monte carlo methods for electron transport

MONTE CARLO METHODSFOR ELECTRON TRANSPORT

Mark J. KushnerUniversity of Illinois

Department of Electrical and Computer Engineering1406 W. Green St.

Urbana, IL 61801 USA217-244-5137 [email protected] http://uigelz.ece.uiuc.edu

May 2002

MCSHORT_02_00

University of IllinoisOptical and Discharge Physics

MONTE CARLO METHODS FOR ELECTRON TRANSPORT

• The Monte Carlo (MC) method was developed during WWII for analysis of neutron moderation and transport.

• MC methods enable direct simulation of complex physical phenomena which may not be amenable to conventional PDE analysis.

• The method relies upon knowledge of probability functions for the phenomena of interest to statistically (randomly) select occurrences of events whose ensemble average is “the answer”.

• These methods are extensively used in simulating electron transport do obtain, for example, electron energy distributions.

MCSHORT_02_01


EXAMPLE: ELECTRON ENERGY DISTRIBUTION IN ICP

• Inductively Coupled Plasma: Ar, 10 mTorr, 6.78 MHz

MCSHORT_02_31

• EED at r= 4.5 cm vsDistance from Window

• Electric field (overlay) and ion density (max = 1.7 x 1011 cm-3)


MONTE CARLO METHOD REFERENCES

• J. P. Boeuf and E. Marode, J. Phys. D 15, 2169 (1982)• G. L. Braglia, Physica 92C, 91 (1977)• S. R. Hunter, Aust. J. Phys. 30, 83 (1977)• S. Lin and J. Bardsley, J. Chem. Phys 66, 435 (1977)• S. Longo, Plasma Source Science Technol. 9, 468 (2000)• J. Lucas, Int. J. Electronics 32, 393 (1972)• J. Lucas and H. T. Saelee, J. Phys. D 8, 640 (1975)• K. Nanbu, Phys. Rev E 55, 4642 (1997)• M. Yousfi, A. Hennad and A. Alkaa, Phys Rev E 49, 3264 (1994)

• Computational Science and Engineering Projecthttp://csep1.phy.ornl.gov, http://csep1.phy.ornl.gov/CSEP/MC/MC.html

MCSHORT_02_02


BASICS OF THE MONTE CARLO METHOD: p(x)

• A physical phenomenon has a known probability distribution function p(x) which, for example, gives the probability of an event occurring at position x.

MCSHORT_02_03

0.0

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p(x)

x

∫∞

=0

1dxxp )(


BASICS OF THE MONTE CARLO METHOD: P(x)

• The cumulative probability distribution function P(x) is the likelihood that an event has occurred prior to x.

MCSHORT_02_04

1P00P

dxxpxPx

o

=∞=

= ∫)(,)(

')'()(

• Since p(x) is always positive, there is a 1-to-1 mapping of r=[0,1] onto P(x = 0 →x = ∞).

0.0

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0.0

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1.0

0 2 4 6 8 10

p(x)

P(x

)x

P(x)

p(x)


RANDOM USE OF P(x) TO REGENERATE p(x)

• By randomly choosing “product values” of P(x) (distributed [0,1]) and binning the occurrences of the argument x, we reproduce p(x).

MCSHORT_02_05

)(rPx 1−=

• The function which, given a random number r=[0,1], provides a randomly selected value of x is:

0.0

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1.0

0 2 4 6 8 10

P(x

)

x


EXAMPLE: RANDOM P(x) TO REGENERATE p(x)

MCSHORT_02_06

],[)ln()(

)exp()(

)exp()(

10rr1xrPx

xx1xP

xx

x1xp

1

=−⋅∆−==

∆−−=

∆−

∆=

−

• WARNING!!! In practical problems, p(x) cannot be analytically integrated for P(x) and/or P(x) cannot be analytically inverted for P-1(x). These operations must be done numerically.

0

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0

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1

0 2 4 6 8 10

p(x)

P(x)

x

p(x)

P(x) 2x =∆


EXAMPLE: RANDOM P(x) TO REGENERATE p(x)

MCSHORT_02_07

• p(x) is reproduced within random statistical error (n-1/2=0.01).

ibins=100itrials=10000deltax=2xmax=10.dx=xmax/ibinsynorm=0.do i=1,itrialsr = random(iseed)x=-deltax*alog(1.-r)ibin=x/dxynorm=ynorm+dxy(ibin)=y(ibin)+1.

end dodo i=1,ibinsy(i)=y(i)/ynorm

end do

0

0.1

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0.5

0 2 4 6 8 10

p(x)regen p(x)

p(x)

x


ELECTRON SCATTERING

MCSHORT_02_08

• An electron with energy ε collides with an atom with differential cross section

providing the likelihood of scattering into the solid angle centered on .

Note: Typically only the explicit dependence on polar angle θis considered. Scattering with azimuthal angle ϕ is usually assumed to be uniform.

( )φθεσ ,,

( )φθ ,


DIFFERENTIAL SCATTERING

MCSHORT_02_09

• for real atomsand molecules can be quite complex (C2F6)

Christophorou, J. Chem. Phys.Ref. Data 27, 1, (1998)

• Accounting for forward scattering at higher energies (> 10s eV) is very important in simulating electron transport.

• Assuming Isotropic scattering in the polar direction yields:

( )θεσ ,

( ) ( ) ( ) ( )1r2d21P

0 oo

o −=== ∫ acos,''sin, θθθσσ

θσθσθ

1. Determine Eularian angles of 2. Rotate frame by so z-, x-axes align with3. Rotate by to yield direction of 4. Account for change in speed5. Rotate frame by to original orientation


COLLISION DYNAMICS

MCSHORT_02_10

• To account for the change in velocity of an electron following a collision:

initialvr

( )φθ , finalvr

( )αβ ,

( )βα −− ,e

2initial

2

final m2vv /ε∆−=rr

( )αβ ,initialvr

initialvr


COLLISION DYNAMICS

MCSHORT_02_11

• End result is the “scattering matrix” which transforms initial velocity to final velocity:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−+−+

⋅=

θαφθαφθβθαβφθαβφθβθαβφθαβ

coscoscossinsinsinsincoscossinsincossincossinsinsinsincossincoscossincoscos

finalfinal vv rr


EXAMPLE: ELECTRON SWARM

MCSHORT_02_12

• A swarm of electrons drifts in a uniform electric field in a gashaving a constant elastic collision frequency and isotropic collisions. What is the average drift velocity?

• For constant collision frequency ν, the randomly selected time between collisions is:

• The change in energy in an elastic collision is

)( r1alog1t −⋅−=∆ν

( )( )⎟⎠⎞

⎜⎝⎛ −−=∆ θε cos1

Mm21 e


EXAMPLE: PROGRAM DRIFT

MCSHORT_02_13a



MCSHORT_02_13b



MCSHORT_02_13c


EXAMPLE: ELECTRON SWARM

MCSHORT_02_14

• Collision frequency = 1.048 x 109 s-1

• Drift distance = 20 cm• E/N (Electric field/gas number density) = 1-10 x 10-17 V-cm3• Electron particles=50-500 per E/N

0

1 107

2 107

3 107

4 107

5 107

0 2 4 6 8 10

50 Particles100 Particles500 Particles

DR

IFT

SPE

ED

(cm

/s)

E/N (10-17 V-cm2)

• Real atoms/molecules have many electron collision processes (elastic, vibrational excitation, electronic excitation, ionization) with separate differential cross sections.

• These processes can be statistically accounted for using MC techniques

C2F6


MUTIPLE COLLISIONS

MCSHORT_02_15

Christophorou, J. Chem. Phys. Ref. Data 27, 1, (1998)


MODEL CROSS SECTIONS

MCSHORT_02_16

0

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0 5 10 15 20 25 30 35 40

Cro

ss S

ectio

n (1

0-16

cm2 )

Electron Energy (eV)

Elastic

ElectronicExcitation

Ionization

• Compute collision frequencies for each process j having collision partner density Nj,

( ) ( ) ( ) ( )ενενεσεεν ∑=⎟⎟⎠

⎞⎜⎜⎝

⎛=

jjtotaljj

21

ej N

m2 ,

/

0 100

5 109

10 109

15 109

20 109

0 5 10 15 20 25 30 35 40

Col

lisio

n Fr

eque

ncy

(s-1

)


Elastic

Electronic

Ionization

Total


CUMULATIVE COLLISION PROBABILITY

MCSHORT_02_17

• Cumulative collision probability is sum of probability of experiencing “yours” and all “previous” collisions. (Note: Order of summation is not important.)

( )( )

( )εν

ενε

total

j

1jij

jp∑

−==

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1.0

0 5 10 15 20 25 30 35 40

Cum

mul

ativ

e C

ollis

ion

Pro

babi

lity


Elastic

Electronic

Ionization


COLLISION SELECTION PROCESS

MCSHORT_02_18

• Choose time between collisions based on total collision frequency.

( ) )log( 1total

r11t −−

=∆εν

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1.0

0 5 10 15 20 25 30 35 40

Cum

mul

ativ

e C

ollis

ion

Pro

babi

lity


Elastic

Electronic

Ionization0.4

0.6

0.29

0.13

0.58

( ) ( )εε j21j prp ≤<−

• The collision which occurs is that which satisfies


NULL COLLISION FREQUENCY

MCSHORT_02_19

• The electron energy and collision frequency can change during the free flight between collisions.

• There is an ambiguity in choosing the time between collisions.

• The ambiguity is eliminated by the “null collision frequency” (NCF).

0 100

5 109

10 109

15 109

20 109

0 5 10 15 20 25 30 35 40

Col

lisio

n Fr

eque

ncy

(s-1

)


Elastic

Electronic

Ionization

Total

FREEFLIGHT


NULL COLLISION FREQUENCY

MCSHORT_02_20

• The NCF is a fictitious process used to make it appear that all energies have the same collision frequency.

( ) ( )[ ] ( )εεε totaltotalnull vvMaxv −=

0 100

5 109

10 109

15 109

20 109

0 5 10 15 20 25 30 35 40

Col

lisio

n Fr

eque

ncy

(s-1

)


Elastic Electronic

Ionization

Total w/ Null

Null

Total w/o Null

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Cum

ulat

ive

Col

lisio

n P

roba

bilit

y


Elastic

Electronic

IonizationNull

Null

• Collision frequencies with Null.

• Cumulative Probabilities with Null.


COLLISION SELECTION PROCESS WITH NULL

MCSHORT_02_21

• Choose time between collisions based on maximum total collision frequency.

( )[ ] )log( 1total

r1Max

1t −−

=∆εν

( ) ( )εε j21j prp ≤<−

• The collision which occurs is that which satisfies.

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Cum

ulat

ive

Col

lisio

n P

roba

bilit

y


Elastic

Electronic

IonizationNull

Null

0.34

0.25

0.41

• If the null is chosen, disregard the collision. Allow the electron to proceed to the next free flight without changing its velocity.


SPATIALLY VARYING COLLISION FREQUENCY

MCSHORT_02_22

• [e] and Cl2 densities in an ICP for etching (Ar/Cl2=80/20, 15 mTorr)

• In many of the systems of interest, the density of the collisionpartner depends on position and time.

• The choice of can be ambiguous.

( ) ( ) ( )txNm2tx jj

21

ej ,,,

/

εσεεν ⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )εν total


EXTENSION OF NULL METHOD TO ACCOUNT FOR N(x,t)

MCSHORT_02_23

• Sample time/space domain to determine .• Compute using .

( )[ ] ( )

( ) ( )

( )( )[ ]

( ) ( )MatrixScattering

rrFscattering

trajectorynextNulltxNMax

txNr

trajectorynextNullj

prpCollisiontavvtvxx

r1xMax

1tTimestep

541

j

j3

j21j

1total

•=•

→→>•

→=•

≤<•∆⋅+=∆⋅+=•

−−

=∆•

−

−

,,

,,

:,

log,

:

φθ

εε

εν

( )],[ txNMax j

( )],[ txNMax j( ) ( ) ( )[ ] ( )ενενενεν nulltotaltotalj Max ,,,


SAMPLING AND INTEGRATION METHODS

MCSHORT_02_24

• Electron distributions are obtained by sampling the particle trajectories; binning particles by energy, velocity, position toobtain .

• How you sample affects the distribution function you derive.• Integration ∆t should be less than: ∆tcol, fraction of 1/νrf,

fraction of or other constraining frequencies. • ∆t can be different for each particle. Particles can “diverge in

time” until they reach a time when they must be coincident. • Recommended sampling and integration strategy:

•Choose t(next collision) = t(last collision) + ∆tcol•Integrate using ∆t ≤ t- t(next collision) •Sample particles for every ∆t weighting the contribution by ∆t .•When reach t(next collision), collide and choose new ∆tcol

( ) ( )trvfortrf ,,,, rrrε

[ ] 1dxdE

Ev −


EXAMPLE: ELECTRON ENERGY DISTRIBUTION

MCSHORT_02_25

• Compute electron energy distribution and rate coefficients for idealized cross sections.

• Conditions:

• E/N: 100 x 10-17 V-cm2 (100 Td)• Drift distance: 3 cm (sample after

0.5 cm)• Number of Particles: 2000 0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Cro

ss S

ectio

n (1

0-16

cm2 )


Elastic

ElectronicExcitation

Ionization



MCSHORT_02_26

• Sampling method• 1: Every ∆tcol• 2: Every ∆t (constant) < ∆tcol

• Rate Coefficients (cm3/s)•Elastic 1.1 x 10-7

•Electronic 2.0 x 10-9

•Ionization 9.9 x 10-14

• Number of Collisions:•Elastic 1.46 x 107

•Electronic 2.21 x 105

•Ionization 10•Null 5.93 x 107

• Lesson!!! Do NOT compute rate coefficients by counting collisions! Directly compute rate coefficients from EED.

( ) ( ) ( ) εεεσεε dm2trftrk 21

21

e

//

,,, ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

rr

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20

F(e) Sampling 1F(e) Sampling 2

Ele

ctro

n E

nerg

y D

istri

butio

n (e

V-3

/2)

Energy (eV)



MCSHORT_02_27

• Required samplings are dictated by the tail of the EED. Rate coefficients for high threshold events are sensitive to the tail.

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20

50 Particles100 Particles500 Particles2000 Particles

Ele

ctro

n E

nerg

y D

istri

butio

n (e

V-3

/2)

Energy (eV)

10-8

10-7

10-6

10-5

10-4

10-3

12 13 14 15 16 17 18 19

50 Particles100 Particles500 Particles2000 Particles

Ele

ctro

n E

nerg

y D

istri

butio

n (e

V-3

/2)

Energy (eV)


EFFICIENCY ISSUES

MCSHORT_02_28

• Create look-up tables where-ever possible (memory and lookups are cheap, computations are expensive).

• Minimize null-collisions by having sub-intervals of energy range with different .

• Be cognizant of “pipelining” opportunities. Perform array operations with stencils to include-exclude indices for particles which are added-removed due to attachment, losses to walls or ionization.

• Take advantage of cyclic conditions to bin particles by phase as opposed to time.

• NEVER hardwire anything!! Define all cross sections, densities from “outside.”

( )[ ]εν totalMax

ADVANCED TOPICS


TYPICAL INDUCTIVELY COUPLED PLASMA FOR ETCHING

• Power is coupled into the plasma by both inductive and capacitive routes.

EIND_0502_04


WALK THROUGH: Ar/Cl2 PLASMA FOR p-Si ETCHING

EIND_0502_05

• The inductively coupled electromagnetic fields have a skin depth of 3-4 cm.

• Absorption of the fields produces power deposition in the plasma.

• Electric Field (max = 6.3 V/cm)

• Ar/Cl2 = 80/20• 20 mTorr• 1000 W ICP 2 MHz• 250 V bias, 2 MHz (260 W)


Ar/Cl2 ICP: POWER AND ELECTRON TEMPERATURE

EIND_0502_06

• ICP Power heats electrons, capacitively coupled power dominantly accelerates ions.

• Ar/Cl2 = 80/20, 20 mTorr, 1000 W ICP 2 MHz,250 V bias, 2 MHz (260 W)

• Power Deposition (max = 0.9 W/cm3) • Electron Temperature (max = 5 eV)


Ar/Cl2 ICP: IONIZATION

EIND_0502_07

• Ionization is produced by bulk electrons and sheath accelerated secondary electrons.


• Beam Ionization(max = 1.3 x 1014 cm-3s-1)

• Bulk Ionization(max = 5.4 x 1015 cm-3s-1)


Ar/Cl2 ICP: POSITIVE ION DENSITY

EIND_0502_08

• Diffusion from the remote plasma source produces uniform ion densities at the substrate.


• Positive Ion Density(max = 1.8 x 1011 cm-3)

MATCH BOX-COIL CIRCUIT MODEL

ELECTRO- MAGNETICS

FREQUENCY DOMAIN

ELECTRO-MAGNETICS

FDTD

MAGNETO- STATICS MODULE

ELECTRONMONTE CARLO

SIMULATION

ELECTRONBEAM MODULE

ELECTRON ENERGY

EQUATION

BOLTZMANN MODULE

NON-COLLISIONAL

HEATING

ON-THE-FLY FREQUENCY

DOMAIN EXTERNALCIRCUITMODULE

PLASMACHEMISTRY

MONTE CARLOSIMULATION

MESO-SCALEMODULE

SURFACECHEMISTRY

MODULE

CONTINUITY

MOMENTUM

ENERGY

SHEATH MODULE

LONG MEANFREE PATH

(MONTE CARLO)

SIMPLE CIRCUIT MODULE

POISSON ELECTRO- STATICS

AMBIPOLAR ELECTRO- STATICS

SPUTTER MODULE

E(r,θ,z,φ)

B(r,θ,z,φ)

B(r,z)

S(r,z,φ)

Te(r,z,φ)

µ(r,z,φ)

Es(r,z,φ) N(r,z)

σ(r,z)

V(rf),V(dc)

Φ(r,z,φ)

Φ(r,z,φ)

s(r,z)

Es(r,z,φ)

S(r,z,φ)

J(r,z,φ)ID(coils)

MONTE CARLO FEATUREPROFILE MODEL

IAD(r,z) IED(r,z)

VPEM: SENSORS, CONTROLLERS, ACTUATORS


HYBRID PLASMA EQUIPMENT MODEL

SNLA_0102_39


ELECTROMAGNETICS MODEL

EIND_0502_10

• The wave equation is solved in the frequency domain using sparsematrix techniques (2D,3D):

• Conductivities are tensor quantities (2D,3D):

( ) ( )t

JEtEEE

∂+⋅∂

+∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∇⋅∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅∇∇−

σεµµ

1 12

2

)))((exp()(),( rtirEtrE rrrrrϕω +−′=

( )m

e2

om

2z

2zrzr

zr22

rz

zrrz2r

2

22

mo

mnq

mqiEj

BBBBBBBBBBBBBBBBBBBBB

B

1q

m

νσνωασ

ααααααααα

αανσσ

θθ

θθθ

θθ

=+

=⋅=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++−+−+++−+−++

⎟⎠⎞⎜

⎝⎛ +

=

,/

rv

r


ELECTROMAGNETICS MODEL (cont.)

EIND_0502_11

• The electrostatic term in the wave equation is addressed using aperturbation to the electron density (2D).

• Conduction currents can be kinetically derived from the ElectronMonte Carlo Simulation to account for non-collisional effects (2D).

⎟⎠⎞

⎜⎝⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⋅⋅−∇=∆

∆==⋅∇ ω

τσ

εερ i

qEnnqE e

e 1/ ,

( ) ( ) ( )( )( ) ( ) ( ) ( )( )( )rtirvrqnrtirJtr veevorrrrrrrr φωφω +−=+= expexp,J e


ELECTRON ENERGY TRANSPORT

where S(Te) = Power deposition from electric fieldsL(Te) = Electron power loss due to collisionsΦ = Electron fluxκ(Te) = Electron thermal conductivity tensorSEB = Power source source from beam electrons

• Power deposition has contributions from wave and electrostatic heating.

• Kinetic (2D,3D): A Monte Carlo Simulation is used to derive including electron-electron collisions using electromagnetic fields from the EMM and electrostatic fields from the FKM.

SNLA_0102_41

( ) ( ) ( ) EBeeeeeee STTkTTLTStkTn +⎟⎠⎞

⎜⎝⎛ ∇⋅−Φ⋅∇−−=∂⎟

⎠⎞

⎜⎝⎛∂ κ

25/

23

( )trf ,, rε

• Continuum (2D,3D):


PLASMA CHEMISTRY, TRANSPORT AND ELECTROSTATICS

• Continuity, momentum and energy equations are solved for each species (with jump conditions at boundaries) (2D,3D).

AVS01_ 05

• Implicit solution of Poisson’s equation (2D,3D):

( ) ( )⎟⎠

⎞⎜⎝

⎛⋅∇⋅∆+=∆+Φ∇⋅∇ ∑∑

iiqt-- i

iiis Nqtt φρε

r

iiii SN

tN

+⋅−∇= )v( r

∂∂

( ) ( ) ( ) ( ) iii

iiiiiii

i

ii BvEmNqvvNTkN

mtvN µ

∂∂

⋅∇−×++⋅∇−∇=rrrrr

r 1

( ) ijjijj

imm

j vvNNm

ji

νrr−−∑

+

( ) 222

2

)()U(UQ E

mqNNP

tN

ii

iiiiiiiii

ii

ωννε

∂ε∂

+=⋅∇+⋅∇+⋅∇+

∑∑ ±−+

++j

jBijjij

ijBijjiji

ijs

ii

ii TkRNNTTkRNNmm

mE

mqN 3)(32

2

ν


FORCES ON ELECTRONS IN ICPs

EIND_0502_09

• Inductive electric field provides azimuthal acceleration; penetrates(1-3 cm)

• Electrostatic (capacitive); penetrates (100s µm to mm)

• Non-linear Lorentz Force rfBvFrr

×= θ

( )( ) 212

eeDDS en8kT10 πλλλ =≈ ,

( )( ) 212

eoe nem µδ =

Eθ

Es

BR, BZ

δ

λs

vθ

BR

vθ x BR

• Collisional heating:

• Anomalous skin effect:

• Electrons receive (positive) and deliver (negative) power from/to the E-field.

• E-field is non-monotonic.


ANAMOLOUS SKIN EFFECT AND POWER DEPOSITION

EIND_0502_12

Ref: V. Godyak, “Electron Kinetics of Glow Discharges”

( ) ( )BvF

dtrdtrEttrrtrJe

skinmfp

rvr

rrrrrrr

×=

=

>

∫∫ ''','',,',),( σ

δλ

( ) ( )trEtrtrJeskinmfp ,,),(, rrrrrσδλ =<


COLLISIONLESS TRANSPORT ELECTRIC FIELDS

• We capture these affects by kinetically deriving electron current.

• Eθ during the rf cycle exhibits extrema and nodes resulting from this non-collisional transport.

• “Sheets” of electrons with different phases provide current sources interfering or reinforcing the electric field for the next sheet.

• Axial transport results fromforces.

EIND_0502_13• Ar, 10 mTorr, 7 MHz, 100 W

rfBvrr

×

( ) ( )( )( ) ( )( )okk

kk

o

ttirvq

dAttirj

−

=⋅−

∑∫

ω

ω

exp

exprr

rr

ANIMATION SLIDE


POWER DEPOSITION: POSITIVE AND NEGATIVE

• The end result is regions of positive and negative power deposition.

SNLA_0102_19

• Ar, 10 mTorr, 7 MHz, 100 W

POSITIVE

NEGATIVE


POWER DEPOSITION vs FREQUENCY

• The shorter skin depth at high frequency produces more layers ofnegative power deposition of larger magnitude.

SNLA_0102_32

• 6.7 MHz(5x10-5 – 1.4 W/cm3)

• 13.4 MHz(8x10-5 – 2.2 W/cm3)• Ar, 10 mTorr, 200 W

• Ref: Godyak, PRL (1997)

MAXMIN


TIME DEPENDENCE OF EEDs: FOURIER ANALYSIS

• To obtain time dependent EEDs, Fourier transforms are performed “on-the-fly” in the Electron Monte Carlo Simulation.

• As electron trajectories are integrated, complex Fourier coefficients and weightings are incremented by.

EIND_0502_15

⎟⎟⎠

⎞⎜⎜⎝

⎛ΨΨ

=Ψ

= −

)],(Re[)],(Im[tan),(,|),(|),( 1

kin

kinkin

ik

kinkin r

rrW

rrA r

rr

rr

εεεφεε

( ) ( ) ,))(())((exp, 21

21 ∑∑ −∆±−∆±∆=∆Ψ ++

kjklkljii

jjjjkin rrrtintwr rrrr δεεεδωε

∑∑ −∆±∆=∆ ++k

jklkljj

jjk rrrtwW ))(( 21 rrrδ

• The Fourier coefficients are then obtained from:


TIME DEPENDENCE OF EEDs: FOURIER ANALYSIS

• The time dependence of the nth harmonic of the EED is then reconstructed

∑=

=N

nn trftrf

0),,(),,( rr εε

[ ]),(sin ),(),,( rntrAtrf nnnrrr εφωεε +=

• …and the total time dependence of the electron distribution function is obtain from summation of the harmonics:

….where f0 is the time averaged distribution function.

EIND_0502_16

EXCITATION RATES: “ON THE FLY”

• In a similar manner, Fourier components of excitation rates can be obtained directly from the Electron MCS

• For the nth harmonic of the mth process,

• The resulting Fourier coefficients then reconstruct the time dependence of electron impact source functions.

( )∑∑ −∆±+→ ++k

jklklj

jjmjnmlnml rrrtinwkk ))((exp)( 21'' rrrδωυεσ

)Re()Im(tan

,)sin(][)(

nml

nml1nml

n

0nnmlnmlmllml

kk

tnkNetSm

−

=

=

+= ∑

φ

φω

University of IllinoisOptical and Discharge PhysicsEIND_0502_17


ALGORITHM FOR E-E COLLISIONS

• The basis of the algorithm for e-e collisions is “particle-mesh”.

• Statistics on the EEDs are collected according to spatial location.

• A collision target is randomly selected from the EED at that location and a random direction is assigned for the target’s velocity.

• The relative speed between the electron and its target electron is used to determine the probability for an e-e collision

SNLA_0102_01

( ) ( ) 21, eeeeeee vvvtvvrnP rrr−=∆⋅⋅⋅= σ

( ) 22

1

22 4,ln14

vmqb

bbv

e

oo

o

Doee

πελπσ =⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

• If a collision occurs, classical collision dynamics determine the change in momentum of the electron.

• The consequences of e-e collisions on the targets are obtained by continuously updating the stored EEDs.


ICP CELL FOR INVESTIGATION

• The experimental cell is an ICP reactor with a Faraday shield to minimize capacitive coupling.

MCSHORT_02_29


TYPICAL CONDITIONS: Ar, 10 mTorr, 200 W, 7 MHz

• On axis peak in [e] occurs in spite of off-axis power deposition and off-axis peak in electron temperature.

MCSHORT_05_30


TIME DEPENDENCE OF THE EED

• Time variation of the EED is mostly at higher energies where electrons are more collisional.

• Dynamics are dominantly in the electromagnetic skin depth where both collisional and non-linear Lorentz Forces) peak.

• The second harmonic dominates these dynamics.

SNLA_0102_10

• Ar, 10 mTorr, 100 W, 7 MHz, r = 4 cm ANIMATION SLIDE


TIME DEPENDENCE OF THE EED: 2nd HARMONIC

• Electrons in skin depth quickly increase in energy and are “launched” into the bulk plasma.

• Undergoing collisions while traversing the reactor, they degrade in energy.

• Those surviving “climb” the opposite sheath, exchanging kinetic for potential energy.

• Several “pulses” are in transit simultaneously.

SNLA_0102_11

• Ar, 10 mTorr, 100 W, 7 MHz, r = 4 cm

• Amplitude of 2nd HarmonicANIMATION SLIDE

HARMONICS IN ICP

• To investigate harmonics an Ar/N2gas mixture was selected as having low and high threshold processes.

• e- + Ar → Ar+ + e- + e-, ∆ε = 16 eV

High threshold reactions capture modulation in the tail of the EED.

• e- + N2 → N2 (vib) + e-, ∆ε = 0.29 eV

Low threshold reactions capture modulation of the bulk of the EED.

• Base case conditions: • Pressure: 5 mTorr• Frequency: 13.56 MHz• Ar / N2: 90 / 10• Power : 650 W

0.18 2.7

Electron density (1011 cm-3)

University of IllinoisOptical and Discharge PhysicsSNLA_0102_25

SOURCES FUNCTION vs TIME: THRESHOLD

• Ionization of Ar6 x 1014 – 3 x 1016 cm-3s-1

MIN MAX

• Excitation of N2(v)1.4 x 1014 – 8 x 1015 cm-3s-1

• Ionization of Ar has more modulation than vibrational excitation of N2 due to modulation of the tail of the EED.


ANIMATION SLIDE

HARMONICS OF Ar IONIZATION: FREQUENCY

• At large ω, both υm/ω and 1/(υmω) are small, and so both collisional and NLF harmonics are small.

• At small ω, both υm/ω and 1/(υmω) are large. Both collisional and NLF contribute to harmonics.

0.0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50Frequency (MHz)

S2

S4

S1, S3S

n / S

0

• Harmonic Amplitude/Time Average• Ar/N2=90/10, 5 mTorr


HARMONICS OF Ar IONIZATION: PRESSURE

Pressure (mTorr)

0.0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60 70S

n / S

0

S2

S1

S3-S8

S4

• Harmonic Amplitude/Time Average

• Ar/N2=90/10, 13.56 MHz

• At large P, υm/ω is large and 1/(υmω) is small. Harmonics result from collisional (or linear) processes.

• At small P, υm/ω is small and 1/(υmω) are large. Harmonics likely result from NLF.


• 5 mTorr6 x 1014 – 3 x 1016 cm-3s-1

• 20 mTorr1.5 x 1014 – 1.7 x 1016 cm-3s-1

TIME DEPENDENCE OF Ar IONIZATION: PRESSURE

MAXMIN

• Although Brf may be nearly the same, at large P, vθ and mean-free-paths are smaller, leading to lower harmonic amplitudes.


ANIMATION SLIDE

monte carlo methods for electron transport

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