Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device...
Transcript of Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device...
![Page 1: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/1.jpg)
1
Physics-based Device Simulation
C. Jungemann
Institut für Theoretische ElektrotechnikRWTH Aachen University
52056 Aachen
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2OutlineI MotivationI Pros & ConsI Semiclassical transportI ResultsI Conclusions
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3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 4: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/4.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 5: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/5.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 6: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/6.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 7: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/7.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 8: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/8.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 9: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/9.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 10: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/10.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 11: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/11.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 12: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/12.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 13: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/13.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 14: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/14.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 15: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/15.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 16: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/16.jpg)
3MotivationSimulation hierarchy
I Quantum transportI Semiclassical transportI TCADI Compact models
Speed
Acc
urac
y
Com
plexity
Why semiclassical transport simulation (Boltzmann equation)?
I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model
![Page 17: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/17.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 18: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/18.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 19: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/19.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 20: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/20.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 21: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/21.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 22: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/22.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 23: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/23.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 24: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/24.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 25: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/25.jpg)
4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models
![Page 26: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/26.jpg)
5
Semiclassical transport
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6Semiclassical transportPath of an electron in a semiconductor
0 100 200 300 400
50
100
200
300
X [nm]
Y [
nm
]
Newton’s equations of motion
d~pdt
= ~F ,d~rdt
= ~v
Two first order ODEsTwo initial conditions (position, momentum)
![Page 28: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/28.jpg)
6Semiclassical transportPath of an electron in a semiconductor
0 100 200 300 400
50
100
200
300
X [nm]
Y [
nm
]
Newton’s equations of motion
d~pdt
= ~F ,d~rdt
= ~v
Two first order ODEsTwo initial conditions (position, momentum)
![Page 29: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/29.jpg)
6Semiclassical transportPath of an electron in a semiconductor
0 100 200 300 400
50
100
200
300
X [nm]
Y [
nm
]
Newton’s equations of motion
d~pdt
= ~F ,d~rdt
= ~v
Two first order ODEsTwo initial conditions (position, momentum)
![Page 30: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/30.jpg)
6Semiclassical transportPath of an electron in a semiconductor
0 100 200 300 400
50
100
200
300
X [nm]
Y [
nm
]
Newton’s equations of motion
d~pdt
= ~F ,d~rdt
= ~v
Two first order ODEsTwo initial conditions (position, momentum)
![Page 31: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/31.jpg)
6Semiclassical transportPath of an electron in a semiconductor
0 100 200 300 400
50
100
200
300
X [nm]
Y [
nm
]
Newton’s equations of motion
d~pdt
= ~F ,d~rdt
= ~v
Two first order ODEsTwo initial conditions (position, momentum)
![Page 32: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/32.jpg)
7Semiclassical TransportParticle ensemble described by a distribution function (~p = ~~k )
f (~r , ~k , t)
Number of particles in a volume of phase space
dN =2
(2π)3 f (~r , ~k , t)d3rd3k
Equilibrium (Boltzmann distribution)
f0 = c exp
(−ε(~r , ~k)
kT
)
![Page 33: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/33.jpg)
7Semiclassical TransportParticle ensemble described by a distribution function (~p = ~~k )
f (~r , ~k , t)
Number of particles in a volume of phase space
dN =2
(2π)3 f (~r , ~k , t)d3rd3k
Equilibrium (Boltzmann distribution)
f0 = c exp
(−ε(~r , ~k)
kT
)
![Page 34: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/34.jpg)
7Semiclassical TransportParticle ensemble described by a distribution function (~p = ~~k )
f (~r , ~k , t)
Number of particles in a volume of phase space
dN =2
(2π)3 f (~r , ~k , t)d3rd3k
Equilibrium (Boltzmann distribution)
f0 = c exp
(−ε(~r , ~k)
kT
)
![Page 35: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/35.jpg)
8Semiclassical TransportBoltzmann Transport Equation
∂f(~r , ~k , t
)∂t
− q~~E(~r) · ∇~k f
(~r , ~k , t
)+ ~v
(~k)· ∇~r f
(~r , ~k , t
)=
Ωsys
(2π)3
∫W(~r , ~k |~k ′
)f(~r , ~k ′, t
)−W
(~r , ~k ′|~k
)f(~r , ~k , t
)d3k ′
Integro-differential equation
Direct discretization of the 6D phase space is not possible
Spherical Harmonics Expansion solver (SHE) for the BTE(Bologna group, Maryland group et al.)
![Page 36: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/36.jpg)
8Semiclassical TransportBoltzmann Transport Equation
∂f(~r , ~k , t
)∂t
− q~~E(~r) · ∇~k f
(~r , ~k , t
)+ ~v
(~k)· ∇~r f
(~r , ~k , t
)=
Ωsys
(2π)3
∫W(~r , ~k |~k ′
)f(~r , ~k ′, t
)−W
(~r , ~k ′|~k
)f(~r , ~k , t
)d3k ′
Integro-differential equation
Direct discretization of the 6D phase space is not possible
Spherical Harmonics Expansion solver (SHE) for the BTE(Bologna group, Maryland group et al.)
![Page 37: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/37.jpg)
8Semiclassical TransportBoltzmann Transport Equation
∂f(~r , ~k , t
)∂t
− q~~E(~r) · ∇~k f
(~r , ~k , t
)+ ~v
(~k)· ∇~r f
(~r , ~k , t
)=
Ωsys
(2π)3
∫W(~r , ~k |~k ′
)f(~r , ~k ′, t
)−W
(~r , ~k ′|~k
)f(~r , ~k , t
)d3k ′
Integro-differential equation
Direct discretization of the 6D phase space is not possible
Spherical Harmonics Expansion solver (SHE) for the BTE(Bologna group, Maryland group et al.)
![Page 38: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/38.jpg)
8Semiclassical TransportBoltzmann Transport Equation
∂f(~r , ~k , t
)∂t
− q~~E(~r) · ∇~k f
(~r , ~k , t
)+ ~v
(~k)· ∇~r f
(~r , ~k , t
)=
Ωsys
(2π)3
∫W(~r , ~k |~k ′
)f(~r , ~k ′, t
)−W
(~r , ~k ′|~k
)f(~r , ~k , t
)d3k ′
Integro-differential equation
Direct discretization of the 6D phase space is not possible
Spherical Harmonics Expansion solver (SHE) for the BTE(Bologna group, Maryland group et al.)
![Page 39: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/39.jpg)
9Spherical Harmonics ExpansionSpherical coordinates for k -space
(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)
Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)
Y0,0 =1√4π
Y1,−1 =
√3
4πsinϑ sinϕ
Y1,0 =
√3
4πcosϑ
Y1,1 =
√3
4πsinϑ cosϕ
∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ
![Page 40: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/40.jpg)
9Spherical Harmonics ExpansionSpherical coordinates for k -space
(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)
Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)
Y0,0 =1√4π
Y1,−1 =
√3
4πsinϑ sinϕ
Y1,0 =
√3
4πcosϑ
Y1,1 =
√3
4πsinϑ cosϕ
∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ
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9Spherical Harmonics ExpansionSpherical coordinates for k -space
(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)
Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)
Y0,0 =1√4π
Y1,−1 =
√3
4πsinϑ sinϕ
Y1,0 =
√3
4πcosϑ
Y1,1 =
√3
4πsinϑ cosϕ
∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ
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9Spherical Harmonics ExpansionSpherical coordinates for k -space
(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)
Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)
Y0,0 =1√4π
Y1,−1 =
√3
4πsinϑ sinϕ
Y1,0 =
√3
4πcosϑ
Y1,1 =
√3
4πsinϑ cosϕ
∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ
![Page 43: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/43.jpg)
9Spherical Harmonics ExpansionSpherical coordinates for k -space
(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)
Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)
Y0,0 =1√4π
Y1,−1 =
√3
4πsinϑ sinϕ
Y1,0 =
√3
4πcosϑ
Y1,1 =
√3
4πsinϑ cosϕ
∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ
![Page 44: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/44.jpg)
10Semiclassical transportSpherical harmonics expansion of the distribution function
gl,m(~r , ε, t) =1
(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k
g(~r , ~k , t) = g(~r , ε, ϑ, ϕ, t) ≈lmax∑l=0
m=l∑m=−l
gl,m(~r , ε, t)Yl,m(ϑ, ϕ)
Electron density
n(~r , t) =2
(2π)3
∫f (~r , ~k , t)d3k = 2
√4π∫ ∞
0g0,0(~r , ε, t)dε
Electron current density (spherical bands)
~j(~r , t) = 2
√4π3
∫ ∞0
v(ε)
g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)
dε
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10Semiclassical transportSpherical harmonics expansion of the distribution function
gl,m(~r , ε, t) =1
(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k
g(~r , ~k , t) = g(~r , ε, ϑ, ϕ, t) ≈lmax∑l=0
m=l∑m=−l
gl,m(~r , ε, t)Yl,m(ϑ, ϕ)
Electron density
n(~r , t) =2
(2π)3
∫f (~r , ~k , t)d3k = 2
√4π∫ ∞
0g0,0(~r , ε, t)dε
Electron current density (spherical bands)
~j(~r , t) = 2
√4π3
∫ ∞0
v(ε)
g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)
dε
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10Semiclassical transportSpherical harmonics expansion of the distribution function
gl,m(~r , ε, t) =1
(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k
g(~r , ~k , t) = g(~r , ε, ϑ, ϕ, t) ≈lmax∑l=0
m=l∑m=−l
gl,m(~r , ε, t)Yl,m(ϑ, ϕ)
Electron density
n(~r , t) =2
(2π)3
∫f (~r , ~k , t)d3k = 2
√4π∫ ∞
0g0,0(~r , ε, t)dε
Electron current density (spherical bands)
~j(~r , t) = 2
√4π3
∫ ∞0
v(ε)
g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)
dε
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10Semiclassical transportSpherical harmonics expansion of the distribution function
gl,m(~r , ε, t) =1
(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k
g(~r , ~k , t) = g(~r , ε, ϑ, ϕ, t) ≈lmax∑l=0
m=l∑m=−l
gl,m(~r , ε, t)Yl,m(ϑ, ϕ)
Electron density
n(~r , t) =2
(2π)3
∫f (~r , ~k , t)d3k = 2
√4π∫ ∞
0g0,0(~r , ε, t)dε
Electron current density (spherical bands)
~j(~r , t) = 2
√4π3
∫ ∞0
v(ε)
g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)
dε
![Page 48: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/48.jpg)
11Semiclassical transportSpherical harmonics expansion of the Boltzmann equation
1(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ) BEd3k
yields balance equations over energy/real space
∂gl,m
∂t− q~E ·
∂~jl,m∂ε
+∇~r ·~jl,m − Γl,m = Wl,mg
with
~jl,m(~r , ε, t) = v(ε)∑l ′,m′
~al,m,l ′,m′gl ′.m′(~r , ε, t)
The balance equations are coupled
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11Semiclassical transportSpherical harmonics expansion of the Boltzmann equation
1(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ) BEd3k
yields balance equations over energy/real space
∂gl,m
∂t− q~E ·
∂~jl,m∂ε
+∇~r ·~jl,m − Γl,m = Wl,mg
with
~jl,m(~r , ε, t) = v(ε)∑l ′,m′
~al,m,l ′,m′gl ′.m′(~r , ε, t)
The balance equations are coupled
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11Semiclassical transportSpherical harmonics expansion of the Boltzmann equation
1(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ) BEd3k
yields balance equations over energy/real space
∂gl,m
∂t− q~E ·
∂~jl,m∂ε
+∇~r ·~jl,m − Γl,m = Wl,mg
with
~jl,m(~r , ε, t) = v(ε)∑l ′,m′
~al,m,l ′,m′gl ′.m′(~r , ε, t)
The balance equations are coupled
![Page 51: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/51.jpg)
11Semiclassical transportSpherical harmonics expansion of the Boltzmann equation
1(2π)3
∫δ(ε− ε(~k)
)Yl,m(ϑ, ϕ) BEd3k
yields balance equations over energy/real space
∂gl,m
∂t− q~E ·
∂~jl,m∂ε
+∇~r ·~jl,m − Γl,m = Wl,mg
with
~jl,m(~r , ε, t) = v(ε)∑l ′,m′
~al,m,l ′,m′gl ′.m′(~r , ε, t)
The balance equations are coupled
![Page 52: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/52.jpg)
12Semiclassical transportH-transform
5eV
25eV
0eV x
H
H(~r , ~k
)= ε
(~k)− qΨ
(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)
Cancels the energy derivative for the stationary state
H
HHHHHH
−q~E ·∂~j ′l,m∂ε
+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′
Captures ballistic transport
![Page 53: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/53.jpg)
12Semiclassical transportH-transform
5eV
25eV
0eV x
H
H(~r , ~k
)= ε
(~k)− qΨ
(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)
Cancels the energy derivative for the stationary state
H
HHHHHH
−q~E ·∂~j ′l,m∂ε
+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′
Captures ballistic transport
![Page 54: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/54.jpg)
12Semiclassical transportH-transform
5eV
25eV
0eV x
H
H(~r , ~k
)= ε
(~k)− qΨ
(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)
Cancels the energy derivative for the stationary state
H
HHHHHH
−q~E ·∂~j ′l,m∂ε
+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′
Captures ballistic transport
![Page 55: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/55.jpg)
12Semiclassical transportH-transform
5eV
25eV
0eV x
H
H(~r , ~k
)= ε
(~k)− qΨ
(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)
Cancels the energy derivative for the stationary state
H
HHHHHH
−q~E ·∂~j ′l,m∂ε
+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′
Captures ballistic transport
![Page 56: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/56.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 57: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/57.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 58: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/58.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 59: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/59.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 60: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/60.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 61: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/61.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 62: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/62.jpg)
13Semiclassical transportNumerics
I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently
![Page 63: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/63.jpg)
14Semiclassical transport
Complex band structure effects
First conduction band of Si Density of states of Si
0 1 2 3 4 5 6
0
1
2
3·1028
Energy [eV]
DO
S[1/
eVcm
3 ]
Calculated by the nonlocal empirical pseudopotential method
Only DOS and v2 are required for SHE
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14Semiclassical transport
Complex band structure effects
First conduction band of Si Density of states of Si
0 1 2 3 4 5 6
0
1
2
3·1028
Energy [eV]
DO
S[1/
eVcm
3 ]
Calculated by the nonlocal empirical pseudopotential method
Only DOS and v2 are required for SHE
![Page 65: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/65.jpg)
15Semiclassical transportElectron-phonon scattering rate of silicon at 300K
0 1 2 3 4 5 6
0
1
2
3·1014
Energy [eV]
Sca
tterin
gra
te[1/s
]
All parameters determined by theory or basic experiments
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16Semiclassical transportElectron mobility
0 100 200 300 400 500
103
104
105
106
Temperature [K]
Low
-fiel
dm
obili
ty[c
m2 /
Vs]
BECanali
Electron velocity at 300Kin < 111 > direction
10−1 100 101 102105
106
107
Electric field [kV/cm]D
riftv
eloc
ity[c
m/
s]
BECanali
Good agreement between simulation and experiment for silicon
![Page 67: Physics-based Device Simulation - TU Dresden · 2015-12-16 · Pros & Cons 4 I Realistic device structure I Microscopic, accurate physics (Boltzmann equation) I Detailed physics I](https://reader034.fdocuments.in/reader034/viewer/2022050408/5f84cc8176a35e72fb29a1f3/html5/thumbnails/67.jpg)
17Semiclassical transport
Impact ionization
Ionization coefficient
2 4 6 8 10100
101
102
103
104
105
Inverse electric field [cm/kV]
IIco
effic
ient
[1/c
m]
BEExp.
Quantum yield
1 1.5 2 2.5 3
10−8
10−6
10−4
10−2
100
Energy [eV]
Qua
ntum
yiel
dBE
DiMaria
Good agreement between simulation and experiment for silicon
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18
Results
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19
Bipolar transistor
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20Bipolar transistor
Doping profile Collector current
SHE can handle high doping concentrations without problems
SHE can handle small currents without problems
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20Bipolar transistor
Doping profile Collector current
SHE can handle high doping concentrations without problems
SHE can handle small currents without problems
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21Bipolar transistor
VBE = 0.55V , VCE = 0.5VEnergy distribution function
VCE = 0.5VElectron density
SHE can handle huge variations in the density without problems
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22Bipolar transistor
Dependence on the maximum order of SHEVBE = 0.85V , VCE = 0.5V
Error in current Electron velocity
Transport in nanometric devices requires at least 5th order SHE
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23
N+NN+ structure
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24N+NN+ structure
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25N+NN+ structure
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26N+NN+ structure
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27N+NN+ structure
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28N+NN+ structure
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29
-0.30 -0.20 -0.10 0.000 0.10 0.20 0.30
0.00
0.05
0.10
0.15
0.20
0.25
Silicon
Bottom oxide
GS D
Top oxide
Partially depleted SOI NMOSFET
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30PDSOI NMOSFET
N+
N+
VSource
VBulk
VDrain
VGate
ISub
Impactionization
event
n−MOSFET
P
−4
−2
0
2
4
6
ε (
eV
)
L Γ X
k
[111] [100]
Every impact ionization event generates a secondary electron/hole pair
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31PDSOI NMOSFET
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
Drain voltage [V]
Dra
incu
rren
t[A
/m]
with IIw/o II
Kink effect due to impact ionization (II) (Vgate = 1.0V )
Exact stationary solutions
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31PDSOI NMOSFET
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
Drain voltage [V]
Dra
incu
rren
t[A
/m]
with IIw/o II
Kink effect due to impact ionization (II) (Vgate = 1.0V )
Exact stationary solutions
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32PDSOI NMOSFET
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
Drain voltage [V]
Cur
rent
[A/m
]ID with IIID w/o II
10−16
10−15
10−14
10−13
10−12
10−11
10−10
II-curr.
About 17 orders of magnitude difference in currents at kink
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33
pn Junction
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34pn Junction
Simulation of avalanche breakdown
35 36 37 38 39
10−17
10−14
10−11
10−8
10−5
Reverse bias [V]
Cur
rent
[A/µ
m2 ]
35 36 37 38 39
0
1
2
3
·10−5
Reverse bias [V]
Cur
rent
[A/µ
m2 ]
Extremely steep increase of current with voltage (20µV/dec)
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34pn Junction
Simulation of avalanche breakdown
35 36 37 38 39
10−17
10−14
10−11
10−8
10−5
Reverse bias [V]
Cur
rent
[A/µ
m2 ]
35 36 37 38 39
0
1
2
3
·10−5
Reverse bias [V]
Cur
rent
[A/µ
m2 ]
Extremely steep increase of current with voltage (20µV/dec)
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35pn JunctionTransient DD simulation of switching a silicon pn-diode from 0 to -39V
100 101 102 10310−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
Time [ps]
Cur
rent
[A/µ
m2 ]
Avalanche breakdown is slow!
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35pn JunctionTransient DD simulation of switching a silicon pn-diode from 0 to -39V
100 101 102 10310−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
Time [ps]
Cur
rent
[A/µ
m2 ]
Avalanche breakdown is slow!
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36pn Junction
Simulation for 37.172V and 10pA/µm2
0
10
20
30
40
Pote
ntia
l[V
]0 1 2 3 4 510−14
10−5
104
1013
1022
Position [µm]
IIge
nera
tion
rate
[a.u
.]
elec.holesPot.
0 1 2 3 4 5101
105
109
1013
1017
Position [µm]
Den
sity
[cm−
3 ]
NANDelec.hole
II generation increases number of particles in the space charge region
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37pn Junction
Increase of current by a factor of 1000 (dashed lines)
0 1 2 3 4 5101
105
109
1013
1017
Position [µm]
Den
sity
[cm−
3 ]
0 1 2 3 4 510−16
10−7
102
1011
1020
Position [µm]
IIge
nera
tion
rate
[µm−
1 C−
1 ]Particle density in the space charge region increases 1000 times
Relevant II generation rate scales with current
Hot electron/hole effects scale with current
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37pn Junction
Increase of current by a factor of 1000 (dashed lines)
0 1 2 3 4 5101
105
109
1013
1017
Position [µm]
Den
sity
[cm−
3 ]
0 1 2 3 4 510−16
10−7
102
1011
1020
Position [µm]
IIge
nera
tion
rate
[µm−
1 C−
1 ]Particle density in the space charge region increases 1000 times
Relevant II generation rate scales with current
Hot electron/hole effects scale with current
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37pn Junction
Increase of current by a factor of 1000 (dashed lines)
0 1 2 3 4 5101
105
109
1013
1017
Position [µm]
Den
sity
[cm−
3 ]
0 1 2 3 4 510−16
10−7
102
1011
1020
Position [µm]
IIge
nera
tion
rate
[µm−
1 C−
1 ]Particle density in the space charge region increases 1000 times
Relevant II generation rate scales with current
Hot electron/hole effects scale with current
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37pn Junction
Increase of current by a factor of 1000 (dashed lines)
0 1 2 3 4 5101
105
109
1013
1017
Position [µm]
Den
sity
[cm−
3 ]
0 1 2 3 4 510−16
10−7
102
1011
1020
Position [µm]
IIge
nera
tion
rate
[µm−
1 C−
1 ]Particle density in the space charge region increases 1000 times
Relevant II generation rate scales with current
Hot electron/hole effects scale with current
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38pn Junction
Increase of current by a factor of 1000 (dashed lines)
Scaled electron/hole distribution function in the middle of the pn junction
0 1 2 3 4 5 6 710−4
10−1
102
105
108
Energy [eV]
Dis
trib
utio
nfu
nctio
n[µ
m−
1 eV−
1 A−
1 ]
Hot electron/hole effects scale with current
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38pn Junction
Increase of current by a factor of 1000 (dashed lines)
Scaled electron/hole distribution function in the middle of the pn junction
0 1 2 3 4 5 6 710−4
10−1
102
105
108
Energy [eV]
Dis
trib
utio
nfu
nctio
n[µ
m−
1 eV−
1 A−
1 ]
Hot electron/hole effects scale with current
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39
Vertical power transistor
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40Power MOSFETAvalanche breakdown of a power transistor
BE simulation for 35.83V and 10pA/µm2
Electrostatic Potential [V]
-0.6 6.8 14 22 36
e-Imp-Ioniz [cm^-3*s^-1]
2.6e+12 2.6e+16 2.6e+22
h-Imp-Ioniz [cm^-3*s^-1]
5.2e+13 5.2e+17 5.2e+23
45 · 106 variables, 2 CPU hours
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41Power MOSFET
BE simulation for 35.83V and 10pA/µm2
Electrostatic Potential [V]
-0.6 6.8 14 22 36
0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−2
101
104
107
1010
1013
Normalized path-length
Hig
h-en
ergy
part
icle
dens
ity[c
m−
3 ]1eV2eV3eV
Injection of electrons and holes into the oxide can be calculated
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42
Structure, DC and AC behaviorof the npn THz SiGe HBT
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43Comparison with experiments
Comparison for an ST-Microelectronics SiGe HBTat 10GHz and VCE = 1.2V
Cutoff frequency
10−1 100 101 102
101
102
Collector current [mA/µm2]
Cut
offf
requ
ency
[GH
z]
BESchröter
Minimum noise figure
100 1010
1
2
3
4
Collector current [mA/µm2]
Noi
sefig
ure
[dB
]
BESchröter
Good agreement between simulation and measurement
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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44Device structure
x
y0.0
10.0
20.0
30.0
90.0
100.00.0 10.0 20.0 30.0 40.0 50.0
OxideEmitter
Base
Collector
Emitter-window width: 2 × 25nm
I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21
spherical harmonics)I Up to 1.42 · 108 unknown
variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)
I Over 200 GB of memoryrequired (for AC)
I Up to 105 CPU-seconds for aDC point
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45Doping and germanium profile
0
5
10
15
20
25
30
Ge
cont
ent[
%]
0.01 0.02 0.03 0.04·10−2
1016
1017
1018
1019
1020
1021
x [µm]
Dop
ing
[cm−
3 ]
NANDGe
Base thickness: 7nm
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46Input characteristic
101
102
103
Cur
rent
gain
0.4 0.5 0.6 0.7 0.8 0.9 1
10−8
10−6
10−4
10−2
100
102
Base/emitter bias [V]
Cur
rent
[mA/µ
m2 ]
ICIBIC/IB
VCE = 1.0VNo SRH recombination
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47Output characteristic
10−3
10−2
10−1
100
101
Bas
ecu
rren
t[µ
A/µ
m2 ]
0 0.5 1 1.5 2 2.5 30
20
40
60
Collector/emitter bias [V ]
Col
lect
orcu
rren
t[µ
A/µ
m2 ]
IC|IB|
VBE = 0.7V
BVCE0 = 1.4V
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48Gains
108 109 1010 1011 1012 1013
0
10
20
30
40
50
Frequency [Hz]
Gai
n[d
B]
|HCB|2UMSG/MAG
108 109 1010 1011 1012 1013
0
10
20
30
40
50
Frequency [Hz]
Gai
n[d
B]
|HCB|2UMSG/MAG
VBE = 0.88V, VCE = 1.0V, IC = 18.2mA/µm2
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49Cutoff and max. oscil. frequencies
10−1 100 101 102
101
102
103
Collector current [mA/µm2]
Freq
uenc
y[G
Hz]
fT QSfT [email protected] [email protected]
VCE = 1.0V , fmax by extrapolation of Upeak fT = 1.2THz, peak fmax = 1.45THz
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49Cutoff and max. oscil. frequencies
10−1 100 101 102
101
102
103
Collector current [mA/µm2]
Freq
uenc
y[G
Hz]
fT QSfT [email protected] [email protected]
VCE = 1.0V , fmax by extrapolation of Upeak fT = 1.2THz, peak fmax = 1.45THz
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50Local contribution to the transit timefT =
12πτ
, τ =
∫dτdx
dx
5 10 15 20 25 30 35
−20
−10
0
10
20Base
x [nm]
dτ/dx
[fs/
nm]
IC = 54.6mA/µm2
IC = 19.6mA/µm2
VCE = 1.0V
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51Minimum noise figure
10−1 100 101 102
0
2
4
6
Collector current [mA/µm2]
Min
imum
nois
efig
ure
[dB
] NFmin @ 1GHzNFmin @ 100GHzNFmin @ 300GHz
VCE = 1.0V
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52Electrothermal Transport
High power densities lead to self-heating
Heating is non-local!
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52Electrothermal Transport
High power densities lead to self-heating
Heating is non-local!
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53Electrothermal Transport
Dirichlet boundary conditions
Scattering between optical and acoustic phonons goes both ways
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53Electrothermal Transport
Dirichlet boundary conditions
Scattering between optical and acoustic phonons goes both ways
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53Electrothermal Transport
Dirichlet boundary conditions
Scattering between optical and acoustic phonons goes both ways
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54Electrothermal Transport
Junction temperature determines the current
Self-heating can lead to thermal runaway
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54Electrothermal Transport
Junction temperature determines the current
Self-heating can lead to thermal runaway
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54Electrothermal Transport
Junction temperature determines the current
Self-heating can lead to thermal runaway
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55Electrothermal Transport
2D electrothermal simulation with SHE
Exact thermal simulation requires 3D simulation including metalization
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55Electrothermal Transport
2D electrothermal simulation with SHE
Exact thermal simulation requires 3D simulation including metalization
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56
Conclusions
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex
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57ConclusionsI Semiclassical simulations give detailed physics at the price of high
CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex