Signals propagated in actual and future IC's interconnections are spreading, in terms of spectrum frequency, from DC to microwave domain because of the continually increasing clock frequencies of circuits. So, predictions of impedance, delay, rise/fall times and crosstalk levels need to be investigated within this whole spectrum. QUEST is a powerful two-dimensional extractor of the RLCG frequency dependent parameters of the transmission lines. This tool is based on a fast and accurate computation so-called 'fictitious domain method' [1]. In this article three examples of its use are presented:

the variations of mutual/self inductance with the geometrical parameters,
the crosstalk,
the skin effect

The reader could compare these results with [2].

1. The Coupling Effects for Different Geometries

The main structure considered in this example consists of two metal wires (_ = 30 MS/m) separated from the substrate by an oxide. The thickness of the oxide and the spacing between the two wires are variables (see Figures 1 and 2). The frequency has been fixed to 10 MHz. Two values have been assigned to each variable, so that QUEST would perform four simulations automatically.

Figure 1. Layout definition with parameterized spacing.

Figure 2. DOE user-defined variation of pre-defined variables.

As outputs QUEST gives the four RLCG matrices. Table 1 summarizes the values of the inductances calculated by QUEST.

Table 1. Simulated values of inductances from QUEST.

The self-inductance increases with the oxide thickness, but it is independent of the spacing between the lines. Indeed the self-inductance characterizes the magnetic energy stored in a closed conductor. This conductor is not necessarily physically closed. In our example, a wire and the substrate make a loop. Therefore, giving the following formulas :

with

B the magnetic field,

the magnetic flux,

I the current in the loop whose surface is S.

The further the line from the substrate, the bigger the area of the loop and the higher the self-inductance. The results of the mutual inductance show that the coupling between the two lines increases when the spacing decreases. Thus in VLSI technology, the lines most prone to mutual induction are the lines far from the substrate and close to each other.

2. Crosstalk

Neighboring wires are capacitively and inductively coupled to each other, such that a transient signal on a wire can have a significant effect on another unconnected wire: this is the diaphony or crosstalk.

The previous example is used with a spacing equal to 1 µm and an oxide thickness equal to 10 µm, so that coupling effects are significant. The simulation is performed at 600 MHz, and the output RLCG matrix is used in SmartSpice for a transient simulation.

Figure 3 shows the circuit and figure 4 the result of the transient simulation.


Figure 3 . Schematic diagram of the test circuit used to examine crosstalk between two wires.	Figure 4. SmartSpice plot of node voltages illustrating crosstalk between two wires.

The curve v(clk) is the clock signal (a pulse). The wire between nodes 3 and 4 is the aggressor line and the wire between nodes 1 and 2 is the victim line. At the end of the aggressor line (node 4, v(4)), one observes oscillations and voltage overshoot due to the self-inductance. On the victim line, the crosstalk is obvious: a parasitic signal propagates from node 1 (v(1) curve) to node 2 (v(2) curve), despite no bias being applied to the line.

It is also well known that the crosstalk increases as the rise/fall times decreases. Figure 5 illustrates this case, the rise time equals to 0.02 ns and the fall time 0.1 ns.

The crosstalk is much larger after the ramp-up bias than after the ramp-down of the clock signal. This can be explained by the higher frequency components of the Fourier transform of the clock signal at the rise time than at the fall time.

Figure 5. SmartSpice plot of node voltages illustarting
enhanced crosstalk with reduced rise and fall times.

3. The Skin Effect

The skin effect can be observed at very high frequencies and large conductors. Indeed the skin depth of aluminum at 1 GHz is 2.8 ?m and so very close to the cross-sectional dimensions of a wire. When the skin effect appears the resistance of the line increases because the current flows principally at the edge of the conductor. For the simulation, two wires are defined, spaced by 4 ?m, their width is 5 ?m, their thickness is 3 ?m. Calculations are performed at 20 Mhz and 20 GHz. Figure 6 shows the cross-section of the two wires and a cutline was performed inside them. The current density is displayed at 20 MHz. For this low frequency, the current density value can easily be calculated using :

E with =30 MS/m the conductivity of aluminum and E=1V/m the applied electric field, J is the current density in A/cm. One obtains J= 3000 A/cm uniformly in the excited left wire. Figure 7 displays the same wires at 20 GHz, the current density is less than half the previous case and the current density is higher at the edges of the conductor: this is the skin effect. At high frequency, the electromagnetic field is greatly attenuated inside the conductor due to the inductive effect. This explains the skin effect, but also this explains the proximity effect : a current flow appears at the edge of the neighboring conductor. This is what one can observe in Figure 7, in the right hand side wire.


Figure 6. Two-dimensional plot (left) and one-dimensional plot (right) of current density inside two wires, one excited at 20 MHz.	Figure 7. Two-dimensional (left) and one-dimensional (right) plots of current density inside two wires, one being excited at 20 GHz. The skin effect is clearly evident.

Conclusion

This article presented some examples of the electromagnetic behavior of two transmission lines: inductive effects function of geometric variables, crosstalk and skin effect. It shows that QUEST is a powerful tool dedicated to characterizing transmission lines for microelectronic design. QUEST gives a good solution to such analyses that become more and more necessary as the clock frequency and integration increase.

References

[1] A Fast and Accurate Computation of Interconnect Capacitance, S. Putot, F. Charlet, P. Witomski, IEDM 1999.

[2] High Frequency VLSI Interconnect Modeling, Dennis Sylvester

Part 2 - Comparison with Experiments will be published in the
Simulation Standard Volume 12, Number 2, May 2002 issue.