HIGH-SPEEDNEXT GENERATIONMULTI-WALL CARBON NANOTUBEINTERCONNECT SIMULATORA COLLABORATIVE PROJECT BY

DALLAS MANN (CE) – HEAD OF PROGRAMMING

MIKE BRUNGARDT (EE) – HEAD OF RESEARCH

SUPERVISING PROFESSOR: SOURAJEET ROY

PRESENTATION OUTLINE

• Motivation of CNT• Moores Law

• Structural, Thermal, and Electrical Properties

• Applications

• Breaking down a CNT simulation• Modified Nodal Analysis/Circuit Stamps

• Resistor & Capacitor example

• Solving MNA Matrices• Frequency Doman

• Time Domain

• Connection to our project• MWCNT vs SWCNT

• MCC to ESC

• Results

• Steps for next semester• Increasing the number of conductors

• Different Dimensions

WHY CARBON NANOTUBES

• Moore’s Law – The observation that the amount

of transistors on a chip will double every two

years since 1965

• What was once in the matter of thousands, is now

in the multi billions, and still increasing

exponentially

• With more a more dense concentration of

transistors using roughly the same amount of

power innovations are needed in both power

consumption, and the thermal and electrical

properties of their interconnects

• To understand why CNT’s have these properties it is

important to understand their structure

CARBON NANOTUBES ARE FASCINATING:STRENGTH, THERMAL, AND ELECTRICAL PROPERTIES

STRENGTHWHAT

• CNT’s are one of the strongest

structures known to man at around

1TPa

• For reference, steal’s strength is

around 200 Gpa

WHY

• Carbon Nanotubes are mainly

composed of Sp2 bonds, which are

stronger then diamond’s Sp3 bonds

THERMAL CONDUCTIVITY

• Thermal Conductivity– [W per m-1K-1] the transport of energy in the form of heat though a body of mass as the result of a temperature gradient.

• Copper has the thermal conductivity of 385 Wm-

1K-1

• A SWCNT has the thermal conductivity of 3500 Wm-1K-1

ELECTRICAL CONDUCTIVITY• Electrical Conductivity – [S/m] the inverse of resistance, or how easy it is

for electricity to transfer across a structure or conduct electricity

• CNT’s display different properties depending on how the sheet is rolled.

• Armchair ɸ = 0

• Treated as metallic

• Can carry an electric Current density of 4GA/cm2

• Copper’s maximum current density is only 4MA/cm2

• Limited by electromigration

• Zig-Zag ɸ = 30

• Treated as though a semiconductor

• Chiral 0 < ɸ < 30

• Treated as though a semiconductor

APPLICATIONS: TODAY AND THE FUTURE

Current Applications

• Atomic-Force Microscopes

• CNT’s have been used to make probing tips

• Anasys Instruments

• Material Sciences

• Creates a composite material composed of epoxy bonded with

CNT that is 20%-30% stronger then other composite materials

• Amroy Europe Oy

• Tissue Engineering

• CNT’s have been used to act as a scaffolding for bone growth

• Rice University and Radboud University (Netherlands)

• Water treatment

• A MineralWater System using Nanomeshn Purification Technology

has created to filter water without the use of heat, chemicals, or

power

• Seldon Technologies

Future Applications:

• Batteries with 10x the life

• Silicon coated CNT in anodes for Lithium-Ion Batteries, preventing the

expansion of Silicon based anodes

• North Carolina State University

• Artificial Muscles

• Are 200x stronger then natural muscles with the same concentration

• University of Texas - Dallas

• Cancer Detection and Treatment• Interconnection of CNTs and gold nanoparticles to create a sensor to indicate the presence of oral Cancer

• In lab tests have been proven to destroy breast cancer tumors

• University of Connecticut

• Microelectronics

• Current copper interconnects are on the order of tens to hundreds of nanometers in

diameter

• One mode of failure in modern electronics is due to electromigration effects on these

copper interconnects

BREAKING DOWN THE SIMULATION

Motivation and Aim of the Simulator

• It is important to note that a carbon nanotube can be made

with a model composed of resistors, inductors, capacitors

known as a lumped sum model

• We wanted to make a simulation similar to PI-SPICE or

Cadence minus the GUI

• Didn’t want to waste time/resources/memory/complexity on

anything we did not need for this project

Internally SPICE is a two step process:

• First – Formulating all circuit elements taken in from a netlist and put

into a system of mathematical equations

• Depends on:

• Type of circuit Element

• Node in

• Node out

• Value of Circuit Element

• Second – Creating a solution for these equations

• Found using

• Initial Voltages

• Initial Currents

EXPLANATION BEHIND MODIFIED NODAL ANALYSIS

The laws of KCL and KVL produce:

• Basic Circuit elements in the Time domain:

• Resistors: v=Ri=i/g

• Capacitors: I = C dv/dt

• Inductors: v=L di/dt

• Basic Circuit elements in the Frequency

domain:

• Resistors: V=Ri=I/g

• Capacitors: I = sCV-CV0

• Inductors: V=sLI-LI0

• Nodal Formulation, made from Nodal

Analysis (KCL) can be easy to model

• Current going in is the same as going out

• This is the Creation of Modified Nodal

analysis, MNA for short

• Modified nodal Analysis can take any circuit

element (Resistors, inductors, capacitors, op-

amps, MOFSET, transistors, etc.)

• Turns each into a series of unique algebraic

elements

• This is called a stamp

HOW MNA MAKES STAMPS: RESISTORS

• Using KCL and Ohm’s Law we can write:

{G(Va – Vb) + …… = 0 X = [Va] ath Row

{G(Vb – Va) + …… = 0 [Vb] bth Row

• MNA Creates the following Matrices for any general

circuit:

[G][X] + [C] d[X]/dt = [B]

• [ G, -G ] [Va] satisfies G(Va - Vb) = 0

• [ -G, G ] [Vb] satisfies G(Vb - Va) = 0

• G, C, B, are the matrices where we insert each

components stamp

• G – non-derivative terms

• C – derivative terms

• B – Set DC Voltages

HOW MNA MAKES STAMPS: CAPACITORS

• Using KCL and Ohm’s Law we can write:

{C(d(Va - Vb))/dt + …… = 0 X = [Va] ath Row

{C(d(Vb - Va))/dt + …… = 0 [Vb] bth Row

• [ C, -C ] [d(Va/dt)] satisfies C(d(Va - Vb))/dt = 0

• [ -C, C ] [d(Vb/dt)] satisfies C(d(Vb - Va))/dt = 0

SOLVING IN THE FREQUENCY DOMAIN: INTRO

• [G][X] + [C][dx/dt] = B(t) Time Domain

• As we have seen sometimes Freq needed

• Get there by taking a Laplace transform

• GX(s)+(xS(s)-x0) = B(s) (G+sC)X(s) = B(s)

• Which turns to the linear system of equations:

AX = B

• One common approach to solve this is Gaussian

Elimination

• It is important to note that A

represents (G+sC) making

A-1 difficult for both us and

the computer

• A is frequency dependent

• 1000 frequency points would

take 1000 inversions of x

LU DECOMPOSITION

• LU Decomposition breaks up the A

matrix into 2 parts L and U

• L – Lower Triangular

• U – Upper Triangular

• O(n3)

• Todays Fastest algorithm is

between O(n3)O(n2.2)

FORWARDS - BACKWARDS SUBSTITUTION

• What do we do with the L and U

matrices?

• Ax=B can be written as LUx=B

• We can treat Ux as its own matrix y

• Our Ax=B is now Ly=B

• Forwards Substitution

• O(n2/2)

• With y known we can now solve Ux=Y

• Backwards Substitution

• O(n2/2)

TIME DOMAIN

• When working in the time domain we use linear

multistep methods to solve [G][X]+[C][dX/dt] = [B]

• Forward Euler (Explicit)

• dx(tk)/dt = (x(tk+1)-x(tk))/(tk+1-tk)

• Backward Euler (Implicit)

• dx(tk)/dt = (x(tk)-x(tk-1))/(tk-tk-1)

• Trapezoidal Rule (Implicit)

• 1/2 [dx(tk)/dt + dx(tk-1)/dt) = (x(tk)-x(tk-1))/(tk-tk-1)

• We only want to use Implicit solving methods

• Trapezoidal rule is guaranteed to be stable

• Implicit methods are more computationally expensive

• Increased computational complexity is a trade off for

increased accuracy

SOLVING IN THE TIME DOMAIN: BACKWARD EULER

• GX(tk) + C dX(tk)/dt = B(tk)

• GX(tk) + C (X(tk)-X(tk-1))/h) = B(tk)

• [G+C/h]X(tk) = B(tk)+C/hX(tk-1)

• Simple(r) version of other numerical integration techniques

• Less accurate then other implicit solving techniques

• Error propagates from the first time point to the last

SOLVING IN THE TIME DOMAIN: TRAPEZOIDAL

• GX(tk) + C dX(tk)/dt = B(tk)

• GX(tk-1) + C dX(tk-1)/dt = B(tk-1)

• (G/2)(X(tk)+X(tK-1)) + C X(tk)-X(tk-1)/n = (B(tk)-B(tk-1))/2

• [G/2 + C/h]X(tk) = (B(tk)+B(tk+1))/2 + (C/h – G/2)X(tk-1)

• More accurate than backward Euler

• More computationally complex

• 2nd order numerical integration technique

MULTIWALLED CARBON NANOTUBE (MWCNT)

• It is important to know that for our

project we are using a MULTI-WALL

carbon nanotube (MWCNT)

• Single Walled Carbon Nanotubes

SWCNT have an array of problems

• Random Chirality

• Brings unwanted semi conductive

properties

• Intrinsic Resistance of 6.5KΩ

• Hard to grow dense amount of bundles

• MWCNT’s can be described as many

concentric shells of cap-less SWCNT

• In our model we use up to 30 shells

BASING OUR RESEARCH

Part 1: Conducting Channels & RLC

• Modeling and Fast Simulation of

Multiwalled Carbon Nanotube Interconnects

• Min Tang, Member, IEEE, and Junfa Mao,

Fellow, IEEE

Part 2: Refining Previous paper + G & σ

• Circuit Modeling and Performance Analysis of

Multi-Walled Carbon Nanotube Interconnects

• Hong Li, Student Member, IEEE, Wen-Yan Yin,

Senior Member, IEEE, Kaustav Banerjee, Senior

Member, IEEE, and Jun-Fa Mao, Senior Member,

IEEE

MODELING MWCNT: EVERY RLC TO ACCOUNT FOR

• As shown a previous slide, a CNT can be represented with

a combination of Resistors, Capacitors and Inductors.

• Each shell has its own RLC

• There are two outside resistances just by connecting the

MWCNT to the probes

• Imperfect contact Resistance (RMC)

• When connecting a MWCNT metal must be deposited to

connect to all the shells

• Quantum Contact Resistance (RQ)

• Intrinsic resistance due to the fabrication process

• Because this deals shells near atomic levels the electrons

make “jumps from one point to another”

• Inside Each MWCNT section there are individual

Resistances, Capacitances, and inductances

• Resistances:

• Scattering Resistance (Rs)

• When connecting the anode to shells, electrons go from

following a single path to scattering

• Inductances

• Kinetic Inductance (LK)

• The inertia of the electron

• Fixed at 8nH/um 40pH/Section

• Magnetic Inductance (LM)

• Magnetic field created by moving electrons

• Incredibly small (2pH/um 100fH/section)

• Capacitances

• Quantum Capacitance (CQ)

• Capacitance between shells effected by the electron density

• Electrostatic Capacitance between shells (CS)

• Capacitance between shells

• Electrostatic Capacitance from the ground plane (CE)

• Capacitance between the dielectric and the MWCNT

MODELING MWCNT: MCC AND ESC MODELSMCC Model – Multi Conductor Circuit (Step 1)

• Shows every circuit element from each

section of each shell

• Pros

• Used for performance prediction

• Accounts for inter-shell conductivity

• Cons

• Slower process, harder to model

• Time and memory issues

ESC Model – Equivalent Single Conductor (Step 2)

• Simplifies each shell into a single one line circuit

• Pros

• Much faster then MCC model

• Simpler then the ESC model, O(n3/3) smaller

• Cons

• Approximations have to be made, thus the circuit is

less accurate

• Does not account for inter-shell conductivity

CONVERTING MCC TO ESCTo convert an MCC to an ESC

• RMC, RQ, RS combine in parallel

• Lk adds in parallel

• Again LM is neglected

• CQ and Cs follow a recursive formula

• There is only one CE and is kept as is

Per Unit Length (p.u.l.)

• It is important to be aware of the units

or these RLC’s

• Often we had numbers seemingly way

out of range

• Numbers were often Ω/m, F/m, H/m

• Need to be Ω/um, F/um, H/um

• OR even better, with 200 sections p.u.l.

• Ω/Sec, F/um, H,um

TIME DELAYSIMULATION: THEORY VS. CADENCE VS. OURS

THEORETICAL:

FROM RESEARCH PAPEROUR CIRCUIT SIMULATOR &

CADENCE MODEL:

WHERE WE’RE AT AND WHERE WE WANT TO BECurrently:

• We have produced a simulator that is able to:

• Solve Complex RLC circuits with transistors in time

and frequency domain

• Simulates a MWCNT with 30 shells using the ESC

model

• Simulation runs on any modern laptop

• Currently runs a simulation for a single

conductor

• Have not used any of our allotted Budget

Next Semester

• Solve MCC model

• Add in the dimension of dielectric conductance

as well as inter-shell conductance

• Run our simulation on a supercomputer for a

accurate, time-efficient, full MCC model

• We wish to run our simulation on 1,2,3…n

multiple conductors

• Will have to allot funds from budget for

supercomputer use

QUESTIONS?