Errors due to process variations Deterministic error –Characterized a priori Over etching, vicinity effects, … –A priori unknown Gradient errors due to thermal, potential, stress, … Random errors –Intrinsic randomness in materials –Randomness in processes –Randomness during operation

Various Capacitor Errors

Common error reduction techniques Use large area to reduce random error Common Centroid layout to reduce linear gradient errors Use unit element arrays Interdigitize for matching Use of symmetry of photolithographic invariance Controlled edge or corner effects Dummy device for similar vicinity Guard rings for isolation Careful floor planning

Gradient errors: Linear gradient Higher order gradient Unknown magnitude and direction Common centroid cancels linear gradient errors, but not nonlinear gradient errors nor local random errors

Edge effects  better

Capacitor layout example

Resistor layout example

Transistor layout example

Layout of a differential pair Which layout is a better implementation of the differential pair? (b) or (c) or (d)?

Tilted implant beam causes asymmetry Aligned gate Parallel gate PLI PLI?

Parallel gate can be PLI Aligned gate can benefit from symmetry Asymmetric metal Add dummy metal Dummy makes it symmetric

Can have both gate alignment and symmetry But still suffers from gradient errors

M2M2 M1M1 M2M2 M1M1 This pattern cancels gradient Can it be made more compact?

Here is a simpler way

Donut shaped MOS to reduce C D and C gd

Avoiding the Antenna Effect Long metal acts as antenna to collect charges that may destroy gates: Break the antenna by going to a different layer:

Reference distribution Converted Iref to Vref to be shared Vulnerable to voltage drops in ground lines

Reference distribution Distribute Iref over longer distances is more robust

Routing long interconnects Parasitic capacitors or even inductors between parallel metal lines Parasitic caps between crossing metal lines

Differential signaling to reduce parasitic coupling effects

Shield or guard critical lines Larger spacing reduces coupling

Example shielding scheme Critical signals shielded

Substrate Coupling Distributed substrate network model Digital activities couple to analog part through substrate coupling

Nearby devices affect each other through effect on V th change Effect is worse at very high speed Example effect:

Techniques for reducing couple Use differential circuit Distribute digital signals in complementary Critical circuit in “quiet area”, far away from noisy (digital) circuit Devices in a well are less sensitive Use trenches for isolation Critical circuit in guard ring Critical operations in “quiet time”

Example use of guard ring

Analog cell array Other circuit H-shape distribution

An N th Order Central Symmetrical Layout Pattern for Nonlinear Gradients Cancellation Xin Dai, Chengming He, Hanqing Xing, Degang Chen, Randall Geiger Iowa State University

Introduction Matching accuracy plays a key role in the performance of precision circuits. Causes of mismatch in a parameter f  Local random variations  Gradient effect Gradient Direction

Introduction Theorem: If only linear gradient effects are present, then the mismatch of a parameter f of two elements will vanish if a common centroid layout is used. Gradient Direction Common Centroid Layout

Introduction Common Centroid Layout with Nonlinear Gradient Hypothesis of common-centroid theorem are not satisfied and AABB

Introduction Common Centroid Layout with Nonlinear Gradient Are there any simple layout strategies that will also cancel nonlinear gradients? AABB

Objective Develop a layout strategy for canceling mismatch of parameters for two matching- critical elements when nonlinear gradients of higher orders are present

Example It was observed by simulations that this structure is insensitive to first-order, second-order and third-order gradients ! What properties does this structure have that provides insensitivity to these gradients and can other structures be constructed that are insensitive to gradients of any order?

Outline Background of Gradient Effects Objective Center-Symmetric Structure Concept Cancellation of higher-order gradients Simulation Results Conclusions

Center-Symmetric Patterns Definition: A pattern P n of two elements relative to a common centroid pattern P is center-symmetric of order n if it can be generated by the following recursive algorithm 1)Map each segment of one of the two elements to +1 and each segment of the second element to -1 2)Define pattern P 1 =P 3)Assume pattern P k has been generated 4)If k<n, create pattern P k+1 by a)Rotating pattern P k by 180 o around any point external to pattern P k b)Multiplying all elements in the rotated pattern by (-1) k+1 to create a rotated constrained complimented pattern c)Define pattern P k+1 to be the union of pattern P k and the rotated constrained complimented pattern

Center-Symmetric Pattern Example Common Centroid Layout 1 1 k=1 1 1 Rotation Point Rotated Pattern k=2

Center-Symmetric Pattern Example Rotated Pattern k= constrained complimented pattern

Center-Symmetric Pattern Example Pattern P Pattern P 1

Center-Symmetric Pattern Example Pattern P Rotated Pattern k=3

Center-Symmetric Pattern Example constrained complimented pattern (no change since k is odd) k=3

Center-Symmetric Pattern Example Pattern P 3

st order 2 nd order 3 rd order 4 th order 5 th order

Center-Symmetric Patterns Many different center-symmetric layouts can be easily generated Different starting common-centroid circuits and different external rotation points will generate different structures

Property of Center-Symmetric Networks Theorem : A parameter f of a layout pattern of two elements that is center- symmetric of order n is insensitive to the k th -order gradients for

Gradient Modeling 1 st order gradient Up to n th order gradient Let (x 0,y 0 ) be any reference point in the neighborhood of the matching critical devices

Gradient Effect Moving the center of the n th order gradient will only introduce lower order components This argument can be repeated for gradient components of orders n-1, n-2, … 1 Let (x A,y A ) be any point in the neighborhood of the matching critical region It can be shown that f n (x,y) can be written as

1 st order 2 nd order 3 rd order 4 th order 5 th order Simulation Results Totally 5 patterns are simulated

Simulation Results Mismatch (%) Highest Order of Gradient Effect 1 st 2 nd 3 rd 4 th 5 th 1 st nd rd th th Pattern’s order Setup: Same total device area are assigned. Large gradient effects are artificially generated. Random variations are neglected.

Summary n th order pattern cancels up to n th order gradient effect using 2 n unit cells for each device. Easy to generate and extend Flexible cell placement Area efficient