EE 508 Lecture 13. Statistical Characterization of Filter Characteristics

Transcription

1 EE 508 Lecture 3 Statistical Characterization of Filter Characteristics

2 Comonents used to build filters are not recisely redictable R L C Temerature Variations Manufacturing Variations Aging Model variations Different aroaches are used to address each of these roblems Manufacturing variations is one of the most challenging and will be the focus of this lecture

3 Wafers are rocessed in batches or lots of 20 to 40 wafers and variations occur over time or over location R(t ) R(t 2 ) These variations are often the major contributor to rocess variability and can be in the range or larger 30% R(t 3 ) These variations often look like random variations

4 Within a batch, individual wafers are subjected to some variability during rocessing Temerature may vary with osition of wafer in the boat during diffusion Environment may vary with osition of wafer in boat during diffusion or other rocessing stes This variation causes characteristics of comonents to vary from wafer-to-wafer These variations often look like random variations

5 Environment may vary across individual wafers due to gradients in environmental variables during rocessing + + This variation causes characteristics of comonents to vary from die to die on a wafer These variations often look like random variations

6 Smaller variations may occur across individual die due to gradients in environmental variables during rocessing + + This variation causes characteristics of comonents to vary across a die These variations often look like random variations

7 Even smaller variations may occur across individual closely laced devices due to local gradients and local random rocessing variations + + This variation introduces local gradients in device characteristics as well as local random variations The direction and magnitude of the local gradients are random variables The local random variations are also random variables

8 Effects of manufacturing variations on comonents R L C A rigorous statistical analysis can be used to analytically redict how comonents vary and how comonent variations imact circuit erformance Montecarlo simulations are often used to simulate effects of comonent variations Requires minimal statistical knowledge to use MC simulations Simulation times may be rohibitively long to get useful results Gives little insight into secific source of roblems Must be sure to correctly include correlations in setu Often key statistical information is not readily available from the foundry

9 Modeling rocess variations in semiconductor rocesses R X = X + xrproc + xrwafer + xrdie + xrlgrad + xrlvar X is the nominal value of the arameter (tyically TT) and is a constant and art of the standard device model x RPROC is a random variable that changes from one lot of wafers to another x RWAFER is a random variable that changes from one wafer to another in a batch x RDIE is a random variable that changes from die to another on a wafer x RLGRAD is a random variable that is comrised of a magnitude and direction which are themselves both random variables and characterizes very local variations on a die x RLVAR is a random variable that characterizes very local variations on a die

10 Acknowledgement of artwork on following slides: htt:// htt:// 03.htm / t://

11 Acknowledgement of artwork on following slides:

12 x RPROC is a random variable that changes from one lot of wafers to another uncorrelated Correlated (and equal) Correlated (and equal) x RWAFER is a random variable that changes from one wafer to another in a batch uncorrelated Correlated (and equal) Correlated (and equal)

13 x RDIE is a random variable that changes from die to another on a wafer uncorrelated Correlated (and equal) Correlated (and equal)

14 x RLGRAD is a random variable that is comrised of a magnitude and direction which are themselves both random variables and characterizes very local variations on a die x RLVAR is a random variable that characterizes very local variations on a die uncorrelated x RLGRAD x RLVAR X RLGRAD Correlated (and equal) X RLVAR Uncorrelated X RLGRAD Correlated (and equal) X RLVAR Uncorrelated

15 Modeling rocess variations in semiconductor rocesses R X = X + xrproc + xrwafer + xrdie + xrlgrad + xrlvar x RPROC, x RWAFER, x RDIE, x RLVAR often assumed to be Gausian with zero mean Magnitude of x RLGRAD is usually assumed Gaussian with zero mean, direction is uniform from 0 o to 360 o LVAR PROC WAFER DIE DIE DIE LVAR GRAD Strongly deendent uon area and layout LVAR LVAR Area Perimeter Relative size between σ LVAR and σ GRAD deendent uon A, P, and rocess

16 Effects of layout on local random variations Actual Channel W L Drawn and Actual Features for MOS Transistor Variations also occur vertically in both oxide thickness and doing levels/rofiles and often these will dominate the lateral effects

17 Modeling rocess variations in semiconductor rocesses R Statistics associated with value of dimensioned arameters (oles, GB, SR,R,C,transresistance gains, transconductance gains, dominated by x RPROC ) Statistics associated with matching/sensitive dimensionless arameters such as voltage or current gains, comonent ratios, ole Q, (almost always closely laced) dominated by x RLGRAD and x RLVAR (because locally x RPROC, x RWAFER, x RDIE are all correlated and equal) Gradients are dominantly linear if sacing is not too large Secial layout techniques using common centroid aroaches can be used to eliminate (or dramatically reduce) linear gradient effects so, if emloyed, matching/sensitive arameters dominated by x RLVAR but occasionally common centroid layouts become imractical or areas become too large so that gradients become nonlinear and in these cases gradient effects will still limit erformance Higher-order gradient effects can be eliminated with layout aroaches that cancel higher moments but area and effort may not be attractive

18 Be sure correct statistical information is available when doing a statistical analysis using either analytical or Montecarlo methods R Some statistics associated with making many measurements over many devices over many lots of wafers Some statistics associated with many measurements in a articular rocess run Some statistics associated with making many measurements across a wafer Some statistics associated with making many measurements on closelylaced devices Some statistics associated with making many measurements on closelylaced devices that have common-centroid layouts Some statistics resented (articularly in literature or occasionally in PDK) with limited information about how data was gathered

19 Statistical Modeling of dimensioned arameters - examle Determine the standard deviation of the ole frequency (or band edge) of the first-order assive filter. Assume the rocess variables are zero mean with standard deviations given by R R PROC C C PROC

20 = RC Since R and C are random variables, the ole is also a random variable Theorem: The sum of uncorrelated Gaussian random variables is a multivariate Gaussian random variable Theorem: If X X m are uncorrelated random variables with standard deviations σ, σ 2, σ m, and a,a 2, a m are constants, then the standard deviation of the random variable y m aix i m 2 2 y ai i i i is given by the exression

21 = RC Since R and C are random variables, the ole is also a random variable = R R C C RAN Unfortunately the df which is the recirocal of the roduct of Gaussian variables is very difficult to obtain RAN Observe can exress as = R RRAN C CRAN RC R RAN C RAN R C

22 = RC = R RRAN C CRAN RC R RAN C RAN R C But R RAN <<R and C RAN <<C It thus follows from a truncated ower series exansion of the two-variable fraction that R RAN C RAN RC R C RRAN C RAN RC R C These oerations were used to linearize in terms of the random variables! Note that is the sum of two Gaussian random variables that are assumed to be uncorrelated so is also Gaussian

23 = RC RRAN C RAN RC R C It thus follows from the theorem that 2 2 R RAN CRAN RC R C But the nominal value of the ole is It thus follows that 2 2 RRAN C R C RAN R C Observe: N,

24 = RC 2 2 RRAN C R C RAN But R RAN and C RAN are aroximately R PROC and C PROC 2 2 RPROC C R C PROC recall R R PROC C C PROC

25 = RC Determine the 3σ range in the ole location 2. Determine the ercent of the rocess lots that will have a ole with mean that is within 0% of the nominal value 3. What can the designer do to tighten the band edge of this filter?

26 = RC Determine the 3σ range in the ole location The 3σ range is simly So, if the nominal ole location is 0KHz, the average value of the ole location from lot to lot will vary between 3.4KHz and 6.6KHz

27 = RC Determine the ercent of the rocess lots that will have a ole with mean that is within 0% of the nominal value Observe a 0% window is Recall N, For a kσ window the robability of being inside that window is the area under the df curve between kσ and +kσ Observe N 0, -kσ f(x) +kσ x

28 = RC Determine the ercent of the rocess lots that will have a ole with mean that is within 0% of the nominal value Observe a 0% window is f(x) For a Gaussian variable, this area is given by θ rob= 2FN(0,) k - = 2FN(0,) x

29 Offset Voltage Distribution Pdf of zero-mean Gaussian distribution f(x) -kσ kσ x Percent between: ±σ 68.3% ±2σ 95.5% ±3σ 99.73%

30

31 = RC Determine the ercent of the rocess lots that will have a ole with mean that is within 0% of the nominal value rob N(0,) θ = 2F 0.45 f(x) θ rob= Thus, aroximately 35% of the wafer lots will have a ole within 0% of the nominal value x

32 = RC What can the designer do to tighten the band edge of this filter?

33 End of Lecture 4