Indi an Journal of Pure & Applied Phys ics Vol. 40, January 2002, pp. 24-3 1

Unified empirical model for collective oscillations of asymmetric positive space charge sheath

R am Prakas h , A Sarma*, C B Dwivedi , U Deka, B Sing ha*, S Bujarbarua & J C Upadhyaya* *

Centre of Pl asma Physic , Di spur, Guwahati , Assam 781 006

*Institute of Advanced Study in Science & Technology, Khanapara. G uwahati 78 1 022

**Agra Coll ege, Agra

Received 19 June 200 I ; rev ised 25 September 200 I; accepted 30 October 200 I

Hypothesis for a comprehensible unifi cation of the previous physical models for the collecti ve oscillation dynamics and its driving mechanism fo r an asymmetric positive space charge sheath has been proposed . Under diode-like c ircui t approximations of the sheath , the asymmetry driven beam-plasma interaction model provides a source mechani sm to produce an internal rf resonator to transform the diode- like sheath into a system of an acti ve oscillator. The LCR components and the resonant frequency of the equival ent diode-like circuit of the sheath are es timated for real experimental plasma parameters. The numerical values of the resonant eigen frequency of the equivalent circuit for different grid bias ing voltages are validated aga inst the real experimental values within some error limit. The resonant eigen frequency agrees well with that of the internal rf source of ion beam-plasma model ori gin.

1 Introduction

The existence of a non-neutral space charge in the vicinity of any boundary surface of the plasma is an estab li shed fact. The problem of sheath form ation is one of the oldest problems in pl asma physics 1

•2

• In spite of several publications on theoreticaJ3·5 and experimental~ -7 in vestigations pertaining to static and dynamical behaviour of the plasma sheath , there are many basic issues which are still not fully understood. The ever-growing importance of the sheath in pl asma science and technology including pl asma process ing and fusion research allows it to remain as a subject of active research program.

According to two-l ayer theory of plasma-sheath potential mode l, the boundary between the two layers lies at a transonic point , which is often referred to as the sonic barri er, where the quasi­neutrality condition breaks down and the ion flow speed exceeds the sonic speedx. The issue of the two layers behaving as a sharp boundary or a diffused (or extended) boundary is of re levance. Mathematically, the two-layer theory predicts the pl asma sheath interface as a singul arity. This is referred to as a two-scale transiti on problemx. Some efforts have been made to address this problem in a few specific cases of the pure-1·6 ·

9•10 and impure

pl asmas 11.12. However, in genera l, the problem of

singularity is a scientifically well-posed problem to be resolved by more theoretical and experimental investigations. It probably needs a new paradigm to define and characteri ze the sheath edge with some unique quality. There is some basic reporting about the like lihood of two different Mach points especia lly in the case of electronegat ive impurity ions 13

• Thi s observation demands a more prec ise and unique definition of the plasma-sheath boundary to specify its nature and exact location esp~cially in the presence of impurity ion s. Intensive investigations including analytical calculations and experi ments are needed to determine and measure the differential flow dynamics of diffe rent ion species in the sheath/pre-sheath region . The ton fl ow measurements and acoustic wave/turbulence measurements in the sheath edge surround ings may complement to understand and resolve the sheath edge singularity.

The dynamica l aspect of a pl asma sheath study may be classified into three categories: (a) tempora l evolution of a plasma sheath itself, (b) de or equilibrium sheath driven oscillations and (c) the ac or rf driven sheath osc illati ons. The first of these categories includes the problems of theoretica l and experimental study of a transient sheath to specify its relaxation behav iour to some other new equil ibrium 14

• The second comprises the osci !lat ions

RAM PRAKASH et a/.: SPACE CHARGE SHEATH 25

an smg from an asymmetric bi-potential space charge region (sheath) formed around a biased grid or e lectrode. The last category considers the cases of rf induced sheath dynamics where ac signals at probes or e lectrodes are observed to exhibit the resonances. In fact, both high 15 and low 16 frequency resonance oscillations have been reported in bas ic experiments of rf probe measurements and characteri zati on.

Efforts were made by several worke rs to develop theoreti ca l mode ls to understand the ori g in and excitati on mechani sm of these osc illations. The high frequency oscill ati ons near e lectron plasma frequency are well-known phenomena of rf probes and antenn as in pl asmas. Thi s is the behaviour of e lectron-rich sheath interpreted as natural consequences of sheath-pl asma resonance. An overview of the earli er experimenta l observati ons 17•

1x

e luc idates the di ss imil ar behaviour of the ion and e lectron resonances at the e lectrodes. The di stincti on lies in the resonance mechani sm; the e lectron resonance ari ses as a consequence of natural sheath-plasma interacti on whereas the ion resonance is of pure ly sheath orig in . The present authors, however, do not agree with the denial of the role of pl asma-sheath interacti on process in low frequency rf resonances5

.

The simil ar behav iour of low frequency ion resonance is also observed in a bi -potenti al ion-rich sheath structure around a negati ve ly bi ased grid in double plasma deviceH 1

Y20

• In sp ite of severa l theoreti ca l mode ls5

·21 for asymmetry induced sheath

driven low frequency instabil iti es (SDLFI), the underl ying phys ics of the orig in and excitati on mechani sm of the SDLFI is still not very clear. The present contribu tion puts forward a hypothetica l model suggesti on to combine the d iode-like sheath mode l of Rosa22 and the beam-plas ma model of Dwivedi et a/.1 to arri ve at a uni fied phys ical model to d iscuss the underl yin g phys ics of the sheath d ri ven low frequency in stabili ty. The earlier knowledge of ac sheath dynamics 1 7 · 1 x·2 ~ has been used to g ive phys ica l insight to our hypothes is.

2 Theoretical Models

An overview of the previous theoret ical models wi ll be g iven to desc ribe the diode- like ion sheath character in response to an app lied rf voltage and the underl ying physics of the asymmetric dri ving mechan ism of the ion sheath driven low frequency

instability. This will highlight the importance of these mode ls for self-consistent interpretation of the observed SDLFI.

2.1 Ion transit time model

It is now well-understood that the sheath prope rti es are dominated by the transit time effect at and near the two pl asma frequenc ies . In order to study the ion tran sit time effect (near the ion pl asma frequency) in a pos itive ion-rich sheath , Rosa 22

deve loped a simplifi ed mode l of sheath in response to an applied rf voltage. The mode l treatment is based on the approximati ons that the e lectron density is negligible inside the sheath and that the e lectri c fi e ld is al so negli gible at the pl asma-sheath edge. Under these approximations, the gove rning equati ons have been reduced to Poisson 's equati on under one-dimensional mode l as:

aE en;

ax E0

and the total current density equation as:

1 . . aE

= ] ;- l e +Eo --a;

... ( I )

... (2)

where E is the sheath assoc iated e lectric fi e ld , e is the e lectronic charge on the ions, n; is the ion

density, £0 is the pe rmitti vity of the vacuum , 1 is the total current density,}; is the ion current density and Jc is the e lectron current density.

Eq. ( I ) describes the ion space charge d istributi on unde r negli gible effect of the e lectron density . Continuity equati ons of the total current density is invoked as:

_a1 =O ax ... (3)

For frequency well be low the e lectron pl asma frequency, it has been argued that, the e lectron current th rough the sheath is a lways in equilibrium with in stantaneous fie ld to obey the cont inuity equati on of the for m22

:

a}e = 0 ax ... (4)

Us ing Eqs ( I to 4 ) under specifi ed approx imations and afte r some mathemat ical manip ul at ion, Llewell yn type equation22 is derived

26 INDIAN 1 PURE & APPL PHYS , VOL 40, JANUARY 2002

in Lagrangian vari ables to describe the kineti cs of ion sheath diode:

. .. (5)

where t0 is the ini t ial time of the ions at the sheath edge, t is the time taken by the ions to reach the pos iti on x and that is measured with respect to initi al time and m; is the ionic mass .

Eq. (5) corresponds to a dynamic situati on where the tempora l vari ation of the partic les acceleration becomes important23

• Hence the exact soluti on demands the knowledge of th ree in itial va lues for phys ical vari ables like pos ition, velocity (current) and accelerati on.

The equilibrium solution gives the width (X0)

and normalized amplitude (11 0=eVG/kTc) of the de sheath at the wall and these are given be low:

( ;.,2 );.,2

and 17o = I + 4 )2

where c,= [T.: is the acoustic speed in bulk pl asma, ~ -;:

A=Wr;'t;0 , Wr; is the ion pl asma oscill ation frequency, 't;o is defined as the ion transit time and VG is the app li ed vo ltage at the grid wa ll.

Now, assuming the fluctuations (as a result of response dynamics of rf vo ltage to ion sheath ) to vary as exp (iwt) , the perturbed soluti on of positi on can be deri ved unde r the considerati on of initi a l va lues as specified in Ref. 23. T hese values are chosen in accordance with the approx imations as outlined earli er in the tex t after Eq . (5). Subsequently, the perturbed potenti a l due to rf source across the sheath can be es timated to find the corre lat ion between the potential (V) and current perturbati ons:

Here, w is the rf frequency, 8 = W't;o and subsc ri pt I denotes the perturbed vari ab les and

lfi (A, 8 ) = i8 +

;.,2 (i (} 3 l (i2 6 + i8{ exp(-i8 ) + I} + 2{ exp- i(}) -I}

F inally, the sheath admittance ( Y= }/V1) can be evaluated in the fo llowing fo rm:

y * (} 2 - I

(

2 l=exp(7Jo - 17o) - - 2 lfl (A_ ,(} ) E0w1" A

c,.

. . . (6)

The floating potenti al (11 0.) IS defi ned as fo ll ows:

• 1 ( m J llo = 21n 27t~e .

It is now clear from Eq . (6) that the response dynamics of an rf sheath can be di scussed in terms of an equivalent circuit whose acti ve e lements can be determined after some mathemat ica l exerc ise. For low frequency consideration (W<'t;o-1

), Eq. (6) can be simplified to find out an express ion for general impedance [Z ( w)= I I Y] of ion-ri ch sheath as fo ll ows:

Now, comparin g thi s ex pression with the known form of the impedance of the conventional seri es resonant LCR c ircuit , the equi va lent sheath assoc iated capacitance ( C; 11) , inductance (L;11) and resis tance (R;11) fo r low frequency rf source of internal orig in 5

·7 (as d iscussed in the next section) or

of ex ternal orig in 22 can be directl y written as :

o ( ~-~- ~I -~ Cw - - x

0

RAM PRAKASH et al.:SPACE CHARGE SHEATH 27

Here, the subscript ilf refers to the internal low frequency , rf source.

The equivalent circuit behaviour of the diode­like ion-rich sheath in the form of resonance oscillations should be observable only when ac source modulation of either external origin or of internal origin is introduced to the sheath as, for instance, in the rf probe sheath resonance at or near the ion plasma oscillation frequency.

2.2 Beam-plasma model

In fact, the ion-rich sheath either floating or biased is always associated with space charge limited ion flow structure as dictated by Bohm condition for a sheath to exist. It is well-established that a sheath potential asymmetry at some threshold value for a given hi-potential positive ion-rich sheath structure around a negatively biased grid drives low frequency instability5·7 • This is found to be predominantly localized near the plasma-sheath edge with an exception where global oscillation was also reported in experimental condition by Barrett and Greaves 19

• A simple beam-plasma model was invoked by Dwivedi et aU to understand the excitation mechanism of the aforesaid instability. The model calculation considers the experimental situation of double plasma device in which the hi­potential positive ion-rich sheath is formed by negative biasing on the grid . The asymmetry in the sheath potential structure was introduced by source plasma biasing with respect to the target plasma or by density imbalance in source and target plasmas7

•

In that situation, a three ion-beam flow structure becomes a reality to drive the SDLFI at lower potential side (target plasma) as described earlier5

•

Under the approximation of quasi-neutral equilibrium ion-beam flow (valid in entire pre­sheath region) , which is maintained by Boltzmannian electron distribution, a linear normal mode analysis gives the following dispersion relation 5

:

where F(k)= [Q52 +2Q5(k. v5o) + (k. Vsor +

(k. vT0)2] I [Q5 +k.(vso+VTo)Jl,

.. . (7)

Q = oo-k. v50, is the Doppler shifted eigen-mode frequency in source ion-beam frame, k is the wave number, v50 is the un-normalised source ion-beam flow velocity and vTo is the un-normalised ion beam flow velocity in target plasma. Here, Q 2s.T =

k2A.200002r.s.T/(I+k2A..2Dc) defines the plasma eigen­mode frequency in source and target plasmas. The dispersion relation (7) has been deduced for resonant mode-mode coupling conditions at

Q- Q s -jk .(vso- v70 )j and k. (v50 -VTo ) < 0.

;--+-e -+-

' I I , -0-7)-~)

( -

0

(a)

(b)

all with if source of w

X

rv Internal if source ( w)

Fig. I - Schematic diagram of an equivalent series resonant LCR circuit associated with ion-rich sheath, showing an ion­rich sheath on a wall with applied rf source of signal w (a), and the associated circuit elements and an internal rf source of frequency w (which is decided by resonant eigen-mode frequency characterized by Eq. (7b)

The defined resonant conditions predict the propagation direction as well as the value of resonant eigen-mode frequency (Q,) of the unstable mode. Numerical analysis of the full dispersion relation [Eq. (3) of Ref. 5] yields the value of

28 INDIAN J PURE & APPL PHYS, VOL 40, JANUARY 2002

normali sed resonant eigen-mode frequency Q , - 0.6 fo r di fferent normali sed v511 values lying in between I< v50 < 1.5. Here, the frequency Q , is normali sed by the ion pl asma oscill ati on frequency Wr; and v50 is normali sed by the acoustic speed c,.

The relevance of the di spersion relation (7) for beam-pl asma interacti on dynamics lies in its importance to describe the conditi on for the onset and dri ving mechani sm of the internal rf source generati on in an asymmetric bi-potenti al ion-rich sheath . The proposed three ion-beams pl asma model considerati on is a pre-requisite cond ition for the excitation of the sheath driven low frequency in stability which is supposed to play the role of intern al rf to act ivate the ion-ri ch sheath assoc iated series resonant LCR circuit as shown in Fi g. I .

A

~ -4 . 0x10~ c til I

·1 Ox1 0'1-'------- ---------_j

Fig. 2 - Variati on of the sheath inductance (L; 11-) with A (defi ned as a physical parameter wit h implic it appearance of the grid-bi asi ng voltage)

I.Ox i O· ~

9.0x10 9

8 Ox 10 9

7 Ox10"1

~ 6.0x10-s

"' 5.0x10"9 (;; lL

u' 4.0x1o·'

3 Ox10 9

2.0x10 9

1 Ox1 0 9

\

~ "'--....__

. ........ ..... .

-·-·-· ......

00 OS 1~ 15 2 0 25 30 35 40 45 50 55

1..

Fig. 3 - Variation of the sheath capacitance (C, 11 ) wi th A (defi ned as a physical parameter wi th impli cit appearance of the grid-bias ing vol tage)

3 Unified Empirical Model for SDLFI

As di scussed in foregoing secti ons, the ion-rich sheath behaves as an equ ivalent electrical ci rcui t under the diode-like ion sheath approximati on (Fig. I ), when it is perturbed by some rf source of external or internal origin . In low frequency limit of the applied rf voltage, the equi valent circui t is characteri sed by L, C and R elements which render the ion sheath to behave as a resonant circuit. Figs 2-4 provide quantitative values of the associated L, C and R elements respecti vely fo r different values of the pl asma parameters. Fig. 5 depicts the va ri ation of sheath ad mittance with the normalised rf frequency. From thi s, one can determine the resonance frequency of the specified circuit.

25x106

2.0x106

"" 1/l

E 1.5x101

.r: 0 '-"

c{ 1.0x106

5 Ox101

0.0

0.0

~

/ 1/

---~~

I I

I I

I I

I I

I I

I

' I

--.-, 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

).,

Fig. 4 - Variation of the sheath res istance (R;11· ) with A (defin ed as a physical parameter with impli cit appearance of the grid-bias ing voltage)

Subsequently, the vari ati on of resonant frequency wi th the applied voltage has been shown in Fig. 6. These plots have been produced by using the expressions of L;1r. C; 11·, R;1r components given in secti on 2. 1 for the real values of the experimental plasma parameters of double pl asma device7

• It is found that the circuit resonant frequencies are in reasonably good agreement (wi th in order of magnitude) with beam-pl asma model prediction5 of the resonant eigen-mode frequency (Fig. 6). Thi s motivates us to put forward the following hypothesis to unify the ion-beam plasma model and

RAM PRAKASH et af.:SPACE CHARGE SHEATH 29

the diode-like sheath c ircuit model to discuss the se lf-consistent underlying physics of SDLFI.

3.1 The proposed hypothesis

The authors propose the hypothesis that the beam-plasma model provides a source mechanism to generate an internal low frequency rf source. The model is based on the equi librium plasma-sheath coupling5. Subsequently, the internal rf source coup les with ion-sheath diode to transform the system into an acti ve monotron osc ill ator. This is what, the authors believe, is occurring in the experimental situations in negatively biased grid with external or internal source of the sheath potential asymmetry.

4.0x10 3 ·- ----------·- · -------

3.6x10.3 - t- ),=409 I

3.2x10·3 - +- A=3.37 - I- A=4 94

2.8x103

2.4x10 3

>-Ul 2.0x10 3

.0 <(

16x10·3

12x10 3

B 0x1o•

4.0x10"

0.0 0.5 10 2.0 2.5 3.0

Fi g. 5- Vari ation of the absol ute sheath admittance (Y) with the normalised source ac signal frequency (w) for different A values

The consideration of kinetic treatment for bounce osc ill ations of the trapped ions in negative potential well around the biased grid may indeed provide a basis to estimate the grid-bi asing threshold value for occurrence of SDLFI. Thi s is not included in the theoretical model of the beam­plasma interaction process5

. Since the grid-bi as ing potential appears in the equivalent circuit interpretation of the diode-like ion-rich sheath , it is

intuitively fe lt that our hypothesis offers a valuable formu lation for comprehensible understanding of the basic physics of SDLFI.

0.6

05

.• 04 _@ 3

·20 -40 ·60 -80 ·100 ·120 -140 ·160 Vg(volts)

Fig. 6 - Variation of the normalised resonant frequency (w )

and A with grid-biasing voltage (Vg); * corresponds to empiri cal ion transit time model as suggested by Barrett and Greaves 1 ~ . In their model, a discrepancy was aris ing due to improper mathematical treatment, which was overcome by in troducing a fitting parameter ~ in the concerned equati on (in our case. we have chosen ~ = 0.67). • corresponds to circuit model22

calcu lations. ... corresponds to real experimental values for double plasma device with Vs (source basing voltage)= 1 OV . The vert ical bars represent the error of I 0%. • corresponds to A val ues

4 Results and Discussion

Utilising the basic properties of the equivalent c ircuit behaviour of an ion-rich sheath as a resonant circuit and that of the asymmetry as a source for internal rf generation, a unified empirical model has been hypothesised for self-consistent discussion of the underlying physics of the SDLFI. Numerical values of the resonant eigen-mode frequency of the equivalen t ion-rich sheath circuit and that of the SDLFI, for real experimental plasma parameters, are found in good agreement (see Fig. 5) and hence the proposed hypothesis seems to li e on sound footing of physical ground. It may be interesting to conjecture the app lication of the proposed hypothesis to understand the presence of osci llations at the sheath edge in an expanding sheath realisable in transient mode of ion-sheath dynamics 14

•

Difference may lie in the ori gin and driwng mechanism of the internal rf source. Thus, the modelling of an expanding ion-sheath in terms of an

30 INDIAN 1 PURE & APPL PHYS, VOL 40, JANUARY 2002

equiva lent LCR ci rcuit activated by plasma driven internal rf source can be an interesting problem of interdi sc iplinary research.

Bas ic idea of the proposed hypothesis for uni fied empirical model of SDLFI could be useful to understand the non-linear dynamics of an acti ve monotron oscill ator assoc iated with an asymmetric ion-ri ch sheath or of an ac modulation of the grid­bi as ing potential in double pl asma device experiments. Th is can serve as the phys ical pri nciple to produce and study wide varieties of collecti ve behaviour of the coupled osci ll ators incl uding parti al synchronisati on, phase trapping, large amplitude Hopf oscill ations and even chaotic behav iour In di ffe rent practi ca l situations of interest24

•25

•

The authors' hypothesis could provide guide lines to set-up pl asma experiments to in vestigate into a variety of the time de lay effects of coupled osci llators24

, which ex ist as natura l course of excitation by asymmetric modi fication of a pos itive space charge region around a negatively bi ased grid . Time delay effect may be monitored by ac modul ati on of the grid-bias ing as is usually done for experimental observati ons of non-linear phenomena in double pl asma device. Intuitive ly, the fo ll owing oscillatory modes could be visuali sed in an asymmetric ion-rich sheath with an ac modulation at the grid . The beam-plasma dri ven instability (supposed to act as an interna l rf source) coupled with diode-like ion sheath can be considered as a self-consistent monotron oscillator of internal origin . The ion bounce oscillations could be another oscillator mechani sm again of in ternal on gm. Thi s (bounce-oscill ations) can be characteri sed by kinetic model of collective plasma dynamics. The external ac modul ation at the sheath grid may act as a dri ver oscill ator to monitor the self-consistent coupling between the internal modes of oscill ations. The driver can be used to manipulate the time de lay control from outside of the internal oscill ators for wide range variation of the plasma parameters and grid-bias ing vo ltage. This may lead one to argue that the observati on (non-observation) of sheath dri ven instability should be interpreted as a consequence of time de lay effect of these coup led oscillati ons to cause in stability (or amplitude death ) of these oscill ati ons.

5 Conclusion

The uni fied approach reduces the sheath­induced phenomena in terms of e lec trical c ircuit resonance of sheath oscill ators. It appears that the potenti al relaxation instability2 1 is associated with the re laxation effect of the e lectron inerti al delay, which cannot be under-estimated when the average ion-beam velocity compares with the Bohm speed. The favourable condi tion for such occurrence is more like ly In grid-e lectrode configurat ion21

•

However, more theoretical and experi mental investigations are needed to resolve the bas ic issues perta ining to the sheath edge singularity and the origin and dri ving mechani sm of the SDLFI.

Acknowledgements

One of the authors, (RP) acknowledges CSIR, Govt of India for providing fin ancia l ass istance as SRF to work at CPP. The authors (BS, RP, AS and UD) acknowledge DST, Govt. of Indi a for provid ing financial ass istance and experi mental fac il ity at IASST under the project D PPD, IASST under the Princ ipa l Investigatorshi p of Prof J Chuti a.

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RAM PRAKASH et al.:SPACE CHARGE SHEATH 31

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