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In this work, we have studied Joule heating in carbon nanotube based very large scale integration (VLSI) interconnects and incorporated Joule heating influenced scattering in our previously developed current transport model. The theoretical model explains breakdown in carbon nanotube resistance which limits the current density. We have also studied scattering parameters of carbon nanotube (CNT) interconnects and compared with the earlier work. For 1 µm length single-wall carbon nanotube, 3 dB frequency in S_{12} parameter reduces to ~120 GHz from 1 THz considering Joule heating. It has been found that bias voltage has little effect on scattering parameters, while length has very strong effect on scattering parameters.

The current complementary metal oxide semiconductor (CMOS) technology in nm- and sub-nm node for very large scale integration (VLSI) is facing challenges due to performance limitation of Cu/low-k dielectric material as an interconnection, because of increased resistivity of Cu, electromigartion and void formation [

Though carbon nanotube has high thermal conductivity, it has been observed experimentally that the conducting carbon nanotube breaks down due to Joule heating which thus limits its current density [

The conducting single-wall, multi-wall and bundle of single-wall carbon nanotube have been considered for possible replacement of Cu/low-k dielectric interconnects used in current CMOS technology depending upon the type of the interconnect, such as whether local or global. In the present work, we have considered Joule heating in a metallic single-wall carbon nanotube for better understanding of breakdown and also for analytical simplicity.

In our earlier work [

Single wall carbon nanotube.

The motion of electrons across the SWCNT in a 1D fluid model can be described by the following equation [_{0} is equilibrium three-dimensional electron density and σ is conductivity. Electron-electron repulsion factor is described by α. The thermodynamic speed of sound is _{e}; _{z} is electric field and

We have modified the relaxation frequency (

In Equation (2), _{F} is Fermi velocity of electron and _{eff}_{op}_{ac}

These two scattering lengths depend on diameter of the carbon nanotube (_{op}_{,abs} has been modeled by the following equation [

In Equation (5), _{op}_{op,abs}

In Equations (5), (7) and (8), _{op}_{B} in Equation (8) is the Boltzmann constant. Carrier scattering path due to optical emission under the influence of electric field (

Assuming SWCNT as a good conductor and considering only one-dimensional flow we can deduce the current density from Equation (1) which is given by,

Combining Equations (12) and (13), we obtain,
_{K} is kinetic inductance in per unit length and _{Q} is quantum capacitance in per unit length. The SWCNT interconnect can then be better explained as a transmission line shown in

In Equation (18), _{0} is quantum resistance taken as 12.906 kΩ and _{M} is described by the Equation (20) as the magnetic inductance.

In thermal modeling of SWCNT interconnect material, _{eff}

Transmission line model of SWCNT interconnect.

Temperature profiles along the SWCNT length can be numerically solved from steady state heat equation given by Carslaw and Jaeger [^{−1} K^{−1}, have been used [

Consider differential length of CNT and calculate mean free path as well as differential resistance for differential element using Equations (10) and (19). Estimate total resistance of SWCNT by summing all differential resistances.

Calculate current from Equation _{c}), where _{c} is the contact resistance 30 KΩ [

Calculate ^{2}

Use current temperature profile as the initial temperature for next iteration. Repeat steps 1 to 4 until convergence is obtained.

_{eff}

Temperature profile of SWCNT of 2 µm length.

Power dissipation due to Joule heating along the SWCNT length.

Using Equations (16)–(18), we have calculated kinetic inductance, _{K} = 3.6 nH/µm, quantum capacitance, _{Q} = 90 aF/µm and electrostatic capacitance _{E} = 70 aF/µm of SWCNT transmission line to study S-parameters. Typically, a SWCNT diameter is ~1 nm and oxide thickness over which SWCNT is deposited is ~100 Å. The calculated value of _{M} ~1 pH/μm which is very small compared to the value of _{K}. The calculated kinetic inductance is consistent with the value calculated in [_{11} is the ratio of power reflected from the transmission line to the incident power. Scattering Parameter S_{12} is the ratio of power transmitted through the transmission line to the incident power. Two port network parameters S_{11} and S_{12} have been calculated considering lumped elements and normalized by 50 ohm impedance. Although transmission line is a distributed device, we have used lumped element model to calculate S-parameters for the sake of efficient computation. _{11} and S_{1}_{2} parameters of SWCNT at 0.1 V bias voltage. At higher frequencies inductive and capacitive terms dominate over the resistive term which results in an oscillatory behavior of S-parameters as observed in _{11} parameter exhibits an oscillatory behavior for 100 µm long interconnect above 10 GHz in the frequency range studied. _{12} parameter shows an oscillatory behavior for both 10 µm and 100 µm long interconnects above 70 GHz and 7 GHz, respectively. However, the short interconnect of 1 µm length does not show any oscillatory behavior for S_{11} and S_{12} parameters.

Plot of S_{11} parameter of SWCNT interconnects at 0.1 V bias voltage.

Plot of S_{12} parameter of SWCNT interconnects at 0.1 V bias voltage.

_{1}_{2} parameter is within −3 dB of its maximum value at 0.1 V. It is noticeable from

S_{1}_{2} parameters of SWCNT.

Length of SWCNT (µm) | Band Width (GHz) without Scattering [ |
Band Width (GHz) with Scattering |
---|---|---|

1 | 1000 | 120 |

10 | 110 | 11 |

100 | 30 | 1.0 |

_{12} on bias voltages and shows an increase in S_{12} with increased bias voltage.

Plot of SWCNT resistance

S_{12} parameter of SWCNT interconnects at different bias voltages.

In this work, we have incorporated Joule heating induced phenomenon in 1D fluid model of CNT interconnects. We have studied scattering parameters of SWCNT for short, local and global interconnect lengths with different biasing voltages. We have observed that the bias voltage does not greatly affect the scattering parameters; on the other hand it significantly influences Joule heating. The breakdown shown in resistance

Part of the work is supported by the United States Air Force Research Laboratory under agreement number FA9453-10-1-0002. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon.