Device Simulation for Single-Event Effects

Mark E. LawEric Dattoli, Dan Cummings

NCAA Basketball Champions - University of FloridaSWAMP Center

Objectives• Provide SEE device simulation environment

• Address SEE specific issues– Physics - strain– Numerics - automatic operation

• Long term:– Simulate 1000’s of events to get statistics

– With SEE appropriate physics– Without extensive human intervention

Outline• Background - FLOODS Code• Numeric Issues and Enhancements– Grid Refinement– Parallel Computing Platforms

• Physical Issues and Enhancements– Transient / Base Materials– Mobility– Coupling to MRED / GEANT

FLOOPS / FLOODS• Object-oriented codes• Multi-dimensional• P = Process / D = Device 90% code shared

• Scripting capability for PDE’s - Alagator

• Commercialized - ISE / Synopsis– Sentaurus - Process is based on FLOOPS

• Licensed at over 200 sites world-wide

What is Alagator?

• Scripting language for PDE’s• Parsed into an expression tree• Assembled using FV / FE techniques

• Stored in hierarchical parameter data base

• Models are accessible, easily modified

What is Alagator?

• Example use of operators for diffusion equation

• Fick’s Second Law of Diffusion– ddt(Boron) - 9.0e-16 * grad(Boron)– ∂C(x,t) / ∂t = D ∂2C(x,t) / ∂x2

Operator Description

“ddt” Time derivative

“grad” Spatial derivative

“sgrad” Scharfetter / Gummel Discretization Operator

“diff”

Returns the magnitude of the derivative of the argument parallel to the edge of evaluation – electric field applications for mobility in a device – returns a scalar

“trans”

Returns the magnitude of the derivative of the argument perpendicular to the edge of evaluation – electric field application for mobility in a device – returns a scalar

“elastic”

Compute elastic forces - FEM balance

Basic Upgrades• FLOODS has been used for:

– Bipolar devices (SiGe)– GaN based heterostructures MEM’s

•Coupled H diffusion to device operation•4 equations , n, p, H

– Noise simulations for RF bipolar devices

• Enhancements for modern MOS– More flexible contacting options (transients)

– Accurate mobility - transverse field– Alternate channel materials

Outline• Background - FLOODS Code• Numeric Issues and Enhancements– Grid Refinement– Parallel Computing Platforms

• Physical Issues and Enhancements– Transient / Base Materials– Mobility– Coupling to MRED

Adaptive Refinement

• Charge Deposition is not on grid lines

Charge Spreads in timeFine grid at zero timeCoarser grid as time goes

Simulate many hits, we can’t have user defined grid

Object Oriented

• Modular - Grid / Operators / Fields

• Code written for elements works in all dimensions

• Example - every element can compute Size

Element Class

Volume

Face

EdgeNode

Example - Isotropic Refinement

• Local Error Estimate - Bank Weiser Based

• Remove– Replace an edge w/ a node– Dose Stays Constant– Position new node at optimal quality position

• Addition– Subdivide an edge– Find effected volumes (Voronoi)– Centroidal positioning

SRC Supported

Anisotropic Grid - Initial• Rectangular region created at the command line

• Remainder of the silicon is smoothed

• Silicon Elements 478

• Joint Quality 0.936

• Average Quality 0.944

SRC Supported

Anisotropic Grid

• Refinement of both extension and deep source / drain

• LevelSet Spacer• Note - etch onto rectangular regions

• Silicon Elements 1150• Joint Quality 0.937• Average Quality 0.961

• Improved Quality on Add!

SRC Supported

Good for Process Simulation

• Device Simulation is Different!– Channel Needs Anisotropic refinement

– Unrefinement difficult– Global Operations and Data Structures

Device Simulation Driven Refinement

• All brick elements (2D example)• Refine and terminate• Unrefinement easier to track

– Glue elements together– Remove excess discretization nodes

• Requires Multi-point Templates– 4, 5, and 6 point square discretization (2D)

– Virtual functions in an Object Oriented Scheme

Object Oriented• Derived Specific Geometry Elements• Working on refinement• Working on Discretization

Element Class

Volume

Face

EdgeNode

2 -Edge

3 -Edge

Face

Quad

Tri

Parallel Computing

• 3D Transient is time consuming

• What can be done to accelerate?

Numerical Approximations• Discretization

– Replace continuous functions w/ piecewise linear approximations

– Grid Spacing, Time

• Linearization– Reduce nonlinear

terms using multi-dimension Newton’s method

– Mobility, Statistics, …

• Linear Matrix Problem– Number of PDE’s x

number of nodes square

– Direct Solver

Nonlinear set of PDENonline

ar algebra

ic equatio

ns

Linear Matrix Problem

Temporal and SpatialDiscretization

PoissonCarrier ContinuityLattice Temperature

Multi-dimensional Newton Linearization

Flux = (n1 - n2) / x12

1

x12

1 1

1 1

n1

n2

F

F

CPU Effort and Time• Assembly of Matrix

– Calculate the large, linear system

– Lots of Data read– Potential for

Overlapping writes– Lots of Parallel

Potential– Linear in number of

elements

• Solution of Matrix– Large Sparse System– Established means for

parallel solve– Leverage Argonne Nat’l

Lab Code– Low power of equations

n1.5

Nonlinear set of PDENonline

ar algebra

ic equatio

ns

Linear Matrix Problem

Temporal and SpatialDiscretization

PoissonCarrier ContinuityLattice Temperature

Multi-dimensional Newton Linearization

Flux = (n1 - n2) / x12

1

x12

1 1

1 1

n1

n2

F

F

Alagator Assembly• Equations are split

– Edge pieces (current, electric field)– Node pieces (recombination, time derivative)

– Element pieces (perpendicular field)

• Pieces are vectorized– 128 pieces in tight BLAS loops for performance

– Operations are broken down in scripting

• Overall CPU linear in # of pieces

Parallel Assembly

• Two Options• High Level Parallel

– Assemble Different PDE’s on Different CPU’s

– Limited Parallel Speedup

• Low Level Parallel– Split Grid, assemble pieces

• Match to Linear Solve

Parallel Assembly

• Partition the work on different processors

• Assemble pieces on processor that will solve

Parallel Performance - Assembly

• High Level Partition

• Poisson on Node 1

• Electrons on Node 0

Edge Assembly Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Serial Node 0 Node 1

Parallel

(s)

Linear Solve Speedup - PETSC Package

• Amdahl’s Law Clearly Visible

3-D Dopant Diffusion, Linear Solve Times

0

102030

40

50

6070

80

0 1 2 3 4 5

Number of Nodes

(s) IBM SP/2 system

Linux, AMD Opteron Cluster

Speedup of Linear Solve

1

1.5

2

2.5

3

3.5

4

4.5

1 2 3 4 5

Number of Nodes

Sp

ee

du

p

IBM SP/2 MeasuredPerformance

Ideal Performance

AMD Opteron Cluster

Linear Solve Speedup - Options

• Ordering Algorithms are not helpful

• Some Parallel Methods increase solve time

Effect of Matrix Ordering on Serial Performance

0102030405060708090

100

Natural Quotient MinimumDegree

Reverse Cuthill-McKee Nested Dissection

Parallel GMRES Performance Problems

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5

Number of Nodes

Se

co

nd

s

Outline• Background - FLOODS Code• Numeric Issues and Enhancements– Grid Refinement– Parallel Computing Platforms

• Physical Issues and Enhancements– Transient / Base Materials– Mobility– Coupling to MRED

Today’s TransistorScaled MOSFETS and alternate materials to extend Moore’s Law

• Technology scaling is driven by cost per transistor

• Channel length scaling is slowing in bulk planar devices

• Limited by leakage current

• Strained Si devices

S. Thompson et al., IEEE EDL. 191-193, 2004.

S. Thompson et al., IEDM Tech. Dig. 61-64, 2003.

Enable Transients for Devices

• Added transient device command• Extended Contacts to allow switching• Contact Templates Available Now

Example NMOSSwitching TransientGate Ramped from3V to 0V in 1ps

Enable Transients for Devices

• 1D Diode• Charge added to depletion region at time 0• Simplest possible SEE

Mobility Modeling• Combination of terms

– Ionized Dopants– Carrier-Carrier– Surface Roughness– Strain

• Combined using Mathiessen’s rule

1

1

b

1

s

Low-Field Mobility• Lots of models - implemented Phillips unified

model• Includes

– Dopant (dependent on dopant type)– Carrier - Carrier scattering– Minority carrier scattering

n 1

N scN sceff

N refN scf

2

n pN sceff

N sc ND NA p N sceff NA GND p

F

0

200

400

600

800

1000

1200

1016 1017 1018 1019 1020 1021 1022

Ele

ctro

n M

obili

ty (

cm2 /V

s)

Doping (cm -3)

Low-Field Mobility - Carrier-Carrier

• In single event simulation• Dominant term can be carrier - carrier• Serious mistakes by ignoring these terms

200

400

600

800

1000

1200

1016 1017 1018 1019 1020 1021

Ele

ctro

n M

obili

ty (

cm2 /V

s)

Hole Concentration (cm -3)

Donor Density of 1016

Surface Scattering• Acoustic Phonons• Surface Roughness• Both depend on

perpendicular field

• Decay factor applies only in channel

• Tuned to measured MOS results

• In progress!

ac B

E

C(N /N0)

E1/ 3(T /T0)k

sr E /E ref A

E

3

1

1

1

b

D

ac

D

sr

D e x /

100

120

140

160

180

200

220

240

260

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Mob

ility

(cm

/Vs)

Normal Electric Field (MV/cm)

Normal Field Computation• Requires element

assembly– Increased computation– More complex matrix

• Compute field perpendicular to an interface– Fixed geometry– Might interact w/ single

event

• Field perpendicular to current flow– Convergence difficulties

at low current– Assumes current is

perpendicular…..– Make sure it doesn’t

apply in bulk

100

120

140

160

180

200

220

240

260

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Mob

ility

(cm

/Vs)

Normal Electric Field (MV/cm)

Current

Field

SiO2

Channel Materials• Heterostructure Boundaries • Fairly Easy, since we had heterostructure experience in FLOODS before

• Development of Ge channel simulations

500Å Ge Channel30Å Gate NitridePoly Gate Bias Swept Up0.1m Channel LengthIdeal Doping Profiles

Note: Concentration Discontinuity at interface

Boundary Conditions• Commercial simulators only allow BC at contacts• FLOODS has large flexibility at boundaries• Example - Sink on sides• pdbSetString ReflectLeft Equation “1.0e-3*(Elec-Doping)

• Simulation as function of device simulation size

• Reflecting boundaries at edges and back change current collected at contacts

Courtesy of Ron Schrimpf, Andrew Sternberg

Finite Element Method Mechanics

• Theory of Elasticity – linear elastic materials - Silicon is modeled as an isotropic material for simplicity

• Enhanced Alagator– Added elastic operator for displacement– Added source term operators

• Elastic(displacement) + BodyStrain(Boron*k)

σ

ε

SRC Supported

(μm)

(μm

)45 nm

140

nm

120

nm

30 nm

Si0.83Ge0.17Si0.83Ge0.17

STI STI

-536

-83403

95

31

Source FLOOPS

MPa

Stress Contours

Future - Strain and SEU Upgrades

• Anisotropic operators– Current direction, strain interaction

– Mobility has an orientation

• Density of States• Recombination• Driving Forces?

Connection to Thompson

Trajectory Read

• Trajectory Read Command

Summary• Numerics

– Started Developing refinement appropriate to SEE

– Parallel Port, Begun Testing

• Physics– Built some basic capability for SEE– Read Tracks

• Next Year– Demonstrate link, run demos on parallel machines