An Introduction to Matching and Layout Alan Hastings Texas Instruments

1. An Introduction to Matching and Layout Alan HastingsTexas Instruments

2. Overview of Matching • Two devices with the same physical layout never have quite the same electrical properties. • Variations between devices are called mismatches. • Mismatches may have large impacts on certain circuit parameters, for example common mode rejection ratio (CMRR). • By default, simulators such as SPICE do not model mismatches. The designer must deliberately insert mismatches to see their effects.

3. Kinds of Mismatch • Mismatches may be eitherrandomor systematic,or a combination of both. • Suppose two matched devices have parameters P1 and P2. • Then let the mismatch between the devices equal P = P2 – P1. • For a sample of units, measure P . • Compute sample mean m(P ) and standard deviation s(P ). • m(P ) is a measure of systematic mismatch. • s(P) is a measure of random mismatch.

4. Random Mismatches • Random mismatches are usually due to process variation. • These process variations are usually manifestations of statistical variation, for example in scattering of dopant atoms or defect sites. • Random mismatches cannot be eliminated, but they can be reduced by increasing device dimensions. • In a rectangular device with active dimensions W by L, an areal mismatch can be modeled as:

5. Random Mismatches (continued) • Random mismatches thus scales as the inverse square root of active device area. • To reduce mismatch by a factor of two, increase area by a factor of four. • Precision matching requires large devices. • Other performance criteria (such as speed) may conflict with matching.

6. Systematic Mismatches • Systematic mismatches may arise from imperfect balancing in a circuit. • Example: A mismatch VCE between the two bipolar transistors of a differential pair generates an input offset voltage VBE equal to: • Simulations will readily show this source of systematic offset. • Usually, the circuit can be redesigned to minimize or even to completely eliminate this type of systematic offset.

7. Gradients • Systematic mismatches may also arise from gradients. • Certain physical parameters may vary gradually across an integrated circuit, for example: • Temperature • Pressure • Oxide thickness • These types of variations are usually treated as 2D fields, the gradients of which can (at least theoretically) be computed or measured. • Because of the way we mathematically treat these variations, they are called gradients.

8. Gradients (continued) • Even subtle gradients can produce large effects. • A 1C change in temperature produces a –2mV in VBE, which equates to an 8% variation in IC. • Power devices on-board an integrated circuit can easily produce temperature differences of 10–20C.

9. Analyzing Gradients • For a simplified analysis of gradients: • Make the following assumptions of linearity: • The gradient is constant over the area of interest. • Electrical parameters depend linearly upon physical parameters. • Although neither assumption is strictly true, they are usually approximately true, at least for properly laid out devices.

10. Analyzing Gradients (continued) • Assuming linearity, • We can reduce each distributed device to a lumped device located at the centroid of the device area. • The magnitude of the mismatch equals the product of the distance between the centroids and the magnitude of the gradient along the axis of separation. • Therefore we can reduce the impact of the mismatch by reducing the separation of the centroids.

11. How to Find a Centroid (Easily) • Rules for finding a centroid (assuming linearity): • If a geometric figure has an axis of symmetry, then the centroid lies on it. • If a geometric figure has two or more axes of symmetry, then the centroid must lie at their intersection.

12. The Centroid of an Array • The centroid of an array can be computed from the centroids of its segments. • If all of the segments of the array are of equal size, then the location of the centroid of the array is the average of the centroids of the segments: • Note that the centroid of an array does not have to fall within the active area of any of its segments. • This suggests that two properly constructed arrays could have the same centroid….

13. Common-centroid Arrays • Arrays whose centroids coincide are called common-centroid arrays. • Theoretically, a common-centroid array should entirely cancel systematic mismatches due to gradients. • In practice, this doesn’t happen because the assumptions of linearity are only approximately true. • Common-centroids don’t help random mismatches at all. Neither are they a cure for sloppy circuit design! • Virtually all precisely matched components in integrated circuits use common centroids.

14. Interdigitation • The simplest sort of common-centroid array consists of a series of devices arrayed in one dimension. • One-dimensional common-centroid arrays are ideal for long, thin devices, such as resistors. • Since the segments of the matched devices are slipped between one another to form the array, the process is often called interdigitation.

15. Interdigitation (continued) • Not all interdigitated arrays are made equal! • Certain arrays precisely align the centroids of the matched devices (A, C). These provide superior matching. • Other arrays only approximately align the centroids (B). These provide inferior matching.

16. 2D Cross-coupled Arrays • A more elaborate sort of common-centroid array involves devices cross-coupled in a rectangular two-dimensional array. • This type of array is ideal for roughly square devices, such as capacitors and bipolars.

17. 2D Cross-coupled Arrays (cont’d) • The simplest two-dimensional cross-coupled array contains four segments. • This type of array is called a cross-coupled pair. • For many devices, particularly smaller ones, the cross-coupled pair provides the best possible layout. • More complicated 2D arrays containing more segments provide better matching for large devices because they minimize the impact of nonlinearities.

18. Practical Common-centroid Arrays • Often, the design of a common-centroid array is complicated by layout considerations. • Sometimes certain devices can be merged, resulting in a smaller overall array. • Unfortunately, such mergers often constrain the layout of the array. • The proper design of a cross-coupled array is often quite difficult, and a certain degree of experience is required to obtain good results.

19. 2:1:2 Bipolar Array • This array matches a 4X and a 1X bipolar transistor using interdigitation.

20. 4:1:4 Bipolar Array • This array matches an 8X and a 1X bipolar transistor using interdigitation.

21. Rules for Common Centroids • The following rules summarize good design practices: • Coincidence: The centroids of the matched devices should coincide, at least approximately. Ideally, the centroids should exactly coincide. • Symmetry: The array should be symmetric about both the X- and the Y-axes. Ideally, this symmetry should arise from the placement of the segments in the array, and not from the symmetry of the individual segments. • Dispersion: The array should exhibit the highest possible degree of dispersion; in other words, the segments of each device should be distributed throughout the array as uniformly as possible. • Compactness: The array should be as compact as possible. Ideally, it should be nearly square.