plasma sources iv: high density sources: icp, ecr, mw · this is not a magic but a legal number by...

1 U. Czarnetzki, ISLTPP, Bad Honnef 2016 U. Czarnetzki, ISLTPP, Bad Honnef 2016

Plasma Sources IV:

High Density Sources:

ICP, ECR, MW

U. Czarnetzki

International School on

Low Temperature Plasma Physics:

Basics and Applications

Physikzentrum Bad Honnef, October 2-8, 2016

2 U. Czarnetzki, ISLTPP, Bad Honnef 2016

Outline

• Some Applications.

• Some general wave properties.

• ICP discharges.

• ECR discharges.

• Microwave discharges.


Some Applications


Application:

Etching of micro structures in silicon

Creation of ions and radicals and synergistic directional etching:

Reactive ion etching and plasma aided chemical vapor deposition.


Application: Deposition of Thin-Films


Some General Wave Properties


Basic Wave Equation General wave equation (linear equation in E!):

2 2

0 0 0 0E E/c / E+ / .extj j

No space charge for transversal waves:

(only this case is considered here) 0/ E E 0.ik

• Only time varying currents contribute.

• External currents jext are source terms.

• They can be included in the boundary conditions.

• Without magnetic field, the conductivity is isotropic,

i.e. it is a scalar, with magnetic field it is a tensor.

.j E

The current in the plasma is proportional to the electric field:

The conductivity follows from the momentum equation.


Dispersion Relation

The conductivity can be complex.

Then also k is complex (for real w).

The wave equation together with the conductivity leads to

the dispersion relation (for simplicity here scalar ):

2 2 2

0 0

0

( ) , using: 1/ck i c

w w

Example of a homogeneous

(unmagnetized) plasma:

22

0 0

pee

e

e n

m i i

w

w w

• The imaginary part of the conductivity contributes to the

real part of the wave vector.

• The real part of the conductivity contributes to the

imaginary part of the wave vector.


Damping of the wave

Complex wave vector:

Electric field: expE i k r tw

Damped wave:

' i k''k k

exp '' exp ' r tE k r i k w

• The imaginary part of k (real part of ) leads to damping

of the wave (k’’ > 0).

• In general, waves can also be excited in a plasma (k’’ < 0).

• This happens in case of instabilities.


Classical Skin Depth

Imaginary part:

Effective parameter:

Real part:

Skin depth (1/e amplitude depletion):

2

2

2

1''

2 1

''' ''

1

m

pe

w

w w wk

c w

kk k

w w

w

w

1/ '' exp /skin skink E z z


Typical Parameters

RF discharge:

Pressure p = 10 Pa, hydrogen:


Skin depth:

Plasma frequency:

8 14 10m s

8 113.56 0.85 10f MHz sw

0.21 1w

(a)

MW discharge:


Skin depth:

10 12.45 1.54 10f GHz sw

385 1w

(b)

11 3 4 1 10 110 5.64 10 1.8 10pen cm s n sw

25.1skin

pe

ccm

w

w

1.6skin

pe

ccm

w


Effect of Collisions

1 1( )

21

0 ( )

peskin

pe

cfor wfor w HF casec

for wfor w DC casew

ww

Special cases:

Collisions (dissipation)

enlarge the skin depth.

m

ww


Electric Field Profile

Note: Also the real part of k depends on w.

• Without dissipation the wave is (classically)

simply evanescent.

• With dissipation, damped oscillations can occur.

(generally no practical consequences due to low amplitudes).

0 2 4 6 8 1010

-4

10-3

10-2

10-1

100

E /

E0

z / Skin

w / m = 0

w / m = 2

w / m = 10


Phase Velocity

• With collisions, the evanescent wave has a

low phase velocity that can be much smaller than c.

• Without collisions, the phase velocity is infinite.

01

23

4

5

04

812

1620

0

>> w

za

mp

litu

de

(a

.u.)

w t

01

23

4

5

04

812

1620

0

<< w

z

am

plit

ud

e (

a.u

.)

w t


Failure of the Local Picture

• The classical approach presented before breaks down in a

collisionless plasma (mean free path l >> skin depth skin).

• Then the conductivity is no longer a local quantity.

• For high collisionality, the local velocity depends only on

the local field at the local time.

• Without collisions, it depends on the entire history,

i.e. the trajectory in space and time.

• The correct description can no longer be performed

in a fluid dynamic picture but requires a kinetic treatment,

since the individual particle velocities matter.

• The entire interaction between wave and particles needs

to be treated in parallel.

• This affects the skin depth and the power deposition!


Principle of the Non-Local Effect • Thermal flux of electrons towards the antenna window.

• Reflection of the electrons at the floating potential sheath

in front of the window.

• Gain and loss of energy occurs in different regions of

space.

• Due to the field inhomogeneity in the skin layer,

the net effect is non-zero.

01

23

4

5

04

812

1620

0

<< w

z

am

plit

ud

e (

a.u

.)

w t

antenna window

bulk plasma skin layer

Important:

The electric field

oscillates perpendicular

to the direction of the

(thermal) particle motion

and the field gradient!


Anomalous Skin Effect

experiment and theory

• Alternation of positive and

negative power deposition.

• Net effect is positive: Heating.


Consequences of the Non-Local Effect

• Net power deposition even without collisions:

Stochastic heating.

• The field decay is not exponential.

• Different scaling of the skin depth (larger than classical): 1/31/3

2

2

6

8 8

9 2 9 2

8.10 /

pe th thskin

pe pe

eth

vc c v

c

kTv typ m s

m

w

w w ww

• Remarks:

- The factor 8/9 is often skipped.

- Requirement for stochastic heating:

- Difficult to realize at low pressures.

- Most effective at lower RF frequencies, e.g. 1 MHz.

10 1

11 3

1, 13.56 :

2 10

1 10

pe th

pe

e

cAt f MHz

v

s

n cm

w

w

w


Frequency and Pressure Dependence

• Experimental result from an

ICP discharge.

• The ratio between the total

(collisional) and the Ohmic

overall power deposition

(volume integrated) is

determined.

• The scaling is similar for

all frequencies.

• The non-local effect

becomes effective only for

pressures of less than 1 Pa.


Some General Technical Consequences • The electron density in metals is much higher.

• Therefore the skin depth is much smaller.

• In copper typical values for RF frequencies are 10 m.

• For microwaves the skin depth is only about 1 m.

• The current is flowing only at the surface.

• Large surfaces are requires.

• The cross section does not matter.

• The surface quality matters.

• Contacts can cause strong losses.

• Polishing and coating with silver are common measures.

• Further: Impedances scale with frequency.

• Dielectric isolators can easily transmit displacement current,

tiny bends in wirings can cause noticeable inductances.

• Without impedance matching strong reflections occur.


ICP discharges


Basic Principles of CCP and ICP Discharges

current in

the plasma

heating

time

sheath

voltage

time

heating

time

+

+

+

+

++

-

--

-VRF

electrode

plasma

sheath

I

I

A

Pl

antenna

plasma

E-field

B-field

CCP-mode:

„piston“-

principle

ICP-mode:

transformer-

principle


Radio Frequency (RF)

• processing of non-conducting dielectrics (CCP)

• extra heating mechanism for electrons (CCP)

(not existing in DC discharges)

• operation at low pressures (< 1 Pa)

• high plasma densities (up to 1012 cm-3)

• large scale homogeneous discharge possible

• high efficiency for the transformer (ICP)

CCP and ICP discharges operate in the radio frequency regime:

The most common frequency is 13.56 MHz (and harmonics).

This is not a magic but a legal number by international

telecommunication regulations (also harmonics).


Some General Characteristics I

• ICP discharges operate usually at an order

of magnitude higher powers than CCPs.

• They produce an order of magnitude more

dense plasmas.

• The plasma potential is low and quiet for

pure ICP operation.

• They can sometimes operate in a hybrid CCP/ICP mode,

since the antenna can also act as an electrode

(especially at low densities).


Some General Characteristics II

• Often they are operated together with a separate CCP:

ICP for plasma generation, CCP for RF-bias.

• This allows independent control of the plasma density

(ion flux) and the bias (ion energy).

• The antenna coil is usually either a cylindrical or planar coil.

• A dielectric window (quartz or aluminium oxide) is required.


Typical Antenna Coils

Faraday shield (spoke like structure):

• only poloidal fields can penetrate

• capacitive coupling is suppressed.

• disadvantage: No self-ignition,

high voltage between coil and the shield

Cylindrical Antenna Planar Antenna


0 2 4 6 8 100,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

XL ()000225a.o

rg

50

40

30

20

C2C1

resistance ()

capacitance (nF)

ICP Matching Unit

C

CL

R

R 2

1

in out

VRF

0


-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120

50

40

30

20

10

0B

ver [-0,009 ~ 0,061 mT/A]

R (mm)h

(m

m)

-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120

50

40

30

20

10

0E

pol [0 ~ 55 (V/m)/A]

R (mm)

h (

mm

)

Calculated antenna fields in vacuum:

Bz and Ej

The relevant induced electric field is in azimuthal direction.


Induced Azimuthal Electric Field

(calculated for vacuum)

The radial current is causing distortion of the symmetry.


-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

150

radial positon (mm)

rad

ial p

osito

n (

mm

)

70.00 -- 75.00

65.00 -- 70.00

60.00 -- 65.00

55.00 -- 60.00

50.00 -- 55.00

45.00 -- 50.00

40.00 -- 45.00

35.00 -- 40.00

30.00 -- 35.00

25.00 -- 30.00

20.00 -- 25.00

15.00 -- 20.00

10.00 -- 15.00

5.000 -- 10.00

0 -- 5.000

Electric Field Induced by a Flat Coil Antenna

Flat coil with four parallel spirals, each 2.5 turns.

(HV in the centre, ground at the outer connectors)

D L Crintea, D Luggenhölscher, V A Kadetov, Ch Isenberg and

U Czarnetzki, J. Phys. D: Appl. Phys. 41 (2008) 082003


Wave Equation for the Induced Field

• Two dimensional problem.

• Displacement current can be neglected since the plasma

frequency is much higher than the RF frequency.

• Analytical solution only for a homogeneous plasma.

• The flat antenna is approximated as a conducting disc

of radius R.

• Plasma is assumed as homogeneous!

• Inhomogeneous plasma: Numerical solution!


Plasma Current Density

Local isotropic conductivity:

Ej

• For mean free paths of the order of the skin depth

or larger the conductivity is no longer local.

• Then the product of conductivity and field becomes

a convolution.


Geometry and Boundary Condition

Conditions are set at the walls

and the interfaces:

generally


Complex Solution for the Electric Field

The coefficients are:

Radially Bessel functions, axially exponentials:

ln: n-th zero of the

Bessel function 2

0pk iw


Relevance of the Individual Terms

Only very few terms contribute, actually mainly 2!

distance from

antenna

number of Bessel

zero

Dominik Winter, Master Thesis, RUB (2010)


Electric Field Depletion

Typical values:

• In reality, mostly a mixture of geometry and skin depth!

• Approximately exponential decay of the field.

• The depletion length has a geometrical and a plasma

contribution: 22

11

geometry skin effect

: 1/ , 3.83pe

L c

wl w l

1 typ. 0.1pe

cO ml

w

largechambers : classical skin depth behaviour

smallchambers : geometrical depletion

L

L


Radial Profile

• Fundamental radial mode profile is a Bessel function.

• Only the first two Bessel functions contribute

significantly.

• Higher modes shift the maximum to smaller radii.

• Rule of thumb: Maximum at r = 2/3 R.

1 1 1 2 1 2

r rE r A J A J

L Ll l

Physics:

• B-field is a dipole.

• Naturally, no electric field

on axis and no field at the wall.

• Zero at the conducting wall due to

eddy currents.


Comparison with numerical integration

analytical solution

numerical solution

(Comsol)



Self consistent numerical solution I

Plasma density and ion flux



Self consistent numerical solution II

Induced power and electron temperature



Radial Profiles of Te

-40 -20 0 20 40 60 80 100 120 1400,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

p = 10 Pa H2

1200 W

800 W

500 W

300 W

Te (

eV

)

radial position (mm)

Collisional losses are strong in hydrogen and the energy is balanced

locally. The electron temperature reproduces schematically the torus

structure of the induced electric field (radial direction).

Victor Kadetov, PhD Thesis, RUB (2004)


Phase Resolved Velocity Distribution

Function in an ICP (Ar, 0.5 Pa, 1 kW, Thomson Scattering)

-6 -4 -2 0 2 4 60

100

200

300

400

500

600

700

t = 60 ns

inte

nsity (

a.u

.)

wavelength shift (nm)

t = 20 ns

Argon ICP discharge, f = 13.56 MHz, P = 1 kW, p = 0.5 Pa.

Accumulation over 90,000 shots (30 minutes) at each phase.

Temporally independent EEPF, oscillating drift velocity.

0 5 10 151

10

100

1000

EE

PF

(a

.u.)

energy (eV)

kTe = 2.6 eV

ne = 1.0x10

11 cm

-3

t = 20 ns, 60 ns, 80 ns




Harmonically Oscillating Drift Velocity

V/cm 0.66Eˆ

v wm

Ee

0 20 40 60 80-10

-5

0

5

10

15

20

dri

ft v

elo

city (

10

4 m

/s)

Phase (ns)

0 v )tsin( v v jw




Self consistent numerical solution III

Electric field amplitude and phase



Measurement (B-dot probe)

Very similar to simulation.



Optically Measured Field Penetration

Intensity and field are radially integrated and the axial dependence

and time evolution are displayed as coloured contour plots.

0,0 0,2 0,4 0,6 0,8 1,050

40

30

20

10

0Pathe Shift

Time/Period

ve

rtic

al p

ositio

n (

mm

)

Electric field


01

23

4

5

04

812

1620

0

>> w

z

am

plit

ud

e (

a.u

.)

w t


Measurement (optical modulation)

Very good agreement throughout.



Transformer Model

2

2

2

2

2

122

LR

LRRS

w

w

2

2

2

2

2

1221

LR

LLLLS

w

wwww

The plasma inductance and

resistance are transformed to

an effective antenna

inductance and resistance.

2ˆ2

1RFSdiss IRP

M. A. Lieberman and A. J. Lichtenberg,

Principles of Plasma Discharges and Materials Processing

(Wiley, New York, 1994).


Low and High Density Operation

Low density operation

(skin depth >> plasma size):

e

P

SPP nR

RLR 1

w

eP

PSPPn

RRLR11

wHigh density (normal) operation

(skin depth << plasma size):

Transformed resistance:

22

2

12

)( PP

PSLR

LRR

w

w

RS

electron density


Dissipated Power and

Operation Threshold

threshold for

ICP operation

Ploss

I3 < I

2

I2 < I

1

I1

Pdiss

electron density

2ˆRFediss InP

2ˆ1RF

e

diss In

P

eloss nP

Low density:

High density:

Power lost by the discharge

(collisions, flux to the wall):

• There is a minimum power (current) for ICP operation.

• Below this value, the discharge operates capacitively.


10-1

100

101

0,001

0,01

0,1

1

S = K (U / d)3 n-3/2

US = U4 / U

p3

= C (U / d)4 / n2

0011

05A

.OR

G P

lot1

US /

U

U / Up

Capacitive Coupling by the Antenna

• The quartz window and the sheath form a non-linear

capacitive voltage divider between the antenna and

the plasma.

• At high densities and/or low antenna voltages,

the capacitive coupling is reduced strongly.


Effect of CCP-ICP Mode Transition on

the Matching

0,1 1 10 1000

200

400

600

800

1000

1200

C1 divergence

guide to the eye

optical emission

H2 gas

po

we

r (W

)

pressure (Pa)

Rs 0 C1

• Divergence of the matching capacitance C1 at

the transition point to CCP operation.

ICP

CCP


s

s


Effect on the Modulation

of the Optical Emission

Discharge in hydrogen at

p = 10 Pa:

mode transition CCP to ICP

at P = 150 W

Observation of the Balmera

emission line (integrated

spatially):

%100I

)t(Imodulation



Observation of the Mode Transition in the

Modulation of the Optical Emission

ICP discharge in hydrogen at p = 10 Pa, Balmera emission

Pulsed mode: P = 300 W, f = 1 kHz, 1:1

0

100

200

300

4000.0

0.5

1.0

1.5

2.0

60

80

100

120

140

160

modula

tion (

%)

power

(W)

time/period


• ICP discharges always ignite as CCPs.

• Then the density rises, the sheath shrinks,

and the mode transition happens.

%100I

)t(Imodulation


• The mode transition between ICP and CCP has a strong

effect on the ion energy distribution function.

• ICP: Quasi mono-energetic, low-energy ions from the

floating potential sheath.

• CCP: High energy wide distributed ions from the RF sheath.

Effect on the Ion Energy



1) Pulsed discharges and afterglow.

2) Electronegative discharges and instabilities.

3) Gas heating and electron pressure effects.

4) Atmospheric pressures.

5) Hybrid discharges ICP – CCP.

6) Many applications…..

Further Topics in ICPs


ECR Discharges

(and a little bit of Whistlers, Helicons)


Addition of a Weak Axial Magnetic Field

Argon p = 0.1 Pa, B = 10 mT, lme = 1 m = 2 x diameter.

ICP,

without

B-field

Helicon,

with

B-field

static magnetic

field coil

antenna

Y. Celik et al., POP 18, 022107 (2011)


RF Modulation Spectroscopy Upstream

oscillatory

velocity (2w)

azimuthal

drift

velocity (1w)

Y. Celik et al., POP 18, 022107 (2011)


RF Modulation Spectroscopy Downstream

phase (1w)

• Modulation can still be observed downstream.

• A helicon wave propagates along the field lines.

• The diamagnetic drift (grad(Te)) displays the RF phase front.

• The phase front is bended due to the inhomogeneous

dispersion by the density profile

drift velocity (1w)

Y. Celik et al., POP 18, 022107 (2011)


Helicon Wave Magnetic Field (B-Dot Probe)

-10 -5 0 5 10 15 20-40

-20

0

20

40

60

80

100

120

Bz (

T

)

r (cm)

ICP (no static B-field) Helicon (with static B-field)

• The antenna field determines

the Bz(r) wave field.

• Maxwell’s equations then

determine the Helicon Bj(r):

)()( rBr

rB z

j

Y. Celik et al., POP 18, 022107 (2011)


Electromagnetic Fields Propagating Along

Static Magnetic Fields

ie tzki

1EE 0

w

• The wave vector k points along the field lines.

• All electric fields are perpendicular to the wave vector k.

• Two fundamental polarization states:

- R-wave (rotating clockwise, with the electrons, “+”)

- L-wave (rotating counter clockwise, with the ions, “-”)

• x- and y-components are 90° out of phase.

• The Lorentz force is important in this case:

e

mv E v Bm

c

e

e B

mw

• Electrons rotate clockwise with the cyclotron frequency:


Current density from the momentum equation:

Solution of the equation (determines also the conductivity):

zc

peej

cji

w

ww

0

2

2E

E

1

12

2

0

w

w

ww

c

pe

cji

Insertion into the wave equation gives the dispersion relation:

w

w

www

w

c

pe

cckji

ck

1

1EE

2

2

2

22

02

22

Determination of the Dispersion Relation


Solving for k2 yields:

Normalization by the electron plasma frequency helps

identifying the characteristics:

2

2 2 " " : , " " :

1

pe

c

ck R wave L wavew

ww

w

cpecpepe

c

kkcmit

k

wwwwwww

w

ww

/,/,/

1

122

Structure of the Dispersion Relation


Cut-off frequencies (k = 0) are easily identified:

• Three intersections with the w-axis.

• R-wave at w = 0 and w = wR.

• L-wave at wL, in between the two cut-offs of the R-wave.

• Only the R-wave has a resonance (k .

• This resonance is at the cyclotron frequency w = wc.

• Natural result with view on the electron gyration.

cLR

ccLRc

wwww

wwww

w

ww

0,42

1

1

10 2

/

2

Cut-Offs and Resonances


• The dispersion relation is shown for wc = 2 wpe.

• Then the L-wave cut-off is below the R-wave resonance

(wc).

• In case wc < wpe, the L-wave cut-off is between wR and wc.

• The R-wave has a band-

gap between wc and wR.

• The L-wave has band-gap

between 0 und wL.

• For large k both waves

behave like waves in

vacuum with a phase

velocity c.

R- and L-Wave Dispersion

0

1

2

3

4

5

0 1 2 3 4 5

R wave

c k / wpe

w /

wp

e

L wave

R wave

wR

wL

wc

resonance, ECR

whistlers (helicon)

Faraday

rotation


• At low frequencies only R-waves exist (cut-off for L-waves).

• The dispersion relation can then be expanded to first order

(w << wc) which provides a simple expression for the phase

velocity:

pe

c

ph ck w

wwwv

• The phase velocity increases monotonically with w.

• Higher frequencies propagate faster and arrive earlier.

• If all frequency start at the same time a x = 0 and detection

is after a distance s, each frequency arrives at its own time:

R-Wave Dispersion at Low w and k:

Whistler Waves

2

0

1 1s

ph

dxt t

v tw

w


• Lightning flashes at the southern hemisphere excite whistler

waves within a broad spectral range (kHz).

• Waves propagate along the earth magnetic field lines.

• Due to dispersion, higher frequencies arrive earlier.

F1

F2

• A falling glissando can be heard (coining the term “whistler”).

• The time dependent frequency spectra scale like 1/t2.

• A fit “by eye” shows very good agreement (red lines).

Natural Whistlers

http://www-pw.physics.uiowa.edu/mcgreevy/ Stanford vfl group

2whis494.wav


• Helicon waves are whistler waves which are radially

bounded.

• This requires a finite kr and the dispersion relation changes.

• The difference is particularly pronounced at R < lz .

• For large radii the Helicon wave eventually converges

to the normal whistler wave.

2

2

2

2 2 2

1 Helicon

Whistler

Note:

z z

r rpe

c rz

r

z r

k k

k k

c kk

k

k k k

ww

w

0.0 0.5 1.0 1.5 2.0

0

1

2

3

4

5

Helicon wave

Whistler wave

w

wc (

wp

e/

c k

r)2

w

kz / k

r R / l

z

Helicon and Whistler Dispersion

Comparison:


• At high frequencies R- and L-waves both travel almost with

the speed of light in vacuum.

• However, still the R-wave is a little faser.

• Therefore, with distance L an increase phase difference j

develops. In second order in wpe/w this difference is:

nBc

LkL

pecRL 0

2

w

wwjjj

Transformation of a linearly polarized wave into a left and right

polarized wave: xyxyx eeieeie

2

Faraday Rotation

• The phase difference between the R- and L-components

rotates the direction of polarization.

• This effect is called Faraday rotation.


The R- and L- parts are recombined after the wave has

experienced a certain phase shift:

2/sin

2/cosE

11

2

EEE

2/

00

j

jjjjj RR iii

LR ei

ei

e

The initially only x-direction polarized wave has been rotated

by an angle -j/2 in direction of the y-axis.

Linear Polarization Rotation

j

z direction (wave propagation)

x,y direction (wave polarization)


Applications:

• Measuring magnetic fields in astro physics.

• Determination of magnetic fields in fusion experiments.

• Building optical diodes for laser applications.

Application of Faraday Rotation

H.R. Koslowski, H. Soltwisch, W. Stodiek,

Plasma Physics and Controlled Fusion 38, 271 (1996)

B.M. Gaensler et al., Science 307m 1610 (2005)

Large Magellian Cloud


• ECR discharges take advantage of the resonance of the

R-wave at the electron cyclotron frequency.

• For commercial applications they are generally operated at

a microwave frequency of f = 2.45 GHz.

• This requires a magnetic field strength of B = 87.5 mT.

• The magnetic field is generally non-uniform.

• Waves are launched from the high field side (wc > w).

Electron Cyclotron Resonance (ECR)

Kouichi Ono, Mutumi Tuda, Hiroki Ootera, and Tatsuo Oomori,

Pure & Appl. Chem., Vol. 66, No. 6, pp. 1327-1334, 1994.


• Ion source for highly charged ions, e.g. at accelerators.

• High density plasma source in the semiconductor industry.

• Satellite propulsion.

THE SACLAY (FRANCE) HIGH-CURRENT

PROTON AND DEUTERON ECR SOURCE

Roth & Rau: commercial ECR source

z.B. for semiconductor treatment,

At P = 1 kW, n = a few 1017 m-3 .

Pressure: p = 0,1 – 10 Pa.

Examples of ECR Sources


POSTECH (Korea)

W. C. Lee, S. I. Moon, H. J. Jang, Y. S. Bae, M. H. Cho, W. Namkung (POSTECH)

* Work supported by KAERI’s KOMAC Project

Control of the Resonance Location

by the Magnetic Field


1 MW at f = 140 GHz

Figure shows a so called

“gyratron”, a high power

microwave source.

So far 10 gyratrons installed

at Wendelstein with a total

power of 10 MW.

ECR Heating (ECRH) at the New

STELLARATOR Wendelstein 7-X (W7-X)


ICRH (Ion Cyclotron Resonant

Heating) at the TOKAMAK

Joint European Torus (JET):

23 – 57 MHz, up to 32 MW

in 16 channels of 2 MW.

Ion Cyclotron Resonance Heating (ICRH)

http://www.jet.efda.org/pages/focus/006heating/j91-08c-18.jpg


The equation of motion for a single particle is solved:

Collisionless Heating in ECR Discharges

0 0 0, , 0mv e E v B v z z t v

• The time varying electric field E is assumed as

homogeneous, i.e.it is only a function of time

(corrections -> later).

• The static magnetic field B points only in the z-direction

and is only a function of z (magnetic mirror effects are

ignored).

• The aim is the calculation of the change in energy

between the initial moment and a later moment.


• The initial condition can always be met by adding the

solution for a free gyrating electron.

Free Gyrating Particle

0 0, , 0f h fmv ev B v z z t v

• This solution cancels out later when calculating the energy

change and averaging over all possible initial phases.

• Solution for a homogeneous B field:

0 0

0 0 0

0

sin cos

sin sin ,

cos

c

f c c

te B

v v tm

w j

w j w

• If B varies with z only the time

dependence changes.

• Note: Ions gyrate counter

clockwise and have a much

higher mass (lower frequency).


• The fields and initial conditions are now:

Homogeneous B-Field

0, , 0 0i t

z x y

e Be E E e i e e v t

m

ww

• The solution is:

0

cos

ˆ ˆsin , ,

0

e Ev v t v

m

j

j j j ww

• The velocity amplitude increases linearly with time!

• The energy increase quadratically:

22ˆ

2

mv j

• Therefore, very high energies can be reached.


• An inhomogeneous B field is now approximated by a

linear variation (first order expansion around the

resonance point):

Inhomogeneous B Field

1 z

e B ze

m Lw

• The solution is slightly more tricky but in the end effectively

the linear term j is replaced by an integral and some phase

shift appears:

0

cos / 4

ˆ ˆ/ 4 , ,

0

e Ev v cos t v

m

j j

j j j j ww

• For typical parameters, the integral converges to a constant:

• The integral reads:

' 20 0

00

' 2 ,2

i z

z

L v ze d

v L L

jj w

j j j j jw

2 10 1 6

0 10 10 10 /zL v m s m sw


Since the velocity along z is constant, phase j

and z-position can be directly converted:

Energy Gain Along the z-Axis

0 0zz v z

L L Lj

w

Naturally, the width of the resonance zone is about L.


• The result for the energy gain is then:

Energy Gain in an Inhomogeneous B-Field

• The gain scales inversely with the initial velocity in

z-direction, i.e. it is proportional to the transit time

through the resonance zone.

• Remarkably, this result is independent of the initial velocity.

• The power per unit area is obtained by multiplying with

the flux density n v0z:

2

0

0 0

ˆˆ2 with 1,

2 z z

mv L L eEv

v v m

w w

w

2

0e E LnS

m

w

• Since the frequency is very high collisions do not play a

significant role up to a quite high pressure of about 100 Pa.


• A correct description of the heating process must take

the propagation and depletion of the wave into account.

Wave Equation Model

• The collisonless wave equation is (R - dispersion relation):

• A solution to this equation was found by K.G. Budden

in1966 but it is rather complex.

• Using again the linear B-field and constant plasma density

approximation and some normalization, the equation

reads:

222

0 021 0,

perr

c

Ek E k

z cz

w w

w w w

22

021 0, ,

perr

LEE zk

c

w

w

K.G. Budden, Radio Waves in the Ionosphere,

Cambrige University Press, UK 1966


Reflection and Transmission

• One of the most interesting aspect about the solution is

that it allows also calculation of the reflection and

transmission coefficients. These results are quite simple:

• Quite unexpected, there is no reflection.

• Typical values are k0 = 50 m-1, L = 0.1 m which leads to

a moderate density condition:

2 2

021 , , 0,

ref pe peabs trans

inc inc inc

S LS Se e k L

S S S c

w w

w w

2

10 3

20.2 1.5 10

pen cm

w

w


Tunneling Through the Band Gap

• The result of Budden implies that the wave is tunneling

through the band gap which exists between wc and wR.

• Since wc and z are interchangeable variables in the model,

it is illustrative to re-plot the dispersion relation in an

alternative representation.

• For low plasma densities

the band gap is small and

the wave can tunnel.

• Within the band gap, the

wave is evanescent.

band gap


Further Aspects

• Doppler shift of the resonance:

z ck z v zw w

• This shifts the resonant magnetic field by typically a few

percent.

• Heating at higher harmonics of the cyclotron frequency.

• Parametric instabilities and non-linear power absorption.

• Confinement of electrons in a mirror field.

• Effect of the chamber geometry on resonances

(cavity effect at longer wavelengths or smaller chambers).

• Effects related to the transversal magnetic field.


Microwave Discharges


Some Advantages of Microwave Sources

• Simplicity of plasma generation at high (>100 W/cm3) and

low (< 1 W/cm3 ) powers.

• Wide range of operating pressures (from 10 Pa up to

atmospheric pressure).

• Control of the density profile by the antenna structure.

• Size of the discharge chamber can be variable.

• No electrodes.

• Large areas / volumes can be treated.

• High power sources at relatively low price available.

Y.A. Lebedev, Journal of Physics:

Conference Series 257, 012016 (2010)


Operation Schemes

For most microwave sources the operation principle is either:

• Standing waves in a resonator.

• Propagation of surface waves (with or without resonator).

A. Schulz et al., Contrib. Plasma Phys. 52, 607 2012) Yi-Hsin Chou, et al. Dalton Trans., 2010,39, 6062-6066

Plasma torch:


Surface Waves

• Dielectric (x > 0) - plasma (x < 0) interface.

• Wave is evanescent on both sides (exponential decay).

• Propagation along the surface (z – direction).

• Wave guide effect.

• Simplification: Homogeneous plasma.

x

y

z plasma

dielectric

Ex

Ez

By

2

0

, , exp exp exp

,

0 : 1 1

0 :

x z y m m

pe

p

D

E E B x i k z i t

m P D

P x ii

D x

w

w

w w w

• Transversal electric and magnetic fields (y – direction).

• Longitudinal (z) electric field due to waveguide effect!

k


Surface Wave Parameters

• The boundary conditions couple the wave properties in the

two regions.

( ) ( )

( ) ( )

( ) ( )

1 1( )

( )

P D

y yP D

z z

p D

P D

y y

B Bi E E

x x

ii B B

2

P Dp D

P D

mm

P D

k k kc

c

w

w

Boundary condition (x = 0):

Solution (Note: The quantities are complex due to collisions!):


Dispersion Relation (collisionless)

Simple case (no collisions):

2

2

2

2

2

2

1

1

1

1:

1

: 1

Resonance : :

1

1

pe

p

pe

peD peD

pe

D

pe

pe

r

r

pe D

c k

k

w

w

w

w

w w

w

w ww

ww w

w w

w

w

surface wave: w < wr

bulk wave

resonance bandgap


Damping Evanescent Wave (collisonless)

• The decay of the wave in the plasma is always faster

than in the dielectric. (This can change in cylindrical

geometry.)

pe

pc

w

Dielectric Plasma

imaginary

(oscillating)

imaginary

(oscillating)


Sources Types I

• Cavity and waveguide sources: Typically for flow-tube

reactors with a quartz or ceramic tube.




Sources Types II

• Surface wave discharges: Flow tube as well as large area

application.




Resonator and Surface Wave: Surfatron

http://sincaf.icmse.csic.es/showhighlight.php?posic=3

https://plasimo.phys.tue.nl/applications/surfatron/index.html

• Often used as atomic

radical source, e.g. H,

N, O etc. (and excited

molecules).


Inverted Geometry: Duo Plasma Line

• Inner conductor, outer plasma.

• MW feeding from both ends to compensate for

damping of the wave along the line.

• Large homogeneous (along the line) plasmas.

• Usually operated with a number of lines in parallel.

W. Petasch, H. Räuchle, H. Mügge, K. Mügge,

Surface and Coatings Technology 93, 112 (1997)

A. Schulz et al., Contr. Plasma Phys. 52, 607 (2012)


Sources Types III

• Large area and large volume source:

Many slots with individual surface waves at each slot.

• Planar or ring shaped geometry.




antenna slot

coupling antenna

field of viewCCD-camera

monitor antenna

shorting plunger

quartz tube

magnetron

ring resonator

laser

circulator

The SLAN Source

Dirk Luggenhölscher, Dissertation RUB, 2004


-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

2500 s Bessel Function

4000 s Bessel Function

030226b.o

pj

ne(1

01

0cm

-3)

distance to the center (cm)

Argon dominated (90 % Ar, 10 % H2): p = 30 Pa , f = 200 Hz, 20 % on

R

r41.2Jnrn 00

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5 1 s 500 s

10 s 1000 s

50 s 2500 s

100 s 4000 s

030226b.o

pj

ne(1

01

1cm

-3)

distance to the center (cm)

Density Decay in the Afterglow

• Source terms only at the wall due to small skin depth (cm).

• Collisional electron cooling due to high pressure (cm).

• Diffusion of recombining plasma leads to hollow profiles.

• In the afterglow, relaxation to the basic diffusion mode.

Dirk Luggenhölscher, Dissertation RUB, 2004


Some Conclusions

• At high frequencies the penetration of the electromagnetic

wave into the plasma is a major issue.

• ECR discharges allow the wave to travel long distances

before the energy is deposited within a relatively short range.

• In ICP and microwave discharges the skin depth is short

and power is deposited in the vicinity of the antenna.

• Particularly microwave discharge have a skin depth of

typically only one cm. However, power can be distributed

by operation of a large number of antenna slots. These are

often connected to one common resonator.

• Microwaves can propagate in the sheath between surfaces

and the plasma via surface waves.

• All discharges can produce dense plasmas in the range

of typically n = 1010 cm-3 to 1013 cm-3 and electron

temperatures of a couple of eV and much higher in ECRs.


Literature M. A. Lieberman and A. J. Lichtenberg,

Principles of Plasma Discharges and Materials Processing

(Wiley, New York, 1994).

P. Chabert and N. Braithwaite,

Physics of Radio-Frequency Plasmas

(Cambridge University press, Cambridge, 2011).

S. Ichimaru

Basic principles of plasma physics: a statistical approach

(Benjamin, Reading, 1973).

Y. Raizer,

Gas Discharge Physics

(Springer, Berlin, 1991)

Rolf Wilhem,

ECR Plasma Sources

(Springer, Heidelberg 1993)

K.G. Budden,

Radio Waves in the Ionosphere,

(Cambrige University Press, Cambridge 1966)


Literature J. Reece Roth,

Industrial Plasma Ingeneering,

(Institute of Physics Publishing, Bristol, 1995)

Yu. Aliev, H. Schlüter and A. Shivarova,

Guided Wave Produced Plasmas,

(Springer, Heidelberg, 2000)

Yuri P. Raizer, Mikhail N. Shneider, and Nikolai A. Yatsenko,

Radio-frequency capacitive discharges,

(CRC Press, 1995)

Thomas Howard Stix

Waves in Plasmas

(American Institute of Physics, New York 1992)

R. Geller

Electron Cyclotron Resonance Ion Sources and ECR Plasmas

(Institute of Physics Publishing, Bristol 1996)

Francis F. Chen

Introduction to Plasma Physics

(Plenum Press, New York, 1984)

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