Semiconductor Device Simulation Using DEVSIM Juan Sanchez July 2, - - PowerPoint PPT Presentation

Semiconductor Device Simulation Using DEVSIM

Juan Sanchez

July 2, 2018

Introduction

• PDE semiconductor device simulator
• Finite volume method
• Solves 1D, 2D, and 3D structures
• External meshing tools or internal mesher
• Symbolic model evaluation
• Visualization using standard output formats

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Examples

Magnetic Potential Capacitance

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Device Equations

• Drift-diffusion equations

∇2ϕ = q(p−n+ND −NA)

(Poisson)

∂n ∂t =

1 q∇·

Jn +Gn −Rn

(Electron Continuity)

∂ p ∂t = −1

q∇·

Jp +Gp −Rp

(Hole Continuity)

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Example – 1D Diode

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Example – 2D MOSFET

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Example – 2D MOSFET

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Introduction

• Project started in 2008
• Open source since 2013 https://devsim.org
• C++ using STL, C++-11, and templates
• Platform Agnostic (Linux, OS X, Windows)
• Uses Python scripting to set up equations

and control simulation

• Approximately 64,000 lines of code

https://www.openhub.net/p/devsim

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Architecture – Analysis

• Nonlinear simulation

– dc – transient

• Linear analysis

– small-signal ac – sensitivity (impedance field) – noise

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Architecture – Scripting

• Models implemented using scripting

– Faster development cycle – Design for efficiency

• Symbolic differentiation

– Faster development time – Add derivatives w.r.t. new variables – Common subexpression elimination

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Architecture – Python

• well defined and consistent
• avoids domain specific languages with

limited debugging

• provides users more control
• has numerous libraries for analysis and

visualization

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Architecture – Numerics

• BLAS and LAPACK

– Used for dense matrix and vector

• perations, geometric processing

– Optimized for most platforms – Called by sparse matrix factorization

• SuperLU, MKL Pardiso used for sparse

matrix factorization

• Iterative Math Library used for GMRES

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SYMDIFF

• Symbolic differentiation library
• Open source https://symdiff.org
• String based approach with dynamic

binding of names to referred quantities – Constants – Independent variables – Models

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SYMDIFF – Parser

• Uses rules of precedence and associativity
• Has simplify algorithm to reduce cost

<<<< diff(a + b + cˆ2, c) (2 * c) <<<< diff(xˆx, x) (((x * (xˆ(-1))) + log(x)) * (xˆx)) <<<< simplify(diff(xˆx,x)) ((1 + log(x)) * (xˆx))

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SYMDIFF – User functions

• Defining functions requires specification of

new function and derivatives w.r.t. each named variable argument > define(sqrt(x),0.5 * xˆ(-0.5)) sqrt(x) > diff(sqrt(x*y),y) ((0.5 * ((x * y)ˆ(-0.5))) * x)

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SYMDIFF – Models

• Models allow

– creation of new PDEs – hierarchy for sub-expression elimination – ability to specify or generate derivatives

• Models dynamically bound by name

– diff(Model,x) is Model:x

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Element Assembly

• Expressions evaluated at run time
• Symbolic derivatives of models for Jacobian

assembly

• Assembles bulk, interface, and contact

equations

• Circuit boundary conditions

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Node Models

NodeVolume EdgeCouple

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Node Models – Shockley Read Hall

USRH = np−n2

i

τp(n+n1)+τn(p+ p1)

USRH="(Electrons*Holes - n_iˆ2)/ \ (taup*(Electrons + n1) + taun*(Holes + p1))" Gn = "-ElectronCharge * USRH" Gp = "+ElectronCharge * USRH" NodeModel("USRH", USRH) NodeModel("ElectronGeneration", Gn) NodeModel("HoleGeneration", Gp) for i in ("Electrons", "Holes"): NodeModelDerivative("USRH", USRH, i) NodeModelDerivative("Gn", Gn, i) NodeModelDerivative("Gp", Gp, i)

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Node Models – Shockley Read Hall

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Edge Models

EdgeLength EdgeCouple n1 n0

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Edge Models

• Electric field (E ) w.r.t potential (ϕ)

edge_model(device=device, region=region, name=’E ’, equation=’(ϕ@n0 - ϕ@n1)*EdgeInverseLength’) edge_model(device=device, region=region, name=’E :ϕ@n0’, equation=’EdgeInverseLength’) edge_model (device=device, region=region, name=’E :ϕ@n1’, equation=’-EdgeInverseLength’)

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Element Edge Models

ElementNodeVolume en1 en2 en0 EdgeLength ElementEdgeCouple

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2D MOSFET Mobility

• Element models are used to simulate mobility with respect to

electric field normal and perpendicular to current flow

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BJT Example

Available

https://github.com/devsim/devsim_bjt_example

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BJT – DC Analysis

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Vbe (V) 10 10

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β Ic Ib

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A/cm 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Vce (V) 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Ic (A/cm)

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BJT – AC Analysis

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Ic (A/cm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 fT (Hz) 1e9 10

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f (Hz) 10

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|β|

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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Density Gradient

• Quantum correction method for carrier

density near interfaces

• Carrier quantization effects

Λe = −bn

∇2√n √n

∇2√n √n =

1 2

• ∇2 logn+ 1

2 (∇logn)2

Using n = exp(u)

• Λe∂v = −bn

2

• ∇u·∂s+ 1

2

• (∇u)2 ∂v
• + bnox

xn σint

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Density Gradient

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Density Gradient

1 1 2 3 4 5 Vg (V) 2 4 6 8 10 12 C (fF/ m)

CV Curves tox=3 (nm)

NA=1017 (#/cm3) DG NA=1018 (#/cm3) DG NA=1019 (#/cm3) DG

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