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

Direct Solution of the Boltzmann Transport Equation

Particle-Based Approaches Spherical Harmonics Numerical Solution of the Boltzmann-Poisson

Problem

In here we will focus on Particle-Based (Monte Carlo) approaches to solving the Boltzmann Transport Equation

Ways of Solving the BTE Using MCT

Single particle Monte Carlo Technique Follow single particle for long enough time to

collect sufficient statistics Practical for characterization of bulk materials

or inversion layersEnsemble Monte Carlo Technique

MUST BE USED when modeling SEMICONDUCTOR DEVICES to have the complete self-consistency built in

Carlo Jacoboni and Lino Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645 - 705 (1983).

http://prola.aps.org/search/field/author/Jacoboni_C

http://prola.aps.org/search/field/author/Jacoboni_C

http://prola.aps.org/search/field/author/Reggiani_L



Path-Integral Solution to the BTE

The path integral solution of the Boltzmann Transport Equation (BTE), where L=Nt and tn=nt, is of the form:

1( )

0

( ) ( ') ( ', ( ) )N

N m tN m eff

m

f t t f p S p p eE N m t e

( ( ) )mg p eE N m t

K. K. Thornber and Richard P. Feynman, Phys. Rev. B 1, 4099 (1970).

The two-step procedure is then found by using N=1, which means that t=t, i.e.:

1 0'

( ) ( ') ( ', ) teff

p

f t t f p S p p eE t e

0 ( )g p eE t

Intermediate function that describesthe occupancy of the state (p+eEt)at time t=0, which can be changeddue to scattering events (SCATTER)

Integration over a trajectory, i.e.probability that no scattering occurred within time integral t (FREE FLIGHT)

+

Monte Carlo Approach to Solving the Boltzmann Transport Equation Using path integral formulation to the

BTE one can show that one can decompose the solution procedure into two components:

1. Carrier free-flights that are interrupted by scattering events

2. Memory-less scattering events that change the momentum and the energy of the particle instantaneously

Particle Trajectories in Phase Space

Particle trajectories in k-space and real space

yk

xk

-e

x

x

x x

x x

x x

x

x

xEy

x

y

x

Carrier Free-Flights The probability of an electron scattering in a small time interval dt is

(k)dt, where (k) is the total transition rate per unit time. Time dependence originates from the change in k(t) during acceleration by external forces

where v is the velocity of the particle. The probability that an electron has not scattered after scattering at t =

0 is:

It is this (unnormalized) probability that we utilize as a non-uniform distribution of free flight times over a semi-infinite interval 0 to . We want to sample random flight times from this non-uniform distribution using uniformly distributed random numbers over the interval 0 to 1.

t

ttd

n etP 0)(k

/0 tet BvEkk

Generation of Random Flight TimesHence, we choose a random number

t

ttd

i er 01,

k

Ith particle first random number

We have a problem with this integral!

We solve this by introducing a new, fictitious scattering process which does not change energy or momentum:

)()()()( kkxx Ek Sss

Generation of Random Flight Times

t

ttd

i er 01,

k

i

i kk )()( The sum runs over all the real scattering processes. To this we add the fictitious self-scattering which is chosen to have a nice property:0new

scatterersreal

iss kk )()( 0

• The use of the full integral form of the free-flight probability density function is tedious (unless k is invariant during the free flight).

• The introduction of self-scattering (Rees, J. Phys. Chem. Solids 30, 643, 1969) simplifies the procedure considerably.

• The properties of the self-scattering mechanism are that it does not change either the energy or the momentum of the particle.

• The self-scattering rate adjusts itself in time so that the total scattering rate is constant. Under these circumstances, one has that:

dtedtedttPtt ttd

self

t

0kk

Self-Scattering

Self-Scattering• Random flight times tr may be generated from P(t) above using

the direct method to get:

where r is a uniform random between 0 and 1 (and therefore r and 1-r are the same).

• must be chosen (a priori) such that > (k(t)) during the entire flight.

• Choosing a new tr after every collision generates a random walk in k-space over which statistics concerning the occupancy of the various states k are collected.

rrter rtr lnln

111

Free-Flight Scatter Sequence for Ensemble Monte Carlo

= collisions

1n23456

N

1,it

1,itHowever, we need a second time scale, which provides the times at which the ensemble is “stopped” and averages are computed.

Particle time scale

Free-FlightScatterSequence

dte=dtau

dte ≥ t?

no yes

dt2 = dte dt2 = t

Call drift(dt2)

dte ≥ t?yes

dte2 = dte

Call scatter_carrier()

Generate free-flight dt3

dtp=t-dte2

dt3 ≤ dtp?

no yes

dt2 = dtp dt2 = dt3

Call drift(dt2)

dte2=dte2+dt3dte=dte2

no

yesdte < t ?

dte=dte-t

dtau=dte

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 1983.

Choice of Scattering Event Terminating Free Flighto At the end of the free flight ti, the type of scattering which ends the

flight (either real or self-scattering) must be chosen according to the relative probabilities for each mechanism.

o Assume that the total scattering rate for each scattering mechanism is a function only of the energy, E, of the particle at the end of the free flight

where the rates due to the real scattering mechanisms are typically stored in a lookup table.

o A histogram is formed of the scattering rates, and a random number r is used as a pointer to select the right mechanism. This is schematically shown on the next slide.

EEEE popaciself

Choice of Scattering Event Terminating Free-Flight

We can make a table of the scattering processes at the energy of the particle at the scattering time:

itE121

321

4321

Self

54321

0

1

32

4

5

Selection process for scattering

0r

21 30EE2E3

E4

Look-up table of scattering rates:

Store the total scattering rates in a table for a grid in energy

………

Choice of the Final State After Scattering

Using a random number and probability distribution function

Using analytical expressions (slides that follow)

0 0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6x 10

4

Polar angle

Arb

itrar

y U

nits

1. Isotropic scattering processescos 1 2 , 2r r

2. Anisotropic scattering processes (Coulomb, POP)

kx

ky

kz

k0

0

Step 1:Determine 0 and 0

kx’

ky’

kz’

kStep 2:Assume rotatedcoordinatesystem

Step 3:perform scattering

0

2

0

21 1 2cos ,

rk k

k k

E E

E E

POP

2 2

2cos 11 4 (1 )D

rk L r

Coulomb

=2r for both

kx’

ky’

kz’

k

k’

Step 4:kxp = k’sin cos, kyp = k’sin*sin, kzp = k’cos

Return back to the original coordinate system:kx = kxpcos0cos0-kypsin0+kzpcos0sin0ky = kxpsin0cos0+kypcos0+kzpsin0sin0kz = -kxpsin0+kzpcos0

k’≠k forinelastic

Representative Simulation Results From Bulk Simulations - GaAs

k-vector

-valley

X-valley [100]L-valley [111]

Conduction bands

Valence bands

-valley table-Mechanism1-Mechanism2- …-MechanismN

L-valley table-Mechanism1-Mechanism2- …-MechanismNL

X-valley table-Mechanism1-Mechanism2- …-MechanismNx

Define scattering mechanisms for each valley

Call specified scattering mechanisms subroutines

Renormalize scattering tablesSimulation Results Obtained byD. Vasileska’s Monte Carlo Code.

parameters initializationreadin()

scattering table constructionsc_table()

histograms calculationhistograms()

Free-Flight-Scatterfree_flight_scatter()


write datawrite()

?

carriers initializationinit()

t t t Time t exceeds maximum simulation time tmax

yes

no

Optional

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy [eV]

Cum

ulat

ive

rate

[1/s

]

-6 -4 -2 0 2 4 6

x 108

0

5

10

15

20

25

30

35

40

Wavevector ky [1/m]

Arb

itrar

y U

nits

Initial Distribution of thewavevector along the y-axisthat is created with thesubroutine init()

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

10

20

30

40

50

60

70

80

90

Energy [eV]

Arb

itrar

y U

nits

Initial Energy Distribution createdwith the subroutine init()

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

10

1011

1012

1013

1014

1015

Energy [eV]

Sca

tterin

g R

ate

[1/s

]

acoustic

polar optical phonons

intervalley gamma to L

intervalley gamma to X

parameters initializationreadin()

scattering table constructionsc_table()


Free-Flight-Scatterfree_flight_scatter()


write datawrite()

?

carriers initializationinit()

t t t Time t exceeds maximum simulation time tmax

yes

no

Optional

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

10

1011

1012

1013

1014

1015

Energy [eV]

Sca

tterin

g R

ate

[1/s

]

acoustic

polar optical phonons

intervalley gamma to L

intervalley gamma to X

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy [eV]

Cum

ulat

ive

rate

[1/s

]

-6 -4 -2 0 2 4 6

x 108

0

5

10

15

20

25

30

35

40

Wavevector ky [1/m]

Arb

itrar

y U

nits

Initial Distribution of thewavevector along the y-axisthat is created with thesubroutine init()

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

10

20

30

40

50

60

70

80

90

Energy [eV]

Arb

itrar

y U

nits

Initial Energy Distribution createdwith the subroutine init()

Transient Data

0 1 2

x 10-11

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

time [s]

velo

city

[m/s

]

Steady-State Results

0 1 2 3 4 5 6 7

x 105

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5

Electric Field [V/m]

Drif

t Vel

ocity

[m/s

]

0 1 2 3 4 5 6 7

x 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Electric Field [V/m]

Con

duct

ion

Ban

d V

alle

y O

ccup

ancy gamma valley occupancy

L valley occupancy

X valley occupancy

k-vector

-valley

X-valley [100]L-valley [111]

Conduction bands

Valence bandsGunn Effect

Particle-Based Device Simulations

In a particle-based device simulation approach the Poisson equation is decoupled from the BTE over a short time period dt smaller than the dielectric relaxation time Poisson and BTE are solved in a time-marching

manner During each time step dt the electric field is

assumed to be constant (kept frozen)

Particle-Mesh CouplingThe particle-mesh coupling scheme consists of the following steps:

- Assign charge to the Poisson solver mesh- Solve Poisson’s equation for V(r) - Calculate the force and interpolate it to the particle locations - Solve the equations of motion:

r1 Ek krk

qdtdtE

dtd ;

Laux, S.E., On particle-mesh coupling in Monte Carlo semiconductor device simulation, Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, Volume 15, Issue 10, Oct 1996 Page(s):1266 - 1277

Assign Charge to the Poisson Mesh

1. Nearest grid point scheme

2. Nearest element cell scheme

3. Cloud in cell scheme

Force interpolation

The SAME METHOD that is used for the charge assignment has to be used for the FORCE INTERPOLATION:

,r r r r F Ei i p pp

q W

xp-1 xp xp+1

epxw

px

wp

ppep

pp

xVV

xVV 11

21

pE

Treatment of the Contacts

From the aspect of device physics, one can distinguish between the following types of contacts:

(1) Contacts, which allow a current flow in and out of the device

- Ohmic contacts: purely voltage or purely current controlled

- Schottky contacts

(2) Contacts where only voltages can be applied

Calculation of the Current

The current in steady-state conditions is calculated in two ways: By counting the total number of particles that

enter/exit particular contact By using the Ramo-Shockley theorem

according to which, in the channel, the current is calculated using

1

( ),N

xi

eI v idL

Current Calculated by Counting the Net Number of Particles Exiting/Entering a Contact

Electrons that naturally came out in time interval dt (N1)Electrons that were deleted (N2)Electrons that were injected (N3)

dq = q(N1+N2-N3), q(t+dt)=q(t) + dq, current equals the slope of q(t) vs. t

Source DrainGate

Mesh nodeElectronDopant

Device Simulation Results for MOSFETs: Current Conservation

0

1000

2000

3000

4000

5000

6000

7000

1 1.5 2 2.5 3

source contactdrain contact

net #

of e

lect

rons

exiti

ng/e

nter

ing

cont

act

time [ps]

WG = 0.5 mm

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150

Cur

rent

ID [

mA

/mm

]Distance [nm]

(b)

VG=1.4 V, VD=1 V

Cumulative net number of particlesEntering/exiting a contact for a 50 nmChannel length device

Drain contactSource contact

Current calculated using Ramo-Schockleyformula

1

( ),N

xi

eI v idL

X. He, MS, ASU, 2000.

Simulation Results for MOSFETs: Velocity and Enery Along the Channel

0

5x106

1x107

1.5x107

2x107

2.5x107

0 50 100 150

Drif

t vel

ocity

[cm

/s]

Distance [nm]

(a)

Mean Drift Velocity Along the Channel

0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150

Ave

rage

ene

rgy

[eV

]Distance [nm]

(b)

Average Kinetic Energy Along the Channel

VD = 1 V, VG = 1.2 V

Velocity overshoot effect observed throughout the whole channel length of the device – non-stationary transport.

For the bias conditions used average electron energy is smaller that 0.6 eV which justifies the use of non-parabolic band model.

Simulation Results for MOSFETs: IV Characteristics

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Dra

in c

urre

nt [

mA

/mm

]

Drain voltage VD [V]

1.2 V

0.8 V

1.0 V

Silvaco simulations

VG = 1.0 V

VG = 1.4 V

2D MCPS

VG = 1.4 V

The differences between the Monte Carlo and the Silvaco simulations are due to the following reasons:

• Different transport models used (non-stationary transport is taking place in this device structure).

• Surface-roughness and Coulomb scattering are not included in the theoretical model used in the 2D-MCPS.

X. He, MS, ASU, 2000.

Simulation Results For SOI MESFET Devices – Where are the Carriers?

Nd = 1019 Na =3-10x 1015 Nd = 1019

Si Substrate

Lg =60, 100nm Lg =60, 100nm

Oxide Layer

Nd = 1019 Nd = 3-10x1015 Nd =1019

Si Substrate

Oxide Layer

SOIMOSFET

SOIMESFET

Applications:

Low-power RF electronics.T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985).

Micropowercircuits basedon weakly inverted

Implantablecochlea and retina

Digital Watch

Pacemaker

MOSFETs

Proper Modeling of SOI MESFET Device

Gate current calculation: WKB Approximation Transfer Matrix Approach

for piece-wise linear potentials

Interface-Roughness: K-space treatment Real-space treatment

Goodnick et al., Phys. Rev. B 32, 8171 (1985)

107

108

109

1010

1011

1012

0.1 1 10 100 1000

Lg =25nm simulated resultsLg =50nm simulated resultsLg = 90nm simulated resultsLg = 0.6um Experimental resultsLg = 50nm Projected Experimental results

Drain Current Id [ µA/µm]

Cut

off F

requ

ency

f T [Hz]

Output Characteristics and Cut-off Frequency of a Si MESFET Device

-1000

-800

-600

-400

-200

0

200

400

-0.2 0 0.2 0.4 0.6 0.8 1

Vgs

= 0.3VV

gs = 0.4V

Vgs

=0.5VV

gs = 0.6V

I d [µA

/µm]

Vds

[Volts]

Tarik Khan, PhD, ASU, 2004.

Output Characteristics and Cut-off Frequency of a Si MESFET Device

-1000

-800

-600

-400

-200

0

200

400

-0.2 0 0.2 0.4 0.6 0.8 1

Vgs

= 0.3VV

gs = 0.4V

Vgs

=0.5VV

gs = 0.6V

I d [µA

/µm]

Vds

[Volts]107

108

109

1010

1011

1012

0.1 1 10 100 1000

Lg =25nm simulated resultsLg =50nm simulated resultsLg = 90nm simulated resultsLg = 0.6um Experimental resultsLg = 50nm Projected Experimental results

Drain Current Id [ µA/µm]

Cut

off F

requ

ency

f T [Hz]

Tarik Khan, PhD, ASU, 2004.

Modeling of SOI Devices

When modeling SOI devices lattice heating effects has to be accounted for

In what follows we discuss the following: Comparison of the Monte Carlo, Hydrodynamic

and Drift-Diffusion results of Fully-Depleted SOI Device Structures*

Impact of self-heating effects on the operation of the same generations of Fully-Depleted SOI Devices

*D. Vasileska. K. Raleva and S. M. Goodnick, IEEE Trans. Electron Dev., in press.

FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]

DD SRHD SRMonte Carlo

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


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%

Source/drain doping = 1020 cm-3 and 1019 cm-3 (series resistance (SR) case) Channel doping = 1E18 cm-3

Overshoot= (IDHD-IDDD)/IDDD (%) at on-state

Silvaco ATLAS simulations performed by Prof. Vasileska.

90 nm


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Drain Voltage [V]

Dra

in C

urre

nt [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

0.5

1

1.5

2

2.5

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


25 nm


0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Drain Voltage [V]

Dra

in C

urre

nt [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 10

0.5

1

1.5

2

2.5

3

Drain Voltage [V]

Dra

in C

urre

nt [m

A/u

m]


14 nm

FD-SOI Devices:Why Self-Heating Effect is Important?

1. Alternative materials (SiGe)2. Alternative device designs (FD SOI, DG,TG, MG, Fin-FET transistors

FD-SOI Devices:Why Self-Heating Effect is Important?

dS

~ 300nm

A. Majumdar, “Microscale Heat Conduction in Dielectric Thin Films,” Journal of Heat Transfer, Vol. 115, pp. 7-16, 1993.

Conductivity of Thin Silicon Films – Vasileska Empirical Formula

300 400 500 600

20

40

60

80

Temperature (K)

Ther

mal c

ondu

ctivi

ty (

W/m

/K)

experimental datafull lines: BTE predictionsdashed lines: empirical modelthin lines: Sondheimer

20nm

30nm50nm

100nm

/ 23

00

2( ) ( ) sin 1 exp cosh2 ( )cos 2 ( ) cos

a a zz T dT T

0( ) (300 / )T T

0 2

135( ) W/m/KTa bT cT

0 2 4 6 8 10105

7.5

10

12.5

15

17.5

2020

Distance from Si/gate oxide interface (nm)

Ther

mal c

ondu

ctivi

ty (

W/m

-K)

300K400K600K

SiBOX

10nm

0 20 40 60 80 1000102030405060708080

Distance from Si/gate oxide interface (nm)

Ther

mal c

ondu

ctivi

ty (

W/m

-K)

20nm30nm50nm100nm

Particle-Based Device Simulator That Includes Heating

Average and smooth: electron density, drift velocity and electron

energy at each mesh point

end of MCPS phase?

Acoustic and Optical Phonon Energy Balance Equations Solver

end of simulation?

end

no

yes

Define device structure

Generate phonon temperature dependent scattering tables

Initial potential, fields, positions and velocities of carriers

t = 0

t = t + t

Transport Kernel (MC phase)

Field Kernel (Poisson Solver)

t = n t?

yes

Average and smooth: electron density, drift velocity and electron

energy at each mesh point

end of MCPS phase?

end of MCPS phase?

Acoustic and Optical Phonon Energy Balance Equations Solver

end of simulation?

end of simulation?

end

no

yes




t = 0

t = t + t



t = n t?




t = 0

t = t + t



t = n t?t = n t?

yes

Heating vs. Different Technology Generation

25 nm FD SOI nMOSFET (Vgs=Vds=1.2V)

10 20 30 40 50 60 70

3

8


20 40 60 80 100 120

3

12


20 40 60 80 100 120 140 160 180

3

15

2727

300

400

500

300

400

500

300

400

500

source contact drain contact

source region drain region


50 100 150 200

3

20


50 100 150 200 250

3

21


50 100 150 200 250 300

3

23

4343

300

400

500

600

300400500

300

400

500


0 120 240 360360

4

28


50 100 150 200 250 300 350 400

4

33

6060

x (nm)


100 200 300 400 500

4

41

7676400

600

400

600

300400500600


10 20 30 40 50 60 70

3

8


20 40 60 80 100 120

3

12


20 40 60 80 100 120 140 160 180

3

15

2727 300

400

500

300

400500

300

400

500

600source contact drain contact


50 100 150 200

3

20


50 100 150 200 250

3

21


50 100 150 200 250 300

3

23

4343 300

400

500

600

400

600

300

400500600


50 100 150 200 250 300 350

4

28


50 100 150 200 250 300 350 400

4

33

6060

x (nm)


100 200 300 400 500

4

41

7676400

600

400

600

300400500600700

T=300K on gate T=400K on gateAcoustic Phonon Temperature Profiles

Higher Order Effects Inclusion in Particle-Based Simulators Degeneracy – Pauli Exclusion Principle

Short-Range Coulomb Interactions

Fast Multipole Method (FMM)V. Rokhlin and L. Greengard, J. Comp. Phys., 73, pp. 325-348 (1987).

Corrected Coulomb ApproachW. J. Gross, D. Vasileska and D. K. Ferry, IEEE Electron Device Lett. 20, No. 9, pp. 463-465 (1999).

P3M MethodHockney and Eastwood, Computer Simulation Using Particles.

Potential, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.

MOTIVATION

Length [nm]

100

110

120

130

140

150

Wid

th [n

m]

60 80 100 120 140

Current Stream Lines, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.

MOTIVATION

Discrete Impurities(Short-Range Interaction)

Efficient 3D PoissonEquation Solvers

3D Monte CarloTransport Kernel

Effective Potential (Space Quantization) 3DMCDS3DMCDS

Discrete Impurities(Short-Range Interaction)

Efficient 3D PoissonEquation Solvers

3D Monte CarloTransport Kernel

Effective Potential (Space Quantization) 3DMCDS3DMCDS

The ASU Particle-Based Device Simulator

(1) Corrected Coulomb Approach

(2) P3M Algorithm (3) Fast Multipole

Method (FMM)

(1) Ferry’s Effective Potential Method

(2) Quantum Field Approach

Statistical Enhancement: Event Biasing Scheme

Short-Range Interactions and Discrete/Unintentional

Dopants

Quantum Mechanical Size-quantization Effects

Boltzmann Transport Equations(Particle-Based Monte Carlo Transport Kernel)

Long-range Interactions(3D Poisson

Equation Solver)

Significant Data ObtainedBetween 1998 and 2002

MOSFETs - Standard Characteristics

0

50

100

150

200

250

300

350

40 60 80 100 120 140

VD=1.0 [V], V

G=1.0 [V]

VD=0.5 [V], V

G=1.0 [V]

Ele

ctro

n en

ergy

[m

eV]

Distance [nm]

LG= 80 nm

0

5x106

1x107

1.5x107

2x107

40 60 80 100 120 140

VD=0.5 [V], V

G=1.0 [V]

VD=1.0 [V], V

G=1.0 [V]

Drif

t vel

ocity

[cm

/s]

Distance [nm]

LG=80 nm

(a)

The average energy of the carriers increases when going from the source to the drain end of the channel. Most of the phonon scattering events occur at the first half of the channel.

Velocity overshoot occurs near the drain end of the channel. The sharp velocity drop is due to e-e and e-i interactions coming from the drain.

W. J. Gross, D. Vasileska and D. K. Ferry, "3D Simulations of Ultra-Small MOSFETs with Real-Space Treatment of the Electron-Electron and Electron-Ion Interactions," VLSI Design, Vol. 10, pp. 437-452 (2000).

MOSFETs - Role of the E-E and E-I

0

100

200

300

400

100 110 120 130 140 150 160 170 180

with e-e and e-imesh force only

Ele

ctro

n en

ergy

[m

eV]

Length [nm]

VD=1 V, V

G=1 V

channel drain

0

100

200

300

400

500

600

700

800

0.12 0.13 0.14 0.15 0.16 0.17 0.18

Ene

rgy

[meV

]

Length [nm]

0

100

200

300

400

500

600

700

800

0.12 0.13 0.14 0.15 0.16 0.17 0.18

Length [nm]

Ene

rgy

[meV

]mesh forceonly

with e-e and e-i

Individual electron trajectories over

time

-1x107

-5x106

0

5x106

1x107

1.5x107

2x107

2.5x107

0 40 80 120 160

with e-e and e-imesh force only

Drif

t vel

ocity

[cm

/s]

Length [nm]

VD=V

G=1.0 V

source drainchannel

MOSFETs - Role of the E-E and E-I Mesh force onlyMesh force only With With ee--ee and and ee--ii

Short-range e-e and e-i interactions push someof the electrons towards higher energies

10-3

10-2

0 50 100 150 200 250 300 350 400

sourcechanneldrain

Ele

ctro

n di

strib

utio

n(a

rb. u

nits

)Energy [meV]

VG=0.5 V, V

D=0.8 V

10-3

10-2

0 50 100 150 200 250 300 350 400

SourceChannelDrain

Ele

ctro

n di

strib

utio

n(a

rb. u

nits

)

Energy [meV]

VG=0.5 V, V

D=0.8 V

D. Vasileska, W. J. Gross, and D. K. Ferry, "Monte-Carlo particle-based simulations of deep-submicron n-MOSFETs with real-space treatment of electron-electron and electron-impurity interactions," Superlattices and Microstructures, Vol. 27, No. 2/3, pp. 147-157 (2000).

Degradation of Output Characteristics

0

10

20

30

40

50

60

70

80

0.0 0.2 0.4 0.6 0.8 1.0 1.2

with corrected Coulombmesh force only

Dra

in c

urre

nt I D

[mA

]


increasing VG

LG = 35 nm, WG = 35 nm, NA = 5x1018 cm-3, Tox = 2 nm, VG = 11.6 V (0.2 V)

The short range e -e and e -i interactions have significant influence on the device output characteristics.

There is almost a factor of two decrease in current when these two inte-raction terms are considered.

LG = 50 nm, WG = 35 nm, NA = 5x1018 cm-3

Tox = 2 nm, VG = 11.6 V (0.2 V)

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2

with corrected Coulombmesh force only

Dra

in c

urre

nt I D [

mA]


increasing VG

W. J. Gross, D. Vasileska and D. K. Ferry, "Ultra-small MOSFETs: The importance of the full Coulomb interaction on device characteristics," IEEE Trans. Electron Devices, Vol. 47, No. 10, pp. 1831-1837 (2000).

Mizuno result:(60% of the fluctuations)

Stolk et al. result:(100% of the fluctuations)

Fluctuations in thesurface potential

Fluctuations in theelectric field

Depth-Distributionof the charges

MOSFETs - Discrete Impurity Effects

0

20

40

60

80

100

0 1 2 3 4 5

sVth

=> approach 1s

Vth => approach 2

sVth

=> our simulation results

s Vth [

mV

]

Oxide thickness Tox

[nm]

05

1015

2025

3035

40

1x1018 3x1018 5x1018 7x1018

sVth

=> approach 1s

Vth => approach 2

sVth


s Vth [

mV

]

Doping density NA [cm-3]

10

20

30

40

50

60

20 40 60 80 100 120 140

sVth

=> approach 1s

Vth => approach 2

sVth


s Vth

[mV

]Device width [nm]

effeff

A

ox

ox

ABSi

BBSiVth WL

NTNqq/Tkq 44 3

434

sApproach 2 [2]:

si

ABB

effeff

A

ox

oxBSiVth n

Nln

qTk

;WLNTq

44 3

2Approach 1 [1]:

[1] T. Mizuno, J. Okamura, and A. Toriumi, IEEE Trans. Electron Dev. 41, 2216 (1994).

[2] P. A. Stolk, F. P. Widdershoven, and D. B. Klaassen, IEEE Trans. Electron Dev. 45, 1960 (1998).

020406080

100120140160180200

160

170

180

190

200

210

220

230

240

250

260

270

Number of Atoms in Channel

Num

ber o

f Dev

ices

5 samples at maximum

5 samples at minimum

5 samples of average

Depth Correlation of sVT To understand the role that the position

of the impurity atoms plays on the threshold voltage fluctuations, statistical ensembles of 5 devices from the low-end, center and the high-end of the distribution were considered.

Significant correlation was observed between the threshold voltage and the number of atoms that fall within the first 15 nm depth of the channel.

0.9

1

1.1

1.2

1.3

1.4

160 180 200 220 240 260 280 300

Thre

shol

d vo

ltage

[V]

Number of channel dopant atoms

low-end

center

high-end

(a)LG=50 nm, W

G=35 nm

NA=5x1018 cm-3, T

ox=3 nm

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

V T cor

rela

tion

Depth [nm]

depth

Moving slab

rangeND ND

(b) LG=50 nm, W

G=35 nm, T

ox=3 nm

NA=5x1018 cm-3

Number of atoms in the channel

Num

ber o

f dev

ices


Depth [nm]

VT

corr

elat

ion

Thre

shol

d vo

ltage

[V

]

Fluctuations in High-Field Characteristics

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

average velocitycorrelationdrain currentcorrelation

Cor

rela

tion

Depth [nm]

(c)

LG=50 nm

WG=35 nm

Tox

=3 nm

NA=5x1018 cm-3

Impurity distribution in the channel also affects the carrier mobility and saturation current of the device.

Significant correlation was observed between the drift velocity (saturation current) and the number of atoms that fall within the first 10 nm depth of the channel.

0

5 106

1 107

1.5 107

160 180 200 220 240 260 280

Drif

t vel

ocity

[cm

/s]


(a)low-end

center

high-endV

G=1.5 V, V

D=1 V

LG=50 nm, W

G=35 nm

NA=5x1018 cm-3

0

5

10

15

20

160 180 200 220 240 260 280

Dra

in c

urre

nt [

mA]


low-end

center

high-end

(b)


Drif

t vel

ocity

[cm

/s] Number of atoms in the channel

Depth [nm]

Cor

rela

tion

Dra

in c

urre

nt [

mA] VG = 1.5 V, VD = 1 V

Current Issues in NovelDevices – Unintentional Dopants

THE EXPERIMENT …

Results for SOI DeviceSize Quantization Effect (Effective Potential):

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

2 4 6 8 10 12 14 16Channel Width [nm]

Thre

shol

d Vo

ltage

[V]

Experimental

Simulation

S. S. Ahmed and D. Vasileska, “Threshold voltage shifts in narrow-width SOI devices due to quantum mechanical size-quantization effect”, Physica E, Vol. 19, pp. 48-52 (2003).

Results for SOI Device

Due to the unintentional dopant both the electrostatics and the transport are affected.

-10000

10000

30000

50000

70000

90000

110000

130000

0 20 40 60 80Distance Along the Channel [nm]

Aver

age

Velo

city

[m/s

]0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Aver

age

Kin

etic

Ene

rgy

[eV]

VelocityEnergy

Dip due to the presence of the impurity. This affects the transport of the carriers.


Unintentional Dopant:

D. Vasileska and S. S. Ahmed, “Narrow-Width SOI Devices: The Role of Quantum Mechanical Size Quantization Effect and the Unintentional Doping on the Device Operation”, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.


Channel Width = 10 nmVG = 1.0 VVD = 0.1 V

32.54%

47.62%

33.01%

26.98%

42.85%

27.18%

11.90%

26.19%

11.11%

01

23

45

67

89

10

0 10 20 30 40 50Distance Along the Channel [nm]

Dis

tanc

e A

long

the

Wid

th [n

m]

.

Sour

ce

Drai

n34.13%

47.62%

42.06%

16.66%

42.85%

26.19%

9.93%

26.19%

19.46%

0

1

2

3

4

5

6

70 10 20 30 40 50

Distance Along the Channel [nm]

Dis

tanc

e A

long

the

Dep

th [n

m]

.

Sour

ce

Dra

in


86.30%

96.76%

86.52%

87.39%

96.09%

86.96%

69.57%

88.26%

67.39%

0

1

2

3

4

5


Dis

tanc

e A

long

the

Wid

th [n

m]

Sou

rce

Dra

in81.09%

96.76%

88.48%

79.78%

96.09%

88.26%

59.78%

88.26%

76.09%

0

1

2

3

4

5

6

70 10 20 30 40 50


Dis

tanc

e A

long

the

Dep

th [n

m]

Sou

rce

Dra

in

Channel Width = 5 nmVG = 1.0 VVD = 0.1 V


Impurity located at the very source-end, due to the availability of Increasing number of electrons screening the impurity ion, has reduced impact on the overall drain current.

0%

10%

20%

30%

40%

50%

60%


Cur

rent

Red

uctio

n

Impurity position varying along the center of the channel

V G = 1.0 VV D = 0.2 V

Source end Drain end


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

2 4 6 8 10 12 14 16Channel Width [nm]

Thre

shol

d Vo

ltage

[V]

ExperimentalSimulation (QM)Discrete single dopants

D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.

S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 – 471.

D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

Results for SOI DeviceElectron-Electron Interactions:

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

0 0.2 0.4 0.6 0.8 1

Electron Kinetic Energy [eV]

Dis

tribu

tion

Func

tion

[a.u

.]

PMFMM

V G = 1.0 VV D = 0.3 V

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

0 20 40 60 80 100


Ele

ctro

n V

eloc

ity [m

/s]

PMFMM

V G = 1.0 VV D = 0.3 V

Source Drain

D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.

S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 – 471.

D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

Summary

Particle-based device simulations are the most desired tool when modeling transport in devices in which velocity overshoot (non-stationary transport) exists

Particle-based device simulators are rather suitable for modeling ballistic transport in nano-devices

It is rather easy to include short-range electron-electron and electron-ion interactions in particle-based device simulators via a real-space molecular dynamics routine

Quantum-mechanical effects (size quantization and density of states modifications) can be incorporated in the model quite easily with the assumption of adiabatic approximation and solution of the 1D or 2D Schrodinger equation in slices along the channel section of the device

Semi-Classical Transport Theory

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Transcript of Semi-Classical Transport Theory