1. Inverse Problems in Semiconductor Devices Martin Burger Johannes Kepler Universität Linz

2. Inverse Problems in Semiconductor Devices Linz, September, 2004 2 Outline Introduction: Drift-Diffusion Model Inverse Dopant Profiling Sensitivities

3. Inverse Problems in Semiconductor Devices Linz, September, 2004 3 Joint work with Heinz Engl, RICAM Peter Markowich, Universität Wien & RICAM Antonio Leitao, Florianopolis & RICAM Paola Pietra, Pavia

4. Inverse Problems in Semiconductor Devices Linz, September, 2004 4 Inverse Dopant Profiling Identify the device doping profile from measurements of the device characteristics Device characteristics:  Current-Voltage map  Voltage-Capacitance map

5. Inverse Problems in Semiconductor Devices Linz, September, 2004 5 Inverse Dopant Profiling Device characteristics are obtained by applying different voltage patterns (space- time) on some contact Measurements:  Outflow Current on Contacts  Capacitance = variation of charge with respect to voltage modulation

6. Inverse Problems in Semiconductor Devices Linz, September, 2004 6 Mathematical Model Stationary Drift Diffusion Model: PDE system for potential V, electron density n and hole density p in  (subset of R 2 ) Doping Profile C(x) enters as source term

7. Inverse Problems in Semiconductor Devices Linz, September, 2004 7 Boundary Conditions Boundary of  homogeneous Neumann boundary conditions on  N and on Dirichlet boundary  D (Ohmic Contacts )

8. Inverse Problems in Semiconductor Devices Linz, September, 2004 8 Device Characteristics Measured on a contact  0 on  D : Outflow current density Capacitance

9. Inverse Problems in Semiconductor Devices Linz, September, 2004 9 Scaled Drift-Diffusion System After (exponential) transform to Slotboom variables (u=e -V n, p = e V p) and scaling: Similar transforms and scaling for boundary conditions

10. Inverse Problems in Semiconductor Devices Linz, September, 2004 10 Scaled Drift-Diffusion System Similar transforms and scaling for boundary Conditions Essential (possibly small) parameters - Debye length - Injection Parameter  -Applied Voltage U

11. Inverse Problems in Semiconductor Devices Linz, September, 2004 11 Scaled Drift-Diffusion System Inverse Problem for full model ( scale  = 1)

12. Inverse Problems in Semiconductor Devices Linz, September, 2004 12 Optimization Problem Take current measurements on a contact  0 in the following Least-Squares Optimization: minimize for N large

13. Inverse Problems in Semiconductor Devices Linz, September, 2004 13 Optimization Problem This least squares problem is ill-posed Consider Tikhonov-regularized version C 0 is a given prior (a lot is known about C) Problem is of large scale, evaluation of F involves N solves of the nonlinear drift-diffusion system

14. Inverse Problems in Semiconductor Devices Linz, September, 2004 14 Sensitivies Define Lagrangian

15. Inverse Problems in Semiconductor Devices Linz, September, 2004 15 Sensitivies Primal equations, with different boundary conditions

16. Inverse Problems in Semiconductor Devices Linz, September, 2004 16 Sensitivies Dual equations

17. Inverse Problems in Semiconductor Devices Linz, September, 2004 17 Sensitivies Boundary conditions on contact  0 homogeneous boundary conditions else

18. Inverse Problems in Semiconductor Devices Linz, September, 2004 18 Sensitivies Optimality condition (H 1 - regularization) with homogeneous boundary conditions for C - C 0

19. Inverse Problems in Semiconductor Devices Linz, September, 2004 19 Numerical Solution If N is large, we obtain a huge optimality system of 6N+1 equations Direct discretization is challenging with respect to memory consumption and computational effort If we do gradient method, we can solve 3 x 3 subsystems, but the overall convergence is slow

20. Inverse Problems in Semiconductor Devices Linz, September, 2004 20 Numerical Solution Structure of KKT-System

21. Inverse Problems in Semiconductor Devices Linz, September, 2004 21 Close to Equilibrium For small applied voltages one can use linearization of DD system around U=0 Equilibrium potential V 0 satisfies Boundary conditions for V 0 with U = 0 → one-to-one relation between C and V 0

22. Inverse Problems in Semiconductor Devices Linz, September, 2004 22 Linearized DD System Linearized DD system around equilibrium (first order expansion in  for U =  ) Dirichlet boundary condition V 1 = - u 1 = v 1 =  Depends only on V 0 : Identify V 0 (smoother !) instead of C

23. Inverse Problems in Semiconductor Devices Linz, September, 2004 23 Advantages of Linearization Linearization around equilibrium is not strongly coupled (triangular structure) Numerical solution easier around equilibrium Solution is always unique close to equilibrium Without capacitance data, no solution of linearized potential equation needed

24. Inverse Problems in Semiconductor Devices Linz, September, 2004 24 Advantages of Linearization Under additional unipolarity (v = 0), scalar elliptic equation – the problem of identifying the equilibrium potential can be rewritten as the identification of a diffusion coefficient a = e V 0 Well-known problem from Impedance Tomography Caution: The inverse problem is always non-linear, even for the linearized DD model !

25. Inverse Problems in Semiconductor Devices Linz, September, 2004 25 Identifiability Natural question: do the data determine the doping profile uniquely ? For a quasi 1D device (ballistic diode), the doping profile cannot be determined, information content of current data corresponds to one real number (slope of the I-V curve)

26. Inverse Problems in Semiconductor Devices Linz, September, 2004 26 Identifiability For a unipolar 2D device (MESFET, MOSFET), voltage-current data around equilibrium suffice only when currents ar measured on the whole boundary (B-Engl-Markowich-Pietra 01) – not realistic ! For a unipolar 3D device, voltage-current data around equilibrium determine the doping profile uniquely under reasonable conditions

27. Inverse Problems in Semiconductor Devices Linz, September, 2004 27 Numerical Tests Test for a P-N Diode Real Doping ProfileInitial Guess

28. Inverse Problems in Semiconductor Devices Linz, September, 2004 28 Numerical Tests Different Voltage Sources

29. Inverse Problems in Semiconductor Devices Linz, September, 2004 29 Numerical Tests Reconstructions with first source

30. Inverse Problems in Semiconductor Devices Linz, September, 2004 30 Numerical Tests Reconstructions with second source

31. Inverse Problems in Semiconductor Devices Linz, September, 2004 31 The P-N Diode Simplest device geometry, two Ohmic contacts, single p-n junction

32. Inverse Problems in Semiconductor Devices Linz, September, 2004 32 Identifying P-N Junctions Doping profiles look often like a step function, with a single discontinuity curve  (p-n junction) Identification of p-n junction is of major interest in this case Voltage applied on contact 1, device characteristics measured on contact 2

33. Inverse Problems in Semiconductor Devices Linz, September, 2004 33 Model Reduction 1 Typically small Debye length Consider limit → 0 (zero space charge) Equilibrium potential equation becomes algebraic relation between V 0 and C - V 0 is piecewise constant - identify junction in V 0 or a = exp(V 0 ) Continuity equations div ( a  u 1 ) = div ( a -1  v 1 ) = 0

34. Inverse Problems in Semiconductor Devices Linz, September, 2004 34 Identifiability Since we only want to identify the junction , we need less measurements For a unipolar diode with zero space charge, the junction is locally unique if we only measure the current for a single applied voltage (N=1) Computational effort reduced to scalar elliptic equation

35. Inverse Problems in Semiconductor Devices Linz, September, 2004 35 Model Reduction 2 If, in addition to zero space charge, there is also low injection (  small), the model can be reduced further (cf. Schmeiser 91) In the P-region, the function u satisfies  u = 0 Current is determined by  u only Inverse boundary value problem in the P-region, overposed boundary values on contact 2 (u = 0 on  u = 1 on contact 2, current flux = normal derivative of u measured)

36. Inverse Problems in Semiconductor Devices Linz, September, 2004 36 Identifiability For a P-N Diode, junction is determined uniquely by a single current measurement (B-Engl-Markowich- Pietra 01)

37. Inverse Problems in Semiconductor Devices Linz, September, 2004 37 Numerical Results For zero space charge and low injection, computational effort reduces to inverse free boundary problem for Laplace equation

38. Inverse Problems in Semiconductor Devices Linz, September, 2004 38 Results for C 0 = 10 20 m -3

39. Inverse Problems in Semiconductor Devices Linz, September, 2004 39 Results for C 0 = 10 21 m -3