1. Variation Aware Gate Delay Models Dinesh Ganesan

2. Motivation Background Finite Point Model for gates Results Future work Outline

3. Motivation Variability is an emerging design concern Design tools are just beginning to address variability and reliability concerns Right combination of better modeling and regular IC fabrics is the best way to resolve IC variability challenges

4. Background Physical models Based on physics of devices Accurate Very slow considering the number of transistors in today's design Eg - TCAD Empirical models Device modeled with equations in various regions of operation. Good accuracy Slow for chip level simulations Eg - BSIM3-the current industry standard, PSP, Gummel-Poon, Ebers Moll Table model Stored as lookup tables Does not scale with change in circuit topology Requires other models for characterization

5. BSIM Given voltages at ports calculates the current/capacitance between the ports Model them using equations with physical meaning Parameter values obtained from the foundry/fab Parameters grow exponentially in today's technology to address the emerging second order effects

6. Circuit Simulation Circuit simulators like SPICE (Simulation Program with Integrated Circuit Emphasis) – a general purpose analog circuit simulator. Solve nonlinear equations iteratively Nodal analysis using Newton Raphson/Secant iteration for convergence Use model file like BSIM for obtaining the voltage/current Use of BSIM models for complete chip simulation takes months Faster models of simulation required

7. Variations As transistor geometry decreases control of device parameters becomes difficult Shift from Static timing analysis to Statistical timing analysis Device model should be robust to process variations 130nm and above SSTA 90nm and below

8. Requirement Fast simulation (including Monte Carlo) Accurate Model device Robust to process variations

9. Current Source Model Idc(Vi,Vo) – Current source Qx_y – Charge at x when y switches i – input o - output Gate Model Idc – current captures static characteristics Q – charge – captures the dynamic characteristics

10. Idc(Vi,Vo) Static I-V characteristics of a gate Obtained using finite point model for pull-up (PUN) and pull down network (PDN) separately Gate Idc(Vi,Vo) = PDN Idc(Vi,Vo) + PUN Idc(Vi,Vo)

11. Idc(Vi,Vo) – PUN/PDN Method similar to finite point model of transistor used, except that Id-Vi is nonlinear in this case Two points for Id-Vo and five points for Id-Vi are sufficient to generate the complete IV Points obtained by single DC simulation Process variations – included in the IV Continuous model in all regions required – continuous model for Idc-Vi Points required for the finite point model Idc - Vo Idc - Vi

12. Idc(Vi,Vo) Simulation results for NAND2 Vout Vin

13. Q(Vi,Vo) Calculate charge at input and output node based on switching at the nodes Requires two transient simulations Charge calculated by monitoring the current at the nodes during the simulation

14. Q(Vi,Vo) Calculation of Qo_i & Qi_i Vo PDN PUN Vi t0 t1 Vi Transient simulation Idc model Repeated for all values of Vi and Vo

15. Q(Vi,Vo) Qo_i calculated from SPICE Idc Qo_i calculated from Idc model Qi_i

16. Q(Vi,Vo) Calculation of Qo_o & Qi_o Vo PDN PUN Vi t0 t1 Vo Transient simulation Idc model Repeated for all values of Vi and Vo

17. Q(Vi,Vo) Qo_o calculated from Idc model Qi_o Qo_i calculated from Idc model Qi_i

18. Q(Vi,Vo) - approximation Qo_o -Actual Qo_o -approximated Qo_o -error Qo_i -Actual Qo_i -approximated Qo_i -error

19. Charge implementation in VerilogA tintout gr Vin = V(tin,gr); Vout = V(tout,gr); Qi_i = f1(Vin,Vout); Qi_o = f2(Vin,Vout); Qo_i = f3(Vin,Vout); Qo_o = f4(Vin,Vout); I(tin,gr) <+ ddt(Qi_is); I(tout,gr) <+ ddt(Qo_os); I(tin,tout) <+ ddt(Qo_is); I(tout,tin) <+ ddt(Qi_os); tintout gr Gate charge

20. Results Delay vs Input slew Output slew vs Input slew

21. Results Output voltage vs timeOutput current vs time

22. Advantages Fast simulation Accurate analysis Process variations included with ease Helps in fast Monte Carlo analysis

23. Questions???