A Theoretical Modeling Analysis of Adapted Composite CNT Bundle for High-Speed VLSI Interconnect

The aroused quest to reduce the delay at interconnect level is the main urge of this 1 paper to come across a configuration of Carbon Nanotube (CNT) bundle namely squarely packed 2 bundle of composite CNTs. The approach, demonstrated in this paper, adapts the composite 3 bundle to adopt for high speed Very Large Scale Integration (VLSI) interconnect with technology 4 sizing down. To reduce the delay of the proposed configuration of composite CNT bundle, the 5 behavioral change of Resistance (R), Inductance (L) and Capacitance (C) has been observed with 6 respect to both width of the bundle and diameter of the CNTs in the bundle. Consequently, 7 the performance of the modified bundle configuration is compared with previously developed 8 configuration namely squarely packed bundle of dimorphic MWCNTs in terms of propagation 9 delay and crosstalk delay at local, semiglobal and global level interconnect. The proposed bundle 10 configuration is ultimately enacted as better one for 32 nm and 16 nm technology node and suitable 11 for 7 nm as well. 12


Introduction
The overwhelming exploitation of interconnect to the device delay makes researcher 15 to weigh Carbon Nanotubes (CNTs) for the possession pertinent to long mean free path 16 [1], electrical properties [1,2], thermal properties [2,3], electromigration and current 17 density [4]. Moreover, crosstalk delay is potential stymie for CNTs due to capacitive 18 coupling between adjacent bundle [5]. As it has already been anticipated in [1] that the 19 performance will be meliorated with further technology scaling, CNT can provide much 20 better performance with exploration of some features. 21 It is claimed in [5,6], that mutual inductance doesn't have a considerable impact on 22 crosstalk-induced delay and glitch instead coupling capacitance along with electrostatic 23 and quantun capacitance puts main contribution. It is also noticed from [7] that the 24 graphitized electron beam-induced deposition (EBID) carbon has capability to produce 25 a low-resistance ohmic contact to multiple shells of MWCNT in the context of making 26 high-performance electrical interconnect structure for the next-generation electronic 27 circuits. 28 While the configuration of the bundle of CNTs is seizing attention, a trade-off 29 between propagation delay and crosstalk delay is conspicuous [8] as it is mentioned in 30 [5] that SWCNT and DWCNT shows poorer performance than Cu based interconnect 31 owing to higher coupling capacitance. Due to this reason, we endeavored to avoid 32 putting any SWCNT and DWCNT on the edge of the bundle in our configuration. 33 To improve the crosstalk delay along with propagation delay, some works [6,[9][10][11][12]   The aim of this paper is to present an innovative diameter controlled configuration 49 to alleviate the propagation delay and crosstalk delay of size shrinking interconnect 50 which is feasible from fabrication aspect. This configuration is presented here with 51 detailed analytical analysis and comparison results with the aim of better investigating 52 its performance and enlightening its advantages. To analyze the delay performance, the 53 analytical delay model has been obtained using the parameters from [13][14][15].

54
The residue of this article is going to follow upcoming course. A modified configu-55 ration is proposed as the successor of the configuration, introduced in [16], in Section 56 2. Section 3 is used to develop the mathematical models for RLC elements for isolated 57 CNTs and eventually for composite bundle based on the configuration exposed in Sec-58 tion 2. Section 4 is dedicated to the interest of simulating and analyzing the performance 59 indicators, propagation delay and crosstalk noise for different technology nodes and 60 showing a comparison with the previously well-developed research work [16]. The 61 conclusion including further research direction is ultimately delimited in Section 5.

63
Our endeavor in this paper is to enhance the performance by altering the configura-64 tion, shown in Fig. 1(a). In this newly introduced configuration, illustrated in Fig. 1(b), 65 the replacement of smaller MWCNTs is taken place with a bunch of SWCNTs which are 66 wrapped up by the larger MWCNTs. This modified approach is virtue of increasing the 67 number of CNTs in the bundle so that we can fill up the unoccupied space with SWCNT 68 more efficiently and densely.

69
The proposed configuration is going to follow the geometric pattern, similar to 70 the previous configuration from [16,17], to accommodate more number of CNTs in the MWCNTs and the number of SWCNTs in the bundle are calculated using (1).

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By considering the one third of the shells of MWCNTs as metallic [18], the average number of conducting channels for a shell can be calculated using (1) the square formed by the MWCNTs in Fig. 1. So, the diameter of the outermost shell of MWCNT is The number of shells of MWCNTs can be calculated using following formula (3) according to [9,12,19].
According to the geometry of circle, we know that the diagonal of the bigger square from To calculate the plausible number of accommodated SWCNTs in the smaller square of Fig. 1, where, Fig. 1

Number of MWCNTs in
where, ⌊X⌋ and ⌈X⌉ signifies that rounds each element of X to the nearest integer 79 less than or equal to that element and more than or equal to that element respectively.

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Number of smaller square in Fig. 1 N Total number of SWCNTs in the bundle  in parallel [20]. According to [21], the quantum resistance of SWCNT can be estimated 91 using the conductance G = 2e 2 h MT where, e is electron charge with the value of

Improved Mathematical Models
On the other hand, in case of single wall nanotube length (l) exceeding the mean free path of electrons (λ SWCNT ), another resistance comes up along with the former one owing to scattering which can be computed from (11).
Finally, the total resistance, emerged from previous two components of the resis-94 tance, for isolated SWCNT is denoted by (12).
The lump resistance (R lump ), having the quantum or intrinsic resistance from (10), engendered by the quantum detainment of electrons in a nano-wire and imperfect metal-nanotube contact resistance (R mc ), may vary from few to several hundreds of kilo-ohms based on the fabrication process [9,12,22]. The lump resistance for different isolated MWCNTs of the proposed bundle configuration using (13), acquired from [4,9,19,20,22,23].
The per unit length (p.u.l) scattering resistance (R s ) emerges for length of the nano-96 wire surpassing the effective mean free path of the electron [20]. The scattering resistance 97 (R s ) for different isolated MWCNTs of the proposed bundle configuration is estimated 98 from (14) based on [4,9,22].
The equivalent resistance of the bundle including both MWCNT and SWCNT can 100 be recokoned using (15).
The characteristics of resistance (R) depends on both the width of the interconnect 102 wire based on the technology node and the diameter of the used CNTs in the bundle.

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The simultaneous impact of both factors is observed in Figure 2. It is obvious that the  To determine the overall inductance for our proposed configuration of the composite 112 bundle, firstly we will calculate the inductance for isolated SWCNT and then for isolated 113 MWCNT, finally the equivalent inductance for the entire bundle will be demonstrated 114 as given in (21). The inductance of SWCNT consists of two components which are 115 denoted as kinetic inductance (L k ) and magnetic inductance (L m ). Considering the 116 ballistic conduction for a 1-D conductor, the kinetic inductance can be approximated by 117 (16). where, v F is Fermi velocity of electron with the value of approximately 8 × 10 5 ms −1 .

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Since, L k is the function of some constant values, the approximate per unit length (p.u.l.) 120 value is 16 nH/¯m [24]. On the other hand, the stored energy of carriers in magnetic 121 field engenders magnetic inductance, approximated in (17), in SWCNT [24].
Now, in case of MWCNT, the magnetic inductance comes from is expressed as (20) 123 for i th shell.
Kinetic and magnetic inductance of the isolated MWCNT in the proposed bundle is calculated using (19) and (20) respectively.
Finally, the overall equivalent inductance of the bundle is estimated in (21) which 125 indicates that kinetic inductance component of SWCNT exists when the length of the 126 interconnect wire exceeds the electron mean free path.
Inductance of the bundle also exhibits the same phenomena as the resistance does.

132
The p. u. l. quantum capacitance for a CNT is estimated in (22) by taking the analogy of the required energy to enclose an extra electron at an acquirable quantum state level beyond the Fermi energy level and effective capacitance. This capacitance comes into notification due to the quantum electrostatic energy stored in the nanotube during carrying the current [24].
It is already mentioned to estimate the inductance for a isolated SWCNT that SWCNT has four conducting channels and these channels form a parallel combination [24]. As a result, the equivalent effective quantum capacitance of a isolated SWCNT can be approximated in (23).
The electrostatic capacitance, expressed in (24) is calculated in [24] by considering the SWCNT as a thin wire with diameter D SW putting at a distance of ′ y ′ away from the ground.
Now the capacitance for isolated MWCNT is calculated using the recursive model. It is recommended in [6] that the quantum capacitance of each shell is mandatory to get determined before estimating the effective capacitance of a single MWCNT. The quantum capacitance is basically the estimation of the finite density of electronic states of quantum wire [20].
According to ESC model of MWCNT, it can be inferred from [23] that a shell-to-shell mutual capacitance between two adjacent shells of MWCNT.
At first, in case of outermost shell, the equivalent capacitance, expressed in (27) represents only the quantum capacitance of that shell. As much as we go toward the inner shell, the quantum capacitance of that particular shell makes a parallel combination with the equivalent capacitance (C q−s ) as shown in (29), a series combination of the capacitance of any shell and the mutual capacitance between that shell and previous shell obtained in (28), and go on until reach the innermost shell. The electrostatic capacitance demonstrated the potential difference between the CNT over the ground plane and ground [20]. The p. u. l. electrostatic capacitance can be approximated in (30).
The conglomerate capacitance of the proposed composite bundle is obtained in (31) by considering the overall effect of SWCNT and MWCNT in the bundle. To estimate this, we considered the effect of electrostatic capacitance of MWCNTs over the ground plane on the effective capacitance in series.
Unlike the resistance and inductance, capacitance shows descending behavior with   Table 1.  Table 1.  Table 1. configuration of squarely packed bundle using the proposed approach. As a conse- impact by following [6,9].
Eventually, the crosstalk performance of our proposed configuration is depicted 169 in Figure 8, Figure 9 and Figure 10 for local, semiglobal and global level interconnect 170 by comparing with the preceding configuration from [16]. It is noticeable from Figure   171 9 that crosstalk performance of the proposed configuration in a substantial manner 172 for both 32 nm and 16 nm technology nodes. On the other hand, Figure 9 illustrates is almost same as it is illustrated in Figure 10. It is also demonstrable that our proposed 177 configuration is appropriate for 7 nm technology node.

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A modified configuration of the squarely packed bundle CNTs is presented by fol-180 lowing the previously developed configuration in [16]. By proposing this configuration, In this approach, CNTs with only two different sizes are used. In the upcoming 187 endeavor, our intention is to advance the work by adding the CNTs with various sizes to 188 make the configuration more convenient in terms of fabrication process.