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Inverse Problems in Semiconductor Devices

Inverse Problems in Semiconductor Devices. Martin Burger. Johannes Kepler Universität Linz. Outline. Introduction: Drift-Diffusion Model Inverse Dopant Profiling Sensitivities. Joint work with. Heinz Engl, RICAM Peter Markowich, Universität Wien & RICAM

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Inverse Problems in Semiconductor Devices

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

  2. Outline • Introduction: Drift-Diffusion Model • Inverse Dopant Profiling • Sensitivities 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 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 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 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 W (subset of R2) • Doping Profile C(x)enters as source term Inverse Problems in Semiconductor Devices Linz, September, 2004

  7. Boundary Conditions Boundary of W : homogeneous Neumann boundary conditions on GN and on Dirichlet boundary GD (Ohmic Contacts) Inverse Problems in Semiconductor Devices Linz, September, 2004

  8. Device Characteristics Measured on a contact G0 on GD: Outflow current density Capacitance Inverse Problems in Semiconductor Devices Linz, September, 2004

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

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

  12. Optimization Problem Take current measurements on a contact G0 in the following Least-Squares Optimization: minimize for N large Inverse Problems in Semiconductor Devices Linz, September, 2004

  13. Optimization Problem This least squares problem is ill-posed Consider Tikhonov-regularized version C0is 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 Inverse Problems in Semiconductor Devices Linz, September, 2004

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

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

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

  17. Sensitivies Boundary conditions on contact G0 homogeneous boundary conditions else Inverse Problems in Semiconductor Devices Linz, September, 2004

  18. Sensitivies Optimality condition (H1 - regularization) with homogeneous boundary conditions for C - C0 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 Inverse Problems in Semiconductor Devices Linz, September, 2004

  20. Numerical Solution Structure of KKT-System 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 V0 satisfies Boundary conditions for V0 with U = 0 →one-to-one relation between C and V0 Inverse Problems in Semiconductor Devices Linz, September, 2004

  22. Linearized DD System • Linearized DD system around equilibrium • (first order expansion inefor U = e F) • Dirichlet boundary condition V1 = - u1 = v1 = F • Depends only on V0: • Identify V0 (smoother !) instead of C 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 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 = eV0 Well-known problem from Impedance Tomography Caution: The inverse problem is always non-linear, even for the linearized DD model ! 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) 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 Inverse Problems in Semiconductor Devices Linz, September, 2004

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

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

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

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

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

  33. Model Reduction 1 • Typically small Debye lengthl • Consider limitl→ 0 (zero space charge) Equilibrium potential equation becomes algebraic relation between V0 and C • - V0is piecewise constant • - identify junction in V0or a = exp(V0 ) • Continuity equations • div ( a  u1 ) = div ( a-1 v1 ) = 0 Inverse Problems in Semiconductor Devices Linz, September, 2004

  34. Identifiability Since we only want to identify the junction G, 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 Inverse Problems in Semiconductor Devices Linz, September, 2004

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

  38. Results for C0 = 1020m-3 Inverse Problems in Semiconductor Devices Linz, September, 2004

  39. Results for C0 = 1021m-3 Inverse Problems in Semiconductor Devices Linz, September, 2004

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