Inverse Problems in Semiconductor Devices Martin Burger Johannes Kepler Universität Linz

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

Inverse Problems in Semiconductor Devices Linz, September, 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, 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, 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, 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

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

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

Inverse Problems in Semiconductor Devices Linz, September, 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

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

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

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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, Sensitivies Define Lagrangian

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

Inverse Problems in Semiconductor Devices Linz, September, Sensitivies Dual equations

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

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

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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, 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, 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 !

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

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

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

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

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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, 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

Inverse Problems in Semiconductor Devices Linz, September, 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)

Inverse Problems in Semiconductor Devices Linz, September, 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, 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, Results for C 0 = m -3

Inverse Problems in Semiconductor Devices Linz, September, Results for C 0 = m -3