Plasma and Fusion Research: Review Articles

Volume 13, 1101014 (2018)

Review of Helicon High-Density Plasma: Production Mechanism
and Plasma/Wave Characteristics
Shogo ISAYAMA, Shunjiro SHINOHARA1) and Tohru HADA2)
Department of Physics, National Central University, Taoyuan 32001, Taiwan
Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei,
Tokyo 184-8588, Japan
2)
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga-Kohen, Kasuga,
Fukuoka 816-8580, Japan
1)(Received 26 December 2017 / Accepted 8 February 2018)
Helicon plasma is one of the radio frequency plasma source that can generate high-density and lowtemperature plasmas by utilizing the helicon wave, i.e., the electromagnetic whistler wave in a bounded geometry.
Helicon plasma is very useful for various applications due to its extremely eﬃcient production of high-density
plasma. Conversely, many unsolved physical issues remain regarding how an eﬃcient production of the helicon
plasma is realized in laboratories and what determines the density maximum. The past decades of the heliconstudies have revealed that the “Trivelpiece-Gould” wave is responsible for the eﬃcient power absorption. In recent years, the drift-wave type and the parametric-decay instabilities have been extensively studied, by using thelinear magnetized helicon plasma sources. The present helicon study considers these non-linear eﬀects. Consequently, it includes many interests for both industry and research fields. The mechanism of the helicon production
is discussed, based on the several critical physical issues. In addition, some recent topics and the eﬀorts to build
a more refined physical model of the helicon plasma are highlighted.
c 2018 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: helicon plasma, helicon wave, TG wave, neutral depletion, drift-wave instability, parametric-decay
instability
DOI: 10.1585/pfr.13.1101014

1. Introduction
Helicon plasma sources attract keen interest worldwide. These types of plasma sources can generate a uniform, high-density (electron density up to 1020 m−3 ) andlow-temperature (electron temperature from a few to several eV) plasma, under a wide range of parameters, e.g.,
device scale, magnetic field, neutral pressure [1–4]. Dueto these distinctive advantages, the helicon source is usedfor various applications including plasma processing, nuclear fusion, electric thrusters, and space laboratory experiments [5–7].
The helicon plasma is generated by using the helicon wave, i.e., electromagnetic whistler wave in a bounded
plasma. Over the past decades, research on the helicon plasma has revealed some important mechanisms ofeﬃcient plasma production. One part of the eﬃcient
power absorption can be explained by the strong collisional damping of the quasi-electrostatic wave or so called“Trivelpiece-Gould” (TG) wave [8]. The TG wave is excited dominantly near the radio frequency (RF) antenna
and deposits its energy near the plasma surface [9–11],
while the helicon wave penetrates into the core region
and deposits its energy via collisions in the core plasma[9,12,13]. The existence of the TG wave is experimentallyauthor’s e-mail: isayamashogo@gmail.com

verified [14], and the dispersion relation and the analysisof the power balance explain, at least qualitatively, somedistinct features observed in the helicon discharge.
Contrariwise, recent studies suggested that not only
the directly excited waves but also some non-linear parametrically excited waves can contribute to the power absorption and ion heating [15–17]. In addition, the driftwave type instabilities have also been observed in the large
linear helicon devices [18–21]. Drift-waves can cause theanomalous plasma transport across the magnetic field [22],which result in notable losses of particles and energy fromthe confinement regions by influencing the equilibrium
states of the particles and the power balances [18]. Therefore, the present helicon studies need to consider these
non-linear eﬀects on the plasma transport and dischargeequilibrium. Furthermore, for developing the RF plasma
sources with higher eﬃciency and high density, it is inevitable to clarify the mechanism of the neutral depletion[23], which is thought to limit the maximum density of thehelicon source [5, 24, 25].
This review intends to describe a basic understandingof the helicon plasma production mechanism, and highlight some recent topics and remaining critical issues. In
Sec. 2, a brief history of the helicon study is overviewed.
Section 3, describes a simple model of the helicon wave,

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c 2018 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Review Articles

Volume 13, 1101014 (2018)with its dispersion relation and experimental results. The
role of the electron inertia, which adds the second dispersion of the TG wave, is then discussed (Sec. 4). Section 5 examines the resonance and anti-resonance properties of the excited waves in the cylindrical bounded system,
which leads to the non-linear power absorption curves as afunction of the plasma density or the magnetic field. In
Sec. 6, one of the distinct features of the density jumps
observed in the experiments are discussed, based on the
non-linear power absorption curves, and the unsolved issue is also suggested. Section 7 describes the experimental
and theoretical studies of neutral depletion. In Sec. 8, oneof the controversial issues of the plasma transport in the
RF sources are overviewed. Recent studies about the instabilities in the helicon source (Sec. 9) and other notable
experiments (Sec. 10) are reviewed. In Sec. 11, the remaining critical issues are summarized and attempts to clarify
these matters by the self-consistent model are suggested inSec. 12. In Sec. 13, the applications of the helicon source
are introduced. Finally, Sec. 14 provides some concluding
remarks.

eﬃcient ionization was considered unimportant.
Shamrai et al. [11, 39–43] suggest that the strong collisional damping of the TG wave can explain the eﬀective
power absorption. In the following frequency regime,

2. History of the Helicon Study

where c is the speed of light and ν is the electron collision
frequency between the neutrals and ions. Two diﬀerent
wave modes exist; the negative and the positive signs correspond to the helicon wave and the TG wave, respectively.
Considering the density profile shown in Fig. 1 [43], one
may note the three-characteristic density regions. The firstis the low-density region (n < nlow and r > rlow ), where
only the TG is propagating while the helicon wave is nonpropagating; the second is the intermediate-density region(nlow < n < nup and rup < r < rlow ), where both the helicon and the TG waves are propagating; and the third is
the high-density region (n > nup and r < rup ) where both
waves become evanescent. In the typical experimental

The term “helicon” was named by Aigrain in 1960
[26], to describe the whistler wave propagating in solid
metal. The helicon wave in metallic sodium was first observed by Bowers, Legendy, and Rose [27], who also investigated its dispersion relation. The linear relationshipbetween the frequency and the strength of the static magnetic field, observed in Kanai’s [28] experiment, showed
good agreement with the theoretical prediction. In 1965,
Lehane and Thonemann [29] conducted the first study ofhelicon wave propagation in a gaseous plasma. The phasevelocity, the structure, and the decay length of the helicon wave corroborated the theory proposed by Klosenberg,
McNamara, and Thonemann [30].
In 1970, Boswell [31] discovered that high-density
plasma can be generated by exciting helicon waves.
Boswell et al. [1,32] investigated the structures of the helicon waves with theoretical fittings and the average electron
density variation due to the magnetic field and gas pressure. In Boswell’s experiments, however, it was found thatthe eﬀective collision frequency had to be at least 1,000
times greater than the theoretical collision rate to explain
the observed eﬃcient ionization and radial wave profiles.
In 1991, Chen suggested that Landau damping could be
the cause of the eﬃcient energy absorption [2, 33]. Thismechanism was adopted by authors, such as [34, 35] andsupported by the detection of fast electrons by Ellingboe etal. [36] and Molvik et al. [37] using RF-modulated emissions of Ar+ light. However, by using an energy analyzer that accounts for RF fluctuations in plasma potential,
Chen and Blackwell [38] quantitatively demonstrated thatthe number of fast electrons was insuﬃcient to explain thehigh eﬃcient ionization. Thus, the Landau damping forωci  ωLH < ω  ωce  ωpe ,(1)

where ω is the RF driving frequency, ωpe and ωce(ci) are
the plasma and the gyro-frequencies of electrons (ions), respectively, and ωLH is the lower hybrid frequency. The dispersion relation of magnetized cold plasma can be written
as [42],k⊥± 2 =

k 2(1−2γα−2γ2 α2 β2 ± 1−4γα), (2)2γ2 α2 β2where k⊥ and k are the perpendicular and parallelwavenumbers in the background magnetic field, respectively, and
ν
ωpe 2
ωωce N 2
,
β
=
,
γ
=
1
+
i
,
ω
ωce 2 N 2
ωpe 2
ck
,
N=
ω

α=

(3)

Fig. 1 The density profile and the radial variation of transverse
wavenumber in the local approximation. Helicon (H) andTrivelpiece-Gould (TG) branches of the dispersion curve[43].

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conditions [1, 44, 45], the damping length of the TG wave
is much shorter than for the plasma column k⊥+,imag r0  1,
where k⊥+,imag is the imaginary part of the k⊥+ , and r0 is theplasma radius, while that of the helicon wave is longer thanthe plasma column k⊥+,imag r0 < 1 except for in the highdensity region (n ∼ nup ), where the damping length of thehelicon wave becomes comparable to that of the TG wave.
Therefore, the collisional damping of the TG wave plays
a dominant role on the surface power absorption while thehelicon wave is important for power absorption in the central high-density region [9, 12, 13, 46].
The quasi-electrostatic TG wave is excited by twomode conversion mechanisms [43]. The first is the mode
conversion that occurs at the plasma surface. Since theradial current jr ≈ Er + i(ω/ωce )Eθ must vanish at theplasma surface due to the boundary condition, the strong
radial electric field Er is generated to balance (ω/ωce )Eθ ,and the TG wave is consequently excited. The second is
the bulk mode conversion, which occurs near the mode
conversion surface (MCS) where the helicon and the TG
wave branches merge at n = nup as shown in Fig. 1. The
helicon wave penetrates into the high-density region due to
its weak damping rate, and when the helicon wave reachesn = nup , it is reflected and naturally transformed to thebackward propagating TG wave from this merging point.
However, when a finite collision is introduced, the eﬃciency of the bulk mode conversion becomes strongly dependent on the magnitude of the collision frequency [13].
The range of the plasma density produced by the helicon source is n ∼ 1016 - 1020 m−3 [47, 48]. There are
believed to be there are three distinct modes of operation for a helicon plasma source: the capacitively coupledplasma (CCP or E) mode [49–51], the inductively coupledplasma (ICP or H) mode [52, 53], and the helicon wavesustained (W) mode. The E–H–W mode transitions areobserved with jumps in the plasma density. Transitions
between the low- and high-density modes are shown by
varying the external magnetic field [1, 44, 54], the RF input power [44, 47, 55–59], and the neutral pressure [58].
The E–H transition occurs at a density of approximately
1016 - 1017 m−3 [48, 52, 60, 61]. This mode transition generally occurs when the collisionless skin depth δ = c/ωpe
is equal to approximately half a plasma radius [48]. The
H–W transition occurs at a density of approximately 1017 1018 m−3 [48]. This mode transition is characterized by thedispersion relation of the helicon wave [47, 48, 57, 59]. Before the density jump, the wave fields are localized near
the antenna with a standing wave character, and the dispersion relation of the helicon wave is not satisfied. After the
jump, the propagating waves are excited and the dispersion
relation of the helicon wave is confirmed.
In recent studies, Kline and Scime [16] suggest a nonlinear coupling mechanism between the helicon wave andthe electrostatic waves. They showed that the helicon
pump wave can decay into the ion-acoustic wave and the
TG wave through parametric decay instability. Altukhov etal. [62] first evidenced this mechanism by the spectral measurements. Krämer et al. [17, 63, 64] presented the threshold of the pump field for this instability, and the growth
rates of the parametric instability, which fit the experimental results reasonably well.
As one of the eminent characteristics of helicon discharge, it is well known that plasma density increases withthe magnetic field strength in an essentially linear fashion [44]. This linear relationship is derived from the simple theory of the bounded helicon wave dispersion relation with fixed k [2]. The observed non-monotonic density
jumps with increases in the magnetic field that arise fromthe mode transition among diﬀerent axial wave modes k
[1, 65]. However, many experiments have observed an unexpected decrease in the plasma density with increases in
the magnetic field strength [55, 66–69]. This may be explained using the framework of low-frequency electrostatic
instability, which can cause anomalous radial diﬀusion andgreatly aﬀect the plasma equilibrium [18]. The candidates
to drive such diﬀusion are the resistive drift-wave instability [70–75] driven by the gradient of the plasma density
and the Kelvin-Helmholtz [76–79] or the Simon-Hoh instabilities [80, 81] driven by the gradient of the spatial potential. Light et al. [18, 82] experimentally demonstrated
good agreement of the theory and verified the existence ofsuch instabilities.

3. Simple Model of the Helicon Wave
The helicon wave is essentially a bounded whistler
wave propagating in a dc magnetic field, and the dispersion relation of the helicon wave is derived from the standard theory of whistler waves in cold plasma. In the basicmodel of the helicon wave, the electron inertia term is neglected in the limit of zero electron mass me = 0, and theaxial electric field Ez also vanishes [2, 83]. The dispersion
relation is given at the frequency range of (1) as follows:
ω2pe
c2 k2
,
=
ωωce cos θω2(4)

where k = (k⊥2 + k2 )1/2 is the wavenumber, θ is the angleof wave propagation relative to B0 , and k = kcosθ is theparallel (axial) wavenumber. The dispersion relation above(4) may be written as
ω n0 eμ0
ω ωpe
=
,
2
k ωce c
k B0
2

k=

(5)

where e and μ0 are the electric charge and the permittivity of free space, respectively. When the wave is confined
to a cylinder of radius a, for cases with an insulating or
conducting boundary at r = a, the boundary conditions forthe waves with k a  1 are derived from the condition of
Br = 0 as
mkJm (k⊥ a) = −k aJ (k⊥ a) ∼ 0 (k a  1),(6)

where Jm is the mth order Bessel’s function. For m = 1, thelowest Bessel root is k⊥ a = 3.83, and therefore, for k 1101014-3

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Fig. 2 Linear relationship between the plasma density and themagnetic field [44].

k⊥ , the dispersion relation can be approximately obtained
as [86]
ω n0
3.83 ω n0 eμ0
=
∝
.
a
k B0
k B0

Fig. 3 Radial profiles of RF current |Jz | at 30 G as measured(dot-solid line), as predicted by theory (solid line), andas calculated from simple helicon theory without the TG
mode (dotted line) [14].

(7)This shows the linear relationship for a given mode (ωand k are fixed) between the density n and the magnetic
field B0 . Figure 2 [44] shows the n − B curve as the RF
power varies with the Boswell-type antenna [35]. When
PRF > 400 - 500 W and B0 > 400 - 500 G, the value of n/B
is approximately constant, as expected from (7). The basicn − B relationship above threshold values of PRF and B0
has been verified by a number of authors such as Boswell,
Porteus, and Ellingboe [47, 84], who suggest that a certain
threshold is related to the discharge mode transition fromthe ICP to the helicon plasma. When the magnetic field isweak and RF power is low in relation to the ICP operation,
the density is not high enough to satisfy (7). If PRF or
B0 is raised, the helicon wave is excited and (7) is satisfied[85]. Chen et al. [86,87] also calculated the wave magnetic
field with the measured density profile, and showed good
agreements with the m = 1 mode radial wave profiles as
expected for the right-hand polarized antenna. Shinohara
et al. [88] also found agreement for the radial wave profilesof the m = 0 helicon mode when excited by a planar spiralantenna.

4. Role of the Electron Inertia
Davis [89] pointed out that, even when the wave frequency is much lower than the electron cyclotron frequency, the electron inertial term must be accounted for in
the generalized Ohm’s law [90]. Boswell [91] first investigated the eﬀects of the electron inertia on the dispersion
relation of helicon waves. The TG wave strongly dissipatesat the collision frequencies typical for the experiments anddeposits its energy in a narrow surface layer, while the helicon waves penetrate the bulk plasma [11,41–43]. The numerical calculations by Mouzouris and Scharer [9], Park et
al. [10], and Shamrai and Shinohara [11] showed that the
major RF power is deposited by the TG wave at the plasma
surface.
Lho et al. [92] detected the TG wave in the low-magnetic field (B0 ∼ 16 - 24 G) helicon discharge andshowed the observed dispersion relation of the TG wave
with theoretical fittings. The radial structures of the TG
wave were clearly detected by Blackwell et al. [14] in anexperiment with a low magnetic field (B0 < 50 G). The
wavelength of the TG wave becomes comparable to that of
the helicon wave with such a low magnetic field, and suchas, the TG wave structure extends far into the interior of
the plasma and its eﬀect on the wave shapes is detectable.
Figure 3 shows the |Jz | profiles measured at B0 = 30 G andn = 3.5 × 1011 cm−3 , compared to curves calculated by the
HELIC code [12, 93] (at the density n = 3.0 × 1011 cm−3 )with and without inclusion of the TG wave. This figure has“wings” structures due to the presence of the TG waves
around the plasma surface regions (|r| > 2), that are notpresent with the helicon wave alone.
The TG wave is excited at the plasma surface (surface
mode conversion) and in the bulk high-density region of
the plasma (bulk mode conversion). The bulk mode conversion occurs near the MCS. Based on the dispersion relation, Kim and Hwang [94] pointed out that for bulk mode
conversion, the collisional dissipation near the MCS plays
an important role. Isayama et al. [13] report the quantitative dependence of the bulk mode conversion eﬃciency
on the collision frequency. Figures 4 (a) and (b) show thespatial variations of the real wavenumber k⊥real and thenormalized imaginary wavenumber k⊥imag x0 (where x0 =7.5 cm is the plasma size) of the helicon wave (red line) andthe TG wave (blue line), respectively. Here, the parabolic
density profile, which peaks at x = 0, is assumed to havethe various values of the collision frequency ν/ω = 0 (solidline), 0.01 (dashed line), and 0.08 (dotted line). Whenthe collision is absent, the dispersion curves of the helicon and the TG wave merge at x = xup , corresponding to
the MCS where the discriminant in (2) is zero. In this case,a helicon wave propagating toward the negative direction
converts to a TG wave as a reflected wave (the group velocity is in the positive direction). When a finite collision

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Fig. 4 (a) Real and (b) normalized imaginary parts of transverse
wavenumbers of the helicon (red) and TG waves (blue)
with ν/ω (ω/2π = 7 MHz) = 0 (solid), 0.01 (dashed), and0.08 (dotted). (c) The dependence of bulk mode conversion eﬃciency Λ on the collision frequency ν/ω [13].

is added, the dispersion curves are separated because thediscriminant in (2) includes the imaginary part and cannot
be zero. Since some amount of the helicon wave energy
directly decays due to the finite collision, the energy thatconverts to the TG wave reduces as ν/ω increases. The
separation between two dispersion curves also increases as
ν/ω does, reducing the eﬃciency of the bulk mode conversion. In this case, some fraction of the helicon wave
energy converts to TG wave energy, and the rest of the helicon wave energy directly dissipates via collision due to its
large imaginary part (k⊥imag x0 ∼ 1 at ν/ω = 0.08) aroundthe MCS. Figure 4 (c) shows the dependence of the bulk
mode conversion eﬃciency Λ = ΓTG /ΓHelicon on the collision frequency ν/ω, where ΓTG and ΓHelicon are the energy
fluxes of the TG and the helicon waves, respectively, whenevaluated near the MCS. One may note that the bulk mode
conversion eﬃciency decreases monotonically when ν/ω
increases.

[39, 40]. The oscillations of any bounded system driven by
an external source can be represented as a superpositionof multiple waves. When the driving frequency is close tosome of the eigenfrequencies of the system, the amplitudeof the specific mode grows sharply. The wave fields are
represented as Fourier series in the longitudinal k and azimuthal m modes. Substituting these wave fields into theMaxwell equations and the cold electron momentum equation give the wave field amplitudes for each component.
The field amplitudes in the cylindrical helicon plasma arerepresented by the superposition of the helicon wave amplitude AH and the TG wave amplitude ATG with the basis
function of the mth order Bessel function Jm (k⊥±r ), wherem is the azimuthal mode number and k⊥± are the transverse wavenumber of the helicon wave (k⊥− ) and the TG
wave (k⊥+ ), respectively.
In typical laboratory experiments, plasma is surrounded by insulating walls made of quarts or glass. Since
the small phase shift introduced by a real glass or quartz
tube does not aﬀect the wave field calculation [95], we cantreat the insulator as having a dielectric constant of unity(vacuum) for simplicity. The waves in the vacuum region
are highly evanescent since the axial wavenumber of thehelicon waves k is typically larger than k0 = ω/c by twoorders of magnitude [93]. Therefore, the wave fields in thevacuum are represented by the superposition of the amplitudes of the TM mode (ATM ) and the TE mode (ATE ),whose profiles are written using the modified Bessel function Im (k r) and McDonnel function Km (k r). The electromagnetic field excited in the plasma by the antenna can beobtained by connecting the wave field in the plasma andin the vacuum across the plasma surface (r = r0 ) where
the antenna current is supplied. The conditions to be imposed are as follows: the continuity of the tangential electric field,
Eθ,z (r = r0 + 0) = Eθ,z (r = r0 − 0),and the Faraday’s law across the plasma surface layer,mBθ (r = r0 + 0) − Bθ (r = r0 − 0) = −i NI,k r(9)Bz (r = r0 + 0) − Bz (r = r0 − 0) = NI,
where I is the antenna current multiplied by some constant.
The four unknown wave amplitudes are the helicon (AH )and TG (ATG ) waves in the plasma and the TM (ATM ) andTE (ATE ) modes in the vacuum, which are found by using
the above four joining conditions. This joining procedureyields the following constants for the helicon and the TG
wave amplitudes [41],AH = −β5. Resonance and Anti-Resonance
Properties of Helicon Discharge

(8)

D+
D−
I, ATG = −β
I.
D
D

(10)

D and D± are the dispersion functions defined as

The concept of the “resonance” wave discharge wasintroduced by Shamrai and Taranov to explain the highpower absorption eﬃciency of the helicon plasma source
1101014-5

D ≡ −D+ DTE + D− Jm (k⊥± r0 ),m
D± = Jm (k⊥± r0 ) +Jm (k⊥± r0 ),
k r0

(11)

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where r = r0 is the plasma radius and DTE is the dispersion
function of the TE mode, which is defined as follows, withthe outer conducting boundary at r = R (R > r0 ) where
tangential electric fields Eθ (r = R) and Ez (r = R) are vanished,
DTE = −δD− + Jm (k⊥− r0 ),δ(r0 , R) = −βΔ01 (r0 , R),
Δ11 (r0 , R)

(12)

where
Km (k r0 )Im (k r0 ),Km (k R)
K (k r0 )Im (k r0 )Δ11 (r0 , R) = Im (k r0 ) − m.
Km (k R)
Δ01 (r0 , R) = Im (k r0 ) −(13)

The dispersion function D in the denominators in (10)determines the resonance properties of the helicon and TG
waves. These fields of the appropriate mode grow nearD = 0; such resonance properties are well established forbounded plasmas. However, the functions D± in the numerators in (10) determine the anti-resonance propertiesof these waves, which are suppressed near D± = 0. In
the absence of the dissipation (ν = 0), when the dispersion functions satisfy D = 0 or D± = 0 in appropriate
modes, these resonance and anti-resonance properties are
displayed in the excited wave fields. However, when a finite dissipation is included (ν  0), the dispersion functions of D and D+ cannot be zero (D  0 and D+  0) due
to the large imaginary part of the transverse TG wavenumber (k⊥+,imag r0  1), destroying the resonance properties of the helicon and TG waves (D = 0) and the antiresonance property of the helicon wave (D+ = 0). Since
the imaginary part of the helicon wavenumber is very small(k⊥+,imag r0  1), only the anti-resonance property of the
TG wave (D− = 0) survives even when the dissipation is
included. This is in the case of a wide gap (R  r0 ) between the plasma (r = r0 ) and outer conducting boundary (r = R), which is normally satisfied in the experiment. Shamrai and Taranov also showed that helicon resonances can survive in case of the narrow gap (R ∼ r0 ) [41].
Arnush and Chen [12, 93] also investigated these properties using the HELIC code, which can calculate the onedimensional wave and power absorption profiles of the helicon and the TG waves for more realistic experimental
conditions, such as non-uniform plasma density, electron
temperature, and neutral gas density with an arbitrary antenna type. These resonance or anti-resonance properties
result in non-monotonic power absorption dependence on
the plasma density and magnetic field strength [41–43].

6. Mode Jumps in Helicon Discharge
In numerous helicon experiments, abrupt changes inthe discharge regime have been observed as the plasma
density jumps. Figure 5 [1] shows an example of the density jump observed by varying the magnetic field. The
plasma density does not change linearly with B0 (along the

Fig. 5 Variation of the average electron density n̄ with B0 . The
solid curve is fitted to the experimental points and thebroken curve is the dispersion relation of whistler wave
[1].

broken curve for plane wave whistler propagation) as indicated by (7). The first jump occurs at B0 ∼ 500 G, where
the electron density is five times greater than before thejump and the central core of the plasma turns light blue
due to strong emission of ArII lines. Further increasing B0
causes a second density jump at B0 = 750 G. Both of these
density jumps accompany the variation of the axial wavelength, meaning that the linear relationship (7) with fixedk is not satisfied.
Chi et al. [59] found the presence of the four jumps byincreasing the RF power, as shown in Fig. 6. The first two
jumps at low RF power are identified as E–H and H–W
transitions. In this case, three diﬀerent W modes (named
W1 , W2 , and W3 ) are seen. These three modes correspondto the diﬀerent helicon wave modes. The wave vectors (k⊥ ,k ) for these modes are (0.33, 0.21), (0.55, 0.21) and (0.80,0.28) in units of cm−3 for W1, W2, and W3, respectively.
They measured the amplitude and phase of the wave magnetic fields along the axis for each mode. For the first twomodes, it was found that wave fields exist near the antenna,
but only small-amplitude axial standing waves were foundin the diﬀusion chamber, indicating that the helicon wave
was not eﬀectively excited. Therefore, in these two discharge modes, the near fields of the antenna play a dominant role on plasma production, as expected for the E andH modes [47]. In the three W modes, large-amplitude axial waves were observed. These wave structures showed
partial modulation in the wave amplitude due to the interference of incident waves reflected from the end plate ofthe diﬀusion chamber, indicating that the three W modes
form a partially traveling and partially standing wave structure. The right panel in Fig. 6 shows the mode transitionsof W1–W2–W3 in k space. The W1 and W2 modes share1101014-6

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Fig. 6 Left panel: ion saturation current (i.e. plasma density) as a function of RF power. Right panel: transition diagram for the wavemodes in k space. The horizontal and vertical axes are total and parallel wavenumbers, respectively. Dispersion relations (2) are
shown for n = 1011 cm−3 to n = 1012 cm−3 in steps of n = 1011 cm−3 [59].

the same axial mode number, but they have diﬀerent perpendicular wavenumbers at diﬀerent plasma densities, as
expected from the dispersion relation (2). In contrast to
the W1 and the W2 modes, the W3 has a higher axial
wavenumber. The E–H mode transition exhibits hysteresis [60, 96, 97], which can qualitatively be understood interms of an electron power balance [60,61,98] wherein the
power absorbed by the plasma has a nonlinear dependence
on the electron density. The equilibrium state plasma densities are determined at the intersections of the two curvesof the absorbed power and the dissipated power, displayedin the plane of the power and density. When there are multiple solutions for a given value of the coil current due tothe nonlinearity of the absorbed power curve, a range ofvalues of the coil current that permit either E or H mode
discharges exist, resulting in the hysteresis feature of themode transition [98]. This dependence is also intrinsic forthe density jump in the helicon plasma. Indeed, the theory predicts that the nonlinearity of the power absorption
curves arises from decreasing the power absorption at theanti-resonance of the TG wave [41–43]. The power balance between the dissipated power and the absorbed power
predicts a stable discharge in several density modes and thepossibility of abrupt transitions between them.
Figure 7 [43] shows the total power absorption anddissipated power dependences of the plasma density (assuming uniformity) for a Nagoya-type antenna at diﬀerent currents that were calculated using the following parameters: background magnetic field B0 = 250 G, RF frequency ωRF /2π = 13.56 MHz, collision frequency ν = 107
1/s (Ar pressure approximately 1 mTorr), plasma radiusr0 = 7.5 cm, conducting cylinder radius R = 10 cm, andthe device length L = 50 cm. The power absorption
curves vary non-linearly with the plasma density. The antiresonance of the TG wave occurs when the boundary valueof the helicon wave Eθ,H has a node at the plasma surface(r = r0 ), that originates from the zero-point of the Bessel
function describing the helicon wave [41]. The boundary
value of the Eθ,H oscillates as density increases since theFig. 7 Absorbed and dissipated power Ploss curves demonstrating direct (2 to 3’) and back (3 to 2’) density jumps, whichshow the origin of hysteresis of density as a function of
RF power which corresponds to antenna current Ia [43].

radial scale of the helicon wave decreases, as confirmed by
the experiment [99]. The two minimums seen in Fig. 7 are
originally related to the first and the second radial modes.
Forming a discharge equilibrium is possible at the intersections of the absorbed power and dissipated power, and thestabilities of the equilibrium states are determined by conducting a stability analysis using a small fluctuation [43].
The stability analysis shows that the equilibrium state isstable against fluctuations increasing the plasma density at
n1 < n < n2 and n3 < n and unstable at n2 < n < n3 . As
seen in Fig. 7, the discharge begins provided that the antenna currents exceed the critical value Icr1 and the plasma
density grows continuously as the antenna current (the RF
power) increases from Icr1 to Icr2 . The dissipation curve
touches the absorption curve from below at point 2, where
the equilibrium state is unstable, resulting in a discontinuous jump to a higher density at point 3’, which is a stable equilibrium point. If the antenna current decreases, the1101014-7

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Volume 13, 1101014 (2018)back transition between 3-2’ occurs at Icr3 < Icr2 . Thus,
hysteresis is intrinsic to density jumps. Multiple transitions at varying input power accompanying hysteresis areobserved in the experiment [99]. A direct experimental andtheoretical comparison of the antenna loading and density
jumps was carried out by Shinohara and Shamrai [100],and showed satisfactorily agreements within 50 % of thethreshold density (just after the density jump) over a wide
range of operation parameters.
Calculations by Chen and Torreblanca [101] using the
HELIC code predict a sudden density jump at a threshold
RF power depending on the ratio of plasma loading resistance to circuit resistance. Figure 8 [101] shows equilibrium densities with varying input power PRF for the threevalues of the circuit resistance Rc . As seen in Fig. 8, whenmeasured experimentally, the threshold power (∼ 40 W)
and plasma density (∼ 1012 cm−3 after the jump) show thegood agreement with the prediction in the case of Rc = 1 Ω.
This is in rough agreement with the experimentally observed value of Rc ∼ 0.7 Ω in the measured density jump.
It was also suggested that the observed density jump is
not from an ICP mode into a helicon mode since the finite magnetic field (B0 = 80 G) precludes the ICP modein the magnetized system and may be a low-density helicon mode. In the magnetized ICP (MICP) modes, there are
two regions called skin (ωRF > ωce ) and wave eﬀect dominant (ωRF < ωce ) regions [102]. The skin eﬀect dominantregion is similar to unmagnetized plasma (ICP). In thisregion, the waves decay inside the skin depth region andelectromagnetic oscillations exist at skin depth only. In the
wave eﬀect dominant region, there is a wave penetration
into the plasma (helicon plasma) [102–106].
In this experiment, the H–W mode transition is understood to be the transition from the axial standing wave
structure (the wavenumber of which is nearly the same as
the fundamental excited antenna wavenumber) to the axial
traveling wave structure (the wavenumber of which satisfies the helicon’s dispersion relation) [57]. Although thisseems to be a reasonable way to distinguish the H mode

Fig. 8 Density versus PRF for three values of circuit resistance
Rc , showing the abrupt jump in density as applied power
is increased. The dashed curve shows density measured
inside the discharge tube [101].

from the W mode, uncertainty still remains, as Chen and
Torreblanca pointed out [101]. In a low-density region, thehelicon wave cannot propagate as shown in Fig. 1; in thisregion, it becomes an evanescent wave (penetrates within
the skin depth δ = 2π/kH, imag ) with wave structures (standing wave structures) similar to those of ICP fields. However, as seen in Fig. 1, the TG wave can propagate andcontribute to the power deposition in low-density plasma.
From (2), we can obtain the approximated TG wavenumber (real part k⊥,TG and imaginary part kTG, imag ) at α  1(that is n  nup ) as follows [42],k⊥,TG = k

ωce
ν
ωce
, kTG, imag = −k.
ωRF |γ|2 ωRF|γ|2 ωRF(14)

With fixed k , when the magnetic field is suﬃciently
strong ωRF  ωce , since the TG wave is highly electrostatic k⊥  k and highly dissipative kTG, imag r0  1, thiswave structure cannot be observed and has no eﬀect on
the observed wave field structures of the magnetic wave
field components. On the other hand, when the magnetic
field is small, the TG wave can be observed. Indeed, thelow-density mode sustained by the TG wave is observedby Lho et al. [92]. Figure 9 shows the comparison of the
wave dispersion related to the experimental data. The observed wave fields at the first harmonic (drive frequency
is f = 13.56 MHz) are identified as TG waves (pseudosurface waves) at a low-density (n ∼ 9.0 × 1010 cm−3 ) andhigh B field (ωce /ω > 2.5 ∼ B0 = 12 G), while waves atthe second harmonic (drive frequency is f = 27.12 MHz)behave as helicon waves at a low B field (ωce /ω < 2.5 ∼B0 = 24 G). By using high-frequency f = 50 MHz and

Fig. 9 Comparison of the wave dispersion relation to experimental data. Waves of the first harmonic ( f =13.56 MHz) follow the pseudosurface wave dispersion
relation, while waves of the second harmonic ( f =27.12 MHz) follow the helicon wave. The deviation ofexperimental data from the helicon dispersion relation
might come from the uncertainty of the density measurement [92].

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Volume 13, 1101014 (2018)very-high-frequency f = 144 MHz, Sakawa et al. [107]also observed low-density (∼ 1010 cm−3 ) discharge being
sustained by the TG wave.

7. Density Limit and Neutral Depletion
A higher-density helicon source is useful for applications such as the plasma process, the nuclear fusion, andthe electric thruster. If a peak density of order 1021 m−3
can be achieved, such plasma would be useful for accelerator applications [108]. One of the advantages of a heliconsource is its wide variation of device scales. Shinohara
and Tanikawa [4, 109, 110] demonstrated excellent particle production eﬃciency Ne /Pinp (Ne is the total numberof electrons in a whole plasma and Pinp is the input RF
power) at various device scales from small (plasma volume of 23 cm3 ) to large (plasma volume of 2.1 m3 ). Figure 10 [110] shows the relationship between Ne /Pinp andthe device cross section a2 (a is the device radius). As seenin Fig. 10, Ne /Pinp is proportional to a2 , as expected fromthe classical diﬀusion (in this case, primarily radial diﬀusion [109]).
Maximum peak densities of order 1020 m−3 are reported in some experiments [35, 44]. Miljak and Chen [24]have tried to produce higher-density helicon plasma with
two helical antennas to excite m = +1 waves, using 2 kW
of RF power on each antenna. They observed a maximum
plasma density of 8.0×1019 m−3 and found that higher density could not be achieved by increasing the pressure andmoving the antennas. They suggest the density limit resultsfrom neutral depletion.
There are two regimes of collisional and collisionless plasma in which neutral depletion occurs in diﬀerent
ways [23]. In collisional plasma, neutral depletion occurs
due to the pressure balance between plasma and neutrals,
which plasma and neutrals exert on each other due to mutual ion-neutral collisions. This process of neutral depletion is called neutral-pumping [111–114]. In collisional

Fig. 10 Relationship between particle production eﬃciency(Ne /Pinp ) vs. device cross section (a2 ) for diﬀerent helicon sources [110].

plasma, neutral depletion is more pronounced with neutral
gas heating since the total pressure balance changes due tothe neutral density being depleted in the high neutral temperature region [103, 115–119]. In collisionless plasma,neutral depletion occurs only due to electron impact ionization followed by rapid movement from the plasma to thechamber wall via the presheath electric field. This processof neutral depletion is called ion-pumping [112, 120, 121].
In RF plasma sources, the eﬀects of the neutral depletion
have been observed in electron cyclotron resonance (ECR)
plasma [103, 115, 122], ICP [117–121, 123, 124], and helicon plasma [25, 86, 120, 125–131].
Gilland et al. [125] observed the eﬀect of neutral depletion in helicon plasma, and showed that the neutral pressure in the bulk of a helicon discharge decreases by a factorof ten compared with the pressure before the discharge.
Aanesland et al. [127] directly measured the neutral gas
density by spatially resolved laser-induced fluorescence
spectroscopy with two-photon excitation (TALIF). Figure 11 (a) shows the neutral depletion D = nXe, on /nXe, oﬀ ,where nXe, on and nXe, oﬀ are the relative neutral densities
when the plasma is on and oﬀ as a function of the radius
for three diﬀerent magnetic fields (0, 95, and 150 G). The
largest neutral depletion is obtained in the center of the discharge for all these cases and increases with an increasing
magnetic field. Figure 11 (b) shows the plasma pressure(Pe = ne kb T e , where kb is the Boltzmann constant) and theneutral pressure (Pg = ng kb T g = Pfill , where Pfill is thefill pressure before discharge, ng is the neutral density, andFig. 11 (a) Neutral depletion as a function of radius, where
squares, diamonds, and circles correspond to 0, 95, and150 [G], respectively. (b) Plasma pressure (squares) andneutral pressure (circles) as a function of radius, thesolid line being the sum of the two [127].

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the neutral gas temperature T g is assumed to be constant).
Both Pe and Pg are nonuniform in the radial direction withgradients pointing inward and outward, respectively. The
ions and neutrals are pushed by these gradients, and mutual
collisions with each other yield an additional drag force
acting in the opposite direction. Therefore, the drag force
on the ions due to collisions with neutrals points inward,
while the drag force on the neutrals by collisions with ionspoints outward, which results in a neutral depletion at thecenter of the discharge. The unbalance of the total pressure in Fig. 11 (b) may result from ignoring the eﬀects of
the neutral heating and magnetic pressure [127].
Clarenbach et al. [128] measured the time-dependenceof the temperature and the density of the Ar∗ (3 P0 )metastable atoms. From the time variation of the Ar∗(3 P0 ) density, the density of the ground-state argon atoms
can be deduced. They showed that, when the dischargechanged from ICP to helicon discharge, argon atom depletion of up to 90% and ionization degrees of up to 65% wereachieved in the antenna zone. Magee et al. [25] also measured the time-resolved neutral density at and downstream
from the RF antenna and observed slower neutral depletion(∼14 ms) compared with the decreasing timescale of theneutral density underneath the antenna (∼15 µs, which may
be due to the high ionization rate). Magee et al. [129] also
showed a hollow neutral density profile such that the coredensity was 20th of the edge density, which may be dueto the ion-pumping eﬀect rather than neutral-pumping. By
using the collisional radiative (CR) model [130], Denning
et al. [131] observed evidence of enhanced axial plasma
velocity as a consequence of reduced drag by the neutrals
due to depletion.
Fruchtman et al. [111] analytically revealed the eﬀectof neutral depletion on unmagnetized plasmas in pressurebalance with neutral gas by self-consistently solving theplasma and neutral dynamics. Figure 12 shows the plasma
and neutral density profile calculated for two cases; the firstcase is a discharge in which the total neutral number NT
is fixed, and in the second case the neutral density at theplasma boundary NW is fixed by using the following parameters: plasma size a = 5 cm (ξ = x/a), the neutral pressure prior to the discharge NW T g = 10 mTorr (assuming T g
is fixed as the room temperature), and two diﬀerent power
levels (low and high). When the input power is low, theneutral density almost does not vary; however, when the
power is high, in both cases, the plasma density sharply
drops to zero and the neutral density sharply peaks to ahigh density near the boundary. These results are qualitatively consistent with those of the experiment [121]. They
also showed that the neutral depletion enhances plasmatransport, which can result in an unexpected decrease of
the plasma density when the power is increased despite theincrease of plasma production with power. In the collisional regime, where the pressure balance is maintained,
Liard et al. [132, 133] and Fruchtman [134] included theeﬀect of the neutral gas heating. They showed that neu-

Fig. 12 Plasma and neutral density profiles. Top panel: fixed
total neutral number with P = 33 (dotted line) and 509(solid line) kW/m2 . Bottom panel: fixed neutral density
at the boundary with P = 3 (dotted line) and 121 (solidline) kW/m2 [111].

tral depletion and plasma transport are enhanced due to gas
heating.
Raimbault et al. [114] extended the steady-state
isothermal model with three neutral pressure regimes according to the three well-established theories: the lowpressure regime (the product of the pressure with thedischarge length pL < 0.1 mTorr·m) with the Tonks
and Langmuir model [135], the high-pressure regime(pL > 10 mTorr·m) with the Schottky model [136], andthe intermediate-pressure regime with the Godyak model[137], which bridges the gap between the low- and highpressure regimes. They predict that the neutral density canpresent a maximum at the center of the discharge when thedrag on neutrals due to collisions with ions is negligible(the Tonks and Langmuir model). In this case, the neutral dynamics is governed by a balance between the inertial
and pressure terms of neutrals rather than the total pressurebalance between plasma and neutrals [114]. Fruchtman etal. [112, 113, 138] call this centrally peaked neutral density
profile neutral repletion and predicts that neutral repletion
will occur when the ionization is intense and the neutrals
are suﬃciently collisionless.
The eﬀect of the magnetic field on neutral depletion
was studied by Liard et al. [139] and Fruchtman [140,141].
Fruchtman [141] points out that the rate of neutral-gas depletion is determined by the plasma outward flux at theboundaries and that the neutral-gas depletion increases
when this flux increases. When the plasma density is kept
constant while the magnetic field is varied, the plasma particle flux decreases as the magnetic field increases, thenneutral depletion decreases with the magnetic field. Whenthe plasma particle flux is kept constant while the magnetic field is varied, neutral depletion does not vary withthe magnetic field. In these cases, the magnetic field isassumed to be uniform and the eﬀect of the diamagnetic
eﬀect is not included.

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Diamagnetism is a fundamental characteristic of highbeta plasmas. Low-temperature weakly magnetized plasmas can have a high beta value, especially in plasma
sources such as helicon plasma [142]. Indeed, the diamagnetic eﬀect in helicon plasma has been observed by Roberson et al. [143] and Takahashi et al. [144–146]. However,some researchers [147–150] report that the magnetic reduction was much smaller than expected with the plasmabeta. Scime et al. [147] claim that the diamagnetic effect was small because the MHD equilibrium was notestablished in his experiment. The cases of a significant diamagnetic eﬀect with neutral depletion have beenstudied by Fruchtman and Shinohara theoretically [142]and experimentally [151]. They concluded that diamagnetism is weakened if the neutral’s pressure is comparable to the plasma pressure and if the coupling of plasma
and neutral’s pressures by ion-neutral collisions is suﬃciently strong. Figure 13 [151] shows the dependence ofthe magnetic field reduction rate R = |ΔB/B0 | on B0 at
the neutral pressure P0 = 0.30 Pa. By considering thepressure balance between the plasma and the magnetic
field, the magnetic
field reduction rate can be written as

RI = 1 − 1 − β. When the neutral pressure is includedin the total pressure balance, the magnetic
field reductionrate can be written as RII = 1 − 1 − β − Δβn , whereΔβn = (Pg − Pg0 )/(B20 /2μ0 ) = βn − βn0 (βn0 and βn are
neutral’s beta at the wall and the center of the discharge,
respectively). The RI fitting curve is represented as a black
dotted-circle line, and the RII fitting is represented as “Cal.”(with standard ion-neutral collision frequency) and “Cal.
(3.5)” (with a collision frequency enhanced by a factorof 3.5) noted lines. The “C” and “C(3.5)” curves indicate whether diamagnetism is expected to be suppressedby neutral depletion (the diamagnetic eﬀect is noticeable
when C  1 and the diamagnetic eﬀect is small, while
the neutral’s pressure gradient due to neutral depletion islarge when C  1). As seen from Fig. 13, despite certain simplifying assumptions of MHD equilibrium, there

Fig. 13 Measurements of the diamagnetic eﬀect in a heliconsource [151].

is good agreement between the experiments and calculated
RII lines. Note that a theoretical RI , neglecting a neutral’spressure eﬀect, can describe the experimentally obtained
diamagnetic eﬀect only in the higher B0 where C < 1 issatisfied.
The eﬀects of the neutral depletion on the plasmatransport and density limit have been accepted by numerous authors. However, since the precise mechanism of neutral depletion remains unclear, further investigations areneeded to overcome this issue to develop higher-eﬃciency
plasma sources.

8. Diﬀusion Processes in ICP and Helicon Discharges
The plasma diﬀusion process in RF plasma still remains a controversial issue, since the observed plasma density is usually flat or peaked on the axis even when the skindepth of the RF power deposition is much smaller than theplasma radius [152].
In the ICP (unmagnetized) discharge, as seen from
Fig. 14 [153], although the RF field from the antenna current penetrates the skin slightly and the electron temperature peaks in the skin layer, the observed profile shows
the minimum in the skin layer and peaks near the center. A mechanism to explain this centrally peaked density
profile in the ICP is the anomalous skin eﬀect [154–160],
which theoretically proposes that electron thermal motions
could transport RF energy past the skin layer when the local Ohm’s law is violated. Godyak et al. [161–163] havediscovered the nonlinear eﬀect of the ponderomotive force
and the second harmonic force caused by the nonlinear
Lorentz force FL = e(vd × B) and the convective part ofthe integral force FI = mvd ∇vd , where vd is the RF drift
velocity and B is the RF magnetic field. Smolyakov etal. [164] showed that the curvature of the magnetic field inplanar ICP and nonlinear electron inertia are direct sourcesof nonlinear poloidal currents (Jr and Jz ) and the azimuthal
magnetic field Bθ . Godyak and Kolobov [165] observed

Fig. 14 Spatial profiles of plasma density n (1011 cm−3 ), electron temperature kb T e , space potential Vs , and axial RF
magnetic field Bz (arbitrary units) in an ICP discharge in
10 mTorr of argon, with PRF = 300 W and fRF = 2 MHz[153].

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the negative power absorption of the electromagnetic field
in a low-pressure ICP discharge. When the electron mean
free path is approximately the size of the discharge, the RF
current transferred by thermal electron motion can be theopposite in phase to the local electric field, resulting in local negative energy transfer from the RF field to electrons.
A simple analytic model describing the negative power absorption was developed by Tyshetskiy et al. [166], whoshowed the qualitative agreement of the power absorption
profile with the experiment. By tracing the electrons’ orbit
over many RF cycles, Chen and Evans [153, 167] showedthat the anomalous skin eﬀect is enhanced by the nonlinear Lorentz force and indeed the density is peaked on
the axis.
In the helicon (magnetized) discharge, most of the RF
power is absorbed by the excitation of the TG mode ina thin layer of the edge boundary. The plasma diﬀusion
across a magnetic field is a long-standing problem and has
been studied extensively for decades [168]. The classicalambipolar diﬀusion across the magnetic field, of which diffusion coeﬃcient D⊥ is inversely proportional to the squareof the magnetic field D⊥ ∝ B−2 [169], is several orders
smaller than the “Bohm diﬀusion” scaling as D⊥ ∝ B−1[22]. When the diﬀusion rate was investigated experimentally [22], the observed value D⊥ = 3 × 103 cm2 /sec wasmuch higher than the expected classical ambipolar diffusion D⊥ = 20 cm2 /sec. As a result of this large discrepancy, Bohm [22] postulated that the fluctuation of thecharge density may produce electric fields that give rise toanomalous diﬀusion across the magnetic field.
The “short-circuit eﬀect” mechanism is suggested by
Simon [170, 171], who claims that the ions can diﬀuse
freely across the magnetic field without the restrictionof ambipolarity, while electrons do not follow the ions
across the magnetic field, but instead travel along the fieldlines and conducting wall, and eventually maintain the
charge neutrality. In his results, the diﬀusion coeﬃcientshows B(H)−2 dependence. Figure 15 [171] shows the observed dependence of the e-folding distance r0 , which isdetermined as the solution of the diﬀusion equation withto thethe diﬀusion coeﬃcient. Since r0 is proportional√square root of diﬀusion coeﬃcient D (r0 ∝ D), the dataclearly indicate D ∝ B(H)−2 dependence. He showedthat the ions’ intrinsic diﬀusion rate can explain the discrepancy between theory and the experiment without additional mechanisms, such as the anomalous diﬀusions in
Bohm diﬀusion. The scaling of particle production eﬃciency Ne /Pinp exhibited in Fig. 10 [110], which is proportional to a2 , is derived from the diﬀusion equation assuming the ions’ intrinsic diﬀusion rate. Fruchtman [140] extended the fluid theory with ambipolar and non-ambipolar(short-circuit eﬀect) diﬀusion limits and showed the differences of the steady-state density profiles with various
collision limits. Chen and Curreli [152, 168] point out thatthe short-circuit eﬀect arises from the sheath adjustment at
the end plates, which allows electrons to eﬀectively move

Fig. 15 Plot of slopes 1/r0 (r0 is reciprocal e-folding distance ofplasma density profile) as a function of magnetic fieldstrength. The e-folding distance r0 is proportional√ toD).
the squared root of diﬀusion coeﬃcient (r0 ∝D ∝ B(H)−2 dependence is plotted as straight line (solidline), and D ∝ B(H)−1 is plotted as curve line (brokenline) [171].

across the magnetic field. Lafleur and Boswell [172] investigated cross-field diﬀusion in low-pressure magnetizedplasmas both in the presence and absence of conducting
axial boundaries thorough two-dimensional PIC simulation. They found that cross-field diﬀusion was ambipolar
without the axial boundary, but when the axial boundary
was added, the diﬀusion became non-ambipolar in whichions and electrons escaping to the transverse and axial wall,
respectively, and total fluxes integrated over all the boundaries were equal.
By using a very long discharge tube in which theshort-circuit eﬀect can be avoided, Hoh and Lehnert [173]and Allen et al. [174] showed that the agreement between
the experiment and classical diﬀusion theory was valid
only up to a certain critical magnetic field (Bc ). Abovethe critical magnetic field, an oscillating helical structureappeared, and higher diﬀusion was present. Bonnal etal. [175] also observed the enhanced transversal flux ofions to the magnetic field above the critical magnetic field.
Kadomtsev and Nedospasov [176] investigated the stability problem for the harmonic perturbations in a diﬀusion
model including magnetized electrons. They found thetheoretical critical value of the magnetic field (i.e., the onset of the anomalous diﬀusion) and oscillation frequencyof the unstable mode above the critical magnetic field.
The relation of the electric and magnetic fields in thisoscillation showed the good agreement both qualitatively

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and quantitatively with the measurements performed by
Hoh and Lehnert [173]. On the other hand, other experiments [177, 178] report diﬀusion with the classical rate(D⊥ ∝ B−2 ) even in short cylindrical discharge and strong
magnetic fields.
In recent years, Bohm diﬀusion is predominantly interpreted as an upper limiting value of cross-field diﬀusion
with low-frequency drift-wave fluctuation [179], whichplays a dominant role in enhancing classical diﬀusion[73,180]. Density and temperature gradients in magnetizedplasma give rise to electron and ion diamagnetic currents
across the magnetic field, and the drift velocities associatedwith these currents excite collective oscillations called drift
waves. The eﬀect of drift-wave transport on the plasma
confinement has been extensively studied in the magnetic
fusion energy research field over past decades [180]. The
collisional (resistive) drift-wave instability was discovered
and identified by Hendel et al. [73] in a long cylindricaldevice called the Q machine [181]. A variety of drift-typeinstabilities, such as the trapped electron modes [182], thetrapped ion instability [183], the collisionless curvaturedriven trapped particle mode [184], and the ion temperature gradient-driven drift mode [185], have been observedin the Columbia Linear Machine.
In this chapter, we overviewed the non-classical behaviors of the anomalous skin eﬀect in ICP discharge andthe anomalous diﬀusion across the magnetic field in helicon discharge. Despite the extensive studies on these physical issues, there is currently no consensus on which theory
to use to describe diﬀusion transport.

9. Instabilities in Helicon Discharge
Enhanced plasma transport aﬀects the equilibrium
plasma behavior [18]. As seen from Fig. 2 and (7), theplasma density is expected to increase monotonically withthe magnetic field. Sakawa et al. [66] report that theplasma density does not increase monotonically, but decreases with the magnetic field in light gases. The bottom
panel of Fig. 16 [18] shows the measured ion density for
four gases, from which one can observe that the light ion[hydrogen (H)] peaks at a low magnetic field and shows
a dramatic decrease of plasma density above the criticalmagnetic field (Bcrit ), while the heavier gases [helium (He),neon (Ne) and argon (Ar)] tend to saturate. Light et al.
[18, 82] explain these saturations and unexpected decreaseof density in the framework of low-frequency electrostatic
instability. The upper panel of Fig. 16 shows a spectrogramof the density perturbation n0 (B0 ) in Ne along with B0 . One
may note that the saturation of n0 in the plasma density
and onset of strong, low-frequency fluctuations in n0 occur
at the same Bcrit . They suggest that the measured density|∇n0 | and the space potential profiles |∇Vs | can drive theresistive drift and Kelvin-Helmholtz instabilities, respectively at these low-frequency ranges, and these instabilities may enhance the plasma losses. They also showed

Fig. 16 Equilibrium plasma density vs. magnetic field strength(bottom panel) and spectrogram of the density perturbation n0 (B0 ) in neon (Ne) along with B0 (upper panel)in a helicon source, with 8 mTorr fill pressure and 2 kW
input power [18].

the theoretical growth rates for diﬀerent ion masses, whichshowed excellent correlations with the observed equilibrium density decrease, maximum of the measured fluctuation amplitude as a function of magnetic field strength, andonset of maximum fluctuation amplitude for low frequencies in spectrograms.
Tynan et al. [19,186] have observed a radially sheared
E × B azimuthal flow driven by the resistive drift-wave turbulence [187] in the controlled shear decorrelation experiment (CSDX) device. Figure 17 (a) [19] shows the measured time-averaged plasma potential (open circle) normalized to the electron temperature T e at r = 3 cm, which is
the location of the maximum density gradient. The plasma
potential peaks at the center of the plasma, has a minimum at approximately r = 4 cm, and then begins to increase in value for r > 4 cm. Thus, a positive radial electric field exists in the inner portion and a negative electric field exists outside of a shear layer (r ∼ 4 cm). The
shape of the time-averaged plasma potential profile associated with a radially sheared azimuthal E × B flow formed
via an inverse energy transfer process was predicted [seesolid line in Fig. 17 (a)] using theory and numerical simulation [188] and shows excellent agreement with the observation. Figure 17 (b) shows the radial profiles of thelow-frequency (i.e., ω < ωci ) broadband (from 3 - 4 kHz
up to 25 - 30 kHz) electrostatic potential fluctuation ampli-

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Fig. 17 Time-averaged radial profiles of (a) measured (open circles) and predicted (solid line) plasma potential and(b) broadband potential fluctuation amplitude (solidline) and low-frequency azimuthally symmetric potential fluctuation amplitude (dash-dot line) [19].

tude (solid line) normalized to the electron temperature at
r = 3 cm. The electrostatic potential fluctuation has a peak
amplitude near the maximum equilibrium pressure gradient at r = 3 cm, which is consistent with expectations forhighly resistive drift-wave turbulence [187]. The potential
fluctuation also contains a significant component over afrequency band of 0 - 2 kHz (dash-dot line), which is seenclearly in the r > 4 cm region, suggesting that there is aradial structure of azimuthally symmetric potential fluctuation. Manz et al. [189] investigated the scenario of transition from an unstable mode to drift-wave turbulence. Antar et al. [190] used a fast-imaging camera to investigate
the spatio-temporal properties of radially convective eventsin the CSDX device. They found that the change of fluctuation structures inside the main plasma column derived
from the excitation of diﬀerent poloidal mode numbers as
a function of time.
In the linear Versatile Instrument for studies on Nonlinearity, Electromagnetism, Turbulence, and Applications(VINETA) [191], Grulke et al. [20, 192–196] identified theresistive drift-wave instability by comparing the drift modedensity fluctuation characteristics with a linear model proposed by Ellis et al. [75], which reproduced the experimentally observed bent eigenmode structures due to the radial
gradients of the collisionality (electron-ion collision frequency) profile. Figure 18 [20] compares the azimuthalmode structure of an m = 5 drift mode as obtained fromthe linear dispersion relation calculation with the experimental measurement. A similar mode pattern is obtained
from the linear model for experimental radial collisionality
profiles. They also found the coupling of the drift mode to
the externally excited Alfvén waves, which resulted in frequency changes in the drift mode. Brochard et al. [197] investigated the nonlinear coupling of drift-wave fluctuationin the wavenumber spectrum and showed that intermittent
structures in weakly developed drift-wave turbulence canbe produced by mode coupling.
Drift-wave measurements with the new probe array
were performed in Large Mirror Device-Upgrade (LMD-

Fig. 18 Azimuthal density fluctuation pattern of a m = 5 drift
mode as measured experimentally (left) and as obtained
from the linear dispersion relation model (right) [20].

Fig. 19 Power spectrum S (m, f ) (arbitrary unit, contour) overplotted with the linear dispersion relation (solid line)
calculated by a numerical simulation code. Error bars
are standard deviations of several results using possible electron density profiles. The dashed line indicatesthe linear dispersion where the mean E × B drift motion
is neglected. The calculated eigenmodes have positive
growth rates [21].

U) [88]. Terasaka et al. [198] observed a change from thecoherent to broadband density fluctuations by increasing
the magnetic field. Figure 19 shows the two-dimensional
power spectrum [S (m, f ), where m is the poloidal mode
number and f is the frequency] of drift-wave turbulencesin the helicon discharge measured by Yamada et al. [21].
The linear dispersion relation calculated by a numericalsimulation code is also plotted with a white line in Fig. 19.
The fluctuation modes A (m = 1 and f = 2.8 kHz) and B(m = 2 and f = 3.4 kHz) satisfy the linear dispersion relation of drift-waves. The peak at m = 1 and f = −0.9 kHz,
which is considered a flute mode, is labeled C, and all otherpeaks appearing in Fig. 19 can be expressed by the addition
and/or subtraction of A, B, and C. Therefore, the formationof turbulences can be regarded as a result of the non-linear
interaction of parent modes that satisfy the linear dispersion relation.

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Akhiezer et al. [199] suggest that the mechanism ofion-sound kinetic instability, which is excited by the parent helicon wave and the absorption of the ion-sound wave
energy by electrons, can be a suﬃciently strong mechanism for power absorption. Kline et al. [15, 16, 200] haveobserved the parametric decay of the helicon wave into
two electrostatic waves. Figure 20 [16] shows the electricfield power spectrum, and one may note the parent wave of
11 MHz, forward ( f0 + f1 ) and backward ( f0 − f1 ) propagating waves, and low-frequency beat wave (a spectral peakat approximately 9 MHz is an experimental artifact). They
also suggest that the TG wave and these parametrically excited waves are responsible for ion heating in the perpendicular direction [15, 16, 200, 201]. Figure 21 [15] showsthe electron temperature, electron density, and perpendicular ion temperature as a function of antenna frequency
and magnetic field strength. As seen from Fig. 21 (a) and(b), there is a clear increase in electron temperature as theantenna frequency nears the lower hybrid frequency. For
antenna frequencies of 8 - 12.5 MHz, the largest plasma
densities occur for ω ∝ ωLH . Figure 21 (c) shows the perpendicular ion temperature peaks for ω < ωLH (0) (on axis)

because of the low plasma density near the plasma edge
(the local lower hybrid frequency drops below 10 MHz).
The radial profile of the perpendicular ion temperature also
shows the strong perpendicular heating at the plasma edge.
This suggests the mechanism of the ion-Landau damping of the electrostatic TG waves at a local lower hybrid
frequency resonance [15]. When the RF wave frequency
equals the local lower hybrid frequency, the perpendicular wavenumber of the electrostatic TG wave goes to infinity. At a large perpendicular number, the wave phasevelocity is suﬃciently low that ion-Landau damping canoccur. Lorenz et al. [17, 63] have demonstrated the fluctuations that originate from the parametric decay of thehelicon waves into the ion-sound and TG waves, whichshowed good agreement of the measured growth rate andthreshold with theoretical prediction for the parametric decay of helicon waves.
In this chapter, the recent studies on the instabilities inhelicon discharge were overviewed. Drift-wave-type instabilities have been observed in linear helicon devices, whichaﬀect the plasma transport and equilibrium state of helicondischarge. The parametric decay instabilities of the helicon pump wave in the ion-sound and TG waves and thelower hybrid resonance eﬀect of the TG waves have beensuggested to play roles in eﬃcient power coupling in helicon discharge. We need further investigations for thesecontributions on the power balance of helicon discharge.

10. Other Experiments

Fig. 20 Electrostatic fluctuation spectrum for a RF frequency of
11 MHz and a magnetic field strength of 845 G [16].

Shinohara et al. [58,202] have investigated the properties of helicon plasma by changing the wavenumber spectrum, diﬀerent kinds of end plates [203], and the device
aspect ratio [203] and in a very strong magnetic field up
to B0 ∼ 10 kG [3]. They have succeeded in controlling
the plasma density profile by inserting the voltage-biasing
electrodes [204, 205] and changing the magnetic field configurations and antenna radiation field patterns [206].
Nisoa et al. [207–209] have produced high-density helicon plasma by exciting the m = 0 standing helicon wave

Fig. 21 (a) Electron temperature, (b) plasma density, and (c) ion temperature versus antenna frequency and magnetic field strength. The
white line indicates where the antenna frequency equals the on-axis lower hybrid frequency [15].

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Fig. 22 Measured plasma density np at B0 = 237 G and the calculated stable np versus PRF . Lines represent np at thepeaks in plasma resistance for N = 1 and 3 modes [209].

and observed the mode transition from N = 1 (N is thenumber of half-wavelengths in the plasma column) to the
N = 3 standing wave mode as the input power (plasma
density) increased. Figure 22 shows the measured plasma
density np versus PRF , and stable np was calculated byusing a 2D wave code (TASK/WF) [210]. The density
jump can be seen clearly due to the mode transition fromN = 1 to N = 3 at PRF ∼ 1.8 kW. A low-density plasma
np ∼ 1011 cm−3 , is thought be produced by the antenna
induction field when PRF < 0.4 kW, but no theoretical prediction is given to explain this low-density mode. In theircalculation, there are two stable np that appear between thelines of the N = 1 and 3 modes, and the observed densities
always show agreements with the lower np values, but thereason for this is unclear. The calculated stable densities
and the density jump point agree well with the measured
ones.
Yun et al. [211, 212] have investigated the helicon
plasmas near the lower hybrid frequency. They found that
abrupt density profile changes occur near the lower hybrid
frequency [212]. The centrally peaked density profile turns
into the hollow density profile at particular frequencies.
Figure 23 shows the measured frequencies when the hollow density profile is observed with various magnetic fields
and ion masses. The theoretical lower hybrid frequency
calculated using the measured density is also plotted in thisfigure. One may note that the measured frequencies have acorrelation between the radial density profile and the lower
hybrid frequencies.
Loewenhardt et al. [213, 214] have investigated helicon wave propagation in toroidal geometry and showedgeometry influences on the propagation of waves by comparing the measured density with predictions from the helicon dispersion relation. Chang et al. [215] numericallystudied the eﬀects of axial gradient in plasma density anda static magnetic field on wave propagations by modelingthe experiment in a magnetized plasma interaction experiment (MAGPIE) [216]. They showed the axial gradientin the magnetic field increased the decay length of heliconFig. 23 Frequencies where the hollow density profile is observed with various magnetic fields and ion masses (atRF power 800 W and neutral pressure 3.5 mTorr) (dotted graph). The theoretical lower hybrid resonance frequencies for each ion are shown as a line graph [212].

Fig. 24 Equilibrium plasma parameter profiles in the poloidal
xy-plane perpendicular to the ambient magnetic field.
Plasma density (top row) and electron temperature (bottom row) for all three discharge modes are shown.
White circles indicate the position of the antenna edge[217].

waves in the downstream region, and the wave amplitude
in non-uniform plasma density was much bigger in both
the downstream and upstream regions than in the case of
uniform plasma density. They also showed that the wave
amplitude became lower and the power deposited into thecore plasma decreased as the Coulomb collision frequency
increased due to the strong edge heating.
Franck et al. [217] report on two-dimensional measurements of plasma parameters and magnetic eigenmode
profiles in the capacitive-, inductive-, and helicon-wavesustained discharge modes of a helicon source with highspatial resolution. Figure 24 shows the plasma parametersof three discharge modes. The capacitive mode (PRF =400 W, PAr = 0.08 Pa, B0 = 56 mT): The plasma density
is in the order of 1017 m−3 , which is relatively high compared with the typical CCP discharge (1015 - 1016 m−3 )[60, 97], and has a peak of n ∼ 2.0 × 1017 m−3 extend-

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ing from the edge region. The electron temperature profile is hollow, and peak values of T e ∼ 2.5 eV are foundat the antenna edge. The inductive mode (PRF = 600 W,
PAr = 0.6 Pa, B0 = 56 mT): The plasma density has analmost flat-top and the peak density n ∼ 3.0 × 1017 m−3 .
The electron temperature follows the flat-top of the plasma
density and has a peak value of T e ∼ 4 eV. The helicon mode (PRF = 3500 W, PAr = 0.5 Pa, B0 = 75 mT):The plasma density and electron temperature are strongly
peaked at the center of the plasma column with peak values of n ∼ 3.0 × 1018 m−3 and T e ∼ 8 eV, respectively.
They also showed diﬀerences between the magnetic eigenmode structures of the helicon and capacitive modes. The
dereference between the inductive and helicon modes is
not discussed in detail.
Corr et al. [218] investigated ion-acoustic wave phenomena in helicon sources. The wave amplitudes are spatially localized near the edge of a plasma column where
there is a significant gradient in the plasma pressure as theamplitudes of the “mirror waves” that are separated fromthe RF frequency by the ion-wave frequency. Although
the wave spectra are indicative of a parametric decay phenomenon, they suggest two possible mechanisms for theion-wave excitation (i.e., the parametric decay and the radial pressure gradient).
Ganguli et al. [219] theoretically presented the damping and energy absorption for the helicon and TG wave
modes, including both Landau and collisional damping inside a conducting waveguide. They also conducted theexperiment [219, 220] with an axial magnetic mirror profile in two separate regimes. The first was a low-pressure(∼ 0.2 - 0.3 mTorr) and low-magnetic-field (∼ 16 - 20 G)
regime where the Landau damping played a dominant rolein the power absorption [219]. The second was a moderatepressure (∼ 10 mTorr) and moderate-magnetic-field (∼ 60 65 G) regime. In the first case, both the wave absorption and warm electron production were shown to proceed
through Landau damping. In the second case, where thepower absorption was collisional, they observed a double
layer with large potential drops [221], and warm electron
production proceeded via high-energy electrons producedby the acceleration of bulk electrons (from neighboring regions) across large potential gradients.
Breizman and Alexey [222] suggest that the radial
density gradient can create a radial potential well for nonaxisymmetric whistlers, allowing radially localized solutions with a slightly lower frequency shift, which is calleda radially localized helicon (RLH) wave. Panevsky andBengtson [223], Chen et al. [224], and Lee et al. [225]showed evidence for the existence of the RHL wave experimentally and numerically.

11. Remaining Critical Issues
Since helicon studies began in 1960, many parts of helicon plasma physics have been clarified owing to numer-

ous theoretical and experimental attempts. However, theexact mechanism responsible for high-eﬃciency plasma
production is not yet completely understood. In recent
studies, many new physical mechanisms, such as driftwave instability and parametric decay instability, havebeen successfully revealed. These mechanisms aﬀect
plasma production and diﬀusion transport. Therefore, thehelicon source should be optimized to include these eﬀects
more precisely. Here, we summarize the critical remaining
issues.
(1) The mode transition from the H-mode to the W-mode
The density jump between the intermediate-density
mode to the high-density mode is believed to be a modetransition from the ICP (H) mode to the helicon (W) mode.
However, as mentioned in Sec. 6, the dispersion relation
predicts that the TG wave will propagate even at a lowdensity, and can contribute to the RF power deposition.
Actually, the induction fields observed at a low-density
in the experiments [47, 48, 57, 59] are evanescent helicon
waves, which oscillate near the antenna (within their skindepth). Therefore, which wave is responsible for RF absorption should be investigated. Indeed, the low-density
plasmas sustained by the TG waves are observed in a low
magnetic field [92] and very high drive frequency range[107], where the wavelength of the TG wave becomescomparable to the plasma radius. The diﬀerence between
the ICP mode and the helicon mode should be investigatedin more detail from the perspective of the power balance.
(2) Density limit in helicon discharge
The major factor that limits the density of helicon discharge seems to be neutral depletion [24]. The mechanismof neutral depletion should be considered with two main
parameters, the collision frequency and the magnetic field.
In collisionless plasma, neutral depletion occurs only due
to electron impact ionization. In collisional plasma, the total pressure balance between the neutral and plasma resultsin neutral depletion due to the drag force of ion-neutral
collision. In high-beta plasma, the total pressure balanceshould contain the magnetic pressure that results from thediamagnetic eﬀect [151]. Besides these parameters, the ion
and neutral temperatures also aﬀect the drag force (the collision frequency) and pressure balance. More detailed investigations are needed both theoretically and experimentally to overcome the density limit.
(3) Centrally peaked density profileDiﬀusion across the magnetic field remains a controversial issue. Chen et al. [6, 152, 168] insist that the shortcircuit eﬀect plays a crucial role in the centrally peaked
density profile. On the other hand, the drift-wave turbulences observed in the helicon source [19–21,192–196] canalso be a candidate for anomalous diﬀusion. There is noconsensus on which plays a dominant role in plasma transport across the magnetic field. A self-consistent model
which includes the instabilities and enhanced plasma trans-

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port, is needed for an improved understanding.

modes) and plasma transport processes should be treatedself-consistently. There are various timescales in helicondischarge and each physical parts aﬀects the other in thetime evolution of helicon discharge.
Figure 25 shows the timescales in helicon discharge,
which are evaluated by using the typical parameters in
HELIX [15]. The upper panel shows the time periods ofthe drive frequency T RF = 2π/ωRF , the electron (ion) cyclotron frequency T ce(ci) = 2π/ωce(ci) , the lower hybrid
frequency T LH = 2π/ωLH , and the collision periods ofelectron-neutral τen , electron-ion τei , electron-electron τee ,ion-neutral τin , ion-electron τie , and ion-ion τii . These collision frequencies are given in [226, 227]. The bottom
panel shows the times scales of the axial diﬀusion τ,e =Lz2 /D,e and perpendicular diﬀusions of τ⊥,i = Lz2 /D⊥,i ,and τ⊥,Bohm = Lz2 /D⊥,Bohm , where D,e , D⊥,i , and D⊥,Bohm
are the diﬀusion coeﬃcients of the electrons along thefield line, perpendicular ion diﬀusion, and Bohm diﬀusion,respectively. Here, we consider non-classical diﬀusions,such as the short-circuit eﬀect and anomalous diﬀusion.
The ionization timescale is evaluated as τion = 1/ng αion ,where αion is the ionization rate [226]. The energy confinement timescale is evaluated as τene = nT e V/P, whereV = πLr2 Lz is the plasma volume. The growth timescaleof the drift-wave τdw is estimated by solving Ellis’s equations [75, 194] with the Gaussian parameter a2 /r02 = 1.75
(where a is the plasma radius and r0 is the scale lengthof the Gaussian), azimuthal mode number m = 1, axial
wavelength λ = 5 m, and collision frequencies in Fig. 25.
Since the drift-wave becomes unstable for the long axial
wavelength [75], the axial wavelength λ = 5 m is used
here. This wavelength is larger than Lz = 1.6 m, but this isreasonable since the HELIX is combined with a large, 2m-diameter, 4-m-long space simulation chamber (LEIA)(4) Instabilities
The saturation and decrease of plasma density above
the critical values of the magnetic field seem to be causedby the drift-wave instability [18, 82]. The eﬀect on the particle balance and plasma transport should be investigated in
more detail. The non-linear RF power coupling caused by
the parametric decay instability can also be a strong mechanism for power absorption [199]. The relative importanceof the power absorption mechanism should be discussed inrelation to normal joule heating.
(5) Ion heating
The heating of perpendicular ion temperature has been
observed in experiments [15, 16, 200, 201]. The TG wave
and these parametrically excited waves are responsible forion heating. The ion-Landau damping caused by the lower
hybrid resonance TG waves seems to be the main mechanism of the ion heating. The ion heating also aﬀects neutral
depletion and plasma transport. Therefore, the ion dynamics and energy balance should also be included in the helicon discharge model.

12. Attempts for Self ConsistentModel
The remaining issues in Sec. 11 are closely related;for example, ion or neutral heating enhances neutral depletion, which leads to enhanced plasma transport and thedramatic change of the power balance and density profile, and this determines the density limit in helicon discharge. Furthermore, the eﬀects of instabilities may critically aﬀect all these processes. In addition, the power
absorptions from the RF antenna via the electrostatic (theE mode) and the electromagnetic fields (the H and W

Fig. 25 Various physical time scales with following typical parameters in HELIX [15]; plasma density n = 2.0 × 1018 m−3 , electron
temperature T e = 8.0 eV, ion temperature T i = 0.5 eV, RF frequency fRF = 10 MHz, RF power PRF = 750 W, magnetic fieldB0 = 0.1 T, Ar pressure PAr = 6.7 mTorr, source tube radius Lr = 0.075 m, and axial length Lz = 1.6 m.

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Volume 13, 1101014 (2018)[147]. The growth timescale of the electrostatic (ES) fluctuation τes ∼ 10−5 s was observed by Krämer et al. [17].
The neutral depletion has a range of timescales: slow depletion, the order of τnd,down ∼ 10 ms, was observed in thedownstream region from the antenna, and fast depletion,
the order of τnd,up ∼ 10 µs, which showed significant depletion of 1% of its initial value, was observed near the antenna in Magee’s experiment [25]. The short timescale depletion of τnd,down ∼ 1 ms was observed in the downstream
region in Liard’s experiment [133]. Magee et al. [25] insisted that the fast neutral depletion (∼ 10 µs) observed
near the antenna resulted from the ion-pumping (ionization) eﬀect rather than the neutral-pumping eﬀect. The
neutral inflow timescale τflow was estimated to be the traveling timescale of the krypton neutral from the gas inlet to
30 cm with thermal velocity [25].
Since the RF period T RF is much faster than the othertimescales, the timescale of the TG and helicon wave perturbations are well separated from those dynamics. Therefore, the wave fields can be treated as steady parts. The
timescale of the ionization τion is also well separated, andthe rapid growth of the plasma density can be treated separately. However, the relaxation timescale for the steadystate plasma density should be determined from the plasma
diﬀusion timescales. After the density grows near the antenna, the plasma density expands to the axial diﬀusion
chamber in the timescale of τ,e ∼ 0.1 ms (with axial length
Lz = 1.6 m), and the radial plasma density profile relaxes
to the steady-state in the timescale of τ⊥,i ∼ 2 ms (withradial length Lr = 0.075 m). Indeed, the plasma density
observed at the expansion chamber relaxed to the steadystate in the timescale of 2 - 5 ms [17,25]. While the plasma
density relaxes to the steady state, the ES and drift-wave instabilities occur within the timescale of ∼ 20 µs, and these
instabilities aﬀect the particle and energy balances via the
non-linear power absorption and amorous diﬀusions, respectively. Therefore, the plasma diﬀusion, energy balance (the energy confinement time is also in the same order as the growth timescales of the instability), and these
instabilities should be treated self-consistently. The neutral
depletion can be considered separately in the upstream region (near the antenna) and in the downstream region. The
former may result from the ion-pumping eﬀect in the fasttimescale of 10 µs, and latter may result from the neutralpumping eﬀect in the slow timescale of 1 - 10 ms. Regarding the neutral depletion in the downstream, the pressurebalance between neutral and plasma should be treated selfconsistently with the plasma diﬀusion. Since the neutral
depletion timescale that results from the ion-pumping effect is much shorter than the neutral inflow timescale, thegas inlet position should be close to the antenna to overcome the significant neutral depletion near the antenna.
The neutral depletion that results from the pressure balance would be improved by controlling the pressure balance by applying the inhomogeneous magnetic field or applying the diamagnetic eﬀect.

The helicon discharge system is so complicatedthat we cannot easily include these physical parts selfconsistently, and this attempt is challenging for developing the next-generation helicon source. Several basic selfconsistent models of helicon discharge have been developed.
Kinder et al. have developed the two-dimensionalmodel [228,229] of the Trikon PMT MØRI helicon source[230] and three-dimensional model [231] of the Wave On
Magnetized Beams and Turbulence (WOMBAT) heliconsource [84, 232] for use in material processing. Their hybrid plasma equipment model (HPEM) [228, 229, 233] calculates the wave fields excited by the RF antenna, electron energy equation in which power absorption by wave
fields and power loss due to collision is included, and fluid
continuity, momentum, and energy equations for ion andneutrals (the drift-diﬀusion formation in which the inertial
term is omitted is used for the electron momentum equation). These calculations are iterated until a converged solution is obtained. This model also includes the ElectronMonte Carlo Simulation (EMCS) for calculating the electron energy distribution (EED). They showed the diﬀerences between the H mode (ignoring the TG wave), the W
mode, and the TG wave-sustained mode.
The left panel of Fig. 26 [229] shows azimuthal, radial, and axial electric field profiles and power absorption
profiles at 20 G and 300 G for the H and W modes, respectively, where the TG wave is omitted in this calculation. As shown in Fig. 26, at B = 20 G, the azimuthal
electric field is predominantly inductively coupled with anenhanced plasma skin depth. At B = 300 G, the electricfield profiles show the traveling wave (the helicon wave),

Fig. 26 The electric field profiles and power absorption profileof the H (B0 = 20 G) and W modes (B0 = 300 G)
without the TG wave (left panel) and low-magneticfield (B0 = 20 G) operation with including the TG wave
(right panel) [229].

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which propagates to the downstream. At B = 20 G, thepower deposition peaks at 2.1 W/cm3 , and at B0 = 300 G,
the same total power is deposited over a large volume, resulting in a lower peak power deposition. The left panel ofFig. 27 [229] shows the electron temperature (T e ) and theplasma density (e) profiles for the H (B = 20 G) and W
modes (B = 300 G). At B = 20 G, the electron temperature and electron density peak at ∼ 4 eV and 4 × 1012 cm−3
near the antenna, respectively, where most power disposition occurs. At B = 300 G, the electron temperature andelectron density peak at ∼ 3.5 eV and 3.2 × 1012 cm−3 ,respectively in the downstream. They also show that the
tail of the EED increases in magnitude in the downstream
chamber for B0 > 150 G and P < 5 mTorr, which indicatessome electron trapping and collisionless heating with thephase matching of thermal electrons to the phase velocityof the wave.
The right panel of Fig. 26 shows the electric field profiles and power absorption profile including the TG wave ata low magnetic field of B0 = 20 G. In this case, the radial
wavelength of the TG wave is comparable to the plasmasize; therefore, the TG wave, which propagates into thecore of the plasma, can be seen. Although the power absorption peaks near the location of the peak electric fields
in the upstream region, significant power deposition also
occurs in the electron cyclotron (ECR) zone where thedrive frequency fRF = 13.56 MHz matches the electron cyclotron frequency at B0 = 4.8 G. The right panel of Fig. 27
shows the electron density perturbation and the electron
density profiles at B0 = 20 G and B0 = 300 G including

Fig. 27 Left panel: electron temperature and plasma density
profiles of the H (B0 = 20 G) and W modes (B0 =300 G) without the TG wave. Right panel: electron density perturbation (Δne ) and electron density (e) profiles
at low-magnetic-field (B0 = 20 G) and high-magneticfield (B0 = 300 G) operations including the TG wave[229].

the TG wave. At B0 = 20 G, the electrostatic term can beseen through the harmonically driven perturbation in theelectron density, and there is significant propagation into
the chamber until the ECR zone appears. Therefore, thelow-density (the peak value of ∼ 2.8 × 1012 cm−3 ) mode issustained by the TG wave. At B0 = 300 G, in this case,the wavelength of the TG wave is so short that the TG
wave is strongly absorbed near the plasma surface. In thiscase, although the electron density profile appears similar
to that of the pure helicon case, the average electron density of 7.7×1011 cm−3 is higher than that of the pure heliconcase (2.4 × 1011 cm−3 ). These results provide evidence ofthe low-density TG wave-sustained mode in the low magnetic field, which is observed in Lho’s experiment [92], andshow the clear diﬀerences in the power absorption, plasma
density and electron temperature profiles between the H-,
W-, and TG wave-sustained modes.
By using the same model, Bose et al. [234] showedthat edge heating due to the TG waves dominated thepower absorption under high external magnetic fields. Cho
et al. [235] also studied the eﬀect of the external magnetic
field on the radial plasma density profile with the onedimensional steady-state fluid model combined with wave
equations. They showed that the plasma density changed
from a relatively flat to steep profile as the external magnetic field increased, and when the TG wave dominatedthe power absorption, the density profile was relatively
flat in the central region compared with the case when
there was significant bulk heating by the helicon wave. In
their results, the linear dependence of the plasma density
on the magnetic field, which is one of the typical characteristics observed in helicon discharges, was also demonstrated. Chen et al. [236] used the two-dimensional steadystate fluid model of Boswell’s experiment [1]. They foundthat the classical heat conductivity was too small to sustain
a centrally peaked density profile under the experimental
conditions, and the resulting density profile showed a hollow density profile. Assuming enhanced radial heat conductivity and particle transport, they also found steadystate solutions that were in good agreement with density profile and density jump observed in the experiment.
Isayama et al. [237] first discussed the temporal behavior of the helicon discharge with the time-integrated fluid
model combined with the wave equations calculated by
the finite-diﬀerence time-domain (FDTD) method. They
showed the two time phases in the time evolution of the helicon discharge. In the early stage of discharge growth, theTG wave dominated the power absorption near the plasmaedge region; thus, the edge-localized electron temperatureand plasma density profile grew over time. After the density grew suﬃciently for the helicon wave to be able topropagate, the helicon wave deposited its power in the coreof the plasma. Eventually, the centrally peaked density
profile was obtained in the steady-state. The combinationof these self-consistent discharge models with instability
models and neutral dynamics is a challenging issue for a

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Fig. 28 Comparison of the numerical and the experimental results. (a) Neutral and ion temperature profiles, (b) neutral, electron and totalpressure profiles, and (c) neutral density profile [119].

through understanding of helicon discharge. There are several models of drift-wave instability and neutral dynamics.
Numerical simulations of the drift-wave turbulence inlinear devices have been carried by Kervalishvili et al.
[238] and Kasuya et al. [239–241]. Kervalishvili et al.
[238] modified the original three-dimensional Gyrofluid
Electro Magnetic (GEM3) code [242] for cylindrical annulus geometry to use it for the simulation of VINETA. They
showed the linear growth of the potential fluctuation andfound strong peaks for the m = 6 mode, which correspondswith the experimentally observed fluctuations of m = 1 8 [193]. Kasuya et al. [240] developed a three-dimensional
numerical simulation code for the resistive drift-wave turbulence in a linear device, called the Numerical Linear
Device (NLD). The selective formation of the zonal flow
and the streamer using ion-neutral collision νin as a control
parameter was observed in their simulation results. The
value of νin represents the strength of the damping forceof the turbulent structures. When the collision frequency
is large, the zonal flow remains stable and the streamer isformed. The bifurcation of these structures was also investigated in their model. Chang et al. [243] applied a twofluid model to interpret plasma oscillations in the WOBAT
device. They showed that the density and space potential
profiles agreed with their experimental results, but the temperature profile was not flat in the observed results, as assumed by the model.
The direct simulation Monte Carlo (DSMC) method[244] is often used for computing the neutral gas distribution. Hartig and Kushner [245] and Kilgore et al. [246]investigated the neutral transport in ECR plasma. The
neutral density was found to be depleted in the volumeof the chamber due to ionization (ion-pumping), and thedepletion was enhanced by neutral heating by electronneutral and ion-neutral collisions [246]. The combination model of the DSMC method with the particle-incell (PIC) method was developed by Vladimir et al. [247]for the DC and ICP discharges. Appreciable gas heating
and depletion in the discharge was demonstrated. Shimada et al. [119] developed a hybrid-type DSMC modelfor the ICP discharge, in which electrons were treated as

background particles using the measured electron densityand electron temperature profiles, and ions and neutrals
were treated as test particles. In this model, ionization,
electron-ion, electron-neutral elastic, ion-neutral elastic,
ion-neutral charge exchange, and neutral-neutral elastic
collisions [244, 248] were all considered. Figure 28 [119]shows the comparison of the numerical and experimental
results [249, 250] for (a) neutral (T n ) and ion (T i ) temperature profiles; (b) neutral (Pn ), electron (Pe ), and total pressure (Pt ) profiles; and (c) the neutral density profile (nn ).
Excellent agreement between the experiment and the simulation for the radial profiles of T n , Pn , Pe , Pt , and nn canbe seen, and these results indicate that the neutral heating
and pressure balance (neutral-pumping) cause the neutral
depletion in ICP discharge.

13. Applications
Microprocessing is necessary for realizing the highdensity integration of semiconductor devices and the highperformance of thin-film transistor liquid crystal panels[251]. One of the techniques for achieving this is a dry process, in which high-density plasma sources with an electron density of 1011 - 1013 cm−3 are applied. The standard
systems are CCP and ICP sources. Helicon sources would
generate higher plasma density, which enables high deposition and etch rates. In the 90s, many helicon sources[230,252–255] were built for commercial use, but the issueof the uniformity of the plasma density remained, which
is needed for uniform etching. In recent studies, Chen etal. [6, 256] have developed multiple helicon sources andsuccessfully produced good uniform plasma with a densityof up to 1012 cm−3 . They have also developed a compact
helicon source using a permanent magnet that is expected
to be a more practical helicon source in the semiconductor
industry.
The large helicon source has been used for a spacecraft intended for travel to Mars, the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) [257]. In thissystem, the generated plasma is energized further by an
ion cyclotron heating (ICH) RF stage [258] that uses left-

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hand polarized slow-mode waves launched from the highfield side of the ion cyclotron resonance, and useful thrustis produced as the plasma accelerates in an expanding
magnetic field [259]. In recent studies, the thruster eﬃciency and thrust power of a high-power VASIMR prototype have been measured at a level of 200 kW RF input
power [258, 260–263].
The permanent magnet helicon current-free doublelayer (DL) thruster (PM-HDLT) has been developed
by Takahashi [144–146, 264–271], Charles [272–275],
Fruchtman [113, 276], and their research team. The magnetic nozzle constitutes the acceleration stage of the helicon thruster, located in the downstream of the chamber.
Theoretical analyses of the helicon plasma thrust performance with the magnetic nozzle have been developed [113,276–278], and direct thrust measurements have also been
performed in a number of devices [266–268, 279–282].
In recent studies, the performance has been improved bymodifying the magnetic field configuration and increasing the magnetic field strength [144, 266]. Takahashi etal. [269, 271, 283] point out that the neutral depletion induces an axial asymmetric density profile, which causes
the decrease in the axial thrust.
A new concept of electrodeless discharge and electromagnetic acceleration has been developed by the heliconelectrodeless advanced thruster (HEAT) project [110, 280,
284–295]. The electrodeless configuration is a promising solution for the erosion and contamination problems inconventional electric propulsion systems [284]. The rotating RF electric field (REF) [280, 285–288, 295] and rotating RF magnetic field (RMF) [110,289,291–293] are basedon the concept that the entire acceleration process can be
conducted without contacts between the electrodes andthe plasma by using RF antennas outside of the dischargechamber. In these methods, an azimuthal electric current
is induced by the additional REF or RMF, and the plasma
is electromagnetically accelerated in the magnetic nozzle.
Ponderomotive acceleration (PA) with the ion cyclotron
resonance (ICR) acceleration method [290, 294] may efficiently heat ions perpendicularly through ICR, and, byapplying the RF waves in such a way that the resonancepoint coincides with the peak of the wave energy density,
the ions can gain parallel acceleration because of the EM
ponderomotive force. The idea of the m = 0 acceleration method [4, 289] is to induce an azimuthal current in
the presence of the divergent magnetic field to produce theJθ × Br axial force, like in the RMF and REF schemes.
In a half-RF wave period (in the following half-period), a
positive (negative) axial force is expected, which leads toa zero net force. Then, if the field driven by the antenna
is suﬃciently strong to push the plasma out of the acceleration region during one half-period, a net positive force
remains.
In the nuclear fusion field, Loewenhardt et al. [34] andZhang et al. [214] have investigated the characteristics ofthe toroidal helicon sources in the Small Heliac Experi-mental Apparatus (SHEILA) [296] and the HEALIC (H1) [297]. Shats et al. [298] report the sudden transition
from the low confinement mode to the high confinementmode (L–H transition) in the H-1 heliac and showed aclear dependence on magnetic configuration. They showedthat the ion temperature increased from 40 to 80 eV in thistransition [299], and suggest the mechanism driving the radial electric field Er at a critical value before the bifurcation [300]. Tripathi et al. [301, 302] have investigatedthe characteristics of the helicon wave in a tight-aspectratio toroidal device and showed the significant diﬀerence
between the toroidal mode and the conventional heliconmode.
For the study of the Earth’s magnetosheath, Scime etal. and their research team [7, 147] have investigated themagnetohydrodynamic waves in HELIX-LEIA. The excellent agreement between the measurements and the theoretical prediction provide strong evidence of the successful
excitation of Aflven waves [303].

14. Concluding Remarks
In this review, we have performed an extensive review
of discussions on helicon plasma production mechanisms
and introduced them in some detail with particular emphasis on physical processes involving helicon and TG waves.
In Secs. 1 and 2, the outline and brief history of the helicon study are described. Research on helicon plasma hasrevealed important mechanisms of eﬃcient plasma production since helicon studies began in the 60 s, when the observation of the helicon wave in metallic sodium was firstannounced.
In Sec. 3, the dispersion relation of the helicon wave
is reviewed. In past decades, the mechanism of helicon
plasma production has been investigated based on the dispersion relation of the helicon wave, ignoring the eﬀect
of electron inertia (the TG wave) [1, 87]. The comparison with the experimental results shows good agreementwith the dispersion relation [1, 59, 92] and the wave profiles [1, 86, 88].
In Secs. 4 - 6, the role of the TG wave and the density jump mechanism are reviewed. Since the role of electron inertia was fully recognized [41], eﬃcient RF power
coupling in helicon discharge has been understood withthe collisional dissipations of the helicon and TG waves.
The TG wave deposits its energy near the plasma surface [9–11], while the helicon wave penetrates into the coreregion and deposits its energy in the core plasma [9, 12].
The resonance and anti-resonance properties of the TG
wave excitation in the bounded cavity result in the nonmonotonic power dependence on the plasma density andthe magnetic field [43]. Some parts of the density jumpobserved in the experiments [1, 59] can be explained basedon the power balance with non-monotonic power absorption curves [43, 101, 209]. However, the discharge mode
before the mode transition into the helicon mode (W) re-

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Volume 13, 1101014 (2018)

mains controversial issue. This low-density mode before
the density jump is believed to be the ICP (H) mode. Whenthe finite magnetic field is applied, the ICP mode is precluded, which could be a low-density TG wave-sustainedmode. Indeed, the low-density plasmas sustained by the
TG waves are observed in a low magnetic field [92] and at avery high drive frequency range [107]. The induction fields
observed at a low density in the experiments [47,48,57,59]
are evanescent helicon waves, which oscillate near the antenna (within their skin depth). Therefore, which wave is
responsible for RF absorption should be determined.
In Sec. 7, the neutral depletion mechanism is reviewed. Neutral depletion is considered to be responsible for the limitation of the plasma density in the helicon source [5, 24, 25]. There are two ideas for overcoming
neutral depletion. One is the proper design of the neutral
pump. The gas inlet position should be close to the antenna
to overcome significant neutral depletion due to the ionpumping eﬀect (ionization) near the antenna. The other isapplying the inhomogeneous magnetic field or using thediamagnetic eﬀect [142, 151]. Neutral depletion resulting
from the neutral-pumping eﬀect (pressure balance) would
be improved by controlling the pressure balance between
the plasma pressure, the neutral pressure, and the magnetic
pressure.
In Secs. 8 - 10, the plasma diﬀusion mechanism, nonlinear mechanisms, and other notable experiments are reviewed. There are two main mechanisms for the anomalous diﬀusion across the magnetic field in helicon discharge. One is the “short-circuit eﬀect” [170], in which
the ions can diﬀuse freely across the magnetic field without the restriction of ambipolarity, while electrons do notfollow the ions across the magnetic field, but instead travel
along the field lines and conducting wall, and eventually
maintain charge neutrality [152, 168, 172]. The other is
Bohm diﬀusion [22], which is predominantly interpreted
as an upper limiting value of cross-field diﬀusion with lowfrequency drift-wave fluctuation. However, there is no current consensus on which theory to use to describe diﬀusiontransport. Recent studies have revealed the considerable
non-linear mechanisms of the drift-wave [18–21] and parametric decay [15,17] instabilities. These non-nonlinear dynamics can aﬀect the plasma transport and power balance.
Other experiments have revealed various remarkable features of helicon discharge (e.g., the eﬀect of the device
aspect ratio [203], the helicon discharge sustained by thestanding waves [207–209], the helicon plasmas near the
lower hybrid frequency [211, 212], helicon wave propagation in the toroidal geometry [213–215]).
In Secs. 11 and 12, the remaining critical issues, andseveral basic self-consistent fluid models of helicon discharge to solve them, are reviewed. There are five main
issues to be further investigated: (1) The mode transition
from the ICP to the helicon mode, (2) the density limit
in helicon discharge, (3) the centrally peaked density profile, (4) instabilities, and (5) ion heating. These issues areclosely related. The combination of the fluid models with
the non-linear instability models and the neutral dynamics
leads to thorough understanding of helicon discharge.
In Sec. 13, various applications of the helicon source
are introduced. The compact helicon source using the permanent magnet [6,256] is expected to be useful in the semiconductor industry. A next-generation electric thruster using the helicon source is now being developed for a spacecraft intended for deep-space travel [145, 257, 291]. The
applications for nuclear fusion [298, 299, 301] and spaceplasma [7] are also being extensively studied.
The current helicon study should take the eﬀects of thespatio-temporal non-linear dynamics and neutral dynamics into consideration, leading to a through understandingof helicon plasma production. The plasma transport andthe equilibrium state in the helicon discharge should be
discussed by including them. We believe that the attempt
at a self-consistent model of helicon discharge will shedlight on these issues, leading to the higher-eﬃciency helicon plasma production for future applications.
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