Variation

2 Sources of Variation 1.Process (manufacturing) (physical) variations:  Uncertainty in the parameters of fabricated devices and interconnects −From die to die −Within a particular die 2.Environmental (operating context) (temporal) (dynamic) variations:  Uncertainty in the operating environment of a particular device during its lifetime −Temperature −Supply voltage −Lifetime wear-out

3 Variation Classification

4 Supply Voltage Variation

5 Temperature Variation Within die temperature variation Temperature Variation:  Both the device and interconnect performance have temperature dependence, −Higher temperature  performance degradation.

6 Process Variation Process variation: Sample space:  Set of manufactured dies  Results in yield loss −Y = # working die / # manuf. Die  A small portion of sample space is allowed to fail timing constraints −CPU/GPU design: Speed/core binning: for different applications −  Lessens the requirement that all or very high percentage of die meets the fastest timing constraint

Process Variation 7 [Cadence]

8 Environmental Variation Environmental Variation: Sample space:  Operational life of a chip  A pessimistic analysis is required −Should ensure correct operation throughout lifetime  Design that operates faster than necessary for much of its operational life −  loss in efficiency  One approach: −Runtime adaptivity of the design Environmental Variation: Treated by worst case analysis Process variation: Treated statistically

9 Process Variation: Sources PV Sources:  Var. in physical parameters (due to imperfect manufacturing): −Gate length or critical dimension (CD) −Gate oxide thickness −Channel doping concentration −Interconnect thickness −Interconnect width −…−… −Dominant factors: CD and channel doping  Var. in electrical parameters of components −V th −Drive strength of transistors −Resistance of wires −Capacitance of wires −….  Var. in circuit characteristics: −Delay −Power −Noise

10 Process Variation Sources x Wafer X Wafer Y 2.0 [IBM, Intel and TSMC]

11 Variation Variations Variation of variation over years % WID/total variation (from mean value) −Gate oxides are so thin that a change of one atom can cause a 25 percent difference in substrate current. −EE Times (04/11/2006) ILD: inter-layer dielectric [

12 Process Variation A physical parameter variation may affect more than one electrical parameter:  Wire width  −Wire capacitance −Wire resistance −Coupling noise  Gate oxide thickness  −Drive current −V th −C g

13 Correlation  Must consider correlation between electrical parameters  If ignore correlation (C w, R w ), −In theory, both may be at worst–case values −Impossible in practice Correlation among physical parameters themselves  An equipment variation (e.g. lens deviation) may impact multiple physical parameter values (all metal layers and poly) −Hard to model due to large number of equipment-related parameters  Most algorithms take physical parameters to be basic random variables

14 Classification Types of physical-parameter variations: 1.Systematic (deterministic): −Show predictable variational trends across a chip −Caused by known physical phenomena during manufacturing −Can be predicted upfront by analyzing the designed layout −Can be avoided in final stages −E.g. Metal fill, optical proximity effects −But at early stages, common to be treated statistically - Most of the time, not available to designers/CAD developers - E.g.,regions with uniform metal densities have more uniform ILD thicknesses

15 Classification Types of physical-parameter variations: 2.Non-systematic (random): −Truly uncertain component of physical-parameter variations −Resulted from processes that are statistically independent of the design implementation −Only the statistical characteristics are known at design time, −  Must be modeled using RVs Common practice:  In earlier stages, both systematic and nonsystematic variations are modeled statistically  As we move through the design process and more detailed information is obtained, the systematic components can be modeled deterministically (if sufficient analysis capabilities are available)

Scaling Effect  A 4nm MOSFET predicted in mass production in 2020,  < 10 Si atoms are expected along the channel (IBM roadmap)  MOS transistors are rapidly becoming truly atomistic devices  Random variations are becoming dominant. 16  A 22nm MOSFET expected in mass production  50 Si atoms along the channel  Large parameter fluctuations

17 Classification Classification of variation:  Die-to-die (inter-die) (global): −Affects all devices on the same die in the same way  Within-die: WID (intra-die) (local) (on-chip: OCV): −Affects each device on the same die differently −E.g. some devices have larger/smaller CDs than nominal

D2D Variation 18 [Menezes07]

19 Classification Types of within-die variation: 1.Spatially-correlated: −Many of the underlying processes that give rise to within-die variation change gradually from one location to the next. −  Affect closely spaced devices in a similar manner −  Make them more likely to have similar characteristics than those placed far apart 2.Independent: −Statistically independent from all other devices −Scaling  Contribution of independent within-die variation is increasing −With SC: −L eff, −Temperature −Supply voltage −No SC: −t ox, −Dopant concentration

20 Inter-die vs. Intra-die Variations Figures are courtesy of IBM, Intel and TSMC Intra-die spatial Correlation Inter-die global Correlation L eff

References [Blaauw08] Blaauw, Chopra, Srivastava, Scheffer, “Statistical Timing Analysis: From Basic Principles to State of the Art,” IEEE Transactions on CAD, Vol. 27, No. 4, April [Forzan09] Forzan, Pandini, “Statistical static timing analysis: A survey,” Integration, The VLSI Journal, 42, [Menezes07] Menezes, “The Good, the Bad, and the Statistical,” Invited talk, ISPD