b drift diffusion hydrodynamic modeling

8/12/2019 B Drift Diffusion Hydrodynamic Modeling

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Semi-Classical Transport

Theory


2/41

Outline:

What is Computational Electronics?

Semi-Classical Transport Theory

Drift-Diffusion Simulations

Hydrodynamic Simulations

Particle-Based Device Simulations

Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators

Tunneling Effect: WKB Approximation and Transfer Matrix Approach

Quantum-Mechanical Size Quantization Effect

Drift-Diffusion and Hydrodynamics: Quantum Correction and QuantumMoment Methods

Particle-Based Device Simulations: Effective Potential Approach

Quantum Transport

Direct Solution of the Schrodinger Equation (Usuki Method) and TheoreticalBasis of the Greens Functions Approach (NEGF)

NEGF: Recursive Greens Function Technique and CBR Approach

Atomistic SimulationsThe Future

Prologue


3/41

Semi-Classical Transport Theory

It is based on direct or approximate solution of theBoltzmann Transport Equation for the semi-classicaldistribution function f(r,k,t)

which gives one the probability of finding a particle inregion (r,r+dr) and (k,k+dk) at time t

Moments of the distribution function give us information

about:Particle Density

Current Density

Energy Density

3

11 1

8k k kk

Fkk k r k k k k k k k

f VE f f d f f f f

t

D. K. Ferry, Semiconductors, MacMillian, 1990.


4/41

Semi-Classical Transport Approaches

1. Drift-Diffusion Method

2. Hydrodynamic Method

3. Direct Solution of the Boltzmann TransportEquation via:Particle-Based ApproachesMonte Carlo

method

Spherical Harmonics

Numerical Solution of the Boltzmann-PoissonProblem

C. Jacoboni, P. Lugli: "The Monte Carlo Method for Semicond uctor Device Simulat ion,

in series "Computational Microelectronics", series editor: S. Selberherr; Springer, 1989, ISBN: 3-211-82110-4.
http://www.iue.tuwien.ac.at/books/mc_method_semihttp://www.iue.tuwien.ac.at/books/mc_method_semi


5/41

1. Drift-Diffusion Approach

Constitutive Equations

Poisson

Continuity Equations

Current Density Equations

1

1

J

J

n n

p p

nU

t q

pU

t q

D AV p n N N

( ) ( )

( ) ( )

n n n

p p p

dnJ qn x E x qD

dx

dnJ qp x E x qD

dx

S. Selberherr: "Analysis and Simulation ofSemiconductor Devices, Springer, 1984.


6/41

Numerical Solution Details

Linearization of the Poisson equation

Scharfetter-Gummel Discretization of the

Continuity equation

De Mari scaling of variables

Discretization of the equations

Finite Differenceeasier to implement but requires

more node points, difficult to deal with curved interfaces

Finite Elementsstandard, smaller number of node

points, resolves curved surfaces

Finite Volume


7/41

Linearized Poisson Equation

+ where = new- old

Finite difference discretization:

Potential varies linearly between mesh points

Electric field is constant between mesh points

Linearization Diagonally-dominant

coefficient matrix A is obtained

2/ / / /

2

2/ / / /

2

/ /

/

/

old old old old T T T T

old old old old T T T T

old old T T

newV V V V V V V V i i

i

newV V V V V V V V newi i

i

T

V V V V oldi

T

new old

en end Ve e C n e e

dxen end V

e e V e e C ndx V

ene e V

V

V V


8/41

Scharfetter-Gummel Discretization of the

Continuity Equation

Electron and hole densities nandpvary exponentially

between mesh points relaxes the requirement of

using very small mesh sizes

The exponential dependence of nandpupon the

potential is buried in the Bernoulli functions

1/ 2 1 1/ 2 1 1/2 1 1/ 2 1

1 12 2 2 2

1/ 2 1 1/ 2 1 1/ 2 1

12 2 2

n n n n

i i i i i i i i i i i i

i i i i

T T T T

n n ni i i i i i i i i

i

T T T

D V V D V V D V V D V VB n B B n B n U

V V V V

D V V D V V D V VB p B B

V V V

1/ 2 1

12

ni i i

i i i

T

D V Vp B p U

V

( )

1x

xB x

e

Bernoulli function:


9/41

Scaling due to de Mari

Variable Scaling Variable Formula

Space Intrinsic Debye length (N=ni)

Extrinsic Debye length (N=Nmax)

2

Bk TLq N

Potential Thermal voltage* B

k TV

q

Carrier concentration Intrinsic concentration

Maximum doping concentration

N=ni

N=Nmax

Diffusion coefficient Practical unit

Maximum diffusion coefficient

2

1cm

Ds

D = Dmax

Mobility

*

DM

V

Generation-Recombination2

DNR

L

Time 2LT

D


10/41


Governing

Equations

ICS/BCS

Discretization

System of

Algebraic

Equations

Equation

(Matrix)

Solver

Approximate

Solution

Continuous

Solutions

Finite-Difference

Finite-Volume

Finite-Element

Spectral

Boundary Element

Hybrid

Discrete

Nodal

Values

Tridiagonal

SOR

Gauss-Seidel

Krylov

Multigrid

i (x,y,z,t)

p (x,y,z,t)

n (x,y,z,t)

D. Vasileska, EEE533 Semiconductor Device and Process SimulationLecture Notes,Arizona State University, Tempe, AZ.


11/41


Poisson solvers:Direct

Gaussian Eliminatioln

LU decompositionIterative

Mesh Relaxation Methods

Jacobi, Gauss-Seidel, Successive over-Relaxation

Advanced Iterative Solvers ILU, Stones strongly implicit method, Conjugate gradient

methods and Multigrid methods

G. Speyer, D. Vasileska and S. M. Goodnick, "Efficient Poisson solver for semiconductor

device modeling using the multi-grid preconditioned BICGSTAB method", Journal ofComputational Electronics, Vol. 1, pp. 359-363 (2002).


12/41

Complexity of linear solvers

2D 3D

Sparse Cholesky: O(n1.5 ) O(n2 )

CG, exact

arithmetic:

O(n2 ) O(n2 )

CG, no precond: O(n1.5 ) O(n1.33 )

CG, modified IC: O(n1.25 ) O(n1.17 )

CG, support trees: O(n1.20 ) -> O(n1+ ) O(n1.75 ) -> O(n1.31 )

Multigrid: O(n) O(n)

n1/2 n1/3

Time to solvemodel problem

(Poissons

equation) on

regular mesh


13/41

LU Decomposition

If LU decomposition exists, then for a tri-diagonal matrix A,resulting from the finite-difference discretization of the 1D

Poisson equation, one can write

where

Then, the solution is found by forward and back substitution:

n

n

nnn

nnn c

c

c

ab

cab

cab

ca

1

22

11

3

2

111

222

11

...

......

1

.........

1

1

.........

nkcab

a kkkkk

kk ,...,3,2,,, 1

111

1,2,...,1,,

,,...,3,2,,

1

111

ni

xcg

x

g

x

nigfgfg

i

iiii

n

nn

iiii


14/41

Numerical Solution Details of the Coupled

Equation Set

Solution Procedures:

Gummels Approachwhen the constitutive

equations are weekly coupled

Newtons Methodwhen the constitutiveequations are strongly coupled

Gummel/Newtonmore efficient approach

D. Vasileska and S. M. Goodnick,Computational Electronics, Morgan & Claypool,

2006.

D. Vasileska, S. M. Goodnick and G. Klimeck, Computational Electronics: From

Semiclassical to Quantum Transport Modeling, Taylor & Francis, in press.


15/41

Schematic Description of the Gummels

Approach

initial guess

of the solution

solve

Poissons eq.

Solve electron eq.

Solve hole eq.

nconverged?

converged?n

y

y

initial guess

of the solution

solve

Poissons eq.

Solve electron eq.

Solve hole eq.

nconverged?

converged?n

y

y

initial guess

of the solution

Solve Poissons eq.

Electron eq.

Hole eq.

Update

generation rate

nconverged?

converged?n

y

y

initial guess

of the solution

Solve Poissons eq.

Electron eq.

Hole eq.

Update

generation rate

nconverged?

converged?n

y

y

Original Gummels scheme Modified Gummels scheme


16/41

Constraints on the MESH Size and TIME

Step

The time step and the mesh size may correlate

to each other in connection with the numerical

stability:

The time step tmust be related to the plasmafrequency

The Mesh size is related to the Debye length

*

2

m

ne

sp

max

s TD

VL

qN


17/41

When is the Drift-Diffusion Model Valid?

Large devices where the velocity of the carriers issmaller than the saturation velocity

The validity of the method can be extended for velocitysaturated devicesas well by introduction of electric fielddependent mobility and diffusion coefficient:

Dn(E) and n(E)

0 1 2 33x 10

-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)

Tgate=300K

Tgate=400K

Tgate=600K

0 1.4 2.8 4.4.

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)

Tgate=300K

Tgate=400K

Tgate=600K

0 2.5 5 7.5x 10

-8

0

0.5

1

1.5

2

2.5x 10

x (m)

25 nm 140 nm100 nm


18/41

What Contributes to The Mobility?

Scattering Mechanisms

Defect Scattering Carr ier-Carr ier Scattering Latt ice Scatter ing

Crystal

Defects Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic Optical

Nonpolar Polar Deformation

potential

Piezo-

electric

Scattering Mechanisms

Defect Scattering Carr ier-Carr ier Scattering Latt ice Scatter ing

Crystal

Defects Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic OpticalAcoustic Optical

Nonpolar Polar Nonpolar Polar Deformation

potential

Piezo-

electric

D. Vasileska and S. M. Goodnick, "Computational Electronics", Materials Scienceand Engineering, Reports: A Review Journal, Vol. R38, No. 5, pp. 181-236 (2002)


19/41

Mobility Modeling

Mobility modeling can be separated in

three parts:

Low-field mobility characterizationfor bulk

or inversion layers

High-field mobility characterizationto

account for velocity saturation effect

Smooth interpolationbetween the low-fieldand high-field regions

Silvaco ATLAS Manual.


20/41

(A) Low-Field Models for Bulk Materials

Phonon scattering:

- Simple power-law dependence of the

temperature

- Sah et al. model:acoustic + optical and intervalley phonons

combined via Mathiessens rule

Ionized impurity scattering:

- Conwell-Weiskopf model

- Brooks-Herring model


21/41

(A) Low-Field Models for Bulk

Materials (contd)

Combined phonon and ionized impurity scattering:

- Dorkel and Leturg model:

temperature-dependent phonon scattering +

ionized impurity scattering + carrier-carrierinteractions

- Caughey and Thomas model:

temperature independent phonon scattering +ionized impurity scattering


22/41


Materials (contd)

- Sharfetter-Gummel model:

phonon scattering + ionized impurity scattering

(parameterized expressiondoes not use the

Mathiessens rule)- Arora model:

similar to Caughey and Thomas, but with

temperature dependent phonon scattering


23/41


Materials (contd)

Carrier-carrier scattering

- modified Dorkel and Leturg model

Neutral impurity scattering:

- Li and Thurber model:

mobility component due to neutral impurity

scattering is combined with the mobility due tolattice, ionized impurity and carrier-carrier

scattering via the Mathiessens rule


24/41

(B) Field-Dependent Mobility

The field-dependent mobility model provides smooth transitionbetween low-field and high-field behavior

vsat

is modeled as a temperature-dependent quantity:

/1

0

0

1

)(

satvE

E = 1 for electrons

= 2 for holes

cm/s

600exp8.01

104.2)(

7

Lsat T

Tv


25/41

(C) Inversion Layer Mobility Models

CVT model:

combines acoustic phonon, non-polar optical

phonon and surface-roughness scattering (as

an inverse square dependence of theperpendicular electric field) via Mathiessens

rule

Yamaguchi model:

low-field part combines lattice, ionized impurity

and surface-roughness scattering

there is also a parametric dependence on the

in-plane field (high-field component)


26/41

(C) Inversion Layer Mobility Models

(contd)

Shirahata model:

uses Klaassens low-field mobility model

takes into account screening effects into the

inversion layer

has improved perpendicular field dependence

for thin gate oxides

Tasch model:the best model for modeling the mobility in

MOS inversion layers; uses universal mobility

behavior


27/41

Generation-Recombination Mechanisms

Classification

Twoparticle

One step

(Direct)

Two-step

(indirect)

Energy-level

consideration

Photogeneration Radiative recombination

Direct thermal generation

Direct thermal recomb.

Shockley-Read-Hall(SRH) generation-

recombination

Surface generation-

recombination

Shockley-Read-Hall(SRH) generation-

recombination

Surface generation-

recombination

Threeparticle

Impact

ionization

Auger

Electron emission

Hole emission

Electron capture

Hole capture

Pure generation process


28/41

Hydrodynamic Modeling

In small devices there exists non-stationary transport and carriers aremoving through the device with velocity

larger than the saturation velocityIn Si devices non-stationary transport occurs

because of the different order of magnitude ofthe carrier momentum and energy relaxation

timesIn GaAs devices velocity overshoot occurs due

to intervalley transfer

T. Grasser (ed.): "Ad vanced Device Model ing and Simulat ion, World Scientific

Publishing Co., 2003, ISBN: 9-812-38607-6 M.M. Lundstrom, Fundamentals of Carrier Transport, 1990.
http://www.iue.tuwien.ac.at/books/adv_dev_modhttp://www.iue.tuwien.ac.at/books/adv_dev_mod


29/41

Velocity Overshoot in Silicon

-5x106

0

5x106

1x107

1.5x107

2x107

2.5x107

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm5 kV/cm

10 kV/cm50 kV/cm

time [ps]

Driftvelocity

[cm/s

]

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm5 kV/cm10 kV/cm50 kV/cm

Energy

[e

V]

time [ps]

Scattering mechanisms:

Acoustic deformation potential scattering

Zero-order intervalley scattering (fand g-

phonons)

First-order intervalley scattering (fand g-

phonons)

g

f

kz

kx

ky

g

f

g

ff

kz

kx

ky

X. He, MS Thesis, ASU, 2000.


30/41

How is the Velocity Overshoot Accounted

For?

In Hydrodynamic/Energy balance

modeling the velocity overshoot effect is

accounted for through the addition of an

energy conservation equation in additionto:

Particle Conservation (Continuity Equation)

Momentum (mass) Conservation Equation


31/41

Hydrodynamic Model due to Blotakjer

Constitutive Equations: Poisson +

coll

d

d

Bdd

coll

d

dddd

colld

t

we

vmw

kn

nw

t

w

tm

e

vnmnwnm

mmt

t

nn

t

n

)(

2

*

3

2

)(

*

*2

1

*3

2)*(

*

)(

2

2

vE

vv

vE

vvv

v


32/41

Closure

To have a closed set of equations, one either:

(a) ignores the heat flux altogether

(b) uses a simple recipe for the calculation of the heat flux:

)(*25,

2

wvmnTkTn B q


33/41

Momentum Relaxation Rate

The momentum rate is determined by a steady-state MCcalculation in a bulk semiconductor under a uniform bias

electric field, for which:

dp

dp

coll

dd

vm

eEw

w

m

e

tm

e

t

*)(

0)(

**

v

EvEv

K. Tomizawa, Numerical Simulation Of

Submicron Semiconductor Devices.


34/41

Energy Relaxation Rate

The emsemble energy relaxation rate is also determined by asteady-state MC calculation in a bulk semiconductor under a

uniform bias electric field, for which:

0

0

)(

0)(

ww

ew

wwe

t

we

t

w

dw

wd

coll

d

vE

vEvE

K. Tomizawa, Numerical Simulation Of

Submicron Semiconductor Devices.


35/41


36/41

Validity of the Hydrodynamic Model

Source Drain

Gate oxide

BOX

tox

tsi

tBOX

LS Lgate LD

feature 14 nm 25 nm 90 nm

Tox 1 nm 1.2 nm 1.5 nm

VDD 1V 1.2 V 1.4 V

Overshoot

EB/HD

233% / 224% 139% / 126% 31% /21%

Overshoot EB/DD

with series resistance

153%/96% 108%/67% 39%/26%

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

Drain Voltage [V]

DrainCurrent[mA/um]

DD

EB

HD

DD SREB SR

HD SR

Silvaco ATLAS simulations performed by Prof. Vasileska.

25 nm device

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

Drain Voltage [V]

DrainCurrent[m

A/um]

DD

HD

EB

DD SR

EB SR

HD SR

SR = series resistance


37/41


38/41

Failure of the Hydrodynamic Model

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

Drain Voltage [V]

DrainCurre

nt[mA/um]

1020cm-3

1019cm-3

0.1 ps

0.3 ps

0.2 ps


90 nm

25 nm

14 nm

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Drain Voltage [V]

DrainCurrent[mA/u

m]

1020

cm-3

1019cm-3

0.1 ps

0.2 ps

0.3 ps

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

Drain Voltage [V]

DrainC

urrent[mA/um]

1019

cm-3

1020

cm-3

0.1 ps

0.2 ps0.3 ps


39/41


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

Drain Voltage [V]

DrainC

urrent[mA/um]

1019

cm-3

1020

cm-3

0.1 ps

0.2 ps0.3 ps


90 nm

25 nm

14 nm

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

Drain Voltage [V]

DrainCurre

nt[mA/um]

1020cm-3

1019cm-3

0.1 ps

0.3 ps

0.2 ps

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Drain Voltage [V]

DrainCurrent[mA/u

m]

1020

cm-3

1019cm-3

0.1 ps

0.2 ps

0.3 ps


40/41


0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Drain Voltage [V]

DrainCurrent[mA/u

m]

1020

cm-3

1019cm-3

0.1 ps

0.2 ps

0.3 ps

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

Drain Voltage [V]

DrainC

urrent[mA/um]

1019

cm-3

1020

cm-3

0.1 ps

0.2 ps0.3 ps


90 nm

25 nm

14 nm

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

Drain Voltage [V]

DrainCurre

nt[mA/um]

1020cm-3

1019cm-3

0.1 ps

0.3 ps

0.2 ps


41/41

Summary

Drift-Diffusion model is good for large MOSFET devices,BJTs, Solar Cells and/or high frequency/high powerdevices that operate in the velocity saturation regime

Hydrodynamic model must be used with caution when

modeling devices in which velocity overshoot, which is asignature of non-stationary transport, exists in the device

Proper choice of the energy relaxation times is aproblem in hydrodynamic modeling
http://en.wikipedia.org/wiki/File:I-V_Curve_T.pnghttp://en.wikipedia.org/wiki/File:Solar_cell.png

b drift diffusion hydrodynamic modeling

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