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Based on the complex effective conductivity method, a closed-form expression for the internal impedance of mixed carbon nanotube (CNT) bundles, in which the number of CNTs for a given diameter follows a Gaussian distribution, is proposed in this paper. It can appropriately capture the skin effect as well as the temperature effect of mixed CNT bundles. The results of the closed-form expression and the numerical calculation are compared with various mean diameters, standard deviations, and temperatures. It is shown that the proposed model has very high accuracy in the whole frequency range considered, with maximum errors of 1% and 2.3% for the resistance and the internal inductance, respectively. Moreover, by using the proposed model, the high-frequency electrical characteristics of mixed CNT bundles are deeply analyzed to provide helpful design guidelines for their application in future high-performance three-dimensional integrated circuits.
Due to their extremely desirable properties of high mechanical and thermal stability, high thermal and electrical conductivity, and large current carrying capacity, carbon nanotubes (CNTs) have been identified as a promising candidate for next generation high-speed interconnect systems.[1–3] According to the number of rolled up graphene sheets, CNTs are classified as single-walled CNTs (SWCNTs, with only one shell) and multi-walled CNTs (MWCNTs, with several concentric shells).[4–6] SWCNTs can be either metallic or semiconducting depending on their chirality, and MWCNTs always behave as metallic conductors.[7] The high intrinsic ballistic resistance associated with an isolated CNT suggests the use of bundles consisting of numerous parallel connected CNTs to realize low-impedance interconnects.[8] It has been proven that CNT interconnects can provide significant performance enhancement with respect to Cu interconnects.[9]
Three-dimensional integrated circuits (3-D ICs) can dramatically enhance chip performance, functionality, and device packing density by stacking multiple layers of active devices vertically.[10,11] Through-silicon vias (TSVs) are essentially cylindrical conducting vias coated with a layer of insulating material residing in silicon substrates for vertical interconnects in 3-D ICs.[12] Recently, TSV arrays filled with CNT bundles were fabricated successfully for 3-D ICs using various methods.[13–16] Meanwhile, the equivalent-circuit models for pure CNT bundles based signal-ground TSV pairs were also proposed.[17–19] However, almost all experiment results show that a realistic CNT bundle is a mixed bundle consisting of SWCNTs and MWCNTs.[20–24] Considering the limitations and cost of the current diameter-controlling methods, adopting pure SWCNT bundles for large-scale integration appears to be impractical in the short term.[25,26] In addition, mixed CNT bundles have the advantages of easy fabrication and the potential benefit of improving the density of the conduction channels. Therefore, they will become the most potential material for high-performance TSVs in future 3-D ICs.
There are few research efforts that have been made to address the modeling and characterization of mixed CNT bundles due to their complexity in structure. The conductance and inductance models for mixed CNT bundles were firstly proposed in Refs. [26] and [27], respectively. However, since the effects of the coupling, frequency, and temperature were not considered, the proposed models are applied only to the conditions of DC and low-frequencies at room temperature. In addition, the models also need numerical calculations, so they are quite inconvenient to use in practice. Furthermore, a similar model for mixed CNT bundles was also proposed, and the effects of the CNT rearrangement inside the bundle on the capacitive and inductive crosstalk were analyzed deeply.[28] More recently, the equivalent-circuit model for mixed CNT bundle based differential TSVs was reported, but the DC conductance model was still used.[29] Therefore, the obtained results are inaccurate.
The objectives of this paper are to propose a closed-form internal impedance model of cylindrical mixed CNT bundles, and study their high-frequency electrical characteristics to explore the feasibility in the application of future high-performance 3-D ICs. The organization is as follows. In Section
Since an SWCNT, which consists of one shell, is a special case of the MWCNT, mixed CNT bundles can be treated as a combination of multiple MWCNTs.[29] The cross-sectional view of a mixed CNT bundle is shown in Fig.
For MWCNTs, the number of conduction channels per shell is[30]
According to the practical manufacturing process of mixed CNT bundles,[24] the number of MWCNTs for a given
The accurate expression for the internal impedance of cylindrical conductors is[32]
Figures
![]() | Fig. 2. (color online) Comparison of the resistances of mixed CNT bundles obtained from the closed-form expression and the numerical calculation. |
For mixed CNT bundles in the actual application, the number of CNTs is large enough so that the effect of the quantum capacitance can be neglected. In addition, their electrostatic capacitance and external inductance are the same as those of Cu interconnects, respectively. Therefore, the primary distinction between mixed CNT bundles and Cu interconnects is the internal impedance including the resistance and internal inductance. In this section, we will mainly analyze the effects of the mean diameter
As the mean diameter
![]() | Fig. 4. (color online) Resistance of mixed CNT bundles obtained from the proposed model versus the mean diameter and frequency. |
Although the standard deviation σ is a dominant factor determining the distribution of the diameter of MWCNTs inside a mixed CNT bundle, its effect on the total number of conduction channels is quite small. Therefore, the resistance decreases slowly as the standard deviation σ increases, as shown in Fig.
![]() | Fig. 5. (color online) Resistance of mixed CNT bundles obtained from the proposed model versus the standard deviation and frequency. |
As the temperature T increases, although the number of conduction channels of larger diameter MWCNTs increases linearly, which can effectively decrease the resistance, the resistance still increases sharply and the increasing-speed is almost frequency-independent, as shown in Fig.
Similar to the case of the resistance, as the mean diameter
![]() | Fig. 7. (color online) Internal inductance of mixed CNT bundles obtained from the proposed model versus the mean diameter and frequency. |
It is obvious that the internal inductance decreases slowly as the standard deviation σ increases, as shown in Fig.
![]() | Fig. 8. (color online) Internal inductance of mixed CNT bundles obtained from the proposed model versus the standard deviation and frequency. |
The internal inductance decreases as the temperature T increases and the rate is almost frequency-independent due to the linear increase of the number of conduction channels of larger diameter MWCNTs inside a mixed CNT bundle, as shown in Fig.
A closed-form expression for the internal impedance of mixed CNT bundles is proposed in this paper. The results of the proposed model agree extremely well with those of the numerical calculation up to 100 GHz, with maximum errors of 1% and 2.3% for the resistance and the internal inductance, respectively. In order to provide helpful design guidelines for mixed CNT bundles in future 3-D ICs, the effects of the mean diameter, standard deviation, and temperature on their internal impedance are analyzed deeply using the proposed model. It is shown that both the resistance and the internal inductance decrease rapidly as the mean diameter increases. Meanwhile, the two corresponding rates remain almost the same and increase as the frequency increases. However, they are decreasing slowly as the standard deviation increases since its effect on the total number of conduction channels is quite small. In addition, the resistance increases rapidly and the internal inductance decreases, respectively, as the temperature increases, and the corresponding rates are both almost frequency-independent.
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