Equivalent Single Conductor Model Research Articles

Analysis of Delay and Dynamic Crosstalk in Spatially Arranged Mixed CNT Bundle Interconnects at Different Technology Nodes

In the present nanoscale regime, mixed carbon nanotube bundles (MCBs) are considered to be highly promising interconnect options. This research paper introduces a spatially arranged mixed carbon nanotubes (CNTs) bundle (MCB), wherein single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) occupy equal halves in the MCB. An equivalent single conductor (ESC) model for MCB is employed to analyze the interconnect performances in terms of signal transmission delay and dynamic crosstalk delay at different technology nodes (i.e., 32nm, 22nm, and 16 nm). Encouragingly, a significant reduction in signal transmission delay and dynamic crosstalk induced delay are observed at 32 nm technology node. It is observed that at 32 nm technology node, the propagation delay and crosstalk induced delay significantly improves by 29.40% and 55.53%, respectively, compared to 22 nm technology node and 187.88% and 185.94%, respectively, compared to 16 nm technology node. The improvement in interconnect performances can primarily be attributed to the improvement in the number of conducting channels inside the MCB at 32 nm, which greatly impacted the interconnect parasitics such as quantum resistance, quantum capacitance, kinetic inductance etc.

Analytical Delay Model and Stability Analysis for MLGNR Interconnects

This paper analyzes the signal delay and relative stability of multilayer graphene nanoribbon (MLGNR) interconnects dependent on temperature at global lengths. A thermally aware driver interconnect load (DIL) system, constituted by equivalent single conductor (ESC) model along with mathematical equations, is proposed to convert multi-conductor transmission circuit into single transmission line of MLGNR. The temperature-dependent analytical delay model of MLGNR is evaluated and the obtained outcomes are compared with the simulated results. The analytical and simulated results are obtained at global interconnect length (2000-[Formula: see text]m) for 32[Formula: see text]nm, 22[Formula: see text]nm and 16[Formula: see text]nm nodes of technology under 200–500[Formula: see text]K temperature range of MLGNR. The simulation and analytical results reveal that the outcomes of the two models correspond well. The trend of the models shows the increase in delay with the rising temperature levels (200–500[Formula: see text]K) for three different nodes of technology. Further, relative stability analysis of MLGNR as interconnect line at three different technological nodes from 500–2000-[Formula: see text]m lengths is examined.

A Bilevel Multi-Fidelity Polynomial Chaos Approach for the Uncertainty Quantification of MWCNT Interconnect Networks With Variable Imperfect Contact Resistances

With on-chip copper interconnects reaching their performance limits at 22 nanometer technology nodes, multi-walled carbon nanotube (MWCNT) interconnects are projected to replace them below this point. A major aspect of MWCNT interconnect design is to perform uncertainty quantification (UQ) in an efficient yet accurate manner. In this paper, a polynomial chaos (PC) based approach is developed for the UQ of MWCNT interconnect networks under the condition that some shells of each conductor of the network are perfectly contacted while others are imperfectly contacted. The key feature of the proposed approach is the development of a bilevel multi-fidelity algorithm where two different low-fidelity models are combined together. The main outcome of using this bilevel approach is to further reduce the computational time cost of state-of-the-art single level multi-fidelity algorithms, especially in the presence of variable imperfect contact resistances where single level multi-fidelity algorithms fail to provide much speedup over conventional PC approaches. The proposed approach adopts a SPICE hybrid model that combines the features of the equivalent single conductor (ESC) model and the rigorous multiconductor circuit (MCC) model of the MWCNT conductors. Then the low-fidelity ESC model, the intermediate-fidelity hybrid model, and the high-fidelity MCC model are exploited in a bilevel multi-fidelity algorithm for the recovery of the PC metamodel of the interconnect network. This proposed bilevel multi-fidelity algorithm is demonstrably 3-5x more numerically efficient than state-of-the-art single level multi-fidelity algorithms while being even more accurate. Once recovered, the PC metamodel is used to derive all statistical information of the network transient responses.

Transient Analysis of Hybrid Cu-CNT On-Chip Interconnects Using MRA Technique

This paper presents the transient analysis of the equivalent single conductor (ESC) model of hybrid Cu-CNT on-chip interconnects for nanopackaging using matrix rational approximation (MRA) modeling technique. The analysis of propagation delay and peak crosstalk noise is carried out for single and coupled Cu-CNT interconnect lines at 14 nm and 22 nm technology nodes. It has been observed that the proposed MRA model provides a speed-up factor of 131 compared to the HSPICE. An error of less than 1% confirms the accuracy of the proposed model compared to the SPICE simulations. It is observed that Cu-CNT lines are more immune to the crosstalk due to lesser coupling effects compared to Cu and CNT interconnects. The efficacy, accuracy, and comprehensive analysis using the proposed model ensures immense application possibility of the proposed model in the VLSI design automation tools at the nanopackaging level.

Modeling and Analysis of Cu-Carbon Nanotube Composites for Sub-Threshold Interconnects

The sub-threshold regime is suited for applications requiring ultra-low power consumption with low to medium frequency (tens to hundreds of MHz) of operation. Therefore, this paper presents electrical modeling and comprehensive analysis of copper-carbon nanotube (Cu-CNT) composite interconnects for sub-threshold circuit design. At lower operating frequencies, the effective complex conductivity of Cu-CNT composites in the nanoscale is formulated by developing an analytical model. Based on the proposed equivalent single conductor model, the frequency-dependent resistance and inductance of composite interconnects are computed. Finally, the sub-threshold crosstalk effect, transfer gain, and Nyquist stability of coupled Cu-CNT composite interconnect are analyzed using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ABCD</i> matrix approach.

Dielectric surface roughness scattering induced crosstalk performance of coupled MCB interconnects

A CMOS inverter driven equivalent single conductor model of capacitively coupled mixed CNT bundle (MCB) interconnects are taken to analyze the crosstalk induced time domain and frequency domain behaviors at 14 nm technology node for a temperature range (300–500 K). Four possible structures of MCB (e.g.S1, S2, S3 & S4) are analyzed by considering the dielectric substrate materials induced scattering. The most typical and consistent dielectric materials like silicon dioxide (SiO2), boron nitride (BN) and silicon carbide (SiC) are taken into account for studying the behavior of MCB depending upon dielectric surface roughness (DSR) scattering. A complete crosstalk (dynamic & functional) effect in terms of positive peaks, time duration and area of rise glitch is studied. The results revealed that MCB structure ‘S4’ is having less delay and more 3-dB frequency bandwidth in comparison to other structures of MCB. Moreover, S4 placed on SiC substrate material is depicting best results among other substrate materials. The bode stability of ‘S4’ on smooth surface and rough surface material is analyzed to validate the performance of ‘S4SiC’ in terms of phase and gain margin. It is also observed that the 3-dB frequency bandwidth of ‘S4SiC'reduces with increase in temperature. Consequently, the amplitude of surface roughness parameter (δsr) plays an important role to analyze the performance of coupled interconnects of MCB in deep sub-micron (DSM) regime.

Performance analysis of MCB-based VLSI interconnects depending on scattering induced by substrate surface roughness

An equivalent single-conductor model is developed for all possible structures of mixed carbon nanotube bundle interconnects at the 14-nm technology node. The results are analyzed by including the effect of scattering caused by the surface roughness of the dielectric substrate material in the temperature range from 300 to 500 K. Widely used dielectrics such as silicon dioxide (SiO2), boron nitride (BN), and silicon carbide (SiC) are compared with respect to their mean free path due to surface roughness scattering. The dynamic and functional crosstalk-induced delay, crosstalk-induced noise area, and frequency spectra of all the structures of mixed CNT bundles (MCB) are analyzed, including the effect of scattering caused by substrate surface roughness, over the complete temperature range from 300 to 500 K. The results reveal that MCB structure 4 (S4) on the SiC substrate outperforms all the other structures built on the other dielectric materials. Also, the results for S4 show a fourfold higher crosstalk-induced delay when built on each substrate material, viz. BN, SiO2, and SiC, as compared with a smooth surface over the complete range of temperature from 300 to 500 K. The crosstalk-induced noise area (CINA) for structure S4 is almost five times higher on the investigated substrate materials (BN, SiO2, and SiC) compared with a smooth surface. Moreover, the bandwidth of S4 on SiC (S4 SiC) is found to be greater than on the other substrate materials (SiO2 and BN) under the dynamic signal switching condition.

Performance analysis of mixed CNT bundle interconnects at 10 nm technology

In recent past, the cross-coupling crosstalk becomes a dominating factor due to the closer proximity of wire that reduces the performance of coupled interconnects at lower technology. To overwhelm interconnect problems, this work demonstrates a comprehensive study of unshielded and active shielded spatially arranged mixed carbon nanotube (CNT) bundle (SMCB) and randomly distributed mixed CNT bundle (RMCB) interconnects at 10 nm technology. Using a driver-interconnect-load setup, a unique multi-conductor transmission line and an equivalent single conductor model is proposed considering the impact of different CNT diameters with their associated line and coupling parasitics. A resistive and CNT field-effect transistor (CNTFET) driver model is considered at 10 nm technology to demonstrate the impact of single line delay, cross-coupling delay, and power dissipation for the densely packed bundle at global lengths. It is observed that a CNTFET-based realistic RMCB exhibits on an average 29.19 and 39.56% reduced single line delay and power dissipation, respectively compared to different SMCB configurations at 700 µm interconnect lengths. Moreover, a shielded RMCB encouragingly provides an improved immunity of cross-coupling impact for the on-chip interconnects at 10 nm technology. Therefore, from fabrication and modelling aspects, a randomly distributed MCB can be proved as emerging interconnect for next-generation on-chip applications.

Temperature-Dependent Circuit Modeling and Performance Evaluation Due to Crosstalk in Capacitively Coupled Interconnects of Intercalation-Doped Multilayer Graphene Nanoribbon

The present work explores the temperature-dependent performance evaluation due to crosstalk in coupled interconnects of intercalation-doped multilayer graphene nanoribbons (ID-MLGNR). A temperature-dependent equivalent single conductor (ESC) model of arsenic pentafluoride (AsF5) ID-MLGNR is considered and is employed to analyze its crosstalk performance (i.e., crosstalk-induced noise and crosstalk-induced delay). Other metallic interconnect materials viz. mixed carbon nanotube bundles (MCB) and copper (Cu) are also analyzed in a similar manner. The temperature-dependent crosstalk performance of ID-MLGNR is compared with that of MCB and Cu, at different temperatures from 300 to 500 K. For the specified temperature range, in case of crosstalk-induced noise, the AsF5 ID-MLGNR-based long interconnects demonstrate superior performance to its MCBs and Cu counterparts. Moreover, the temperature-dependent crosstalk-induced delay is less with ID-MLGNR in comparison with both MCB and Cu interconnects. The large values of the mean free path (MFP) and conductivity of AsF5 ID-MLGNR contribute to its improved performance. In addition, the impact of variation in MFP, interlayer spacing and Fermi level is studied on the crosstalk delay performance of ID-MLGNR. From the obtained results, it is revealed that variation in these parameters results in the significant variation of 62 ps, 51 ps and 169 ps, respectively, in the odd mode (OM) crosstalk delay. Hence, for emerging nano-technology nodes (e.g., 14 nm), ID-MLGNR interconnects with its improved crosstalk performance are proposed as a potential alternative to its MCBs and Cu counterparts.

Analysis of Temperature‐Dependent Functional and Dynamic Crosstalk Noise in Adjacent Interconnects of Doped Multilayer Graphene Nanoribbon with Armchair and Zigzag Edges

Multilayer graphene nanoribbon (MLGNR) is a potential candidate in nanoscale very large scale integration (VLSI) interconnects. A detailed analysis emphasizing the impact of GNR's edge shape in an armchair (ac) and zigzag (zz) structures on crosstalk performance considering undoped and doped (intercalated with AsF5 and FeCl3) MLGNR interconnects is presented. A capacitively‐coupled driver‐interconnect‐load (DIL) line configuration is used to analyze both the functional and dynamic crosstalk at 14 nm technology node for global interconnects. A temperature‐dependent equivalent single conductor (TD‐ESC) model is considered. It is observed that over a temperature range from 300 to 500 K, crosstalk‐induced low noise peaks in doped MLGNRs are obtained with zz‐edges as compared with undoped MLGNR, whereas, the time duration of crosstalk‐induced noise is small for doped MLGNRs with ac‐edges. Similar findings are obtained in terms of propagation delay and crosstalk‐induced delay for single and coupled interconnects, respectively, that is, the delays obtained in both kinds are small for doped MLGNR with ac‐edges. It is also observed that AsF5‐doped MLGNR outperforms FeCl3‐doped MLGNR in terms of crosstalk‐induced noise and delay. The results of MLGNR are also compared with the mixed carbon‐nanotube bundle (MCB).

A Predictor–Corrector Algorithm for Fast Polynomial Chaos-Based Uncertainty Quantification of Multi-Walled Carbon Nanotube Interconnects

In this article, a predictor–corrector algorithm for the fast polynomial chaos (PC)-based uncertainty quantification (UQ) of multi-walled carbon nanotube (MWCNT) interconnect networks is presented. The proposed algorithm intelligently combines the numerical efficiency of the approximate equivalent single conductor (ESC) model of the MWCNT interconnect network with the rigor and accuracy of a multi-conductor circuit (MCC) model. Consequently, this algorithm significantly accelerates the generation of the PC surrogate models (or metamodels) of the network responses for minimal loss in accuracy. These metamodels can be probed efficiently and repeatedly to quantify the impact of manufacturing and fabrication process uncertainty on the MWCNT network responses.

Transient Analysis of Crosstalk Induced Effects in Mixed CNT Bundle Interconnects Using FDTD Technique

The nanoscale regime in current IC technology considers mixed carbon nanotube bundles (MCBs) as a potential on-chip interconnect alternative to the conventional copper interconnects. This paper presents a temperature-dependent equivalent single conductor (ESC) model for on-chip MCB interconnect structure. A finite-difference time-domain (FDTD) technique is used to model the Telegrapher's equations. To the best of authors' knowledge, it is for the first time that temperature-dependent ESC modeling and crosstalk for coupled-MCB interconnect line structures have been reported using FDTD. The proposed numerical model is highly time as well as memory efficient and accurate to analyze the crosstalk induced propagation delay, and peak voltage for different cases of input switching. The performance parameters obtained for the proposed model are validated with HSPICE simulations. The crosstalk induced propagation delay shows less than 1% error for the proposed model and HSPICE simulations. The robustness of the proposed model is verified against a range of test cases. The proposed model is of high importance as it can be used to determine the crosstalk induced performance metrics for on-chip MCB interconnects.

Dielectric Surface Roughness Scattering Limited Performance of MLGNR Interconnects

Based on an equivalent single conductor model incorporating surface roughness scattering, the effect of the surface roughness of an underlying dielectric material on the performance of multilayer graphene nanoribbon (MLGNR) interconnects with side contacts is investigated. The performance of MLGNR interconnects with different doping concentrations on the most common dielectrics [SiO2, boron nitride, and silicon carbide (SiC)] is compared with that of Cu counterpart in terms of the time delay and the bandwidth. Results show that the surface roughness of the dielectric material can severely degrade the performance of MLGNR interconnects; hence, it must be taken into account in design considerations. While MLGNR interconnects with perfect edges on the SiC substrate outperform Cu counterpart both in terms of time delay and bandwidth, for other substrate materials, the performance of MLGNR interconnects with respect to Cu counterpart depends on doping concentration and interconnect length.

Crosstalk analysis of coupled MLGNR interconnects with different types of repeater insertion

This paper investigates crosstalk (XT) analysis of emerging multi-layer graphene nanoribbons (MLGNR) as chip interconnect material with different types of repeater insertions. The performance of repeaters comprising of FinFET and carbon nanotube field effect transistor (CNFET) have been evaluated based on the equivalent single conductor (ESC) model. Using three-line bus architecture, the worst-case XT delay on the victim line for armchair (ac) and zig-zag (zz) based MLGNR interconnects have been computed for local/ intermediate and global levels. The detailed crosstalk investigations include in-phase (IP), out-of-phase (OP), and net XT induced delay. Also, the peak crosstalk noise voltage and average power consumption have been computed for MLGNR interconnects. The SPICE simulation results further indicate reduced crosstalk delay at local/intermediate levels for MLGNR based interconnects with degradation at global level. The reductions observed in worst-case XT induced delay for MLGNR interconnects with FinFET repeaters are 41.54, 34.57, and 19.71% and with CNFET repeaters are 47.47, 30.91, and 20.65% as compared to Copper (Cu) interconnects for interconnect lengths of 100, 500, and 1000 µm, respectively. Significant reductions in peak crosstalk noise voltage amplitude have been observed for local/intermediate level MLGNR interconnects as compared to Copper.

Analytical Study of Bundled MWCNT and Edged MLGNR Interconnects: Impact on Propagation Delay and Area

This research letter presents an accurate and efficient model for rough-edged multi-layered graphene nanoribbon (MLGNR) and multi-walled carbon nanotube (MWCNT) bundled interconnects. To address the impact of delay and area, an analytical finite-difference time-domain and real-time simulation based approach are considered. In order to obtain the propagation delay, an equivalent single conductor (ESC) model is accurately presented by simplifying a multi-conductor transmission line (MTL) of bundled MWCNT and MLGNR interconnects. The proposed equivalent ESC model is in good agreement to the MTL with an average deviation of 4.6% only. In order to validate the proposed model, the equivalent resistance, inductance, and capacitance values are compared with standard experimental results. It is observed that the model expression possesses a good agreement with experimental data with an average error of below 9.93%. Using the ESC-based driver-interconnect-load setup, an approximate number of MWCNTs in a densely packed bundle is obtained for a rough-edged MLGNR at 22 nm technology to obtain an equivalent delay. In order to acquire the equivalent propagation delay, a fewer number of MWCNTs in a bundle can provide similar performance as of MLGNR for below 200 μm interconnect length. Furthermore, the number of MWCNTs significantly increases for longer interconnects. Therefore, on an average, an edged MLGNR requires 97.98% lesser area compared to the bundled MWCNT for global interconnects. Hence, a rough-edged MLGNR can be treated as a most desired interconnect material as compared to MWCNT bundle for the future interconnect technology.

Crosstalk Delay and Stability Analysis of MLGNR Interconnects on Rough Surface Dielectrics

The crosstalk delay and frequency stability of MLGNR interconnects on dielectric materials with rough surfaces are investigated. Two interacting parallel interconnect lines (aggressor and victim) are considered. ABCD matrix model of two port networks is employed to obtain an equivalent single conductor (ESC) circuit model for decoupled victim line. The effect of substrate surface roughness is appropriately included in ESC model. The most common dielectric materials (SiO2, BN and SiC) are considered. By setting up a driver-interconnect-load (DIL) system the crosstalk delay and stability analysis of MLGNR and Cu interconnects are performed. Results show that the crosstalk delay and the stability of MLGNR interconnects oppositely response to the surface roughness of underlying dielectric material. Namely, dielectric materials with rougher surfaces translate to longer crosstalk delays while more stable systems. While MLGNR interconnects on SiO2 have longer crosstalk delay in compare to Cu counterpart, MLGNRs on BN and SiC have shorter crosstalk delay with respect to Cu ones. Therefore, to hold advantage of MLGNR interconnects on SiO2 over Cu counterpart either the SiO2 surface must be more precisely polished or MLGNRs with higher doping concentrations shall be used.

Performance Analysis of Mixed Carbon Nano Tubes as VLSI Interconnects

Background/Objectives: This paper presents various possible structures of Mixed Carbon Nano Tubes bundle which consists of different topology of Single Wall Carbon Nano Tubes (SWCNTs) and Multi wall Carbon Nano Tubes (MWCNTs). Methods/Statistical Analysis: It is assumed in this paper that SWCNTs and MWCNTs are densely packed. An equivalent single conductor model is derived for all the bundles of Mixed CNT. Findings: Performance is analyzed by considering parameters like Propagation delay, Power dissipation and Crosstalk delay of the interconnects having different lengths. Applications: Carbon Nanotubes finds applications in many fields such as biomedical applications, air and water filtration, structural applications etc.

Study of Crosstalk Effect on the Propagation Characteristics of Coupled MLGNR Interconnects

Multi-layer graphene nanoribbons (MLGNRs) material has been a potential solution to replace conventional Cu for next-generation on-chip interconnects. Based on equivalent single conductor model, this paper extracts the equivalent resistance-inductance-capacitance parameters for MLGNRs with consideration of edge roughness and Fermi level. A distributed circuit model for a pair of coupled MLGNR interconnects is provided, with both capacitive and inductive coupling taken into account, and validated against the Spice results and the predicted experiment results in frequency and time domains. Using the proposed model, the impact of various dimensional and technology parameters on the transfer gain and crosstalk delay is investigated for the global MLGNR interconnects at different phase modes. It is demonstrated that MLGNR interconnects with smooth edge exhibit higher transfer gain and lower crosstalk delay in comparison to its Cu counterpart at same dimension. However, edge roughness in the present fabrication technologies is inevitable that significantly deteriorates the propagation performance and the performance difference due to edge roughness is relatively less in wider MLGNR interconnects. Moreover, it is shown that side contact MLGNR interconnects have better electrical performance than that of top contact MLGNR interconnects at short interconnect length in comparison to that at long interconnects due to the domination of in-layer resistance. The results presented in this paper would be helpful to fully understand the propagation characteristics and provide guidelines for signal integrity analysis of MLGNR interconnects.

Meshless analysis of bilayer graphene nanoribbon for radio frequency interconnects

Numerical analysis of a bilayer graphene nanoribbon (BLGNR) using the reproducing kernel particle method (RKPM) – a true meshless approach – is reported. Electrostatic analysis of the BLGNR is performed using the RKPM approach. Validation of the structure is executed by comparing the result with an equivalent single conductor model and shows better accordance. Radio frequency (RF) analysis of the BLGNR as the interconnect is studied using advanced design systems with the quasi-static analysis derived through the RKPM method. The simulation results in a return loss of −89.69 dB and transmission loss of −0.00028 dB at 1 THz which is suitable for RF interconnect applications.

Time and Frequency Domain Analysis of MLGNR Interconnects

Multilayer graphene nanoribbons (MLGNRs) have potentially provided attractive solutions in an intensely growing researched area of interconnects. However, for MLGNR interconnects, the doping is inevitable since the conductivity of neutral MLGNR is much lower than even Cu. Therefore, a doped MLGNR can potentially exhibits smaller resistance in comparison to Cu wires. This paper analyzes and compares the power, delay, and bandwidth performance of Cu and doped MLGNR using an equivalent single conductor model. For similar dimensions, the overall delay and power dissipation of doped MLGNR is substantially smaller by 86.13% and 43.72%, respectively, in comparison to the Cu interconnects. Moreover, MLGNR demonstrates prominently improved bandwidth and relative stability at global interconnect dimensions. However, a narrow width MLGNR in a realistic scenario exhibits rough edges that significantly reduces the mean free path and, thereby, raises its resistance. Considering these facts, this paper for the first time analyzes and compares the performance of Cu and MLGNR interconnects with different edge roughness conditions.