1. Errors due to process variations
Deterministic error within a die
Characterized a priori
Over etching, vicinity effects, …
A priori unknown
Gradient errors due to thermal, potential, stress, …
Random errors
Intrinsic randomness in materials
Use better materials, use large area for averaging
Randomness in processes
Use better processes, use large area for averaging
Randomness during operation
Use differential signaling, shielding, isolation, etc

2. L

3. Various Capacitor Errors

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

5. Unknown magnitude and direction
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
Common centroid

6. Edge effects
Edge control 
better

7. Better A-B matching
Vicinity control

8. Capacitor layout example

9. Are there any thing else you can improve?
Is there full symmetry between C1 and C2?

11. Resistor layout example

12. OK for really non-critical resistors. Accuracy is quite poor.
Should use unit cell arrays for improved accuracy.

13. Positives: use of dummy, use of unit cells, intedigitized, outfold jumps
Negatives: fold-in jumps, centroids not co-incidental
Improvements: ???

14. Transistor layout example

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

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

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

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

20. This pattern cancels gradientM2 M1 M2 M1
Can it be made more compact?

21. This pattern cancels gradientM2 M1 M2 M1
This one does.

24. Does the layout of M1 and M2 have common centroids?
Does the layout have symmetry?
Does it have PLI?

25. 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:

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

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

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

29. Differential signaling to reduce parasitic coupling effects

30. Shield or guard critical lines
Larger spacing reduces coupling

31. Example shielding scheme
Critical signals shielded

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

33. Nearby devices affect each other through effect on Vth change
Effect is worse at very high speed
Example effect:

34. Techniques for reducing coupling
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 inside guard ring
Critical operations in “quiet time”

35. Example use of guard ring

38. 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.
Common Centroid Layout
Gradient
Direction

39. Introduction A B B A Common Centroid Layout with Nonlinear Gradient
Hypothesis of common-centroid theorem are not satisfied and

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

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

42. Center-Symmetric Pattern Example
0-th order pattern P0:
To generate 1st order pattern P1: take odd symmetry, that is, rotate about a point of symmetry
point of symmetry
Rotate
180deg

43. Center-Symmetric Pattern Example
To generate 2nd order pattern P2: take even symmetry, that is, flip about one side
Pattern P1
Pattern P2

44. Center-Symmetric Pattern Example
To generate 3nd order pattern P3: take oddsymmetry, that is, rotate 180 deg about one point
Pattern P2
Pattern P3

45. alternatively
Pattern P2
Pattern P3

46. Center-Symmetric Pattern Example
To generate 4th order pattern P4: take even symmetry, that is, flip about one line
Pattern P3
Pattern P4

47. alternative
Pattern P3
Pattern P4

48. 2nd alternative
Pattern P3
Pattern P4

49. 5th order pattern

50. Alternative 5th order pattern

51. 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

52. 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 kth-order gradients for

53. Gradient Modeling 1st order gradient Up to nth order gradient
Let (x0,y0) be any reference point in the neighborhood of the matching critical devices
1st order gradient
Up to nth order gradient

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

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

56. Highest Order of Gradient Effect
Simulation Results
Setup: Same total device area are assigned.
Large gradient effects are artificially generated.
Random variations are neglected.
Mismatch (%)
Highest Order of Gradient Effect
1st
2nd
3rd
4th
5th
2.77
5.22
7.43
10.39
0.24
0.87
1.70
0.01
0.068
0.0023
Pattern’s order

57. Test structure <0.002% systematic mismatch
3rd order central symmetry
<0.04% systematic mismatch
2nd order central symmetry

58. Measurement results

59. Measurement results

60. Matching performance Conference papers: Journal papers:
C. He, et al., ‘Nth order circular symmetry pattern and hexagonal tessellation: two new layout techniques canceling nonlinear gradient,’ ISCAS 2004
X. Dai, C. He, et al., ’an Nth order central symmetrical layout pattern for nonlinear gradients cancellation,’ ISCAS 2005
Journal papers:
C. He, et al., ‘New layout strategies with improved matching performance,’ Analog integrated circuits and signal processing, vol. 49 , issue 3, pp ,2006  

61. 1 1
Binary matching 1 1

62. 1 2 1
Binary matching 1 1 2 2 1 1 1 2 1 2 1 1 1 2 1 1 2 1 1 1 2

63. 1 2 3 1
Binary matching 1 1 2 2 1 1 1 3 2 1 2 1 1 3 1 2 1 1 2 1 3 1 1 2

64. 1 2 3 1
Binary matching 4 1 1 2 2 1 1 1 3 2 1 2 1 1 3 1 5 2 1 1 2 4 1 3 1 1 2

65. 1 2 3
3-way match 2 1 2 3 2 3 1 3 1