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Outline

- Introduction: Drift-Diffusion Model
- Inverse Dopant Profiling
- Sensitivities

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

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

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

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

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

Device Characteristics

Measured on a contact G0 on GD:

Outflow current density

Capacitance

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

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

Scaled Drift-Diffusion System

Inverse Problem for full model ( scale d = 1)

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

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

Sensitivies

Primal equations,

with different boundary conditions

Inverse Problems in Semiconductor Devices

Linz, September, 2004

Sensitivies

Boundary conditions on contact G0

homogeneous boundary conditions else

Inverse Problems in Semiconductor Devices

Linz, September, 2004

Sensitivies

Optimality condition (H1 - regularization)

with homogeneous boundary conditions for C - C0

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

Numerical Solution

Structure of KKT-System

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

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

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

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

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

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

Numerical Tests

Test for a P-N Diode

Real Doping Profile Initial Guess

Inverse Problems in Semiconductor Devices

Linz, September, 2004

Numerical Tests

Different Voltage Sources

Inverse Problems in Semiconductor Devices

Linz, September, 2004

Numerical Tests

Reconstructions with first source

Inverse Problems in Semiconductor Devices

Linz, September, 2004

Numerical Tests

Reconstructions with second source

Inverse Problems in Semiconductor Devices

Linz, September, 2004

The P-N Diode

Simplest device geometry, two Ohmic contacts, single p-n junction

Inverse Problems in Semiconductor Devices

Linz, September, 2004

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

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

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

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

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

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

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