ABSTRACT
In one-dimensional fluid model [22], we regard the graphene sheet that is rolled to form a CNT is to be infinitesimally thin. The conduction electrons are then distributed over the lateral surface of the CNT cylinder shell and electrons are embedded in a rigid uniform positive charge background with a uniform surface number density. Thus, the motion of electrons is confined to the surface. Furthermore, electrical charge neutrality requires that in equilibrium, the conduction electron charge density precisely cancel with that of the background positive ions. Since the van der Waals force between the carbon atoms in different shells in MWCNTs is negligible compared to valence band between the carbon atoms in the same shell [23], the one-dimensional fluid model described by Eq. (4.9) in Chapter 4 can be applied to each shell of the MWCNT with modification because the electron-electron interaction in the MWCNT is different from that in the SWCNT, which means the parameter a needs to be recalculated.In addition, two assumptions are made: The electrons can only move along the z-axis; all other fluid variables, such as the tangential component of the electric field to the nanotube surface, are almost uniform in the cross section plane of the shells in MWCNT. These two assumptions are valid if both the nanotube length and
greater than the nanotube radius [24,25]. Furthermore, the analysis assumes room temperature operation and on distribution of elections in other directions than in z-direction for the MWCNT and SWCNT bundle interconnections. We assume that the velocity of these electrons equals the Fermi velocity. As a result, the kinetic energy is given by
2K F1= 4 × 7 eV2E M mv M (5.2) There are two channels in each shell of the MWCNT and two different spin electrons in each channel. So we consider that there are four electrons at the same point in each shell of the MWCNT. We can then calculate the potential energy by moving these 4M electrons from ∞ to the same point of the MWCNT. We first consider moving every four electrons into one shell of the MWCNT. The potential energy can be obtained as follows [22,26]:
4 2 2P 0 0=1 =2 =11 1 ,= ( –1) = 62 2M Mj jj i je eE i Md d e e (5.3)where dj is diameter of shell number j. We then consider moving all shells from ∞ to adjacent shells to construct an MWCNT. The potential energy can be calculated using following equation:
Following the derivation in [22], we can obtain an equation for each shell in an MWCNT: E K Q1 ,= + +i qRi L t C z (5.5)where R LK sgn (l)v is the resistance of each shell in an MWCNT per unit length. 2K F4L e v is the kinetic inductance per unit length
of each shell. ue = vF/ √______ 1-a is the thermodynamic speed of sound of the electron fluid under a neutral environment. The magnetic inductance per unit length of each shell can also be calculated using Eq. (4.37). In an SWCNT, magnetic inductance is neglected compared with kinetic inductance; therefore, it can also be neglected in each shell of an MWCNT.The outermost shell shields inner shells from the ground plane; therefore, the electrostatic capacitance, CE, does not exist in inner shells. However, there exists electrostatic capacitance, CS, between the neighboring metallic shells and its value is given by [26,27]
0S 2 ,= ln( )i jC D De (5.6)where e0 is the permittivity of vacuum, Di and Dj are the diameters of the i-th and j-th metallic shells, respectively and I < j.We assume that the outermost shell is metallic. In a recent work [27], we have derived an equivalent circuit of a metallic MWCNT interconnect as shown in Fig. 5.1. It is simplified as shown in Fig. 5.2 by considering that the RLC parts of all inner shells are identical. If we assume that there are no variation in distributed parameters, R and LK, then R and LK are same for each shell. The potential across components of each shell in an MWCNT is equal. As a result, a simplified equivalent circuit of an MWCNT interconnect can be derived as shown as Fig. 5.3. RC in Figs. 5.1-5.3 is the contact resistance and its ideal quantum value is 3.2 k per shell [12].
