Semi-Classical Transport Theory. Outline: What is Computational Electronics? Semi-Classical...

Semi-Classical Transport Theory

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 Quantum Moment Methods

Particle-Based Device Simulations: Effective Potential Approach

Quantum Transport Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical

Basis of the Green’s Functions Approach (NEGF) NEGF: Recursive Green’s Function Technique and CBR Approach Atomistic Simulations – The Future

Prologue

Semi-Classical Transport Theory

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

which gives one the probability of finding a particle in region (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

8 k 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.

Semi-Classical Transport Approaches

1.Drift-Diffusion Method2.Hydrodynamic Method3.Direct Solution of the Boltzmann Transport

Equation via: Particle-Based Approaches – Monte Carlo

method Spherical Harmonics Numerical Solution of the Boltzmann-Poisson

Problem

C. Jacoboni, P. Lugli: "The Monte Carlo Method for Semiconductor Device Simulation“, in series "Computational Microelectronics", series editor: S. Selberherr; Springer, 1989, ISBN: 3-211-82110-4.

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

dxdn

J qp x E x qDdx

S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.

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 Difference – easier to implement but requires more node points, difficult to deal with curved interfaces

Finite Elements – standard, smaller number of node points, resolves curved surfaces

Finite Volume

Linearized Poisson Equation φ→φ + δ where δ= φnew - φold

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 oldT T T T

old old old oldT T T T

old oldT T

newV V V V V V V Vi i

i

newV V V V V V V Vnewi i

iT

V V V V oldi

T

new old

en end Ve e C n e e

dx

en end Ve e V e e C n

dx V

ene e V

V

V V

Scharfetter-Gummel Discretization of the Continuity Equation

Electron and hole densities n and p vary exponentially between mesh points → relaxes the requirement of using very small mesh sizes

The exponential dependence of n and p upon the potential is buried in the Bernoulli functions

1/ 2 1 1/ 2 1 1/ 2 1 1/ 2 11 12 2 2 2

1/ 2 1 1/ 2 1 1/ 2 112 2 2

n n n ni i i i i i i i i i i i

i i i iT T T T

n n ni i i i i i i i i

iT 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/ 2 112

ni i i

i i iT

D V Vp B p U

V

( )1x

xB x

e

Bernoulli function:

Scaling due to de MariVariable Scaling Variable Formula

Space Intrinsic Debye length (N=ni)

Extrinsic Debye length (N=Nmax)

2Bk T

Lq N

Potential Thermal voltage * Bk T

Vq

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-Recombination 2

DNR

L

Time 2LT

D


Governing Equations ICS/BCS

DiscretizationSystem of Algebraic Equations

Equation (Matrix) Solver

ApproximateSolution

Continuous Solutions

Finite-Difference

Finite-Volume

Finite-Element

Spectral

Boundary Element

Hybrid

Discrete Nodal Values

Tridiagonal

SOR

Gauss-Seidel

Krylov

Multigrid

φi (x,y,z,t)

p (x,y,z,t)

n (x,y,z,t)

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


Poisson solvers: Direct

Gaussian Eliminatioln LU decomposition

Iterative Mesh Relaxation Methods

• Jacobi, Gauss-Seidel, Successive over-Relaxation Advanced Iterative Solvers

• ILU, Stone’s 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 of Computational Electronics, Vol. 1, pp. 359-363 (2002).

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 solve model problem (Poisson’s equation) on regular mesh

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

cc

abcab

cabca

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

nixcg

xg

x

nigfgfg

i

iiii

n

nn

iiii

Numerical Solution Details of the Coupled Equation Set

Solution Procedures: Gummel’s Approach – when the constitutive

equations are weekly coupled Newton’s Method – when the constitutive

equations are strongly coupled Gummel/Newton – more 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.

Schematic Description of the Gummel’s Approach

initial guessof the solution

solvePoisson’s eq.

Solve electron eq.Solve hole eq.

nconverged?

converged?n

y

y


solvePoisson’s eq.

Solve electron eq.Solve hole eq.

nconverged?

converged?n

y

y


Solve Poisson’s eq.Electron eq.

Hole eq.

Updategeneration rate

nconverged?

converged?n

y

y


Solve Poisson’s eq.Electron eq.

Hole eq.

Updategeneration rate

nconverged?

converged?n

y

y

Original Gummel’s scheme Modified Gummel’s scheme

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 t must be related to the plasma frequency

The Mesh size is related to the Debye length

*

2

m

ne

sp

max

s TD

VL

qN

When is the Drift-Diffusion Model Valid? Large devices where the velocity of the carriers is

smaller than the saturation velocity

The validity of the method can be extended for velocity saturated devices as well by introduction of electric field dependent mobility and diffusion coefficient:

Dn(E) and μn(E)

0 0.8 1.6 2.42.4

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)

Velo

cit

y (

m/s

)

80 nm FD SOI nMOSFET

0 0.9 1.8 2.72.7

x 10-7

0

0.5

1

1.5

2

2.5x 10


x (m)0 1 2 33

x 10-7

0

0.5

1

1.5

2

2.5x 10


x (m)

Tgate=300KTgate=400KTgate=600K

0 1.2 2.4 3.63.6

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)

Velo

cit

y (

m/s

)


0 1.4 2.8 4.24.2

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)


0 1.8 3.6 5.45.4

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)



0 2.5 5 7.5

x 10-8

0

0.5

1

1.5

2

2.5x 10

5

x (m)

Velo

cit

y (

m/s

)


0 0.45 0.9 1.351.35

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x (m)


0 0.6 1.2 1.81.8

x 10-7

0

0.5

1

1.5

2

2.5x 10

5

x(m)



sourcechannel

drain

25 nm 140 nm100 nm

What Contributes to The Mobility?

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic Optical

Nonpolar PolarDeformationpotential

Piezo-electric

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic OpticalAcoustic Optical

Nonpolar PolarNonpolar PolarDeformationpotential

Piezo-electric

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

Mobility Modeling

Mobility modeling can be separated in three parts: Low-field mobility characterization for bulk

or inversion layers High-field mobility characterization to

account for velocity saturation effect Smooth interpolation between the low-field

and high-field regions

Silvaco ATLAS Manual.

(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 Mathiessen’s rule

Ionized impurity scattering:

- Conwell-Weiskopf model

- Brooks-Herring model

(A) Low-Field Models for Bulk Materials (cont’d)Combined phonon and ionized impurity scattering:

- Dorkel and Leturg model:

temperature-dependent phonon scattering +ionized impurity scattering + carrier-carrier interactions

- Caughey and Thomas model:

temperature independent phonon scattering + ionized impurity scattering

(A) Low-Field Models for Bulk Materials (cont’d)

- Sharfetter-Gummel model:

phonon scattering + ionized impurity scattering (parameterized expression – does not use the Mathiessen’s rule)

- Arora model:

similar to Caughey and Thomas, but with temperature dependent phonon scattering

(A) Low-Field Models for Bulk Materials (cont’d)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 to lattice, ionized impurity and carrier-carrier scattering via the Mathiessen’s rule

(B) Field-Dependent Mobility

The field-dependent mobility model provides smooth transition between 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

(C) Inversion Layer Mobility Models

CVT model: combines acoustic phonon, non-polar optical

phonon and surface-roughness scattering (as an inverse square dependence of the perpendicular electric field) via Mathiessen’s 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)

(C) Inversion Layer Mobility Models (cont’d)Shirahata model:

uses Klaassen’s 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

Generation-Recombination MechanismsClassification

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

Impactionization

Auger

• Electron emission• Hole emission• Electron capture• Hole capture

Pure generation process

Hydrodynamic Modeling

In small devices there exists non-stationary transport and carriers are moving through the device with velocity larger than the saturation velocity In Si devices non-stationary transport occurs

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

In GaAs devices velocity overshoot occurs due to intervalley transfer

T. Grasser (ed.): "Advanced Device Modeling and Simulation“, World Scientific Publishing Co., 2003, ISBN: 9-812-38607-6 M.M. Lundstrom, Fundamentals of Carrier Transport, 1990.

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/cm10 kV/cm50 kV/cm

time [ps]

Drift

velo

city

[c

m/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

Ene

rgy

[eV

]

time [ps]

Scattering mechanisms:

• Acoustic deformation potential scattering• Zero-order intervalley scattering (f and g-

phonons)• First-order intervalley scattering (f and g-

phonons)

g

f

kz

kx

ky

g

f

g

ff

kz

kx

ky

X. He, MS Thesis, ASU, 2000.

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 addition to: Particle Conservation (Continuity Equation) Momentum (mass) Conservation Equation

Hydrodynamic Model due to Blotakjer

Constitutive Equations: Poisson +• More convenient set of balance equations is in terms of n, vdand w:

colld

d

Bdd

coll

d

dddd

colld

tw

e

vmw

kn

nw

tw

tme

vnmnwnm

mmt

tn

ntn

)(

2

*

32

)(

*

*21

*32

)*(*

)(

2

2

vE

vv

vE

vvv

v

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:

)(*2

5,

2

wvm

nTkTn B q

• Substituting T with the density of the carrier energy, the momentum and energy balance equations become:

Momentum Relaxation Rate• The momentum rate is determined by a steady-state MC

calculation in a bulk semiconductor under a uniform bias electric field, for which:

dp

dpcoll

dd

vmeE

w

wme

tme

t

*)(

0)(**

v

EvEv

K. Tomizawa, Numerical Simulation Of Submicron Semiconductor Devices.

Energy Relaxation Rate• The emsemble energy relaxation rate is also determined by a

steady-state MC calculation in a bulk semiconductor under a uniform bias electric field, for which:

0

0

)(

0)(

ww

ew

wwetw

etw

dw

wdcoll

d

vE

vEvE

K. Tomizawa, Numerical Simulation Of Submicron Semiconductor Devices.

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]

Dra

in C

urr

en

t [m

A/u

m]

DDEBHDDD SREB SRHD SR

Silvaco ATLAS simulations performed by Prof. Vasileska.

90 nm device

SR = series resistance


Source Drain

Gate oxide

BOX

tox

tsi

tBOX

LS Lgate LD



VDD 1V 1.2 V 1.4 V

Overshoot EB/HD

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


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]

Dra

in C

urr

en

t [m

A/u

m]



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]D

rain

Cu

rre

nt [

mA

/um

]

DDHDEBDD SREB SRHD SR


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]

Dra

in C

urr

en

t [m

A/u

m]



Source Drain

Gate oxide

BOX

tox

tsi

tBOX

LS Lgate LD



VDD 1V 1.2 V 1.4 V

Overshoot EB/HD

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


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

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

Drain Voltage [V]D

rain

Cu

rre

nt [

mA

/um

]

DDHDEBDD SREB SRHD SR


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]


14 nm device


Failure of the Hydrodynamic Model

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-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]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-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]

Dra

in C

urr

en

t [m

A/u

m]

1019 cm-3

1020 cm-3

0.1 ps

0.2 ps0.3 ps


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

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 10

2

4

6

8

10

12

14

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-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]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-3

0.1 ps

0.2 ps

0.3 ps


0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-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]

Dra

in C

urr

en

t [m

A/u

m]

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 10

2

4

6

8

10

12

14

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-3

0.1 ps

0.3 ps

0.2 ps

Summary

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

Hydrodynamic model must be used with caution when modeling devices in which velocity overshoot, which is a signature of non-stationary transport, exists in the device

Proper choice of the energy relaxation times is a problem in hydrodynamic modeling

Semi-Classical Transport Theory. Outline: What is Computational Electronics? Semi-Classical...

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Transcript of Semi-Classical Transport Theory. Outline: What is Computational Electronics? Semi-Classical...