106 J. Meichsner

5.3 Plasma Surface Transition

5.3.1 Plasma Boundary Sheath, Bohm Criterion

We consider a stationary, uniform low temperature plasma consisting of sin-gle charged positive ions and electrons with Maxwell energy distributions(Te Ti). The discharge electrodes, surrounding surfaces or immersed sub-strates/probes will be negatively charged in respect to the plasma poten-tial (VPl) because of higher mobility of electrons in respect to the ions( [mi Te/(me Ti)]1/2). The negative charged surface is shielded by a positivespace charge sheath in front of the surface. This plasma boundary sheath, thepotential of which is shown in Fig. 5.4, determines the charge carrier trans-port to the surface and may inuence the discharge mechanism by secondaryparticle emission from the surface.

pote

ntia

l plasmapresheath

sheath

VV

Vn+ nen+ne n+=ne

surface

Bohm

Pl

sheathd zFig. 5.4. Plasma boundary sheath. Typical potential change in dependence on thedistance from the surface

More in detail, the plasma surface transition is characterised by a pre-sheath and the space charge sheath. Coming from the quasi-neutral plasmaat the potential VPl, a small potential drop over the quasi-neutral pre-sheathaccelerates positive ions to the Bohm velocity.

Potential drop : VBohm =12

kBTee

, (5.21)

Bohm velocity : vBohm =(

kBTemi

)1/2, (5.22)

Bohm criterion : mi2

v2Bohm 12kB Te . (5.23)

In the space charge region the ions will be accelerated and electrons retardedaccording their kinetic energy. For an insulating surface the net current insteady state will be zero:

5 Low Temperature Plasmas 107

je + ji = 0 , (5.24)

je = e ne0 exp(1/2)(

kBTe2me

)1/2exp

(eVsh

kBTe

), (5.25)

ji = e ni0 exp(1/2)(

kBTemi

)1/2; ni0 = ne0 . (5.26)

Then, the potential drop across the space charge sheath is

Vsh =12kBTe

eln

(mi

2me

) 3 4 kBTe

e, (5.27)

Vfl = VPl VBohm Vsh . (5.28)Typical electron temperature of a few 104 K (few eV) leads to a oat-ing potential Vfl of about 1020V negative to the plasma potential. UsingPoissons equation and a plane geometry the positive ion current density canbe calculated in the collision free and collision dominated case:

i dsh collision free : ji = 490(

2emi

)1/2V

3/2sh

d2sh,

(Child Langmuir) (5.29)

i dsh collision dominated : ji = 980 iV 2shd3sh

. (5.30)

Furthermore, taking into calculation the ion current density ji according tothe Bohm criterion at the sheath boundary, the space charge sheath thicknesscan be expressed in terms of the Debye length D [see ( ) in Chap. 1] andthe sheath voltage (5.34).

In a similar way the plasma sheaths are formed in front of dischargeelectrodes (dc cathode layer, rf electrodes), immersed samples for plasmadiagnostics or samples for material surface treatment, and walls.

In low pressure dc discharges, the ion current at the cathode representsnearly the total discharge current. Davis and Vanderslice [8] have presentedrstly an analytical expression for the ion energy distribution function at thecathode. Rickards [9] has modied this model.

Linear increasing electric eld strength towards the cathode surface, noionisation in the cathode layer, and charge transfer collisions with constantcross section are included in the model.

F () =dNid

=Ni02

L

ex(1 )1/2 exp

[ L

ex+

L

ex(1 )1/2

],

(5.31)where ex is the mean free path for charge transfer, L thickness of the cathodelayer, = /(eVc) the ion energy relative to the maximal ion energy forcollision free transport in the cathode layer ( = 0 . . . 1).

1.7

108 J. Meichsner

5.3.2 RF Plasma Sheath

More complicated is the situation at the electrodes in rf discharges, charac-terised by excitation frequencies between ion and electron plasma frequency.In a capacitively coupled, unconned and for that reason asymmetrical rf dis-charge, a negative self bias voltage is created at the powered electrode whichis little less than half the peak to peak rf voltage. Assuming capacitive sheathmodel the simplied potential change with time is represented in Fig. 5.5.

Pote

ntia

l in V VRF

Vself bias

VPl

Vsheath

Grounded electrode

Vsheath

RF-electrode

t

Fig. 5.5. Simplied potential change with time and resulting sheath voltage at therf and grounded electrode. The dierence between the plasma potential VPl and Vrfrespectively ground potential (V = 0) represents the sheath voltage at the poweredand grounded electrode

VSB 12VPP = Vrf0 ; Vrf = VSB + Vrf0 sin (t) ; (5.32)

VPl = Vfl +12(VSB + Vrf0) [1 + sin (t)] . (5.33)

The negative self bias voltage results from the fact that no net dc current canbe ow over one rf cycle at capacitive coupling. The electrons can follow thealternating electric eld and leads to a moving electron front which modulatesthe positive space charge region, respectively the voltage and the thickness ofthe sheath. Increasing rf voltage amplitude mainly aect the self bias voltageand have only little inuence on the plasma potential in strong asymmetricrf discharges. As the result the self bias voltage at the powered electrodeincreases with rf voltage, whereas at the grounded electrode no inuence canbe seen. The sheath properties at the grounded electrode are comparablewith a dc plasma sheath.

5 Low Temperature Plasmas 109

The transport of positive ions is inuenced by the oscillating plasmasheath voltage, charge transfer collisions and elastic collisions.

The sheath thickness is determined by the Debye length D and the volt-age drop across the sheath Vsh. For collisionless regime the sheath thicknessis given by (5.34). The factor Ce represents a correction term due to the timemodulated electron density [10].

dsh =25/4

3CeD

(e VshkB Te

)3/4with 1 Ce

(5027

)1/2. (5.34)

The value Ce = 1 is given for the dc sheath.The energy of impinging positive ions at the rf electrodes depends on

the ratio between the ion transition time in the rf sheath and the rf-cycle.Therefore, the ion mass, rf frequency, and collisions in the sheath regionshave signicant inuence on the shape of the ion energy distribution func-tion. Direct ion extraction at the discharge electrode and energy selectivemass spectrometry can be used for experimental determination of ion energydistributions. Figure 5.6 (a) and Fig. 5.6 (b) show the time averaged Ar+ en-ergy distributions at the driven and grounded electrode of an rf discharge inargon. As expected from the potential structure, the ion energy at the pow-ered electrode is strongly coupled to the rf voltage due to the self bias voltage.The corresponding maximal ion energy is of the order of several hundred eV,whereas the energy of positive ions at the grounded electrode is lower then20 eV.

0 100 200 300 400 500 600

*5

2.5*10

0

5

5

5.0*10

pp

pp

pp

pp

pp

pp

175 V

350 V

525 V

700 V

875 V

1050 V

Ion Energy [ eV ]

Lin

. Int

ensi

ty [

C/ s

]

0 5 10 15 20 25 30

*10

0

6

5*105

1*10

pp

pp

pp

pp

pp

pp

175 V

350 V

525 V

700 V

875 V

1050 V

Ion Energy [ eV ]

Lin

. Int

ensi

ty [

C/ s

]

Fig. 5.6. Ion energy distribution function (Ar+) at the rf electrode (a) andgrounded electrode (b) (13.56MHz-discharge in argon at 5Pa, parameter: peak-to-peak voltage [11])

110 J. Meichsner

Signicant dierences in the shape of the ion energy distribution at thetwo electrodes are observed. Ions coming from the bulk plasma need severalrf cycles for transition to the rf electrode. In Fig. 5.6(a) the arrows markthe situation for entering the ions from the bulk plasma into the sheathat the low and high sheath voltage, respectively. In result a saddle shapedstructure in the ion energy distribution is found. The observed multiple peakstructure in the low energetic part comes from charge transfer collisions inthe sheath region. In the time averaged ion energy distribution the saddleshaped structures overlap from ions directly from bulk plasma and ions fromcharge transfer collisions. At the grounded electrode the single peak at thehigh energy end is seen, only.

With increasing pressure the elastic collisions will have more inuence.This is connected with increasing ion intensity at the low energy part anddisappearing (multiple) peak structure.

In microwave discharges and inductively coupled rf discharges (ICP) thelower sheath voltages represents conditions similar to the conditions in frontof surfaces at oating potential. Using magnetized plasmas the cyclotronmotion must be taken into the consideration.

5.3.3 Electric Probes

Electric probes are widely applied to determine experimentally the electronenergy distribution function f(), or Te in the case of Maxwellian distribution,electron and ion density and potentials. The I/U -characteristic is taken froma circuit consisting of a small metallic electrode (probe) of plane, cylindricalor spherical shape immersed in the plasma, a reference electrode, and variableexternal dc power supply. In single probe measurement (Langmuir probe) thereference electrode is one of the discharge electrodes, and in double probemeasurements a second immersed probe is applied, respectively. Generally,the immersed electric probe represents a disturbance of the plasma aroundthe probe position due to space charge sheath in front of the probe surfaceand the extraction of charged particles. Despite the relative simple technique,specic conditions have to be fullled for accurate analysis of the plasmaparameters derived from the I/U -characteristic [12].

Single (Langmuir) Probe

We consider an uniform low temperature plasma consisting of single chargedpositive ions and electrons at Maxwellian energy distribution with Te Ti.A typical I/U -characteristic of a cylindrical Langmuir probe is shown inFig. 5.7. Principally, the characteristic can be divided into three parts:

I: Ion current saturation region Ii Ie (V Vfl)II: Transition regionIII: Electron current saturation region Ii Ie (V > Vpl)

5 Low Temperature Plasmas 111

Probe voltage U = V-VPl

Prob

e cu

rrent

II

Ionsaturation current

IITransition

region

IIIElectron

saturation current

Vfl VPl

Probe voltage U = V-VPl

Prob

e cu

rrent

II

Ionsaturation current

IITransition

region

IIIElectron

saturation current

Vfl VPl

Fig. 5.7. Dierent regions of a single probe characteristic

The working regime of the electric probe is determined by the parametersprobe radius rP , the mean free path length of charge carrier i/e for ions andelectrons respectively, and the Debye length D. For the classic collisionlesssingle probe theory the two working regimes are dened by

i/e rP D (thin sheath)i/e D rP (thick sheath, orbital motion limit OML )

Ion Current Saturation Region (I)

A simple solution is given for the OML case, and energy and angular mo-mentum conservation law. Not all ions entering the sheath are collected bythe probe. For cylindrical probes the ion saturation currents is given by:

Ii = ni e2 rp lp

(kBTemi

)1/2 (1 eU

kBTe

)1/2(5.35)

with U = V Vpl. Furthermore, Sonin [13] has introduced the so-calledSonin-plot for calculation the ion density. In this plot the term 2P ii is plottedagainst the dimensionless current ii for a pre-selected value of fl 10 [14].Taking into calculation the ion current:

Ii = ni eAp

(kBTemi

)1/2ii . (5.36)

The multiplication of (5.36) with 2P = (rp/D)2 and conversion results in

2P ii =

(r2pe

) (2mie

)1/2 (e

kBTe

)3/2 [Ii (fl 10)

Ap

](5.37)

112 J. Meichsner

with fl = (eVfleVpl)/(kBTe). The right side of (5.37) includes experimentaldata, only. From the determined value 2P ii the dimensionless current ii canbe found in the Sonin-plot, and from (5.36) the ion density.

Transition Region (II)

The electron current increases due to the reduced retarding negative probepotential in respect to the plasma potential. Therefore, the information aboutthe EEDF is included in this part of the probe characteristic. AssumingMaxwellian distribution of the electron energy, the electron current is

Ie = ne eAp

(kBTe2me

)1/2exp

( eU

kBTe

). (5.38)

The electron temperature can be calculated from the slop of the semi-logarithmic plot according to

Te = ekB

ddU

ln(

IeIe0

)(5.39)

and the electron density from the electron current at plasma potential:

ne =Ie(U = 0)

Ap e

(2mekBTe

)1/2. (5.40)

If the electron energy distribution function is non-Maxwellian, the Druyvesteynmethod can be used by second derivative of the electron current:

d2IedU2

=(

2eme

)1/2eAp4

ne (U)1/2 f()V . (5.41)

with U = V Vpl.

Electron Current Saturation Region (III)

The electron saturation current in the OML case can be described similar tothe ion saturation current. The electron current is expressed by

Ie = ne eAp2

(kBTe2me

)1/2 (1 +

eU

kBTe

)1/2. (5.42)

Symmetrical Double Probe

The I/U -characteristic is taken from two small symmetric probes immersedin the plasma. This characteristic resembles the ion current part of the singleprobe characteristic in both direction of the applied voltage, see Fig. 5.8.

5 Low Temperature Plasmas 113

Probe voltage U

Prob

e cu

rrent

I

fliI ,

fldU

dI

sat

i

dU

dI

sat

i

dU

dIfliI ,

Probe voltage U

Prob

e cu

rrent

IPr

obe

curre

ntI

fliI ,

fldU

dI

sat

i

dU

dI

sat

i

dU

dIfliI ,

Fig. 5.8. I/U-Characteristic of a symmetric double probe

The double probe theory was rstly given by Johnson and Malter [15].The advantage is that the complete probe system is on oating potential, andit can be applied in electrode-less generated plasmas without a big referenceelectrode. The disadvantage is that no information can be obtained aboutthe complete electron energy distribution function. Taking into calculation aMaxwellian EEDF the electron temperature is given by:

kBTee

= Ii,fl

[2(

dIdU

)fl

(

dIidU

)sat

]1. (5.43)

The ion density is obtained similar to the procedure at single probe measure-ment.

Special Cases, Problems in Probe Diagnostics

The analysis of I/U -characteristics and calculation of plasma parameters ismore complicated because of the inuence by:

1. collisions in the sheath,2. dierent kinds of positive ions,3. presence of negative ions,4. magnetized plasmas (anisotropic transport),5. rf plasmas (modulated plasma potential needs compensation methods and

damping of the rf amplitude in the probe circuit) and6. deposition plasmas.