inverse problems in semiconductor devices

Inverse Problems in Semiconductor Devices

Martin Burger

Johannes Kepler Universität Linz


Linz, September, 2004 2

OutlineIntroduction: Drift-Diffusion Model

Inverse Dopant Profiling

Sensitivities



Joint work withHeinz Engl, RICAM

Peter Markowich, Universität Wien & RICAM

Antonio Leitao, Florianopolis & RICAM

Paola Pietra, Pavia



Inverse Dopant ProfilingIdentify the device doping profile from measurements of the device characteristics

Device characteristics:

Current-Voltage map

Voltage-Capacitance map



Inverse Dopant ProfilingDevice 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



Mathematical ModelStationary Drift Diffusion Model:

PDE system for potential V, electron density n and hole density p

in (subset of R2)

Doping Profile C(x) enters as source term



Boundary ConditionsBoundary of homogeneous Neumann boundary conditions on N and

on Dirichlet boundary D (Ohmic Contacts)



Device CharacteristicsMeasured on a contact 0 on D :Outflow current density

Capacitance



Scaled Drift-Diffusion SystemAfter (exponential) transform to Slotboom

variables (u=e-V n, p = eV p) and scaling:

Similar transforms and scaling for boundary

conditions



Scaled Drift-Diffusion SystemSimilar transforms and scaling for boundary

Conditions

Essential (possibly small) parameters

- Debye length - Injection Parameter -Applied Voltage U



Scaled Drift-Diffusion SystemInverse Problem for full model ( scale = 1)



Optimization ProblemTake current measurements on a contact 0 in the following

Least-Squares Optimization: minimize

for N large



Optimization ProblemThis least squares problem is ill-posed

Consider Tikhonov-regularized version

C0 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



SensitiviesDefine Lagrangian



SensitiviesPrimal equations,

with different boundary conditions



SensitiviesDual equations



SensitiviesBoundary conditions on contact 0

homogeneous boundary conditions else



Sensitivies

Optimality condition (H1 - regularization)

with homogeneous boundary conditions for C - C0



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



Numerical Solution

Structure of KKT-System



Close to Equilibrium

For small applied voltages one can use linearization of DD system around U=0Equilibrium potential V0 satisfies

Boundary conditions for V0 with U = 0→ one-to-one relation between C and V0



Linearized DD System Linearized DD system around equilibrium

(first order expansion in for U = )

Dirichlet boundary condition V1 = - u1 = v1 = Depends only on V0:

Identify V0 (smoother !) instead of C



Advantages of LinearizationLinearization 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



Advantages of LinearizationUnder additional unipolarity (v = 0), scalar elliptic equation – the problem of identifying the equilibriumpotential 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 !



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)



IdentifiabilityFor 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



Numerical TestsTest for a P-N Diode

Real Doping Profile Initial Guess



Numerical Tests

Different Voltage Sources



Numerical Tests

Reconstructions with first source



Numerical Tests

Reconstructions with second source



The P-N Diode

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



Identifying P-N JunctionsDoping 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



Model Reduction 1Typically small Debye length Consider limit → 0 (zero space charge)

Equilibrium potential equation becomes algebraic relation between V0 and C

- V0 is piecewise constant

- identify junction in V0 or a = exp(V0 )

Continuity equations

div ( a u1 ) = div ( a-1 v1 ) = 0



IdentifiabilitySince 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



Model Reduction 2If, 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)



Identifiability

For a P-N Diode, junction is determined uniquely by a single current measurement (B-Engl-Markowich-Pietra 01)



Numerical Results

For zero space charge and low injection, computational effort reduces to inverse free boundary problem for Laplace equation



Results for C0 = 1020m-3



Results for C0 = 1021m-3

inverse problems in semiconductor devices

Documents

contact measurements

electron density n

pde system

linearization of dd

c c0numerical solutionif

large scale

v0 smoother

huge optimality system