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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 - PowerPoint PPT Presentation
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Inverse Problems in Semiconductor Devices
Martin Burger
Johannes Kepler Universität Linz
Inverse Problems in Semiconductor Devices
Linz, September, 2004 2
OutlineIntroduction: Drift-Diffusion Model
Inverse Dopant Profiling
Sensitivities
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Joint work withHeinz Engl, RICAM
Peter Markowich, Universität Wien & RICAM
Antonio Leitao, Florianopolis & RICAM
Paola Pietra, Pavia
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Inverse Dopant ProfilingIdentify the device doping profile from measurements of the device characteristics
Device characteristics:
Current-Voltage map
Voltage-Capacitance map
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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
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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
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Boundary ConditionsBoundary of homogeneous Neumann boundary conditions on N and
on Dirichlet boundary D (Ohmic Contacts)
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Device CharacteristicsMeasured on a contact 0 on D :Outflow current density
Capacitance
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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
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Scaled Drift-Diffusion SystemSimilar transforms and scaling for boundary
Conditions
Essential (possibly small) parameters
- Debye length - Injection Parameter -Applied Voltage U
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Scaled Drift-Diffusion SystemInverse Problem for full model ( scale = 1)
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Optimization ProblemTake current measurements on a contact 0 in the following
Least-Squares Optimization: minimize
for N large
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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
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SensitiviesDefine Lagrangian
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SensitiviesPrimal equations,
with different boundary conditions
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SensitiviesDual equations
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SensitiviesBoundary conditions on contact 0
homogeneous boundary conditions else
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Sensitivies
Optimality condition (H1 - regularization)
with homogeneous boundary conditions for C - C0
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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
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Numerical Solution
Structure of KKT-System
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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
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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
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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
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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 !
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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)
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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
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Numerical TestsTest for a P-N Diode
Real Doping Profile Initial Guess
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Numerical Tests
Different Voltage Sources
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Numerical Tests
Reconstructions with first source
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Numerical Tests
Reconstructions with second source
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The P-N Diode
Simplest device geometry, two Ohmic contacts, single p-n junction
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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
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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
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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
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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)
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Identifiability
For a P-N Diode, junction is determined uniquely by a single current measurement (B-Engl-Markowich-Pietra 01)
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Numerical Results
For zero space charge and low injection, computational effort reduces to inverse free boundary problem for Laplace equation
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Results for C0 = 1020m-3
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Results for C0 = 1021m-3
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