2. Wim Schoenmaker ©magwel2005 Electromagnetic Modeling of Back-End Structures on Semiconductors Wim Schoenmaker

3. ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Putting vector potentials on the grid Static results High-frequency results Conclusions

4. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Vector potentials: their ‘physical’ meaning Static results High-frequency results Conclusions

5. Wim Schoenmaker ©magwel2005 Introduction -interconnects On-chip connections between transistors Dimensions/pitches still decreasing Increasing clock frequencies Frequency dependent factors become more and more important: e.g. Cross-talk, skin effect, substrate currents Transistors (gates, sources and drains)

6. Wim Schoenmaker ©magwel2005 Introduction-integrated passives Passive structures in RF systems – e.g. antennas, switches, … Cost reduction by integration in IC’s Simulation of RF components Increases reliability Increases production yield Decreases developing cycle RF section WLAN receiver - 5.2 GHz

7. Wim Schoenmaker ©magwel2005 Introduction - problem definition

8. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Putting vector potentials on the grid Static results High-frequency results Conclusions

9. Wim Schoenmaker ©magwel2005 Designer’s needs - Problem definition constraints: –Use the ‘language’ of designers not electric and magnetic fields (E,B) but Poisson field V and at high frequency also vector potential A –Full 3D approach exploit Manhattan structure, i.e. 3D grid –Include high frequency consideration work in frequency space provide designers with R(  ), C(  ), L(  ),G(  ) parameters (resistance R, capacitance C, inductance L, conductance G)

10. Wim Schoenmaker ©magwel2005 Designer’s needs - numerical approach WARNING!! Engineers write:

11. Wim Schoenmaker ©magwel2005 Constitutive laws Conductors Semi-conductors with... gives... V, A, n and p as independent variables J n is the electron current density J p is the hole current density U is recombination – generation term p local hole concentration n electron concentration T is lattice temperature k is Boltzmann’s constant Q is elementary charge

12. Wim Schoenmaker ©magwel2005 Designer’s needs - numerical approach Question: How to put V and A on a discrete grid ? –Answer (1): V is a scalar and could be assigned to nodes –Answer (2): A= (A x,A y,A z ) and with we obtain a 3-fold scalar equation (A x,A y,A z ) on nodes too! BUT VECTOR = 3-FOLD SCALAR

13. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Putting vector potentials on the grid Static results High-frequency results Conclusions

14. Wim Schoenmaker ©magwel2005 Continuum vs discrete So far A.dx infinitesimal 1-form: On the computer dx :  x are the distances between grid nodes i.e. connections on the grid A is a connection A exists between the grid nodes

15. Wim Schoenmaker ©magwel2005 Implementation for numerical simulation Equations to be solved Discretization of V is standard: V is put in nodes A is put on the links V i =V(x i ) V j =V(x j ) A ij =A(x i,x j )=A.e m

16. Wim Schoenmaker ©magwel2005 Discretizing A Stokes theorem: Stokes theorem once more:

17. Wim Schoenmaker ©magwel2005 Discretizing A Now 4 times  geometrical factor

18. Wim Schoenmaker ©magwel2005 Counting nodes & links & equations Grid with N 3 nodes N 3 unknowns (V i ) 3N 3 (1-1/N) links3N 3 (1-1/N) unknowns (A l ) N 3 equations for V there are 3N 3 (1-1/N) equations for A BUT not all A are independent PROBLEM! # equations = # unknows

19. Wim Schoenmaker ©magwel2005 Gauge condition Solution: Select a gauge condition –Coulomb gauge –Lorentz gauge –…. Coulomb gauge Each node induces a constraint between A-variables total =N 3

20. Wim Schoenmaker ©magwel2005 Implementation of gauge condition Old proposal: build a ‘gauge tree’ in the grid –highly non-local procedure –difficult to program New proposal: force # equations to match # unknowns –introduce extra fieldsuch that ‘ghost’ field Local procedure sparse matrices easy to program N 3 variables  i

21. Wim Schoenmaker ©magwel2005 Old system of equations New system of equations  Ghost field Core Idea

22. Wim Schoenmaker ©magwel2005 Easy to program creates a Regular matrix Local procedure Sparse Diagonal dominant So.. we do not solveBut... Gauge implementation

23. Wim Schoenmaker ©magwel2005 Gauge implementation Exploit the fact that We can make a Laplace operator for one-forms on the grid by This is an alternative for the ghost field method

24. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Vector potentials: their physical meaning Classical ghosts: a new paradigm in physics Static results High-frequency results Conclusions

25. Wim Schoenmaker ©magwel2005 Why a paradigm ? What is a paradigm ? – paradigm shift = change of the perception of the world (Thomas Kuhn) Examples –Copernicus view on planetary orbits –Einstein’s view on gravity ~curvature of space-time Scientific revolutions with periodicity of –1 year ? Management (pep) talk –software vs 3.4 --> vs 3.5 –300 years ? Ok, see examples above –25 years ? Acceptable and operational use of the word “paradigm ”

26. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Vector potentials: their physical meaning Static results High-frequency results

27. Wim Schoenmaker ©magwel2005 Spiral inductor E magn1 =1.16E-12 J E magn2 =1.20E-12 J L=2.041E-11 H E magn1 =1.82E-19 J E magn2 =1.87E-19 J L=3.69E-13 H Static B-field of spiral Static B-field of ring

28. Wim Schoenmaker ©magwel2005 Outline Introduction and problem definition Designer’s needs: the numerical approach Vector potentials: their physical meaning Classical ghosts: a new paradigm in physics Static results High-frequency results Conclusions

29. Wim Schoenmaker ©magwel2005 Results:Cylindrical wire (Al) 2 a = 3  m 100 GHz50 GHz 25 GHz15 GHz4 GHz

30. Wim Schoenmaker ©magwel2005 Results:Cylindrical wire Resistance (analytical) Resistance (solver) Reactance (analytical) Reactance (solver) 85DC 5530 4 14 GHz Analytical result Solver

31. Wim Schoenmaker ©magwel2005 Proximity effect 1 GHz 3 GHz Current density

32. Wim Schoenmaker ©magwel2005 Problem: Alternating currents alternating fields alternating currents ….. Substrate Current

33. Wim Schoenmaker ©magwel2005 ~ V Results: ring Boundary conditions: A-field on boundary vanishes (DC = no perp B-field) dV/dn perp to edge of simulation domain vanishes (DC = no perp E-field) On the contacts a harmonic signal for V Consider only first harmonic variables

34. Wim Schoenmaker ©magwel2005 Results: current densities and the ring