Quantum Capacitance Extraction for Carbon Nanotube Interconnects
© The Author(s) 2010
Received: 2 March 2010
Accepted: 19 May 2010
Published: 8 June 2010
Electrical transport in metallic carbon nanotubes, especially the ones with diameters of the order of a few nanometers can be best described using the Tomanaga Luttinger liquid (TL) model. Recently, the TL model has been used to create a convenient transmission line like phenomenological model for carbon nanotubes. In this paper, we have characterized metallic nanotubes based on that model, quantifying the quantum capacitances of individual metallic single walled carbon nanotubes and crystalline bundles of single walled tubes of different diameters. Our calculations show that the quantum capacitances for both individual tubes and the bundles show a weak dependence on the diameters of their constituent tubes. The nanotube bundles exhibit a significantly large quantum capacitance due to enhancement of density of states at the Fermi level.
Recently carbon nanotubes have acquired importance as a material with a wide variety of potential applications in nanoelectronics. A significant amount of interest has been generated in metallic carbon nanotubes for their application as an on-chip interconnect, replacing the traditional copper wires which are nearing their performance limits. The International Technology Roadmap for Semiconductors (ITRS) has already placed carbon nanotubes as a potential candidate interconnect material for technology nodes beyond 22 nm . The propagation speed of a signal on a transmission line is related to distributed inductance and capacitance of the system as . For mesoscopic systems, the capacitance term C comprises a second “quantum” component apart from the Maxwellian capacitance. This parameter is related to the electronic structure of the material. In this paper, we present calculations that illustrate how the quantum capacitance of different carbon nanotubes vary with size and chirality. This information is necessary to construct a simulation model that will be able to characterize nanotube performance accurately.
It has been long known that the Fermi liquid model is not able to describe transport properties of one dimensional metals. The presence of strong electron–electron interactions prevents the formation of a sharp Fermi surface as would be conventionally expected in a regular bulk metal. The Tomanaga Luttinger (TL) model is used to describe electronic transport in one dimensional systems such as a 1D electron gas as present in a carbon nanotube. The TL model attempts to describe transport properties of a 1D electron gas taking into account strong electron–electron correlation, for energies in the vicinity of the Fermi level. The model is constructed by linearizing the energy-wavevector dispersion of the nanotube around the Fermi wave vector and mapping it onto an equivalent system of boson quasiparticles . Recently, the use of the concepts of Luttinger liquid theory was suggested by authors in [3, 4] to build a phenomenological model of microwave transport in these nanotubes. The authors created a transmission line model mapping a Luttinger model-based Lagrangian to a conventional LC lossless transmission line model. This model is used as a starting point in this paper and has four principal components. These are the classical Maxwellian capacitance (C es ) and inductance (L m ), a quantum component of capacitance (C q ) and a kinetic inductance term (L k ). The quantum capacitance is a manifestation of finite size quantization effects in nanotubes. The electrostatic capacitance C es is related to the electron–electron interactions within the nanotube. The kinetic inductance is nothing but a measure of the kinetic energy of the electrons. Typically for conductors as small as carbon nanotubes L k is several orders of magnitude larger than is magnetic counterpart. In this paper we have quantified the quantum capacitance (C q ) of a variety of carbon nanotubes including different chiralities (armchair and metallic zigzag tubes) and systems of both isolated nanotubes and their bundles. The data obtained will be used to form a detailed transmission line simulation model for ULSI interconnects based on carbon nanotube technology.
This rest of this paper is organized as follows. Section 2 describes the concept of quantum capacitance and how they will be evaluated for the carbon nanotube systems under consideration. In Sect. 3, we describe the methodology employed in obtaining first-principles data. Finally in Sect. 4, we discuss the results obtained from our calculations and provide a discussion in context of VLSI interconnections.
where the Fermi velocity is given by and η is the number of bands contributing at a given energy. The band structure of single-walled metallic carbon nanotubes exhibit linear energy-wave vector dispersion in the vicinity of the Fermi level. Hence, it becomes convenient to compute quantum capacitance using (6), since is a constant. When considering more complex systems like carbon nanotube bundles, this is no longer true and the Fermi velocity becomes energy dependent. It is much convenient to extract the quantum capacitance using Eqs. (4) and (5). The quantum capacitance calculations for the nanotubes bundles are at E = E f .
The calculations presented in this paper are strongly dependant on the electronic structure, the nanotube systems under consideration. The bandstructure was calculated through ab-initio computations calculated using the plane wave codes implemented in PWscf 3.2 distribution  on a 2.8 GHz Intel Core 2 CPU based machine with 1GB of physical memory.
This section describes a brief account of the employed methodology and the simulation parameters used in our work. The electronic structure calculations were preceded with an optimization of the carbon nanotube unit cell geometries. First, the approximate coordinates of the carbon atoms in the nanotube unit cell were calculated using simple formulae available in published literature . The unit cells were then subjected to a Broyden-Fletcher-Goldfarb-Shanno (BFGS) nonlinear optimization procedure and relaxed to their most stable geometry. The optimization procedure essentially involved varying the unit cell dimensions in such a way so as to find a minimum of the total energy, which was calculated self-consistently. To ensure that the supercell of the individual tube used was big enough to ignore intercell interaction, the relaxation runs were performed with a hexagonal and a cubic lattice similar to the methodology employed in .
Simulation parameters used in relaxation calculations
Von Barth Car USPP-PZ
Kinetic energy cutoff
Charge density cutoff
Charge mixing β
Gaussian (0.02 Ry.)
Our studies have included a variety of nanotube systems with diameters ranging from 5 to 12 nm. We have performed calculations for single walled tubes and bundles that constitute of these individual tubes. The calculations for SCF were done with a denser K-point mesh compared to that used for the structural relaxation runs. For the nanotube bundles, an 8 × 8 × 8 MP grid and a 50–80 Ry. kinetic energy cutoff was found sufficient for numerical convergence of total energy. Other than that, all other simulation parameters were similar to that in Table 1. Band structure calculations for the single walled tubes were performed using 20 linearly spaced k-points along the z-direction of the tube i.e. for armchair tubes and for zigzag tubes, for armchair tubes and for zigzag tubes, where a 0 = 2.47 Å is the lattice constant of Graphene.
Results and Discussion
Single Walled Carbon Nanotubes
Quantum capacitances for metallic SWNTs
C q (fF/μm)
Luttinger interaction parameter ‘g’
Here, C es is the electrostatic capacitance of the system under consideration. These numbers for the electrostatic capacitance were drawn from our previous research work on electrostatic capacitance extraction for different nanotube interconnect configurations . We get g to range between 0.14 and 0.33. Both zigzag tubes show a similar ‘g’ values that were considerably smaller (g = 0.14) than those for the armchair varieties. Readers must note that ‘g’ values were calculated by linearizing the dispersion curves near the Fermi levels. The linearizing around the Fermi level is especially important for the smaller (4,4), (5,5) tubes in the system for which our calculations show small band gaps opening up as the result of tube curvature. Our results compare well with experiments reported by authors in [15–18]. An experiment by , however, suggests a much smaller observed quantum capacitance value for one of their metallic specimens. It may be of interest to note that ‘g’ values indicate faster plasmon propagation speeds. The propagation velocity is related to ‘g’ as v p = v f /g. To compare with copper (v f ≈1.57 × 106 m/s) a (12,0) tube is predicted to have a plasmon velocity v p ≈ 6.02 × 106 m/s. This is what would make SWNT-based interconnects extremely competitive as interconnects for nanoscale integrated circuits.
Bundled Carbon Nanotubes
Quantum capacitance for crystalline SWNT bundles
Intertube spacing (Å)
C q (nFμm −3)
C′ q (fF tube −1μm −1)
In this paper, we have characterized individual metallic carbon nanotubes and crystalline nanotube bundles for their quantum capacitance, to model the high-frequency transmission line interconnects comprised of these nanotubes. We have seen that the quantum capacitance of individual tubes have a very weak dependence on chirality. Zigzag tubes owing to the presence of degenerate bands around the Fermi level exhibit almost twice the quantum capacitance compared to the armchair varieties. The value of the Luttinger parameter ‘g’ was estimated between 0.14 and 0.33.
The zigzag varieties exhibit a much smaller interaction parameter (g = 0.14). Consequently, they have an advantage over armchair tubes and even bulk Copper in terms of signal propagation delay. When put in a bundle, the electronic density of states shows a significant increase around the Fermi level, due to electronic coupling between 2p orbitals oriented normal to the tube surface, thus markedly increasing the value of C q per unit cell when compared to the constituent nanotube. Bundled nanotubes also show a poor C q dependence on the chirality of its constituent tubes.
Acknowledgements and Disclaimer
The research reported in this document was performed in connection with contract DAAD17-03-C-0115 with the US Army Research Laboratory. The views and conclusions contained in this document are those of the authors and should not be interpreted as presenting the official policies or position, either expressed or implied, of the US Army Research Laboratory or the US Government unless so designated by other authorized documents. Citation of manufacturer’s or trade name does not constitute an official endorsement or approval of the use thereof. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
- International technology roadmap for semiconductors, interconnect 2005.
- Haldane FDM: Luttinger liquid theory of one-dimensional quantum fluids. i. properties of the luttinger model and their extension to the general 1d interacting spinless fermi gas. J. Phys. C: Solid State Phys. 1981,14(19):2585–2609. Bibcode number [1981JPhC...14.2585H] Bibcode number [1981JPhC...14.2585H] 10.1088/0022-3719/14/19/010View ArticleGoogle Scholar
- Bockrath M: Carbon nanotubes: electrons in one dimension. UC Berkeley; 1999. PhD thesis PhD thesisGoogle Scholar
- Burke P: Luttinger liquid theory as a model of the gigahertz electrical properties of carbon nanotubes. Nanotechnol., IEEE Trans. 2002, 1: 129–144. Bibcode number [2002ITNan...1..129B] Bibcode number [2002ITNan...1..129B] 10.1109/TNANO.2002.806823View ArticleGoogle Scholar
- Büttiker M, Thomas H, Pretre A: Mesoscopic capacitors. Phys. Lett. A 1993, 180: 364–369. Bibcode number [1993PhLA..180..364B] Bibcode number [1993PhLA..180..364B] 10.1016/0375-9601(93)91193-9View ArticleGoogle Scholar
- Luryi S: Quantum capacitance devices. Appl. Phys. Lett. 1988,52(6):501–503. Bibcode number [1988ApPhL..52..501L] Bibcode number [1988ApPhL..52..501L] 10.1063/1.99649View ArticleGoogle Scholar
- John DL, Castro LC, Pulfrey DL: Quantum capacitance in nanoscale device modeling. J. Appl. Phys. 2004,96(9):5180–5184. COI number [1:CAS:528:DC%2BD2cXptFGmsLw%3D]; Bibcode number [2004JAP....96.5180J] COI number [1:CAS:528:DC%2BD2cXptFGmsLw%3D]; Bibcode number [2004JAP....96.5180J] 10.1063/1.1803614View ArticleGoogle Scholar
- Giannozzi P[http://www.quantum-espresso.org]
- Saito R, Dresselhaus G, Dresselhaus MS: Physical properties of carbon nanotubes. World Scientific Publishing Company; 1998. September SeptemberView ArticleGoogle Scholar
- Kürti J, Kresse G, Kuzmany H: First-principles calculations of the radial breathing mode of single-wall carbon nanotubes. Phys. Rev. B 1998, 58: R8869-R8872. Bibcode number [1998PhRvB..58.8869K] Bibcode number [1998PhRvB..58.8869K] 10.1103/PhysRevB.58.R8869View ArticleGoogle Scholar
- We used the pseudopotential c.pz-vbc.upf from, [http://www.quantum-espresso.orgdistribution] We used the pseudopotential c.pz-vbc.upf from
- Perdew JP, Zunger A: Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 1981, 23: 5048–5079. COI number [1:CAS:528:DyaL3MXktFejurk%3D]; Bibcode number [1981PhRvB..23.5048P] COI number [1:CAS:528:DyaL3MXktFejurk%3D]; Bibcode number [1981PhRvB..23.5048P] 10.1103/PhysRevB.23.5048View ArticleGoogle Scholar
- Monkhorst HJ, Pack JD: Special points for brillouin-zone integrations. Phys. Rev. B 1976, 13: 5188–5192. Bibcode number [1976PhRvB..13.5188M] Bibcode number [1976PhRvB..13.5188M] 10.1103/PhysRevB.13.5188View ArticleGoogle Scholar
- Parkash V, Goel A: Electrostatic capacitance extraction for carbon nanotube interconnects. in Circuits and Systems, 2008. MWSCAS 2008. 51st Midwest Symposium on 2008, 834–837. Aug. Aug.View ArticleGoogle Scholar
- Bockrath M, Cobden DH, Rinzler AG, Smalley RE: Luttinger-liquid behavior in carbon nanotubes. Nature 1998, 397: 598.Google Scholar
- Ilani S, Donev LAK, Kindermann M, McEuen PL: Measurement of the quantum capacitance of interacting electrons in carbon nanotubes. Nat. Phys. 2006, 2: 687–691. COI number [1:CAS:528:DC%2BD28XhtFelur7N] COI number [1:CAS:528:DC%2BD28XhtFelur7N] 10.1038/nphys412View ArticleGoogle Scholar
- Ishii H, Kataura H, Shiozawa H, Yoshioka H, Otsubo H, Takayama Y, Miyahara T, Suzuki S, Achiba Y, Nakatake M, Narimura T, Higashiguchi M, Shimada K, Namatame H, Taniguchi M: Direct observation of tomonaga-luttinger-liquid state in carbon nanotubes at low temperatures. Nature 2003,426(6966):540–544. COI number [1:CAS:528:DC%2BD3sXpsVejsbw%3D]; Bibcode number [2003Natur.426..540I] COI number [1:CAS:528:DC%2BD3sXpsVejsbw%3D]; Bibcode number [2003Natur.426..540I] 10.1038/nature02074View ArticleGoogle Scholar
- Dai J, Li J, Zeng H, Cui X: Measurements on quantum capacitance of individual single-walled carbon nanotubes. Appl. Phys. Lett 2009, 94: 093114. Bibcode number [2009ApPhL..94i3114D] Bibcode number [2009ApPhL..94i3114D] 10.1063/1.3093443View ArticleGoogle Scholar