#an Efficient Method For Substrate Impedance Extraction

An Efficient Method for Substrate Impedance Extraction Q. Wang and N.P. van der Meijs Department of EEMCS, Delft University of Technology, The Netherlands Abstract This paper introduces an efficient method for extraction of frequency dependent substrate impedances. The method uses a standard Green’s function approach for resistance and capacitance separately and combines the resulting networks to get the overall frequency dependent model. We show that our method is accurate enough via some simulations and comparisons. I. I NTRODUCTION With the increasing operation frequencies of silicon integrated circuits and the increasing integration density of analog and digital circuits on a chip, circuit isolation has become a more and more serious problem and the study of the substrate behavior has become more and more important. Up to now, the silicon substrate has mainly been treated as purely resistive [1], [2]. This assumption is valid up to a few GHz. At higher frequencies, the dielectric behavior of the substrate becomes more and more important, and the intrinsic substrate capacitances must also be taken into account in addition to the intrinsic substrate resistances. Several papers have been published on the high frequency behavior of the substrate. For example [3] and [4] are fitting EM simulations to semi-physical RC equivalent circuits. In [5], a boundary element based approach was used with a complex Green’s function and extraction at multiple frequencies followed by fitting to simple one-pole RC models. An in principle more comprehensive method was presented in [6]. This method conceptually solves for both the scalar and the vector potential in the substrate with the purpose of frequency dependent noise analysis. The above mentioned methods can be reasonably, or very, accurate, but at a relatively high computational cost. For example, a complex Green’s function leads to more involved computations, might allow smaller classes of fast solvers and requires analysis at multiple frequencies. Thus, these methods are not well suited for extraction of large circuits. Therefore, we propose a simpler method that only approximates the frequency-dependent substrate impedance but in a well controlled way. In fact, the method ensures

the accuracy of both the low and the high frequency asymptotic behavior, both in magnitude and phase of the impedance. There will be small deviations for intermediate frequencies, but by comparing with more accurate techniques we show that this error is relatively small. The rest of this paper is structured as follows. First, Section 2 explains our substrate model and our extraction technique. Subsequently, Section 3 compares our method to a FEM based approach and a Complex Green’s function based approach, which is done for two types of substrates. We compare all methods for 2contact configurations as well as for a ring oscillator that produces noise in the substrate and which is evaluated at some point in its neighborhood. Finally, Section 4 concludes. II. M ODEL

AND

T ECHNIQUE

A. Substrate modeling In this paper, the substrate is modeled as a two layered half space domain with infinite lateral dimensions and thickness and with contacts on the top surface, as shown in Figure 1. If necessary, the number of layers can be increased without changing the principle of this method. Each layer is treated as a lossy dielectric material with a conductivity σ and a permittivity . When the frequency is comparable with fc where fc = σ/2π is a crossover frequency characterizing the substrate, the substrate is no longer purely resistive and the capacitive effect of the substrate needs to be included. In such a situation a complex conductivity σ 0 = σ + jω can be defined, where ω is the radian frequency. The top surface (z=0) is the Si − SiO2 interface. The contacts on the surface are treated as equipotential regions. The other part of the surface is taken into account by stating that no current may flow through the surface, i.e. a so-called homogeneous Neumann boundary condition is applied along this part. B. Green’s function We use a particular form of the Green’s function to capture the resistive and capacitive effect of the substrate. By the method of images, the Green’s function in our case can be written as [7]:

r top surface

z=0 source point q

d

observation poin p

σ1  1

PSfrag replacements

layer interface

σ2  2 ∞ Figure 1. Substrate layered model.

G=

∞ X n=0

2K n

p

r2 + (2nd)2

(1)

with 0

K=

0

σ1 − σ 2 0 0 σ1 + σ 2

(2)

Here, σ 0 i = σi + jωi (σi and i are the conductivity and the permittivity of the ith layer of the substrate, i=1,2) , r is the lateral Euclidian distance between source point q and observation point p, and d is the depth of the top layer. Since the Green’s function is complex, we will refer to a method using this Green’s function as the Complex Green’s function method which will be a reference of our approach. For actual computation, the contact regions are subdivided into elemental areas Γj , j = 1, ..., N . Following the methodology of [7], in order to compute the contactto-contact impedance matrix, a coefficient-of-potential matrix G is generated first, using the substrate Complex Green’s function for integration. Here Gij is the average potential of element Γi caused by the source element Γj . Then, the admittance matrix can be found as Y = F T G−1 F , where F is an incidence matrix (see e.g. [7] for more detailed information). C. Separate Extraction method The matrix G in the case of our Complex Green’s function is complex. Therefore, the computation time becomes very long, and the results are frequency dependent. In order to avoid the long computation time and get frequency independent results, an approximation is used to the Complex Green’s function. Namely, the mutual influence of resistance and capacitance is ignored. That means that we can get resistance and capacitance between contacts separately. After that, these two networks can be combined together to get the final network. Because this method will extract resistance and capacitance separately, we refer to it as the ”Separate Extraction” method.

Although this method makes some approximations, it is still asymptotically correct in the limit of low frequency (σ  |jω|) and in the limit of high frequency (σ  |jω|). In the transitional region, the result is also acceptable as we will show using frequency sweeps in Section 3. This method is very simple and easy to implement. Any existing substrate extraction tool based on the BE method can be used to obtain the resistance network. For a single-layer (homogeneous) substrate,  can be treated as a scaled σ to obtain the capacitance network. Both networks will then have exactly the same topology, so they can be combined in a straight-forward fashion. For a multi-layer substrate, one normal resistance extraction should be done and a second extraction using a homogeneous substrate to obtain the capacitance model in exactly the same way as for the homogeneous substrate. Both models can again be combined in the same way. III. S IMULATIONS A prototype software tool is developed using MATLAB to realize the Separate Extraction method and the Complex Green’s function method as a reference. We have validated this tool against an independent reference simulator (FEMLAB). The simulations have been run for two kinds of typical substrate with two contacts which are 0.24µm × 0.24µm with a 5µm separation (centerto-center) on the top. One is a low ohmic substrate with high-ohmic epi layer on top. The σ of the top layer is 10S/m with a depth of 4µm, and the σ of the bottom layer is 1000S/m. The other one is a high ohmic substrate with channel stopper doping at the top. The σ of the top layer is 1000S/m with a depth 1µm, and the σ of the bottom layer is 10S/m. The same relative permittivity r 11.9 is set for both. The simulations are run for a frequency sweep, from 20kHz to 270THz 1 . The comparisons are divided into two parts. First of all, we the compare Complex Green’s function method with FEMLAB to validate our implementation. Subsequently, we compare our Separate Extraction method with the Complex Green’s function method using exactly the same discretization and other computation setup. Hence, the differences in results are due to the effect of Separate Extraction only. A. Validation of the implementation The domain size used for FEMLAB is 55µm×50µm× 60µm, which is relatively large to eliminate as much 1 The reason for such a high frequency is to show the asymptotic matching for high frequency to the Complex Green’s function, thereby showing the consistency of the method. It is acknowledged that the model does not have physical relevance much beyond f=100GHz.

6

0.03

5% 0

magnitude error 5

phase −0.2

4%

0.02

−0.4

3%

0.01

phase error

−0.6

3

2%

2

1%

1

0

0

−0.8

−1 −0.01

−1.2

the epi substrate the channel stopper substrate

the epi substrate the channel stopper substrate 0 4

6

8

−0.02 −1.4

10

12

14

−1% 16

−1.6 4

6

8

10

12

14

−0.03 16

Freq (Hz) in log scale

Freq (Hz) in log scale

Figure 3. Phase and phase error versus frequency. Figure 2. Magnitude and magnitude error versus frequency. Table 1 M AXIMUM

as possible the effects of a fundamental difference between the BEM (Boundary Element Method, used by the Complex Green’s function method) and the FEM (Finite Element Method, used by FEMLAB). This difference entails that the FEM uses a bounded domain while the BEM uses a half infinite space domain. Consequently, the results obtained by FEMLAB will not match those obtained by both BEM-based methods, unless the FEMLAB domain is ’large enough’. Unfortunately, we cannot reach this point in the simulations that follow because of size limitations imposed by FEMLAB. Under such a setting, the results show that the magnitude errors are no more than 5% and the phase errors are no more than 0.005 radians for both substrates over the complete frequency range. Given this, we consider the match accurate enough to prove the validity of our implementation.

B. Accuracy of the Separate Extraction method For sufficiently low and high frequencies, the Separate Extraction method and the Complex Green’s function method have exactly the same results. For intermediate frequencies, there are some intrinsic differences between them, but these remain acceptable. Figure 2 (Figure 3) shows the magnitude (phase) obtained with the Separate Extraction method and the magnitude error (phase error) compared with the Complex Green’s function method varying with the frequency. Actually, the errors are so small that the magnitude and the phase obtained from the Complex Green’s function would completely overlap, in Figure 2 and Figure 3, the curves obtained from the Separate Extraction method.

ERRORS OF THE MAGNITUDE AND PHASE AT THE

DIFFERENT DEPTHS OF THE TOP LAYER

depths magnitude phase (rad) epi magnitude phase (rad) cs

1 µm 3.8% 0.018 4.2% 0.02

2 µm 0.8% 0.005 1.2 % 0.005

4 µm 0.1 % 0.001 0.26 % 0.001

C. Consistency of the Separate Extraction method Some other simulations have also been done for different substrates, with different conductivities and depths of the top layer. The results show the same behavior, just slightly different error values. Table 1 shows the maximum errors of the magnitude and phase at the different depths of the top layer for the channel stopper substrate and the epi substrate. Table 2 shows the maximum errors of the magnitude and phase at the different conductivity ratios between the top and the bottom layers for the channel stopper substrate and the epi substrate. From these experiments, we can see that the accuracy of our method is dependent on the doping profile of the substrate. When the depth of the top layer is smaller or the conductivity ratio between the two layers is larger, the error will be larger. But even in the worst case, the accuracy is still acceptable, so the consistency of our method is verified. D. Ring oscillator After verifying the consistency of our method, we apply this method to a more practical circuit. We again compare the results to reference results obtained with a

Phase Error (rad)

4

Phase (rad) Magnitude Error (%)

Magnitude (Ω) in log scale

magnitude

Table 2 M AXIMUM

−4

x 10

ERRORS OF THE MAGNITUDE AND PHASE AT THE

DIFFERENT CONDUCTIVITY RATIOS ( TOP LAYER LAYER

6

(S/ M )/ BOTTOM 4

(S/ M ))

2

100/10 2.5% 0.01 10/500 0.25% 0.0013

1000/10 3.8% 0.018 10/1000 0.26% 0.001

3000/10 4.0% 0.02 10/2000 0.28% 0.0008

0

Vsens (V) →

conductivity ratios cs magnitude phase (rad) conductivity ratios epi magnitude phase (rad)

−2 −4 −6 −8 without capacitance with cap. from SEP with cap. from FEMLAB

−10

9

9.05

9.1

9.15

9.2

9.25

9.3

9.35

time (s) →

9.4

9.45

9.5 −10

x 10

Figure 5. Simulated noise waveforms on the sensor node ”sens”

itance, 4.68e-4 for the FEMLAB extraction and 4.84e-4 for our method. The latter two are very close, and they both differ substantially from the no-capacitance result. IV. C ONCLUSION

Figure 4. Layout of 3-section ring oscillator

combination of FEMLAB (for the parasitic substrate resistance and capacitance) and SPACE [7] (for the active device and interconnect parameters). The circuit is a 3-section ring oscillator with 100 nm gate length transistors on a channel stopper substrate. The conductivity and depth of the top layer are 1000 S/m and 1 µm respectively, the conductivity of the bottom layer is 10 S/m. The noise generated by the ring oscillator is captured by the sensor node ”sens”, as shown in Figure 4. The simulation results for the noise waveforms are shown in Figure 5, the dash-dot line demonstrates the noise with only the resistive effect of the substrate, the solid line demonstrates the noise with both resistive and capacitive effects of the substrate extracted by FEMLAB, the dash line demonstrates the noise as predicted by our Separate Extraction method. From the figure, we can find that the noise is quite different with or without capacitive effect of the substrate, and that the result of our method matches FEMLAB results (extracted at 20 GHz) very well. From 1ns to 4ns, the RMS value of the noise is 2.89e-4 for the case without capac-

A technique for extraction of substrate impedance from low frequency to high frequency has been presented. The Complex Green’s function is also applied as a reference and FEMLAB has been used as an independent validation. Some simulations have been done to show that our method is asymptotically correct at sufficiently low and high frequencies and acceptable at intermediate frequencies. Our contribution is to show that an approach based on just combining separately obtained R and C models for low and high frequency situations results in relatively small errors and can thus be highly effective. The advantage lies in a reduced computational effort for extraction and simulation (the model is frequency independent). R EFERENCES [1] R. Gharpurey et al. Modeling and analysis of substrate coupling in integated circuits. IEEE journal of solid-state circuits, 31(3):344–353, Mar 1996. [2] T. Smedes, N.P. van der Meijs, and A.J. van Genderen. Extraction of circuit models for substrate cross-talk. In Proc. ICCAD, pages 199–206, 1995. [3] M. Pfost and H. M. Rein. Modeling and measurement of substrate coupling in si-bipolar ic’s up to 40 ghz. IEEE Solid-State Circuits, 33(4):582–591, Apr 1998. [4] W. Steiner et al. Methods for measurement and simulation of weak substrate coupling in high-speed bipolar ICs. In IEEE Transactions on Microwave Theory and Techniques, pages 1705–1713, July 2002. [5] R. Gharpurey and S. Hosur. Transform domain techniques for efficient extraction of substrate parasitics. In ICCAD, pages 461–467, Nov 1997. [6] H. Li et al. Comprehensive frequency-dependent substrate noise analysis using boundary element methods. In ICCAD, pages 2–9, Nov 2002. [7] N.P. van der Meijs. Space for substrate resistance extraction. In Stephane Donnay and Georges Gielen, editors, Substrate Noise Coupling in Mixed-Signal ASICs, chapter 4. Kluwer Academic Publishers, 2003.