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Robust Parameter Extraction for the R3 Nonlinear Resistor Model for Diffused and Poly Resistors Colin C. McAndrew, Fellow, IEEE, and Tamara Bettinger, Member, IEEE
Abstract—This paper presents robust algorithms to determine the parameters of the R3 nonlinear resistor model for both diffused and poly resistors. Extraction of many R3 parameters is simplified if it is based on the zero-bias conductance G0 , because G0 is not affected by velocity saturation or self-heating. We present techniques to reliably determine G0 , even when measured data are noisy or highly nonlinear, and show how to extract basic resistance parameters, temperature coefficients, and depletion pinching parameters using G0 . We describe how to determine thermal conductance parameters, and present a final parameter optimization strategy that prevents imprecision in modeling G0 from compromising the accuracy of fitting resistor nonlinearity. Index Terms—Resistors, semiconductor device modeling, SPICE.
I. Introduction
T
HIN-FILM resistors are important components in integrated circuit (IC) design. IC manufacturing processes can include two types of resistors: polysilicon (or poly) resistors, and diffused resistors (which are really JFETs). To accurately simulate distortion the nonlinearity of both types of resistors must be accurately modeled. Nonlinearity in poly resistors comes from self-heating and MOS action. Self-heating is a significant source of nonlinearity in poly resistors because they are encased in SiO2 , or a similar dielectric material, which has a low thermal conductivity. The MOS action depends on the potential difference between the poly resistor body and the bulk silicon region underneath it; because the separating dielectric is generally thick the nonlinearity caused by the MOS action is small, but is nevertheless observable in experimental data. Depending on the polarity of the body to bulk potential difference, the MOS action can either modulate the depth of the depletion region at the bottom of the polysilicon, which alters the thickness and hence resistance of the conducting portion of the resistor, or can modulate accumulation charge at the bottom of the polysilicon, which again alters the overall device resistance. Diffused resistor nonlinearity also comes from self-heating and depletion region modulation with bias, although from the p–n junction rather than MOS physics, and additionally from velocity saturation. Because diffused resistors are formed in
Manuscript received November 14, 2011; revised February 27, 2012; accepted April 1, 2012. Date of publication April 9, 2012; date of current version October 25, 2012. The authors are with Freescale Semiconductor, Inc., Tempe, AZ 85284 USA (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSM.2012.2194170
a bulk silicon region, which has a relatively high thermal conductivity compared to the dielectric that encases poly resistors, the nonlinearity from self-heating is less pronounced in diffused than in poly resistors, but it can still be significant. The magnitudes of each of these physical causes of nonlinearity vary with resistor width W and length L, with temperature T , with physical process parameters such as the film depth tb and doping level N (here assumed uniform throughout the body of a resistor), and with the potential differences across the body of a resistor V and between the resistor body and the silicon region it sits in or above VC . A resistor model, R3, that accounts for all of these sources of nonlinearity has previously been derived [1], [2], and this model captures in a single physical formulation both the p–n junction and MOS depletion region physics. Parameter extraction for R3 requires care. A useful general extraction strategy is “divide and conquer:” identify subsets of measured data that primarily depend on subsets of the model parameters, then develop techniques to determine those parameters from the reduced data sets. The parameter values can, if needed, be refined by subsequent numerical optimization. For R3 a significant number of parameters can be extracted from the conductance at V = 0 I(V, VC ) . V →0 V
G0 (VC ) = lim G(V, VC ) = lim V →0
(1)
G0 is affected by neither self-heating nor velocity saturation, and can be used to determine the basic resistance parameters, temperature coefficients, and depletion pinching parameters of R3. However, G0 cannot be directly measured but must be determined from G versus V for each VC , and such data can be noisy for low V (especially for highly linear resistors) and can be significantly nonlinear for high V , so accurate calculation of G0 is difficult. Further, even if G0 is known accurately there is insufficient information in G0 (VC ) alone to determine the R3 depletion pinching parameters for poly resistors. This paper presents robust parameter extraction techniques for R3. We describe reliable algorithms to determine G0 for diffused and poly resistors, detail how basic resistance parameters and temperature coefficients can be determined from Rz = 1/G0 (VC = 0) (note that Rz does not depend on bias, it denotes the resistance for V = VC = 0, whereas G0 is a function of VC ), and show how to determine the R3 depletion pinching parameters for both diffused resistors and, in a manner that avoids numerical problems, poly resistors. Knowing the depletion pinching parameters, we show how
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the depletion pinching nonlinearity can be factored out of measured data so that the velocity saturation and thermal conductance parameters, which model self-heating, can be characterized. We also present a final numerical optimization strategy that prevents errors in modeling Rz from compromising fitting of resistor nonlinearity over bias and geometry. The algorithms described here provide enhancements over, and expanded details of, the procedures presented in [3] and [4]. Bias polarities below are for an n-type resistor body.
II. R3 Model Description The R3 model for the current in a resistor is [2] I(V, VC )
= G(V, VC )V√
G
=
(2)
−2VC +V 1 1 1−df dp√ Rz 1+rμ 1−df dp
Fig. 1.
where dp is an effective depletion potential, df is an effective depletion pinching factor, and rμ is an effective mobility reduction factor. rμ accounts for both velocity saturation and self-heating, is zero for V = 0, and from symmetry of these physical mechanisms is an even function of V ; it therefore imparts a parabolic shape to G(V ) curves around V = 0. Velocity saturation is modeled via the velocity–field v(E) relation μ0 E (3) v= 1 + rμvs where μ0 is low field mobility, E = V/L is the average electric field in a resistor, and rμvs is the contribution to rμ from velocity saturation. Rather than using the conventional forms for 1+rμvs [5], [6], R3 uses the more accurate empirical model of [2] rμvs
=
E−Ece 2Ecr
+
2
E+Ece 2Ecr
where Ece =
+ 2
dμ Ece Ecr
+
−
Ece Ecr
2 +
4dμ Ece Ecr
(4)
dμ Ece Ecr
2 + 4d 2 E2 − 2d E . Eco μ cr μ cr
(5)
Eco is a parameter that defines the corner field at which rμvs starts to increase with increasing |E|, Ecr is a critical field parameter (the reciprocal of the slope of rμvs versus E at high field), and dμ is a fitting parameter for the “hardness” of the transition around E = Eco (see Fig. 1). For small |V | rμvs ≈
2dμ E2 = Sμvs E2 2 (1 + 4d E /E )3/2 Ece μ cr ce
(6)
and as velocity saturation is a basic physical phenomenon the coefficient Sμvs of E2 in (6) does not depend on device geometry. Self-heating is handled in R3 by introducing a power source (of value IV ) that drives a thermal conductance and a thermal resistance connected in parallel [7]; this generates the local temperature rise of the device, which is then fed through the temperature variation equations of the model. Conceptually,
Velocity saturation model (4), default R3 parameters.
for low and moderate fields, this is equivalent to introducing an effective mobility degradation factor for self-heating rμsh =
1 TC1 TC1 IV ≈ W GTH Rz L GTHA + 2GTHP L1 +
1 W
E2
(7)
where TC1 is the first-order temperature coefficient of resistance and GTH is thermal conductance (modeled through area GTHA and perimeter GTHP components). The coefficient of E2 in (7) depends on both material properties and resistor geometry and so varies from device to device. Physical analysis [2] shows that (2) is valid for depletion region pinching of both the MOS system (for poly resistors) and p–n junctions (for diffused resistors). However, the conductivity of poly resistors can also be enhanced if VC induces accumulation in the poly at the oxide interface. This would at first seem to require a different form of model than (2) as accumulation is a different physical phenomenon than the depletion pinching analysis (2) is based on, but that turns out not to be the case. The sheet resistance of a thin film is ρs = 1/(μ|Q |), where μ is mobility and |Q | is the magnitude of the (conducting) charge per unit area of the film. In accumulation the surface potential at the bottom of a poly resistor (with V = 0 for simplicity) is pinned to approximately zero, so |Q | ≈ Cox (VC −VFB ) where Cox is the capacitance per unit area of the oxide under the resistor and VFB is the flatband voltage. The rate of change of conductivity with respect to VC in accumulation is, therefore, ∂|Q |/∂VC = Cox . Physical analysis for operation in depletion gives [2]
|Q | = qNtb
s 2s 1+ − tb Cox
γ 2 /4 − VFB − VC γ tb Cox
(8)
where q is the magnitude of the √ electronic charge, s is the permittivity of silicon, and γ = 2qs N/Cox . Therefore ∂|Q | qNs = 2 ∂VC γCox γ /4 − VFB − VC
(9)
and as γ is large for poly resistors (because the oxide between the poly resistor body and the silicon is thick so Cox is small)
McANDREW AND BETTINGER: ROBUST PARAMETER EXTRACTION FOR THE R3 NONLINEAR RESISTOR MODEL
Fig. 2.
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Typical Kelvin contacted resistor layout, L/W = 63 μm/63 μm.
∂|Q |/∂VC ≈ Cox which is the same as in accumulation; the form (2) is based on analysis of MOS depletion behavior but is also valid if the lower surface of the poly is accumulated; this has not been pointed out previously. [This also follows from [8, p. 108]; for large γ the capacitance between the poly and the silicon is essentially independent of VC , therefore the rate of change of |Q | with VC is constant and (2) applies when the poly is in depletion or accumulation—inversion does not occur for practical values of VC as the threshold voltage of the poly-oxide-silicon system is large because of the thick oxide.]
Fig. 3. G(V ) for a 2500 /ⵧ diffused resistor, L/W = 63 μm/2.1 μm.
III. Test Structures and Measurements We have developed our parameter extraction techniques based on multiple process technologies, resistor types, and resistor geometries. Fig. 2 shows a typical resistor layout, with Kelvin contacts to remove the effect of cabling, probe-to-pad, and metal interconnect parasitic series resistance. Typically, we have a large selection of W and L values for each type of resistor; only data from selected geometries are presented here. The maximum applied V was adjusted for each resistor to avoid excessively high current. Measurements were done using standard source-measure units. The relative accuracy of G calculated from measured data decreases as |V | approaches zero; G(V ) data at low |V | become visibly “noisy.” In the results presented below, raw measured data are filtered to remove such noisy data based on criteria for minimum acceptable I and V derived from inspection of the measured data. IV. Zero-Bias Conductance Extrapolation for Diffused Resistors As noted above, determining G0 can be difficult. A conventional approach is to fit a low-order polynomial to G(V ) over some range of V and then use the polynomial to extrapolate G0 ; unfortunately, that does not work for diffused resistors because the shape of their G(V ) curves is not well approximated by a polynomial. Serendipitously, for low and moderate E both (4) and (7) have a similar, quadratic dependence on E. This means that fitting G(V ) data (for each VC and T individually) with a simplified form of R3 that includes only depletion
Fig. 4. G(V ) for a 780 /ⵧ diffused resistor, L/W = 63 μm/63 μm.
pinching and velocity saturation effects gives a functional fit that can be reliably used to calculate G0 . Fitting of the simplified R3 model is restricted to data for which G is greater than 80% of its maximum value, to avoid data that are “too” nonlinear. The simplified R3 model is easily coded to give an analytical solution without requiring circuit simulation using the full R3 model, and the parameters are determined using a robust nonlinear least-squares optimizer [9]. Fig. 3 shows measured and modeled G(V ) for data that are relatively noise free and come from a long device for which both velocity saturation and self-heating are small; the nonlinearity is primarily from depletion pinching, and the simplified model fits the data well and gives a reliable extrapolated value for G0 for each VC . Fig. 4 shows measured and modeled G(V ) for a resistor of the same length and body doping polarity as that of Fig. 3, but with a wider width and less than one third of the sheet resistance. The velocity saturation effect should be similar between the two devices because the average field E is the same and the doping type is the same; the qualitative difference in the shape of G(V ), the “droop” at high V in Fig. 4, is therefore from increased self-
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Fig. 5.
IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 25, NO. 4, NOVEMBER 2012
G(V ) for a 780 /ⵧ diffused resistor, L/W = 2.1 μm/4.2 μm.
heating, which is due both to the decreased sheet resistance, which increases the current and hence the power dissipation at a given V , and to the increased width, which reduces the relative contribution of GTHP to GTH . Despite the qualitative change in shape and the increased noise in the G(V ) data in Fig. 4 (see Fig. 3), G0 is still extrapolated reliably. Note that, somewhat surprisingly, even for the relatively long device of Fig. 4 self-heating is the dominant cause of nonlinearity. Fig. 5 shows results for a short, narrow resistor of the same type as that of Fig. 4. Because of the small length this resistor is more affected by both self-heating and velocity saturation than the device of Fig. 4, so G(V ) is more nonlinear, and has visibly greater levels of noise at low V ; extrapolation of the simplified model to give G0 is still reliable. Note that even though Figs. 4 and 5 show that self-heating can be accurately modeled by the R3 velocity saturation model this does not mean it should be used for that purpose. For the greater level of self-heating seen in poly resistors this approach is not feasible, and self-heating is a function of frequency whereas velocity saturation is not [2], [4]. The use of velocity saturation to model self-heating should be done only to help extrapolate G0 , not to model actual resistors with R3. V. Zero-Bias Conductance Interpolation for Poly Resistors G(V ) data are usually noisier for poly than for diffused resistors, because they are more linear. In addition, velocity saturation does not affect poly resistors, and because of their low thermal conductance they have greater self-heating than equivalent size diffused resistors. Unlike for diffused resistors there is no parasitic p–n junction that can be forward biased in poly resistors; therefore, G(V ) can, and should, be measured for both negative and positive V . The depletion pinching effect is essentially linear in poly resistors, and can therefore be characterized by the coefficient 1√ ∂G l1 = G1 ∂V = √df C dp 1−df dp V =VC =0 (10) = −2 G1 ∂G . ∂V V =V =0 C
Fig. 6. G(V ) for an 87 /ⵧ poly resistor, L/W = 42 μm/4.2 μm.
Fig. 7.
Enlarged view of Fig. 6 around V = 0.
From (2) and (7), for low and moderate fields self-heating adds a parabolic nonlinearity to G(V ), in addition to the linear variation characterized by (10). Rather than just fitting a quadratic polynomial to measured G(V ), it is better to fit G(V ) = b0 + b1 V + b2 V 2 + b4 V 4
(11)
for each VC , because self-heating can be so pronounced in poly resistors that the second-order temperature coefficient TC2 , which contributes the fourth-order term in (11), can be important. This is manifest as an upside-down “w” shape of G(V ), see Fig. 6 (the initial increase in G as |V | increases away from zero is from TC1 , which is negative for the device of Fig. 6, and the decrease in G with increased |V | for higher |V | is from TC2 , which is positive). As we only want to fit data that are primarily quadratic in nature, the fourth-order term is included so that deviations from near-quadratic behavior can be detected and filtered out; if b2 b4 < 0 then there are inflexion √ points at V = ± −b2 /(6b4 ) and only data for V within the middle half of this range are retained (and the polynomial coefficients are then refitted to the reduced data set). Fig. 6 shows measured and modeled G(V ) data for a poly resistor, and Fig. 7 shows a blowup of Fig. 6 around V = 0.
McANDREW AND BETTINGER: ROBUST PARAMETER EXTRACTION FOR THE R3 NONLINEAR RESISTOR MODEL
The data are fairly noisy for the fitting range selected based on the criterion in the previous paragraph; however, the noise averages over a significant number of data points, and unlike the procedure for diffused resistors we are interpolating rather than extrapolating, so G0 is determined reliably. VI. Resistance and Temperature Coefficient Extraction For nondogbone-shaped resistors, ignoring the finite dopant effect which is only important for well resistors [10] L Lm + L + LW /Wm Rz = ρs (12) = ρs W Wm + W + WW /Wm where Lm and Wm are the mask (or design) length and width, respectively, L and W are fixed offsets, LW accounts for end resistance effects [2], and WW models the width dependence of the width offset [10]. Equation (12) cannot be directly solved for the five model parameters; however, it can be recast in the form ρs −
R z W Rz WW ρs L ρs LW Wm − + + = Rz Lm Lm Wm Lm Lm Wm Lm
(13)
and given Rz for five appropriate selections of Lm and Wm (13) can be directly solved for ρs , W , WW , ρs L , and ρs LW , and from the last two of these L and LW are calculated knowing ρs . Here, “appropriate” geometry selections mean the coefficient matrix from (13) is well-conditioned [11]; if data from more than five geometries are available, (13) can be solved in a least-squares sense using the Moore–Penrose pseudoinverse [11]. The temperature dependence of zero bias resistance is modeled in R3 as Rz (T ) = Rz (Tn ) 1 + TC1 (T − Tn ) + TC2 (T − Tn )2
(14)
where Tn is the nominal temperature, and TC1 and TC2 are determined by a least-squares fit of a quadratic polynomial to Rz (T ) data for three or more temperatures. Omitting the subscripts 1 and 2 for simplicity (the relation below applies identically to both first and second-order temperature coefficients), the geometry dependence for the coefficients can be written as follows: TCW TCL TCWL TC∞ + (15) + + = TC W L WL where TC∞ , TCW , TCL , and TCWL are parameters. Note that (15) is formulated in terms of effective and not mask geometries, so the geometry offsets need to be computed, as detailed in the previous paragraph, before the temperature coefficient geometry dependence parameters are determined. With the temperature coefficients from four appropriate geometry selections, (15) can be directly solved for the parameters (where “appropriate” has the same meaning as in the last paragraph). If data from five or more geometries are available, (15) can again be solved in a least-squares sense using the Moore– Penrose pseudoinverse [11]. Contact resistance cannot be determined from direct measurements of resistors because L effects cannot be separated
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from contact resistance effects [2]. If contact resistance and its temperature coefficients are known, from measurements of separate, dedicated test structures [12], then the total contact resistance should be calculated and subtracted from Rz prior to the characterization steps described in this section. There can also be additional resistance at the ends of a resistor from nonuniform current flow (current “spreading,” both vertical and lateral) and from a highly doped contact enhancement implant for diffused resistors or from a silicided contact region for poly resistors with unsilicided bodies. These resistances can be modeled via L and LW [2] and if they have different temperature coefficients than the resistor body this can be modeled via the TCL term in (15) [2]. VII. Depletion Pinching Parameter Extraction for Diffused Resistors There are many possible strategies to determine df and dp from G(V ) data that are affected only by depletion pinching, i.e., that are affected neither by velocity saturation nor selfheating. From (4) and (7) these last two effects vary as V/L (for low and moderate fields), so it would be expected that the nonlinearity in G(V ) for sufficiently long devices depends primarily on depletion pinching. Unfortunately, as Fig. 4 shows, even what would be considered to be a “long” device can be significantly affected by self-heating, and df and dp depend on geometry and so need to be determined separately for short as well as long devices. Determining df and dp reliably therefore requires G(V ) data that are either not affected by velocity saturation or self-heating or have the nonlinearity from these effects removed. If G0 can be reliably extrapolated from G(V ) data for three or more values of VC , then with gf = 1/[Rz (1 − df dp )] (2) can be manipulated to give, for V = 0 2 1 1 − d p − 2 G + 2 2 G2 . (16) −2VC = 2 df d f gf df gf Performing a regression of b0 + b1 G + b2 G2 on −2VC gives the coefficients b0 , b1 , and b2 , and from (16) √ b1 1/b2 1 df = dp = 2 − b 0 . gf = − (17) 2b2 gf df Fig. 8 shows the results of this procedure for the data of Fig. 3; G0 (VC ) is fitted well, and df = 0.0673 and dp = 0.724. If it is not possible to reliably extrapolate G0 (which we have rarely found to be the case using the approach of Section IV), or if G(V ) data are only available for two values of VC , an alternative is needed. For a given device, velocity saturation depends on V only and not on VC . In addition, if a resistor is fairly linear then the current for a given V does not change much with VC , so neither does the degree of self-heating. The effects of velocity saturation and self-heating are, therefore, close to the same for a given V independent of VC , so forming the ratio of G for different values of VC cancels these two effects, i.e., the 1 + rμ factor in (2) cancels giving dp + V − 2VC2 − dp − 2VC1 + V G(V, VC1 ) −1 = . (18) G(V, VC2 ) (1/df ) − dp + V − 2VC2
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locity saturation and self-heating so the 1 + rμ cancelation that leads to (18) is less accurate, but if G(V ) sweeps for only two values of VC are available the ratio technique must be used. VIII. Depletion Pinching Parameter Extraction for Poly Resistors Theoretically, from (2) ∂G/∂VC = −dp ∂2 G/∂VC2 V =VC =0
Fig. 8.
G0 (VC ) for the device of Fig. 3.
Fig. 9.
Ratio (18) for the device of Fig. 3.
The ratio on the left-hand side of (18) is determined from data (using interpolation for Kelvined measurements so V are the same for the two VC values) and df and dp are calculated to minimize the sum of squares of residuals between the left and right-hand sides of (18) [9]. As Fig. 9 shows, the results are quite reasonable, clearly show a depletion nature that decreases as |VC1 | increases, as expected, and give df = 0.0684 and dp = 0.641. The original ratioing procedure of [4] used VC1 = 0, but using adjacent VC pairs averages out noise for different sweeps and better cancels the influence of self-heating (currents for adjacent sweeps are closer than if VC1 is always zero). The results from the two techniques described above are slightly different; this is not an issue as the df and dp values from each device are subsequently used to fit the geometry dependence model for each parameter, using a procedure identical to that described for fitting the geometry dependence of temperature coefficients (15), and then are refined by numerical optimization to fit G(V ) from all measured devices. The need is for reliable initial values, and both approaches work well for this purpose. Fitting G0 can work better for short devices, which are more affected than longer devices by ve-
(19)
and this, along with (10), should enable both dp and df to be calculated from measured data. However, for poly resistors G is too linear in VC to enable (19) to be determined reliably from measured data. Consequently, there is only one piece of information, the linearity coefficient l1 in (10), from which to determine the two parameters dp and df , so multiple solutions are possible. In practice, dp just has to be large enough so that over the allowable bias range for a resistor |V − 2VC | dp , which makes G(V, VC ) linear; unfortunately, rearranging (10) gives l 1 dp df dp = (20) 1 + l 1 dp and for a given l1 if dp is too large then df dp approaches unity, which can cause numerical problems with the 1−df dp factor in the denominator of (2). Instead, computing G from (8) and equating its derivative with respect to VC to that of G of (2), at V = 0, gives 1 df dp = (21) / 1 + tb Cox s which guarantees that df dp < 1 and therefore avoids any possible numerical problems. From (20) and (21) s 1 dp = (22) l tb Cox 1 where l1 is determined from a linear regression of G0 (VC ); df then follows from the physical relation (21). IX. Velocity Saturation and Thermal Conductance Characterization From the analyses of Section II, in particular (6) and (7), it is apparent that the slope of rμ versus E2 provides information about the velocity saturation and thermal conductance parameters. From (2) 1 1 1 − df dp − 2VC + V vs sh rμ = rμ + rμ = −1 (23) G Rz 1 − d f dp therefore using the depletion pinching parameters determined as described in Section VII for diffused resistors or Section VIII for poly resistors, the measured G can be manipulated using (23) to remove the effects of depletion pinching and give rμ (E2 ), which characterizes self-heating and, for diffused resistors, velocity saturation. Counterintuitively, although both velocity saturation and self-heating have a greater effect for short than for long resistors, it is preferable to generate initial extracted values
McANDREW AND BETTINGER: ROBUST PARAMETER EXTRACTION FOR THE R3 NONLINEAR RESISTOR MODEL
Fig. 10. rμ versus E2 from 780 /ⵧ diffused resistors, L = 63 μm.
Fig. 11. rμ versus E2 from 87 /ⵧ poly resistors, L = 42 μm.
for thermal conductance parameters from long devices. This is because the quadratic forms (6) and (7) are most accurate at low fields, so using long L prevents E = V/L from becoming too large, and also because the difference between effective electrical length and mask length can be ignored. In addition, the contacts at the ends of a resistor contribute to the overall thermal conductance of a device, often significantly for poly resistors, and using long resistors means the influence of the contact component of thermal resistance can be ignored. Narrow devices are significantly less affected by self-heating than wide resistors (because of the perimeter component of thermal conductance). For diffused resistors, a (fairly crude) estimate of Sμvs of (6) is simply the slope of rμ versus E2 for the narrowest long resistor. As the thermal conductance of a wide and long resistor is dominated by the area component, from (6) and (7) without the perimeter component of thermal conductance we have ∂rμ TC1 L = Sμvs + 2 ∂E GTHA Rz W
(24)
and from the measured value of this slope for the widest long device and the Sμvs estimate from the narrowest long
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device (24) can be solved for GTHA , given TC1 and Rz from previous extraction steps. The difference in slopes between the second narrowest and narrowest long devices, which removes the influence of Sμvs , can then be used to determine GTHP based on (7). These initial estimates of Sμvs , GTHA , and GTHP are then refined using nonlinear least-squares optimization [9] to fit the rμ versus E2 data. Fig. 10 shows measured and modeled results from this procedure for diffused resistors; the data are not perfectly linear, but the model fit is reasonable and gives good starting values for the final numerical optimization. Note that the slope of rμ (E2 ) in Fig. 10 increases as the device width increases; if the nonlinearity [after the depletion pinching effect is removed using (23)] were predominantly from velocity saturation the slope should be independent of W; Fig. 10 shows this is clearly not that case and that even for relatively high sheet resistance devices (which are least affected by self-heating) the self-heating effect is dominant for wide devices. The curves rotate counterclockwise as W increases because TC1 in (7) is positive. (Although Sμvs is determined by the above procedure it is only used for getting initial estimates of GTHA and GTHP ; the R3 velocity saturation model is physical and does not vary significantly between types of device, so the velocity saturation parameters for the final optimization are initialized to the default values, Ecr = 4.0 V/μm, Eco = 0.4 V/μm, and dμ = 0.02, for p-body resistors and to Ecr = 1.2 V/μm, Eco = 0.12 V/μm, and dμ = 0.02 for n-body resistors.) For poly resistors, Sμvs is set to zero, (24) with that term removed is used to estimate GTHA from the slope of rμ (E2 ) for the widest long device (which is least influenced by GTHP ), then the slope of rμ (E2 ) for the narrowest long device is used to calculate GTHP based on (7). These initial estimates are then refined by adjusting GTHA and GTHP to fit rμ (E2 ) for long devices of all widths using nonlinear least-squares optimization [9]. Fig. 11 shows the results of such a fit. In contrast to the diffused resistor results of Fig. 10, the curves for the poly resistors in Fig. 11 have a negative slope and rotate clockwise as W increases because TC1 in (7) is negative. In addition, the curves in Fig. 11 do not “rotate” to have a positive slope for the narrowest W, but maintain a negative slope; this indicates that self-heating is the major contributor to rμ for poly resistors, and that velocity saturation, which would cause the slope to be positive, is not observable. The thermal conductance of short resistors is influenced by the thermal conductance of each contact, gTHC . Rather than analyzing rμ (E2 ) from a short resistor to estimate this, which can be difficult as wide and short resistors have a low resistance and can be hard to measure accurately for a reasonable range of V , we have found that setting gTHC = 10 μW/C works well for both diffused and poly resistors. The initial estimates of the thermal conductance parameters are refined by a final global optimization. X. Final Optimization If there are more than five geometries used for parameter extraction (12) cannot fit Rz exactly for all devices, for two reasons. First, the model (12) is not perfect. Second, any real
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Fig. 12.
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R3 G(V ) for a 780 /ⵧ diffused resistor, L/W = 2.1 μm/4.2 μm.
meas before computing the G from R3 by the ratio RR3 z /Rz residuals between the measured and modeled values. Fig. 12 shows an example of an R3 model fit to measured data from a diffused resistor after a final global optimization. The resistor of Fig. 12 is significantly affected by depletion pinching, velocity saturation, and self-heating; clearly, R3 is able to model all of these effects accurately. Fig. 13 shows an R3 model fit for a polysilicon resistor after a final global optimization. The modeled self-heating temperature rise T − Tn is also shown (Tn = 27 °C); it approaches 500 °C, which makes the device temperature T well above the value (248 °C) at which ρs changes from decreasing with increased T to increasing with increased T because of the influence of TC2 . The modeling of nonlinearity due to both self-heating and depletion pinching is reasonable; for the significant temperature rise observed the thermal conductance will change from its extracted value at Tn , but this is not taken into account in the R3 model.
XI. Conclusion
Fig. 13. R3 G(V ) for an 87 /ⵧ poly resistor, L/W = 42 μm/4.2 μm. Inset shows temperature rise.
devices are affected by local variation, i.e., by mismatch, and measurements are affected by noise, so even if (12) were perfect the extracted values of Rz for a given set of devices would have some unavoidable and unknowable statistical variation. Conceptually, the R3 model parameters for one temperature can be classified as either modeling Rz or modeling the shape of the nonlinearity of G(V, VC ). For highly linear resistors, the error in modeling Rz can be a nonnegligible fraction of the variation of G over all biases. Therefore, in a final numerical optimization of the R3 model parameters to fit measured data for all available device geometries, imprecision in modeling Rz can lead to a compromise in fitting nonlinearity, as the optimization balances the residuals in fitting G across all device geometries and biases. Solving (13), in fact, already provides a least-squares fit of Rz . The final optimization should therefore keep the Rz modeling parameters fixed, and only optimize those parameters that influence the shape of the nonlinearity in G (the depletion pinching, thermal conductance, and, for diffused resistors, velocity saturation parameters). This requires removing errors from any imprecision in modeling Rz , which is done by scaling
We presented improved, robust parameter extraction procedures for the R3 model. This included reliable techniques to determine G0 for both diffused and poly resistors, calculation of basic resistance parameters and temperature coefficients based on Rz = 1/G0 (VC = 0), and calculation of depletion pinching parameters of diffused resistors based on G0 (VC = 0). For poly resistors we showed how the physical relationship (21) can be combined with polynomial fitting of G0 (VC ) to avoid numerical problems in extracting the depletion pinching parameters. Knowing the depletion pinching parameters, the R3 model can then be used to remove the effects of depletion pinching from measured data to compute rμ , and we showed how the ensuing rμ (E2 ) characteristic can be used to generate initial estimates of thermal conductance parameters. Finally, we presented a numerical optimization procedure that does not compromise fitting of device nonlinearity to compensate for inaccuracies in modeling of Rz over geometry, which can be a problem for highly linear resistors.
References [1] R. V. H. Booth and C. C. McAndrew, “A 3-terminal model for diffused and ion-implanted resistors,” IEEE Trans. Electron Devices, vol. 44, no. 5, pp. 809–814, May 1997. [2] C. C. McAndrew, “Integrated resistor modeling,” in Compact Modeling: Principles, Techniques and Applications, G. Gildenblat, Ed. Berlin, Germany: Springer, 2010. [3] Z. Yu and C. C. McAndrew, “RF MOS is more than CMOS: Modeling of RF passive components,” in Proc. IEEE CICC, Sep. 2009, pp. 407–414. [4] C. C. McAndrew and T. Bettinger, “Improved parameter extraction procedures for the R3 model,” in Proc. IEEE ICMTS, Apr. 2011, pp. 43–48. [5] F. N. Trofimenkoff, “Field-dependent mobility analysis of the field-effect transistor,” Proc. IEEE, vol. 53, no. 11, pp. 1765–1766, Nov. 1965. [6] D. M. Caughey and R. E. Thomas, “Carrier mobilities in silicon empirically related to doping and fields,” Proc. IEEE, vol. 55, no. 12, pp. 2192–2193, Dec. 1967. [7] R. Vogelsong and C. Brzezinski, “Simulation of thermal effects in electrical systems,” in Proc. IEEE APEC, Mar. 1989, pp. 353–356. [8] Y. Tsividis and C. McAndrew, Operation and Modeling of the MOS Transistor, 3rd ed. New Work: Oxford Univ. Press, 2011.
McANDREW AND BETTINGER: ROBUST PARAMETER EXTRACTION FOR THE R3 NONLINEAR RESISTOR MODEL
[9] J. E. Dennis, D. M. Gay, and R. E. Welsch, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Softw., vol. 7, no. 3, pp. 348–368, Sep. 1981. [10] C. C. McAndrew, S. Sekine, A. Cassagnes, and Z. Wu, “Physically based effective width modeling of MOSFETs and diffused resistors,” in Proc. IEEE ICMTS, Mar. 2000, pp. 169–174. [11] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. [12] W. M. Loh, S. E. Swirhum, T. A. Schreyer, R. M. Swanson, and K. C. Saraswat, “Modeling and measurement of contact resistances,” IEEE Trans. Electron Devices, vol. 34, no. 3, pp. 512–524, Mar. 1987.
Colin C. McAndrew (S’82–M’84–SM’90–F’04) received the B.E. (Hons.) degree in electrical engineering from Monash University, Melbourne, Victoria, Australia, in 1978, and the M.A.Sc. and Ph.D. degrees in systems design engineering from the University of Waterloo, Waterloo, ON, Canada, in 1982 and 1984, respectively. From 1978 to 1980, and from 1984 to 1987, he was with the Herman Research Laboratories, State Electricity Commission of Victoria, Richmond, Australia. From 1987 to 1995, he was with the AT&T Bell Laboratories, Allentown, PA. Since 1995, he has been with Freescale Semiconductor, Inc. (formerly Motorola), Tempe, AZ. His work primarily focuses on compact and statistical modeling and characterization for circuit simulation. Dr. McAndrew received the Ian Langlands Medal from the Institute of Engineers of Australia in 1978, the Best Paper Awards for ICMTS in 1993 and 2012 and for CICC in 2002, and the BCTM Award in 2005. He was an editor of the IEEE Transactions on Electron Devices from 2001 to 2010, and is or has been on the technical program committees for the IEEE BCTM, ICMTS, CICC, and BMAS conferences.
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Tamara Bettinger (M’84) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1983, 1985, and 1990, respectively. While attending Purdue University, she was a Cooperative Education Student with the IBM Government Systems Division, Manassas, VA. She was a Senior Project Engineer with Delphi-Delco Electronics, Kokomo, IN, until 1998. She has been with Freescale Semiconductor, Inc. (formerly Motorola), Tempe, AZ, since 1998, and is currently a Senior Member of the Technical Staff working on semiconductor device characterization and model integration into process design kits.
