1. 2007 International Symposium on EMC
Reduced Order Modeling for Transient
Analysis of Carbon Nanotubes Interconnects
G. Antonini, A. Orlandi
UAq EMC Laboratory
Dept. of Electrical Engineering
University of L’Aquila, Italy
July 12, 2007 UAq EMC Laboratory 1

2. Introduction
•As semiconductor device dimensions become increasingly
smaller, the density of off-chip interconnects continues to
increase.
•One approach to keeping up with the semiconductor
processing industry’s ever shrinking device dimensions, is the
nano-scale interconnect technology:
• small dimensions, enabling high speed and high
functional density (nanoelectronics, lab-on-chip)
• lightweight devices and sensors (smart dust)
• high sensitivity (sensors)
• FPGA programmable interconnects
July 12, 2007 UAq EMC Laboratory 2

3. Introduction
In both TOP-DOWN
nanostructures are made through scaling and miniaturization
or BOTTOM-UP
building nanostructures through atom-by-atom or molecule-by
molecule engineering.
technologies there is a clear need of simulation techniques.
In the state-of-the-art EDA tools, it is common to use a
rational macromodel to allow the transient analysis of complex
electromagnetic systems.
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4. Introduction
This work proposes an
efficient technique to develop
a state-equation based
macromodel of CNT
interconnect, starting from
their equivalent per unit length
(p.u.l) parameters.
This macromodel is suited for a reliable and accurate transient
solution.
An aside product of this technique is the fitting of the transfer
function by means of a rational function defined by its poles and
residues. The macromodel can be implemented directly in the
time domain and linked to non-linear Spice-like solvers.
July 12, 2007 UAq EMC Laboratory 4

5. CNT main features
• Very good conductors (up to 109 A/cm2 while Cu
interconnects tolerate at most 106 A/cm2)
• Meccanically robust
• Excellent thermal conductivity (1700-3000 W/mK
while for Cu it is 400 W/mK) which may mitigate
the problem of heat dissipation in scaled
technologies
July 12, 2007 UAq EMC Laboratory

6. Approach
Quasi- TEM propagation is assumed-MTL model is adopted
July 12, 2007 UAq EMC Laboratory

7. Introduction
For the development of transient analysis algorithms for
lossy transmission lines, a new method is presented
based on half-T ladder network (HTLN) theory and
special polynomials.
Z1 Z1 Z1 Z1
β
β-1
I1 In
1 n
0
Vin Y2 Y2 Y2 Y2 Vout
closed-
The ladder network has a closed-form two port representation
in terms of rational functions.
functions.
A reduced order model can be developed.
developed.
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8. Rational model of the HTLN
It allows the extraction of poles and residues of
the half-T ladder network approximating the
MTL which is well suited for developing an
efficient model order reduction technique.
The method can be extended to frequency
dependent per unit length parameters (FDPUL)
as well.
The proposed procedure automatically
generates a poles/residues representation that
can be readily imported in the design flow.
July 12, 2007 UAq EMC Laboratory

9. Single-Walled Nanotube
Characterization
The NT interconnect can be treated with the multiconductor
transmission lines (MTLs) formalism,described by the
Telegrapher's equations:
The solution can be written in the Laplace
domain using an exponential matrix function as:
July 12, 2007 UAq EMC Laboratory 9

10. Single-Walled Nanotube
Characterization
The global p.u.l. matrices to be used in the MTL equations
are given by:
If the medium is homogenous and the conductors are not
closely spaced, matrix Ce can be computed analitycally
and, then, matrix Le is evaluated as:
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11. Single-Walled Nanotube
Characterization
where:
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12. Single-Walled Nanotube
Characterization
Rational model for transient analysis
It is useful to define the following quantities:
A half-T ladder network can be analytically characterized
in terms of Chebyshev polynomials.
The polynomial based approach can be adopted for general
interconnects with frequency dependent p.u.l. parameters.
To this aim let us define the half-T cell factor K(s) as:
July 12, 2007 UAq EMC Laboratory 12

13. Single-Walled Nanotube
Characterization
Z1(s) and Y2(s) matrices can be approximated using
standard fitting techniques, leading to the following
rational representations:
The zero-pole form reads:
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14. Single-Walled Nanotube
Characterization
The electrical characteristics of a half-T LN can be expressed
in terms of two polynomials depending on K(s):
Two port A,B,C and D parameters can be expressed in
terms of Pbn and Pcn polynomials as:
July 12, 2007 UAq EMC Laboratory 14

15. Single-Walled Nanotube
Characterization
The Y matrix can be evaluated by computing Y11 and Y21 in
terms of ABCD parameters and then enforcing the reciprocity
and the symmetry of the transmission line:
Polynomials Pbn(K), Pbn-1(K) and Pcn(K) can be factored into
zero-pole pairs:
July 12, 2007 UAq EMC Laboratory 15

16. Single-Walled Nanotube
Characterization
Y11 e Y21, taking into account that K(s) · Y2(s) = Z1(s) = R + sL,
can be factored in the following way:
Rational functions sharing the same poles
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17. Single-Walled Nanotube
Characterization
a) Computation of poles of Y matrix
Poles of Y matrix functions are obtained as the zeros of the
following equation:
The total number of poles for the N-conductor MTL
approximated with an order n half-T ladder network is:
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18. Single-Walled Nanotube
Characterization
b) Computation of residues of Y matrix
c) Generation of the rational macromodel
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19. Numerical Results
The proposed methodology is applied to the transient
analysis of the CNT interconnect sketched in Fig. 1:
r = 0.5 nm
∆ = 1.5 nm
h = 1 µm
L = 10 µm
lmfp = 1 µm
A ladder network of order n = 60 has been considered for
modeling the interconnect.
The model order reduction has been applied, among the 238 poles
only 54 have been retained as dominant providing a satisfactory
accuracy in the 0-10 GHz frequency range.
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20. Numerical Results
Fig. 2 shows the location
of all the poles of the
rational model and those
selected as dominant in
the complex plane.
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21. Numerical Results
The magnitude of residues
of admittance Y12 are shown
in Fig. 3.
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22. Numerical Results
The rational model
has been validated by
comparing the
admittance evaluated
in the frequency domain.
A good agreement is
achieved in the 0-10 GHz
frequency range as seen
in Fig. 4 where the
magnitude spectrum of
the transfer admittance
Y24 is shown.
July 12, 2007 UAq EMC Laboratory 22

23. Numerical Results
One of the lines has been
driven by a pseudo-random
bit sequence,with a 250 MHz
bit rate, rise and fall times
τr = τf = 1.25 ns.
Fig. 5 shows the output
current which confirms the
accuracy of the proposed
macromodel.
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24. Numerical Results
1 Gbps PRBS with added
pre-emphasis (2 Taps)
July 12, 2007 UAq EMC Laboratory

25. Ultra high speed CNT interconnect
a
r = 0.5nm
∆y
∆ = 1.5 nm
h = 1 µm
∆x l
Hc
L = 100 µm
lmfp = 1 µm
εr
Order HTLN 10
l Ground plane
Poles 88
Dominant poles 15
July 12, 2007 UAq EMC Laboratory

26. Impulsive voltage source
-9
1 10
0.9
-10
0.8 10
0.7
Magnitude of Vs [V]
-11
0.6 10
Vs [V]
0.5
-12
0.4 10
0.3
-13
0.2 10
0.1
-14
0 10
0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 90 100
Time [ns] Frequency [GHz]
Rise time 1 ps
July 12, 2007 UAq EMC Laboratory

27. Impulsive voltage source
10
10 10
10
Data
Data
9 Proposed
10
Proposed
8
10
9
10 7
10
|Z18| [Ω ]
|Z11| [ Ω ]
6
10
5
10
8
10
4
10
3
10
2
7
10
10
0 10 20 30 40 50 60 70 80 90 100
0 10 20 30 40 50 60 70 80 90 100
Frequency [GHz]
Frequency [GHz]
Magnitude spectra of Z11 and Z18
July 12, 2007 UAq EMC Laboratory

28. Impulsive voltage source
TLT
TLT
Proposed
0.5
0.5 Proposed
0.4 0.4
Voltage [V]
Voltage [V]
0.3 0.3
0.2 0.2
0.1 0.1
0 0
-0.1 -0.1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time [ns]
Time [ns]
Input and output port voltages
July 12, 2007 UAq EMC Laboratory

29. Pseudo random bit sequence excitation
3.5
3
2.5
2
Vs [V]
1.5
1
0.5
0
-0.5
0 1 2 3 4 5 6 7 8 9 10
Time [ns]
Bit rate: 10 Gbps, bit time 100 ps, rise time 1 ps
July 12, 2007 UAq EMC Laboratory

30. Pseudo random bit sequence excitation
15
x 10
4
Order 10 ladder network
3
Complete model: 88 poles
2
Reduced order model: 15
1
poles
Imag
0
-1
-2
-3
-4
-8.05 -8 -7.95 -7.9
Real 11
x 10
July 12, 2007 UAq EMC Laboratory

31. Pseudo random bit sequence excitation
1.8
TLT
TLT
2
Proposed
Proposed 1.6
1.4
1.5
1.2
1
Voltage [V]
Voltage [V]
1
0.8
0.6
0.5
0.4
0.2
0
0
-0.2
-0.5
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time [ns]
Time [ns]
Input and output voltages
July 12, 2007 UAq EMC Laboratory

32. Experimental Set-Up
• In order to characterize the electrical performances of the
nanotubes a test procedure is developed
•In collaboration with
TECHNOLABS (V. Ricchiuti)
Prof. S. Santucci (SEM images, Univ.of L’Aquila)
a test board has been designed.
•The procedure is based on the VNA TRL calibration
technique
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33. Experimental Set-Up
Test board
Contacts
Test section
soldering
substrate
calibration
Calibration lines for TRL
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34. Experimental Set-Up
TRL Reference
Wire
Planes
Bonding
Calibration lines for TRL
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35. Experimental Set-Up
•NT grown @ 500 °C
Less regular spatial distribution
•NT grown @ 700°C
more regular spatial ditribution
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36. To Be Done
De-embedding of the wire
bonds
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37. Conclusions
A rational model of CNT interconnect is proposed which is
based on polynomials which allows to analytically compute
poles and residues of the admittance Y matrix.
The knowledge of poles allows to select the dominant ones,
thus leading to a more compact model while preserving
accuracy.
The results of the numerical computations confirms the
capability of the proposed compact model to properly capture
the physics of CNT interconnects.
An experimental set-up is developed in order to
characterize the electrical properties of the nanotubes.
July 12, 2007 UAq EMC Laboratory 37

38. Future steps
• Frequency dependent effects are expected coming into
the game
• The proposed method is able to capture frequency
dependent phenomena
• Large bundles of CNTs will be considered (hundreds of
CNTs)
July 12, 2007 UAq EMC Laboratory

39. Numerical Results
July 12, 2007 UAq EMC Laboratory