr})\end{array}.$$

(25d)

Combining the total ABCD parameter matrix described in Equation (25), the relationship between the voltage and current of input–output ports for the MLGNR victim interconnect depicted in Figure 2, can be deduced as,

$$\left[\begin{array}{c}{V}_{vi}(s)\\ I_{vi}(s)\end{array}\right]=\left[\begin{array}{cc}{A}_{T(in,out)}& B_{T(in,out)}\\ C_{T(in,out)}& D_{T(in,out)}\end{array}\right]\left[\begin{array}{c}{V}_{vo}(s)\\ I_{vo}(s)\end{array}\right].$$

(26)

Substituting the expression of load capacitance I_vo = sC_lV_vo shown in Figure 2 into Equation (26), the transfer functions of the decoupled MLGNR victim interconnect under different phase modes are derived as,

$$H{(s)}_{in}=\frac{1}{A_{T(in)}+s{C}_l{B}_{T(in)}}={\left(1+{\displaystyle \sum _{i=1}^5(a_i{s}^i)}\right)}^{-1},$$

(27)

$$H{(s)}_{out}=\frac{1}{A_{T(out)}+s{C}_l{B}_{T(out)}}={\left(1+{\displaystyle \sum _{i=1}^5(b_i{s}^i)}\right)}^{-1}.$$

(28)

Herein, in order to ensure the signal integrity characteristics at the output port of MLGNR victim interconnect, the transfer functions are approximated by adopting a fifth-order pade’s expansion.

The crosstalk noise is usually defined as the functional crosstalk. It can be defined that the aggressor line switches from logic 0 to logic 1 while the victim line keeps in a quiescent state of logic 0. Based on the principle of the functional crosstalk, a peak voltage will be observed at output port of the victim line when the aggressor line switches owing to the coupled crosstalk. The noise output of the MLGNR victim interconnect induced by the switching of aggressor line can be obtained as follows [32],

$$V_{noise}(s)=\frac{1}{2}{V}_{agg}(s)\left(H{(s)}_{in}-H{(s)}_{out}\right),$$

(29)

where V_noise(s) represents the noise output signal of the MLGNR victim interconnect in the Laplace domain. Here the input port of MLGNR aggressor line is defined into an ideal step-response signal V_agg(s) = 1/s. We can obtain the noise output signal in the time domain by applying the inverse Laplace transform for the Equation (29) as,

$$V_{noise}(t)=L^{-1}\left[\frac{1}{2}{V}_{agg}(s)\left(H{(s)}_{in}-H{(s)}_{out}\right)\right].$$

(30)

5. Results and Discussions

This section investigates the impacts of different dielectric materials on crosstalk delay, transfer gain, and noise output signal of coupled MLGNR interconnect at global level of 7 nm technology node. All geometrical and physical electrical parameters were extracted from references [33,34] as follows, W = 11.5 nm, T_gnr = 26.91 nm, S = 11.5 nm, T_ox = 17.25 nm, E_f = 0.3 eV, ε₀ = 1.95 × 10⁻¹¹ F/m, µ₀ = 4π × 10⁻⁷, P = 0, R_d = 20.51 kΩ, C_d = 0.063 fF, C_l = 0.2 fF. Herein, R_d, C_d, and C_l are the equivalent values of minimum-sized gate. In general, the sizes of driver and load are 100 times larger than that of the minimum-sized gate at global level (100 µm ≤ L_gnr ≤ 10 mm) interconnects [35], then their values can be rewritten as, R_d’ = R_d/100, C_d’ = C_d × 100 and C_l’ = C_l × 100. All the numerical simulation results presented in the next section were obtained by carrying out the MATLAB R2013a.

In order to compare the impacts of different dielectric mediums on delay time of coupled MLGNR interconnect, the crosstalk delay of victim line versus interconnect length under different phase modes were obtained by the Equation (20), as displayed in Figure 3.

As shown in Figure 3, it can be observed that the crosstalk delay for the coupled MLGNR interconnect with the nanoglass (ε_r = 1.3) as the dielectric medium is less than that of the conventional SiO₂ dielectric medium under in-phase and out-of-phase modes. Taking the interconnect length of L_gnr = 2000 µm as an instance, the delay time under out-of-phase mode for the nanoglass medium is 6.886 ns while for the SiO₂ dielectric medium is 18.115 ns. Similarly, the delay time at in-phase mode for nanoglass and SiO₂ dielectric mediums are 2.277 ns and 4.286 ns in the same length as the former, respectively. The reason behind this is that the electrostatic capacitance and coupling capacitance will reduce as the relative dielectric constant ε_r decrease, thereby, leading to a lesser total equivalent capacitance C_T. In combination with Equation (20), it is evident that the crosstalk delay is approximately in positive proportion with the total equivalent capacitance C_T. In addition, according to our numerical simulation results, the maximum difference of delay time between SiO₂ and nanoglass dielectric mediums can reach to 25.202 ns for an interconnect length of L_gnr = 3000 µm at the out-of phase crosstalk. Thus, replacing the traditional SiO₂ with the ultra-low-k dielectric material is an efficient way to reduce crosstalk delay of coupled interconnects.

Moreover, it is clearly shown from Figure 3 that the crosstalk delay of coupled MLGNR interconnects at out-of-phase mode is significantly higher than that of in-phase crosstalk mode for all dielectric materials. Giving the interconnect length of L_gnr = 2500 µm as an example, the delay time for SiO₂ dielectric medium under in-phase and out-of-phase modes are 6.308 ns and 27.883 ns. The corresponding values for p-SiCOH dielectric medium are 4.525 ns and 17.802 ns, and the case for nanoglass dielectric medium are 3.173 ns and 10.364 ns. This can be explained by the Miller coupling capacitance, which only exists in out-of-phase crosstalk mode, causing the total equivalent capacitance of the MLGNR victim interconnects under out-of-phase mode to be greater than that of in-phase mode.

Figure 4 shows the frequency response of MLGNR victim line with the interconnect length L_gnr = 1000 µm under in-phase and out-of-phase modes for different dielectric materials. Transfer gain represents the magnitude of frequency response of the interconnect system and is the ratio of amplitude between the output and input signal at different frequencies. The transfer gain under in-phase and out-of-phase can be obtained by the Equations (27) and (28), respectively. As shown in Figure 4, in the high frequency region, it is obvious that the transfer gain increases as the relative dielectric constant ε_r decreases for in-phase and out-of-phase modes. This is due to the fact that the coupled MLGNR interconnects system can be considered as the RC low pass filter and its cut-off frequency is approximately expressed as: 1/(2π × C_T × R_E) [16], and the total capacitance C_T of the victim line decreases with the decrease of ε_r. Thus the MLGNR victim interconnect for using the nanoglass has a larger cut-off frequency compared with the p-SiCOH and SiO₂ cases.

Moreover, it can be found from Figure 4 that transfer gain of coupled MLGNR interconnects under in-phase mode is evidently greater than that of out-of-phase crosstalk mode for any dielectric materials. The reason for this phenomenon is that the total capacitance of the victim MLGNR interconnect under out-of-phase mode is larger than that of in-phase mode. Therefore, the former will have a lesser cut-off frequency compared with the latter.

Based on the Equation (30), the effect of different dielectric mediums on crosstalk noise of victim MLGNR interconnect is illustrated in Figure 5a, meanwhile the peak noise voltage regarding the indispensable noise parameter for different dielectric mediums versus interconnect length is described in Figure 5b.

It is remarkable from Figure 5a that the peak noise voltage will decrease with the decrease of the relative dielectric constant ε_r. For instance, the peak noise voltage with the interconnect length L_gnr = 1000 µm for SiO₂, p-SiCOH and nanoglass dielectric mediums are 0.270 V, 0.234 V, and 0.183 V, respectively. It can be explained that the crosstalk noise is induced by the coupling capacitance C_c and mutual inductance M_m existing on the position between aggressor and victim MLGNR interconnect, as shown in Figure 2. On the other hand, it is noteworthy that the Miller coupling capacitance is the dominant factor of forming the crosstalk noise. Certainly, there is no doubt according to the Equations (17)–(19) that a lesser relative dielectric constant ε_r can result in a smaller Miller coupling capacitance.

Moreover, as shown in Figure 5b, it is obviously found that the peak noise voltage of coupled MLGNR interconnect increases as the interconnect length increases for all different dielectric mediums. This is due to the fact that the increases of interconnect length give rise to Miller coupling capacitance. In addition, it is shown from Figure 5b that applying the ultra-low-k dielectric material to reduce the peak noise voltage is very obvious at global level interconnects. Based on the numerical simulation results, the maximum difference of peak noise voltage between the SiO₂ and nanoglass dielectric mediums can reach to 0.102 V when the interconnect length L_gnr = 1000 µm, and the corresponding value for the minimum difference is 0.038 V when the interconnect length is chosen as L_gnr = 6000 µm.

6. Conclusions

Based on the transmission line model, an equivalent distributed circuit of coupled MLGNR interconnects was established and the coupling capacitance and mutual inductance were taken into consideration. By using the extracted parameters, the impacts of different dielectric materials on crosstalk delay, noise output voltage, and transfer gain were predicted. The numerical simulation results showed that substituting the conventional SiO₂ dielectric medium with the ultra-low-k dielectric material for the coupled MLGNR interconnects has a greater performance advantage in terms of the crosstalk delay, noise output, and transfer gain at the same conditions. Furthermore, it was found that the coupled MLGNR interconnect under out-of-phase mode has a greater crosstalk delay and a lesser transfer gain compared with under in-phase mode for all dielectric material, and the peak noise voltage of coupled MLGNR interconnect increases as the interconnect length increases. In the light of our simulation results, it can be expected that the ultra-low-k dielectric materials may be an emerging technology to improve the performance of crosstalk delay, noise output and transfer gain of the coupled MLGNR interconnects.