AC Conductance Modeling and Analysis of Graphene Nanoribbon Interconnects Deblina Sarkar, Chuan Xu, Hong Li and Kaustav Banerjee

Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, US phone: +1-805-8933978 , e-mail: {deblina, chuanxu, hongli, kaustav}@ece.ucsb.edu

Abstract

This paper presents the first accurate AC impedance extraction methodology for the evaluation of high-frequency behavior of graphene nanoribbon (GNR) structures targeted for interconnect and inductor applications. To overcome the simplifying ���������� �� ����� ����that is invalid for high-frequency analysis of GNRs and to take into account the electric field variation within a mean free path, the current density is derived starting from the basic Boltzmann equation and combining the unique E-k dispersion relation and the concept of two equivalent valleys in the Brillouin zone of graphene. This is followed by self-�������� ��������� ���������� �� ��������� ������ ����� �������function approach using the concept of vector potentials. Using the developed method the intricate high-frequency effects in GNR such as Anomalous Skin Effect (ASE), high-frequency resistance and inductance saturation, intercoupled relation between edge specularity and ASE and the influence of linear dimensions on impedance are investigated in details for the first time.

Introduction As a prospective interconnect material [1], besides circuit performance in terms of delay, a thorough investigation of the high-frequency effects of Graphene Nanoribbons (GNRs, shown in Fig.1) is equally compelling. Recently it has been shown that CNTs can be very attractive for high-frequency applications due to their large momentum relaxation time [2]. However fabrication of such inductor structures with long CNT bundles in horizontal direction is very challenging [3]. GNRs, on the other hand, not only enjoy a large momentum relaxation time but at the same time are more controllable from the fabrication point of view. This is due to the planar nature of graphene which can be patterned horizontally using high resolution lithography. Thus GNRs pose to be a potential candidate for high-frequency interconnects and inductor applications.

High-Frequency Issues in GNR GNRs (Fig. 1) can be either monolayered or multilayered (doped or undoped). The monolayer and undoped multilayer GNRs have high resistivity and are not of much interest for low loss interconnect applications. Hence, the focus of our work will be on doped multilayer GNRs and we consider the stage-2 AsF5 intercalated GNR as a representative case, since it has been shown to have in-plane conductivity greater than Cu [4]. However our methodology is general and applicable to other types of doping as well. However, the high-frequency effects in doped multi-layer GNRs are very complicated. Each GNR layer cannot be treated as one element because of the current density redistribution due to skin effect. Moreover, because of its large mean free path (MFP), GNR is susceptible to anomalous skin effect (ASE) [5-7]. ASE occurs when the MFP becomes comparable to the skin depth and hence, the electric field varies significantly within a MFP. The variation of skin depth of Cu and GNRs is shown in Fig. 2. From Fig. 3, it is clear that GNRs will be susceptible to ASE at a much lower frequency compared to Cu due to their large MFP. Though CNTs also enjoy large MFP, they will not have any ASE as explained in Fig. 4.

Impedance Extraction based on Boltzmann Equation ASE renders the simplifying assumptions �� ����� ���� ������� ���hence, for accurate extraction of the impedance, a more generic approach based on Boltzmann equation is adopted. The Boltzmann equation for the GNR structure is given by (1), which can be solved to obtain the distribution function of carriers (2) using the boundary conditions given by (3). Derivation of current density from the distribution function for GNR is shown in steps (4) and (5). Incorporating the band structure of graphene (6), we arrive at the final expression (7) where the prefactor of 2 arises from the two equivalent valleys in the hexagonal Brillouin zone of graphene (Fig 6b). The current density is calculated from (7) with an initial assumed electric field distribution. Then the GNR structure is divided into several filaments and new values of electric field are calculated using Green�s function method as shown in (8)-(11).

Subsequently, the calculation of current and electric field is carried out self-consistently till convergence is achieved. Finally the impedance is extracted as in (12). At lower frequencies, ASE can be neglected and current can be taken to be a local function of the electric field. Thus, if only normal skin effect (NSE) is considered, the distribution function simplifies to (13). In this case, ������aw can be taken to be valid and effective complex conductivity considering kinetic inductance is calculated as in (14).

High-Frequency Analysis of GNR Structures The methodology described above can be used not only for the evaluation of high-frequency impedance, but also for accurate extraction of low frequency as well as d.c. impedance. To gauge the impact of edge specularity, the two extreme cases of perfectly diffusive (p=0) and perfectly specular (p=1) reflection are considered, where p is the fraction of carriers scattered elastically at the edges. In Fig. 9 the d.c conductance extracted from this exact procedure is compared with the approximate results based on the ������������� ����� ���� and the validity of ������������� ����� ��� ���� ��������� �����is is proved. From the current distribution graphs shown in Fig. 10 it is clear that consideration of ASE is required to capture the realistic high-frequency behavior of the GNR structure. In Fig. 11, the effect of kinetic inductance (KI) on the impedance is investigated. It is seen that the large KI of graphene tends to saturate the resistance as well as inductance at higher values of frequency for both p=0 and p=1. From Fig. 12 it is clear that ASE causes considerable increase in resistance compared to the NSE case at high frequencies. Also the impact of ASE for p=0 is found to be greater than that for p=1 as is shown in Fig. 13. Since p=0 is related to completely diffusive scattering, the corresponding resistance is always greater than that for p=1. At low frequencies the difference is small. As the frequency increases, the current gets confined more and more towards the edges and hence the effect of edge scattering becomes increasingly important and thus the difference between the resistances for p=0 and p=1 also increases significantly. However, ASE has very little impact on inductance as shown in Fig. 14. Variation of resistivity (�) with width, keeping the thickness fixed reveals some very important physics (Fig. 15). Resistivity decreases as width is decreased due to the reduction in skin effect. The inset figures show the percentage increase in resistivity due to ASE compared to the NSE case. At very small widths, this percentage is low since the skin effect itself is weak. The percentage increases as the width increases, but with further increase in width it decreases again. This is because at very large widths, the distribution function becomes almost independent of width and hence the term containing the derivative of f1 with respect to x in (1), becomes less important. Thus the difference between ASE and NSE basically reduces. In Fig. 16 it is shown that the percentage increase in resistivity for p=0 with respect to p=1 decreases as width is increased, indicating that the effect of edge specularity reduces with increasing width. When the height is varied keeping the width fixed (Fig. 17), at very small heights the difference in resistivity for NSE and ASE cases is very low as in the case of width variation. However, with the increase in the height, the difference continues to increase and does not show any decreasing trend, which is obvious since the �f1/�x term in (1) is not affected. ASE or edge scattering is found to have very little effect on the inductance (Fig. 18).

Conclusion An accurate methodology based on Boltzmann equation for high-frequency impedance extraction of GNRs is presented for the first time followed by rigorous analysis of the various intricate processes occurring at high frequencies in GNRs. It is shown that ASE leads to significant increase in the resistance of GNR while the high KI of graphene results in some saturation effects. The complex inter-coupled relation between edge specularity, ASE, and GNR dimensions is elaborately discussed. The results indicate possible high-frequency applications of GNRs in interconnects and inductor designs.

978-1-4244-7677-0/10/$26.00 ©2010 IEEE978-1-4244-7678-7/10/$26.00 ©2010 IEEE

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References [1]C. Xu, H. Li and K. Banerjee, IEEE TED, vol. 56, no. 8, pp. 1567-1578, 2009. [2]H. Li and K. Banerjee, IEEE TED, vol. 56, no. 10, pp. 2202-2214, 2009. [3]Y. Awano, IEICE Trans. Electronics., vol. E89-C, pp. 1499-1503, Nov. 2006. [4] L. R. Hanlon, E. R. Falardeau and J. E. Fischer, Solid State Commn., vol.24, pp. 377-381, Sep. 1977. [5] E.H. Sondheimer, Advances in Physics, vol. 50, no. 6, pp. 499-537, Sept. 2001. [6] G.E.H.Reuter and E.H. Sondheimer, Proc. The Royal Society, no. 195, pp.336-364, 1948. [7] A.B. Pippard, Proc. The Royal Society, no. 191, pp. 385-399, 1947.

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Fig. 1 Schematic view of (a) mono-layer GNRs, and (b) neutral and (c) intercalation doped multi-layer GNRs. The solid lines indicate graphene layers, while the dots in (c) indicate intercalation dopant layers.

Fig. 5 (a) GNR structure for RF applications showing selection of axis and direction of electric field. (b) Carrier transport elaborated on a single layer. Carriers in GNR move within a 2D layer, get scattered by the edges and finally contribute to current along y. (c) The Boltzmann equation for the GNR structure reduces to (1) as the inter-layer impedance is very high and velocity along the z direction is negligible. Here f0 and f1 are the equilibrium and non-equilibrium distribution function, vx and vy represent the components of velocity along x and y axis, is the momentum relaxation time, � is the angular frequency, is the electric field, E is the energy, v and r are the velocity and radial vectors, Ef is the Fermi level.

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Fig. 16 Percentage increase in resistivity (�) as a result of perfectly diffuse scattering (p=0) with respect to that of perfectly elastic scattering (p=1) for both NSE and ASE cases show a decreasing trend with width.

Fig. 10 Current density distribution graphs across the cross-section of GNR(H=W=2µm) at f=120GHz. Due to skin effect J is confined mainly towardsthe boundaries. For NSE the current density J is symmetrical about the centre�����������%���������#� ������������������������������������� For NSEno visible difference is obtained between p=0 and p=1. The impact of edgescattering effect can only be appreciated when ASE is considered where current at the edges (i.e x=0 and x=W) is reduced compared to the boundaries z=0 and z=H (which are the bottom and top layers). For p=0, the edgescattering is more effective as is clear from the figure with ASE which is actually expected physically.

Fig. 11 Effect of kinetic inductance on (a) resistance and (b) inductance as afunction of frequency. The large KI of graphene leads to saturation of both resistance and inductance at high frequencies.

Fig. 12 Variation in resistance of a 2μm x 2μm GNR structure with frequency for (a) p=0 and (b) p=1, for the two cases: with and without considering the anomalous skin effect. Difference in resistance due to ASE becomes apparent after about 100 GHz and this difference increases as frequency is increaseddue to the decrease in skin depth with increasing frequency.

Fig. 18 Variation of inductance at a frequency of 400GHz with (a) width at fixed height (H=2μm) and (b) height at fixed width (W=2μm). Inductance decreases as width or height is increased. ASE or edge scattering is found to have negligible effect on the inductance values.

Fig. 17 Variation of resistivity withheight (W=2μm, f=400GHz ). For bothp=0 and p=1, the difference inresistivity between NSE and ASE cases increases with height.

Fig. 13. Percentage increase in resistance due to ASE with respect to NSE case as a function of frequency illustrating the effect of edge scattering. ASE has more impact for p=0 than that for p=1 and this difference increases with increasing frequency because of the confinement of current towards the edges making the effect of edge scattering significant

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