Introduction
Signal integrity analyses often involve a form of copper-based interconnect. Copper is a common metal in commercial interconnect (e.g., cables, ribbons, printed circuit boards, connectors), is well characterized, and is readily modeled in modern signal integrity (SI) tools. Being a normal conductor, it has non-zero resistance, and it thus attenuates and reduces the amplitude of signals it carries. Copper is correspondingly subject to skin effect that creates frequency-dependent loss and associated dispersion that changes the waveform’s shape. These effects contribute to the limited bandwidth and finite transmission distance of signals propagating through copper-based interconnect.
In contrast to normal conductors, superconductors conceptually have zero resistance across the entire applicable frequency spectrum. In principle, superconducting transmission lines maintain the signal’s amplitude and waveform shape across their entire length, which leads to visions of tremendous bandwidths over infinite physical distances.
It is relatively easy to model the zero-resistance behavior of superconductors by setting metal conductivity to an extraordinarily high value or using a perfect electrical conductor (PEC) model. Much can be learned with such models, but they neglect an important parameter that is unique to superconductors and affects key SI metrics like characteristic impedance, propagation velocity, and crosstalk. This article summarizes kinetic inductance and discusses its effect on signal integrity. To aid in understanding, the article compares normal-conducting interconnect to superconducting interconnect with supporting simulations.
Kinetic Inductance
To understand kinetic inductance, it is useful to start with understanding the behavior of normal-conducting electrons and magnetic inductance. In normal conductors, magnetic inductance (LM) is related to the magnetic flux generated by current through a structure.1 The magnetic inductance is dependent on the structure’s geometric shape, and it describes the opposition to change in current within the said structure. That is, a change in current produces a voltage potential across the structure due to the inductive reactance intrinsic to that structure.
Also in normal conductors, resistance is attributed to the scattering of electrons as they propagate through metal.2 In superconductors, the lack of resistance comes with a corresponding lack of scattering, yielding a zero-resistance path for the electrons within the superconducting metal.3 As a result, the effective group velocity of superconducting electrons (called Cooper pairs) is significant enough to give the electrons impactful levels of kinetic energy. Kinetic energy is the amount of work needed to accelerate an object from rest4, and it also opposes a change in current. This opposition to change in current is akin to, but distinct from, magnetic inductance, and is termed kinetic inductance (LK).
All moving electrons have kinetic energy, and thus electrons in normal conductors also have kinetic inductance. However, the value in normal conductors is negligible due to electron scattering. As a result, kinetic inductance most often is only relevant to superconducting interconnect.2,3
Within superconductors, there remains small amount of normally-conducting electrons propagating among the superconducting electrons. Early pioneers recognized this phenomenon and formalized it in what is often called the Two-Fluid Model.5,6 This model treats the normal and superconducting electrons as separate fluids (currents) within the conductor. The behavior of the separate currents in the model maintains the distinction between LM and LK and yields a practical model for engineering analysis.
Describing the opposition to change in superconducting electrons as an inductance can lead to some confusion, as LK is not a conventional inductance. LK is a proxy for representing the macroscopic behavior of superconducting currents from a circuit perspective. LK is not a magnetic or a frequency-dependent phenomenon and is therefore distinct in effect from LM. LK is not due to magnetic fields produced by the currents, but it is instead due to the physical kinetic energy of the superconducting electrons. The contribution from LK to internal and external inductances is completely independent of LM, but LK does affect the effective loop inductance of the signal and return paths.
Normal conductors are subject to skin effect that crowds current near the outer surface of the conductor.7 Skin effect is frequency dependent and thus contributes to frequency-dependent loss and dispersion.
While superconductors are subject to the same magnetic skin depth, the zero-resistance property drives the magnetic skin depth to 0, analogous to PEC. Instead of magnetic skin depth, the depth of the propagating currents in superconductors is dictated by the London penetration depth,2 often annotated as l. Unlike magnetic skin depth that is frequency dependent, the London penetration depth is dependent on the material and is independent of frequency. Like magnetic skin effect, the currents in superconductors crowd immediately near the conductor’s surface with exponential decay inward from the surface — with the London penetration depth defined as the depth at which the current density reaches 1/e= 36.8% of the outer maximum. This current-density profile is maintained in superconductors from DC to hundreds of GHz.
It is important to distinguish between magnetic-based skin depth that is dependent on both frequency and electrical conductivity of the metal, and London penetration depth that is solely dependent on the superconducting material. The two skin depths are independent of one another. In superconductors, the effects of magnetic skin depth are largely ignored in SI analysis, because the superconductor has zero resistance and thus the magnetic skin depth resembles PEC. In superconductors, the primary importance of London penetration depth to SI is its effect on kinetic inductance, as shown in the following.
Like magnetic inductance, kinetic inductance can be estimated with three-dimensional (3D) electromagnetic (EM) modeling tools. A handful of commercial tools account for kinetic inductance and thus support superconductors, such as COMSOL1, InductEx2, and High Frequency Structure Simulator3. An approximate value for the partial inductance of a length of interconnect can be calculated with the following equation:
(1)
with additional parameters permeability (µ0), physical length of the interconnect (len), and cross-section area (A) in which the superconducting current flows as defined by the London penetration depth.
From Equation 1, it becomes clear that kinetic inductance is dependent on both the London penetration depth (l) and cross-sectional geometries (A). Generally speaking, thicker London penetration depths and smaller cross-sectional geometries result in greater kinetic inductance. Anecdotally, kinetic inductance can range between 0% to 200% of magnetic inductance.
Effect on Key Metrics
From a circuit perspective, kinetic inductance adds to magnetic inductance, as shown in Equation 2 where LK and LM are respective contributions to the loop inductance that includes both the signal and return paths. As a convenient example, if it is arbitrarily assumed that LK has the same value as LM, then the total effective self-inductance (LSelf) doubles relative to that of a normal conductor. The increase in self-inductance correspondingly affects multiple key metrics used by SI engineers, as discussed in this section.
(2)
The effect of kinetic inductance is readily seen in characteristic impedance (Z0), which increases relative to the magnetic inductance alone. As an example, if LK is equal to LM, then the characteristic impedance will increase by 41% relative to a normal-conducting interconnect of the same dimensions.
(3)
Similarly, the propagation velocity (u) slows with added kinetic inductance due to the increased magnitude in the denominator in Equation 4. Slower velocities result in increased propagation delays.
(4)
Less obvious is the effect on crosstalk. As described, the backward and forward coupling factors Kb and Kf are dependent on the ratios of mutual-to-self inductance and capacitance.8,9,10
(5)
(6)where TD is the propagation delay, Cmutual and Lmutual are the traditional mutual capacitance and inductance, Cself is the traditional self-capacitance, and Lself is the total self-inductance due to both the magnetic and kinetic inductances.
The addition of LK to LM in Lself changes both coupling factors and thus affects crosstalk. Because the self-to-mutual ratios are added in the backward coupling factor, backward crosstalk in superconductors tends to be reduced relative to the same geometries in normal conductors.
The same is not necessarily true with the forward coupling factor. Because the ratios are subtracted from one another, the forward coupling factor (and thus crosstalk) can decrease or increase, depending on the balance of the two ratios.
As a classic example, forward crosstalk in a pair of striplines made with normal conductors in a homogenous dielectric is theoretically zero, because the capacitive and inductive ratios are equivalent and thus cancel to zero. With superconducting interconnect, the addition of LK imbalances the ratios, which produces a non-zero Kf, and thus creates forward crosstalk between superconducting striplines that would not ordinarily exist in normal-conducting striplines.
Effect on Data Links
It is informative to compare the performance between a handful of example data links with normal-conducting and superconducting interconnect. While the mechanics of such comparisons are relatively simple, meaningful comparisons require a bit of engineering consideration.
For example, SI engineers commonly analyze printed circuit boards (PCBs) made with copper having cross-sectional geometries often measured in mils or millimeters. Unfortunately, the physical cross-sectional geometries of PCB traces are sufficiently large to drive LK down to negligible levels due to the increased area A in Equation 1. By using PCB-like geometries, links made with superconductors would certainly exhibit the benefit of zero resistance, but they would also obscure the effects of kinetic inductance, which is the primary topic of this paper. In addition, superconducting PCBs are uncommon, thus providing a poor contextual example to reference.
In contrast, superconducting interconnects of relevance to this paper are often on the scale of integrated circuits (ICs) — with cross-sectional geometries measured in microns (mm) or nanometers (nm). Unfortunately, normal-conducting links with IC-like geometries have tremendous resistance and corresponding loss, which produce nearly-undetectable waveforms at the receiver. These considerations obscure direct comparisons between normal-conducting and superconducting data links that use similar geometries.
To clearly demonstrate the effect of kinetic inductance with relevant geometries, one compares superconducting interconnect to PEC interconnect using IC-like geometries. The PEC model has zero resistance and correspondingly zero kinetic inductance, thus emulating an ideal normal conductor, whereas the superconducting model includes kinetic inductance. The resulting PEC-to-superconductor comparison clearly demonstrates the effects of kinetic inductance.
The physical geometries and schematic of the basic link are shown in Figure 1. The line width is adjusted to obtain 50 Ω characteristic impedance with the PEC material. The superconducting metal is assumed to have a London penetration depth of 100 nm.
Figure 1. Example data link. Cross-sectional geometries are shown on the left, and the circuit schematic is shown on the right.Using 3D EM simulation tools, the self-inductance is estimated to be 313 nh/m for the PEC interconnect and 704 nh/m for the superconducting interconnect. The effective kinetic inductance is the difference between these two values at 391 nh/m, which represents approximately 125% of the magnetic inductance. The capacitance and conductance values are the same between the two interconnects.
Figure 2 demonstrates the effect of kinetic inductance on characteristic impedance by simulating a time-domain reflectometry (TDR) measurement of the interconnect by itself. The effect of the kinetic inductance is observed through the increased impedance. In this example, LK within the superconducting interconnect increased impedance to nearly 75 Ω, which is approximately 50% greater than the PEC interconnect’s 50 W value.
Figure 2. TDR simulation demonstrating impact on characteristic impedance.Figure 3 compares the simulated waveforms of a single bit transmitted in the example data link of Figure 1. The decrease in propagation velocity due to kinetic inductance is indicated by the increase in propagation delay. In this example, LK increased the propagation delay in the superconducting interconnect by approximately 50% relative to the PEC interconnect.
Figure 3. Simulated waveforms of single pulse transmitted in example data link.To evaluate crosstalk, the pair of striplines shown in Figure 4 were simulated surrounded with a homogeneous dielectric. To avoid obscuring the results due to impedance mismatch, the termination resistors in each case are set to match the characteristic impedance of the respective traces.
Figure 4. Example crosstalk model. Cross-sectional geometries are shown on the left. On the right, the circuit schematic is illustrated.Figure 5 shows the simulated crosstalk. As discussed earlier, the backward crosstalk in the superconducting interconnect is slightly lower than in the normal-conducting interconnect, as shown in the left graph of Figure 5. The reduction in backward crosstalk is not due to the zero-resistance property of superconductors, but it is instead due to the existence of kinetic inductance (Equation 5).
Figure 5. Simulated crosstalk. Backward crosstalk is shown on the left, and forward crosstalk is shown on the right.The right graph in Figure 5 demonstrates the simulated forward crosstalk. Here, the crosstalk noticeably increases with the superconducting interconnect. In particular, the forward crosstalk in the normal-conducting interconnect is near the theoretical nil value due to the balance of the capacitive and inductive ratios (Equation 6). The inclusion of LK in the superconducting interconnect unbalances the ratios, which correspondingly creates appreciable forward crosstalk.
Summary
Superconductors show promise of enabling faster data rates over longer distances. As superconductors expand into high-speed data applications, SI engineers are faced with new terms, tools, and challenges.
Of the numerous parameters that are unique to superconductors (e.g., critical temperature, critical current, critical field strength, Meissner shielding, etc.), kinetic inductance is of significant importance to signal integrity analyses. This article aims to highlight the impact of kinetic inductance to help SI engineers address critical design challenges unique to superconducting interconnects and thus improve analysis outcomes.
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