#substrate Resistance Modeling For Noise Coupling Analysis

I

Substrate Resistance Modeling for Noise Coupling Analysis Simon Kristiansson, Shiva

P. Kagganti,

Tony Ewert, Fredrik Ingvarson, Jorgen Olsson a n d Kjell 0. Jeppson

Abrtraer- Accurate substrate modeling is of utmost importance for substrate noise coupling analysis in mixed-signal circuits. In this paper we present a two-port Z-parameter model based on a physical description of the substrate surface potential. The Z-parameter model is expressed using B one-port semiempirical resistance model. This resistance model accurately describes the observed initial increase in resistance followed by the observed saturation as the contact separation increases. The Zparameter model was compared to measurement data obtained S . model fits measured from a set of CMOS test S ~ F U C ~ U ~ CThe results well, in contrary to when resistive networks are used to represent the substrate. Furthermore, we show that the substrate coupling between a digital circuit and an andog circuit does not have to become zero as the distance between the circuit blocks increases, instead the coupling between the circuits approaches a constant non-zero value.

I. INTRODUCTION

I

N SYSTEM-ON-CHIP solutions analog circuit performance can be seriously degraded due to the presence of substrate noise generated by the digital circuits on the chip [I]. At each switching event of the digital circuitry, noise is injected into the substrate and transmitted throughout the chip. To investigate the influence of switching-induced noise on analog circuit performance an interface between digital and analog circuitry is required. Figure I shows a schematic cross section of two MOSFETs, one in a digital circuit and the other in an analog circuit. The coupling between transistors can be represented by equivalent circuits describing the substrate close to the transistor and the substrate between devices and circuits. Different coupling networks and techniques for characterizing and reducing substrate crass-talk have been proposed [2]-[4]. The interdevicelintercircuit model in Fig. I is usually represented by different equivalent circuits such as a single resistor, a Il or a T resistor network. Capacitors can be added to these networks to account for the increasingly dielectric behavior of the substrate at very high frequencies (for simplicity any capacitive effects are neglected in the following sections). When applying a single resistor, a n or a T network, or any other equivalent circuit for coupling analysis using circuit simulation, all components in the equivalent circuit must be assigned proper parameter values. Finding such values for any given device geometly, interdevice distance and substrate S. Kristianrson. S. P, Kagganti. F. Ingvarron and K . 0. Jcppson are with the Solid State Elcctronics Laboratory, Dcpt. of Microelectmnics, Chalmcrs University of Technology. SE-412 96 Cbteborg. Sweden. E-mail: [email protected]. T. Ewe* and 1. Olrson arc with the hngstr6m Laboratory, Solid State Electronics, Uppsala University, SE-751 21 Uppsala, Swedcn.

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Fig. I. Schematic cross section of MOSFETs in digital and analog circuits. The coupling between digital and analog circuits is modeled using transistor substrate models and interdcviceiintcrcircuit models.

doping profile, is not a straightforward process. The reason for this is that the substrate between devices and circuits to be represented by lumped components is not geometrically well defined. Each component in the coupling network can therefore not be given a complete physical interpretation. Ozis et al. [4] showed that all resistors in the coupling network need to be assigned new resistance values for proper coupling modeling when more interacting devices are added. They suggested that the substrate coupling be described by a 2-parameter representation to avoid this problem. This representation has the advantage that the interaction between each pair of devices is independent of the presence of other devices. However, their model to predict the Z-parameters was empirical and not based on physical assumptions. In this paper we present a physics based two-port Zparameter model for substrate coupling analysis. The model is based on a physical description of the substrate surface potential rather than on an equivalent resistor network. The surface potential arises from currents injected into the substrate due to substrate noise. This description results in a physical twoport model. Even if the model is not based on an equivalent resistor network, the Z-parameters can be predicted using a substrate resistance model which is obtained from the same surface potential analysis. This paper is organized as follows. In Section I1 the one-port resistance model is presented which is used in the two-port model described in Section 111. Section IV shows experimental results from a 0.35 p m CMOS process for both the one- and two-porl models. Finally, in Section V an example of noise coupling analysis is presented where our new model is used and compared with a classical Il network. II. RESISTANCE MODEL In this section the resistance model used for expressing the Z-parameter model is presented. The Z-parameter model will then he used directly in a circuit simulator to analyze noise coupling between a digital circuit and an analog circuit.

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Our new model for the resistance between two contacts separated a distance d (contact edge to contact edge) reads

where R, is the resistance for very large contact separations, n is a fitting parameter, and do is a constant which denotes how large the contact separation has to be for the resistance to reach 213 of its maximum value R,. The parameters in ( I ) are extracted by fitting the model to measurement data. Joardar [ 2 ] used a similar resistance model reading

Fig. 2. The two-pon sfruofure Contact B can r~prcsenlthe bulk node ofthe digital aggressor transistor and contact C the bulk node of the analog victim transistor.

not a good solution. Instead, the ground is supplied through

substrate contacts placed on the topside of the chip. The Zparameters are in this case clearly dependent on where the where cl,2,3 are fitting parameters and L is the length of the substrate contacts are placed. We have measured and modeled contacts. We have found that fitting ( 2 ) to measurement data the Z-parameters for the two-port shown in Fig. 2 consisting can result in extracted model parameter values giving complex of four parallel (and aligned) rectangular contacts labelled A. R ( d ) and in some cases the fitting parameters are such that a B , C , and D. Port one (between contact B and ground A) negalive d is required for R = 0 (i.e. when e2 # 1). These can be visualized as representing the digital circuitry and its properties are clearly undesirable. Our model in (1) does not digital ground supply, and port two (between contact C and ground D ) as representing the analog victim circuitry and exhibit such a behavior. Both our model and the model by Joardar are based on its analog ground supply. The substrate coupling between the the physic01 model by Schumann and Gardner [ 5 ] , which digital aggressor and the analog victim is modeled by the twogives the resistance between two circular contacts on a semi- port and in particular by its Zzi-parameter since, by definition, infinite substrate with a layered doping profile. This model is 2 2 1 is the open-circuit voltage measured at port two as a result calculated by superposition of the surface potentials arising of the current injected into pori one. from the currents flowing through the contacts. The model The structure in Fig. 2 is often modeled as a ll network is expressed in integral form, and only when the substrate is with one resistor each between contacts A, B , C and D [SI. uniformly doped can the integral he analytically solved [ 6 ] For a ll network consisting of linear (with respect to contact resulting in the resistance separation) resistors the following expressions can easily be obtained for the 2-parameters' (3) In this expression p i s the substrate resistivity, a the contact radius, and d the center-to-center distance between the contacts. By modifying (3) as in ( I ) , the resistance between rectangular contacts on both uniform substrates, and substrates with e.g. a channel-stop layer, can be modeled. This is discussed further in Appendix A. A very interesting property of the above models is that when the contact separation increases, the resistance asymptotically approaches a constant value (denoted R , in our model). For two circular contacts on a uniformly doped semi-infinite substrate, (3) gives this limiting resistance as p l ( 2 a ) . This is also discussed in [7]. 111. Z-PARAMETER MODELIM~ To model the coupling between circuit blocks with many contacts a Z-parameter formulation can be used. Previous work has presented Z-parameter models for epitaxial-type wafers with a grounded backplane [4], [8]. These substrates consist of a thin lightly-doped epitaxial layer on top of a thick heavilydoped hulk, However, in mixed-signal circuits uniform lightly-doped substrates are also used. The ground can also in this case be supplied through a metallized backplane, but since the resistive path to ground is rather high for these substrates this is usually

However, for substrate networks consisting of nonlinear (with respect to contact separation) resistors the Z-parameters must be calculated differently. In this case, the total resistance between two nodes cannot be calculated from the sum of any partial series resistances since the resistances saturate for long distances between the contacts. Instead we must use nonlinear resistance models of the type discussed in Section 11, e.g. (I). Such models can be obtained by superposition of the surface potential contributions from each of the different contacts in the two-port. This is the same method applied to four (or more) contacts as was used for two contacts when the nonlinear resistance model was originally derived. The details are shown in Appendix B. There are many ways to express the Z-parameters obtained using nonlinear resistances, hut here we have chosen a way

'All cumenis are defined as ~oritivewhen Rowing in

to

the substrate.

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Fig. 3 . The i r ~ chip. l The magnification s h o w 1x0 SO @m widc transi~tor~ in !he center surmundcd by ten pads. fire for each transistor. The Iraniirlais have one 1 x SO pm' suhstrafc m n t m on each ride. There substrate c o n f a ~ t ~ are used in thc resistance and two-pon measurements.

that makes possible easy identification with the linear network: IA nidi) - I D [?id+&) z,,= - __ +IB 2 Is 2

2

(9)

The expressions for Z12 and 222 are obtained by making the following substitutions in equations (4) through ( I O ) :

I* Is

--

d:! ID

( 1 1)

IC.

If dl = d 2 , then Zll = 2 2 2 and ZZI = Z12 which is expected since then the two-port is symmetric. When the distance d goes to 2ero Z,, approaches ZI,, which is expected since the potential in contact C approaches the potential in contact B. This is easily seen by inspecting ( 6 ) and (7). Hence, we have developed a model by which the Z-parameters can be predicted from the simulated or measured one-volt resistances between the contacts in the network. The nonlinearity (in distance) of the resistances is investigated and its influence on the Zparameters are made fully clear. ~

.

(6)

The main difference to the linear network is that the resistance between any two nodes is no longer expressed as the sum of any partial resistances, and that a corrective resistance term R,,,, has been added. It is easily seen that the corrective resistance is reduced to zero for linear resistors. The results in ( 6 ) and (7) can be obtained by adding the corrective resistance R,,,, to R(d1) and R(d2) in (4) and ( 5 ) and using nonlinear resistances. Under the assumption that I'a = VD the current fractions -I.AJIB and - I D J I B are found to be

dl

. 1200,

(hl

fie. 4. Measured (dots), d e s a i s 3D simulation (circles). and modeled (solid line = OUT model. (1)) R ( d ) for a) square contacts (1 X t urn2) and b ) rectangular cantacri ( 1 Y 50 g m 2 1 .

IV. E X P E R I M E N TRAELS U L T S

A set of test structures consisting of rectangular p + contacts in a p-type substrate were included on a test chip, see Fig. 3, manufactured in a 0.35 p n CMOS process. A nuniber of different contact separations were included. I ( V ) characteristics for these structures were acquired using an HP4156B Semiconductor Parameter Analyzer and the corresponding resistances were determined. Figure 4 shows the measured resistance for different contact separations. The solid lines in Fig. 4 are (I)fitted to measurement data. The agreement between measurement and data is very good. The extracted parameters R,, do and n for two contact geometties are shown in Table I. In order IO verify that the resistance between two surface contacts exhibits a saturating behavior when the separation is increased, a series of numerical device simulations in 3 0 were performed using the dessis simulator from ISE. The substrate doping profile in the simulations was estimated from TABLE I THE EXTRACTED RESISTANCE PARAMETERS

1x1"~ I x snp

1 Rm (Rl 1 11 4643 I d jl 1193 I

do (Nm)

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Circuit for investigating the coupling bctueen an aggressor ctrcuif

(CMOS invencrj and a victim device (NMOSFET).

Fig. 7. Simulated peak-to-peak valrags ar lhc output node or the victim transistor for di Nerenl aggressor-victim separations.

(bi Fig. 5. =

Mcasured (dots) and modeled (solid line = OUT model. dashed line and Z u against the distance d for the rwa-pon.

n network) Z n

SIMS measurements to consist of a 0.38 pm thick channel-stop layer with resistivity 0.1 Rcm on top o f a 10 n c m bulk. The contacts were set to 1 x 1 p n z . The results from the simulation are shown by the circles in Fig. 4(a). The agreement between the measured and simulated data is good thus verifying that the assumed doping profile results in a saturating one-port resistance. The measured and modeled Z-parameters for the two-port network are shown in Fig. 5. The results from using the Z-parameters, (4) and ( 5 ) , are shown by the dashed lines. As can be seen in the figure the match between OUT new model and the measurements is good. Using (4) and ( 5 ) gives the right qualitative results, but not the correct quantitative results. Most importantly it can be seen that Zzi is saturating for large distances, which implies that the coupling between the ports is saturating when increasing the distance.

n

V. SUBSTRATE NOISE C O U P L I N G

To illustrate the use of the new coupling model a simple circuit was simulated in Agilent ADS to show the effects of noise coupling. Figure 6 shows a CMOS inverter acting as an aggressor circuit and an NMOSFET biased in saturation acting as a victim circuit. Our Z-parameter two-port representing the substrate was implemented in ADS and used for connecting

the bulk nodes of the two NMOSFETs (the injected noise from the PMOS transistor was not considered for simplicity).' The width ofthe NMOS transistors was set to 50 p m and the PMOS width was selected proportionately to get a symmetrical inverter output. The circuit was simulated in ADS using the MOSFET parameters provided by the foundry which manufactured the test structures. A transient analysis was performed to show that the switching of the inverter causes the bulk node voltage of the victim transistor to fluctuate. Hence, the victim output voltage is changed because of the back-gate effect. The peak-to-peak voltage V p - pat the victim output was determined for different distances between the bulk nodes of the aggressor and the victim. Figure 7 shows Vp-p at the output node of the victim transistor for different aggressor-victim separations. A clear decrease in coupling for increased separation is observed, but for large separations the coupling saturates. This is expected since 2 2 1 becomes constant for large device separations. The observed saturation in ,721 means that the coupling is not much reduced by increasing the aggressor-victim separation beyond a certain distance. The parameter do in ( I ) is a measure of such a separation. In order to compare the Z-parameter model with a simple Il network representing the substrate, further simulations were performed. Our Z-parameter two-port in Fig. 6 was replaced with a classical lI network. The vertical resistors in the n network represents the resistance between the bulk node and

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coordinates (r, $?z j , thc surfacc potential (f = 0) caused by the current I flowing through the disc is given by [IO] when r 5 a

V ( r . 0 )- V,,,

Fig. 8. Surface potentialr from a circular contact with radius e = 1 and unit potential on a substrate consisting of an epitaxial layer. resistivity 15 I k m and thickness 7 pm. on a 300 pm thick bulk. The backside ofthe substrate i s grounded. The bulk resistivity going from the top dotted c u n e and down a n : I S ncm, I ffcm. and 0.08 glcm. The top solid CYWC is (12) and the bottom solid E U W shows ~ the result for a zero bulk reiistivily

the substrate contact of each transistor. These resistors are equal and constant (since the distance d l = d2 is fixed). The horizontal resistor is varied according to the separation distance between the bulk nodes of the two NMOS transistors The resistor values are obtained from the one-port model ( I ) . The simulation results obtained by replacing OUT two-polt with the simple network are also shown in Fig. 7. The magnitude of the coupling shown at the output of the victim transistor is almost three times higher when the n network is used to represent the substrate. This is in agreement with the 2 2 1 parameter results in Fig. Xb).

n

VI. CONCLUSIONS We have presented a physically based Z-parameter model of representative two-port wirh a l l conracrs on t h e surface of a lightly doped substrate. This Z-parameter model is expressed using a one-port resistance model, Both the resistance and the Z-parameter models were validated with experimental results. The Z-parameter model was then used for predicting the noise coupling in a simple mixed-signal circuit. Simulations show that the coupling does not go to zero as the distance between the circuit blocks increase, instead a saturating coupling is observed. Hence, the coupling is not much reduced by increasing the aggressor-victim separation beyond a certain distance. We have also shown that a simple l l network cannot quantitatively model the coupling between two devices. Further work is needed for deriving Z-parameter models for the mixed case where both surface ground contacts and a grounded backplane is present. The coupling models must also be generalized to include more contacts.

a

THE

APPENDIX A POTENTIAL

SURFACE

The substrate resistance model and the 2-parameter model are based on superposition of the surface potentials arising from the currents flowing- through . each of the contacts on the substrate. The potential around a circular charged disc of radius a on a semi-infinite uniformly doped substrate with resistivity p is a classical problem [IO]. In cylindrical

(12) ($1 -arcsiii when 1' > a. 2na The reference-potential V,,, is the substrate potential at infinity and the currents are defined as positive flowing in to the substrate. Note that when T >> a , (12) reduces to V ( r , O )V,,, = pI/(Znr) as from a point contact. Superpositiori ofthe surface potentials from one contact with the current I flowing into the substrate, and the other contact with the same current flowing out of the substrate, results in the resistance expression given in (3). In general, the potential around a charged circular disc on substrates with a layered doping profile can be expressed as an infinite integral of a function which is most often found to be rather complicated [ I I]. Only for uniform substrates can this function be simplified enough to be integrated analytically [ 6 ] , with the result given in (12). However, for nonuniform substrates the function is much too complicated to be integrated analytically. One way to handle this is to generalize the result in (3) by adding empirical parameters, like we have done in (I). In particular, this has been found useful for uniform, lightly doped substrates with an implanted channel-stop surface layer. However, the surface potential cannot be approximated by an arcsin-function on every substrate type. There are two important cases where (12) is not applicable. For uniform substrates with a grounded backplane, the surface potential falls like a r c s i n ( a 1 r ) as long as T << t,,b (where t s u h is the total substrate thickness). When T sz t,,b and T > t s u h the =

(:)

surface potential decreases exponentially w h i c h i s i l l w t m t c d

by the bottom solid line in Fig. 8. The other case is a substrate composed of a thin lightly doped epitaxial layer an a thick bulk with a grounded backplane. Here T zz teD;or r > t,,, (where t,,, is the thickness of the epitaxial layer) for all interesting distances, and the surface potential depends mostly on the relation between the epi-layer and the bulk resistivity. Some results are shown in Fig. 8. where the transition from a potential decreasing as &rcsin(air) to an exponentially decreasing potential is clearly seen. The results shown in Fig. 8 specifically means that (12) cannot be used for heavily doped substrates with a grounded backplane like those used in [4] and [9]. For such substrates the integral in [ I I] must be solved numerically. resulting in an exponentially decreasing surface potential as illustrated in Fig. 8. This is in agreement with the empirical Z2,-model, Z2, =~nezp(-@), used by Ozis et al. for a similar type of substrate. The results in Fig. 8 were calculated using the model by Schumann and Gardner [ 5 ] .

-

APPtNUIX 0

D E R I V A T I OON F T H E Z-PARAMETER MODEL

In our model the Z-parameters are derived by calculating the potential at each contact. This is done by summing the surfacepotential contributions due to the currents flowing through

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Using (16), (I7) and by defining resistances between pairs of contacts as R ( d z g ) = 2 [f(oj f ( d z u ) l I (18) we obtain the following expressions for the Z-parameters

d

.II.
Fig. 9. Surface-pofentialr for the two-pon.At each confact. A , 8, C, and D. there four potent id^ IE summed. The reference-potential V,,, is Shawn by the thick dashcd line. Note that in this figure IC = 0.

each of the contacts on the chip (where a current is defined as positive when flowing into the substrate). It is assumed that every current flowing through a contact contributes with a surface potential which is a fnnction of the distance from the contact, i.e. V ( T , O )- V,,f = f ( r ) I . An example of a surface-potential function is given by (12). The structure in Fig. 2 consists of totally four contacts. When deriving the expressions for Zll and Zz1 the current through contact C is zero, and the currents I A , I B = - I A - I D , and I D result in the qualitative surface potential picture shown in Fig. 9. The contact-potentials (relative to the reference-potential V,,,) and the contact-currents are related through

where the coupling matrix C can be written as

In (14) d,, is the distance between contacts z and y. Since all the contacts in this case are parallel and of the same size, the matrix C is symmetrical, in general this is not so. By shorting terminals A and D (that is setting Va = VD),and regarding the system of equations as a two-port, (13) can be reduced to VB - VA

(vC-VD)=(2

"z::)(?>.

(I5)

Since V, = VDthe potential differences can be written in the following symmetrical way

(23)

-

-

Observethat ( - I A / I B ) + ( - I D / I B ) = 1 as required. I f d l = d p , then - I A / I B 112 and - I A / I B + 112 as d 0. Note also that when d + m, I D I I B does not go to zero. This can be understood since the resistance between the B and D contacts is saturating for large distances. The expressions for 2 2 2 and 2 1 2 are obtained by making the substitutions described by (11) in equations (19) through (23).

REFERENCES ( I ] M. Xu. D. K. Su, D. K. Shaeffer. T. H. Lee. and E. A. Wooley. "Measuring and modeling the effects of substrate misc on the LNA for a CMOS GPS receiver," IEEE Journol qfSolid-Store Circuits, vol. 36, no. 3. pp. 473485. March 2001. [2] K. JoBidar. "A simple approach to modeling cross-talk in integrated circuitr."lEEE Journolo/Solid-Siure Cvcuirr. vol. 29. no. 10. pp. 12121219, October 1994. [3] D. Szmyd, L. Gambus. and W. Wilbankr. "Stralcgier and test ~ f m c t ~ r e s for improving isolation between ciicuif blocks," in IEEE hr Gun/ on Microelectronic Teest Stmcrurer, 2002. pp. 89-93. [4] D. Ozir. K. Mayaram. and T,Fiez, "An ehicient modeling approach for substrate noise coupling analysts," fn IEEE Inr Symp. on Circuils ond Sysrem, vol. 5, 2002. pp. 237-240. (51 P Schumann Jr. and E. Gardner. "Application of multilayer potential distribution to spreading rcsisfance correction factor$," Journol o/ the Elemochemiral Society. vol. 116, no. I,pp. 87-91. 1969. (61 M. Abramowitz and I. Slsgun, Handbook o/ morhemoricd /unctions. Dover Publieations,lnc.. New York. 1965. p. 487. eq. 11.4.35. [71 S. Choo. M. Lcanp. C . Liem, and K. Kong, "Extraction of semiconductor dopant profiles h m spreading resistance data: an inverse problem:' Solid.Slare Eleerronier. vol. 33. no. 6, pp. 783-791, 1990. [XI N. Verghew, D. Allnot. and M. Wolfe. "Verification techniques for substrate coupling and their application to mired-signal IC design:' l E E E J ~ ~ ~ ~ ~ l ~ / SCircuit$. ~ l i d vol. - S 31. r ~ no. ~ ~3. pp. 354-365, March 1996. [9] A. Ssmavsdam. A. Sadate. K . Mayaram. and T. Fie& "A ralsable substrate noise coupling model for design of mixed-signal IC's:' IEEE Journal o/Sulid Safe Circuits. vol. 35. pp. 895-904, 2000. [IO] I. N. Sneddon. Mixdhoundory vvluepmblemr inporenrialrheoq. John Wilcy and Sons. Inc.. 1966. [ I I ] S. Choo, M. Leong. and K. Kuan, "On the c d ~ ~ l a t of i ~ spreading n re~istancecorrection factars:' Solid.Sme Electmnirr. vol. 19. p. 561, 1976.

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