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Mathematical Simulation and Optimization of Semiconductor Devices Mathematical Simulation and Michael D ohler Zheng He Optimization of Semiconductor Devices Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor:

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Mathematical Simulation and Optimization of Semiconductor Devices

Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram

20th ECMI Modeling week @DTU, Denmark

24th August 2006

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

Solid states material can be grouped into three classes:

◮ Insulators ◮ Semiconductors ◮ Conductors

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

Solid states material can be grouped into three classes:

◮ Insulators ◮ Semiconductors ◮ Conductors

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ Electrons occupy different energy levels

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ Electrons occupy different energy levels ◮ Conduction/Valence Band

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ Electrons occupy different energy levels ◮ Conduction/Valence Band ◮ Different conductivities result from different ’Band Gaps’

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ Electrons occupy different energy levels ◮ Conduction/Valence Band ◮ Different conductivities result from different ’Band Gaps’ ◮ Semiconductors typically have a gap size of about 1eV

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ Electrons occupy different energy levels ◮ Conduction/Valence Band ◮ Different conductivities result from different ’Band Gaps’ ◮ Semiconductors typically have a gap size of about 1eV

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ To achieve a desired band gap, ’doping’ is used ◮ Doping means adding impurities (different elements);

donors (n-type) or acceptors (p-type)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Physics

◮ To achieve a desired band gap, ’doping’ is used ◮ Doping means adding impurities (different elements);

donors (n-type) or acceptors (p-type)

(a) intrinsic (b) n-type (c) p-type

Figure: Doping

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductors devices

Advantages

◮ Very small length (e.g. ≈ 10nm for a transistor used in a

CPU)

◮ Controlable conductivity

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductors devices

Advantages

◮ Very small length (e.g. ≈ 10nm for a transistor used in a

CPU)

◮ Controlable conductivity

Applications of semiconductor devices

◮ Camcorders, solar cells (Photonic devices)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductors devices

Advantages

◮ Very small length (e.g. ≈ 10nm for a transistor used in a

CPU)

◮ Controlable conductivity

Applications of semiconductor devices

◮ Camcorders, solar cells (Photonic devices) ◮ LEDs (Optoelectric emitters)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductors devices

Advantages

◮ Very small length (e.g. ≈ 10nm for a transistor used in a

CPU)

◮ Controlable conductivity

Applications of semiconductor devices

◮ Camcorders, solar cells (Photonic devices) ◮ LEDs (Optoelectric emitters) ◮ Flat panel displays

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductors devices

Advantages

◮ Very small length (e.g. ≈ 10nm for a transistor used in a

CPU)

◮ Controlable conductivity

Applications of semiconductor devices

◮ Camcorders, solar cells (Photonic devices) ◮ LEDs (Optoelectric emitters) ◮ Flat panel displays ◮ Integrated circuits (MOSFETs) and many more...

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductor devices

◮ We consider the following (called diode)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductor devices

◮ We consider the following (called diode)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Semiconductor devices

◮ We consider the following (called diode) ◮ In our case input parameter is applied voltage and output

parameter is the electric current

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Drift-Diffusion Equations (DDE)

∂tn − 1 q div Jn = −R(n, p) (1) ∂tp + 1 q div Jp = −R(n, p) (2) Jn = qµn(UT∇n − n∇V) (3) Jp = qµp(UT∇p + p∇V) (4) εs∆V = q(n − p − C) (5)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Thermal Equilibrium State

◮ Boundary Conditions

Assuming that semiconductor domain consists of: Dirichlet part n = nD, p = pD, V = VD on ΓD

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Thermal Equilibrium State

◮ Boundary Conditions

Assuming that semiconductor domain consists of: Dirichlet part n = nD, p = pD, V = VD on ΓD Neumann part Jn · ν = Jp · ν = ∇V · ν = 0 on ΓN

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Thermal Equilibrium State

◮ Boundary Conditions

Assuming that semiconductor domain consists of: Dirichlet part n = nD, p = pD, V = VD on ΓD Neumann part Jn · ν = Jp · ν = ∇V · ν = 0 on ΓN

◮ Some calculations yield the (analytical) solution

εs∆Ve = q

  • 2ni sinh Ve

UT − C

  • in Ω

Ve = UT ln nD ni

  • n ΓD,

∇Ve · ν = 0 on ΓN

◮ and

ne = nie

Ve UT , pe = nie −Ve UT

in Ω

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Final model

We simplify by assuming

◮ 1D ◮ Time independance

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Final model

We simplify by assuming

◮ 1D ◮ Time independance ◮ With scaling our DDEs become:

n′′ − n′V ′ − nV ′′ = R µ′

n

p′′ + p′V ′ + pV ′′ = R µ′

p

V ′′ = 1 λ2 (n − p − C)

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Solving DDE & Results

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Solving DDE & Results

◮ MATLAB

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Solving DDE & Results

◮ MATLAB ◮ FEMLAB

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Solving DDE & Results

◮ MATLAB ◮ FEMLAB

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Inverse Problem

◮ Now we want to compute the doping profile C = C(x) that

causes desired currents Ij for given voltages Uj: F(C, Uj) = Ij ∀ j = 1, ..., N

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Inverse Problem

◮ Now we want to compute the doping profile C = C(x) that

causes desired currents Ij for given voltages Uj: F(C, Uj) = Ij ∀ j = 1, ..., N

◮ Approximate ill-posed inverse problem by ”neighbouring”

well-posed problem with Tikhonov regularization: min

C

  

N

  • j=1
  • F
  • C, Uj
  • − Iδ

j

  • 2 + α C − C∗2

   with F (C, U) = Jn + Jp whereas Jn and Jp can be computed with n, p and V from the solution of the DDE.

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SLIDE 31

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Compute the derivative of

f(C, n, p, V) =

  • F
  • C, Uj
  • − Iδ

j

  • 2 + α C − C∗2

with respect to C using the adjoint gradient method:

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Compute the derivative of

f(C, n, p, V) =

  • F
  • C, Uj
  • − Iδ

j

  • 2 + α C − C∗2

with respect to C using the adjoint gradient method:

◮ Function to be optimized depends on C, n, p and V,

whereas n, p and V must fulfil the DDE

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Compute the derivative of

f(C, n, p, V) =

  • F
  • C, Uj
  • − Iδ

j

  • 2 + α C − C∗2

with respect to C using the adjoint gradient method:

◮ Function to be optimized depends on C, n, p and V,

whereas n, p and V must fulfil the DDE

◮ Set up the Lagrangian L with Langrange multipliers λ1(x),

λ2(x) and λ3(x) for each equation of the DDE

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Compute the derivative of

f(C, n, p, V) =

  • F
  • C, Uj
  • − Iδ

j

  • 2 + α C − C∗2

with respect to C using the adjoint gradient method:

◮ Function to be optimized depends on C, n, p and V,

whereas n, p and V must fulfil the DDE

◮ Set up the Lagrangian L with Langrange multipliers λ1(x),

λ2(x) and λ3(x) for each equation of the DDE

◮ It holds that

∂f ∂C h = ∂L ∂C h ∀ h ∈ L2[0, 1], so it is sufficient to look for KKT points of L: ∂L ∂n = ∂L ∂p = ∂L ∂V = 0

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SLIDE 35

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Leads to adjoint PDE

−λ1 − (λ2 + λ3)∂R ∂n + µn (V ′λ′

2 + λ′′ 2)

= λ1 − (λ2 + λ3)∂R ∂p − µp (V ′λ′

3 − λ′′ 3)

= λ2λ′′

1 − µn (λ′′ 2n + λ′ 2n′) + µp (λ′′ 3p + λ′ 3p′)

= with λ2(1) = −2(Jn + Jp − Ij) λ3(1) = 2(Jn + Jp − Ij) and homogenous boundary conditions otherwise.

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SLIDE 36

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Leads to adjoint PDE

−λ1 − (λ2 + λ3)∂R ∂n + µn (V ′λ′

2 + λ′′ 2)

= λ1 − (λ2 + λ3)∂R ∂p − µp (V ′λ′

3 − λ′′ 3)

= λ2λ′′

1 − µn (λ′′ 2n + λ′ 2n′) + µp (λ′′ 3p + λ′ 3p′)

= with λ2(1) = −2(Jn + Jp − Ij) λ3(1) = 2(Jn + Jp − Ij) and homogenous boundary conditions otherwise.

◮ Hence,

∂f ∂C (x) = λ1(x) + 2α|C(x) − C∗(x)|

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SLIDE 37

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Optimization

◮ Looking for optimum using gradient based method starting

with C0 = C∗, Ck+1 = Ck + τkpk whereas τk is step length and pk is a descent direction

◮ pk computed with Quasi-Newton method (BFGS) ◮ τk computed with linesearch

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Initial guess for doping profile C* x C(x)

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SLIDE 38

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck

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Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

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SLIDE 40

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE

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SLIDE 41

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE ◮ Calculate ∂f ∂C using λ1

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SLIDE 42

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE ◮ Calculate ∂f ∂C using λ1 ◮ Calculate Ck+1 with BFGS and linesearch using the

derivative

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SLIDE 43

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE ◮ Calculate ∂f ∂C using λ1 ◮ Calculate Ck+1 with BFGS and linesearch using the

derivative Example of Computation:

◮ Calculating I-U-characteristics using doping profile C as a

step function

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SLIDE 44

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE ◮ Calculate ∂f ∂C using λ1 ◮ Calculate Ck+1 with BFGS and linesearch using the

derivative Example of Computation:

◮ Calculating I-U-characteristics using doping profile C as a

step function

◮ Using eight pairs of these (Uj, Ij) values

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SLIDE 45

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Algorithm for inverse problem

◮ Start with Ck ◮ Calculate n, p, V and hence Jn + Jp by solving the

’forward’ PDE

◮ Calculate λ1, λ2 and λ3 by solving the ’adjoint’ PDE ◮ Calculate ∂f ∂C using λ1 ◮ Calculate Ck+1 with BFGS and linesearch using the

derivative Example of Computation:

◮ Calculating I-U-characteristics using doping profile C as a

step function

◮ Using eight pairs of these (Uj, Ij) values ◮ Choose C∗ as a linear function and use the inverse solver

to calculate a doping profile that fits our (Uj, Ij) values

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SLIDE 46

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0.5 1 1.5 x C(x) Doping profiles actual doping profile initial guess calculated doping profile

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SLIDE 47

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 x 10

−3

I−U−characteristics U/V approximated current measured current

slide-48
SLIDE 48

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Conclusions

Accomplished:

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SLIDE 49

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Conclusions

Accomplished:

◮ Numerical forward solution of the simplified model (1D and

time-independent)

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SLIDE 50

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Conclusions

Accomplished:

◮ Numerical forward solution of the simplified model (1D and

time-independent)

◮ Numerical solution of the inverse problem (given I, U,

searching for C)

slide-51
SLIDE 51

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Conclusions

Accomplished:

◮ Numerical forward solution of the simplified model (1D and

time-independent)

◮ Numerical solution of the inverse problem (given I, U,

searching for C) Outlook:

◮ Improvement of the algorithm (computing time)

slide-52
SLIDE 52

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Conclusions

Accomplished:

◮ Numerical forward solution of the simplified model (1D and

time-independent)

◮ Numerical solution of the inverse problem (given I, U,

searching for C) Outlook:

◮ Improvement of the algorithm (computing time) ◮ Generalisation of the model (2D and time-dependent)

slide-53
SLIDE 53

Mathematical Simulation and Optimization of Semiconductor Devices Michael D¨

  • hler

Zheng He Maike Lorenz Philipp Monreal Aleksandra Rakic Sapna Sharma Instructor: Marie-Therese Wolfram Physics Semiconductor Devices Drift-Diffusion Equation (DDE) Solving DDE & Results Inverse Problem Optimization Algorithm for Inverse Problem Numerical Results Conclusions

Thank you for your attention!