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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 >> T i , m e << m i , n e = n i so z e >> z i. z e. z i.

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electron and ion currents
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 currents1
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

steady state
Steady-State
  • 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

(steady-state)

-

-

-

-

-

-

eGe

Insulated

object

eGi

E

sheath region
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

-

-

-

-

-

-

ne

Insulated

object

Sheath region

dark spaces
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
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

-

-

-

-

-

-

eGe

Insulated

object

eGi

E

sheath currents1
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) ]

-

-

-

-

-

-

eGe

Insulated

object

eGi

E

sheath currents2
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 currents3
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 currents4
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

-

-

-

-

-

-

eGe

Insulated

object

eGi

E

sheath currents5
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
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
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
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

-

-

-

Ge

Vf

Gi

floating potential1
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
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
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
“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
Langmuir Probe
  • Can measure I-V curve of plasma using a Langmuir probe

from Mahan, colorplate VI.18

langmuir probe1
Langmuir Probe
  • From the measured I-V curve, can determine :
    • Floating potential
    • Plasma potential

from Manos, Fig. 18, p. 31

conducting surfaces1
Conducting Surfaces

“Electron Retardation” Regime

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

from Manos, Fig. 18, p. 31

langmuir probe2
Langmuir Probe
  • Electron temperature,
  • Te ~ e / [ k dln(J)/dV ]
  • = 47 840 K

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

langmuir probe3
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
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 fall1
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
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
Sheath Width
  • What is the width of the sheath region ?

Cathode

Vp

Va

sheath

region

plasma

body

sheath width1
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 width2
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 width3
Sheath Width
  • The potential in the plasma may be determined by solving Poisson’s equation:
  • 2V(r) = – r(r)/eo
  • 2 = Laplacian operator
sheath width4
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
Debye Length
  • 2V(r) = - (e2n/eokTe) V(r)
    • Solving gives,
    • V (r) = Vo exp (-r/LD)
    • LD = Debye length
    • = (eokTe / e2n)½
debye shielding
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 length1
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
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 fall2
Cathode Fall

from Mahan, colorplate VI.18

plasma reactions
Plasma Reactions

Homogeneous

Reactions

(occur within

the plasma)

Heterogeneous

Reactions

(occur on

a surface)

homogeneous reactions
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 reactions1
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 reactions2
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
Heterogeneous Reactions
  • Reactions that occur on the surface
    • Sputtering
    • Secondary electron emission
    • Reactive etching/deposition
slide42

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

slide43

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

slide44
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

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