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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


    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


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