M.A. Lieberman PHYSICS OF - EECS at UC Berkeleylieber/dualfreqphys8May05.pdf · University of California, Berkeley P L A S M A SHEATH PHYSICS UNDERSTANDING • Analytical solution

University of California, Berkeley PLASMA

PHYSICS OFDUAL FREQUENCY/HIGH FREQUENCY

CAPACITIVE DISCHARGES

M.A. Lieberman

Department of Electrical Engineering and Computer SciencesUniversity of California

Berkeley, CA 94720

DualHighFreqPhys8May05 1


OUTLINE OF TALK

• Introduction to dual frequency capacitive discharges• Independent control of ion flux and ion energy• Global models for discharge equilibrium• Control of ion energy distribution• Standing wave and skin effects at high frequencies• Transmission line analysis for standing waves• Conclusions



INTRODUCTION



WHY DUAL FREQUENCY CAPACITIVE DISCHARGE?

• Motivation for capacitive discharge– Low cost– Robust uniformity over large area– Control of dissociation (fluorine)

• Motivation for dual frequency– Independent control of ion flux and ion energy

High frequency voltage controls ion fluxLow frequency controls ion energy

• A critical application for dielectric etch



TYPICAL OPERATING CONDITIONS

• R ∼ 15–30 cm, L ∼ 1–3 cm• p ∼ 30–300 mTorr, C4F8/O2/Ar feedstock• High frequency fh ∼ 27.1–160 MHz, Vh ∼ 200–500 V• Low frequency fl ∼ 2–13.56 MHz, Vl ∼ 500–1500 V• Absorbed powers Ph, Pl ∼ 500–3000 W

~

~

Vl

Vh

+

–

+

–

~

~

Vl

Vh

+

–

+

–

Diode (2 electrodes) Triode (3 electrodes)



INDEPENDENT CONTROL OFION FLUX AND ION ENERGY



CONTROL OF PLASMA DENSITY

• Particle balance =⇒ electron temperature Te

(independent of plasma density)=⇒ Te ∼ 2–6 eV

• Electron power balance =⇒ plasma density n(once electron temperature Te is known)

n ∝ Pe

• In these dischargesPe = power absorbed by electrons ∝ ω2Vrf

=⇒ n ∝ ω2Vrf

• Make ω2hVh � ω2

l Vl

Vh controls plasma density (ion flux)



CONTROL OF ION ENERGY

• Ion bombarding energy is total dc bias voltage across sheath

Ei ∼ |Vh + Vl|• Make Vl � Vh

Vl controls ion energy

• Combined condition for independent control of ion flux and energy

ω2h

ω2l

� Vl

Vh� 1

1. M.A. Lieberman, J. Kim, J.P. Booth, J.M. Rax and M.M. Turner,

SEMICON Korea Etching Symposium, p. 23 (2003)

2. H.C. Kim, J.K. Lee, and J.W. Shon, Phys. Plasmas 10, 4545 (2003)3. P.C. Boyle, A.R. Ellingboe, and M.M. Turner, J. Phys. D: Appl. Phys.

37, 697 (2004)



GLOBAL MODELSFOR DISCHARGE EQUILIBRIUM



SHEATH PHYSICSDual frequency

sheathBulk plasma

High frequencyhomogeneoussheath motion

Low frequency Child lawsheath motion

Ions Electrons

Stochasticheating

Ohmic heatingin sheath Ohmic heating

in bulk plasma

Wall

• High frequency sheath operates at nearly constant ion density⇒ homogeneous model

• Low frequency sheath sets ion density profile⇒ Child law model

• High frequency current is modulated by low frequency sheaths• Three high frequency electron heating terms:

stochastic, ohmic in sheath (two parts), ohmic in bulkDualHighFreqPhys8May05 10


SHEATH PHYSICS UNDERSTANDING

• Analytical solution for current-driven, uniform ion density model(bulk plasma and both high and low frequency sheaths)H.C. Kim, J.K. Lee, and J.W. Shon, Phys. Plasmas 10, 4545 (2003)

• Approximate solutions for self-consistent Child law sheath dynamicsfor Vh < Vl

1. J. Robiche, P.C. Boyle, M.M. Turner, and A.R. Ellingboe,

J. Phys. D: Appl. Phys.36, 1810 (2003)

2. R.N. Franklin, J. Phys. D: Appl. Phys. 36, 2660 (2003)

3. P.C. Boyle, J. Robiche, and M.M. Turner, J. Phys. D: Appl. Phys.

37, 1451 (2004)



EXAMPLE OF GLOBAL MODEL

• Asymmetric diode (plate areas Aa and Ab)

~ Vl

+

–

~ Vh

+

– Lsa2sa1

sb1sb2

Ab

Aa

(ωh)

(ωl)

• Low frequency Child law sheaths• High frequency homogeneous sheaths• Ion particle balance, and electron and ion power balance• All high and low frequency heating terms

M.A. Lieberman, J. Kim, J.P. Booth, J.M. Rax and M.M. Turner,

SEMICON Korea Etching Symposium, p. 23 (2003)



RESULTS FOR 27.1/2 AND 60/2 MHz

• Aa = 544 cm2, Ab = 707 cm2, L = 1.6 cm, p = 190 mTorr argonfh = 27.1 MHz fh = 60 MHz

0 100 200 300 400 5000

5

10

15

x 1016

Ph (W)

n0 (

m-3

)

Vl = 900 VVl =1200 VVl =1500 VVl =1800 V

0 100 200 300 400 5000

200

400

600

800

1000

1200

1400

Ph (W)

Eia

(V

)

0 100 200 3000

0.5

1

1.5

2

x 1017

n0 (

m-3

)

Vl = 900 VVl =1200 VVl =1500 VVl =1800 V

0 100 200 3000

200

400

600

800

1000

1200

Eia

(V

)

Ph (W)

Ph (W)

The high/low frequencydecoupling is goodbut not perfect

The high/low frequencydecoupling is betterbut not perfect



THREE UNRESOLVED PHYSICS ISSUES



1. DUAL FREQUENCY STOCHASTIC HEATING?

• High frequency sheath oscillates over nearly uniform ion density

Bulk plasma

High frequency homogeneoussheath motion Ions

Electrons

Wall

Stochasticheating

Bulk plasma

High frequency Child lawsheath motion

Ions Electrons

Wall

Stochasticheating

Bulk plasma

High frequency step modelsheath motion

Ions

Electrons

Wall

Stochasticheating

(Lieberman, 1988) (Lieberman, 1988) (Kaganovich, 2002)Sstoc = 0 Sstoc ≠ 0 Sstoc ≠ 0

Dual frequencysheath

Bulk plasma

High frequencynear-homogeneous

sheath motion

Ions Electrons

Stochasticheating???

Wall



2. COUPLING OF VOLTAGES?

• Additive assumption?Ei ≈ 0.41 (Vl + Vh)

• Current-driven homogeneous model gives

Ei =38

Vl + Vh − 23

VlVh

Vl + Vh︸︷︷︸

cross termWorst case for crossterm is 1

6 (Vl + Vh) when Vl = Vh

• Cross term for Child law sheaths?Voltage-driven discharges?Real systems driven through matching networks?



3. ION POWER SUPPLIED BY SOURCES?

• For single frequency capacitive discharges

Electron power =Pe ∝ ω2Vrf ∝ n

Ion power =Pi ∝ ω2V 2rf ∝ nVrf � Pe

⇒ RF power source supplies Pe and Pi

• For dual frequency discharges

Electron power =Pe ∝ ω2hVh ∝ n

Ion power =Pi ∝ ω2Vh (Vl + Vh) � Pe

High frequency source supplies Pe

Which sources supply the ion power? Is Pil/Pih = Vl/Vh?



CONTROL OF ION ENERGY DISTRIBUTION



TRIPLE FREQUENCY RF DISCHARGE

• IEDF depends on rf bias period/ion transit time across sheath

~

~

Vl1

Vh

+

–

+

–

~ Vl2

+

–

Power

Bias

(60–160 MHz)

(13.6 MHz)

(1 MHz)

Ions

f(E)

E 1

∆

EE 2

E i

Coburn and Kay (1972)



WIDTH OF ION ENERGY DISTRIBUTION

(Argon, 600 V rf bias, 1011 cm−3 density)

0.0 0.5 1.0 1.5 2.0 2.5

0.00

0.25

0.50

0.75

1.00

1.25

theory

simulations

theo

ry

x

x

13.56 MHz

1 MHz

RF bias period/ion transit time across sheath

Ion

ener

gy w

idth

Ion

ener

gy m

axim

um

1. V. Georgieva, A. Bogaerts, and R. Gijbels, Phys. Rev. E 69, 026406 (2004)

2. J.K. Lee, O.V. Manuilenko, N.Y. Babaeva, H.C. Kim, and J.W. Shon,

Plasma Sources Sci. Technol. 14, 89 (2005)3. S. Shannon, D. Hoffman, J.G. Yang, A. Paterson, and J. Holland,

submitted to Journal of Applied Physics (May 2005)



STANDING WAVE AND SKIN EFFECTS



HIGH FREQUENCY ELECTROMAGNETIC EFFECTS

• High frequency and large area ⇒ standing wave effects

• High frequency ⇒ high density ⇒ skin effects

• Previous studies of capacitive discharges mostly based on electro-statics, not full set of Maxwell equations

=⇒ no standing wave or skin effects

M.A. Lieberman, J.P. Booth, P. Chabert, J.M. Rax, and M.M. Turner,

Plasma Sources Sci. Technol. 11, 283 (2002)



CYLINDRICAL CAPACITIVE DISCHARGE

Consider only the high frequency source

~+–

2R

s

s

2d 2lPlasma

Vh

z

r

Sheath

Sheath

Fields cannot pass through metal plates

(1) Vs excites radially outward wave in top vacuum gap(2) Outward wave excites radially inward wave in plasma



BASIC PHYSICS

• Plasma is (weakly) lossy dielectric slab

κp = 1 −ω2

p

ω(ω − jνm)where

ωp = (e2ne/ε0m)1/2 = plasma frequencyνm = electron-neutral collision frequency

• TM modes with Hφ ∼ ejωt

• Maxwell’s equations∂Hφ

∂z= −jωε0κpEr (inductive field)

1r

∂(rHφ)∂r

= jωε0κpEz (capacitive field)

∂Er

∂z− ∂Ez

∂r= −jωµ0Hφ

• Choose uniform density ne and sheath width s (arbitrary choice!)• Solve with appropriate boundary conditions



SURFACE WAVE MODE

• Power enters the plasma via a surface wave mode:

PlasmaStandingwave

Decay(weak)Decay

Surface Wave Mode

λδ

s2ds

• Radial wavelength for surface wave (low density limit):

λ ≈ λ0√1 + d/s

∼ λ0

3

with λ0 = c/f the free space wavelength• Axial skin depth for surface wave:

δ ∼ c

ωp∝ 1√

n

• There are also evanescent modes leading to edge effects near r = RDualHighFreqPhys8May05 25


POWER DEPOSITION VERSUS RADIUS AT 13.56 MHz

• R = 50 cm, d = 2 cm, s = 0.4 cm (λ ≈ 9–10 m)• Pcap (dash), Pind (dot) and Ptot (solid) as a function of r

ne = 109 cm−3

δ = 16.7 cmne = 1010 cm−3

δ = 5.3 cm

Pow

er/a

rea

r (cm)0 25 50

0.5

1

0

Total

Capacitive

Inductive

Edgeeffect

Pow

er/a

rea

r (cm)0 25 50

0.5

1

0

Total

Capacitive

Inductive

Edgeeffect

Small stand-ing wave andskin effects

Large skin ef-fect



POWER DEPOSITION VERSUS RADIUS AT 40.7 MHz

• R = 50 cm, d = 2 cm, s = 0.4 cm (λ ≈ 3 m)• Pcap (dash), Pind (dot) and Ptot (solid) as a function of r

ne = 109 cm−3

δ = 15.9 cmne = 7×109 cm−3

δ = 6.3 cm

Pow

er/a

rea

r (cm)0 25 50

0.5

1

0

Total

Capacitive

Inductive

Edgeeffect

Pow

er/a

rea

r (cm)0 25 50

0.5

1

0

Total

Capacitive Inductive

Edgeeffect

Large stand-ing wave ef-fect

Standing waveand skin ef-fects cancel



EXPERIMENTAL RESULTS FOR STANDING WAVES

20×20 cm dischargep = 150 mTorr50 W rf power

The standing wave ef-fect is seen at 60 MHzand is more pronouncedat 81.36 MHz

A. Perret, P. Chabert, J-P Booth, J. Jolly, J. Guillon and Ph. Auvray,

Appl. Phys. Lett. 83, 243 (2003)



TRANSMISSION LINE MODELSFOR STANDING WAVES



TRANSMISSION LINE MODELS

• Parallel plane transmission line model with local (in x) particle andenergy balance to determine density ne(x) and sheath width sm(x)

x = lx = 0

~ Vrf

+

–

EdgeMidplane

+

–

V

I

Z' = jωL'

Y'=G'(V)+jωC'(V)

Sheath capacitance Ion energy loss

Stochastic heating

Ohmic heating

Plasma inductance

+

–

V(x)

dx

P. Chabert, J.L. Raimbault, J.M. Rax, and M.A. Lieberman,

Phys. Plasmas 11, 1775 (2004)



STANDING WAVE EFFECTS

• Transmission line equations are nonlinear:dV

dx= −Z ′I

dI

dx= −Y ′(|V |)V

• Most important nonlinearity is sheath thicknessvariation with voltage

• Example of collisionless sheath with stochastic heating

C ′(V ) ∝ V −1/4

⇒ wavelength λ of the standing wave

λ ∝ V1/8rf

ω3/2 l1/2



SUPPRESSION OF STANDING WAVE EFFECTS

• Shaped electrode (and diel plate) eliminate standing wave effects

• Increased overall thickness in center compared to edge keeps voltageacross discharge section constant

• The electrode shape is Gaussian, independent of the plasma prop-erties (for R � λ0)1. L. Sansonnens and J. Schmitt, Appl. Phys. Lett. 82, 182 (2003)2. P. Chabert, J.L. Raimbault, J.M. Rax, and A. Perret,

Phys. Plasmas 11, 4081 (2004)



EXPERIMENTAL CONFIRMATION

• 5–250 mTorr argon, 50–300 W

H. Schmitt et al, J. Appl. Phys. 95, 4559 (2004)



SKIN EFFECTS

• Skin effects =⇒ radial nonuniformities at high densities when

δ <∼ 0.45√

d R

δ ∝ 1√n

= collisional or collisionless skin depth

d = bulk plasma half-thicknessR = discharge radius

• Control of skin effects?

1. M.A. Lieberman, J.P. Booth, P. Chabert, J.M. Rax, and M.M. Turner,

Plasma Sources Sci. Technol. 11, 283 (2002)

2. P. Chabert, J.L. Raimbault, P. Levif, J.M. Rax, and M.A. Lieberman,

submitted to Physics of Plasmas, May 2005



CONCLUSIONS

• Dual frequency capacitive discharges are used extensively for inte-grated circuit processing

• The conditions for independent control of ion flux and ion energyhave been determined

• Global (volume-averaged) discharge models have been used to de-termine the discharge equilibrium

• Ion energy distributions might be controlled using two different lowfrequency bias sources

• High frequency standing wave and skin effects can produce plasmanonuniformities

• Transmission line models can be used to understand and controlstanding wave nonuniformities


Documents

M.A. Lieberman PHYSICS OF - EECS at UC Berkeleylieber/dualfreqphys8May05.pdf · University of California, Berkeley P L A S M A SHEATH PHYSICS UNDERSTANDING • Analytical solution