Asenov, Asen

Georgiev, Vihar P.

Lee, Jaehyun

Sadi, Toufik

Todri-Sanial, Aida

English

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A hierarchical model for CNT and Cu-CNT composite interconnects: from density functional theory to circuit-level simulations Jaehyun Lee1, Toufik Sadi1, Jie Liang2,  Vihar P. Georgiev1, Aida Todri-Sanial2, and Asen Asenov1* School of Engineering, University of Glasgow, Glasgow, UK1 CNRS-LIRMM, University of Montpellier, Montpellier, France2  Metallic carbon nanotubes (CNTs) and Cu-CNT composites have been regarded for years as one of the promising candidates for the future interconnects to replace Cu. Due to CNTs’ small effective mass and 1-D structure, and strong C-C bonding, they have a very high ampacity, a large mean free path (𝜆), and excellent mechanical properties [1]. In this work, we have investigated the transport properties of CNT interconnects based on density functional theory (DFT) and compact model analysis. We also propose a hierarchical model to connect the DFT with circuit-level simulations.   We have used ATK [2] for DFT calculations involving the generalized gradient approximation (GGA). As the first step, a single-wall CNT (SWCNT) bundle as shown in Fig. 1 has been considered. In order to get optimal lattice constants, we have performed geometric optimization. All atoms are fully relaxed until the forces of any atom become less than 0.01 eV/Å. With the optimized atomistic structure, we have calculated the ballistic conductance (𝐺𝑏𝑎𝑙) at 300 K using the non-equilibrium Green’s function (NEGF) formalism.   Fig. 2 shows 𝐺𝑏𝑎𝑙 of SWCNT bundles and stand-alone SWNCTs. The difference between them, caused by the interaction with the adjacent SWCNTs, decreases as the diameter of CNTs (𝐷𝐶𝑁𝑇) increases.   Fig. 3 describes atomistic structures of iodine-doped SWCNT(24,0) and Cu-CNT(6,0) composite. In this work, the width (W) and height (H) of the local interconnect are assumed to be 8 and 16 nm, respectively, with reference to the ITRS node 2024 [3].  The resistance of the interconnect ( 𝑅𝑤 ) made of SWCNT bundles, the doped SWCNT, bulk Cu, and the Cu-CNT composite is plotted in Fig. 4. For bulk Cu, we do not consider the surface and grain boundary scattering effects. To calculate the diffusive conductance (𝐺𝑑𝑖𝑓 ) from 𝐺𝑏𝑎𝑙 , we applied the mean free path approximation; 𝐺𝑑𝑖𝑓 = 𝐺𝑏𝑎𝑙(1.0 + 𝐿/𝜆)−1 , where 𝐿  is the interconnect length. As can be seen in this figure, bulk Cu has smaller 𝑅𝑤  for short interconnects. Due to the large 𝜆  of CNT, however, 𝑅𝑤  of the SWCNT(6,0) becomes better as 𝐿 increases. It is noticeable that Cu-CNT composite is characterized by combining the advantage of both Cu and CNT.   To calculate the propagation time delay ( 𝑡𝑑 ), we have considered a driver-interconnect-load system, as shown in Fig. 5. 𝑅𝑠, 𝐶𝑠, and 𝐶𝐿 are assumed to be 35 kΩ, 7 aF, and 14 aF, respectively [3]. The capacitance of the interconnect ( 𝐶𝑤 ) is defined by 𝐶𝑤−1 = 𝐶𝐸−1 + 𝐶𝑄−1 , where 𝐶𝐸  and 𝐶𝑄  are the classical electrostatic and quantum capacitances, respectively. 𝐶𝑄 was extracted using the following relationship [4]; 𝐶𝑄 =𝑒24𝑘𝑇∫ 𝐷(𝐸) sech2𝐸2𝑘𝑇𝑑𝐸, where 𝐷(𝐸) is the density-of-states obtained from DFT-GGA calculations. The calculated 𝐶𝑄 values are summarized in Table 1.    We have calculated 𝑡𝑑 by using the Elmore formula; 𝑡𝑑 = 0.69{𝑅𝑠(𝐶𝑠 + 𝐶𝐿) + (𝑅𝑤𝐶𝐿 +𝑅𝑠𝐶𝑤)+𝑅𝑤𝐶𝑤}. Fig. 6 shows the dependence of 𝑡𝑑  on 𝐿  and 𝐶𝐸 . In this calculation, we assumed that 𝐶𝐸  does not rely on the interconnect material. As expected, long interconnects have large 𝑡𝑑  due to large 𝑅𝑤 and 𝐶𝑤 . When 𝐶𝐸 = 10−3  pF/μm, the SWCNT(6,0) bundle has the smallest 𝑡𝑑 because of its small 𝑅𝑤 (See Fig. 4). As 𝐶𝐸 increases (the feature size decreases), however, 𝑡𝑑  of the SWCNT(24,0) bundle becomes smaller than that of the SWCNT(6,0) bundle because of the small 𝐶𝑄 of SWCNT(24,0).   References [1] C. Subramaniam et al., Nat. Commu. 4, 3202 (2013) [2] http://quantumwise.com/ [3] A. Alizadeh and R. Sarvari, IEEE Trans. VLSI Syst. 24, 2, 803-807 (2016) [4] C. Zhan et al., J. Phys. Chem. 119, 22297-22303 (2015)  Fig 1. Atomic structure of the SWCNT(6,0) bundle. The black box indicates the primitive unit cell.  Fig 2. Dependence of 𝐺𝑏𝑎𝑙 on 𝐷𝐶𝑁𝑇 of zigzag and armchair SWCNTs.  Fig. 3 Atomic structures of the iodine-doped SWCNT and the Cu-CNT composite.  Fig. 4 𝑅𝑤 vs 𝐿 with W = 8 nm and H = 16 nm.  Fig. 5 Schematic representation of a typical interconnect.  Material CQ (pF/μm) SWCNT(6,0) bundle 0.3227 SWCNT(24,0) bundle 0.0146 Iodine-doped SWCNT(24,0) 0.0321 Cu-CNT(6,0) 0.7370 Bulk Cu 0.7821 Table 1. 𝐶𝑄 of the interconnect with W = 8 nm and H = 16 nm at 300 K.   Fig. 6 Dependence of 𝑡𝑑 on 𝐿 and 𝐶𝐸. W and H of the interconnect are assumed to be 8 and 16 nm, respectively. 

A Hierarchical Model for CNT and Cu-CNT Composite Interconnects: From Density Functional Theory to Circuit-Level Simulations

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A Hierarchical Model for CNT and Cu-CNT Composite Interconnects: From Density Functional Theory to Circuit-Level Simulations

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