Electron and Ion Currents

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# Electron and Ion Currents - PowerPoint PPT Presentation

Electron and Ion Currents. From kinetic theory of gases, impingement rates of electrons and ions within a plasma are: z e = n e (kT e / 2 p m e ) ½ z i = n i (kT i / 2 p m i ) ½ These are called diffusion currents T e &gt;&gt; T i , m e &lt;&lt; m i , n e = n i so z e &gt;&gt; z i. z e. z i.

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Presentation Transcript
Electron and Ion Currents
• From kinetic theory of gases, impingement rates of electrons and ions within a plasma are:
• ze = ne (kTe / 2pme)½
• zi = ni (kTi / 2pmi)½
• These are called diffusion currents
• Te >> Ti , me << mi , ne = ni
• so ze >> zi

ze

zi

Electron and Ion Currents
• For example,
• ne = ni = 1010 cm-3
• Te = 23000 K
• Ti = 500 K
• Then ze = 2.35 x 1017 cm-2s-1
• Je = eze = 37.6 mAcm-2
• zi = 1.28 x 1014 cm-2s-1
• Ji = ezi = 0.0205 mAcm-2

Je = 37.6 mAcm-2

Ji = 0.0205 mAcm-2

Je >> Ji

• No net current can flow through an insulator
• Negative charge will build-up on the object repelling electrons and attracting ions (drift currents develop)
• A steady-state is achieved when the electron and ion currents are equal

diffusion currents (initial)

Je = eze

Insulated

object

Ji = ezi

diffusion + drift currents

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-

eGe

Insulated

object

eGi

E

Sheath Region
• A positive space-charge region is created that is depleted of electrons, leaving predominantly gas atoms and ions (e.g., Ar, Ar+).
• This region is called the sheath region and is similar to the depletion region formed in a semiconductor device such as a p-n junction diode

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ne

Insulated

object

Sheath region

Dark Spaces
• The sheath regions are also called “dark spaces” due to their visual appearance
• Fewer electrons result in less optical emission

from Mahan, colorplate VI.18

Sheath Currents
• At steady-state the impingement rates at the surface are:
• For electrons, Ge = -meneE – Dene
• For ions, Gi = miniE – Dini
• = mobility

D = diffusion coefficient

Drift

Term

Diffusion

Term

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-

eGe

Insulated

object

eGi

E

Sheath Currents
• In 1-D,  = d/dx, giving:
• For electrons, Ge = – meneE – Dedne/dx
• For ions, Gi = miniE – Didni/dx
• Using ni = ne = n at the edge of the plasma sheath and Ge = Gi (steady-state) gives:
• – menE – Dedn/dx = minE – Didn/dx
• Solving for E gives:
• E = [(dn/dx)/n] [ (Di – De) / (mi + me) ]

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eGe

Insulated

object

eGi

E

Sheath Currents
• Substituting this expression for E into the ion flux equation gives:
• Gi = – Da dni/dx
• Da = (miDe + meDi)/(me + mi)
• (ambipolar diffusion coefficient)
• Since me >> mi, we have
• Da = Di + (mi/me)De
Sheath Currents
• Next we can use the Einstein relation between mobility and diffusion, D/m = kT/q, to give:
• Da = Di (1 + Te / Ti)
• Since Te >> Ti, we have
• Da = DiTe/Ti
• We see that Da >> Di
Sheath Currents
• The effect of the electrons is to establish an electric field that pulls the ions and increases it’s effective diffusion from Di (the unaided diffusion at E = 0) to Da
• This effect is known as ambipolar diffusion

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eGe

Insulated

object

eGi

E

Sheath Currents
• The ion current increases to
• Gi~ ni √(kTe/mi)
• For example, for ni = 1010 cm-3, Te = 23000 K, and Ar gas, we have
• Gi = 2 x 1015 cm-2s-1
• eGi = 0.35 mA/cm2
• The enhanced ion current is much greater than the unaided diffusive flux calculated previously (ezi = 0.0205 mAcm-2)
• Gi ~ surface atom density in 1 sec
Growth Rate Example

Gi ~ 1 mAcm-2

= 6.2 x 1015 ions s-1cm-2

Y (1 keV Ar+ ions on Al) ~ 1.5

Sputter rate of Al = 9.3 x 1015 atoms cm-2s-1

Surface density of Al = 6.07 x 1022 atoms cm-3

The deposition rate would be

15 Å s-1 = 5.4 mm/hr

Plasma Potential
• Since charged particles are abundant in the plasma, it is a fairly good conductor
• The plasma is at an equipotential, Vp, called the plasma potential

Insulated

object

Vp

?

sheath

region

plasma

body

Floating Potential
• An insulating object placed in a plasma will develop a negative charge
• A “floating potential develops” (Vf) until steady-state is achieved (Ge = Gi)

Insulated

object

Vp

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-

Ge

Vf

Gi

Floating Potential

Insulated

object

Vp

-

-

-

Ge

Vf

Gi

• M-B distribution of energies:
• Gi = Ge = ze exp [ -e (Vp – Vf)/kTe ]
• Rearranging gives
• Vp – Vf = (kTe/e) ln ( ze / Gi )
• = (kTe/2e)ln(mi/2pme)
• e.g., if Te = 23000 K, eze = 37.6 mAcm-2, and eGi = 0.35 mAcm-2 as calculated previously then Vp – Vf = 9.3 V
Conducting Surfaces
• A conducting surface at the plasma potential draws the diffusion currents

Plasma Potential

Cathode

Vp

Va = Vp

ze > zi

Va

eze

ezi

• A conducting surface at the floating potential draws no net current

Floating Potential

Cathode

Vp

eGe

Va = Vf

Ge = Gi

Va

eGi

Saturation Regions (Conducting Surfaces)

“Ion saturation” regime

Va << Vp: all electrons are repelled

Vp

-

-

-

-

-

Va

eGi= 0.35 mAcm-2

“Electron saturation” regime

Va >> Vp: all ions are repelled

+

+

+

+

+

eze = 37.6 mAcm-2

Va

Vp

“Diode” Plasma
• Since the electron current is much greater than the ion current, an I-V curve of a conducting surface in the plasma shows rectifying behavior
• Hence, the term “diode” plasma

from Manos, Fig. 18, p. 31

Langmuir Probe
• Can measure I-V curve of plasma using a Langmuir probe

from Mahan, colorplate VI.18

Langmuir Probe
• From the measured I-V curve, can determine :
• Floating potential
• Plasma potential

from Manos, Fig. 18, p. 31

Conducting Surfaces

“Electron Retardation” Regime

J< = eGi - eze exp [ -e (Vp – Va)/kTe ]

from Manos, Fig. 18, p. 31

Langmuir Probe
• Electron temperature,
• Te ~ e / [ k dln(J)/dV ]
• = 47 840 K

from Mahan, Fig. VI.7, p. 166

Langmuir Probe
• Electron density can be determined from diffusion current:
• ne = (eze) / [e(kTe / 2pme)½ ]
• = 5.1x109 cm-3

from Mahan, Fig. VI.7, p. 166

Cathode Fall
• The sheath region has low conductivity
• Most of the applied potential is dropped across the cathode sheath
• Cathode fall ~ Va ~ breakdown voltage

cathode

Vp

-

cathode

fall

V = Va

Cathode Fall
• The cathode fall is the kinetic energy gained by ions striking the cathode and of secondary electrons entering the plasma (ignoring collisions in the sheath)
• Cathode fall ~ 100’s Volts

electrons

ions

cathode

Vp

-

cathode

fall

V = Va

Energy Distribution of Sputtered Particles
• The sputtered particle energies are much greater than thermal energies
• This helps in producing conformal films

from Powell, Fig. 2.9, p. 33

Sheath Width
• What is the width of the sheath region ?

Cathode

Vp

Va

sheath

region

plasma

body

Sheath Width
• The width of the sheath (depletion region) can be estimated by calculating the potential that results from a test charge placed within the plasma

from Manos, Fig. 2, p. 189

Sheath Width
• The charge creates a potential, which in free space (no plasma) would be:
• Vo(r) = e / (4peor)
• r = distance from the test charge
Sheath Width
• The potential in the plasma may be determined by solving Poisson’s equation:
• 2V(r) = – r(r)/eo
• 2 = Laplacian operator
Sheath Width

r(r) = local charge density

= e [ ni(r) – ne(r) ]

Boltzmann’s law:

ne(r) = ne exp [ eV(r) / kTe ]

~ ne [ 1 + eV(r) / kTe ]

ni(r) ~ ni since ions are too slow to respond relative to the electrons

ni, ne = n = plasma density

r(r) ~ – (e2n/kTe) V(r)

Debye Length
• 2V(r) = - (e2n/eokTe) V(r)
• Solving gives,
• V (r) = Vo exp (-r/LD)
• LD = Debye length
• = (eokTe / e2n)½
Debye Shielding
• In free-space, Vo(r) = e / (4peor)
• In a plasma, V(r) = Vo exp (-r/LD)
• The plasma electrons rearrange to shield the potential causing its attenuation with a decay length equal to LD
• The plasma is expelled within a region ~ LD (sheath region)

Unscreened

potential

Vo(r) = Q / (4peor)

Q

Shielded

potential

V(r) = Voexp(-r/LD)

r

Debye Length
•  LD = Debye length
• = (eokTe/e2n)½
• = 6.93 [ Te(K) / ne (cm-3) ] ½
• = 743 [ Te(eV) / ne(cm-3) ] ½
• For example, for Te = 1 eV, ne = 1010 cm-3, we get LD = 74 mm
Child’s Law
• A more exact treatment for a planar surface (cathode) gives:
• Ls = (4eo/9eGa)½(2e/mi)¼(Vp-Va)¾
• Substituting Gias determined previously gives:
• Ls ~ 0.8 ¾ LD
•  = e(Vp - Va)/ kTe
• Hence, the sheath thickness is on the order of 10’s of LD (mm’s)
• Electrode spacing ~ cm’s
Cathode Fall

from Mahan, colorplate VI.18

Plasma Reactions

Homogeneous

Reactions

(occur within

the plasma)

Heterogeneous

Reactions

(occur on

a surface)

Homogeneous Reactions
• Reactions that occur within the plasma
• Excitation :
• Electrons produce vibrational, rotational, and electronic states leaving the atom or molecule in an excited state
• e- + O2 e- + O2*
• e- + Ar  e- + Ar*
• e- + O  e- + O*
• Glow discharge:
• O2*  O2 + hn
• O*  O + hn
Homogeneous Reactions
• Ionization :
• Responsible for ion & electron formation which sustains the plasma
• Produces ions for sputtering
• e- + Ar  Ar+ + 2e-
• e- + O2 O2+ + 2e-
Homogeneous Reactions
• Dissociation :
• A molecule is broken into smaller atomic or molecular fragments (radicals) that are generally much more chemically active than the parent molecule
• This is important in reactive sputtering (e.g., reactive ion etching) and plasma-enhanced CVD
• e- + O2 O + O + e-
• e- + CF4 e- + CF3* + F*
Heterogeneous Reactions
• Reactions that occur on the surface
• Sputtering
• Secondary electron emission
• Reactive etching/deposition

Reactive Ion Sputtering (Deposition)

• Excited species (particularly radicals) can react with the surface to deposit nitrides and oxides

Reactive sputter deposition :

From Ohring, p. 126

Reactive Ion Etching (RIE)

• A reactive gas (e.g., N2, O2, CF4) is mixed with the inert gas (e.g., Ar)
• The reactive gases are broken down in the plasma into ions, fragments, radicals, excited molecules, etc.

from Powell, Fig. 3.18, p. 77

RIE
• Acceleration of ions across sheath region results in anisotropic etching

Wet chemical etching

Plasma etching

Ion

bombardment

adapted from Manos, Fig. 8, p. 12