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February 13, 2013

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Landau resonant heating using standing waves excited by a distributed circuit for

electron velocity distribution control

K. J. Bowers and C. K. Birdsall

Electrical Engineering and Computer Science Department

University of California at Berkeley

Berkeley, CA 94720

(Dated: September 21, 2001 - Submitted; December 18, 2001 - Revised)

The use of waves applied by a periodic electrode to controllably modify the particle velocity

distribution in a plasma is explored. The applied waves induce a strong wave-particle interaction

analogous to Landau damping. However, as the waves are powered by an external circuit, the

interaction continues until a new plasma equilibrium is established. At saturation, the velocity

distribution is enhanced above the phase velocity of the applied wave and suppressed below. The

use of this technique to selectively heat the electron high energy tail and to enhance the electron-

impact ionization rate is demonstrated using particle-in-cell simulation.

PACS numbers: 52.40.Fd, 52.35.Fp, 52.20.Fs, 52.65.Rr

Keywords: Landau damping, resonant heating, electron-impact ionization, electron velocity distribution,

periodic circuit, slow wave structure

If the wave is imposed on the plasma such that it does

I. INTRODUCTION

not damp, the wave-particle interaction continues until

a new equilibrium is established. Here it is shown that

It is well appreciated that a particle moving nearly syn-

such an applied wave may be used to controllably ma-

chronous with a wave gains energy if its velocity is just

nipulate the electron velocity distribution function at se-

below the wave phase velocity. Conversely, just above

lective energies. This process is referred to as Landau

the phase velocity, a particle loses energy. This wave-

heating in this article. As many important reactions in

particle interaction is the underlying mechanism for Lan-

low-temperature plasma discharges are very sensitive to

dau damping. Landau damping is extensively discussed

the distribution function (for example, electron-impact

in many texts see for example Nicholson.1 Useful re-

ionization and excitation)2, such an applied wave may

sults from the theory of Landau damping are given in

be used to electrically modify reaction dynamics.

this article as necessary.

In an unmagnetized driftless uniform Maxwellian If the applied wave is resonant with natural plasma

plasma, Landau damping predicts that Bohm-Gross waves, Landau heating is greatly enhanced Landau res-

waves (Langmuir oscillations) will spontaneously damp onant heating (LRH). For selectively heating electrons

(except at the trivial point k = 0), even in the absence of in an unmagnetized uniform plasma, the obvious wave

collisions. The damping goes as e d t and, for waves with to use is the Bohm-Gross wave. A minimal 1d particle-

a positive phase velocity, is given by: in-cell (PIC) simulation is conducted here which demon-

strates LRH of the electron high energy tail in a low-

p f temperature plasma. The simulation ignores collisional

d = (1)

2 v

2k e ects. Over two orders of magnitude enhancement of

v = /k

the high energy tail about the target heating energy is

Here, f is the electron velocity distribution function observed in this idealized model.

( f dv = 1), v is the velocity parallel to the wave vec- In bounded non-uniform plasmas as found in real de-

tor, is the wave frequency, k is the wave number and vices, it is shown here that the most favorable wave

p is the electron plasma frequency. A small ionic con- to use for LRH is the electron series resonance (ESR)

tribution to the damping was neglected and this assumes surface wave. Two 2d PIC-MCC (Particle-In-Cell with

the wavelength is much larger than the electron Debye Monte-Carlo Collisions) simulations of a bounded argon

length ( d ). The damping occurs because there are more plasma are conducted to demonstrate LRH in a more

electrons gaining energy from the wave than losing en- realistic model. In the rst simulation, a periodic elec-

ergy to it. It should be noted that only slow waves (that trode designed to enhance the ionization rate with the

is, waves whose phase velocity is slower than light) can ESR surface wave is used to sustain a plasma discharge.

directly participate in a wave-particle interaction. The second simulation is conducted using identical ini-

tial plasma conditions, device dimensions, frequencies

and amplitudes but no periodic electrode. Here, the

ionization rate is greatly reduced (so much so that the

Electronic address: *****.*.******@*******.***; Presently at

plasma quickly extinguishes). Unlike the 1d simula-

Agere Systems, 700 Mountain Avenue Room 1E332, Murray Hill,

tion, these simulations incorporate important collisional

NJ 07974

Electronic address: ********@****.********.*** processes namely, electron-impact ionization and exci-

tation, electron-neutral scattering, ion-neutral scattering

f(v) (a) Initial

and ion-neutral charge exchange. The cross sections for

these processes were taken from experimental data so ef-

fects such as the Ramsauer minimum are included.

PIC-MCC simulation3 5 is used as it captures critical

kinetic e ects. Fluid simulation techniques have di -

culty modeling kinetic e ects like Landau damping. PIC-

MCC simulation allows Landau damping to be modeled

self-consistently with realistic atomic physics.

Landau heating of electrons in low-temperature un-

magnetized planar plasma devices is emphasized in this

article as such plasmas are useful for materials process- v

ing purposes. This technique could be easily applied to

f(v) (b) Saturated

a cylindrical geometry for enhancing or suppressing ion-

ization or excitation reactions in a ourescent lighting

discharges (possibly yielding devices with improved e -

ciencies). This technique could also be fruitfully applied

to other species and other waves in a plasma. Such may

have applications in fusion, wake eld and space plasma

devices.

II. WAVE-PARTICLE INTERACTION IN AN

APPLIED WAVE

f(v) (c) Thermal Eq.

A. Traveling Wave

The interaction of electrons in a plasma with a trav-

eling wave has been extensively studied. For illustra-

tive purposes, consider an unmagnetized uniform charge

neutral plasma in which the electrons are initially drift-

less Maxwellian distributed. A slow phase velocity wave

( /k

the plasma by some unspeci ed external circuit.

In this idealized model, the applied wave potential may

be expressed as:

/k v

V (z, t) = V0 cos ( t kz ) (2)

Over short time scales (relative to the Landau bounce FIG. 1: Evolution of the Velocity Distribution in a Travel-

time given below), particles just below(above) the wave ing Wave: These graphs qualitatively show how an initially

Maxwellian velocity distribution function reacts to an applied

phase velocity gain(lose) energy. Since there are more

wave. The thermal equilibrium state is usually not reached

particles below the phase velocity than above, overall,

for the plasmas of interest here.

the electrons gain kinetic energy from the wave. This

process is directly analogous to the linear stage of Landau

damping.

However, as the individual potential wells are not har-

Over medium time scales (comparable to the Landau

monic oscillators, trapped electrons with di erent initial

bounce time), the e ect of particle trapping must be

positions and velocities have di erent oscillation frequen-

considered. In a frame moving synchronous with the

cies. This causes the trapped electrons to phase-mix. In

wave, the wave appears as a stationary periodic poten-

the lab frame this leads to a attening of the velocity

tial. Nearly synchronous electrons are trapped in the

distribution function about the wave phase velocity. The

potential wells and oscillate. The Landau bounce time is

width of the attened region is dictated by the amplitude

the period for linear oscillations in the potential wells:

of the wave eld (which determines the velocity range of

trapped particles):

4 2 me

= (3)

ek 2 V0 16eVo

v,tr = (4)

Here, e is the magnitude of electron charge and me is the

electron mass. Once the velocity distribution function slope is zero

at the wave phase velocity, the wave-particle interaction a attened velocity distribution about the positive and

saturates and no net energy transfer occurs between the negative phase velocities associated with the oppositely

applied wave and the plasma. Equivalently, the Landau directed waves. The width of the attened regions is

damping rate (1) is zero. This process is analogous to suitably reduced (but there are two such regions):

the non-linear stage of Landau damping.

If su ciently strong collisional mechanisms are present 8eVo

v,st = (7)

to allow the electrons to come to thermal equilibrium me

with the traveling wave, equilibrium thermodynamics

Thus, the interaction of electrons in a plasma with a

predicts the long term evolution of the electrons. The

standing wave is expected to result in a non-thermal equi-

electron distribution function is proportional to the

librium electron velocity distribution. For positive elec-

Boltzmann factor in the wave frame. Thus, in the

tron velocities, the electron velocity distribution function

lab frame, the electrons have a periodic spatially non-

just below the wave phase velocity is suppressed com-

uniform density which travels with the applied wave. The

pared to the initial Maxwellian while the distribution

electrons also acquire a net drift from the momentum

function is enhanced just above the phase velocity. The

transferred by the wave eld:

equivalent distortion also occurs for negative velocities.

me (v /k ) 2eV0 cos ( t kz )

Ne (z, v, t) exp

2kB Te

C. Resonant Waves

(5)

Here, Ne is the electron phase-space density, kB is Boltz-

mann s constant and Te is the nal electron temperature. If the applied standing wave frequency and wavelength

Figure 1 shows qualitative graphs of the evolution of is chosen carefully, the applied wave resonantly excites

the distribution function in an applied wave. Figures natural waves in the plasma. The net impact of this is

1a, 1b and 1c correspond to the short, medium and long that V0 is actually much larger than the eld applied by

time scales. In the partially-ionized low-temperature low- the external circuit.

pressure plasma discharges that are the focus here, colli- This is very useful if a standing wave is to be em-

sions with the background neutral species (usually at rest ployed in a practical device as very small applied elds

in the lab frame) and the low electron collision frequency may be used to e ect velocity distribution modi cations.

relative to the high electron loss rate to the boundaries Also, the high e ective V0 has the impact of reducing the

strongly inhibit thermalization of the electrons with the bounce time. Shorter bounce times are preferable as it

traveling wave. Thus, the thermal equilibrium state is reduces the size of the plasma device necessary for LRH

usually not reached. (if the plasma device is too small, the electrons interact

with walls before they have su cient time to phase-mix

and atten the velocity distribution function).

B. Standing Wave

A standing wave is the sum of two oppositely directed III. DEMONSTRATION OF LANDAU

traveling waves. RESONANT HEATING

V (z, t) = V0 sin t sin kz

Figure 2 shows the wave number and frequency to in-

1 1

duce Bohm-Gross wave LRH about 3vt in a uniform un-

= V0 cos ( t kz ) V0 cos ( t + kz ) (6)

2 2 magnetized plasma. vt is the thermal velocity of the

electrons ( kB Te /me ). Due to the large-k asymptotic

Given the same initial plasma and wave frequency con-

behavior of Bohm-Gross waves, these waves may only in-

ditions as above, over short and medium time scales, the

duce LRH for velocities v > vt 3. Since modifying the

electron behavior is largely unchanged from the travel-

energetic tail of the velocity distribution is the goal, this

ing wave discussion. The same processes occur for the

presents no problem.

z traveling wave (v /k ) as for the + traveling

wave. The trapping process for z wave is not signif-

A 1d periodic electrostatic PIC simulation of a plasma

with average density n0 = 0.745 1011 /cm3 was con-

icantly in uenced by the z traveling wave as the fre-

quency of the z wave in a frame moving synchronous

ducted. The initial electron velocity distribution was

with the z wave is well above the electron plasma fre-

Maxwellian with a temperature of Te = 2eV . Corre-

quency. Consequently, electrons trapped by the z wave

spondingly, for a target Landau heating velocity of 3vt

(target energy of E = 9eV ), the necessary wavelength and

do not respond strongly to the z wave.

For a standing wave, there is no frame in which the frequency are = 0.588mm and f = 3.00GHz respec-

total wave eld is time independent. Thus, the ther- tively. The amplitude of the applied wave is V0 = 0.2V ;

modynamic argument for the traveling wave case does this is much smaller than the mean electron kinetic en-

ergy ( Te /2). The ions are singly-ionized hydrogen.

not apply for a standing wave. As a result, it is ex-

pected that the wave-particle interaction saturates with The initial ion temperature is Ti = 300K .

Landau Resonant Interaction ( /k = 3vt) Average Kinetic Energy Per Electron

1.35

Bohm Gross Dispersion

1.3

Target Phase Velocity

1.4

1.25

Applied Wave Frequency

Energy (eV)

and Wave Number to Use 1.2

/ p

1.2

1.15

Landau

1.1 Bounce

Time

1.05

**-**-**-**-**-**

0 0.2 0.4 0.6 pt / 2

k d

FIG. 4: Average Kinetic Energy per Particle: This gure

FIG. 2: Determination of the Wave Frequency and Number: shows the average kinetic energy per particle during LRH.

This graph shows how to determine the wave number and fre- The wave-particle interaction rapidly heats the plasma and

quency to apply to enhance the electron velocity distribution quickly saturates.

function using a Bohm-Gross wave in a uniform plasma.

Electron Vel. Dist.

odic potential Poisson solver is described in Hockney and

10 Eastwood4 . A thorough numerical analysis of this type

simulation may be found in either Birdsall and Langdon3

or Hockney and Eastwood4 .

Figure 3 shows the electron velocity distribution be-

Arb. units

10 fore and after the LRH saturates. Flattened regions

about the phase velocities of the applied standing wave

are seen. The widths of the two regions are much larger

than the applied wave amplitude alone would suggest.

Final

As expected, the applied eld excites a large standing

Initial

10 Bohm-Gross wave. At the high energy edge of the at-

Maxwellian Fit

tened regions, the density of high energy tail electrons

Target Velocity

is enhanced by over two orders of magnitude. The bulk

electrons are actually mildly colder after the LRH sat-

3 2 1 0 1 2 3

urates than the initial and the Maxwellian t velocity

Velocity (m/s) 6

x 10 distributions.

Figure 4 shows the average kinetic energy per particle

FIG. 3: Landau Resonant Heating, Before and After: This

versus time. LRH rapidly heats the plasma and saturates

graph shows the initial and nal electron velocity distribu-

tion functions. As expected from Landau damping theory, in a couple of hundred wave cycles. The rapid initial

the velocity distribution function is attened about the wave rise corresponds to the linear stage of Landau damping.

phase velocity. Over two orders of magnitude enhancement Once the velocity distribution function is attened at the

of the high energy tail for electrons is seen. wave phase velocity, the wave-particle interaction satu-

rates and the kinetic energy growth levels o . It should

be noted that if collisional and boundary interaction pro-

The simulation is periodic with a length of 1cm (17 cesses were included in the simulation, this graph would

periods of the applied wave). The simulation timestep have di erent characteristics as superthermal particles

is t = 0.2/ p and the mesh spacing is x d . The would be lost quickly to the boundaries and/or lose their

macro-particle density was 10000 particles per cell on av- energy by collisions. Here, kinetic energy gained from

erage which gives very good resolution of the high en- the standing wave is not dissipated as the trapped syn-

ergy tail. The simulation code uses linear interpolation chronous electrons oscillate incoherently in the potential

and weighting for the explicit leap-frog particle advance wells. The ions do not participate strongly in this inter-

(the same algorithms in the code PDP16 ). The peri- action given their high mass.

0o 180o Electrode

p ESR

V V

Surface

Wave

v = vt 3 2

Target

v = 2Eiz me Interaction

/ k = 2Eiz / me

180

FIG. 6: Landau Resonant Heating for Enhanced Ionization:

d These graphs show the interaction of the periodic electrode

with the ESR surface wave7 use to pull electrons above the

ionization threshhold. These gures are not to scale. In par-

FIG. 5: Planar Device with a Periodic Electrode: This gure

ticular, the periodic electrode resonance frequency r greatly

shows a planar device utilizing a periodic electrode at the

exceeds p such that the two possible interaction points are

top. The two voltage sources are driven 180 out-of-phase.

e ectively degenerate and occur at k 2 / .

One period of the corresponding 2d3v PIC-MCC simulation

is shown below. A snapshot of the applied electric eld in

vacuum is also qualitatively shown. The eld is well suited to

exciting the ESR surface wave. between the sources driving the two teethed electrode

segments. Since only a few periods of the device are sim-

ulated, an electrostatic simulation is appropriate. How-

ever, if the entire device is modeled, the slow wave struc-

IV. ENHANCEMENT OF ELECTRON-IMPACT

ture nature of the periodic electrode needs to be taken

IONIZATION THROUGH STANDING SURFACE

WAVES EXCITED BY A PERIODIC into account as does the electromagnetic long wavelength

ELECTRODE behavior of surface waves.7

The ESR surface wave is favorable for LRH as it is

Figure 5 shows a qualitative schematic of a planar the dominant high frequency natural mode in a bounded

plasma device with a periodic electrode and one period plasma and it has elds which are concentrated near the

of the corresponding 2d3v PIC-MCC simulation. This is bounding walls. Thus this wave interacts strongly with

not the only structure useful for LRH but it does allow properly designed periodic bounding electrodes. The

for a direct comparison to a discharge without a peri- mode cuto is largely insensitive to the plasma temper-

ature; vgr vt 3/2 is the mode group velocity.7 Thus

odic electrode. This is done by changing the phase shift

(a) Ionization (Regular Electrode) x 10

0.7

Cumulative Rate (#/cm3s)

0.6

1.5

0.5

z (cm)

0.4 1

0.3

0.2 0.5

0.1

0 0

0 0.5 1 1.5 2 2.5 3

x (cm)

(b) Ionization (Periodic Electrode) x 10

0.7

Cumulative Rate (#/cm3s)

0.6

1.5

0.5

z (cm)

0.4 1

0.3

0.2 0.5

0.1

0 0

0 0.5 1 1.5 2 2.5 3

x (cm)

FIG. 7: Comparison of the Cumulative Ionization Rates: The periodic electrode simulation shows much more ionization than

the regular electrode simulation (note the range of the scales too). The ionization rates shown are accumulated over an ion

sound speed transit time. Since the regular electrode plasma is rapidly decaying due to insu cient ionization, the cumulative

ionization rate is skewed from the instantaneous ionization rate.

this mode is preferable to use over Tonks-Dattner sur- such that particles below the ionization energy in argon

(Eiz 15.76eV ) are pushed above it. The target phase

face wave modes (whose cuto frequencies are very tem-

velocity is 2Eiz /me = 2.35 106 m/s. From f = v,

perature sensitive) or Bohm-Gross body modes (whose

elds are concentrated in the plasma bulk). Sustaining the electrode period is 0.75cm. Since the applied

a plasma with an ESR surface wave is analogous to the elds decay into the bulk plasma and since the electrode

ESR discharges investigated by Bowers et al .8 also supplies the power necessary to sustain the plasma,

a larger applied eld is used than the 1d simulation; the

Figure 6 shows graphically how to design the periodic

periodic electrode was driven with 10V (still very small

electrode to modify the distribution function such that

compared to other types of discharges).

electron-impact ionization is enhanced. A 2d3v PIC-

A second simulation for comparison purposes was also

MCC simulation of an initially decaying di usive pro le

conducted using the same initial plasma, device dimen-

argon plasma was conducted to demonstrate this. The

sions, wave frequencies and wave amplitudes. However,

initial peak plasma density was 2 1010 /cm3 with an ap-

the two teethed electrode plates were operated in-phase.

proximately sinusoidal pro le between the two bounding

Thus, this discharge is the same as a planar capacitively

planes separated by 3cm. The initial electron tempera-

coupled discharge operated near the ESR frequency.

ture was 2eV and the initial ion / neutral temperature

was 300K . The neutral pressure was a low 3mT orr. For The code used for this simulation is the same code de-

scribed in Bowers (but includes Monte-Carlo collisions).9

a sinusoidal pro le, the ESR surface wave has a frequency

of p /4.7 Accordingly, the applied frequency was cho- Bi-linear interpolation and weighting is used for the ex-

sen to be 315M Hz . The electrode period was chosen plicit leap-frog particle advance. The elds are solved

using a 5-point nite di erencing of Poisson s equation. cies and wavelengths to excite a Bohm-Gross wave in a

These algorithms are similar to the code PDP2.10,11 uniform plasma and an electron series resonance surface

Figure 7 shows the cumulative ionization rate within wave in a non-uniform bounded plasma were calculated

one electrode period by location for the two simulations approximately. Further work remains to fully optimize

accumulated over an ion sound speed transit time. As the interactions and electrode structure. Such an opti-

is seen, essentially no ionization takes place in the sim- mization should take into account the change in plasma

ulation without the periodic electrode. Not surprisingly density and velocity distribution due to the applied wave

then, the plasma in that simulation rapidly decays; over eld instead of computing the electrode parameters from

half the plasma is lost in an ion sound speed transit time an assumed initial Maxwellian plasma. Another intrigu-

and the electron temperature drops to 1.3eV as the ing optimization is to use a multi-frequency drive to a

electrons thermalize with the ions and neutrals. The ion- exert very precise control over the electron velocity dis-

ization rate in the simulation with the periodic electrode tribution.

is su cient to maintain the plasma though; after a short LRH of the high energy tail for the electrons was suc-

period of decay, the periodic electrode comes into reso- cessfully demonstrated with a 1d PIC simulation un-

nance with the ESR surface wave (analogous to the lock- magnetized high-density low-temperature plasma. A 2d

on process8 ). The average kinetic energy per electron PIC-MCC simulation successfully demonstrated the use

more than doubles; an e ective electron temperature of of a periodic electrode to greatly enhance the rate of

4.1eV is observed. The velocity distribution is not electron-impact ionization collisions in low-temperature

Maxwellian but does not show as clear a plateau as the low-pressure unmagnetized plasmas.

collisionless 1d simulation. This is because ionization col- It should be noted that to scale LRH to even higher

lisions deplete the tail while LRH populates the tail. Also density low-temperature plasmas than used here requires

velocity components transverse to the wave do not un- impractically small electrode periods. For example, to

dergo LRH and must be considered in a full analysis of enhance ionization in argon using the same electrode

the ionization enhancement. structure used in this article requires a electrode period

on the order of tenths of millimeters for plasma densities

of 1012 /cm3 and temperatures of a few eV . However, it

V. SUMMARY may be possible to use higher order spatial harmonics of

a periodic electrode to induce LRH. Alternatively, Lan-

dau heating at these high densities and low temperatures

In this article, the use of waves applied by a periodic

can still be pursued if lower RF frequencies (with corre-

electrode to modify the particle velocity distribution in a

spondingly larger electrode periods) are used. Such an

plasma was explored. The applied waves induce a strong

interaction is non-resonant and requires a larger applied

wave-particle interaction analogous to Landau damping.

eld.

However, as the waves are powered by an external cir-

cuit, this Landau heating continues until saturation. At

saturation, the particle velocity distribution is enhanced

Acknowledgments

above the wave phase velocity and suppressed below.

Landau resonant heating (LRH) was emphasized. In

LRH, the applied wave is designed to excite natural slow This work was supported by the Fannie and John Hertz

phase velocity modes in a plasma. The necessary frequen- Foundation Fellowship Program.

D. R. Nicholson, Introduction to Plasma Theory (John Wi- Birdsall, J. Comp. Phys. 104, 321 (1993).

ley and Sons, Inc., New York, NY, 1983). K. J. Bowers, Slow Phase Velocity Electron Surface Waves

M. A. Lieberman and A. J. Lichtenberg, Principles of in Unmagnetized Bounded Plasmas: Part I: Modeling,

Plasma Discharges and Materials Processing (John Wiley submitted to Phys. Plasmas.

and Sons, New York, NY, 1994). K. J. Bowers, W. D. Qiu, and C. K. Birdsall, Electron

C. K. Birdsall and A. B. Langdon, Plasma Physics via Series Resonant Discharges: Part II: Simulations of Initia-

Computer Simulation (McGraw-Hill Inc., New York, NY, tion, submitted to Plasma Sources Sci. Technol.

1985). K. J. Bowers, Slow Phase Velocity Electron Surface Waves

R. W. Hockney and J. W. Eastwood, Computer Simulation in Unmagnetized Bounded Plasmas: Part II: Simulation,

Using Particles (Institute of Physics Publishing, Philadel- submitted to Phys. Plasmas.

phia, PA, 1988). V. Vahedi, C. K. Birdsall, M. A. Lieberman, G. DiPeso,

V. Vahedi and M. Surendra, Comp. Phys. Comm. 87, 179 and T. D. Rognlien, Phys. Fluids B 5, 2719 (1993).

(1995). V. Vahedi and G. DiPeso, J. Comp. Phys. 131, 149 (1997).

J. P. Verboncoeur, M. V. Alves, V. Vahedi, and C. K.

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