1. 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

3. Various Capacitor Errors

4. 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

5. 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

6. Edge effects  better

11. Capacitor layout example

14. Resistor layout example

17. Transistor layout example

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

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

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

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

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

24. Here is a simpler way

28. Donut shaped MOS to reduce C D and C gd

29. 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:

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

31. Reference distribution Distribute Iref over longer distances is more robust

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

33. Differential signaling to reduce parasitic coupling effects

34. Shield or guard critical lines Larger spacing reduces coupling

35. Example shielding scheme Critical signals shielded

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

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

38. 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”

39. Example use of guard ring

41. Analog cell array Other circuit H-shape distribution

43. 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

44. 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

45. 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

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

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

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

49. 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?

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

51. 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

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

53. Center-Symmetric Pattern Example 1 1 1 1 Rotated Pattern k=2 1 1 1 1 constrained complimented pattern

54. Center-Symmetric Pattern Example 1 1 1 1 Pattern P 2 1 1 Pattern P 1

55. Center-Symmetric Pattern Example 1 1 1 1 Pattern P 2 1 1 1 1 1 1 1 1 Rotated Pattern k=3

56. Center-Symmetric Pattern Example 1 1 1 1 1 1 1 1 constrained complimented pattern (no change since k is odd) k=3

57. Center-Symmetric Pattern Example 1 1 1 1 1 1 1 1 Pattern P 3

58. 712 1 139 5 1115 314 2 64 10 8 16 1 23 4 52 8316 74 1 st order 2 nd order 3 rd order 4 th order 5 th order

60. 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

61. 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

62. 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

63. 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

64. 1 st order 2 nd order 3 rd order 4 th order 5 th order Simulation Results Totally 5 patterns are simulated

65. Simulation Results Mismatch (%) Highest Order of Gradient Effect 1 st 2 nd 3 rd 4 th 5 th 1 st 02.775.227.4310.39 2 nd 000.240.871.70 3 rd 0000.010.068 4 th 00000.0023 5 th 00000 Pattern’s order Setup: Same total device area are assigned. Large gradient effects are artificially generated. Random variations are neglected.

66. 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