High-Frequency Effects in Carbon Nanotube ... - Semantic Scholar
High-Frequency Effects in Carbon Nanotube Interconnects and Implications for On-Chip Inductor Design Hong Li and Kaustav Banerjee Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA e-mail: {hongli, kaustav}@ece.ucsb.edu
Abstract This paper presents a rigorous investigation of high-frequency effects in carbon nanotube interconnects and their implications for the design and performance analysis of high-quality on-chip inductors. An accurate method is developed to calculate the frequency-dependent resistance and inductance of both singlewalled (SWCNT) and multi-walled carbon nanotube (MWCNT) bundle interconnects. Our analysis reveals for the first time that skin effect (current redistribution) in CNT bundles is negligible compared to that in conventional metal conductors, which make them a very attractive and promising material for high-frequency applications, including on-chip inductor design in highperformance RF/mixed-signal circuits. It is subsequently shown that CNT based inductors can achieve nearly 4× higher Q factor than Cu based inductors.
I. Introduction Carbon nanotube (CNT) bundles have been proposed for VLSI interconnect applications [1] due to their outstanding electrical and thermal properties [2]. Previous CNT interconnect modeling works [3, 4] have only developed RC models for CNT interconnect analysis and have neglected inductance. Since inductance plays a critical role in high-frequency applications, it is important to extract inductance of CNT interconnects to understand their high-frequency characteristics and investigate their potential applications. For CNT inductance modeling, the only existing method is the “equivalent conductivity” method proposed in [5]. However, there are several fundamental issues that need to be justified in that method, where realistic CNT bundle with discrete conductors is replaced by a single solid conductor of identical dimension with an equivalent conductivity. As shown in Fig. 1, there could be very large differences in the current redistributions between a discrete bundle structure and a single solid conductor, which indicates that the method in [5] is flawed. Moreover, this method does not consider the impact of CNT kinetic inductance (LK) [7] when calculating the magnetic inductance. As will be shown later in this paper, kinetic inductance plays a critical role in determining the current redistribution, which in turn, affects the inductance of CNT bundle interconnects significantly. Furthermore, SWCNT based on-chip inductors have been analyzed without correctly capturing the resistance and inductance of CNT interconnects at high frequencies [8], leading to incorrect quantitative results and conclusions. As highlighted in this paper for the first time, CNT bundle interconnects have very different characteristics at high frequencies compared to conventional metals, based on which the performance of CNT based inductors is further investigated. Our analysis quells existing misunderstandings and provides insights that could potentially open up new vistas in RF/mixed-signal and off-chip applications of CNT interconnects as inductors.
II. Impedance Analysis of CNT Bundle In order to overcome the limitations of previous inductance extraction method, an accurate method for extracting the inductance of both SWCNT and MWCNT bundles (as shown in Fig. 2) has been developed. The realistic hollow cylindrical
structures and discreteness of the CNTs in the bundle are considered, and the kinetic inductance is taken into account. By employing the geometric mean distance (GMD) concept, magnetic self- and mutual- inductance of SWCNT and MWCNT can be derived based on the procedure in [10] as shown in Fig. 3. Having the equations for magnetic (self and mutual) inductance, the impedance matrix for a CNT bundle can be obtained as shown in Fig. 4. Note that self-impedance of each CNT consists of resistive and inductive impedance (both kinetic and magnetic). The resistance can be calculated by the equations in Fig. 4. Fig. 5(a) shows the d.c. resistivity of different type of CNTs [9]. Due to large number of conducting channels, SWCNT bundle with metallic fraction Fm = 1 has lower resisitivty than that of Cu. For MWCNTs, the conducting channel number is lower (for a given cross-sectional area) but the mean free path is longer than that of SWCNTs due to their large diameters [9]. Therefore, for long lengths (> 50 μm), the resistivity of MWCNT could be lower than that of Cu and even become comparable with that of an ideal SWCNT bundle (Fm=1) as shown in Fig. 5(a). Based on the resistance and inductance calculation method, the effective total impedance (thus effective resistance and inductance) of the bundle (Zeff = Reff + jωLeff) can be obtained by solving the last matrix equation in Fig. 4. Fig. 5(b) shows the fraction of kinetic inductance in the total effective inductance of CNT bundle interconnects for different cross-sections. As can be observed, kinetic inductance is important when the interconnect crosssection is ≤ 100 nm, which is the relevant range for on-chip interconnects in nanometer scale VLSI. Moreover, for large diameter MWCNT bundles, since the number of conducting channels is small compared to that of SWCNT bundles for a given cross-section [9], kinetic inductance forms a more significant fraction of the total inductance.
III. High-Frequency Analysis of CNT Interconnects Based on the method outlined above, the effective resistance and inductance of CNT bundle interconnects at high frequencies has been extracted. Fig. 6 shows the resistance and inductance of SWCNT and MWCNT bundle interconnects as a function of frequency as compared with identical cross-section Cu wire. It can be observed that the skin effect in the Cu wire is significant at high frequencies. However, for the CNT interconnects, their resistance and inductance almost remain constant with frequency. The reason behind this phenomenon can be attributed to the presence of large kinetic inductance (more than 2 orders larger than the magnetic component) in CNTs [9]. As illustrated in Fig. 7(a), due to the large kinetic inductance, the self-impedance of each CNT is always much larger than the mutual-impedance (because jωLkinetic >> jωM) even at very high frequencies. Hence, the self-impedance (Z iself) dominates the total impedance (Zi) for each CNT in the bundle. As a result, the current distribution is restricted, which implies that “skin effect” in CNT bundles at high frequencies is negligible. This explanation is verified in Fig. 8, where the current distribution in a CNT bundle interconnect is simulated for two cases: (a) without considering kinetic inductance, (b) taking kinetic inductance into account. It can be observed that the current distributions are significantly different
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in these two cases. After taking kinetic inductance into account, the current distribution becomes mostly uniform. Fig. 6 also shows the results obtained from the method in [5] for comparison. At low frequencies, the results using the method in [5] are in agreement with the ones derived using the analysis presented in this paper. However, as can be observed, because the kinetic inductance is not included in that method, the high-frequency properties of CNT bundles can not be correctly captured using that method. Since kinetic inductance is critical for highfrequency properties of CNT interconnect, any change in its value will affect the analysis. The value of kinetic inductance is determined by the density of states, and some works (such as [12]) have shown that external magnetic fields could modify CNT’s density of states, especially for large diameter MWCNTs. Hence, the effect of various kinetic inductance values on the resistance of a MWCNT interconnect is analyzed as shown in Fig. 7(b). It can be observed that even if the kinetic inductance value is reduced by one order of magnitude, the skin effect in CNT bundle interconnects would still be very small compared to that in a Cu wire.
IV. CNT Based Inductor Design The high-frequency behavior in CNT bundles (displaying negligible skin effect), as discussed above, is very promising for various high-frequency applications, such as high quality on-chip inductor design (shown in Fig. 9). In Fig. 9(a), a metal block at each corner is employed as contact for CNT bundles along perpendicular directions. Similar structure, in which CNT bundle is grown along perpendicular directions from a common block, has been fabricated in [13], which indicates that the structure shown in Fig. 9(a) can be possibly fabricated in the near future. Quality (Q) factor (defined as Q = Im(Z11)/Re(Z11) in our work, where Z11 is the input impedance of the network in Fig. 9(b)) is a very important performance metric for inductor design. For CNT based inductors, Q factor can be divided into two parts: Qmagnetic and Qkinetic, as shown in Fig. 10(a). It can be observed that although kinetic inductance of each CNT in a bundle has very large value, the value of Qkinetic is negligible ( 10), the magnetic field induced inductance still needs to be utilized. Since magnetic inductance depends on the geometry, one can not shrink the footprint of the inductor. Using the π model defined in Fig. 9(b) for on-chip inductors, the Q factor for CNT based inductors is analyzed and compared to that of Cu inductor as shown in Fig. 11 for different types of CNTs and substrates. For the low-loss (high-resistivity) substrate, the maximum Q factor of CNT based inductors (all metallic SWCNTs and MWCNTs with D = 40 nm) can be nearly 4 times higher than that of the identical size Cu inductor. This significant enhancement in Q factor arises not only because of the lower d.c. resistance of CNTs, but also because of the negligible skin effect in CNT interconnects. It can be clearly observed that even the worst case of CNTs (SWCNT bundle with Fm = 1/3) with much higher resistivity than Cu (see Fig. 5(a)), can achieve better performance than that of the Cu based inductor. For the high-loss substrate (low-resistivity), although the advantage of CNT’s highfrequency property is shadowed by the large substrate eddy current loss (see Fig. 9(c)), CNT based inductors can still achieve as large as 65% increase in maximum Q factor. The resistance and inductance for the inductor at high-frequencies is further shown in Fig. 12, which again highlights the negligible skin effect in CNT bundles. Fig. 13 shows that the impact of imperfect contact resistance (Rmc) is negligible for SWCNT based inductors,
but significant for large diameter MWCNT based inductors due to lower number of conducting channels of MWCNT bundles for a given cross-sectional area [9]. Since fabricating densely packed CNT interconnect is still challenging [16], Fig. 14 illustrates the impact of the volume density of CNT bundles on the performance of inductors. It can be observed that the CNT bundle density affects the performance dramatically. For the high-loss substrate applications, the volume density requirement of CNT bundles is higher than that for the low-loss substrate in order to outperform the Cu inductor. The maximum Q factor enhancement of CNT inductors compared to Cu inductor as a function of the number of turns is also displayed in Fig. 15. For the low-loss substrate, the percentage enhancement of maximum Q factor of CNT based inductor with respect to that of Cu decreases with the number of turns because an inductor with higher number of turns has lower operating frequency, and hence, the Cu based inductors are less affected by skin effect. For the high-loss substrate, the results shown in Fig. 15(b) arise from the relative impact of the substrate loss (Reddy and Leddy) and current redistribution (affecting RS and LS) on the Q factor of the inductors. Recently, very high frequency RF circuits have generated significant interest (such as 60 GHz applications) [17], Fig. 16 shows the Q factor of inductors for several tens of gigahertz applications. It can be observed that at very high frequencies, even the worst case CNT bundle can achieve better performance than that of the Cu inductor, which implies that CNT bundles would be highly promising for on-chip inductor design.
V. Conclusion Our analysis in this work illustrates for the first time that due to the presence of large kinetic inductance in each CNT, the resistance and inductance of CNT bundles vary slightly with frequency. This preferable high-frequency property is then explored in the design and analysis of high-performance on-chip inductors. It is demonstrated that the Q factor of CNT based inductors could be nearly as high as 4× compared to those based on Cu. From a processing perspective, imperfect contact resistance, volume density of CNT bundles, and metallic fractions of SWCNT bundles have significant impact on CNT inductors’ performance. Since fabricating ideal densely packed SWCNT bundles (100% metallic) remains challenging, large diameter MWCNT interconnects could be very promising for future highfrequency interconnect applications including inductor design.
Acknowledgment This research is being supported by the National Science Foundation, Grant No. CCF-0811880.
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4.0
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Fig. 5. (a) D.C. resistivity comparison among SWCNT, MWCNT and Cu interconnects for different diameters and lengths. Fm represents the fraction of metallic CNTs in SWCNT bundles (assuming d = 0.34 nm). (b) Ratio of kinetic inductance ( Lkinetic CNT / ntot ) and total effective inductance of CNT bundles as a function of interconnect width. The interconnect height is set equal to the width, interconnect length is set to be 500 μm.
⎡W − D ⎤ nW = ⎢ ⎥+1 ⎣D+d ⎦
0.68
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Cu wire SWCNT D=1nm, Fm=1 MWCNT D=20nm MWCNT D=40nm 10 Length = 500 μm 8 Width=2μm, Height=1μm
Inductance [nH]
Resistance [Ω]
⎡ ⎤ H−D nH = ⎢ ⎥+1 ⎣ ( 3 2)(D + d ) ⎦ ⎡ nH ⎤ ntot = nw ⋅ nH − ⎢ ⎥ ⎣ 2 ⎦
SWCNT Fm=1 Fm=1/3
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MWCNT D=40nm
0.5 10
Fig. 1. Current distribution differences between a single solid conductor and a bundle (several discrete conductors) structure. (a) single solid 320 nm × 500 nm cross-section Cu conductor, (b) bundle of discrete solid conductors (each with 20 nm square cross-section, and 10 nm interval). The bundle has identical d.c. resistance and total current as the Cu conductor in (a). Simulation is implemented using the electromagnetic field solver Maxwell [6] at 100 GHz.
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1 8 nH / μ m, N ch : number of conducting channels N ch μ0 ⎡ 2L AMD ⎤ LSelf L ⋅ ln , ln GMD = ln Dout − ln χ , AMD ≈ 2 Dout −1+ CNT = 2π ⎣⎢ GMD L ⎦⎥ π ln χ = 0.1(a − bγ − cγ 2 + d γ 3 ), γ = Din / Dout , a = 2.51, b = 0.31, c = 3.81, d = 1.61 Lkinetic CNT =
Din
S
Dout
⎡ ⎛L μ L2 ⎞ S2 S ⎤ M = 0 ⋅ L ⋅ ⎢ln ⎜ + 1 + 2 ⎟ − 1 + 2 + ⎥ ⎜ ⎟ 2π S L⎥ S L ⎠ ⎣⎢ ⎝ ⎦
(LCNTkinetic),
Fig. 3. Equations for kinetic inductance magnetic self-inductance (LCNTSelf), and mutual inductance (M) between CNTs in a bundle. Nch is the number of conducting channels of CNTs, L is the length of CNT bundle, S is the center point distance between two adjacent CNTs. The thickness of SWCNT shell is assumed to be the diameter of carbon atom ( = 0.142 nm). For MWCNTs, γ is assumed to be 1/2. The equations are derived using the concept of Geometric Mean Distance (GMD) and Arithmetic Mean Distance (AMD) [10]. For SWCNT, Nch = 2; for MWCNT, Nch can be obtained from [9]. 1 ⎡ ZCNT ⎢ j ω M 21 [ Z matrix ] = ⎢⎢ ⎢ n1 ⎣⎢ jω M
jω M 21 2 ZCNT
jω M n 2
Z = jω ( L +L )+R jω M n1 ⎤ i ⎥ RSWCNT = RQ (1 + L / λ ) / N chSWCNT jω M n 2 ⎥ , i ⎥ RMWCNT = 1/ ∑ N chshell / ( RQ + RQ ⋅ L / λ ) ⎥ shell n Z CNT ⎦⎥ where RQ = 12.9 K Ω，λ = 1000 D
[V ] = [ Z matrix ][ I ] ⇒ Z eff
i CNT
Self CNTi
kinetic CNTi
i CNT
= Reff + jω Leff
Fig. 4. Impedance matrix of a CNT bundle. Note that the self-impedance (ZiCNT) not only includes resistance (RiCNT) and magnetic self-inductive impedance (jωLCNTSelf), but also includes inductive impedance due to kinetic inductance (jωLCNTkinetic). The equations for both SWCNT and MWCNT resistance are also shown, where λ is the mean free path of CNTs. RQ is the quantum resistance per channel. Total effective resistance (Reff) and inductance (Leff) of CNT bundle can be computed by solving the matrix equation, [V] = [Z][I], to extract [I] for a known voltage across the bundle.
10
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Fig. 6. Total effective (a) resistance and (b) inductance of SWCNT and MWCNT bundle interconnects as a function of frequency, compared to identical cross-section Cu wire. The Cu wire is simulated using FastHenry [11]. The broken lines are the results obtained from the “equivalent conductivity” method in [5], which show that their model can not correctly capture CNT high-frequency effects. 20
Cu LK=0.1 LK0 LK=0.5 LK0 LK= LK0
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i i Z i = Z self + Z mutual i kinetic self Z = RCNT + jω ( LCNTi + LCNTi ) Ij i Z mutual = jω ∑ M i , j Ii j ≠i i self
∵ Lkinetic CNTi >> M i , j， i dominates ⇒ Z1 ≈ Z 2 ∴ Z self
Resistance [Ω]
Fig. 2. Cross-sectional view of CNT bundles, width W and height H. The diameter of each CNT is D and the interval between neighboring CNTs is d (for densely packed bundles, d = 0.34 nm, which is the Van der Waal’s gap). nW and nH are the number of CNTs along the width and height, respectively. Equation for the total number of CNTs (ntot) is shown alongside, where the operator “[·]” indicates that only the integer part is taken into account. For MWCNT, the diameter of innermost shell is assumed to be D/2. The equivalent circuit model [9] of each CNT shell is also shown alongside, where Rmc is the imperfect metal-CNT contact resistance, RQ is quantum resistance, RS is ohmic resistance, LK is kinetic inductance, LM is magnetic inductance, CQ is quantum capacitance, and CE is electrostatic capacitance.
Width=2μm, Height=1μm
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MWCNT, D=40nm Length=500μm Width=2μm, Height=1μm
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Fig. 7. (a) Illustration of impedance among different CNTs in a bundle. Due to the presence of large kinetic inductance, self-impedance always dominates total impedance, which induces negligible current redistribution. (b) The resistance of MWCNT bundle as a function of frequency with different kinetic inductance values. LK0 = 8 nH/μm per channel is the nominal value [7]. Compared to the Cu wire, the resistance increase in the MWCNT bundle structure is very small even if the kinetic inductance value is reduced by one order of magnitude. 3
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Fig. 8. Current distribution in a CNT bundle interconnect (a) without, and (b) with kinetic inductance being taken into account at frequency of 50 GHz. The height and width of the CNT bundle are 3 μm and 2 μm, respectively. These figures are generated by MATLAB® using the method shown in Fig. 4.
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MWCNT D=40nm
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MWCNT D=20nm
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0.06 frequency 100GHz frequency 50GHz frequency 10GHz
( Lmagnetic + Lkinetic ) R = Qmagnetic + Qkinetic
Qkinetic = ω
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Length [μm]
Fig. 10. (a) Schematic view of the Q factor curve of a spiral CNT inductor. The upper-bound of Q factor is ωL/R as shown by the broken line. Due to the presence of parasitic capacitance and conductor/substrate loss, the actual Q factor curve drops down at high frequencies. (b) Qkinetic as a function of frequency and length. The equation used for calculating Qkinetic is shown in the inset. λ is assumed to be 1 μm. 120
80 60
ρsub= 10 Ω-cm
40 20 0 0.1
Cu SWCNT Fm=1/3 SWCNT Fm=1
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Fig. 14. The maximum Q factor of SWCNT and MWCNT bundle based inductors of Fig. 11 as a function of volume density of CNT bundles, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. The volume density is normalized with respect to the densely packed case (d = 0.34 nm). 300 SWCNT Fm=1
250 200
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MWCNT D=40nm MWCNT D=20nm
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Fig. 12. Total series resistance and inductance (LS and RS in Fig. 9(b)) of Cu-, SWCNT-, and MWCNT- based inductors of Fig. 11 as a function of frequency.
SWCNT Fm=1
60 50 40 30
MWCNT D=40nm ρsub= 0.01 Ω-cm Rmc= 10 KΩ
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ρ sub= 10 Ω-cm Rmc= 10 KΩ
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Fig. 11. Quality factor of inductors based on Cu, SWNCT, and MWCNT interconnects as a function of frequency, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. All inductors have outer diameter of 200 μm, 4 turns, wire width W = 10 μm, wire thickness H = 2 μm, wire spacing S = 1 μm. Oxide and substrate thickness are assumed to be 10 μm and 300 μm, respectively. CNT bundles are assumed to be densely packed (d = 0.34 nm) and have perfect contacts (Rmc = 0). 8
10
Rmc per Channel [KΩ]
Normalized Volume Density
LK × L / N ch LK =ω RQ (1 + L / λ ) N ch RQ (1/ L + 1/ λ )
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ω
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R
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Maximum Quality Factor Increase [%]
=
ω Ltotal
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SWCNT Fm=1/3
7
120
Fig. 15. The percentage increase in maximum Q factor of SWCNT and MWCNT bundle based inductors compared to Cu based inductor for different turn numbers, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. The outer diameter of the inductor is fixed at 200 μm, other parameters are the same as those specified in Fig. 11, except the turn number. ρsub= 10 Ω-cm Rmc= 10 KΩ
100
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Qup bound =
QKinetic
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Rmc per Channel [KΩ]
MWCNT D=10nm
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Cu SWCNT Fm=1/3 SWCNT Fm=1 MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
Quality Factor
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0
MWCNT D=20nm
9
Fig. 13. The maximum Q factor of SWCNT and MWCNT bundle based inductors of Fig. 11 as a function of imperfect metal-CNT contact resistance per channel (Rmc), for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. Impact of Rmc is negligible for SWCNT inductors, but needs to be considered for MWCNT inductors.
Maximum Quality Factor
Fig. 9. (a) The schematic view of a 4-turn spiral CNT inductor, with a metal contact at each corner. (b) Equivalent π model for on-chip spiral inductors. CS, Cox, and Csub are the inter-turn conductor capacitance, oxide capacitance, and substrate capacitance, respectively (calculated using distributed capacitance model [14]). LS and RS are the frequency dependent series inductance and resistance, respectively. For CNT, LS and RS are calculated using the method shown in Fig. 4, which includes Lk and RQ. For Cu, LS and RS are extracted using FastHenry [11]. The substrate eddy currents induced parameters, Leddy and Reddy, can be obtained using the complex image theory [15]. (c) Schematic view of substrate eddy currents (opposite to the direction of inductor currents).
MWCNT D=40nm
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Dout
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ρsub= 0.01 Ω-cm
ρsub= 10 Ω-cm
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ρsub= 0.01 Ω-cm Rmc= 10 KΩ
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MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
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Fig. 16. Quality factor of inductors based on Cu, SWNCT, and MWCNT interconnects as a function of frequency for very high-frequency applications, such as 60 GHz application. The substrate is assumed to be (a) low-loss (ρ = 10 Ω·cm), (b) high-loss (ρ = 0.01 Ω·cm). The simulated inductor is 0.75 turn (as shown in the inset figure) and has outer diameter of 100 μm, width 5 μm, thickness 1 μm. The thickness of oxide and substrate are assumed to be 6 μm and 200 μm, respectively.
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Abstract This paper presents a rigorous investigation of high-frequency effects in carbon nanotube interconnects and their implications for the design and performance analysis of high-quality on-chip inductors. An accurate method is developed to calculate the frequency-dependent resistance and inductance of both singlewalled (SWCNT) and multi-walled carbon nanotube (MWCNT) bundle interconnects. Our analysis reveals for the first time that skin effect (current redistribution) in CNT bundles is negligible compared to that in conventional metal conductors, which make them a very attractive and promising material for high-frequency applications, including on-chip inductor design in highperformance RF/mixed-signal circuits. It is subsequently shown that CNT based inductors can achieve nearly 4× higher Q factor than Cu based inductors.
I. Introduction Carbon nanotube (CNT) bundles have been proposed for VLSI interconnect applications [1] due to their outstanding electrical and thermal properties [2]. Previous CNT interconnect modeling works [3, 4] have only developed RC models for CNT interconnect analysis and have neglected inductance. Since inductance plays a critical role in high-frequency applications, it is important to extract inductance of CNT interconnects to understand their high-frequency characteristics and investigate their potential applications. For CNT inductance modeling, the only existing method is the “equivalent conductivity” method proposed in [5]. However, there are several fundamental issues that need to be justified in that method, where realistic CNT bundle with discrete conductors is replaced by a single solid conductor of identical dimension with an equivalent conductivity. As shown in Fig. 1, there could be very large differences in the current redistributions between a discrete bundle structure and a single solid conductor, which indicates that the method in [5] is flawed. Moreover, this method does not consider the impact of CNT kinetic inductance (LK) [7] when calculating the magnetic inductance. As will be shown later in this paper, kinetic inductance plays a critical role in determining the current redistribution, which in turn, affects the inductance of CNT bundle interconnects significantly. Furthermore, SWCNT based on-chip inductors have been analyzed without correctly capturing the resistance and inductance of CNT interconnects at high frequencies [8], leading to incorrect quantitative results and conclusions. As highlighted in this paper for the first time, CNT bundle interconnects have very different characteristics at high frequencies compared to conventional metals, based on which the performance of CNT based inductors is further investigated. Our analysis quells existing misunderstandings and provides insights that could potentially open up new vistas in RF/mixed-signal and off-chip applications of CNT interconnects as inductors.
II. Impedance Analysis of CNT Bundle In order to overcome the limitations of previous inductance extraction method, an accurate method for extracting the inductance of both SWCNT and MWCNT bundles (as shown in Fig. 2) has been developed. The realistic hollow cylindrical
structures and discreteness of the CNTs in the bundle are considered, and the kinetic inductance is taken into account. By employing the geometric mean distance (GMD) concept, magnetic self- and mutual- inductance of SWCNT and MWCNT can be derived based on the procedure in [10] as shown in Fig. 3. Having the equations for magnetic (self and mutual) inductance, the impedance matrix for a CNT bundle can be obtained as shown in Fig. 4. Note that self-impedance of each CNT consists of resistive and inductive impedance (both kinetic and magnetic). The resistance can be calculated by the equations in Fig. 4. Fig. 5(a) shows the d.c. resistivity of different type of CNTs [9]. Due to large number of conducting channels, SWCNT bundle with metallic fraction Fm = 1 has lower resisitivty than that of Cu. For MWCNTs, the conducting channel number is lower (for a given cross-sectional area) but the mean free path is longer than that of SWCNTs due to their large diameters [9]. Therefore, for long lengths (> 50 μm), the resistivity of MWCNT could be lower than that of Cu and even become comparable with that of an ideal SWCNT bundle (Fm=1) as shown in Fig. 5(a). Based on the resistance and inductance calculation method, the effective total impedance (thus effective resistance and inductance) of the bundle (Zeff = Reff + jωLeff) can be obtained by solving the last matrix equation in Fig. 4. Fig. 5(b) shows the fraction of kinetic inductance in the total effective inductance of CNT bundle interconnects for different cross-sections. As can be observed, kinetic inductance is important when the interconnect crosssection is ≤ 100 nm, which is the relevant range for on-chip interconnects in nanometer scale VLSI. Moreover, for large diameter MWCNT bundles, since the number of conducting channels is small compared to that of SWCNT bundles for a given cross-section [9], kinetic inductance forms a more significant fraction of the total inductance.
III. High-Frequency Analysis of CNT Interconnects Based on the method outlined above, the effective resistance and inductance of CNT bundle interconnects at high frequencies has been extracted. Fig. 6 shows the resistance and inductance of SWCNT and MWCNT bundle interconnects as a function of frequency as compared with identical cross-section Cu wire. It can be observed that the skin effect in the Cu wire is significant at high frequencies. However, for the CNT interconnects, their resistance and inductance almost remain constant with frequency. The reason behind this phenomenon can be attributed to the presence of large kinetic inductance (more than 2 orders larger than the magnetic component) in CNTs [9]. As illustrated in Fig. 7(a), due to the large kinetic inductance, the self-impedance of each CNT is always much larger than the mutual-impedance (because jωLkinetic >> jωM) even at very high frequencies. Hence, the self-impedance (Z iself) dominates the total impedance (Zi) for each CNT in the bundle. As a result, the current distribution is restricted, which implies that “skin effect” in CNT bundles at high frequencies is negligible. This explanation is verified in Fig. 8, where the current distribution in a CNT bundle interconnect is simulated for two cases: (a) without considering kinetic inductance, (b) taking kinetic inductance into account. It can be observed that the current distributions are significantly different
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in these two cases. After taking kinetic inductance into account, the current distribution becomes mostly uniform. Fig. 6 also shows the results obtained from the method in [5] for comparison. At low frequencies, the results using the method in [5] are in agreement with the ones derived using the analysis presented in this paper. However, as can be observed, because the kinetic inductance is not included in that method, the high-frequency properties of CNT bundles can not be correctly captured using that method. Since kinetic inductance is critical for highfrequency properties of CNT interconnect, any change in its value will affect the analysis. The value of kinetic inductance is determined by the density of states, and some works (such as [12]) have shown that external magnetic fields could modify CNT’s density of states, especially for large diameter MWCNTs. Hence, the effect of various kinetic inductance values on the resistance of a MWCNT interconnect is analyzed as shown in Fig. 7(b). It can be observed that even if the kinetic inductance value is reduced by one order of magnitude, the skin effect in CNT bundle interconnects would still be very small compared to that in a Cu wire.
IV. CNT Based Inductor Design The high-frequency behavior in CNT bundles (displaying negligible skin effect), as discussed above, is very promising for various high-frequency applications, such as high quality on-chip inductor design (shown in Fig. 9). In Fig. 9(a), a metal block at each corner is employed as contact for CNT bundles along perpendicular directions. Similar structure, in which CNT bundle is grown along perpendicular directions from a common block, has been fabricated in [13], which indicates that the structure shown in Fig. 9(a) can be possibly fabricated in the near future. Quality (Q) factor (defined as Q = Im(Z11)/Re(Z11) in our work, where Z11 is the input impedance of the network in Fig. 9(b)) is a very important performance metric for inductor design. For CNT based inductors, Q factor can be divided into two parts: Qmagnetic and Qkinetic, as shown in Fig. 10(a). It can be observed that although kinetic inductance of each CNT in a bundle has very large value, the value of Qkinetic is negligible ( 10), the magnetic field induced inductance still needs to be utilized. Since magnetic inductance depends on the geometry, one can not shrink the footprint of the inductor. Using the π model defined in Fig. 9(b) for on-chip inductors, the Q factor for CNT based inductors is analyzed and compared to that of Cu inductor as shown in Fig. 11 for different types of CNTs and substrates. For the low-loss (high-resistivity) substrate, the maximum Q factor of CNT based inductors (all metallic SWCNTs and MWCNTs with D = 40 nm) can be nearly 4 times higher than that of the identical size Cu inductor. This significant enhancement in Q factor arises not only because of the lower d.c. resistance of CNTs, but also because of the negligible skin effect in CNT interconnects. It can be clearly observed that even the worst case of CNTs (SWCNT bundle with Fm = 1/3) with much higher resistivity than Cu (see Fig. 5(a)), can achieve better performance than that of the Cu based inductor. For the high-loss substrate (low-resistivity), although the advantage of CNT’s highfrequency property is shadowed by the large substrate eddy current loss (see Fig. 9(c)), CNT based inductors can still achieve as large as 65% increase in maximum Q factor. The resistance and inductance for the inductor at high-frequencies is further shown in Fig. 12, which again highlights the negligible skin effect in CNT bundles. Fig. 13 shows that the impact of imperfect contact resistance (Rmc) is negligible for SWCNT based inductors,
but significant for large diameter MWCNT based inductors due to lower number of conducting channels of MWCNT bundles for a given cross-sectional area [9]. Since fabricating densely packed CNT interconnect is still challenging [16], Fig. 14 illustrates the impact of the volume density of CNT bundles on the performance of inductors. It can be observed that the CNT bundle density affects the performance dramatically. For the high-loss substrate applications, the volume density requirement of CNT bundles is higher than that for the low-loss substrate in order to outperform the Cu inductor. The maximum Q factor enhancement of CNT inductors compared to Cu inductor as a function of the number of turns is also displayed in Fig. 15. For the low-loss substrate, the percentage enhancement of maximum Q factor of CNT based inductor with respect to that of Cu decreases with the number of turns because an inductor with higher number of turns has lower operating frequency, and hence, the Cu based inductors are less affected by skin effect. For the high-loss substrate, the results shown in Fig. 15(b) arise from the relative impact of the substrate loss (Reddy and Leddy) and current redistribution (affecting RS and LS) on the Q factor of the inductors. Recently, very high frequency RF circuits have generated significant interest (such as 60 GHz applications) [17], Fig. 16 shows the Q factor of inductors for several tens of gigahertz applications. It can be observed that at very high frequencies, even the worst case CNT bundle can achieve better performance than that of the Cu inductor, which implies that CNT bundles would be highly promising for on-chip inductor design.
V. Conclusion Our analysis in this work illustrates for the first time that due to the presence of large kinetic inductance in each CNT, the resistance and inductance of CNT bundles vary slightly with frequency. This preferable high-frequency property is then explored in the design and analysis of high-performance on-chip inductors. It is demonstrated that the Q factor of CNT based inductors could be nearly as high as 4× compared to those based on Cu. From a processing perspective, imperfect contact resistance, volume density of CNT bundles, and metallic fractions of SWCNT bundles have significant impact on CNT inductors’ performance. Since fabricating ideal densely packed SWCNT bundles (100% metallic) remains challenging, large diameter MWCNT interconnects could be very promising for future highfrequency interconnect applications including inductor design.
Acknowledgment This research is being supported by the National Science Foundation, Grant No. CCF-0811880.
References [1] F. Kreupl et al., IEDM Tech. Dig., 2004, pp. 683-686. [2] P. L. McEuen et al., IEEE Trans. Nanotech., vol. 1, no. 1, pp. 78, 2002. [3] N. Srivastava et al., IEEE/ACM Intl. Conf. on CAD, 2005, pp. 383-390. [4] A. Naeemi et al., IEEE Trans. Elect. Dev., vol. 54, no. 1, pp. 26, 2007. [5] A. Nieuwoudt et al., IEEE Trans. Nanotech., vol. 5, no. 6, pp. 758, 2006. [6] Maxwell SV, Ansoft Corp., http://www.ansoft.com/maxwellsv/. [7] P. J. Burke, IEEE Trans. Nanotech., vol. 1, no. 3, pp. 129-144, 2002. [8] A. Nieuwoudt et al., IEEE Trans. Elect. Dev., vol.55, no.6, pp.298, 2008. [9] H. Li et al., IEEE Trans. Elect. Dev., vol. 55, no. 6, pp. 1328, 2008. [10] F. W. Grover, Inductance Calculations. Dover publications, 2004. [11] M. Kamon et al., IEEE Trans. Microwave Theory Tech., vol.42, no. 9, pp. 1750, 1994. [12] N. Nemec et al., Pys. Rev. B, 74(165411), 2006. [13] Y. Awano, IEICE Trans. Elect., vol. E89-C, pp. 1499-1503, 2006. [14] C. Wu et al., IEEE J. of Solid-State Cir., vol. 38, no. 6, pp. 1040, 2003. [15] A. Weisshaar et al., IEEE Trans. Adv. Pack., vol. 27, no. 1, pp.126, 2004. [16] M. Nihei et al., Proc. of Intl. Interconnect Conf. 2007, pp. 204-206. [17] B. Razavi, IEEE J. of Solid-State Circuits, vol. 41, no. 1, pp. 17, 2006.
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4.0
1.0
3.0 2.5 2.0
MWCNT D=10nm Cu
MWCNT D=20nm
1.5 1.0
0.6 0.4
SWCNT Fm=1
100
1000
Length [μm]
0.0 10
100
Width of Interconnect [nm]
1000
Fig. 5. (a) D.C. resistivity comparison among SWCNT, MWCNT and Cu interconnects for different diameters and lengths. Fm represents the fraction of metallic CNTs in SWCNT bundles (assuming d = 0.34 nm). (b) Ratio of kinetic inductance ( Lkinetic CNT / ntot ) and total effective inductance of CNT bundles as a function of interconnect width. The interconnect height is set equal to the width, interconnect length is set to be 500 μm.
⎡W − D ⎤ nW = ⎢ ⎥+1 ⎣D+d ⎦
0.68
20
Cu wire SWCNT D=1nm, Fm=1 MWCNT D=20nm MWCNT D=40nm 10 Length = 500 μm 8 Width=2μm, Height=1μm
Inductance [nH]
Resistance [Ω]
⎡ ⎤ H−D nH = ⎢ ⎥+1 ⎣ ( 3 2)(D + d ) ⎦ ⎡ nH ⎤ ntot = nw ⋅ nH − ⎢ ⎥ ⎣ 2 ⎦
SWCNT Fm=1 Fm=1/3
0.2
MWCNT D=40nm
0.5 10
Fig. 1. Current distribution differences between a single solid conductor and a bundle (several discrete conductors) structure. (a) single solid 320 nm × 500 nm cross-section Cu conductor, (b) bundle of discrete solid conductors (each with 20 nm square cross-section, and 10 nm interval). The bundle has identical d.c. resistance and total current as the Cu conductor in (a). Simulation is implemented using the electromagnetic field solver Maxwell [6] at 100 GHz.
MWCNT D=10nm D=20nm D=40nm
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SWCNT Fm=1/3
LK/Leff
Resistivity [μΩ-cm]
3.5
6 4
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MWCNT D=40nm MWCNT D=20nm SWCNT D=1nm Fm=1 Cu wire
0.64
0.62 Length = 500 μm 2
1 8 nH / μ m, N ch : number of conducting channels N ch μ0 ⎡ 2L AMD ⎤ LSelf L ⋅ ln , ln GMD = ln Dout − ln χ , AMD ≈ 2 Dout −1+ CNT = 2π ⎣⎢ GMD L ⎦⎥ π ln χ = 0.1(a − bγ − cγ 2 + d γ 3 ), γ = Din / Dout , a = 2.51, b = 0.31, c = 3.81, d = 1.61 Lkinetic CNT =
Din
S
Dout
⎡ ⎛L μ L2 ⎞ S2 S ⎤ M = 0 ⋅ L ⋅ ⎢ln ⎜ + 1 + 2 ⎟ − 1 + 2 + ⎥ ⎜ ⎟ 2π S L⎥ S L ⎠ ⎣⎢ ⎝ ⎦
(LCNTkinetic),
Fig. 3. Equations for kinetic inductance magnetic self-inductance (LCNTSelf), and mutual inductance (M) between CNTs in a bundle. Nch is the number of conducting channels of CNTs, L is the length of CNT bundle, S is the center point distance between two adjacent CNTs. The thickness of SWCNT shell is assumed to be the diameter of carbon atom ( = 0.142 nm). For MWCNTs, γ is assumed to be 1/2. The equations are derived using the concept of Geometric Mean Distance (GMD) and Arithmetic Mean Distance (AMD) [10]. For SWCNT, Nch = 2; for MWCNT, Nch can be obtained from [9]. 1 ⎡ ZCNT ⎢ j ω M 21 [ Z matrix ] = ⎢⎢ ⎢ n1 ⎣⎢ jω M
jω M 21 2 ZCNT
jω M n 2
Z = jω ( L +L )+R jω M n1 ⎤ i ⎥ RSWCNT = RQ (1 + L / λ ) / N chSWCNT jω M n 2 ⎥ , i ⎥ RMWCNT = 1/ ∑ N chshell / ( RQ + RQ ⋅ L / λ ) ⎥ shell n Z CNT ⎦⎥ where RQ = 12.9 K Ω，λ = 1000 D
[V ] = [ Z matrix ][ I ] ⇒ Z eff
i CNT
Self CNTi
kinetic CNTi
i CNT
= Reff + jω Leff
Fig. 4. Impedance matrix of a CNT bundle. Note that the self-impedance (ZiCNT) not only includes resistance (RiCNT) and magnetic self-inductive impedance (jωLCNTSelf), but also includes inductive impedance due to kinetic inductance (jωLCNTkinetic). The equations for both SWCNT and MWCNT resistance are also shown, where λ is the mean free path of CNTs. RQ is the quantum resistance per channel. Total effective resistance (Reff) and inductance (Leff) of CNT bundle can be computed by solving the matrix equation, [V] = [Z][I], to extract [I] for a known voltage across the bundle.
10
100
1
Frequency [GHz]
10
100
Frequency [GHz]
Fig. 6. Total effective (a) resistance and (b) inductance of SWCNT and MWCNT bundle interconnects as a function of frequency, compared to identical cross-section Cu wire. The Cu wire is simulated using FastHenry [11]. The broken lines are the results obtained from the “equivalent conductivity” method in [5], which show that their model can not correctly capture CNT high-frequency effects. 20
Cu LK=0.1 LK0 LK=0.5 LK0 LK= LK0
15
i i Z i = Z self + Z mutual i kinetic self Z = RCNT + jω ( LCNTi + LCNTi ) Ij i Z mutual = jω ∑ M i , j Ii j ≠i i self
∵ Lkinetic CNTi >> M i , j， i dominates ⇒ Z1 ≈ Z 2 ∴ Z self
Resistance [Ω]
Fig. 2. Cross-sectional view of CNT bundles, width W and height H. The diameter of each CNT is D and the interval between neighboring CNTs is d (for densely packed bundles, d = 0.34 nm, which is the Van der Waal’s gap). nW and nH are the number of CNTs along the width and height, respectively. Equation for the total number of CNTs (ntot) is shown alongside, where the operator “[·]” indicates that only the integer part is taken into account. For MWCNT, the diameter of innermost shell is assumed to be D/2. The equivalent circuit model [9] of each CNT shell is also shown alongside, where Rmc is the imperfect metal-CNT contact resistance, RQ is quantum resistance, RS is ohmic resistance, LK is kinetic inductance, LM is magnetic inductance, CQ is quantum capacitance, and CE is electrostatic capacitance.
Width=2μm, Height=1μm
1
MWCNT, D=40nm Length=500μm Width=2μm, Height=1μm
10
5
0
1
10
Frequency [GHz]
100
Fig. 7. (a) Illustration of impedance among different CNTs in a bundle. Due to the presence of large kinetic inductance, self-impedance always dominates total impedance, which induces negligible current redistribution. (b) The resistance of MWCNT bundle as a function of frequency with different kinetic inductance values. LK0 = 8 nH/μm per channel is the nominal value [7]. Compared to the Cu wire, the resistance increase in the MWCNT bundle structure is very small even if the kinetic inductance value is reduced by one order of magnitude. 3
10
2.5
8 2
6 [µA]
1.5
[µm] 1
4
0.5
2
0 0
0.5
1
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2 0
0.5
1
1.5
2
0
[µm]
Fig. 8. Current distribution in a CNT bundle interconnect (a) without, and (b) with kinetic inductance being taken into account at frequency of 50 GHz. The height and width of the CNT bundle are 3 μm and 2 μm, respectively. These figures are generated by MATLAB® using the method shown in Fig. 4.
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120
100
MWCNT D=40nm
90 80
MWCNT D=20nm
70 60
MWCNT D=10nm
50
SWCNT Fm=1/3
11
0.06 frequency 100GHz frequency 50GHz frequency 10GHz
( Lmagnetic + Lkinetic ) R = Qmagnetic + Qkinetic
Qkinetic = ω
0.01
10
100
1000
Length [μm]
Fig. 10. (a) Schematic view of the Q factor curve of a spiral CNT inductor. The upper-bound of Q factor is ωL/R as shown by the broken line. Due to the presence of parasitic capacitance and conductor/substrate loss, the actual Q factor curve drops down at high frequencies. (b) Qkinetic as a function of frequency and length. The equation used for calculating Qkinetic is shown in the inset. λ is assumed to be 1 μm. 120
80 60
ρsub= 10 Ω-cm
40 20 0 0.1
Cu SWCNT Fm=1/3 SWCNT Fm=1
12
Cu SWCNT Fm=1/3 SWCNT Fm=1 MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
Quality Factor
Quality Factor
100
10
ρsub= 0.01 Ω-cm
8 6 4
MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
2 1
Frequency [GHz]
10
0 0.1
1
Frequency [GHz]
7
4.42
Cu SWCNT Fm=1/3 SWCNT Fm=1 MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
6 5 4
4.40 4.38 4.36 4.34 4.32
3
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1
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Frequency [GHz]
30
40
ρsub = 0.01Ω-cm Rmc = 10 KΩ
40 20
Cu
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 14. The maximum Q factor of SWCNT and MWCNT bundle based inductors of Fig. 11 as a function of volume density of CNT bundles, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. The volume density is normalized with respect to the densely packed case (d = 0.34 nm). 300 SWCNT Fm=1
250 200
ρsub= 10 Ω-cm Rmc= 10 KΩ
MWCNT D=40nm MWCNT D=20nm
150
MWCNT D=10nm
100
SWCNT Fm=1/3
50 3
4
5
6
70
10
10
Fig. 12. Total series resistance and inductance (LS and RS in Fig. 9(b)) of Cu-, SWCNT-, and MWCNT- based inductors of Fig. 11 as a function of frequency.
SWCNT Fm=1
60 50 40 30
MWCNT D=40nm ρsub= 0.01 Ω-cm Rmc= 10 KΩ
MWCNT D=20nm
20 MWCNT D=10nm
10 0 -10
SWCNT Fm=1/3
2
3
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Number of Inductor Turns
Number of Inductor Turns
Inductance w/o Substrate [nH]
Resistance w/o Substrate [Ω]
4.44
20
ρ sub= 10 Ω-cm Rmc= 10 KΩ
60
2
Fig. 11. Quality factor of inductors based on Cu, SWNCT, and MWCNT interconnects as a function of frequency, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. All inductors have outer diameter of 200 μm, 4 turns, wire width W = 10 μm, wire thickness H = 2 μm, wire spacing S = 1 μm. Oxide and substrate thickness are assumed to be 10 μm and 300 μm, respectively. CNT bundles are assumed to be densely packed (d = 0.34 nm) and have perfect contacts (Rmc = 0). 8
10
Rmc per Channel [KΩ]
Normalized Volume Density
LK × L / N ch LK =ω RQ (1 + L / λ ) N ch RQ (1/ L + 1/ λ )
1
0
80
0
0.02
ω
6
40
Maximum Quality Factor Increase [%]
R
0.03
30
SWCNT Fm=1 MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
100
Maximum Quality Factor Increase [%]
=
ω Ltotal
20
SWCNT Fm=1/3
7
120
Fig. 15. The percentage increase in maximum Q factor of SWCNT and MWCNT bundle based inductors compared to Cu based inductor for different turn numbers, for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. The outer diameter of the inductor is fixed at 200 μm, other parameters are the same as those specified in Fig. 11, except the turn number. ρsub= 10 Ω-cm Rmc= 10 KΩ
100
Quality Factor
Qup bound =
QKinetic
0.04
10
Rmc per Channel [KΩ]
MWCNT D=10nm
8
60GHz
80 60
10
40 20 0
Cu SWCNT Fm=1/3 SWCNT Fm=1
12
Cu SWCNT Fm=1/3 SWCNT Fm=1 MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
Quality Factor
0.05
0
MWCNT D=20nm
9
Fig. 13. The maximum Q factor of SWCNT and MWCNT bundle based inductors of Fig. 11 as a function of imperfect metal-CNT contact resistance per channel (Rmc), for (a) low-loss substrate ρ = 10 Ω·cm, and (b) high-loss substrate ρ = 0.01 Ω·cm. Impact of Rmc is negligible for SWCNT inductors, but needs to be considered for MWCNT inductors.
Maximum Quality Factor
Fig. 9. (a) The schematic view of a 4-turn spiral CNT inductor, with a metal contact at each corner. (b) Equivalent π model for on-chip spiral inductors. CS, Cox, and Csub are the inter-turn conductor capacitance, oxide capacitance, and substrate capacitance, respectively (calculated using distributed capacitance model [14]). LS and RS are the frequency dependent series inductance and resistance, respectively. For CNT, LS and RS are calculated using the method shown in Fig. 4, which includes Lk and RQ. For Cu, LS and RS are extracted using FastHenry [11]. The substrate eddy currents induced parameters, Leddy and Reddy, can be obtained using the complex image theory [15]. (c) Schematic view of substrate eddy currents (opposite to the direction of inductor currents).
MWCNT D=40nm
10
40 30
SWCNT Fm=1
12
SWCNT Fm=1
Maximum Quality Factor
Dout
Maximum Quality Factor
ρsub= 0.01 Ω-cm
ρsub= 10 Ω-cm
110
ρsub= 0.01 Ω-cm Rmc= 10 KΩ
8 6 4
MWCNT D=10nm MWCNT D=20nm MWCNT D=40nm
2
1
10
Frequency [GHz]
100
60GHz
0
1
10
Frequency [GHz]
100
Fig. 16. Quality factor of inductors based on Cu, SWNCT, and MWCNT interconnects as a function of frequency for very high-frequency applications, such as 60 GHz application. The substrate is assumed to be (a) low-loss (ρ = 10 Ω·cm), (b) high-loss (ρ = 0.01 Ω·cm). The simulated inductor is 0.75 turn (as shown in the inset figure) and has outer diameter of 100 μm, width 5 μm, thickness 1 μm. The thickness of oxide and substrate are assumed to be 6 μm and 200 μm, respectively.
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